13 Chap Heat Exchangers
Review Problems
13-111 Hot oil is cooled by water in a multi-pass shell-and-tube heat exchanger. The overall heat transfer coefficient based on the inner surface is to be determined. Assumptions 1 Water flow is fully developed. 2 Properties of the water are constant. Properties The properties of water at 300 K 25 C are (Table A-9)
k Pr 0.607 W/m. C / 6.14 0.894 10
6
m 2 /s
Analysis The Reynolds number is
Re Vm D (3 m/s)(0.013 m) 0.894 10
6
m 2 /s
43,771
Outer surface D0, A0, h0, U0 Inner surface Di, Ai, hi, Ui
which is greater than 10,000. Therefore, we assume fully developed turbulent flow, and determine Nusselt number from
Nu 0.023 Re 0.8 Pr 0.4 0.023(43,771) 0.8 (6.14) 0.4 245
and
hi k Nu D 0.607 W/m. C (245) 11,440 W/m 2 . C 0.013 m
The inner and the outer surface areas of the tube are
Ai Ao Di L Do L (0.013 m)(1 m) (0.015 m)(1 m) 0.04084 m 2 0.04712 m 2
The total thermal resistance of this heat exchanger per unit length is
R 1 hi Ai ln( Do / Di ) 2 kL 1 (11,440 W/m 2 . C)(0.04084 m 2 ) 0.609 C/W 1 ho Ao ln(1.5 / 1.3) 2 (110 W/m. C)(1 m) 1 (35 W/m 2 . C)(0.04712 m 2 )
Then the overall heat transfer coefficient of this heat exchanger based on the inner surface becomes
R 1 U i Ai Ui 1 RAi 1 (0.609 C/W )(0.04084 m 2 ) 40.2 W/m2 . C
13-87
Chap 13 Heat Exchangers 13-112 Hot oil is cooled by water in a multi-pass shell-and-tube heat exchanger. The overall heat transfer coefficient based on the inner surface is to be determined. Assumptions 1 Water flow is fully developed. 2 Properties of the water are constant. Properties The properties of water at 300 K 25 C are (Table A-9)
k Pr 0.607 W/m. C / 6.14 0.894 10
6
m 2 /s
Analysis The Reynolds number is
Re Vm D (3 m/s)(0.013 m) 0.894 10
6
m 2 /s
43,771
Outer surface D0, A0, h0, U0 Inner surface Di, Ai, hi, Ui
which is greater than 10,000. Therefore, we assume fully developed turbulent flow, and determine Nusselt number from
Nu 0.023 Re 0.8 Pr 0.4 0.023(43,771) 0.8 (6.14) 0.4 245
and
hi k Nu D 0.607 W/m. C (245) 11,440 W/m 2 . C 0.013 m
The inner and the outer surface areas of the tube are
Ai Ao Di L Do L (0.013 m)(1 m) (0.015 m)(1 m) 0.04084 m 2 0.04712 m 2
The total thermal resistance of this heat exchanger per unit length of it with a fouling factor is
R 1 hi Ai ln( Do / Di ) 2 kL 1 (11,440 W/m . C)(0.04084 m ) 0.0004 m 2 . C/W 0.04712 m 0.617 C/W
2 2 2 2
R f ,o Ao
1 ho Ao ln(15 / 13) 2 (110 W/m. C)(1 m) 1
(35 W/m . C)(0.04712 m 2 )
Then the overall heat transfer coefficient of this heat exchanger based on the inner surface becomes
R 1 U i Ai Ui 1 RAi 1 (0.617 C/W )(0.04084 m 2 ) 39.7 W/m 2 . C
13-88
Chap 13 Heat Exchangers 13-113 Water is heated by hot oil in a multi-pass shell-and-tube heat exchanger. The rate of heat transfer and the heat transfer surface area on the outer side of the tube are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 The overall heat transfer coefficient is constant and uniform. Properties The specific heats of the water and oil are given to be 4.18 and 2.2 kJ/kg. C, respectively. Analysis (a)The rate of heat transfer in this heat exchanger is
Q mh C ph (Th,in Th,out ) (3 kg/s)(2.2 kJ/kg. C)(130 C 60 C) = 462 kW
(b) The outlet temperature of the cold water is
Q mc C pc (Tc,out Tc,in ) Tc,out Tc,in Q mc C pc 20 C 462 kW (3 kg / s)(4.18 kJ / kg. C) 56.8 C
The temperature differences at the two ends are
T1 T2
Th,in Tc,out Th,out Tc,in
130 C 56.8 C = 73.2 C 60 C 20 C = 40 C
Hot Oil 130 C 3 kg/s
The logarithmic mean temperature difference is
Tlm,CF T1 T2 ln( T1 / T2 ) 73.2 40 ln( 73.2 / 40) 54.9 C
Cold Water 20 C 3 kg/s
and
P R t 2 t1 T1 t1 T2 T1 t 2 t1 56.8 20 130 20 130 60 56.8 20 0.335 F 1.90 0.96
(20 tube passes) 60 C
