Computational Heat Transfer
and Fluid Flow
A lecture at the
Asian Symposium ASCHT-2007
Oct 17-21, 2007
Xian, China
CFD: Progress and
Prospects
by
Brian Spalding,
of CHAM, Ltd
Computational Heat Transfer
and Fluid Flow
1. Introduction
1.1 Purpose
Oct 17-21, 2007
Xian, China
Computational fluid dynamics started half a century ago. In this
lecture, I review its progress and seek to indicate how it may
profitably develop further.
I direct my words to research
workers seeking problems which it
is possible and beneficial to solve.
I
address
also
engineers,
especially
those
working
in
process
industries,
whose
designs can be improved if the
indicated developments are carried
out.
Computational Heat Transfer
and Fluid Flow
1.2 Patterns of analysis
Oct 17-21, 2007
Xian, China
The problems facing applied science are multi-dimensional; and
they can be approached in various ways.
The main dimensions of variation are in:
• time,
• space, and
• population (to be explained below).
Variations in time are easiest to handle,
because we all grow older at the same
rate: one day per day.
Variations in space are more complex,
but easy to understand; for some of us
can run faster than others.
Computational Heat Transfer
and Fluid Flow
1.2 Patterns of analysis
Oct 17-21, 2007
Xian, China
Variations in population?
Here is a one-dimensional histogram
representing the distribution of the
age of persons for a particular
community at a particular time;
and here is a picture to show that
histograms can be two-dimensional.
Populations which are relevant to CFD include those of:
•
•
•
•
liquid droplets with differing diameters;
solid particles with differing velocities;
gas ‘fragments’ with differing compositions, or temperatures; and
radiation fluxes with differing directions.
Computational Heat Transfer
and Fluid Flow
1.2 Patterns of analysis
Oct 17-21, 2007
Xian, China
I shall further distinguish the three main approaches to nonuniformity, whether in time, space or population dimensions,
namely:
• neglect,
• presume,
which means in effect, guess, and
• calculate;
and I shall argue that, in respect of calculation, the methods which
are used for spatial variations can be applied to population
variations also.
Computational Heat Transfer
and Fluid Flow
1.2 Patterns of analysis
Oct 17-21, 2007
Xian, China
I shall not argue that 'neglect' is always bad, or that 'calculate' is
always best.
Indeed, most successful approaches are hybrid; thus:
• even the most extreme of the calculators neglect something; and
• nearly all presume rather than calculate some non-uniformities.
What is necessary is to make wise decisions about
• what to neglect,
• what to presume,
• what to calculate, and
• when to do each.
Computational Heat Transfer
and Fluid Flow
1.3 The structure of
the lecture
In part 2, I shall explain my 3-dimension ~ 3approach classification; and I shall illustrate it by
way of examples from science and engineering.
In part 3, I shall recommend that CFD specialists
should provide:
• heat-exchanger designers with software based
on less presumption and more calculation;
• chemical-reactor operators with prediction tools
which calculate the distribution of fluid fragments
in composition space; and
• mechanical engineers with computer codes
which calculate the flow of fluids and the stresses
in solids simultaneously.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
2. Examples of engineering analysis
2.1 Piston engines; space-direction
variations
Oct 17-21, 2007
Xian, China
The steam engine
For this example, the 'neglect' approach is quite
satisfactory, because the variations of steam temperature
and pressure with position in the space above the piston
are small at any instant of time.
Computational Heat Transfer
and Fluid Flow
2. Examples of engineering analysis
2.1 Piston engines; space-direction
variations
Oct 17-21, 2007
Xian, China
Internal-combustion engines
Here the 'neglect' approach is not
satisfactory,because flames spread
slowly.
The 'presume' approach is best,
especially when flame speed or spray
burning rates are based on experimental
observations.
The 'calculate' approach, i.e. conventional CFD, is often
employed; with limited success. Why? Because
it neglects 'population' aspects of:
(1) turbulent combustion and (2) droplet vaporisation.
Computational Heat Transfer
and Fluid Flow
2. Examples of engineering analysis
2.2 Simpler turbulent flows
Oct 17-21, 2007
Xian, China
.
