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Extending the Boundaries of Heat Transfer
by
Brian Spalding
The 13th International Heat Transfer Conference
August 16, 2006, Sydney, Australia
James P.Hartnett Lecture
Abstract
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In keeping with Jim Hartnett's breadth
of vision, and of his readiness to be
controversial, this lecture questions
some common assumptions about the
subject of Heat Transfer.
Specifically, it is argued that:
1. Heat Transfer and its effects is our
proper field of study.
Abstract
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2. Among the not-to-be neglected effects are the
resulting Stresses in Solids.
3. Numerical-Heat-Transfer techniques require
corresponding extension to displacements and
stresses, but without the needless complications
of finite-element methodology.
4. CFD ( i.e. Computational Fluid dynamics ) requires
extension to SFT ( i.e. Solid-Fluid-Thermal analysis ,
for which its finite-volume methods are fully
sufficient.
Abstract
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4. Heat-exchanger designers should move from guess-theflow-pattern to compute-the-flow-pattern methods.
5. But conventional (detailed-geometry) CFD techniques are
inadequate for this; only space-averaged formulations
are practicable.
6. Still, data-input obstacles remain formidable. Heatexchanger designers need software which can:
(a) understand formulae, and
(b) accept data in the form of relations.
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Contents of this lecture
Part1
1. What is 'Heat Transfer'?
1. The received view
2. Reasons for enlargement
3. Some details by way of example.
2. Extending numerical heat transfer
1. Conventional methods for heat conduction
2. Simple extensions to chemical reaction
3. Extensions to displacements in solids
4. The research opportunities
3. CFD to SFT
1. Essential ideas
2. A simple example
3.A choice to be made
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Contents of this lecture
Part 2
4. How not to design heat exchangers
1. What the Handbooks Say
2. Can CFD assist?
3. Why conventional packages fail to satisfy
5. Improving the input procedures
1. Input of formulae
2. Input of relations
3. Optimization
6. Concluding remarks
7. Acknowledgements
8. References
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What is ‘Heat Transfer’?
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1. What is ‘Heat Transfer’?
1.1 The received view
The conventional answer to this question is given by the
chapter headings in the popular textbooks; they follow
the century-old pattern set by Nusselt and Jakob in
Germany.
1. conduction;
2. convection;
3. radiation;
then perhaps:
4. melting and freezing.
5. boiling;
6. condensation.
What is ‘Heat Transfer’?
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But it need not have been so; for
• the action-at-a-distance laws of radiation are unlike the
close-contact laws of conduction and convection;
they might have been rtreated as belonging to optics; and
• the phase-change topics (melting, freezing, etc) might
have been left to thermodynamicists;
they concern more the effects of heat transfer than the
process itself.
Conversely, if some of the effects of heat transfer are to be
included, why not others? for example:
• ignition and extinction of flames? or
• stresses in solids?
They are surely of sufficient practical importance.
The argument
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The existing boundaries of the subject of Heat Transfer are
historical rather than rational.
In the 1960s we added Mass Transfer to our territory, as
witness:
• IJHMT, the Journal published by Robert Maxwell, of which
AV Luikov, Jim Hartnett and I were editors at launch time.
• ICHMT, the Centre proposed by Naim Afgan and Zoran
Zaric and created with help from Jim and me.
I shall argue that it is time to extend the boundaries
further, so as to cover:
HMT and its chemical and mechanical effects.
Reasons for enlargement
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Reason 1
Heat transfer is for engineers, who design
equipment; and this must both:
• meet performance requirements, and
• ensure safety.
They must therefore predict both the desired and
the undesired consequences of their actions.
Examples are:
 chemical effects (explosions)
and
 mechanical effects (distortions and fractures).
Reasons for enlargement
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Reason 2
The necessary additional ideas are few, namely:
• that combustion phenomena result from temperaturedependent heat sources;
• that thermal stresses occur when heated bodies are
mechanically constrained;
• That stress is proportional to strain (Hooke's Law);
Heat-transfer engineers need not, however, become chemists
or metallurgists;
they need just enough extra knowledge, but no more.
