Solved: A thin electrical heater is inserted between a

Consider the plane wall of Figure 3.1, separating hot and cold fluids at temperatures T
,1 and T
,2, respectively. Using surface energy balances as boundary conditions at x
0 and x
L (see Equation 2.34), obtain the temperature distribution within the wall and the heat flux in terms of T
,1, T
,2, h1, h2, k, and L.

A new building to be located in a cold climate is being designed with a basement that has an L
200-mm-thick wall. Inner and outer basement wall temperatures are Ti
20 C and To
0 C, respectively. The architect can specify the wall material to be either aerated concrete block with kac
0.15 W/m
K, or stone mix concrete. To reduce the conduction heat flux through the stone mix wall to a level equivalent to that of the aerated concrete wall, what thickness of extruded polystyrene sheet must be applied onto the inner surface of the stone mix con crete wall? Floor dimensions of the basement are 20 m 30 m, and the expected rental rate is $50/m2 / month. What is the yearly cost, in terms of lost rental income, if the stone mix concrete wall with polystyrene insulation is specified?

The rear window of an automobile is defogged by passing warm air over its inner surface. (a) If the warm air is at T
,i
40 C and the corresponding convection coefficient is hi
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 C and the associated convection coefficient is ho
65 W/m2
K? (b) In practice T
,o and ho vary according to weather conditions and car speed. For values of ho
2, 65, and 100 W/m2
K, compute and plot the inner and outer surface temperatures as a function of T
,o for 30 T
,o 0 C.

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. (a) For 4-mm-thick window glass, determine the electrical power required per unit window area to maintain an inner surface temperature of 15 C when the interior air temperature and convection coefficient are T
,i
25 C and hi
10 W/m2
K, while the exterior (ambient) air temperature and convection coefficient are T
,o
10 C and ho
65 W/m2
K. (b) In practice T
,o and ho vary according to weather conditions and car speed. For values of ho
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 C. From your results, what can you conclude about the need for heater operation at low values of ho? How is this conclusion affected by the value of T
,o? If h  Vn , where V is the vehicle speed and n is a positive exponent, how does the vehicle speed affect the need for heater operation?

A dormitory at a large university, built 50 years ago, has exterior walls constructed of Ls
25-mm-thick sheathing with a thermal conductivity of ks
0.1 W/m
K. To reduce heat losses in the winter, the university decides to encapsulate the entire dormitory by applying an Li
25-mm-thick layer of extruded insulation characterized by ki
0.029 W/m
K to the exterior of the original sheathing. The extruded insulation is, in turn, covered with an Lg
5-mm-thick architectural glass with kg
1.4 W/m
K. Determine the heat flux through the original and retrofitted walls when the interior and exterior air temperatures are T
,i
22 C and T
,o
20 C, respectively. The inner and outer convection heat transfer coefficients are hi
5 W/m2
K and ho
25 W/m2
K, respectively.

In a manufacturing process, a transparent film is being bonded to a substrate as shown in the sketch. To cure the bond at a temperature T0, a radiant source is used to provide a heat flux q 0 (W/m2 ), all of which is absorbed at the bonded surface. The back of the substrate is maintained at T1 while the free surface of the film is exposed to air at T
and a convection heat transfer coefficient h. (a) Show the thermal circuit representing the steady-state heat transfer situation. Be sure to label all elements, nodes, and heat rates. Leave in symbolic form. (b) Assume the following conditions: T

20 C, h
50 W/m2
K, and T1
30 C. Calculate the heat flux q 0 that is required to maintain the bonded surface at T0
60 C. (c) Compute and plot the required heat flux as a function of the film thickness for 0 L 1 mm. (d) If the film is not transparent and all of the radiant heat flux is absorbed at its upper surface, determine the heat flux required to achieve bonding. Plot your results as a function of L for 0 L 1 mm.

The walls of a refrigerator are typically constructed by sandwiching a layer of insulation between sheet metal panels. Consider a wall made from fiberglass insulation of thermal conductivity ki
0.046 W/m
K and thickness Li
50 mm and steel panels, each of thermal conductivity kp
60 W/m
K and thickness Lp
3 mm. If the wall separates refrigerated air at T
, i
4 C from ambient air at T
,o
25 C, what is the heat gain per unit surface area? Coefficients associated with natural convection at the inner and outer surfaces may be approximated as hi
ho
5 W/m2
K

A t
10-mm-thick horizontal layer of water has a top surface temperature of Tc
4 C and a bottom surface temperature of Th
2 C. Determine the location of the solidliquid interface at steady state.

A technique for measuring convection heat transfer coefficients involves bonding one surface of a thin metallic foil to an insulating material and exposing the other surface to the fluid flow conditions of interest. By passing an electric current through the foil, heat is dissipated uniformly within the foil and the corresponding flux, P elec, may be inferred from related voltage and current measurements. If the insulation thickness L and thermal conductivity k are known and the fluid, foil, and insulation temperatures (T
, Ts, Tb) are measured, the convection coefficient may be determined. Consider conditions for which T

Tb
25 C, P elec
2000 W/m2 , L
10 mm, and k
0.040 W/m
K. (a) With water flow over the surface, the foil temperature measurement yields Ts
27 C. Determine the convection coefficient. What error would be incurred by assuming all of the dissipated power to be transferred to the water by convection? (b) If, instead, air flows over the surface and the temperature measurement yields Ts
125 C, what is the convection coefficient? The foil has an emissivity of 0.15 and is exposed to large surroundings at 25 C. What error would be incurred by assuming all of the dissipated power to be transferred to the air by convection? (c) Typically, heat flux gages are operated at a fixed temperature (Ts), in which case the power dissipation provides a direct measure of the convection coefficient. For Ts
27 C, plot P elec as a function of ho for 10 ho 1000 W/m2
K. What effect does ho have on the error associated with neglecting conduction through the insulation?

The wind chill, which is experienced on a cold, windy day, is related to increased heat transfer from exposed human skin to the surrounding atmosphere. Consider a layer of fatty tissue that is 3 mm thick and whose interior surface is maintained at a temperature of 36 C. On a calm day the convection heat transfer coefficient at the outer surface is 25 W/m2
K, but with 30 km/h winds it reaches 65 W/m2
K. In both cases the ambient air temperature is 15 C. (a) What is the ratio of the heat loss per unit area from the skin for the calm day to that for the windy day? (b) What will be the skin outer surface temperature for the calm day? For the windy day? (c) What temperature would the air have to assume on the calm day to produce the same heat loss occurring with the air temperature at 15 C on the windy day?

Determine the thermal conductivity of the carbon nanotube of Example 3.4 when the heating island temperature is measured to be Th
332.6 K, without evaluating the thermal resistances of the supports. The conditions are the same as in the example

A thermopane window consists of two pieces of glass 7 mm thick that enclose an air space 7 mm thick. The window separates room air at 20 C from outside ambient air at 10 C. The convection coefficient associated with the inner (room-side) surface is 10 W/m2
K. (a) If the convection coefficient associated with the outer (ambient) air is ho
80 W/m2
K, what is the heat loss through a window that is 0.8 m long by 0.5 m wide? Neglect radiation, and assume the air enclosed between the panes to be stagnant. (b) Compute and plot the effect of ho on the heat loss for 10 ho 100 W/m2
K. Repeat this calculation for a triple-pane construction in which a third pane and a second air space of equivalent thickness are added.

A house has a composite wall of wood, fiberglass insulation, and plaster board, as indicated in the sketch. On a cold winter day, the convection heat transfer coefficients are \(h_{o}=60 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(h_i=30\mathrm{\ W}/\mathrm{m}^2\cdot\mathrm{K}\). The total wall surface area is \(350 \mathrm{m}^{2}\) . (a) Determine a symbolic expression for the total thermal resistance of the wall, including inside and outside convection effects for the prescribed conditions. (b) Determine the total heat loss through the wall. (c) If the wind were blowing violently, raising \(h_{o}\) to \(300\mathrm{\ W}/\mathrm{m}^2\cdot\mathrm{K}\), determine the percentage increase in the heat loss. (d) What is the controlling resistance that determines the amount of heat flow through the wall?

Consider the composite wall of Problem 3.13 under conditions for which the inside air is still characterized by T
,i
20 C and hi
30 W/m2
K. However, use the more realistic conditions for which the outside air is characterized by a diurnal (time) varying temperature of the form with ho
60 W/m2
K. Assuming quasi-steady conditions for which changes in energy storage within the wall may be neglected, estimate the daily heat loss through the wall if its total surface area is 200 m2 .

Consider a composite wall that includes an 8-mm-thick hardwood siding, 40-mm by 130-mm hardwood studs on 0.65-m centers with glass fiber insulation (paper Consider a composite wall that includes an 8-mm-thick hardwood siding, 40-mm by 130-mm hardwood studs on 0.65-m centers with glass fiber insulation (paper What is the thermal resistance associated with a wall that is 2.5 m high by 6.5 m wide (having 10 studs, each 2.5 m high)? Assume surfaces normal to the x-direction are isothermal.

Work Problem 3.15 assuming surfaces parallel to the x-direction are adiabatic.

Consider the oven of Problem 1.54. The walls of the oven consist of L
30-mm-thick layers of insulation characterized by kins
0.03 W/m
K that are sandwiched between two thin layers of sheet metal. The exterior surface of the oven is exposed to air at 23 C with hext
2 W/m2
K. The interior oven air temperature is 180 C. Neglecting radiation heat transfer, determine the steady-state heat flux through the oven walls when the convection mode is disabled and the free convection coefficient at the inner oven surface is hfr
3 W/m2
K. Determine the heat flux through the oven walls when the convection mode is activated, in which case the forced convection coefficient at the inner oven surface is hfo
27 W/m2
K. Does operation of the oven in its convection mode result in signifi- cantly increased heat losses from the oven to the kitchen? Would your conclusion change if radiation were included in your analysis?

The composite wall of an oven consists of three materials, two of which are of known thermal conductivity, kA
20 W/m
K and kC
50 W/m
K, and known thickness, LA
0.30 m and LC
0.15 m. The third material, B, which is sandwiched between materials A and C, is of known thickness, LB
0.15 m, but unknown thermal conductivity kB. Under steady-state operating conditions, measurements reveal an outer surface temperature of Ts,o
20 C, an inner surface temperature of Ts,i
600 C, and an oven air temperature of T

800 C. The inside convection coefficient h is known to be 25 W/m2
K. What is the value of kB?

The wall of a drying oven is constructed by sandwiching an insulation material of thermal conductivity k
0.05 W/m
K between thin metal sheets. The oven air is at T
,i
300 C, and the corresponding convection coeffi- cient is hi
30 W/m2
K. The inner wall surface absorbs a radiant flux of qrad
100 W/m2 from hotter objects within the oven. The room air is at T
,o
25 C, and the overall coefficient for convection and radiation from the outer surface is ho
10 W/m2
K. (a) Draw the thermal circuit for the wall and label all temperatures, heat rates, and thermal resistances. (b) What insulation thickness L is required to maintain the outer wall surface at a safe-to-touch temperature of To
40 C?

The t
4-mm-thick glass windows of an automobile have a surface area of A
2.6 m2 . The outside temperature is T
,o
32 C while the passenger compartment is maintained at T
,i
22 C. The convection heat transfer coefficient on the exterior window surface is ho
90 W/m2
K. Determine the heat gain through the windows when the interior convection heat transfer coefficient is hi
15 W/m2
K. By controlling the air- flow in the passenger compartment the interior heat transfer coefficient can be reduced to hi
5 W/m2
K without sacrificing passenger comfort. Determine the heat gain through the window for the reduced inside heat transfer coefficient

The thermal characteristics of a small, dormitory refrigerator are determined by performing two separate experiments, each with the door closed and the refrigerator placed in ambient air at T

25 C. In one case, an electric heater is suspended in the refrigerator cavity, while the refrigerator is unplugged. With the heater dissipating 20 W, a steady-state temperature of 90 C is recorded within the cavity. With the heater removed and the refrigerator now in operation, the second experiment involves maintaining a steady-state cavity temperature of 5 C for a fixed time interval and recording the electrical energy required to operate the refrigerator. In such an experiment for which steady operation is maintained over a 12-hour period, the input electrical energy is 125,000 J. Determine the refrigerators coeffi- cient of performance (COP).

In the design of buildings, energy conservation requirements dictate that the exterior surface area, As, be minimized. This requirement implies that, for a desired floor space, there may be optimum values associated with the number of floors and horizontal dimensions of the building. Consider a design for which the total floor space, Af , and the vertical distance between floors, Hf , are prescribed. (a) If the building has a square cross section of width W on a side, obtain an expression for the value of W that would minimize heat loss to the surroundings. Heat loss may be assumed to occur from the four vertical side walls and from a flat roof. Express your result in terms of Af and Hf . (b) If Af
32,768 m2 and Hf
4 m, for what values of W and Nf (the number of floors) is the heat loss minimized? If the average overall heat transfer coefficient is U
1 W/m2
K and the difference between the inside and ambient air temperatures is 25 C, what is the corresponding heat loss? What is the percentage reduction in heat loss compared with a building for Nf
2?

