GHTMDD - 270Received: February 20, 1995

Bulletin of the Chemists and Technolo~ts of Macedonia, Vol. 15, No. 1, pp. 39 - 44 (1996)

ISSN 350 - 0136UDC 66. 011 : 66.045.1

Original scielltific paper

OPTIMUM DESIGN OF SHELL-AND-TUBE HEAT EXCHANGER

Liljana Markovska, Vera Mesko~Radmila Kiprijanova, Aleksandar GrizoFaculty of Technology and Metallurgy, The "Sv. Kiril & M etodij" University, 91000 Skopje, Republic of Macedonia

Optimization of shell-and-tube-heat exchanger is accomplished by use of the OPTIMIZER software package.The objective function is defined together with the implicit constraint. Thesimultaneous equation solving method

is used to solve the equations that describe the process. The advantages of the simultaneous equations solving approachare that: (1) it is a natural way to specify a problem since the design problem is by nature an optimization problem andthe engineer does not have any other criterion for specifying many arbitrary variables, (2) it is easy to specify variablesand constraints, and (3) it can handle highly integrated systems since all equations are solved simultaneously.

The Extended Complex algorithm is chosen for such optimization study. The optimal value of the objectivefunction and appropriate design variables are obtained.

Key words: optimum design; shell-and-tube heat exchanger; optimization; Complex method

INTRODUCTION

An optimum design is based on the best or mostfavorable conditions. In almost every case, these op-timum conditions. can ultimately be reduced to aconsideration of costs or profits. Thus an optimumeconomic design could be based on conditions givingthe least cost per unit of time or the maximum profitper unit of production. When one design variable ischanged, it is often found that some costs increase andothers decrease. Under these conditions, the total costmay go through a minimum at one value of theparticular design variable, and this value would beconsidered as an optimum.

'!Wotypes of quantitative problems are common-ly encountered by the design engineer when he isdealing with heat-transfer calculations. In the firsttype, all of the design variables are set, and thecalculations involve only the determination of theindicated non variant quantities. By choosing variousconditions, the engineer could ultimately arrive at afinal design that would give the least total cost for fixed

charges and operation. Thus, the second type ofquantitative problem involves conditions in which atleast one variable is not fixed, and the goal is to obtainan optimum economic design.

In general, increased fluid velocities result inlarger heat-transfer coefficients and, consequently,less heat-transfer area and exchanger cl;)stfor a givenrate of heat transfer. On the other hand, the increasedfluid velocities cause an increase in pressure drop andgreater pumping costs. The optimum economic designoccurs at the conditions where the total cost is a

minimum. The basic problem, therefore, is to mini-mize the sum of the variable annual costs for the

exchanger and its operation.The objective function is the total annual cost for

heat exchanger.

In this paper the optimum design of a heatexchanger is developed by use of a new technique.

SOFTWARE REQUIREMENTS

The optimization of the heat exchanger networkis accomplished using the ChemEng Software OP-TIMIZER [1].

This package consists a set of ,optimizationroutines specially selected for solving the chemicalengineering problems. Real engineering optimizationproblems are characterized by being highly con-strained, both with explicit and implicit constraints,and by having many discontinuities [2,3].

OPTIMIZER consists of four algorithms withdifferent characteristics which are suitable for dif-

ferent types or different stages in the optimization ofengineering systems - particularly the optimization ofprocesses [4,5]. .

The most suitable method for optimizing a heatexchanger is the Extended Complex Method. Themethod is used for nonlinear optimization problemswith constraints [6]. But application of this method tounconstrained problems is also possible. This method

40 L. Markovska, V. Mesko, R Kiprijanova, A. Grizo

has the advantage of easy implementation on a digitalcomputer, of easy handling capability for implicitinequality constraints, and of not requiring computa-tion of any derivatives.

The simultaneous equation-solving method isused to solve the equations that describe the process.The basic idea of this method is simply to collect all theequations describing the flowsheet and solve them asa large system of non-linear algebraic equations.

Mathematically the problem can be stated as

solvef(x,u) = 0

with g(X,u) :s 0where

x is vector of dependent (state) variables

u is vector of independent (decision) variables

f(x, u) is the set of process model equations

g(x, u) is the set of inequality or equality con-straints.

