Solidfluidthermal analysis of heat exchangers
by
Brian Spalding
Keynote Lecture at 2005 ASME Heat Transfer Conference
July 1722, 2005, San Francisco, California
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Abstract
Traditional heatexchanger design methods predict steadystate
uniformproperty performance imperfectly; and
transient, varyingproperty
flow, heat transfer, chemical reaction and stresses in solids not at
all.
Traditional CFD techniques, relying on bodyfitting grids and
dubious turbulence models, handle only smallscale phenomena
such as fewtube subsections of tube banks.
Intermediate CFD methods, which replace tubebank details by
spaceaveraged representations and
heattransfer correlations, have been successfully used for
years; but they are little favoured by designers.
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One reason is that heattransfer correlations and other formulae
needed by designers are too numerous and varied to be supplied
with commercial codes; so users should provide Fortran
themselves; but they lack time or skill.
What has long been needed is software which will itself
understand formulae.
The lecture exemplifies how such software has been applied to a
shellandtube heat exchanger.
It also shows how the stresses in the solid components of the heat exchanger can
be computed at the same time.
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Contents of the lecture
 Introduction
 Early successes
 Obstacles to further progress
 A way forward
 The Requirements of a HeatExchangerDesign Method
 Geometrical input data
 Materialproperty data
 Thermal and massflow boundary conditions
 Empirical correlations
 Predicting the flow pattern and temperature distribution
 Three ways of satisfying the requirements
 Method 1: 'Usersupplied subroutines'
 Method 2: 'Automated subroutine writing'
 Method 3: When the program understands formulae
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 Practical Example
 The heat exchanger and computer code in question
 Distributions of velocity, temperature, pressure and related properties
 Distributions of Reynolds, Prandtl and Nusselt number
 Variations of heattransfer coefficient
 How the input data were introduced
 Discussion
 Simultaneous calculation of stresses in solids
 A common misconception
 The truth of the matter
 Fluidstructure interaction examples
 Exemplification for the heat exchanger
 Concluding remarks
 Acknowledgement
 References
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The spaceaveragedCFD technique applied to heat exchangers was first
described than thirty years ago by Patankar and Spalding [1]. They concluded:
"It therefore seems that a tool of considerable practical utility is in
embryonic existence".
At first their expectations appeared to be fulfilled: application to twophase flows helped to
resolve difficulties encountered by the then growing nuclearpower industry.
Specifically, the shellside steamwater mixture circulating in boilers,
heated by pressurized water from the nuclear reactor, caused tubes
to vibrate and baffles to corrode. Consequently, first US and German companies
finally EPRI, sponsored the development of flowsimulating computer programs.
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The work was pioneering and did not always proceed
as rapidly as desired.
So one joker suggested that the name
adopted for the EPRIsponsored code, URSULA, was an acronym for:
Urgently Required Solution Unacceptably Late Arriving.
Despite the implied criticism,
the work was successful; and it was followed by the development of
further computer codes for simulating steam condensers and cooling
towers.
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Nevertheless, the heatexchangerdesign community has shown little
enthusiasm; and a recent
paper [2] on the subject concluded "very few applications can be found
of using CFD technique as a tool for heatexchanger design optimization".
Instead, designers still prefer to use methods, for example those of
Tinker [3] or Bell [4], in which the flow patterns are guessed ,
not calculated.
The reasons are not entirely clear. However, that
they are in part psychological is suggested by the remarks of
J Taborek [5] in the Hemisphere Handbook of Heat Exchanger Design.
He there opines: "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".
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The approach recommended here may be congenial; for it enables
designers to insert the same formulae, including TinkerBell
'correction factors', as they would supply to the 'handheld calculators'
preferred by Taborek.
The method to be described can be applied to heat exchangers of all
types, to any participating fluids, and to any conditions of operation.
However, in order to focus on essentials, discussion will henceforth be
limited to:
 a baffled shellandtube heatexchanger,
 singlephase nonreacting fluids,
 steadystate operation,
 thermal and pressuredrop performance; and
 longituginal stresses in the tubes.