When raised to very high temperatures, many conventional liquid fuels dissociate into hydrogen and other components. Thus the advantage of a solid oxide fuel cell is that such a device can internally reform readily available liquid fuels into hydrogen that can then be used to produce electrical power in a manner similar to Example 1.5. Consider a portable solid oxide fuel cell, operating at a temperature of Tfc
800 C. The fuel cell is housed within a cylindrical canister of diameter D
75 mm and length L
120 mm. The outer surface of the canister is insulated with a low-thermal-conductivity material. For a particular application, it is desired that the thermal signature of the canister be small, to avoid its detection by infrared sensors. The degree to which the canister can be detected with an infrared sensor may be estimated by equating the radiation heat flux emitted from the exterior surface of the canister (Equation 1.5; Es
s
Ts 4 ) to the heat flux emitted from an equivalent black surface, (Eb

Tb 4 ). If the equivalent black surface temperature Tb is near the surroundings temperature, the thermal signature of the canister is too small to be detectedthe canister is indistinguishable from the surroundings. (a) Determine the required thickness of insulation to be applied to the cylindrical wall of the canister to ensure that the canister does not become highly visible to an infrared sensor (i.e., Tb Tsur  5 K). Consider cases where (i) the outer surface is covered with a very thin layer of dirt (s
0.90) and (ii) the outer surface is comprised of a very thin polished aluminum sheet (s
0.08). Calculate the required thicknesses for two types of insulating material, calcium silicate (k
0.09 W/m
K) and aerogel (k
0.006 W/m
K). The temperatures of the surroundings and the ambient are Tsur
300 K and T

298 K, respectively. The outer surface is characterized by a convective heat transfer coeffi- cient of h
12 W/m2
K. (b) Calculate the outer surface temperature of the canister for the four cases (high and low thermal conductivity; high and low surface emissivity). (c) Calculate the heat loss from the cylindrical walls of the canister for the four cases.

A firefighters protective clothing, referred to as a turnout coat, is typically constructed as an ensemble of three layers separated by air gaps, as shown schematically. Representative dimensions and thermal conductivities for the layers are as follows. Layer Thickness (mm) k (W/m
K) Shell (s) 0.8 0.047 Moisture barrier (mb) 0.55 0.012 Thermal liner (tl) 3.5 0.038 The air gaps between the layers are 1 mm thick, and heat is transferred by conduction and radiation exchange through the stagnant air. The linearized radiation coefficient for a gap may be approximated as, , where Tavg represents the average temperature of the surfaces comprising the gap, and the radiation flux across the gap may be expressed as . (a) Represent the turnout coat by a thermal circuit, labeling all the thermal resistances. Calculate and tabulate the thermal resistances per unit area (m2
K/W) for each of the layers, as well as for the conduction and radiation processes in the gaps. Assume that a value of Tavg
470 K may be used to approximate the radiation resistance of both gaps. Comment on the relative magnitudes of the resistances. (b) For a pre-flash-over fire environment in which fire- fighters often work, the typical radiant heat flux on the fire-side of the turnout coat is 0.25 W/cm2 . What is the outer surface temperature of the turnout coat if the inner surface temperature is 66 C, a condition that would result in burn injury?

A particular thermal system involves three objects of fixed shape with conduction resistances of R1
1 K/W, R2
2 K/W and R3
4 K/W, respectively. An objective is to minimize the total thermal resistance Rtot associated with a combination of R1, R2, and R3. The chief engineer is willing to invest limited funds to specify an alternative material for just one of the three objects; the alternative material will have a thermal conductivity that is twice its nominal value. Which object (1, 2, or 3) should be fabricated of the higher thermal conductivity material to most significantly decrease Rtot? Hint: Consider two cases, one for which the three thermal resistances are arranged in series, and the second for which the three resistances are arranged in parallel

A composite wall separates combustion gases at 2600 C from a liquid coolant at 100 C, 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 0.05 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

Approximately 106 discrete electrical components can be placed on a single integrated circuit (chip), with electrical heat dissipation as high as 30,000 W/m2 . The chip, which is very thin, is exposed to a dielectric liquid at its outer surface, with ho
1000 W/m2
K and T
,o
20 C, and is joined to a circuit board at its inner surface. The thermal contact resistance between the chip and the board is 104 m2
K/W, and the board thickness and thermal conductivity are Lb
5 mm and kb
1 W/m
K, respectively. The other surface of the board is exposed to ambient air for which hi
40 W/m2
K and T
,i
20 C. (a) Sketch the equivalent thermal circuit corresponding to steady-state conditions. In variable form, label appropriate resistances, temperatures, and heat fluxes. (b) Under steady-state conditions for which the chip heat dissipation is q c
30,000 W/m2 , what is the chip temperature? (c) The maximum allowable heat flux, q c,m, is determined by the constraint that the chip temperature must not exceed 85 C. Determine q c,m for the foregoing conditions. If air is used in lieu of the dielectric liquid, the convection coefficient is reduced by approximately an order of magnitude. What is the value of q c,m for ho
100 W/m2
K? With air cooling, can significant improvements be realized by using an aluminum oxide circuit board and/or by using a conductive paste at the chip/board interface for which Rt, c
105 m2
K/W?

Two stainless steel plates 10 mm thick are subjected to a contact pressure of 1 bar under vacuum conditions for which there is an overall temperature drop of 100 C across the plates. What is the heat flux through the plates? What is the temperature drop across the contact plane?

Consider a plane composite wall that is composed of two materials of thermal conductivities kA
0.1 W/m
K and kB
0.04 W/m
K and thicknesses LA
10 mm and LB
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 C for which h
10 W/m2
K, and material B adjoins a fluid at 40 C 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

The performance of gas turbine engines may be improved by increasing the tolerance of the turbine blades to hot gases emerging from the combustor. One approach to achieving high operating temperatures involves application of a thermal barrier coating (TBC) to the exterior surface of a blade, while passing cooling air through the blade. Typically, the blade is made from a high-temperature superalloy, such as Inconel (k 25 W/m
K), while a ceramic, such as zirconia (k 1.3 W/m
K), is used as a TBC. Consider conditions for which hot gases at T
,o
1700 K and cooling air at T
,i
400 K provide outer and inner surface convection coefficients of ho
1000 W/m2
K and hi
500 W/m2
K, respectively. If a 0.5-mm-thick zirconia TBC is attached to a 5-mmthick Inconel blade wall by means of a metallic bonding agent, which provides an interfacial thermal resistance of Rt,c
104 m2
K/W, can the Inconel be maintained at a temperature that is below its maximum allowable value of 1250 K? Radiation effects may be neglected, and the turbine blade may be approximated as a plane wall. Plot the temperature distribution with and without the TBC. Are there any limits to the thickness of the TBC?

A commercial grade cubical freezer, 3 m on a side, has a composite wall consisting of an exterior sheet of 6.35-mm-thick plain carbon steel, an intermediate layer of 100-mm-thick cork insulation, and an inner sheet of 6.35-mm-thick aluminum alloy (2024). Adhesive interfaces between the insulation and the metallic strips are each characterized by a thermal contact resistance of . What is the steady-state cooling load that must be maintained by the refrigerator under conditions for which the outer and inner surface temperatures are 22 C and 6 C, respectively?

Physicists have determined the theoretical value of the thermal conductivity of a carbon nanotube to be kcn,T
5000 W/m
K. (a) Assuming the actual thermal conductivity of the carbon nanotube is the same as its theoretical value, find the thermal contact resistance, Rt,c, that exists between the carbon nanotube and the top surfaces of the heated and sensing islands in Example 3.4 . (b) Using the value of the thermal contact resistance calculated in part (a), plot the fraction of the total resistance between the heated and sensing islands that is due to the thermal contact resistances for island separation distances of 5 m s 20 m.

Consider a power transistor encapsulated in an aluminum case that is attached at its base to a square aluminum plate of thermal conductivity k
240 W/m
K, thickness L
6 mm, and width W
20 mm. The case is joined to the plate by screws that maintain a contact pressure of 1 bar, and the back surface of the plate transfers heat by natural convection and radiation to ambient air and large surroundings at T

Tsur
25 C. The surface has an emissivity of 
0.9, and the convection coefficient is h
4 W/m2
K. The case is completely enclosed such that heat transfer may be assumed to occur exclusively through the base plate (a) If the air-filled aluminum-to-aluminum interface is characterized by an area of Ac
2 104 m2 and a roughness of 10 m, what is the maximum allowable power dissipation if the surface temperature of the case, Ts,c, is not to exceed 85 C? (b) The convection coefficient may be increased by subjecting the plate surface to a forced flow of air. Explore the effect of increasing the coefficient over the range 4 h 200 W/m2
K.

Ring-porous woods, such as oak, are characterized by grains. The dark grains consist of very low-density material that forms early in the springtime. The surrounding lighter-colored wood is composed of highdensity material that forms slowly throughout most of the growing season. Assuming the low-density material is highly porous and the oak is dry, determine the fraction of the oak crosssection that appears as being grained. Hint: Assume the thermal conductivity parallel to the grains is the same as the radial conductivity of Table A.3

A batt of glass fiber insulation is of density
28 kg/m3 . Determine the maximum and minimum possible values of the effective thermal conductivity of the insulation at T
300 K, and compare with the value reported in Table A.3.

Air usually constitutes up to half of the volume of commercial ice creams and takes the form of small spherical bubbles interspersed within a matrix of frozen matter. The thermal conductivity of ice cream that contains no air is kna
1.1 W/m
K at T
20 C. Determine the thermal conductivity of commercial ice cream characterized by 
0.20, also at T
20 C.

Determine the density, specific heat, and thermal conductivity of a lightweight aggregate concrete that is composed of 65% stone mix concrete and 35% air by volume. Evaluate properties at T
300 K

A one-dimensional plane wall of thickness L is constructed of a solid material with a linear, nonuniform porosity distribution described by (x)
max(x/L). Plot the steady-state temperature distribution, T(x), for ks
10 W/m
K, kf
0.1 W/m
K, L
1 m, max
0.25, T(x
0)
30 C and q x
100 W/m2 using the expression for the minimum effective thermal conductivity of a porous medium, the expression for the maximum effective thermal conductivity of a porous medium, Maxwells expression, and for the case where keff(x)
k

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 qx.

A truncated solid cone is of circular cross section, and its diameter is related to the axial coordinate by an expression of the form D
ax3/2, where a
1.0 m1/2. 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 C, x2
0.225 m, and T2
20 C?

From Figure 2.5 it is evident that, over a wide temperature range, the temperature dependence of the thermal conductivity of many solids may be approximated by a linear expression of the form k
ko aT, where ko is a positive constant and a is a coefficient that may be positive or negative. Obtain an expression for the heat flux across a plane wall whose inner and outer surfaces are maintained at T0 and T1, respectively. Sketch the forms of the temperature distribution corresponding to a  0, a
0, and a  0.

Consider a tube wall of inner and outer radii ri and ro, whose temperatures are maintained at Ti and To, respectively. The thermal conductivity of the cylinder is temperature dependent and may be represented by an expression of the form k
ko(1 aT), where ko and a are constants. Obtain an expression for the heat transfer per unit length of the tube. What is the thermal resistance of the tube wall?

Measurements show that steady-state conduction through a plane wall without heat generation produced a convex temperature distribution such that the midpoint temperature was To higher than expected for a linear temperature distribution. Assuming that the thermal conductivity has a linear dependence on temperature, k
ko(1 T), where is a constant, develop a relationship to evaluate in terms of To, T1, and T2.

A device used to measure the surface temperature of an object to within a spatial resolution of approximately 50 nm is shown in the schematic. It consists of an extremely sharp-tipped stylus and an extremely small cantilever that is scanned across the surface. The probe tip is of circular cross section and is fabricated of polycrystalline silicon dioxide. The ambient temperature is measured at the pivoted end of the cantilever as T
25 C, and the device is equipped with a sensor to measure the temperature at the upper end of the sharp tip, Tsen. The thermal resistance between the sensing probe and the pivoted end is Rt
5 106 K/W. (a) Determine the thermal resistance between the surface temperature and the sensing temperature. (b) If the sensing temperature is Tsen
28.5 C, determine the surface temperature. Hint: Although nanoscale heat transfer effects may be important, assume that the conduction occurring in the air adjacent to the probe tip can be described by Fouriers law and the thermal conductivity found in Table A.4

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 Ts,1
800 K and Ts,2
490 K, respectively, what is the heat loss per unit length (q) of the pipe? (b) We wish to explore the effect of insulation thickness on the heat loss q and outer surface temperature Ts,2, with the inner surface temperature fixed at Ts,1
800 K. The outer surface is exposed to an airflow (T

25 C) that maintains a convection coefficient of h
25 W/m2
K and to large surroundings for which Tsur
T

25 C. 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 r1)/(r2 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 r1) 0.14 m. 3

Consider the water heater described in Problem 1.48. We now wish to determine the energy needed to compensate for heat losses incurred while the water is stored at the prescribed temperature of 55 C. The cylindrical storage tank (with flat ends) has a capacity of 100 gal, and foamed urethane is used to insulate the side and end walls from ambient air at an annual average temperature of 20 C. The resistance to heat transfer is dominated by conduction in the insulation and by free convection in the air, for which h 2 W/m2
K. If electric resistance heating is used to compensate for the losses and the cost of electric power is $0.18/kWh, specify tank and insulation dimensions for which the annual cost associated with the heat losses is less than $50

To maximize production and minimize pumping costs, crude oil is heated to reduce its viscosity during transportation from a production field. (a) Consider a pipe-in-pipe configuration consisting of concentric steel tubes with an intervening insulating material. The inner tube is used to transport warm crude oil through cold ocean water. The inner steel pipe (ks
35 W/m
K) has an inside diameter of Di,1
150 mm and wall thickness ti
10 mm while the outer steel pipe has an inside diameter of Di,2
250 mm and wall thickness to
ti . Determine the maximum allowable crude oil temperature to ensure the polyurethane foam insulation (kp
0.075 W/m
K) between the two pipes does not exceed its maximum service temperature of Tp,max
70 C. The ocean water is at T
,o
5 C and provides an external convection heat transfer coefficient of ho
500 W/m2
K. The convection coefficient associated with the flowing crude oil is hi
450 W/m2
K. (b) It is proposed to enhance the performance of the pipe-in-pipe device by replacing a thin (ta
5 mm) section of polyurethane located at the outside of the inner pipe with an aerogel insulation material (ka
0.012 W/m
K). Determine the maximum allowable crude oil temperature to ensure maximum polyurethane temperatures are below Tp,max
70 C

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 C. 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 Rt,c
0.01 m
K/W. The outer surface of the heater is exposed to a fluid with T

10 C and a convection coefficient of h
100 W/m2
K. Determine the heater power per unit length of tube required to maintain the heater at To
25 C.