The plane turbulent mixing layer; non-uniformity in space
I start with the simplest of
all turbulent flows; the
plane mixing layer.
The task is to predict the angle of the wedge-shaped layer of
turbulent fluid at the edge of a jet injected into fluid at rest.
Computational Heat Transfer
and Fluid Flow
Shape functions and weighting functions
Oct 17-21, 2007
Xian, China
The 'neglect' approach is not applicable here; for nonuniformity is of the essence.
The presumed-profile approach involves:
• Guess the shapes of the velocity and effective-viscosity
profies, e.g. as sloping or horizontal straight lines
• Multiply the differential equations by weighting functions.
• Integrate across the layer analytically.
• Deduce the angle by algebra.
Advantage: quick and easy.
Disadvantage: accuracy is uncertain.
Computational Heat Transfer
and Fluid Flow
The plane turbulent mixing layer;
the Finite-Volume Method
Oct 17-21, 2007
Xian, China
The ‘calculate’ approach
(version of Patankar and myself, 1967):
• presumes only that the velocity profile is a
histogram, with unknown column heights;
• uses weighting functions of 1, i.e. none at all;
• integrates across each histogram interval;
• deduces the unknowns numerically.
This is now known as the 'finite-volume' method' (FVM),the
general form of its equations being:
value in the volume = sum for all faces of coefficient * value in
neighbour volume + sum of additional sources
wherein the coefficients express diffusion and convection.
Computational Heat Transfer
and Fluid Flow
Other steady-state turbulent jets,
wakes, plumes and boundary layers
The early days of CFD; a condensed history
The FVM was soon applied to these flows
which:
• had already been extensively studied
experimentally, and by presumed-profile
methods;
• are 'parabolic' (i.e. downstream events
do not influence upstream ones);
• therefore permitted solution by 'marching'
methods' on memory-scarce computers;
• allowed turbulence models to be tested;
• gave us confidence to extend the FVM to
recirculating,
three-dimensional,
unsteady, compressible and chemicallyreacting flows
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
2.3 Steady flow around solid bodies
immersed in fluid streams
Oct 17-21, 2007
Xian, China
Streamlined objects
Before CFD,
• aircraft design was based mainly
on a 'neglect' approach, in that the
variations of stagnation pressure
were neglected.
The aerodynamic forces on the aircraft were
then computed by way of ideal-fluid theory.
• The effects of viscosity, and indeed turbulence were expressed by
the supposition that the 'displacement thickness' of thin boundary
layers enveloping wings and fuselage made these, in effect, rather
thicker than they truly were.
• The presumption
approach was used, however, to calculate
the displacement-thickness distribution; so the whole method can be
characterised as being 'hybrid'.
Computational Heat Transfer
and Fluid Flow
Current practice
Oct 17-21, 2007
Xian, China
Now that CFD exists,
• the calculation' approach is adopted for the whole of the
space occupied by the fluid; which allows also the small
regions of 'separated flow’ to be simulated.
• However, an accurate calculation of the frictional
forces on the solid surface can be made only by the use
a very fine grid in the boundary layer;
• so, for economy, some element of profilepresumption is retained, by way of wall functions.
Computational Heat Transfer
and Fluid Flow
Flows around and inside buildings
Oct 17-21, 2007
Xian, China
• Before CFD, flow prediction was
based on experiments with small
geometrically similar physical
models;
• but this was unreliable , because
the similarity criteria of Reynolds
(viscosity) and Froude (buoyancy)
could not both be satisfied.
• Neither the neglect nor presume approaches had
anything to offer. Therefore, engineers concerned with
heating, ventilating, air-conditioning and
fireprotection of buildings were among the first to turn to
CFD.
Computational Heat Transfer
and Fluid Flow
Flows around and inside buildings
Oct 17-21, 2007
Xian, China
• CFD has satisfied their requirements; and
• it is for widely used for simulating fires in car-parks and other
buildings;
• BUT, for phenomena such as the fire-ball, it needs to take
account of variations in hot-gas-population space.
