Qureshi 2014

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Heat exchanger inventory cost optimization for power cycles with one

feedwater heater

Bilal Ahmed Qureshi, Mohamed A. Antar, Syed M. Zubair

Mechanical Engineering Department, KFUPM Box # 1474, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

a r t i c l e i n f o

Article history:

Received 6 March 2014Accepted 7 May 2014

Keywords:

Inventory

Heat exchanger

Thermoeconomic

Optimization

Feedwater heater

a b s t r a c t

Cost optimization of heat exchanger inventory in power cycles with one open feedwater heater is under-

taken. In this regard, thermoeconomic analysis for an endoreversible power cycle with an open feedwater

heater is shown. The scenarios of constant heat rejection and addition rates, power as well as rate of heat

transfer in the open feedwater heater are studied. All cost functions displayed minima with respect to the

high-side absolute temperature ratio (h1). In this case, the effect of the Carnot temperature ratio (U1),

absolute temperature ratio (n) and the phase-change absolute temperature ratio for the feedwater heater

(U2) are qualitatively the same. Furthermore, the constant heat addition scenario resulted in the lowest

value of the cost function. For variation of all cost functions, the smaller the value of the phase-change

absolute temperature ratio for the feedwater heater (U2), lower the cost at the minima. As feedwater

heater to hot end unit cost ratio decreases, the minimum total conductance required increases.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

Using feedwater heaters, also called regenerators, to enhance

efficiency of steam power plants is a standard practice in industry

and, therefore, thermoeconomic analysis of such systems is

important. Feedwater heaters can either be open or closed type.

When the heat is transferred from the steam (bled from the tur-

bine) to the feedwater by mixing, it is considered an open type.

The advantages offered include a higher efficiency of the power

plant due to a rise in the average temperature of heat addition.

Next, it helps to prevent boiler corrosion by providing an easy

way to remove air that leaks into the condenser. Also, it lowers

the volume flow in the final turbine stages. For further details,

the work of Babcock & Wilcox [1] may be consulted along with

textbooks on Thermodynamics such as Cengel and Boles [2].

Authors such as Bejan[3,4]addressed the issue of heat exchan-ger inventory allocation for different situations such as maximizing

the efficiency. The Carnot model developed by Bejan[4]was used

by Antar and Zubair[5]to study cost optimization of power plant

heat exchanger inventory for a specified power output. The total

inventory reached a minimum when the unit cost ratio attained

unity. Sahin and Kodal [6] carried out thermoeconomic optimiza-

tion of endoreversible heat engines using a new thermoeconomic

optimization criterion i.e. power output per unit total cost.

Analytical equations for optimum working fluid temperatures,

specific power output, thermal efficiency and distribution of heat

exchanger areas were determined. The effect of relative fuel cost

was also discussed. This new criteria was later used by the authors

for irreversible heat engines[7]as well. Using profit maximization

as the objective function, exergoeconomic performance optimiza-

tion of a finite-time irreversible Carnot engine was investigated

by Chen et al.[8]. The authors derived relevant formulae concern-

ing profit and efficiency for this purpose.

Bandyopadhyay et al. [9] studied combined cycle power plant

cost optimization using irreversible Carnot-like heat engines. It

was found that the yearly plant cost rose along with a decline in

power output as the number of stages was increased. For off-

design conditions in combined cycle gas turbine power plants,

Rovira et al.[10]performed thermoeconomic optimization in heat

recovery steam generators. Based on design conditions, negligibledifference was found in the optimization results when compared

to those obtained from usual thermoeconomic models except for

the fact that, with the new model, a minor decrease was seen in

the amortization cost and design efficiency. For combined cycle

power plants, genetic algorithms have also been used for the pur-

pose of thermoeconomic optimization [1113]. Using the Second

Law of Thermodynamics, Silveira and Tuna[14]presented a ther-

moeconomic functional analysis method. Four cogeneration sys-

tems were analyzed and the system consisting of a gas turbine

with heat recovery steam generator only was found to have the

lowest exergetic production cost. Al-Sulaiman et al. [15,16]

http://dx.doi.org/10.1016/j.enconman.2014.05.028

0196-8904/2014 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +966 3 860 3135.

