second law efficiencyanalysis

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Second Law Efficiency Analysis of Heat Exchangers

Journal: Heat Transfer - Asian Research

Manuscript ID: Draft

Wiley - Manuscript type: Original Article

Date Submitted by the Author: n/a

Complete List of Authors: K, Manjunath; Delhi Technological University, Department of MechanicalEngineeringKaushik, S.C.; Indian Institute of Technology, Centre for Energy Studies

Keywords:heat exchangers, exergy analysis, energy efficiency, condenser, heattransfer, pressure drop

Abstract:

Analytical analysis of unbalanced heat exchangers is carried out to study

the second law thermodynamic performance parameter through second lawefficiency by varying length-to-diameter ratio for counter flow and parallelflow configurations. In a single closed form expression, three importantirreversibilities occurring in the heat exchangers namely due to heattransfer, pressure drop and imbalance between the mass flow streams are

considered which is not possible in the first law thermodynamic analysis.

The study is carried out by giving special influences on geometriccharacteristic like tube length-to-diameter dimensions; working conditionlike changing heat capacity ratio, changing the value of maximum heatcapacity rate as hot stream and cold stream separately and fluid flow typei.e. laminar and turbulent flows for fully developed condition. Optimumheat exchanger geometrical dimension namely length-to-diameter ratio canbe obtained from the second law analysis corresponding to lower totalentropy generation and higher second law efficiency. Further, second lawefficiency analysis is carried out for condenser and evaporator heatexchangers by varying effectiveness and number of heat transfer units fordifferent values of inlet temperature to reference temperature ratio by

considering heat transfer irreversibility.

Heat Transfer - Asian Research


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K. Manjunath1,*

and S.C. Kaushik2

1Department of Mechanical Engineering, Delhi Technological University, Bawana Road, New Delhi 110042, India

*Corresponding author, Email: [email protected]

2Centre for Energy Studies, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi - 110016, India

Email: [email protected]

Abstract

Analytical analysis of unbalanced heat exchangers is carried out to study the second law thermodynamic

performance parameter through second law efficiency by varying length-to-diameter ratio for counter flow

and parallel flow configurations. In a single closed form expression, three important irreversibilities

occurring in the heat exchangers namely due to heat transfer, pressure drop and imbalance between the

mass flow streams are considered which is not possible in the first law thermodynamic analysis. The study

is carried out by giving special influences on geometric characteristic like tube length-to-diameter

dimensions; working condition like changing heat capacity ratio, changing the value of maximum heat

capacity rate as hot stream and cold stream separately and fluid flow type i.e. laminar and turbulent flows

for fully developed condition. Optimum heat exchanger geometrical dimension namely length-to-diameter

ratio can be obtained from the second law analysis corresponding to lower total entropy generation and

higher second law efficiency. Further, second law efficiency analysis is carried out for condenser and

evaporator heat exchangers by varying effectiveness and number of heat transfer units for different values

of inlet temperature to reference temperature ratio by considering heat transfer irreversibility.

Keywords:Heat exchangers, second law efficiency, irreversibilities, streams imbalance loss, condenser,

evaporator

1. Introduction

Heat Exchangers have got wide applications and play a major role in energy conservation opportunity.

Energy waste in any form results in reduction of available work from the energy resources considered. The

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losses due to process irreversibility can be calculated using second law analysis. The property exergy

serves as a valuable tool in determining the quality of energy and comparing work potentials of different

energy sources or systems. There exists a direct proportionality between irreversibility, quantity of entropy

generated and the amount of available work lost in the process. Second law analysis seeks to minimize

these losses by keeping the entropy generated to a minimum. A realistic design approach for systems is to

base the design on minimum entropy production. In other words, entropy generation can be used to

determine quantitatively the quality of thermal energy transformation(Bejan [1]). Therefore, in the analysis

and design of heat exchangers, it is essential to give due consideration to the rate of irreversible entropy

generation as well as available energy or exergy destruction process.

Exergy analysis of heat exchangers has been carried out by several investigators. In these investigations,

dimensionless exergy measure has been used in different forms. Second law efficiency of heat exchanger

was introduced by Bruges [2] and defined as the ratio of availability (exergy) gained by the cold stream to

availability (exergy) donated by the warm stream. Golem and Brzustowski [3] examined the irreversibility

of heat exchangers using the rational effectiveness, and extended this concept to the local level. Mukherjee

et al. [4] proposed the use of merit functionto evaluate heat exchangers, and defined the merit function as

the ratio of exergy transferred to the sum of exergy transferred and exergy destroyed. The exergy

destruction number, which is the ratio of the non-dimensional exergy destruction number of the augmented

surfaces to that of the unaugmented one, can be used to evaluate heat transfer enhancement devices as

analyzed by Prasad and Sen [5]. The exergy analysis method including the non-dimensional exergy

destruction, the exergy destruction number, and the heat transfer improvement number was used to

determine the performance of several wire-coils inserts in forced convection heat transfer by Prasad and

Sen [6]. The analysis of counter flow heat exchanger using the relations of rational efficiency and

effectiveness has been carried out by Cornelissen et al. [7] using heat transfer irreversibility.