Reasons for enlargement
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Reason 3
Not equipping the heat-transfer engineer with the
necessary skills is:
• at best, uneconomical, and
• at worst, dangerous.
The alternative, calling in specialists is expensive,
time-consuming, and sometimes too late.
They speak different languages; and
misunderstandings are frequent.
1.3 What heat-transfer engineers
should know about combustion
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Flame-propagation speeds of fuel-air mixtures vary thus:
Experimental combustion data can then be
correlated thus:
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This is from the 1954 thesis of Barry Tall, my first Australian student
1.4 What heat-transfer engineers
should know about stress analysis
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:
• That the three material properties of importance are:
– Young's modulus,
– Poisson's ratio,
– thermal expansion coefficient.
• That (a few) formulae exist for stresses and strains in
solids when the boundary conditions are simple.
• Otherwise, numerical methods of calculation are
available.
• These can be of the 'finite-difference' or 'finite-volume'
kinds, familiar from studies of heat conduction;
• There is no need to learn the 'foreign language' associated
with 'finite elements'.
2. Extending Numerical Heat
Transfer
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2.1 Numerical methods for heat conduction
Analytical formulae exist only for heatconduction problems which are simple in
respect of:
 geometry (rectangular, cylindrical or
spherical),
 boundary conditions (constant, or linear in
temperature),
 material properties (uniform);
but these conditions prevail so seldom that
numerical methods are almost always used for
calculating temperature distributions.
Extending Numerical Heat
Transfer
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The figure and
equation shown
here will be
familiar to all
users of such
methods.
How to solve the equations
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There is one such equation for every volume into
which the space is divided.
The complete set of equations is soluble by
successive-substitution methods.
Before we had computers, the graphical method
pioneered by Ernst Schmidt was often used.
It was laborious, but profoundly educative.
I luckily encountered it early in my career as shown
by the following ‘reminiscence’. It concerns one
of the chemical effects of HMT, namely flame
propagation.
2.2 Numerical heat transfer
with chemical reaction
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I used the Schmidt method for calculating the speed of
laminar flame propagation, 50 years ago.
Extending Numerical Heat
Transfer
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The graphs on the left show successive temperature
distributions after two bodies of hot (burned) and cold
(unburned) gas are brought into contact
Extending Numerical Heat
Transfer
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The graph on the right shows the source (horizontal) versus
temperature (vertical) function which represents (sufficient
of) the laws of chemical reaction.
Extending Numerical Heat
Transfer
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When computers came along, of course, pencils and rulers
were pushed aside; but I am glad that I started work before
then.
I would want every student in my imagined "HMT and Its
Effects" course to have 'flame ignition and propagation' as
an obligatory homework item.
2.3 Extension to
displacements in solids
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The numerical methods used for heat-conduction problems
can also be extended to the calculation of stresses and
strains in solids.
There are many ways of doing so; but probably the simplest
is to solve the equations for the displacement
components.
The Figure and Equation shown below are a little more
complex than those for temperature; but not much.
Control-volume for vertical
displacement v
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First the Figure
Extending Numerical Heat
Transfer
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then the equation
The slight complication of the displacement-component
problem is that there are three sets of equations ( for U,
V and W); and they are linked together in special (but
easily-formulated) ways.
Solving the equations
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I now show some results of solving the
equations by the same successivesubstitution method as is used for heat
conduction.
It is applied to the case of a square-sectioned
beam having a square hole, filled with
fluid, along its axis.
Contours and vectors of displacement are
shown.
1/4 of square beam with fluid
in square hole
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When the outer-wall temperature is raised;
1/4 of square beam with fluid
in square hole
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When the inner-duct pressure is raised;
1/4 of square beam with fluid
in square hole
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When both changes are made simultaneously.
Consequential stresses
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From the
displacement
fields may be
deduced the
distributions of
the direct
stresses in the
horizontal
direction...
Consequential stresses
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… and in the vertical direction.
