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Effect of ambient heat-in-leak on the performance of a three fluid heat

exchanger, for cryogenic applications, using finite element method

V. Krishna a,, Pradeep G. Hegde a, N. Subramanian b, K.N. Seetharamu a

a Department of Mechanical Engineering, P.E.S. Institute of Technology, Bangalore 560085, Indiab Department of Mechanical Engineering, B.M.S. College of Engineering, Bangalore 560004, India

a r t i c l e i n f o

Article history:

Received 1 August 2011

Received in revised form 31 March 2012

Accepted 10 April 2012

Available online 7 June 2012

Keywords:

Heat exchangers

Effectiveness

Finite element method

Ambient heat in leak

a b s t r a c t

In most cryogenic applications, heat in leak from the ambient is a significant factor for the degradation in

the performance of heat exchangers. The effect of heat in leak to the cold fluid in a three-fluid heat

exchanger, for a cryogenic application, involving thermal interaction between all the three fluids, has

been investigated using both the analytical and finite element methods. Cooling of the hot fluid has been

identified as the objective of the three fluid heat exchanger. Seven non-dimensional parameters, includ-

ing one to account for ambient heat in leak to the cold fluid, have been identified and their effects on hot

fluid behaviour temperature profile, effectiveness and degradation factor have been studied. The

results presented give valuable inputs towards better understanding of the behaviour of the hot fluid

in this class of heat exchangers.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

Three fluid heat exchangers, involving all the three fluids in

thermal communication, are used in several applications found in

aerospace, petro-chemical and chemical industries to name a

few. Systems that deal with ammonia gas synthesis, purification

and liquefaction of hydrogen, air separation systems and helium-

air separation units are typical applications which make use of

three fluid heat exchangers [1].

The general analytical procedure to obtain the temperature dis-

tribution in all of the fluid streams in multi-stream, one-dimen-

sional heat exchangers assuming that there are no multiple

eigenvalues to the solution have been presented by several

researchers [2,3]. Others have presented explicit/iterative flow

direction dependent solutions for this class of three-fluid heat

exchangers [47] assuming that multiple eigenvalues do not exist.

Three-fluid models involving three thermal communications and

also multiple zero eigenvalues have been developed by Aulds

[18] and Aulds and Barron [8]. Sekulic and Shah [1] have provided

an extensive review of the work related to three-fluid heat

exchangers. A model of a two-fluid counter current heat exchanger

where both fluids are subjected to external heating has been devel-

oped by Ameel and Hewavitharana [9]. Another model of a two-

fluid parallel flow heat exchanger where again both fluids can

interact with the ambient has been developed by Ameel [10]. Both

models [9,10] have been developed assuming that multiple eigen-

values do not exist. As the ambient can be considered a third fluid

with infinite thermal capacity, both models [9,10] can be consid-

ered special cases of three-fluid heat exchangers with three ther-

mal communications. Barron [11] has also developed a model

where one of the fluids in a two-fluid heat exchanger is interacting

with the ambient. A unified, flow direction independent, non-

dimensional model for three-fluid heat exchangers with two ther-

mal communications has been developed for all possible fluid flow

cases by Sekulic and Shah [1]. The need for a general model for a

three-fluid heat exchanger with three thermal communications is

expressed in their paper. This has been addressed by Shrivastava

and Ameel [12] who have developed a three-fluid HX model with

three thermal communications that is insulated from the ambient.

Their model considers all possible thermal interactions and flow

arrangements. It is also mentioned by Sekulic and Shah [1] that

further studies should be conducted on the overall performance

of the three-fluid heat exchanger as well as on reconsideration of

the overall three-fluid heat exchanger effectiveness. This concern

is also addressed in a second paper by Shrivastava and Ameel

[13]. Saeid and Seetharamu [15] have presented a finite element

method (FEM) model for a three fluid heat exchanger (HX) with

two thermal communications. They have compared the effective-

ness values obtained from their model with those obtained from

standard 2 NTU equations and obtained accurate results. Krishna

et al [16] have extended this FEM analysis and have proposed an

FEM model to predict the performance of a two-fluid counter-flow

HX with heat leak to the evaporator considering the effect of

0017-9310/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.005

Corresponding author. Tel.: +91 80 26722108; fax: +91 80 26720886.

E-mail addresses: [email protected] (V. Krishna), [email protected] (P.G.

Hegde), [email protected] (N. Subramanian), knseetharamu@yahoo.

com (K.N. Seetharamu).

International Journal of Heat and Mass Transfer 55 (2012) 54595470

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer

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 / i j h m t
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.005mailto:[email protected]:[email protected]:[email protected]:knseetharamu@yahoo.%20%20commailto:knseetharamu@yahoo.%20%20comhttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.005http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.005mailto:knseetharamu@yahoo.%20%20commailto:knseetharamu@yahoo.%20%20commailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.005

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longitudinal heat conduction in the separating wall. This method-

ology has been further extended, in the present paper, to a three

fluid HX, with thermal communication between all the fluids, in

addition to, heat leak in from the ambient to the cold fluid.

