41

5.1 Shell-and-Tube Heat Exchangers

The most common type of heat exchanger in industrial applications is shell-and-tube heat

exchangers. The exchangers exhibit more than 65% of the market share with a variety of

design experiences of about 100 years. Shell-and tube heat exchangers provide typically

the surface area density ranging from 50 to 500 m2/m

3 and are easily cleaned. The design

codes and standards are available in the TEMA (1999)-Tubular Exchanger Manufacturers

Association. A simple exchanger, which involves one shell and one pass, is shown in

Figure 5.18.

Shell inlet

Shell outletTube inlet

Tube outlet

Baffles Endchannel

TubeShell

Shell sheet

Figure 5.18 Schematic of one-shell one-pass (1-1) shell-and-tube heat exchanger.

Baffles

In Figure 5.18, baffles are placed within the shell of the heat exchanger firstly to support

the tubes, preventing tube vibration and sagging, and secondly to direct the flow to have a

higher heat transfer coefficient. The distance between two baffles is baffle spacing.

Multiple Passes

Shell-and-tube heat exchangers can have multiple passes, such as 1-1, 1-2, 1-4, 1-6, and

1-8 exchangers, where the first number denotes the number of the shells and the second

number denotes the number of passes. An odd number of tube passes is seldom used

except the 1-1 exchanger. A 1-2 shell-and-tube heat exchanger is illustrated in Figure

5.19.

42

Shell inlet

Shell outletTube inlet

Tube outlet

Baffles

Endchannel

Passpartition

TubeShell

Figure 5.19 Schematic of one-shell two-pass (1-2) shell-and-tube heat exchanger.

Lt

Ds

B

Figure 5.20 Dimensions of 1-1 shell-and-tube heat exchanger

Dimensions of Shell-and-Tube Heat Exchanger

Some of the following dimensions are pictured in Figure 5.20.

L = tube length

tN = number of tube

pN = number of pass

sD = Shell inside diameter

bN = number of baffle

B = baffle spacing

The baffle spacing is obtained

1+=

b

t

N

LB (5.128)

43

Shell-Side Tube Layout

Figure 5.21 shows a cross section of both a square and triangular pitch layouts. The tube

pitch tP and the clearance tC between adjacent tubes are both defined. Equation (5.30) of

the equivalent diameter is rewritten here for convenience

heated

c

eP

AD

4= (5.129)

From Figure 5.21, the equivalent diameter for the square pitch layout is

( )o

ote

d

dPD

ππ 44

22 −= (5.130a)

From Figure 5.21, the equivalent diameter for the triangular pitch layout is

2

84

34

22

o

ot

ed

dP

Dπ

π

−

= (5.130b)

The cross flow area of the shell cA is defined as

T

tsc

P

BCDA = (5.131)

Pt

di

do

FlowPt

(a) (b)

do

di

CtCt

Figure 5.21 (a) Square-pitch layout, (b) triangular-pitch layout.

The diameter ratio dr is defined by

i

or

d

dd = (5.132)

Some diameter ratios for nominal pipe sizes are illustrated in Table C.6 in Appendix C.

The tube pitch ratio Pr is defined by

44

o

tr

d

PP = (5.133)

The tube clearance Ct is obtained from Figure 5.21.

ott dPC −= (5.134)

The number of tube Nt can be predicted in fair approximation with the shell inside

diameter Ds.

( )ShadeArea

DCTPN s

t

42π= (5.135)

where CTP is the tube count constant that accounts for the incomplete coverage of the

shell diameter by the tubes, due to necessary clearance between the shell and the outer

tube circle and tube omissions due to tube pass lanes for multiple pass design [1].

CTP = 0.93 for one-pass exchanger

CTP = 0.9 for two-pass exchanger (5.136)

CTP = 0.85 for three-pass exchanger

2

tPCLShadeArea ⋅= (5.137)

where CL is the tube layout constant.