Das and Roetzel [8] presented a second law analysis for thermally dispersive flow through a plate heat

exchanger using the specific irreversibility. San and Jan [9] studied a second law analysis of a wet cross

flow heat exchanger for various weather conditions. Mahmud et al. [10] analytically investigated the first

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and second law characteristics of fully developed non-Newtonian fluid flow and heat transfer inside a

cylindrical annular space. Durmus [11] experimentally studied the heat transfer and exergy loss in a

concentric heat exchanger with snail entrance. The effects on heat transfer, friction factor and

dimensionless exergy loss were investigated experimentally by mounting helical wires of different pitch in

the inner pipe in a double pipe heat exchanger by Akpinar [12]. Gupta et al. [13] defined the fractional

exergy loss differently by dividing the irreversibility by maximum heat that can be transferred ideally so

that one can get better understanding about the quantitative values of exergy loss in respect of maximum

heat transferred. San [14] considered the exergy change rate in an ideal gas flow or an incompressible flow

to analyze heat exchanger. Experimental and theoretical investigations on the entropy generation, exergy

loss of a horizontal concentric micro-fin tube heat exchanger are presented by Naphon [15]. The effects of

the technical and economical parameters on the general and optimal thermoeconomical performances have

been discussed.

A multi-objective exergy-based optimization through a genetic algorithm method is conducted to study and

improve the performance of shell-and-tube type heat recovery heat exchangers, by considering two key

parameters, such as exergy efficiency and cost by Hajabdollahi et al. [16]. Comparison of a constructal heat

exchanger and normal heat exchanger is analyzed by Manjunath and Kaushik [17] using second law

analysis. Analysis is carried out by considering the three irreversibilities due to heat transfer, pressure drop

and production of the materials and the construction of the heat exchanger. Based on constructal theory,

entropy generation minimization and second law efficiency equations are formulated by Manjunath and

Kaushik [18] for tree-shaped counter flow imbalanced heat exchanger for fully developed laminar and

turbulent fluid flow. Entropy generation number, rational efficiency and effectiveness behavior with respect

to changes in number of pairing levels and different tube length-to-diameter ratios of constructal heat

exchanger are analyzed analytically

For heat exchangers analysis, we use performance criteria based on exergy analysis, known generally as

exergetic efficiency. A yardstick of this type can be used to compare the performance of different types of

production processes with the same form of output and can help in selecting the one with the least exergy



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destruction as provided by Kotas [19]. Exergetic efficiency is also known as rational efficiency or second

law efficiency, in the case of heat exchangers, is defined as the ratio of exergy rate at the outlet of heat

exchanger to that at the inlet of the heat exchanger. Exergetic efficiency gives an indication of the degree of

thermodynamic perfection. The higher values of the exergetic efficiency mean that the heat exchanger

operates closer to ideal processes. In this work equations relating rational efficiency (second law efficiency)

with effectiveness and number of heat transfer units were considered to study the thermal performance of

heat exchangers of different configurations based on exergy analysis. The analysis has been carried out in

two different cases by considering the value of maximum heat capacity rate as hot stream and cold stream

separately.To appreciate the difference between first law and second law analysis, a comparison has been

carried out between effectiveness and rational efficiency. Rational efficiency takes care of all the

irreversibilities occurring in the system in a single closed form expression. Also, second law analysis of

condensers and evaporators is carried out using rational efficiency expressions by considering heat transfer

irreversibility.

Nomenclature

A surface area, m2

Ac cross-sectional area of tube, m2

C heat capacity ratio

cp specific heat, j/kg K

D tube diameter, m

.

E exergy rate, W

f friction factor

g acceleration due to gravity, m/s2

G mass velocity, kg / m2

s

G* dimensionless mass velocity

h enthalpy, j/kg

.

I irreversibility, W

L tube length, m



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.

m mass flow rate, kg/s

NTU number of heat transfer units

Ns entropy generation number

Nu Nusselt number

P pressure, Pa

Pr Prandtl number

P pressure drop, Pa

R gas constant, j/kg K

Re Reynolds number

gen

.