E-mail address:[email protected](S.M. Zubair).

Energy Conversion and Management 86 (2014) 379387

Contents lists available at ScienceDirect

Energy Conversion and Management

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n c o n m a n
http://dx.doi.org/10.1016/j.enconman.2014.05.028mailto:[email protected]://dx.doi.org/10.1016/j.enconman.2014.05.028http://www.sciencedirect.com/science/journal/01968904http://www.elsevier.com/locate/enconmanhttp://www.elsevier.com/locate/enconmanhttp://www.sciencedirect.com/science/journal/01968904http://dx.doi.org/10.1016/j.enconman.2014.05.028mailto:[email protected]://dx.doi.org/10.1016/j.enconman.2014.05.028http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://crossmark.crossref.org/dialog/?doi=10.1016/j.enconman.2014.05.028&domain=pdfhttp://-/?-

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performed thermoeconomic optimization of three trigeneration

systems using organic Rankine cycles. Formulations were pre-

sented and the systems examined which revealed that, from the

three systems, the solar trigeneration system offered the best ther-

moeconomic performance. Abusoglu and Kanoglu [17] reviewed

exergoeconomic optimization and analysis for combined heat

and power systems. A comparison of the advantages and disadvan-

tages regarding important thermoeconomic methodologies found

in the literature were made.

Xiong et al.[18]performed thermoeconomic optimization of a

600 MWe pulverized-coal-fired power plant using oxy-combus-

tion. A 10% increase in unit thermoeconomic product costs was

seen due to the extra power utilization for the oxy-combustion

system and another 10% due to its other related costs such as oper-

ation and maintenance, investment as well as interest. Bassily[19]

performed cost optimization of commercial combined cycle power

plants with triple-pressure reheating. It was determined that, for a

400 MW power plant, an annual saving of about $29.2 million

could be obtained by optimizing the net revenue.

Rosen and Dincer [20] thermoeconomically examined an electri-

calgeneratingstationfueled by coal basedon capital cost only. They

emphasized that the reason for this is that the capital cost is often

the most significant cost component and costs other than that are

often proportional to it. Thus, qualitative agreement is expected.

Forthe designand analysis of energysystems, Silveiraet al.[21] pre-

sented a thermoeconomic optimization methodology using the

exergetic production cost as the objective function. Depending on

the energy system analyzed, the various costs included operationalandcapital cost fora given amount andtype of exergy. Seyyedi et al.

[22] provided a new approach for optimization of thermal power

plants based on the exergoeconomic analysis and structural optimi-

zation method. Important advantages of this new method are its

applicability to large complex thermal systems and rapid conver-

gence. Considering various objective functions based on finite-time

thermodynamics and thermoeconomics, Durmayaz et al.[23]pre-

sented an extensive review on optimization of thermal systems.

The conclusion of the authors was that finite-time thermoeconomic

analysisneededmore workin fundamentaltheorydevelopmentand

applications as it was still in its early stages.

It was found through the literature review that, for the endore-

versible case of a power cycle with one feedwater heater, cost opti-

mization has not been considered for design and performance

evaluation purpose. Therefore, this paper aims to develop the rel-

evant endoreversible models and then perform thermoeconomic

analysis of this system. The scenarios of constant heat addition

and rejection rates, power as well as rate of heat transfer in the

open feedwater heater will be studied.

