Proceedings of the Seventieth European
Study Group with Industry
Limerick, Ireland, June 28th − July 3rd 2009
Editors:
Stephen O’Brien
Martina O’Sullivan
Peg Hanrahan
William Lee
Joanna Mason
Jean Charpin
Marguerite Robinson
Andrei Korobeinikov
ii
Contents
Contents
iii
Preface
v
List of delegates
vii
List of problems
xv
Partial Wetting Phenomenon in Superhydrophobic Microchannels
xvii
The Effect of Mechanical Loading on the Frequency of an Oscillator Circuit
xix
Designing a Green Roof for Ireland
xxi
Uplift Quadratic Program in Irish Electricity Price Setting
Particle Impact Analysis of Bulk Powder During Pneumatic Conveyance
Improvement of Energy Efficiency for Wastewater Treatment
xxiii
xxv
xxvii
Initiation of Guinness
xxix
Solar Reflector Design
xxxi
iii
Technical reports
1
Partial Wetting Phenomenon in Superhydrophobic Microchannels
3
The Effect of Mechanical Loading on the Frequency of an Oscillator Circuit
29
Designing a Green Roof for Ireland
45
Uplift Quadratic Program in Irish Electricity Price Setting
65
Particle Impact Analysis of Bulk Powder During Pneumatic Conveyance
85
Improvement of Energy Efficiency for Wastewater Treatment
107
Initiation of Guinness
141
Solar Reflector Design
183
iv
Preface
The second European Study Group with Industry (ESGI) workshop in Ireland was held by the
Mathematics Applications Consortium for Science and Industry (MACSI) in the Department
of Mathematics and Statistics at the University of Limerick, Ireland, from 28th June to 3rd
July 2009.
The 70th ESGI workshop was opened by Professor Brian Fitzgerald, the Vice President of
Research at the University of Limerick.
The ESGI workshop followed the usual format of presentations from industry on the Monday
with intensive working sessions by leading applied mathematicians from across Europe throughout the week and concluded with presentations from the academics on their recommendations
to the companies.
Delegates were free to choose which problem or problems they contributed to; each problem
was assigned a work room as a base for the week. Some participants opted to stay in one
room throughout the week, contributing to their problem right through from defining the
problem, designing the model, finding and analysing results and writing the presentation and
report. Others moved between each room giving fresh insight to all problems, commenting on
the approaches adopted and making suggestions for improvements. Postgraduates often find
it difficult to contribute to the early modelling stages when the speed of discussions can be
very fast and are in areas of mathematics and applications outside of their immediate field of
expertise. This usually changes throughout the week when they are able to work on analysis
and numerical simulations and contribute to the presentations and reports.
For academics, study groups provide an excellent opportunity to work on industrial mathematics problems outside of their usual areas of research with researchers from postgraduates
to professors. The highly interactive nature of these events allows participants to exchange
ideas and form long lasting collaborative links with other academics and companies to a much
greater extent than with a regular conference.
For industry, study groups offer the opportunity of calling on the combined and varied expertise
of leading specialists in industrial mathematics to make significant progress on problems in a
short space and time. They kick start the research process and provide answers quickly at
minimal cost.
v
Over 90 participants registered for ESGI 70 from Ireland, England, Scotland, Denmark, Poland,
Spain, Sweden, Germany and USA. All of the MACSI postgraduates and postdoctoral fellows
attended the study group and contributed to work on the problems.
The problems were varied and were largely representative of the type of industrial mathematics
that usually appears in study groups across Europe.
Events throughout the week included a guest lecture on Thursday morning by Professor Chris
Budd on Geometric integration: how to compute solutions of differential equations which have
the right qualitative features.
The MACSI study group was a resounding success. Excellent feedback was received from
both university researchers and industrial representatives. All of the companies involved have
indicated their strong intention to continue work on the problems with MACSI and action plans
are already in place to take the projects forward.
Thank you to all of the participants in ESGI 70, who made it an enjoyable and successful week
of collaborative problem solving. These proceedings illustrate the results of all their expertise,
hard work, energy and enthusiasm. We look forward to seeing you at next year’s Irish Study
Group!
The sponsors of the ESGI70 were:
• Science Foundation Ireland (SFI)
• National Development Plan (NDP) Ireland
MACSI would like to thank the sponsors without whom the European Study Group with
Industry workshop could not have taken place.
MACSI is funded by the Science Fundation Ireland Mathematics Initiative grant 06/MI/005.
vi
List of delegates
Academics and Post-Docs
Adley, Catherine - Department of Chemical and Environmental Science, University of Limerick, Ireland.
Bell, Christopher - Department of Bioengineering, Imperial College London, United Kingdom.
Borg, Karl - MACSI, Department of Mathematics and Statistics, University of Limerick,
Ireland.
Budd, Chris - School of Mathematical Sciences, University of Bath, United Kingdom.
Burke, Mark - Department of Mathematics and Statistics, University of Limerick, Ireland.
Chapwanya, Michael - MACSI, Department of Mathematics and Statistics, University of
Limerick, Ireland.
Charpin, Jean - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Cooker, Mark - School of Mathematics, University of East Anglia, United Kingdom.
Dellar, Paul - OCIAM, Mathematical Institute, University of Oxford, United Kingdom.
Doherty, Kevin - National University of Ireland Galway, Ireland.
Enright, Ryan - Stokes Institute, University of Limerick, Ireland.
Evatt, Geoff - University of Manchester, United Kingdom.
vii
Fowler, Andrew - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Gaburro, Romina - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Gleeson, James - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Hayes, Martin - Department of Electronic and Computer Engineering, University of Limerick, Ireland.
Hegarty, Alan - Department of Mathematics and Statistics, University of Limerick, Ireland.
Hjorth, Poul - Department of Mathematics, Technical University of Denmark, Denmark.
Houghton, Conor - Department of Mathematics, Trinity College Dublin, Ireland.
Jabłońska, Matylda - Department of Mathematics, Lappeenranta University Finland, Finland.
Kinsella, John - Department of Mathematics and Statistics, University of Limerick, Ireland.
Kirwan, Laura - Agri-Environment Research Department, Teagasc Environment Research
Centre, Ireland.
Kopteva, Natalia - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Korobeinikov, Andrei - MACSI, Department of Mathematics and Statistics, University of
Limerick, Ireland.
Korostynska, Olga - Microelectronic and Semiconductor Research Centre, University of Limerick, Ireland.
Lacey, Andrew - Department of Mathematics, University Heriot-Watt, United Kingdom.
Lee, William - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Lessels, Gordon - Department of Mathematics and Statistics, University of Limerick, Ireland.
Linss, Torsten - Department of Mathematics, Czech Academy of Sciences, Dresden, Germany.
viii
Lukaschuk, Sergei - Department of Engineering, University of Hull, United Kingdom.
Mackey, Dana - School of Mathematical Sciences, Dublin Institute of Technology, Ireland.
Mason, Joanna - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
McGuinness, Mark - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland / School of Mathematics, Statistics and Operations Research, Victoria University
of Wellington, New Zealand.
Meere, Martin - Department of Applied Mathematics, National University of Ireland Galway,
Ireland.
Mitchell, Sarah - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
O’Brien, Stephen - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
O’Sullivan, Catherine - Department of Civil and Environmental Engineering, Imperial College, United Kingdom.
O’Sullivan, Martina - MACSI, Department of Mathematics and Statistics, University of
Limerick, Ireland.
Piiroinen, Petri - Department of Applied Mathematics, National University of Ireland Galway, Ireland.
Power, Oliver - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Rachinskii, Dmitrii - Department of Applied Mathematics, University College Cork, Ireland.
Rebrova, Natalia - Tyndall Institute, Ireland.
Richardson, Giles - School of Mathematics, University of Southampton, United Kingdom.
ix
Robinson, Marguerite - MACSI, Department of Mathematics and Statistics, University of
Limerick, Ireland.
Ryan, Gearóid - School of Mathematical Sciences, University College Cork, Ireland.
Schwartz, Len - Department of Mechanical Engineering, University of Delaware, United States
of America.
Soussi, Sofiane - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Vynnycky, Michael - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Ward, Jonathan - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Waters, Thomas - Department of Applied Mathematics, National University of Ireland Galway, Ireland.
Wilson, Eddie - Department of Engineering Mathematics, University of Bristol, United Kingdom.
Zhelev, Toshko - Department of Chemical and Environmental Science, University of Limerick, University of Limerick, Ireland.
Postgraduate students
Carey, Michelle - MACSI, Department of Mathematics and Statistics, University of Limerick,
Ireland.
Cregan, Vincent - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Fay, Gemma - OCIAM, Mathematical Institute, University of Oxford, United Kingdom.
Gonzalez, Maria - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
x
Hanley, Kevin - College of Science, Engineering and Food Science, University College Cork,
Ireland.
Healy, Timothy - University College Cork, Ireland.
Hewitt, Ian - OCIAM, Mathematical Institute, University of Oxford, United Kingdom.
Hurley, Julie - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Lapin, Vladimir - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Liu, Yupeng - School of Mathematical Sciences, Dublin Institute of Technology, Ireland.
Mellgren, Niklas - Department of Mechanics, KTH Royal Institute of Technology, Sweden.
Murphy, Thomas - Department of Chemical and Environmental Science, University of Limerick, Ireland.
O’Brien, Mick - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
O’Donovan, Celine Kemmy Business School, University of Limerick, Ireland.
O’Keeffe, Niall - Stokes Institute, University of Limerick, Ireland.
Pimenov, Alexander - College of Science, Engineering and Food Science, University College Cork, Ireland.
Rojas, Jaime - Department of Chemical and Environmental Science, University of Limerick, University of Limerick, Ireland.
Sedakov, Roman - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Simboeck, Hildegard - University of Limerick, Ireland.
Tuoi, Vo Thi Ngoc - MACSI, Department of Applied Mathematics, National University
of Ireland Galway, Ireland.
Viscor, Martin - MACSI, School of Mathematical Sciences, University College Cork, Ireland.
xi
Wang, Dengli - Business School, Dublin City University, Ireland.
Wang, Qi - School of Mathematical Sciences, Dublin Institute of Technology, Ireland.
Winstanley, Henry - Mathematical Institute, University of Oxford, United Kingdom /
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Witheephanich, Kritchai - Wireless Access Research Centre, University of Limerick, Ireland.
Zubkov, Vladimir - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Undergraduate students
Bandeira, Afonso - Universidade de Coimbra, Portugal.
Delaney, Niamh - University of Limerick, Ireland.
Ganchev, Galin - University of Limerick, Ireland.
Kirby, Grainne - University of Limerick, Ireland.
Mayrhofer, Arno - University of Limerick, Ireland.
McCarthy, Stephen - University College Cork, Ireland.
Menzies, Colin - University of Limerick, Ireland.
Zhezherun, Andrey - University College Cork.
Industrial representatives and guests
Bourke, Sile - Bord Gáis Energy, First Floor (BGE), City Quarter, Lapps Quay, Cork, Ireland
Browne, Conor - Diageo Ireland, St. James’s Gate, Dublin 8, Ireland
Carnus, Tim - Teagasc, Johnstown Castle, Co. Wexford, Ireland
Healy, Sandra - Analog Devices, Raheen Industrial Estate, Raheen, Limerick, Ireland
xii
Hoolan, Sean - Erin Energy Ltd, Applied Research into Solar Technologies, MIRC Building, Athlone Institute of Technology, Dublin Road, Athlone, Co Westmeath, Ireland
Hurley, Gavin - Bord Gáis Energy, First Floor (BGE), City Quarter, Lapps Quay, Cork,
Ireland
Jordan, Rory - Science Foundation Ireland, Wilton Park House, Wilton Place, Dublin 2,
Ireland.
Kirwan, Billy - GFM Systems, Waterway House, Crag Crescent, Clondalkin Ind. Estate,
Dublin 22, Ireland
O’Hara, John - Landtech Soils Ltd, 20 Kenyon Street, Nenagh, Co. Tipperary, Ireland
O’Mahony, Seamus - Wyeth Nutritionals Ireland, Askeaton, Co. Limerick, Ireland
O’Mara, Brian - Analog Devices, Raheen Industrial Estate, Raheen, Limerick, Ireland
Swallow, Richard - VGraph UK Ltd, 28 Hudson Way, Radcliffe On Trent, Nottingham,
NG12 2PP, United Kingdom
xiii
Organising committee
O’Brien, Stephen - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
O’Sullivan, Martina - MACSI, Department of Mathematics and Statistics, University of
Limerick, Ireland.
Hanrahan, Peg - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Lee, William - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Mason, Joanna - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Charpin, Jean - MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland.
Robinson, Marguerite - MACSI, Department of Mathematics and Statistics, University
of Limerick, Ireland.
Korobeinikov, Andrei - MACSI, Department of Mathematics and Statistics, University of
Limerick, Ireland.
xiv
List of problems
xvi
Partial Wetting Phenomenon in
Superhydrophobic Microchannels
Industrial Partner: Stokes Institute, University of Limerick
Industry representatives
• Ryan Enright, Stokes Institute, University of Limerick, Ireland.
• Niall O’Keeffe, Stokes Institute, University of Limerick, Ireland.
• Cormac Eason, Stokes Institute, University of Limerick, Ireland.
Description of the problem
Superhydrophobic (SH) surfaces are a class of materials that display extreme wetting behaviour
that can result in almost spherical droplets, i.e. the contact angle, theta>180deg, when a liquid
is deposited on such a surface. This behaviour is a direct result of a favourable surface potential
combined with surface micro/nano-structure and was famously described by Cassie and Baxter
[1]. These surfaces have received much attention in the last decade due to the proliferation of
microfabrcation techniques allowing large areas of geometrically well defined surface structures
to be produced.
One application of SH surfaces is as liquid flow boundaries in microfluidics. It has been shown
analytically, numerically, and experimentally that SH surfaces can produce an apparent slip
effect due to reduced liquid/solid interaction and the presence of a trapped gas/vapour layer at
the SH surface. This segregated two-phase flow exists due to surface tension and the stability of
the flow structure is controlled primarily by the geometry of the SH surface structures. Energybased calculations performed assuming equilibrium conditions provide criteria for the stability
of the wetting regime and have been validated experimentally in droplet experiments. However,
recent experimental investigation performed by our group have demonstrated the existence of
xvii
a ‘partial’ wetting state on pillar-structured SH surfaces that is not predicted by equilibrium
analysis. The partial wetting state has been observed using direct imaging of the SH surfaces
located in a Hele-Shaw microchannel using laser-scanning confocal microscopy and shows that
the gas/liquid/solid contact line forms some distance from the top of the pillar structures upon
filling of the microchannel. Investigation of the pillar structures using SEM has shown that
the pillar sides are not smooth, but instead demonstrate scallop features, which are a result
of the fabrication process. These scallop features represent contact-line pinning sites that are
not captured in existing energy calculations, which can explain why the pillars remain partially
wetted. However, it is unknown why this partial wetting phenomenon should occur in the first
place and it does not appear to have been described in the literature prior to this work. An
understanding of possible mechanisms, developed through modelling, is of particular interest
for informing the design of future devices.
[1] A.B.D. Cassie and S. Baxter. Wettability of porous surfaces. Trans. Faraday Soc., 40:546551, 1944.
xviii
The Effect of Mechanical Loading on the
Frequency of an Oscillator Circuit
Industrial Partner: Analog Devices
Industry representatives
• Sandra Healy, Analog Devices, Raheen Industrial Estate, Raheen, Limerick, Ireland.
Description of the problem
A particular type of silicon chip produced by Analog Devices is loaded with an arbitrary force
of the order of 100N during testing. A key component in the chip is an oscillator circuit whose
frequency must reach a 1% accuracy specification over a temperature range of -40-105řC. At
present the oscillators do not meet the specification during the loading test. We suspect that
during the test, deformation of the chip leads to a piezo-resistive effect that alters the oscillator
frequency. This problem has come to light during a critical stage in production and a solution
is required urgently. Redesigning the chip is not an option at this stage in the development,
however the testing configuration may be altered to achieve the required specification. On a
much longer timescale, a better understanding of the mechanics of the chip during loading will
aid improvements in future designs to prevent this problem from occurring again.
Objectives
There are two main goals for this project:
• Develop a mechanical model of the chip under loading,
• Relate the deformation of the chip to the oscillator frequency.
xix
Further Information
Chip Geometry
A detailed diagram of the chip geometry will be supplied. There are two silicon wafers (or die)
in the chip, the smaller of which sits on top of the larger. These are cast in a plastic mould
with connector pins on the base.
Testing Rig
An example of the testing rig will be brought to the study group and details of its design
(including material properties) will be supplied. The chip is placed in a well in the plastic rig
and rests on metal pins that contact the connectors on the base of the chip. A plunger is then
pressed against the top of the chip and force is applied via a screw mechanism. The size of this
force is not known and may vary from test to test. Preliminary experimental data consisting
of coarse measurements of applied force versus oscillator frequency will be available. Details of
the plunger head design will also be supplied.
Oscillator Circuit
The oscillator circuit consists of resistors, inductors and capacitors. These components are built
up in thin layers inside the silicon wafer.
xx
Designing a Green Roof for Ireland
Industrial Partners: Landtech Soils Ltd and GFM Systems
Industry representatives
• Catherine Adley, Department of Chemical and Environmental Science, University of Limerick, Ireland.
• Billy Kirwan, GFM Systems, Waterway House, Crag Crescent, Clondalkin Ind. Estate,
Dublin 22, Ireland.
• Landtech Soils Ltd, 20 Kenyon Street, Nenagh, Co. Tipperary, Ireland.
Description of the problem
Green roofs are becoming increasingly popular in many countries. The direct benefit of a green
roof is it provides an environmentally friendly form of roof insulation for homes and commercial
buildings. They also provide significant indirect benefits, such as contributing to: increasing
biodiversity and wildlife; water management; reducing air pollution and the urban heat island
effect; and, most importantly, they act to recycle carbon dioxide. Aesthetically a green area is
pleasing and relaxing.
The development of an optimised green roof for Ireland is part of a collaborative research
programme in UL. The task to be addressed by the study group is to develop a mathematical
model of a green roof focusing on energy and water balances. Energy balances determine the
effectiveness of the roof as an insulator, while getting the drainage properties of the roof correct
is essential to keeping the integrity of the roof structure; preventing water leakage to the interior;
and managing water availability for plant growth and sustenance. This model will be used to
help design the experimental phase of the research project by indicating the most important
parameters to be measured and, once the model is fully parameterised, used to determine an
optimal design.
xxi
The model must contain descriptions of the following:
• The Irish climate: inputs will include sunlight and rainfall.
• Energy exchange with the house below.
• The biota (plants, microbes and fungi) living on the roof - one aspect of optimal roof
design will be choosing the right combination of biota.
• The properties of the layers making up the roof: typically soils, plant mix drainage layers,
root barriers and waterproof sheeting.
xxii
Uplift Quadratic Program in Irish
Electricity Price Setting
Industrial Partner: Bord Gáis Energy
Industry representatives
• Gavin Hurley, Bord Gáis Energy, First Floor (BGE), City Quarter, Lapps Quay, Cork,
Ireland.
• Sile Bourke, Bord Gáis Energy, First Floor (BGE), City Quarter, Lapps Quay, Cork,
Ireland.
Description of the problem
Energy prices in the Irish Electricity Market are composed of a Shadow Price and an Uplift
Component. The Shadow Price is the cost of meeting a marginal unit of demand, as determined
by solving the Unit Commitment Problem. The Unit Commitment Problem is a Mixed Integer
Program which determines which units (generators) to commit (turn on), so as to minimize
the total system cost of generation, subject to meeting demand in each period and respecting
technical characteristics of the units (e.g. ramp-rates, min on/off times, minimum generation).
Participants declare their true marginal costs and therefore the Shadow Price alone does not
guarantee that generators will recover their fixed running costs, e.g. start-up and no-load. The
Uplift component of prices addresses this. It is derived via a Quadratic Program (QP) which
ensures that the generators recover those costs. The QP aims to minimize a parameterized
combination of
• total additional cost due to Uplift
• volatility of Uplift
xxiii
The focus of this study will be on the Uplift Quadratic Program and approaches may include
• Analysis of necessary conditions for optimality (Karush-Kuhn-Tucker conditions)
• Analysis of Lagrangian multipliers on the constraints
• Analysis of market data via MATLAB implementation
• Study of impact of different parameterizations
• Data mining on ultimate drivers of uplift, e.g. demand, wind variability, etc.
We will be able to bring extensive market data and also a MATLAB implementation of the
quadratic program described.
xxiv
Particle Impact Analysis of Bulk Powder
During Pneumatic Conveyance
Industrial Partners: UCC and Wyeth
Industry representatives
• Kevin Cronin, University College Cork (UCC), Ireland.
• Seamus O’Mahony, Wyeth Nutritionals Ireland, Askeaton, Co. Limerick, Ireland.
Description of the problem
Particle Impact Analysis
In many of Ireland’s process industries, the product is manufactured and conveyed in powder
form. This is especially true of food and pharmaceutical materials. During processing and
handling, the particles are usually immersed in a flowing gas (air) stream. One important
aspect of the behaviour of these products is their response to impact. This can be impact
against the walls of the process equipment or impact against each other.
Particle geometry is generally (approximately) spherical. Size as measured by diameter is
dispersed but is usually within the range of 10 microns (0.01 mm) up to 1000 microns (1 mm).
An individual spherical particle can either be solid or hollow and if hollow the wall of the sphere
may be solid or porous. In some cases, the bulk product consists of billions of these individual
particles. In other cases, the powder is in an agglomerated form as granules where each granule
consists of between 10 and 100 individual particles bonded together.
The actual micro-structure of the material can be complex; for powders of biological origin
(such as foods or bio-active chemicals), the structure consists of some matrix of protein, sugar
and fat. However we usually take the structure to be homogeneous.
xxv
At UCC, we analyse these particles in a number of different systems such as fluidised bed granulation, spray drying, rotary drum processing, pneumatic conveying, etc. For some experiments
we use real materials or powders while for others we used model materials such as glass beads
(corresponding to hard elastic solid behaviour) or plasticine balls (soft, visco-elastic behaviour).
In certain industrial systems, the rheology of the particles changes during processing from soft
to hard as a result of heat and mass transfer. In addition, sometimes the particle is coated with
a layer of viscous liquid which affects its impact behaviour.
A standard or reference case that we wish to understand is the normal impact of two spherical
particles (not granules) of different diameters with dry surfaces. Both particles are moving in a
gas stream of known density and viscosity. The collision will be defined by a velocity dependent
coefficient of restitution. We wish to know the force versus deflection behaviour, particularly
the maximum force that is achieved, that occurs during the impact. Knowledge of the force
enables us to determine whether the collision will result in rebound, coalescence or breakage of
the particles.
If the two particles are considered as solid elastic spheres, then Hertzian contact stress provides
an analytical expression for the force that is developed. However once the spheres are hollow
(even if they are still elastic), then we cannot find a solution.
xxvi
Improvement of energy efficiency for
wastewater treatment
Industrial Partner: Department of Chemical and Environmental Science, University of Limerick
Industry representative
• Toshko Zhelev, Department of Chemical and Environmental Science, University of Limerick, Ireland.
Description of the problem
The autothermal thermophilic aerobic digestion process for wastewater treatment uses the
heat generated by thermophilic bacteria to sterilise wastewater. These bacterial require high
temperatures (60 -65 degrees centigrade) and oxygen to operate most efficiently. During (batch)
processing of wastewater the temperature in the treatment tank is regularly lowered below this
temperature by the introduction of new material. Restoring the tank to an optimal temperature
is a time and energy consuming process requiring increased aereation over several hours.
There are two parts to this study group problem. The first part is the formulation of a mathematical model of the treatment process. The second part is to use this model to develop an
optimised plant design and treatment strategy.
The model must be able to describe changes to the plant and the way it is operated, for instance:
• Using heat exchangers to increase the temperature of pretreated material using heat
recovered from treated material,
• Changing the feeding pattern,
• Changing the size or number of reactors.
xxvii
The optimisation problem must take include the following aspects of the problem:
• Efficiency criteria (contains energy usage, processing time and sterrilisation),
• Practicality constraints,
• The nature of the solution space, which contains both discrete variables (due to the
batched nature of the processing) and continuous variables,
• Account for problem complexity: formulation of problem of parametrical optimisation
expanded to structural-parametrical optimisation (optimum regime parameters and process flow structure optimisation = number and size of reactors, number and sizes of heat
exchangers).
xxviii
Initiation of Guinness
Industrial Partner: Diageo
Industry representatives
• Conor Browne, Diageo Ireland, St. James’s Gate, Dublin 8, Ireland.
• Richard Swallow, VGraph UK Ltd, 28 Hudson Way, Radcliffe On Trent, Nottingham,
NG12 2PP, United Kingdom.
Description of the problem
Unlike carbonated beers which foam spontaneously, Guinness contains a mixture of nitrogen
and carbon dioxide which will not spontaneously exsolve. Diageo use various strategies to
promote foaming. In canned and bottled Guinness a pressurised plastic container with a small
hole in it, known as a widget, is used to trigger exsolution. The inclusion of the widget adds a
small but significant cost to the manufacture of the product which Diageo wishes to minimise.
To assist with this, the study group is asked to develop a mathematical description of the way
the widget promotes foaming, possibly taking the bottle widget as an specific example.
Ideally this model should be sufficiently general to be able to shed light on other methods of
foam initiation such as ultrasound or nucleating surfaces.
xxix
xxx
Solar Reflector Design
Industrial Partner: Erin Energy
Industry representative
• Sean Hoolan, Erin Energy Ltd, Applied Research into Solar Technologies, MIRC Building,
Athlone Institute of Technology, Dublin Road, Athlone, Co Westmeath, Ireland.
Description of the problem
Objective
To develop a design methodology for solar tracking reflector curve generation that will deliver
optimal energy capture, such that a low absorber/reflector ratio is achieved within a shallow
construction.
Parameters
Absorber/reflector aperture ratio. Currently, absorber elements are more expensive than the
reflector elements. Therefore the aim is to have a low ratio of absorber area to that of the
reflector and thus achieve an optimal cost/performance balance point.
Distance between absorber centres.
• The optimal pitch between absorbers. There are several examples with the absorbers in
close proximity to each other, thus casting a shadow over each other reducing the unit
effectiveness.
• Impacts on the relative height of the mirror wall relative to the absorber height. (Better
to lower mirror wall height so as to allow direct hit on adjacent absorber.)
xxxi
Depth of curve/reflector. Surface mount markets require that the depth of the reflector curve
plus the absorber must be maintained as shallow as possible.
Incident angle. The greater the acceptance angle, the more of the daily yield will be captured.
xxxii
Technical reports
2
Partial Wetting Phenomenon in
Superhydrophobic Microchannels
Report Contributors: Karl Borg1 , Vincent Cregan1,2,
Andrew Fowler1 , Mark McGuinness1,3, Stephen B.G. O’Brien1 ,
Leonard W. Schwartz4 and Vladimir Zubkov1
Industry Representatives:
Niall O’Keefe5, Ryan Enright5 and Cormac Eason5
1
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
Report coordinator, vincent.cregan@ul.ie
3
School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, New Zealand
4
Department of Mechanical Engineering, University of Delaware, United States of America
5
Stokes Institute, University of Limerick, Ireland
2
1
Introduction
Superhydrophobic surfaces are a class of materials that display extreme wetting behaviour that
can result in almost spherical liquid droplets. For superhydrophobic surfaces the contact angle between the liquid and the surface may be much greater than 90o . This extreme wetting
behaviour is a direct result of a favourable surface potential combined with surface micro/nanostructures. A high degree of surface roughness can lead to a significant increase in the contact
angle compared to a smooth surface[1].
Figure 1: Overview of various liquid drop orientations on a surface.
(a) Lotus leaf
(b) Water strider
Figure 2: Examples of superhydrophobic surfaces in nature.
The leaves of the lotus plant[2] and the lady’s mantle[3] are well-known examples of superhydrophobic surfaces in nature. Microstructues on the surface of the lotus leaf allows the leaf
4
to be self-cleaning and strongly repellent to water droplets. Superhydrophobicity is not restricted just to the plant world. Water striders posses superhydrophobic legs which allow them
to walk along the surface of water and the wings of many butterflies have superhydrophobic
properties[3].
An understanding of superhydrophobic wetting behaviour is of huge importance in industry
and the phenomenon has received much attention in the last decade due to the proliferation
of microfabrication techniques allowing large areas of geometrically well defined surface structures to be produced. Superhydrophobic surfaces are used in the design of waterproof clothing
raincoats, windscreens and satellite dishes. Superhydrophobic surfaces are currently playing a
major role in the fabrication of silicon wafers[3] via fluidic self-assembly. Another application
of these surfaces is as liquid flow boundaries in microfluidics. It has been shown analytically,
numerically, and experimentally that superhydrophobic surfaces can produce an apparent slip
effect due to reduced liquid/solid interaction and the presence of a trapped gas/vapour layer
at the superhydrophobic surface.
The Stokes Institute are currently using superhydrophobic surfaces in a number of their experiments. In one particular experimental process, a syringe pump is used to force deionised
water through a parallel plate microchannel. The microchannel inlet and outlet are connected
to two large fluid reservoirs. The upper wall is a flat no-slip surface while the lower wall is a
superhydrophobic surface made up of small cylindrical pillars.
(a) Microchannel schematic
(b) Experimental apparatus
Figure 3: Overview of experimental setup.
The breadth, length and height of the microchannel is 10mm, 10mm and 80µm, respectively.
Pillar dimensions vary depending on the type of experimental setup but for this report we will
use 7.5µm and 25µm as values for the pillar radius and height, respectively. The distance between two pillars (centre to centre) is 20µm. The velocity of the water as it passes through the
microchannel is 0.012ms−1 . [4] contains a detailed description of the experimental apparatus.
As the water passes through the microchannel, air gets trapped between the pillars and the
5
water. This segregated two-phase flow exists due to surface tension. The stability of the
flow structure is controlled primarily by the geometry of the superhydrophobic surface structures. Assuming equilibrium conditions, energy-based calculations performed provide criteria
for the stability of the wetting regime and have been validated experimentally in droplet experiments. However, recent experimental investigation performed by Stokes have demonstrated
the existence of a ‘partial’ wetting state on pillar-structured superhydrophobic surfaces that
is not predicted by equilibrium analysis. The partial wetting state has been observed using
direct imaging of the superhydrophobic surfaces located in a Hele-Shaw microchannel using
laser-scanning confocal microscopy and shows that the gas/liquid/solid contact line forms some
distance from the top of the pillar structures upon filling of the microchannel. Typically, the
contact line has been found 10 − 15µm below the pillar tops. In addition, the height of the contact line varies from pillar to pillar. Investigation of the pillar structures using SEM (Scanning
electron microscope) has shown that the pillar sides are not smooth, but instead demonstrate
scallop features, which are a result of the fabrication process. These scallop features represent
contact-line pinning sites that are not captured in existing energy calculations, which can explain why the pillars remain partially wetted. However, it is unknown why this partial wetting
phenomenon should occur in the first place and it does not appear to have been described in
the literature prior to Stokes’ experimental work.
(a) Pillars on superhydrophobic
surface
(b) Pillar scallops
(c) Confocal image of liquid interface in
contact with pillars.
Stokes are interested in mathematical modelling their experimental set-up and understanding
6
the physical mechanisms involved in their experiment. An understanding of these mechanisms,
developed through modelling, is of particular interest for informing the design of future devices.
The key problems to be studied may be summarised as follows:
• Problem 1: Describe the mechanism behind liquid penetration of the pillars. What is
the cause of the liquid going 10 − 15µm down along the pillars? Why does the liquid drop
height vary from pillar to pillar?
• Problem 2: How do the scallops affect the contact line pinning of the liquid?
The modelling work performed during the study group is summarised in this report. The report
is split into six main sections and suggestions for future work are found in the concluding section.
The content of the sections may be summarised as follows:
• Section 2: Review of possible surface wetting surface states
• Section 3: Time scales involved in problem
• Section 4: Height of water-air interface drop in contact with single cylindrical pillar
• Section 5: Partial wetting phenomenon in superhydrophobic microchannels in two dimensions
• Section 6: Poiseuille flow across pillars
7
2
Surface wetting states
Young’s equation relates the thermodynamic equilibrium contact angle, θc of a liquid drop to
the specific energies of the surface-liquid, surface-air and liquid-air interfaces[5]
γSG = γSL + γLG cos θC
γij denotes the specific energy between medium i and medium j. However, Young’s equation
is restricted to smooth, flat surfaces. It was recognised early that surface roughness may lead
to deviations in the contact angle predicted by Young’s equation. In the literature, Wenzel’s
seminal work was to first to describe the impact of surface roughness and superhydrophobic
wetting behaviour in the context of waterproofing of knitted fabrics[7]. Wenzel illustrated how
a rough surface can increase the apparent contact angle at the boundary between a liquid and
a surface. Wenzel related the standard Young contact angle, θC to the contact angle on the
rough surface, θW by the formula
cos θW = r cos θC
where r is a roughness factor. When a liquid drop occupies the spaces between the surface
projections, the drop is said to be in the Wenzel state.
Figure 4: Overview of various liquid drop orientations on a superhydrophobic surface with
topography.
Cassie and Baxter extended Wenzel’s work by considering the wettability or water-repellency
of porous surfaces such as natural and artificial clothing[8]. In the Cassie-Baxter (or Fakir)
state, a liquid drop remains balanced on the surface projections with air trapped underneath.
8
The associated equilibrium condition is
cos θCB = Φ (cos θC + 1) − 1
where Φ is the solid fraction of the surface. [6] contains a detailed description of the various
wetting states associated with superhydrophobic surfaces.
9
3
Problem timescales
In this section we discuss some of the timescales relevant to the problem. The Stokes Institute
advise that it takes about one second for the water to traverse the top of the pillars, with an
associated velocity of 10mm s−1 . It is of interest to consider how quickly air that is caught
between the water and the bottom of the pillar array can escape through the pillar array.
(a)
(b)
Figure 5: (a) As the water moves through the microchannel, air is free to move around the
pillars and escape. (b) Air gets trapped between the water and the pillars once the water has
passed completely through the microchannel.
Treating the array as a porous medium, with a porosity of 0.6 calculated from the geometry,
Darcy’s law gives the volume flux (ms−1 )
q=
k
∇p
µg
where the dynamic viscosity of air is µg = 1.8 × 10−5 kgm−1 s−1 at 20◦ C, and the pressure difference used in the experiments to drive the water, taken to be also driving air flow, is typically
about 100Pa over 10mm, so that ∇p ≈ 104 Pam−1 . The permeability is the most uncertain
parameter. Here we will use two different ways to estimate k, permeability-porosity plots from
hydrogeology and the more formal Carmen-Kozeny relationship.
10
Figure 6: A porosity-permeability plot, from the Bureau of Economic Geology, University of
Texas at Austin. The lines indicate pore sizes.
Using the permeability-porosity cross plot illustrated in figure (6), the relationship between
porosity and permeability for rocks depends on the average pore dimension also. Using the line
corresponding to a 20 µm pore size, a porosity of 0.6 is outside the data range, but looks to
have a permeability that is k ≈ 2 − 5 × 10−12 m2 .
An alternative approach is to use the Carman-Kozeny equation
k=
1 φ3 D2p
72τ (1 − φ)2
where τ ≈ 2 is tortuosity, the square of the mean path length taken by the air over the total
path length, and Dp is the particle diameter, 20µm. Using φ = 0.6 gives a similar value to the
above one, k = 4 × 10−12 m2 .
Using k = 4 × 10−12 m2 gives
q ≈ 0.002 m s−1
The time for air to move across a 10mm array at this speed is about 5 seconds, comparable to
11
but a little slower than the time taken by the water.
Another time scale of interest is the time taken for water to penetrate down the pillars, which
may be estimated by considering the experimental results of Moulinet and Bartolo. Their
experiment consisted of depositing a small droplet of water in the Fakir state on a superhydrohpobic surface and monitoring the droplet profile evolution. As evaporation of the droplet
ensues the droplet radius decreases. The contact lines recedes and the transition from the Fakir
state to Wenzel occurs in less than 20ms. This is relatively fast compared to the speed of the
water and the air.
The implication of these timescales is that it is possible for the water to trap air underneath
it, as it passes over the array of pillars and then seals off the escape route at the exit. This
motivates the more careful examination of the interaction between water and air that appears
later in this report.
12
O
4
Liquid in contact with a single cylindrical pillar
Consider a single cylindrical pillar and a liquid in contact with the curved surface of the cylinder.
In particular we are interested in the shape of the liquid meniscus. The motivation behind
considering this problem is that we can get an approximation for how much the water-air will
drop in a distance of 10µm which is half the distance between the centre of the pillars in Stokes’
experimental setup. The radius of the cylinder, R is 7.5µm and the capillary length for water
is 2.73 × 10−3 m. [10] and [11] derived asymptotic solutions to the Young-Laplace for the liquid
profile in contact with a single cylindrical pillar. Of particular interest is the inner solution in
[11] which is valid in an O(R) boundary layer near the cylinder.
(a)
(b)
Figure 7: (a) Schematic of liquid meniscus. (b) Dimensionless plot of liquid height y against
distance from cylinder centre, r. We multiply dimensionless quantities by R to convert to
dimensional quantities. The contact angle was taken to be 170◦ .
For the given parameters, the plot in figure 7 suggests that the water-air interface can drop
approximately 12µm in a distance of 10µm.
13
5
Partial wetting phenomenon in superhydrophobic microchannels in two dimensions
In this section we study a two-dimensional, Poiseuille flow through a microchannel. Via some
geometrical arguments we establish a criterion for which the liquid will go down in between
the pillars. Consider the profile of the liquid moving through the channel with velocity u at
various instances in time, t0 , t1 , t2 and t3 .
Figure 8: Liquid profile at various times as it flows through the microchannel.
α and β are the liquid-surface contact angles related to the lower and upper channel surfaces,
respectively. The equilibrium condition of the free surface is given by
∆p = γ
1
R
where ∆p∗ is the pressure difference across the liquid-air interface and R is the radius of curvature. If we consider the circular liquid front in figure (9) and via some geometrical arguments
it may be shown that the center of the circle in figure (9) is
(x0 , y0 ) = (−R cos β, −R cos α)
Hence, the resulting equation of the circle for the liquid front is
(x + R cos β )2 + ( y + R cos α )2 = R2
Rewriting (5.1) for point C = (d, D), we obtain
14
(5.1)
d = −R cos β +
p
R2 − ( D + R cos α )2
(5.2)
Points A and B have coordinates A = (xA , 0) and B = (0, yB) where
xA = −R cos β + R sin α,
yB = −R cos α + R sin β
(5.3)
Figure 9: Schematic of liquid going through the microchannel.
The condition necessary for the liquid front to move down the gap between the pillars is xA > d
which may be rewritten using (5.2) and (5.3) to give
d
cos β − sin α
<
D
2 cos α
(5.4)
For the case of β = 105o and α = 105o , (5.4) yields
d < 2.36D
15
(5.5)
Result (5.5) gives us a condition for which the advancing liquid front will go down between the
pillars. However, this result only applies to a two-dimensional system and it does not address
the influence of the surrounding pillars on the liquid-air interface in three-dimensions.
16
6
6.1
Poiseuille flow across pillars
Local equilibrium
A mathematical model is now developed for a pressure driven liquid flow through the microchannel across the hydrophobic pillars behind the advancing liquid front. Air can escape
between the pillars under the advancing liquid. Assuming that the liquid-air surface is at local
equilibrium, we balance the pressure difference across the surface with the surface curvature to
get
p − Π = 2γκ
(6.1)
where p is the liquid pressure, Π is the air pressure, γ is the surface tension and κ is the mean
curvature of the liquid-air interface. When the interface is in equilibrium a pressure difference
is induced across the interface. We also assume that the liquid-air interface evolves in time as
shown in the figure below.
Figure 10: Possible liquid-air interface evolution at times T1 , T2 , T3 and T4 .
h is the average height of the interface above the base of the pillars and D is the distance
between the pillars. At times T1 and T2 , the liquid-air interface is pinned to top of the pillars
at z = h0 . At time T1 the liquid pressure is less than the air pressure. Hence, the liquid-air
interface is curved upwards. At some point the liquid pressure becomes greater than the air
pressure and the interface becomes curved downwards (as shown at time T2 in figure (10)).
17
Increasing liquid pressure causes the interface to move down between the pillars until it reaches
the bottom surface between the pillars. At time T4 the liquid-air interface is pinned to the
bottom surface of the microchannel.
6.2
Local equilibrium in relation to theoretical function f(h)
Assuming that the liquid-air interface progresses with time as shown in figure (10), we theorize
that a suitable equilibrium condition for the interface is
p−Π =
γ
f(h)
D
(6.2)
where f(h) is an unknown function of h. With some suitable choice of geometry, interfaces with
constant curvature and some physical arguments, the function f(h) can be estimated. Based
on figure (10), a plot of f(h) against h will take the form
Figure 11: General shape of function f(h).
where h0 = (2.5 × 10−5 m) is the top of the pillar and h1 is near the bottom of the pillar. h1 is
the point near the bottom of the pillar where the water-air interface goes from being concave
to convex. Figure (11) is just a qualitative description of the function f(h). The main features
of the function are briefly discussed here and a more detailed derivation of the function will be
18
outlined in future work.
At time T1 , f(h) is negative because the pressure difference, p − Π is negative. The change of
signs of f(h) at h0 and h1 are related to the change of signs in the curvature of the liquid-air
interface. In the time interval from T2 to T3 , f(h) and p − Π reach their maximum value, β.
This occurs while the liquid-air interface is concave and moving down the surface of the pillar.
The function in figure (11) was derived via a combination of smooth approximations to the
Heaviside function, H(h) of the form
H(h) ≈
1
1 + exp (−2 s h)
(6.3)
where s is a steepness factor used to vary the transition from T1 to T2 and T3 to T4 . Values
of h0 , depend on the experimental setup and are easily obtainable. However, exact values of
h1 are not directly available. Further experimental work by the Stokes institute is needed to
attain a range of values for h1 .
6.3
Poiseuille flow
We assume that the flow of the fluid through the microchannel takes the form of a Poiseuille
flow. d is the height of the channel and l is the length of the channel. A pressure gradient in
the x direction is set up in the microchannel due to the pump used in the experiment.
Figure 12: Basic microchannel dimensions.
Deoionised water passes through the microchannel. Water may be classified as a Newtonian,
visocous, incompressible fluid. The motion of a Newtonian fluid can be described via the
19
Navier-Stokes equations. The Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of a Newtonian fluid. The complexity of the Navier
equations can be reduced significantly via lubrication theory to obtain the thin film equations.
Lubrication theory is applicable to the flow of fluids in a geometry in which one length scale
is significantly smaller than the other length scale. As the channel height, d is much smaller
than the channel length, l, lubrication theory may be used. The resulting governing equation
for the velocity in the x direction, u is
∂p
∂2 u
= µl 2
∂x
∂z
(6.4)
where µl is the dynamic viscosity of the liquid. Integrating (6.4) twice with respect to z and
using the no-slip boundary conditions at z = 0 and z = d yields
u =
px
2 µl
(z
2
− dz
)
(6.5)
The vertical velocity component, w is found by substituting (6.5) into the incompressibility
condition
∂u
∂w
+
= 0
∂x
∂z
and using the no penetration boundary condition at z = d, to give
w = −
pxx
2 µl
( z3
3
−
dz2
d3
+
2
6
)
(6.6)
At the surface of the pillars the downward averaged velocity is
w = −
d3
pxx
12 µl
(6.7)
If we consider the pillars to be a porous medium we can use Darcy’s Law to relate the vertical
fluid velocity to the pressure difference across the liquid-air interface. Darcy’s law describes
the flow of a liquid through a porous medium. The law relates the volume flux per unit area
(having units of velocity), q to an applied pressure gradient. Hence,
q = −
k Πx
h
µg
20
(6.8)
where k is the effective permeability between the pillars and µg is the dynamic viscosity of the
air. The downward velocity at the surface of the pillars can be related to the rate at which h
is evolving and the pillar-free fraction, φ by the equation
w = φ
∂h
∂t
(6.9)
If w is negative the fluid is moving downward and the height of the liquid above the pillar
base, h is decreasing with time. Whereas, if w is positive the fluid is moving upward and h is
increasing with time. The continuity equation for h requires that φ ht = −qx . Hence, we have
(6.10)
w = φ ht = − qx
Substituting (6.8) into (6.10) yields
−
( )
k ∂
d3
pxx = φ ht =
hΠx
12 µl
µg ∂x
(6.11)
We have the boundary conditions for the liquid pressure and the air pressure
Πx = 0 ; p = p0 + ∆p ;
Π = p = p0 ; at x = l
at x = 0
(6.12)
(6.13)
where ∆p is the water pressure difference across the length of the channel. Integrating (6.11)
once with respect to x leads to
−
d3
k
px =
hΠx + C1
12 µl
µg
(6.14)
where C1 is an integration constant. Via the boundary condition for the derivative of the air
pressure at x = 0, (6.14) reduces to
−
d3
px = C1
12 µl
21
Hence, we get
C1 = A
d3 ∆p
12 µl l
where A is an O(1) constant. (6.14) becomes
−
k
d3 ∆p
d3
px =
hΠx + A
12 µl
µg
12 µl l
(6.15)
and dividing both sides by d3 /(12 µl) gives
−px = M h Πx + A
∆p
l
(6.16)
where M = (12 k µl)/(d3µg ). Rewriting (6.2) in terms of p and differentiating with respect to
x leads to
p x = Πx +
γ
fx
D
(6.17)
Substituting this expression into (6.16) and rearranging in terms of Πx yields
Πx
(
= −
A∆p
l
+
γ
D
fx
)
(6.18)
1 + Mh
Substituting (6.18) into (6.11) gives
φ ht = −
k ∂
µg ∂x
γ
( 1 +hM h [ A∆p
+
f (h)h ])
l
D
′
x
(6.19)
which is a nonlinear Richards type equation in h. Richards type equations are typically used
to describe water movement in unsaturated soils[12]. Via (6.18) and boundary condition (6.12)
we get
γ
A∆p
= − f ′ (h)hx
l
D
22
at x = 0
(6.20)
Now rearranging in (6.17) in terms of Πx yields
Πx = p x −
γ
fx
D
(6.21)
and putting this in (6.16) leads to
px = −A
γ Mh
∆p
1
+
fx
l 1 + Mh
D 1 + Mh
(6.22)
Integrating (6.22) over 0 to l gives
Zl
0
∆p
px dx = −A
l
Zl
0
γ
dx
+
1 + Mh
D
Zl
0
Mh
fx dx
1 + Mh
and since M h ∼ 0.1 << 1, the above expression reduces to
Zl
0
∆p
px dx = −A
l
Zl
dx
0
Via the boundary conditions for the liquid pressure, we get A = 1. Hence, (6.20) becomes
∆p
γ
= − f ′ (h)hx
l
D
at x = 0
(6.23)
Summary of dimensional problem
The full dimensional problem for the average liquid height may be summarised as follows.
φ
k ∂
∂h
= −
∂t
µg ∂x
( h [ ∆pl + Dγ f (h)h ])
′
γ
∆p
= − f ′ (h)hx
l
D
h = h0
at x = l
23
at x = 0
x
(6.24)
(6.25)
(6.26)
Non-dimensionalisation
We define the dimensionless variables
^;
h = Dh
x = l^
x;
t =
φ D µg l2
^t
kγ
and substituting them into (6.24) leads to
^
^
∂h
∂ ^
^ ∂h )
= −
h α + f ′ (h)
∂^
x
∂^
x
∂^t
( [
])
(6.27)
where α = (∆p D)/γ is a dimensionless parameter. Boundary conditions (6.25) and (6.26)
become
^
^ ∂h = α
−f ′ (h)
∂^
x
^ = h
^0
h
at x
^=0
at x
^=1
(6.28)
(6.29)
^ 0 = h0 /D.
where h
Steady state solution
^ the time derivative term in
Assuming there is a steady state solution for the liquid height, h,
(6.27) vanishes to give
^
d ^
^ dh
h α + f ′ (h)
d^
x
d^
x
( [
]) = 0
(6.30)
Integrating both sides of the above equation with respect to x
^ leads to
^
h
( α + f (h)^ dd^hx^ ) = C
′
2
(6.31)
where C2 is an integration constant. Via boundary condition (6.28) we get that C2 equals zero
24
to give
^
h
( α + f (h)^ dd^hx^ ) = 0
′
(6.32)
which implies that
^ = 0;
h
or
^
^ dh = 0
α + f ′ (h)
d^
x
(6.33)
If follows that the solution to (6.33) is
f = −α x
^ + C3
(6.34)
where C3 is an integration constant. Using (6.29) we get that
^ = f(h^0) = 0
f(h)
at x
^=1
which gives the following two steady state solutions
^ = 0
h
f(^
x) = α (1 − x
^)
(6.35)
(6.36)
The first of the steady state solutions simply states that the water-air interface becomes pinned
to the top of the pillars and that the interface is horizontal. The second of the steady state
solutions is a linear, negative relationship between the function f and the dimensionless length
along the microchannel, x
^. From (6.2) we have that f(h) ∼ p − Π. Hence, solution (6.36)
suggests as x
^ decreases, f increases and hence, p − Π increases. This increase in the function
f is associated with the interval from T1 to T2 in figure (10). Thus, solution (6.36) also only
explains a steady state scenario where the water-air interface is pinned to the top of the pillar.
In addition, while the interface is pinned the curvature of the interface, changes sign. In other
words, the interface goes from being concave to convex. Moreover, solution (6.36) does not
provide any information about f as a function of h and how the interface evolves as it moves
down between the pillars.
To summarise, in the case of α < β (were β is the max value of f(h) and the pressure difference across the water-air interface), the steady state solutions only describe cases where the
25
interface is pinned to the top of the pillar. If ∆p, the pressure difference across the length of
the microchannel becomes sufficiently large, then α > β and the interface will move down in
between the pillars. A full time dependent solution of (6.27) is necessary to characterise the
interface evolution between the pillars.
7
Conclusions and future work
This report outlines a mathematical model used to describe an experimental setup presented
by the Stokes Institute. In particular we are interested in modelling the interaction between
a microchannel whose surfaces are superhydrophobic and deionised water flowing through the
microchannel subject to a pressure gradient. In section (2) we reviewed the various possible
wetting states associated with superhydrophobic surfaces. Section (3) summarises the time
scales involved in the process. The key result in this section being that it is possible for air
to get trapped amongst the microchannel pillars underneath the advancing water front. In
section (4) classical asymptotic results for the shape of liquid meniscus in contact with the
curved surface of a cylinder are used to estimate the distance the water-air interface will drop
between two pillars. Simple two-dimensional, geometrical arguments in section (5) are used
to establish a criteria for which the advancing water front will go down between the pillars.
In the final section we considered a pressure driven Poiseuille flow over the top of the pillars.
Lubrication theory provided the downward velocity of the water at the top of the pillars. This
result along with Darcy’s Law was used to derive a nonlinear Richards type equation for the
average water-air interface height. Steady state solutions were derived and the validity of these
solutions was discussed.
Future work will include a more thorough derivation of the function f(h) which is proportional
to the pressure difference across the water-air interface, p − Π. f(h) may be approximated via
hyperbolic functions of the form (6.3). More careful analysis and collaboration with the Stokes
Institute is required to determine the exact nature of the function f(h). A full time dependent
