Page 1

Journal of Robotics and Mechanical Engineering Research

Effect of Heat Exchangers Connection on Effectiveness

Voitto W. Kotiaho1, Markku J. Lampinen1 and M. El Haj Assad*2

1Aalto University, School of Science and Technology, P O Box 14100, FIN-00076, Finland

2Australian College of Kuwait, P O Box 1411Safat-13015 Kuwait

www.verizonaonlinepublishing.com

J Robot Mech Eng Resr 1(1).

Page | 11

*Corresponding author: Dr. Mamdouh El Haj Assad, Associate Professor/Head of Mechanical Engineering Department, Australian

College of Kuwait, Mechanical Engineering Department, PO Box 1411, Safat-13015Kuwait, Tel: +965-1828225 Ext: 4179, Fax: +965-

2537-6222; Email: m.assad@ack.edu.kw

Article Type: Research, Submission Date: 19 February 2015, Accepted Date: 04 May 2015, Published Date: 25 May 2015.

Citation: Voitto W. Kotiaho, Markku J. Lampinen and M. El Haj Assad (2015) Effect of Heat Exchangers Connection on

Effectiveness. J Robot Mech Eng Resr 1(1): 11-17.

Copyright: © 2015 Dr. Mamdouh El Haj Assad. This is an open-access article distributed under the terms of the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and

source are credited.

Vol: 1, Issue: 1

Abstract

In this work, the performance of the recuperative type heat

exchangers connected in series is studied. General expressions

for the effectiveness of counter flow and parallel flow connections

for any type of heat exchangers have been derived. It is proved

that the effectiveness of the heat exchanger is only function of

the thermal conductance and heat capacity flows. A numerical

example is given in order to find how many cross flow heat

exchangers connected in series would give the same effectiveness

as that of single counterflow heat exchanger.

Introduction

Heat exchangers are devices used to transfer heat between two

fluids at different temperatures. The goal of heat exchanger

design is to relate the inlet and outlet temperature, the overall

heat transfer coefficient, and the geometry of the heat exchanger

to the rate of heat transfer between the two fluids.

Heat exchangers are extensively used in power plants as boilers,

condensers, feedwater heaters, superheaters, economizers and

air heaters; in refrigeration and air-conditioning equipments as

evaporators and condensers; and in many other applications.

Effect of the maximum and minimum heat capacitance on the

performance of heat exchangers from entropy generation point of

view has been investigated [1]. Theoretical analysis of counterflow

heat exchangers [2] and parallel flow heat exchangers [3] with

a heat source within the hot fluid has been studied. Moreover

heat exchangers of any type can be connected in series for certain

purposes. More details on heat exchanger design and application

can be found in literature [4-7].

Heat exchangers can be connected in series in counter flow

connection as shown Figure 1a or in parallel flow connection

as shown in Figure 1b, while the heat exchangers connected in

series in both connections can be any type of heat exchanger.

In order to avoid any misunderstandings between concepts

counter-flow unit and counter connection as well as between

parallel-flow unit and parallel connection, a counter connection

of three parallel flow units (parallel flow heat exchangers) is

presented in Figure 2.

In series connection, the overall conductance of the connected

units is equal to the sum of the conductances of all individual

units.

The main objective of this work is to derive a general expression

for the overall effectiveness of the series connection as a function

of the effectiveness’s of whole units in that connection. Moreover,

this study purpose is to obtain the best performance of the heat

exchanger based on their connection.

Figure1: Counter- and parallel connection.

Figure 2: Parallel connection with parallel flow units.

Counter connection

Let us designate by T and T

Cx as the temperature and heat capacity

of hot flow; and by t and T

Cx as the temperature and heat capacity

of cold flow, respectively.

Consider the counter connection as shown in Figure 3 and

assume first that

t

T

C

C

x

x >

.

Figure 3 is a counter connection with counter flow heat exchangers

(counter units). However units can be any type of heat exchanger

since the same mathematical formulation is valid for them; this

J Robot Mech Eng Resr 1(1).