In Problem 3.48, the electrical power required to maintain the heater at To
25 C depends on the thermal conductivity of the wall material k, the thermal contact resistance Rt,c and the convection coefficient h. Compute and plot the separate effect of changes in k (1 k 200 W/m
K), Rt,c (0 Rt,c 0.1 m
K/W), and h (10 h 1000 W/m2
K) on the total heater power requirement, as well as the rate of heat transfer to the inner surface of the tube and to the fluid.

A stainless steel (AISI 304) tube used to transport a chilled pharmaceutical has an inner diameter of 36 mm and a wall thickness of 2 mm. The pharmaceutical and ambient air are at temperatures of 6 C and 23 C, respectively, while the corresponding inner and outer convection coefficients are 400 W/m2
K and 6 W/m2
K, respectively. (a) What is the heat gain per unit tube length? (b) What is the heat gain per unit length if a 10-mmthick layer of calcium silicate insulation (kins
0.050 W/m
K) is applied to the tube?

Superheated steam at 575 C is routed from a boiler to the turbine of an electric power plant through steel tubes (k
35 W/m
K) of 300-mm inner diameter and 30-mm wall thickness. To reduce heat loss to the surroundings and to maintain a safe-to-touch outer surface temperature, a layer of calcium silicate insulation (k
0.10 W/m
K) is applied to the tubes, while degradation of the insulation is reduced by wrapping it in a thin sheet of aluminum having an emissivity of 
0.20. The air and wall temperatures of the power plant are 27 C. (a) Assuming that the inner surface temperature of a steel tube corresponds to that of the steam and the convection coefficient outside the aluminum sheet is 6 W/m2
K, what is the minimum insulation thickness needed to ensure that the temperature of the aluminum does not exceed 50 C? What is the corresponding heat loss per meter of tube length? (b) Explore the effect of the insulation thickness on the temperature of the aluminum and the heat loss per unit tube length

A thin electrical heater is inserted between a long circular rod and a concentric tube with inner and outer radii of 20 and 40 mm. The rod (A) has a thermal conductivity of kA
0.15 W/m
K, while the tube (B) has a thermal conductivity of kB
1.5 W/m
K and its outer surface is subjected to convection with a fluid of temperature T

15 C and heat transfer coefficient 50 W/m2
K. The thermal contact resistance between the cylinder surfaces and the heater is negligible. (a) Determine the electrical power per unit length of the cylinders (W/m) that is required to maintain the outer surface of cylinder B at 5 C. (b) What is the temperature at the center of cylinder A?

A wire of diameter D
2 mm and uniform temperature T has an electrical resistance of 0.01 /m and a current flow of 20 A. (a) What is the rate at which heat is dissipated per unit length of wire? What is the heat dissipation per unit volume within the wire? (b) If the wire is not insulated and is in ambient air and large surroundings for which T

Tsur
20 C, what is the temperature T of the wire? The wire has an emissivity of 0.3, and the coefficient associated with heat transfer by natural convection may be approximated by an expression of the form, h
C[(T T
)/D] 1/4, where C
1.25 W/m7/4
K5/4. (c) If the wire is coated with plastic insulation of 2-mm thickness and a thermal conductivity of 0.25 W/m
K, what are the inner and outer surface temperatures of the insulation? The insulation has an emissivity of 0.9, and the convection coefficient is given by the expression of part (b). Explore the effect of the insulation thickness on the surface temperatures.

A 2-mm-diameter electrical wire is insulated by a 2-mm-thick rubberized sheath (k
0.13 W/m
K), and the wire/sheath interface is characterized by a thermal contact resistance of . The convection heat transfer coefficient at the outer surface of the sheath is 10 W/m2
K, and the temperature of the ambient air is 20 C. If the temperature of the insulation may not exceed 50 C, what is the maximum allowable electrical power that may be dissipated per unit length of the conductor? What is the critical radius of the insulation?

Electric current flows through a long rod generating thermal energy at a uniform volumetric rate of . The rod is concentric with a hollow ceramic cylinder, creating an enclosure that is filled with air. The thermal resistance per unit length due to radiation between the enclosure surfaces is , and the coefficient associated with free convection in the enclosure is h
20 W/m2
K. (a) Construct a thermal circuit that can be used to calculate the surface temperature of the rod, Tr. Label all temperatures, heat rates, and thermal resistances, and evaluate each thermal resistance. (b) Calculate the surface temperature of the rod for the prescribed conditions.

The evaporator section of a refrigeration unit consists of thin-walled, 10-mm-diameter tubes through which refrigerant passes at a temperature of 18 C. Air is cooled as it flows over the tubes, maintaining a surface convection coefficient of 100 W/m2
K, and is subsequently routed to the refrigerator compartment. (a) For the foregoing conditions and an air temperature of 3 C, what is the rate at which heat is extracted from the air per unit tube length? (b) If the refrigerators defrost unit malfunctions, frost will slowly accumulate on the outer tube surface. Assess the effect of frost formation on the cooling capacity of a tube for frost layer thicknesses in the range 0  4 mm. Frost may be assumed to have a thermal conductivity of 0.4 W/m
K. (c) The refrigerator is disconnected after the defrost unit malfunctions and a 2-mm-thick layer of frost has formed. If the tubes are in ambient air for which T

20 C and natural convection maintains a convection coefficient of 2 W/m2
K, how long will it take for the frost to melt? The frost may be assumed to have a mass density of 700 kg/m3 and a latent heat of fusion of 334 kJ/kg.

A composite cylindrical wall is composed of two materials of thermal conductivity kA and kB, which are separated by a very thin, electric resistance heater for which interfacial contact resistances are negligible. Liquid pumped through the tube is at a temperature T
,i and provides a convection coefficient hi at the inner surface of the composite. The outer surface is exposed to ambient air, which is at T
,o and provides a convection coefficient of ho. Under steady-state conditions, a uniform heat flux of q h is dissipated by the heater. (a) Sketch the equivalent thermal circuit of the system and express all resistances in terms of relevant variables. (b) Obtain an expression that may be used to determine the heater temperature, Th. (c) Obtain an expression for the ratio of heat flows to the outer and inner fluids, q o /qi . How might the variables of the problem be adjusted to minimize this ratio?

An electrical current of 700 A flows through a stainless steel cable having a diameter of 5 mm and an electrical resistance of 6 104 /m (i.e., per meter of cable length). The cable is in an environment having a temperature of 30 C, and the total coefficient associated with convection and radiation between the cable and the environment is approximately 25 W/m2
K. (a) If the cable is bare, what is its surface temperature? (b) If a very thin coating of electrical insulation is applied to the cable, with a contact resistance of 0.02 m2
K/W, what are the insulation and cable surface temperatures? (c) There is some concern about the ability of the insulation to withstand elevated temperatures. What thickness of this insulation (k
0.5 W/m
K) will yield the lowest value of the maximum insulation temperature? What is the value of the maximum temperature when this thickness is used?

A 0.20-m-diameter, thin-walled steel pipe is used to transport saturated steam at a pressure of 20 bars in a room for which the air temperature is 25 C and the convection heat transfer coefficient at the outer surface of the pipe is 20 W/m2
K. (a) What is the heat loss per unit length from the bare pipe (no insulation)? Estimate the heat loss per unit length if a 50-mm-thick layer of insulation (magnesia, 85%) is added. The steel and magnesia may each be assumed to have an emissivity of 0.8, and the steam-side convection resistance may be neglected. (b) The costs associated with generating the steam and installing the insulation are known to be $4/109 J and $100/m of pipe length, respectively. If the steam line is to operate 7500 h/yr, how many years are needed to pay back the initial investment in insulation?

An uninsulated, thin-walled pipe of 100-mm diameter is used to transport water to equipment that operates outdoors and uses the water as a coolant. During particularly harsh winter conditions, the pipe wall achieves a temperature of 15 C and a cylindrical layer of ice forms on the inner surface of the wall. If the mean water temperature is 3 C and a convection coefficient of 2000 W/m2
K is maintained at the inner surface of the ice, which is at 0 C, what is the thickness of the ice layer?

Steam flowing through a long, thin-walled pipe maintains the pipe wall at a uniform temperature of 500 K. The pipe is covered with an insulation blanket comprised of two different materials, A and B. The interface between the two materials may be assumed to have an infinite contact resistance, and the entire outer surface is exposed to air for which T

300 K and h
25 W/m2
K. (a) Sketch the thermal circuit of the system. Label (using the preceding symbols) all pertinent nodes and resistances. (b) For the prescribed conditions, what is the total heat loss from the pipe? What are the outer surface temperatures Ts,2(A) and Ts,2(B)?

A bakelite coating is to be used with a 10-mm-diameter conducting rod, whose surface is maintained at 200 C by passage of an electrical current. The rod is in a fluid at 25 C, and the convection coefficient is 140 W/m2
K. What is the critical radius associated with the coating? What is the heat transfer rate per unit length for the bare rod and for the rod with a coating of bakelite that corresponds to the critical radius? How much bakelite should be added to reduce the heat transfer associated with the bare rod by 25%?

A storage tank consists of a cylindrical section that has a length and inner diameter of L
2 m and Di
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

Consider the liquid oxygen storage system and the laboratory environmental conditions of Problem 1.49. To reduce oxygen loss due to vaporization, an insulating layer should be applied to the outer surface of the container. Consider using a laminated aluminum foil/glass mat insulation, for which the thermal conductivity and surface emissivity are k
0.00016 W/m
K and 
0.20, respectively. (a) If the container is covered with a 10-mm-thick layer of insulation, what is the percentage reduction in oxygen loss relative to the uncovered container? (b) Compute and plot the oxygen evaporation rate (kg/s) as a function of the insulation thickness t for 0 t 50 mm

A spherical Pyrex glass shell has inside and outside diameters of D1
0.1 m and D2
0.2 m, respectively. The inner surface is at Ts,1
100 C while the outer surface is at Ts,2
45 C. (a) Determine the temperature at the midpoint of the shell thickness, T(rm
0.075 m). (b) For the same surface temperatures and dimensions as in part (a), show how the midpoint temperature would change if the shell material were aluminum

In Example 3.6, an expression was derived for the critical insulation radius of an insulated, cylindrical tube. Derive the expression that would be appropriate for an insulated sphere

A hollow aluminum sphere, with an electrical heater in the center, is used in tests to determine the thermal conductivity of insulating materials. The inner and outer radii of the sphere are 0.15 and 0.18 m, respectively, and testing is done under steady-state conditions with the inner surface of the aluminum maintained at 250 C. In a particular test, a spherical shell of insulation is cast on the outer surface of the sphere to a thickness of 0.12 m. The system is in a room for which the air temperature is 20 C and the convection coefficient at the outer surface of the insulation is 30 W/m2
K. If 80 W are dissipated by the heater under steady-state conditions, what is the thermal conductivity of the insulation?

A spherical tank for storing liquid oxygen on the space shuttle is to be made from stainless steel of 0.80-m outer diameter and 5-mm wall thickness. The boiling point and latent heat of vaporization of liquid oxygen are 90 K and 213 kJ/kg, respectively. The tank is to be installed in a large compartment whose temperature is to be maintained at 240 K. Design a thermal insulation system that will maintain oxygen losses due to boiling below 1 kg/day.

A spherical, cryosurgical probe may be imbedded in diseased tissue for the purpose of freezing, and thereby destroying, the tissue. Consider a probe of 3-mm diameter whose surface is maintained at 30 C when imbedded in tissue that is at 37 C. A spherical layer of frozen tissue forms around the probe, with a temperature of 0 C existing at the phase front (interface) between the frozen and normal tissue. If the thermal conductivity of frozen tissue is approximately 1.5 W/m
K and heat transfer at the phase front may be characterized by an effective convection coefficient of 50 W/m2
K, what is the thickness of the layer of frozen tissue (assuming negligible perfusion)?

A spherical vessel used as a reactor for producing pharmaceuticals has a 10-mm-thick stainless steel wall (k
17 W/m
K) and an inner diameter of l m. The exterior surface of the vessel is exposed to ambient air (T

25 C) for which a convection coefficient of 6 W/m2
K may be assumed. (a) During steady-state operation, an inner surface temperature of 50 C is maintained by energy generated within the reactor. What is the heat loss from the vessel? (b) If a 20-mm-thick layer of fiberglass insulation (k
0.040 W/m
K) is applied to the exterior of the vessel and the rate of thermal energy generation is unchanged, what is the inner surface temperature of the vessel?

The wall of a spherical tank of 1-m diameter contains an exothermic chemical reaction and is at 200 C when the ambient air temperature is 25 C. What thickness of urethane foam is required to reduce the exterior temperature to 40 C, assuming the convection coefficient is 20 W/m2
K for both situations? What is the percentage reduction in heat rate achieved by using the insulation?

A composite spherical shell of inner radius r1
0.25 m is constructed from lead of outer radius r2
0.30 m and AISI 302 stainless steel of outer radius r3
0.31 m. The cavity is filled with radioactive wastes that generate heat at a rate of
5 105 W/m3 . It is proposed to submerge the container in oceanic waters that are at a temperature of T

10 C and provide a uniform convection coefficient of h
500 W/m2
K at the outer surface of the container. Are there any problems associated with this proposal?