Computational Heat Transfer
and Fluid Flow
2.4 Chemical-engineering equipment
Heat exchangers; non-uniformities in space
No designer can 'neglect' the
temperature variations in heat
exchangers.
Instead, most guess them as
being similar to that calculated for
idealised counter-flow systems.
Since they know that the flow
patterns must differ, they multiply
their calculated heat-transfer rates
by correction factors like those
on the right.
But these are still guesses, none
the less.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
Heat exchangers; non-uniformities in
space (end)
Oct 17-21, 2007
Xian, China
These presumption practices derive from the pre-CFD age.
However, it was shown more than thirty years ago (by
Patankar and myself, as it happens), that the calculate
approach is practicable and indeed easy.
It is strange therefore that most heat exchangers
today are still based on presumption rather than
calculation.
Therefore, in section 3.1, below, I shall be
recommending a change of practice.
Computational Heat Transfer
and Fluid Flow
Stirred chemical reactors, showing
variations in both space and
population
The process:
Many chemicals products are
created by pumping feedstock
materials (A and B) into a reactor
vessel, where they are stirred
together by a paddle, in order to
react chemically.
The task is to predict how the rate of
production of C from reactants A and B
depends upon the power consumed by
stirring and the rate when mixed in a testtube, where: rate/(concA*concB) = k_tube .
Oct 17-21, 2007
Xian, China
Stirred chemical reactors
Computational Heat Transfer
and Fluid Flow
Variations of time-averaged concentration
Oct 17-21, 2007
Xian, China
Before CFD,
the 'neglect' approach had to be used for variations
with position; and it was not bad; for, if the stirring is
vigorous enough, the time-average values of concA and
concB will indeed be almost uniform.
But what about moderate stirring?
The 'presume' approach is not usable in this case; for no
guidance exists as to what profiles should be presumed.
Nowadays, CFD is employed; but it is not enough;
for, if R_ave / (concA_ave * concB_ave)= k_reactor ,
it is found experimentally is that
k_reactor is much less than k_tube. Why is this?
Stirred chemical reactors
Computational Heat Transfer
and Fluid Flow
Variations in population space
The answer: non-uniformity in population
space, also called unmixedness, shown here ->
At any point in the reactor, fluid fragments of
many different concentrations can be found.
To calculate their time-average values, one must
know for what proportion of time each is
present.
That means that one needs a probabilitydensity function, like this
--->
Can one calculate it? Yes, as I shall explain later;
and for each location and stirring rate too.
From it can be deduced the C- production rate.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
Furnaces and other combustors;
more variations in space and
population
General description
A coal-fired furnace is a special kind of
chemical reactor; and the processes
taking place in it present a severe
challenge to computer simulation,
because of the importance of:
• chemical reactions (coal pyrolysis,
volatilisation, combustion, NOX formation)
• solid-fluid interaction (diffusion of
oxygen to the surface);
• thermal radiation; and
• particle-wall impact.
Oct 17-21, 2007
Xian, China
Furnaces and other combustors
Computational Heat Transfer
and Fluid Flow
Variations in position and population
Oct 17-21, 2007
Xian, China
Which approach should be used for space variations?
Only the calculate approach has any hope of representing the
distributions of temperature, velocity, and pressure throughout the
volume; and it has indeed been used for many years.
And for population non-uniformity?
• Of coal-particle size: often neglected but sometimes
presumed to vary in accordance with the empirical formula
of Rosin and Rammler;
• of radiation angle: often neglected ( in conduction model)
sometimes presumed (in six-flux model) , and less often
calculated (discrete-ordinates formulation);
• of radiation wavelength: nearly always neglected;
• of gas concentrations : nearly always neglected.
To recommend calculate for all would be too ambitious.
Computational Heat Transfer
and Fluid Flow
2.5 Simpler non-uniformities in
population: droplet-size
Oct 17-21, 2007
Xian, China
Vaporization of fuel sprays (in Diesels
or gas turbines) consisting of droplets of
various diameters, D, which change size
at a rate governed by :
- dD/dT = const * (1/D) * ln(1+B)
where B, the driving force for mass
transfer, depends upon (e.g.) local
. temperatures and other gas properties.