Page | 12

Citation: Voitto W. Kotiaho, Markku J. Lampinenand M. El Haj Assad (2015) Effect of Heat Exchangers Connection on Effectiveness. J Robot

Mech Eng Resr 1(1): 11-17.

is because we only consider the temperatures of the hot and cold

flows at the inlet and exit of the unit.

Consider unit 1 of Figure 3.

The heat balance can be written as

1

T

1t

TCtC

∆

=

∆

x

x

and

effectiveness of the unit is ε1 = ∆t1/θo. From these expressions it

follows that ∆t1 = ε1

θ

o and ∆T1 = (

T

t

C/Cx

x

)∆t1 = Rε1

θ

o, where

T

t

max

min

C/C

C/

CR

x

x

x

x

=

≡

. Hence θ2 = θo - ε1

θ

o and θ1 = θo -

R ε1

θ

o, we obtain

1

1

1

2

R1

1

ε

−

ε

−

=

θ

θ

. Correspondingly for the other

units

and

3

3

3

4

R1

1

ε

−

ε

−

=

θ

θ

. Multiply

we get

3

3

2

2

1

1

1

4

R1

1

R1

1

R1

1

ε

−

ε

−

ε

−

ε

−

ε

−

ε

−

=

θ

θ

.

The total heat balance of the connection is

)

t

(CtC

1

4

T

t

θ

−

∆

+

θ

=

∆

x

x

, where ∆t = ∆t1 + ∆t2 + ∆t3. Hence R

∆t = θ4 + ∆t - θ1 and

1R

t

1

4

−

θ

−

θ

=

∆

for R ≠ 1, and θ4 = θ1 = θ2 =

θ

3. for R = 1

On the other hand the effectiveness of the whole connection

is

and thus

4

4

t

1

θ

+

∆

θ

=

ε

−

. Now let us define an

ε

≡

δ

1− ε

The auxiliary variable is obtained as

R1

1

1R

1

)1R(

t

1

4

1

4

1

4

1

4

4

−

−

θ

θ

=

−

θ

θ

−

=

−

θ

θ

−

θ

=

θ

∆

=

ε

−

ε

=

δ

Substituting the expression of θ4/θ1 in the above equation, we

get

R1

1

1

R1

1

R1

1

R1

3

3

2

2

1

1

−

−

















ε

−

ε

−

ε

−

ε

−

ε

−

ε

−

=

δ

or in general form

















−

















ε

−

ε

−

−

=

δ

∏

=

n

1k

k

k

1

1

R1

R1

1

.

Then by using Eq. (1), the effectiveness of the whole connection

for 0 ≤ R < 1 is obtained as

∏

∏

=

=

−

















ε

−

ε

−

−

















ε

−

ε

−

=

ε

n

1k

k

k

n

1k

k

k

R

1

R1

1

1

R1

(2)

In the same way as above, a general expression for the whole

connection (counter connection) effectiveness for

t

T

C

C

x

x <

, i.e.

R =

t

T

C/Cx

x

, can be obtained, which is the same as that of Eq. (2).

So the heat capacitance ratio, R, does not change mathematically

the general expression for the effectiveness.

For R = 1, ∆t1 = ∆T1, ∆t2 = ∆T2 and ∆t3 = ∆T3, hence we

obtain

3

2

1

4

3

2

1

t

t

t

t

t

t

∆

+

∆

+

∆

+

θ

∆

+

∆

+

∆

=

ε

and thus we obtain

3

2

1

4

4

t

t

t

1

∆

+

∆

+

∆

+

θ

θ

=

ε

−

.