Alternatively, the problem may be formulatedmathematically as an optimization problem,

minimize h(x, u)

with f(x, u) = 0

g(x, u) :s 0

where h(X, u) is the objective function andf(x, u) andg(x, u) have the same meaning as before. The equalityconstraints are the same set of equations describedabove, but rather than specify the decision variablesarbitrarily, they are selected to minimize the objectivefunction.

HEAT EXCHANGER PROBLEM

The design of a heat exchanger involves initialconditions in which the following variables are known:

- process-fluid rate of flow

- change in temperature of process fluid

- inlet temperature of utility fluid (for cooling orheating).

With this information, the engineer must preparea design for the optimum exchanger that will meet therequired process conditions. Ordinarily, the followingresults must be determined:

- heat-transfer area;

- exit temperature and flow rate of utility fluid;

- number, length, diameter, and arrangement oftubes;

- tube-side and shell-side pressure drops.The variable annual costs of importance are the

fixed charges on the equipment, the cost for the utilityfluid, and the power cost for pumping the fluidsthrough the exchanger. The total annual cost foroptimization, therefore, can be represented by thefollowing equation [7]:

Tv = AnTpCAn + QfI,CS + Arf/i,Ce + ArfJIFn

(1)

The terminology used in solving the problem andthe values of the variables are defined at the end of thepaper.

Under ordinary circumstances,the effect of tubediameter on total cost at the optimum operatingconditions is not great, and a reasonable choice of tubediameter, wall thickness, and the tube spacing can bespecified at the start of the design. Similarly, thenumber of tubes is usually specified. If a change inphase of one of the fluids occurs (for example, if the

utility fluid is condensing steam), solution of Eq. (1)for optimum conditions can often be simplified. Forthe case of no change in phase, the solution canbecome complex, because the velocities and resultingpower costs and heat-transfer coefficients can bevaried independently over a wide range of values. Inthe following analysis, the general case of steady-stateheat transfer in shell-and-tube exchangers with nochange in fluid phase is considered, and a specifiedtube diameter, wall thickness, number of passes, andarrangement of baffles and tubes are assumed.Simplifications are indicated for the common condi-tions of turbulent flow.

Choice of independent variables

The heat-transfer area An can be related to theflow rates and the temperature changes by an overallheat balance and tt!e rate equation.

Q = QScpit2 - tl) = Qrcpr(tl' - t2') = KnAn !:!..tsr

(2)

From Eq. (2)

- Q

Qs - cps [ (!:!..tl- !:!..t2)+ (tl' - t2

where!:!..tI = t2' - tI and!:!..t2 = tI' - t2' Since Q, cps'!:!..tI'tI', and'tz' are constant, Qs is a function only ofthe independent variable !:!..tz.The area An is known ifKn and !:!..tsrare fixed. The overall coefficient is knownif the inside and outside film coefficients, av and an'are fixed, and for a given number of tube passes, !:!..tsrvaries only with changes in !:!..tz.Therefore, An is afunction of av ' a,l' !:!..tsr'as shown by the followingequations:

(3)

!:!..tsr=~=~ (~+-L+RIQ KnAn An lDvav an )

(4)

BulI.Chem.TechnoI.Macedonia, 15, 1, p.39-44(I996)

Optimum design of shell-and-tube heat exchanger 41

FT(!!.tZ - !!.t1)= ~ = ~ (~ + -.l. + R

]KnAn An ~Dvav an(5)

With no change in phase av depends on mass flowrate Gv inside the tubes, as it can be seen from thefollowing equation for turbulent flow (Re> 10000):

~~a,(D~G'r (;f (:XI4The relevant equation for an depending on shell-

side mass velocity is:

(6)

( )

n

( )

0.33

~=;:D~Gn if (7)

The Eq. (6) is valid for 10000 < Re < 120000,0.7 < Pr < 120 and LID = 60. In this rangesav = 0.023, n = 0.8 and m = 0.3. The Eq. (7) is validfor 20000 < Re < 40000, an = 0.33 if the tubes arestaggered, an = 0.26 if the tubes are in line andn = 0.6.