Such a heat exchanger is shown from the outside
here and in a sectioned view, but without the
tubes, here.
The software package allows the various components to be moved, resized,
and viewed from various angles.
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Whatever the calculation technique,
the apparatus in question must first be described
in geometrical terms, including (for the simplest cases):
 inside shell diameter
 inside shellnozzle diameter
 tube outside diameter
 tubewall thickness
 tubelayout pitch
 tubelayout characteristic angle
 tube length
 baffle cut
 baffle spacing
 number of tubes
 number of tube passes
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Specification must be made of:
 the thermal conductivity of the tube material
 the thermal conductivities of the shell and tubeside fluids
 the specific heats of both fluids
 the densities of both fluids, and
 the viscosities of both fluids.
However, for most materials, these properties are known to vary with
temperature; and this knowledge is expressed either by way of:
 formulae,
 tables of numbers, or
 graphs of various kinds.
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If graphs are in question, their content must be converted into formulae
or
tables before it can be communicated to a computer program.
However, even
when this has been done, the problem of using the information
remains; for the whole point of a heat exchanger is to change temperature;
and it is not
known in advance what temperatures will prevail at any chosen point within
the tubes or shell.
Therefore either:
 a guessed 'average' temperature is taken, as in traditional heatexchanger
design, or
 some means must be found of communicating to the computer program
the whole content of the formulae or tables, together with the instruction:
"YOU work out values of conductivity and density etc for each
point."
The second is the main theme of the present paper.
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Also needed, of course, are the (known):
 massflow rate and temperature of the shellside fluid in the inlet nozzle;
and
 massflow rate and temperature of the tubeside fluid in its inlet header.
The main task of performance prediction is to determine what will be the
(massflowweighted average) temperatures of the shell and tubeside fluids
at their outlets from the heat exchanger.
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If:
 the geometry in question were extremely simple (e.g. with unbaffled shell
of large length/diameter, and one central tube),
 the flow were laminar, and
 the temperature variations were small,
a traditional CFD code could predict performance well.
However:
 industrial heat exchangers have very many tubes;
 baffles are present; and
 the flow is usually turbulent.
This entails that,
if performance were to be predicted purely from CFD,
a very fine grid would have to be needed (see Prof. Sunden's lecture); and
the performanceprediction calculation would take more time
than any designer could afford to wait.
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Moreover, for flow in tube banks, the
knowledge of turbulence
is still rudimentary; so the reliability of the
predictions far from one hundred per cent.
The only practical solution is therefore to
introduce additional information,
derived from such experimental data as can be found, concerning the rates of
heat and momentum transfer per unit area of solidfluid interface.
This
information is usually expressed in the form of
mathematicallyexpressed relationships between the wellknown
'dimensionless parameters'
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namely:
 Nusselt or Stanton number, for the heattransfer coefficient;
 Reynolds number, to characterize the state of the fluid, and
 Prandtl number, to characterize the relative ease of heat and mass transfer
in the fluid.
All these parameters involve the material properties listed in section 2.1.
Therefore the use of empirical correlations still requires
the computer program to work out the
property values from the given formulae and the temperatures which it
finds at every point.
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The outlet temperatures of the heatexchanging fluids are the main
quantities which it is desired to predict; however, these depend on the
temperatures just upstream of the outlet; these justupstreamofoutlet
temperatures depend on the temperatures upstream of them;
and so on.
Therefore, the whole temperature distribution has to be computed.
When the flow pattern is not onedimensional, what point lies
'just upstream of' a given point is not obvious a priori;
therefore
ability
to calculate the temperature distribution depends on ability to
calculate the whole threedimensional flow distribution giving rise to it.
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This therefore is what the computer program must additionally do, providing
incidentally further information needed by the designer:
the pressure losses suffered by the two streams.
Fortunately, computer programs (the socalled CFD codes) do exist for
computing both the flow fields and the temperature distributions
simultaneously.