Comparison with
solutions made
by the finiteelement code
Elcut showed
close agreement,
of course;
for the finitevolume and finiteelement methods
solve the same
differential
equations.
2.4 The research
opportunities
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The computer time needed for solving the 3 displacement
equations is more than 3 times that needed for the temperature
equation.
The reason is that the equations for the 3 displacement
components are inter-linked.
Naive sequential solution procedures may (depending on geometry)
converge rather slowly.
More refined procedures are needed, and are being developed;
but there is still much to do.
Researchers seeking little-exploited territories may therefore find
them here; and the world still awaits compilation and publication
of the definitive textbook.
Why? The numerical-stress-analysis field was devastated in the
1960's by the finite-element tsunami. Recovery takes time.
3. Extending Computational
Fluid Dynamics to SFT
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3.1 Essential Ideas
When Numerical Heat Transfer concerns
itself with convection as well as
conduction, it becomes a part of CFD..
This also came into existence in the late
1960s.
It uses equations similar to those governing
heat conduction, shown above, with
additional features, namely:
The additional features of
the CFD equations
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 the dependent variables include the components of
velocity;
 the coefficients (aN, aS, etc). account for convective as
well as diffusive interactions between adjacent control
volumes;
 the sources include pressure gradients, gravity, centrifugal
and Coriolis forces; and
 the effective transport properties vary with position over
many orders of magnitude.
The CFD equations is thus more complex than the
thermal-stress problem; yet satisfactory iterative solution
procedures have been in widespread use since the early
1970s.
Use of CFD procedures for
solid-stress problems
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•
CFD solution procedures have been successfully applied to solid-stress
problems. Both Steven Beale and I independently showed this in 1990,
as did Demirdzic and Mustaferija soon after.
•
Mark Cross's group at Greenwich University has also made significant
use of such methods for fluid-solid-interaction problems.
•
Since the fluids and the solids occupy geometrically separate
volumes, a single computer program can predict the behaviour of
both solids and fluids simultaneously.
•
This possibility has not been widely exploited because of the popular
misconception that solid-stress problems must be solved by finiteelement methods.
•
It is therefore high time that CFD should enlarge to become SFT, i.e.
Solid-Fluid-Thermal.
3.2 A simple example
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Let us consider a primitive counterflow heat exchanger,
consisting of two concentric tubes.
Let us also suppose that because of:
• natural convection in the cross-stream plane, or
• non-uniformity of external surface temperature, or
• turbulence-promoting baffles within one or both of the
tubes ,
the distributions of temperature and pressure, and therefore
also of stress and strain in the tubes, are not
axisymmetrical.
The concentric-tube heat
exchanger
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How are the stresses and strains to be computed?
Numerically, of course; and, if (misguided !) common practice
is followed, one computer code will be used for the
fluids and another for the solids.
Then means must be devised for transferring information
between them.
How much more convenient it will be to use one computer
code for the whole job!
Extending CFD to SFT
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A true SFT code can do just that by:
 solving for velocities and pressure in the space occupied
by fluid;
 solving for displacements and strains in that occupied by
solid;
 solving simultaneously for temperature in both spaces.
The following images relate to the heat exchanger in question,
with the radial dimension magnified four-fold.
Concentric tube heat
exchanger
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1. Pressures in the two fluids causing mechanical stresses;
Concentric tube heat
exchanger
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2. The temperature distribution, causing thermal stresses.
Concentric tube heat
exchanger
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The circumferential variation of temperature imposed on the
outer surface has produced 3D variations of
temperature, stress and strain, as follows:
3. radial-direction
strains (positive
being extensions,
negative
compressions);
Concentric tube heat
exchanger
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4. circumferential-direction strains;
Concentric tube heat
exchanger
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5. radial-direction stresses (positive being tensile, and
negative compressive);
Concentric tube heat
exchanger
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6. circumferential-direction stresses;
Concentric tube heat
exchanger
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7. axial-direction stresses.
Extending CFD to SFT
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Three questions:
1. Are the predictions correct?
Probably, because:
• the code produces the analytically-derived exact solutions for
all cases in which these exist;
• the displacement equations, are, after all, very simple.