Effectiveness is a measure of the performance of any heat ex-

changer. In three-fluid heat exchangers, with three thermal com-

munications, the heat exchanged between the cold and

intermediate temperature streams or the hot and intermediate

temperature streams cannot be neglected in comparison to the

heat exchanged between the cold and hot fluid streams. Several

effectiveness definitions have been proposed in the past to assess

the performance of three-fluid heat exchangers. Most of these def-initions give the temperature effectiveness of a particular stream

and are defined as the ratio of the actual temperature difference

to the maximum temperature difference that the stream of inter-

est can attain [16]. These definitions assess the performance of a

three-fluid heat exchanger by its ability to achieve a maximum

temperature difference for a selected stream. The temperature

effectiveness definition is based on the actual temperature change

of a single stream. Thus, at least three separate and different tem-

perature effectiveness definitions of this kind are possible for

three-fluid heat exchangers [13]. The definition of effectiveness

for any three fluid HX depends on its objective. Aulds and Barron

[8] defined effectiveness for three fluid heat exchangers with three

thermal communications as the ratio of the actual heat transferred

to the cold and intermediate fluids to the maximum heat that

could be transferred to both of these streams. A specific objective

of their heat exchanger has not been mentioned. However, given

their definition, the objective could be identified as to cool the

hot fluid. It can be concluded from the above discussion that there

can be no single definition to evaluate the general performance of

three-fluid heat exchangers with three thermal communications.

Shrivastava and Ameel [13] identify five different objectives for

three-fluid heat exchangers:-(1) heating the cold fluid, (2) cooling

the hot fluid, (3) cooling the intermediate fluid, (4) heating the

intermediate fluid, and (5) maximizing the enthalpy change of

the central fluid stream or the two lateral fluid streams.

The model of the three fluid HX proposed in this paper is a gen-

eral model andcan be appliedfor all three fluid, single pass, parallel

flow heat exchangers considering all possible thermal interactions

and flow arrangements. In the present paper, governing equations

have been solved for the flow arrangement in Case (2) with objec-

tive of determining the deviation in the behaviour of the hot fluid

due to ambient heat in leak to the cold fluid. The equations have

been solved by both theanalytical method and FEM. Both the meth-

ods give matching results with no deviations. The results have been

further validated by comparing them with values published earlier

by neglecting the effect of the ambient [1,8,12,13]. A degradation

factor, s, is defined to evaluate the deterioration in the performance

of the heat exchanger due to heat in leak from the ambient. The re-

sults obtained for the degradation factor have been validated by

comparing them with values published by Gupta and Atrey [14]

for a coiled tube-in-tube two-fluid heat exchanger withheat in leak.Thevalues obtained by themodel proposed matchvery closely with

the values published. After validating the methodology, the effect of

ambient heat in leak to the cold fluid and how it affects the perfor-

mance of a three fluid HX, has been studied. The objective of the

three fluid HX has been identified as the cooling of the hot fluid.

The effect of ambient heat in leak has been studied for its effect

on the temperature profile, effectiveness and degradation factor

of the hot fluid, and how these vary with respect to various non-

dimensional parameters on which the performance of the HX

depends.

Many of the simplifying assumptions that are required for ana-

lytical solutions are not required in the FEM analysis adopted in

this paper. The methodology presented is very versatile and can

accommodate most real time situations since it is element based.

In most cryogenic applications, heat in leak from the ambient is

a significant factor for the degradation in the performance of heat

exchangers. The effect of ambient heat in leak on the performance

of a three fluid HX, with all three fluids in thermal communication

with each other, has not been examined earlier. This has been

examined in the present paper to arrive at the extent of degrada-

tion in the performance of the three fluid HX.

2. Model formulation

A three fluid, single pass, parallel flow heat exchanger involving

thermal communication between all the three fluids hot, inter-

mediate and cold has been considered. The pipe configuration

for the HX appears as shown in Fig. 1. Each of the fluids interacts

Nomenclature

HX heat exchanger_m mass flow rate, (kg/s)_Q heat transfer rate, (W)

cp specific heat at constant pressure, (J/kg-K)T temperature (K)

C heat capacity rate of the fluids defined by the product of_m and cp (W/K)

P wetted perimeter for any contact area (m)P1, P2, P3, P4 wetted perimeters corresponding to areas A1, A2, A3,

A4 respectivelyL heat exchanger length (m)Le effective length of heat exchanger as defined by L/num-

ber of elements (m)A surface area for heat transfer as defined by the product

of P and Le(m2)

A1, A2, A3, A4 areas as illustrated in Fig. 1U overall heat transfer coefficient (W/m2-K)U1, U2, U3, U4 overall heat transfer coefficients as illustrated in

Fig. 1

H1, H2, H3 dimensionless parameters as defined in Eq. (4)R1, R2 ratio of heat capacity rates as defined in Eq. (5)x axial co-ordinate (m)NTU number of transfer units as defined in Eq. (4)X non-dimensional axial co-ordinate as defined in Eq. (5)

N1 & N2 shape functions as defined in Eq. (13)ih, ii, ic Directional constants as defined in Eqs. (1)(3)Greeksh dimensionless temperature as defined in Eq. (5)2 thermal effectivenessm temperature effectivenesss degradation factor as defined in Eq. (33)Subscriptsc cold fluidh hot fluidi intermediate fluidin inletout outlet1 Ambient

5460 V. Krishna et al. / International Journal of Heat and Mass Transfer 55 (2012) 54595470

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with the other two while the cold fluid interacts with the ambient

in addition to the hot and the intermediate fluids. The HX consid-

ered is for a cryogenic application and as such the effect of the

ambient will be a heat leak in to the cold fluid. Depending on the

flow directions, four different flow arrangements Cases 14 are

possible [1,12], as shown in Fig. 2. The following assumptions have

been made for the analysis: (a) The HX is in a steady state (b) All

properties are constant with time and space. (c) There is no axial

conduction in the pipes or the fluids (d) Within a stream the tem-

perature distribution is uniform in the transverse direction and

equal to the average temperature of the fluid. (e) There is no heat

source or sink in the HX or in any of the fluids (f) There is no phase

change in the fluid streams (g) The heat transfer area is constant

along the length of the HX.