solution to equation (6.27) is also desirable as the steady state solutions do not provide a sufficient description of the water-air interface progression between the pillars. Finally, this report
does not account for the effect of the scallops on the surface of the pillars in determining the
water-air contact line. Any future model must address the interaction between the scallops and
the water-air interface.
Acknowledgments
We acknowledge the support of the Mathematics Applications Consortium for Science and
Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics initiative
grant 06/MI/005.
26
Nomenclature
(x, y, z)
u = (u, v, w)
(p, Π)
κ
h
φ
t
q
(b, l, d)
(R, h0)
D
U
Lc
γ
θ
(µl , µg)
k
∆p
α
β
^0
h
Cartesian coordinates vector
Velocity vector
(Water/air) pressure
Curvature
Average height of liquid above pillars
Pillar free fraction
Time
Height flux
Microchannel (breadth/length/height)
Pillar (radius/height)
Gap between pillars (centre to centre)
Water velocity
Capillary length of water
Surface tension of water
Contact angle
(Water/air) dynamic viscosity
Permeability
Water pressure difference across
the array
(∆p D)/γ
Maximum f(h) value
h0 /D
27
(10−2 /10−2 /8 × 10−5 )
(7.5 × 10−6 /2.5 × 10−5 )
2 × 10−5
0.012
2.73 × 10−3
0.073
(8.9 × 10−4 /1.8 × 10−5 )
4 × 10−12
100
100
0.0274
1.25
m
ms−1
Nm−2
m−1
m
ND
s
ms−1
m
m
m
ms−1
m
Nm−1
ND
kg m−1 s−1
m2
Pa
Pa
ND
ND
Bibliography
[1] J .Zhang and D.Y. Kwok. Contact line and contact angle dynamics in superhydrophobic
channels Langmuir, 22(11):4998-5004, 2006.
[2] A. Dupuis and Y.M. Yeomans, Modeling droplets on superhydrophobic surfaces: equilibrium states and transitions. Langmuir, 21(6):2624-2629, 2005.
[3] C. Dorrer and Ruḧe. Some thoughts on superhydrophobic wetting. Soft Matter, 5(1):51-61,
2008.
[4] R. Enright, T. Dalton, T.N. Krupenkin et al. Effects of interfacial position on drag reduction in a superhydrophobic microchannel. 6th Int. Conf. Nanochannels, Microchannels,
and Minichannels, June 23-25:835-845, 2008.
[5] T. Young, An essay on the cohesion of fluids, Philos. T. Roy. Soc. Lon., 95:65-87, 1805.
[6] S. Wang and L. Jiang, Definition of superhydrophobic states. Adv. Mater., 19:3423-3424,
2007.
[7] R.N.Wenzel. Resistance of solid surfaces to wetting by water. Ind. Chem. Eng., 28(8):988995, 1936.
[8] A.B.D. Cassie and S. Baxter. Wettability of porous surfaces. Trans. Faraday Soc., 40:546551, 1944.
[9] S. Moulinet and D. Bartolo, Life and death of a fakir droplet: Impalement transitions on
superhydrophobic surfaces. Eur. Phys. J. E, 24(3):251-260, 2007.
[10] D. James. The meniscus on the outside of a small circular cylinder. J. Fluid Mech.,
63(4):657-664, 1974.
[11] L.L. Lo. The meniscus on a needle - a lesson in matching. J. Fluid Mech., 132:65-78, 1983.
[12] L.A. Richards. Capillary conduction of liquids through porous mediums. Physics 1,
318:318-333, 1931.
28
The Effect of Mechanical Loading on the
Frequency of an Oscillator Circuit
Report Contributors: Jonathan A. Ward1,2,
Vladimir Lapin1 and William Lee1
Study Group Contributors: Chris Budd3, Mark Cooker4 ,
Paul J. Dellar 5, Martin Hayes6 , Poul Hjorth7 ,
Olga Korostynska8 and Arno Mayrhofer1
Industry Representative: Sandra Healy9
1
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
Report coordinator, jonathan.ward@ul.ie
3
School of Mathematical Sciences, University of Bath, United Kingdom
4
School of Mathematics, University of East Anglia, United Kingdom
5
OCIAM, Mathematical Institute, University of Oxford, United Kingdom
6
Department of Electronic and Computer Engineering, University of Limerick, Ireland
7
Department of Mathematics, Technical University of Denmark, Denmark
8
Microelectronic and Semiconductor Research Centre, University of Limerick, Ireland
9
Analog Devices, Raheen Industrial Estate, Raheen, Limerick, Ireland
2
Abstract
We investigate the effect of mechanical strain on the frequency of an electronic oscillator embedded in an integrated circuit. This analysis is aimed at explaining a 1% inaccuracy in the
oscillator frequency under test conditions prescribed by a leading supplier of semi-conductor
devices. During the test the package containing the oscillator was clamped to a circuit board
by mechanical pressure. By considering the nature of the oscillator we show that tensile strains
of the order of 10−4 could explain the observations via the piezoresistance effect. Both a simple
one-dimensional analysis based on the beam equation and an elastic finite element simulation
show that strains of this magnitude can be generated during the test.
30
1
Introduction
Silicon chips, or Integrated Circuits (ICs), perform the underlying functions of most modern
electronic devices. These miniature circuit boards consist of millions of electronic devices such
as transistors, capacitors, resistors and diodes. During manufacturing, the components in ICs
are built up and connected together in layers on the surface of a silicon substrate, known as a
die, via etching, deposition and photo-lithography [1]. The die are then moulded into a plastic
case, which we refer to as a package.
Chips are typically tested before being sold and may be moved mechanically between a variety
of test sites. During testing, a downward force of the order of 100 N is exerted on the chip to
ensure good electrical contact. A particular chip manufactured by a leading semi-conductor
supplier (who we refer to simply as “the supplier”), contains an oscillator circuit whose frequency
must reach a ±1% accuracy specification over a temperature range of −40 to +105◦ C and 100%
of the chips are tested. However, the supplier observed that the loads experienced by the chip
could lead to significant errors in the accuracy of the oscillator circuit, possibly due to some
sort of piezo effect.
In this paper we investigate how a vertical load placed on such a chip might result in errors in
the oscillator frequency. This problem was presented at the 70th European Study Group with
Industry (ESGI), hosted by MACSI at the University of Limerick in 2009, funded by the Science
Foundation Ireland mathematics initiative grant. In Section 2 we give a detailed description
of the problem including the structure of the package, the test-rig set-up, data concerning the
frequency response to loading and how the chips are “trimmed” to the required frequency. We
describe in Section 3 how the key components in the oscillator circuit, namely capacitors and
resistors, might affect the oscillator frequency when subjected to a vertical load. In Section 4
we discuss mechanical modelling of the elastic response of the package to loading using the
beam equation and finite element elasticity calculations. Finally we summarise our results in
Section 5.
2
2.1
Detailed Problem Description and Data
Package Design
Due to commercial sensitivity, details of the specific package and chip design cannot be presented in this paper. A sketch of the package is illustrated in Figure 1(a). There are two die
in each package, one of which is smaller and sits on top of the larger (see Figure 1(b)). These
are placed on a copper base known as the lead frame. The three layers are aligned at their
centres and attached using epoxy adhesives. There are 8 contacts along each side of the base
of the package that are connected to the die via small wires. The die, lead frame, contacts and
connections are moulded into a plastic case.
31
(c)
(a)
6mm
6mm
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Bottom Die
(b)
Oscillator
circuit
Contact
Lead Frame
(d)
Plastic Case
Top Die
11111111111111111111111111111111111111
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00000000000000000000000000000000000000
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Contact Glue
Test-Rig Base
1mm
Socket
Lead Frame Contact
Figure 1: (a) Sketch of the base of the package (not to scale) illustrating the lead frame and
contact positions. The relative positions inside the package of the top and bottom die (dashed
boxes) and the oscillator circuit (shaded grey box) are indicated. (b) Cross-section of the
package (larger scale than (a)). Again the positions of the lead frame, contacts and the top
and bottom die are marked. The epoxy glues are also labelled. (c) Photo of the base of the
test-rig in which the package is placed during testing. (d) Magnification of the socket in which
the package is placed and the electrical contact pins on which it sits.
2.2
Test Rig Set-Up
During testing, each package is transported between test sites (see Figure 1(c)) using a mechanical test rig arm. This device places a small cylindrical nozzle on top of each package and
then applies a suction force to attach the package to the arm before it is relocated. In each
test, an individual package is placed in a plastic well on the test circuit board, where it rests
on top of a set of copper pins that connect with the contacts on the base of the package (see
Figure 1(d)). There are three pins positioned slightly off centre that contact the lead frame
and one for each of the contacts around the edge of the package. The test rig arm exerts a
vertical force on the package to ensure good electric contact, but the magnitude of this force is
not known accurately and varies between different test rigs.
2.3
Problems with Frequency Response Due to Loading
As stated, the oscillator circuit in the chip must oscillate at a frequency of 128 kHz with a
±1% tolerance. However the supplier found that this error could not be reduced below 1.5%,
which they suspected was due to the loads placed on the chip. This was confirmed using a
hand test rig consisting of the standard test rig base and a locking top that applies a vertical
32
% Frequency Change
% Frequency Change vs. Applied Mass
Force (kg equivalent)
Figure 2: Data recorded by Analog Devices relating percentage change in frequency to applied
mass (i.e. the load placed on top of the package). A decreasing linear response is observed.
Note that a load of the order of 100 N (i.e. 10 kg) causes approximately a −0.3% error. Thus
the testing process alone wastes a significant amount of the specified tolerance.
force to the chip via a hand operated screw mechanism. Although variations in frequency were
observed during the hand test, the amount of force applied during the test was not quantifiable
in any way. Furthermore, the hand test rig is not used during the actual testing of the chips,
so is only an indication that loading of the chip affects the oscillator frequency.
To quantify the effect caused by loading the package, the supplier collected a small amount of
preliminary quantitative data, illustrated in Figure 2, by applying a range of vertical loads to
a given package. The results of this experiment suggest a linear relationship between the load
applied and the frequency response of the oscillator circuit, although it is stressed that the data
is only indicative. Note that these experiments indicate that for a load of the order of 100 N
(i.e. 10 kg), a −0.3% error is observed. Thus a significant amount of the specified tolerance is
wasted on the testing process alone, which must be compensated for by higher manufacturing
precision.
2.4
Circuit Trimming
Banks of components (including resistors and capacitors) are used to tune the oscillator to the
required frequency over the given temperature range. This process is known as trimming, and
in the traditional sense would mean that particular components were removed permanently
from the active circuit. In this instance, a flash memory stores the identities of the trimmed
components.
33
3
Piezo Effects in the Oscillator Circuit
We now describe how the response of the key components in the oscillator circuit might be
affected by mechanical loading. An oscillator circuit provides a continuous, synchronised trigger
signal to the rest of the devices on the chip from which information is processed [2]. Due to
commercial sensitivity (and its shear complexity), the specific design of the oscillator circuit
in question was not provided. However, there are two main types of electronic oscillators, RC
circuits (composed of Resistors and C apacitors) and LC circuits (composed of inductors and
capacitors). Inductors are not common in ICs, hence we focus on the first type of oscillator.
The oscillation frequency of an RC circuit is f = 1/(2πRC) [1], where R and C are the effective
resistance and capacitance of the banks of resistors and capacitors on the chip respectively.
Typical values of resistance and capacitance are R = 10kΩ and C = 100pF. A small fractional
increase, δ, in the product of the resistance and capacitance results in a fractional decrease in
frequency, f, of the same amount,
f=
1
1
≈
(1 − δ).
2πRC(1 + δ)
2πRC
(3.1)
Thus a 1% drop in frequency requires a 1% increase in capacitance or resistance (i.e. δ = 0.01).
We now consider whether the kind of mechanical load experienced by the chip can give rise to
such changes in the capacitance or resistance.
3.1
Capacitor Response
Capacitors consist of two parallel conducting plates with a dielectric material sandwiched between. The plane of the conducting plates is perpendicular to the applied load in this case.
The value of capacitance is given by
εAC
,
(3.2)
C=
d
where ε is the permittivity of the dielectric, AC is the area of the plates and d is the distance
between them. Thus the capacitance increases in response to a decrease in distance between
plates, which might be expected when subjected to a vertical load as in this scenario.
However, the dielectric material is typically silicon dioxide, which is a very stiff material. Given
a 100N force placed on a package approximately 6mm×6mm, the resulting pressure P is 4 MPa.
The Young’s modulus E of silicon dioxide is approximately 100 GPa, thus considering Hooke’s
law, the resulting strain (and hence the change in distance between plates) is
P
≃ 3 × 10−5 ,
E
which is much smaller then the 1.5% change required.
34
(3.3)
Thus the change in capacitance due to the plates being forced closer together cannot be responsible for the observed 1.5% change in frequency. The deformation of the dielectric does
not change its permittivity either, since silicon dioxide does exhibit any piezoelectric effects [3].
3.2
Resistor Response
Piezoresistance is a well known effect that occurs when resistors are subject to mechanical
deformation [4]. The resistance of a conducting element with a fixed cross section of area AR
and length l is given by
l
R0 = ρ0
,
(3.4)
AR
where ρ0 is resistivity. If the shape of the resistor is changed, the subsequent change in resistance
is related to the strain ǫ = ∆l/l via
∆R
= Gǫ,
(3.5)
R0
where G is known as the gauge factor,
G := (1 + 2ν) +
(∆ρ/ρ0)
,
ǫ
(3.6)
and ν is the Poisson’s ratio of the resistor material, which accounts for the change in crosssectional area due to the change in length. It is known that the resistivity change (∆ρ/ρ0 )/ǫ for
semiconductors is much larger than the dimensional change (1 + 2ν) [5]. In fact gauge factors
as large as 100 have been measured for p-type silicon and as low as −100 in n-type silicon [4, 6].
Thus to observe a 1% decrease in frequency due to piezoresistance, we would need strains of
the order of 10−4 .
4
Mechanical Modelling
To calculate the strains induced by mechanical loading, we first consider a simple method in
which the package is modelled using a one-dimensional beam equation. This approach neglects
inhomogeneities in the package and corner effects that arise in the plane perpendicular to the
loading due to the stacking of the die. Thus we develop a more sophisticated two-dimensional
axisymmetric model of the composite package that we solve using finite element software in
COMSOL, from which we obtain the strain field in a cross-section of the package.
4.1
One-Dimensional Model
A simple approach to calculating approximate values of the strain in the package is to model
the entire assembly using the one-dimensional beam equation [7]. Thus we consider the package
35
Figure 3: Geometry in which the package is modelled as an elastic beam. The pins are modelled
as Hookean springs, while the die is modelled as two localised forces.
to be elastically homogeneous and made entirely of silicon. The geometry used in this case is
shown in Figure 3. The width of the package is 2A and the load is represented by two localised
forces F imposed at x = ±B. The pins are modelled as Hookean springs, positioned at the edges
of the package (x = ±A) and the centre (x = 0) with spring constants k1 and k2 respectively.
The spring constant of the outer pins is chosen to be five times larger than at the centre to
reflect that there are more pins around the edge of the package. However, the spring constants
are not known accurately,thus we consider two cases: (a), stiff springs, k1 = 500 MN/m and
k2 = 100 MN/m ; and (b), flexible springs, k1 = 50 MN/m and k2 = 10 MN/m .
As noted, we model the vertical displacement u(x) of the package using the one-dimensional
beam equation with point forcing,
d4 u
EI 4 = F [δ(x + B) + δ(x − B)] − k2 δ(x) u(x),
dx
(4.1)
where E is the Young’s modulus of the package and I is the second moment of area. The
boundary conditions at the edges of the package are
d3 u EI 3 = −k1 u(±A) and
(4.2)
dx x=±A
d2 u = 0.
(4.3)
dx2 x=±A
Note that at each discontinuity, the zeroth, first and second derivatives must be equal. These
equations form a linear system whose solution is a piecewise cubic polynomial in x. The
polynomial coefficients in each of the four regions along the beam can be easily calculated using
computer algebra, however they are somewhat complicated and hence we do not reproduce them
here.
The parameters used in the calculation are given in Table 1. The displacement along the beam
u(x) is illustrated in Figure 4(a) for the stiff springs case and in Figure 4(b) for the case flexible
springs case. These illustrate that the stiffness of the pins can affect the overall shape of the
package: stiffer pins result in a “W”–shape, whereas flexible pins give rise to a concave shape.
36
Parameter
A
B
h
E
F
Value
3 mm
2 mm
1 mm
150 GPa
50 N
Table 1: Values of parameters used in the calculation. In addition, two different types of spring
constant are considered: stiff springs, k1 = 500 MN/m and k2 = 100 MN/m ; and flexible
springs, k1 = 50 MN/m and k2 = 10 MN/m .
Figure 4: Displacement calculated from equations 4.1–4.3 for the cases (a) (stiff springs) and
(b) (flexible springs).
Given the displacement and the height of the package h, we can calculate the tensile strain,
ǫxx
h d2 u
,
=
2 dx2
(4.4)
at the base of the package. This is plotted in Figure 5(a) and (b) for the stiff and flexible pin
cases respectively. In both instances, the magnitude of the strains are large enough to account
for the observed frequency variations due to piezoresistive effects. The stiffer springs used in
Figure 4(a) result in a region of compression at the centre of the package. Note that in both
cases, the largest strains occur in regions of extension (positive strain) where the downward
load is applied. The strain is zero at the edges due to the boundary conditions (4.3).
4.2
Axisymmetric Finite Element Model
Although the simple approach using the beam equation shows some promising results, it is
based on a very restrictive set of assumptions, in particular that the package is homogeneous
and can be represented as thin rod. A linear elasticity model of the composite package can be
37
Figure 5: Tensile strain at the base of the package for cases (a) (stiff springs) and (b) (flexible
springs).
computed numerically using finite element methods. To facilitate this, we used the Structural
Mechanics Module of COMSOL 3.4 software package [8]. This includes a stationary linear
solver for the equilibrium equation from linear elasticity theory,
σij,j = 0
where σij,j is the derivative with respect to j of the ij component of the stress tensor. This is
supplemented with the linear stress-strain and strain-displacement relationships,
σij = λǫkk δij + 2µǫij ,
ǫij = 1/2 (ui,j + uj,i ) ,
respectively where ǫ is the strain tensor, u is the displacement and λ and µ are Lamé’s parameters, which can be expressed in terms of the Poisson’s ratio and Young’s modulus. We used
2nd-order Lagrange finite elements with ideal constraints.
The simulation set-up is illustrated in Figure 6. We consider an axisymmetric model in cylindrical coordinates to reduce the complexity of the simulation. Consequently the model of the
package is disc shaped. In this geometry, we model the full composite structure of the package
including the copper lead frame and contacts, the silicon die, the epoxy glues and the plastic
casing. Material properties of each of the component parts are listed in Table 2. The load is
applied over a 1mm wide annular region on top of the package. The upward forces exerted by
the pins are positioned under the contact on the outside edge of the bottom of the package and
at the centre under the lead frame. We distribute the downward load between the inner and
outer pins with a 1:5 ratio respectively to reflect the fact that there are more pins around the
outside of the package. This is an approximation since the pins are not completely stiff and
hence may distribute the load differently. The point at the base of the edge of the package is
held fixed.
In Figure 7, we illustrate the deformation under loading of a cross-section of the package. Note
that the scale of the displacements have been increased to accentuate the change in shape. We
38
z
P
(g)
(e)
(a)
(f)
(b)
(c)
(d)
F0
FL
r
Figure 6: Cross-section of the numerical simulation set-up for the linear elasticity calculation
in axisymmetric coordinates. The composite package is modelled, including: (a) top die, (b)
bottom die, (c) lead frame, (d) contact, (e) and (f) epoxy glues, (g) plastic case. The applied
load is labelled P and the reaction forces from the pins on the contacts and lead frame are
labelled FL and FL respectively. These forces are distributed over the coloured regions.
observe that the maximum displacement (shaded in dark red) occurs where the downward force
is applied and, due to the boundary conditions, there is no displacement at the bottom of the
outer edge (shaded dark blue). The centre of the package (r = 0) is also displaced vertically
since there is less support from the pins. Thus a slice through the full diameter of the package
would resemble Figure 4(a), i.e a W–shape. Furthermore, if we reduce the load supported at the
centre of the package, we find that the deformed package becomes concave, as in Figure 4(b).
The ǫrr strain field in the lead frame, top and bottom die are illustrated in Figure 8. We have
omitted the strain field in the rest of the package since the circuitry lies in a thin layer on
the surfaces of the two die, i.e. in the (r, θ) plane. Furthermore, the plastic casing and epoxy
glues are much softer than the die and lead frame and consequently the magnitudes of their
strains are much larger. Note that the strains on the top surfaces of the die where the electrical
components are placed are much less on the top die than on the bottom. This accounts for the
fact that the problems with the accuracy of the frequency of the oscillator circuit came to light
after it was moved from the top die to the bottom die.
39
Young’s Modulus
E, GPa
Lead frame (Cu)
127
Contacts (Cu)
127
Die (Si)
150
Epoxy Glue - bottom 3.1
Epoxy Glue - top
0.3
Plastic case
0.2
Table 2: Material properties used in linear elasticity simulation.
nm
z
r
Figure 7: Deformation of the package under loading. The black out line illustrates the original
position and the shaded region shows the subsequent deformation. The colour correspond to
the magnitude of the total displacement. These have been greatly enhanced in this picture to
illustrate the qualitative behaviour of the package.
40
z
r
Figure 8: The coloured shading indicates the ǫrr strain field in the lead frame, top and bottom
die. Note that the lead frame is stretched, however the bottom die is compressed.
(i)
ǫrr
ǫθθ
r
(ii)
r
Figure 9: The strains ǫrr (a) and ǫθθ (b) along the top surface of the bottom die in the rdirection.
41
In Figures 9(a) and (b) we plot the ǫrr and ǫθθ strains respectively against r at the z–position
corresponding to the top surface of the bottom die. The oscillator circuit lies in this plane,
between the edge of the top die (r = 0.75) and the edge of the bottom die (r = 1.5). The
magnitude of the strains there are large enough to account for the observed change in frequency
due to the piezoresistive effect. Note that because of the W–shape of the package, there is a
change in the sign of the strain. Furthermore, toward the outer edge of the package, the
ǫθθ strain is less than the ǫrr strain. Thus a qualitative feature of our simulation results
is that an appropriate choice of position and orientation of the components of the oscillator
circuit on the bottom die would minimise the strain they experience. We stress however that
computing the strains within the package with enough precision to achieve this would require
extending the geometry to three dimensions in rectangular coordinates (which requires far
greater computational power) as well as a detailed knowledge of the components themselves.
5
Conclusion and Recommendations
This report investigates the possibility that inaccuracies in the frequency of an electronic oscillator measured under test conditions may be due to mechanical deformation of the chip
containing the oscillator that occur during testing. We show that the most likely explanation
is that the oscillator frequency is modified by the piezoresistive effect and estimate that the
observed variations could be produced by tensile strains of the order of 10−4 . We carried out
two elasticity calculations: a highly simplified calculation based on the beam equation and a
second, considerably more sophisticated, treatment using finite element analysis. These simulations both demonstrated that strains the of the order of 10−4 in magnitude occur within the
die, which explains the drift in oscillator frequency during test conditions.
From our simulations, we observe that the strain is largest directly underneath the nozzle that
pushes down on the package and that the strains are significantly lower in the top die. Thus
one might consider relocating the oscillator circuit or changing the shape of the nozzle that
applies the downward force in order to reduce the strain in the package. Another possible
solution could involve tuning the method in which the oscillator circuit is trimmed such that
components which undergo lower values of strain are chosen preferentially.
Acknowledgements
All contributors would like to thank Sandra Healy from Analog Devices for introducing the
problem and assisting in answering questions during the entire week. We acknowledge the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie)
funded by the Science Foundation Ireland mathematics initiative grant 06/MI/005.
42
Bibliography
[1] A.R. Hambley. Electronics. Prentice-Hall, second edition, 2000.
[2] Colin D. Simpson. Industrial Electronics. Prentice-Hall, first edition, 1996.
[3] L.D. Landau, E.M. Liftshitz, and L.P. Pitaevskii. Electrodynamics of Continuous Media.
Butterworth-Heinemann, second edition, 1984.
[4] S. Middelhoek and S.A. Audet. Silicon Sensors. Academic Press Ltd., first edition, 1989.
[5] Yozo Kanda. Piezoresistance effect of silicon. Sensor Actuat. A, 28:83–91, 1991.
[6] Charles S. Smith. Piezoresistance effect in germanium and silicon. Phys. Rev., 94(1):42–49,
1954.
[7] L.D. Landau and E.M. Liftshitz. Theory of Elasticity. Butterworth-Heinemann, third
edition, 1986.
[8] COMSOL AB. COMSOL Multiphysics Modeling Guide. COMSOL Ltd, 3.4 edition, 2007.
43
44
Designing a Green Roof for Ireland
Report Contributors: Ian Hewitt1 , Andrew Lacey2,
Niklas Mellgren 3, Michael Vynnycky4,
Marguerite Robinson4,5 and Mark Cooker6
Study Group Contributors: Andrew Fowler4 , Maria Gonzalez4,
Hilda Simboek4, Thomas Murphy 7, Michael Devereux4, Roman
Sedakov4 , Grainne Kirby 4, Gemma Fay 1 and Qi Wang 8
Industry Representatives:
Catherine Adley 7, Billy Kirwan
1
9
and John O’Hara
10
OCIAM, Mathematical Institute, University of Oxford, United Kingdom
Department of Mathematics, University Heriot-Watt, United Kingdom
3
Department of Mechanics, KTH Royal Institute of Technology, Sweden
4
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
5
Report coordinator, marguerite.robinson@ul.ie
6
School of Mathematics, University of East Anglia, United Kingdom
7
Department of Chemical and Environmental Science, University of Limerick, Ireland
8
School of mathematical sciences, Dublin Institute of Technology, Ireland
9
GFM Systems, Waterway House, Crag Crescent, Clondalkin Ind. Estate, Dublin 22, Ireland
10
Landtech Soils Ltd, 20 Kenyon Street, Nenagh, Co. Tipperary, Ireland
2
Abstract
A model is presented for the gravity-driven flow of rainwater descending through the soil layer
of a green roof, treated as a porous medium on a flat permeable surface representing an efficient
drainage layer. A fully saturated zone is shown to occur. It is typically a thin layer, relative
to the total soil thickness, and lies at the bottom of the soil layer. This provides a bottom
boundary condition for the partially saturated upper zone. It is shown that after the onset of
rainfall, well-defined fronts of water can descend through the soil layer. Also the rainwater flow
is relatively quick compared with the moisture uptake by the roots of the plants in the roof.
In a separate model the exchanges of water are described between the (smaller-scale) porous
granules of soil, the roots and the rainwater in the inter-granule pores.
46
1
Introduction
Green roofs are becoming increasingly popular around the world. The many benefits of a green
roof include assistance in the management of storm water, pollution control, building insulation,
recycling of carbon dioxide, in addition to being aesthetically pleasing. A green roof is subject
to various stresses from the weather, in particular wind-loading, which we ignore in this report,
and rainfall: it is the flow, drainage and uptake of rainwater that we model. An understanding
of where the water goes is essential to design a roof able to achieve sustained healthy plants
and loads that lie within the safe capacity of the supporting structure.
The main focus of this report is on the transport of water through the green roof structure.
Inadequate drainage can lead to the undesirable occurrence of a fully saturated soil which will
cut off the air supply to the plants. Conversely, if the saturation levels are too low plants will
die from lack of water. Ideally a degree of saturation that is less than eighty per cent should
be maintained at all times. Our goal is to model the distribution of the degree of saturation
through the depth of the soil layer, and to see how it changes due to spells of rain, and under
the influence of moisture-uptake by plant roots.
The basic structure of a common green roof is shown in Fig. 1. A waterproof root barrier
protects the underlying roof structure. A drainage layer sits atop this barrier. The typical
thickness of this layer is 8/15/20 mm depending on the type of roof. The soil and drainage
layers are separated by a thin sheet of perforated hard plastic containing holes approximately 2
mm in diameter and spaced 2 cm apart. There are two layers of soil at the top of the structure
separated by a layer of felt. A thin layer (< 2 cm) of refined rooting soil contains the plant
life, mainly sedum for thinner roofs and, for thicker ones, low growing grasses such as common
bent grass and/or other plants, such as cowslip and ladies bedstraw. Beneath the rooting soil
are pellets of lightweight expanded clay. This layer is 5-10 cm thick. Grain sizes are typically
< 2 mm for rooting soil and 4-8 mm for expanded clay pellets.
2
The Model
We model the dynamics of water flow through the soil layer. We consider a single soil layer
with thickness L ≈ 10−1 m and we ignore the presence of the felt layer. We assume that the
soil-drainage-layer interface is located at z = 0 and the soil surface at z = L. We consider two
possible scenarios: (i ) the entire region 0 ≤ z ≤ L is unsaturated, so that the soil saturation S is
always less than 1 and (ii ) a saturated region forms at the bottom of the soil layer for 0 ≤ z ≤ h.
Note that the model as presented here is one-dimensional, and represents a horizontal roof, but
can be easily extended to two (or three) dimensions, and to account for sloping roofs.
47
Figure 1: Green roof structure.
2.1
The Unsaturated Region
We first assume the entire region 0 ≤ z ≤ L is unsaturated (S < 1). The basic model for this
region follows that outlined in [1] and [2]. The one-dimensional Richards’ equation for water
flow in the unsaturated soil is
∂S
∂
∂S
D0 D(S)
=
+ K0 K(S) − R,
(2.1)
φ
∂t
∂z
∂z
where S = S(z, t), φ is the constant porosity of the soil, taken here to be 0.25, D0 D(S) and
K0 K(S) are the water diffusivity and hydraulic conductivity respectively, with the functions
D(S) and K(S) given by
K(S) = S1/2 [1 − (1 − S1/m )m ]2 ,
(2.2)
D(S) =
[1 − (1 − S1/m )m ]2
,
S1/m−1/2 (1 − S1/m )m
(2.3)
where 0 < m < 1. The value of m for the expanded-clay soil was not known but, for later use
in simulations and analysis of the model, was taken to be m = 12 . Likewise the values of the
constants K0 , the conductivity for saturated soil, and D0, a representative value of diffusivity,
were not known. The water flux in the z direction is q = −(D0 D(S)Sz +K0 K(S)). Water uptake
by the plant roots is incorporated into the model through the last term in (2.1) and is given by
R = 2πakr ld (pa − pc f(S) − pr ) ,
(2.4)
where kr is the radial conductivity of water, a is the root radius, ld is the root length density,
pa is atmospheric pressure, pr is an effective pressure in the roots (although it can be negative),
pc f(S) is the capillary pressure in the soil, with pc another constant characterising the partly
saturated pellets, and
1
f(S) = (S− m − 1)1−m .
(2.5)
48
We take parameter values from Roose and Fowler [2] and let 2πakr = 7.85 × 10−16 m2 s−1
Pa−1 , ld = 5 × 103 m−2 and pc = 104 N m−2 . The root pressure pr will be determined from
conservation of water within the root. Finally we must prescribe boundary conditions at the
top and bottom of the soil layer. At the soil surface we take
D0 D(S)
∂S
+ K0 K(S) = Qin (t) at z = L,
∂z
(2.6)
where Qin is the rainfall rate averaged over the surface area of the ground. We assume, in this
unsaturated case, no outflow at the base of the soil layer and set
D0 D(S)
∂S
+ K0 K(S) = 0 at z = 0.
∂z
(2.7)
We nondimensionalise the equations by scaling
z = L^
z,
pr = |P|p^r ,
t=
L
^t,
K0
p = pa + pc p
^,
^
R = 2πakr ld |P|R,
^
Qin = Qtyp Q,
(2.8)
where P is the (negative) root pressure at the soil surface and we set |P| = 106 N m−2 . The
time scale used here is that for flow though the soil layer under the action of gravity. The
dimensionless Richards’ equation (2.1) then has the form
∂S
∂S
∂
φ
δD(S)
+ K(S) − η(θ − εf(S) − p
^ r ),
(2.9)
=
∂^
z
∂^
z
∂^t
where
δ=
D0
≈ 10−4 ,
LK0
η=
2πakr ld |P|L
≈ 4 × 10−6 ,
K0
θ=
pa
≈ 10−1 ,
|P|
ε=
pc
≈ 10−2 . (2.10)
|P|
Roose and Fowler [2] give values of D0 for different soil types and we can reasonably take D0 =
10−6 m2 s−1 . However, the value of K0 is more difficult to determine as it varies significantly
with different soil types. The parameter values in (2.10) are given for K0 = 10−1 m s−1 . We note
that η ≪ 1 suggesting that water uptake by the roots is negligible over the chosen timescale.
The dimensionless forms of the boundary conditions are given by
δD(S)
∂S
^ at z^ = 1,
+ K(S) = νQ
∂^
z
(2.11)
∂S
+ K(S) = 0 at z^ = 0,
∂^
z
(2.12)
δD(S)
where
ν=
Qtyp
≈ 3 × 10−6 ,
K0
(2.13)
with Qtyp taken to be some typical rainfall. We set Qtyp = 3 × 10−7 m s−1 for a “wet day" in
Ireland.
49
2.2
The Saturated Region
When the soil becomes saturated we assume that a moving boundary forms at z = h(t) and
the soil saturation is identically one for z ∈ [0, h]. Our governing equation can now be written
in the (dimensionless) form
∂
1 ∂^
p
1+
− η (θ + ε^
p−p
^ r ) = 0,
(2.14)
∂^
z
γ ∂^
z
where
ρgL
≈ 10−1 .
(2.15)
pc
The flux through the membrane at z = 0 is prescribed to occur at a rate proportional to the
pressure difference across it; Qmem = κ(p − pa ) dimensionally, where pa is the atmospheric
pressure in the drainage layer beneath, p is the pressure at z = 0, and κ ≈ 10−5 m s−1 Pa−1
(determined experimentally in the next sub-section). This gives the dimensionless condition
γ=
1+
p
1 ∂^
= α^
p at z^ = 0,
γ ∂^
z
c
^ ^t) ≡
where α = κp
≈ 1. At the saturation front z^ = h(
K0
continuity of fluid flux requires
K(S) + δD(S)
∂S
1 ∂^
p
=1+
∂^
z
γ ∂^
z
(2.16)
h(t)
,
L
p
^ = 0 (atmospheric), and
^
at z^ = h.
(2.17)
^ along with
Neglecting the η term in (2.14) for this saturated region, and using p
^ = 0 at z^ = h
(2.16), gives
^ ^t) − z^)
γ(h(
p
^ (^
z, ^t) =
,
(2.18)
^ ^t)
1 + αγh(
so that (2.17) becomes
K(S) + δD(S)
^
αγh
∂S
=
^
∂^
z
1 + αγh
^
at z^ = h.
(2.19)
^ ^t) determine h
^ in terms
In principle equation (2.19) and boundary condition S = 1 at z^ = h(
of the flux from the unsaturated region. However, we can simplify things if we notice from
(2.11) that the dimensionless flux will in general be small, of order ν (due to the rainfall). If
^ required to satisfy (2.19) will be small. Physically, this is
this is the case, then the value of h
because for the typical size of fluid flux considered, the pressure required to force it through
the membrane according to (2.16) is provided by the hydrostatic head of a very thin layer of
water (dimensionally, h is calculated to be much less than 1 mm).
Thus if a saturated region is created at the bottom of the soil layer, it will quickly grow to a
depth which is sufficient to drain exactly the same amount of water through the membrane as
50
is arriving from the unsaturated region above. Provided this depth is substantially less than
the depth of the soil, the saturated region can be ‘collapsed’ (mathematically) onto the line
z^ = 0, and the boundary condition applied to the problem in the unsaturated zone for some of
the numerical solutions of sub-section 2.5 is then
∂
∂S
K(S) + δD(S)
= 0 at z^ = 0, 11
(2.20)
∂^
z
∂^
z
which states that whatever flux arrives there from above is allowed through the membrane (the
build-up of the small saturated layer which would physically enable this is therefore assumed
to happen instantaneously), with the uptake of water by the roots being neglected. We apply
(2.20) while K(S) + δD(S) ∂S
> 0 at z^ = 0 since water is trying to exit, not enter, the soil layer.
∂^
z
2.3
Experimental measurement of κ
The value of κ was deduced from a simple experiment, which involved puncturing a 2 mm
diameter hole in a plastic bottle, made with material similar to what we believe the drainage
membrane is made of. The rate of drainage through the hole driven by the hydraulic head
in the bottle was measured, and used to determine the coefficient of proportionality between
pressure difference across the membrane ∆p and the water flux through it q. Writing
q = k∆p,
(2.21)
where ∆p = ρgh, the water depth in the bottle, h, satisfies the equation
Abottle
dh
= −kρgh,
dt
(2.22)
where Abottle is the cross-sectional area of the bottle. Thus
log h = −
kρg
t.
Abottle
(2.23)
Measurements of h against t made during the experiment are in Fig. 2, and the best fit value
of the time constant tc = Abottle /kρg was 74 seconds. The flux through an individual hole can
be converted into an average velocity through a membrane, using the area of the membrane
Amembrane that is drained by each hole. Thus
ū = κ∆p,
κ=
Abottle
.
Amembrane ρgtc
(2.24)
Taking Amembrane = π cm2 , and using the cross-sectional area of the bottle Abottle = 25 cm2 ,
ρ = 103 kg m−3 , and g = 10 m s−2 , gives κ ≈ 10−5 m s−1 Pa−1 .
11
More accurately the bottom condition might be specified in linear complementary form (1 − S)q = 0 with
1 − S ≥ 0 (for no super-saturation) and q ≤ 0 (for upward flux). We use (2.20) for its convenience in some of
the numerical simulations, but for others we approximate the more correct linear complementary form, as in
(2.33).
51
1
h (cm)
10
0
20
40
60
t (s)
80
100
Figure 2: Experimental measurements of h against t.
2.4
The Root Pressure
To determine the root pressure pr in equation (2.9), we assume that the root extends through
the full thickness of the soil layer of depth L. Conservation of water inside the root yields
kz
d2 pr
+ 2πakr (pa − pc f(S) − pr ) = 0,
dz2
(2.25)
where kz = 10−14 m6 s−1 N−1 is the root axial conductivity and f(S) is defined in equation (2.5).
Zero axial flux at the root tip implies
dpr
+ ρg = 0 at z = 0.
dz
(2.26)
In addition we prescribe a driving pressure at the root base yielding
pr = pa + P
at z = L.
(2.27)
In dimensionless form the root pressure will satisfy
d2 p
^r
+ τ (θ − εf(S) − p
^ r ) = 0,
d^
z2
(2.28)
d^
pr
= −εγ at z^ = 0,
d^
z
(2.29)
subject to
52
p
^ r = θ − 1 at z^ = 1,
where
τ=
(2.30)
2πakr L2
≈ 10−3 .
kz
2
The parameters τ ≪ 1 and εγ ≪ 1 which implies dd^zp^2r ≈ 0 subject to
root pressure is thus given by
p
^ r = θ − 1.
(2.31)
d^
pr
d^
z
= 0 on z^ = 0. The
(2.32)
The complete model is now given by (2.9), with the definitions (2.2), (2.3), (2.5) and (2.32),
with boundary condition (2.11) at z^ = 1 and (2.12) if S < 1, or (2.20) if S = 1, at z^ = 0. An
initial condition is also needed.
The diffusion term in (2.9) is small, so the equation is essentially a first order non-linear wave
equation; the boundary condition (rainfall) is transmitted downwards as a wave. If rain starts
suddenly, there is a shock front that propagates quickly down to the bottom of the soil; if it
stops suddenly there is an expansion fan.
2.5
Numerical Solutions
The governing equation for the unsaturated region (2.9) was solved subject to boundary
conditions (2.11) and (2.12). As a first approach the η term in (2.9) is neglected so that
we are just considering drainage of the soil layer under gravity. The initial saturation was
taken to be uniform throughout the soil layer. Three different initial values of the saturation
Sinit : 0.05, 0.1, 0.15 were considered. The profiles obtained for S in each case when the computation was stopped are shown in Fig. 3; a corresponding semilog plot is shown in Fig. 4, in
order to demonstrate the boundary layer of thickness δ1/2 in S at z^ = 0 that is predicted by an
asymptotic analysis, and which is captured by the numerical solution, but which is not visible
in Fig. 3. For Sinit = 0.1 and 0.15, computations were stopped when the value of S at z^ = 0,
Sbottom , reached 1; for Sinit = 0.05, Sbottom is still far from 1, even for the value of dimensionless
time (100) shown here. The time evolution of Sbottom is shown in Fig. 5, while that for S at
z^ = 1, Stop , is shown in Fig. 6.
Thus, the results suggest an appreciable difference in the time at which complete saturation
is achieved at the bottom of the soil when Sinit is increased from 0.05 to 0.1. The effect of
the rainfall boundary condition (2.11) has (by the end of the simulations) only affected the
tiny region at the right of Fig. 3, where there is the beginning of a shock front propagating
downwards from z^ = 1; since δ has been taken to be very small, the shock looks very sharp,
and the values on either side of it are the initial condition (below, or left, of the shock), and the
^ (above, or right, of the shock - this value is expectedly independent
value given by K(S) = νQ
of the initial condition, as shown in Fig. 6).
53
1
sinit=0.05
0.9
sinit=0.1
sinit=0.15
0.8
0.7
s
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
z^
Figure 3: S vs. z^ for three different initial conditions (Sinit = 0.05, 0.1, 0.15) at either dimensionless time 100 (Sinit = 0.05) or when S reaches 1 at z^ = 0 (Sinit = 0.1, 0.15). Parameter
values are m = 1/2, δ = 10−4 , ν = 3 × 10−6 .
1
sinit=0.05
0.9
sinit=0.1
sinit=0.15
0.8
0.7
s
0.6
0.5
0.4
0.3
0.2
0.1
0
−10
10
−8
10
−6
−4
10
10
−2
10
0
10
z^
Figure 4: A semilog plot of S vs. z^ for three different initial conditions (Sinit = 0.05, 0.1, 0.15)
at either dimensionless time 100 (Sinit = 0.05) or when S reaches 1 at z^ = 0 (Sinit = 0.1, 0.15).
Parameter values are m = 1/2, δ = 10−4 , ν = 3 × 10−6 .
54
1
sinit=0.05
0.9
sinit=0.1
sinit=0.15
0.8
sbottom
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
Dimensionless time
Figure 5: Sbottom vs. dimensionless time for three different initial conditions (Sinit =
0.05, 0.1, 0.15). Parameter values are m = 1/2, δ = 10−4 , ν = 3 × 10−6 .
0.16
sinit=0.05
sinit=0.1
0.14
sinit=0.15
stop
0.12
0.1
0.08
0.06
0.04
0
20
40
60
80
100
Dimensionless time
Figure 6: Stop vs. dimensionless time for three different initial conditions (Sinit =
0.05, 0.1, 0.15). Parameter values are m = 1/2, δ = 10−4 , ν = 3 × 10−6 .
55
The complete problem, with a small saturated region allowed for by using boundary condition
(2.20), and with η 6= 0, was also solved by discretising in space and solving with the method
of lines using ode15s in Matlab. To apply the switch in boundary conditions smoothly, the
condition
q0 = q1 e−1000(1−S) ,
(2.33)
was applied for the flux at the bottom node q0 in terms of the flux at the node above q1 ; thus
when S is close to 1 this becomes ∂^
q/∂^
z = 0, and when S is less than 1 it becomes q0 = 0. The
diffusion coefficient is infinite when S = 1, but this does not cause any issues in the numerics,
possibly because the above boundary condition ensures S never quite reaches 1.
This seems to allow for steady states when rainfall is constant; if there is more rainfall than is
taken up by the roots, the saturation at the bottom is 1 and there is a Rboundary layer of width
^ − η L R (Fig. 7). If there is
δ1/2 in which it adjusts to the value as determined by K(S) ≈ νQ
0
less rainfall than is taken up by the roots, the saturation at the bottom decreases almost to 0.
^ = 0.1 to Q
^ = 10, which shows the
Fig. 7 shows the result of a sudden increase in rainfall from Q
initial shock front travelling down into the soil and the eventual steady state. The saturation
at the bottom does not increase towards 1 until the shock front arrives there. Fig. 8 shows the
^ = 0.1. Note that the time intervals shown are longer;
result of a sudden decrease back to Q
there is an initial expansion fan on the timescale of the gravity drainage which reduces the
^ followed by a slower decay (on the timescale
saturation to the value given by K(S) = νQ,
1
O( η )) as the roots take up the remaining water.
3
Two-Porosity Model
The expanded clay pellets used in green roof construction are quite large but contain lots of
pore space. The difference in pore sizes between these, and the inter-pellet space means water
can be drawn into the pellets and retained there for longer than it would otherwise remain in
the soil. Thus a two-porosity model would seem appropriate.
This is an outline of a “box" or “lumped" model for water storage in the macro-pores between
soil particles, which have saturation S, and in the micro-pores within the particles, which have
saturation SP . Transport of water into or out of the particles is parameterised to occur at
a rate proportional to the saturation difference S − SP .12 The roots do not penetrate into
individual particles so provide a sink term R only from the macro-pores. This root uptake R(S)
is primarily due to the large negative pressure in the root system, but as saturation decreases
a large capillary pressure acts to counteract this; thus R(S) is roughly constant for S close to 1
but decreases at small S (as in the model above).
12
A variant of this model might assume that water transfer into the particles occurs at a rate proportional to
the pressure difference pcP f(SP ) − pc f(S); since the capillary pressure in the micropores would be larger than
in the macropores (pcP > pc ), this would cause more water to be transferred into the micropores, and a larger
supply would be maintained there for the roots to take up.
56
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
z/L
z/L
1
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
0
1
S
0
0.2
0.4
0.6
R/R
0.8
1
0
Figure 7: Profiles of saturation and root uptake at time intervals of 1 (in the dimensionless
units); the arrow shows the direction of increasing time. This is the result of a sudden increase
^ = 10, from the steady state when Q
^ = 0.1, and the dashed line shows the
in rainfall to Q
steady state that results. Parameter values are m = 1/2, δ = 10−2 , η = 4 × 10−4 , ν = 3 × 10−4 ,
ε = 10−2 , γ = 10−1 .
57
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
z/L
z/L
1
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
0
1
S
0
0.2
0.4
0.6
R/R
0.8
1
0
Figure 8: Profiles of saturation and root uptake at time intervals of 10 (in the dimensionless
units); the arrow shows the direction of increasing time. This is the result of a sudden decrease
^ = 10, to Q
^ = 0.1. Parameter values are m = 1/2, δ = 10−2 , η = 4 × 10−4 ,
in rainfall from Q
−4
−2
ν = 3 × 10 , ε = 10 , γ = 10−1 .
58
The following equations are dimensionless, and the time scale has been chosen to be that due to
uptake by the roots (the time scale differs from that used previously by a factor η). Drainage
from the volume of soil is supposed to occur due to gravity at a rate K(S), and occurs on
a time scale η compared to the uptake by the roots (see above). Rainfall provides a source
which is scaled to be the same size as the gravity drainage (note this is different to above the scale for the rainfall here is large and is intended to represent the size of heavy showers;
the dimensionless r(^t) will be 0 most of the time, when it is not raining, and O(1) when it is
raining heavily).
dS
1
1
= r(^t) − K(S) − λ(S − SP ) − R(S),
η
η
d^t
dSP
(1 − φ)φP
= λ(S − SP ),
d^t
^ φP is the porosity of the pellets and
where r = νQ,
φ
K(S) = S1/2 [1 − (1 − S2 )1/2 ]2 ,
which comes from equation (2.2) with m =
1
2
(3.1)
(3.2)
(3.3)
and
R(S) = 1 − ε
(1 − S2 )1/2
.
S
(3.4)
The use of K(S) for the gravity drainage in equation (3.1) is motivated by the fact that the
water flow in the earlier part of the report is (since δ is small) essentially determined by this
hydraulic conductivity. The time scale for water to diffuse into individual particles is estimated
using their dimensions LP ∼ 1 cm and a diffusion coefficient DP ∼ 10−9 m2 s−1 . L2P /DP is
comparable to the time scale for uptake by the roots (∼ 105 s), so the parameter λ is order 1. In
equation (3.1) η is very small and in equation (3.4) ε is also small, and we consider especially
the distinguished case ε ∼ η2/7 .
The behaviour of solutions to this model is quite straightforward, and an example solution
for a large rain storm followed by dry weather is in Fig. 9. When it is raining, r is order
1, and on a fast time scale, ^t ∼ O(η), the saturation S relaxes towards the equilibrium given
by K(S) = r(^t). This causes water to then transfer into the particles on an O(1) time scale
according to (3.2). By balancing terms in equation (3.1) we seek a similarity solution for small
9
S for which K(S) ≈ 41 S 2 . When it stops raining r = 0, and the saturation S decreases quickly
due to gravity drainage on an O(η) time scale; approximately
φ
dS 1
∼ K(S).
d^t η
(3.5)
2
Thus, neglecting the small O(η 7 ) terms suggests S tends towards 0 as
2/7
2φη
S∼
.
7^t
59
(3.6)
1
0.8
0.4
S
r / r0
0.6
0.4
0.2
0.2
0
0
0.5
1
1.5
t / t0
2
2.5
0
3
0
0.5
1
1.5
t / t0
2
2.5
3
0
0.5
1
1.5
t / t0
2
2.5
3
1
1
0.8
p
0.6
0.6
S
R / R0
0.8
0.4
0.4
0.2
0.2
0
0
0.5
1
1.5
t / t0
2
2.5
0
3
Figure 9: Solutions for macro-scale and particle-scale saturations S and SP , and root uptake R,
as a result of rainfall r(t) which represents a large rain shower. Dashed lines show the limiting
behaviour. Parameter values are η = 10−4 , λ = 1, ε = 10−2 .
On the O(1) time scale, therefore, S ∼ η2/7 , and the equations are
ε
2/7
2/7
2/7
,
O(η ) = O(η ) − λ(O(η − SP ) − 1 −
S
(3.7)
dSP
= λ(O(η2/7 ) − SP ).
(3.8)
^
dt
Thus, ignoring the O(η2/7 ) terms, Sp decays exponentially and the water coming out into the
macropores is immediately taken up by the roots: R = λSP = −(1 − φ)φP dSP /d^t, as shown in
figure 9. Root uptake is maintained for a much longer period after it ceases to rain than would
be the case with no micropores, when S decreases rapidly towards 0.
(1 − φ)φP
3.1
A Model for Fast Saturation
Assuming instead fast saturation of the pellets,
SP = SO H(S) ,
(3.9)
where H denotes the Heaviside function, SP denotes the saturation of the individual pellets, SO
is the porosity of an individual pellet, and S is the saturation of the inter-pellet pores. The
required short time scale can arise from high capillary pressures associated with the very small
pores within the pellets.
60
Taking now φ = 41 to be the total proportion of space occupied by air and water within the
1
soil, and SO = 15 , the inter-pellet porosity is ϕ = 16
(given by ϕ + 15 (1 − ϕ) = 41 ). Equation
(2.9) can then be replaced by
1 ∂
∂
∂S
^,
(S + 3H(S)) =
K(S) + δD(S)
− ηR
(3.10)
16 ∂^t
∂^
z
∂^
z
^ ∼ 1, from (2.9) and (2.32). (Equation (3.10) might be better written in terms of the total
with R
water content, ST = 161 (S + 3H(S)), so that S on the right-hand side is replaced by S(ST ) = 0
3
for 0 ≤ ST ≤ 163 , S(ST ) = 16(ST − 163 ) for 16
≤ ST ≤ 14 .)
Where the pellets are saturated, S > 0 and H(S) = 1, the equations are as in Section 2. Here,
for simplicity, an initially dry soil is considered, so that at ^t = 0, S ≡ SP ≡ 0. For ^t > 0, a
^ ^t) < z^ < 1 has become wet:
region W(
^,
S = H(S) = 0 in 0 < z^ < W
^ < z^ < 1 .
S > 0 and H(S) = 1 in W
(3.11)
To obtain an order-one sized wet region, the relevant time scale must be that for the rainfall
(days) so that time has to be rescaled:
^t = t̃/ν .
(3.12)
Note that this time scale is similar to that for the up-take of water by the plants’ roots. It is
also appropriate, from the top boundary condition, to rescale the saturation:
S = ν2/9 S̃ ,
(3.13)
where, since we have assumed that m = 21 , K(S) ∼ 41 S9/2 and D(S) ∼ 14 S5/2 for small S.
Neglecting the time-derivative term (now effectively of order ν2/9 ), the partial differential equation becomes
!
∂
S̃
1 ∂
^.
= η̃R
(3.14)
S̃9/2 + δ̃S̃5/2
4 ∂^
z
∂^
z
1
Here η̃ = η/ν ≈ 31 and δ̃ = δ/ν2/9 ≈ 600
, using the values of Section 2. Although the value of
δ̃ is small here, because of the uncertainty in the values of the physical parameters describing
water transport through the soil, it could conceivably be of order one and it is therefore retained
in (3.14), for the present.
The differential equation is subject to the top boundary condition
!
∂
S̃
1
^ in at z^ = 1
=Q
S̃9/2 + δ̃S̃5/2
4
∂^
z
(3.15)
and, assuming that the diffusive, δ̃, term is retained, a lower boundary condition
^ t̃) .
S̃ = 0 at z^ = W(
61
(3.16)
^ ^t) between dry and wet soil, conservation
Finally, to fix the position of the free boundary z^ = W(
of mass of water at this point, where SP jumps from 0 to SO , leads to
!
^
∂
S̃
4
dW
^ t̃) .
at z^ = W(
(3.17)
S̃9/2 + δ̃S̃5/2
=−
3
∂^
z
dt̃
(Since, for δ̃ > 0, S̃ =
point, the second term on the right-hand side should then be
0 at this
∂S̃
interpreted as δ̃ lim S̃5/2
.)
^
∂^
z
z^→W
Of course, if the pellets were already partially saturated, (3.17) would be suitably modified,
leading to a faster-moving free boundary.
^ in + z^ − 1
Note also that if the diffusion can be neglected, (3.14) and (3.15) lead to 41 S̃9/2 = Q
so (3.17) gives
^
dW
16 ^
^ −1 .
−
Qin + W
(3.18)
=
3
dt̃
^ t̃ ≤ 0. An
The free-boundary condition (3.17) only applies for an advancing wet region, dW/d
alternative form is needed for when this region contracts, which will happen when the rainfall
decreases sufficiently. In any part of the soil between the lowest location of the free boundary
and its current position, the roots can continue to remove water from the pellets, thereby reducing SP .
As described in this report we could could now have at least four types of region within the
soil layer:
1. Dry zone, where S = SP = 0;
2. Damp or moist (unsaturated) zone I, where S = 0, 0 < SP < SO ;
3. Damp or moist (unsaturated) zone II, where 0 < S < 1, SP = SO ;
4. Wet (saturated) zone, where S = 1, SP = SO .
4
Conclusions
In this report a one-dimensional time dependent mathematical model has been described for
the development of the saturation in the soil layer of a flat green roof. Our model suggests that
a fully-saturated (S = 1) region forms at the base of the soil layer and this region can be thin
relative to the total soil thickness.
62
From an initial dry state and from the onset of persistent rain, fronts of saturation were computed to descend through the layer. The decrease of saturation from unity following a decrease
in rainfall was also described. The end result is that most of the rainwater falls through the
soil layer and exits through the network of holes in the bottom supporting sheet.
On a smaller scale, the pellets and soil particles are themselves porous and made up of micropores. The water flow in and out of a typical particle is modelled using the flux between (a) the
macropores (whose saturation is as modelled above) and (b) the root system. This two-porosity
model suggests that during the time between spells of rain the micropores can retain (for long
periods of time) water that is available to be taken up by the roots.
Further work might include adapting the soil thickness L to rainfall at the site of the building
with the aim of making L as small as possible, while avoiding problems with saturation and
aridity. A first step towards this goal would be to carry out experiments to more accurately
determine the values of the constants. Further simulations using more extensive rainfall data
could then be carried out to determine the optimum soil thickness. In addition small modifications could be made to include the influence of a sloped roof.
Acknowledgements
We acknowledge the support of the Mathematics Applications Consortium for Science and
Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics initiative