2. Mathematical framework

Following the methodology of Bejan[4] and Antar and Zubair

[5], the endoreversible form of the power cycle with one open

feedwater heater is now considered. The schematic of system

under consideration is shown in Fig. 1(a) while Fig. 1(b) shows

its T-s diagram. The purpose of the current study is to determinethe minimum of the total cost of conductance (UA) for the follow-

ing scenarios: constant rate of heat addition, power, heat transfer

in the preheater and heat rejection. The Heat Exchanger Inventory

Cost Equation (HEICE) can be written in terms of heat exchanger

unit cost parameters as[5]:

C cHUAH cLUAL cOFHUAOFH 1wherecH, cL, and cOFHare unit cost of conductance for the boiler,condenser and preheater, respectively, such that C has units of

dollars. Next,

_QH UAHTH THC 2

_QL

UA

L

T01

TL

3

_QOFH UAOFHT02 TOFH 4a

whereTOFH is the average preheating temperature and is given by:

TOFH T02 DTOFH;avg 4bwhereDTOFH,avgis the average amount of preheating and considered

as half of the total achieved. Putting Eqs. (2)(4)in Eq.(1)results in

C cH_QH

TH THC cL_QL

T01 TL cOFH_QOFH

DTOFH;avg5

Dividing throughout bycH, we get

C

cH

_QH

TH THCcL

cH

_QL

T01 TL cOFH

cH

_QOFH

DTOFH;avg 6

Nomenclature

A area (m2)F non-dimensional cost function ()G unit cost conductance ratio ()K non-dimensional quantity defined by Eq.(15f)()k1 ratio of feedwater heater to condenser entropy

change ()_m mass flow rate (kg s1)_Q rate of heat transfer (kW)s specific entropy (kJ kg1 K1)T absolute temperature (K)U overall heat transfer coefficient (kW m2 K1)

GreekC total cost ($)c unit conductance cost ($kW1 K)U1 Carnot temperature ratio ()U2 phase-change absolute temperature ratio for the

feedwater heater ()

h1 high-side absolute temperature ratio ()h2 average preheating absolute temperature ratio ()n absolute temperature ratio ()

Subscripts01 at condenser02 at phase-change in feedwater heatera constant rate of workb constant rate of heat rejection from the condenserC reversible compartmentc constant rate of heat addition in the boilerd constant rate of heat transfer in the open feedwater

heaterH hot endL cold endmin minimumOFH open feedwater heatertot total

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Let cLcH GL and

cOFHcH

GOFH, thus Eq.(6) becomes

C

cH _QH

TH THC GL_QL

T01 TL GOFH_QOFH

DTOFH;avg 7Factoring out _QH, we get

C

cH _QH 1

TH THC GL_QL=_QHT01 TL GOFH

_QOFH=_QHDTOFH;avg

" # 8

Now, fromFig. 1(b), we see that

_QOFH _m01TOFHs3 s2 9a

_Q36 _m02T02s6 s3 9b

_QL _m01T01s7 s1 9c

But _

QOFH _

Q36 since they are exchanging heat with each other

exclusively in the feedwater heater. Thus,

_QOFH_QL

_m02T02s6 s3_m01T01s7 s1 10

We see fromFig. 1(b) that, in general, (s6 s3) = k1(s7 s1) wherek1can be any number less than one. Now, Eq. (10)becomes

_QOFH_QL

k1_m02T02_m01T01

11

Applying the Clausis inequality to the internally reversible cycle

gives us

_QHTHC

_QLT01

; 12a

_QL_QH

T01THC

12b

Combining Eqs.(11), (12a), (12b), we get

_QOFH_QH

k1_m02T02_m01THC

13

Putting Eqs. (12b) and (13) into Eq. (8), we get after dividing the

right hand side by TH/TH,

C

cH

_QHTH

1

1 THCTH

GLT01THC

T01TH

TLTH

GOFHk1

_m02T02_m01THC

DTOFH;avgTH

24

35 14

Introducing the following non-dimensional quantities:

h1 THCTH

15a

h2 DTOFH;avgTH

15b

U1

T01THC

15c

U2T02

THC15d

n TLTH

15e

K k1_m02_m01

15f

Putting Eqs.(15a), (15b), (15c), (15d), (15e), (15f)into Eq.(14), and

multiplying both sides by THgives us

C

cHTH _QH 1

1

h1

GL U1U1h1

n

GOFHKU2h2 16

Beginning from Eq. (16), the next section discussesthe following

scenarios: constant power, heat addition and rejection capacities as

well as heat transfer rate in the open feedwater heater.