In the flow arrangements indicated, Cases (2)(4) are similar, in

the sense that, in each of these cases, two fluids flow in one direc-

tion while the other flows in the opposite direction. It can be

shown that fluid streams in these three cases are thermally

identical if proper values of the non dimensional parameters are

chosen [12]. Thus there are only two major cases Case (1) and

Case (2) that need to be studied to understand the behaviour

of three-fluid heat exchangers [12]. Case (1) corresponds to a case

of parallel, co-current flow involving all three fluids, while in Case

(2) the cold and the intermediate fluids flow in one direction while

the hot fluid flows in the opposite direction. It is also observed that

the effect of the different parameters on the Cases (1) and (2) are

similar [12]. Case (2) is more widely used and more effective since

it involves the hot and cold fluids flowing in opposite directions.

Thus the present model has been analyzed only for Case (2).

The governing equations for the hot, cold and the intermediate

fluids, obtained by energy balance, are as follows:

Hot Fluid:

ihdhHdX

NTU

R2hh hc

NTU H2R2

hh hi 0 1

Intermediate Fluid:

iidhidX

NTU R1 H1

R2hi hc

NTU R1 H2R2

hh hi 0 2

Cold Fluid:

ic

dhc

dXNTU

hh

hc

NTUH

1hi

hc

NTUH

3h

1 hc

03

In the above expressions hh, hi and hc represent the dimensionless

temperatures of the hot fluid, intermediate fluid and the cold fluid

respectively. Directional constants ih, ii & ic are introduced to

the governing equations to make them applicable for all four flow

arrangements. Their values are +1 for the positive x direction and

1 for negative x direction. For the flow arrangement of Case (2)

analysed in this paper, ih =1, ic = +1 and ii = +1. The different

non-dimensional terms used in the analysis are defined as men-

tioned below:

NTU U1P1LeCc

; H1 U2P2U1P1

; H2 U3P3U1P1

; H3 U4P4U1P1

; 4

R1 ChCi

; R2 ChCc

; h T Tc;in

Th;in Tc;in; X

x

Le5

Fig. 1. Tubular arrangement of the three fluid heat exchanger chosen for analysis

with ambient heat in leak to the cold fluid.

Fig. 2. Flow arrangements.

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3. Analytical solution

The governing equations, Eqs. (1)(3), are rearranged and writ-

ten in matrix form as

d

dX

hc

hh

hi

264

375 A

hc

hh

hi

264

375 f 6

where

A

NTUic

1 H2 H3NTUic

NTUicH2

NTUihR2

NTUihR2

1 H2NTUihR2

H1NTUR1iiR2

NTUR1H1iiR2

NTUR1iiR2

H2 H1

26643775

and f =

NTUH3haic

00

24

35

This system of differential equations is solved using the method

of decoupling transformations.

A is diagonalisedas x1Ax = D where D is a diagonal matrix con-

taining the Eigen values of A and x contains the Eigenvectors of A.

The Eigen values and Eigenvectors are directly computed using in-

built functions in MATLAB.

i.e.,

D

D1 0 0

0 D2 0

0 0 D3

264

375 and x

x11 x12 x13

x21 x22 x23

x31 x32 x33

264

375

Eq. (6) is written as

x1d

dX

hc

hh

hi

264

375 Dx1

hc

hh

hi

264

375 x1f

d

dX

z1

z2

z3

264

375

D

z1

z2

z3

264

375

g 7

where

z1

z2

z3

264

375 x1

hc

hh

hi

264

375 and x1f

g1

g2

g3

264

375 8

Eq. (7) represents a set of 1st order linear equations which can be

solved by simple integration.

We have dzidX Di zi

giDi

h iwhere i = 1, 2, 3 from which we get

zi eDiXCi gi

Di9

Boundary Conditions:

hc;in 0 at X1ic

2

hh;in 1 at X1ih

2

hi;in any value between 0 1 at X1ii

2

9>>=>>;

10

In this paper the value of hi,in has been taken to be 0.5 to compare

the results with those published by Ameel and Shrivastava [12].

From Eq. (8) we have

x

z1

z2z3

264

375

hc

hh

hi

264

375

11

Substituting for z from Eq. (9) and incorporating boundary condi-

tions from Eq. (10) into Eq. (11) we get a system of linear equations

with three variables eD1C1 ; eD2C2 & eD2C2 which are solved to give

;

;1

;2

;3

264

375

eD1C1

eD2C2

eD3C3

264

375

i.e.,

Ci 1

Diln ;i 12

Substituting Eq. (12) and (9) in Eq. (11) we get the temperature pro-file equations for the three fluids:

hc X3i1

x1i eDiXln;i

giDi

hh X3i1

x2i eDiXln;i

giDi

hi X3i1

x3i eDiXln;i

giDi

This method is applicable when the three eigen values D1, D2 and D3are different from each other. Combinations of design parameters

which give multiple eigen values have little importance in real

world problems [12] and hence this has not been considered.

4. Finite element method

The heat exchanger is discretized into a number of elements. A

linear variation is assumed for the hot, intermediate and the cold

fluids in a single element. The fluid temperature at any point, for

co-current arrangement Case 1, is given by the following

equations:

hh N1hh;in N2hh;out 13

hi N1hi;in N2hi;out 14

hc N1hc;in N2hc;out 15

where N1 and N2 are the shape functions and given by

Fig. 3. Effect of H2 on the hot fluid temperature profile for a three fluid HX with no

ambient heat in leak. Comparison of Present values (Analytical & FEM values) with

Shrivastava and Ameels values [12]. Values of other non dimensional parameters:

H1 = 1.5, R1 = 2, R2 = 1.25, NTU= 1, hi,in = 0.5.


