An electric resistance heater is embedded in a long

The thermal conductivity of a sheet of rigid, extruded insulation is reported to be k
0.029 W/m
K. The measured temperature difference across a 20-mm-thick sheet of the material is T1 T2
10 C. (a) What is the heat flux through a 2 m  2 m sheet of the insulation? (b) What is the rate of heat transfer through the sheet of insulation?

The heat flux that is applied to the left face of a plane wall is . The wall is of thickness L
10 mm and of thermal conductivity k
12 W/m
K. If the surface temperatures of the wall are measured to be 50 C on the left side and 30 C on the right side, do steady-state conditions exist?

A concrete wall, which has a surface area of 20 m2 and is 0.30 m thick, separates conditioned room air from ambient air. The temperature of the inner surface of the wall is maintained at 25 C, and the thermal conductivity of the concrete is 1 W/m
K. (a) Determine the heat loss through the wall for outer surface temperatures ranging from 15 C to 38 C, which correspond to winter and summer extremes, respectively. Display your results graphically. (b) On your graph, also plot the heat loss as a function of the outer surface temperature for wall materials having thermal conductivities of 0.75 and 1.25 W/m
K. Explain the family of curves you have obtained.

The concrete slab of a basement is 11 m long, 8 m wide, and 0.20 m thick. During the winter, temperatures are nominally 17 C and 10 C at the top and bottom surfaces, respectively. If the concrete has a thermal conductivity of 1.4 W/m
K, what is the rate of heat loss through the slab? If the basement is heated by a gas furnace operating at an efficiency of f
0.90 and natural gas is priced at Cg
$0.02/MJ, what is the daily cost of the heat loss?

Consider Figure 1.3. The heat flux in the x-direction is , the thermal conductivity and wall thickness are k
2.3 W/m
K and L
20 mm, respectively, and steady-state conditions exist. Determine the value of the temperature gradient in units of K/m. What is the value of the temperature gradient in units of C/m?

The heat flux through a wood slab 50 mm thick, whose inner and outer surface temperatures are 40 and 20 C, respectively, has been determined to be 40 W/m2 . What is the thermal conductivity of the wood?

The inner and outer surface temperatures of a glass window 5 mm thick are 15 and 5 C. What is the heat loss through a 1 m  3 m window? The thermal conductivity of glass is 1.4 W/m
K.

A thermodynamic analysis of a proposed Brayton cycle gas turbine yields P
5 MW of net power production. The compressor, at an average temperature of Tc
400 C, is driven by the turbine at an average temperature of Th
1000 C by way of an L
1-m-long, d
70-mmdiameter shaft of thermal conductivity k
40 W/m
K. (a) Compare the steady-state conduction rate through the shaft connecting the hot turbine to the warm compressor to the net power predicted by the thermodynamics-based analysis. (b) A research team proposes to scale down the gas turbine of part (a), keeping all dimensions in the same proportions. The team assumes that the same hot and cold temperatures exist as in part (a) and that the net power output of the gas turbine is proportional to the overall volume of the device. Plot the ratio of the conduction through the shaft to the net power output of the turbine over the range 0.005 m  L  1 m. Is a scaled-down device with L
0.005 m feasible?

A glass window of width W
1 m and height H
2 m is 5 mm thick and has a thermal conductivity of kg
1.4 W/m
K. If the inner and outer surface temperatures of the glass are 15 C and 20 C, respectively, on a cold winter day, what is the rate of heat loss through the glass? To reduce heat loss through windows, it is customary to use a double pane construction in which adjoining panes are separated by an air space. If the spacing is 10 mm and the glass surfaces in contact with the air have temperatures of 10 C and 15 C, what is the rate of heat loss from a 1 m  2 m window? The thermal conductivity of air is ka
0.024 W/m
K.

A freezer compartment consists of a cubical cavity that is 2 m on a side. Assume the bottom to be perfectly insulated. What is the minimum thickness of styrofoam insulation (k
0.030 W/m
K) that must be applied to the top and side walls to ensure a heat load of less than 500 W, when the inner and outer surfaces are 10 and 35 C?

The heat flux that is applied to one face of a plane wall is q
20 W/m2 . The opposite face is exposed to air at temperature 30 C, with a convection heat transfer coef- ficient of 20 W/m2
K. The surface temperature of the wall exposed to air is measured and found to be 50 C. Do steady-state conditions exist? If not, is the temperature of the wall increasing or decreasing with time?