3. Results and discussion

In those situations in which the unit cost conductance ratios are

considered unity, it is expected that minimization of the dimen-

sionless HEICE will result in the same for the total conductance

(UAtot).

3.1. Constant power

The non-dimensional equation resulting from dividing Eq.(16)by the power is:

Fig. 1. Endoreversible power cycle with an open feedwater heater: (a) Schematic

and (b)T sdiagram.

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Fa CcH

_WTH

_QH_W

1

1 h1 GLU1

U1h1 n GOFHKU2h2

17

Keeping Fig. 1(a) in mind and using the First Law of

Thermodynamics, we get

_W _QH _QL 18

where _

Wis the total power consumed by the pumps. Now, dividingboth sides by _QHgives us

_W_QH

1 _QL_QH

19

Putting Eq.(12)in Eq.(19)gives us

_W_QH

1 T01THC

20a

The reciprocal of the above equation gives us

_QH_W

11 T01

THC

20b

Combining Eqs. (15c) and (20b) and putting them into Eq. (17)results in

Fa 11 U1

1

1 h1 GLU1

U1h1 n GOFHKU2h2

21

Eq. (21) is the non-dimensional HEICE for an endoreversible

power cycle with one open feedwater heater for the constant

power output scenario. We see that there is a direct relationship

between Fa and n as well as U2 and an inverse one with respect

to h2 and, thus, these parameters do not exhibit minima. It is

unclear whether minima exist with respect to U1 and h1. Deter-

mining the derivative ofFa with respect to U1 and then equating

it to zero gives us:

@Fa

@U1 1

1 U11

1 h1 GL

U1

U1h1 n GOFH

KU2

h2

GL nU1h1 n2" #

0 22aor

U1;min cn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 c cn

p1 ch1 22b

where

c 11 h1 GOFH

KU2h2

1

GL

Fig. 2(a) shows a plot of the cost function Faagainst U1for differentvalues ofn. It is found that the minimum value ofU1as well as the

cost rise asn increases. The reason for the increasing cost is evident

from Eq.(21)wheren is seen in the second term inside the brackets

only. As n increases, this term increases in value and, thus, Faas well.

Regardingthe minimum value ofU1 (that provides a minimum cost),

it can be seen from Eq.(22b)that this is due to the fact that U1,minis

directly proportionalto n. Furthermore,thisshowsthata lowerambi-

ent temperature would result in a lower cost if all other quantities

remained the same. Fig. 2(b) shows a plot of the cost function Faagainst U1 for different values ofh1. It is found that the minimum

value ofU1 decreases ash1 increases but the cost rises. The reason

for the increasing cost is understood from Eq. (21)where h1is seen

in the first and second terms inside the brackets. The first term con-

tainingh1is the dominant term, therefore, as h1increases, this termincreasesin value and, thus, Fa as well. Regardingthe minimum value

ofU1, it can be seen from Eq.(22b)that this is due to the fact that

U1,min is inversely proportional to h1. Also, the (1 +c) term, which

contains h1 as well, dominates the quantities containing c in the

numerator. Furthermore, this shows that a higher furnace tempera-

ture may result in a lower cost if all other quantities remained the

same. Fig. 2(c) shows a plot of the cost function Faagainst U1for dif-

ferent values ofU2. It was noted that lower values ofU2produced

lower costs while the minimum value ofU1 was not affected by

change in U2. The reason for the decreasing cost is understood from

Eq.(21)where U2 is seen in the last term inside the bracket only

and directly proportional to Fa. Regarding the minimum value ofU1not changing significantly, it can be explained from the fact that

the effect of variation in U2 is very small on c. Furthermore, this

shows that a lower feedwater heater temperature (T02) may resultin a lower cost if all other quantities remained the same.