The variables in Eq. (1) areAn, Q, El" En' andtheir values are set if av ' an and !!.t2are known. Theoptimizing procedure is simplified by retaining thefollowing variables: av' an !!.t2andAn'

Defining relevant equations for optimization study

The first step is to express Eq. (1) in terms of thefundamental variables. The following relationships forpower loss inside tubes and power loss outside tubesare suitable for conditions of turbulent flow andshell-side fluid flowing in a direction normal to thetubes [7,8]:

E =.It a3.5v 't'v v (8)

E = "It a4.75v 't'n n (9)

where

-

[

12000D\~.5 ,u~.83(;.lzl,uv )°,63

]1jJv- Bv D ZA2.33 1.17 (10)

ge nPv v cp.'

B N N

[

2a D DO.75p.75 Il1.4Z

]

rp = n r 0 n c n s fn (11)n 11 Ne ;TT;a4.75 g p2 A3.17 c1.58

p n c n fn Pro

All the terms in the brackets are set by the design

conditions. The values of Bv and Bn l11p are notcompletely independent of the film coeffIcients, butthey do not vary enough to be critical. As a firstapproximation, Bv is usually close to 1,and Bll is takento be equal to the number of baffle passes~. Thevalueof the safety factor Fs is taken as 1.6 for designestimates. The ratio NrN/Nc depends on the tubelayout and baffle arrangement. For rectangular tubebundles and segmental baffles, the ratio is equal to one.For other tube layouts and segmental baffles, the ratiois usually in the range of 0.6 to 1.2.

The relevant equations for the optimizationstudy are:

Objective function:

QH.CT =A T C + 1 s +

v n F An C (!!.t -!!.t + t I - t I)ps 1 2 1 2

+AnHjCe1jJva~.5 +Anrpna~.75 HjCn (12)

Implicit constraint:

FT(!!.t2 - !!.t1) 1

(

D n 1

J

-- -+-+R =0Q In (!!.tZI!!.t1) An Dvav an

(13)

OPTIMIZATION

The optimization of the objective function Eq.(12) with implicit constraint Eq. (13) is possible usingclassical methods of searching the optimum. TheLagrange multiplier method is used for developmentof the optimum design for a shell-and-tube heatexchanger [7,8]. The optimum values of the variablesare obtained by partial differentiation of the objectivefunction incorporating the Lagrange multiplier,equating the derivatives to zero and solving theseequations.

In the present paper the optimization of shell-and-tube-heat exchanger is accomplished by use ofOPTIMIZER software package. The advantage ofusing this software is the saved time and efforts for suchanalysis.

rnac.xeM.TeXHon.MaKe,[{oHHja, 15, 1, c.39-44(1996)

Ewmple

A gas under pressure with properties equivalentto air must be cooled from 65°C to 38 Qc. Coolingwater is available at a temperature of 21°C. On thebasis of the following data and specifications, it isnecessarily to be determined the tube length and thenumber oftubes for the optimum exchanger which willhandle 10000 kg/h and will operate 8760 h per year.

Exchanger specification:

- steel shell-and-tube exchanger with cross-flowbaffling,

- cooling water passes through shell side of unit,

- one tube pass and counter current flow,

- outside diameter of the tubes 25 mm,

42 L. Markovska, V. Mesko, R Kiprijanova, A. Grizo

- inside diameter of the tubes 20 mm,

- 24 mm triangular pitch. Tubes are staggered.

Costs:

- purchased cost per surface unit is 8500 den/m2,

- installation cost equals 15% of purchased cost,

- annual fixed charges including maintenanceequal 20 % of installed cost,

- cost for cooling water (not including pumpingcost) is 1.5 den/m3,

- cost for energy supplied to force the coolingwater and the gas through exchanger (including effectof pump and motor efficiency and cost) is 5 den/kWh

Operating conditions:

- aver~e absolute pressure of gas in exchanger is98.1 10 N/m ,

- correction factors Bv and Bn are: Bv = 1.2Bn =np ,

- safetyfactor Fs for the outside filmcoefficientis 1.3,

- fouling coefficient for cooling water is 8400W/mzK,

- fouling coefficient for gas is 11200 W/mZ K,

- at the optimum conditions flow on tube sideand shell side is turbulent,

- the factor N,NoINc = 1.