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N1 1 X and N2 X: 16

For the counter flow arrangement of Case 2, the equations for hi and

hc do not change. However, the equation for hh changes and is given

by:

hh N1hh;out N2hh;in 17

Using the Galerkins method of minimizing the weighted residual

(Lewis et al. [17]), the governing equations are reduced to a set ofalgebraic equations. The discretized governing equations are writ-

ten in matrix form for each element as:

Kfhg ffg 18

where [K] is known as the stiffness matrix and it is a (6 6) matrix

for each element, {h} is the non-dimensional temperature vector

and {f} gives the loading terms. The expressions for [K], {h} and {f}

for Case 2 are provided in the Appendix. The stiffness matrix is

assembled for all the elements in the solution domain to get the glo-

bal stiffness matrix. The boundary conditions are enforced and the

system of equations is solved by MATLAB to get the dimensionless

temperatures along the heat exchanger.

4.1. Boundary conditions

The boundary conditions for the four different flow conditions

are provided in the Table 1. It can benoticedthat in eachcase three

boundary conditions are specified. These are enforced in the global

stiffness matrix before solving the equations.

5. Effectiveness

Cooling of the hot fluid has been identified as the objective of

the three fluid HX adopted for analysis. Cooling effectiveness of

the hot fluid for any three fluid HXcan be defined based on its tem-

perature effectiveness or its ability to release thermal energy to the

other two streams [13]. Cooling temperature effectiveness mh may

be defined as the ratio of the actual temperature difference be-tween the hot fluid inlet and outlet to the maximum possible tem-

perature difference that the hot fluid stream can attain.

mh Th;in Th;outTh;in Tc;in

19

In the normal sense, cooling thermal effectiveness of the hot fluid

may be defined as the ratio of the actual heat transferred from

the hot fluid to the maximum possible heat that can be transferred.

h Qh;actualQh;max

20

Qh;actual ChTh;in Th;out 21

When the thermal capacity of the hot fluid is greater than the ther-

mal capacities of theother two fluids and when the hot fluid is flow-

ing counter to the other two streams, the maximum heat transfer

from the hot fluid is given by

Qh;max CcTh;in Tc;in CiTh;in Tc;in 22

For all other possible combinations of thermal capacities of the three

fluids, the maximum heat transfer from the hot fluid is given by

Qh;max ChTh;in Tc;in 23

On the same lines it is possible to arrive at expressions for the effec-

tiveness of the cold and intermediate fluids [13].

6. Degradation factor

Degradation factor, s, is defined to evaluate the extent of dete-

rioration in the performance of the heat exchanger due to heat in

leak from the ambient to the cold fluid. The degradation factor

for the hot fluid is defined as the ratio of the loss in thermal effec-

tiveness due to ambient heat in leak to the thermal effectiveness

under no loss conditions and given by

sh eh;no loss eh;with loss

eh;no loss24

7. Results and discussion

7.1. Validation of the present methodology

The model of the three fluid HX proposed in this paper is a gen-

eral model and can be appliedfor all three fluid, single pass, parallel

flow heat exchangers considering all possible thermal interactions

and flow arrangements. In the present paper, the governing equa-tions have been solved for the flow arrangement in Case (2) with

the objective of determining the deviation in the behaviour of the

hot fluid due to the effect of ambient heat in leak to the cold fluid.

For the flow arrangement in Case (2), the directional constants take

the values ih =1, ic = +1 and ii = +1. The equations have been

solved by both the analytical method and FEM. Both the methods

give matching results with no deviations. These have been shown

in Table 2. The results have been further validated by comparison

with similar models on heat exchangers reported previously. The

comparisons have been made by selecting appropriate values for

the non-dimensional parameters to simulate the conditions exist-

ingin themodelschosen for comparison. The FEM analysis hasbeen

made by increasing the number of elements while adopting the

Galerkins method. It has been observed that the results convergedTable 1Boundary Conditions for different flow arrangements.

Flow arrangement Case 1

X hh hc hi0 1 0 hi,in1


X hh hc hi0 0 hi,in1 1


X hh hc hi0 0

1 1 hi,inFlow arrangement Case 4

X hh hc hi0 1 0

1 hi,in

Table 2

Comparison of FEM and Analytical values for temperature profiles of hot, cold and

intermediate fluids for a three fluid heat exchanger with ambient heat-in-leak.

(Values of non dimensional parameters: H1 = 1.5, H2 = 2, H3 = 0.1, R1 = 2, R2 = 1.25,

NTU = 1, hi,in = 0.5, hc,in = 0, h1 = 1).

X hh hc hi

FEM Analytical FEM Analytical FEM Analytical

0.0 0.4354 0.4354 0.0000 0.0000 0.5000 0.5000

0.2 0.5104 0.5104 0.1861 0.1861 0.3809 0.3809

0.4 0.6138 0.6138 0.3067 0.3067 0.4188 0.4188

0.6 0.7314 0.7314 0.4120 0.4120 0.5060 0.5060

0.8 0.8601 0.8601 0.5170 0.5170 0.6120 0.6120

1.0 1.0000 1.0000 0.6268 0.6268 0.7303 0.7303


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for 128 elements. As such, all values presented have been obtained

with 128 elements along the length of the HX.

