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/m3 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.
Tube outlet
Shell inlet
Tube
Shell sheet
Shell
Baffles
Shell outlet
End
channel
Tube inlet
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
Tube outlet
Shell inlet
Tube
Pass
partition
Shell
End
channel
Baffles
Shell outlet
Tube inlet
Figure 5.19 Schematic of one-shell two-pass (1-2) shell-and-tube heat exchanger.
Lt
B
Ds
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
N t = number of tube
N p = number of pass
Ds = Shell inside diameter
N b = number of baffle
B = baffle spacing
The baffle spacing is obtained
B=
Lt
Nb +1
(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 Pt and the clearance Ct between adjacent tubes are both defined. Equation (5.30) of
the equivalent diameter is rewritten here for convenience
De =
4 Ac
Pheated
(5.129)
From Figure 5.21, the equivalent diameter for the square pitch layout is
De =
(
4 Pt − πd o 4
πd o
2
2
)
(5.130a)
From Figure 5.21, the equivalent diameter for the triangular pitch layout is
 3Pt 2 πd o 2 

4
−

4
8

De = 
πd o 2
(5.130b)
The cross flow area of the shell Ac is defined as
Ac =
Ds C t B
PT
(5.131)
do
do
di
di
Flow
Ct
Ct
Pt
(a)
Pt
(b)
Figure 5.21 (a) Square-pitch layout, (b) triangular-pitch layout.
The diameter ratio dr is defined by
dr =
do
di
(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
Pr =
Pt
do
(5.133)
The tube clearance Ct is obtained from Figure 5.21.
Ct = Pt − d o
(5.134)
The number of tube Nt can be predicted in fair approximation with the shell inside
diameter Ds.
N t = (CTP )
πDs2 4
(5.135)
ShadeArea
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
CTP = 0.9
CTP = 0.85
for one-pass exchanger
for two-pass exchanger
for three-pass exchanger
ShadeArea = CL ⋅ Pt 2
(5.136)
(5.137)
where CL is the tube layout constant.
CL = 1
for square-pitch layout
CL = sin(60°) = 0.866 for triangular-pitch layout
(5.138)
Plugging Equation (5.137) into (5.135) gives
π  CTP  Ds2

 2 = 

4  CL  Pt
4  CL  Pr2 d o2
(5.139)
Table 5.1 Summary of shell-and-tube heat exchangers
Description
Equation
q = m& 1c p1 (T1i − T1o )
Basic Equations
q = m& 2 c p 2 (T2o − T2i )
(5.140)
Nt =
π  CTP  Ds2
Heat transfer areas of the
inner and outer surfaces
of an inner pipe
Ai = π ⋅ d i ⋅ N t ⋅ L
Ao = π ⋅ d o ⋅ N t ⋅ L
(5.141)
(5.142a)
(5.142b)
45
Overall Heat Transfer
Coefficient
Uo =
1 Ao
d 
ln o 
d
1
1
+  i +
hi Ai
ho Ao
2πkL
(5.143)
Tube side
Reynolds number
ρu m d i m& d i
=
µ
Ac µ
Re D =
Ac =
(5.144)
πd i2 N t
(5.144a)
4 Np
1
Laminar flow
(Re < 2300)
hd
 d Re Pr  3  µ
Nu D = i = 1.86 i
 
kf
L   µ s




0.14
(5.145)
0.48 < Pr < 16,700
0.0044 < (µ µ s ) < 9.75
Use Nu D = 3.66 if Nu D < 3.66
Turbulent flow
(Re > 2300)
Nu D =
hd i
( f / 2)(Re D − 1000) Pr
=
12
kf
1 + 12.7( f / 2 ) Pr 2 3 − 1
(
)
(5.146)
3000 < Re D < 5 × 10 6 [4]
0.5 ≤ Pr ≤ 2000
Friction factor
Shell side
Square pitch layout
(Figure 5.21)
Triangular pitch layout
(Figure 5.21)
Cross flow area
Reynolds number
f = (1.58 ln (Re D ) − 3.28)
(
4 Pt − πd o 4
De =
πd o
2
2
)
 3Pt 2 πd o 2 

−
4

4
8

De = 
πd o 2
DCB
Ac = s t
Pt
ρu D
m& De
Re D = m e =
µ
Ac µ
−2
(5.147)
(5.148a)
(5.148b)
(5.149)
(5.150)
46
Nusselt number
 µ
h D
Nu = o e = 0.36 Re 0.55 Pr 1 3 
kf
 µs
6
2000 <Re < 1 x 10



0.14
(5.151)
ε-NTU Method
Heat transfer unit
(NTU)
NTU =
Capacity ratio
Cr =
(5.152)
U o Ao
(m& c p )min
(m& c )
(m& c )
p min
(5.153)
p max
Effectiveness
One shell (2, 4,.. passes)