Assumptions:

- exit temperature of exit water 44°C,

-average M over cooling water film 10 % of total!1t,

- average!1t over air film 85 % of total M,

- temperatures: tl' = 65°C, tz' = 38°C,tl = 21°C, tz = 44°C, !1tl = tz' - tl = 17°C,

average bulk water temperature 32.5 DC,

average bulk gas temperature 51.5 DC,

average water film temperature 33.4 DC,

inside wall temperature 35.4 DC,

- physical properties for water and air areselected due to the correspond temperatures.

The dimension of the investigated problem isfour and it is assumed:

!1tz -+Xl, av -+Xl, an -+ X3, An-+X4

Thking into account all specifications and as-sumptions the objective function Eq. (12) takes theform:

OBJFN =1955*X4+886950/( 44- Xl)+

+ 56781.8*X23.5*X4+0.33hX34.75*X4(14)

with implicit constraint:

Xl - 17 1

(

1.25 1

)78.571n(Xl/17) - X4 X2 + X3 + 0.29 = 0(15)

The objective function Eq. (14) and the implicitconstraint Eq. (15) have to be incorporated in theprogram segment OPTMOD*.FOR in FORTRANnotation. The way of introducing the implicit con-straint in the subroutine OPTMOD* .FOR is impor-tant. The software is designed for constraints ofunequality type. In this paper the constraint (Eq. 15)has to be written in the form X4=G(Xl, X2, X3),where X4 denotes the heat transfer area. The bounds

of the constraint in the optimization procedure aretaken 0 and 0.01 which is satisfactorily error of estima-tion the heat transfer area. The FORTRAN subroutine

source code must be compiled with the same compileras the other object segments and then the program canbe linked.

RESULTS OF OPTIMIZATION

The response surface methodology is used foroptimization the objective function and the mostsuitable a~gorithm is the Extended Complex method.This algorithm consists of iteration steps, the maximalnumber of which is chosen as a stopping criterion. Inthis investigation the number of iterations is 20.

The input data of the design variables (lowerbounds, upper bounds and starting points) used duringthe optimization are selected due to the physicalunderstanding of the process. The optimization isaccomplished by use of ten different starting points.The obtained optimum points depends on the valuesof the variables at the starting points as it is shown inthe Thble 1. In this table the results from only fivedifferent points are given. The applied algorithm is

repeated 2-5 times to reach reproducible results. Sothe number of iterations to reach the optimum pointsis at least 100.

Studding the obtained results in Thble 1, theresult at the starting point 5 isacceptable. The minimalvalue of the objective function is 96027.2 and thevariables are Xl =20.2391, X2=0.306, X3=3.4653 andX4=19.721.

Introducing the appropriated assignments theoptimum design variables are: An = 19.721 mZ, av =0.306 kW/mz K, an =3.4653 kW/mz K, !1tz = 20.2391Dc. For shell-and-tube heat exchanger the number oftubes and their length might be calculated. For thisexchanger the total annual variable cost and its opera-tion is 96027.2 deniyear.

BulI.Chem.TechnoI.Macedonia, 15, 1, p.39-44(1996)

Optimum design of shell-and-tube heat exchanger 43

Ta b Ie I

Results of optimization of shell-and-tube heatexchanger

NOTATION

Tv total annual variable cost for heat exchanger andits operation, den/year

An area of heat transfer outside of tubes, m2

TF annual fixed charges including maintenance, ex-pressed as a fraction of initial cost for comple-tely installed unit

CAn installed cost of heat exchanger per unit of outsi-de-tube heat-transfer area, den/m2

rmc.xeM.TexHoJl.MaKeJ1oHl1ja, 15, 1, c.39-44(1996)

Cs cost of utility fluid, den/kg

Ce cost for supplying 1 kW energy to pump fluid flo-wing through inside of tubes, den/m kg

Cn cost for supplying 1 kW energy to pump fluid flo-wing through shell side of unit, den/m kg