The energy transferred from the anterior chamber of the eye through the cornea varies considerably depending on whether a contact lens is worn. Treat the eye as a spherical system and assume the system to be at steady state. The convection coefficient ho is unchanged with and without the contact lens in place. The cornea and the lens cover one-third of the spherical surface area. Values of the parameters representing this situation are as follows: r1
10.2 mm r2
12.7 mm r3
16.5 mm T
,o
21 C T
,i
37 C k2
0.80 W/m
K k1
0.35 W/m
K ho
6 W/m2
K hi
12 W/m2
K (a) Construct the thermal circuits, labeling all potentials and flows for the systems excluding the contact lens and including the contact lens. Write resistance elements in terms of appropriate parameters. (b) Determine the heat loss from the anterior chamber with and without the contact lens in place. (c) Discuss the implication of your results.

The outer surface of a hollow sphere of radius r2 is subjected to a uniform heat flux q 2. The inner surface at r1 is held at a constant temperature Ts,1. (a) Develop an expression for the temperature distribution T(r) in the sphere wall in terms of q 2, Ts,1, r1, r2, and the thermal conductivity of the wall material k. (b) If the inner and outer tube radii are r1
50 mm and r2
100 mm, what heat flux q 2 is required to maintain the outer surface at Ts,2
50 C, while the inner surface is at Ts,1
20 C? The thermal conductivity of the wall material is k
10 W/m
K

A spherical shell of inner and outer radii ri and ro, respectively, is filled with a heat-generating material that provides for a uniform volumetric generation rate (W/m3 ) of . The outer surface of the shell is exposed to a fluid having a temperature T
and a convection coeffi- cient h. Obtain an expression for the steady-state temperature distribution T(r) in the shell, expressing your result in terms of ri , ro, , h, T
, and the thermal conductivity k of the shell material

A spherical tank of 3-m diameter contains a liquifiedpetroleum gas at 60 C. Insulation with a thermal conductivity of 0.06 W/m
K and thickness 250 mm is applied to the tank to reduce the heat gain. (a) Determine the radial position in the insulation layer at which the temperature is 0 C when the ambient air temperature is 20 C and the convection coeffi- cient on the outer surface is 6 W/m2
K. (b) If the insulation is pervious to moisture from the atmospheric air, what conclusions can you reach about the formation of ice in the insulation? What effect will ice formation have on heat gain to the LP gas? How could this situation be avoided?

A transistor, which may be approximated as a hemispherical heat source of radius ro
0.1 mm, is embedded in a large silicon substrate (k
125 W/m
K) and dissipates heat at a rate q. All boundaries of the silicon are maintained at an ambient temperature of T

27 C, except for the top surface, which is well insulated. Obtain a general expression for the substrate temperature distribution and evaluate the surface temperature of the heat source for q
4 W.

One modality for destroying malignant tissue involves imbedding a small spherical heat source of radius ro within the tissue and maintaining local temperatures above a critical value Tc for an extended period. Tissue that is well removed from the source may be assumed to remain at normal body temperature (Tb
37 C). Obtain a general expression for the radial temperature distribution in the tissue under steady-state conditions for which heat is dissipated at a rate q. If ro
0.5 mm, what heat rate must be supplied to maintain a tissue temperature of T  Tc
42 C in the domain 0.5 r 5 mm? The tissue thermal conductivity is approximately 0.5 W/m
K. Assume negligible perfusion

The air inside a chamber at T
,i
50 C is heated convectively with hi
20 W/m2
K by a 200-mm-thick wall having a thermal conductivity of 4 W/m
K and a uniform heat generation of 1000 W/m3 . To prevent any heat generated within the wall from being lost to the outside of the chamber at T
,o
25 C with ho
5 W/m2
K, a very thin electrical strip heater is placed on the outer wall to provide a uniform heat flux, q o. (a) Sketch the temperature distribution in the wall on T x coordinates for the condition where no heat generated within the wall is lost to the outside of the chamber. (b) What are the temperatures at the wall boundaries, T(0) and T(L), for the conditions of part (a)? (c) Determine the value of q o that must be supplied by the strip heater so that all heat generated within the wall is transferred to the inside of the chamber. (d) If the heat generation in the wall were switched off while the heat flux to the strip heater remained constant, what would be the steady-state temperature, T(0), of the outer wall surface?

Consider cylindrical and spherical shells with inner and outer surfaces at r1 and r2 maintained at uniform temperatures Ts,1 and Ts,2, respectively. If there is uniform heat generation within the shells, obtain expressions for the steady-state, one-dimensional radial distributions of the temperature, heat flux, and heat rate. Contrast your results with those summarized in Appendix C.

A plane wall of thickness 0.1 m and thermal conductivity 25 W/m
K having uniform volumetric heat generation of 0.3 MW/m3 is insulated on one side, while the other side is exposed to a fluid at 92 C. The convection heat transfer coefficient between the wall and the fluid is 500 W/m2
K. Determine the maximum temperature in the wall.

Large, cylindrical bales of hay used to feed livestock in the winter months are D
2 m in diameter and are stored end-to-end in long rows. Microbial energy generation occurs in the hay and can be excessive if the farmer bales the hay in a too-wet condition. Assuming the thermal conductivity of baled hay to be k
0.04 W/m
K, determine the maximum steady-state hay temperature for dry hay (q .
1W/m3 ), moist hay (q .
10 W/m3 ), and wet hay (q .
100 W/m3 ). Ambient conditions are T

0 C and h
25 W/m2
K

Consider the cylindrical bales of hay in Problem 3.82. It is proposed to utilize the microbial energy generation associated with wet hay to heat water. Consider a 30-mm diameter, thin-walled tube inserted lengthwise through the middle of a cylindrical bale. The tube carries water at T
,i
20 C with hi
200 W/m2
K. (a) Determine the steady-state heat transfer to the water per unit length of tube. (b) Plot the radial temperature distribution in the hay, T(r). (c) Plot the heat transfer to the water per unit length of tube for bale diameters of 0.2 m D 2 m

Consider one-dimensional conduction in a plane composite wall. The outer surfaces are exposed to a fluid at 25°C and a convection heat transfer coefficient of 1000 \(W/m^2.K\). The middle wall B experiences uniform heat generation \(\dot{q}_{\mathrm{B}}\), while there is no generation in walls A and C. The temperatures at the interfaces are \(T_1\) = 261°C and \(T_2\) = 211°C. (a) Assuming negligible contact resistance at the interfaces, determine the volumetric heat generation \(\dot{q}_{\mathrm{B}}\) and the thermal conductivity \(k_B\). (b) Plot the temperature distribution, showing its important features. (c) Consider conditions corresponding to a loss of coolant at the exposed surface of material A (h = 0). Determine \(T_1\) and \(T_2\) and plot the temperature distribution throughout the system.

Consider a plane composite wall that is composed of three materials (materials A, B, and C are arranged left to right) of thermal conductivities kA
0.24 W/m
K, kB
0.13 W/m
K, and kC
0.50 W/m
K. The thicknesses of the three sections of the wall are LA
20 mm, LB
13 mm, and LC
20 mm. A contact resistance of Rt,c
102 m2
K/W exists at the interface between materials A and B, as well as at the interface between materials B and C. The left face of the composite wall is insulated, while the right face is exposed to convective conditions characterized by h
10 W/m2
K, T

20 C. For Case 1, thermal energy is generated within material A at the rate q . A
5000 W/m3 . For Case 2, thermal energy is generated within material C at the rate q . C
5000 W/m3 . (a) Determine the maximum temperature within the composite wall under steady-state conditions for Case 1. (b) Sketch the steady-state temperature distribution on T x coordinates for Case 1. (c) Sketch the steady-state temperature distribution for Case 2 on the same T x coordinates used for Case 1.

An air heater may be fabricated by coiling Nichrome wire and passing air in cross flow over the wire. Consider a heater fabricated from wire of diameter D
1 mm, electrical resistivity e
106 
m, thermal conductivity k
25 W/m
K, and emissivity 
0.20. The heater is designed to deliver air at a temperature of T

50 C under flow conditions that provide a convection coefficient of h
250 W/m2
K for the wire. The temperature of the housing that encloses the wire and through which the air flows is Tsur
50 C. If the maximum allowable temperature of the wire is Tmax
1200 C, what is the maximum allowable electric current I? If the maximum available voltage is E
110 V, what is the corresponding length L of wire that may be used in the heater and the power rating of the heater? Hint: In your solution, assume negligible temperature variations within the wire, but after obtaining the desired results, assess the validity of this assumption.

Consider the composite wall of Example 3.7. In the Comments section, temperature distributions in the wall were determined assuming negligible contact resistance between materials A and B. Compute and plot the temperature distributions if the thermal contact resistance is

Consider uniform thermal energy generation inside a one-dimensional plane wall of thickness L with one surface held at Ts,1 and the other surface insulated. (a) Find an expression for the conduction heat flux to the cold surface and the temperature of the hot surface Ts,2, expressing your results in terms of k, q . , L, and Ts,1. (b) Compare the heat flux found in part (a) with the heat flux associated with a plane wall without energy generation whose surface temperatures are Ts,1 and Ts,

A plane wall of thickness 2L and thermal conductivity k experiences a uniform volumetric generation rate q . . As shown in the sketch for Case 1, the surface at x
L is perfectly insulated, while the other surface is maintained at a uniform, constant temperature To. For Case 2, a very thin dielectric strip is inserted at the midpoint of the wall (x
0) in order to electrically isolate the two sections, A and B. The thermal resistance of the strip is R t
0.0005 m2
K/W. The parameters associated with the wall are k
50 W/m
K, L
20 mm, and . (a) Sketch the temperature distribution for Case 1 on T  x coordinates. Describe the key features of this distribution. Identify the location of the maximum temperature in the wall and calculate this temperature. (b) Sketch the temperature distribution for Case 2 on the same T  x coordinates. Describe the key features of this distribution. (c) What is the temperature difference between the two walls at x
0 for Case 2? (d) What is the location of the maximum temperature in the composite wall of Case 2? Calculate this temperature.

A nuclear fuel element of thickness 2L is covered with a steel cladding of thickness b. Heat generated within the nuclear fuel at a rate q . is removed by a fluid at T
, which adjoins one surface and is characterized by a convection coefficient h. The other surface is well insulated, and the fuel and steel have thermal conductivities of k and ks, respectively. (a) Obtain an equation for the temperature distribution T(x) in the nuclear fuel. Express your results in terms of q . , k, L, b, ks, h, and T
. (b) Sketch the temperature distribution T(x) for the entire system

Consider the clad fuel element of Problem 3.90. (a) Using appropriate relations from Tables C.1 and C.2, obtain an expression for the temperature distribution T(x) in the fuel element. For kf
60 W/m
K, L
15 mm, b
3 mm, ks
15 W/m
K, h
10,000 W/m2
K, and T

200 C, what are the largest and smallest temperatures in the fuel element if heat is generated uniformly at a volumetric rate of
2 107 W/m3 ? What are the corresponding locations? (b) If the insulation is removed and equivalent convection conditions are maintained at each surface, what is the corresponding form of the temperature distribution in the fuel element? For the conditions of part (a), what are the largest and smallest temperatures in the fuel? What are the corresponding locations? (c) For the conditions of parts (a) and (b), plot the temperature distributions in the fuel element

In Problem 3.79 the strip heater acts to guard against heat losses from the wall to the outside, and the required heat flux q o depends on chamber operating conditions such as q . and T
,i . As a first step in designing a controller for the guard heater, compute and plot q o and T(0) as a function of q . for 200 q . 2000 W/m3 and T
,i
30, 50, and 70 C

The exposed surface (x
0) of a plane wall of thermal conductivity k is subjected to microwave radiation that causes volumetric heating to vary as (x)
o
1 x L q where q . o (W/m3 ) is a constant. The boundary at x
L is perfectly insulated, while the exposed surface is maintained at a constant temperature To. Determine the temperature distribution T(x) in terms of x, L, k, q . o, and T

A quartz window of thickness L serves as a viewing port in a furnace used for annealing steel. The inner surface (x
0) of the window is irradiated with a uniform heat flux q o due to emission from hot gases in the furnace. A fraction, , of this radiation may be assumed to be absorbed at the inner surface, while the remaining radiation is partially absorbed as it passes through the quartz. The volumetric heat generation due to this absorption may be described by an expression of the form q . (x)
(1 )q o e x where is the absorption coefficient of the quartz. Convection heat transfer occurs from the outer surface (x
L) of the window to ambient air at T
and is characterized by the convection coefficient h. Convection and radiation emission from the inner surface may be neglected, along with radiation emission from the outer surface. Determine the temperature distribution in the quartz, expressing your result in terms of the foregoing parameters.

For the conditions described in Problem 1.44, determine the temperature distribution, T(r), in the container, expressing your result in terms of q . o, ro, T
, h, and the thermal conductivity k of the radioactive wastes

A cylindrical shell of inner and outer radii, ri and ro, respectively, is filled with a heat-generating material that provides a uniform volumetric generation rate (W/m3 ) of q . . The inner surface is insulated, while the outer surface of the shell is exposed to a fluid at T
and a convection coefficient h. (a) Obtain an expression for the steady-state temperature distribution T(r) in the shell, expressing your result in terms of ri , ro, q . , h, T
, and the thermal conductivity k of the shell material. (b) Determine an expression for the heat rate, q(ro), at the outer radius of the shell in terms of q . and shell dimensions

The cross section of a long cylindrical fuel element in a nuclear reactor is shown. Energy generation occurs uniformly in the thorium fuel rod, which is of diameter D
25 mm and is wrapped in a thin aluminum cladding (a) It is proposed that, under steady-state conditions, the system operates with a generation rate of q .
7 108 W/m3 and cooling system characteristics of T

95 C and h
7000 W/m2
K. Is this proposal satisfactory? (b) Explore the effect of variations in q . and h by plotting temperature distributions T(r) for a range of parameter values. Suggest an envelope of acceptable operating conditions.