The present model was at the outset compared with a standard

two-fluid single pass parallel, co-current and counter-current

model with no thermal interaction with the ambient. The results

obtained for effectiveness were compared with those obtained

from the standard 2 NTU formulae provided in Eqs. (25) and

(26). Concurrent results were obtained with no deviations. For

two-fluid parallel co-current flow

1 eNTU1Rmin

1 Rmin25

For two-fluid parallel counter-current flow

1 eNTU1Rmin

1 Rmin eNTU1Rmin 26

Aulds and Barron [8] have presentedpredictions for a three fluid HX

model involving three thermal communications. The flow arrange-

ment they have chosen is the same as Case (2) presented in this

paper. They have assumed that the HX is completely insulated with

no thermal interaction with the ambient. The model presented in

this paper was compared with that presented by Aulds and Barron[8], by neglecting the effect of the ambient, and closely matching re-

sults were obtained. Sekulic and Shah [1] have predicted the non-

dimensional outlet temperatures for various NTU values for the

flow arrangement of Case (1), with no thermal interaction with

the ambient. The results obtained from the model presented in this

paper, have compared perfectly with those presented by Sekulic

and Shah [1] and have been shown in Table 3. Shrivastava and

Ameel [12,13] have presented results for a three-fluid, single pass,

parallel flow HX model with three thermal communications which

considers all possible thermal interactions and flow arrangements.

They have presented results for the flow arrangement of Case (2)

showing the effect of various non-dimensional design parameters

such as R1, R2, H1, H2, NTU and hi,in on the non-dimensional temper-

ature distributions of the three streams and the different effective-

nesses. They have also assumed that the HX is completely insulated

with no thermal interaction with the ambient. The present model

has been compared with that of Shrivastava and Ameel [12,13] by

neglecting the effect of the ambient. The comparisons have been

shown for non-dimensional temperature distribution of the hot

fluid in Fig. 3 and they match perfectly. Gupta and Atrey [14] have

presented results for a two-fluid counter-flow, coiled, tube-in-tube

HX, considering the effect of heat in leak from the ambient and lon-

gitudinal heat conduction through the wall separating the two flu-

ids, separately and together. Their results have been presented in

terms of the degradation factor, defined earlier. One of their results

includes only ambient heat-in-leak while neglecting the effect of

longitudinal heat conduction through the wall. Degradation values,

obtained from the present model, have been compared with their

results, to account for ambient heat-in-leak, for two different valuesof ambient temperature and presented in Fig. 4. The comparisons

show that for hamb = 4.67, a maximum deviation of 0.04% is

observed at a Cc/Ch value of 2.0, where sh = 0.32% from their meth-

odology while sh = 0.36% from our methodology. For hamb = 1.0, a

maximum deviation of 0.01% is observed at a Cc/Ch value of 1.6,

where sh = 0.09% from their methodology while sh = 0.10% from

our methodology. From the comparisons shown and the close

match of the present results with those published earlier, the pres-

ent methodology has been validated.

7.2. Effect of ambient heat in leak

The effect of ambient heat in leak to the cold fluid is to increase

the temperature of the cold fluid. The ambient heat in leak is ac-

counted for by the parameter H3 which is given by:

H3 U4P4U1P1

1

U1P1h i

1U4P4

h i

Heat Transfer Resistancebetween hot and cold fluids

Heat Transfer Resistancebetween cold fluid and the ambient

27

The values of H3 taken in this paper are in the range 00.1. These

values are taken to show the effect of heat in leak. The actual values

ofH3 have to be deduced accurately through repeated and rigorous

experimentation under various operating conditions. Increased val-

ues of H3 reflect a decreased value of heat transfer resistance be-

tween the cold fluid and ambient and/or increased value of heat

transfer resistance between the hot and the cold fluids. Thus an in-

crease in the value ofH3 results in greater ambient heat in leak and/

or decreased heat transfer between the hot and the cold fluids. Thecombined effect would be to increase the temperature of the hot

fluid throughout the HX.

Table 3

Comparison of non-dimensional outlet temperatures between present values (Analytical & FEM values) and Sekulic and Shahs values [1] for a three fluid HX (Case 1) with no

ambient heat in leak. Values of other non dimensional parameters: H1 = 0, H2 = 0.3, H3 = 0, Cc/Ch = 2, Ci/Ch = 0.8 and hi,in = 0.5.

NTU hh,out hi,out hc,out

Sekulic & Shahs values [1] Present Values Sekulic & Shahs values [1] Present Values Sekulic & Shahs values [1] Present Values

0 1 1 0.5 0.5 0 0

1 0.377 0.377 0.501 0.501 0.311 0.311

2 0.365 0.365 0.43 0.43 0.346 0.346

3 0.367 0.367 0.396 0.396 0.358 0.358

4 0.368 0.368 0.381 0.381 0.364 0.364

5 0.368 0.368 0.374 0.374 0.366 0.366

Fig. 4. Effect of Cc/Ch onthe hot fluiddegradation factor for a two fluidHX withheat

in leak. Comparison of Present values (Analytical & FEM values) with Gupta and

Atreys values [14]. Values of other non dimensional parameters: H1 = 0, H2 = 0,

H3 = 0.001, R1 = 0, R2 = 1, NTU= 5, hh,in = 1, hc,in = 0, hi,in = 0.


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The effect ofH1 and ambient heat in leak is shown in Figs. 5ac.