The heat transfer surface area on the outer side of the tube is then determined from
Q UAs F Tlm As Q UF Tlm 462 kW (0.3 kW/m . C)(0.96)(54.9 C)
2
29.2 m2
13-89
Chap 13 Heat Exchangers 13-114E Water is heated by solar-heated hot air in a double-pipe counter-flow heat exchanger. The required length of the tube is to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 The overall heat transfer coefficient is constant and uniform. Properties The specific heats of the water and air are given to be 1.0 and 0.24 Btu/lbm. F, respectively. Analysis The rate of heat transfer in this heat exchanger is
Q mh C ph (Th,in Th,out ) (0.7 lbm/s)(0.24 Btu/lbm. F)(190 F 135 F) = 9.24 Btu/s
The outlet temperature of the cold water is
Q mc C pc (Tc,out Tc,in ) Tc,out Tc,in Q mc C pc 70 F 9.24 Btu/s (0.35 lbm/s)(1.0 Btu/lbm. F) 96.4 F
The temperature differences at the two ends are
T1 T2 Th,in Tc,out Th,out Tc,in 190 F 96.4 F = 93.6 F 135 F 70 F = 65 F
Hot Air 130 F 0.7 lbm/s 135 F Cold Water 70 F 0.35 lbm/s
The logarithmic mean temperature difference is
Tlm T1 T2 ln( T1 / T2 ) 93.6 65 ln( 93.6 / 65) 78.43 F
The heat transfer surface area on the outer side of the tube is determined from
Q UAs Tlm As Q U Tlm 9.24 Btu/s (20 / 3600 Btu/s.ft 2 . F)(78.43 F) 21.21 ft 2
Then the length of the tube required becomes
As DL L As D 21.21 ft 2 (0.5 / 12 ft) 162.0 ft
13-115 It is to be shown that when T1 = T2 for a heat exchanger, the Tlm relation reduces to Tlm = T1 = T2. Analysis When T1 = T2, we obtain
Tlm T1 T2 ln( T1 / T2 ) 0 0
This case can be handled by applying L'Hospital's rule (taking derivatives of nominator and denominator separately with respect to T1 or T2 ). That is,
Tlm d ( T1 T2 ) / d T1 d [ln( T1 / T2 )] / d T1 1 1 / T1 T1 T2
13-90
Chap 13 Heat Exchangers 13-116 Refrigerant-134a is condensed by air in the condenser of a room air conditioner. The heat transfer area on the refrigerant side is to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 The overall heat transfer coefficient is constant and uniform. Properties The specific heat of air is given to be 1.005 kJ/kg. C. Analysis The temperature differences at the two ends are
T1 T2 Th,in Th,out Tc,out Tc,in 40 C 35 C = 5 C 40 C 25 C = 15 C
R-134a 40 C
The logarithmic mean temperature difference is
Tlm T1 T2 ln( T1 / T2 ) 5 15 ln(5 / 15) 9.1 C
Air 25 C
35 C
The heat transfer surface area on the outer side of the tube is determined from
Q UAs Tlm As Q U Tlm (15,000 / 3600) kW (0.150 kW/m 2 . C) (9.1 C) 3.05 m2
40 C
13-117 Air is preheated by hot exhaust gases in a cross-flow heat exchanger. The rate of heat transfer is to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 The overall heat transfer coefficient is constant and uniform. Properties The specific heats of air and combustion gases are given to be 1.005 and 1.1 kJ/kg. C, respectively. Analysis The rate of heat transfer is simply
Q [mC p (Tin Tout )]gas. (1.1 kg/s)(1.1kJ/kg. C)(180 C 95 C) = 102.9 kW
13-91
Chap 13 Heat Exchangers 13-118 A water-to-water heat exchanger is proposed to preheat the incoming cold water by the drained hot water in a plant to save energy. The heat transfer rating of the heat exchanger and the amount of money this heat exchanger will save are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat is exchanger well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. Properties The specific heat of the hot water is given to be 4.18 kJ/kg. C. Analysis The maximum rate of heat transfer is
Qmax mh C ph (Th,in Tc,in )
Hot water 60 C 8 kg/s Cold Water 14 C
(8 / 60 kg/s)(4.18 kJ/kg. C)(60 C 14 C) 25.6 kW
Noting that the heat exchanger will recover 72% of it, the actual heat transfer rate becomes
Q
Qmax
(0.72)(256 kJ / s) = 18.43 kW .