S entropy generation rate, W/K

St Stanton number

T temperature, K

To reference temperature, K

U overall heat transfer coefficient, W/m2 K

Greek symbols

effectiveness

viscosity, N s/m2

density, kg/m3

rational efficiency

Subscripts

c cold stream

h hot stream

in inlet

max maximum

min minimum

out outlet

T heat transfer component only

Superscripts

P pressure exergy component

T heat transfer exergy component



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2. Analysis

Some of the assumptions made while carrying out the analysis are 1. the flow is fully developed, 2. the

pressure drops are mainly due to friction along the straight cross section of the heat exchanger, 3.

neglecting the local pressure drops associated in the joints of tubes, 4. one tube of hot stream is right next to

its counterpart in the cold stream and have excellent thermal contact. 5. The working fluid is assumed to be

ideal gas.

The type of exergetic efficiency called the rational efficiency is defined by Kotas [19] as the ratio of

desired exergy output to exergy used,

used

.

putdesiredout

.

E

E= (1)

putdesiredout

.

E is the sum of all exergy transfers from the system, which is regarded as constituting the desired

output, plus any by-product, which is produced by the system. The desired output is determined by

examining the function of the system. used.

E is the required exergy input for the process to be performed

which can be expressed in terms of irreversibilities as,

.

putdesiredout

.

used

.

IEE +=

Alternative form of the rational efficiency can be obtained as,

.

putdesiredout

.

putdesiredout

.

IE

E

+

= (2)

From the consideration of the heat exchanger, we can consider that the desired exergy output is the increase

of the thermal component of exergy of the cold stream, i.e. [7],

.Tcputdesiredout

.

EE =

where,

T

in,c

.T

out,c

.T

c

.

EEE =



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With reference to equation (1), in which rational efficiency is formulated, we can identify the required

exergy input, used.

E as,

P

h

.P

c

.T

h

.

used

.

EEEE ++=

By using equations (1) and (2), the rational efficiency of the heat exchanger is obtained as,

.T

c

.

T

c

.

P

h

.P

c

.T

h

.

T

c

.

IE

E

EEE

E

+

=

++

= (3)

The exergy change of the hot and cold streams can be written with the help of ideal gas relations as follows,

[ ]

)P/Pln(RTm)T/Tln(cmT)TT(cm

)ss(ThhmEE

inouto

.

inoutP

.

oinoutP

.

inoutoinout

.

in

.

out

.

+=

+= (4)

The desired exergy output which is the increase of the thermal component of exergy of the cold stream is

obtained by equation (4) as,

( )

=

in,c

out,cpoin,cout,c

c

p

.T

c

.

T

TlncTTTcmE (5)

The effectiveness expression of heat exchanger is defined as,

)TT()cm(

)TT()cm(

)TT()cm(

)TT()cm(

in,cin,hminP

.

out,hin,hhP

.

in,cin,hminP

.

in,cout,ccP

.

=

= (6)

The effectiveness and number of heat transfer units (NTU) relationship for counter flow and parallel flow

heat exchangers respectively are given as [20],

)C/C1(NTUmaxmin

)C/C1(NTU

maxmin

maxmin

e)C/C(1

e1

= (7)

)C/C(1

e1

maxmin

)C/C1(NTU maxmin

+

=

+

(8)

Expressing the outlet temperature in terms of inlet temperature and effectiveness in equation (5),

( )

+= 1

T

T

C

C1lnTTT

C

CCE

in,c

in,h

c

minoin,cin,h

c

minc

T

c

.

(9)



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Defining the heat capacity rate ratio as,

min

max

C

CC= (10)

Where, Cmaxis the maximum heat capacity and Cminis minimum heat capacity of the fluids.

Case 1:Considering Cmax=Chand Cmin= Cc,

c

h

C

CC=

The equation (9) becomes,

( )

+= 1

T

T1lnTTTCE

in,c

in,hoin,cin,hc

T

c

.

(11)

The irreversibility term in the rational efficiency expression of equation (3) is obtained as product of

entropy generation rate and reference temperature. Expressing entropy generation rate in terms of entropy

generation number from equation (18)which will be defined latter and minimum capacity rate as,

( )minso.

CNTI= (12)

Finally, the rational efficiency is obtained in terms of increase of the thermal component of exergy of the

cold stream and total irreversibility terms by substituting equations (11) and (12) into equation (3) as,

( )

( ) ( )minsoin,c

in,hoin,cin,hc

in,c

in,hoin,cin,hc

CNT1T

T1lnTTTC

1T

T1lnTTTC

+

+

+

=

(13)

Neglecting pressure drop terms, we are able to derive the rational efficiency defined by heat transfer term

only from equation (13) as follows [7],

+

+

=

)T

T1(

C

11lnC)T/TT/T(

)1T

T(1ln)T/TT/T(

in,h

in,coin,coin,h

in,c

in,hoin,coin,h

T

(14)

The entropy generation rate in the heat exchanger is given by Bejan [21] using first and second law

statements as,



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+

=

in,h

out,h

h

.

in,c

out,c

c

.

in,h

out,h

h

.

pin,c

out,c

c

.

pgen

.

p

plnRm

p

plnRm

T

Tlnmc

T

TlnmcS

(15)

Defining entropy generation number by dividing entropy generation by minimum heat capacity rate i.e.