Then the auxiliary variable is obtained as

3

3

2

2

1

1

4

3

2

1

t

t

t

t

t

t

1

θ

∆

+

θ

∆

+

θ

∆

=

θ

∆

+

∆

+

∆

=

ε

−

ε

=

δ

,

since θ4 = θ1 = θ2 = θ3 for R = 1. On the other hand, the effectiveness

of unit 1 is

1

1

1

1

2

1

1

t

t

t

t

∆

+

θ

∆

=

∆

+

θ

∆

=

ε

, then

1

1

1

1

t

1

∆

+

θ

θ

=

ε

−

and

1

1

1

1

1

t

1

θ

∆

=

ε

−

ε

=

δ

. Correspondingly,

2

2

2

2

2

t

1

θ

∆

=

ε

−

ε

=

δ

and

3

3

3

3

3

t

1

θ

∆

=

ε

−

ε

=

δ

, hence we get δ = δ1 + δ2 + δ3 or

generally

∑

=

δ

=

δ

n

1k

k .

The effectiveness of the whole connection for R = 1 can be written

in general form as

Figure 3: Counter connection with counter flow units.

auxiliary variable as

(1)

J Robot Mech Eng Resr 1(1).

Page | 13

Citation: Voitto W. Kotiaho, Markku J. Lampinenand M. El Haj Assad (2015) Effect of Heat Exchangers Connection on Effectiveness. J Robot

Mech Eng Resr 1(1): 11-17.

∑

∑

=

=

ε

−

ε

+

ε

−

ε

=

ε

n

1k

k

k

n

1k

k

k

1

1

1

(3)

When the units are identical and we assume, that the conductance

of a unit does not change if its geometry does not change, it is

valid that ε1 = ε2 = ε3 = ...= εu. That is because it can be proven

that the effectiveness ε of a heat exchanger is only function of

heat capacity flows and conductance as will be shown later, and

the heat capacity flows are the same for all the units. Hence it

follows from Eqs. (2) and (3) that

R

R

R

n

u

u

n

u

u

−

















−

−

−

















−

−

=

ε

ε

ε

ε

ε

1

1

1

1

1

for 0 ≤ R < 1

(4)

and

( ) u

u

1n1

n

ε

−

+

ε

=

ε

for R = 1

(5)

where R is either

T

t

C/Cx

x

or

t

T

C/Cx

x

(the minimum heat

capacity flow is in the nominator) and n is the number of units.

In a similar way it is easy to show that Eqs. (2) and (4) are valid

also for R = 0.

Equation (5) is obtained by talking the limit of Equation (4) as R

tends to 1 and then using L’Hopitals rule as R tends to 1.

Now we consider the case where n→∞. The heat balance of an

individual unit can be written as

θ

′

=

∆

=

∆

u

u

t

u

T

AG

tC

T

C

x

x

,

where G’’ is conductance per unit heat transfer area of the heat

exchanger walls, Au is area of the heat exchangers walls and θ is

the average temperature difference between cold and hot fluids.

As n → ∞ and the size of the connection still remains finite, the

size of the units becomes differentially small and it follows that

, where θ = T - t. But this is the heat

balance of a differential unit of a counter-flow exchanger that

means that we end up to equations of the ordinary counter-flow

exchanger. Thus, as the number of units is very large, we can

consider the whole connection as one counter-flow exchanger.

Parallel connection

Consider the parallel connection where the individual units

consist of parallel flow heat exchangers as shown in Figure 4.

Let us first consider the case where

t

T

C

C

x

x >

, which i.e. R =

T

t

C/Cx

x

. Considering unit 1, the heat balance is

1

T

1

t

T

C

tC

∆

−

=

∆

x

x

since ∆T1 < 0. The effectiveness of unit 1 is ε1 = ∆t1/θo = ∆t1/

θ

1. From these equations it follows that ∆t1 = ε1

θ

1 and ∆T1 = - (

T

t

C/Cx

x

)∆t1 = -Rε1

θ

1.

