Open access peer-reviewed chapter

# Basic Design Methods of Heat Exchanger

By Cüneyt Ezgi

Submitted: March 23rd 2016Reviewed: February 14th 2017Published: April 27th 2017

DOI: 10.5772/67888

## Abstract

Heat exchangers are devices that transfer energy between fluids at different temperatures by heat transfer. These devices can be used widely both in daily life and industrial applications such as steam generators in thermal power plants, distillers in chemical industry, evaporators and condensers in HVAC applications and refrigeration process, heat sinks, automobile radiators and regenerators in gas turbine engines. This chapter discusses the basic design methods for two fluid heat exchangers.

### Keywords

• log-mean temperature difference (LMTD)
• effectiveness-number of transfer units (ε − NTU)
• dimensionless mean temperature difference (Ψ − P) and (P1 – P2) effectiveness-modified number of transfer units (ε − NTUo)
• reduced length and reduced period (Λ − π)

## 1. Introduction

Heat exchangers (HE) are devices that transfer energy between fluids at different temperatures by heat transfer. Heat exchangers may be classified according to different criteria. The classification separates heat exchangers (HE) in recuperators and regenerators, according to construction is being used. In recuperators, heat is transferred directly (immediately) between the two fluids and by opposition, in the regenerators there is no immediate heat exchange between the fluids. Rather this is done through an intermediate step involving thermal energy storage. Recuperators can be classified according to transfer process in direct contact and indirect contact types. In indirect contact HE, there is a wall (physical separation) between the fluids. The recuperators are referred to as a direct transfer type. In contrast, the regenerators are devices in which there is intermittent heat exchange between the hot and cold fluids through thermal energy storage and release through the heat exchanger surface or matrix. Regenerators are basically classified into rotary and fixed matrix models. The regenerators are referred to as an indirect transfer type.

This chapter discusses the basic design methods for two fluid heat exchangers. We discuss the log-mean temperature difference (LMTD) method, the effectiveness εNTUmethod, dimensionless mean temperature difference (ΨP) and (P1P2) to analyse recuperators. The LMTD method can be used if inlet temperatures, one of the fluid outlet temperatures, and mass flow rates are known. The ε – NTU method can be used when the outlet temperatures of the fluids are not known. Also, it is discussed effectiveness-modified number of transfer units (εNTUo) and reduced length and reduced period (Λπ) methods for regenerators.

## 2. Governing equations

The energy rate balance is

dEcvdt=Q˙W˙+im˙i(hi+Vi22+gzi)em˙e(he+Ve22+gze)E1

For a control volume at steady state, dEcvdt=0. Changes in the kinetic and potential energies of the flowing streams from inlet to exit can be ignored. The only work of a control volume enclosing a heat exchanger is flow work, so W˙=0and single-stream (only one inlet and one exit) and from the steady-state form the heat transfer rate becomes simply 

Q˙=m˙(h2h1)E2

For single stream, we denote the inlet state by subscript 1 and the exit state by subscript 2.

For hot fluids,

Q˙=m˙(hh1hh2)E3

For cold fluids,

Q˙=m˙(hc2hc1)E4

The total heat transfer rate between the fluids can be determined from

Q˙=UAΔTlmE5

where U is the overall heat transfer coefficient, whose unit is W/m2 oC and ΔTlmis log-mean temperature difference.

## 3. Overall heat transfer coefficient

A heat exchanger involves two flowing fluids separated by a solid wall. Heat is transferred from the hot fluid to the wall by convection, through the wall by conduction and from the wall to the cold fluid by convection.

UA=UoAo=UiAi=1RtE6

where Ai=πDiLand Ao=πDoLand U is the overall heat transfer coefficient based on that area. Rt is the total thermal resistance and can be expressed as 

Rt=1UA=1hiAi+Rw+RfiAi+RfoAo+1ho AoE7

where Rf is fouling resistance (factor) and Rw is wall resistance and is obtained from the following equations.

For a bare plane wall

Rw=tkAE8

where t is the thickness of the wall

For a cylindrical wall

Rw=ln(rori)2πLkE9

The overall heat transfer coefficient based on the outside surface area of the wall for the unfinned tubular heat exchangers,

Uo=1rori1hi+rori Rfi+rokln(rori)+Rfo+1hoE10

where Rfi and Rfo are fouling resistance of the inside and outside surfaces, respectively.

or

Uo=1rori1hi+Rft+rokln(rori)+1hoE11

where Rft is the total fouling resistance, given as

Rft=AoAiRfi+RfoE12

For finned surfaces,

Q˙=ηhAΔTE13

where η is the overall surface efficiency and

η=1AfA(1ηf)E14

where Af is fin surface area and ηf is fin efficiency and is defined as

ηf=Q˙fQ˙f,maxE15

Constant cross-section of very long fins and fins with insulated tips, the fin efficiency can be expressed as

ηf,long=1mLE16
ηf,insulated=tanh(mL)mLE17

where L is the fin length.