2. Did solving for stress and strain increase the computer time?
Not noticeably. Calculating finite values of displacement is not
much more expensive then setting velocities to zero; and
convergence of the velocity and pressure fields dictated
how many iterations were needed.
3. Could the same result have been achieved by coupling a finitevolume and a finite-element code?
Certainly, but with much greater difficulty; so why bother?
3.3 A choice to be made
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Which forms the better method for SFT? Finite-volume or
finite-element?
The printed version of the lecture discusses the question at
length. Here I summarise thus:
 The general-purpose SFT codes needed by heat-transfer
engineers could be based on finite-element methods). But..
 The highly-demanding F part of SFT, is handled so much
better by finite-volume methods than finite-element ones
[Why else did Ansys buy Fluent and CFX?],
that the best SFT codes are likely to be FV-based.
 Early arguments that FE methods are better for awkward
geometries lost their force more than twenty years ago.
 It is only mental and commercial inertia that keeps the finiteelement juggernaut in motion.
Final examples
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1. distortions of a sea-bed structure by ocean
waves,
Final examples
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2. flapping of a wing, courtesy of K Pericleous:
Part 2. How not to design
heat exchangers
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4.1 What the handbooks say
AC Mueller, in Hartnett and Rohsenow's 'Handbook of Heat
Transfer' states:
"Heat exchangers are designed by the usual equation:; q =
U*A*MTD"
wherein:
 U is the overall heat-transfer coefficient,
 A is the area of the heat-exchange surface, and
 MTD is the Mean Temperature Difference.
The area, A, is fairly easy to estimate; otherwise we can be
sure only that:
 U is not a constant, and that
 MTD can be determined only for simple flow patterns
which never exist in practice.
How not to design heat
exchangers
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Ah! But that’s why we have ‘correction factors.’
Yes, we do; and we have all seen, and perhaps used, such
charts as this from Hartnett and Rohsenow; but they based
on unrealistic idealised flow patterns.
How not to design heat
exchangers
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The Tinker-Bell-Devore corrections
Then there are allowances for leakages between baffles and
shell , and for 'by-pass streams', based on experiments
carried out long ago, at the University of Delaware and
elsewhere.
How not to design heat
exchangers
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But the experiments are of course too few. Indeed to carry
out enough experiments, and then to express their results
as formulae, is an impossible task.
Nowadays, few designers use the charts and correction
formulae directly; for they have been embodied in software
which reduces labour.
Alas, it also reduces the doubt which their users ought to
maintain; for the underlying concepts are based on
fictions, not physics.
4.2 Can CFD assist?
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Computational Fluid Dynamics is based on physics. Can CFD
then be a better basis for heat-exchanger design? My
answers are:
1. Yes, in principle , but heat exchangers have many closetogether solid-fluid interfaces;
2. Therefore flow details can not be simulated.
3. However, the space-averaged (also called porous medium)
approach works well, especially for 'difficult' equipment,
e.g. power-station steam condensers and nuclear boilers.
4. Its lack of adoption by the heat-exchanger fraternity may
have resulted from data-input difficulties, which are now
being removed.
Before turning to the difficulties, I show results from a recent
study of a baffled shell-and-tube heat exchanger.
Computed flow patterns
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The baffles produce a complex three-dimensional flow,
different for each configuration.
Computed temperature
distributions
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No handbook 'correction factor' can represent temperature
distributions like this.
Computed fluid property
distributions
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Material properties vary throughout; and so must heattransfer coefficients.
Computed Nusselt numbers
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Note the wide variation of values of the dimensionless heattransfer coefficient.
Space-averaged CFD is
needed; and it’s available
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In Summary
 Hand-book methods of heat-exchanger design make
assumptions about:
 uniformity of properties;
 uniformity of heat-transfer coefficient;
 existence of idealised flow patterns;
 calculability therefrom of the mean temperature
difference.
 Every physics-based numerical simulation of practical heat
exchangers shows that the assumptions are wrong.