An inexpensive food and beverage container is fabricated from 25-mm-thick polystyrene (k
0.023 W/m
K) and has interior dimensions of 0.8 m  0.6 m  0.6 m. Under conditions for which an inner surface temperature of approximately 2 C is maintained by an ice-water mixture and an outer surface temperature of 20 C is maintained by the ambient, what is the heat flux through the container wall? Assuming negligible heat gain through the 0.8 m  0.6 m base of the cooler, what is the total heat load for the prescribed conditions?

What is the thickness required of a masonry wall having thermal conductivity 0.75 W/m
K if the heat rate is to be 80% of the heat rate through a composite structural wall having a thermal conductivity of 0.25 W/m
K and a thickness of 100 mm? Both walls are subjected to the same surface temperature difference

A wall is made from an inhomogeneous (nonuniform) material for which the thermal conductivity varies through the thickness according to k
ax b, where a and b are constants. The heat flux is known to be constant. Determine expressions for the temperature gradient and the temperature distribution when the surface at x
0 is at temperature T1.

The 5-mm-thick bottom of a 200-mm-diameter pan may be made from aluminum (k
240 W/m
K) or copper (k
390 W/m
K). When used to boil water, the surface of the bottom exposed to the water is nominally at 110 C. If heat is transferred from the stove to the pan at a rate of 600 W, what is the temperature of the surface in contact with the stove for each of the two materials?

A square silicon chip (k
150 W/m
K) is of width w
5 mm on a side and of thickness t
1 mm. The chip is mounted in a substrate such that its side and back surfaces are insulated, while the front surface is exposed to a coolant. If 4 W are being dissipated in circuits mounted to the back surface of the chip, what is the steady-state temperature difference between back and front surfaces?

For a boiling process such as shown in Figure 1.5c, the ambient temperature T
in Newtons law of cooling is replaced by the saturation temperature of the fluid Tsat. Consider a situation where the heat flux from the hot plate is q
20  105 W/m2 . If the fluid is water at atmospheric pressure and the convection heat transfer coefficient is hw
20  103 W/m2
K, determine the upper surface temperature of the plate, Ts,w. In an effort to minimize the surface temperature, a technician proposes replacing the water with a dielectric fluid whose saturation temperature is Tsat,d
52 C. If the heat transfer coefficient associated with the dielectric fluid is hd
3  103 W/m2
K, will the technicians plan work?

Youve experienced convection cooling if youve ever extended your hand out the window of a moving vehicle or into a flowing water stream. With the surface of your hand at a temperature of 30 C, determine the convection heat flux for (a) a vehicle speed of 35 km/h in air at 5 C with a convection coefficient of 40 W/m2
K and (b) a velocity of 0.2 m/s in a water stream at 10C with a convection coefficient of 900 W/m2
K. Which condition would feel colder? Contrast these results with a heat loss of approximately 30 W/m2 under normal room conditions.

Air at 40 C flows over a long, 25-mm-diameter cylinder with an embedded electrical heater. In a series of tests, measurements were made of the power per unit length, P , required to maintain the cylinder surface temperature at 300 C for different free stream velocities V of the air. The results are as follows: Air velocity, V (m/s) 1 2 4 8 12 Power, P (W/m) 450 658 983 1507 1963 (a) Determine the convection coefficient for each velocity, and display your results graphically. (b) Assuming the dependence of the convection coeffi- cient on the velocity to be of the form h
CVn , determine the parameters C and n from the results of part (a).

A wall has inner and outer surface temperatures of 16 and 6 C, respectively. The interior and exterior air temperatures are 20 and 5 C, respectively. The inner and outer convection heat transfer coefficients are 5 and 20 W/m2
K, respectively. Calculate the heat flux from the interior air to the wall, from the wall to the exterior air, and from the wall to the interior air. Is the wall under steady-state conditions?

An electric resistance heater is embedded in a long cylinder of diameter 30 mm. When water with a temperature of 25 C and velocity of 1 m/s flows crosswise over the cylinder, the power per unit length required to maintain the surface at a uniform temperature of 90 C is 28 kW/m. When air, also at 25 C, but with a velocity of 10 m/s is flowing, the power per unit length required to maintain the same surface temperature is 400 W/m. Calculate and compare the convection coefficients for the flows of water and air.

The free convection heat transfer coefficient on a thin hot vertical plate suspended in still air can be determined from observations of the change in plate temperature with time as it cools. Assuming the plate is isothermal and radiation exchange with its surroundings is negligible, evaluate the convection coefficient at the instant of time when the plate temperature is 225 C and the change in plate temperature with time (dT/dt) is 0.022 K/s. The ambient air temperature is 25 C and the plate measures 0.3  0.3 m with a mass of 3.75 kg and a specific heat of 2770 J/kg
K.