grant 06/MI/005.
63
Bibliography
[1] M. T. Van Genuchten. A closed-form equation for predicting the hydraulic conductivity
of unsaturated soil. Soil Sci. Soc. Am. J., 44:892-898, 1980.
[2] T. Roose and A.C. Fowler. A model for water uptake by plant roots. J. Theoret. Biol.,
228:155-171, 2004.
64
Uplift Quadratic Program in Irish Electricity
Price Setting
Report Contributors: Michelle Carey1,2, Conor Houghton 3,
Matylda Jabłońska4 and John Kinsella5
Study Group Contributors: James Gleeson1, Gavin Hurley1 ,
Julie Hurley1 , Arno Mayrhofer1 , Colin Menzies1,
Celine O’Donovan6 , Gearóid Ryan7 ,
Dengli Wang8 and R. Eddie Wilson9
Industry Representatives: Gavin Hurley10 and Sile Bourke10
1
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
Report coordinator, michelle.carey@ul.ie
3
Department of Mathematics, Trinity College Dublin, Ireland
4
Department of Mathematics, Lappeenranta University Finland, Finland
5
Department of Mathematics and Statistics, University of Limerick, Ireland
6
Kemmy Business School, University of Limerick, Ireland
7
School of Mathematical Sciences, University College Cork, Ireland
8
Business School, Dublin City University , Ireland
9
Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, United Kingdom
10
Bord Gáis Energy, First Floor (BGE), City Quarter, Lapps Quay, Cork, Ireland
2
Abstract
This report summarises progress made towards the problem submitted by Bord Gáis at the 70th
European Study Group with Industry organised by Mathematics Applications and Consortium
for Science and Industry (MACSI) and held in the University of Limerick from the 28thJune 3rd July 2009. Bord Gáis required a deeper insight into the dynamics of Uplift prices. Therefore,
the aim of the group was to apply a variety of analytical tools to the problem in order to satisfy
Bord Gáis requirements. The group conducted a KKT Optimality Analysis of the quadratic
program used to determine the Uplift prices, performed statistical analysis to identify the
binding constraints and their sensitives to the Uplift prices, simulated a synthetic stochastic
process that is consistent with the Uplift pricing series and investigated alternative objective
functions for the quadratic program.
66
Introduction
Energy prices in the Irish Electricity Market are composed of a Shadow Price and an Uplift
Component. The Shadow Price is the cost of meeting a marginal unit of demand, as determined
by solving the Unit Commitment Problem. The Unit Commitment Problem is a Mixed Integer
Program which determines which units (generators) to commit (turn on), so as to minimize
the total system cost of generation, subject to meeting demand in each period and respecting
technical characteristics of the units (e.g. ramp-rates, min on/off times, minimum generation).
Participants declare their true marginal costs and therefore the Shadow Price alone does not
guarantee that generators will recover their fixed running costs, e.g. start-up and no-load. The
Uplift component of prices addresses this. It is derived via a Quadratic Program (QP) which
ensures that the generators recover those costs. The QP aims to minimize a parameterized
combination of total additional cost due to Uplift and the volatility of Uplift. The problem
involves investigating the quadratic program used to determine the Uplift component of the
prices and to examine the behaviour of the this Uplift component. Our aim for the week was
to exploit the expertise of the group to develop analytical tools that provided Bord Gáis with
a deeper insight into the uplift process.
The report begins by describing the structure of the quadratic program, followed by analysis
of the optimal Uplift components using the Karush_Kuhn_Tucker (KKT) conditions. One
can also employ statistical analysis to identify the generators contributing to Uplift prices, this
method is described and the results produced. The subsequent sections involves simulating a
stochastic process that reproduces the Uplift series to a quantitatively comparable level and
investigating other objective functions for the QP. Finally, the report concludes with results
and suggestions for further research.
67
1
Problem Formulation
The “System Marginal Price" (SMP), the incremental cost of supplying the next MWh of energy
demanded for each half-hourly period h, of day t is
SMPh = SPh + UPh .
Where SPh represents the “Shadow prices”, the marginal price of electricity per MWhr in
each half-hour based on the information provided by the generators and UPh symbolizes the
“Uplift”, the correction factor that is applied retrospectively to the shadow prices to cover the
fixed running costs.
The SEMO (Single Electricity Market Operator) determines the Uplift function by solving
a quadratic program that minimises Uplift revenues (the Cost objective) and minimises the
shadow price distortion (the Profile objective)
"
#
X
X
X
(SPh + UPh )
Qgh + β
min F(UPh ) ≡ α
UP2h ,
(1.1)
UPh ,h=1...48
|
subject to,
X
h
h
{z
Cost objective
[(SPh + UPh )Qgh ] ≥ CRg ,
UPh ≥ 0,
g
}
|
h
{z
}
Profile objective
for g = (1, . . . , G),
(1.2)
for h = (1, . . . , 48).
(1.3)
Given that Qgh denotes the volume (quantity) of electricity provided by each generator g in
each half-hour period h and CRg is the total cost of running for generator g, given by
X
CRg =
[Qgh Cu + NLCg I(Qgh > 0)] + STg ,
(1.4)
h
Where Cu is the variable (fuel) cost per unit, NLCg is the no-load cost of generator g, which is
the element of operating costs which is invariant with the actual level of output and STg is the
start-up cost of generator g, the cost of turning on generator g at any time h during day t.
The first constraint given in equation (1.2) ensures that each generator g covers its costs CRg
and the second constraint given in equation (1.3) certifies that all Uplift values are positive.
For convenience equations (1.1,1.2,1.3) were converted into Matlab-like notation as follows;
min F(u) ≡ αeT Q(s + u) + βuT u,
u∈R48
(1.5)
subject to,
Q(s + u) ≥ c,
u ≥ 0.
68
(1.6)
(1.7)
Please refer to the conversion table in the appendix for the details.
The problem was attacked using four separate but complementary approaches. These approaches all provide a insight into the composition of the Uplift process and our outlined in the
subsequent sections. Section one examines the optimality of the quadratic program using the
Karush_Kuhn_Tucker(KKT) conditions, section two describes a statistical analysis of both
the actual and re-constructed data set, section three develops a stochastic model that mimics
the underlying dynamics of the Uplift prices and finally section five investigates the validility
of alternative objective functions.
2
KKT Optimality Analysis of the QP
The KKT 1st -order necessary conditions for a general inequality-constrained problem are:
X
λi ∆u ci (u) = 0
(2.1)
∆u L = 0 ≡ ∆u F(u) −
i∈I
where I is the index set for the inequality constraints.
Computing the gradients with respect to u;
αQT e + 2βu − QT λ1 − λ2 = 0,
(2.2)
where λ1,g ≥ 0 for g = (1, . . . , G) when the corresponding constraints are active and zero
otherwise, and λ2,h ≥ 0 for h = (1, . . . , 48) when the corresponding constraints are active and
zero otherwise. As the QP is convex, the KKT conditions are both necessary and sufficient for
optimality (i.e the lagrange multipliers are unique). Moreover, the lagrange multiplier λ2 must
be zero for all h. Therefore one can rewrite equation (2.2) as;
2βu = −αQT e + QT λ1 .
(2.3)
Under the Single Electricity Market (SEM) trading and settlement code the regulatory authorities determine the weighting parameters α, β in equation (2.2) for each year. The first
parameter α governs the importance of the Uplift cost objective, such that 0 ≤ α ≤ 1. The
second parameter β governs the importance of the Uplift profile objective, such that 0 ≤ β ≤ 1
and such that α + β = 1. The regulatory authorities have found that setting the parameters to α = 0 and β = 1 provides the most appropriate balance of cost and price stability.
Therefore applying these conditions to equation (2.2), given that several generation constraints
represented by g ∈ Ag0 are binding yields an optimal set of positive uplifts u which are a
non-negative linear combination of the columns of QT (the rows of Q) corresponding to the
binding constraints.
X
λ1,g (qg ) .
(2.4)
2u =
g∈Ag0
69
replacemen
The binding constraints now take the form (Qu)g = cg − (Qs)g for g ∈ Ag0 .
2.1
Results
Equation (2.4), motivated an analysis of historical data to determine which generation constraints have been binding in each 24-hour period. This is illustrated in figure (1), which
clearly shows that the generators have an intrinsic band structure.
Figure 1: Recurrence of Binding Generators
Two small subsets of generators can be identified from figure (1). A set of the same generators
that are clearly binding from day to day and a set of generators that are never binding. This
provoked a further examination of the classification and frequency of the binding generators.
The classification of generators that are binding and therefore influencing Uplift prices are
shown in figure (2).
It appears that it is almost always the thermal plants (gas, oil, distillate and coal) that have
binding constraints. The frequency of generators that are typically binding is given in figure
(3).
It is clear that in general there is only one (45% of all occurrences) or two generators (34% of
all occurrences) binding with very few occasions with more than four generators binding (.01%
of all occurrences).
70
Figure 2: Classification of binding generators
Figure 3: Frequency of binding by generator type
Figure 4: Frequency of binding by generator type
3
Factor Analysis
Using statistical analysis the group tried to identify factors that were contributing to Uplift
prices, in order to state the sensitivity of Uplift prices to certain known factors. The Uplift
71
prices are an adjustment price and therefore can have zero values, in this case 40% of the Uplift
price values are equal to zero.
A factor model of Uplift prices was employed to decompose the random prices into factor related
and specific prices.
u = α + β1 f1 + . . . + βk fk + ε,
where u denotes the time series of Uplift prices, α is a single coefficient introduced in order
to ensure E[ε] = 0, βi is the sensitivity of Uplift to the factor fi and ε represents the error or
Uplift specific price’s.
If the vector of factors f is known one can model this using a macroeconomic factor model,
however if they are unknown and therefore must be estimated one uses a statistical factor
model.
3.1
Macroeconomic model
The results from the KKT conditions shown in equation (2.2) are structurally similar to a
macroeconomic factor model. The equation states that the factors contributing to Uplift prices
u are qg , the quantity of electricity provided by each generator g in each half-hour period h.
Using the macroeconomic factor model, one can verify the results from the KKT conditions
with minimal assumptions on the underlying dynamics of the problem. The advantage to using
this method is that it can be applied to both the reconstructed Uplift data and the actual Uplift
data, producing the binding generation constraints in both cases.
From figures (2) and (3), one can establish that 45% of the generators bind only once, 86% of
generators have less than three binding generators and that wind, pumped storage, peat and
the interconnector generators don’t contribute to Uplift prices. Exploiting this and the fact
that all the parameters in equation (2.2) are positive one can identify the binding generators
by analyzing the correlation structure between the output of each of the 85 generators that
influence Uplift prices and the Uplift prices themselves. The results are shown in figures (5,6).
In figure (5,6), the positive correlation values represented by the light green, yellow, orange and
red give the set Ag of all generators that could be binding in any given day t, for the 385 days
considered. From this one can identify how many generators are binding, which generators are
binding and the sensitivity of Uplift prices to the binding generators on each day t. For the
reconstructed data, there are 76 days in which one generator is binding this corresponds to
generators which have a correlation coefficient value greater than or equal to 0.9999. To identify the binding generators when more than one generator is considered one would construct
a macroeconomic factor model with the Uplift regressed against all possible generator’s identified by the correlation coefficients (> 0 and < 0.9999), the max number of possible binding
generators in this case is 42. Therefore one can develop a stepwise algorithm which selects the
combination of generators that are binding based on the AIC (Akaike’s information criterion) of
each combination. AIC makes comparisons between a number of possible models on the basis
72
Figure 5: Correlations between generator’s and Uplift prices for reconstructed data
Figure 6: Correlations between generator’s and Uplift prices for reconstructed data
of the following statistic: AIC= −2 × log(maximum partial likelihood) + 2 × b, where b is the
number of β coefficients in each model under consideration. The AIC is based on maximum
likelihood and a penalty for each parameter. The best model has the smallest AIC value. This
algorithm produces the binding generators and their sensitivities to Uplift prices on any given
day t.
This procedure can also be run on the actual Uplift prices. Thus producing the binding constraints and their sensitivities to actual Uplift prices. Figures (7 and 8) illustrate the results.
There are only 5 days for which only one generator is binding. From figures (5) and (7) one
can clearly identify a band structure indicating that there is a recurring group of generators
that are binding.
73
Figure 7: Correlations between generator’s and Uplift prices for actual data
Figure 8: Correlations between generator’s and Uplift prices for actual data
4
4.1
Uplift as an individual stochastic process
Motivation for stochastic analysis
In terms of forecasting, electricity prices are one of the most difficult types of financial time
series. This is due to their highly volatile behavior. However, the Uplift prices exhibit some
specific statistical features that can be exploited in order to reconstruct the series up to some
qualitatively comparable level.
The aim of the group was to reproduce a synthetic stochastic process which would behave
similarly to the real Uplift prices, that would be suitable for Monte Carlo simulations or some
other stochastic purposes. The following subsections present specific Uplift features that were
identified and employed in simulations to generate the synthetic process and an examination
of the synthetic process produced.
74
4.2
Identified features of the Real Uplift process
Initially the group examined the path of the Uplift data over consecutive days. This is illustrated
in figure (9). One can clearly identify a jump structure inherent in the data.
Figure 9: Path covering a few consecutive days of Uplift half-hourly data
Considering that by definition Uplift must be positive for all time t and the length of the
plateaus may be similar to waiting times. One can assume a Poisson distribution to model
the lengths of plateaus in the Uplift prices and an investigation of the empirical Uplift jump
distributions could be used to verify the jump heights.
Since electricity is highly dependant on half-hourly demand (contributing to half-hourly, daily
and annual seasonality) it is reasonable to expect the Uplift prices to also reflect some seasonality features. These features can be clearly identified by the autocorrelation function (ACF) and
partial autocorrelation function (PACF) for the Uplift price series presented in figure (10). As
desired, we clearly see peaks in ACF repeating with 48−lag regularity. Additionally, they are
slightly locally maximal points for every 336th lag (48 half-hours times 7 week days), portraying
weekly seasonality but this is not as significant as the daily seasonality. Thus, in the simulation, in order to reproduce the daily seasonality the current observation is regressed against the
observation that occurred 48 half-hour periods earlier. One would suspect that there is also an
annual seasonality unfortunately there is an insufficient amount of data to verify this claim.
4.3
Uplift jump waiting times
A Poisson distribution was employed to model jump waiting times in the Uplift process. However, due to time limitations, the parameters were set using heuristic methods. It is evident
from Figure (9), that waiting times are not distributed uniformly, as there is a clear dependency
on the current price level. In particular, the process zero level components are considerably
long, whereas at high levels (40-100), the process jumps down almost immediately. Therefore,
75
Figure 10: Autocorrelation and partial autocorrelation functions for the Uplift price series.
it seems sensible to set four different price levels, defined by various values of the Poisson λ
parameter. These values are as follows;
• λ = 50 for uh = 0.
• λ = 10 for 0 < uh ≤ 40.
• λ = 2 for 40 < uh ≤ 100.
• λ = 0.05 for uh > 100.
4.4
Jump up/down probability and Jump heights
One can also clearly identify a jump structure in figure (9). The higher the value of current
Uplift, the higher the probability of a negative jump. Another trait that becomes apparent from
examining the data and figure (9) is that if the current price level crosses some considerably
high price level (in this case uh > 50), then the process exhibits continuous negative jumps
until it reaches zero. Sometimes, these negative jumps occur in a few time steps but often
within one time step only.
Two empirical cumulative distribution functions conveying jump direction are derived from the
occurrences of positive and negative movements in the Uplift series for the purpose of random
number generations. Figure (11) is a histogram of the occurrences of jumps up and down for a
given Uplift level. The histograms structure conveys an exponential curve.
The distribution of the jump magnitude conditional on the price level is derived by computing
the density function of the jump magnitudes. This is illustrated in figure (12).
76
Figure 11: Histograms for occurrences of jumps up/down depending on current Uplift level.
Figure 12: Density function of the jump magnitudes.
4.5
Plateaus within night hours
Plateaus during the night half-hour intervals are also apparent in the data these portray a high
probability of constant (usually zero) Uplift during this period. Once the base process was
simulated one could now induce this attribute by setting the night half-hour intervals equal to
zero, based on a uniformly distributed probability of zero level continuing for 12 to 20 half-hour
periods.
4.6
Algorithm
The general algorithm for the simulation approach is as follows;
1. For h = (1, . . . , 48) set ua,h = us,h . Where ua,h is the actual Uplift price for the half-hour
interval h and us,h is the simulated Uplift price for the half-hour interval h.
2. Given us,h for a particular half hour h, identify the corresponding Poisson parameter.
77
Using this parameter generate the length l of the plateaus (amount of half hour intervals
h for which the process remains constant).
3. Set the current time point hc , to the previous time point hp plus the length of the plateau
l. hc = hp + l.
4. For l > 1 then h = (hp + 1, . . . , hp + l) let us,h = us,h−1 and for l = 1, then h = (hp + 1)
let us,h = us,h−1 .
5. Then, Given that us,h is the last price value after the last jump.
• Set a limiting threshold T . If us,h > T , force the process into negative jumps until
us,h = 0.
• If us,h = 0, force the process to only allow positive jump’s generated from the
empirical distribution.
• If 0 < us,h < T , determine from the empirical distribution of jump direction whether
the process jumps up or down, and then based on the empirical distribution of jump
magnitude determine the magnitude of the jump. Sum the sampled jump magnitude
and the regressed price value that occurred 48 periods ago. Set hc = hc + 1.
6. Repeat until the hc exceeds the target process length to be simulated.
4.7
Results
The simulation was implemented to construct a sample of n half-hour intervals of the Uplift
series. Figure (13) presents the simulated Uplift series which is quantitatively (by distribution
parameters) comparable with the original Uplift series. Moreover, it is clear that our simulation
is also able to produce values significantly differing from the process mean, which is also reflected
in the real data.
Figure 13: The real and simulated Uplift process.
78
In order to identify if the structural elements of the simulated Uplift process mimic those
evident in the actual process, one would compare the two data sets microstructure. This is
illustrated in figure (14). The figure confirms that the jump structure is accurately modelled
by the simulated process and that the plateaus (more persistent when price at zero level) occur
with different lengths at different price levels in both the actual and simulated data.
Figure 14: The real and simulated Uplift process.
Finally, figure (15) illustrates the ACF and PACF of the actual and simulated Uplift data. The
half-hourly seasonality, apparent in the actual data has been reconstructed in the simulated
data up to a significant level. Additionally, the PACF’s of both the original and simulated data
have similar features.
Figure 15: Autocorrelation and partial autocorrelation functions of the real and simulated
Uplift process.
79
5
Other Objectives
The group noticed that the objective formula equation (1.5) is not dimensionally consistent,
the cost objective function has units of money whereas the profile objective function has units
of money per load by time, all squared. This is dissatisfying, it means for example the currency
has to specified as well as α and β and it has the practical problem that there is no reason to
expect the cost objective and the profile objective to have a similar scale in fact, for the uplifts
calculated with the β = 1 objective
C̄ = 684651
P̄ = 13478.
(5.1)
Where C̄ is the mean of the cost objective and P̄ is the mean of the profile objective, taken
over all the days considered. Of course, even if α and β were fine tuned to make these similar,
the changes as UPh varies would not be comparable.
In order to solve this problem of inconsistency one would have to identify an objective function
that explicitly referred to total cost and to variance so that the balance between the two could
be decided more transparently as a matter of policy by the industry. Hopefully this would also
allow the actual economic cost of reducing variance to be computed as the mixture of these two
terms was varied.
To do this in a dimensionally consistent way seems to require that the same method of averaging
is used for both terms, at the moment C the cost objective effectively averages over load while
P the profile objective averages over time; for a given quantity xt , lets use
hxt it =
for the time average and
for the load average; at the moment
1 X
xt
48 t
(5.2)
P
gt Qgt xt
hxt iQ = P
gt Qgt
(5.3)
C ≈ hut iQ
P ≈ hu2t it
(5.4)
There are, correspondingly, two consistent objectives which would separate out cost and variance
E2t = µhut i2t + (1 − µ) hu2t it − hut i2t
(5.5)
and
E2Q = µhut i2Q + (1 − µ) hu2t iQ − hut i2Q
(5.6)
Both are quadratic objectives, both are dimensionally consistent and both have separate variance and cost terms.
80
Another approach to the objective is to decide that the Uplift price should be chosen so that the
time course of the Uplift reflects the actual time course of the cost of generation; this may have
the advantage of communicating economic signals to the consumers about the actual half-hour
by half-hour cost of producing electricity. In short, imagine there wasn’t a single Uplift, but
instead a time-constant Uplift ug for each generator period. This would spread the economic
cost of g’s electrical production evenly across g’s load:
cg
t Qgt
ug = P
(5.7)
The most obvious way to use this generator-period Uplift to calculate ut would be to take the
maximum ut at every half-hour. This has the aesthetic advantage of matching this method to
the method used to calculate the shadow price; it is however very expensive, it has an average
cost of
C̄ = 2690010
(5.8)
An alternative would be to calculate a load weighted average of the ug :
P
g Qgt ug
ũt = P
g Qgt
(5.9)
This Uplift would be very cheap;
C̄ = 72627
(5.10)
and would give an Uplift time course well matched to the time course of the electricity generation
cost, however, it does not satisfy constraints. The idea, therefore, would be to use this to pose
an objective
X
Ẽ =
(ũt − ut )2
(5.11)
t
Other profiling goals could be posed; for example related to the shadow price.
Similarly, it might be possible to find a linear objective; C is already linear, all that would be
required would be a linear approach to imposing the profiling goal. One method would be to
specify a restricted space for ut ; it could be required that it tracks the total load, or the loads
of the relevant generator periods or the shadow price or some positive weighted combination of
these.
5.1
Conclusion
This report uses various modelling approaches to gain a deeper insight into the underlying
dynamics of the Uplift prices. Initially, the group examined the optimality of the quadratic
program used to derive the Uplift prices using the KKT conditions. This resulted in equation
(2.4) which states that the optimal set of positive uplifts u are a non-negative linear combination of the columns of QT (the rows of Q), corresponding to the binding constraints. This
81
result motivated an analysis of historical data to determine which generation constraints have
been binding in each 24-hour period. In order to identify these binding generation constraints
one can compute brute force searches or employ statistical analysis. The statistical analysis in
section (3) exploits the results computed from section (2) and utilizes correlation coefficients
and factor analysis to identify the binding generation constraints for each day t and the sensitives of Uplift prices to these binding generation constraints. While the statistical and brute
force searches are both very separate approaches, the results from both methods were equivalent for the reconstructed data. Surprising results were that there is only a small amount of
binding generators and that hydro plants were contributing to Uplift prices. Both methods
also identified a band structure to the binding generation constraints, implying that there is a
recurring group of generators that are binding. The advantage of using the statistical analysis is
that the algorithm can be applied to the actual Uplift prices, producing the binding generation
constraints and the sensitives of actual Uplift prices to these binding generation constraints.
The group then developed a synthetic stochastic process which behaves similarly to the Uplift
pricing series. By identifying specific statistical features of the Uplift prices the group was able
to simulate a stochastic process which is qualitatively comparable with the actual Uplift prices,
conveys structural elements that are inherent in the actual Uplift prices and portrays the halfhourly seasonality evident in the actual Uplift prices. Finally, the group investigated possible
alternative objective functions for the quadratic program used to compute Uplift prices as the
objective function used by the SEMO is dimensionally inconsistent. Alternative objectives were
posed and discussed.
Propositions for further research; a statistical factor analysis, where the parameters of interest
are proxies which are influencing Uplift prices such as Oil prices, weather conditions etc, an
investigation into the parameters of the Poisson distribution for modelling the waiting times
and the validation of a linear profiling objective. A distributional analysis of Uplift for each
day t and an examination of the effect of annual seasonality with the aid of extra data can be
found in ([3]).
Acknowledgments
All contributors would like to thank Gavin Hurley and Sile Bourke for introducing the problem
and assisting in answering questions and formulating the models.
We also acknowledge support of the Mathematics Applications Consortium for Science and
Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative
Grant 06/MI/005.
82
Bibliography
[1] Single Electricity Market Committee, SMP Uplift Parameters 2010, Consultation Paper,
7 June 2009, SEM-09-066.
[2] All Island Project, SMP Uplift Methodology and Parameters, Decision paper, 15 March
2007, AIP-SEM-07-51.
[3] Matylda Jablonska, Arno Mayrhofer and James Gleeson, Title, Journal (in preparation).
Appendix
Notation In industry standard notation the QP is:
"
#
X
X
X
UPh2
(SPh + UPh )
Qgh + β
min F(UPh ) ≡ α
UPh ,h=1...48
g
h
(5.12)
h
subject to
X
h
[(SPh + UPh )Qgh] ≥ CRg , for g = 1 . . . G
(5.13)
UPh ≥ 0, for h = 1 . . . 48
(5.14)
Here the range of h is h = 1 . . . 48 and the range of g is g = 1 . . . G.
Writing the above in a more abbreviated notation where:
u = (u1 , . . . , u48 ), uh ≡ UPh , h = 1 . . . 48
s = (s1 , . . . , s48 ), sh ≡ SPh , h = 1 . . . 48
c = (c1, . . . , cG ), cg ≡ CRg , g = 1 . . . G
we have
min
uh ,h=1...48
F(uh ) ≡ α
X
h
"
(sh + uh )
X
g
83
#
Qgh + β
X
h
u2h
(5.15)
subject to
X
h
[(sh + uh )Quh] ≥ cg , for g = 1 . . . G
(5.16)
uh ≥ 0, for h = 1 . . . 48
(5.17)
The above can be written in a Matlab-like notation as
min F(u) ≡ αeT Q(s + u) + βuT u
(5.18)
Q(s + u) ≥ c
u≥0
(5.19)
(5.20)
u∈R48
subject to
where e is a column vector of size G of ones.
84
Particle Impact Analysis of Bulk Powder During
Pneumatic Conveyance
Report Contributors: Giles Richardson1 , Mark Cooker2 ,
Niamh Delaney3 , Kevin Hanley4 , Poul Hjorth5 , Dana Mackey6,
Joanna Mason3,7, Sarah Mitchell3 and Catherine O’Sullivan8
Industry Representative: Seamus O’Mahony9
1
School of Mathematics, University of Southampton, United Kingdom
School of Mathematics, University of East Anglia, United Kingdom
3
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
4
Department of Process and Chemical Engineering, University College Cork, Ireland
5
Department of Mathematics, Technical University of Denmark, Denmark
6
School of Mathematical Sciences, Dublin Institute of Technology, Ireland
7
Report coordinator, joanna.mason@ul.ie
8
Department of Civil and Environmental Engineering, Imperial College London, United Kingdom
9
Wyeth Nutritionals Ireland, Askeaton, Co. Limerick, Ireland
2
Abstract
Fragmentation of powders during transportation is a common problem for manufacturers of
food and pharmaceutical products. We illustrate that the primary cause of breakage is due to
inter-particle collisions, rather than particle-wall impacts, and provide a statistical mechanics
model giving the number of collisions resulting in fragmentation.
86
1
Introduction
Pneumatic conveyance of food and pharmaceutical products in dehydrated, powdered form, is
very common in Ireland’s process industries. During conveyance, powder particles may break
into smaller pieces, due to impacts, resulting in a change in overall bulk density. A leading
manufacturer of infant formula approached the 70th European Study Group with Industry to
investigate the change in bulk density that they observe during transportation of this powder.
The powder is inhomogeneous and consists of approximately six main ingredients, and around
twenty additional micro-nutrients, which are smaller particles. An average particle is approximately spherical and around 100 microns in diameter. Particles consist of proteins, sugars and
fats and can be either solid or hollow. If hollow, the size of the wall is typically one fifth of the
diameter, and the walls can be solid or porous see Figure 1a.
Initially the powder is produced from a liquid in the spray dryer. During this phase the particles
can coalesce into clusters (see Figure 1b), which is desirable as agglomerates of particles are
better at dissolving in water, when reconstituting the infant formula. The particles are blown
along pipes by an air flow, and then intermediately stored in silos, before packing into cans.
During transportation particles impact both the walls of the pipe and also each other. A
schematic diagram of an industrial pneumatic conveyance line is shown in Figure 2. Typical
parameters for both the particles, and the pneumatic conveyance line are listed in Table 3.
Infant formula is usually measured out by volume using a scoop. Therefore bulk density of
the powder is an essential quality control parameter for infant formula manufacturers. If there
is too much breakage during transportation, the average size of the particle clusters decreases
because the particles can pack together more tightly, and as a result the bulk density increases.
This results in the formula becoming too nutrient rich when reconstituted. Conversely, if there
is unusually little breakage during transportation, the final bulk density could be too low, and
the rehydrated formula will be insufficient in meeting the infant’s nutritional requirements.
As a certain amount of breakage during transportation is inevitable, manufacturers currently
estimate the change in bulk density from empirical data. The aim of the study group is to devise
a mathematical model for the amount of breakage during transportation, and thus quantify the
bulk density change.
The purpose of this report is twofold: to outline the main mechanisms of impact, and to provide
an analytical expression for the rate of particle breakage during transportation. The outline
is as follows. We begin in Section 2 by describing Dean flow, which is the standard way of
describing flow in a curved pipe. In Sections 3 and 4 we consider particle flow in the pipe.
We initially consider a toy two-dimensional problem, and then extend our analysis to the full
three-dimensional problem. An overview of Hertzian contact laws, which we can use to estimate
the energy at which particle impacts result in fracture, is presented in Section 5. For Hertzian
theory to apply, it is assumed that the impacting bodies are solid. In Section 5.2 we outline an
argument to illustrate that the walls of the particles under consideration are sufficiently thick
that we may justifiably model them as solid. Section 6 focuses on constructing a statistical
87
M
Figure 1: SEM images of a typical particle and clusters of particles. A scale bar is shown below
each image. Photographs reproduced with permission from Kevin Hanley.
3m
Tall−form
Intermediate
spray
storage silo
dryer
Air out
20m
0.1m
Fluidised bed
5m
Air in
Rotary feeder
To
Air input
Pneumatic conveying line
canning
line
Figure 2: A schematic diagram of an industrial pneumatic conveyance line for the transport of
infant formula from the spray drying phase to the storage silos.
88
Quantity
Particle radius
Radius of pipe
Radius of curvature of pipe
Estimated density of particles in the pipe
Estimated velocity of particles in the pipe
Poisson’s ratio
Young’s Modulus
Coefficient of restitution
Yield stress
symbol
a0
R
κ0
ρ
v0
ν
E
σf
value(s)
5 × 10−6 –5 × 10−4
0.05
1
250–350
0.5–1
0.3
3 × 109 (from [10])
0.2
5 × 105
units
m
m
m
kg/m3
m/s
Pa
Pa
Table 3: Typical particle and pipe parameters provided by Kevin Hanley, unless otherwise
stated.
mechanics model of collisions resulting in fragmentation, which implicitly incorporates Hertzian
theory. Finally, in Section 7 we provide some concluding remarks and outline areas for future
work.
2
The flow in the pipe
The pipe has an approximately circular cross-section and has a centreline that is weakly curved
(i.e. has typical radius of curvature L ≫ 2σ where 2σ is the diameter of the pipe cross section.
Flow in curved pipes is usually described as a Dean flow [4] and reviewed in [3]. Such flows are
characterised by the so-called Dean number De and δ the ratio of the typical pipe radius (or halfwidth in the case of non-circular pipes) σ to its typical radius of curvature L. Unfortunately
the Dean number is not defined in the same manner in all works. Here we shall adopt the
definitions
ρGσ3 (2σ/L)1/2
De =
,
µ2
δ=
σ
,
L
(2.1)
where G is a typical pressure gradient along the pipe, µ is the viscosity of the fluid and ρ
its density. At low Dean numbers (below about 100) this definition is equivalent to another
commonly used definition namely
1/2
2σρW̄0
^ = 4Re 2σ
De
,
(2.2)
where the Reynolds number Re =
L
µ
where W̄0 is the mean velocity along the pipe. However these two definitions do not coincide
for larger Dean numbers.
89
At low Dean numbers De ≪ 1, and for loosely coiled pipes δ ≪ 1, the flow is, to a good
approximation, a Poiseuille flow along the axis of the pipe with a small circulatory component
in the transverse cross-section of the pipe. In the study group we initially made the assumption
that significant flow (as far as motion of particles in the flow is concerned) occurs mainly in the
axial direction. However, as we shall briefly outline here, this assumption is probably invalid
so that further investigation should be directed to considering flow in the cross-section of the
pipe. In fact the high Dean number limit turns out to be of more relevance to the flows in the
conveyance line than the low Dean number limit.
The structure of the flow at high Dean number. The high Dean number limit has been
considered previously by [5, 6, 14] by using a mixture of numerical and asymptotic techniques.
They define the flow velocity u = wt + v(sin αb − cos αn) + u(cos αb + sin αn) (here t is the
unit vector along the axis of the pipe, n is the unit vector in the plane of the pipe centreline
directed towards the inside of the this curve, b is the other unit normal vector (see Figure 3)).
In the limit De ≫ 1 δ ≫ 1 [5, 6, 14] have shown that in the bulk of the flow the scalings for
the various velocity components are
1/2 !
ν
ν
ν
L
u = O De1/3
, v = O De1/3
, w = O De2/3
.
(2.3)
σ
σ
σ 2σ
In addition they identify a boundary layer of thickness O(σDe−1/3 ) around the edge of the pipe
in which the solution is such that
1/2 !
ν
ν
ν
L
u = O De2/3
, v = O De1/3
, w = O De2/3
.
(2.4)
σ
σ
σ 2σ
At these high Dean numbers the Dean number is related to the mean axial velocity W̄ by (2.3c),
that is by
3/2 9/4 W̄ σ
.
(2.5)
De = O
ν3/2 L3/4
Here we give no further details of this solution except to say that it has to be calculated
numerically, even in the boundary layer.
Turbulent pipe flow. The conveyance line flow occurs at high Reynolds number. So for
example, we estimated the Reynolds number as 20,000 by taking the viscosity of the flow to be
that of air. In practice the net viscosity of the mixture of air and particles will be more than
this, so the Reynolds number will be less than for air alone. Nevertheless the flow may still be
turbulent. The boundary layer structure of a turbulent flow in a straight pipe is described to a
reasonable approximation by the so-called Law of the Wall (see for example [1, 16]). It is less
clear however what the equivalent structure is in a curved pipe.
90
M
b
u
v
α
n
(a)
t
(b)
Figure 3: Illustration of the coordinates used to describe the flow in the pipe. Panel (a)
shows the definitions of the vectors {t, n, b} while panel (b) shows the definition of the velocity
components (here t is directed into the paper).
3
Can collisions of particles with the wall of the pipe account for particle fragmentation?
Here we consider particle motion in the pipe flow. Since particles are small such motion is
usually dominated by the fluid drag, rather than inertia. This motivates us to ask the question
can particles ever collide with the side of the pipe with sufficient velocity in order to cause their
fragmentation (i.e. velocities of O(1ms−1 ))? In particular we note that it is the centrifugal
force on the particle that is likely to be the cause of motion across streamlines towards the edge
of the pipe and, further, that since the flow velocity at the boundary of the pipe is zero we
would not expect the centrifugal force on a particle close to the edge of the pipe to be large.
However it is still possible to envisage a situation in which there is a narrow boundary layer
in the vicinity of the pipe wall, across which the fluid velocity drops rapidly to zero, where
the particle has sufficient inertia to cross from the region of high fluid velocity to the pipe wall
without losing most of its kinetic energy. In order to investigate this possibility we will consider
the toy problem of a particle moving in a two-dimensional flow.
3.1
The 2D problem
Here we consider a toy two-dimensional problem noting that where the two-dimensional boundary layer structure is similar to that of the real flow the resulting particle trajectories will be
qualitatively similar. Since the particle Reynolds number is typically small we can approximate
91
its equation of motion by balancing particle inertia against linear Stokes drag to give
d2 x
dx
m 2 = 6πµrp v −
.
dt
dt
(3.1)
Here m is the mass of one particle, rp is the radius of the particle, x and v are the position
and velocity of the particle respectively, and µ is the fluid viscosity. Note that the Stokes drag
formula is assumed because the flow of air relative to the moving particle has a low (local)
Reynolds number, quite different from the high Re for the air flow down the pipe.
We would like to consider two particular scenarios. In the first (i) the particle moves in a
pipe flow around a circular bend in which the predominant flow is in the direction of the
tangent to the centreline t (i.e. takes the form u = wt). The second scenario (ii) aims to
model the situation where the secondary Dean flows in the cross-section of the pipe are of more
significance to particle-wall collisions than the primary flow along the axis of the pipe. In order
to model these situations we introduce polar coordinates for the particle describing its position
by x = R(t)er so that, on differentiation with respect to t we find
ẋ = Ṙer + Rθ̇eθ ,
ẍ = (R̈ − Rθ̇2 )er + (2Ṙθ̇ + Rθ̈)eθ ,
where er and eθ are the unit radial and azimuthal vectors in the coordinate directions of the
plane polars r and θ, whose origin is at the pipe centreline. Furthermore in both scenarios
(i) and (ii) the flow is predominantly in the azimuthal direction; this motivates us to write
v = g(R, θ)eθ .
We non-dimensionalise (3.1) by writing
x = R0 x∗ , R = R0 R∗ , v = V0 v∗ , t =
R0 ∗
t,
V0
which gives, on dropping ′ ∗s, substituting x∗ = R∗ er and equating components in the radial
and azimuthal directions,
R̈ − Rθ̇2 = −Γ̃ Ṙ,
2Ṙθ̇ + Rθ̈ = Γ̃ (g(R, θ) − Rθ̇).
(3.2)
where
Γ̃ =
6πµrpR0
.
mV0
In scenario (i) we are typically interested in a pipe with radius of curvature which is much
greater than the pipe radius whereas in scenario (ii) we are interested in flow occurring in a
narrow boundary layer around the edge of the pipe. This suggests that we should further write
R = 1 − εz where R = 1 denotes the outer edge of the pipe and where the small parameter
ε = XRbl0 gives the ratio of the boundary layer thickness to the typical radius of curvature of the
92
pipe. On making this substitution in (3.2), writing θ̇ = Ω and on rescaling time by writing
t = εΓ̃ τ we obtain the following system
d2 z
dz
= −(1 − εz)Ω2 − Υ 2
dτ dτ
dΩ
dz
Υ (1 − εz)
=g
^(z, θ) − (1 − εz)Ω
− 2εΩ
dτ
dτ
dθ
= ε1/2 Υ−1/2 Ω.
dτ
(3.3)
(3.4)
(3.5)
where Υ = ε−1 Γ̃ −2 . The key parameter in this problem is Υ (recall that ε ≪ 1). This parameter
corresponds to the ratio of to the timescale for drag to accelerate a particle to the velocity of the
flow to that for a particle to traverse the boundary layer. Thus if Υ ≪ 1 the particle moves, to
a good approximation, with the velocity of the flow. If however Υ = O(1) the particle can cross
the boundary layer to the wall of the pipe without being decelerated to almost zero velocity at
the wall. In other words if Υ ≪ 1 it will impact the wall (at z = 0) with very small velocity
(an asymptotic analysis reveals that the dimensionless velocity normal to the wall on impact
will at most be O(Υ2 ) corresponding to a dimensional velocity of at most O(V0 ε1/2 Υ5/2 )). For
Υ = O(1), however, the particle impacts the wall with O(1) dimensionless velocity normal to
the wall corresponding to a dimensional velocity of O(V0 ε1/2 ). 10
In summary we expect that the chance of particle impacts on the wall leading to particle
fracture is negligible if
Υ=
m2 V02
≪ 1,
(6πµrp)2 R0 Xbl
where Xbl is the width of the flow boundary layer and R0 is the typical radius curvature of
the problem (in case (i) flow along the axis of the pipe R0 is the typical radius of curvature of
the pipe centreline while in case (ii) in which secondary flows dominate the process R0 is the
radius of the pipe) . Where Υ = O(1) the particle impacts the wall with a velocity of O(V0 ε1/2 )
and there is a significantly greater chance of particle fracture on impact. Even where Υ ≪ 1
inter-particle collisions give another possible mechanism giving rise to particle fracture. This
will be considered further in §4.
Relevance to Dean flow. Consider first the flow along the pipe, parallel to its centreline,
corresponding to case (i). Here we can identify R0 with the radius of curvature of the pipe
centreline L, the velocity can be read off from (2.3c) and (2.4c) so that the key parameters are
1/2
L
2/3 ν
R0 = L,
V0 = De
,
Xbl = σDe−1/3 ,
σ 2σ
10
Note that where Υ ≫ 1 the asymptotic analysis reveals an inertial particle trajectory in which both w and
dz/dτ are unchanged to leading order; in order to retain more information it is necessary to rescale distance
normal to the boundary to give an O(1) value of Υ.
93
u
u
R
θ
R
θ
(i)
(ii)
Figure 4: The flows considered for the toy problem
and thus give
Υ=
m2 ν2 De5/3
(6πµrp)2 2σ4
in case (i).
(3.6)
In case (ii) we can identify R0 with the radius of the pipe σ, while the typical velocity is that
in the azimuthal direction (which can be read off from (2.4a)). The key parameters are thus
R0 = σ,
ν
V0 = De2/3 ,
σ
Xbl = σDe−1/3 ,
from which it follows that
m2 ν2 De5/3
Υ=
(6πµrp)2 σ4
in case (ii).
(3.7)
We can relate these expressions for Υ to the mean axial velocity in the pipe W̄ via (2.5). It
is immediately apparent that Υ is the same order of magnitude in both cases signifying that
the azimuthal and longitudinal velocities occurring in Dean flow have comparable effects on
the motion of particles across the boundary layer adjacent to the pipe boundary. Thus any
quantitative analysis requires that we consider the full three-dimensional problem (rather than
the toy model treated here).
94
4
Particle transport in the pipe
The particle Reynolds number is relatively small and so we can approximate drag force by the
linear Stokes relation. Balancing inertia with the drag force gives
d2 x
dx
m 2 = 6πµa v −
,
(4.1)
dt
dt
where a is the radius of the particle, x and v are the position and velocity of the particle
respectively, and µ is the fluid viscosity.
Since the radius of the pipe is generally considerably smaller than its radius of curvature it is
helpful to introduce a local coordinate system based about the centre line of the pipe, r = q(s),
defined by
(4.2)
x = q(s) + ξn(s) + ηb,
where n(s) and b are the principal unit normal
p and unit binormal vectors respectively, s
measures arclength along the centre line, and η2 + ξ2 measures the distance of the point
from the centre line.
We use the standard Serret-Frenet formulae (see, for example [12]) to relate the various quantities in (4.2), namely
dq
= t,
ds
dt
= κn,
ds
dn
= τb − κt,
ds
(4.3)
where κ is the pipe curvature, τ is the torsion and t is the tangent vector. To make the equations
complete we also have db/dt = τn. We assume a planar pipe and so τ = 0, therefore in the
subsequent analysis we can ignore the n component of the equations of motion.
In order to track an individual particle at position x(t) we rewrite (4.2) as x(t) = q(s(t)) +
ξ(t)n(s(t)) + ηb and use (4.1) to find differential equations for ξ and s. Before substituting
x(t) into (4.1) we nondimensionalise by choosing
x=
x∗
,
κ0
q=
q∗
,
κ0
r = Rr∗ ,
ξ = Rξ∗ , a = a0 a∗ , t =
η = Rη∗ , κ = κ0 κ∗ ,
t∗
, v = v0 v∗ ,
κ0 v0
s=
s∗
,
κ0
(4.4)
where R is the radius of the pipe, κ0 is the typical curvature of the pipe centreline, a0 is a
typical particle radius and v0 is a typical fluid velocity. On dropping the ∗’s the dimensionless
equations (4.1) and (4.2) become
d2 x
dx
Γ
,
(4.5)
= 2 g(r)t −
dt2
a
dt
95
and
x = q(s) + ǫ(ξn(s) + ηb),
(4.6)
where we have assumed that the flow is primarily
p parallel to the centreline of the pipe and has
the nondimensional form v = g(r)t, with r = η2 + ξ2 . We assume that the coordinate η is
constant while the particle moves. Note that equation (4.3) remains unchanged.
The dimensionless parameters in (4.5) and (4.6) are given by
Γ=
9µ
2a20 v0 κ0 ρ
ǫ = Rκ0 .
,
(4.7)
Typical sizes of these parameters, using Table 3, are Γ ≈ 11 and ǫ ≈ 0.1, which allows us to
set Γ = γ/ǫ where γ = O(1).
Using the Serret-Frenet equations (4.3) we differentiate x in (4.6) to give
dx
= ṡ(1 − ǫξκ)t + ǫξ̇n
dt d2 x
dκ
2
s̈(1
−
ǫξκ)
−
ǫ
ṡ
2
ξ̇κ
+
ξ
=
ṡ
t
+
κ(1
−
ǫξκ)
ṡ
+
ǫ
ξ̈
n.
dt2
ds
(4.8)
(4.9)
Upon substitution into (4.5) and equating coefficients of t and n we obtain the following
equations relating ξ and s
γ
dκ
[g(r) − ṡ(1 − ǫξκ)]
(4.10)
s̈(1 − ǫξκ) − ǫṡ 2ξ̇κ + ξ ṡ =
ds
ǫa2
γ
κ(1 − ǫξκ)ṡ2 + ǫξ̈ = − 2 ξ̇.
(4.11)
a
Assuming nǫ ≪ 1 we find that the leading order terms satisfy
p
a2 κ(s) 2
ṡ .
(4.12)
ṡ = g( η2 + ξ2 ),
ξ̇ = −
γ
Eliminating ṡ gives the expression
i2
a2 κ(s) h p 2
g( η + ξ2 ) ,
γ
(4.13)
i2
2a2 κ(s)v20 ρ h p
g( (η/R)2 + (ξ/R)2) .
9µ
(4.14)
ξ̇ = −
or in dimensional form
ξ̇ = −
Further work could treat a particle on the median plane η = 0. Also quite generally, since
decreases to zero as the particle
g(r = 1) = 0 at the wall, we can at least be sure that dξ
dt
2
approaches the wall. Although we can’t set g(r) = 1 − r , because the Reynolds number is
high, such a dependence in the boundary layer near r = 1 tells us that 1 − r = 1 − ξ(t) (on
η = 0) decreases to zero with exponential decay, hence there is no impact velocity.
96
5
5.1
Hertz theory for binary collisions
Critical Energy
Hertz theory is an analysis of stresses at the contact of elastic solids. It was initially developed
for static loading, but subsequently extended to quasi-static impacts (e.g. of spheres). We use
Hertz theory to give an estimate for the critical energy, i.e., the energy needed to break a given
particle of some radius a, here we give a brief overview of the key equations.
As in [15], consider a (collinear) collision between two particles, modelled as smooth, homogeneous, solid, elastic spheres, of radii a1 and a2 respectively. The elastic properties of the
spheres are given by their Young moduli Ei and Poisson’s ratios νi (i = 1, 2). We look at a
quasi-static process and thus consider an instant where the mutual force of compression is P.
The spheres compress elastically, with a contact area of radius ac , see Figure 5.
P
z
Figure 5: From left to right: elastic sphere forced onto an elastic plane with a force P, and two
spherical particles of radii a1 and a2 colliding collinearly. z is the amount of compression, and
ac is the radius of the contact disk.
The distribution of pressure across the contact area, as a function of distance r away from the
centreline, is
r
p = p(r) = p0 1 −
,
(5.1)
ac
where ac is the radius of the contact zone between the particles, and the maximum pressure p0
is given by
3P
p0 =
.
(5.2)
2πa2c
97
The effective values of the Young’s modulus E∗ and radius a∗ are defined via
1 − ν1 2 1 − ν2 2
1
+
,
=
E∗
E1
E2
1
1
1
+
,
=
a∗
a1 a2
and
respectively.
From Hertz theory, we then have the following relations between the applied force P, the
effective Young’s modulus E∗ , the effective radius a∗ , and the compression distance z:
4
P = E∗ a∗1/2 z3/2 ,
3
where z is given by a2c = a∗ z, and
ac =
3Pa∗
4E∗
13
.
(5.3)
(5.4)
The critical compression distance, at which fracture occurs, is denoted zf . For a collision that
just reaches this critical compression length zf , the work done by the compression force just
equals the initial kinetic energy Er , and thus:
Z zf
1
2
Er = mVf =
P(z)dz.
(5.5)
2
0
V is the relative impact velocity, and Vy is defined as the yield velocity. Provided V < Vy the
interaction is assumed to be elastic. Inserting the above expression for P(z) we get
Er =
8 E∗ z5f
,
15 a∗2
(5.6)
We can relate the critical compression distance zf to the contact yield stress σf of the material
by
σf ≡ p0 (zf ),
From equation (5.2) and (5.3) we have
σf =
2E∗ zf
.
πa∗
(5.7)
Rearranging for zf , and substituting this expression into (5.5) we obtain the energy at which
the particle is estimated to fracture:
Er =
π5 a∗3 σ5f
.
60 E∗4
(5.8)
We shall return to this expression in Section 6, where we shall consider the statistics of particle
collisions leading to fragmentation.
98
5.2
Particle structure
Hertz theory, as mentioned in the previous section, assumes that the two impacting spheres are
solid. However, the particles that make up infant formula, see Figure 1 are clearly hollow. The
study group was asked to investigate how far into the particle there are significant increases in
stress, and therefore whether modelling the spheres as hollow should be incorporated into the
model. Here, we consider the simpler case of two impacting cylinders, although it should be
straightforward to extend the result to two spheres.
Recall that z is the distance from the contact point. If we take the origin of the z-axis to be
at the point where the two cylinders first contact, then z = 0 is the location of a plane of
contact between the two cylinders i.e., the point at which the contact force is distributed over
a rectangle of width 2ac and length equal to the length of the shortest cylinder. This is the
smallest possible area over which the contact force is distributed, and therefore the point at
which the stress has its maximum value.
As the compression increases, the rectangular area becomes larger, because the width of the
rectangle constantly increases due to the curvature of the circular cross-section. Since stress is
force/area, the stresses decrease as z increases.
From [7] we have an expression for σz , the stress in the z direction
ac
σz
,
= −p
p0
a2c + z2
(5.9)
∗ 1
2
where p0 is the maximum pressure, and for two cylinders in contact is given by p0 = PE
πR
(see [7]). (Note that the convention in mechanical engineering is to take compressive stresses
to be negative.)
In Figure 6 we plot a graph of the stress divided by the maximum pressure ( pσ0z ) against the
distance from the contact point, zac , for a constant ac and typical particle parameters.
At z = 0, the compressive stress is at its maximum. We also observe that the stresses diminish
extremely rapidly as you move further away from the contact point. We expect an entirely
analogous result in the case of two impacting spheres. If the stress changes are non negligible
at a depth that is close to the wall thickness, then Hertz theory is not applicable. If the extent
of the zone where there is a measurable increase in stress is small, then Hertz theory can be
used to represent the contact mechanics of these particles. As the walls of the infant formula
particles are approximately a fifth of the diameter, we conclude that approximating the particles
as solid spheres is a reasonable assumption.
99
z 300
ac
250
200
150
100
50
0
−1
−0.8
−0.6
−0.4
σz
p0
−0.2
0
Figure 6: We consider two cylinders in contact. The graph shows the stress divided by the
maximum pressure ( pσ0z ) plotted against the distance from the contact point, ( azc ), for a constant
ac .
6
The statistics of collisions leading to fragmentation
Consider a particle size distribution such that the proportion of particles with radius lying in
the range (a, a + da) is n(a)da. Furthermore take the probability of breaking a particle of
size a when it undergoes a collision in which an energy in the range (E, E + dE) is released to
it is pb (E, a)dE. We note that at present both these distributions will have to be derived from
experiment, though an ultimate goal of this type of work might be to formulate a Smoluchowski
model of the evolution of the size distribution n(a).
In §4 we calculated the transverse velocity ξ̇, due to centrifugal force, of a particle being
advected in a curved pipe as a function of its radius. Notably this was proportional to the
square of the radius of the particle (bigger particles travel more quickly than smaller ones). We
also saw that this velocity slowed markedly as the particle approached the outer edge of the
pipe, a consequence of the slowing of the flow along the pipe in the immediate vicinity of the
pipe wall. This lead us to conclude that collisions of particles with the pipe wall give rise to
insignificant particle fragmentation and to hypothesise that the main cause of particle breakage
is collisions between particles of different radii travelling with different transverse velocities ξ̇.
It is the aim of this section to quantify this process.
We start by considering the total number of collisions between particles of sizes in the range
(a, a + da) with those in the range (α, α + dα). In the direction of the principle normal n they
approach each other with velocity |ξ̇(a)− ξ̇(α)|, see Figure 7. Since they will only collide if their
100
a
|ξ̇(a) − ξ̇(α)|
b
α
Figure 7: A schematic diagram of a particle with radius a approaching a particle with radius
α. b denotes the distance between their centres, and |ξ̇(a) − ξ̇(α)| their approach velocity.
centres come within a distance (a + α) of each other the average number of collisions occurring
to a particle of radius a with particles with radii in the range (α, α + dα) in a small time dt
is Nπ(a + α)2 |ξ̇(a) − ξ̇(α)|n(α)dαdt, where N is the total number density of all particles. It
follows that