which is the heat transfer rating. The operating hours per year are The annual operating hours = (8 h/day)(5 days/week)(52 week/year) = 2080 h/year The energy saved during the entire year will be Energy saved = (heat transfer rate)(operating time) = (18.43 kJ/s)(2080 h/year)(3600 s/h) = 1.38x108 kJ/year Then amount of fuel and money saved will be
Fuel saved Energy saved Furnace efficiency 1.38 10 8 kJ/year 1 therm 0.78 105,500 kJ 1677 therms/year
Money saved = (fuel saved)(the price of fuel) = (1677 therms/year)($ 0.54/therm) = $906/year
13-92
Chap 13 Heat Exchangers 13-119 A shell-and-tube heat exchanger is used to heat water with geothermal steam condensing. The rate of heat transfer, the rate of condensation of steam, and the overall heat transfer coefficient are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 Fluid properties are constant. Properties The heat of vaporization of geothermal water at 120 C is given to be hfg = 2203 kJ/kg and specific heat of water is given to be Cp = 4180 J/kg. C. Analysis (a) The outlet temperature of the water is
Tc,out Th,out 46 120 C 46 C = 74 C
Steam 120 C
Then the rate of heat transfer becomes
Q [mC p (Tout = 847.7 kW Tin )]water 22 C)
(3.9 kg/s)(4.18 kJ/kg. C)(74 C
(b) The rate of condensation of steam is determined from
Q 847.7 kW (mh fg ) geothermal
steam
22 C Water 3.9 kg/s 120 C
14 tubes
m(2203 kJ/kg)
m 0.385 kg/s
(c) The heat transfer area is
Ai n Di L 14 (0.024 m)(3.2 m) = 3.378 m 2
The logarithmic mean temperature difference for counter-flow arrangement and the correction factor F are
T1 T2 Th,in Tc,out Th,out Tc,in 120 C 74 C = 46 C 120 C 22 C = 98 C
46 98 ln( 46 / 98)
0.53 F 0 1
Tlm,CF
t 2 t1 T1 t1
T1 T2 ln( T1 / T2 )
74 22 120 22 120 120 74 22
68.8 C
P R
T1 T2 t 2 t1
Then the overall heat transfer coefficient is determined to be
Q U i Ai F Tlm,CF Ui Q Ai F Tlm,CF 847,700 W (3.378 m )(1)(68.8 C)
2
3648 W/m 2 . C
13-93
Chap 13 Heat Exchangers 13-120 Water is heated by geothermal water in a double-pipe counter-flow heat exchanger. The mass flow rate of the geothermal water and the outlet temperatures of both fluids are to be determined. Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 The overall heat transfer coefficient is constant and uniform. Properties The specific heats of the geothermal water and the cold water are given to be 4.25 and 4.18 kJ/kg. C, respectively. Analysis The heat capacity rates of the hot and cold fluids are
Ch Cc m h C ph mc C pc m h (4.25 kJ/kg. C) = 4.25m h (1.2 kg/s)(4.18 kJ/kg. C) = 5.016 kW/ C
Cold Water 12 C 1.2 kg/s Geothermal water 95 C
Cmin
and
Cc
C
5.016 kW/ C
Cmin Cmax 5.016 4.25mh 1.1802 mh
The NTU of this heat exchanger is
NTU
UAs C min
(0.480 kW/m 2 . C)(25 m 2 ) 5.016 kW/ C
2.392
Using the effectiveness relation, we find the capacity ratio
1 exp NTU(1 C ) 1 C exp NTU(1 C )
0.823
1 exp 2.392(1 C ) 1 C exp 2.392(1 C )
C
0.494
Then the mass flow rate of geothermal water is determined from
C
1.1802 mh
0.494
1.1802 mh
mh
2.39 kg/s
The maximum heat transfer rate is
Q max C min (Th,in Tc,in ) (5.016 kW/ C)(95 C - 12 C) 416.328 kW
Then the actual rate of heat transfer rate becomes
Q Qmax (0.823)(416.328 kW) 342.64 kW
The outlet temperatures of the geothermal and cold waters are determined to be