A transmission case measures W
0.30 m on a side and receives a power input of Pi
150 hp from the engine If the transmission efficiency is 
0.93 and airflow over the case corresponds to T

30 C and h
200 W/m2
K, what is the surface temperature of the transmission?

A cartridge electrical heater is shaped as a cylinder of length L
200 mm and outer diameter D
20 mm. Under normal operating conditions, the heater dissipates 2 kW while submerged in a water flow that is at 20 C and provides a convection heat transfer coefficient of h
5000 W/m2
K. Neglecting heat transfer from the ends of the heater, determine its surface temperature Ts. If the water flow is inadvertently terminated while the heater continues to operate, the heater surface is exposed to air that is also at 20 C but for which h
50 W/m2
K. What is the corresponding surface temperature? What are the consequences of such an event?

A common procedure for measuring the velocity of an airstream involves the insertion of an electrically heated wire (called a hot-wire anemometer) into the airflow, with the axis of the wire oriented perpendicular to the flow direction. The electrical energy dissipated in the wire is assumed to be transferred to the air by forced convection. Hence, for a prescribed electrical power, the temperature of the wire depends on the convection coef- ficient, which, in turn, depends on the velocity of the air. Consider a wire of length L
20 mm and diameter D
0.5 mm, for which a calibration of the form V
6.25  105 h2 has been determined. The velocity V and the convection coefficient h have units of m/s and W/m2
K, respectively. In an application involving air at a temperature of T

25 C, the surface temperature of the anemometer is maintained at Ts
75 C with a voltage drop of 5 V and an electric current of 0.1 A. What is the velocity of the air?

A square isothermal chip is of width w
5 mm on a side and is mounted in a substrate such that its side and back surfaces are well insulated; the front surface is exposed to the flow of a coolant at T

15 C. From reliability considerations, the chip temperature must not exceed T
85 C. If the coolant is air and the corresponding convection coefficient is h
200 W/m2
K, what is the maximum allowable chip power? If the coolant is a dielectric liquid for which h
3000 W/m2
K, what is the maximum allowable power?

The temperature controller for a clothes dryer consists of a bimetallic switch mounted on an electrical heater attached to a wall-mounted insulation pad. The switch is set to open at 70 C, the maximum dryer air temperature. To operate the dryer at a lower air temperature, sufficient power is supplied to the heater such that the switch reaches 70 C (Tset) when the air temperature T is less than Tset. If the convection heat transfer coefficient between the air and the exposed switch surface of 30 mm2 is 25 W/m2
K, how much heater power Pe is required when the desired dryer air temperature is T

50 C?

An overhead 25-m-long, uninsulated industrial steam pipe of 100-mm diameter is routed through a building whose walls and air are at 25 C. Pressurized steam maintains a pipe surface temperature of 150 C, and the coefficient associated with natural convection is h
10 W/m2
K. The surface emissivity is
0.8. (a) What is the rate of heat loss from the steam line? (b) If the steam is generated in a gas-fired boiler operating at an efficiency of f
0.90 and natural gas is priced at Cg
$0.02 per MJ, what is the annual cost of heat loss from the line?

Under conditions for which the same room temperature is maintained by a heating or cooling system, it is not uncommon for a person to feel chilled in the winter but comfortable in the summer. Provide a plausible explanation for this situation (with supporting calculations) by considering a room whose air temperature is maintained at 20 C throughout the year, while the walls of the room are nominally at 27 C and 14 C in the summer and winter, respectively. The exposed surface of a person in the room may be assumed to be at a temperature of 32 C throughout the year and to have an emissivity of 0.90. The coefficient associated with heat transfer by natural convection between the person and the room air is approximately 2 W/m2
K.

A spherical interplanetary probe of 0.5-m diameter contains electronics that dissipate 150 W. If the probe surface has an emissivity of 0.8 and the probe does not receive radiation from other surfaces, as, for example, from the sun, what is its surface temperature?

An instrumentation package has a spherical outer surface of diameter D
100 mm and emissivity
0.25. The package is placed in a large space simulation chamber whose walls are maintained at 77 K. If operation of the electronic components is restricted to the temperature range 40  T  85 C, what is the range of acceptable power dissipation for the package? Display your results graphically, showing also the effect of variations in the emissivity by considering values of 0.20 and 0.30.