Although their accuracy depends on the fineness of
computational grid which is employed, and desirably fine grids do
increase computer times and therefore costs, the
requirements relating to shellandtube heat exchangers are usually
affordable;
but only when the spaceaveraged approach is adopted.
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However, the heattransfer and friction correlations require:
 not only material properties which vary with temperature according to formulae which must be
made known to the code; but also
 other quantities which can not be
specified a priori, namely the
three components of the shellside velocity.
It follows that, even if the temperature variations were small enough not to
affect material properties, the need for the code to evaluate formulae from
values which varied from place to place would remain.
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Thus the Reynolds Number
enters most pressuredrop and convectiveheattransfer formulae; and its
value depends on the local velocity, which varies with position in ways that
are not known at the start.
Moreover, the angle between the meanflow direction and the tube axis is
probably important. It should appear in the formulae.
In summary, predicting the performance of shellandtube heat exchangers
necessitates use of a program which is not restricted, as conventional
design methods are, to the presumptions that:
 material properties can safely be based on guessed average temperatures,
and
 mass flow rates can be based on guessed average velocities and densities.
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Of course, many CFD codes have some builtin correlationevaluation
sequences, representing friction and heattransfer; and they
also contain computercoding modules which express the variation with
temperature and pressure of some properties of some
materials.
In principle, there is no limit to the extent to which these provisions can
be extended. But in practice, however much is provided, some users of the
code will:
 require different correlations
 want them at once;
 be unwilling to pay the codedeveloper to provide them.
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From the earliest years of commercial CFD, therefore, developers have
allowed users to add coding modules of their own, usually in the form of
Fortran or C subroutines, which would supplement the builtin correlations
in the desired respect.
Users of the 1981 PHOENICS code, for example, will remember what clever
use some users made of the socalled 'GROUNDcoding' facility, which
indeed many oldstagers continue to use. The paper by Stevanovich et al [2] is an excellent
example of the use of this technique for heat exchangers.
However, the proportion of CFDcode users with the necessary skills is
constantly diminishing; and few heatexchanger designers
either possess or have the time to acquire them.
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In order to enable PHOENICS users to benefit from the features of
'GROUNDcoding' without themselves having to be familiar with either
Fortran or C, the socalled 'PLANT' feature was introduced in 1997.
This enabled the user to express his wishes by way of formulae, written in
accordance with prescribed rules; whereupon PHOENICS itself:
 interpreted the formulae;
 created corresponding Fortran subroutines;
 compiled them;
 rebuilt the executable; and
 carried out the required flowsimulating calculation.
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This was a big step forward; and it did, at least potentially, satisfy
the 'formulaprocessing' requirement which has been pointed out above.
However, because it was not adequately presented to them, it did
not convert many heatexchanger designers into CFD users.
Perhaps also the 'prescribed rules' were shaped by those thinking too much
of the Fortran to be written, and not sufficiently of the prospective user.
PLANT remains as a valuable feature of the current PHOENICS; but the
feature now to be described has rendered it almost redundant.
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The new feature has the acronym "InForm", standing for "Input (or Intake)
of Formulae". Its concept is very simple, namely:
 let the user, in language which he or she can easily understand, write the
formulae according to which he wants the solver to compute values of, say,
heattransfer coefficient;
 let this be sent to the equationsolving module in the form of a character
string;
 let the latter module first inspect and 'parse' all the strings which it
receives, converting them into settings of 'internal switches' which dictate
how the calculation shall be conducted; and then
 let the calculation proceed.
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The formulae may provide CFD inputs of any kind, whether:
 initial values,
 sources and sinks,
 boundary conditions,
 numerical criteria.
The formulae most relevant to heatexchanger design are those which:
 compute the local material properties in shellside fluid,
tubeside fluid and metal, from the local temperatures and (if appropriate)
pressure;
 compute the corresponding Reynolds and Prandtl numbers;
 compute the corresponding Nusselt or Stanton numbers and thence the local
heattransfer and friction coefficients;
 compute the resulting local sources and sinks of heat and momentum, and
 supply these to the equations on which the solver operates.