 The numerical simulations also rest on assumptions; but
these, being local rather than global, are far more reliable.
 The computer time needed for calculating rather than
presuming the flow and temperature distributions is trivial,
 Heat-exchanger-design software should therefore embody
physics-based space-averaged CFD flow simulations.
4.3 Why conventional packages
fail to satisfy
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1.CFD specialists distrust conventional heat-exchangerdesign packages because the packages lack physics.
2. Some experienced heat-exchanger designers distrust
them for other reasons. Thus, J Taborek [5] in the
Hemisphere Handbook of Heat Exchanger Design, states:
"Only if calculations are performed manually will the
engineer develop a 'feel' for the design process as
compared to the impersonal 'black box' calculations of a
computer program".
3. The package designers seem to distrust their users: they
treat them as capable only of making selections by
mouse-clicks on tick boxes.
The mouse’s revenge
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Being restricted to the choices provided by the package
designer is indeed to be a ‘prisoner of the mouse’, in fact
rather like this:
Heat exchanger design is for
men not mice
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Engineers who prefer 'manual calculation‘ do so because they
like to decide for themselves what formulae for:
 heat-transfer coefficients;
 pressure-drop coefficients;
 fouling factors;
 etc.
are to be used in the various parts to exchanger.
What is needed is software which respects their experience,
and enables them to use it, freeing them from the
constraints which mouse-click codes impose.
But the software should also allow them to used calculated
flow patterns, not out-dated guesses.
5. Improving the input
procedures
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5.1 Input of formulae; the history
 Early '80s CFD codes contained built-in modules for
calculating, say:
 viscosity from temperature, pressure and composition of
fluids;
 Nusselt from Reynolds and Prandtl numbers for specific
geometries.
 There were never enough of these; so provision was
made for users to add their own Fortran or C coding.
 Mid-'90s codes contained self-programming features, to
which users simply supplied formulae.
Input of formulae
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 The latest codes react to formulae directly:

If the user writes lines like:
Nusselt is 0.023*Reynolds**0.8*Prandtl**0.33
the computer code works out for itself what to do.
 The formulae can be of arbitrary complexity.
 Therefore anyone who can write a formula can "do
CFD".
Input of formulae was reported at the 2005 ASME Summer
Heat Transfer Conference in San Francisco. I therefore turn
to a newer development: the input of relations.
5.2 Input of relations
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The main steps in setting up a heat-exchanger simulation
are:
a. assemble all component objects (shell, nozzles, headers,
baffles, tubes, etc);
b. specify their proper dimensions and positions;
c. assign the property formulae to the various solids and
fluids;
d. select the heat-transfer and friction formulae to be used;
e. assign the inlet flows and temperatures, and any other
relevant thermal, or mechanical conditions;
f. let the computer work out the consequential 3D
temperature distributions (and stresses) as functions of
time.
I shall now show some parts of the process, conducted by
way of the relational input module, PRELUDE.
Shell-and-tube heatexchanger in PRELUDE
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Objects, position, size and attributes
The shell-and-tube exchanger (one half only) might, in the
course of assembly, look like this:
The family of objects
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• It is a collection of inter-linked objects, having names on
the left of this picture which shows them linked as 'parent'
and 'child'.
Attributes of objects; the
dialogue box for the shell
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•Each object has attributes, expressed as numbers,
variables, relationships or file-names.
The size- and position
dialogue box
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Each object has also size and position which may be
similarly expressed.
Further details of the
relational-input module
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• Attributes, position and size may be:
 created by a generic shell-and-tube heat-exchanger
script; or
 read in from a particular shell-and-tube heatexchanger file (e.g. one of those which the desiner has
used before); or
 entered interactively.
 As soon as any value or relationship is changed
interactively, all consequential changes, for all objects, are
made, and seen, at once.
 At the end of the interactive session, all positions, sizes
and attributes, including relations, are saved, into a file, for
later re-use.
How the relations and formulae
appear in the file
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Here, in italics, are the some of the relations governing 'bundle'.
