solucionario capitulo 18 - paul e. tippens

8/2/2019 Solucionario Capitulo 18 - Paul E. Tippens

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Chapter 18. Transfer of Heat Physics, 6th Edition

Chapter 18. Transfer of Heat

Note: Refer to Tables 18-1 and 18-2 for thermal conductivities and other constants needed in the

solution of the problems in this section.

Thermal Conductivity

18-1. A block of copper has a cross-section of 20 cm2 and a length of 50 cm. The left end is

maintained at 00C and the right end is a 1000C. What is the rate of heat flow in watts?

k = 385 W/m K; ∆t = 1000C – 00C = 100 C0; 20 cm2 = 20 x 10-4 m2

0 -4 2 o(385 W/m C )(20 x 10 m )(100 C )

0.50 m

kA t H

L

∆= = ; H = 154 W

18-2. In Problem 18-1, what is the heat flow if the copper block is replaced with an block of glass

having the same dimensions?

k G = 0.80 W/m K; A = 20 x 10-4 m2; L = 0.50 m; ∆t = 100 C0

(6 cm)3 cm

2

B A B

B

A B

k L k L

k k

== = ; H = 0.320 W

18-3. A 50-cm long brass rod has a diameter of 3 mm. The temperature of one end is 76 C0 higher

than the other end. How much heat is conducted in one minute?

2 2-6 2(0.003 m)

7.07 x 10 m4 4

D A

π π = = = ; L = 0.50 m; ∆t = 76 C0

0 -6 2 o(109 W/m C )(7.07 x 10 m )(76 C )

0.50 m

kA t H

L

∆= = ; H = 0.117 W

(6 cm)3 cm

2

B A B

B

A B

k L k L

k k = = = ; Q = 7.03 J

249




18-4. A pane of window glass is 10 in. wide, 16 in. long, and 1/8 in. thick. The inside surface is at

600F and the outside surface is at 200F. How many Btu are transferred to the outside in a

time of 2 h? [ L = (1/8) in. = 0.125 in.; A = (10 in.)(16 in.) = 160 in.2 ]

22 2

2

1 ft(160 in. ) 1.11 ft

144 in. A

= =

; ∆t = 600F – 200F = 40 F0

2 0 2 0(5.6 Btu in./ft h F )(1.11 ft )(40 F )(2 h)

0.125 in.

kA t Q

L

τ ∆= = ; Q = 3980 Btu

18-5. One end of an iron rod 30 cm long and 4 cm2 in cross-section is placed in a bath of ice and

water. The other end is placed in a steam bath. How many minutes are needed to transfer

1.0 kcal of heat? In what direction does the heat flow?

-2 0 -4 2 0

(1 kcal)(0.30 m)

(1.2 x 10 kcal/m s C )(4 x 10 m )(100 C )

QL

kA t τ = =

∆; τ = 625 s; τ = 10.4 min

The direction of heat flow is from the stem bath to the ice and water.

18-6. A steel plate of thickness 20 mm has a cross-section of 600 cm2. One side is at 1700C and

the other is at 1200C. What is the rate of heat transfer?

-4 2 0 0(50.2 W/m K)(600 x 10 m )(170 C - 120 C)

0.020 m H = ; H = 7530 W

18-7. How much heat is lost in 12 h through a 3-in. brick firewall 10 ft2 area if one side is at 3300F

and the other is at 78

0

F?

∆t = 3300F – 780F = 252 F0; A = 10 ft2; τ = 12 h; L = 3 in.

2 0 2 0(5 Btu in./ft h F )(10 ft )(252 F )(12 h)

3 in.Q = ; Q = 50,400 Btu

250




*18-8. A composite wall 6 m long and 3 m high consists of 12 cm of concrete joined with 10 cm

of corkboard. The inside temperature is 100C and the outside temperature is 400C. Find

the temperature at the interface of the two materials. [ ∆t = 400C – 100C = 30 C0 ]

H steel = H silver ;( 6 c m )

3 c2

B A B

B

A B

k L k L

k k =

0 0 0 0(50.2 kcal/m s C )(95 C - ) (406 kcal/m s C )( - 5 C )

0.30 m 0.60 m

t t = ; t = 22.80C

*18-9. What is the steady-state rate of heat flow through the composite wall of Problem 10-8?

H 1 = H 2 =0 0 0 -4 2

(406 kcal/m s C )(22.8 C - 5 C)(4 x 10 m )0.60 m

; H = 4.83 J/s

Thermal Radiation

18-10. What is the power radiated from a spherical blackbody of surface area 20 cm2 if its

temperature is 2500C?

4 ; P R e T A

σ = = T = 2500C + 2730 = 523 K; 20 cm2 = 20 x 10-4 m2

4 -8 2 4 -4 2 4(1)(5.67 x 10 W/m K )(20 x 10 m )(523 K) P e AT σ = = ; P = 8.48 W

18-11. What is the rate of radiation for a spherical blackbody at a temperature of 3270C? Will this

rate change if the radius is doubled for the same temperature?