Thus θ2 = θ1 - ∆t1 -(- ∆T1) = θ1 - ε1

θ

1 - Rε1

θ

1 = θ1(1 - ε1 - Rε1)

which means that

1

1

1

2

R

1

ε

−

ε

−

=

θ

θ

. Correspondingly for the other units

2

2

2

3

R

1

ε

−

ε

−

=

θ

θ

and

3

3

3

4

R

1

ε

−

ε

−

=

θ

θ

. Multiplying the

temperature difference ratio as

⋅ , the following

temperature difference ratio is obtained as

heat balance of the connection is

which means that R∆t = θ1 - ∆t - θ4 and

1R

t

4

1

+

θ

−

θ

=

∆

.

The effectiveness of the whole connection is

1R

)R

1)(R

1)(R

1(1

1R

1

1R

1t

t

3

3

2

2

1

1

1

4

4

1

1

1

o

+

ε

−

ε

−

ε

−

ε

−

ε

−

ε

−

−

=

+

θ

θ

−

=

+

θ

−

θ

θ

=

θ

∆

=

θ

∆

=

ε

or in a general form

(

)

1R

R

1

1

n

1k

k

k

+

ε

−

ε

−

−

=

ε

∏

=

.

(6)

In a similar way we end up with Eq. (6) for the case where

t

T

C

C

x

x <

. It is easy to show that Eq. (6) is valid also when R

= 1. If all the units of a parallel connection are identical with

effectiveness εu, it follows from Eq. (6) that

(

)

1R

R

11

n

u

u

+

ε

−

ε

−

−

=

ε

,

(7)

Figure 4: Parallel connection with parallel flow units.

The overall

Citation: Voitto W. Kotiaho, Markku J. Lampinenand M. El Haj Assad (2015) Effect of Heat Exchangers Connection on Effectiveness. J Robot

Mech Eng Resr 1(1): 11-17.

J Robot Mech Eng Resr 1(1).

Page | 14

which is valid for all values of R (when 0 ≤ R ≤ 1).

Let us again consider the case where n→∞. The heat balance

of one unit can be written as

θ

′

=

∆

=

∆

−

u

u

t

u

T

AG

tC

T

C

x

x

. As n→∞ and the size of the connection still remains finite,

the size of the units is differentially small and it follows that

, where θ = T - t. However, this is the

heat balance of a differential unit of a parallel-flow exchanger,

hence we end up with the equations of the ordinary parallel-

flow exchanger. Thus as the number of units is very large, we can

consider the whole connection as one parallel-flow exchanger.

A proof that the effectiveness of a heat exchanger is a

function of R and NTU only

This proof is based on the Buckingham’s Π-theorem and on

such a reasonable assumption that the change of temperature

of the smaller heat capacity flow in the heat exchanger, ∆Tmax,

is function of four variables which are temperature difference of

entering flows θo (which is the maximum temperature difference

in the whole heat exchanger), minimum heat capacity flow min

Cx

, maximum heat capacity flow max

Cx

, and conductance of the

exchanger G. Hence we can write

∆Tmax = ∆Tmax(θo, min

Cx

, max

Cx

, G).

Mathematically there is a function F that satisfies the following

condition F (∆Tmax, θo, min

Cx

, max

Cx

, G) = 0. The dimensions of

the variables are: [∆Tmax] = [θo] = K and [ min

Cx

] = [ max

Cx

] = [G]

= W/K. We can choose dimensions K and W as basic units, that

means there are two linearly independent dimensions in function

F. This can be done even though W = kg m2 s-3 is not a basic unit

in the standard unit systems, because the linear independence of

K and W is all that is demanded. Thus the amount of variables is

5 and the amount of independent dimensions is 2.

According to Buckingham’s Π-theorem no information is lost if

the function F is derived to a function of dimensionless groups,

that is Π-groups, so that the amount of Π-groups in the new

function is 3.

We can form, from the variables of function F, for example the

following Π-groups: Π1 = ∆Tmax/θo ≡ e, Π2 = min

Cx

/ max

Cx

≡ R and

Π

3 = G / min

Cx

≡ NTU. Hence we get a new function f so that f(Π1,

Π

2, Π3) = 0. But this can be derived directly to form Π1 = Π1(Π2,

Π

3) or e = e (R, NTU), which was to be proved.