For straight triangular fins,

ηf,triangular=1mLI1(2mL)I0(2mL)E18

For straight parabolic fins,

ηf,parabolic=21+(2mL)2+1E19

For circular fins of rectangular profile,

ηf,rectangular=CK1(mr1)I1(mr2c)I1(mr1)K1(mr2c)I0(mr1)K1(mr2c)K0(mr1)I1(mr2c)E20

where the mathematical functions I and K are the modified Bessel functions and

m=2h/ktE21

where t is the fin thickness.

and

C=2r1/mr2c2r12E22

where

r2c=r2+t/2E23

For pin fins of rectangular profile,

ηf,pin,rectangular=tanhmLcmLcE24

where

m=4h/kDE25

and corrected fin length, Lc, defined as

Lc=L+D/4E26

where L is the fin length and D is the diameter of the cylindrical fins. The corrected fin length is an approximate, yet practical and accurate way of accounting for the loss from the fin tip is to replace the fin length L in the relation for the insulated tip case.

A is the total surface area on one side

A=Au+AfE27

The overall heat transfer coefficient is based on the outside surface area of the wall for the finned tubular heat exchangers,

Uo=1AoAi1ηihi+AoAiRfiηi+AoRw+Rfoηo+1ηohoE28

where Ao and Ai represent the total surface area of the outer and inner surfaces, respectively.

## 4. Thermal design for recuperators

Four methods are used for the recuperator thermal performance analysis: log-mean temperature difference (LMTD), effectiveness-number of transfer units (εNTU), dimensionless mean temperature difference (ΨP) and (P1P2) methods.

### 4.1. The log-mean temperature difference (LMTD) method

The use of the method is clearly facilitated by knowledge of the hot and cold fluid inlet and outlet temperatures. Such applications may be classified as heat exchanger design problems; that is, problems in which the temperatures and capacity rates are known, and it is desired to size the exchanger.

#### 4.1.1. Parallel and counter flow heat exchanger

Two types of flow arrangement are possible in a double-pipe heat exchanger: parallel flow and counter flow. In parallel flow, both the hot and cold fluids enter the heat exchanger at the same end and move in the same direction, as shown in Figure 1. In counter flow, the hot and cold fluids enter the heat exchanger at opposite end and flow in opposite direction, as shown in Figure 2.

The heat transfer rate is

Q˙=UAΔTlmE29

where ΔTlmis log-mean temperature difference and is

ΔTlm=ΔT1ΔT2ln(ΔT1ΔT2)E30

Then,

Q˙=UAΔT1ΔT2ln(ΔT1ΔT2)E31

where the endpoint temperatures, ΔT1and ΔT2, for the parallel flow exchanger are

ΔT1=ThiTciE32
ΔT2=ThoTcoE33

where Thi is the hot fluid inlet temperature, Tci is the cold fluid inlet temperature, Tho is the hot fluid outlet temperature and Tco is the cold fluid outlet temperature.

The endpoint temperatures, ΔT1and ΔT2, for the counter flow exchanger are

ΔT1=ThiTcoE34
ΔT2=ThoTciE35

#### 4.1.2. Multipass and cross-flow heat exchanger

In compact heat exchangers, the two fluids usually move perpendicular to each other, and such flow configuration is called cross-flow. The cross-flow is further classified as unmixed and mixed flow, depending on the flow configuration, as shown in Figures 3 and 4.

Multipass flow arrangements are frequently used in shell-and-tube heat exchangers with baffles (Figure 5).

Log-mean temperature difference ΔTlmis computed under assumption of counter flow conditions. Heat transfer rate is

Q˙=UAFΔTlm,cfE36

where F is a correction factor and non-dimensional and depends on temperature effectiveness P, the heat capacity rate ratio R and the flow arrangement.

P=Tc2Tc1Th1Tc1E37
R=Th1Th2Tc2Tc1E38

The value of P ranges from 0 to 1. The value of R ranges from 0 to infinity. If the temperature change of one fluid is negligible, either P or R is zero and F is 1. Hence, the exchanger behaviour is independent of the specific configuration. Such would be the case if one of the fluids underwent a phase change.