Cmin[21],

min

gen

.

SC

SN = (16)

Following the procedure from Bejan [21], consider the total entropy generation number equation (15)

which includes pressure drop irreversibility along with heat transfer irreversibility. With the ideal gas

assumption and assuming the relative pressure drops (P/P) along each stream are sufficiently small and

conductive thermal resistance of the wall which separates fluids is negligible we are having,

in,c

c

in,c

out,c

P

P

P

Pln

=

and

in,h

h

in,h

out,h

P

P

P

Pln

=

Pressure drop expression is given as,

( )2*GD

L4f

P

P=

(17)

Where, ( )P2

G*G 2

= is the dimensionless mass velocity.

Expressing the outlet temperature in terms of inlet temperature and effectiveness using equation (6) and

substituting equation (15), total entropy generation number for flow imbalance heat exchanger of equation

(16) becomes,

( ) ( )h

2

h,pc

2

c,pin,h

in,c

in,c

in,hS

D

L4*Gf

c

RC

D

L4*Gf

c

R

T

T1

C

11lnC1

T

T1lnN

+

+

+

+=

(18)

The first two terms in the above equation represents heat transfer entropy generation number while

third and fourth terms represent pressure drop entropy generation number. As provided in Ordonez and

Bejan [22], the overall thermal resistance is the sum of the resistances of the two sides of the heat

exchanger surface,



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( ) ( )hc UA1

UA

1

UA

1+= (19)

Here it is assumed that the resistance to the conduction across the solid wall separating the two streams is

negligible. The NTU is defined as,

minC

UANTU= (20)

Combining with NTU equation (20), the equation (19) becomes,

( ) ( )hc NTU1

NTU

1

NTU

1+= (21)

Stanton number is defined as,

Gc

U

Stp=

With the NTU and Stanton number relations, the overall NTU equation (21) for cold and hot streams

becomes as,

( ) ( )hhc

h

c

cc

c

St

1

A

A

C

C

St

1

A

A

NTU

1

+

=

Where Acis the cross sectional area and A is the surface area of each tube. Making assumptions that same

type of fluid flows in both the streams and both sides of the streams have same geometrical dimensions, the

above equation becomes,

+=

1C

CSt

D

L4NTU (22)

Case 2:Considering Cmax=Ccand Cmin= Ch,

h

c

C

CC=

The equation (9) becomes,

( )

+= 1

T

T

C

11lnTTT

C

1CE

in,c

in,hoin,cin,hc

T

c

.



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Following the procedure as mentioned in the case 1, we obtain the expressions for rational efficiency,

rational efficiency defined by heat transfer term only and total entropy generation number respectively as

follows,

( )

( ) ( )minsoin,c

in,hoin,cin,hc

in,c

in,hoin,cin,hc

CNT1T

T

C

11lnTTT

C

1C

1TT

C11lnTTT

C1C

+

+

+

=

(23)

+

+

=

)T

T1(1ln)T/TT/T(

)1T

T(

C

11lnC)T/TT/T(

in,h

in,coin,coin,h

in,c

in,hoin,coin,h

T

(24)

( ) ( )h

2

h,pc

2

c,pin,h

in,c

in,c

in,hS

D

L4*Gf

c

R

D

L4*Gf

c

RC

T

T11ln1

T

T

C

11lnCN

+

+

+

+=

(25)

2.1 For fully developed laminar flow in the heat exchanger

In fully developed laminar flow (L/D 100), the heat transfer coefficient is constant, that is, independent of

longitudinal position. The Nusselt number values for fully developed laminar flow for circular duct with

uniform heat flux is equal to 4.364 as provided in Bejan [23]. Heat transfer in heat exchangers frequently

occurs with a constant heat flux. At Pr 0.7 there is very low significant difference between heat transfer

for uniform heat flux and uniform surface temperature cases [20]. The Stanton number can be calculated in

terms of Nusselt number, Reynolds number and Prandtl number as

Pr.Re

NuSt=

The friction factor expressed in terms of Reynolds number is given as,

Re

16f =



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NTU is related to Stanton number as mentioned earlier in the equation (22). The NSequation (18) for the

heat exchanger varies with the L/D ratio through the equation (22) and effectiveness equations for

particular configuration such as counter flow equation (7) and parallel flow equation (8).