Comparison between a counterflow exchanger and

a counterflow connection made of crossflow elements.

Considering the three types of heat exchangers, counter-, cross-

and parallel flow exchangers, it is well known that the counterflow

exchanger has the best effectiveness. Hence, the effectiveness of

the counter flow exchanger can be used as a reference. The closer

the effectiveness of a connection to the reference one, the better

is the connection. An important question is how many elements

needed in order that a connection practically performs as counter

flow exchanger with the same heat transfer area?

In order to study this, we use the following approximate equation

for cross-flow exchangers with both flows unmixed [8]:

(8)

Consider a counter flow exchanger and a counter connection

made of cross-flow units so that they both have the same G,

min

Cx

and max

Cx

and hence the same R and NTU. Let n be the

number of the cross flow units. For a cross flow unit

NTUu = NTU/n

(9)

Substituting Eq. (9) into Eq (8), we get the effectiveness, εu for

an individual cross flow unit. Using Eqs. (4) and (5), we get the

effectiveness εcon for counter connection.

For a counter-flow exchanger [1], the effectiveness is

(10)

Then the percentage difference between both effectivenesses

for various values of R, NTU and n can be calculated by

%100

cou

con

cou

ε

ε−

ε

.

For example, with values R = 0.75, NTU = 5 and n = 4 we get ecro

= 0.828 (Eq.(8)), NTUu = 1.25 (Eq.(9)), eu = 0.563 (Eq.(8)), econ =

0.892 (Eq.(4)),

econ = 0.909 (Eq.(10)) and

%2

%100

cou

con

cou

=

ε

ε

−

ε

.

Results and Discussion

For a counter connection the effectiveness of the connection can

be calculated from Eq. (2) or Eq. (3) and especially from Eq. (4)

or Eq. (5) when the units are identical.

The corresponding equations for a parallel-flow connection are

Eqs. (6) and (7).

The data calculated according to the above procedure is shown

graphically in Figures 5a - 5d.

Effectivenesses of counterflow-and crossflow heat exchangers

and of counter connections made of crossflow units with

different values of R. In each of the figures, the uppermost curve

represents the counter flow heat exchanger and the lowermost

represents a single cross flow heat exchanger and between the

upper- and lowermost curves there are 5 curves, which represent

the counter connections made of 2, 4, 6, 8 and 10 with cross flow

heat exchanger units. The larger the number of units the closer

the curve is to the uppermost curve.

Figures 5 a - 5 d illustrate that the larger R value the larger the

differences of e between individual cases. In case R = 0 which

takes place in a phase change, there would be no difference at all

and there would be only a single curve in the figure.

Thus we can make a conclusion that the differences are the

largest for R = 1, i.e. when the heat capacity flows are the same.

This conclusion can be reasoned also in the following way: let

us consider a situation where

max

min

C

C

x

x

<

. In this case, the

temperature change of max

Cx

is very small hence there is almost

constant temperature on the other side of the wall. So it is almost

insignificant to which direction min

Cx

flows compared to max

Cx

. Thus the temperature change of min

Cx

is less sensitive to the

relative flow directions, i.e. the geometry, the larger the difference

between min

Cx

and max

Cx

, i.e. the closer the R is to zero. Hence

the temperature change of min

Cx

, and by definition the e is more

sensitive to the geometry the closer R value to unity.

Thus, the largest differences between effectivenesses appear in the

case R = 1. The largest differences between a single counterflow

exchanger and counterflow connections with various number of

units of cross flow heat exchangers have been found from the

numerical data of Figure 5a. The main results are given in Table 1.

The largest differences of e between a single counterflow

exchanger and counterflow connections when R = 1. In the table

n = number of units, NTU (max) = the NTU-value where the

largest difference occurs, % (max) = largest difference (%), %

(NTU = 1) = the difference of e when NTU = 1 (%).

The numerical data was calculated with the accuracy of two

decimals and with some values of n there appeared to be two

maximum points. In most cases, the maximum point was very

close to the point NTU = 1 and for this reason there is an extra

row which shows the difference for NTU = 1.