ε = 21 + C r + 1 + C r 2

(
[ ( ) ]
)
1 − exp[− NTU (1 + C ) ] (5.154)
12
1 + exp − NTU 1 1 + C r2
12
2 12
r
1
NTU 1 = NTU N p
Heat transfer unit (NTU)
(
where E =
)
 E −1 
ln

 E +1
2 ε − (1 + C r )
NTU = − 1 + C r
2 −1 2
(5.155)
(1 + C )
2 12
r
Effectiveness
Heat transfer rate
ε=
q
qmax
=
(m& c )(T − T ) = (m& c )(T
(m& c ) (T − T ) (m& c ) (T
1 p1
p min
1i
1i
1o
2i
2 p2
p min
2o
1i
− T2i )
− T2i ) (5.156)
q = ε (m& c p )min (T1i − T2i )
(5.157)
 f ⋅ Lt
 1
∆P = 4
+ 1 N p ρ ⋅ v 2
2
 di

(5.158)
Tube Side Pressure
Drop
Pressure drop
Laminar flow
Turbulent flow
Shell Side Pressure
Drop
f = 16 Re D
f = (1.58 ln (Re D ) − 3.28)
−2
(5.159)
(5.160)
47
Ds
(N b + 1) 1 ρ ⋅ v 2
De
2
f = exp(0.576 − 0.19 ln (Re s ))
∆P = f
(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 m2K/W for engine oil and 0.353x10-3 m2K/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.
Toil :=
( 120°C + 115°C)
2
= 117.5⋅ °C
Tcool :=
( 90°C + 100°C)
2
= 95⋅ °C
(E5.2.1)
49
Engine oil (subscript 1)-tube side
kg
ρ1 := 828
3
m
cp1 := 2307
50% Ethylene glycol (subscript 2)-shell side
kg
ρ2 := 1020
3
m
J
cp2 := 3650
kg⋅ K
W
k1 := 0.135
m⋅ K
(E5.2.2)
J
kg⋅ K
W
k2 := 0.442
m⋅ K
− 2 N⋅s
µ 1 := 1.027⋅ 10
2
− 2 N⋅ s
µ 2 := 0.08⋅ 10
2
m
Pr1 := 175
m
Pr2 := 6.6
The thermal conductivity for the tube material (Chromium alloy) is given
W
kw := 42.7
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 := 0.23
kg
mdot2 := 0.47
s
kg
s
(E.5.2.5)
The fouling factors for engine oil and 50% ethylene glycol are given as
2
− 3 m ⋅K
Rfi := 0.176⋅ 10
W
2
− 3 m ⋅K
Rfo := 0.353⋅ 10
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
Lt := 15in
Tube length
d o :=
1
8
in
Ds = 50.8mm
Lt = 381mm
Tube outside diameter
(E5.2.9)
(E5.2.10)
do = 3.175⋅ mm
(E5.2.11)
The diameter ratio (dr = do/di) is given as suggested in the problem description.
d r := 1.3
di :=
1
⋅d
dr o
di = 2.442⋅ mm
(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.
B :=
8
8
in
Lt
Nb :=
−1
B
B = 25.4mm
(E5.2.15)
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 = 0.794⋅ mm
(E5.2.18)
51
From Equation (5.136), the tube count calculation constants (CTP) up to three-passes are
given
CTP :=
0.93 if Np
1
0.9 if Np
2
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.
D
π
2
CTP 
s
Ntube ( Ds , d o , Pr) := ⋅ 
⋅
4  CL  2
P ⋅d
r
(
(
(
)
Ntube Ds , d o , Pr = 138.189
2
o
(E5.2.21)
))
Nt := round Ntube Ds , d o , Pr
Nt = 138
(E5.2.22)
Tube side (Engine oil)
The crossflow area, velocity and Reynolds number are defined as
2
A c1 :=
v 1 :=
π ⋅ di
4
⋅
Nt
Np
mdot1
ρ1⋅ A c1
Re1 :=
ρ1⋅ v 1⋅ d i
µ1
−4 2
A c1 = 6.465 × 10
m
(E5.2.23)
v 1 = 0.43
m
s
(E5.2.24)
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
16
ReD
if ReD > 2300
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
 2 
f
Re
(
)



