第三章习题

4.Yovanovich, M. M., "Recent Developments in Thermal Contact, Gap and Joint

Conductance Theories and Experiment," in C. L. Tien, V. P. Carey, and J. K. Ferrel, Eds., Heat Transfer-1986, V ol. 1, Hemisphere, New York, 1986, pp. 35-45.

PROBLEMS

1 The rear window of an automobile is defogged by passing warm air over its inner surface. If the warm air is at T∞,i=40℃and the corresponding convection coefficient is h i=30 W/m2·K, what are the inner and outer surface temperatures of 4-mm-thick window glass, if the outside ambient air temperature is T∞,o=-10℃ and the associated convection coefficient is h o=65 W/m2·K? （3.2）

2 The rear window of an automobile is defogged by attaching a thin, transparent, film-type heating element to its inner surface. By electrically heating this element, a uniform heat flux may be established at the inner surface. (3.3)

(a) For 4-mm-thick window glass, determine the electrical power required per unit window area to maintain an inner surface temperature of 15℃when the interior air temperature and convection coefficient are T∞,i=25℃ and h i=10 W/m2·K, while the exterior (ambient) air temperature and convection coefficient are T∞,o=-10℃ and h o=65 W/m2·K.

(b) In practice T∞,o and h o vary according to weather conditions and car speed. For values of h o =2, 20, 65, and 100 W/m2·K, determine and plot the electrical power requirement as a function of T∞,o for -30≤T∞,o≤0℃. From your results, what can you conclude about the need for heater operation at low values of h o? How is this conclusion affected by the value of T∞,o? If h o∝V n, where V is the vehicle speed and n is a positive exponent, how does the vehicle speed affect the need for heater operation?

3 The composite wall of an oven consists of three materials, two of which are of known thermal conductivity, λA = 20W/m·K and λC =50 W/m·K, and known thickness, L A=0.30 m and L C=0.15 m. The third material, B, which is sandwiched between materials A and C, is of known thickness, L C=0.15 m, but unknown thermal conductivity λB. Under steady-state operating conditions, measurements reveal an outer surface temperature of T s,o =20℃, an inner surface temperature of T s,i =600℃, and an oven air temperature of T

∞=800℃. The

inside convection coefficient h is known to be 25

W/m2·K. What is the value of λB? (3.9)

4 A composite wall separates combustion gases at 2600℃from a liquid coolant at l00℃, with gas- and liquid-side convection coefficients of 50 and 1000 W/m2·K. The wall is composed of a 10-mm-thick layer of beryllium oxide on the gas side and a 20-mm-thick slab of stainless steel (AISI 304) on the liquid side. The contact resistance between the oxide and the steel is 0.0

5 m2·K/W. What is the heat loss per unit surface area of the composite? Sketch the temperature distribution from the gas to the liquid. (3.20)

5 Consider a plane composite wall that is composed of two materials of thermal conductivities λA =0.1 W/m·K and λB =0.04 W/m·K and thicknesses L A=10 mm and L B=20 mm. The contact resistance at the interface between the two materials is known to be 0.30 m2·K/W. Material A adjoins a fluid at 200℃ for which h=l0W/m2·K, and material B adjoins a fluid at 40℃ for which h=20 W/m2·K.

(a) What is the rate of heat transfer through a wall that is 2 m high by 2.5 m wide?

(b) Sketch the temperature distribution. (3.22)

6 The diagram shows a conical section fabricated from pure aluminum. It is of circular cross section having diameter D=ax1/2, where a=0.5 m1/2. The small end is located at x1=25 mm and the large end at

x2=125 mm. The end temperatures are T1=600

K and T2=400 K, while the lateral surface is

well insulated.

(a) Derive an expression for the temperature

distribution T(x) in symbolic form, assuming

one-dimensional conditions. Sketch the

temperature distribution.

(b) Calculate the heat rate q x. (3.29)

7 A truncated solid cone is of circular cross section, its diameter is related to the axial coordinate by an expression of the form D=ax3/2, where a=1.0 m-1/2. The sides are

well insulated, while the top surface of the cone at x1 is

maintained at T1 and the bottom surface at x2 is

maintained at T2. (a) Obtain an expression for the

temperature distribution T(x). (b) What is the rate of heat

transfer across the cone if it is constructed of pure

aluminum with x1=0.075 m, T1=100℃, x2=0.225 m, and

T2=20℃? (3.30)

8 A steam pipe of 0.12-m outside diameter is insulated with a layer of calcium silicate.

(a) If the insulation is 20 mm thick and its inner and outer surfaces are maintained at T s,1=800 K and T s,2=490 K, respectively, what is the heat loss per unit length (Φ') of the pipe?