CL = 1 for square-pitch layout (5.138)

CL = sin(60°) = 0.866 for triangular-pitch layout

Plugging Equation (5.137) into (5.135) gives

22

2

2

2

44 or

s

t

st

dP

D

CL

CTP

P

D

CL

CTPN

=

=ππ

(5.139)

Table 5.1 Summary of shell-and-tube heat exchangers

Description Equation

Basic Equations ( )oip TTcmq 1111 −= & (5.140)

( )iop TTcmq 2222 −= & (5.141)

Heat transfer areas of the

inner and outer surfaces

of an inner pipe

LNdA tii ⋅⋅⋅= π

LNdA too ⋅⋅⋅= π

(5.142a)

(5.142b)

45

Overall Heat Transfer

Coefficient

oo

i

o

ii

o

o

AhkL

d

d

Ah

AU

1

2

ln1

1

+

+

=

π

(5.143)

Tube side

Reynolds number µµρ

c

iim

DA

dmdu &==Re

p

tic

N

NdA

4

2π=

(5.144)

(5.144a)

Laminar flow

(Re < 2300)

14.03

1

PrRe86.1

==

s

i

f

i

DL

d

k

hdNu

µµ

0.48 < Pr < 16,700

0.0044 < ( )sµµ < 9.75

Use 66.3=DNu if 66.3<DNu

(5.145)

Turbulent flow

(Re > 2300)

Friction factor

( )( )( ) ( )1Pr2/7.121

Pr1000Re2/3221 −+

−==

f

f

k

hdNu D

f

i

D

6105Re3000 ×<< D [4]

2000Pr5.0 ≤≤

( )( ) 228.3Reln58.1

−−= Df

(5.146)

(5.147)

Shell side

Square pitch layout

(Figure 5.21) ( )

o

ote

d

dPD

ππ 44

22 −=

(5.148a)

Triangular pitch layout

(Figure 5.21)

2

84

34

22

o

ot

ed

dP

Dπ

π

−

=

(5.148b)

Cross flow area

t

tsc

P

BCDA =

(5.149)

Reynolds number µµρ

c

eem

DA

DmDu &==Re

(5.150)

46

Nusselt number

14.0

3155.0 PrRe36.0

==

sf

eo

k

DhNu

µµ

2000 <Re < 1 x 106

(5.151)

εεεε-NTU Method

Heat transfer unit

(NTU)

( )minp

oo

cm

AUNTU

&=

(5.152)

Capacity ratio ( )( )

max

min

p

p

rcm

cmC

&

&

=

(5.153)

Effectiveness

One shell (2, 4,.. passes)

( ) ( )[ ]( )[ ]212

1

212

1212

1exp1

1exp1112

+−−

+−++++=

r

rrr

CNTU

CNTUCCε

pNNTUNTU =1

(5.154)

Heat transfer unit (NTU) ( )

+−

+−=−

1

1ln1

212

E

ECNTU r

where ( )

( ) 2121

12

r

r

C

CE

+

+−=

ε

(5.155)

Effectiveness ( )( )

( ) ( )( )( )( ) ( )iip

iop

iip

oip

TTcm

TTcm

TTcm

TTcm

q

q

21min

2222

21min

1111

max −

−=

−

−==

&

&

&

&ε

(5.156)

Heat transfer rate ( ) ( )iip TTcmq 21min−= &ε (5.157)

Tube Side Pressure

Drop

Pressure drop 2v

2

114 ⋅

+

⋅=∆ ρp

i

t Nd

LfP

(5.158)

Laminar flow Df Re16= (5.159)

Turbulent flow ( )( ) 228.3Reln58.1

−−= Df

(5.160)

Shell Side Pressure

Drop

47

( ) 2v2

11 ⋅+=∆ ρb

e

s ND

DfP

( )( )sf Reln19.0576.0exp −=

(5.161)

(5.162)

48

Example 5.2 Miniature Shell-and-Tube Heat Exchanger

A miniature shell-and-tube heat exchanger is designed to cool engine oil in an engine

with the engine coolant (50% ethylene glycol). The engine oil at a flow rate of 0.23 kg/s

enters the exchanger at 120°C and leaves at 115°C. The 50% ethylene glycol at a rate of

0.47 kg/s enters at 90°C. The tube material is Cr alloy (kw = 42.7 W/mK). Fouling factors

of 0.176x10-3 m

2K/W for engine oil and 0.353x10

-3 m

2K/W for 50% ethylene glycol are

specified. Route the engine oil through the tubes. The permissible maximum pressure

drop on each side is 10 kPa. The volume of the exchanger is required to be minimized.

Since the exchanger is custom designed, the tube size can be smaller than NPS 1/8 (DN 6

mm) that is the smallest size in Table C.6 in Appendix C, wherein the tube pitch ratio of

1.25 and the diameter ratio of 1.3 can be applied. Design the shell-and-tube heat

exchanger.