An increase in the value ofH1 leads to decrease in the heat transfer

resistance between the intermediate and coldfluids and/or increase

in the heat transfer resistance between the hot and cold fluids. Due

to this an increase in H1 results in increased heat transfer betweenthe intermediate and cold fluids and/or decreased heat transfer

between the hot and the cold fluids. It is observed from Fig. 5a that

the hot fluid shows a marginal increase in the temperature differ-

ence between the inlet and exit with increasing values of H1. The

effect of ambient heat in leak is to again increase the hot fluid tem-

perature throughout the HX in each case. The effect ofH1 and H3 on

hot fluid effectiveness is shown ifFig. 5b. An increase in H1 leads to

lower values of hot fluid exit temperature leading to an increase in

effectiveness both temperature and thermal effectiveness. An in-

crease in ambient heat in leak (increase in the value of H3) leads to

increased hot fluid temperatures leading to a decrease in the hot

fluid effectiveness. The effect of H1 and H3 on degradation factor is

shown in Fig. 5c. AtanyvalueofH3, it isobserved thatthereis a mar-

ginal increase in the degradation values with increasing values ofH1.Degradation values are almost constant for H1 values beyond 3.

The effect ofH2 and ambient heat in leak is shown in Figs. 6ac.

The effect of increase in H2 may be explained on the same lines as

H1 and an increase in H2 results in increased heat transfer between

the hot and the intermediate fluids and/or decreased heat transfer

between the hot and the cold fluids. It is observed from Fig. 6a thatthe hot fluid will show an increasing temperature difference be-

tween the inlet and the exit with increasing values of H2. In each

case however the effect of heat in leak is to enhance the hot fluid

temperatures throughout the HX as indicated. An increase in the

value of H2 leads to increased values of hot fluid effectiveness as

shown in Fig. 6b. This is due to increased heat transfer between

the hot and the intermediate fluids. However an increase in the

value of H3 leads to greater ambient heat in leak which results in

greater temperatures of the hot fluid leading to a decrease in the

hot fluid effectiveness both temperature effectiveness and ther-

mal effectiveness as indicated. A decrease in effectiveness leads

to an increase in the value of degradation as observed in Fig. 6c.

It is observed that with increasing values of H2 the degradation

tends to decrease for each value of H3 and later becomes almosta constant for H2 values greater than 5.

Fig. 5a. Effect of H1 on the hot fluid temperature profile for a three fluid HX with

ambient heat in leak, values of other non-dimensional parameters: NTU = 1, H2 = 2,

R1 = 2, R2 = 1.25. hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.

Fig. 5b. Effect ofH1 on the hot fluid effectiveness for a three fluid HX with ambient

heat in leak, values of other non-dimensional parameters: NTU = 1, H2 = 2, R1 = 2,

R2 = 1.25. hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.

Fig. 5c. Effect of H1 on the hot fluid degradation factor for a three fluid HX with

ambient heat in leak, values of other non-dimensional parameters: NTU = 1, H2 = 2,


Fig. 6a. Effect of H2 on the hot fluid temperature profile for a three fluid HX with

ambient heat in leak, values of other non-dimensional parameters: NTU= 1,

H1 = 1.5, R1 = 2, R2 = 1.25. hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.


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Fromits definition it is clear that a change in NTU will affect the

thermal resistance between the hot and cold fluids, and/or the

thermal capacity of the cold fluid. If the values of H1 and H2 are

fixed, an increase in NTU will not result in any change in the ther-

mal resistance between the hot and the cold fluids. This will only

result in a decrease in the thermal capacity of the cold fluid. If

the specific heat is assumed constant then this results in a decrease

in the mass flow rate of the cold fluid. Since R1 and R2 are also con-

stant, this results in the decrease in the flow rates of the hot and

the intermediate fluids as well. The effect of all this is steeper ther-

mal gradients for all three fluids and increased differences between

the inlet and exit temperatures, with increasing values of NTU. The

effect of NTU and ambient heat in leak is shown in Figs. 7ac. As

NTU is increased from 0.1 to 1, gradual changes in the temperature

distributions occur. As NTU is increased from 110, the decrease in

the mass flow rates of all the three streams results in a sharp fall in

the hot fluid temperature at the inlet and subsequent increase to-

wards the exit as seen in Fig. 7a. An increased difference between

the hot fluid inlet and exit temperatures is also observed. For an

NTU value of 10 the ambient heat in leak is very pronounced and

this is evident by the manner in which the hot fluid temperature

is enhanced across the length of the HX for the parameters chosen.

An increase in the value of NTU results in a decrease in the exit

temperature of the hot fluid which leads to an increase in the effec-

tiveness both temperature and thermal effectiveness as shown in

Fig. 7b. However, an increase in the value of H3 leads to increased

hot fluid temperatures throughout the length of the HX decreasing

effectiveness. With increasing values of NTU the size of the HX

would increase and more area is available for ambient heat in leak.

Further, an increase in the value ofH3 results in greater heat in leak

and hence higher values of degradation factor. Thus degradation

factor increases with NTU for each value ofH3 as shown in Fig. 7c.

When other parameters are fixed, an increase in the value of

Cc/Ch leadsto anincreasein thethermal capacity ofthe cold fluidrel-

ative to that ofthe hot fluid. If the specific heats of the two fluids are

assumed to be constant, an increase in Cc/Ch would mean that the

mass flow rate of the cold fluid increases relative to that of the hot

fluid. The effect of Cc/Ch and ambient heat in leak is shown in

Fig. 6b. Effect ofH2 on the hot fluid effectiveness for a three fluid HX with ambient

heat in leak, values of other non-dimensional parameters: NTU = 1, H1 = 1.5, R1 = 2,

R2 = 1.25. hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.

Fig. 6c. Effect of H2 on the hot fluid degradation factor for a three fluid HX with

ambient heat in leak, values of other non-dimensional parameters: NTU = 1,

H1 = 1.5, R1 = 2, R2 = 1.25. hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.