Although they have their own vocabulary, it is easy to learn,
and use.
 position and size:
xmid(bundle) = Xmidcoord(SHELL)
ymid(bundle) = Ymidcoord(SHELL)
zmin(bundle) = Zmaxcoord(HEAD1)
radius(bundle) = inradius
 shape:
disk bundle ! Disk is an object type; bundle is one of them
 shell-side heat-transfer coefficient:
nuss at bundle is 0.2*reys^0.6*prns^0.33) ! shell-side Nu
coes at bundle is aoverv*nuss*cond/diam) ! and coeff.
Formulae for Reynolds,
Prandtl & Nusselt numbers
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 tube-side coefficient:
reyt at bundle is diam*tubvel/enut ! tube-side Re
prnt at bundle is cpt*rho2*enu2/cont ! and Pr
nust at bundle is max(2.0,0.328*(reyt*prnt)^0.33) ! and Nu
coet at bundle is aoverv*nust*cont/diam) ! and coefficient
 overall coefficient:
coeU at bundle is 1/(1/coes+1/coet)
 the heat flux:
flux at bundle is coeu*(temt-tems) ! *temperature difference
These statements may be edited manually or interactively.
Doing so gives the engineer the freedom which he needs,
and which the wretched mouse-prisoner can never enjoy.
Co-ordinated changes
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Changing the number of baffles
When the user changes the baffle number from 3 to 4, they
jump into their new positions at once; and the outlet nozzle
moves from the top to the bottom, as seen here
A deeper-level script
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This is because of lines in the set-up script like this:
if {$oddeven>0} { ! oddeven refers to baffle number
$baff1 setposition [list wallthick/2. ysize($parname)/2.0\
$ic*(Zmincoord($d2)-Zmaxcoord($d1))/$nmax ]
} else { $baff1 setposition [list Xsize($parname)-wallthick/2.
\ ysize($parname)/2.0 $ic*(Zmincoord($d2)Zmaxcoord($d1))/$nmax ]
$baff1 setzrot 180.
}
Heat-exchanger designers would NOT be expected to look at
such details; but their computer-specialist colleagues could
do so, if some new functionality were required.
A common difficulty
concerned with re-use
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Most CFD packages have graphical user interfaces which
enable:
 flow-simulation scenarios to be set up;
 objects to be brought in from solid-modelling packages;
 material properties to be assigned to the objects;
 boundary conditions to be attached to them; and
 computation-controlling settings to be made.
Many also allow for the data-input files to be stored and reused.
However, when re-use involves changing the numbers,
materials, sizes, shapes or positions of the objects, the
labour required for the second scenario is nearly as great
as for the first.
The advantage of relational
input modules
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A code equipped with a relational input module greatly reduces
that labour; for it remembers why the objects in the first
scenario were placed where they were, recording these in its
'Book of Rules'
Then, unless instructed otherwise, it will apply the same rules
for the second scenario as were laid down for the first.
For example, if the shell-length of a heat exchanger is increased,
the headers will move appropriately further apart.
Any desired relationship can be built in, including those linking
geometric with thermal or computational conditions.
Relational input modules are especially useful for handling SFT
problems, in which objects, their supports and their
applied loads must move together.
5.3 Optimization
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Finally, for completeness, I mention that the designer's true
task is not 'merely' that of predicting the performance of a
prescribed heat exchanger.
What is needed is the ability to determine the dimensions
and configuration of the best-possible heat-exchanger for
the prescribed duty, with prescribed constraints.
Provided that a parameterised input procedure is
available, of the PRELUDE kind, computers can be
instructed systematically to search for the optimal
parameter set.
This is rarely done at present; but it can and should become
the norm.
6. Concluding Remarks, 1
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In remembrance of Jim Hartnett, I have sought to be
controversial, having asserted that:
• the territory of 'Heat Transfer' should be enlarged so as to
include more of its 'Effects';
• CFD should become SFT;
• inclusion of stress analysis is best done without finite
elements;
• heat-exchanger design should be based on physics, not
fiction;
• software packages should allow input of arbitrary formulae;
• objects are best assembled via algebraic relations which
packages must understand;
• enforced restriction to mouse-clicking can damage one's
mental health..