Correction factor F charts for common shell-and-tube and cross-flow heat exchangers are shown in Figures 610.

#### 4.1.3. The procedure to be followed with the LMTD method

1. Select the type of heat exchanger.

2. Calculate any unknown inlet or outlet temperatures and the heat transfer rate.

3. Calculate the log-mean temperature difference and the correction factor, if necessary.

4. Calculate the overall heat transfer coefficient.

5. Calculate the heat transfer surface area.

6. Calculate the length of the tube or heat exchanger

### 4.2. The ε – NTU method

If the exchanger type and size are known and the fluid outlet temperatures need to be determined, the application is referred to as a performance calculation problem. Such problems are best analysed by the NTU-effectiveness method [4, 5].

Capacity rate ratio is

C*=CminCmaxE39

where Cmin and Cmax are the smaller and larger of the two magnitudes of Ch and Cc, respectively, and Ch and Cc are the hot and cold fluid heat capacity rates, respectively.

Heat exchanger effectiveness εis defined as

ε=Q˙Q˙max=Actual heat transfer rateMaximum possible heat transfer rateE40

where

Q˙max=(m˙cp)c(Th1Tc1)ifCc<ChE41

or

Q˙max=(m˙cp)h(Th1Tc1) ifCh<CcE42

where Cc=mc˙cpcand Ch=mh˙cphare the heat capacity rates of the cold and the hot fluids, respectively, and m˙is the rate of mass flow and cp is specific heat at constant pressure.

Heat exchanger effectiveness is therefore written as

ε=Ch(Th1Th2)Cmin(Th1Tc1)=Cc(Tc2Tc1)Cmin(Th1Tc1)E43

The number of transfer unit (NTU) is defined as a ratio of the overall thermal conductance to the smaller heat capacity rate. NTU designates the non-dimensional heat transfer size or thermal size of the exchanger [4, 5].

NTU=UACmin=1CminUdAE44

In evaporator and condenser for parallel flow and counter flow,

C*=CminCmax=0E45

and

ε=1eNTUE46

The effectivenesses of some common types of heat exchangers are also plotted in Figures 1116.

#### 4.2.1. The procedure to be followed with the ε – NTU method

1. For the rating analysis:

1. Calculate the capacity rate ratio

2. Calculate NTU.

3. Determine the effectiveness.

4. Calculate the total heat transfer rate.

5. Calculate the outlet temperatures.

2. For the sizing problem:

1. Calculate the effectiveness.

2. Calculate the capacity rate ratio.

3. Calculate the overall heat transfer coefficient.

4. Determine NTU.

5. Calculate the heat transfer surface area.

6. Calculate the length of the tube or heat exchanger

### 4.3. The Ψ – P method

The dimensionless mean temperature difference is 

ψ=ΔTmThiTci=ΔTmΔTmaxE47
ψ=εNTU=P1NTU1=P2NTU2E48

where P is the temperature effectiveness and the temperature effectivenesses of fluids 1 and 2 are defined as, respectively

P1=T1,oT1,iT2,iT1,iE49

P2=T2,iT2,oT2,iT1,iE50
ψ={FP1(1R1)ln[(1R1P1)(1P1)] for R11F(1P1) for R1=1E51

where 1 and 2 are fluid stream 1 and fluid stream 2, respectively, and R is the heat capacity ratio and defined as

R1=C1C2=T2,iT2,oT1,oT1,iE52
R2=C2C1=T1,oT1,iT2,iT2,oE53
R1=1R2E54

Non-dimensional mean temperature difference as a function for P1 and R1 with the lines for constant values of NTU1 and the factor is shown in Figure 17. Figure 17.Non-dimensional mean temperature difference as a function for P1 and R1.

The heat transfer rate is given by

q=UAΨ(ThiTci)E55

#### 4.3.1. The procedure to be followed with the Ψ – P method

1. Calculate NTU1.

2. Calculate F factor.

3. Calculate R1 with the lines for constant values of NTU1 and the F factor superimposed in Figure 17.

4. Plot the dimensionless mean temperature Ψ as a function of P1 and R1 in Figure 17.

5. Calculate the heat transfer rate.

### 4.4. The Pl – P2 method

The dimensionless mean temperature difference is 

ψ=εNTU=P1NTU1=P2NTU2E56

P1 P2 chart for 1–2 shell and tube heat exchanger  with shell fluid mixed is shown in Figure 18. Figure 18.P1 – P2 chart for 1–2 shell and tube heat exchanger with shell fluid mixed.

where 1 and 2 are one shell pass and two tube passes, respectively.