2.2 For fully developed turbulent flow in the heat exchanger

In fully developed turbulent flow (L/D60), the Nusselt number expression is given by Dittus-Boelter

equation as [23],

4.05/4 PrRe023.0Nu=

The Stanton number is obtained in terms of Nusselt number and Reynolds number as mentioned earlier.

The friction factor is expressed in terms of Reynolds number as,

4/1Re078.0f = 2300 < Re < 2 x 104

5/1Re046.0f = 2 x 104< Re < 106

Further, the procedure is followed as mentioned in laminar flow case. Cold stream mass velocity is

calculated by using the given input values of Reynolds number and diameter of the tube as,

D

ReGc

=

Hot stream mass velocity is calculated by using capacity ratio as,

ch G.CG =

These mass velocity values along with the friction factor are used to calculate the pressure drop entropy

generation terms.

2.3 Condenser

In the case of condenser, to define the rational efficiency, the desired exergy output term is taken as

increase of exergy of cold stream and exergy input term is taken as decrease of exergy of hot stream if we

neglect pressure drop terms. In condensation, Th,in= Th,out(saturation temperature), therefore hot and cold

capacity rate becomes,



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Ch = Cmaxand Cc= Cmin.

The capacity ratio becomes,

===c

h

min

max

C

C

C

CC

Defining the ratio of inlet cold fluid temperature to reference temperature as,

o

in,cc

T

TR =

and defining the ratio of inlet hot fluid temperature to reference temperature as,

o

in,hh

T

TR =

Following the approximation procedure as provided in Hesselgreaves [24], rational efficiency given by

equation (14) for condenser becomes,

( )

( )

+

=

h

cch

c

hch

T

R

R1RR

1R

R1lnRR

(26)

2.4 Evaporator

In the case of evaporator, to define the rational efficiency, the desired exergy output term is taken as

decrease of exergy of hot stream and exergy input term is taken as increase of exergy of cold stream if we

neglect pressure drop terms.

In evaporation, Tc,in= Tc,out(saturation temperature), therefore cold and hot capacity rate becomes,

Cc= Cmaxand Cmin= Ch.

The capacity ratio becomes,

0C

C

C

CCh

c

min

max ===

Following the approximation procedure as provided in Hesselgreaves [24], rational efficiency given by

equation (24) for evaporator becomes,



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+

=

1R

R)RR(

)R

R1(1ln)RR(

c

hch

h

cch

T

(27)

The effectiveness and number of heat transfer units (NTU) relationship for condenser and evaporator is

given as [18],

NTUe1 = (28)

3. Results and discussion

The input values considered for the analysis as referred from Mohamed [25] are: inlet hot to cold streams

temperature ratio, Th,in/Tc,in= 1.5; inlet hot to cold streams pressure ratio, Ph,in/Pc,in= 10; inlet cold stream

temperature, Tc,in = 300 K; inlet cold stream pressure, Pc,in = 105 Pa; tube length, L = 1 m and taking

reference temperature, To= Tc,in. The two stream fluids in the heat exchanger are considered as air and its

thermo physical properties are referenced at average temperature from [20]. The analysis is carried out

using EES software [26] and results are generated. Length-to-diameter ratio, L/D for fully developed

laminar flow is varied from 100 to 600 and for fully developed turbulent flow is varied from 60 to 300 [23].

Considering capacity ratio value as 1 and 2, the following results are obtained for different fluid flow types

and heat exchanger configurations.

3.1 Unbalanced heat exchanger analysis using second law efficiency

3.1.1 Counter flow heat exchanger

Figure 1 shows the results of rational efficiency () and rational efficiency defined by heat transfer terms

only (T) versus length-to-diameter ratio (L/D) for counter flow (laminar flow) heat exchanger by varying

capacity ratio (C) and for the two cases i.e. for Cmax= Chand Cmax= Cc. As the value of L/D ratio increases,

increases, attains maximum value and then decreases, but T increases continuously. This maximum



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value of rational efficiency can be used to obtain optimum value of L/D ratio for lower irreversibilities.

Initially, for some value of L/D ratio, the values of and Tare same and then onwards the value of

decreasesdue to the inclusion of pressure drop irreversibility along with heat transfer irreversibility in the

rational efficiency expression. This provides more realistic second law analysis and we can obtain optimum

value of L/D ratio for lower irreversibilities considered in a single closed form expression. As we increase

the value of C, both and Tdecreases indicating increase in the loss due to imbalance irreversibility. In

the definition of capacity ratio, when Cmax = Cc, both the values of and T becomes lower when we

compare with Cmax= Ch. This is because, when Cmax= Ch, temperature difference of hot stream will be less,

so exergy increase of cold stream will be more and when Cmax= Cc, temperature difference of hot stream

will be more leading increase of heat transfer irreversibility, so exergy increase of cold stream will be less.

