16.2 It is desired to transport liquid metal through a pipe embedded in a wall at a point where the temperature is 650 K. A 1.2-m-thick wall constructed of a material having a thermal conductivity varying with temperature according to k = 0.0073 (1 + 0.0054 T), where T is in K and k is in W/m · K, has its inside surface maintained at 925 K. The outside surface is exposed to air at 300 K with a convective heat-transfer coefficient of 23 W/ m2 · K. How far from the hot surface should the pipe be located? What is the heat flux for the wall?
Get solution
16.3 A stainless steel plate 1.6 cm thick rests on top of a hot plate, which is maintained at 250°C. Air flows over the top surface of the plate to provide a convective heat-transfer coefficient of h = 50W/m2 · K. The air temperature is maintained at 20°C.
a. What is the heat flux through the stainless steel plate, in W/m2?...
b. What is the temperature at the top surface, T1, of the stainless steel plate?
c. Based on the analysis above, what can you conclude about the heat-transfer resistance offered by the hydrodynamic boundary layer?
Get solution
16.4 A furnace wall is to be designed to transmit a maximum heat flux of 200 Btu/h ft2 of wall area. The inside and outside wall temperatures are to be 2000°F and 300°F, respectively. Determine the most economical arrangement of bricks measuring 9 by 4 ½ by 3 in. if they are made from two materials, one with a k of 0.44 Btu/h ft °F and a maximum usable temperature of 1500°F and other with a k of 0.94 Btu/h ft °F and a maximum usable temperature of 2200°F. Bricks made of each material cost the same amount and may be laid in any manner.
Get solution
16.5 A furnace wall consisting of 0.25 m of fire clay brick, 0.20 m of kaolin, and a 0.10-m outer layer of masonry brick is exposed to furnace gas at 1370 K with air at 300 K adjacent to the outside wall. The inside and outside convective heat- transfer coefficients are 115 and 23 W/m2 · K, respectively. Determine the heat loss per square foot of wall and the temperature of the outside wall surface under these conditions.
Get solution
16.6 A heater composed of Nichrome wire wound back and forth and closely spaced is covered on both sides with1/8-in.thickness of asbestos (k = 0.15 Btu/h ft °F) and then with a1/8-in.thickness of stainless steel (k = 10 Btu/h ft °F). If the center temperature of this sandwich construction is considered constant at 1000°F and the outside convective heat-transfer coefficient is 3 Btu/h ft2 °F, how much energy must be supplied in W/ft2 to the heater? What will be the outside temperature of the stainless steel?
Get solution
16.7 A composite wall is to be constructed of ¼ in. of stainless steel (k = 10 Btu/h ft °F), 3 in. of corkboard (k = 0.025 Btu/h ft °F), and ½ in. of plastic (k = 1.5 Btu/h ft °F). Determine the thermal resistance of this wall if it is bolted together by ½-in: -diameter bolts on 6-in. centers made of
a. stainless steel
b. aluminum (k = 120 Btu/h ft °F)
Get solution
16.8 A 2-in. schedule-40 steel pipe carries saturated steam at 60 psi through a laboratory that is 60 ft long. The pipe is insulated with 1.5 in. of 85% magnesia that costs $0.75 per foot. How long must the steam line be in service to justify the insulation cost if the heating cost for the steam is $0.68 per 105 Btu? The outside-surface convective heat-transfer coefficient may be taken as 5 Btu/h ft2 °F.
Get solution
16.9 A computer IC chip consumes 10 W of power, which is dissipated as heat. The chip measures 4 cm by 4 cm on a side and is 0.5-cm thick. Currently the IC chip is packaged into an electronic device as shown in Figure 17.5. The base of the chip is in contact with an inert aluminum plate that is 0.3-cm thick. The IC chip and its aluminum base are mounted within a thermally insulating ceramic material, which you may be assumed to act as a perfect thermal insulator. The top side of the IC chip is exposed to air, which provides a convective heat- transfer coefficient of 100 W/m2 · K. The ambient air tempera- ture (T∞) is maintained at 30°C. The chip is operated for a sufficient time that steady-state operation can be assumed. Within the process temperature range of interest, the thermal conductivity of the IC chip material is kIC = 1 W/m · K, and the thermal conductivity of aluminum is kAl = 230 W=m · K.
a. What is the surface temperature, T1, of the IC chip exposed to the air?...Figure 17.5 Annular element in a long, circular cylinder with internal heat generation....
b. What is the temperature, T2, of the IC chip package at the base of the IC chip (x = L1) resting on top of the aluminum plate?
Get solution
16.10 Copper wire having a diameter of 3/16 in. is insulated with a 4-in. layer of material having a thermal conductivity of 0.14 Btu/h ft °F. The outer surface of the insulation is maintained at 70°F. How much current may pass through the wire if the insulation temperature is limited to a maximum of 120°F? The resistivity of copper is 1.72 × 10–6 ohm – cm.