A nuclear reactor fuel element consists of a solid cylindrical pin of radius r1 and thermal conductivity kf . The fuel pin is in good contact with a cladding material of outer radius r2 and thermal conductivity kc. Consider steady-state conditions for which uniform heat generation occurs within the fuel at a volumetric rate q . and the outer surface of the cladding is exposed to a coolant that is characterized by a temperature T
and a convection coefficient h. (a) Obtain equations for the temperature distributions Tf(r) and Tc(r) in the fuel and cladding, respectively. Express your results exclusively in terms of the foregoing variables. (b) Consider a uranium oxide fuel pin for which k
2 W/m
K and r1
6 mm and cladding for which kc
25 W/m
K and r2
9 mm. If q .
2 108 W/m3 , h
2000 W/m2
K, and T

300 K, what is the maximum temperature in the fuel element? (c) Compute and plot the temperature distribution, T(r), for values of h
2000, 5000, and 10,000 W/m2
K. If the operator wishes to maintain the centerline temperature of the fuel element below 1000 K, can she do so by adjusting the coolant flow and hence the value of h?

Consider the configuration of Example 3.8, where uniform volumetric heating within a stainless steel tube is induced by an electric current and heat is transferred by convection to air flowing through the tube. The tube wall has inner and outer radii of r1
25 mm and r2
35 mm, a thermal conductivity of k
15 W/m
K, an electrical resistivity of e
0.7 106 
m, and a maximum allowable operating temperature of 1400 K. (a) Assuming the outer tube surface to be perfectly insulated and the airflow to be characterized by a temperature and convection coefficient of T
,1
400 K and h1
100 W/m2
K, determine the maximum allowable electric current I. (b) Compute and plot the radial temperature distribution in the tube wall for the electric current of part (a) and three values of h1 (100, 500, and 1000 W/m2
K). For each value of h1, determine the rate of heat transfer to the air per unit length of tube. (c) In practice, even the best of insulating materials would be unable to maintain adiabatic conditions at the outer tube surface. Consider use of a refractory insulating material of thermal conductivity k
1.0 W/m
K and neglect radiation exchange at its outer surface. For h1
100 W/m2
K and the maximum allowable current determined in part (a), compute and plot the temperature distribution in the composite wall for two values of the insulation thickness (
25 and 50 mm). The outer surface of the insulation is exposed to room air for which T
, 2
300 K and h2
25 W/m2
K. For each insulation thickness, determine the rate of heat transfer per unit tube length to the inner airflow and the ambient air.

A high-temperature, gas-cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (k 57 W/m
K) is encased in graphite (k 3 W/m
K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium temperature is T

600 K and the convection coefficient at the outer surface of the graphite is h
2000 W/m2
K. (a) If thermal energy is uniformly generated in the fuel element at a rate q .
108 W/m3 , what are the temperatures T1 and T2 at the inner and outer surfaces, respectively, of the fuel element? (b) Compute and plot the temperature distribution in the composite wall for selected values of q . . What is the maximum allowable value of q . ?

A long cylindrical rod of diameter 200 mm with thermal conductivity of 0.5 W/m
K experiences uniform volumetric heat generation of 24,000 W/m3 . The rod is encapsulated by a circular sleeve having an outer diameter of 400 mm and a thermal conductivity of 4 W/m
K. The outer surface of the sleeve is exposed to cross flow of air at 27 C with a convection coeffi- cient of 25 W/m2
K. (a) Find the temperature at the interface between the rod and sleeve and on the outer surface. (b) What is the temperature at the center of the rod?

A radioactive material of thermal conductivity k is cast as a solid sphere of radius ro and placed in a liquid bath for which the temperature T
and convection coeffi- cient h are known. Heat is uniformly generated within the solid at a volumetric rate of q . . Obtain the steadystate radial temperature distribution in the solid, expressing your result in terms of ro, q . , k, h, and T

Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuniformly according to the relation q .
q . o[1 (r/ro) 2 ] where q . is the local rate of energy generation per unit volume, q . is a constant, and ro is the radius of the container. Steadystate conditions are maintained by submerging the container in a liquid that is at T
and provides a uniform convection coefficient h. Determine the temperature distribution, T(r), in the container. Express your result in terms of q . o, ro, T
, h, and the thermal conductivity k of the radioactive wastes.

Radioactive wastes (krw
20 W/m
K) are stored in a spherical, stainless steel (kss
15 W/m
K) container of inner and outer radii equal to ri
0.5 m and ro
0.6 m. Heat is generated volumetrically within the wastes at a uniform rate of q .
105 W/m3 , and the outer surface of the container is exposed to a water flow for which h
1000 W/m2
K and T

25 C. (a) Evaluate the steady-state outer surface temperature, Ts,o. (b) Evaluate the steady-state inner surface temperature, Ts,i . (c) Obtain an expression for the temperature distribution, T(r), in the radioactive wastes. Express your result in terms of ri , Ts,i , krw, and q . . Evaluate the temperature at r
0. (d) A proposed extension of the foregoing design involves storing waste materials having the same thermal conductivity but twice the heat generation (q .
2 105 W/m3 ) in a stainless steel container of equivalent inner radius (ri
0.5 m). Safety considerations dictate that the maximum system temperature not exceed 475 C and that the container wall thickness be no less than t
0.04 m and preferably at or close to the original design (t
0.1 m). Assess the effect of varying the outside convection coefficient to a maximum achievable value of h
5000 W/m2
K (by increasing the water velocity) and the container wall thickness. Is the proposed extension feasible? If so, recommend suitable operating and design conditions for h and t, respectively.

Unique characteristics of biologically active materials such as fruits, vegetables, and other products require special care in handling. Following harvest and separation from producing plants, glucose is catabolized to produce carbon dioxide, water vapor, and heat, with attendant internal energy generation. Consider a carton of apples, each of 80-mm diameter, which is ventilated with air at 5 C and a velocity of 0.5 m/s. The corresponding value of the heat transfer coefficient is 7.5 W/m2
K. Within each apple thermal energy is uniformly generated at a total rate of 4000 J/kg
day. The density and thermal conductivity of the apple are 840 kg/m3 and 0.5 W/m
K, respectively. (a) Determine the apple center and surface temperatures. (b) For the stacked arrangement of apples within the crate, the convection coefficient depends on the velocity as h
C1V 0.425, where C1
10.1 W/m2
K
(m/s)0.425. Compute and plot the center and surface temperatures as a function of the air velocity for 0.1 V 1 m/s.

Consider the plane wall, long cylinder, and sphere shown schematically, each with the same characteristic length a, thermal conductivity k, and uniform volumetric energy generation rate q . . (a) On the same graph, plot the steady-state dimensionless temperature, [T(x or r) T(a)]/[(q . a2 )/2k], versus the dimensionless characteristic length, x/a or r/a, for each shape. (b) Which shape has the smallest temperature difference between the center and the surface? Explain this behavior by comparing the ratio of the volumeto-surface area. (c) Which shape would be preferred for use as a nuclear fuel element? Explain why.

Consider the plane wall, long cylinder, and sphere shown schematically, each with the same characteristic length a, thermal conductivity k, and uniform volumetric energy generation rate q . . (a) On the same graph, plot the steady-state dimensionless temperature, [T(x or r) T(a)]/[(q . a2 )/2k], versus the dimensionless characteristic length, x/a or r/a, for each shape. (b) Which shape has the smallest temperature difference between the center and the surface? Explain this behavior by comparing the ratio of the volumeto-surface area. (c) Which shape would be preferred for use as a nuclear fuel element? Explain why. Obtain the differential equation that determines T(r), the temperature distribution in the foil, under steady-state conditions. Solve this equation to obtain an expression relating T to qi . You may neglect radiation exchange between the foil and its surroundings

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 Tw
60 C. Under steady-state operating conditions for which the net radiation heat flux to the surface is qrad
800 W/m2 , what is the maximum temperature on the plate and the heat transfer rate per unit length of tube? Note that qrad 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

One method that is used to grow nanowires (nanotubes with solid cores) is to initially deposit a small droplet of a liquid catalyst onto a flat surface. The surface and catalyst are heated and simultaneously exposed to a higher-temperature, low-pressure gas that contains a mixture of chemical species from which the nanowire is to be formed. The catalytic liquid slowly absorbs the species from the gas through its top surface and converts these to a solid material that is deposited onto the underlying liquid-solid interface, resulting in construction of the nanowire. The liquid catalyst remains suspended at the tip of the nanowire. Consider the growth of a 15-nm-diameter silicon carbide nanowire onto a silicon carbide surface. The surface is maintained at a temperature of Ts
2400 K, and the particular liquid catalyst that is used must be maintained in the range 2400 K Tc 3000 K to perform its function. Determine the maximum length of a nanowire that may be grown for conditions characterized by h
105 W/m2
K and T

8000 K. Assume properties of the nanowire are the same as for bulk silicon carbide

Consider the manufacture of photovoltaic silicon, as described in Problem 1.42. The thin sheet of silicon is pulled from the pool of molten material very slowly and is subjected to an ambient temperature of T

527 C within the growth chamber. A convection coefficient of h
7.5 W/m2
K is associated with the exposed surfaces of the silicon sheet when it is inside the growth chamber. Calculate the maximum allowable velocity of the silicon sheet Vsi. The latent heat of fusion for silicon is hsf
1.8 106 J/kg. It can be assumed that the thermal energy released due to solidification is removed by conduction along the sheet

Copper tubing is joined to a solar collector plate of thickness t, and the working fluid maintains the temperature of the plate above the tubes at To. There is a uniform net radiation heat flux qrad to the top surface of the plate, while the bottom surface is well insulated. The top surface is also exposed to a fluid at T
that provides for a uniform convection coefficient h (b) Obtain a solution to the differential equation for appropriate boundary conditions.

A thin flat plate of length L, thickness t, and width W  L is thermally joined to two large heat sinks that are maintained at a temperature To. The bottom of the plate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniform value of q o. (a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks.

Consider the flat plate of Problem 3.112, but with the heat sinks at different temperatures, T(0)
To and T(L)
TL, and with the bottom surface no longer insulated. Convection heat transfer is now allowed to occur between this surface and a fluid at T
, with a convection coefficient h. (a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks. (c) For q o
20,000 W/m2 , To
100 C, TL
35 C, T

25 C, k
25 W/m
K, h
50 W/m2
K, L
100 mm, t
5 mm, and a plate width of W
30 mm, plot the temperature distribution and determine the sink heat rates, qx(0) and qx(L). On the same graph, plot three additional temperature distributions corresponding to changes in the following parameters, with the remaining parameters unchanged: (i) q o
30,000 W/m2 , (ii) h
200 W/m2
K, and (iii) the value of q o for which qx(0)
0 when h
200 W/m2
K

The temperature of a flowing gas is to be measured with a thermocouple junction and wire stretched between two legs of a sting, a wind tunnel test fixture. The junction is formed by butt-welding two wires of different material, as shown in the schematic. For wires of diameter D
125 m and a convection coefficient of h
700 W/m2
K, determine the minimum separation distance between the two legs of the sting, L
L1 L2, to ensure that the sting temperature does not influence the junction temperature and, in turn, invalidate the gas temperature measurement. Consider two different types of thermocouple junctions consisting of (i) copper and constantan wires and (ii) chromel and alumel wires. Evaluate the thermal conductivity of copper and constantan at T
300 K. Use kCh
19 W/m
K and kAl
29 W/m
K for the thermal conductivities of the chromel and alumel wires, respectively.

A bonding operation utilizes a laser to provide a constant heat flux, q o, across the top surface of a thin adhesivebacked, plastic film to be affixed to a metal strip as shown in the sketch. The metal strip has a thickness d
1.25 mm, and its width is large relative to that of the film. The thermophysical properties of the strip are
7850 kg/m3 , cp
435 J/kg
K, and k
60 W/m
K. The thermal resistance of the plastic film of width w1
40 mm is negligible. The upper and lower surfaces of the strip (including the plastic film) experience convection with air at 25 C and a convection coefficient of 10 W/m2
K. The strip and film are very long in the direction normal to the page. Assume the edges of the metal strip are at the air temperature (T
). (a) Derive an expression for the temperature distribution in the portion of the steel strip with the plastic film (w1/2 x w1/2). (b) If the heat flux provided by the laser is 10,000 W/m2 , determine the temperature of the plastic film at the center (x
0) and its edges (x
w1/2). (c) Plot the temperature distribution for the entire strip and point out its special features.