Figure 2 shows the behavior of rational efficiency () and rational efficiency defined by heat transfer terms

only (T) versus length-to-diameter ratio (L/D) for counter flow (turbulent flow) heat exchanger by varying

capacity ratio (C). There is increasing, attaining maximum value and then decreasing behavior of for

increase in the value of L/D ratio, but Tincreases continuously. The behavior of is due to inclusion of

pressure drop irreversibility to the rational efficiency expression which gives more realistic value of

optimum L/D ratio for the lower irreversibilities. In the case of turbulent flow, the value of is lower

compared to the case of laminar flow. The optimum L/D ratio for maximum is lower for the case of

turbulent flow compared to laminar flow. As the value of C is increased, after a particular value of L/D

ratio, rational efficiency decreases. This is due to increase of imbalance irreversibility in the case of C = 2.

Same result follows as in laminar flow for the case of capacity rate Cmax= Chand Cmax= Cc. That is when

Cmax= Cc, both the values of and Tbecomes lower compared to the case when Cmax= Ch. The reason for

this behavior is same as mentioned earlier.

3.1.2 Parallel flow heat exchanger

Figure 3 shows the result of rational efficiency () and rational efficiency defined by heat transfer terms

only (T) versus length-to-diameter ratio (L/D) for parallel flow (laminar flow) heat exchanger by varying



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capacity ratio (C) and for the two cases i.e. for Cmax= Chand Cmax= Cc. The behaviors of the curves are

same as counter flow case. But the values of rational efficiency for all the curves are much lower compared

to counter flow case. This is because of lower value of effectiveness in the case of parallel flow compared

to counter flow.

Figure 4 shows the result of rational efficiency () and rational efficiency defined by heat transfer terms

only (T) versus length-to-diameter ratio (L/D) for parallel flow (turbulent flow) heat exchanger by varying

capacity ratio (C) and for the two cases i.e. for Cmax= Chand Cmax= Cc. Here also, behavior of curves are

same with lower values of rational efficiency as compared to counter flow case except for the case of

rational efficiency () curves when C = 2, which is having same value. This is due to that when C = 2, the

values of pressure drop entropy generation for both the cases are same and it is having higher value

compared to heat transfer entropy generation values.

The reason for this is provided in Bejan [1] and Ratts and Raut [27] that after some particular value of

Reynolds number, the viscous loss increase will be more compared to heat transfer loss. The trade-off

between competing irreversibilities is to determine thermodynamically optimal size or operating regime of

an engineering system, where by optimal we mean the condition in which the system destroys the least

exergy while still performing its fundamental engineering function. In many systems, the various

mechanisms and design features that account for irreversibility compete with one another. In the heat

exchanger, we are having mainly two types of irrversibilities namely, heat transfer and fluid flow

irreversibilities. The relative importance of the two irreversibility mechanisms is described by the

irreversibility distribution ratio, which is defined as the ratio of fluid flow irreversibility versus heat transfer

irreversibility [1].

From the above results and discussion of counter flow and parallel flow heat exchangers, we are able to

compare the second law and first law thermodynamic analysis referring to the values compiled in the table

1. In the case of laminar flow, optimum L/D ratio value corresponding to higher rational efficiency, (for

lower total irreversibilities) is 410 for counter flow and is 280 for parallel flow for the input values



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considered. This optimum value cannot be easily obtained from first law effectiveness result only. This is

because as referred from the table 1, the maximum effectiveness is obtained when the L/D ratio is having

maximum value considerd for the analysis i.e. L/D = 600 for laminar flow. Corresponding to this higher

L/D ratios value, the rational efficiency, will be lower (total irreversibilities will be higher). Same result

is followed for turbulent flow case. In parallel flow configuration, we can observe that the rational

efficiency will be lower compared to counter flow. This is because of lower value of effectiveness in the

case of parallel flow configuration compared to counter flow.

3.2 Condenser and evaporator analysis using second law efficiency

In the analysis, the values for the inlet temperature to reference temperature ratios are considered according

to refrigeration system condenser and evaporator working temperatures.

The behavior of rational efficiency versus effectiveness for the condenser is shown in the figure 5. As the

effectiveness increases, rational efficiency increases. By keeping the value R c=1 constant and increasing Rh

value will results in increase of rational efficiency. There is not much difference between the two curves

belonging to Rh=1.2 and 1.3, because of Rcvalue equal to reference temperature. By increasing the value of

Rcabove the reference temperature, rational efficiency increases due to lowering the temperature difference

between the streams which reduces the value of heat transfer entropy generation.

