Abstract

Traditional CFD techniques, relying on body-fitting grids and dubious turbulence models, handle only small-scale phenomena such as few-tube sub-sections of tube banks.

Intermediate CFD methods, which replace tube-bank details by space-averaged representations and heat-transfer correlations, have been successfully used for years; but they are little favoured by designers.

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One reason is that heat-transfer 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 shell-and-tube 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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1. Introduction
   1. Early successes
   2. Obstacles to further progress
   3. A way forward
2. The Requirements of a Heat-Exchanger-Design Method
   1. Geometrical input data
   2. Material-property data
   3. Thermal and mass-flow boundary conditions
   4. Empirical correlations
   5. Predicting the flow pattern and temperature distribution
3. Three ways of satisfying the requirements
   1. Method 1: 'User-supplied sub-routines'
   2. Method 2: 'Automated sub-routine writing'
   3. Method 3: When the program understands formulae
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4. Practical Example
   1. The heat exchanger and computer code in question
   2. Distributions of velocity, temperature, pressure and related properties
   3. Distributions of Reynolds, Prandtl and Nusselt number
   4. Variations of heat-transfer coefficient
   5. How the input data were introduced
   6. Discussion
5. Simultaneous calculation of stresses in solids
   1. A common misconception
   2. The truth of the matter
   3. Fluid-structure interaction examples
   4. Exemplification for the heat exchanger
6. Concluding remarks
7. Acknowledgement
8. References

1. Introduction

1.1 Early successes

At first their expectations appeared to be fulfilled: application to two-phase flows helped to resolve difficulties encountered by the then growing nuclear-power industry.

Specifically, the shell-side steam-water 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 flow-simulating 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 EPRI-sponsored 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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1.2 Obstacles to further progress

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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1.3 A way forward

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 shell-and-tube heat-exchanger,
• single-phase non-reacting fluids,
• steady-state operation,
• thermal and pressure-drop performance; and
• longituginal stresses in the tubes.

2. The Requirements of a Heat-Exchanger Design Method

2.1 Geometrical input data

• inside shell diameter
• inside shell-nozzle diameter
• tube outside diameter
• tube-wall thickness
• tube-layout pitch
• tube-layout characteristic angle
• tube length
• baffle cut
• baffle spacing
• number of tubes
• number of tube passes

2.2 Material-property data

• the thermal conductivity of the tube material
• the thermal conductivities of the shell- and tube-side fluids
• the specific heats of both fluids
• the densities of both fluids, and
• the viscosities of both fluids.

• formulae,
• tables of numbers, or
• graphs of various kinds.

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 heat-exchanger 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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2.3 Thermal and mass-flow boundary conditions

• mass-flow rate and temperature of the shell-side fluid in the inlet nozzle; and
• mass-flow rate and temperature of the tube-side fluid in its inlet header.

2.4 Empirical correlations

• 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,

• industrial heat exchangers have very many tubes;
• baffles are present; and
• the flow is usually turbulent.

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 solid-fluid interface.

This information is usually expressed in the form of mathematically-expressed relationships between the well-known 'dimensionless parameters'
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• Nusselt or Stanton number, for the heat-transfer 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.

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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2.5 Predicting the flow pattern and temperature distribution

Therefore, the whole temperature distribution has to be computed.

When the flow pattern is not one-dimensional, 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 three-dimensional 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 so-called 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 shell-and-tube heat exchangers are usually affordable; but only when the space-averaged approach is adopted.

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However, the heat-transfer 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 shell-side velocity.

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Thus the Reynolds Number enters most pressure-drop and convective-heat-transfer 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 mean-flow direction and the tube axis is probably important. It should appear in the formulae.

In summary, predicting the performance of shell-and-tube 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.

3. Three Ways of Satisfying the Requirements

3.1 Method: 'User-supplied sub-routines'

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 code-developer to provide them.