D 
 ⋅  Pr 3 − 1
1 + 12.7⋅ 
 2 
if ReD > 2300
0.5
1
 Dh ⋅ ReD⋅ Pr 
1.86⋅ 

Lt


3
otherwise
(E5.2.27)
(
)
Nu1 := NuD d i , Lt , Re1 , Pr1
h 1 :=
Nu1 ⋅ k1
di
Nu1 = 8.484
(E5.2.28)
h 1 = 468.972⋅
W
2
m ⋅K
(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
A c2 :=
v 2 :=
Ds ⋅ Ct ⋅ B
Pt
mdot2
ρ2⋅ A c2
−4 2
A c2 = 2.581 × 10
m
(E5.2.30)
v 2 = 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
 P 2⋅ 3 π ⋅ d 2 
o 
 t
−
8 
 4
De := 4⋅
  π ⋅d  
  o 
  2  
Re2 :=
De = 2.295⋅ mm
(E5.2.32)
ρ2⋅ v 2⋅ De
3
Re2 = 5.225 × 10
µ2
(E5.2.33)
The Nusselt number is given in Equation (5.152) and the heat transfer coefficient is
obtained.
1
0.55
Nu2 := 0.36⋅ Re2
h 2 :=
⋅ Pr2
3
(E5.2.34)
Nu2 ⋅ k2
4
h 2 = 1.442 × 10 ⋅
De
W
2
m ⋅K
(E5.2.35)
The total heat transfer areas for both fluids are obtained as
2
A i := π ⋅ d i⋅ Lt ⋅ Nt
A i = 0.403m
A o := π ⋅ d o ⋅ Lt ⋅ Nt
A o = 0.524⋅ m
(E5.2.36)
2
(E5.2.37)
The overall heat transfer coefficient is calculated using Equation (5.143) with the fouling
factors as
1
Ao
Uo :=
1
h 1⋅ A i
+
Rfi
Ai
 do 

 d i  + Rfo +
W
Uo = 209.677⋅
2
m ⋅K
ln
+
2⋅ π ⋅ kw⋅ Lt
Ao
1
h 2⋅ A o
(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
W
C1 = 530.61⋅
K
C1 := mdot1⋅ cp1
3 W
C2 = 1.716 × 10 ⋅
K
C2 := mdot2⋅ cp2
(
(E5.2.39)
)
(
Cmin := min C1 , C2
(E5.2.40)
)
Cmax := max C1 , C2
(E5.2.41)
The heat capacity ratio is defined as
Cmin
Cr :=
Cmax
Cr = 0.309
(E5.2.42)
The number of transfer unit is defined as
NTU :=
Uo ⋅ A o
NTU = 0.207
Cmin
(E5.2.43)
The effectiveness for shell-and-tube heat exchanger is give using Equation (5.154) as
NTU
NTU1 :=
Np
Since
(E5.2.44a)
0.5 


2
0.5 1 + exp−NTU ⋅  1 + C   

1
r  
2

ε hx := 2⋅ 1 + Cr +  1 + Cr  ⋅



0.5


2

1 − exp−NTU1 ⋅  1 + Cr  



 
−1
ε hx = 0.182
(E5.2.44b)
Using Equation (5.156), the effectiveness is expressed as
ε hx
(
)
q
C1⋅ T1i − T1o
q max
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 − ε hx⋅
Cmin
C1
(
)
⋅ T1i − T2i
T1o = 114.544°C
⋅
(E5.2.46)
Cmin
T2o := T2i + ε hx⋅
⋅ T1i − T2i
C2
(
)
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
(
)
3
q := ε hx⋅ Cmin⋅ T1i − T2i
q = 2.895 × 10 W
(E5.2.48)
The pressure drops for both fluids are obtained using Equations (5.158) and (5.161) as
 f ( Re1) ⋅ Lt
∆P 1 := 4⋅ 
di


1
2
+ 1  ⋅ Np ⋅ ⋅ ρ1⋅ v 1
2

(E5.2.49)
Ds
1
2
∆P 2 := f Re2 ⋅
Nb + 1 ⋅ ⋅ ρ2⋅ v 2
De
2
( )
(
∆P 1 = 9.325⋅ kPa
)
∆P 2 = 5.141⋅ kPa
(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.
β 1 :=
Ao
 π ⋅ D 2 
s
 4  ⋅ Lt


2
β 1 = 679.134⋅
m
3
m
(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 = 0.23
kg
mdot2 = 0.47
kg
mass flow rate of engine oil
s
mass flow rate of 50% ethylene glycol
s
−4
Rfi = 1.76 × 10
2 K
⋅m ⋅
−4
Rfo = 3.53 × 10
fouling factor of engine oil
W
2 K
⋅m ⋅
fouling factor of 50% ethylene glycol
W
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
d o = 3.175⋅ mm
tube outer diameter
d i = 2.442⋅ mm
tube inner diameter
Lt = 381⋅ mm
tube length
Nt = 138
number of tube
Ct = 0.794⋅ mm
tube clearance
B = 25.4⋅ mm
baffle spacing
Nb = 14
number of baffle
T1o = 114.544°C
⋅
engine oil outlet temperature
T2o = 91.687⋅ °C
50% ethylene glycol outlet temperature
q = 2.895⋅ kW
heat transfer rate
2
β 1 = 679⋅
m
surface density
3
m
∆P 1 = 9.325⋅ kPa
pressure drop for engine oil
Ds = 2 in
Lt = 15in
B = 1 in
57
∆P 2 = 5.141⋅ kPa
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 m2K/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