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InForm has many more capabilities than the heatexchanger designer is
likely to need. Therefore only the few relevant ones will be illustrated
here, as follows:
 specific fluids may be introduced by statements such as:
tube_fluid is SAE_5W30_engine_oil
which dictates use of a preexisting set of formulae for density,
viscosity, specific heat and thermal conductivity;
 individual material properties can be set by such statements as:
property RHO1 at SHELL is CONST1+CONST2*TEM1+CONST3*TEM1^2
where:
 RHO1 stands for density,
 TEM1 stands of temperature, and
 ^ is the symbol used for exponentiation;
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 the local Reynolds number can be defined by a statement such as:
Stored var Reyn is TUBEDIAM*RHO1*AbsV/Visc
with 'Stored var' meaning 'compute and place in 3D store', and
AbsV and Visc signifying absolute velocity and viscosity;
 then Stanton Numbers can be specified by the statement:
Stored var STAN is CONST1*Reyn^CONST2*PRNO^CONST3
with PRNO meaning Prandtl number;
 whereafter a local heattransfer coefficient can be specified by way
of:
Stored var COEF is STAN*RHO1*CP1*ABSV
with CP1 meaning specific heat; and finally
 the heat source can be specified via:
Source of TEM1 is COEF*AREA(TMETTEM1)
with TMET meaning metal temperature.
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The relevance of these statements to heatexchanger design should be
obvious; for they say (nearly) all that is to be said about how local heat
transfer from tube to fluid is to be calculated.
Once they have been typed into the input file, there is nothing further for
the designer to do than to wait (for a minute or two)
until the calculation has been completed.
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A heat exchanger will be considered which has:
 a single tube pass,
 a shell of 4m length and 1m diameter with
 three baffles.
The shellside fluid is water and the tubeside fluid SAE 5W30 engineoil ;
these enter at 10 and 60 degrees Celsius respectively.
The mass flow rates of both fluids are 100 kg/s.
Textbook formula, connecting Nusselt, Reynolds and Prandtl numbers, have
been adopted, with modifications made by the user, for the shellside and tubeside heat transfer coefficients.
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The computer code
The distributions of pressure, velocity and temperature of the two fluids
have been computed by the generalpurpose CFD code, PHOENICS, without
activation of any heatexchangerspecific special features.
Therefore any other CFD code equipped with a formulaunderstanding module
could have been used.
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The results to be displayed
The computer runs of which the results will be presented have been chosen in
order to illustrate that:
 the variations in space of properties, flow patterns, Reynolds numbers,
Prandtl numbers, Nusselt numbers and heattransfer coefficients, which are
neglected in conventional design methods, are rather large;
 it is however as easy to calculate, and to take account of, these
variations as it is to neglect them; and
 changing the formulae used,
the output required or
any other specification, and
 then observing the consequences, is a matter of minutes only.
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The distributions are of course threedimensional. However it is sufficient
for present purposes to consider the central plane only.
Results will be represented by way of vector and contour plots on this plane.
After presentation of the results, some of the formulae used for
generating them will be shown and discussed.
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The velocity distribution in the shell, on the central plane is shown in
Fig. 1.
The shellfluid inlet is at the bottom on the left, and its outlet at the top
on the right. The influences of the two baffles is clearly seen.
The temperature distributions in the shell and tubeside fluids are shown
in Fig.2 and Fig. 3.
The latter conforms to that to be expected in an ideal
counterflow heatexchanger, but the former (understandably) does not.
The pressure distribution in the shell is shown in
Fig.4. It shows the
tobeexpected discontinuities caused by the baffles.
As has been emphasized above, the properties of most fluids depend upon
temperature. The variation of the viscosity of the tubeside fluid, for
example, is shown in Fig. 5.
Evidently (see scale on the right), it varies
by a factor of more than two.