#### 4.4.1. The procedure to be followed with the P1– P2 method

1. Calculate NTU1 or NTU2.

2. Calculate R1 or R2.

3. Plot P1 as a function of R1 with NTU1 or P2 as a function of R2 with NTU2 in Figure 18.

4. Calculate the dimensionless mean temperature Ψ.

5. Calculate the heat transfer rate.

## 5. Thermal design for regenerators

Two methods are used for the regenerator thermal performance analysis: εNTUoand Λπmethods, respectively, for rotary and fixed matrix regenerators.

### 5.1. The ε – NTUo method

The ε – NTUo method was developed by Coppage and London in 1953. The modified number of transfer units is 

NTUo=1Cmin[11(hA)h+1(hA)c]E57
C*=CminCmaxE58
Cr*=CrCminE59
Cr=MwcwNE60

where cw is the specific heat of wall material, N is the rotational speed for a rotary regenerator and Mw is matrix mass and determined as

Mw=ArcHrρmSmE61

where Arc is the rotor cross-sectional area, Hr is the rotor height, ρm is the matrix material density and Sm is the matrix solidity.

The convection conductance ratio is

(hA)*=(hA) Cmin(hA)CmaxE62

Most regenerators operate in the range of 0.25(hA)*<4. The effect of (hA)*on the regenerator effectiveness can usually be ignored.

A is the total matrix surface area and given as

A=ArcHrβFrfaE63

where Arc is the rotor cross-sectional area, Hr is the rotor height, β is the matrix packing density and Frfa is the fraction of rotor face area not covered by radial seals.

The hot and cold gas side surface areas are proportional to the respective sector angles.

Ah=(αh360°)AE64
Ac=(αc360°)AE65

where αhand αcare disk sector angles of hot flow and cold flow in degree, respectively.

The regenerator effectiveness is

ε=qqmaxE66
qmax=Cmin(ThiTci)E67

#### 5.1.1. The counter flow regenerator

The regenerator effectiveness for ε0.9is

ε=εcf(119Cr*1.93)E68

where εcfis the counter flow recuperator effectiveness and is determined as

εcf=1exp[NTUo(1C*)]1C*exp[NTUo(1C*)]E69

The counter flow regenerator effectiveness as a function of NTUo and for C* = 1 is presented in Figure 19. The regenerator effectiveness increases with Cr*for given values of NTUo and C*. The range of the optimum value of Cr*is between 2 and 4 for optimum regenerator effectiveness. Figure 19.The counter flow regenerator effectiveness as a function of NTUo and for C* = 1.

#### 5.1.2. The parallel flow regenerator

The parallel flow regenerator effectiveness as a function of NTUo and for C* = 1 and (hA)* = 1 is presented in Figure 20. Figure 20.The parallel flow regenerator effectiveness as a function of NTUo and for C* = 1 and (hA)* = 1.

#### 5.1.3. The procedure to be followed with the ε – NTUo method

1. Calculate the capacity rate ratio.

2. Calculate (hA)*.

3. Calculate (Cr)*.

4. Calculate NTUo.

5. Determine the effectiveness.

6. Calculate the total heat transfer rate.

7. Calculate the outlet temperatures.

### 5.2. The Λ – π method

This method is generally used for fixed matrix regenerators. The reduced length designates the dimensionless heat transfer or thermal size of the regenerator. The reduced length is 

Λ=bLE70

The reduced lengths for hot and cold sides, respectively, are

Λh=(hAC)h=ntuhE71
Λc=(hAC)c=ntucE72

The reduced period is

π=cPh or cPcE73

where b and c are constants.

The reduced periods for hot and cold sides, respectively, are

πh=(hACr)hE74
πc=(hACr)cE75

Designations of various types of regenerators are given in Table 1. For a symmetric and balanced regenerator, the reduced length and the reduced period are equal on the hot and cold sides:

Regenerator
BalancedΛhΠh=ΛcΠcorγ=1
UnbalancedΛhΠhΛcΠc
Symmetricπh=πc
Unsymmetricπhπc
Symmetric and balancedΛh=Λc, πh=πc
Unsymmetric and balancedΛhΠh=ΛcΠc
LongΛ/Π>5

### Table 1.

Designation of various types of regenerators for ΛΠ method.