This shows that droplets diminish in size
at different rates, the smaller ones
disappearing the more rapidly.
The task is to calculate the overall rate of vaporization.
This necessitates knowing the droplet-size distribution at each
location and each time.
Computational Heat Transfer
and Fluid Flow
Vaporization of a spray; droplet-size
population
Oct 17-21, 2007
Xian, China
The usual three ways are:
1. Neglect variations, i.e. suppose that all the droplets at a
single location in the spray have the same diameter.
2. Presume that the profile is
constant (e. g.) of RosinRammler form, which cannot
be very accurate.
3. Calculate the ordinates of the
histogram by way of a
standard finite-volume
equation, with the source term
dD/dT above.
Use calculate if droplet size is
critical, as in fire extinction.
Computational Heat Transfer
and Fluid Flow
The turbulent diffusion flame;
fuel-air-ratio population
Oct 17-21, 2007
Xian, China
Experimentally-observed unmixedness
Hottel, Weddell and Hawthorne drew attention in 1949
to the 'unmixedness' of the gases in a flame
produced by a jet of fuel gas injected into air.
They measured finite time-average concentrations of
both fuel and oxygen at the same location.
That could never be found in a laminar flame.
The first CFD analyses
It was not until 1971 that the first attempt to simulate
this unmixedness numerically was made, on the basis
of a very simple profile presumption.
The turbulent diffusion flame;
Computational Heat Transfer
and Fluid Flow
presumed fuel-air-ratio population
Oct 17-21, 2007
Xian, China
The guess was that, at a
point where the timeaverage fuel-air ratio was
F,
say,
the
gases
actually present there
had the ratio
F+ g for half the time,
and
F- g for the other half.
Standard CFD calculated F easily.
For g, a new differential equations was invented, having sources
guessed as being proportional to gradients of F- and velocity.
This approach, when appropriate empirical constants were
introduced, allowed turbulent diffusion flames to be simulated.
Computational Heat Transfer
and Fluid Flow
Confined pre-mixed flame;
reactedness population
Oct 17-21, 2007
Xian, China
In the turbulent diffusion flame,
fuel and air enter separately, and
must be mixed before chemical
reaction can occur, at a rate
limited by the rate of that mixing.
I now consider a flow in which the fuel and air are mixed before they
enter, at uniform and constant velocity, a plane-walled duct in which
is placed a bluff-body 'flame- holder'.
A turbulent wedge-shaped flame spreads across the duct, as the
sketch indicates; and the profile of longitudinal velocity is roughly as
shown.
What then limits its rate? A different kind of mixing: that between
burned and unburned gases.
Computational Heat Transfer
and Fluid Flow
Confined pre-mixed flame;
the near-constancy of its angle
When first investigated, this flame showed some
puzzling features, namely that the wedge angle
was almost independent of:
• inlet velocity
• fuel-air ratio;
• inlet temperature;
• pressure; and
• inlet turbulence intensity.
But why?
H.S. Tsien, while at CalTech, explained the
shape of the profile; but what governed
its angle remained a mystery.
We learned only later
• non-uniformity in space depends on
• non-uniformity in population.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
Confined pre-mixed flame;
the first population presumption
Oct 17-21, 2007
Xian, China
The guessed profile
The first idea, embodied in the so-called eddy-break-up model ,
was that the gas population consisted of two components, namely:
(1) fragments of wholly un-burned gas which were
too cold to burn; and
(2) fragments of hot wholly-burned gas which also
could not burn because either all the fuel or all the
oxygen had been consumed.
The histogram representing the
presumed population therefore
consisted of two spikes; and
their relative heights dictated
what would be measured as the
time-average temperature.
Confined pre-mixed flame;
Computational Heat Transfer
and Fluid Flow
collision between burned and unburned gas
fragments
Oct 17-21, 2007
Xian, China
These two elements of the population were
thought of as colliding with one another and
thereby producing sub-fragments of
intermediate temperature and composition.
These latter, being sufficiently hot and also containing
reactants, could burn; and did so very rapidly, thereby increasing
the height of the right-hand spike. Their actual concentration
was considered, implicitly, to be negligibly small.