Fig. 7a. Effect of NTU on the hot fluid temperature profile for a three fluid HX with

ambient heat in leak, values of other non-dimensional parameters: H1 = 1.5, H2 = 2,


Fig. 7b. Effect of NTU on the hot fluid effectiveness for a three fluid HX with




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Figs. 8ac. An increase in Cc/Ch leads to an increase in the tempera-

ture difference between the inlet and exit for the hot fluid as indi-

cated by Fig. 8a. At high values of Cc/Ch, as observed by Shrivastava

and Ameel [13] and verified by our methodology as well, the cold

fluid temperature is close to zero across the heat exchanger length.This results in high temperature gradients for the hot fluid at the

ends of the HX. It may also be observed that for Cc/Ch = 10, the hot

fluid temperature profile almost reduces to zero in the mid-section

of the HX indicating high levels of heat transfer for the hot fluid.

Anincrease in the value ofCc/Ch results in a decrease in the hot fluid

exit temperature leading to increased values of temperature effec-

tiveness of the hot fluid. The effect ofCc/Ch and H3 on hot fluid effec-

tivenessis shown in Fig.8b. Whenthe values ofCc/Ch are lower than 1

the thermal effectiveness of the hot fluid is directly proportional to

(1hh,out) and inversely proportional to Cc/Ch [13]. For low values

ofCc/Ch, (1hh,out) grows faster than Cc/Ch and hence thermal effec-

tiveness increases initially and decreases subsequently. But from a

value ofCc/Ch = 1 onwards, higher values ofCc/Ch lead to lesser tem-

perature changein thecoldfluidandgreaterchangesin thevaluesof

hot fluid temperature. Due to this hot fluid exit temperature will

decrease leading to higher effectiveness. Ambient heat in leak de-

creases the effectiveness albeit marginally. The degradation factor

is highest for Cc/Ch = 1 and reduces with increasing values of Cc/Chas shown ifFig. 8c. This is because the ambient heat in leak is maxi-

mum for the conditions chosen for a balanced flow condition i.e.Cc = Ch. Such trends have been reported earlier [14]. Higher values

ofH3 give rise to higher values of degradation due to increased heat

in leak.

The effect of heat-in-leak with varying values of Ci/Ch (R11)

have been shown in Figs. 9ac. The effect of a change in Ci/Chmay be explained on lines similar to that of Cc/Ch. An increase in

Ci/Ch would mean that the mass flow rate of the intermediate fluid

increases relative to that of the hot fluid indicating greater heat

transfer from the hot fluid and the temperature profile undergoes

a marginal change as indicated in Fig. 9a. The temperature profile

is almost linear for Ci/Ch = 0.1 and for higher values of Ci/Ch, the

profiles show slightly increasing temperature gradients at the hot

end with a subsequent increase in exit temperature at higher val-

ues ofCi

/Ch

. The effect of heat-in-leak is observed to be slightly

higher at lower values of Ci/Ch. The effect of Ci/Ch on effectiveness

Fig. 7c. Effect of NTU on the hot fluid degradation factor for a three fluid HX with



Fig. 8a. Effect of Cc/Ch on the hot fluid temperature profile for a three fluid HX with


H1 = 1.5, H2 = 2, R1 = 2, hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.

Fig. 8b. Effect of Cc/Ch on the hot fluid effectiveness for a three fluid HX with



Fig. 8c. Effect of Cc/Ch on the hot fluid degradation factor for a three fluid HX with




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follows on the samelines as Cc/Ch as shown in Fig. 9b with a shift in

the effectiveness at values close to Ci/Ch = 1, since Ci starts playing a

dominant role. This results in a steep increase in hot fluid effective-

ness. But an increase in the value of Ci/Ch beyond 1 does not have

any significant effect on the hot fluid effectiveness. Ambient heat inleak decreases the effectiveness as indicated. The degradation fac-

tor reduces with increasing values ofCi/Ch, due to reduced heat-in-

leak, as shown in Fig. 9c. As seen the degradation factor becomes

almost constant for values of Ci/Ch values more than 5.

The effect of heat-in-leak with varying values of hi,in have been

shownin Figs.10ac. Anincreasein thevalueofhi,in leadsto more heat

transfer to the cold fluid and as such the temperature difference be-

tween inlet and exit for both the hot and cold fluids will get reduced.

This manifests in the increasing exit temperatures for the hot fluid

with increasing values ofhi,in as shown in Fig. 10a. The ambient heat

in leak is marginal enhancement of temperature for all values ofhi,in.

Increasing values of hi,in lead to decreasing values of hot fluid

effectiveness. The effect of ambient heat-in-leak is to further decrease

the effectiveness as shownin Fig. 10b. Thedecrease in effectiveness isaround 2% for all values of hi,in. This is crucial in heat exchangers

Fig. 9a. Effect ofCi/Ch on the hot fluid temperature profile for a three fluid HX with

ambient heat in leak, values of other non-dimensional parameters: NTU = 1,

H1 = 1.5, H2 = 2, R2 = 1.25, hh,in = 1, hi,,in = 0.5, hc,in = 0, h1 = 1.

Fig. 9b. Effect of Ci/Ch on the hot fluid effectiveness for a three fluid HX with

ambient heat in leak, Values of other non-dimensional parameters: NTU = 1,

H1 = 1.5, H2 = 2, R2 = 1.25. hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.

Fig. 9c. Effect of Ci/Ch on the hot fluid degradation factor for a three fluid HX with

ambient heat in leak, values of other non-dimensional parameters: NTU = 1,H1 = 1.5, H2 = 2, R2 = 1.25. hh,in = 1, hi,in = 0.5, hc,in = 0, h1 = 1.