The number of collisions in time dt per unit
 volume between particles with radii in range (a, a + da) 
with those with radii in range (α, α + dα)
= N2 π(a + α)2 |ξ̇(a) − ξ̇(α)|n(α)n(a)dαdadt
Of these collisions we need to work out how many lead to fracture. We have already calculated
the energy released in an impact between a particle radius a and one with radius α when their
centres are offset by a distance b (see Figure 7). Such collisions release energy
1
b2
m(a)m(α)
2
Er (a, α, b) =
(v(a) − v(α)) 1 −
.
2 m(a) + m(α)
(a + α)2
Of this we hypothesise that the energy divides between the particles proportional to their mass,
that is
Energy available for
fracture of particle a
Energy available for
fracture of particle α
m(a)
Er (a, α, b),
m(a) + m(α)
m(α)
Er (a, α, b),
=
m(a) + m(α)
=
(6.1)
(6.2)
where Er is the energy at which the particle is estimated to fracture, and is given in Section 5.
The fraction of collisions for which the offset distance between the centres of the particles (at
collision) lies in (b, b + db) is (2πbdb)/(π(a + α)2 ) (see Figure 8b) and the energy released in
such a collision is Er (a, α, b)db.
101
α
b+db b
α
a
a+ α
Figure 8: Here the first figure shows a large particle of size a approaching a smaller particle
size α. The second figure shows the end view of the cylinder with the smaller particle lying at
the centre. Provided the centre of the larger particle (with radius a) lies within the cylinder
the particles will collide.
Referring to (6.2) we find that
Z
Number of particles size
2
range (a, a + da) per unit time = N π (a + α)2 |ξ̇(a)
per unit volume which fracture
Z Emax m(a)
m(a)
pb
−ξ̇(α)|n(a)n(α)
Er d
Er dα. (6.3)
m(a) + m(α)
m(a) + m(α)
0
Translating (6.3) into terms of the separation b between the particle centres (perpendicular to
the velocity) upon impact this can be written as
Number of particles size
Fracture rate = range (a, a + da) per unit time = F(a, x)
per unit volume
Z∞
m(a)(a + α)2
2
|ξ̇(a, x)
= Nπ
0 m(a) + m(α)
Z a+α m(a)
∂Er
−ξ̇(α, x)|n(a)n(α)
pb
Er (a, α, b)
db dα. (6.4)
m(a) + m(α)
∂b
0
7
Conclusions
We have provided a model for the flow in the curved pipe which we used to show that the
velocity of particles decreases rapidly as they approach the pipe wall. We believe that particle102
wall collisions give rise to insignificant particle fragmentation, and that the main mechanism
for particle breakage is collisions between particles of different sizes.
We have analysed inter-particle collisions, using Hertzian contact laws to provide a criterion for
when a binary collision leads to breakup. In addition we have justified modelling these particles
as solid, rather than hollow. Finally, we used a statistical mechanics approach to generate an
expression for the total rate of particle breakup in the flow.
Future work would involve experimental verification of this model.
Acknowledgements
This work was carried out at the 70th European Study Group with Industry, hosted by the
Mathematics Applications Consortium for Science and Industry (MACSI), funded by the Science Foundation Ireland (SFI) Mathematics initiative 06/MI/005. We would like to thank
William Lee for his comments on an earlier draft.
103
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Lond. A 451, 165–188, (1995).
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systems. Phys. A, 321, 417–424, (1996).
[3] S.A. Burger, L. Talbot and L.-S. Yao, Flow in curved pipes. Ann. Rev. Fluid Mech. 15,
461–512, (1983).
[4] W.R. Dean, The stream-line motion of fluid in a curved pipe. Phil. Mag. 5, 673 (1928).
[5] S.C.R. Dennis and N. Riley, On the fully developed flow in a curved pipe at large Dean
number. Proc. R. Soc. Lond. A 434, 473–478, (1991).
[6] H. Ito, Laminar flow in curved pipes. Z. agnew Math. Mech. 49, 653–663, (1969).
[7] K.L. Johnson, Contact Mechanics, Cambridge University Press, (1985).
[8] F. Kun and H.J. Herrmann, Transition from damage to fragmentation in collision of solids.
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(2003).
[10] M. Perkins, S.J. Ebbens, S. Hayes, C.J. Roberts, C.E. Madden, S.Y. Luk, N. Patel, Elastic
modulus measurements from individual lactose particles using atomic force microscopy.
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in Physics, (2003).
[12] John Roe, Elementary Geometry, Oxford University Press, (1993).
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077102 (2005).
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(1976).
104
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[16] http://www.maths-in-industry.org/miis/105/
105
106
Improvement of Energy Efficiency for
Wastewater Treatment
Report Contributors: Mark Burke1 , Michael Chapwanya2,3,
Kevin Doherty4 , Ian Hewitt 5, Andrei Korobeinikov2 ,
Martin Meere6, Stephen McCarty 7, Mick O’Brien2 ,
Vo Thi Ngoc Tuoi6 and Henry Winstanley2,5
Industry Representatives: Toshko Zhelev8 and Jaime Rojas8
1
Department of Mathematics and Statistics, University of Limerick, Ireland
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
3
Report coordinator, m.chapwanya@ul.ie
4
National University of Ireland Galway, Ireland
5
OCIAM, Mathematical Institute, University of Oxford, United Kingdom
6
Department of Applied Mathematics, National University of Ireland Galway, Ireland
7
University College Cork, Ireland
8
Department of Chemical and Environmental Science, University of Limerick, Ireland
2
Abstract
Wastewater treatment requires the elimination of pathogens and reduction of organic matter in
the treated sludge to acceptable levels. One process used to achieve this is Autothermal Thermophylic Aerobic Digestion (ATAD), which relies on promoting non-pathogenic thermophilic
bacteria to digest organic matter and kill pathogens through metabolic heat generation. This
process requires continuous aeration that may be energy consuming, and the final aim of the
study is to identify how the process design can minimize the energy input per mass of treated
sludge. Appropriate modeling of the reactor process is an essential ingredient, so we explore
properties of an existing model and propose a simplified alternative model.
108
(a) A single-reactor design. An example of this design is found in Spain.
(b) A two-reactor design. Semi continuous or continuous treatment requires multiple reactors
to achieve pasteurization. An example of this design is found in Killarney, Ireland.
109
1
Introduction
1.1
Background
Wastewater treatment employs a variety of processes to reduce the environmental and health
impacts of effluent. Treatment is an environmental necessity but is itself very energy-intensive,
and this energy use is one of the principal costs and environmental impacts of the treatment.
Wastewater impurities include a wide range of inorganic and organic chemical species and
microorganisms. A filtered, concentrated ‘sludge’ is produced at initial stages of treament. A
key subsequent stage is to eliminate potential pathogenic organisms and reduce the organic
chemical content which might act as a substrate for further microbial growth. One process that
is used to achieve these treatment goals is autothermal thermophilic aerobic digestion (ATAD).
ATAD makes use of bacterial growth within the sludge both to reduce organic chemical content
and to kill pathogenic bacteria. Aeration of the sludge promotes the growth of aerobic bacteria
which feed on and reduce the organic substrates available in the sludge. This metabolic activity
generates heat and raises the temperature of the sludge. Thermophilic bacteria in the sludge
thrive at high temperatures (Fig. 1). Most pathogens, meanwhile, are mesophiles: their total
metabolic activity reduces rapidly at temperatures above 40 – 45 o C, and they are eventually
killed by sufficiently long exposure (Fig. 2). Aerobic thermophiles proliferate and dominate in
the hot sludge, and ATAD uses their metabolic activity and heat generation to achieve the
treatment goals of pasteurization and reduction of organics.
8
Thermophilic decay rate, [1/day]
Thermophilic growth rate, [1/day]
25
20
15
10
µ
−6.086/(T−34)
HT
= 155.4e
−7.734/(T−34)
− 132.8e
5
Experimental data
Numerical fitting
0
35
40
45
50
55
o
Temperature, T [ C]
60
HT
= 0.001T2 + 0.11T − 2.8
6
5
4
3
2
65
b
7
Experimental data
Numerical fitting
35
40
45
50
55
o
Temperature, T [ C]
60
65
Figure 1: Temperature dependence of the thermophilic growth and decay rate (adapted from
[6]).
The time and cost implications of running full-scale experiments for different plant arrangements
and operating conditions render such an approach infeasible for process design and operational
management. Laboratory scale experiments have provided understanding (to varying degree)
of many of the microbiological and chemical processes involved. By encapsulating much of this
110
′
Mesophilic decay rate, [1/day]
Mesophilic growth rate, [1/day]
80
Experimental data
Numerical fitting
18
16
14
12
−42.95/T
µ = 72e
10
H
−135.6/T
− 201.5e
70
−159.9/T
60
b =1075e
H
50
40
30
20
10
8
25
30
35
40
45
50
o
Temperature, T [ C]
55
60
0
65
Experimental data
Numerical fitting
25
30
35
40
45
50
o
Temperature, T [ C]
55
60
65
Figure 2: Temperature dependence of the mesophilic growth and decay rate (adapted from [6]).
understanding in mathematical models, engineers have provided tools which allow computer
simulations to inform the design process. The ‘Activated Sludge Model 1’ (ASM1) [5] and
subsequent variants [3, 4] have achieved a broad level of acceptance in the wastewater treatment
world. Based on the dominance of the ASM1 model (and family) in the field, its extension to
the ATAD process as set out by [6, 2] appears to have created a de facto standard for modeling
this process, despite drawbacks summarized below. We identify this existing model with the
label ASM1.
1.2
Objective
The intended purpose of the Study Group problem was to use a model of the ATAD treatment
process to provide insights into the optimization of both plant design and operating conditions
in order to reduce the energy consumed per mass of treated sludge while achieving the required
treatment goals.
Approaching the problem as a multidimensional constrained optimization based on the ASM1
model, the problem presenters had identified some generic indications for performance improvements, but encountered numerical difficulties due to the high dimensionality and sensitivity to
parameters inherent in the ASM1 model. Their next intention was to identify rational means
of reducing the dimensionality of the optimization.
1.3
Report Overview
This report addresses the problem in the following way:
• Section 2 provides a more detailed overview of the problem and the considerations raised
111
during the Study Group.
• In Section 3 we discuss the ASM1 model, perform a nondimensionalization based on
typical parameter values (adapted from [2] and [6]), and identify certain regimes within
the timecourse of a single typical batch reaction for which asymptotic approximations
may be appropriate.
• In Section 4 we present a simplified ‘ESGI70’ model proposed by the Study Group with
the intention of being the minimal model required to describe the observed process. The
model comprises a system of ODEs with four dependent variables.
• In Section 5 we compare our simulations to experiments and present a sensitivity analysis
of the ESGI70 model.
• In Section 6 we present an optimal strategy for pasteurization.
2
2.1
Problem Overview
Treatment Goals
Legal requirements for sludge treatment in Ireland and USA set minimum standards for the
pasteurization and stabilization of the treated outflow sludge ([8]). Stabilization refers to the
reduction of volatile solids concentration between sludge intake Xfeed
and outflow Xout
, and the
vs
vs
required minimum reduction is set at 38%:
− Xout
Xfeed
vs
vs
≥ 0.38.
Xfeed
vs
(2.1)
Pasteurization is achieved by the sludge being subjected to high temperature for a prolonged
period of time. Rather than requiring direct microbiological testing of the outflow, the legal
requirement for pasteurization specifies a minimum value of 1.0 for a derived ‘lethality’ statistic
which is an empirically-based function of sludge temperature and time elapsed in the ATAD
reaction process:
Zt
1 bT (t)
L(T, t) =
10
dt ≥ 1,
(2.2)
t0 a
where temperature T is measured in Celsius, elapsed time t in days, and a = 5.007 × 107 and
b = 0.14 ◦C−1 are defined parameter values. Lethality L is only calculated in continuous time
intervals in which temperature T > 50 ◦C, and it is equal to zero when T ≤ 50 ◦ C.
112
Air
In: Raw
sludge
Out: Pathogen
free solids
Figure 3: Schematic representation of a reactor. The ATAD process must have a feed control
mechanism which ensures that the organic feed is above a certain minimum concentration for
adequate heat production. The solids must also be below a maximum concentration to allow
adequate mixing. See for example [8].
2.2
Treatment Plant and Process
ATAD is operated as a batch or semi-batch process. A large reactor containing sludge receives
an additional volume of untreated sludge at the start of a batch via a feed inlet (see schematic
in Fig. 3). During the batch reaction air is pumped continuously into the reactor and provides
both the oxygenation required for aerobic bacterial growth and the physical mixing of the sludge
in the whole reactor. Digestion of organic substrates proceeds hand in hand with bacterial
growth, predominantly of thermophiles due to the elevated temperature. At the end of the
batch reaction time, a volume of treated outflow sludge is removed and is immediately replaced
by the next batch of intake sludge. Thus the outflow sludge has the same composition as the
mixed sludge in the reactor at the end of the batch reaction time tbatch . Batch outflow and
inflow cannot be open at the same time since untreated inflow sludge might be drawn off with
and taint the outflow sludge. The time between successive batch intakes is typically fixed at
24 hours for staffing reasons. In order to achieve the desired treatment outcomes, treatment
plant designs may include a single ATAD reactor stage, or two reactors in series with outflow
from the first reactor being inflow into the second. In some cases, the treatment plant consist
of an ATAD reactor and an anaerobic mesophilic digester in series (cf. [8]). The method of
operation and the number of reactors used depends on the fluctuation of hydraulic loading and
on the quantity of sludge requiring treatment.
113
2.3
Optimization
As indicated in Section 1.2, the primary objective is to optimize the plant design and operating
conditions so that the minimum energy consumed per mass of treated sludge. The plant’s
energy consumption is dominated by the submerged aeration pumps’ energy requirements. For
a sludge aeration rate A (air volumes per reactor sludge volume per day), reactor volume V,
and batch inflow (and outflow) volume Vin , the optimization corresponds to finding the set of
controls U which yield
Z
V
min
A(t)dt,
(2.3)
U Vin
subject to the treatment requirement constraints Eqs. (2.1),(2.2). The full range of optimization
controls U comprises both plant design issues and operating conditions. Design choices include
reactor volume and the use of single or series reactors. Other non-standard possibilities can also
be assessed such as parallel heat exchangers to pre-warm the sludge inflow from sludge outflow,
reactor, or exhaust air heat sources. Controls on operating conditions in the reactor are limited
to the aeration rate and batch reaction time tbatch , and controls on the batch inflow volume,
temperature and substrate concentration. For staffing reasons, current standard operating
conditions generally use a 24-hour batch cycle and maintain a constant aeration rate A during
the whole process. The inflow volume can be expressed as a reactor batch exchange fraction
Q = Vin /V ∈ (0, 1] and is typically in the range 0.1 – 0.2. The inflow is usually at ambient
temperature and existing plants are not set up for inflow heating either directly or by heat
exchangers. The substrate concentration of inflow sludge can be increased by addition of a
bulking organic component at minimal expense.
2.4
Study Group Approach
As with many biological systems, there is considerable uncertainty in inferring both the functional description of processes and the accuracy of parameterizations when applying laboratoryderived models to the real system. The current models in the wastewater literature are motivated by microbiological and engineering interest and have been devised to incorporate as
much as possible of the apparent understanding of underlying processes. As a result they tend
to include a relatively large number of variables, processes, and parameters.
This stands in contrast to the paucity of data accessible from ATAD plants in operation:
aeration rate and sludge temperature are easily monitored, but data on sludge composition at
inlet and outlet and during the course of the batch reaction is typically limited to the total dry
weight of solids and the chemical oxygen demand of the ‘volatile solids’ (by controlled chemical
oxidation of the sludge). Volatile solids include the biomass of bacteria in the sludge together
with organic growth substrates and metabolically inert organic compounds, so no distinction
is made between these components in the measured data.
In addition to the known model being over-parameterized, commonly-used values of several
114
parameters are inferred from curve-fitting predicated on the model itself. In such an exercise,
the discrepancy between the large numbers of variables and parameters in the model and their
poor representation in the available data clearly motivates the use of a simplified model of the
process.
The key focus of the Study Group was modeling the core process of a single reactor system
during a batch reaction. Such a model is trivially set up to run successive batches and can
easily be extended later to a multi-reactor system and other design variants. Two approaches
were followed: one being to investigate the existing ASM1 model for mathematical simplifications (Section 3), the other being to construct a new model using a plausible minimal set of
interactions to see if it can reproduce the observed behavior (Section 4).
The optimization of the process based on such a model was not fully addressed in the Study
Group, though some sensitivity analysis and initial investigations into its optimization were
tackled. The focus on the core process model and the relative lack of suitably realistic estimates
for many of the plant-specific parameters suggested greater priority be placed on single reactor
operating conditions as optimization controls. Thus the initial focus was on a single reactor
system of fixed design, with plant design optimization left as a task for future work.
2.5
Basic Modeling Assumptions
Some simplifying assumptions permit us to reduce the complexity of the model while at the
same time retaining the essential aspects of the processes:
• There is complete mixing inside the reactor(s).
• All the biological activity takes place only in the reactors.
• The batch outflow and inflow stages are of sufficiently short duration that biological
activity during them is negligible.
• Aeration is sufficient that anaerobic metabolic activity is negligible.
3
The Existing ASM1 Model
The ASM1 model incorporates all the processes identified by microbiologists as being relevant
to modeling the ATAD process. A transfer diagram of the ASM1 model is shown in Fig. 4
with the solid curve cycle showing the processes included in the original ASM1 model (prior to
the ATAD extension). Here XBH denotes the concentration of active mesophilic biomass, SO is
the concentration of oxygen and XP is the concentration of inert organic matter. Organic substrate is separated into readily biodegradable substrate SS and slowly degradable substrate XS .
Untreated sludge in the batch inflow contains a certain concentration of both these substrates.
115
The slow substrate XS consists of organic macromolecules which can be hydrolyzed to produce
the fast substrate SS . Microbial decay adds to the pool of XS : cell lysis on death releases the
cell’s constituents into the extracellular space, a fraction fP of which is inert matter and (1 −fP )
is hydrolyzable macromolecular substrate. Fast substrate SS and oxygen SO are consumed in
growth of the bacteria. All the metabolic process rates are assumed to depend nonlinearly on
substrate and oxygen concentrations through dual Michaelis–Menten kinetics terms.
The ATAD-specific modifications to the ASM1 model are (broken curve cycle in Fig. 4) via
the introduction of a new component: thermophilic biomass, in active XBHT and inactive XSP
forms. While mesophiles are relevant to batch reactions starting from cold, typical ATAD
plants are operated with batches following in rapid succession precisely in order to maintain the
temperature of the reactor. Under these conditions, the mesophiles respire but the temperatures
are beyond their optimum level and so they decay very fast and thus do not make noticeable
contribution to the process, see Fig. 2.
XSP
Activation
XBHT
Thermophilic
decay
Thermophilic
growth
Thermophilic
respiration
S
XBH
O
Decay
Growth
XI
Respiration
S
S
Thermophilic
hydrolysis
XS
Hydrolysis
Figure 4: Transfer diagram of the processes in the ASM1 model (solid lines) and the proposed
extended model (broken lines). In Section 3.1 we present the mathematical representation of
the extended model. (adapted from [7])
3.1
The ASM1 Model at Thermophilic Temperatures
The ASM1 model at thermophilic temperatures tracks the evolution in time of eight quantities in the reactor, these being: SS (t) the readily biodegradable substrate, XS (t) the slowly
biodegradable substrate, XSP (t) the inactivate thermophilic biomass, XBH (t) the active mesophilic biomass, XBHT (t) the active thermophilic biomass, XP (t) inert organic matter from decay,
SO (t) the dissolved oxygen, and T (t) the temperature.
116
For simplicity, we disregard the temperature dependence here, and so do not display an equation
for the temperature. We take fixed values for the model parameters and choose these values for
reactor operation at 50o C. We also neglect the effect of thermophilic activation, on the basis
that it occurs rapidly at the start of the batch reaction, and so do not consider inactive biomass
XSP (t) in the model. Under these assumptions, the governing equations from [6] are
SS
1
SS
1
dSS
SO
SO
XBH −
µHT
XBHT
= − µH
dt
YH KS + SS KO + SO
YHT
KST + SS KOT + SO
XS
SO
XS
SO
+kH
XBH + kHT
XBHT ,
KX XBH + XS KO + SO
KXT XBHT + XS KOT + SO
XS
SO
dXS
= bH (1 − fP )XBH + bHT(1 − fPT )XBHT − kH
XBH
dt
KX XBH + XS KO + SO
XS
SO
−kHT
XBHT ,
KXT XBHT + XS KOT + SO
SS
SO
dXBH
XBH − bH XBH ,
(3.1)
= µH
dt
KS + SS KO + SO
dXBHT
SS
SO
XBHT − bHT XBHT ,
= µHT
dt
KST + SS KOT + SO
(1 − YH )
SS
(1 − YHT )
SS
SO
SO
dSO
=−
µH
XBH −
µHT
XBHT
dt
YH
KS + SS KO + SO
YHT
KST + SS KOT + SO
+kLa S¯O − SO ,
dXP
XS
XS
SO
SO
= fP kH
XBH + fPT kHT
XBHT .
dt
KX XBH + XS KO + SO
KXT XBHT + XS KOT + SO
The parameters appearing in the model are explained in Table 5. We see that the first five of
these equations can be solved independently of the sixth, and so we do not display the equation
for the inert organic matter XP again.
We model the operation of a single tank system for one batch, and so solve (3.1)1 – (3.1)5
subject to initial conditions
SS = SoS , XS = XoS , XBH = XoBH , XBHT = XoBHT , SO = SoO at t = 0.
(3.2)
For the illustrative numerical results used in the current analysis, we take
SoS = 5 gl−1 , XoS = 15 gl−1 , XoBH = 1 gl−1 , XoBHT = 1 gl−1 , SoO = 10−3 gl−1 .
We scale the system of ordinary differential equations by choosing
t ∼ tbatch , SS ∼ SoS , XS ∼ XoS , XBH ∼ XoBH , XBHT ∼ XoBHT , SO ∼ SoO ,
where tbatch = 1 d (one day). With this choice of scaling, the dimensionless form for (3.1)1 –
117
(3.1)5 , (3.2) is given by,
dSS
SS
SS
SO
SO
= −δ1
XBH − δ4
XBHT
dt
δ2 + S S δ3 + S O
δ5 + S S δ6 + S O
XS
SO
SO
XS
XBH + δ9
XBHT ,
+ δ7
δ8 XBH + XS δ3 + SO
δ10 XBHT + XS δ6 + SO
dXS
XS
XS
SO
SO
XBH − δ14
XBHT ,
= δ11 XBH + δ12 XBHT − δ13
dt
δ8 XBH + XS δ3 + SO
δ10 XBHT + XS δ6 + SO
SS
SO
dXBH
= δ15
XBH − δ16 XBH ,
(3.3)
dt
δ2 + S S δ3 + S O
dXBHT
SS
SO
= δ17
XBHT − δ18 XBHT ,
dt
δ5 + S S δ6 + S O
SS
SS
SO
SO
dSO
= −δ19
XBH − δ20
XBHT + δ21 (δ22 − SO ) ,
dt
δ2 + S S δ3 + S O
δ5 + S S δ6 + S O
subject to initial conditions
SS = 1, XS = 1, XBH = 1, XBHT = 1, SO = 1 at t = 0,
where the definition of, and values for, the non-dimensional parameters δi (1 ≤ i ≤ 22) are
given in Table 5.
In Figs. 5 and 6, we show the solution to the system of equations (3.3) for the parameter values
listed in Table 5. It is noteworthy from the parameter values chosen, the readily degradable
material is depleted over a period of approximately one half of a day, but that the slowly
degradable material persists for approximately five days. We should note, however, that that
some of the parameter values used are uncertain, and that they can depend on temperature
and time, as well as other factors, such as the character of the inflow sludge and the reactor
system. Since we have assumed the parameter values to be fixed, the limitations of such an
approach are clear. However, our goal in this section is to gain some insight into aspects of the
ASM1 model, rather than attempting to model a particular reactor system in detail.
3.2
A Subcase: Asymptotic Analysis of the ASM1 Model
As mentioned above, under normal ATAD operating conditions a relatively high temperature
is maintained in the reactor. Commonly, the temperature after adding a new inflow batch
and mixing remains higher than the mesophilic temperature range, and hence their metabolic
activity is negligible. Therefore we neglect mesophiles in the model and set XBH = 0; equations
118
1.4
←I
1.2
Concentrations
1
III →
II
0.8
IV
0.6
0.4
S (t)
S
X (t)
S
XBH(t)
0.2
X
(t)
BHT
S (t)
O
0
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
1
Figure 5: Plot of the solution to system of equations (3.3) over one day and for the parameter
values displayed in Table 5.
4.5
4
Concentrations
3.5 ← I
← III
3
2.5
IV
V
VI
II
2
1.5
S (t)
S
X (t)
1
S
XBH(t)
X
0.5
(t)
BHT
S (t)
O
0
0
1
2
3
4
5
6
7
Time
Figure 6: The same solution as in Figure 5, plotted over one week. Over this time, the slowly
biodegradable substrate is removed, and the dissolved oxygen achieves its saturation value.
119
Table 4: The set of parameters for the ASM1 model, physical parameters. Experiments have
shown that these parameters can depend, among other things, on temperature and time (see
Figs. 1 and 2).
Symbol
KS
KO
KST
KOT
S¯O
µH
µHT
bH
bHT
kH
kHT
kLa
YH
YHT
fP
fPT
KX
KXT
Description
Value
Physical parameters
mesophilic half saturation constant for SS
mesophilic half saturation constant for SO
thermophilic half saturation constant for SS
thermophilic half saturation constant for SO
oxygen saturation concentration
maximum growth rate of XBH
maximum growth rate of XBHT
decay rate of mesophiles
decay rate of thermophiles
maximum mesophilic hydrolysis rate
maximum thermophilic hydrolysis rate
oxygen mass transfer coefficient
mesophilic yield
thermophilic yield
inert fraction of mesophilic biomass
inert fraction of thermophilic biomass
half saturation constant for mesophilic hydrolysis
half saturation constant for thermophilic hydrolysis
120
0.02
2 × 10−4
0.03
2 × 10−4
5 × 10−3
16.8
24.3
43
5.2
10
9.2
1000
0.6
0.6
0.3
0.3
0.03
0.03
Units
g/l
g/l
g/l
g/l
g/l
1/d
1/d
1/d
1/d
1/d
1/d
1/d
–
–
–
–
–
–
Table 5: The set of parameters for the ASM1 model, dimensionless parameters. Experiments
have shown that these parameters can depend, among other things, on temperature and time
(see Figs. 1 and 2). Here we assume these parameters to be constant (i.e., temperature
independent).
Symbol
δ1
δ2
δ3
δ4
δ5
δ6
δ7
δ8
δ9
δ10
δ11
δ12
δ13
δ14
δ15
δ16
δ17
δ18
δ19
δ20
δ21
δ22
Description
Value
Dimensionless parameters
µH tbatch XoBH /(YHT SoS )
KS /SoS
KO /SoO
µHT tbatch XoBHT /(YHT SoS )
KST /SoS
KOT /SoO
kH tbatch X∗BH /S∗S
KX XoBH /XoS
(kHT tbatch ) (X∗BHT /SoS )
KXT XoBHT /XoS
(1 − fP )bH tbatch XoBH /XoS
(1 − fPT )bHT tbatch XoBHT /XoS
kH tbatch XoBH /XoS
kHT tbatch XoBHT /XoS
µH tbatch
bH tbatch
µHT tbatch
bHT tbatch
(1 − YH )µH tbatch XoBH /(YH SoO )
(1 − YHT )µHT tbatch XoBHT /(YHT SoO )
kLa tbatch
S¯O /SoO
121
5.60
4 × 10−3
0.2
8.10
6 × 10−3
0.2
2.0
2 × 10−3
1.84
2 × 10−3
2.01
0.25
0.67
0.61
16.8
43.0
24.3
5.2
11.2 × 103
16.2 × 103
103
5.0
Units
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
(3.3) then reduce to
dSS
SS
XS
SO
SO
= −δ4
XBHT + δ9
XBHT ,
dt
δ5 + S S δ6 + S O
δ10 XBHT + XS δ6 + SO
XS
SO
dXS
= δ12 XBHT − δ14
XBHT ,
dt
δ10 XBHT + XS δ6 + SO
dXBHT
SS
SO
XBHT − δ18 XBHT ,
= δ17
dt
δ5 + S S δ6 + S O
SS
SO
dSO
= −δ20
XBHT + δ21 (δ22 − SO ) ,
dt
δ5 + S S δ6 + S O
SS = 1, XS = 1, XBHT = 1, SO = 1 at t = 0.
(3.4)
In Figs. 7 and 8, we display numerical solutions of system of equations (3.4) for the parameter
values given in Table 5. Comparing these with the solutions given in Figs. 5 and 6, one notes
that there is good qualitative agreement and reasonable quantitative agreement, as would be
expected.
Motivated by the relevant parameter values listed in Table 4, we denote
δ5 = ε, δ10 = θ1 ε, δ20 = θ2 /ε, δ21 = θ3 /ε,
where θ1 , θ2 , θ3 = O(1), and consider the limit ε → 0. For reference, it is worth re-displaying
equations (3.4) now in terms of the parameter ε
dSS
SS
SO
XS
SO
= −δ4
XBHT + δ9
XBHT ,
dt
ε + S S δ6 + S O
θ1 εXBHT + XS δ6 + SO
XS
SO
dXS
= δ12 XBHT − δ14
XBHT ,
dt
θ1 εXBHT + XS δ6 + SO
SS
SO
dXBHT
= δ17
XBHT − δ18 XBHT ,
dt
ε + S S δ6 + S O
dSO
SS
SO
ε
XBHT + θ3 (δ22 − SO ) ,
= −θ2
dt
ε + S S δ6 + S O
SS = 1, XS = 1, XBHT = 1, SO = 1 at t = 0.
(3.5)
The limit ε → 0 is singular, and the asymptotic structure is indicated in Figs. 7 and 8. We
now briefly discuss some of the asymptotic regions (time-scales) arising.
3.3
Region I, t = O(ε)
This very short initial time-scale, which is not even visible in Fig. 7, appears at t = O(ε) and
marks a region of rapid change in the dissolved oxygen concentration. We rescale t = ε^t, and
at leading order we have
SS ∼ 1, XS ∼ 1, XBHT ∼ 1,
122
1.4
←I
1.2
Concentrations
1
III →
II
0.8
IV
0.6
0.4
S (t)
S
X (t)
S
0.2
X
(t)
BHT
S (t)
O
0
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
1
Figure 7: Plot of the solution to system of equations (3.4) over one day for the parameter values
displayed in Table 5. The asymptotic regions for ε → 0 are indicated.
4.5
4
Concentrations
3.5 ← I
← III
3
2.5
IV
V
VI
II
2
1.5
S (t)
S
1
X (t)
S
X
0.5
(t)
BHT
S (t)
O
0
0
1
2
3
4
5
6
7
Time
Figure 8: The same solution as in Figure 7, but plotted over one week. The asymptotic regions
for ε → 0 are indicated.
123
and pose SO ∼ SIO (^t), to obtain
dSIO
SIO
+ θ3 (δ22 − SIO ) ,
= −θ2
δ6 + SIO
d^t
where
SIO = 1 at ^t = 0,
and dSIO /d^t → 0, SIO → ρ1 as ^t → ∞, with ρ1 given by