The rate of collision per unit volume was guessed as proportional
to the rate of dissipation of turbulence energy.
This explained why the flame angle remained almost unchanged
when the inflow velocity was increased.
Confined pre-mixed flame;
Computational Heat Transfer
and Fluid Flow
the next presumed reactedness profile
Oct 17-21, 2007
Xian, China
The four-fluid model
The EBU, published in 1970, became very popular; so much so that
25 years passed before the obvious next step was taken;:
to increase the number of presumed components from 2 to 4 !
Collisions between fluids
1 and 3 created fluid 2,
2 and 4 created fluid 3,
1 and 4 created fluid 2
and also fluid 3.
Reaction of fluid 3
created fluid 4
at a chemistrycontrolled rate.
Fluids: 1
2
3
4
Computational Heat Transfer
and Fluid Flow
Confined pre-mixed flame;
applications of the four-fluid model
Oct 17-21, 2007
Xian, China
The chemistry-controlled step (fluid 3 creates fluid 4) explained:
why:
1. the flame angle remained nearly constant, and
2. the flame could be suddenly extinguished by a velocity increase.
The four-fluid model was
used successfully for
simulating flame spread in a
baffled duct and for oilplatform explosion
simulation.
It has been little used; but it
was the first step towards
calculating the reactedness
population,
Computational Heat Transfer
and Fluid Flow
From four fluids to many:
the multi-fluid model
Oct 17-21, 2007
Xian, China
In conventional CFD, we
divide space and time into as
many intervals as we need.
Why not do the same for the
reactedness at each point?
The height of each column can
then be deduced from a
Finite-Interval equation’ like this:
height of interval= sum for all faces of coefficient *
height of neighbour interval +
sum of additional sources +
sum for all other intervals of coefficient *
height of other interval )
Computational Heat Transfer
and Fluid Flow
What the terms in the finite-interval
equation represent
Oct 17-21, 2007
Xian, China
In:
height of interval= sum for all faces of coefficient *
height of neighbour interval +
the coefficients express rates of convection and diffusion, as in
the the finite-volume equations of conventional CFD.
But in:
sum for all other intervals of coefficient *
height of other interval
the coefficients express the physical and chemical processes:
• collision between members of the fluid population, and
• chemical conversion of one member into another.
The finite-interval method is thus merely a natural extension of
the finite-volume method; and its equations can be solved in the
familiar successive-substitution manner.
The calculation of population distributions is easy.
Computational Heat Transfer
and Fluid Flow
How material is distributed after
collision
Oct 17-21, 2007
Xian, China
Here is a diagram from one
of the earliest publications.
It depicts one of the possible
hypotheses, called
'Promiscuous Mendelian'.
The 'colliders' are treated as 'mother' and 'father’; and the word
'promiscous' implies that any two members of the population may
collide.
The word Mendelian, a reference to Gregor
Mendel, the Austrian "father of modern
genetics", implies that the offspring may
appear with equal probability in any
interval between those of the parents.
Computational Heat Transfer
and Fluid Flow
A calculated probability-density
function
Oct 17-21, 2007
Xian, China
This hypothesis has been embodied in
the
PHOENICS
computer
code.
Here is one reactedness histogram,
computed with its aid.
As in the the eddy-break-up guess, there
are indeed spikes at zero and unity
reactedness;
but calculation has shown that the
intervals in-between are alsopopulated.
Such probability distributions can to be computed for
each location in the flame. Then the desired reaction
rate for the whole flame can be deduced.
Computational Heat Transfer
and Fluid Flow
Application to gas-turbine
combustion
A three-dimensional gaseous-fuel combustor
I show here one sector of a simple combustor
proposed by Professor Wu Chung-Hua in the
early days of PHOENICS.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
Smoke formation rate is influenced
by turbulent fluctutions
Much later, I used this combustor to
show how one must not neglect
fluctuations of fuel-air ratio when
predicting smoke formation.