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The dimensionless parameters which feature in the calculation of heattransfer
coefficients vary through the heat exchanger partly, because of temperature
variations and partly because of the nonuniformities of velocity.
The
distributions of Reynolds, Prandtl and Nusselt number calculated in the
present case are shown, for the shell and tubeside fluids in turn, in
Fig. 6,
Fig. 7,
Fig. 8,
Fig. 9,
Fig. 10 and
Fig. 11.
In respect of Reynolds number, variations are greatest on the shell side;
the presence of the baffles accounts for this.
The consequential variation
of Nusselt number is correspondingly great, being highest near the edges of
the baffles and fifty times smaller in lowvelocity regions, near the baffle
roots and in the corners opposite inlet and outlet.
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From the Nusselt numbers, the computation of the corresponding shell and
tubeside heattransfer coefficients per unit volume of shell is an easy
step; and the overall heattransfer coefficient, U, follows as their harmonic mean.
Their distributions are displayed in
Fig.12,
Fig.13 and
Fig.14.
The shellside coefficients show wide variations but, their
effect on U is diminished by the fact
that the tubeside ones are the smaller; so the maximum U is
less than 50 % greater than the minimum U.
This observation might be thought to blunt somwhat the above criticism of conventional design methods;
but it does not truly do so.
The objection to conventional methods is not that
that they use a constant value of U for the whole heat exchanger; it is
that the means of determining that value are based on guesswork and not on
detailed analysis of the flow field.
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The results displayed so far will have come as no surprise to any
heatexchanger designer who has thought even superficially about what is
truly happening inside his equipment.
What may interest him more may be to
learn:
 how easily they were obtained, and
 how quickly the effects of changes to the input can be investigated.
The ability of the computer program to understand easilywritten formulae is
the key requirement.
Examples of such formulae will now be presented,
starting with the following extract from the engineer's datainput file:
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 (stored var REY1 is DIAM*VABS/ENUL)
 (stored var PRN1 is CPS*RHO1*ENUL/COND)
 (stored var NUS1 is 0.2*REY1^0.6*PRN1^0.33)
 (stored var COE1 is AOVERV*NUS1*COND/DIAM)
 (stored var REY2 is DIAM*TUBVEL/ENU2)
 (stored var PRN2 is CPT*RHO2*ENU2/CON2)
 (stored var NUS2 is MAX(2.0,0.328*(REY2*PRN2)^0.33))
 (stored var COE2 is AOVERV*NUS2*CON2/DIAM)
 (stored var COEU is 1/(1/COE1+1/COE2))
 (stored var TEMM is (COE1*TEM2+COE2*TEM1)/(COE1+COE2))
The lowercase words ('stored', 'var', 'is') tell the code that it must
compute some new variables everywhere within the 'virtual heat exchanger'
with which it is concerned.
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The uppercase words have been chosen by the engineer, who appears to have
found:
 REY1 an adequate name for 'Reynolds number of shellside fluid',
 REY2 likewise for 'Reynolds number of tubeside fluid',
 PRN1, PRN2, NUS1 and NUS2 as adequate for their Prandtl and Nusselt
numbers,
 that AOVERV will serve as 'area/volume',
 COE1, COE2 and COEU remind him of the shellside, tubeside and overall
heattransfer coefficients,
 and so on.
The engineer has also chosen the formulae, most of which are rather easy to
interpret. Thus:
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 'DIAM*TUBVEL/ENUL' stands for 'diameter times tubefluid velocity divided
by viscosity, i.e. the tubeside Reynolds number,
wherein the first two
variables (the code is clever enough to work out) are constants, whereas
the third must be computed for each location;
 'NUS2 is MAX(2.0,0.328*(REY2*PRN2)^0.33)' represents a formula which some
text books recommend for pipe flow at modest Reynolds numbers, with a
lower limit of 2.0 inserted by the engineer for some reason of his own;
 '1/(1/COE1+1/COE2))' shows how the overall coefficient is to be computed
from the individual coefficients; and
 the formula for the tubemetal temperature, TEMM, shows that it is a
suitablyweighted mean of the fluid temperatures, all varying, like
the coefficients themselves, from place to place.