A thin metallic wire of thermal conductivity k, diameter D, and length 2L is annealed by passing an electrical current through the wire to induce a uniform volumetric heat generation q . . The ambient air around the wire is at a temperature T
, while the ends of the wire at x
L are also maintained at T
. Heat transfer from the wire to the air is characterized by the convection coefficient h. Obtain expressions for the following: (a) The steady-state temperature distribution T(x) along the wire, (b) The maximum wire temperature. (c) The average wire temperature

A motor draws electric power Pelec from a supply line and delivers mechanical power Pmech to a pump through a rotating copper shaft of thermal conductivity ks, length L, and diameter D. The motor is mounted on a square pad of width W, thickness t, and thermal conductivity kp. The surface of the housing exposed to ambient air at T
is of area Ah, and the corresponding convection coefficient is hh. Opposite ends of the shaft are at temperatures of Th and T
, and heat transfer from the shaft to the ambient air is characterized by the convection coefficient hs. The base of the pad is at T
. (a) Expressing your result in terms of Pelec, Pmech, ks, L, D, W, t, kp, Ah, hh, and hs, obtain an expression for (Th T
). (b) What is the value of Th if Pelec
25 kW, Pmech
15 kW, ks
400 W/m
K, L
0.5 m, D
0.05 m, W
0.7 m, t
0.05 m, kp
0.5 W/m
K, Ah
2 m2 , hh
10 W/m2
K, hs
300 W/m2
K, and T

25 C?

Consider the fuel cell stack of Problem 1.58. The t
0.42-mm-thick membranes have a nominal thermal conductivity of k
0.79 W/m
K that can be increased to keff,x
15.1 W/m
K by loading 10%, by volume, carbon nanotubes into the catalyst layers. The membrane experiences uniform volumetric energy generation at a rate of q .
10 106 W/m3 . Air at Ta
80 C provides a convection coefficient of ha
35 W/m2
K on one side of the membrane, while hydrogen at Th
80 C, hh
235 W/m2
K flows on the opposite side of the membrane. The flow channels are 2L
3 mm wide. The membrane is clamped between bipolar plates, each of which is at a temperature Tbp
80 C. (a) Derive the differential equation that governs the temperature distribution T(x) in the membrane. (b) Obtain a solution to the differential equation, assuming the membrane is at the bipolar plate temperature at x
0 and x
2L. (c) Plot the temperature distribution T(x) from x
0 to x
L for carbon nanotube loadings of 0% and 10% by volume. Comment on the ability of the carbon nanotubes to keep the membrane below its softening temperature of 85 C.

Consider a rod of diameter D, thermal conductivity k, and length 2L that is perfectly insulated over one portion of its length, L x 0, and experiences convection with a fluid (T
, h) over the other portion, 0 x L. One end is maintained at T1, while the other is separated from a heat sink at T3 by an interfacial thermal contact resistance . T = 20C h = 500 W/m2K Insulation L 0 +L x T1 T2 T3 Rod R"t,c = 4 104 m2K/W D = 5 mm L = 50 mm k = 100 W/mK R (a) Sketch the temperature distribution on T x coordinates and identify its key features. Assume that T1  T3  T
. (b) Derive an expression for the midpoint temperature T2 in terms of the thermal and geometric parameters of the system. (c) For T1
200 C, T3
100 C, and the conditions provided in the schematic, calculate T2 and plot the temperature distribution. Describe key features of the distribution and compare it to your sketch of part (a).

A carbon nanotube is suspended across a trench of width s
5 m that separates two islands, each at T

300 K. A focused laser beam irradiates the nanotube at a distance  from the left island, delivering q
10 W of energy to the nanotube. The nanotube temperature is measured at the midpoint of the trench using a point probe. The measured nanotube temperature is T1
324.5 K for 1
1.5 m and T2
326.4 K for 2
3.5 m. Determine the two contact resistances, Rt,c,L and Rt,c,R at the left and right ends of the nanotube, respectively. The experiment is performed in a vacuum with Tsur
300 K. The nanotube thermal conductivity and diameter are kcn
3100 W/m
K and D
14 nm, respectively. 3.

A probe of overall length L
200 mm and diameter D
12.5 mm is inserted through a duct wall such that a portion of its length, referred to as the immersion length Li , is in contact with the water stream whose temperature, T
,i , is to be determined. The convection coefficients over the immersion and ambient-exposed lengths are hi
1100 W/m2
K and ho
10 W/m2
K, respectively. The probe has a thermal conductivity of 177 W/m
K and is in poor thermal contact with the duct wall. (a) Derive an expression for evaluating the measurement error, Terr
Ttip T
,i , which is the difference between the tip temperature, Ttip, and the water temperature, T
,i . Hint: Define a coordinate system with the origin at the duct wall and treat the probe as two fins extending inward and outward from the duct, but having the same base temperature. Use Case A results from Table 3.4. (b) With the water and ambient air temperatures at 80 and 20 C, respectively, calculate the measurement error, Terr, as a function of immersion length for the conditions Li/L
0.225, 0.425, and 0.625. (c) Compute and plot the effects of probe thermal conductivity and water velocity (hi ) on the measurement error.

A rod of diameter D
25 mm and thermal conductivity k
60 W/m
K protrudes normally from a furnace wall that is at Tw
200 C and is covered by insulation of thickness Lins
200 mm. The rod is welded to the furnace wall and is used as a hanger for supporting instrumentation cables. To avoid damaging the cables, the temperature of the rod at its exposed surface, To, must be maintained below a specified operating limit of Tmax
100 C. The ambient air temperature is T

25 C, and the convection coefficient is h
15 W/m2
K.geometrical parameters. The rod has an exposed length Lo, and its tip is well insulated. (b) Will a rod with Lo
200 mm meet the specified operating limit? If not, what design parameters would you change? Consider another material, increasing the thickness of the insulation, and increasing the rod length. Also, consider how you might attach the base of the rod to the furnace wall as a means to reduce T

A metal rod of length 2L, diameter D, and thermal conductivity k is inserted into a perfectly insulating wall, exposing one-half of its length to an airstream that is of temperature T
and provides a convection coefficient h at the surface of the rod. An electromagnetic field induces volumetric energy generation at a uniform rate q . within the embedded portion of the rod. (a) Derive an expression for the steady-state temperature Tb at the base of the exposed half of the rod. The exposed region may be approximated as a very long fin. (b) Derive an expression for the steady-state temperature To at the end of the embedded half of the rod. (c) Using numerical values provided in the schematic, plot the temperature distribution in the rod and describe key features of the distribution. Does the rod behave as a very long fin?

A very long rod of 5-mm diameter and uniform thermal conductivity k
25 W/m
K is subjected to a heat treatment process. The center, 30-mm-long portion of the rod within the induction heating coil experiences uniform volumetric heat generation of 7.5 106 W/m3 . The unheated portions of the rod, which protrude from the heating coil on either side, experience convection with the ambient air at T

20 C and h
10 W/m2
K. Assume that there is no convection from the surface of the rod within the coil (a) Calculate the steady-state temperature To of the rod at the midpoint of the heated portion in the coil. (b) Calculate the temperature of the rod Tb at the edge of the heated portion.

From Problem 1.71, consider the wire leads connecting the transistor to the circuit board. The leads are of thermal conductivity k, thickness t, width w, and length L. One end of a lead is maintained at a temperature Tc corresponding to the transistor case, while the other end assumes the temperature Tb of the circuit board. During steady-state operation, current flow through the leads provides for uniform volumetric heating in the amount q . , while there is convection cooling to air that is at T
and maintains a convection coefficient h. (a) Derive an equation from which the temperature distribution in a wire lead may be determined. List all pertinent assumptions. (b) Determine the temperature distribution in a wire lead, expressing your results in terms of the prescribed variables.

Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at T

1200 C and maintains a convection coefficient of h
250 W/m2
K over the blade The blades, which are fabricated from Inconel, k 20 W/m
K, have a length of L
50 mm. The blade profile has a uniform cross-sectional area of Ac
6 104 m2 and a perimeter of P
110 mm. A proposed blade-cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of Tb
300 C. (a) If the maximum allowable blade temperature is 1050 C and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory? (b) For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant?

In a test to determine the friction coefficient associated with a disk brake, one disk and its shaft are rotated at a constant angular velocity , while an equivalent disk/shaft assembly is stationary. Each disk has an outer radius of r2
180 mm, a shaft radius of r1
20 mm, a thickness of t
12 mm, and a thermal conductivity of k
15 W/m
K. A known force F is applied to the system, and the corresponding torque  required to maintain rotation is measured. The disk contact pressure may be assumed to be uniform (i.e., independent of location on the interface), and the disks may be assumed to be well insulated from the surroundings. (a) Obtain an expression that may be used to evaluate from known quantities. (b) For the region r1 r r2, determine the radial temperature distribution T(r) in the disk, where T(r1)
T1 is presumed to be known. (c) Consider test conditions for which F
200 N,
40 rad/s, 
8 N
m, and T1
80 C. Evaluate the friction coefficient and the maximum disk temperature.

Consider an extended surface of rectangular cross section with heat flow in the longitudinal direction. In this problem we seek to determine conditions for which the transverse (y-direction) temperature difference within the extended surface is negligible compared to the temperature difference between the surface and the environment, such that the one-dimensional analysis of Section 3.6.1 is valid. (a) Assume that the transverse temperature distribution is parabolic and of the form where Ts(x) is the surface temperature and To(x) is the centerline temperature at any x-location. Using Fouriers law, write an expression for the conduction heat flux at the surface, , in terms of Ts and To. (b) Write an expression for the convection heat flux at the surface for the x-location. Equating the two expressions for the heat flux by conduction and convection, identify the parameter that determines the ratio (To Ts)/(Ts T
). (c) From the foregoing analysis, develop a criterion for establishing the validity of the onedimensional assumption used to model an extended surface.

A long, circular aluminum rod is attached at one end to a heated wall and transfers heat by convection to a cold fluid. (a) If the diameter of the rod is tripled, by how much would the rate of heat removal change? (b) If a copper rod of the same diameter is used in place of the aluminum, by how much would the rate of heat removal change?

A brass rod 100 mm long and 5 mm in diameter extends horizontally from a casting at 200 C. The rod is in an air environment with T

20 C and h
30 W/m2
K. What is the temperature of the rod 25, 50, and 100 mm from the casting?

The extent to which the tip condition affects the thermal performance of a fin depends on the fin geometry and thermal conductivity, as well as the convection coefficient. Consider an alloyed aluminum (k
180 W/m
K) rectangular fin of length L
10 mm, thickness t
1 mm, and width w  t. The base temperature of the fin is Tb
l00 C, and the fin is exposed to a fluid of temperature T

25 C. (a) Assuming a uniform convection coefficient of h
100 W/m2
K over the entire fin surface, determine the fin heat transfer rate per unit width qf , efficiency f , effectiveness f , thermal resistance per unit width Rt, f , and the tip temperature T(L) for Cases A and B of Table 3.4. Contrast your results with those based on an infinite fin approximation. (b) Explore the effect of variations in the convection coefficient on the heat rate for 10  h  1000 W/m2
K. Also consider the effect of such variations for a stainless steel fin (k
15 W/m
K).

A pin fin of uniform, cross-sectional area is fabricated of an aluminum alloy . The fin diameter is , and the fin is exposed to convective conditions characterized by . It is reported that the fin efficiency is . Determine the fin length and the fin effectiveness . Account for tip convection.

The extent to which the tip condition affects the thermal performance of a fin depends on the fin geometry and thermal conductivity, as well as the convection coeffi- cient. Consider an alloyed aluminum (k
180 W/m
K) rectangular fin whose base temperature is Tb
100 C. The fin is exposed to a fluid of temperature T

25 C, and a uniform convection coefficient of h
100 W/m2
K may be assumed for the fin surface. (a) For a fin of length L
10 mm, thickness t
1 mm, and width w  t, determine the fin heat transfer rate per unit width q f , efficiency f , effectiveness f , thermal resistance per unit width Rt,f, and tip temperature T(L) for Cases A and B of Table 3.4. Contrast your results with those based on an infinite fin approximation. (b) Explore the effect of variations in L on the heat rate for 3  L  50 mm. Also consider the effect of such variations for a stainless steel fin (k
15 W/m
K)

A straight fin fabricated from 2024 aluminum alloy (k
185 W/m
K) has a base thickness of t
3 mm and a length of L
15 mm. Its base temperature is Tb
100 C, and it is exposed to a fluid for which T

20 C 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.

Triangular and parabolic straight fins are subjected to the same thermal conditions as the rectangular straight fin of Problem 3.134. (a) Determine the length of a triangular fin of unit width and base thickness t
3 mm that will provide the same fin heat rate as the straight rectangular fin. Determine the ratio of the mass of the triangular straight fin to that of the rectangular straight fin. (b) Repeat part (a) for a parabolic straight fin.

Two long copper rods of diameter D
10 mm are soldered together end to end, with solder having a melting point of 650 C. The rods are in air at 25 C with a convection coefficient of 10 W/m2
K. What is the minimum power input needed to effect the soldering?

Circular copper rods of diameter D
1 mm and length L
25 mm are used to enhance heat transfer from a surface that is maintained at Ts,1
100 C. One end of the rod is attached to this surface (at x
0), while the other end (x
25 mm) is joined to a second surface, which is maintained at Ts,2
0 C. Air flowing between the surfaces (and over the rods) is also at a temperature of T

0 C, and a convection coefficient of h
100 W/m2
K is maintained. (a) What is the rate of heat transfer by convection from a single copper rod to the air? (b) What is the total rate of heat transfer from a 1 m 1 m section of the surface at 100 C, if a bundle of the rods is installed on 4-mm centers?