s
2
θ2
1
θ2
+ 4δ22 δ6  .
+
ρ1 = δ22 − δ6 −
δ22 − δ6 −
2
θ3
θ3
For the parameter values given in Table 5, ρ1 = 0.087, which is in good agreement with the
numerical solution displayed in Fig. 7. We have to note that ρ1 > 1 is also possible, so that
the dissolved oxygen is not necessarily depleted over this time-scale; the competing effects of
aeration and oxygen consumption by the biomass enter at leading order here.
3.4
Region II, t = O(1) and t < tS
In t = O(1), we pose
SS ∼ SIIS (t), XS ∼ XIIS (t), XBHT ∼ XIIBHT (t), SO ∼ SIIO (t),
for ε → 0, to obtain
SIIO
dSIIS
XII ,
= − (δ4 − δ9 )
dt
δ6 + SIIO BHT
SIIO
dXIIS
XII ,
= δ12 XIIBHT − δ14
dt
δ6 + SIIO BHT
dXIIBHT
SIIO
XII − δ18 XIIBHT ,
= δ17
dt
δ6 + SIIO BHT
SIIO
XII = θ3 (δ22 − SIIO ) .
θ2
δ6 + SIIO BHT
(3.6)
We do not pursue to solve the system of equations (3.6) here, and confine ourselves instead
to considering the system’s qualitative behavior. It is clear, that the growth of the biomass is
not limited by the availability of SS , and that hydrolysis is not limited by the availability of
XS . However, both biomass growth and hydrolysis do depend on the available dissolved oxygen
here.
From (3.6)1 , it is clear that dSIIS /dt ≤ 0 for δ4 > δ9 , or, in dimensional terms, µHT/YHT > kHT .
This corresponds to the case of SS being consumed faster by the biomass than it is generated
by hydrolysis. We restrict our discussion here to this case. For this case, there is tS = O(1)
such that SIIS (tS ) = 0, and this indicates a region that we will discuss next.
124
3.5
Region III, t∗ = O(1)
This time-scale corresponds to t∗ = O(1) where t = tS + εt∗ , and gives the location of a region
where the availability of SS becomes a limiting factor in the further growth of the biomass.
Also, the oxygen concentration undergoes a rapid change over this time-scale. In our numerical
solutions (see Fig. 8), t = tS gives the time when the biomass XBHT achieves its maximum. For
t∗ = O(1), we have
XS ∼ XIIS (tS ), XBHT ∼ XIIBHT (tS ),
where both these quantities are determined as part of the solution to the leading order problem
in Region II. In t∗ = O(1), we pose
SS ∼ εSIII
(t∗ ), SO ∼ SIII
(t∗ ),
S
O
to obtain
SIII
SIII
SIII
dSIII
O
S
O
S
II
X
(t
)
+
δ
XIIBHT (tS ),
=
−δ
S
9
4
BHT
III
III
dt∗
1 + SIII
δ
+
S
δ
+
S
6
6
S
O
O
III
III
dSIII
S
S
O
S
O
XIIBHT (tS ) + θ3 (δ22 − SIII
).
= −θ2
O
III
dt∗
1 + SIII
δ
+
S
6
S
O
(3.7)
∗
III
∗
III
III
As t∗ → ∞, dSIII
S /dt → 0, dSO /dt → 0, SS → ρ2 , SO → ρ3 , where
ρ2 =
δ9
,
δ4 − δ9
(recall that we are considering δ4 > δ9 here), and