Figure E5.2.1 Shell and tube heat exchanger

MathCAD format solution:

Design concept is to develop a MathCAD modeling for a miniature shell-and-tube heat

exchanger and then seek the solution by iterating the calculations by varying the

parameters to satisfy the design requirements. It is reminded that the design requirements

are the engine oil outlet temperature less than 115°C and the pressure drop less than 10

kPa in each side of the fluids.

The properties of engine oil and ethylene glycol are obtained using the average

temperatures from Table C.5 in Appendix C.

Toil120°C 115°C+( )

2117.5 °C⋅=:= Tcool

90°C 100°C+( )

295 °C⋅=:=

(E5.2.1)

49

Engine oil (subscript 1)-tube side 50% Ethylene glycol (subscript 2)-shell side

ρ1 828kg

m3

:= ρ2 1020kg

m3

:=

(E5.2.2)

cp1 2307J

kg K⋅:= cp2 3650

J

kg K⋅:=

k1 0.135W

mK⋅:= k2 0.442

W

mK⋅:=

µ1 1.027102−

⋅N s⋅

m2

:= µ2 0.08 102−

⋅N s⋅

m2

:=

Pr1 175:= Pr2 6.6:=

The thermal conductivity for the tube material (Chromium alloy) is given

kw 42.7W

m K⋅:=

(E5.2.3)

Given information:

The inlet temperatures are given as

T1i 120°C:= T2i 90°C:= (E5.2.4)

The mass flow rates are given as

mdot1 0.23kg

s:= mdot2 0.47

kg

s:=

(E.5.2.5)

The fouling factors for engine oil and 50% ethylene glycol are given as

Rfi 0.176103−

⋅m2K⋅

W:= Rfo 0.35310

3−⋅

m2K⋅

W:=

(E5.2.6)

Design requirement:

The engine oil outlet temperature must be less than 115°C.

T1o 115°C≤ (E5.2.7)

The pressure drop on each side must be

∆P 10kPa≤ (E5.2.8)

50

Design parameters to be sought by iterations

Initially, estimate the following boxed parameters and then iterate the calculations with

different values toward the design requirements.

Ds 2.0in:= Shell inside diameter Ds 50.8mm= (E5.2.9)

Lt 15in:= Tube length Lt 381mm= (E5.2.10)

do1

8in:= Tube outside diameter do 3.175mm⋅=

(E5.2.11)

The diameter ratio (dr = do/di) is given as suggested in the problem description.

dr 1.3:= di1

dr

do⋅:= di 2.442mm⋅=

(E5.2.12)

The tube pitch ratio (Pr = Pt/do) is given as suggested in the problem description.

Pr 1.25:= (E5.2.13)

The tube pitch is then obtained from Equation (5.133).

Pt Pr do⋅:= (E5.2.14)

The baffle spacing is assumed and may be iterated, and the baffle number from Equation

(5.128) is defined.

B8

8in:= B 25.4mm=

(E5.2.15)

Nb

Lt

B1−:=

Nb 14= (E5.2.16)

The number of passes is defined by

Np 1:= (E5.2.17)

The tube clearance Ct is obtained from Figure 5.21 as

Ct Pt do−:= Ct 0.794mm⋅= (E5.2.18)

51

From Equation (5.136), the tube count calculation constants (CTP) up to three-passes are

given

CTP 0.93 Np 1if

0.9 Np 2if

0.85 otherwise

:=

(E5.2.19)

From Equation (5.138), the tube layout constant (CL) for a triangular-pitch layout is

given by

CL 0.866:= (E5.2.20)

The number of tubes Nt is estimated using Equation (5.139) and rounded off in practice.

Note that the number of tubes in the shell inside diameter defined earlier indicates the

compactness of a miniature exchanger. A 253-tube exchanger in a 2.25-inch shell outside

diameter is commercially available for a 2-inch shell diameter.

Ntube Ds do, Pr, ( ) π

4

CTP

CL

⋅Ds

2

Pr2do

2⋅

⋅:= Ntube Ds do, Pr, ( ) 138.189=

(E5.2.21)

Nt round Ntube Ds do, Pr, ( )( ):= Nt 138= (E5.2.22)

Tube side (Engine oil)

The crossflow area, velocity and Reynolds number are defined as

Ac1

π di2

⋅

4

Nt

Np

⋅:= Ac1 6.465 104−

× m2

=

(E5.2.23)

v1

mdot1

ρ1 Ac1⋅:= v1 0.43

m

s=

(E5.2.24)

Re1

ρ1 v1⋅ di⋅

µ1:= Re1 84.603=

(E5.2.25)

The Reynolds number indicates very laminar flow. The velocity in the tubes appears

acceptable when considering a reasonable range of 0.5 – 1.0 m/s in Table 5.4 for the

engine oil.

