HEAT TRANSFER
MCB 3033
4Jun2012
Problem 3-59 1. Consider a 5-m-high, 5-m-high, 8-m-long, 8-m-long, and and 0.22-m-thick wall wall whose whose representative cross section is as shown below. below. The thermal conductivities of various materials used, in W/m·°C, are k A = k F = 2, k B = 8, k C = 20, k D = 15, and k E = 35. The left and right surfaces of the wall are maintained at uniform temperatures of 300 °C and 100°C, respectively. Assuming heat transfer through the wall to be onedimensional, determine: (a) the rate of heat heat transfer transfer through through the wall, wall, (b) the temperatu temperature re at the point point where the sections sections B, D, and E meet, and (c) the tempera temperature ture drop drop across across the section section F . Disregard any contact resistances at the interfaces.
Problem 3-59 1. Consider Consider a 5-m-high, 5-m-high, 8-m-long, 8-m-long, and 0.22-m-th 0.22-m-thick ick wall wall whose representat representative ive cross section section is as shown below. below. The thermal conductivities of various materials used, in W/m·°C, are k A = k F = 2, k B = 8, k C = 20, k D = 15, and k E = 35. The left and right surfaces of the wall are maintained at uniform temperatures of 300°C and 100°C, respectively. respectively. Assuming heat transfer through the wall to be one-dimensional, determine: (a) the rate rate of of heat heat transf transfer er throu through gh the the wall wall,, (b) the temperatur temperature e at the point where where the sections sections B, D, and E meet, and (c) the temper temperatu ature re drop drop acro across ss the the secti section on F . Disregard any contact resistances at the interfaces. R 2 R 1
R 3
T 1 R 4
R 5
R 7 T 2
A 0.12 1 0.12 m
2
R 6
0.01 m L 0.04 C/W kA A (2 W/m C)(0.12 m 2 )
R1 R A
0.05 m L 0.06 C/W kA C (20 W/m C)(0.04 m 2 )
R2 R4 RC
0.05 m L 0.16 C/W R3 R B kA B (8 W/m C)(0.04 m 2 )
1
Rmid ,1
1 0.1 m L R5 R D 0.11 C/W kA D (15 W/m o C)(0.06 m 2 ) 0.1 m L R6 R E 0.05 o C/W 2 kA E (35 W/m C)(0.06 m ) 0.06 m L 0.25 C/W kA F (2 W/m C)(0.12 m 2 )
R7 R F
1 R2
Rmid , 2
1 R5
1
R3
1
R6
1 R4
1
1 0.11
0.06
1
1
0.05
0.16
1 0.06
Rmid ,1 0.025 C/W
Rmid , 2 0.034 C/W C/W
Rtotal R1 Rmid ,1 Rmid , 2 R7 0.04 0.025 0.034 0.25 0.349 C/W
Q
T 1 T 2 Rtotal
(300 100)C 0.349 C/W C/W
Q total (572 W)
572 W (for a 0.12 m 1 m section)
(5 m)(8 m) 0.12 m
2
1.91 10 5 W
Problem 3-59 1. Consider Consider a 5-m-high, 5-m-high, 8-m-long, 8-m-long, and 0.22-m-th 0.22-m-thick ick wall wall whose representat representative ive cross section section is as shown below. below. The thermal conductivities of various materials used, in W/m·°C, are k A = k F = 2, k B = 8, k C = 20, k D = 15, and k E = 35. The left and right surfaces of the wall are maintained at uniform temperatures of 300°C and 100°C, respectively. respectively. Assuming heat transfer through the wall to be one-dimensional, determine: (a) the rate rate of heat heat trans transfer fer thro through ugh the the wall, wall, (b) the tempe temperatur rature e at the point wher where e the sectio sections ns B, D, and E meet meet,, and (c) the temper temperatu ature re drop drop acro across ss the the secti section on F . Disregard any contact resistances at the interfaces. R 2 R 1
R 5
R 3
T 1
T 2 R 6
R 4
Rtotal R1 Rmid ,1 Rmid ,2 R7
Q
T 1 T 2 Rtotal
0.04
( 300 100) C 0.349 C/W
Q total (572 W)
R 7
(5 m)(8 m) 0.12 m
2
0.025
0.034
0.25
0.349 C/W
572 W (for a 0.12 m 1 m section)
1.91 10 5 W
A 0.12 1 0.12 m 2
025 5 0.06 065 5 C/W Rtotal R1 Rmid ,1 0.04 0.02
Q
T 1 T Rtotal
T T 1 Q Rtotal 300C (572 W)(0.065 C/W) 263C
Problem 3-59 1. Consider Consider a 5-m-high, 5-m-high, 8-m-long, 8-m-long, and 0.22-m-th 0.22-m-thick ick wall wall whose representat representative ive cross section section is as shown below. below. The thermal conductivities of various materials used, in W/m·°C, are k A = k F = 2, k B = 8, k C = 20, k D = 15, and k E = 35. The left and right surfaces of the wall are maintained at uniform temperatures of 300°C and 100°C, respectively. respectively. Assuming heat transfer through the wall to be one-dimensional, determine: (a) the rate rate of heat heat trans transfer fer thro through ugh the the wall, wall, (b) the temperatur temperature e at the point where where the sections sections B, D, and E meet, and (c) the tem temper peratu ature re drop drop acro across ss the the secti section on F . Disregard any contact resistances at the interfaces. interfaces. R 2 R 1
R 5
R 3
T 1
T 2 R 6
R 4
Rtotal R1 Rmid ,1 Rmid ,2 R7
Q
T 1 T 2 Rtotal
Q total (572 W)
R 7
0.04
( 300 100) C 0.349 C/W (5 m)(8 m) 0.12 m
2
0.025
0.034
0.25
0.349 C/W
572 W (for a 0.12 m 1 m section)
1.91 10 5 W
A 0.12 1 0.12 m 2
Q
T R F
572 2 W)(0.25 C/W) 143C T Q R F (57
Problem 3-54 A 4-m-high and 6-m-wide 6-m-wide wall consists of a long 15-cm X 25-cm cross section of horizontal bricks (k = = 0.72 W/m·°C) separated by 3-cm-thick plaster layers (k = = 0.22 W/m·°C). There are also 2-cm-thick plaster layers on each side of the wall, a 2-cm-thick rigid foam (k = = 0.026 W/m·°C) on the inner side of the wall. The indoor and the outdoor temperatures are 22 °C and – 4°C, and the convection heat transfer coefficients on the inner and the outer sides are h1 = 10 W/m2·°C and h2 = 20 W/m2·°C, respectively. Assuming 1-D heat transfer and disregarding radiation, determine the rate of heat transfer through the wall.