s
2
θ2 δ9 II
1
θ2 δ9 II
XBHT (tS ) +
X (tS ) + 4δ22 δ6  .
δ22 − δ6 −
ρ3 = δ22 − δ6 −
2
θ3 δ4
θ3 δ4 BHT
These predictions are in good agreement with the numerical solution displayed in Fig. 7.
3.6
Region IV, t = O(1), tS < t < tX
This is the time-scale over which most of the XS is degraded. In this region, we have
SS ∼ ε
δ9
,
δ4 − δ9
and we have XS , XBHT , SO = O(1). For brevity, we do not display or discuss the leading order
equations here, and simply note that there is a tX = O(1) such that XS (tX ) = 0, which
determines the time when the availability of XS first becomes a limiting factor for hydrolysis.
We omit discussion of Regions V and VI, other than to give their locations and the scalings.
Region V is located at t† = O(1), where t = tX + εt† , and in which SS , XS = O(ε), XBHT , SO =
O(1). Region VI is at t = O(1), t > tX and here we also have SS , XS = O(ε), XBHT , SO = O(1).
125
4
The ESGI70 Model
The full ASM1 model has large numbers of variables and interactions, and consequently many
parameters. The motivation for including these in the model is predominantly microbiological,
and seems to arise from the urge to include all possible components about which there is some
knowledge. However, many of the parameter values are poorly known: they are dependent on
the chemical and microbiological make-up of the particular sludge, and therefore typically used
as fitting parameters. With such a highly parameterized model, sets of parameter values can be
found to fit most data but it is unclear if the model remains reliable in such situations. Applying
a parameterized model under different (but physically reasonable) operating conditions may
give spurious modeling artifacts. And this may also cause numerical difficulties in attempting
any optimization.
These considerations motivate us to seek a minimal model which should be capable of describing
the essential mechanisms of the ATAD process, but ignores issues which — though microbiologically ‘known’ — are not material to the process within the likely range of operating conditions.
It is hoped that such a model might describe the process sufficiently accurately while providing
a more suitable basis for optimization.
As indicated in our analysis of the ASM1 model, we opt to ignore the presence of mesophiles
since the mixed sludge is hot enough to render their metabolic activity negligible. A biomass
concentration variable XB corresponds to the concentration of thermophiles in the sludge. Any
mesophiles in the inflow sludge can effectively be considered as already lyzed into additional
contributions to substrate and inerts.
We also choose to ignore the distinction between the two pools of substrate SS and XS , and
consider a single substrate pool X. This is motivated by the urge to simplify to a minimal
model, rather than through an assumption that one or other pool may be neglected because
it is never rate-limiting under normal operating conditions. The effect of this simplification
is greatest after substrate is depleted and becomes limiting (e.g. from Region III of Fig. 7 in
the ASM1 model): in our model substrate availability for metabolism is directly dependent on
biomass decay, while in the ASM1 model it is buffered by the slow substrate reservoir. The size
of this effect remains to be demonstrated as well as its importance in relation to achieving the
treatment goals.
For a single batch and a single reactor, we introduce the following governing ordinary differential
equations,
126
dXB
dt
dX
dt
dS
dt
dT
dt
= G − b(T )XB ,
= −αG + b(T )(1 − f)XB ,
(4.1)
= A(S̄ − S) − βG,
= λG − h(T ),
where biomass concentration XB (t) has units [g/l], substrate concentration X(t) has units [g/l],
S(t) is the oxygen concentration [mg/l], and T (t) is temperature [o C]. Biomass is assumed
here to have a growth rate G given by the dual Michaelis–Menten kinetics expression
G = µ(T )
S
X
XB .
Ks + S Kx + X
(4.2)
The biomass is also subject to linear decay with rate constant b which may be temperaturedependent. Growth of biomass is proportional to the substrate and oxygen consumptions with
the consumption efficiency 1/α and 1/β respectively. Substrate X is also produced as a result
of decay of biomass, of which a fraction (1 − f) is recycled as substrate.
Dissolved oxygen is replenished through aeration that is subject to the saturation concentration
S̄. The sludge temperature increases at a rate proportional to the biomass growth rate and is
decreased by heat loss from the reactor. This heat loss occurs through the reactor walls, via
the exhaust gas stream from aeration and in the sludge outflow. The heat loss is dominated by
losses through latent heat of evaporation in the exhaust gas stream. And while the other heat
loss pathways are proportional to T −Tamb , this heat loss pathway is less temperature-dependent
so we model it with a simple constant loss rate.
The parameters for this model are summarized in Table 6. Most of the parameters relate to the
thermophile metabolism and sludge properties alone, but those dependent on reactor design
are aeration A, heating rate coefficient λ, and heat loss rate h. In the lack of specific plant
information, it was deemed appropriate at this stage to focus on operating conditions rather
than plant design in any optimization approaches.
This model describes the batch reaction period from just after a batch intake is mixed into the
reactor until batch outflow. The required initial conditions are those of the sludge immediately
following mixing of the inflow with the sludge remaining in the reactor from the previous batch,
since the reactor volume is only partially replaced at each batch. The pre-existing sludge
conditions depend on the history of previous batches in the reactor. With repeated batches
of the same type, we expect the batch cycle to settle to state in which successive batches are
identical and the pre-mixing reactor sludge conditions are effectively independent of the initial
conditions of the first batch reactor sludge. We therefore provide initial conditions for the inflow
sludge added at each batch and also reactor sludge initial conditions for the first batch. The
model can be run to simulate successive batches until these settle to a periodic batch cycle, as
127
Table 6: Set of parameters for the ATAD model. Some of the units have been converted from
their COD (chemical oxygen demand) equivalent.
Symbol
Description
Value
Units
Reference
Kx
Ks
µ
b
α
β
λ
h
S̄
Physical parameters
half-saturation constant for substrate 0.1
half-saturation constant for oxygen
0.17
growth rate
10
decay rate
5
substrate transformation
1.5
oxygen transformation
0.5
heat production
6.5
heat loss rate
7
oxygen saturation constant
10
g/l
mg/l
1/d
1/d
–
–
o
Cl/g
o
C/d
mg/l
[1]
"
"
"
"
"
"
–
–
A
Xfeed
B
Xfeed
Sfeed
Vin /V
T feed
Baseline case parameters
aeration rate
200
biomass input/feeding rate
1
sludge input/feeding rate
15
oxygen input/feeding rate
1
ratio of volume input
0.1
temperature of the feed
13
1/d
g/l
g/l
mg/l
–
o
C
[1]
"
"
"
"
–
128
indicated in Fig. 9. The linking of successive batches and initial mixing of inflow sludge for
each batch can be described by adding additional terms into the system:
dXB
dt
dX
dt
dS
dt
dT
dt
= G − bXB + Qδ (Xfeed
− XB ),
B
= −αG + b(1 − f)XB + Qδ (Xfeed − X),
(4.3)
= A(S̄ − S) − βG + Qδ (Sfeed − S),
= λG − h + Qδ (T feed − T ),
where we define the Qδ as
Qδ =
Vin
δ(t − tin ),
V
and δ(t − tin ) is a Dirac delta function.
The treatment goals for stabilization and pasteurization can be calculated at any solution time
after solving the system up to that time. The stabilization target (2.1) is defined in terms of
the volatile solids Xvs , which can be calculated by integrating
dXvs
= G(1 − α) + Qδ (Xfeed
− Xvs ).
vs
dt
(4.4)
The pasteurization target can be calculated by performing the integration (2.2) while T > 50o C
and setting L = 0 at other times.
Our first approach is to neglect the temperature dependence of the parameters so that the
temperature equation decouples from the system. As typical ATAD operating temperatures
are higher than 45o C, we note from Fig. 1 that this will tend to give overestimates of biomass
growth and temperature.
4.1
Notes on Optimization
The main operating optimization controls are the aeration rate A, the exchanged volume fraction Q, and the batch reaction time tbatch . The inflow sludge properties are largely fixed by
the nature of the waste rather than being optimization controls. The exception to this is the
possibility of enriching the inflow with additional substrate.
Operational restrictions mean that the aeration rate is generally fixed as constant throughout
a batch and the batch time is fixed to give a daily cycle. If we adhere to these restrictions,
the controls are reduced to the constant aeration rate A and the exchanged volume fraction Q,
and optimization will correspond to the treatment goals being achieved at the end of the daily
cycle. In this situation, the energy consumption per mass of treated sludge is proportional to
A/Q.
129
0.7
15
Volatile solids (g/l)
Biomass (g/l)
0.6
0.5
0.4
14.5
14
0.3
0.2
25
26
27
28
29
30
31
32
13.5
25
33
1.6
4
1.4
3.5
Growth rate (g/l/day)
Substrate (g/l)
1.2
1
0.8
0.6
0.4
26
27
28
29
30
31
32
Lethality, L(T,t)
Oxygen (mg/l)
30
31
32
33
26
27
28
29
30
31
32
33
26
27
28
29
30
31
32
33
26
27
28
29
30
time (days)
31
32
33
3
2
1.5
2.5
6
4
2
2
1.5
1
0.5
26
27
28
29
30
31
32
0
25
33
0.46
59
0.45
out feed
/X
vs
vs
60
58
Stabilisation, 1−X
Temperature (degC)
29
3
8
57
56
55
54
53
25
28
2.5
0.5
25
33
10
0
25
27
1
0.2
0
25
26
26
27
28
29
30
time (days)
31
32
0.44
0.43
0.42
0.41
0.4
25
33
Figure 9: Multiple batch steady-state numerical simulations of the ESGI70-ATAD model.
130
The effect of aeration rate A is apparently straightforward: additional oxygenation promotes
faster biomass growth and therefore quicker stabilization and temperature increase, giving
quicker pasteurization. For a fixed batch time and an exchanged volume fraction, we would
reduce the aeration rate until the more stringent treatment goal is achieved at the end of
the batch. While energy consumption increases linearly with aeration rate, the growth rate’s
increase with oxygenation is less than linear, approaching linearity near zero oxygen concentration. This would suggest that it may be desirable to maintain the lowest aeration rate which
provides sufficient mixing and limits anaerobic metabolism, though other considerations such
as plant throughput may become overriding.
The effect of the exchanged volume fraction Q is less clear. The inflow sludge brings new substrate into the reactor, which leads to increased growth rate, digestion, and heating. However,
it also cools the sludge mixture which slows the pasteurization process. These two effects have
competing impacts on the time to achieve the pasteurization goal. The stabilization treatment goal is expressed in terms of the reduction of volatile solids between inflow and outflow
(which is extracted directly from the reactor sludge mixture). Similarly, while inflow brings
fresh substrate to fuel the metabolism and promote growth, increasing the exchanged fraction
also decreases the apparent stabilization just after inflow mixing which can be expressed as
∗
(1 − Q)(1 − X∗vs /Xfeed
vs ), where Xvs relates to the pre-mixing reactor sludge remaining from the
previous batch.
A typical ATAD batch reaction follows a timecourse similar to that shown in Fig. 5. The
influx of new substrate on initial mixing leads to a rapid increase in metabolic activity and
concomitant depletion of oxygen. Substrate is initially in abundance and is digested through
the batch reaction, while oxygen is constantly supplied through aeration. At the start of a batch
reaction oxygen rapidly becomes limiting while the substrate remains abundant. In this phase
the biomass growth rate is mostly determined by the aeration rate and is largely independent of
the substrate. As substrate concentration starts to become depleted the control of the process
transfers to the diminishing substrate level.
The effect of inflow substrate enrichment is to delay the depletion of substrate, extending
the duration of the initial oxygen-depleted phase. It has minimal effect on the growth rate or
temperature (and hence treatment goals) until the time at which substrate would otherwise have
started to become limiting. As substrate enrichment directly increases inflow volatile solids, it
means that a greater quantity of volatile solids must be digested to achieve stabilization.
Once substrate is sufficiently depleted that it is limiting, the oxygen level under constant
aeration will generally increase. At this stage, additional substrate comes only from biomass
decay (in the ASM1 scheme, via the remaining slow substrate pool), so the growth rate must
decrease from this point until decay dominates and then finally the temperature will start to
drop. It would therefore appear to continue the reaction (and aeration) significantly into the
stage of a decreasing reaction rate except to achieve an unsatisfied treatment goal.
As a broad guideline for optimization, it would therefore appear ideal to adjust the aeration
rate and exchanged volume fraction so that the substrate should become depleted close to the
131
end of the batch time, and this should coincide with the two treatment goals being achieved.
It is uncertain whether this is achievable with just these controls in a fixed plant design.
Increasing aeration accelerates both treatment goals, but directly increases energy consumption.
Substrate enrichment enhances pasteurization more than stabilization. Increasing exchanged
volume fraction probably delays achievement of both treatment goals, though pasteurization
may be greater affected if the post-mixing temperature drops significantly; it also decreases
energy consumption by increasing the sludge throughput.
5
Numerical Simulations
The derived model consists of a system of first order ordinary differential equations which are
integratated in time using Matlab’s stiff solver, ODE15s. We validate and test the practical
applicability of the model by comparing the numerical simulations with experimental data for
a specific choice of parameters given in Table 6. A comparison between the experimental data
and numerical simulations is given in Figs. 10 and 11 for dynamical operational conditions,
i.e., the input volume is not the same for each batch, cf. [2]. Note that here we ignore any
temperature dependence of the kinetic parameters. The experiments were carried out under
different hydraulic loading which is consistent with the Killarney site, (see for example [8]).
Despite the limitations in the available data, the comparison shows a very good fit.
70
68
Experimental data
Numerical simulation
Temperature, T [oC]
66
64
62
60
58
56
54
52
50
0
2
4
6
8
10
12
Time, [days]
14
16
18
20
22
Figure 10: A comparison between experimental data and numerical simulations of the ESGI70ATAD model for multiple batches.
In the next sections we run the simulations until steady-state conditions are attained. This is
done by checking that the solutions between two consecutive batches are within some threshold
value. This occurs within 28 days using the baseline case parameters. The numerical simulations
for a base case run (and multiple batches) are shown in Fig. 9 where all the parameters are
fixed as given in Table 6. Results presented here are for a single reactor and we assume that
stabilization is the main goal for bio-treatment. However, for completeness we will also show
132
20
19
Experimental data
Numerical simulation
Volitile solids, Xvs [g/l]
18
17
16
15
14
13
12
11
10
0
2
4
6
8
10
12
Time, [days]
14
16
18
20
22
Figure 11: A comparison between experimental data and numerical simulations of the ESGI70ATAD model for multiple batches.
simulations for the case when pasteurization is the main goal. For the purpose of comparison;
all simulations are shown for day 28, whether there is stabilization or not, and baseline case
simulations are highlighted with a ∗ in the figures.
5.1
Sensitivity to the Feed Temperature, T feed
In this section, we study the sensitivity of the model to variations of feed temperature to
investigate the impact of this on the bio-treatment goals, i.e., pasteurization and stabilization.
The fresh sludge can be preheated before loading into the reactor using recovered heat. We
observe from Fig. 12 that, for the given parameters, stabilization is not sensitive to the feed
temperature. However, pasteurization is very sensitive to feed temperature which is consistent
with plants with more than one reactor.
5.2
Sensitivity to Sludge Feed, Xfeed
In this section we consider variations of the sludge feeding pattern to investigate the impact
of this on the bio-treatment goals. Fig. 13 suggests the organic feed must be within certain
concentrations to attain the required levels of pasteurization and stabilization.
5.3
Sensitivity to Aeration Rate, A
Since there is some uncertainty in the choice of the aeration rate, we run numerical simulations
for different values of A. In fact it is an open question whether continuous aeration is necessary
133
′
10
τ feed
=5 C
τ feed
= 13* C
τ feed
= 30 C
τ feed
5
out feed
/X
vs
vs
0.46
o
o
Stabilisation, 1−X
Lethality, L(T,t)
15
o
o
= 50 C
0.45
0.44
0.43
0.42
τ feed
=5 C
τ feed
o
= 13*oC
τ feed
= 30 C
τ feed
= 50 C
o
o
0.41
0
0
0.2
0.4
0.6
Time [days]
0.8
0.4
0
1
0.2
0.4
0.6
Time [days]
0.8
1
Figure 12: Lethality and stabilization for different feed temperatures.
5
Xfeed = 5g/l
= 10g/l
feed
= 15*g/l
feed
= 20g/l
X
4
X
3
Stabilisation, 1−X
Lethality, L(T,t)
X
0.5
out feed
/X
vs
vs
feed
2
1
0
0
0.2
0.4
0.6
Time [days]
0.8
0.45
0.4
0.35
0.3
0.25
0
1
feed
X
= 5g/l
feed
X
= 10g/l
feed
X
= 15*g/l
feed
X
= 20g/l
0.2
0.4
0.6
Time [days]
0.8
1
Figure 13: Lethality and stabilization for different sludge feed concentration.
(such as in the Killarney plant). Numerical results (for a single reactor) suggest that the
required treatment goals cannot be achieved for aeration rate lower that 100 1/d, see Fig. 14.
5.4
Sensitivity to Oxygen Feed, Sfeed
In this section we consider varying the oxygen feed concentration. We run simulations for
values in the range Sfeed = [0.01 − 10] mg/l and results (not shown here) suggest stabilization
and pasteurization are not sensitive to the oxygen feed concentration. This suggests that the
plant can operate efficiently at levels lower than Sfeed = 1.0 mg/l while maintaining the required
treatment goals.
134
′
3.5
0.45
out feed
/X
vs
vs
2.5
A = 50 1/d
A = 100 1/d
A = 200* 1/d
A = 300 1/d
2
Stabilisation, 1−X
Lethality, L(T,t)
3
0.5
1.5
1
0.5
0
0
0.4
0.35
A = 50 1/d
A = 100 1/d
A = 200* 1/d
A = 300 1/d
0.3
0.25
0.2
0.2
0.4
0.6
Time [days]
0.8
1
0
0.2
0.4
0.6
Time [days]
0.8
1
Figure 14: Lethality and stabilization for different aeration rates.
6
Optimal Strategy for Pasteurization
In this section we provide an alternative discussion of the optimization problem. We mentioned
that the treatment has two different Objectives, namely:
1. Killing the pathogen by pasteurization, and
2. Stabilization; that is reducing the organic matter contents to an acceptable level (66–80%
of the initial).
However, one can expect that when one of these Objectives is achieved, another is already
accomplished in the process as well. For example, when the treated sludge is pasteurized and
the pathogen are killed or reduced to safe level, this portion of sludge is also already stabilized
(that is the organic content is reduced to on acceptable level). This observation allows us to
substitute one difficult problem that has two Objectives by two simple separable problems that
have a single Objective each. In this Section we consider an optimal control policy that is aimed
to pasteurization, assuming that the stabilization will be reached in the process of pasteurizing.
Pasteurization implies exposing the sludge to high temperature for a prolonged time. To achieve
this objective in an optimal way, we may consider a simple practical analogue. The optimal
policy for pasteurization will be
1. Firstly, heat the contents to as high as possible temperature, and do it as quickly as
possible.
2. After heating, maintain this temperature for the necessary time.
The temperature of the sludge and the heat influx for this process are shown in Fig. 15.
135
Stage 2
Heat Influx
Stage 1
0
t1
Time
t2
Figure 15: Sludge temperature (given by the curve) and the proposed heat influx (given by the
rectangles).
This involved four questions, namely:
Question 1 : What is the temperature of the sludge should be?
Answer : The temperature should be as high as it can be maintained for prolonged time. It is
obvious that the highest temperature that can be maintained is the temperature when the rate
of metabolism balances the bacteria rate of decay (Fig. 16)
Question 2 : How long Stage 1 (heating) should be?
Answer : As long as needed to reach the pasteurization temperature at the highest possible
heating rate
Question 3 : How much heat is needed to maintain the temperature through Stage 2?
Answer : By the model,
dT
= Heat Influx − Heat Losses.
dt
At equilibrium dT/dt = 0, and hence Heat Influx = Heat Losses. The heat losses are know
for a given temperature, and hence we have the heat influx. For instance, if the heat losses
are linearly proportional to the temperature, that is Heat Losses = hT , and the heat influx is
proportional to the rate of metabolism,
Heat Influx = µ
S
X
XB
Ks + S Kx + X
136
e
r at
m
lis
bo
et a
M
y
ca
te
Ra
e
fd
o
Temperature
Figure 16: The equilibrium of the metabolism rate and the rate of decay.
then the task is to maintain the oxygen concentration S at such a level that
S
≈ 1.
Ks + S
Question 4 : How long Stage 2 should be?
Answer : As long as needed to reach the required level of pasteurization (this time is well known
and is given by a know formula).
6.1
Recommendations
1. Thermal-isolation and utilizing the waste heat. Thermal-isolation reduces heat losses and
hence the heat influx can be also reduced.
2. Splitting the process by stages. We suggest that the process should be divided into
two stages, namely the heating and the maintaining the temperature. It is advisable to
conduct these two processes in different reactors.
3. Preheating the water (utilizing waste heat). Preheating the water would shortening the
first stage, the heating and the thermal shock will be smaller.
4. Start every round of the process at the highest possible initial concentration of bacteria.
This again should shortening the first stage (the heating) and hence reduce the energy
losses..
137
7
Concluding remarks
During the 70th European Study Group, the participants mathematically addressed a problem
of optimizing the process of waste water bio-remediation. It was noted by the participants that
the models that currently exist are greatly oversized, while the reliable data are rather scarce.
Furthermore, it was also noted that the existing models are very sensitive to the parameter
values. It was suggested, therefore, that under the circumstance a simpler model is needed.
Such a model, that includes only the essence of the process, has been developed during the
Study Group. This model includes four crucial variables, namely the substrate concentration
(the nutrient), the biomass, the oxygen concentration and the temperature of the sludge. The
model describes the process of bio-treatment for the case of a single reactor. It has been shown,
that for correctly re-calibrated parameters the ESGI model and the original large-scale models
give effectively the same outcome. Development and verification of this model was the major
outcome of this study. An extension of this model to the case of a few sequential reactors is
reasonable straightforward and possesses no difficulty.
In the analysis that are presented in this report, we applied this model to consider two different
regimes, namely, when either oxygen, or the substrate (the nutrient) are limiting factors. It
appears that one possible operating strategy can be applying a low aeration rate to preserve
the oxygen concentration a limiting factor for longer, with the aim that the two objective of
the bio-treatment were met at the same moment at the end of the cycle.
We would also like to remark that the participant realize that the problem of optimizing of
the bio-treatment, as it was presented to the Study Group, is essentially a problem of optimal
control. However, taking into consideration that the problem is fundamentally non-linear,
addressing this optimal control problem analytically is a highly non-trivial task. This problem
in such circumstances may be explored numerically, applying the technique that is known as
dynamic control. However, a reliable and reasonably simple model is needed for this at the
first instance, and we believe that the model that was developed can serve as the basis for such
study.
Study of the model also gives an important insight into the nature of the process, and enables
us to gain a number of practically relevant recommendations. Thus, it appears that pre-heating
the input sludge will be beneficial, provided it can be done in an energy efficient way (recycling
the wasted heat that is generated in the process).
One avenue for further research is an extension of the ESGI70 model to reactors in series. This
study would be of assistance for a more complex physical plant design.
138
Acknowledgement
We would like to thank Jaime Rojas and Prof. Toshko Zhelev for providing us with the valuable experimental data and background information in formulating a mathematical model for
the autothermal thermophylic aerobic digestion process, especially being available throughout
the week. We acknowledge the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics
initiative grant 06/MI/005.
139
Bibliography
[1] J. M. Gómez. Digestión Aerobic Termófila Autosostenida (ATAD) de fangos. Estudio
experimental a escala real y modelización matemática del reactor. PhD thesis, Universidad
de Navarra, San Sebastińa, Spain, Octubre 2007.
[2] J. M. Gómez, M. de Gracia, E. Ayesa, and J. L. Garcia-Heras. Mathematical modelling of
Autothermal Thermophilic Aerobic Digesters. Water Res., 41:1859–1872, 2007.
[3] W. Gujer, M. Henze, T. Mino, T. Matsuo, M. C. Wentzel, and G. R. Marais. The Activated
Sludge Model No. 2: biological phosphorus removal. Water Sci. Technol., 31(2):1–11, 1995.
[4] W. Gujer, M. Henze, T. Mino, and M. van Loosdrecht. Activated Sludge Model No. 3.
Water Sci. Technol., 39(1):183–193, 1999.
[5] M. Henze, C. P. L. Grady Jr., W. Gujer, G. R. Marais, and T. Matsuo. Activated Sludge
Model No. 1. IAWPRC, London, 1987.
[6] R. Kovács and P. Miháltz. Untersuchungen der kinetik der aerob-thermophilen klaerschlammstabilisierung - einfluss der temperatur, 17. Fruehlingsakademie und Expertentagung. Balatonfuered, Germany. May, 2005.
[7] R. Kovács, P. Miháltz, and Z. Csikor. Kinetics of Autothermal Thermophilic Aerobic
Digestion – Application and extension of Activated Sludge Model No. 1 at thermophilic
temperatures. Water Sci. Technol., 56(9):137–145, 2007.
[8] N. M. Layden. An evaluation of Autothermal Thermophilic Aerobic Digestion (ATAD) of
municipal sludge in Ireland. J. Environ. Eng. Sci., 6:19–29, 2007.
140
The initiation of Guinness
Report Contributors: Oliver A. Power1,2, William Lee1,
Andrew C. Fowler1 , Paul J. Dellar3 , Len W. Schwartz4 ,
Sergei Lukaschuk5 , Gordon Lessells6 , Alan F. Hegarty6 ,
Martina O’Sullivan1 and Yupeng Liu7
Industry Representatives:
Conor Browne8 and Richard Swallow9
1
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
Report coordinator, oliver.power@ul.ie
3
OCIAM, Mathematical Institute, University of Oxford, United Kingdom
4
Department of Mechanical Engineering, University of Delaware, United States of America
5
Department of Engineering, University of Hull, United Kingdom
6
Department of Mathematics and Statistics, University of Limerick, Ireland
7
School of Mathematical Sciences, Dublin Institute of Technology, Dublin, Ireland
8
Diageo Ireland, St. James’s Gate, Dublin 8, Ireland
9
VGraph UK Ltd, 28 Hudson Way, Radcliffe On Trent, Nottingham, NG12 2PP, United Kingdom
2
Abstract
Guinness Draught is a distinctive beer, having a black body and a thick white head of foam. It
is unusual in that it is always dispensed with a gas mixture of about 21 mole % carbon dioxide
and 79 mole % nitrogen. Consequently, it does not foam spontaneously but must instead
be ‘initiated’. The initiation methods employed by Diageo in canned and bottled Guinness
Draught add a significant cost to the final product. Hence, Diageo asked the Study Group to
develop a mathematical model for the initiation of Guinness, focusing on any of the current
initiation methods.
We decided to focus on the ‘surger’ unit, which vibrates ultrasonically to initiate Guinness.
It does so mainly by the process of rectified diffusion, although concepts from passive and
convective diffusion are also important.
In this paper, we briefly review the initiation methods currently employed, the theory of bubble
nucleation and growth, and passive, rectified and convective diffusion. A qualitative explanation
of how Guinness is initiated is given. Finally, a mathematical model for the initiation of
Guinness by the surger unit is developed, the solutions of which are quantitatively reasonable.
142
1
The problem with draught Guinness
Guinness is a distinctive beer that originated in the brewery of Arthur Guinness (1725–1803) at
St. James’s Gate, Dublin, Ireland [1]. It is based on the ‘porter’ that first appeared in London
in the early 18th century. Porter incorporated roasted barley for the first time and it is this
ingredient which endows Guinness with its characteristic burnt or roasted flavour.
The perfect pint of draught Guinness is black, or officially a very dark shade of ruby, with
a thick ‘creamy’ head of foam. The head is an integral part of the experience of drinking
Guinness. It is created as a result of gas bubbling out of the beer when it is being served (the
famous ‘surge and settle’). Guinness Draught is unusual in that it is packaged and dispensed
using a gas mixture of carbon dioxide (CO2 ) and nitrogen (N2 ), in contrast to Guinness Extra
Stout, and indeed most beers, which use only carbon dioxide. There seems to have been two
motivating factors for the use of nitrogen. First, the introduction of metal kegs allowed beer to
be stored and served at high pressures (2 − 4 times atmospheric pressure). For most beers this
pressure is achieved using only CO2 and, consequently, the CO2 concentration can be relatively
high in the beer (∼ 6 g l−1 ). An acidic ‘fresh’ taste is associated with the dissolved CO2 (as it
reacts with water to form carbonic acid), which is not to everyone’s liking [2]. In contrast, the
gas mixture used to dispense Guinness is mainly N2 . In particular, at 3.8 bar and 8 ◦ C the gas
mixture is about 79 mole % N2 and 21 mole % CO2 . This yields the desired CO2 concentration
of ∼ 2 g l−1 and the desired N2 concentration of ∼ 0.072 g l−1 (Section 8). For comparison, a beer
dispensed at the same pressure using only CO2 would have a CO2 concentration of ∼ 9.5 g l−1 .
Clearly, the Guinness would have the less acidic flavour, the nitrogen being tasteless. In fact,
to avoid over-carbonation when high gas pressures are required, such as when pushing beer
through long lines or up vertical lifts, a gas mixture of nitrogen and carbon dioxide is used to
dispense all beers, the proper gas blend being chosen from the supplier’s blend selection chart.
The second advantage to using nitrogen is related to its low solubility. N2 is 100 times less
soluble than CO2 at 8 ◦ C. Consequently, the N2 concentration in Guinness is very low. The CO2
concentration is also low but this is because the gas mixture is mainly N2 . Once bubbles form in
Guinness, they remain small due to the low dissolved gas concentrations and, consequently, low
growth rates. The small bubbles give Guinness Draught a finer ‘creamier’ head of foam than
that on a carbonated beer like Guinness Extra Stout. Furthermore, the head is longer-lasting
since Ostwald ripening, the main process by which foam becomes coarser, is slowed down by
a factor of 100 due to the lower solubility of N2 [3]. The relative concentrations of CO2 and
N2 in air also helps: air is about 79 mole % N2 and only 0.03 mole % CO2 , meaning that the
tendency for gas molecules to escape from the foam bubbles into the air is weaker for N2 than
for CO2 .
However, a major disadvantage of using nitrogen is that, unlike carbonated beers which foam
spontaneously, the head on Guinness does not form spontaneously and must be ‘initiated’. A
good example of this is the Guinness from a ‘surger-ready’ can. If such a can is carefully opened
and poured into a clean glass, little or no head forms. However, if the glass is now placed on
a ‘surger’ unit, which vibrates ultrasonically, a surge of bubbles appears in the body of the
143
Guinness and a thick creamy head forms as these bubbles collect on top of the Guinness.
As well as being available in kegs and surger-ready (widget-free) cans, draught Guinness is
also sold in widget cans and widget bottles, the ‘widget’ being a plastic capsule that initiates
the Guinness when the seal on the container is broken. In all these forms, the container is
pressurised to 3.8 bar at 8 ◦C with a 79 mole % nitrogen and a 21 mole % carbon dioxide gas
mixture.
In order to better understand the existing initiation methods and motivate new ones, Diageo
asked us, at the 70th European Study Group with Industry, to develop a mathematical model
describing bubble formation and growth in Guinness. First, we will review the existing methods
of initiation.
2
Existing methods of initiation
1. Keg Draught Guinness: Draught Guinness first became available to licensed premises
around 1959. To dispense the Guinness, a tap with two lines is attached to the keg. One
line leads to a premixed gas cylinder containing nitrogen and carbon dioxide. Pressure
gauges on this line allow the pressure in the system to be set. The other line leads to the
dispensing tap. On its way to the spout of this tap, the Guinness is forced at high speed
through an orifice plate with five small holes. From Bernoulli’s theorem, the pressure
drops in the beer as it passes through the holes. This allows existing microbubbles to
expand and also induces cavitation in the beer, as evidenced by the characteristic surge
of bubbles seen in the dispensed beer. This initiation method is very effective, the result
being a thick creamy head.
2. Canned and Bottled Draught Guinness: To meet the demand for draught Guinness that
could be poured at home, bottled draught Guinness was launched in 1979. To initiate
the Guinness, a syringe, supplied with the bottles, was used to suck up some of the
Guinness. This was then squirted back into the glass of (headless) Guinness creating
a surge of bubbles and, consequently, the head. The syringe initiator was replaced by
the can ‘widget’ in 1989. The widget is a plastic capsule with a tiny hole connecting
its interior to the surroundings. When the can is opened, a small amount of beer and
nitrogen, previously trapped in the widget, shoots out into the beer. Originally, the widget
was used only in canned draught Guinness and was in the form of an insert wedged into
the bottom of the can. In 1997, the widget became a floating spherical capsule (it can be
heard rattling in the can). A floating bottle widget in the shape of a rocket was introduced
in 1999. The bottle widget is optimised for drinking straight from the bottle, while the
can widget is optimised for pouring into a glass.
The operation of the widget deserves further explanation. In the case of the can, a
controlled amount of liquid nitrogen is added just before it is sealed. When this liquid
nitrogen evaporates it pressurises the can to 3.8 bar. The pressure equalises in the widget,
144
the headspace and the beer, forcing some beer into the widget as it does so. When
the can is opened, the pressure in the headspace and beer rapidly drops to about 1 bar
(atmospheric pressure). The contents of the widget then decompress by squirting gas
(mainly N2 ) and some beer into the Guinness. In the case of the bottle, no liquid nitrogen
is added: the bottle is sealed in an environment already at the desired gas pressure.
3. Surger-Ready Draught Guinness: The ‘surger’ was launched in Tokyo in 2004. It is an
electrical device that recreates the ‘surge and settle’ effect without the need for the kegand-tap dispensing system. The interface of the surger unit is a metal plate that vibrates
ultrasonically. First a little water is poured onto a plate to ensure good mechanical
contact. Then a can of surger-ready (widget-free) Guinness is poured into a glass and
placed on the plate. The Guinness initially has little or no head. When a button is
pressed, the plate vibrates ultrasonically. The associated pressure fluctuations in the
Guinness almost immediately create a surge of bubbles and, consequently, the head.
3
Bubble nucleation and growth
Many good introductions to the formation and growth of bubbles in supersaturated liquids can
be found in the literature (e.g. [4, 5, 6, 7, 8]). Let us consider a container (keg, can or bottle)
filled to the desired level with Guinness and pressurised by some method. In the case of the
can, the container is sealed after a controlled amount of nitrogen has been dropped onto the
liquid surface, while the bottle is sealed in a closed environment already at the desired gas
pressure.
In either case, after the container is sealed, an equilibrium is established between the dissolved
gas in the liquid and the gas in the headspace (and in the widget, if present). This equilibrium obeys Henry’s law, which states that the concentration cl of gas dissolved in a liquid is
proportional to the partial pressure p of that gas above the liquid:
cl = Hp.
(3.1)
Clearly, the solubility of a gas in a liquid increases as the pressure of that gas over the liquid increases. The proportionality constant H is known as the Henry’s law constant. It is
temperature dependent, decreasing as the temperature increases; that is, for a fixed partial
pressure, gas solubility decreases as the temperature increases. This is the reason that bubbles
are seen to form on the surface of a glass of water when it is left in the sun and the reason that
effervescence (bubbling) is initially greater in a hot carbonated beverage than a cold one.
When the container is opened, the pressure p in the headspace falls abruptly and Henry’s law is
no longer satisfied. The liquid is said to be supersaturated with gas molecules. To reestablish
equilibrium or, equivalently, to satisfy Henry’s law (3.1), the dissolved gas concentration cl
must decrease. The gas molecules can escape either by diffusing through the free (top) surface
145
of the liquid or by forming bubbles1 .
Although the laws of thermodynamics favour the formation of gas bubbles, there is a kinetic
barrier to the production of gas bubbles. To see why, we note that the gas pressure pb inside
a spherical bubble of radius R is given by Laplace’s law:
pb = pl +
2γ
,
R
(3.2)
where pl is the pressure in the liquid and γ is the surface tension of the liquid in contact with
the gas. Laplace’s law states that the pressure inside a bubble is greater by 2γ/R than that in
the surrounding liquid due to the fact that the surface tension γ of the liquid tends to contract
the bubble surface. Clearly, the gas in the bubble is in equilibrium with its concentration cl in
the liquid if Henry’s law is satisfied:
cl = Hpb ≡ cb ,
(3.3)
If Henry’s law is not satisfied then the bubble either shrinks or grows. If cl < cb then the
bubble shrinks as the gas in it redissolves, whereas if cl > cb then the bubble grows as gas
diffuses into it from the liquid. Using Laplace’s law (3.2) to eliminate pb yields
2γ
.
c l = H pl +
R
Approximating pl by pa 2 and rearranging terms yields
R=
2γ
≡ Rc ,
cl /H − pa
where Rc is called the critical radius. If R = Rc then the gas bubble is in equilibrium with the
gas concentration in the liquid (i.e. cl = cb ) and it neither shrinks nor grows. If R < Rc then
cl < cb and the bubble shrinks. If R > Rc then cl > cb and the bubble grows (Figure 1). The
critical radius is often given in terms of the supersaturation ratio S which is defined by
S=
cl
− 1,
Hpa
so that
2γ
.
(3.4)
pa S
If p0 is the gas pressure over the liquid in the sealed container, then the gas concentration in
the liquid is initially cl = Hp0 . Immediately after opening the container, the gas concentration
Rc =
1
What are widely called ‘bubbles’ in liquids are actually cavities: a true bubble is a region in which gas is
trapped by a thin film, whereas a cavity is a gas-filled hole in a liquid.
2
Strictly speaking, pl is the sum of the atmospheric pressure pa and the hydrostatic pressure ρl g (h − z),
the bubble being at a depth h − z. Since pa ≈ 105 Pa and ρl gh . 103 Pa, the hydrostatic pressure can be
neglected.
146
Figure 1: The (idealised) dissolved gas concentration close to a postcritical (R > Rc ) bubble. ∆c
is the difference between the dissolved gas concentration cl in the liquid bulk and the dissolved
gas concentration cb = Hpb in the bubble ‘skin’, which is a thin layer next to the bubble surface
and in equilibrium with the gas pressure in the bubble. The concentration falls from cl to cb
over a diffusion boundary layer of thickness δN .
in the liquid is still cl = Hp0 so that the initial supersaturation ratio is S = p0 /pa − 1. For beer
(γ ≈ 40 mN m−1 [3]) and an initial supersaturation ratio of 2.8, which is typical for draught
Guinness, the critical radius is 0.3 µm. If enough gas molecules cluster together to form an
embryonic bubble with a radius greater than this critical radius, then that bubble will survive
and grow. The random thermal motion of the gas molecules is the process that provides these
clusters. However, in order to cluster together, the gas molecules must first push their way
through the polar water molecules that are strongly attracted to each other. This places a
considerable energy barrier before bubble formation. This energy barrier can be lowered by
reducing the critical radius, which also increases the likelihood that enough gas molecules will
randomly cluster together. To this end, the critical radius can be reduced by lowering the
surface tension γ3 or by increasing the supersaturation ratio S. However, calculations, verified
by experiment, show that the random nucleation of bubbles, called homogeneous nucleation,
requires a supersaturation ratio of several hundred to over a thousand to be observable [9, 10].
In practice, bubbles form in even weakly supersaturated liquids, such as carbonated soft drinks
(S ∼ 1 [11]) or champagne (S ∼ 5 [12]). When these liquids are poured into a glass, it is
easy to observe that bubbles form on the surface of the glass rather than in the bulk of the
liquid. When the bubbles reach a critical size, they detach from the glass surface and rise
up through the liquid. Over most of the glass surface, the bubble production is very slow,
produces large (∼ 1 mm) bubbles, and soon stops altogether. However, at a few spots on the
3
E.g. we found that when washing-up liquid is added to a carbonated soft drink and then placed on the
surger unit, a head consisting of small bubbles is formed. Without the washing-up liquid, the head consists of
much larger bubbles and also collapses much faster.
147
Figure 2: Bubbles in a glass of carbonated beverage appear to rise in streams from spots on
the surface of the glass. As they rise, they grow larger and spread further apart.
glass surface, the bubble production is very rapid and a continuous stream of small (< 1 mm)
bubbles can be seen rising from these spots. As the bubbles rise, they grow larger and spread
further apart (Figure 2). The surface of the glass is assisting bubble formation by providing
nucleation sites, which are usually in the form of preexisting gas pockets [13]. This is termed
heterogeneous nucleation. The conventional view is that when a glass is filled with liquid,
air can become trapped in crevices on the glass surface [13]. If the radius of curvature of
any of these gas pockets is greater than the critical radius, then that gas pocket grows into
a macroscopic bubble as dissolved gas diffuses into it (Figure 3). Eventually, the increasing
buoyancy force on the bubble exceeds the capillary force holding it to the glass surface and it
detaches. Some gas may be left behind in the crevice so that the process of bubble production
can repeat itself. However, recent work has shown that cellulose fibres, which either fall out
of the air or are left after drying with a towel, can be much better nucleation sites than glass
crevices [12, 14]. These cellulose fibres are roughly cylindrical and hollow, about 100 µm long
and several µm wide. When a liquid is poured into a glass, air is trapped inside these fibres.
Since the radius of curvature of these gas pockets is greater than the critical radius, they grow
as dissolved gas diffuses into them. When a gas pocket reaches the tip of a fibre, a bubble is
released, but some of the gas is left behind in the fibre so that the process of bubble production
can repeat itself (Figure 4). This results in a characteristic bubble stream (Figure 2). As the
gas concentration in the liquid decreases, the frequency of bubble formation decreases and the
critical radius increases. When the critical radius is equal to the radius of curvature of a gas
pocket, bubble production from that gas pocket stops. In summary then, the bubbles in a glass
containing a carbonated beverage originate from gas pockets trapped either in glass crevices or
cellulose fibres, the bubbles from cellulose fibres being much smaller and more frequent, and
thereby forming bubble streams. In passing, we note that a dramatic example of heterogeneous
nucleation is the addition of a spoonful of sugar to a carbonated soft drink. The sugar grains
most likely contain many small gas pockets.
148
Figure 3: Bubble production cycle from a crevice in a glass wall. (a) Gas diffuses into trapped
air pocket. (b) Bubble forms. (c) Buoyancy force FB eventually matches the capillary force Fc .
(d) Bubble detaches. Adapted from [15].
(a)
(b)
(d)
(c)
Figure 4: Bubble production cycle from a cellulose fibre. (a) Gas diffuses into trapped air
pocket. (b) Pocket reaches tip of fibre. (c) Bubble forms. (d) Bubble detaches.
149
After a bubble is released from its nucleation site, it grows as it makes its way to the surface.
This is due to the continuous diffusion of dissolved gas into the bubble (the drop ρl gh in the
liquid pressure makes a minor contribution to the expansion of the bubble). The buoyancy
force increases as the bubbles grow, causing them to accelerate and spread further apart. The
rising bubbles also experience a friction force (i.e. a drag) in the viscous liquid. If the radius
of a bubble were fixed, then the bubble would rapidly reach a constant velocity, termed the
terminal velocity, at which the upward buoyancy force would be balanced by the downward
drag force. However, for a growing bubble, the buoyancy force (∝ R3 ) always remains slightly
larger than the drag force (∝ R) so that the bubble continuously accelerates as it rises.
The final size of a bubble is affected by three main factors: (i) the distance the bubble has travelled, (ii) the concentration of dissolved gas, and (iii) the concentration of organic compounds.
The first two are obvious factors, whereas the third needs further explanation. Beer contains
many organic compounds which have a water-soluble (hydrophilic) part and a water-insoluble
(hydrophobic) part. Hence, they like to gather around the surface of a bubble with their hydrophobic ends pointing into the gas bubble and their hydrophilic ends in the liquid. They
are thus called surfactants (SURFace-ACTive AgeNTS). As a bubble rises, it collects these
surfactants. According to fluid mechanics, a rigid (i.e. solid) sphere rising through a liquid
experiences a greater drag than a fluid (e.g. gas) sphere. Essentially, the no-slip condition
at the surface of a solid sphere retards its movement, whereas the circulation of gas within
a gas bubble ‘lubricates’ its movement through the liquid. However, surfactants reduce this
lubricating effect, causing the drag on the bubble to increase. The bubble continues to grow as
it rises and if it grows fast enough then the fraction of the bubble surface that is contaminated
by surfactants remains small. This is the case for clear liquids like champagne, which has a
relatively high CO2 content (∼ 12 g l−1 [12]) and thus bubble growth rate, and a low surfactant
concentration (∼ 1 mg l−1 [12]), so that champagne bubbles experience a drag similar to that
which would be experienced by clean bubbles [16]. In contrast, beer bubbles experience a drag
similar to that which would be experienced by a solid sphere of the same radius, the combination of low CO2 content (∼ 6 g l−1 ) and high surfactant concentration (∼ 100 mg l−1 [12])
eliminating the lubricating effect associated with a clean bubble [5, 16]. Furthermore, in liquids
with zero or low surfactant concentrations, such as pure water, carbonated soft drinks and
champagne, bubbles burst almost immediately on reaching the surface. In beer, the surfactant
coating allows the bubbles to collect at the surface of the liquid to form a foam, called the head.
Guinness, being weakly supersaturated, will have no observable homogeneous nucleation within
the body of the liquid. Since surger-ready Guinness can be poured with very little or no head
forming, the heterogeneous nucleation rate on the surface of the glass must also be very low.
It can be made appreciable by adding a spoonful of sugar. The sugar crystals provide many
nucleation sites. Furthermore, as the sugar dissolves into the liquid, it readily liberates the
bubbles forming at these sites. The result is a surge of small bubbles. Bubbles do form on the
surface of the glass but they seem reluctant to detach, probably because the low dissolved gas
concentrations in Guinness prevent them from becoming large enough to detach.
150
4
Qualitative explanation of initiation
There are basically two ways to promote bubble formation: (I) we can introduce postcritical
(R > Rc ) bubbles or gas pockets (‘seeding’), or (II) we can grow the precritical (R < Rc ) bubbles
that are popping into and out of existence. The methods for achieving type I bubble formation
are
• Entrapment of gas bubbles: If a container is shaken, then some of the gas above the
liquid level is mixed into the liquid as small bubbles. When the container is opened,
the pressure falls and these bubbles expand. Perhaps more importantly, dissolved gas
will diffuse into these bubbles if R > Rc . This vigorous bubble growth can cause shaken
carbonated beverages to spray out of their containers. Splashing a liquid into a glass,
rather than pouring it slowly down the side of the glass, can also entrap air bubbles with
R > Rc , as can banging a glass of liquid down or swirling the glass.
By opening a can of surger-ready Guinness at an appropriate angle, it is possible to
initiate the Guinness. It is likely that as the gas rushes out of the can, it causes waves to
form on the liquid surface via the Kelvin-Helmholtz instability. These waves then entrap
gas bubbles as they crash against the tab, the walls of the can or back into the liquid.
Ocean foam has similar origins, wind creating the crashing waves.
• Cavitation: In the keg-and-tap system, the Guinness is forced through an orifice plate.
In accordance with Bernoulli’s theorem, the pressure decreases and the velocity increases
as the liquid passes through the holes. If the pressure decrease is large enough, then
the liquid can be torn apart and cavities (i.e. bubbles) can form in the Guinness. The
distinction between type I and type II bubble formation is blurred here since a liquid is
more likely to be torn apart where it is weakest, such as where a precritical bubble exists.
• Injection of gas: In widget cans and bottles, a jet of gas shoots from the widget into the
Guinness when the container is opened. This jet breaks up into bubbles. If the jet is
too short, then some of the Guinness at the bottom of the container remains uninitiated.
When the Guinness is poured out of the container, the uninitiated beer passes through
or by the foam, providing a fresh supply of nitrogen to existing bubbles and thereby
refreshing the head; that is, pouring completes initiation of the Guinness.
The method for achieving type II bubble formation is
• Pressure fluctuations: Precritical bubbles will expand if the local liquid pressure drops.
For example, if the expansion is isothermal, then a drop in liquid pressure from 1 bar
to 0.5 bar causes a precritical Guinness bubble with R = 0.25 µm to expand until R ≈
0.31 µm, while the critical radius drops from 0.29 µm to 0.24 µm. Hence, dissolved gas in
the liquid diffuses into the bubble. If the liquid pressure then returns to its original value,
the bubble may or may not survive. It will survive if enough gas has diffused into the
151
Initiation method
Keg-and-tap
Syringe
Widget
Shaking container
Banging glass down
Opening can at an angle
Ultrasound
Method of bubble formation
Cavitation
Cavitation; entrapment of air bubbles
Injection of gas bubbles
Entrapment of gas bubbles
Entrapment of gas bubbles
Entrapment of gas bubbles
Pressure fluctuations
Type
I
I
I
I
I
I
II
Table 7: Initiation methods classified according to the method and type of bubble formation
employed.
bubble to keep its radius greater than the critical radius. Even if this is not the case, the
bubble can be encouraged to grow by repeatedly subjecting it to pressure fluctuations.
The Guinness surger unit employs this method by vibrating at ultrasonic frequencies.
In table 7, we have classified the initiation methods for Guinness according to the method
and type of bubble formation employed. Since the surger unit is the ‘cleanest’ experiment, we
decided at the Study Group to focus on this initiation method.
5
The mathematical models
First, we describe a mathematical model for the diffusion of dissolved gas into a bubble which
is not rising in the liquid. We call this passive diffusion. By assuming that the gas bubbles are
purely nitrogen, we can introduce a small parameter which allows us to obtain some analytical
results. These results emphasise the role played by the critical radius. Next, we introduce the
effect of a sound field and apply analytical results found in the literature. It is shown that
above a critical threshold for the amplitude of the sound field, precritical bubbles grow. This is
termed rectified diffusion. Next, we consider how diffusion is altered when the bubble is rising,
as it does in practice. We call this convective diffusion, since convection is now assisting the
transfer of mass to the bubble. Finally, a mathematical model for describing the initiation of
Guinness by the surger unit is described and solved numerically.
5.1
Bubble growth by passive diffusion
Consider a spherical bubble of radius R (t) at rest in an infinite incompressible inviscid liquid
(Figure 5). Far away from the bubble the pressure in the liquid is p∞ (t) and this is taken to
152
Liquid
r
Gas
R(t)
Bubble
surface
Figure 5: A spherical bubble of radius R (t) in a liquid.
be a known input which regulates the growth or collapse of the bubble. We assume that, as its
radius changes in time t, the bubble remains spherical so that the bubble surface moves only
in the radial direction. It follows that the motion induced in the fluid is also radial:
ur = u (r, t) ,
uθ = uφ = 0,
where ur , uθ , uφ are the components of the fluid velocity expressed in spherical coordinates
(r, θ, φ). The origin of the coordinate system is taken at the centre of the bubble so that
r is distance from the centre of the bubble. Because of the spherical symmetry, all physical
quantities are functions of at most r and t. Heat transfer is neglected by assuming that the
temperature T is constant in time and uniform in space. Furthermore, the pressure pb in the
bubble is taken to be a function only of t. We also assume that only one gas species (i.e.
nitrogen) is present in the liquid and the bubble.
The governing equations are the incompressibility condition:
1 ∂ 2 r u = 0,
r2 ∂r
(5.1)
the radial component of the Navier-Stokes equation governing the motion of the liquid:
∂u
∂u
1 ∂pl
+u
=−
,
∂t
∂r
ρl ∂r
(5.2)
and the diffusion-convection equation governing the gas concentration c (r, t) in the incompressible liquid:
∂c
∂c
D ∂
2 ∂c
r
.
(5.3)
+u
= 2
∂t
∂r
r ∂r
∂r
The mass flux of gas into the bubble is given by
∂c 2
,
q = 4πR D
∂r r=R(t)
153
and can be equated to the rate of increase of the bubble mass:
∂c d 4 3
2
.
πR ρb = 4πR D
dt 3
∂r r=R(t)
(5.4)
At the bubble surface r = R (t), we also have the following boundary conditions:
u = Ṙ at r = R (t) ,
pl = pb −
2γ
R
(5.5)
at r = R (t) ,
at r = R (t) .
c = Hpb
(5.6)
(5.7)
The kinematic boundary condition (5.5) holds as the fluid motion is induced by the growth
or collapse of the bubble. Equation (5.6) is simply Laplace’s law (3.2), and (5.7) is simply
an application of Henry’s law (3.1). Far away from the bubble at r = r∞ , the pressure and
concentration are prescribed by
pl = p∞ (t)
at r = r∞ ,
(5.8)
c = c∞ (t)
at r = r∞ .
(5.9)
We also require the following initial conditions:
R = R0
at t = 0,
(5.10)
pl = p0
at t = 0,
(5.11)
c = Hp0
at t = 0.
(5.12)
Here, p0 corresponds to the gas pressure in the sealed container so that (5.12) follows from an
application of Henry’s law. The contribution of gravity to the liquid pressure is neglected so
that the initial liquid pressure can be set equal to p0 . From (3.4), the critical radius is
2 4 × 10−2 N m−1
2γ
=
≈ 0.3 µm,
Rc =
∆p
2.8 × 105 N m−2
where ∆p = p0 − pa .
Solving (5.1) for u and applying the boundary condition (5.5) yields
u=
R2 Ṙ
.
r2
(5.13)
Substituting (5.13) into (5.2) yields
∂
∂t
R2 Ṙ
r2
!
R2 Ṙ ∂
+ 2
r ∂r
154
R2 Ṙ
r2
!
=−
1 ∂pl
.
ρl ∂r
Hence,
2
2
2
d
R
Ṙ
2
R
Ṙ
1 ∂pl
1
−
=−
.
2
5
r
dt
r
ρl ∂r
Integrating from r = R (t) to r = r∞ yields
Z
2 Z r∞ 1
d R2 Ṙ Z r∞ 1
1 r∞ ∂pl
2
dr − 2 R Ṙ
dr = −
dr,
2
5
dt
ρl R ∂r
R r
R r
and so, applying the boundary condition (5.8) and assuming that r∞ ≫ R,
2
1 d R Ṙ
Ṙ2
p∞ (t) − pl (R)
−
=−
.
R dt
2
ρl
This can be rewritten as
pl (R) = p∞ (t) + ρl
3Ṙ2
RR̈ +
2
!
.
Using (5.6) to eliminate pl (R), the liquid pressure at the bubble surface, we obtain
!
3Ṙ2
2γ
.
+ ρl RR̈ +
pb (t) = p∞ (t) +
R
2
(5.14)
This is the well-known Rayleigh-Plesset (RP) equation. In the steady state, it reduces to
Laplace’s law.
Substituting (5.13) into (5.3) yields
∂c R2 Ṙ ∂c
D ∂
+ 2
= 2
∂t
r ∂r
r ∂r
2 ∂c
r
.
∂r
(5.15)
Assuming that the gas in the bubble behaves as a perfect gas, we have
ρb =
pb
,
Rs T
(5.16)
where Rs is the specific gas constant. Substituting (5.16) into (5.4) yields
1 d R3 pb
∂c 2
.
=R D
3Rs T
dt
∂r r=R(t)
Using (5.14) to eliminate the bubble pressure pb finally yields
"
!#
2
1 d
2γ
∂c 3
Ṙ
3
2
.
R p∞ +
=R D
+ ρl RR̈ +
3Rs T dt
R
2
∂r r=R(t)
155
(5.17)
Equations (5.15) and (5.17) govern c (r, t) and R (t). The remaining initial and boundary
conditions are
R = R0 at t = 0,
(5.18)
at t = 0,
!#
3Ṙ2
RR̈ +
2
(5.19)
c = Hp0
"
c = H p∞ +
2γ
+ ρl
R
c ≈ c∞ (0) = Hp0
at r = R (t) ,
at r = r∞ .
(5.20)
(5.21)
In condition (5.21), we have assumed that for a single bubble in an infinite liquid, the concentration far away from the bubble remains unchanged from its initial value.
Initial bubble growth
We now nondimensionalise the governing equations by letting
R∗ =
R
,
Rc
r∗ =
r
,
Rc
t
t∗ = ,
τ
c∗ =
c − Hpa
,
H∆p
(5.22)
where asterisks denote dimensionless variables and
τ=
R2c
,
Dε
ε=
H∆p
,
ρa
ρa =
pa
.
Rs T
The time scale τ corresponds to the time it takes a mass 4πρaR3c /3 to enter a sphere of radius
Rc if the mass flux density is DH∆p/3Rc . For nitrogen τ ≈ 0.8 ms and ε ≈ 0.06. Substituting
(5.22) into the governing equations (5.15) and (5.17) yields (asterisks omitted)
!
2
∂c R Ṙ ∂c
1 ∂
2 ∂c
ε
= 2
r
,
(5.23)
+ 2
∂t
r ∂r
r ∂r
∂r
1 d
3 dt
where
R
3
"
p∞
∆p
3Ṙ
+
+ δ RR̈ +
pa
pa R
2
∆p
≈ 2.8,
pa
δ=
2
!#
∂c .
=R
∂r r=R(t)
2
(5.24)
ρl D2 ε2
≈ 10−9 .
pa R2c
Omitting the small terms in (5.23) and (5.24) yields
1 ∂
2 ∂c
r
= 0,
r2 ∂r
∂r
p∞
∆p
1 d
3
2 ∂c .
R
+
=R
3 dt
pa
pa R
∂r r=R(t)
156
(5.25)
(5.26)
Dropping the terms of order ε in (5.23) means that we can no longer satisfy the initial condition
(5.19). The other initial and boundary conditions in dimensionless form are (asterisks omitted)
R=
pa
c=
∆p
R0
Rc
at t = 0,
(5.27)
(5.28)
p∞
∆p
+
−1
pa
pa R
at r = R (t) ,
r∞
,
(5.29)
Rc
where we have omitted terms of order δ in (5.28). Since r∞ /Rc ≫ 1, we can rewrite (5.29) as
c = 1 at r =
c → 1 as r → ∞.
(5.30)
Solving (5.25) for c and applying the boundary conditions (5.28) and (5.30) yields
Rpa p∞ ∆p 1
+
− 1 − 1 + 1.
c=
r∆p pa
pa R
Hence,
pa p∞ ∆p 1
∂c =−
+
−1 −1 .
∂r r=R
R∆p pa
pa R
Substituting (5.31) into (5.26) yields
p∞
Rpa p∞ ∆p 1
∆p
1 d
3
R
=−
+
+
−1 −1 .
3 dt
pa
pa R
∆p pa
pa R
(5.31)
(5.32)
When the container is opened, the liquid pressure rapidly equilibrates with the atmospheric
pressure so that p∞ = pa . Then (5.32) becomes
∆p
1 d
3
R 1+
= R − 1,
3 dt
pa R
or
dR
R−1
= 2
.
dt
R + 2R∆p/ (3pa )
(5.33)
If R < 1 then Ṙ < 0 and the bubble collapses as the gas in it redissolves. If R > 1 then Ṙ > 0
and the bubble grows as gas in the liquid diffuses into it. If R = 1 then Ṙ = 0 and the bubble
is in an unstable equilibrium state; that is, a perturbation from this state results in the bubble
either growing or collapsing, depending on whether the perturbation is an increase or decrease
in R. This confirms that Rc is the critical radius, determining whether the bubble will grow or
collapse by diffusion. Solving (5.33) for R and applying the initial condition (5.27) yields
2∆p
R20
R0
R−1
1
2
R − 2 + 1+
R−
+ ln
= t.
2
Rc
3pa
Rc
R0 /Rc − 1
As shown
in Figure 6, the bubble collapses if R0 < Rc and grows if R0 > Rc . For R0 > Rc ,
√
R → 2t as t → ∞.
157
1.8
1.6
1.4
R/R
c
1.2
1
0.8
R =1.2R
0.6
0
c
R =1.1R
0
0.4
c
R =1.0R
0
c
R =0.9R
0.2
0
c
R =0.8R
0
0
0
1
2
3
4
c
5
t/τ
Figure 6: The bubble collapses if R0 < Rc and grows if R0 > Rc .
Bubble maturation
Now consider a finite amount of liquid with many (identical) bubbles in it. Initially, the
dissolved gas concentration is c = Hp0 , but as the bubbles grow, the dissolved gas concentration
falls. Eventually, it is too low and the bubbles stop growing. At this stage, the gas in each
bubble is in equilibrium with its concentration in the liquid so that Henry’s law is satisfied:
2γ
,
c = Hpb = H pa +
R∞
where R∞ is the ultimate radius of each bubble. Also at this stage, the gas in each bubble
has been extracted from a volume 34 πr3∞ of the liquid, where r∞ is the catchment radius.
Ignoring the diffusion of gas across the free surface of the liquid and recalling that the liquid is
incompressible, conservation of mass implies
 