I used a 10-fluid model,
with fuel-air-ratio as the
population-defining
attribute. Each cell had
its own computed
histogram
The differences, although
small. are significant when
CFD is being used to optimise
the design.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
Concluding remarks for Part 2
Oct 17-21, 2007
Xian, China
It has been shown that:
1. variations in population space should not be neglected
especially when chemical reaction is involved;
2. they can be presumed;
3. but it is better to calculate them.
Why are not crowds of
researchers pouring into
this scarce-explored
territory?
Perhaps because they are
waiting for less-timid
crowds to do so first.
Computational Heat Transfer
and Fluid Flow
3. Recommendations
3.1 To heat-exchanger designers
Oct 17-21, 2007
Xian, China
So far, I have been discussing general ideas. Now I wish to make
three specific recommendations.
Current practice
I have already mentioned that heat exchangers are still
designed in the basis of presumption.
A shell-and-tube heat
exchanger looking like
this (tubes not shown)
can be expected to have
a rather complex flow in
the shell.
Computational Heat Transfer
and Fluid Flow
or
Oct 17-21, 2007
Xian, China
Yet the software used by designers presumes that the flow in the
shell can be conceptualized thus, and described by very few
parameters.
But why presume when one
can calculate, as was shown
to be possible by the 35-yearold publication in which this
image appeared?
Computational Heat Transfer
and Fluid Flow
3.1 To heat exchanger designers
The solution
Oct 17-21, 2007
Xian, China
The solution is:
1. do not attempt to calculate the flow pattern
between the tubes in detail, because current
computers are not large or fast enough to
handle the necessary fine grids except for a
few tubes at a time.
2. Instead, use the space-averaged approach, with empiricallybased formulae for:
 heat-transfer coefficients per unit volume, and
 friction factors per unit volume,
as functions of local Reynolds and Prandtl numbers.
3. Then solve the finite-volume equations for (space-averaged)
velocity, pressure, temperature for the shell- and tube-side fluids,
treating both as interpenetrating continua, as is easily possible.
Computational Heat Transfer
and Fluid Flow
3.1 To heat exchanger designers
The solution (contd)
Oct 17-21, 2007
Xian, China
I now show some (not new) results for (the central plane of
symmetry of) a particular shell-and-tube heat exchanger.
(a) The shell-side velocity
vectors, when calculated,
appear thus
(b) The consequential shell-side temperatures, are not, as
presumed, a succession of vertical stripes; although the calculated
tube-side temperatures are (very nearly).
Computational Heat Transfer
and Fluid Flow
3.1 To heat exchanger designers
The solution (end)
Oct 17-21, 2007
Xian, China
(c) The conventional heat-exchanger-design packages presume
that the shell-side, tube-side and overall heat-transfer coefficients
are uniform throughout; but calculation reveals that they are not,
as the next pictures clearly demonstrate.
Corresponding non-uniformities are exhibited by the calculated
Reynolds- and Prandtl-number values, and the temperaturedependent fluid properties, from which the heat-transfer
coefficients have been computed.
Computational Heat Transfer
and Fluid Flow
Recommendation number 1
Oct 17-21, 2007
Xian, China
My conclusions are…
• that the conventional presumptions are evidently incorrect;
• that therefore software which is based on them will generate
unsafe designs; and
• that the calculate approach, using experimentally-based data for
the space-averaged heat-transfer and friction coefficients, is the
only sound basis for the design of heat-exchanger equipment.
I declared at the beginning that I had something to say to
engineers.
This first recommendation is addressed to them:
Demand that the suppliers of your heat-exchangerdesign software build into it the
calculate approach.
Computational Heat Transfer
and Fluid Flow
3.2 To stirred-reactor designers and
operators
Oct 17-21, 2007
Xian, China
The calculation required by my first recommendation concerned
non-uniformities in space. There are therefore many CFD
specialists who will know how to implement it.
My second concerns non-uniformities in population; experts in
these are harder to find.
The task
is to predict how stirring-rate influences the conversion rate of
reactants A and B into C in reactors of the kind which I discussed
in Part 2.
Computational Heat Transfer
and Fluid Flow
3.2 To stirred-reactor designers and
operators (contd)
Oct 17-21, 2007
Xian, China
An example
I turn to a ten-year old work [Ref ], in order to emphasise that the
idea is not new, merely neglected. It concerns, for simplicity,
reactants for which the rate constant measured in a laboratory test
tube (i.e. k_lab) is very large.