During the initial stages of the growth of the nanowire of Problem 3.109, a slight perturbation of the liquid catalyst droplet can cause it to be suspended on the top of the nanowire in an off-center position. The resulting nonuniform deposition of solid at the solid-liquid interface can be manipulated to form engineered shapes such as a nanospring, that is characterized by a spring radius r, spring pitch s, overall chord length Lc (length running along the spring), and end-to-end length L, as shown in the sketch. Consider a silicon carbide nanospring of diameter D
15 nm, r
30 nm, s
25 nm, and Lc
425 nm. From experiments, it is known that the average spring pitch s varies with average temperature T by the relation ds/dT
0.1 nm/K. Using this information, a student suggests that a nanoactuator can be constructed by connecting one end of the nanospring to a small heater and raising the temperature of that end of the nano spring above its initial value. Calculate the actuation distance L for conditions where h
106 W/m2
K, T

Ti
25 C, with a base temperature of Tb
50 C. If the base temperature can be controlled to within 1 C, calculate the accuracy to which the actuation distance can be controlled. Hint: Assume the spring radius does not change when the spring is heated. The overall spring length may be approximated by the formula

Consider two long, slender rods of the same diameter but different materials. One end of each rod is attached to a base surface maintained at 100 C, while the surfaces of the rods are exposed to ambient air at 20 C. By traversing the length of each rod with a thermocouple, it was observed that the temperatures of the rods were equal at the positions xA
0.15 m and xB
0.075 m, where x is measured from the base surface. If the thermal conductivity of rod A is known to be kA
70 W/m
K, determine the value of kB for rod B

A 40-mm-long, 2-mm-diameter pin fin is fabricated of an aluminum alloy (k
140 W/m
K). (a) Determine the fin heat transfer rate for Tb
50 C, T

25 C, h
1000 W/m2
K, and an adiabatic tip condition. (b) An engineer suggests that by holding the fin tip at a low temperature, the fin heat transfer rate can be increased. For T(x
L)
0 C, determine the new fin heat transfer rate. Other conditions are as in part (a). (c) Plot the temperature distribution, T(x), over the range 0 x L for the adiabatic tip case and the prescribed tip temperature case. Also show the ambient temperature in your graph. Discuss relevant features of the temperature distribution. (d) Plot the fin heat transfer rate over the range 0 h 1000 W/m2
K for the adiabatic tip case and the prescribed tip temperature case. For the prescribed tip temperature case, what would the calculated fin heat transfer rate be if Equation 3.78 were used to determine qf rather than Equation 3.76?

An experimental arrangement for measuring the thermal conductivity of solid materials involves the use of two long rods that are equivalent in every respect, except that one is fabricated from a standard material of known thermal conductivity kA while the other is fabricated from the material whose thermal conductivity kB is desired. Both rods are attached at one end to a heat source of fixed temperature Tb, are exposed to a fluid of temperature T
, and are instrumented with thermocouples to measure the temperature at a fixed distance x1 from the heat source. If the standard material is aluminum, with kA
200 W/m
K, and measurements reveal values of TA
75 C and TB
60 C at x1 for Tb
100 C and T

25 C, what is the thermal conductivity kB of the test material?

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, qf,o and qf,L, in terms of the base temperatures, To and TL. (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 To
400 K and TL
350 K, what are the corresponding maximum power dissipations if h
150 W/m2
K and T

300 K?

The fin array of Problem 3.142 is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider conditions for which the second fluid maintains equivalent temperatures at the parallel plates, To
TL, thereby establishing symmetry about the midplane of the fin array. The heat exchanger is 1 m long in the direction of the flow of air (first fluid) and 1 m wide in a direction normal to both the airflow and the fin surfaces. The length of the fin passages between adjoining parallel plates is L
8 mm, whereas the fin thermal conductivity and convection coefficient are k
200 W/m
K (aluminum) and h
150 W/m2
K, respectively. (a) If the fin thickness and pitch are t
1 mm and S
4 mm, respectively, what is the value of the thermal resistance Rt,o for a one-half section of the fin array? (b) Subject to the constraints that the fin thickness and pitch may not be less than 0.5 and 3 mm, respectively, assess the effect of changes in t and S

An isothermal silicon chip of width W
20 mm on a side is soldered to an aluminum heat sink (k
180 W/m
K) of equivalent width. The heat sink has a base thickness of Lb
3 mm and an array of rectangular fins, each of length Lf
15 mm. Airflow at T

20 C is maintained through channels formed by the fins and a cover plate, and for a convection coefficient of h
100 W/m2
K, a minimum fin spacing of 1.8 mm is dictated by limitations on the flow pressure drop. The solder joint has a thermal resistance of . Air T, h W Lb Lf t S Chip, Tc, qc Solder, Rt ",c Heat sink, k Cover plate R t, c
2 106 m2
K/W (a) Consider limitations for which the array has N
11 fins, in which case values of the fin thickness t
0.182 mm and pitch S
1.982 mm are obtained from the requirements that W
(N 1)S t and S t
1.8 mm. If the maximum allowable chip temperature is Tc
85 C, what is the corresponding value of the chip power qc? An adiabatic fin tip condition may be assumed, and airflow along the outer surfaces of the heat sink may be assumed to provide a convection coefficient equivalent to that associated with airflow through the channels. (b) With (S t) and h fixed at 1.8 mm and 100 W/m2
K, respectively, explore the effect of increasing the fin thickness by reducing the number of fins. With N
11 and S t fixed at 1.8 mm, but relaxation of the constraint on the pressure drop, explore the effect of increasing the airflow, and hence the convection coefficient. 3.1

As seen in Problem 3.109, silicon carbide nanowires of diameter D
15 nm can be grown onto a solid silicon carbide surface by carefully depositing droplets of catalyst liquid onto a flat silicon carbide substrate. Silicon carbide nanowires grow upward from the deposited drops, and if the drops are deposited in a pattern, an array of nanowire fins can be grown, forming a silicon carbide nano-heat sink. Consider finned and unfinned electronics packages in which an extremely small, 10 m 10 m electronics device is sandwiched between two d
100-nm-thick silicon carbide sheets. In both cases, the coolant is a dielectric liquid at 20 C. A heat transfer coefficient of h
1 105 W/m2
K exists on the top and bottom of the unfinned package and on all surfaces of the exposed silicon carbide fins, which are each L
300 nm long. Each nano-heat sink includes a 200 200 array of nanofins. Determine the maximum allowable heat rate that can be generated by the electronic device so that its temperature is maintained at Tt  85 C for the unfinned and finned packages.

As more and more components are placed on a single integrated circuit (chip), the amount of heat that is dissipated continues to increase. However, this increase is limited by the maximum allowable chip operating temperature, which is approximately 75 C. To maximize heat dissipation, it is proposed that a 4 4 array of copper pin fins be metallurgically joined to the outer surface of a square chip that is 12.7 mm on a side (a) Sketch the equivalent thermal circuit for the pin chipboard assembly, assuming one-dimensional, steady-state conditions and negligible contact resistance between the pins and the chip. In variable form, label appropriate resistances, temperatures, and heat rates. (b) For the conditions prescribed in Problem 3.27, what is the maximum rate at which heat can be dissipated in the chip when the pins are in place? That is, what is the value of qc for Tc
75 C? The pin diameter and length are Dp
1.5 mm and Lp
15 mm.

A homeowners wood stove is equipped with a top burner for cooking. The D
200-mm-diameter burner is fabricated of cast iron (k
65 W/m
K). The bottom (combustion) side of the burner has 8 straight fins of uniform cross section, arranged as shown in the sketch. A very thin ceramic coating (
0.95) is applied to all surfaces of the burner. The top of the burner is exposed to room conditions (Tsur,t
T
,t
20 C, ht
40 W/m2
K), while the bottom of the burner is exposed to combustion conditions (Tsur,b
T
.b
450 C, hb
50 W/m2
K). Compare the top surface temperature of the finned burner to that which would exist for a burner without fins. Hint: Use the same expression for radiation heat transfer to the bottom of the finned burner as for the burner with no fins

In Problem 3.146, the prescribed value of ho
1000 W/m2
K is large and characteristic of liquid cooling. In practice it would be far more preferable to use air cooling, for which a reasonable upper limit to the convection coefficient would be ho
250 W/m2
K. Assess the effect of changes in the pin fin geometry on the chip heat rate if the remaining conditions of Problem 3.146, including a maximum allowable chip temperature of 75 C, remain in effect. Parametric variations that may be considered include the total number of pins N in the square array, the pin diameter Dp, and the pin length Lp. However, the product N1/2Dp should not exceed 9 mm to ensure adequate airflow passage through the array. Recommend a design that enhances chip cooling.

Water is heated by submerging 50-mm-diameter, thinwalled copper tubes in a tank and passing hot combustion gases (Tg
750 K) through the tubes. To enhance heat transfer to the water, four straight fins of uniform cross section, which form a cross, are inserted in each tube. The fins are 5 mm thick and are also made of copper (k
400 W/m
K). If the tube surface temperature is Ts
350 K and the gas-side convection coefficient is hg
30 W/m2
K, what is the rate of heat transfer to the water per meter of pipe length?

As a means of enhancing heat transfer from highperformance logic chips, it is common to attach a heat sink to the chip surface in order to increase the surface area available for convection heat transfer. Because of the ease with which it may be manufactured (by taking orthogonal sawcuts in a block of material), an attractive option is to use a heat sink consisting of an array of square fins of width w on a side. The spacing between adjoining fins would be determined by the width of the sawblade, with the sum of this spacing and the fin width designated as the fin pitch S. The method by which the heat sink is joined to the chip would determine the interfacial contact resistance, Rt,c. Consider a square chip of width Wc
16 mm and conditions for which cooling is provided by a dielectric liquid with T

25 C and h
1500 W/m2
K. The heat sink is fabricated from copper (k
400 W/m
K), and its characteristic dimensions are w
0.25 mm, S
0.50 mm, L
6 mm, and Lb
3 mm. The prescribed values of w and S represent minima imposed by manufacturing constraints and the need to maintain adequate flow in the passages between fins. (a) If a metallurgical joint provides a contact resistance of Rt,c
5 106 m2
K/W and the maximum allowable chip temperature is 85 C, what is the maximum allowable chip power dissipation qc? Assume all of the heat to be transferred through the heat sink. (b) It may be possible to increase the heat dissipation by increasing w, subject to the constraint that (S w)  0.25 mm, and/or increasing L (subject to manufacturing constraints that L 10 mm). Assess the effect of such changes.

Because of the large number of devices in todays PC chips, finned heat sinks are often used to maintain the chip at an acceptable operating temperature. Two fin designs are to be evaluated, both of which have base (unfinned) area dimensions of 53 mm 57 mm. The fins are of square cross section and fabricated from an extruded aluminum alloy with a thermal conductivity of 175 W/m
K. Cooling air may be supplied at 25 C, and the maximum allowable chip temperature is 75 C. Other features of the design and operating conditions are tabulated. Determine which fin arrangement is superior. In your analysis, calculate the heat rate, efficiency, and effectiveness of a single fin, as well as the total heat rate and overall efficiency of the array. Since real estate inside the computer enclosure is important, compare the total heat rate per unit volume for the two designs

Consider design B of Problem 3.151. Over time, dust can collect in the fine grooves that separate the fins. Consider the buildup of a dust layer of thickness Ld, as shown in the sketch. Calculate and plot the total heat rate for design B for dust layers in the range 0 Ld 5 mm. The thermal conductivity of the dust can be taken as kd = 0.032 W/m
K. Include the effects of convection from the fin tip

A long rod of 20-mm diameter and a thermal conductivity of 1.5 W/m
K has a uniform internal volumetric thermal energy generation of 106 W/m3 . The rod is covered with an electrically insulating sleeve of 2-mm thickness and thermal conductivity of 0.5 W/m
K. A spider with 12 ribs and dimensions as shown in the sketch has a thermal conductivity of 175 W/m
K, and is used to support the rod and to maintain concentricity with an 80- mm-diameter tube. Air at T

25 C passes over the spider surface, and the convection coefficient is 20 W/m2
K. The outer surface of the tube is well insulated. We wish to increase volumetric heating within the rod, while not allowing its centerline temperature to exceed 100 C. Determine the impact of the following changes, which may be effected independently or concurrently: (i) increasing the air speed and hence the convection coefficient; (ii) changing the number and/or thickness of the ribs; and (iii) using an electrically nonconducting sleeve material of larger thermal conductivity (e.g., amorphous carbon or quartz). Recommend a realistic configuration that yields a significant increase in q

An air heater consists of a steel tube (k
20 W/m
K), with inner and outer radii of r1
13 mm and r2
16 mm, respectively, and eight integrally machined longitudinal fins, each of thickness t
3 mm. The fins extend to a concentric tube, which is of radius r3
40 mm and insulated on its outer surface. Water at a temperature T
,i
90 C flows through the inner tube,while air at T
,o
25 C flows through the annular region formed by the larger concentric tube. (a) Sketch the equivalent thermal circuit of the heater and relate each thermal resistance to appropriate system parameters. (b) If hi
5000 W/m2
K and ho
200 W/m2
K, what is the heat rate per unit length? (c) Assess the effect of increasing the number of fins N and/or the fin thickness t on the heat rate, subject to the constraint that Nt  50 mm

Determine the percentage increase in heat transfer associated with attaching aluminum fins of rectangular pro- file to a plane wall. The fins are 50 mm long, 0.5 mm thick, and are equally spaced at a distance of 4 mm (250 fins/m). The convection coefficient associated with the bare wall is 40 W/m2
K, while that resulting from attachment of the fins is 30 W/m2
K

Heat is uniformly generated at the rate of 2 105 W/m3 in a wall of thermal conductivity 25 W/m
K and thickness 60 mm. The wall is exposed to convection on both sides, with different heat transfer coeffi- cients and temperatures as shown. There are straight rectangular fins on the right-hand side of the wall, with dimensions as shown and thermal conductivity of 250 W/m
K. What is the maximum temperature that will occur in the wall?

Aluminum fins of triangular profile are attached to a plane wall whose surface temperature is 250 C. The fin base thickness is 2 mm, and its length is 6 mm. The system is in ambient air at a temperature of 20 C, and the surface convection coefficient is 40 W/m2
K. (a) What are the fin efficiency and effectiveness? (b) What is the heat dissipated per unit width by a single fin?

An annular aluminum fin of rectangular profile is attached to a circular tube having an outside diameter of 25 mm and a surface temperature of 250 C. The fin is 1 mm thick and 10 mm long, and the temperature and the convection coefficient associated with the adjoining fluid are 25 C and 25 W/m2
K, respectively. (a) What is the heat loss per fin? (b) If 200 such fins are spaced at 5-mm increments along the tube length, what is the heat loss per meter of tube length?