Fig. 2. Dimensionless HEICE for specified power versus U1: (a) effect of varying n,

(b) effect of varying h1and (c) effect of varying U2.


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The function Fa seems to have a minimum with respect to the

parameter h1. Putting oFa/ oh1 equal to zero and solving, we get

h1;min 1U1

n ffiffiffiffiffiGLp 1 U1GL ffiffiffiffiffiGLp GL 1

23

It should be noted that Eq.(23)is identical to the respective deriva-

tive found by Antar and Zubair [5] though they did not plot it.

Fig. 3(a) shows the effect of different values ofn as Favaries againsth1. It is found that the optimum point ofh1shifts to a greater value

as n increasesas well as thecost.Thisbehavior issimilar to that found

in Fig. 2(a). As can be seen from Eq. (23), thisis simply due to the fact

thatn is directlyproportional toh1,min. Fig.3(b)shows theeffectof dif-

ferentvaluesofU1 as Fa varies against h1. It isseenthatthe costfunc-

tion andh1,min shift to a lower value as U1increases. This is because

U1is inversely proportional toh1,min (See Eq.(23)) and, as far as the

cost is concerned, it can be explained from the fact that, in Eq. (21),

thesecond term insidethe brackets isdominantand theterm outside

is not significantly affected by the variation in U1. It shows that

increasing the condenser temperature may result in a lower cost if

all other quantities remained the same. Fig. 3(c) shows a plot of the

cost functionFaagainsth1for different values ofU2. It is found that

the minimum value ofU1

is not affected by change in U2

and this is

because the term U2 does not exist in the expression for h1,min. It

wasnoted that highervalues ofU2 producedhigher costs and the rea-

sons are the same as was explained earlier for Fig. 2(c). Furthermore,

it shows that a lower feedwater heater temperature may result in a

lower cost if all other quantities remained the same.

3.2. Constant heat rejection rate

The non-dimensional cost equation resulting from dividing Eq.

(16)by the condenser heat transfer rate is:

Fb CcH

_QLTH

_QH_QL

1

1 h1 GLU1

U1h1 n GOFHKU2h2

24

After the relevant substitutions, we get

Fb 1U1

1

1 h1 GLU1

U1h1 n GOFHKU2h2

25

Eq.(25)is the non-dimensional HEICE for an endoreversible power

cycle with one open feedwater heater for the constant heat rejec-

tion rate scenario. We wish to find out whether the function Fbhas minima with respect to n, U1, U2, h1 and h2. Fb is directly pro-

portional to n as well as U2 and inversely proportional to h2 and,

thus, no minima exist. Determining the derivative ofFbwith respect

to U1 and then equating it to zero gives us:

@Fb@U1

1U1

1

1 h1 GLU1

U1h1 n GOFHKU2h2

GL nU1h1 n2" #

0 26In the above equation, all terms are positive and, therefore, a prac-

tical minimum is not possible. The result of taking the derivative of

Fbwith respect toh1, in this case, is identical to Eq.(23).Fig. 4(a)(c)

are plotted for conditions identical to those of Fig. 3(a)(c). It is

found that the behavior for this cost function is qualitatively the

same as Fa and the only difference is in the values. It can be

explained from the fact that the terms inside the brackets for both

the cost functions are identical while there is a minor difference in

the term outside it for both the cost functions are identical while

there is a minor difference in the second bracketed term.

3.3. Constant heat addition rate


(16)by the boiler heat transfer rate is:

Fc CcH

_QHTH 1

1 h1 GLU1

U1h1 n GOFHKU2h2

27

Eq.(27)is the non-dimensional HEICE for an endoreversible power

cycle with one open feedwater heater for the constant heat addition

rate scenario. No minima exist for Fcwith respect to n , U2 and h2.

Determining the derivative ofFcwith respect to U1and then equat-

ing it to zero gives us:

@Fc

@U1 n

n U1h120

28

Fig. 3. Dimensionless HEICE for specified power versus h1: (a) effect of varyingn, (b)effect of varying U1 and (c) effect of varying U2.