Get solution
16.11 Liquid nitrogen at 77 K is stored in an insulated spherical container that is vented to the atmosphere. The container is made of a thin-walled material with an outside diameter of 0.5 m; 25 mm of insulation (k=0.002 W/m · K) covers its outside surface. The latent heat of nitrogen is 200 kJ/kg; its density, in the liquid phase, is 804 kg/m3. For surroundings at 25°C and with a convective coefficient of 18 W/m2 · K at the outside surface of the insulation, what will be the rate of liquid nitrogen boil-off?
Get solution
16.12 A thin slab of material is subjected to microwave radiation that causes volumetric heating to vary according to...where qo has a constant value of 180 kW/m3 and the slab thickness, L, is 0.06 m. The thermal conductivity of the slab material is 0.6 W/m · K.The boundary at x = L is perfectly insulated, while the surface at x = 0 is maintained at a constant temperature of 320 K.
a. Determine an expression for T(x) in terms of X, L, k, q0, and T0.
b. Where in the slab will the maximum temperature occur?
c. What is the value of Tmax?
Get solution
16.13 Radioactive waste (k = 20 W/m · K) is stored in a cylindrical stainless steel (k = 15 W/m · K) container with inner and outer diameters of 1.0 and 1.2 m, respectively. Thermal energy is generated uniformly within the waste material at a volumetric rate of 2 × 105 W/m3. The outer container surface is exposed to water at 25°C, with a surface coefficient of 1000 W/m2 · K. The ends of the cylindrical assembly are insulated so that all heat transfer occurs in the radial direction. For this situation determine
a. the steady-state temperatures at the inner and outer surfaces of the stainless steel
b. the steady-state temperature at the center of the waste material
Get solution
16.14 Radioactive waste is stored in a cylindrical stainless steel container with inner and outer diameters of 1 m and 1.2 m, respectively, so that R0 = 0.5 m and R1 = 0.6 m. Thermal energy is generated uniformly within the waste material at a volumetric rate of 2 × 105 W/m3. The outer container surface is exposed to water at 25°C (T∞ = 25°C), and the outer surface convective heat-transfer coefficient is h = 1000 W=m2 · K). The ends of the cylinder are insulated so that heat transfer is only in the r-direction. You may assume that over the temperature range of interest the thermal conductivity of stainless steel, ksteel, is 15 W/m · K and the thermal conductivity of the radioactive waste, krw, is 20 W/m · K.
What are the temperatures of the outer surface, T0, and inner surface, Ti, of the stainless-steel container?What the maximum temperature within the radioactive waste?Where will this maximum temperature be located?
Get solution
16.15 Consider the composite solid shown. Solid A is a thermally conductive material that is 0.5-cm thick and has a thermal conductivity, kA = 50 W=m · K. The back side of solid A (x = 0) is thermally insulated. Electrical current is applied to solid A such that 20 W per cm3 is generated as heat. Solid B is 0.2-cm thick and has a thermal conductivity of kB = 20 W/m · K. The surface of solid B is exposed to air. The surface temperature, Ts, of solid B is 80°C. The bulk air temperature is constant at 30°C. The process is at steady state.
a. What is the heat-transfer rate per unit area (flux) at x = L2? What is the temperature T1 at x = L1, the boundary between solid A and solid B?...
b. What is the required convective heat-transfer coefficient, h, for the flowing air?
c. What is the temperature T0 at x = 0, the insulating side of solid A?
Get solution
16.16 Consider a section of muscle tissue of cylindrical shape with a radius of 1.5 cm. During highly rigorous exercise, metabolic processes generate 15 kW/m3 of bulk tissue. The outer surface of the tissue is maintained at 37°C.The thermal conductivity of the muscle tissue is km=0.419W=m · K.
a. Develop a mathematical model to predict the steady-state temperature distribution, T(r), in the tissue along the radial position r. What is the temperature of the tissue at a distance 0.75 cm from the surface?...
b. Develop a mathematical model to predict the maximum temperature inside the muscle tissue. What is the maximum temperature?
c. If the length of the muscle, L, is 10 cm, what is the heat flux (W/cm2) leaving the muscle tissue, assuming end effects can be neglected?
d. Now consider that a sheath 0.25-cm thick surrounds the muscle tissue. This sheath has a thermal conductivity of 0.3W=m?K and does not generate any heat. If the outer surface of this sheath is maintained at 37°C, what is the new maximum temperature within the muscle tissue?
Get solution
16.17 A 1-in.-OD steel tube has its outside wall surface maintained at 250°F. It is proposed to increase the rate of heat transfer by adding fins of 3/32-in. thickness and 3/4-in. long to the outside tube surface. Compare the increase in heat transfer achieved by adding 12 longitudinal straight fins or circular fins with the same total surface area as the 12 longitudinal fins. The surrounding air is at 80°F, and the convective heat- transfer coefficient is 6 Btu/h ft2 °F.