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Users of the 1981 PHOENICS code, for example, will remember what clever use some users made of the so-called 'GROUND-coding' facility, which indeed many old-stagers 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 CFD-code users with the necessary skills is constantly diminishing; and few heat-exchanger designers either possess or have the time to acquire them.

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3.2 Method 2: 'Automated sub-routine writing'

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;
• re-built the executable; and
• carried out the required flow-simulating calculation.

However, because it was not adequately presented to them, it did not convert many heat-exchanger 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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3.3 Method 3: When the program understands formulae

• 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, heat-transfer coefficient;

• let this be sent to the equation-solving 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.

• initial values,
• sources and sinks,
• boundary conditions,
• numerical criteria.

• compute the local material properties in shell-side fluid, tube-side 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 heat-transfer 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.

• specific fluids may be introduced by statements such as:
  tube_fluid is SAE_5W-30_engine_oil
  which dictates use of a pre-existing 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 heat-transfer 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(TMET-TEM1)
  with TMET meaning metal temperature.

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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4. Practical Example

4.1 The heat exchanger and computer code in question

• a single tube pass,
• a shell of 4m length and 1m diameter with
• three baffles.

The shell-side fluid is water and the tube-side fluid SAE 5W-30 engine-oil ; these enter at 10 and 60 degrees Celsius respectively.

The mass flow rates of both fluids are 100 kg/s.

Text-book formula, connecting Nusselt, Reynolds and Prandtl numbers, have been adopted, with modifications made by the user, for the shell-side and tube-side heat transfer coefficients.
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The computer code

Therefore any other CFD code equipped with a formula-understanding module could have been used.

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The results to be displayed

• the variations in space of properties, flow patterns, Reynolds numbers, Prandtl numbers, Nusselt numbers and heat-transfer 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.

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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4.2 Distributions of velocity, temperature, pressure and related properties

The temperature distributions in the shell- and tube-side fluids are shown in Fig.2 and Fig. 3. The latter conforms to that to be expected in an ideal counter-flow heat-exchanger, but the former (understandably) does not.

The pressure distribution in the shell is shown in Fig.4. It shows the to-be-expected 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 tube-side 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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4.3 Distributions of Reynolds, Prandtl and Nusselt number

The distributions of Reynolds, Prandtl and Nusselt number calculated in the present case are shown, for the shell- and tube-side 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 low-velocity regions, near the baffle roots and in the corners opposite inlet and outlet.
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4.4 Variations of heat-transfer coefficient

Their distributions are displayed in Fig.12, Fig.13 and Fig.14.

The shell-side coefficients show wide variations but, their effect on U is diminished by the fact that the tube-side 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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4.5 How the input data were introduced

What may interest him more may be to learn:

1. how easily they were obtained, and
2. how quickly the effects of changes to the input can be investigated.

Examples of such formulae will now be presented, starting with the following extract from the engineer's data-input 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 upper-case words have been chosen by the engineer, who appears to have found:

• REY1 an adequate name for 'Reynolds number of shell-side fluid',
• REY2 likewise for 'Reynolds number of tube-side 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 shell-side, tube-side and overall heat-transfer coefficients,
• and so on.

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• 'DIAM*TUBVEL/ENUL'   stands for 'diameter times tube-fluid velocity divided by viscosity, i.e. the tube-side 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 tube-metal temperature, TEMM, shows that it is a suitably-weighted mean of the fluid temperatures, all varying, like the coefficients themselves, from place to place.

• (property RHO2 is 1052.3-0.6420*TEM2)
• (stored var CPT is 753.7+3.65*TEM2)
• (stored var CON2 is 0.1447-2.3073E-5*TEM2)
• (stored var ENU2 is 10.0^(POL4(TEM2,58.2987,-.53817,1.92827e-3, -3.16448E-6,1.97922E-9)-2)

Fortunately, many formulae are provided as items to be selected from a library. Nevertheless, it is not difficult to translate a new formula into 'In-Form-speak': users are not confined to what is on the code-vendor's menu.
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4.6 Discussion

Others however, more like those referred to in the above quote from J.Taborek, will welcome the power which In-Form provides to "develop a 'feel' for the design process'.