52

The friction factor is determined automatically whether it is either laminar or turbulent

using the following program as

f ReD( ) 1.58 ln ReD( )⋅ 3.28−( ) 2−ReD 2300>if

16

ReD

otherwise

:=

(E5.2.26)

The Nusselt number for turbulent or laminar flow is defined using Equations (5.145) and

(5.146) with assuming that µ changes moderately with temperature. The convection heat

transfer coefficient is then obtained.

NuD Dh Lt, ReD, Pr, ( )f ReD( )

2

ReD 1000−( ) Pr⋅

1 12.7f ReD( )

2

0.5

⋅ Pr

2

31−

⋅+

⋅ ReD 2300>if

1.86Dh ReD⋅ Pr⋅

Lt

1

3

⋅ otherwise

:=

(E5.2.27)

Nu1 NuD di Lt, Re1, Pr1, ( ):= Nu1 8.484= (E5.2.28)

h1

Nu1 k1⋅

di

:= h1 468.972W

m2K⋅

⋅=

(E5.2.29)

Shell side (50% ethylene glycol)

The free-flow area is obtained using Equation (5.131) and the velocity in the shell is also

calculated

Ac2

Ds Ct⋅ B⋅

Pt

:= Ac2 2.581 104−

× m2

=

(E5.2.30)

v2

mdot2

ρ2 Ac2⋅:= v2 1.786

m

s=

(E5.2.31)

53

The velocity of 1.786 m/s in the shell is acceptable because the reasonable range of 1.2 –

2.4 m/s for the similar fluid shows in Table 5.4. The equivalent diameter for a triangular

pitch is given in Equation (5.148b) as

De 4

Pt2

3⋅

4

π do2

⋅

8−

π do⋅

2

⋅:= De 2.295mm⋅=

(E5.2.32)

Re2

ρ2 v2⋅ De⋅

µ2:= Re2 5.225 10

3×=

(E5.2.33)

The Nusselt number is given in Equation (5.152) and the heat transfer coefficient is

obtained.

Nu2 0.36 Re20.55

⋅ Pr2

1

3⋅:=

(E5.2.34)

h2

Nu2 k2⋅

De

:= h2 1.442 104

×W

m2K⋅

⋅=

(E5.2.35)

The total heat transfer areas for both fluids are obtained as

Ai π di⋅ Lt⋅ Nt⋅:= Ai 0.403m2

= (E5.2.36)

Ao π do⋅ Lt⋅ Nt⋅:= Ao 0.524m2

⋅= (E5.2.37)

The overall heat transfer coefficient is calculated using Equation (5.143) with the fouling

factors as

Uo

1

Ao

1

h1 Ai⋅

Rfi

Ai

+

lndo

di

2 π⋅ kw⋅ Lt⋅+

Rfo

Ao

+1

h2 Ao⋅+

:= Uo 209.677W

m2K⋅

⋅=

(E5.2.38)

54

ε -NTU method

The heat capacities for both fluids are defined and then the minimum and maximum heat

capacities are obtained using the MathCAD built-in functions as

C1 mdot1 cp1⋅:= C1 530.61W

K⋅=

(E5.2.39)

C2 mdot2 cp2⋅:= C2 1.716 103

×W

K⋅=

(E5.2.40)

Cmin min C1 C2, ( ):= Cmax max C1 C2, ( ):= (E5.2.41)

The heat capacity ratio is defined as

Cr

Cmin

Cmax

:= Cr 0.309=

(E5.2.42)

The number of transfer unit is defined as

NTUUo Ao⋅

Cmin

:= NTU 0.207=

(E5.2.43)

The effectiveness for shell-and-tube heat exchanger is give using Equation (5.154) as

NTU1NTU

Np

:=

(E5.2.44a)

εhx 2 1 Cr+ 1 Cr2

+

0.5 1 exp NTU1− 1 Cr2

+

0.5

⋅

+

1 exp NTU1− 1 Cr2

+

0.5

⋅

−

⋅+

1−

⋅:= εhx 0.182=

Since (E5.2.44b)

Using Equation (5.156), the effectiveness is expressed as

εhxq

qmax

C1 T1i T1o−( )⋅

Cmin T1i T2i−( )⋅

C2 T2o T2i−( )⋅

Cmin T1i T2i−( )⋅ (E5.2.45)

The outlet temperatures are rewritten for comparison with the outlet temperatures.