initial
mass
of



  remaining mass 
mass of gas
gas in liquid of
of gas in liquid
−
.
(5.34)
=
in bubble

  (of same volume) 
 volume 4 πr3 
3 ∞
or, in mathematical notation,
4 3
4 3
4 3
2γ
.
πR ρb = πr∞ Hp0 − πr∞ H pa +
3 ∞
3
3
R∞
Solving for r∞ and assuming that 2γ/R∞ ≪ pa , we obtain
r∞ =
R∞
.
ε1/3
158
(5.35)
To study how the bubble grows to its final radius R∞ , we rescale distance and time by letting
r∗ → Λr∗ ,
R∗ → ΛR∗ ,
t ∗ → Λ2 t ∗ ,
(5.36)
where Λ = R∞ /Rc is a large number (Λ = 200 for a 60 µm bubble). The length scale is then
R∞ (≈ 60 µm, say) and the time scale is R2∞ / (Dε) (≈ 30 s). Substituting (5.36) into (5.23) and
(5.24) yields (asterisks omitted)
!
∂c R2 Ṙ ∂c
1 ∂
2 ∂c
ε
= 2
r
,
+ 2
∂t
r ∂r
r ∂r
∂r
1 d
3 dt
R
3
"
3Ṙ2
RR̈ +
2
p∞
∆p
δ
+
+ 2
pa
pa ΛR Λ
!#
Omitting the small terms in these equations, we obtain
1 ∂
2 ∂c
r
= 0,
r2 ∂r
∂r
1 d R3 p∞
2 ∂c .
=R
3 dt
pa
∂r r=R(t)
∂c =R
.
∂r r=R(t)
2
The appropriate boundary conditions are (asterisks omitted)
pa p∞
c=
−1
at r = R (t) ,
∆p pa
c=
c∞ (t) − Hpa
H∆p
at r =
r∞
1
= 1/3 ,
R∞
ε
(5.37)
(5.38)
(5.39)
(5.40)
where we have neglected terms of order Λ−1 and δΛ−2 in (5.39). Note that we can no longer
approximate c∞ (t) by c∞ (0) as we did in (5.21) as we are now considering a finite liquid in
which there are many bubbles and the concentration is everywhere falling from its initial value
Hp0 . Although ε−1/3 ≈ 2.6, we will assume that it is much larger so that (5.40) can be replaced
by
c → c∗∞ (t) as r → ∞,
(5.41)
where
c∞ (t) − Hpa
.
H∆p
Solving (5.37) for c and applying the boundary conditions (5.39) and (5.41) yields
R ∗
Rpa p∞
c∞ .
−1 + 1−
c=
r∆p pa
r
c∗∞ (t) =
Thus
1 pa p∞
∂c ∗
=−
− 1 − c∞ .
∂r r=R
R ∆p pa
159
(5.42)
(5.43)
Substituting (5.43) into (5.38) yields
pa p∞
1 d R3 p∞
∗
= −R
− 1 − c∞ .
3 dt
pa
∆p pa
(5.44)
Applying the conservation of mass (5.34) when the radius of the bubble is R (t), we obtain, in
mathematical notation and dimensional variables,
Z (r3∞ +R3 )1/3
4 3
4 3
πR ρb = πr∞ Hp0 − 4π
c (r, t) r2 dr.
3
3
R
Using the perfect gas law to eliminate ρb and the RP equation to eliminate pb , we obtain
"
4 3
2γ
ρl
p∞
πR ρa
+
+
3
pa
pa R pa
3Ṙ2
RR̈ +
2
!#
Z (r3∞ +R3 )1/3
4 3
= πr∞ Hp0 − 4π
c (r, t) r2 dr.
3
R
In terms of dimensionless variables (asterisks omitted):
R
3
"
∆p
δ
p∞
+
+ 2
pa
pa ΛR Λ
3Ṙ2
RR̈ +
2
!#
Z (r3∞ /R3∞ +R3 )1/3 r3∞ εp0
pa
= 3
r2 dr.
c+
− 3ε
R∞ ∆p
∆p
R
Using (5.35) to eliminate r∞ and then neglecting small terms yields
p∞ R3
p0
=
− 3ε
pa
∆p
Z ε−1/3 pa
c+
r2 dr.
∆p
R
Substituting (5.42) for c yields
p0
p∞ R3
=
− 3ε
pa
∆p
Z ε−1/3 R
Rpa
r∆p
pa 2
p∞
R ∗
c∞ +
r dr.
−1 + 1−
pa
r
∆p
Upon integrating, we obtain
p0
p∞
1
Rpa
R
pa
p∞ R3
∗
=
−1 +
− 3ε
−
c∞ +
pa
∆p
2ε2/3 ∆p pa
3ε 2ε2/3
3ε∆p
3 R pa p∞
R3 ∗
R3 pa
+ 3ε
.
− 1 − c∞ +
2∆p pa
6
3∆p
160
1
R/Λ R c
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
t/Λ2τ
Figure 7: For large times t ∼ Λ2 τ ∼ 30 s, the bubble radius approaches a limiting value ΛRc =
R∞ .
Neglecting terms of order ε1/3 and smaller (ε) and solving for c∗∞ yields
c∗∞ (t) = 1 −
p∞ 3
R.
pa
Thus, substituting (5.45) into (5.44) yields
pa p∞
p∞ 3
1 d R3 p∞
= −R
−1 −1+
R .
3 dt
pa
∆p pa
pa
(5.45)
(5.46)
If p∞ = pa then (5.46) yields
dR
1 − R3
=
.
dt
R
Solving (5.47) for R and applying the initial condition R → 0 as t → 0, we obtain
2R + 1
1
1
π
1
−1
2
√
− ln (1 − R) + √ = t.
ln R + R + 1 − √ tan
6
3
3
3
6 3
(5.47)
As t → ∞, R → 1 as can be seen in Figure 7.
5.2
Bubble growth by rectified diffusion
Consider a bubble with R0 < Rc . In the absence of pressure fluctuations (i.e. a sound field), it
would gradually dissolve and disappear (Figure 6). However, if a sound field is introduced, then
it can cause the bubble to pulsate and this may lead to the diffusion of gas into the bubble by a
process called ‘rectified diffusion’. This process competes with the normal diffusion of gas out of
161
the precritical bubble. If the amplitude of the sound waves is greater than some threshold value,
then rectified diffusion dominates and the bubble grows. Rectified diffusion is composed of three
effects [17]: (i) When the bubble contracts, the concentration of gas in the bubble increases
and gas tends to diffuse out of the bubble. When the bubble expands, the concentration of gas
decreases and gas tends to diffuse into the bubble; (ii) Since the surface area of the bubble is
greater during the expansion, more gas will enter during the expansion of the bubble than will
leave during the contraction; (iii) During the expansion, the thickness of the diffusion boundary
layer adjacent to the bubble decreases. As a result, the concentration gradient across this layer
increases. Since the mass flow into the bubble is proportional to the concentration gradient, it
is enhanced during expansion. The opposite is true during contraction.
To model the effect of a sound field on a bubble at rest in a liquid, we can set p∞ (t) = pa +ps (t),
where ps = Ps sin (ωt) is the sound field. As shown by Eller and Flynn (1965), progress can
then be made by first assuming that the instantaneous bubble radius R (t) is periodic with a
bubble period Tb and then averaging the equations over one bubble period. This was found to
be a valid approximation if Re ≫ (D/ω)1/2 , where Re is the ‘equilibrium’ radius of the pulsating
bubble such that
3
2γ
Re
pb = pa +
.
Re
R
For D = 2 × 10−9 m2 s−1 and ω = 250π × 103 rad s−1 , this requires Re ≫ 0.05 µm. The final
solution for the rate of change of Re with time is [18]


2γ 
R  p0
DHRs T
hR/Re i
dRe
E 1+
,
(5.48)
=
−D
4
dt
Re [1 + 4γ/ (3pa Re )] Re
pa
p
R
a
e
(R/R )
e
where angle brackets denote a time average over one bubble period, and
2
R
Ps
2
= 1 + Kα
,
Re
pa
* +
2
4
R
Ps
2
= 1 + (3 + 4K) α
,
Re
pa
1
,
3 [1 + 4γ/ (3pa Re ) − β2 ]
4 − β2 /4 + 4γ/ (3pa Re )
,
K=
1 + 4γ/ (3pa Re )
α=
β2 =
ρl ω2 R2e
.
3pa
(5.49)
(5.50)
(5.51)
(5.52)
(5.53)
In deriving (5.49) and (5.50), it is assumed that that the ratio Ps /pa is small, an assumption
that does not appear to be satisfied in the case of the surger unit.
162
60
50
Ps,T/pa
40
30
20
10
0
0
0.2
0.4
0.6
R
/R
e
0.8
1
c
Figure 8: Ps,T versus Re for p0 = 3.8 bar, pa = 1 bar, γ = 40 mN m−1 , ρl = 103 kg m−3 ,
ω = 250π × 103 rad s−1 .
The threshold amplitude Ps,T for rectified diffusion is obtained by setting Ṙe = 0 and solving
for Ps to obtain
s
1 + 2γ/ (pa Re ) − p0 /pa
pa
Ps,T =
.
(5.54)
α (3 + 4K) p0 /pa − [1 + 2γ/ (pa Re )] K
The threshold amplitude increases as Re decreases, as shown in Figure 8. Given R0 , the initial
bubble radius, and Ps , (5.48) can be solved for Re (t) using the Matlab ODE solver ode15s.
Some examples are shown in Figure 9. Those precritical bubbles for which Ps > Ps,T grow,
while those for which Ps < Ps,T shrink.
5.3
Bubble growth by convective diffusion
Since the diffusion coefficient D is small (D ≈ 2 × 10−9 m2 s−1 for N2 in water), the rate of mass
transfer into a bubble at rest is also small. In practice, the bubble rises in the liquid due to
its buoyancy and this motion enhances the rate of mass transfer, just as stirring tea enhances
the dissolution of any added sugar. This transport of matter by the combined processes of
convection and diffusion will be called convective diffusion.
Consider then a bubble with radius R (t) ≫ Rc rising steadily through a supersaturated liquid.
The mass flux q into the bubble would be expected to be proportional to the surface area 4πR2
of the bubble and to the concentration drop ∆c = cl − cb between the liquid bulk and the
bubble ‘skin’ (Figure 1):
q = k4πR2 ∆c,
where the proportionality constant k is known as the mass transfer coefficient. This can be
163
2.5
R =1.2R
0
c
R =1.1R
0
c
R =1.0R
2
0
c
R =0.9R
0
c
R =0.8R
0
1.5
c
R =0.7R
/Rc
0
c
R =0.6R
c
R
e
0
1
0.5
0
0
1
2
3
4
5
t/τ
Figure 9: Re versus t for Ps = 2 bar, p0 = 3.8 bar, pa = 1 bar, γ = 40 mN m−1 , ρl = 103 kg m−3 ,
ω = 250π × 103 rad s−1 .
equated to the rate of increase of the bubble mass:
d 4 3
πR ρb = k4πR2 ∆c.
dt 3
(5.55)
Empirically, k is proportional to the diffusion coefficient of the gas in the liquid, so that (5.55)
can be rewritten as
4πR2D∆c
d 4 3
.
(5.56)
πR ρb =
dt 3
δN
Comparing (5.4) and (5.56), the Nernst diffusion layer thickness δN is seen to be given by
∂c
∆c
=
.
∂r r=R
δN
As shown in Figure 1, δN is the thickness which the diffusion layer would have if the concentration profile were a straight line with slope (∂r c)r=R .
If we assume that the gas obeys the perfect gas law ρb = pb/ (Rs T ) with pb ≈ pa as R ≫ Rc ,
and if we also assume that T is constant, then ρb is also approximately constant. Then
d 4 3
dR
πR ρb = 4πR2 ρb .
(5.57)
dt 3
dt
Substituting (5.57) into (5.55) yields
dR
k∆c
=
.
dt
ρb
(5.58)
Experimental evidence suggests that k is approximately constant [5, 12]. During the rise time of
the bubble, the gas concentration cl in the liquid is also approximately constant. Furthermore,
164
since cb = Hpb and pb ≈ pa , cb is constant. Integrating (5.58) then yields
R = R0 + κt,
where R0 is the initial (t = 0) radius of the bubble and the growth rate of the bubble is
k∆c
.
ρb
κ=
(5.59)
For champagne bubbles originating from cellulose fibres, R0 ≈ 20 µm and κ ≈ 350 − 400 µm s−1
[12]. For a carbonated beer, values of κ ≈ 100 − 150 µm s−1 [12] and κ ≈ 40 µm s−1 [5] have
been reported. The low gas concentrations in Guinness imply smaller values of κ, as we will
see.
In the literature, the mass transfer coefficient k is usually expressed in terms of the Sherwood
number (also known as the Nusselt mass transfer number):
Sh =
2kR
,
D
which is itself a function of the Reynolds number
Re =
2ρlRU
,
µ
and the Peclet number
2RU
.
D
If Pe ≪ 1, then mass transfer is dominated by diffusion. If Pe ≫ 1, then mass transfer is
dominated by convection. In order to determine k, we first need to estimate U, which we can
do by considering the forces on a rising bubble. By Archimedes principle, the buoyancy force
on a body immersed in a liquid is equal to the weight of the liquid displaced by that body.
Thus, for a spherical bubble, the buoyancy force is
Pe =
4
FB = πR3ρl g.
3
Since ρb ≪ ρl , the weight of the bubble is negligible in comparsion to the buoyancy force and
the bubble rises. As the bubble rises it experiences resistance due friction between its surface
and the surrounding viscous liquid. This friction force is expressed as
Fd = Cd
ρl U2 2
πR ,
2
where Cd is the friction factor or drag coefficient. The buoyancy force is nearly always balanced
by the drag force so that we can equate them to obtain
4 3
ρl U2 2
πR ρl g = Cd
πR ,
3
2
165
or, solving for Cd ,
8Rg
.
(5.60)
3U2
This equation can be used to determine Cd experimentally. It is found that beer bubbles
ultimately experience a drag similar to that experienced by solid spheres, for which [16]
Cd =
Cd =
24
1 + 0.15 Re0.687 ,
Re
Re < 800.
Over the rise time of the bubble though, a good approximation is Cd ≈ 48/ Re [12]. Solving
(5.60) for U then yields
ρl gR2
U=
.
(5.61)
9µ
As R increases, U increases and the bubble accelerates. Small bubbles rise slower than large
bubbles. Hence, they can be entrained in the downward liquid flow at the glass walls and
actually sink. These sinking bubbles are easily seen in a pint of Guinness. For bubbles with
radii of order 50 µm in Guinness (ρl ≈ 103 kg m−3 , µ ≈ 2 × 10−3 Pa s [19]), U ≈ 0.1 cm s−1 ,
Re ≈ 0.07 and Pe ≈ 68. In carbonated beers, U is a couple of cm s−1 in good accordance with
Equation (5.61) [5].
For the case of stationary (U, Re, Pe = 0) bubbles, Sh = 2 as can be verified by rewriting (5.32)
in dimensional form. For rising bubbles (nonzero U, Re, Pe), flow inside and outside the bubble
affects the mass transfer. The Sherwood number can be calculated from the semiempirical
expression [8]
!
1/3
0.096
Re
, Re < 105 .
1+
Sh = 1 + 1 + Pe1/3
1 + 7 Re−2
For large Pe, Sh ≈ Pe1/3 so that, from the definitions of Sh and Pe,
2/3
D 1/3
D
k=
Pe =
U1/3 .
2R
2R
Substituting (5.61) for U yields
k=
D
2
2/3 ρl g
9µl
1/3
.
Because of the R2 dependence of U, k becomes independent of R and hence time, as expected.
The growth rate (5.59) becomes
2/3 1/3
D
∆c
ρl g
κ=
2
9µl
ρb
Over time cl drops, so that κ drops. For pure nitrogen bubbles in Guinness, we obtain κ ≈
3 µm s−1 , much smaller than the corresponding values for champagne and carbonated beers.
Although Guinness bubbles rise slower and therefore have longer to grow, they remain smaller
than those in carbonated beverages because of this very low growth rate.
166
5.4
The surger
In the case of the surger, precritical bubbles are initially grown by rectified diffusion. Once they
become postcritical, they grow by convective diffusion as they rise through the Guinness. The
equations governing the growth of any individual bubble are much like those we have already
seen, except now we must consider the combined effects of a sound field and convection. We must
also consider the whole distribution of bubbles within the Guinness, as well as the dissolved
gas concentration, if we want to model the formation of the head.
Consider a (pure nitrogen) bubble which is rising steadily through the liquid as it grows. We
assume that the rate of increase of mass in the bubble is equal to the sum of the diffusive and
the convective fluxes into the bubble:

 
 

 rate of increase   rate of addition   rate of addition 
of mass of gas
of mass by
of mass by
=
+
,

 
 

in bubble
diffusion
convection
or, in mathematical notation,
d
dt
4 3
πR ρb
3
= 4πR2
D∆c
+ πR2 fU∆c.
R
(5.62)
The factor f is unknown initially; through trial-and-error, a value which makes the results
quantitatively reasonable is chosen. Of course, if the Sherwood number were known then
Equation (5.55) could be used in place of (5.62). Using the perfect gas law to eliminate ρb and
Henry’s law to eliminate cb , we obtain
1 fU
1 d R3 pb
2
(cl − Hpb ) .
(5.63)
=R
+
3Rs T
dt
R
4
The bubble is assumed to be rising with the Stokes’ terminal velocity:
U=
2ρl gR2
,
9µ
(5.64)
which seems appropriate if wall effects are negligible [16], as they would be for bubbles which
originate in the liquid bulk. Neglecting the inertial terms in the Rayleigh-Plesset equation, we
have
2γ
.
(5.65)
pb = p∞ +
R
We set p∞ (t) = pa + ps (t), where ps = Ps sin (ωt) is the ultrasound field. Ps can be estimated
as follows: Assuming the plate of the surger unit oscillates sinusoidally with amplitude A,
its position is given by z = z0 − A sin (ωt). The corresponding acceleration is g = z̈ =
ω2 A sin (ωt). This creates the pressure field
ps = ρl g (h − z) = ρl ω2 A (h − z) sin (ωt) ,
167
Figure 10: The fluxes into the shaded region ∆R∆z in phase space (R, z).
inside the liquid (as verified experimentally in a vibrating vessel [20]). Inserting appropriate
values (ρl = 103 kg m−3 , ω = 250π × 103 rad s−1 , A = 89 nm, h = 10 cm), we find that Ps varies
from 55 bar at the bottom (z = 0) of the liquid to 0 bar at the top (z = h). From Figure 8,
we can see that 55 bar is above the threshold amplitude for most of the precritical bubbles.
It does, however, seem like an unusually large pressure and there is uncertainty regarding the
exact value of A. For simplicity, we ignored the dependence of ps on z.
Now suppose that the bubble distribution in the liquid is given by F (R; z, t) such that FdRdz
is the number of bubbles with radii from R to R + dR at heights from z to z + dz at time t.
As shown in Figure 10, the rate of increase of F (R; z, t) ∆R∆z, the number of bubbles in the
shaded region, is given by

rate of increase



of number of
bubbles in



shaded region






 rate of addition 
of bubbles due to
=




bubbles growing



rate of addition 


 




 of precritical 
 rate of addition  
bubbles by
of bubbles due
,
+
+


 


homogeneous 
to bubbles rising





nucleation
168
or, in mathematical notation,
∂F
∆R
∆R
∆R∆z = F R −
; z, t − F R +
; z, t Ṙ∆z
∂t
2
2
∆z
∆z
+ F R; z −
, t − F R; z +
, t U∆R + s (R; t) ∆R∆z,
2
2
where s (R, t) ∆R∆z is the number of precritical bubbles being created or destroyed in the shaded
region per unit time. Dividing both sides by ∆R∆z and taking the limit as ∆R, ∆z → 0 yields
∂F
∂F
∂F
= − Ṙ − U + s (R; t) .
∂t
∂R
∂z
The dissolved gas concentration is given by
 

rate
of
increase
 rate of addition





 

of mass of gas
of mass of gas
=
per unit volume 
 per unit volume




 

by diffusion
of liquid
or, in mathematical notation,

rate of loss of



mass of gas per
−
 unit volume by



 
bubble growth




(5.66)