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As all terms in the equation are clearly positive on one side, there-

fore, no minimum exists. The result of taking the derivative of Fcwith respect to h1, in this case, is identical to Eq. (23).Fig. 5(a)(c)

are plotted for conditions identical to those of Fig. 3(a)(c). It is

found that the behavior for this cost function is qualitatively the

same as Fa and the only difference is in the values. The reason for

this is that the quantities inside the brackets for both the cost

functions (i.e. Fa and Fc) are identical while there is a small

difference outside it which is a multiplying factor only.

3.4. Constant heat transfer rate in feedwater heater


(16)by the rate of heat transfer in the open feedwater heater is:

Fd C

cH _QOFHTH

_QH

_QOFH

1

1 h1 GLU1

U1h1 n GOFHKU2h2

29a

Substituting Eqs.(13), (15d) and (15f)in Eq.(29a), we get

Fd CcH

_QOFHTH 1

KU2

1

1 h1 GLU1

U1h1 n GOFHKU2h2

29b

The functionFdis also found to have no minimum with respect to n

and h2 as it is clearly inversely proportional to them. Determining

the derivative ofFdwith respect to U1and then setting it equal to

zero results in Eq. (28) and, therefore, no minimum exists. Now,

determining the derivative of Fd with respect to U2 and equating

it to zero gives:

@Fd@U2

GOFH 1h2U2

1KU2

2

1

1 h1 GLU1

U1h1 n GOFHKU2h2

0

30a

or, after simplification, gives

Fig. 4. Dimensionless HEICE for specified heat rejection rate versus h1: (a) effect of

varying n, (b) effect of varying U1 and (c) effect of varying U2.

Fig. 5. Dimensionless HEICE for specified heat addition rate versus h1: (a) effect of

varying n, (b) effect of varying U1 and (c) effect of varying U2.


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@Fd@U2

11 h1 GL

U1U1h1 n 0 30b

As all terms in the equation are clearly positive, therefore, no min-

imum exists. Finally, oFd/oh1was found to be the same as in Eq. (23).

Fig. 6(a)(c) are plotted for conditions identical to those ofFig. 3(a)

(c). It is found that the behavior for this cost function is qualitatively

the same asFaand the only difference is in the values. This is due to

the fact that the quantities inside the brackets for both the costfunctions (i.e. Fa and Fd) are the same and, although the term out-

side it is different, it only acts a multiplying factor resulting in a

change of value but not the behavior.

3.5. Effect of unit cost ratios

The purpose of the analysis in this section is determining the

minimum of the total conductance for specified power production

and this scenario is chosen due to its practical nature. Optimum

values for U1 and h1 will be determined from Eqs. (22b) and (23),

respectively.

In order to provide an illustrative example for the purpose of

showing model applicability, it is required that ratios of the con-

ductance costs of each heat exchanger to the total cost be deter-

mined. For the sake of brevity, only the final expressions are

shown below:cHUAH

C 1 GLU11 h1

U1h1 n GOFHKU21 h1

h2

131

cLUALC

1GL

U1h1 nU11 h1 1

GOFHGL

KU2h2

U1h1 nU1

132

Fig. 6. Dimensionless HEICE for specified heat transfer rate in the feedwater heaterversush1: (a) effect of varying n, (b) effect of varying U1and (c) effect of varying U2.

Fig. 7. Example of all conductances versus unit cost ratio of cold to hot end: (a) atGOFH= 1, (b) at GOFH= 0.5 and (c) at GOFH= 0.1.