The geometry, and the body-fitted-co-ordinate grid used in the CFD
calculation, are shown below.
Computational Heat Transfer
and Fluid Flow
3.2 To stirred-reactor designers and
operators (contd)
Oct 17-21, 2007
Xian, China
But what about the mixture-ratio population grid?
Two distinct cases were considered, namely that:
1. the materials from the entering streams of reactants A and B
were fully mixed at each point in the reactor, which would
correspond to presuming
•
that its pdf was the single spike shown on the following
diagram, and that
•
the amount of product C was as indicated by its horizontal
location;
2. alternatively, at each point there
could be found varying amounts
of 'fluids' (in the multi-fluid sense)
having one of eleven distinct
mixture ratios, so that its pdf
could be that of the histogram
Computational Heat Transfer
and Fluid Flow
3.2 To stirred-reactor designers and
operators (contd)
Oct 17-21, 2007
Xian, China
Case 1 is the conventional-CFD approach which presumes the
state of the mixture-ratio population; and Case 2 represents what is
done by those who recognise that non-uniformities in population
space can be calculated.
The results of the two
approaches are different. This
is demonstrated by the
following
two
contour
diagrams showing the product
(i.e. C) concentrations after 10
revolutions.
The general patterns are not very dissimilar; but their scales are:
3.2 for the presumption approach and only
2.4 for the calculation approach, at this moment of time.
Computational Heat Transfer
and Fluid Flow
3.2 To stirred-reactor designers and
operators (contd)
Oct 17-21, 2007
Xian, China
The explanation for the difference is to be found in the calculated
mixture-ratio histograms, of which a few will be shown,
corresponding to a single instant of time, a single vertical height and
circumferential angle, and at six different radii, starting near the axis
and moving outward.
These pdf histograms show that:
• detailed information about the micro-mixing can indeed be
obtained by calculation;
• the pdfs vary is shape in a manner that it would be impossible to
guess;
Computational Heat Transfer
and Fluid Flow
3.2 To stirred-reactor designers and
operators (contd)
Oct 17-21, 2007
Xian, China
• their shapes are utterly unlike the single spike which neglecting
the micro-mixing implies;
• they will assuredly imply different mixture-average product
concentrations.
Inspecting them may lead to questions such as:
• Did these calculations consume much computer time?
Answer: about the same as did the hydrodynamic calculations.
• Was eleven intervals too few? Or too many?
Answer: One has to repeat calculations with finer and coarser
'population grids' to find out, just as for spatial grids.
• Is the ratio of 3.2 to 2.4 typical?
Answer: No; values much closer to and farther from unity can be
encountered.
• Do the predictions agree with experiment?
Answer: I expect so, qualitatively; but no serious investigations
have yet been made.
Computational Heat Transfer
and Fluid Flow
Recommendation number 2
Oct 17-21, 2007
Xian, China
My second recommendation therefore, to
researchers and to their engineering managers is:
• waste no more time on CFD simulations of stirred
reactors unless they calculate the fluid population;
• do not wait for the necessary physical constants to be
accurately determined; for even approximate ones will
be better than the current neglect
of the micromixing phenomenon.
Computational Heat Transfer
and Fluid Flow
3.3 To researchers and engineers
concerned with fluid-solid
interactions
Oct 17-21, 2007
Xian, China
A historical accident
In section 2.2.1, I mentioned 'weighting functions'; and I said that
the finite-volume method uses unity as its weighting function
whereas the finite-element method uses something else.
This is came about because the FEM originators, R.Clough and
O.Zienkiewicz, were mainly concerned with stresses and strains in
solids; and, in that field, methods using weighting functions, such as
those of Galerkin had a long pre-computer history.
R.Clough
O.Zienkiewicz
Galerkin
Computational Heat Transfer
and Fluid Flow
A missed opportunity and its
consequences
Oct 17-21, 2007
Xian, China
When computers arrived, stress analysts simply
carried some of their old baggage with them,
not recognising that it was no longer needed.