Λh=Λc=Λ=Λm=hAm˙cp=ntuE76
πh=πc=π=πm=hAPMwcwE77

The actual heat transfer during one hot or cold gas flow period is

Q=ChPh(ThiTho)=CcPc(TcoTci)E78

The maximum possible heat transfer is

Qmax=(CP)min(ThiTci)E79

The effectiveness for a fixed-matrix regenerator is

ε=QQmax=(CP)h(ThiTho)(CP)min(ThiTci)=(CP)c(TcoTci)(CP)min(ThiTci)E80

The effectiveness chart for a balanced and symmetric counter flow regenerator is given in Figure 21. Figure 21.The effectiveness chart for a balanced and symmetric counter flow regenerator.

The effectiveness chart for a balanced and symmetric parallel flow regenerator is given in Figure 22. Figure 22.The effectiveness chart for a balanced and symmetric parallel flow regenerator.

#### 5.2.1. The procedure to be followed with the Λ – π method

1. Calculate the reduced length.

2. Calculate the reduced period.

3. Calculate C*.

4. Calculate (Cr)*.

5. Calculate NTUo.

6. Determine the effectiveness.

7. Calculate the total heat transfer rate.

8. Calculate the outlet temperatures.

## 6. Conclusion

This chapter has discussed the basic design methods for two fluid heat exchangers. The design techniques of recuperators and regenerators, which are two main classes, were investigated.

The solution to recuperator problem is presented in terms of log-mean temperature difference (LMTD), effectiveness-number of transfer units (εNTU), dimensionless mean temperature difference (ΨP) and (P1 P2) methods. The exchanger rating or sizing problem can be solved by any of these methods and will yield the identical solution within the numerical error of computation. If inlet temperatures, one of the fluid outlet temperatures, and mass flow rates are known, the LMTD method can be used to solve sizing problem. If they are not known, the (εNTU) method can be used. (ΨP) and (P1 P2) methods are graphical methods. The (P1 P2) method includes all major dimensionless heat exchanger parameters. Hence, the solution to the rating and sizing problem is non-iterative straightforward.

Regenerators are basically classified into rotary and fixed matrix models and in the thermal design of these models two methods: effectiveness-modified number of transfer units (εNTUo) and reduced length and reduced period (Λπ) methods for the regenerators. (Λπ) method is generally used for fixed matrix regenerators.

## Nomenclature

A

Total heat transfer surface area of heat exchanger, total heat transfer surface area of all matrices of a regenerator, m2

Af

Fin surface area, m2

Arc

Rotor cross-sectional area, m2

C

Flow stream heat capacity rate, W/K

CW

Matrix heat capacity rate, W/K

cp

Specific heat at constant pressure, J/kgK

cw

Specific heat of wall material, J/kgK

d, D

Diameter, m

E

Total energy, kJ

Frfa

Fraction of rotor face area not covered by radial seals.

Hr

Rotor height

h

Specific enthalpy, kJ/kg

k

Thermal conductivity, W/mK

L

Length of heat exchanger, m

m˙

Mass flow rate, kg/s

Mw

The total mass of all matrices of a regenerator, kg

N

Rotational speed for a rotary regenerator, rev/s, rpm

NTU

Number of transfer units

ntuc

Number of transfer units based on the cold fluid side

ntuh

Number of transfer units based on the hot fluid side

P

Temperature effectiveness for one fluid stream

Q˙

Heat transfer rate, kW

r

R

Thermal resistance, m2K/W

Rf

Fouling resistance, fouling factor, m2K/W

Sm

Matrix solidity

T

Temperature, °C, K

Tc

Cold fluid temperature, °C, K

Th

Hot fluid temperature, °C, K

t

Wall thickness, m

ΔTlm

Log-mean temperature difference, °C, K

U

Overall heat transfer coefficient, W/m2K

V

Velocity, m/s

W˙

Power, kW

z

Elevation, m

## Greek symbols

β

packing density for a regenerator, m2/m3

Δ

Difference

ε

Effectiveness

ρm

Matrix material density, kg/m3

η

Efficiency

Λ

Reduced length for a regenerator

π

Reduced period for a regenerator

## Subscripts

c

Cold fluid

cf

Counter flow

cv

Control volume

e

Exit conditions

f

Fin, finned, friction

h

Hot

i

Inlet conditions, inner, inside

lm

logarithmic mean

max

Maximum

min

Minimum

o

Outer, outside, overall

u

Unfinned

1

Initial or inlet state, fluid 1

2

Final or exit state, fluid 2

## More

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Cüneyt Ezgi (April 27th 2017). Basic Design Methods of Heat Exchanger, Heat Exchangers - Design, Experiment and Simulation, S M Sohel Murshed and Manuel Matos Lopes, IntechOpen, DOI: 10.5772/67888. Available from:

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