T = 3270C + 2730 = 600 K; e = 1; R = eσ T 4

-8 2 4 4(1)(5.67 x 10 W/m K )(600 K) R = ; R = 7.35 kW/m2

4 P R e T

Aσ = = so that changing the radius does NOT change the rate of radiation.

251

21 silver steel

t 50C950C




*18-12. The emissivity of a metallic sphere is 0.3 and at a temperature of 500 K its radiates a

power of 800 W. What is the radius of the sphere?

4 -8 2 4 4 2(0.3)(5.67 x 10 W/m K )(500 K) 1063 W/m R e T σ = = = ; P

R

A

=

2

2

800 W;

(1063 W/m )

P P A r r

R Rπ

π π = = = = ; r = 0.489 m

*18-13. A certain body absorbs 20 percent of the incident thermal radiation, what is its emissivity?

What energy will be emitted by this body in 1 min if its surface area is 1 m2 and its

temperature is 7270C?

e = 20% = 0.2; T = 2730 + 7270 = 1000 K; τ = 60 s;

R = eσ T 4 = (0.2)(5.67 x 10-8 W/m2 K 4)(1000 K)4; R = 11,340 W/m2

2 2; (11,340 W/m )(1 m )(60 s) E

R E RA A

τ τ

= = = ; E = 680 kJ

*18-14. The operating temperature of the filament in a 25-W lamp is 27270C. If the emissivity is

0.3, what is the surface area of the filament? [ T = 27270C + 2730 = 3000 K ]

4

-8 2 4 4

25 W;

(0.3)(5.67 x 10 W/m K )(3000 K) P e AT Aσ = = ;

A = 1.81 x 10-5 m2 or A = 0.181 cm2

Challenge Problems

18-15. A glass window pane is 60 cm wide 1.8 m tall and 3 mm thick. The inside temperature is

200C and the outside temperature is -100C. How much heat leaves the house through this

window in one hour? A = (0.6 m)(1.8 m) = 1.08 m2; ∆t = [20 – (-10)] = 30 C0

2 2 0(0.8 W/m K)(1.08 m )(30 C )(3600 s)

0.003 m

kA t Q

L

τ ∆= = ; Q = 3.11 x 107 J

252




18-16. A 20-cm thickness of fiberglass insulation covers a 20 m by 15 m attic floor. How many

calories of heat are lost to the attic per second if the temperatures on the sides of the

insulation are +100C and +240C. [ ∆t = 24 – 10 = 14 C0; A = (20 m)(15 m) = 300 m2 ]

-5 0 2 0(1 x 10 kcal/m s C )(300 m )(14 C )

0.20 m

Q kA t

Lτ

∆= = ; Q = 0.210 kcal/s

18-17. The bottom of an aluminum pan is 3 mm thick and has a surface area of 120 cm2. How

many calories per minute are conducted through the bottom of the pan if the temperature of

the outer and inner surfaces are 1140C and 1170C. [ ∆t = 1170 – 1140 = 3 C0 ]

0 -4 2 0(0.05 kcal/m s C )(120 x 10 m )(3 C )

0.003 mQ = ; Q = 36 kcal

18-18. A solid wall of concrete is 80 ft high, 100 ft wide and 6 in. thick. The surface temperatures

on the sides of the concrete are 300F and 1000F. What time is required for 400,000 Btu of

heat to be transferred? [ A = (80 ft)(100 ft) = 8000 ft2; ∆t = 1000 – 300 = 70 F0 ]

2 0 2 0

(400,000 Btu)(6 in.)(5.6 Btu in./ft h F )(8000 ft )(70 F )

QLkA t

τ = =∆

; τ = 0.765 h or 45.9 min

18-19. The bottom of a hot metal pan has an area of 86 cm2 and a temperature of 980C. The pan is

placed on top of a corkboard 5 mm thick.. The Formica table top under the corkboard is

maintained at a constant temperature of 200C. How much heat is conducted through the

cork in 2 min?

∆t = 980 – 200 = 78 C0; A = 86 cm2 = 86 x 10-4 m2; τ = 2 min = 120 s

0 -4 2 0(0.04 W/m C )(86 x 10 m )(78 C )(120 s)

0.005 mQ = ; Q = 644 J

253




*18-20. What thickness of copper is required to have the same insulating value as 2 in. of

corkboard? First find the R-value for 2 in. of corkboard .

R = (2 in.)(3.3 ft2 h F0/Btu) = 6.6 ft2 h F0/Btu

Copper: L(0.00038 ft2 h F0/Btu) = 6.6 ft2 h F0/Btu; L = 17,370 in. or 1450 ft

*18-21. What thickness of concrete is required to have the same insulating value as 6 cm of

fiberglass? For the same insulating value, the ratio’s (L/k) must be the same.