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How were the material properties calculated? Also by way of formulae drawn
from textbooks. Thus, the formulae for the properties of the engineoil in
question were found in a book by K.Hagen [6] and translated thus into the
form understood by (at least) the PHOENICS computer code:
 (property RHO2 is 1052.30.6420*TEM2)
 (stored var CPT is 753.7+3.65*TEM2)
 (stored var CON2 is 0.14472.3073E5*TEM2)
 (stored var ENU2 is 10.0^(POL4(TEM2,58.2987,.53817,1.92827e3, 3.16448E6,1.97922E9)2)
wherein POL4(TEM2,.....) signifies a fourthpower polynomial, the
coefficients in which are the numbers which follow the comma.
Fortunately, many formulae are provided as items to be selected from a
library. Nevertheless, it is not difficult to translate a new formula into
'InFormspeak': users are not confined to what is on the codevendor's menu.
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There are engineers who are unwilling to do more, when designing equipment,
than make selections, by way of mouseclicks, from what so many have done
before that the code provider has had time and inducement to adopt as
builtin options.
Others however, more like those referred to in the above quote from J.Taborek,
will welcome the power which InForm provides to "develop a 'feel' for the
design process'.
Example
Noting the considerable temperature differences between
the metal temperature and the tubefluid temperature, a designer might wonder whether
he ought to multiply the relevant coefficient, as some textbooks recommend,
by the ratio (wall viscosity/ bulk viscosity) raised to the power 0.14).
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He could set his doubts at rest in a few moments by introducing the following
lines into his input file
(by copying; God forbid that he should have to type it
again!):
(stored var ENUM is: 10.^(POL4(TEMM,58.2987,.53817,1.92827e3, 3.16448,E6,1.97922E9)2)
so as to compute the viscosity of the engineoil at the wall temperature, TEMM;
then he should modify the formula for coe2 so as to reflect the
viscosityratio effect, thus:
(stored var COE2 is (ENUM/ENU2)^0.14*AOVERV*NUS2*CON2/DIAM).
If he did this, he would learn, after inspecting the computed results, that
the total heat transferred would increase by only 0.5 %, so adding to his
stock of 'engineering feel'.
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Alternatively, he might wish to know how the heattransfer rate per unit
volume varied through the shell; then he could satisfy his curiosity by
writing the single line:
(stored var flux is COEU*(TEM2TEM))
and, one minute later, view the results presented in
Fig. 15.
Comparing this with other images which he has seen before he will be able to
decide whether he is satisfied with the nearlytwofold variation there
disclosed, or will seek by redesign to achieve greater uniformity.
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Lastly, he might suddenly think: "What about fouling?" Then he has only to
insert one line in his input file such as:
(stored var FOUL is
.... whatever function of position and temperature he invents)
and to change the overallcoefficient line to:
(stored var COEU is 1/(1/COE1+1/COE2+FOUL))
The subsequent computer run will inform him of the consequences, both by
way of numbers and of graphical displays.
There is no limit to the easilyeffected variations of input and output
which the InForm facility allows; therefore, the main points having already
been made, no further examples will be presented.
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5.1 A common misconception
It is widely believed that the problem of computing the stresses in solids differs
from that of calculating fluid flow so much that different methods
are needed for each problem.
In particular, it is often supposed that the socalled finiteelement
method must be used for the solidstress problem.
Both these beliefs are erroneous.
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5.2 The truth of the matter
In fact, the differential equations governing displacements in solids
are very similar to (but simpler than) those governing velocities in fluids.
It is true that the displacements in the three coordinate directions are connected, via
Poisson's Ratio, in a manner of which velocities know nothing.
Further, equations for the three components of rotation must be solved in addition
However, the finitevolume methods employed in CFD codes are well able to solve these equations,
without any of the leftovers of precomputer days (e.g. energy theorems and Galerkin weighting
procedures) which clutter the finiteelement formulation.