The results of rational efficiency versus NTU for the condenser are shown in the figure 6. As NTU

increases, rational efficiency increases. By keeping the value Rc=1 as constant and increasing the value of

Rh, will results in increase of rational efficiency. As we increase the value of R cequal to 1.1, the rational

efficiency increases. In this case, as Rhvalue is increased, rational efficiency decreases due to increase in

the heat transfer entropy generation resulting from increase of temperature difference between the streams.

Figure 7 shows the behavior of rational efficiency versus effectiveness for the evaporator. As effectiveness

increases, rational efficiency increases. By keeping the value of Rhconstant and decreasing the value of Rc



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will lead to decrease of rational efficiency. This is due to increase of heat transfer entropy generation

number due to increase of temperature difference between the streams. As the value of R hdecreases, the

value of rational efficiency increases. This is because of decrease of temperature difference between the

streams.

The variation of rational efficiency with NTU is shown in the figure 8. As NTU increases, the value of

rational efficiency increases. Here also by keeping the value of Rhconstant and reducing the value of Rc

will decrease rational efficiency. As the value of Rhdecreases, rational efficiency increases due to reduction

in temperature difference between the streams.

4. Conclusions

The following conclusions can be obtained by the exergy analysis of unbalanced heat exchangers.

a. The total rational efficiency which also includes the irreversibility due to pressure drop is having

lower value as compared to rational efficiency defined by heat transfer term only and having

increasing trend, attaining maximum value and then showing decreasing behavior. This value of

maximum second law efficiency gives more realistic optimum value of length-to-diameter ratio

for lower value of all the irreversibilities. This value of length-to-diameter ratio is having lower

value in the case of turbulent flow as compared to the laminar flow.

b. As we increase the value of heat capacity ratio, the second law efficiency decreases because of the

addition of imbalance irreversibility. But in the first law analysis, from the result of effectiveness

versus number of heat transfer units (NTU), as the value of capacity ratio increases, the value of

effectiveness also increases.

c. In both the types of laminar and turbulent flows, the behavior of both the total rational efficiency

and rational efficiency defined by heat transfer term only will be having higher value when the

maximum capacity rate considered is equal to hot stream capacity rate as compared to cold stream

capacity rate case.



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d. By having all the irreversibilties combined in a single closed form expression, second law

thermodynamic analysis of heat exchanger provides realistic optimum values of different

parameters such as geometrical dimensions, working condition like changing heat capacity ratio

and fluid flow type i.e. laminar and turbulent flows for fully developed fluid flow as compared to

first law thermodynamic analysis.

e. Compared to counter flow, parallel flow heat exchanger is having lower values of rational

efficiency. This is due to the lower value of effectiveness of parallel flow heat exchanger. But in

the case of turbulent flow when heat capacity ratio is equal to 2, the total rational efficiency is

having same value as counter flow heat exchanger because of having same higher value of

pressure drop entropy generation compared to heat transfer entropy generation. That is, when

imbalance between the fluid flow streams is more, then the second law efficiency is same for both

counter flow and parallel flow heat exchangers.

f. In condenser, by keeping condensing temperature constant and increasing the cold stream

temperature, the rational efficiency increases. This is because of less exergy loss between the

streams due to reduction in temperature difference between the two streams.

g. As we decrease the condensing temperature, the rational efficiency decreases if cold stream

temperature is equal to reference temperature.

h. In evaporator, by keeping evaporating temperature constant and increasing the hot stream

temperature, the rational efficiency decreases. This is due to increase in the temperature difference

between the streams which leads to more exergy loss.

i. As we decrease the evaporating temperature, the rational efficiency decreases.

The above results are beneficial for the thermal design of heat exchangers based on exergy analysis.

References

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3. Golem PJ, Brzustowski TA (1976) Second law analysis of energy processes. Part II The performance of

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cylindrical duct. J. Heat Transfer 308313

5. Prasad RC, Shen J (1993) Performance evaluation of convective heat transfer enhancement devices using

exergy analysis. Int. J. of Heat and Mass Transfer 36: 41934197

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in forced convection heat transfer. Int. J. of Heat and Mass Transfer 37: 22972303

7. Cornelissen RL, Hirs GG, Smeding SF, Raas JL (1997) Performance criteria for a heat exchanger.

Achema97: Frankfurt

8. Das SK, Roetzel W (1998) Second law analysis of a plate exchanger with an axial dispersive wave,

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Transfer, 50: 4754-4766

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Using NSGA-II. Heat Transfer Engrg., 33: 618-628



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17. Manjunath K, Kaushik SC (2013) Entropy generation and thermo-economic analysis of constructal heat

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exchangers. Int. J. Energy Res. 24: 843-864