(0.8 W/m K)(6 cm);

(0.04 W/m K)

fib con fibconcon

con fib fib

L k L L L

k k k = = = ; Lcon = 120 cm

18-22. A plate-glass window in an office building measures 2 by 6 m and is 1.2 cm thick. It’s

outer surface is at 230C and its inner surface is at 250C. How many joules of heat pass

through the glass in an hour?

A = (2 m)(6 m) = 12 m2; ∆t = 250 – 230 = 2 C0

0 2 0(0.8 W/m C )(12 m )(2 C )(3600 s)

0.012 m

Q = ; Q = 5760 kJ

18-23. What must be the temperature of a black body if its rate of radiation is 860 W/m2?

24 4 10 4

-8 2 4

860 W/m; 1.517 x 10 K

(1)(5.67 x 10 W/m K )

R R e T T

eσ

σ = = = = ; T = 351 K

18-24. A gray steel ball has an emissivity of 0.75 and when its temperature is 5700C, the power

radiated is 800 W. What is the surface area of the ball? [ T = 2730 + 5700 = 843 K ]

4

4 -8 2 4 4

(800 W);

(0.75)(5.67 x 10 W/m K )(843 K)

P P e AT A

e T σ

σ = = =

A = 0.0373 m2 or 373 cm2

254




18-25. A 25-W lamp has a filament of area 0.212 cm2. If the emissivity is 0.35, what is the

operating temperature of the filament?

(6cm)3cm

2

B A B

B

A B

kL k L

k k

===

13 44 5.94 x 10 K T = ; T = 2776 K

Critical Thinking Questions

*18-26. The wall of a freezing plant consists of 6 in. of concrete and 4 in. of corkboard. The

temperature of the inside cork surface is -150F and the temperature of the outside surface is

700F. What is the temperature at the interface between the cork and concrete? How much

heat is conducted through each square foot

of wall in one hour?

For steady state: H con = H cork

0 0

1 2

1 2

(70 C - ) [ - (-15 C)]i ik A t k A t

L L=

2 0 0 2 0 0(5.6 Btu in./ft h F )(70 C - ) (0.30 Btu in./ft h F )( + 15 C )

6 in. 4 in.

i it t =

(0.933 Btu/ft2 h)(700C – t i) = (0.075 Btu/ft2 h)(t i + 150F)

Solving for t i , we obtain: t i = 63.70F

The heat conducted per square foot is the same. Suppose we choose the concrete, then

2 0 0 0

(5.6 Btu in./ft h F )(70 C 63.7 C)(6 in.)

H A

−= ; H/A = 5.92 Btu/ft2 h

After some time the heat per unit area per unit of time is the same throughout the wall.

255

t i

Cork Concrete

+700F -150F

4in.6 in.




*18-27. A wooden icebox has walls that are 4 cm thick, and the overall effective surface area is 2

m2. How many grams of ice will be melted in 1 min if the inside temperatures of the walls

are 40C and the outside temperatures are 200C?

Inside and outside temperatures are surface temperatures:

; i f

kA t kA t Q Q m L

L L

τ τ ∆ ∆= = =

2 4 2 0 0

3

(0.10 W/m K )(2 m )(60 s)(20 C - 4 C)

(0.04 m)(334 x 10 J/kg)i

f

kA t m

LL

τ ∆= = ; mi = 14.4 g

Note: If the inside and outside temperatures were air temperatures, this would involve

both convection and conduction, making the solution much more complicated.

*18-28. The filament in a lamp operates at a temperature of 7270C and is surrounded by an

envelope at 2270C. If the filament has an emissivity of 0.25 and a surface area of 0.30 cm2,

what is the operating power of the lamp?

P = eσ A(T 14 – T 2

4 ) = eσ A[(1000 K)4 – (500 K)4] = eσA(9.375 x 1011 K 4)

P = (0.25)(5.67 x 10-8 W/m2 K 4)(3 x 10-5 m2)(9.375 x 1011 K 4)

P = 0.399 W

*18-29. The thermal conductivity of material A is twice that for material B. Which has the greater

R value? When 6 cm of material A is between two walls the rate of conduction is 400

W/m2. What thickness of material B is needed to provide the same rate of heat transfer if

other factors remain constant?

k A = 2 k B; The R-value is (L/k), so that

1

2 2 A B B

L L L

k k k

= =

Thus, R A = ½R B

256

40C

C

200

C

A = 2 m2




If the same rates of heat transfer are to be realized, then the thicknesses must be

chosen so that the R values are the same:

A B

A B

L L

k k

= (6 cm)

3 cm

2

B A B B

A B

k L k L

k k

= = = ; LB = 3.00 cm

Material B needs to be only half the thickness of Material A to provide the same insulating

value.

257

solucionario capitulo 18 - paul e. tippens

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