Get solution
16.18 Heat is to be transferred from water to air through an aluminum wall. It is proposed to add rectangular fins 0.05-in. thick and 3/4-in. long spaced 0.08 in. apart to the aluminium surface to aid in transferring heat. The heat-transfer coefficients on the air and water sides are 3 and 25 Btu/h ft2 °F, respectively. Evaluate the percent increase in heat transfer if these fins are added to (a) the air side, (b) the water side, (c) and both sides. What conclusions may be reached regarding this result?
Get solution
16.19 An iron bar used for a chimney support is exposed to hot gases at 625 K with the associated convective heat-transfer coefficient of 740 W/m2 · K. The bar is attached to two opposing chimney walls, which are at 480 K. The bar is 1.9 cm in diameter and 45-cm long. Determine the maximum temperature in the bar.
Get solution
16.20 A 13 cm by 13 cm steel angle with the dimensions shown is attached to a wall with a surface temperature of 600 K. The surrounding air is at 300 K, and the convective heat-transfer coefficient between the angle surface and the air is 45 W/m2 · K.
a. Plot the temperature profile in the angle, assuming a negligible temperature drop through the side of the angle attached to the wall....
b. Determine the heat loss from the sides of the angle projecting out from the wall.
Get solution
16.21 A steel I-beam with a cross-sectional area as shown has its lower and upper surfaces maintained at 700 and 370 K, respectively.
a. Assuming a negligible temperature change through both flanges, develop an expression for the temperature variation in the web as a function of the distance from the upper flange....
b. Plot the temperature profile in the web if the convective heat-transfer coefficient between the steel surface and the surrounding air is 57 W/m2 · K. The air temperature is 300 K.
c. What is the net heat transfer at the upper and lower ends of the web?
Get solution
16.22 Heat from a flat wall is to be enhanced by adding straight fins, of constant thickness, made of stainless steel. The following specifications apply:
| h | = 60 W/m2 K |
| Tb(base) | = 120°C |
| T∞ (air) | = 20°C |
| Fin base thickness; t | = 6 mm |
| Fin length; L | = 20 mm |
Get solution
16.23 A 2-in.-OD stainless steel tube has 16 longitudinal fins spaced around its outside surface as shown. The fins are 1/16-in. thick and extend 1 in. from the outside surface of the tube.
a. If the outside surface of the tube wall is at 250°F, the surrounding air is at 80°F, and the convective heat-transfer coefficient is 8 Btu/h ft2 °F, determine the percent increase in heat transfer for the finned pipe over that for the un finned pipe....
b. Determine the same information as in part (a) for values of h of 2, 5, 15, 50, and 100 Btu/h ft2 °F. Plot the percent increase in q vs. h. What conclusions can be reached concerning this plot?
Get solution
16.24 A computer graphics chip measures 5 cm on a side and is 3-mm thick. The chip consumes 15 W of power, which is dissipated as heat from the top side of the chip. A fan blows air over the surface of the chip to promote convective heat transfer. Unfortunately, the chip fails to operate after a certain time, and the temperature of the chip is suspected to be the culprit. A plot of chip failure rate vs. surface temperature is presented in the figure on the next page. An electrical engineer suggests that an aluminum extended heat-transfer surface could be mounted to the top surface of the chip to promote heat transfer, but does not know how to design it. The electrical engineer asks for your help.
a. If the heat-transfer coefficient in air is 50 W/m2 · K, what is the surface temperature of the chip at steady state if the bulk air temperature is maintained at 20°C (293 K)? What is the failure rate of the chip?...
b. An aluminum extended surface heat-transfer device is next mounted on the top of the chip. It consists of a parallel array of five rectangular fins 1 cm by 5 cm wide by 0.3 mm thick. The new convective heat-transfer coefficient over the fin surfaces is 20 W/m2 · K. Is this configuration suitable? What is the new failure rate of the chip?
Get solution
16.25 A cylindrical tunnel with a diameter of 2 m is dug in permafrost (k = –0.341 W/m2 · K) with its axis parallel to the permafrost surface at the depth of 2.5 m.Determine the rate of heat loss from the cylinder walls, at 280 K, to the permafrost surface at 220 K.
Get solution
16.26 Determine the heat flow per foot for the configuration shown, using the numerical procedure for a grid size of ... ft. The material has a thermal conductivity of 0.15 Btu/h ft °F. The inside and outside temperatures are at the uniform values of 200°F and 0°F, respectively....
Get solution
16.27 Saturated steam at 400°F is transported through the 1-ft pipe shown in the figure, which may be assumed to be at the steam temperature. The pipe is centered in the 2-ft-square duct, whose surface is at 100°F. If the space between the pipe and duct is filled with powdered 85% magnesia insulation, how much steam will condense in a 50-ft length of pipe?...
Get solution