Consider the conditions of Problem 1.22. However, now the plate is in a vacuum with a surrounding temperature of 25 C. What is the emissivity of the plate? What is the rate at which radiation is emitted by the surface?

If in Equation 1.9, the radiation heat transfer coefficient may be approximated as where . We wish to assess the validity of this approximation by comparing values of hr and hr,a for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum (
0.05) or black paint (
0.9), whose temperature may exceed that of the surroundings (Tsur
25 C) by 10 to 100C. Also compare your results with values of the coefficient associated with free convection in air , where h(W/m2
K)
0.98 T1/3. (b) Consider initial conditions associated with placing a workpiece at in a large furnace whose wall temperature may be varied over the range . According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error, (hr hr,a)/hr, as a function of the furnace temperature. 1

A vacuum system, as used in sputtering electrically conducting thin films on microcircuits, is comprised of a baseplate maintained by an electrical heater at 300 K and a shroud within the enclosure maintained at 77 K by a liquid-nitrogen coolant loop. The circular baseplate, insulated on the lower side, is 0.3 m in diameter and has an emissivity of 0.25. (a) How much electrical power must be provided to the baseplate heater? (b) At what rate must liquid nitrogen be supplied to the shroud if its heat of vaporization is 125 kJ/kg? (c) To reduce the liquid nitrogen consumption, it is proposed to bond a thin sheet of aluminum foil (
0.09) to the baseplate. Will this have the desired effect?

An electrical resistor is connected to a battery, as shown schematically. After a brief transient, the resistor assumes a nearly uniform, steady-state temperature of 95 C, while the battery and lead wires remain at the ambient temperature of 25 C. Neglect the electrical resistance of the lead wires. (a) Consider the resistor as a system about which a control surface is placed and Equation 1.12c is applied. Determine the corresponding values of in(W), g(W), out(W), and st(W). If a control surface is placed about the entire system, what are the values of in, g, out, and st? (b) If electrical energy is dissipated uniformly within the resistor, which is a cylinder of diameter D
60 mm and length L
250 mm, what is the volumetric heat generation rate, (W/m3 )? (c) Neglecting radiation from the resistor, what is the convection coefficient?

Pressurized water (pin
10 bar, Tin
110 C) enters the bottom of an L
10-m-long vertical tube of diameter D
100 mm at a mass flow rate of
1.5 kg/s. The tube is located inside a combustion chamber, resulting in heat transfer to the tube. Superheated steam exits the top of the tube at pout
7 bar, Tout
600 C. Determine the change in the rate at which the following quantities enter and exit the tube: (a) the combined thermal and flow work, (b) the mechanical energy, and (c) the total energy of the water. Also, (d) determine the heat transfer rate, q. Hint: Relevant properties may be obtained from a thermodynamics text

Consider the tube and inlet conditions of Problem 1.36. Heat transfer at a rate of q
3.89 MW is delivered to the tube. For an exit pressure of p
8 bar, determine (a) the temperature of the water at the outlet as well as the change in (b) combined thermal and flow work, (c) mechanical energy, and (d) total energy of the water from the inlet to the outlet of the tube. Hint: As a first estimate, neglect the change in mechanical energy in solving part (a). Relevant properties may be obtained from a thermodynamics text.

An internally reversible refrigerator has a modified coefficient of performance accounting for realistic heat transfer processes of COPm
qin W
qin qout qin
Tc,i Th,i Tc,i where qin is the refrigerator cooling rate, qout is the heat rejection rate, and is the power input. Show that COPm can be expressed in terms of the reservoir temperatures Tc and Th, the cold and hot thermal resistances Rt,c and Rt,h, and qin, as COPm
Tc qinRtot Th Tc qinRtot where Rtot
Rt,c Rt,h. Also, show that the power input may be expressed as
qin Th Tc qinRtot Tc qinRto

A household refrigerator operates with cold- and hot-temperature reservoirs of Tc
5 C and Th
25 C, respectively. When new, the cold and hot side resistances are Rc,n
0.05 K/W and Rh,n
0.04 K/W, respectively. Over time, dust accumulates on the refrigerators condenser coil, which is located behind the refrigerator, increasing the hot side resistance to Rh,d
0.1 K/W. It is desired to have a refrigerator cooling rate of qin
750 W. Using the results of Problem 1.38, determine the modified coefficient of performance and the required power input W under (a) clean and (b) dusty coil conditions.