Example

(stored var ENUM is: 10.^(POL4(TEMM,58.2987,-.53817,1.92827e-3, -3.16448,E-6,1.97922E-9)-2)

so as to compute the viscosity of the engine-oil at the wall temperature, TEMM; then he should modify the formula for coe2 so as to reflect the viscosity-ratio 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 heat-transfer rate per unit volume varied through the shell; then he could satisfy his curiosity by writing the single line:

(stored var flux is COEU*(TEM2-TEM))

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 nearly-two-fold variation there disclosed, or will seek by re-design 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 overall-coefficient 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.

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5. Simultaneous calculation of stresses in solids

In particular, it is often supposed that the so-called finite-element method must be used for the solid-stress problem.

Both these beliefs are erroneous.

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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 finite-volume methods employed in CFD codes are well able to solve these equations, without any of the leftovers of pre-computer days (e.g. energy theorems and Galerkin weighting procedures) which clutter the finite-element formulation.

One of the first to recognise this was Stephen Beale [Reference 8].
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Results from an early study

(1) velocity vectors,

(2) displacement vectors, computed at the same time, and

(3) horizontal-direction stresses, obtained by post-processing.

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A beam bent by fluid forces is shown here.

Flow in a turn-around duct

But FV techniqes serve just as well, even when the grid is cartesian, as shown here for flow in a turn-around duct.

In this case there are heat sources in the solids, giving rise to these thermal-expansion contours.

Because of the unified-computational-mechanics technique, we are able to compute simultaneously the velocity and displacement vectors.
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Forces on an under-water structure

Here it is essential to be able to simulate solids and fluids at the same time.

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5.4 Exemplification for the heat exchanger

Fig. 16 shows the results of the calculations in terms of vector diagrams.

Vectors of what?

1. The left half of the diagram shows vectors of shell-side-fluid velocity; and
2. the right-half shows vectors of displacement in the tube metal.

• flows computed on the first grid,
• displacements computed on the second grid, and
• metal temperatures forced to be the same for both grids by In-Form statements.

Why do the displacement vectors have the distributions which have been shown? Because of the arbitrarily-introduced 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 right-hand 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 general-purposecomputer code;
• the additional computational effort was trivial;
• alternative (and more realistic) mechanical constraints can be introduced via user-written In-Form statements.

6. Concluding remarks

• when the heat exchanger is of any other configuration;

• when its operating conditions vary with time (e.g. when the inlet mass-flow rates and/or temperatures vary in accordance with piece-wise-linear formulae);

• when either or both fluids are two- or multi-phase, In-Form 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.

7. Acknowledgement

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8. References

1. 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
2. Z.Stevanovic, G Ilic, N Radojkovic, M Vukic, V Stefanovic and G Vukovic "Design of shell and tube heat exchangers by using CFD technique - part one: thermo-hydraulic calculation". Facta Universitatis Series Mechanical Engineering vol 1 no 8, 2001, pp1091 -1105
3. T Tinker J. Heat Transfer vol 80 pp 36-52 1958
4. KJ Bell "Final report of the cooperative research program on shell-and-tube heat exchangers" University of Delaware Exp.Sta.Bull. 5 1993
5. J Taborek "Recommended method: principles and limitations" in "Hemisphere Handbook of Heat Exchanger Design" ed. by GF Hewitt, Hemisphere, New York 1983
6. K Hagen "Heat transfer, with applications", Prentice Hall, 1999
7. D Chisholm. "Two-phase flow in shell-and-tube heat exchangers" in "Heat Exchanger Technology" ed. by D.Chisholm, Elsevier, 1988
8. S B Beale and SR Elias "Numerical solution of Two-Dimensional Elasticity problems by means of a 'SIMPLE-based' finite-difference scheme". NRC Report No 32090, 1991/02

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