55

T1i 120 °C⋅= T2i 90 °C⋅=

T1o T1i εhx

Cmin

C1

⋅ T1i T2i−( )⋅−:= T1o 114.544°C⋅=

(E5.2.46)

T2o T2i εhx

Cmin

C2

⋅ T1i T2i−( )⋅+:= T2o 91.687°C⋅=

(E5.2.47)

The engine oil outlet temperature of 114.544°C is close enough to the requirement of

105°C. The heat transfer rate is obtained

q εhx Cmin⋅ T1i T2i−( )⋅:= q 2.895 103

× W= (E5.2.48)

The pressure drops for both fluids are obtained using Equations (5.158) and (5.161) as

∆P 1 4f Re1( ) Lt⋅

di

1+

⋅ Np⋅1

2⋅ ρ1⋅ v1

2⋅:= ∆P 1 9.325kPa⋅=

(E5.2.49)

∆P 2 f Re2( )Ds

De

⋅ Nb 1+( ) 1

2⋅ ρ2⋅ v2

2⋅:= ∆P 2 5.141kPa⋅=

(E5.2.50)

Both the pressure drops calculated are less than the requirement of 10 kPa. The iteration

between Equations (E5.2.9) and (E5.2.46) is terminated. The surface density β for the engine oil side is obtained using the relationship of the heat transfer area over the volume

of the exchanger.

β1

Ao

π Ds2

⋅

4

Lt⋅

:= β1 679.134m2

m3

⋅=

(E5.2.51)

Summary of the design of the miniature shell-and-tube heat exchanger

Given information

T1i 120 °C⋅= engine oil inlet temperature

T2i 90 °C⋅= 50% ethylene glycol inlet temperature

kg

56

mdot1 0.23kg

s= mass flow rate of engine oil

mdot2 0.47kg

s= mass flow rate of 50% ethylene glycol

Rfi 1.76 104−

× m2 K

W⋅⋅= fouling factor of engine oil

Rfo 3.53 104−

× m2 K

W⋅⋅= fouling factor of 50% ethylene glycol

Requirements for the exchanger

T1o 115°C≤ Engine outlet temperature

∆P 1 10kPa≤ Pressure drop on both sides

Design obtained

Np 1= number of passes

Ds 50.8 mm⋅= shell inside diameter Ds 2in=

do 3.175mm⋅= tube outer diameter

di 2.442mm⋅= tube inner diameter

Lt 381 mm⋅= tube length Lt 15in=

Nt 138= number of tube

Ct 0.794mm⋅= tube clearance

B 25.4mm⋅= baffle spacing B 1in=

Nb 14= number of baffle

T1o 114.544°C⋅= engine oil outlet temperature

T2o 91.687°C⋅= 50% ethylene glycol outlet temperature

q 2.895kW⋅= heat transfer rate

β1 679m2

m3

⋅= surface density

∆P 1 9.325kPa⋅= pressure drop for engine oil

57

∆P 2 5.141kPa⋅= pressure drop for 50% ethylene glycol

The design satisfies the requirements.

Problems (corrected)

Shell-and-Tube Heat Exchanger

5.3 A miniature shell-and-tube heat exchanger is designed to cool glycerin with cold

water. The glycerin at a flow rate of 0.25 kg/s enters the exchanger at 60°C and leaves

at 50°C. The water at a rate of 0.54 kg/s enters at 18°C, which is shown in Figure

P5.3. The tube material is Cr alloy (kw = 60.5 W/mK). Fouling factors of 0.253x10-3

m2K/W for water and 0.335x10

-3 m

2K/W for glycerin are specified. Route the

glycerin through the tubes. The permissible maximum pressure drop on each side is

30 kPa. The volume of the exchanger is required to be minimized. Since the

exchanger is custom designed, the tube size may be smaller than NPS 1/8 (DN 6 mm)

that is the smallest size in Table C.6 in Appendix C, wherein the tube pitch ratio of

1.25 and the diameter ratio of 1.3 can be applied. Design the shell-and-tube heat

exchanger.

Figure P5.3 Shell-and tube heat exchanger