Qs flow rate of utility fluid, kg/h

Hi hours of operation per year, hiyear

Ev power loss inside tubes per unit of outside tubearea, kW/h m2

En power loss outside tubes per unit of outside tubearea, kW/h m2

cps heat capacity of utility fluid, W/kg °c

cp, heat capacity of process fluid, W/kg °cQ, flow rate of process fluid, kg/h

t2' - tl' temperature-difference of process fluid, °c

t2 - tl temperature-difference of utility fluid, °c

Kn overall coefficient of heat transfer based on out-side tube area, W/m2 °c

Msr mean temperature-difference

Dn outside diameter of tubes, m

Dv inside diameter of tubes, m

av inside film coefficient, W/m2oC

an outside film coefficient, W/m2oC

R combined resistance of tube wall and scaling ordirt factors, W/m2oC

FT correction factor on logarithmic-mean tempera-ture difference for counterflow to a,ccount fornumber of passes; FT = 1 if unit is counterflowand single pass on shell and tube sides, dimensi-onless

Gv mass flow rate inside the tube, kg/h m2

Gfl shell-side mass velocity across the tubes, kg/h m2

cp heat capacity of fluid, KJ/kg °cA thermal conductivity, W/m2oC

f.1 viscosity of fluid (subscript z indicates evaluationat wall temperature), Ns/m2

Fs safety factor to account for bypassing effects

1/Jv factor for evaluationEv' dimensionless1/Jfl factor for evaluation En' dimensionlessBv correction factor, dimensionless

Bfl correction factor, dimensionless

np number of baffles plus 1,dimensionlessNr number of rows of tubes with flow across shell

axis, dimensionless

Ne total number of tubes in exchanger, dimensionless

No number of clearances between tubes for flow ofshell-side fluid across shell axis, dimensionless

Dc clearance between tubes to gtve smallest free areaacross shell axis, dimensionless

LOWER UPPER STARTING OPTIMUMBOUND BOUND POINT (1) POINT

Xl 18 25 19.5 18.5603

X2 0.1 0.5 0.3 0.3418

X3 2 5 2.8 2.9308

X4 16 25 19,5 18.9504

OBJFN 98089.1

STARTING OPTIMUMPOINT (2) POINT

Xl 18 25 20 20.1853

X2 0.1 0.5 0.3 0.3440

X3 2 5 2.8 2.7672

X4 19 25 19 18.1430

OBJFN 98064.2

STARTING OPTIMUMPOINT (3) POINT

Xl 18 25 21 21.4853

X2 0.1 0.5 0.3 0.3062

X3 2 5 4 3.8977

X4 16 25 18 18.9770

OBJFN 97632.3

STARTING OPTIMUMPOINT (4) POINT

Xl 18 25 21 19.5623

X2 0.1 0.5 0.25 0.3250

X3 2 5 3.4 2.7797

X4 16 25 22 19.3453

OBJFN 96431.1

STARTING OPTIMUMPOINT (5) POINT

Xl 18 25 21.5 20.2391

X2 0.1 0.5 0.3 0.3060

X3 2 5 3.5 3.4653

X4 16 25 20.5 19.7210

OBJFN 96027.2

44 L. Markovska, V.Mdko, R Kiprijanova, A. Grizo

REFERENCES

[1] L. M. Rose, OPTIM IZER, Optimization Package for ChemicalEngineers; ChemEng Software & Services. Ltd., Version 5.1,Beaminster, England (1992).

[2] L. M. Rose: The Application of Mathematical Modeling toProcess Development arid Design, Applied Science Publishers,London (1974).

[3] D. M. Himmelblau, K. B. Bischoff, Process Analysis andSimulation -Detenninistic Systems, lohn and Wiley & Sons, Inc.,New York (1968).

[4] D. l. Wilde, C. S. Beightler, Foundation of Optimization,Prentice - Hall. Inc. (1967).

[5] R. lain, The art of Computer Systems Pelfonnance Ana~ysis, lohnWiley & Sons, New York (1991).

[6] Y. Bard, Non-linear Parameter Estimation, Academic Press(1974).

[7] M. S. Peters, K. D. Timmerhous, Plant Design and Economiesfor Chemical Engineers, McGraw-Hill Book Company, NewYork, 1980.

[8] A. Grizo, V. Mdko, Proektiranje i ekonomika na hemiskiposrrojki, Nasa kniga, Skopje, 1990.

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BuII.Chem.TechnoI.Macedoni3, 15, 1, p.39-44(1996)