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**Extending the Boundaries of Heat Transfer James P.Hartnett Lecture**

by Brian Spalding The 13th International Heat Transfer Conference August 16, 2006, Sydney, Australia James P.Hartnett Lecture

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**Abstract 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: Heat Transfer and its effects is our proper field of study.

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Abstract 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.

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Abstract 4. Heat-exchanger designers should move from guess-the- flow-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. Heat- exchanger 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’?**

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.

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**What is ‘Heat Transfer’?**

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.

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The argument 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.

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**Reasons for enlargement**

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).

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**Reasons for enlargement**

The necessary additional ideas are few, namely: that combustion phenomena result from temperature- dependent 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.

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**Reasons for enlargement**

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.

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**1.3 What heat-transfer engineers should know about combustion**

Flame-propagation speeds of fuel-air mixtures vary thus:

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**Experimental combustion data can then be correlated thus:**

This is from the 1954 thesis of Barry Tall, my first Australian student

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**1.4 What heat-transfer engineers should know about stress analysis**

: 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'.

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**2. Extending Numerical Heat Transfer**

2.1 Numerical methods for heat conduction Analytical formulae exist only for heat- conduction 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.

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**Extending Numerical Heat Transfer**

The figure and equation shown here will be familiar to all users of such methods.

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**How to solve the equations**

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.

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**2.2 Numerical heat transfer with chemical reaction**

I used the Schmidt method for calculating the speed of laminar flame propagation, 50 years ago.

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**Extending Numerical Heat Transfer**

The graphs on the left show successive temperature distributions after two bodies of hot (burned) and cold (unburned) gas are brought into contact

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**Extending Numerical Heat Transfer**

The graph on the right shows the source (horizontal) versus temperature (vertical) function which represents (sufficient of) the laws of chemical reaction.

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**Extending Numerical Heat Transfer**

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.

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**2.3 Extension to displacements in solids**

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.

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**Control-volume for vertical displacement v**

First the Figure

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**Extending Numerical Heat Transfer**

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.

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Solving the equations I now show some results of solving the equations by the same successive- substitution 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.

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**1/4 of square beam with fluid in square hole**

When the outer-wall temperature is raised;

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**1/4 of square beam with fluid in square hole**

When the inner-duct pressure is raised;

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**1/4 of square beam with fluid in square hole**

When both changes are made simultaneously.

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**Consequential stresses**

From the displacement fields may be deduced the distributions of the direct stresses in the horizontal direction...

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**Consequential stresses**

… and in the vertical direction. Comparison with solutions made by the finite-element code Elcut showed close agreement, of course; for the finite-volume and finite-element methods solve the same differential equations.

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**2.4 The research opportunities**

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.

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**3. Extending Computational Fluid Dynamics to SFT**

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 s. It uses equations similar to those governing heat conduction, shown above, with additional features, namely:

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**The additional features of the CFD equations**

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 s.

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**Use of CFD procedures for solid-stress problems**

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 finite-element methods. It is therefore high time that CFD should enlarge to become SFT, i.e. Solid-Fluid-Thermal.

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3.2 A simple example 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.

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**The concentric-tube heat exchanger**

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!

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**Extending CFD to SFT 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.

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**Concentric tube heat exchanger**

1. Pressures in the two fluids causing mechanical stresses;

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**Concentric tube heat exchanger**

2. The temperature distribution, causing thermal stresses.

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**Concentric tube heat exchanger**

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);

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**Concentric tube heat exchanger**

4. circumferential-direction strains;

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**Concentric tube heat exchanger**

5. radial-direction stresses (positive being tensile, and negative compressive);

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**Concentric tube heat exchanger**

6. circumferential-direction stresses;

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**Concentric tube heat exchanger**

7. axial-direction stresses.

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**Extending CFD to SFT 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 finite- volume and a finite-element code? Certainly, but with much greater difficulty; so why bother?

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**[Why else did Ansys buy Fluent and CFX?],**

3.3 A choice to be made 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 finite- element juggernaut in motion.

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Final examples 1. distortions of a sea-bed structure by ocean waves,

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Final examples 2. flapping of a wing, courtesy of K Pericleous:

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**Part 2. How not to design heat exchangers**

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.

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**How not to design heat exchangers**

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.

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**How not to design heat exchangers**

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.

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**How not to design heat exchangers**

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.

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4.2 Can CFD assist? 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 close- together 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.

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**Computed flow patterns**

The baffles produce a complex three-dimensional flow, different for each configuration.

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**Computed temperature distributions**

No handbook 'correction factor' can represent temperature distributions like this.

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**Computed fluid property distributions**

Material properties vary throughout; and so must heat- transfer coefficients.

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**Computed Nusselt numbers**

Note the wide variation of values of the dimensionless heat- transfer coefficient.

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**Space-averaged CFD is needed; and it’s available**

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.

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**4.3 Why conventional packages fail to satisfy**

1.CFD specialists distrust conventional heat-exchanger- design 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.

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The mouse’s revenge Being restricted to the choices provided by the package designer is indeed to be a ‘prisoner of the mouse’, in fact rather like this:

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**Heat exchanger design is for men not mice**

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.

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**5. Improving the input procedures**

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.

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**Input of formulae The latest codes react to formulae directly:**

If the user writes lines like: Nusselt is 0.023*Reynolds**0.8*Prandtl** 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.

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5.2 Input of relations 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.

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**Shell-and-tube heat-exchanger in PRELUDE**

Objects, position, size and attributes The shell-and-tube exchanger (one half only) might, in the course of assembly, look like this:

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The family of objects It is a collection of inter-linked objects, having names on the left of this picture which shows them linked as 'parent' and 'child'.

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**Attributes of objects; the dialogue box for the shell**

Each object has attributes, expressed as numbers, variables, relationships or file-names.

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**The size- and position dialogue box**

Each object has also size and position which may be similarly expressed.

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**Further details of the relational-input module**

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 heat- exchanger 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.

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**How the relations and formulae appear in the file**

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.

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**Formulae for Reynolds, Prandtl & Nusselt numbers**

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.

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**Co-ordinated changes 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

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A deeper-level script 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 } 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.

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**A common difficulty concerned with re-use**

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 re- used. 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.

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**The advantage of relational input modules**

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.

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5.3 Optimization 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.

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6. Concluding Remarks, 1 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..

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**!!!! Thank you for your attention !!!!**

Concluding Remarks, 2 These recommendations now appear to be such obvious commonsense as to be totally non- controversial. 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 !!!!

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**Acknowledgements 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,

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**References 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.

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**References Regarding ignition, propagation and extinction of flames**

Botha JP and Spalding DB, 1954, Proc Poy Soc A vol pp 71-96 Spalding DB and Tall BS, 1954, vol 5 p 195 Spalding DB 1955, "Some Fundamentals of Combustion", Butterworths, London

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**References 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 (1972)

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**References Regarding the finite-volume approach to stress-analysis**

Spalding, D B, 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 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,

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References 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 KJ Bell "Final report of the cooperative research program on shell-and-tube heat exchangers" University of Delaware Exp.Sta.Bull J Taborek "Recommended method: principles and limitations" in "Hemisphere Handbook of Heat Exchanger Design" ed. by GF Hewitt, Hemisphere, New York 1983

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**References 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 !!!

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