Learning Outcome
To solve heat transfer tran sfer problems proble ms using thermal resistance network for network for cylinders and spheres.. spheres
HEAT HEAT CONDUCTION IN CYLINDERS AND SPHERES Heat transfer through a pipe can be modeled as steady and one-dimensional . T = T (r ).
This can be used for long cylindrical pipes cylindrical pipes and spherical containers.
is the conduction resistance of the cylinder layer .
A spherical shell with specified inner and outer surface temperatures T 1 and T 2. A
is the conduction resistance of the spherical layer .
2
4 r
where
for a cylindrical layer, and
for a spherical layer
Multilayered Cylinders and Spheres The thermal resistance network for heat transfer through a three-layered composite cylinder subjected to convection on both sides.
Problem 3-77 Steam at 250°C is flowing through a steel pipe (k = 15.5 W/m·°C) whose inner and outer diameters are 10 cm and 12 cm, respectively, respectively, in an environment at 15°C. The pipe is insulated with 7-cm-thick fiberglass insulation ( k = 0.033 W/m·°C). If the heat transfer t ransfer coefficients on the inside and the outside of the pipe are 180 and 40 W/m 2·°C, respectively, respectively, determine the rate of heat loss from the steam per meter length of the pipe. What is the error involved in neglecting the thermal resistance of the steel pipe in calculations?
250°C
Problem 3-77 Steam at 250°C is flowing through a steel pipe ( k = 15.5 W/m·°C) whose inner and outer diameters are 10 cm and 12 cm, respectively, respectively, in an environment at 15°C. The pipe is insulated with 7-cm-thick 7- cm-thick fiberglass insulation (k = 0.033 W/m·°C). If the heat transfer coefficients on the inside and the outside of the pipe are 180 and 40 W/m2·°C, respectively, respectively, determine the rate of heat loss from the steam per meter length of the pipe. What is the error involved in neglecting the thermal resistance of the steel pipe in calculations? Rpipe
Ri
Rinsulation
Ro T 2
T 1
Ri
1 hi Ai
R1 R pipe
1 (180 W/m 2 .C)(0.314 m 2 ) ln(r 2 / r 1 ) 2 k pipe L
R2
Rinsulation
Ro
1 ho Ao
2 k ins L
( 40 W/m . C)(0.8168 m 2 )
Ao
Do L (0.26 m)(1 m) 0.8168 m
3.73 C/W
0.0306 C/W
3.78 C/W
62.2 W
If the thermal resistance of the steel pipe is neglected, the new value of total thermal resistance will be
250°C
0.0177 0.00187 3.73 0.0306 3.78 C/W
(250 15)C
Rtotal Ri R2 Ro
2
0.00187 C/W
2 (0.033 W/m.C)(1 m)
2 o
ln(13 / 6)
1
T 1 T 2 Rtotal
2 (15.5 W/m.C)(1 m)
Di L (0.1 m)(1 m) 0.314 m
0.0177 C/W
ln(6 / 5)
ln(r 3 / r 2 )
Rtotal Ri R1 R2 Ro
Q
2
Ai
0.0177 3.73 0.0306 3.778 C/W
error %
(3.78 3.778)C/W 3.78 C/W
100 0.053%
Problem Consider a 2-m-high electric hot water heater that has a diameter of 40 cm and maintains the hot water at 55 °C. The tank is located in a small room whose average temperature is 27°C, and the heat transfer coefficients on the inner and outer surfaces of the heater are 50 and 12 W/m 2·°C, respectively. The tank is placed in another 46-cm-diameter sheet metal tank of negligible thickness, and the space between the two tanks is filled with foam insulation ( k = 0.03 W/m·°C ). The thermal resistances of the water tank and the outer thin sheet metal shell are very small and can be neglected. The price of electricity is $0.08/kWh, and the home owner pays $280 a year for water heating. Determine the fraction of the hot water energy cost of this household that is due to the heat loss from the tank.
Exercise Study Ex am p les 33-1, 1, 33-2, 2, 3-6 3- 6 and 3-35 35 and 33-58 58 solve Pro b lem s 3.
Test 1 Date: Thursday, 3rd July 2014 Venue: Multi-Purpose Hall Time: 4 – 5 PM
Thank Thank You!