This tiny difference in starting point has led to enormous
differences of practice and language between the stress-analysis
and fluid-flow communities; and it has given rise to
totally false ideas, namely:
1. that the finite-volume and finite-element methods are essentially
different;
2. that the FEM must be used for the calculation of solid stress; and
3. that therefore different methods and computer programs must
be used for solid-stress simulation from those which are used for
fluid flow.
Computational Heat Transfer
and Fluid Flow
What we now know ,
and should act upon
In fact, however, a weighting function of unity
works just as well for solid stress as for fluid
flow.
Therefore a single method and
a single computer program can be used for
both;
and they should be used, for economy,
whenever the problem in question involves the
interaction between solids and fluids.
I will now explain why.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
Why one method can suffice for
both classes of problem
Oct 17-21, 2007
Xian, China
The reasons are:
1.
The differential equations for velocities in fluids are very
similar to those for displacements in solids, from which the
stresses can be deduced. Thus
[del**2]* u - [d/dx]* [ p*c1 ] + fx*c2 + convection terms= 0
for velocity, and
[del**2]* U + [d/dx]* [ D*C1 - Te*C3 ] + Fx*C2 = 0
for displacement.
2. The solid-stress equations are indeed the simpler, being linear
where the former are non-linear.
3. Since the solid-stress problem is simpler than the fluid-flow one,
computer codes written for the latter can easily serve for the
former also, as many publications have proved.
Computational Heat Transfer
and Fluid Flow
A thermal-stress example
Oct 17-21, 2007
Xian, China
The three examples which I shall show are several years
old; for I wish to emphasise that my message is not a new
one. But it has suffered from neglect.
First, a cooling fluid flows through
a pressurised curved duct in
a solid block.
The block is heated at
various several points so
that its thermal expansion
is non-uniform.
Computational Heat Transfer
and Fluid Flow
Thermal and mechanical fluidstructure interactions
The equations for velocity and
displacement and velocity are so similar
that PHOENICS solves both sets at the
same time. Here the solutions are
presented in terms of vectors.
In my second example, the fluidstructure interaction is mechanical
rather than thermal. A thin partition
bends as a consequence of the
differences of fluid pressure on
its two sides.
Oct 17-21, 2007
Xian, China
Computational Heat Transfer
and Fluid Flow
A periodic fluid-structure interaction
Oct 17-21, 2007
Xian, China
The final example is also one of mechanical interaction.
It shows the transient deflection of
an under-water structure under
the influence of the wave motion
of the ocean.
PHOENICS computes the
displacements in the solid and
the velocities of the fluid
simultaneously, as a single set
of vectors.
A question worth asking.
Why, since these and many other examples have been available for
many years, do most vendors still offer separate software packages
for the calculation of fluid flow and solid stress?
Computational Heat Transfer
and Fluid Flow
Why do the false ideas persist?
Oct 17-21, 2007
Xian, China
Neglect of the evidence plays a part.
Presumption that what is done must be done contributes.
But if anyone were to calculate the cost of current practices,
surely the argument for change would become irresistible.
Perhaps the laws of the
market will respond in
the long run; but it is
taking
a long time.
In the meantime… I will
let this picture speak for
itself.
Computational Heat Transfer
and Fluid Flow
Recommendation number 3
Oct 17-21, 2007
Xian, China
My third recommendation is therefore that:
• Researchers should develop and refine the finitevolume method for simultaneous fluid-flow and solidstress calculation;
and
• Engineers concerned with fluid-structure interactions
should demand computer codes which embody
those methods.
Computational Heat Transfer
and Fluid Flow
The last slide
With thanks for your attention
Oct 17-21, 2007
Xian, China
The message of this lecture has been that
the world of CFD is wider
than most of its inhabitants conceive.
Time and space form only four of its dimensions.
Others include:
• reactedness & fuel / air ratio of gas fragments
• size temperature composition velocity of particles
• angle and wavelength of radiation.
Populations
must be
considered,
They should
seldom be
neglected
and probabilitydensity functions
employed, 1D or 2D.
They may
sometimes
be guessed
But the
best is
calculation