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cOFHUAOFHC

1GOFH

h2

KU21 h1 GLGOFH

h2

KU2

U1U1h1 n 1

133

A comparison of the above equations with the work of Antar and

Zubair[5]clearly shows that, due to the presence of the feedwater

heater unit conductance ratio (GOFH) term, Eqs.(31) and (32)do not

reduce to mere dependence onGL. Now, we use the same values for

TH, cL, n and _W; as used by Antar and Zubair [5], for our example(SeeFig. 7(a)(c)). It should be noted that, in power systems, the

unit cost of the boiler (cH) would be higher than the other heatexchangers. Therefore, it is appropriate to focus on values less than

unity for GOFHin our investigation. It is noted that, in Fig. 7(a)(c),

compared to Antar and Zubair [5], the total conductance curve isslightly asymmetric. This is due to the fact that the conductance

of the feedwater heater is larger than the other heat exchangers

and varies non-linearly with GL. The difference at the ends is

1.32 kW/K when GOFHis unity; though it increase to 4.25 kW/K at

GOFH= 0.1. See Table 1 for a more detailed comparison. It is also

noted that when all unit cost ratios are at unity, the minimum total

conductance is also obtained at unity. The reason is that, in this

case, the unit cost of each heat exchanger becomes the same which,

in turn, results in all conductances influencing the total cost (See Eq.

(1) by equal weightage. Another observation is that, as GOFHdecreases, the minimum of the total conductance required

increases and that it is obtained at lower values ofGL. These varia-

tions are found to be non-linear such that when GOFH decreases

from 1 to 0.5, there is only a 0.88% increase in the minimum totalconductance while it increases by 5.7% when GOFH decreases from

0.5 to 0.1.

4. Conclusions

The optimization problem studied for power cycles with one

feedwater heater has two significant minima i.e. h1 and U1. Some

important conclusions are as follows:

Since theF-value is based on an endoreversible power cycle, for

specified temperatures and flow rates, it predicts the minimum

initial heat exchanger cost.

All cost functions displayed a minimum for the parameter h1

that was the same for all cases. Only the value of the cost func-tions was different.

For variation of all F-values with respect to h1and U1, the smal-

ler the value ofU2, lower the cost at the minima.

For variation of all F-values with respect to h1, the effect of dif-

ferent values ofn is qualitatively the same. This is found to be

respectively true for the effect of different values of U1 and

U2. Furthermore, in these cases, the constant heat addition sce-

nario resulted in the lowest value cost function.

For the system investigated, the minimum with respect toU

1exists only for the scenario of specified power cost function

i.e.Fa.

No minima were found with respect to n, h2and U2.

While taking into account the effect of the unit cost ratios, the

following conclusions can be made:

Ratios of the conductance costs of each heat exchanger to the

total cost were attained.

If cold and hot end heat exchanger unit costs are the same i.e.

GL= 1, the total conductance was minimized at GOFH= 1.

At optimum conditions, the total conductance is not equally

divided among the heat exchangers. But when all unit cost

ratios are unity, the hot and cold end conductances are equal

in value.

Acknowledgements

The authors acknowledge the support provided by King Fahd

University of Petroleum & Minerals through the project IN121042.

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Table 1

Comparison of Antar and Zubair[5]with current work for total conductance required.

GL cLcH

UAtota UAtot

b (GOFH= 1) UAtotb (GOFH= 0 .5) UAtot

b (GOFH= 0.1)

0.1 26.783 79.403 76.757 78.084

0.2 22.895 73.261 71.932 74.580

0.3 21.380 70.878 70.175 73.390

0.4 20.599 69.654 69.345 72.876

0.5 20.148 68.949 68.922 72.650

0.6 19.875 68.525 68.713 72.5720.7 19.711 68.270 68.630 72.577

0.8 19.616 68.123 68.625 72.632

0.9 19.568 68.050 68.672 72.719

1 19.555 68.029 68.756 72.828

2 20.148 68.922 70.220 74.218

3 21.067 70.272 71.808 75.579

4 21.999 71.610 73.263 76.800

5 22.895 72.870 74.580 77.904

6 23.748 74.047 75.783 78.915

7 24.559 75.149 76.890 79.851

8 25.333 76.183 77.918 80.726

9 26.073 77.160 78.880 81.550

10 26.783 78.084 79.785 82.331

a Antar and Zubair [5].b Current work.

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