One of the first to recognise this was Stephen Beale [Reference 8].
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5.3 Fluidstructure interaction examples
Results from an early study
are shown below for a twosolidmaterial
block, heated by radiation from above, and cooled by a stream of air:
(1) velocity vectors,
(2) displacement vectors,
computed at the same time, and
(3) horizontaldirection stresses,
obtained by postprocessing.
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A bent beam
A beam bent by fluid forces is shown
here.
Flow in a turnaround duct
One of the early arguments for preferring FE methods was that they
handled curved boundaries more easily.
But FV techniqes serve just as well, even when the grid is
cartesian, as shown here for flow in a
turnaround duct.
In this case there are heat sources in the solids, giving rise to these
thermalexpansion contours.
Because of the unifiedcomputationalmechanics technique, we are able
to compute simultaneously the velocity and displacement
vectors.
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Finally, deflection is shown of a seabed structure, resulting from the
action of
surface waves.
Here it is essential to be able to simulate solids and fluids at
the same time.
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In order to illustrate these assertions, calculations have been performed for the longitudinal
stresses in the tubes of the present shellandtube heat exchanger.
Fig. 16 shows the results of the calculations in terms of vector diagrams.
Vectors of what?
 The left half of the diagram shows vectors of shellsidefluid velocity; and
 the righthalf shows vectors of displacement in the tube metal.
The latter have been shifted to the right of the former for ease of display. However
PHOENICS specialists might recognise that the MUSES (MultiplyUSedSpace) trick has been played,
whereby a doublesize grid has been employed, with:
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 flows computed on the first grid,
 displacements computed on the second grid, and
 metal temperatures forced to be the same for both grids by InForm statements.
Why do the displacement vectors have the distributions which have been shown? Because
of the arbitrarilyintroduced fixings postulated for the tube ends within the shell.
The stressescan be deduced from the strains, which are themselves deduced by differentiating the displacements.
Fig. 17 shows contours of the stresses, of course in
the righthand part of the display region.
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The higher (compressive) stresses appear in the lower part of the shell, where the tube ends have been rigidly fixed.
The importance of Fig. 17 in the present context is that:
 the stresses were computed at the same time as the fluid flow and temperature distributions, by means of a single generalpurposecomputer code;
 the additional computational effort was trivial;
 alternative (and more realistic) mechanical constraints can be introduced via userwritten InForm statements.
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It should be stressed that the techniques employed can be applied
also:
 when the heat exchanger is of any other configuration;
 when its operating conditions vary with time (e.g. when the
inlet massflow rates and/or temperatures vary in accordance with
piecewiselinear formulae);
 when either or both fluids are two or multiphase, InForm being used to
introduce any of the complex formulae which are commonly used for the
design of such equipment [7];
 when chemical reaction takes place within the tubes, for example in accordance with a
formula which depends not only on local temperature but also on the
changes in composition to which the reaction itself gives rise.
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It is therefore argued that:
 the time has come for heatexchanger
designers to exploit computational fluid dynamics with spaceaveraged
representations of heattransfer and friction processes;
 they should consider calculating solid stresses at
the same time;
 specifically, the
difficulty of introducing the necessary empirical knowledge has been
removed by the formulaparsing capability;
 this enables the computer code to 'understand'
the formulae which designers are accustomed to use in their
traditional methods;

thus all that formerly could be done only by specialists capable of
exploiting the 'usersubroutine' facility
,
can now be done by any flexiblyminded heattransfer engineer.
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The author gratefully acknowledges the assistance of Mr. Nikolay Pavitsky,
of CHAM's office in Moscow, Russia, in developing the InForm facility
during the last five years.
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 SV Patankar and DB Spalding "A calculation procedure for the
transient and steady behavior of shell and tube heat exchangers"
in "Heat exchangers: design and theory sourcebook"
edited by N Afgan and E Schluender, McGraw Hill 1974
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The End !!!