Concluding Remarks, 2
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These recommendations now appear to be such
obvious commonsense as to be totally noncontroversial.
Sorry, Jim!
But probably I have not explained my meaning
well enough for some of you; so you may
disagree with what you think that I said.
Perhaps that will produce controversy after all.
!!!! Thank you for your attention !!!!
Acknowledgements
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The author gratefully acknowledges the assistance of:
 Dr Valeriy Artemov of the Moscow Power Engineering
Institute in developing and testing the SFT technique,
 Dr Elena Pankova of the Moscow Baumann Institute in the
preparation of diagrams,
 Dr Geoff Michel of CHAM in developing PRELUDE, the
'relational input module‘; and of
 My sons Peter and Jeremy in ‘Power-Pointing’ this lecture,
References
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Regarding the subject of Heat Transfer
 Bosch M, Ten 1936 "Die Waermeuebertragung, 3rd Ed",
Springer, Berlin
 Jakob M , 1949, Heat Transfer, John Wiley, New York
 Ganic, E, Rohsenow, W. M. and Hartnett, JP (Eds), 1973,
Handbook of Heat Transfer Fundamentals, McGraw Hill.
 Rohsenow, WM and Hartnett, JP (Eds), 1973, Handbook of
Heat Transfer, McGraw Hill.
References
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Regarding ignition, propagation and extinction of flames
 Botha JP and Spalding DB, 1954, Proc Poy Soc A vol 225
pp 71-96
 Spalding DB and Tall BS, 1954, vol 5 p 195
 Spalding DB 1955, "Some Fundamentals of Combustion",
Butterworths, London
References
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Regarding numerical methods generally
 Richardson LF ,1910, Trans Roy Soc A, vol 210, p 307
 Schmidt, E, 1924, "On the application of the calculus of
finite differences to technical heating and cooling
problems", August Foeppl Festschrift, Springer
 Minkowicz, W M, Sparrow, E, Schneider, G E and Pletcher,
R H, (Eds), 1988, Handbook of Numerical Heat Transfer,
John Wiley
 Patankar SV, Spalding DB, "A calculation procedure for
heat, mass and momentum transfer in three-dimensional
parabolic flows"; Int J Heat Mass Transfer vol 15 p 1787
(1972)
References
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Regarding the finite-volume approach to stress-analysis
 Spalding, D B, 1993. Simulation of Fluid Flow, Heat
Transfer and Solid Deformation Simultaneously, NAFEMS
Conference no 4, Brighton.
 Demirdzic, I. and Muzaferija, S., 1994, Finite-Volume
Method for Stress Analysis in Complex Domains, Int J for
Numerical Methods in Engineering vol 37, pp 3751-3766.
 Bailey C, Cross M, Lai C-H, 1995, "A finite-volume
procedure for solving the elastic stress-strain equations on
an unstructured mesh."
Int. J. Num. Meth. in Eng. vol 38,1757-1776
References
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Regarding the currently-used methods of heat-exchanger
design
• Devore, A., 1961, Try this simplified method for rating
baffled exchangers, Pet. Refiner, vol 40, p 221.
 T Tinker J. Heat Transfer vol 80 pp 36-52 1958
 KJ Bell "Final report of the cooperative research program
on shell-and-tube heat exchangers" University of Delaware
Exp.Sta.Bull. 5 1993
 J Taborek "Recommended method: principles and
limitations" in "Hemisphere Handbook of Heat Exchanger
Design" ed. by GF Hewitt, Hemisphere, New York 1983
References
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Regarding the use of formulae in heat-exchanger design
 Spalding DB 2005 "Solid-fluid-thermal analysis of heat
exchangers", ASME Summer Heat Transfer Conference,
San Francisco
Regarding the use of relational input procedures
 Michel GM and Spalding DB 2006 "PRELUDE User Guide",
unpublished
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The End !!!