Annular aluminum fins of rectangular profile are attached to a circular tube having an outside diameter of 50 mm and an outer surface temperature of 200 C. The fins are 4 mm thick and 15 mm long. The system is in ambient air at a temperature of 20 C, and the surface convection coefficient is 40 W/m2
K. (a) What are the fin efficiency and effectiveness? (b) If there are 125 such fins per meter of tube length, what is the rate of heat transfer per unit length of tube?

It is proposed to air-cool the cylinders of a combustion chamber by joining an aluminum casing with annular fins (k
240 W/m
K) to the cylinder wall (k
50 W/m
K). The air is at 320 K and the corresponding convection coefficient is 100 W/m2
K. Although heating at the inner surface is periodic, it is reasonable to assume steady-state conditions with a time-averaged heat flux of qi
105 W/m2 . Assuming negligible contact resistance between the wall and the casing, determine the wall inner temperature Ti , the interface temperature T1, and the fin base temperature Tb. Determine these temperatures if the interface contact resistance is Rt, c
104 m2
K/W.

Consider the air-cooled combustion cylinder of Problem 3.160, but instead of imposing a uniform heat flux at the inner surface, consider conditions for which the time-averaged temperature of the combustion gases is Tg
1100 K and the corresponding convection coeffi- cient is hg
150 W/m2
K. All other conditions, including the cylinder/casing contact resistance, remain the same. Determine the heat rate per unit length of cylinder (W/m), as well as the cylinder inner temperature Ti , the interface temperatures T1,i and T1,o, and the fin base temperature Tb. Subject to the constraint that the fin gap is fixed at 
2 mm, assess the effect of increasing the fin thickness at the expense of reducing the number of fins

Heat transfer from a transistor may be enhanced by inserting it in an aluminum sleeve (k
200 W/m
K) having 12 integrally machined longitudinal fins on its outer surface. The transistor radius and height are r1
2.5 mm and H
4 mm, respectively, while the fins are of length L
r3 r2
8 mm and uniform thickness t
0.8 mm. The thickness of the sleeve base is r2 r1
1 mm, and the contact resistance of the sleeve-transistor interface is Rt,c
0.6 103 m2
K/W. Air at T

20 C flows over the fin surface, providing an approximately uniform convection coeffficient of h
30 W/m2
K. (a) When the transistor case temperature is 80 C, what is the rate of heat transfer from the sleeve? (b) Identify all of the measures that could be taken to improve design and/or operating conditions, such that heat dissipation may be increased while still maintaining a case temperature of 80 C. In words, assess the relative merits of each measure. Choose t what you believe to be the three most promising measures, and numerically assess the effect of corresponding changes in design and/or operating conditions on thermal performance.

Consider the conditions of Problem 3.149 but now allow for a tube wall thickness of 5 mm (inner and outer diameters of 50 and 60 mm), a fin-to-tube thermal contact resistance of 104 m2
K/W, and the fact that the water temperature, Tw
350 K, is known, not the tube surface temperature. The water-side convection coefficient is hw
2000 W/m2
K. Determine the rate of heat transfer per unit tube length (W/m) to the water. What would be the separate effect of each of the following design changes on the heat rate: (i) elimination of the contact resistance; (ii) increasing the number of fins from four to eight; and (iii) changing the tube wall and fin material from copper to AISI 304 stainless steel (k
20 W/m
K)?

A scheme for concurrently heating separate water and air streams involves passing them through and over an array of tubes, respectively, while the tube wall is heated electrically. To enhance gas-side heat transfer, annular fins of rectangular profile are attached to the outer tube surface. Attachment is facilitated with a dielectric adhesive that electrically isolates the fins from the current-carrying tube wall (a) Assuming uniform volumetric heat generation within the tube wall, obtain expressions for the heat rate per unit tube length (W/m) at the inner (ri ) and outer (ro) surfaces of the wall. Express your results in terms of the tube inner and outer surface temperatures, Ts,i and Ts,o, and other pertinent parameters. (b) Obtain expressions that could be used to determine Ts,i and Ts,o in terms of parameters associated with the water- and air-side conditions. (c) Consider conditions for which the water and air are at T
,i
T
,o
300 K, with corresponding convection coefficients of hi
2000 W/m2
K and ho
100 W/m2
K. Heat is uniformly dissipated in a stainless steel tube (kw
15 W/m
K), having inner and outer radii of ri
25 mm and ro
30 mm, and aluminum fins (t

2 mm, rt
55 mm) are attached to the outer surface, with Rt,c
104 m2
K/W. Determine the heat rates and temperatures at the inner and outer surfaces as a function of the rate of volumetric heating q . . The upper limit to q . will be determined by the constraints that Ts,i not exceed the boiling point of water (100 C) and Ts,o not exceed the decomposition temperature of the adhesive (250 C)

Consider the conditions of Example 3.12, except that the person is now exercising (in the air environment), which increases the metabolic heat generation rate by a factor of 8, to 5600 W/m3 . At what rate would the person have to perspire (in liters/s) to maintain the same skin temperature as in that example?

Consider the conditions of Example 3.12 for an air environment, except now the air and surroundings temperatures are both 15 C. Humans respond to cold by shivering, which increases the metabolic heat generation rate. What would the metabolic heat generation rate (per unit volume) have to be to maintain a comfortable skin temperature of 33 C under these conditions?

Consider heat transfer in a forearm, which can be approximated as a cylinder of muscle of radius 50 mm (neglecting the presence of bones), with an outer layer of skin and fat of thickness 3 mm. There is metabolic heat generation and perfusion within the muscle. The metabolic heat generation rate, perfusion rate, arterial temperature, and properties of blood, muscle, and skin/fat layer are identical to those in Example 3.12.The environment and surroundings are the same as for the air environment in Example 3.12. (a) Write the bioheat transfer equation in radial coordinates. Write the boundary conditions that express symmetry at the centerline of the forearm and specified temperature at the outer surface of the muscle. Solve the differential equation and apply the boundary conditions to find an expression for the temperature distribution. Note that the derivatives of the modified Bessel functions are given in Section 3.6.4. (b) Equate the heat flux at the outer surface of the muscle to the heat flux through the skin/fat layer and into the environment to determine the temperature at the outer surface of the muscle. (c) Find the maximum forearm temperature.

For one of the M
48 modules of Example 3.13, determine a variety of different efficiency values concerning the conversion of waste heat to electrical energy. (a) Determine the thermodynamic efficiency, therm PM
1/q1. (b) Determine the figure of merit for one module, and the thermoelectric efficiency, TE using Equation 3.128. (c) Determine the Carnot efficiency, Carnot
1 T2/T1. (d) Determine both the thermoelectric efficiency and the Carnot efficiency for the case where h1
h2 l
. (e) The energy conversion efficiency of thermoelectric devices is commonly reported by evaluating Equation 3.128, but with T
,1 and T
,2 used instead of T1 and T2, respectively. Determine the value of TE based on the inappropriate use of T
,1 and T
,2, and compare with your answers for parts (b) and (d).

One of the thermoelectric modules of Example 3.13 is installed between a hot gas at T
,1
450 C and a cold gas at T
,2
20 C. The convection coefficient associated with the flowing gases is h
h1
h2
80 W/m2
K while the electrical resistance of the load is Re,load
4 . (a) Sketch the equivalent thermal circuit and determine the electric power generated by the module for the situation where the hot and cold gases provide convective heating and cooling directly to the module (no heat sinks). (b) Two heat sinks (k
180 W/m
K; see sketch), each with a base thickness of Lb
4 mm and fin length Lf
20 mm, are soldered to the upper and lower sides of the module. The fin spacing is 3 mm, while the solder joints each have a thermal resistance of R t,c
2.5 106 m2
K/W. Each heat sink has N
11 fins, so that t
2.182 mm and S
5.182 mm, as determined from the requirements that W
(N 1)S t and S t
3 mm. Sketch the equivalent thermal circuit and determine the electric power generated by the module. Compare the electric power generated to your answer for part (a). Assume adiabatic fin tips and convection coefficients that are the same as in part (a). 3.

Thermoelectric modules have been used to generate electric power by tapping the heat generated by wood stoves. Consider the installation of the thermoelectric module of Example 3.13 on a vertical surface of a wood stove that has a surface temperature of Ts
375 C. A thermal contact resistance of R t,c
5 106 m2
K/W exists at the interface between the stove and the thermoelectric module, while the room air and walls are at T

Tsur
25 C. The exposed surface of the thermoelectric module has an emissivity of 
0.90 and is subjected to a convection coef- ficient of h
15 W/m2
K. Sketch the equivalent thermal circuit and determine the electric power generated by the module. The load electrical resistance is Re,load
3 .

The electric power generator for an orbiting satellite is composed of a long, cylindrical uranium heat source that is housed within an enclosure of square cross section. The only way for heat that is generated by the uranium to leave the enclosure is through four rows of the thermoelectric modules of Example 3.13. The thermoelectric modules generate electric power and also radiate heat into deep space characterized by Tsur
4 K. Consider the situation for which there are 20 modules in each row for a total of M
4 20
80 modules. The modules are wired in series with an electrical load of Re,load
250 , and have an emissivity of 
0.93. Determine the electric power generated for and 100 kW. Also determine the surface temperatures of the modules for the three thermal energy generation rates.

Rows of the thermoelectric modules of Example 3.13 are attached to the flat absorber plate of Problem 3.108. The rows of modules are separated by Lsep
0.5 m and the backs of the modules are cooled by water at a temperature of Tw
40 C, with h
45 W/m2
K transfer rate to the flowing water. Assume rows of 20 immediately adjacent modules, with the lengths of both the module rows and water tubing to be Lrow
20W where W
54 mm is the module dimension taken from Example 3.13. Neglect thermal contact resistances and the temperature drop across the tube wall, and assume that the high thermal conductivity tube wall creates a uniform temperature around the tube perimeter. Because of the thermal resistance provided by the thermoelectric modules, it is no longer appropriate to assume that the temperature of the absorber plate directly above a tube is equal to that of the water.

Determine the conduction heat transfer through an air layer held between two 10 mm 10 mm parallel aluminum plates. The plates are at temperatures Ts,1
305 K and Ts,2
295 K, respectively, and the air is at atmospheric pressure. Determine the conduction heat rate for plate spacings of L
1 mm, L
1 m, and L
10 nm. Assume a thermal accommodation coefficient of t
0.92.

Determine the parallel plate separation distance L, above which the thermal resistance associated with molecule-surface collisions Rt,ms is less than 1% of the resistance associated with moleculemolecule collisions, Rt,mm for (i) air between steel plates with t
0.92 and (ii) helium between clean aluminum plates with t
0.02. The gases are at atmospheric pressure, and the temperature is T
300 K.

Determine the conduction heat flux through various plane layers that are subjected to boundary temperatures of Ts,1
301 K and Ts,2
299 K at atmospheric pressure. Hint: Do not account for micro- or nanoscale effects within the solid, and assume the thermal accommodation coefficient for an aluminumair interface is t
0.92. (a) Case A: The plane layer is aluminum. Determine the heat flux q x for Ltot
600 m and Ltot
600 nm. (b) Case B: Conduction occurs through an air layer. Determine the heat flux q x for Ltot
600 m and Ltot
600 nm. (c) Case C: The composite wall is composed of air held between two aluminum sheets. Determine the heat flux q x for Ltot
600 m (with aluminum sheet thicknesses of 
40 m) and Ltot
600 nm (with aluminum sheet thicknesses of 
40 nm). (d) Case D: The composite wall is composed of 7 air layers interspersed between 8 aluminum sheets. Determine the heat flux q x for Ltot
600 m (with aluminum sheet and air layer thicknesses of 
40 m) and Ltot
600 nm (with aluminum sheet and air layer thicknesses of 
40 nm).

The Knudsen number, Kn
mfp/L, is a dimensionless parameter used to describe potential micro- or nanoscale effects. Derive an expression for the ratio of the thermal resistance due to moleculesurface collisions to the thermal resistance associated with moleculemolecule collisions, Rt,ms/Rt,mm, in terms of the Knudsen number, the thermal accommodation coefficient t, and the ratio of specific heats , for an ideal gas. Plot the critical Knudsen number, Kncrit, that is associated with Rt,ms/Rt,mm
0.01 versus t , for
1.4 and 1.67 (corresponding to air and helium, respectively). 3.1

7 A nanolaminated material is fabricated with an atomic layer deposition process, resulting in a series of stacked, alternating layers of tungsten and aluminum oxide, each layer being 
0.5 nm thick. Each tungstenaluminum oxide interface is associated with a thermal resistance of R t,i
3.85 109 m2
K/W. The theoretical values of the thermal conductivities of the thin aluminum oxide and tungsten layers are kA
1.65 W/m
K and kT
6.10 W/m
K, respectively. The properties are evaluated at T
300 K. (a) Determine the effective thermal conductivity of the nanolaminated material. Compare the value of the effective thermal conductivity to the bulk thermal conductivities of aluminum oxide and tungsten, given in Tables A.1 and A.2. (b) Determine the effective thermal conductivity of the nanolaminated material assuming that the thermal conductivities of the tungsten and aluminum oxide layers are equal to their bulk values.

Gold is commonly used in semiconductor packaging to form interconnections (also known as interconnects) that carry electrical signals between different devices in the package. In addition to being a good electrical conductor, gold interconnects are also effective at protecting the heat-generating devices to which they are attached by conducting thermal energy away from the devices to surrounding, cooler regions. Consider a thin film of gold that has a cross section of 60 nm 250 nm. (a) For an applied temperature difference of 20 C, determine the energy conducted along a 1-mlong, thin-film interconnect. Evaluate properties at 300 K. (b) Plot the lengthwise (in the 1-m direction) and spanwise (in the thinnest direction) thermal conductivities of the gold film as a function of the film thickness L for 30 L 140 nm. 2