Q Cc (Tc,out Tc,in ) 342.64 kW = (5.016 kW/ C)(Tc,out 12) Tc,out 80.3 C
Q
m h C ph (Th,in Th,out ) Th,out 61.3 C
342.64 kW = (2.39 kg/s)(4.25 kJ/kg. C)(95 Th,out )
13-94
Chap 13 Heat Exchangers 13-121 Air is to be heated by hot oil in a cross-flow heat exchanger with both fluids unmixed. The effectiveness of the heat exchanger, the mass flow rate of the cold fluid, and the rate of heat transfer are to be determined. .Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 The overall heat transfer coefficient is constant and uniform. Properties The specific heats of the air and the oil are given to be 1.006 and 2.15 kJ/kg. C, respectively. Analysis (a) The heat capacity rates of the hot and cold fluids are
Ch Cc
mh C ph mc C pc
0.5mc (2.15 kJ/kg. C) = 1.075mc mc (1.006 kJ/kg. C) = 1.006mc
Cc 1.006mc
0.936
Oil 80 C
Therefore, Cmin and
C Cmin Cmax
1.006mc c 1.075m
Air 18 C
The effectiveness of the heat exchanger is determined from
Q Qmax C c (Tc,out Tc, in ) 58 18 80 18 0.645
58 C
C c (Th,in Tc,in )
(b) The NTU of this heat exchanger is expressed as
NTU
UAs C min
(0.750 kW/ C) 1.006mc
0.7455 mc
The NTU of this heat exchanger can also be determined from
NTU ln C ln(1 C ) 1 ln 0.936 ln(1 0.645) 1 0.936 3.724
Then the mass flow rate of the air is determined to be
NTU
UAs C min
3.724
(0.750 kW/ C) 1.006mc
mc
0.20 kg/s
(c) The rate of heat transfer is determined from
Q m c C pc (Tc,out Tc,in ) (0.20 kg/s)(1.006 kJ/kg. C)(58 - 18) C 8.05 kW
13-95
Chap 13 Heat Exchangers 13-122 A water-to-water counter-flow heat exchanger is considered. The outlet temperature of the cold water, the effectiveness of the heat exchanger, the mass flow rate of the cold water, and the heat transfer rate are to be determined. .Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the heat transfer to the cold fluid. 3 Changes in the kinetic and potential energies of fluid streams are negligible. 4 The overall heat transfer coefficient is constant and uniform. Properties The specific heats of both the cold and the hot water are given to be 4.18 kJ/kg. C. Analysis (a) The heat capacity rates of the hot and cold fluids are Cold Water 20 C Hot water 95 C
Ch Cc
mh C ph mc C pc
Cc
1.5mc (4.18 kJ/kg. C) = 6.27mc mc (4.18 kJ/kg. C) = 4.18mc
4.18mc
0.667
Therefore, Cmin and
C Cmin Cmax
4.18mc c 6.27m
The rate of heat transfer can be expressed as
Q Q C c (Tc, out
C h (Th,in
Tc, in )
Th,out )
(4.18m c )(Tc,out
20)
15) (6.27m c )(80 Tc,out )
(6.27m c ) 95 (Tc,out
Setting the above two equations equal to each other we obtain the outlet temperature of the cold water
Q 4.18mc (Tc,out 20) 6.27mc (80 Tc,out ) Tc,out 56 C
(b) The effectiveness of the heat exchanger is determined from
Q Qmax C c (Tc,out Tc,in ) C c (Th,in Tc,in ) 4.18mc (56 20) 4.18mc (95 20) 0.48
(c) The NTU of this heat exchanger is determined from
NTU
1 ln C 1
1 C 1
1 0.48 1 ln 0.667 1 0.48 0.667 1
0.805
Then, from the definition of NTU, we obtain the mass flow rate of the cold fluid:
NTU
UAs C min
0.805
1.400 kW/ C 4.18mc
mc
0.416 kg/s
(d) The rate of heat transfer is determined from
Q mc C pc (Tc,out Tc,in ) (0.416 kg/s)(4.18 kJ/kg. C)(56 20) C 62.6 kW
13-123 . . . 13-129 Design and Essay Problems
13-96