,
Z
∂2 c l
d 4 3
∂cl
=D 2 − F
πR ρb dR.
∂t
∂z
dt 3
Using the perfect gas law to eliminate ρb , we obtain
Z
d R3 pb
∂cl
∂2 c l
4π
F
=D 2 −
dR.
∂t
∂z
3Rs T
dt
(5.67)
Equations (5.63) to (5.67) form a closed set for {R, U, pb, F, cl} and govern the growth of an initial
distribution of bubbles in the presence of an ultrasound field ps and a source s of precritical
bubbles.
However, solving these equations is complicated by the problem of choosing the time step. A
naïve simulation would use the ultrasound period 2π/ω ≈ 8 µs over the duration of the surge
and settle (∼ 1 min). To speed up the simulation, we instead split it into two parts, the first part
of which computes the rate at which postcritical bubbles are being produced by the ultrasound
field, and the second part of which uses this rate to study the behaviour of the postcritical
bubbles. The final results from this part are shown in Figures 11, 12, and 13. Figure 11 shows
the bubble radii as a function of time, each bubble corresponding to one cross. Initially, all the
bubbles are precritical and thus their radii are clustered together. As time progresses, the radii
increase with the maximum radius being about 400 µm. As bubbles reach the top of the liquid,
they are removed from the graph. As time progresses further, the mean bubble radius drops
to about 50 µm because the bubble growth rate decreases as the dissolved gas concentration in
the liquid falls. In Figure 12, the corresponding heights of the bubbles are shown. Finally, the
height of the head is shown in Figure 13; it is approximately 12 mm after 5 s.
169
Figure 11: Bubble radii as a function of time.
Figure 12: Bubble heights as a function of time.
170
Figure 13: Head height as a function of time.
6
Conclusions
Diageo asked us to develop a mathematical model describing bubble formation and growth in
Guinness. To this end, we considered the growth of single bubbles by passive, rectified and
convective diffusion. In the passive case, the role of the critical radius Rc was emphasised: in the
absence of a sound field, precritical (R < Rc ) bubbles shrink. In the presence of a sound field,
we showed that precritical bubbles will grow by rectified diffusion provided that the amplitude
of the sound field is above the threshold value given by (5.54). We also considered how the
growth of bubbles by convective diffusion is modelled mathematically.
Applying these ideas to the surger unit, and some new ideas regarding the evolution of the
bubble distribution and the dissolved gas concentration, we successfully modelled the initiation
of Guinness. The results, however, are more qualitative than quantitative since
1. We have completely ignored the dissolved CO2 . Since the CO2 concentration is much
higher than the N2 concentration, it must surely have a significant effect on the growth
of the bubbles (cf. Section 8). In particular, a model taking both gases into account
should be able to explain quantitatively why the head on Guinness does not form spontaneously. Furthermore, there may also be some water vapour in the bubble, although
for low temperatures the ratio of vapour to gas should remain small.
The surger equations, Equations (5.63) to (5.67), are easily modified to take the CO2 into
171
account by replacing Equation (5.63) by
1 d R3 xi pb
= ki R2 (cl,i − Hi xi pb ) ,
3Rs,i T
dt
where we are now using the mass transfer coefficient ki , and Equation (5.67) by
Z
d R3 xi pb
∂2 cl,i
4π
∂cl,i
F
= Di
−
dR,
∂t
∂z2
3Rs,i T
dt
(6.1)
(6.2)
where xi is the mole fraction of gas species i in the bubble so that
X
xi = 1.
Equations (6.1) and (6.2) constitute four equations, two each for the CO2 (i = CO2 ) and
the N2 (i = N2 ).
2. We have not considered the diffusion of gas across the top surface of the Guinness and
into the atmosphere. This could be significant: for champagne, 80% of the dissolved CO2
molecules escape by this method [12].
3. For sufficiently large bubbles and high frequencies, the isothermal (T = const) approximation becomes questionable [21].
4. Assuming that the rise velocity is simply the Stokes velocity is an over-simplification.
In practice, the high number of bubbles creates a complex flow pattern, with liquid and
bubbles rising in the centre of the glass and liquid and small bubbles descending at the
edges.
5. The bubble distribution and dissolved gas concentration will vary horizontally as well as
vertically.
6. The addition of surfactants significantly affects the accuracy of theoretical predictions
regarding rectified diffusion [18].
7. We studied noninteracting spherical bubbles, but bubbles can deform, oscillate, break up
and coalesce.
A noticeable feature of using the surger unit is the slight delay (∼ 1 s) before the surge of
nitrogen bubbles is seen. It is possible that this delay is due to the so-called Bjerknes force
holding the bubbles in place while the surger unit is vibrating [20]. When the surger unit stops
vibrating (it only vibrates for 1 − 2 s after pressing the activation button), there is no Bjerknes
force and the bubbles are free to rise. As they rise, some bubbles move towards the glass walls
where they become visible.
Although we focused on the surger, the equations we have used can be applied to other methods
of initiation simply by modifying the input pressure p∞ (t) and the source term s (R; t).
172
At the Study Group, some ideas for new initiation methods were discussed. Perhaps the most
interesting is the possibility of using a ‘nanowidget’; that is, an insert at the bottom of the can
which has nanometer features, making it superhydrophobic. When liquid is poured over such
a surface, gas is trapped in the cavities between the features. Upon pressurising the can, some
of the gas will dissolve. Upon opening the can, the surviving gas pockets will act as nucleation
sites, initiating the Guinness. Of course, the effectiveness and cost of such a device would have
to be investigated. Another interesting idea is to coat the inside of the can with a surfactant
to promote bubble nucleation. For example, it is known that the surfactant coating on Mentos
(gum arabic) is partly responsible for the explosivity of the Diet Coke and Mentos reaction [22].
7
List of symbols
In the following list of symbols, the corresponding SI unit, if any, is given in brackets.
A amplitude of oscillation of surger plate (m)
Cd drag coefficient
c gas concentration in liquid (kg m−3 )
cb gas concentration in liquid at bubble surface (kg m−3 )
cl gas concentration in liquid bulk (kg m−3 )
c∞ gas concentration in liquid at r = r∞ (kg m−3 )
D diffusivity of gas in liquid (m2 s−1 )
F bubble distribution function (m−2 )
FB buoyancy force (N)
Fc capillary force (N)
Fd drag force (N)
f function defined in Equation (5.62)
g acceleration due to gravity (m s−2 )
H Henry’s law constant (kg m−3 Pa−1 )
h height of liquid (m)
K function defined in Equation (5.52)
k mass transfer coefficient (m s−1 )
L circumference of nucleation site (m)
173
M molar mass (kg mol−1 )
nb = pb Vb / (Rg T ) number of moles of gas in bubble (mol)
Ps amplitude of sound field (Pa)
Ps,T threshold amplitude of sound field (Pa)
Pe = 2RU/D Peclet number
p partial pressure of gas above liquid (Pa)
pa atmospheric pressure (Pa)
pb gas pressure in bubble (Pa)
pl liquid pressure (Pa)
ps = Ps sin (ωt) sound field (Pa)
p0 gas pressure in sealed container (Pa)
p∞ liquid pressure at r = r∞ (Pa)
q mass flux (kg s−1 )
R bubble radius (m)
Ṙ = dR/dt time derivative of R (m s−1 )
Rc critical radius (m)
Re ‘equilibrium’ radius of a pulsating bubble (m)
R0 initial radius (m)
R∞ ultimate radius (m)
Re = 2ρlRU/µ Reynolds number
Rg universal gas constant (J K−1 mol−1 )
Rs = Rg /M specific gas constant (J K−1 kg−1 )
r distance from centre of bubble (m)
r∞ catchment radius (m)
S supersaturation ratio
Sh = 2kR/D Sherwood number
s precritical bubble source (m−2 s−1 )
T temperature (K)
174
t time (s)
(ur , uθ , uφ) liquid velocity components (m s−1 )
u = ur radial component of liquid velocity (m s−1 )
U velocity of a rising bubble (m s−1 )
Vb = 43 πR3 volume of bubble (m3 )
x mole fraction of gas
z distance measured upward from bottom (z = 0) of liquid (m)
α function defined in Equation (5.51)
β function defined in Equation (5.53)
γ surface tension (N m−1 )
∆p = p0 − pa (Pa)
∆c = cl − cb (kg m−3 )
δ = ρl D2ε2 / pa R2c small parameter
δN Nernst diffusion layer thickness (m)
ε = H∆p/ρa small parameter
θ colatitude coordinate in spherical coordinates (r, θ, φ) (rad)
θc contact angle (rad)
κ bubble growth rate (m s−1 )
Λ = R∞ /Rc large parameter
µ dynamic viscosity of liquid (Pa s)
ρa = pa /Rs T density scale (kg m−3 )
ρb gas density in bubble (kg m−3 )
ρl liquid density (kg m−3 )
τ = R2c / (Dε) time scale (s)
φ longitude coordinate in spherical coordinates (r, θ, φ) (rad)
ω angular frequency of oscillation of surger plate (rad s−1 )
175
8
Gas concentrations in Guinness
Dissolved gas concentrations are frequently given in g l−1 (grams of dissolved gas per litre of
solution) or g kg−1 (grams of dissolved gas per kilogram of solution). They can also be expressed
in parts per million (ppm), Volumes (Vol.), or as a volume over volume percentage (% v/v).
These three units are defined by
ppm of dissolved gas = ( g kg−1 of dissolved gas) × 103.
volume of dissolved gas at 0 ◦ C and 1 atm
,
volume of solution
% v/v of dissolved gas = (Vol. of dissolved gas) × 100.
Vol. of dissolved gas =
Thus, to convert from g kg−1 to ppm, we multiply by 103 . For example, the desired CO2
concentration in Guinness is about 2 g l−1 . If 1 l of Guinness has a mass of 1 kg, then this is
equivalent to 2 g kg−1 or 2000ppm. To convert to Vol., we must first determine the conversion
factor from g l−1 to Vol.. From the perfect gas law, one mole of gas occupies 22.4 litres at 0 ◦ C
and 1 atm. Thus, one gram of gas occupies 22.4/M litres, where M is the molar mass of the
gas in g mol−1 . Hence,
Vol. of dissolved gas = g l−1 of dissolved gas ×
Since the molar mass of CO2 is 44.0 g mol−1 ,
22.4
.
M
Vol. CO2 = g l−1 CO2 × 0.509.
Thus, to convert from g l−1 CO2 to Vol. CO2 , we multiply by 0.509, so that 2 g l−1 CO2 is
equivalent to 1.02Vol. CO2 . This means that the CO2 in one litre of Guinness would occupy
1.02 litres if brought to a temperature of 0 ◦C and a CO2 gas pressure of 1 atm. Since the molar
mass of N2 is 28.0 g mol−1 ,
Vol. N2 = g l−1 N2 × 0.800.
To obtain more accurate conversion factors, the deviation from perfect gas behaviour must be
taken into account. To convert from Vol. to % v/v, we multiply by 100. Thus, the desired CO2
concentration is 102% v/v.
Henry’s law relates the solubility c of a gas to its partial pressure p above the solution:
c = Hp,
where H is the Henry’s law constant. The Henry’s law constants of CO2 and N2 in pure water
are given in table 8 for different temperatures.
The desired CO2 concentration is about 2 g l−1 From Henry’s law this requires a partial pressure
of
cCO2
2 g l−1
pCO2 =
= 0.8 bar .
=
HCO2
2.5 g l−1 bar−1
176
Temperature (◦ C)
0
8
15
20
HCO2 (g l−1 bar−1 )
3.42
2.50
2.05
1.76
HN2 (g l−1 bar−1 )
0.0283
0.0241
0.0209
0.0192
Table 8: Henry’s law constants for CO2 and N2 in pure water. Sources: [23] and [24].
In the sealed container the total pressure is 3.8 bar. Thus the partial pressure of N2 in the
headspace (and widget) is pN2 = 3 bar. From Henry’s law this yields a solubility of
cN2 = HN2 pN2 = 0.0241 g l−1 bar−1 (3 bar) = 0.0723 g l−1 ,
or 5.78% v/v. If 1 l of Guinness has a mass of 1 kg, then the solubility of N2 is 72.3ppm. The
mole fractions in the gas phase are
xCO2 =
0.8 bar
= 0.21,
3.8 bar
xN2 =
3 bar
= 0.79,
3.8 bar
so that the gas mixture is 21 mole % CO2 and 79 mole % N2 .
Suppose now that the container is opened and the gas concentrations are allowed to come to
equilibrium with their partial pressures in the atmosphere at 15 ◦ C. The partial pressures of
CO2 and N2 in air are about 3 × 10−4 bar and 0.78 bar, respectively. Hence, from Henry’s law,
the corresponding solubilities are
cCO2 = 2.05 g l−1 bar−1 3 × 10−4 bar = 0.0006 g l−1 ,
cN2 = 0.0209 g l−1 bar−1 (0.78 bar) = 0.016 g l−1 .
It follows that about 0.0282 g of N2 and 1 g of CO2 must escape from every pint (∼ 500 ml)
of Guinness to bring the dissolved gas concentrations into equilibrium with their atmospheric
partial pressures. This corresponds to about one billion 50 µm radius bubbles, 50 µm being
close to the average bubble radius [19]:
total number of excess moles of gas in liquid
total number
=
of bubbles
total number of moles in one bubble
1 g /MCO2 + 0.0282 g /MN2
=
≈ 109 bubbles.
nb
This number of bubbles would create a head 14 cm thick! In practice, most of the CO2 cannot
escape as bubbles and must instead diffuse through the free liquid surface. To see why, let us
consider what the CO2 partial pressure, xCO2 pb , in a postcritical bubble tends towards. From
Henry’s law CO2 will diffuse into the bubble if the dissolved CO2 concentration cCO2 in the
liquid is greater than HCO2 xCO2 pb . Noting that initially cCO2 = HCO2 (0.8 bar) and that for a
postcritical bubble pb ≈ 1 bar, we find that dissolved CO2 diffuses into the bubble if
xCO2 < 0.8.
177
1.4
R/(50µm)
x
1.2
CO
2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
t/0.94s
Figure 14: R and xCO2 versus t for R (0) = 1 µm and xCO2 (0) = 0.0003.
The CO2 mass flux into the bubble will be greater than the N2 mass flux because of the higher
concentration of CO2 . Hence, the CO2 mole fraction in the bubble will tend towards 0.8. In fact,
solving (6.1) for the case of stationary bubbles (Sh = 2, ki = Di /R) and constant dissolved gas
concentrations (cl,i = const), we find that the CO2 mole fraction tends towards 0.7 (figure 14).
It is therefore appropriate to consider how many bubbles, each with xN2 = 0.3 and R = 50 µm,
can be created before the N2 runs out:
total number of excess N2 moles in liquid
total number
=
of bubbles
total number of N2 moles in one bubble
0.0282 g /MN2
≈ 1.5 × 108 bubbles.
=
xN2 nb
This reduces the height of the head to about 2 cm, which is reasonably close to the observed
value of about 1 cm considering the approximations made. The amount of excess CO2 left
dissolved in the liquid when the N2 runs out is given by
CO2 mass in
initial excess
dissolved mass of
−
=
one bubble
CO2 mass
excess CO2 left
number
×
of bubbles
= 1 g − (MCO2 xCO2 nb ) 1.5 × 108 ≈ 0.9 g .
Hence, 90% of the excess CO2 remains and can only escape by diffusing through the free surface.
178
Acknowledgements
The authors would like to thank Diageo, and especially their representatives Conor Browne and
Richard Swallow, for bringing this interesting problem to the Study Group and for answering
all our questions. The authors would also like to acknowledge the support of the Mathematics
Applications Consortium for Science and Industry (MACSI) funded by the Science Foundation
Ireland mathematics initiative grant 06/MI/005.
179
Bibliography
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excellence. J. Inst. Brew., 110 (4), 259-266.
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124-132.
[5] Shafer N.E. and Zare R.N. (1991). Through a beer glass darkly. Physics Today, 44 (10),
48-52.
[6] Weiss P. (2000). The physics of fizz. Sci. News, 157 (19), 300-302.
[7] Liger-Belair G. (2003). The science of bubbly. Sci. Am., 288 (1), 80-85.
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[9] Wilt P.M. (1986). Nucleation rates and bubble stability in water-carbon dioxide solutions.
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[12] Liger-Belair G. (2005). The physics and chemistry behind the bubbling properties of champagne and sparkling wines: A state-of-the-art review. J. Agric. Food Chem., 53, 2788-2802.
[13] Jones S.F., Evans G.M. and Galvin K.P. (1999). Bubble nucleation from gas cavities - a
review. Adv. Colloid Interface Sci., 80, 27-50.
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champagne bubbles. Chem. Soc. Rev., 37, 2490-2511.
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1294-1301.
180
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and Jeandet P. (2000). On the velocity of expanding spherical gas bubbles rising in line in
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[20] Crum L.A. (1975). Bjerknes forces on bubbles in a stationary sound field. J. Acoust. Soc.
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[22] Coffey T. (2008). Soda pop fizz-ics. The Physics Teacher, 46, 473-476.
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2, 100-107.
181
182
Solar Reflector Design
Report Contributors: Afonso Bandeira1 , Christopher G. Bell 2,
Jean P.F. Charpin3,4, Romina Gaburro3 ,
Sofiane Soussi3 and R. Eddie Wilson5
Study Group Contributors:
Giles Richardson6 and Galin Ganchev3
Industry Representative: Sean Hoolan7
1
Universidade de Coimbra, Portugal
Department of Bioengineering, Imperial College London, United Kingdom
3
MACSI, department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
4
Report coordinator, jean.charpin@ul.ie
5
Department of Engineering Mathematics, University of Bristol, United Kingdom
6
School of Mathematics, University of Southampton, United Kingdom
7
Erin Energy Ltd, Applied Research into Solar Technologies, MIRC Building, Athlone Institute of Technology, Dublin Road, Athlone, Co Westmeath, Ireland
2
Abstract
The design of solar panels is investigated. Different aspects of this problem are presented. A
formula averaging the solar energy received on a given location is derived first. The energy received by the collecting solar panel is then calculated using a specially designed algorithm. The
geometry of the device collecting the energy may then be optimised using different algorithms.
The results show that for a given depth, devices of smaller width are more energy efficient than
those of wider dimensions. This leads to a more economically efficient design.
184
Mirror
Energy absorber
Figure 1: Typical configuration
1
Introduction
In recent years, considerable research was carried out on renewable sources of energy. Designing
cheap and efficient devices has become a priority for the industry. Erin Energy, based in
Athlone, Westmeath, Ireland, is focusing on solar energy. A typical example of design they
study may be seen on Figure 1. This device should be placed on the roof. The solar energy
is collected by the energy absorber, transported and then stored in a chemically based energy
tank. Ideally, the roof should be completely covered with energy collectors but they are quite
expensive. To reduce the amount of collectors necessary, the roof is covered with mirrors.
The solar energy may then be collected either directly or after one or more reflections on the
mirrors. The shapes of the mirrors and collectors are key to the efficiency of the system. Erin
Energy currently investigates a cylindrical collector coupled with parabolic or involute shaped
mirrors. They would like to evaluate the efficiency of their current design and if possible, find
more optimised shapes. This problem was submitted to the the 70th European Study Group
in Industry held in Limerick in June-July 2009. This report reflects the work of the group.
Different directions were investigated to tackle this problem. First of all, the incoming solar
energy was considered. This parameter varies with the position of the device, its inclination,
the time of the day and the time of the year. These different aspects were considered and are
detailed in Section 2. The reflections of rays and energy over the mirror was then considered. An
exact and fast ray tracing algorithm was developed in MATLAB and is presented in Section 3.
Several methods were then considered to optimise the shape of the mirrors. An approach based
on inverse methods is introduced in Section 4. An alternative was developed to maximise
the direct and first reflections. This aspect is detailed in Section 5. Finally a more general
optimisation based on a pattern search method was used and the results may be found in
Section 6.
185
Aperture
Mirror
Absorber
Figure 2: Sketch of a trough-like solar concentrator.
2
Model for annual energy collection
To calculate the annual energy collected by a stationary trough-like solar concentrator such
as that pictured in Figure 2 requires two steps. Firstly we must find the energy incident on
the aperture as a function of the position of the sun, and then we must calculate how much
of this energy is transmitted to the absorber via reflections from the mirror. In all of this
work, we only consider direct beam radiation and ignore the effects of diffuse radiation. For a
concentrator situated at latitude angle φ, a simple model for the total annual energy collected
may be written as
Z δmax Z ωlim (δ, φ)
F(δ, ω, φ) T (δ, ω, φ) dω dδ.
(2.1)
Eannual = A
−δmax
0
The energy function F(δ, ω, φ) models the energy striking a unit area of the aperture, and
multiplying by the area A of the aperture gives the total energy entering the aperture. The
transmission function T (δ, ω, φ) (0 ≤ T ≤ 1) models the fraction of the energy entering the
aperture which hits the absorber. Integrating over δ and ω sums the total energy collected
over the year.
The parameter δ is the declination angle [1, 2], which varies between ±δmax = ±0.40928
radians and determines the height of the Sun above the equatorial plane throughout the year.
The variation of δ as a function of day number N, with N = 1 being the 1st January, is depicted
in Figure 3.
The rotation of the Earth is measured by the hour angle, ω [1, 2], and the range of integration [0, ωlim(δ, φ)] is determined by the hours of daylight available to the concentrator. This
186
Solstice, Jun 21st/22nd
δ, radians
0.4
0.3
Equinox, Sep 23rd
0.2
0.1
0
100
K
200
300
ν
0.1
N
K
0.2
K
0.3
K
0.4
Equinox, Mar 22nd
Solstice, Dec 21st/22nd
Figure 3: Graph of how the declination angle δ varies as a function of the day number N.
depends both on the height of the Sun in the sky (which depends on δ and φ) and on the
orientation and angle of tilt of the concentrator.
The function F(δ, ω, φ) models the intensity of the light striking the aperture of the concentrator as a function of δ and ω, corresponding to particular times of day and season. A general
formula for F(δ, ω, φ) may be written as [3]
43200 × 365.25
×
2π2
0.034
2π
1+
cos
× 173 sin δ ×
sin(δmax )
365.25
k
×
a0 + a1 exp −
cos δ cos ω cos φ + sin δ sin φ
s(δ, ω, φ) · n×
F(δ, ω, φ) = 4 × Isc ×
cos δ
q
. (2.2)
sin2 (δmax ) − sin2 δ
In this formula, the first line quantifies the amount of radiation striking the outer edge of
the earth’s atmosphere. The parameter Isc = 1367 W m−2 is the solar constant [2] and the
multiplicative factor results from changing the integration parameters from time to angle. The
second line adjusts for the ellipticity of the earth’s orbit around the Sun [2]. The third line
models the reduction in the intensity of the light striking the earth’s surface depending on how
far it has travelled through the atmosphere. We use the Hottel model [4], and we take the 23km
Visibility Haze parameters at zero altitude:
a0 = 0.1281,
a1 = 0.7569,
k = 0.3872.
(2.3)
The fourth line measures the ‘cosine effect’ and modifies the energy striking the aperture
according to the angle of incidence. Here s(δ, ω, φ) is a vector pointing in the direction of the
187
Sun and n is the normal vector to the aperture of the collector. Finally the last line comes
from the Jacobian due to the change in variables of integration from time to angle.
For trough-like concentrators, the problem of calculating the fraction of energy, T (δ, ω, φ),
transmitted from the aperture to the absorber is essentially two-dimensional. Away from the
ends of the trough, the component of the energy flux along the longitudinal axis may be ignored
and the energy collected is simply a function of the angle of incidence in the plane orthogonal
to the longitudinal axis of the concentrator. We call this angle ξ(δ, ω, φ) (−π/2 ≤ ξ ≤ π/2),
and measure it relative to the normal to the aperture of the concentrator as shown in Figure 4.
This angle of incidence is a function of the position of the Sun (through δ, ω and φ), and
also depends on the orientation and tilt of the concentrator. For a particular shape of mirror
and position of absorber, it is easy to calculate T (ξ) for any angle of incidence ξ by using ray
tracing, and if we know how ξ depends on δ, ω and φ, we may calculate the energy integral.
n
ξ
s1
Aperture
Figure 4: Sketch showing the angle of incidence ξ in the plane orthogonal to the longitudinal
axis of the concentrator. The vector s1 is the projection of the vector s pointing in the direction
of the Sun into this plane and n is the normal to the aperture.
To assess the performance of a concentrator, there are therefore four functions to determine
before the annual energy may be calculated
• ωlim (δ, φ) - the hours of daylight available to the concentrator;
• s(δ, ω, φ) · n - the ‘cosine effect ’ due to the angle of incidence of the energy onto the
aperture;
• ξ(δ, ω, φ) - the angle of incidence in the plane orthogonal to the longitudinal axis of the
concentrator;
• T (ξ) - the fraction of incident energy transmitted to the absorber as a function of the
angle of incidence ξ.
The first three functions depend on the orientation and tilt of the concentrator, while the last
depends on the shape of the mirror and the position of the absorber.
188
In reference [3], we have considered the above problem for a horizontal east-west aligned concentrator tilted at an angle θn from the vertical. If we restrict the location of the concentrator
to be between the Antarctic and Arctic circles so that
π
π
(2.4)
δmax − ≤ φ ≤ − δmax ,
2
2
and we restrict the angle of tilt θn so that
π
π
−φ −
− δmax ≤ θn ≤ −φ +
− δmax ,
2
2
then it is possible to show that the hours of daylight are specified by the upper limit
ωlim (δ, φ, θn ) = min arccos − tan δ tan(φ + θn ) , arccos − tan δ tan φ .
(2.5)
(2.6)
The restriction on the angle of tilt θn is reasonable, since as a general rule of thumb using the
polar mount (or θn = −φ) for an east-west concentrator is close to maximizing the annual
energy incident on the aperture [1]. The ‘cosine effect’ is specified by the relationship
s(δ, ω, φ) · n = cos δ cos ω cos(φ + θn ) + sin δ sin(φ + θn ),
and finally the angle ξ may be calculated through the relationship [5]
tan δ
−1
ξ(δ, ω, φ) = −θn − φ + tan
.
cos ω
(2.7)
(2.8)
For further information on determining the above functions for different orientations and tilts,
see references [1, 2].
As a final comment, in reference [3] for a horizontal tilted east-west concentrator, we show how
to change integration variables from (δ, ω) to (δ, ξ) so that the energy integral may be written
in the form
ZZ
Eannual = A
F(δ, ξ, φ) T (ξ) dδ dξ.
(2.9)
It is then easy to substitute in for the transmission function as a direct function of ξ. In the
next section we show how to develop a ray tracing algorithm to calculate T (ξ).
3
3.1
Ray-tracing package
Motivation and scope
Later in this report we shall propose several different methods for generating reflector shapes.
These are based on either heuristic techniques, or on the formal optimisation of energy return
in some highly simplified setting — for example, we may neglect all secondary reflections or
189
consider only a reduced model for the trajectory of the sun through the sky and its effective
intensity as a function of inclination.
Whatever approach is used to generate reflector shapes, there is a need to definitively evaluate
the return in a realistic setting which captures the full details of energy collection throughout
the year and which may also cope with extra effects such as diffuse radiation (which is thought to
be important in an Irish setting). This definitive evaluation can only be carried out numerically,
and the computational framework that we implemented is based on the classical approach of
ray-tracing.
There are in fact many ray-tracing packages already in the public domain which are equal to our
task, but none that we found are specialised to the particular geometry etc. of solar reflector
design, yet flexible enough for our computational experiments. In summary, it seemed best
to implement our own ray-tracing package during the Study Group itself, using the high-level
computational language Matlab.
Our code took only about one man-day to complete and forms the basis of a very pleasing
tool with which reflector design can be investigated. The user has access to all the powerful
programming and visualisation features of the Matlab environment, yet at the same time, the
computational kernel is sufficiently fast that meaningful calculations can be carried out almost
instantaneously on even modest laptop hardware.
Later in the report, we describe how we drove the ray-tracing code in the inner loop of a
black box optimisation algorithm, thus enabling formal optimisation of the reflector profile
even for realistic models of incident radiation. This is an extremely demanding problem from
the computational point of view, yet the optimisation proved tractable provided the search
was restricted to relatively simple reflector designs. Of course, it will be possible to accelerate
run times significantly, and thus attempt more sophisticated optimisation (over wider search
spaces and resolving more detail of the reflector profile), by either translating the computational
kernel into a low-level language such as C or Fortran, or porting the code to specialised parallel
hardware such as GPU machines. These tasks remain for future work, as does the development
of a polished graphical front-end.
Since we consider only reflectors with a fixed cross-section, we may simplify matters by only
tracing rays constrained to the plane of the cross-section. In reality, at most times of day
real-world rays will have an out-of-plane component, but this may be compensated for by a
reduction in the intensity factor. Here we assume that the reflector is much longer than it is
wide so that end-effects may be neglected.
3.2
Interface to the computational routine
In summary, we need only perform ray-tracing in two dimensions, where the reflector is modelled by a curve and the absorber element by a circle. As a final simplification, we disretise
the reflector by approximating it by a set of straight line segments, since the intersection of
190
straight lines (rays) with straight lines (reflector segments) is a particularly simple computational problem. The accuracy of results may then be refined by increasing the number of line
segments used in the discretisation: order several hundred to a thousand segments is tractable
with standard hardware.
The inputs to our routine are thus:
• The radius R > 0 of a single circular absorber element krk = R, with centre placed at the
origin, so that all other coordinates are expressed relative to it.
• A collection of i = 1, 2, . . . , m straight line reflector segments, described in terms of the
position vectors a(i) and b(i) of their end-points.
– In practice, we will often have a(2) = b(1) , a(3) = b(2) , . . ., so that the segments join
together to give a contiguous curve, but these conditions are not required by the
code, so reflectors consisting of several disconnected components may also be tested.
(j)
• A collection of j = 1, 2, . . . , n rays, prescribed in terms of their starting points r0 and
(j)
unit directions ^t . In practice, this ensemble may consist of:
1. Sets of parallel rays with different starting points arranged uniformly along a straight
line segment above the reflector so that the complete set of rays is incident across
the width of the reflector’s aperture. This set-up tests the energy collection for a
single incident angle.
2. Ensembles of ensembles of type 1., so many incident angles can be tested simultaneously.
3. Ensembles of non-parallel rays to test diffuse radiation.
• The maximum number p of reflections that should be computed for each individual ray.
The output of the routine is:
• The eventual ‘fate’ of each ray. The possibilities are:
1. Ray hit absorber element after k reflections on the solar reflector, 0 ≤ k ≤ p.
2. Ray ‘escapes to infinity’ after k reflections, 0 ≤ k ≤ p, so that it could not be
incident upon the absorber even if the parameter p were increased.
3. Ray is ‘in-play’ after p reflections, in that it has yet to hit the absorber or escape to
infinity. The fate of the ray can only be decided by experimenting with larger values
for p. In practice we fix p at a sufficiently large value (e.g. 5 or 6) so that only a
very small proportion of rays fall in this category.
191
β
Figure 5: Simple output from the ray-tracing routine. A small ensemble of parallel rays are
shown incident upon a parabolic reflector discretised into 200 straight line segments. Reflections
to infinity are not shown. In this configuration no ray is reflected more than twice, but larger
numbers of reflections may be tracked. Here the circular absorber intercepts the caustic of the
reflector and so is particularly efficient at capturing the incident radiation.
In verbose mode, the user also has access to diagnostic information giving the full itinerary of
each ray in terms of the indices of the reflector segments that it hit, and the points at which
it was reflected. The latter data may be used to drive a plotting routine: example output is
shown in Fig. 5.
An energy collection calculation must then weight the output of the routine according to the
nature of each ray (position of the sun etc., incorporating the factors from Section 2) which
hits the absorber and by αk , where k is the number of reflections it has suffered in the mean
time and α is the reflectivity. Rays which are of types 2. or 3. are given zero weight. The total
energy return is then just the scalar obtained by summing over all rays.
3.3
Computational method
We now give a brief description of how the computational kernel works. In essence, it operates
serially over the reflection number k = 1, 2, . . . , p but for a given k the code is vectorised over
all rays and reflector segments. This vectorisation means all the hard number crunching is
performed internally in Matlab commands rather than by interpreted loops — and this is the
key factor in the code’s speed. This structure also means that the code should port to parallel
architectures rather simply.
• For reflection numbers from k = 0 to k = p in sequence:
– For all combinations of admissible reflector segments i = 1, 2, . . . , m and rays j =
1, 2, . . . , n, find potential intersections by solving the 2 × 2 system
(j)
(j)
λ(i,j) a(i) + (1 − λ(i,j) )b(i) = r0 + s(i,j)^t
for λ(i,j) and s(i,j) simultaneously.
192
(3.1)
– Mark intersections as allowable if 0 ≤ λ(i,j) < 1 and s(i,j) > 0, since each reflector
(j)
component is a finite line segment and each ray is in fact a half-line starting at r0 .
Here we consider half-open reflector segments to avoid double counting problems
where the segments are knotted together at their end-points.
– For each ray j, compute
s∗(j) = min s(i,j) ,
(3.2)
allowed i
(j)
and let i∗ denote the arg min. This means that ray j will next hit reflector segment
(j)
(j)
i∗ after distance s∗ , unless it hits the absorber first.
– For each ray j, also consider intersections with the absorber by solving the quadratic
(j)
(j)
kr0 + s̃(j)^t k2 = R2
(3.3)
for s̃(j) . Here we are of course interested only in real solutions with s̃(j) > 0. In the
case of two such solutions, we take the least (corresponding to the first intersection).
In fact, the case of real solutions with opposite signs will generally not occur in
practice since this corresponds to rays that start inside the absorber element!
– Some control logic is now applied to each ray (but this is also vectorisable). If there
is an allowable s̃(j) > 0 then either
(j)
- s̃(j) < s∗ , in which case ray j next hits the absorber element, and the number
of reflections k(j) suffered so far is recorded. No further computation is required
for this ray and it should be removed from the computation: however, the
vectorisation works most efficiently if it is left in the computation with a dummy
flag indicating that the control logic steps should not be applied to it in future.
(j)
(j)
- s̃(j) > s∗ , in which case ray j next hits reflector segment i∗ . In this case, the
(j)
new starting point r0 of the ray is re-set to
(j)
(j)
r0 + s∗(j)^t ,
(j)
and its new unit direction ^t
(3.4)
is then set to
(j)
(j)
^t(j) − 2(^t(j) · n
^ (i∗ ) )^
n(i∗ ) ,
(3.5)
corresponding to reflection, where n
^ (i) is the unit normal of reflector segment
i, which is computed one-shot at the start of the routine from a(i) and b(i) .
(j)
Finally, reflector segment i∗ is marked as an inadmissible hit for ray j at the
next iteration and remains so for further iterations until ray j hits some other
reflector segment. This fix is to avoid ‘double hit’ problems due to rounding
(j)
error, where r0 is placed marginally on one (the wrong) side of the reflector
segment, so that it almost instantaneously next hits the same segment. The
consequent spurious effect is that some rays appear to pass straight through
reflector segments without deviation — hence the admissibility flag which fixes
this problem is of key importance.
193
- There are also a number of ‘empty’ cases which must be covered but which
we shall not enumerate in detail. For example, in the case where there is no
allowable hit with the absorber, the first hit with the reflector should still be
processed. If there are no allowable / admissible hits with the reflector, the
first hit with the absorber should still be processed. If there are no allowable /
admissible hits with either the absorber or the reflector, then the ray in question
has escaped to infinity and is marked as such.
• Loop over reflection number k.
Note. Here the sense n
^ (i) of the reflector segment is not considered, so in effect each reflector
segment is a two-sided mirror. The generalisation to one-sided mirrors with a matt reverse side
is easily achievable, but requires extra control logic which we have yet to implement.
Using the models and algorithms developed in this section and Section 2, work will now focus
on the shape of the mirror. Three approaches will be presented: the inverse problem will be
detailed first, followed by two optimisation methods.
3.4
Computational cost
The run-time of our routine scales like O(mnp), i.e., it is linearly proportional to each of
the number of reflector segments, rays, and reflections considered. In addition, the memory
constraints scale like O(mn) due to the Matlab vectorisation techniques used, but a C or
Fortran code could be optimised to scale like O(m) in this respect. In practice, we have found
that the routine takes just a couple of seconds on a laptop when mnp ∼ 106 .
Clearly the bottleneck in our approach is the consideration at each iteration of all possible
intersections between all rays and all reflector segments. It may be possible to speed-up matters
by reducing the search space: for example, we may segment the plane into rectangular subregions and index reflector segments by which rectangle they belong to. The itinerary of each
ray through the sub-regions may then be used to seek intersections
√ with reduced lists of reflector
segments. It seems plausible that this approach may give a O( m) or O(log m) speed-up, but
with a significant coding burden which we have yet to undertake. Moreover, this refinement
would break the Matlab vectorisation so would only be worthwhile for a low-level language
implementation. These features remain for future work.
4
Inverse problem
In this section we consider the 2-dimensional case in which the absorber is a point source
located at the origin of the xy axes in R2 and the position of the Sun is given by an angle θ,
for 0 ≤ θ ≤ π (anticlockwise). Although in a more realistic situation the absorber would be
194
described by a circle centred at the origin (0, 0), the case of a point source collector seems the
most appropriate simplified situation to be investigated. The modification in the mathematical
model required for the more realistic circle case will be considered later on.
Let us recall that the goal is to find the best shape of a mirror placed nearby the source in
order to maximise the number of reflections of rays coming from the Sun into the absorber. In
the inverse problem we consider the following approach. The case in which radiation is emitted
from the absorber is considered. The electromagnetic radiation is emitted in all directions and
reflected from the nearby mirror. The goal is to model the reflected rays in term of the incident
field coming from the absorber and the shape of the mirror. The inversion consists then in two
steps: i) image the mirror in terms of the reflected rays, ii) find the best shape for the mirror
by prescribing the reflected rays with some conditions that maximise the number of reflections
from the mirror to the general receiver’s location (sun’s location). Let us specify our problem.
4.1
The mathematical model
The incident field
Let us denote by X = (x, y) any point in R2 . We consider the simple scalar wave equation to
model the wave propagation. We make the following assumption.
Assumption 4.1. We assume that the mirror is well separated from the region where the sensors
are located and that in the intervening region c(x) = c0 , where c0 is the (assumed constant)
speed of light in air.
The above assumption is reasonable since the sensors location coincide with the sun’s location.
The incident field due to a delta source located at (0, 0) at time t = 0 satisfies
1
2
∇ − 2 ∂t Uin (t, X) = −δ(X)δ(t),
c0
(4.1)
therefore, at the point X, at the time t′ , it is simply the Green’s function G0 associated to the
wave operator appearing on the left hand side of (4.1)
Uin (t′ , X) = G0 (t′ , X) =
The reflected field
We start by giving two definitions.
195
δ t′ −
|X|
c0
4π|X|
.
DEFINITION 4.1. We denote by M = {(x, f(x)) | x ∈ R} the graph of some function f : R → R
which represents the shape of the mirror.
DEFINITION 4.2. We denote by R(θ) = (Rx (θ), Ry(θ)) the location of the receiver (the sun) at
the angle θ from the ground.
If c0 is the speed of light in the air introduced above, we assume that the perturbation in wave
speed c takes the form
V(X) :=
1
1
= Ṽ(x)δ(y − f(x)),
− 2
2
c0 c (X)
(4.2)
where V is the reflectivity function and it represents the change in speed of the ray when it hits
the mirror at the point X = (x, f(x)). We write the total field U as the sum
U = Uin + Urf ,
where Urf represents field reflected from the mirror. The total field U satisfies
∇2 −
1
∂t U(t, X) = −δ(X)δ(t)
c2 (X)
(4.3)
By combining (4.1) and (4.3) we obtain
∇2 −
1
∂t Urf (t, X) = −V(X)∂2t U.
c2 (x)
(4.4)
Linearised problem
We rewrite (4.4) in integral form and make use of the Born Approximation (single scattering
approximation) to replace the total field U on the right hand side of (4.4) with the incident
field Uin
rf
U (t, θ) = FV(t, θ) =
Z
e−iω(t−(|R(θ)−X|+|X|)/c0 ) 2
ω V(X)dω dX.
(4π)2 |R(θ) − X||X|
(4.5)
This approximation linearises the problem because the product of unknowns VU on the right
hand side of (4.4) is replaced by the product of the unknown V with the known incident field
Uin . If we denote by
196
A(X, θ, t, ω) :=
ω2
,
(4π)2|R(θ) − X||X|
we obtain
rf
U (t, θ) = FV(t, θ) =
Z
e−iω(t−(|R(θ)−X|+|X|)/c0 ) A(X, θ, t, ω) V(X)dω dX,
(4.6)
where ω denotes the angular frequency and F is the forward map which maps the reflectivity
function into the reflected ray collected at different location θ and different travel time t.
Remark 4.2. Notice that the support of V, supp(V) satisfies
supp(V(x, y)) ⊂ {(x, y) ∈ R2 | y = f(x)},
i.e. finding V is equivalent of finding the curve y = f(x).
4.2
Inversion
Microlocal analysis
F is a Fourier Integral Operator (FIO) given by the oscillatory integral (4.6) with phase function
h
i
φ(θ, t, x, ω) = −ω t − (|R θ − X| + |X|)/c0
(4.7)
FIOs are well-studied operators and the imaging scheme planned to be used here will be based
on FIO theory, which is part of microlocal analysis [6, 7, 8]). Singularity in the scene are
mapped into singularity in the data by the so-called canonical relation of F, ΛF
ΛF =
(θ, t; σ, τ), (X, ξ) |
Dω (θ, t, X, ω) = 0
σ = Dθ φ
τ = Dt φ
ξ = DX φ .
For F given by (4.6) we have
197
(4.8)
ΛF =
(θ, t; σ, τ), (X, ξ) |
1
(|R(θ) − X| + |X|)
c0
τ = −ω
τ \
− X · Ṙ(θ)
σ = R(θ)
c0
τ \
b .
ξ = (R(θ)
− X − X)
c0
t=
(4.9)
The first condition in (4.9) is a travel time condition
t2 = |R(θ) − X|2 + |X|2
= (h cos(θ) − x)2 + (h sin(θ) − f(x))2 + x2 + f2 (x)
and it represents the set of points of intersection between the ellipse with foci the sun’s location
R(θ) and the absorber’s location (0, 0) with the curve y = f(x) we want to reconstruct. The
second condition of (4.9) (σ-condition)
σ=
τ \
R(θ) − X · Ṙ(θ)
c0
gives us the correct point we want to image X and its reflection across the velocity field Ṙ(θ):
this is not relevant since the mirror is obviously located below the R(θ), therefore this condition
will not introduce artefacts in the image.
Formation of the image
We use a filtered back projection operator to form the image, i.e. we apply a filtered adjoint of
the forward operator to the data: this operator has the same phase as the adjoint of F, with
the filter to be chosen later. The image of X is given by
Z
I(X) := eiω(t−(|R(θ)−X|+|X|)/c0 ) B(X, θ, t, ω)Urf(θ, t)dωdθdt.
(4.10)
If we combine (4.10) together with (4.6) we obtain
I(X) =
Z
K(X, X′)V(X′ )dX′ ,
where
198
(4.11)
′
K(X, X ) =
Z
′
e−iω((|R(θ)−X|+|R(θ)−X |)/c0 ) A(Z, θ, |X′ − R(θ)|, ω)B(X, θ, |X − R(θ)|, ω)dωdθ.
The kernel K(X, X′) is the imaging point spread function and it arises because it is the image that
would result from a delta function located at the point X′ . If we perform a large-ω stationary
phase analysis of K, it turns out that the main contribution to K does indeed arise from the
point X = X′ as predicted in subsection 2.1. This is quite a well known machinery that have
been used in the community working on this kind of problem [9, 10, 11, 12, 13, 14]).
By making use of the above method we can therefore obtain an image of the mirror y = f(x) in
terms of the reflected rays. The work that needs to be done in the future is the understanding
of how optimising the shape of the mirror would influence the general reflected ray Urf (θ, t).
We believe that an approach to investigate should be the one of maximising the number of
possible travel times of the total field arriving at the same location θ.
5
One reflection construction
In this section the shape of the mirrors is determined to maximise the amount of energy collected
by the energy collector for a given width of the aperture. The optimisation is carried out by
considering direct illumination and first reflections through the entire day. The position of
the collector is determined first, then the amount of reflected energy is calculated. Finally, a
simple optimisation algorithm is used to determine the shape of the mirror. The geometry is
determined using the following assumptions:
• The length of the aperture is fixed,
• The collector is a cylinder of given radius,
• The device is optimised for a flat position,
• The position of the collector is maximised for a Sun angle of incidence θ0 < θ < π − θ0 ,
where θ0 is a small given angle.
5.1
Position of the collector
To maximise direct energy hits, the collector should be positioned as high as possible above the
roof surface. However, if placed too high, a collector will create shadow on the next element
and limit its efficiency. A typical configuration is shown on Figure 6.
199
2R
h
θ1
θ2
2L
Figure 6: Position of the collector.
A simple geometry calculation shows that the height H of the centre of the collector may be
expressed as:
h1 =
L sin θ1 + R
,
cos θ1
h2 =
L sin θ2 − R
cos θ2
where L denotes half the length of the aperture and R is the radius of the collector cylinder. If
the collector is positioned to maximise angles θ0 < θ < π − θ0 , as is obvious from the figure,
the shadow of the cylinder on the next element, described by the angle θ2 , will be the limiting
factor and the height of the cylinder should be:
h=
5.2
L sin θ0 − R
.
cos θ0
(5.1)
First reflection
The shape of the mirror will now be defined. To start with, the energy reflected by one element
of mirror will be calculated and then a simple optimisation algorithm will be developed.
Energy reflected
We consider a cylindrical absorber of radius R placed above a reflector having some arbitrary
shape extending between edges A± with the coordinates (±d, h). The problem is supposed to
be invariant by translation along the absorber axis which allows us to consider the problem in
a 2D geometry. We use the Cartesian coordinate system (O, x, y) as shown on Figure 7.
The Sun position is given by the angle θ between the horizontal and the line going through
the Earth and the Sun centres. The Sun is assumed to have a uniform circular motion so that
there is an equivalence between the variable θ and the time. For a general point M taken on
200
the reflector surface, we denote by (xM , yM ) its coordinates in our coordinate system. , τ the
tangent vector oriented from A− to A+ , and ν the normal vector that is τ rotated by π/2.
Finally we call α the algebraic angle between the x-axis and the vector τ.
S
y
θ
−d
R
xM
d
O
−A
x
A
h
ν
yM
α
M
τ
Figure 7: Problem statement.
A point M of the mirror will be illuminated by the for angles of Sun verifying
(5.2)
θ ∈ (β− , β+ ) \ (γ− , γ+).
where all angles are defined on Figure 8. This corresponds to the red area. The four angles
may be expressed as function of the point position and other geometry parameters:
!
±d − xM
β± = arccos p
,
(±d − xM )2 + (h − yM )2
!
!
xM
R
γ± = π − arccos p 2
± arctan p 2
.
2
xM + yM
xM + y2M
The absorber will collect energy from M when M is lit and the Sun rays are reflected on the
absorber. A necessary condition for the Sun rays to be reflected on the absorber by M is that
θ ∈ (γ−′ , γ+′ ).
where
γ±′
= 2α + arccos
xM
p
x2M + y2M
!
± arctan
p
R
x2M + y2M
!
.
This is illustrated by Figure 9. Combining this result with (5.2), the absorber will collect solar
energy from M exactly for
θ ∈ IM := (β− , β+) \ (γ− , γ+) ∩ (γ−′ , γ+′ ).
201
S
y
θ
−d
R
xM
d
O
−A
ω
β+
ν
γ0
γ+
β−
M
x
A
h
γ−
yM
α
τ
Figure 8: Enlightenment of M.
S
y
O′
θ
γ0′
ω
−d
R
xM
d
O
−A
ω
β+
ν
β−
M
h
γ+
γ−
x
A
γ0
yM
α
τ
Figure 9: Image of the absorber by the reflector.
5.3
Optimisation algorithm
The mirror will now be determined using the following algorithm:
• The mirror is (numerically) divided into planar elements of width ds,
• The algorithm starts at the extremity of the aperture,
• Each element is positioned to maximise the energy reflected towards the collector.
202
50
Depth (mm)
0
-50
-100
-150
-200
-200 -150 -100 -50
0
50
Length (mm)
100 150 200
Figure 10: Mirror shape.
The energy reflected by an element of size ds inclined at an angle α to the horizontal as defined
on Figure 9 is:
Z
dE = P
sin (θ − α)dθ ,
θ∈IM
where P is the energy radiated in the normal cross section of the solar beam. The value of dE
should be maximised for each mirror element with respect to the inclination angle α. Note this
parameter appears in the integral but also in the limits of the interval IM .
Figure 10 shows the position of the collector and the shape of the mirror for a collector radius
R = 24mm and an aperture of 440mm. As could be expected, most of the collector is located
outside the mirror and this guarantees maximum direct energy hits for most of the day. The
mirror shape resembles a parabola but its width differs by up to 20%. The shape is about
200mm deep. This is far from ideal from a building perspective but the model only includes first
reflections. The shape of the mirrors should be significantly modified when further reflections
are considered, particularly at the vertical of the collector and this would certainly reduce
the depth of the mirror. The shape is also dependent on the optimisation method. Another
optimisation method could lead to a different mirror shape. Further investigations are required
to complete the study.
6
Optimisation
In this final section, the mirror shape is optimised for a device with a circular absorber located
in Limerick, at latitude 50◦ N40’. The concentrator is assumed to be horizontal, aligned eastwest and polar mounted, i.e. tilted south by the latitude angle. For each shape, the annual
energy collected by the device is calculated using the theory in Section 2 and the ray tracing
algorithm in Section 3, which determines the transmission function, T (ξ). Optimisation of
203
the mirror shape was performed using a swarm optimisation algorithm [15] coupled with basic
Monte Carlo methods, and the depth of the device was constrained to be less than or equal
to the typical depth of a frame installation. The algorithm was initialised with a given shape
of the mirror, such as a parabola, involute or any other adequate geometry, and specified by a
specific number of points. The optimisation process is extremely time consuming as the annual
energy collected must be recalculated for every change in the shape, which involves recomputing
the transmission function using the ray tracing algorithm at every step. Hence in practice, the
mirror shape was only defined by 20 points. The results presented in this section are calculated
for the standard element width of 440mm. The figures are scaled with the collector radius; 1
unit on the figure corresponds to 24mm. The designs are evaluated using the four following
parameters:
• OOD is the Output Of Driver2_line is the ratio the energy collected and the actual energy
available in the aperture.
• EPA is the Energy Per Absorber. This may be expressed as the OOD multiplied by the
size of a frame and divided by the numbers of absorbers.
• TE stands for Total Energy. This is the product of the EPA and the number of absorbers.
In the present study, the total size of the frame is 10.
• EPC describes the total Energy Per unit Cost. This is the parameter to be maximised
and may be expressed as the Total Energy TE divided by the cost of the unit frame and
absorbers (40 euros for the frame and 20 euros for the absorber)
6.1
Example 1
In this first example, the half aperture corresponds to 10 times the radius of the collector.
There is one collector for each standard 440mm frame. The shape of the mirror may be seen
in Figure 11.
The coordinates of the points are
x
y
x
y
coordinate
coordinate
coordinate
coordinate
-10
2.6557
-4.4
-1.0434
-8.9
3.2461
-3.3
-2.0939
-7.8
1.0585
-2.2
-1.8143
-6.7
-0.7331
-1.1
-2.1132
This geometry leads to the following results
OOD
0.2955
EPA
2.955
204
TE
2.955
EPC
4.925
-5.5
-2.0026
0
-2.0893
8
Depth (ND)
6
4
2
0
-2
-10
-5
0
Length (ND)
5
10
Figure 11: Collector 1.
6.2
Example 2
The half aperture corresponds now to 5 times the radius of the collector. There are two
collectors for each standard 440mm frame. The shape of the mirror may be seen in Figure 12.
4
3
Depth (ND)
2
1
0
-1
-2
-4
-2
0
Length (ND)
Figure 12: Collector 2.
205
2
4
The coordinates of the points are
x
y
x
y
coordinate
coordinate
coordinate
coordinate
-5
3.6882
-2.2
-0.6145
-4.4
3.6881
-1.7
-1.0438
-3.9
1.9397
-1.1
-1.2990
-3.3
0.8423
-0.6
-1.2802
-2.8
-0.0158
0
-1.0180
This geometry leads to the following results
OOD
0.5974
6.3
EPA
2.987
TE
5.974
EPC
7.4675
Example 3
In this example, the half aperture corresponds to 3.3 times the radius of the collector. There
are three collectors for each standard 440mm frame. The shape of the mirror may be seen in
Figure 13.
2.5
Depth (ND)
2
1.5
1
0.5
0
-0.5
-1
-1.5
-3
-2
-1
0
1
Length (ND)
Figure 13: Collector 3.
206
2
3
The coordinates of the points are
x
y
x
y
coordinate
coordinate
coordinate
coordinate
-3.3
2.7680
-1.5
-0.8910
-3
1.3661
-1.1
-1.0479
-2.6
0.4321
-0.7
-1.0910
-2.2
-0.1994
-0.4
-1.0156
-1.9
-0.6211
0
-1.3600
This geometry leads to the following results
OOD
0.7189
6.4
EPA
2.396
TE
7.189
EPC
7.189
Example 4
In this final example, the half aperture corresponds to 2.5 times the radius of the collector.
There are four collectors for each standard 440mm frame. The shape of the mirror may be seen
in Figure 14.
3
2.5
Depth (ND)
2
1.5
1
0.5
0
-0.5
-1
-3
-2
-1
0
1
Length (ND)
Figure 14: Collector 4.
207
2
3
The coordinates of the points are
x
y
x
y
coordinate
coordinate
coordinate
coordinate
-2.5
2.4792
-1.1
-0.8888
-2.2
1.1390
-0.8
-1.0460
-1.9
0.1641
-0.55
-1.0977
-1.6
-0.3419
-0.3
-1.0092
-1.4
-0.6071
0
-1.1762
This geometry leads to the following results
OOD
0.7361
EPA
1.840
TE
7.361
EPC
6.314
The EPC is maximum for two or three collectors per standard frame. The length of the aperture
should therefore be reduced to maximise the cost efficiency of the device when a shallow depth
is imposed.
7
Conclusion and future work
This reports details several aspects of the modelling problem submitted by Erin Energy. The
solar radiation was studied and a ray tracing algorithm was developed. Several techniques were
then presented to calculate the shape of the device. The optimisation process shows that wide
(and shallow) reflectors are not as energy efficient as smaller devices: for the standard 440mm
length, two to three devices should be more economically efficient than a single optimised device
when the depth of the mirror is forced to remain shallow.
Future work could be done in the following directions:
• The optimisation should be performed much more systematically. The present study
limited the research to 20 points in the last section, many more should be used.
• The optimisation process was initialised using a limited number of shapes. More should
be considered, such as the shapes calculated using the inverse method or the one reflection
approach.
• Finally the optimisation should be performed for other orientations of the concentrator,
e.g. with north-south alignment, and inclined by the angle of pitch of a typical roof.
Acknowledgements
The contributions of Giles Richardson and Galin Ganchev are here kindly acknowledged. All
contributors would like to thank Mr Sean Hoolan from Erin Energy Ltd for introducing the problem and assisting in answering questions during the entire week. J.P.F. Charpin, R. Gaburro
208
and S. Soussi acknowledge the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland mathematics
initiative grant 06/MI/005.
209
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