Fig. 10a. Effect ofhi,in on the hot fluid temperature profile for a three fluid HX with


H1 = 1.5, H2 = 2, R1 = 2, R2 = 1.25. hh,in = 1, hc,in = 0, h1 = 1.

Fig. 10b. Effect of hi,in on the hot fluid effectiveness for a three fluid HX with


H1 = 1.5, H2 = 2, R1 = 2, R2 = 1.25. hh,in = 1, hc,in = 0, h1 = 1.


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operating in the cryogenic temperature ranges. A decrease in effec-

tiveness leads to a higher value of degradation factor. Thus, with

increasing values ofhi,in degradation factor gets enhanced as shown

in Fig. 10c. The effect of greater ambient heat in leak is to further in-

crease the degradation factor as observed. For a H3 value of 0.1, the

degradation factor increases by almost 1% as hi,in is increased from

01.

8. Conclusions

The effect of ambient heat in leak to the cold fluid in a three-

fluid HX, for a cryogenic application, involving heat interaction be-

tween all three fluids, has been investigated using the analytical

method and FEM. Cooling of the hot fluid has been identified as

the objective of the three fluid HX. Seven non-dimensional param-

eters NTU, H1, H2, H3, R1, R2 and hi,in have been used to present

the results on three counts (i) hot fluid temperature profile (ii) hot

fluid effectiveness both temperature and thermal effectiveness

(iii) degradation factor. The effect of ambient heat in leak to the

cold fluid is to increase the hot fluid temperature in all cases. While

degradation factor increases with increase in the value ofH1, there

is a reduction in the degradation factor with increasing values ofH2and NTU. In general, degradation factor increases for all cases, with

increase in H3 due to greater heat in leak. The effect of the ratio of

specific heat capacities Cc/Ch on the performance indicates that

maximum degradation factor is for the balanced flow condition

Cc = Ch, i.e., Cc/Ch = 1. There is a pronounced effect of ambient heat

in leak at NTU values of 10 and above. This clearly indicates that

larger the HX, more pronouned is the effect of ambient heat in leak.

For the conditions chosen, the ambient heat in leak is maximum for

low values ofCi

and goes on decreasing with increasing values ofCi

.

An increase in the value ofhi,in leads to greater exit temperatures of

the hot fluid, consequent drop in effectiveness and greater values

of degradation factor. In most cryogenic applications, heat in leak

from the ambient is a significant factor for the degradation in the

performance of heat exchangers. The present analysis has taken

this factor into account. The validation of the finite element meth-

odology, by comparison with the analytical solution presented, and

also with the previously published results, establishes the versatil-

ity of this methodology, which can account for most real time sit-

uations with relative ease and arrive at accurate results. The results

presented give valuable inputs towards better understanding of

the behaviour of the hot fluid in this class of heat exchangers.

Acknowledgements

The authors wish to thank Prof. D. Jawahar, C.E.O., P.E.S. Institu-

tions, Dr. K.N.B. Murthy, Principal and Director, P.E.S. Institute of

Technology, Bangalore, the Head and the faculty of the Department

of Mechanical Engineering, P.E.S.I.T., Bangalore, for the support ex-

tended for this work. The first author wishes to thank the Manage-

ment and the Principal, P.E.S. Institute of Technology, Bangalore,

for providing study leave, which was utilized for this study, leading

to this publication. The authors also wish to thank Miss Spoorthi S.

and Prof. Babu Reddy, Department of Mechanical Engineering and

Prof. N. Narahari, Department of Science and Humanities, P.E.S.

Institute of Technology, for their assistance.

Appendix A

A.1. Details of the matrices in Eq. (18) for the Galerkins Method, for

the flow arrangement Case (2) are as follows

References

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Fig. 10c. Effect ofhi,in on the hot fluid degradation factor for a three fluid HX with


H1 = 1.5, H2 = 2, R1 = 2, R2 = 1.25. hh,in = 1, hc,in = 0, h1 = 1.

K

12

NTU6R2

NTUH26R2

NTUH2

6R2

NTU6R2

12

NTU3R2

NTUH23R2

NTUH2

3R2

NTU3R2

NTUH2R16R2

1

2 NTUH1 R1

6R2 NTUH2 R1

6R2

NTUH1R1

6R2

NTUH2R1

3R2

12

NTUH1R13R2

NTUH2 R13R2

NTUH1 R1

3R2

NTU

6

NTUH1

6

1

2 NTU

6 NTUH1

6 NTUH3

6

NTU

3

NTUH1

3

12

NTU3

NTUH13

NTUH33

1

2 NTU

3R2 NTUH2

3R2

NTUH2

3R2

NTU3R2

12

NTU6R2

NTUH26R2

NTUH2

6R2

NTU6R2

NTUH2R13R2

1

2 NTUH1 R1

3R2 NTUH2 R1

3R2

NTUH1R1

3R2

NTUH2R1

6R2

12

NTUH1R16R2

NTUH2 R16R2

NTUH1 R1

6R2

NTU3

NTUH1

3

1

2 NTU

3 NTUH1

3 NTUH3

3

NTU

6

NTUH1

6

12

NTU6

NTUH16

NTUH36

266666666666664

377777777777775

fhg

hh;out

hi;in

hc;in

hh;in

hi;out

hc;out

8>>>>>>>>>>>>>>>:

9>>>>>>>>=

>>>>>>>>;

ffg

0

0

NTU H3 h1

hi;in

hc;in

hh;in

8>>>>>>>>>>>>>>>:

9>>>>>>>>=

>>>>>>>>;


8/22/2019 fem30

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[3] X. Luo, M. Li, W. Roetzel, A general solution for one-dimensional multi-stream

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Institute, Ruston, La, 1966.


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