(b) We wish to explore the effect of insulation thickness on the heat loss Φ' and outer surface temperature T s,2, with the inner surface temperature fixed at T s,1=800 K. The outer surface is exposed to an airflow (T∞=25℃) that maintains a convection coefficient of h=25 W/m2·K and to large surroundings for which T sur= T∞=25℃. The surface emissivity of calcium silicate is approximately 0.8. Compute and plot the temperature distribution in the insulation as a function of the dimensionless radial coordinate, (r一r1)/( r2一r1), where r1=0.06 m and r2 is a variable (0.06< r2 < 0.20 m). Compute and plot the heat loss as a function of the insulation thickness for 0≤(r2一r1)≤0.14m. (3.35)

9 A thin electrical heater is wrapped around the outer surface of a long cylindrical tube whose inner surface is maintained at a temperature of 5℃. The tube wall has inner and outer radii of 25 and 75 mm, respectively, and a thermal conductivity of 10 W/m·K. The thermal contact resistance between the heater and the outer surface of the tube (per unit length of the tube) is R t,c'=0.01 m·K/W. The outer surface of the heater is exposed to a fluid with T∞=一10℃and a convection coefficient of h=l00 W/m2·K. Determine the heater power per unit length of tube required to maintain the heater at T o=25℃. (3.37)

10 A storage tank consists of a cylindrical section that has a length and inner diameter of L=2 m and D i=1 m, respectively, and two hemispherical end sections. The tank is constructed from 20-mm-thick glass (Pyrex) and is exposed to ambient air for which the temperature is 300 K and the convection coefficient is 10 W/m2·K. The tank is used to store heated oil, which maintains the inner surface at a temperature of 400 K. Determine the electrical power that must be supplied to a heater submerged in the oil if the prescribed conditions are to be maintained. Radiation effects may be neglected, and the Pyrex may be assumed to have a thermal conductivity of 1.4 W/m·K. (3.54)

11 Copper tubing is joined to the absorber of a flat-plate solar collector as shown. The aluminum alloy (2024-T6) absorber plate is 6 mm thick and well insulated on its bottom. The top surface of the plate is separated from a transparent cover plate by an evacuated space. The tubes are spaced a distance L of 0.20 m from each other, and

water is circulated through the tubes to remove the collected energy. The water may be assumed to be at a uniform temperature of T w=60℃. Under steady-state operating conditions for which the net radiation heat flux to the surface is q’’=800 W/m2, what is the maximum temperature on the plate and the heat transfer rate per unit length of tube? Note that q’’ represents the net effect of solar radiation absorption by the absorber

plate and radiation exchange between the

absorber and cover plates. You may assume

the temperature of the absorber plate directly

above a tube to be equal to that of the water.

(3.99)

12 A straight fin fabricated from 2024 aluminum alloy (λ=185 W/m·K) has a base thickness of t=3 mm and a length of L=15 mm. Its base temperature is T b=100℃, and it is exposed to a fluid for which T∞=20℃ and h=50 W/m2·K. For the foregoing conditions and a fin of unit width, compare the fin heat rate, efficiency, and volume for rectangular, triangular, and parabolic profiles. (3.123)

13 Finned passages are frequently formed between parallel plates to enhance convection heat transfer in compact heat exchanger cores. An important application is in electronic equipment cooling, where one or more air-cooled stacks are placed between heat-dissipating electrical components. Consider a single stack of rectangular fins of length L and thickness t, with convection conditions corresponding to h and T∞.

(a) Obtain expressions for the fin heat transfer rates, Φf,o and Φf,L, in terms of the base temperatures, T o and T L. (b) In a specific application, a stack that is 200 mm wide and 100 mm deep contains 50 fins, each of length L=12 mm. The entire stack is made from aluminum, which is everywhere 1.0

mm thick. If temperature limitations

associated with electrical components joined

to opposite plates dictate maximum

allowable plate temperatures of T o =400 K

and T L =350 K, what are the corresponding

maximum power dissipations if h=150

W/m2·K and T∞=300 K?