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Mass Transfer 43: 41894204

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100 200 300 400 500 600

0.2

0.3

0.4

0.5

0.6

0.7

L/D

,,,,

T

(C=2) [Cmax=Ch] (C=2) [Cmax=Ch]

Laminar flow (Counter Flow)

(C=1) (C=1)

T(C=2)[Cmax=Ch]T(C=2)[Cmax=Ch]

T(C=1)T(C=1)

T(C=2) [Cmax=Cc]T(C=2) [Cmax=Cc]

(C=2) [Cmax=Cc] (C=2) [Cmax=Cc]

Figure 1Rational efficiency and rational efficiency defined

by heat transfer terms only versus length-to-diameter ratio forlaminar flow (counter flow) heat exchanger for different

values of capacity ratio, C and for two cases, Cmax= Chand

Cmax= Cc.

100 200 300 400 500 600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

L/D

,,,,

T


Turbulent flow (Counter flow)

(C=1) (C=1)

T(C=2) [Cmax=Ch]T(C=2) [Cmax=Ch]

T

(C=1)T

(C=1)




by heat transfer terms only versus length-to-diameter ratio for

turbulent flow (counter flow) heat exchanger for differentvalues of capacity ratio, C and for two cases, C max= Ch and

Cmax= Cc.

100 200 300 400 500 600

0.2

0.3

0.4

0.5

0.6

0.7

L/D

,,,,

T


Laminar flow (Para llel Flow)

(C=1) (C=1)

T(C=2)[Cmax=Ch]T(C=2)[Cmax=Ch]T(C=1)T(C=1)





laminar flow (parallel flow) heat exchanger for differentvalues of capacity ratio, C and for two cases, C max= Ch and

Cmax= Cc.

100 200 300 400 500 600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

L/D

,,,,

T


Turbulent flow (Parallel flow)

(C=1) (C=1)

T(C=2) [Cmax=Ch]T(C=2) [Cmax=Ch]

T(C=1)T(C=1)T(C=2) [Cmax=Cc]T(C=2) [Cmax=Cc]




turbulent flow (parallel flow) heat exchanger for different

values of capacity ratio, C and for two cases, C max= Ch and

Cmax= Cc.

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

T

Rc=1, R

h=1.2R

c=1, R

h=1.2

Condenser

Rc=1, R

h=1.3R

c=1, R

h=1.3

Rc=1.1,R

h=1.2R

c=1.1,R

h=1.2

Rc=1.1,R

h=1.3R

c=1.1,R

h=1.3

Figure 5 Rational efficiency versus effectiveness of

condenser for different values of inlet temperature toreference temperature ratios.

0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

NTU

T

Condenser

Rc=1, R

h=1.2R

c=1, R

h=1.2

Rc=1, R

h=1.3R

c=1, R

h=1.3

Rc=1.1,R

h=1.2R

c=1.1,R

h=1.2

Rc=1.1,R

h=1.3R

c=1.1,R

h=1.3

Figure 6 Rational efficiency versus NTU of condenser fordifferent values of inlet temperature to reference temperature

ratios.



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0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

T

Rc=0.9, Rh=1Rc=0.9, Rh=1

Evaporator

Rc=0.8, Rh=1Rc=0.8, Rh=1

Rc=0.8, Rh=0.9Rc=0.8, Rh=0.9

Rc=0.7, Rh=0.9Rc=0.7, Rh=0.9

Figure 7 Rational efficiency versus effectiveness of

evaporator for different values of inlet temperature toreference temperature ratios.

0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

NTU

T

Rc=0.9, Rh=1Rc=0.9, Rh=1

Evaporator

Rc=0.8, Rh=1Rc=0.8, Rh=1

Rc=0.8, Rh=0.9Rc=0.8, Rh=0.9

Rc=0.7, Rh=0.9Rc=0.7, Rh=0.9

Figure 8 Rational efficiency versus NTU of evaporator for

different values of inlet temperature to reference temperature

ratios.



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Table 1 Comparison of second law and first law thermodynamic analysis of counter flow and parallel flow heat exchangers.

_______________________________________________________________________________________L/D ratio Rational efficiency, (%) Effectiveness, Remarks

Counter Parallel Counter Parallel Counter Parallel

flow flow flow flow flow flow_______________________________________________________________________________________

For laminar flow

410 280 64 45 0.9 0.65 Optimum L/D ratio

100 100 41 38 0.6 0.54 Minimum L/D ratio

600 600 59 39 0.95 0.65 Maximum L/D ratio

For turbulent flow

100 80 30 25 0.55 0.45 Optimum L/D ratio

60 60 23 20 0.4 0.37 Minimum L/D ratio

300 300 10 6 0.85 0.65 Maximum L/D ratio

_______________________________________________________________________________________

(Capacity ratio, C = 2)


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