The results indicated by Table 1 can be presented shortly so that

two units are needed in order that the difference in all conditions

is less than 10 %, for 3 units less than 5 %, for 8 units less than 2

% and for 28 units less than 1 %. But it must be remembered that

we considered here the limit case R = 1 and with smaller values

of R usually smaller amount of units is needed. Table 1 indicates

also that the difference of e between a counterflow and a single

crossflow exchanger is never larger than 11%.

Citation: Voitto W. Kotiaho, Markku J. Lampinenand M. El Haj Assad (2015) Effect of Heat Exchangers Connection on Effectiveness. J Robot

Mech Eng Resr 1(1): 11-17.

J Robot Mech Eng Resr 1(1).

Page | 15

It can be proven by the dimensional analysis that for any heat

exchanger the effectiveness is a function of NTU and R only and

does not depend on the incoming temperatures.

Practically not so many crossflow units are needed in order that

the effectiveness of a counterflow connection is practically not

worse than that of a single counterflow exchanger. It was proved

that 8 units are needed in order to have only 2 % difference in

effectiveness.

Conclusions

A theoretical formulation was presented in order to obtain

the effectiveness of heat exchangers connected in series. The

effectiveness was derived for counter and parallel flow heat

exchangers connected in series. If the heat exchangers are

connected in series they should be connected in counter flow

connection in order to achieve the best possible effectiveness.

It was shown that the effectiveness of the heat exchanger is only

function of the heat capacitance ratio of the two fluids and the

thermal conductance.

Usually only a few units are needed in order that the effectivenes

is almost the same as a single counterflow exchanger.

Nomenclature

A

area of heat transfer wall, m2

cp

specific heat, J/kgk

Cx

heat capacity rate, W/K, (

p

cmC

x

x =

)

G

conductance, W/K

mx

mass flow, kg/s

n

number of units

NTU number of heat transfer units, (

min

C/G

NTU

x

=

)

R ratio of heat capacitance, (

max

min C/

CR

x

x

=

)

t cold fluid temperature, K

T hot fluid temperature, K

Greek Letters

δ auxiliary variable

ε effectiveness of a heat exchanger

θ temperature difference, K

Subscripts

cro crossflow exchanger

cou counterflow exchanger

con counterflow connection made of crossflow units

Min minimum

max maximum

t cold fluid

T

hot fluid

u

unit

Citation: Voitto W. Kotiaho, Markku J. Lampinenand M. El Haj Assad (2015) Effect of Heat Exchangers Connection on Effectiveness. J Robot

Mech Eng Resr 1(1): 11-17.

n

1

2

3

4

5

6

7

8

9

10

NTU(max)

10.0

17.9

25.6

33.1

40.7

1.1

1.1

1.1

1.1

1.2

%(max)

11.0

6.1

4.2

3.2

2.6

2.3

2.1

2.0

2.0

1.8

%(NTU=1)

6.3

4.1

3.3

2.5

2.5

2.3

2.1

2.0

2.0

1.8

n

11

12

13

14

15

16

17

18

19

20

NTU(max)

1.2

1.2

1.2

1.2

1.3

1.3

1.3

1.3

1.3

1.3

%(max)

1.7

1.6

1.5

1.5

1.4

1.4

1.3

1.3

1.3

1.2

%(NTU=1)

1.7

1.6

1.5

1.5

1.4

1.4

1.3

1.3

1.2

1.2

n

21

22

23

24

25

26

27

28

29

30

NTU(max)

1.3

1.4

1.4

1.4

1.4

1.4

1.4

1.4

1.4

1.4

%(max)

1.2

1.2

1.2

1.1

1.1

1.0

1.0

1.0

1.0

1.0

%(NTU=1)

1.2

1.1

1.1

1.1

1.1

1.0

1.0

1.0

1.0

0.9

Table 1: Main Results

J Robot Mech Eng Resr 1(1).

Page | 16

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