Chips of width L
15 mm on a side are mounted to a substrate that is installed in an enclosure whose walls and air are maintained at a temperature of Tsur
25 C. The chips have an emissivity of
0.60 and a maximum allowable temperature of Ts
85 C. (a) If heat is rejected from the chips by radiation and natural convection, what is the maximum operating power of each chip? The convection coefficient depends on the chip-to-air temperature difference and may be approximated as h
C(Ts T
) 1/4, where C
4.2 W/m2
K5/4. (b) If a fan is used to maintain airflow through the enclosure and heat transfer is by forced convection, with h
250 W/m2
K, what is the maximum operating power?

Consider the transmission case of Problem 1.23, but now allow for radiation exchange with the ground/ chassis, which may be approximated as large surroundings at Tsur
30 C. If the emissivity of the case is
0.80, what is the surface temperature?

One method for growing thin silicon sheets for photovoltaic solar panels is to pass two thin strings of high melting temperature material upward through a bath of molten silicon. The silicon solidifies on the strings near the surface of the molten pool, and the solid silicon sheet is pulled slowly upward out of the pool. The silicon is replenished by supplying the molten pool with solid silicon powder. Consider a silicon sheet that is Wsi
85 mm wide and tsi
150 m thick that is pulled at a velocity of Vsi
20 mm/min. The silicon is melted by supplying electric power to the cylindrical growth chamber of height H
350 mm and diameter D
300 mm. The exposed surfaces of the growth chamber are at Ts
320 K, the corresponding convection coefficient at theexposed surface is h
8 W/m2
K, and the surface is characterized by an emissivity of s
0.9. The solid silicon powder is at Tsi,i
298 K, and the solid silicon sheet exits the chamber at Tsi,o
420 K. Both the surroundings and ambient temperatures are T

Tsur
298 K. (a) Determine the electric power, Pelec, needed to operate the system at steady state. (b) If the photovoltaic panel absorbs a time-averaged solar flux of
180 W/m2 and the panel has a conversion efficiency (the ratio of solar power absorbed to electric power produced) of 
0.20, how long must the solar panel be operated to produce enough electric energy to offset the electric energy that was consumed in its manufacture?

Heat is transferred by radiation and convection between the inner surface of the nacelle of the wind turbine of Example 1.3 and the outer surfaces of the gearbox and generator. The convection heat flux associated with the gearbox and the generator may be described by q conv,gb
h(Tgb T
) and q conv,gen
h(Tgen T
), respectively, where the ambient temperature T
 Ts (which is the nacelle temperature) and h
40 W/m2
K. The outer surfaces of both the gearbox and the generator are characterized by an emissivity of
0.9. If the surface areas of the gearbox and generator are Agb
6 m2 and Agen
4 m2 , respectively, determine their surface temperatures. 1.

Radioactive wastes are packed in a long, thin-walled cylindrical container. The wastes generate thermal energy nonuniformly according to the relation
[1 (r/ro) 2 ], where is the local rate of energy generation per unit volume, is a constant, and ro is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid that is at T
and provides a uniform convection coefficient h. Obtain an expression for the total rate at which energy is generated in a unit length of the container. Use this result to obtain an expression for the temperature Ts of the container wall.

An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface is well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density  and specific heat c of aluminum are known to be 2700 kg/m3 and 900 J/kg
K, respectively. (a) Consider conditions for which the plate is at a temperature of 25 C and its top surface is suddenly exposed to ambient air at T

20 C and to solar radiation that provides an incident flux of 900 W/m2 . The convection heat transfer coefficient between the surface and the air is h
20 W/m2
K. What is the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? (c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for 0.05   1, with all other conditions remaining as prescribed. Repeat your calculations for values of S
0.5 and 1.0, and plot the results with those obtained for S
0.8. If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation? 1

A blood warmer is to be used during the transfusion of blood to a patient. This device is to heat blood taken from the blood bank at 10 C to 37 C at a flow rate of 200 ml/min. The blood passes through tubing of length 2 m, with a rectangular cross section 6.4 mm  1.6 mm At what rate must heat be added to the blood to accomplish the required temperature increase? If the fluid originates from a large tank with nearly zero velocity and flows vertically downward for its 2-m length, estimate the magnitudes of kinetic and potential energy changes. Assume the bloods properties are similar to those of water.

Consider a carton of milk that is refrigerated at a temperature of Tm
5 C. The kitchen temperature on a hot summer day is T

30 C. If the four sides of the carton are of height and width L
200 mm and w
100 mm, respectively, determine the heat transferred to the milk carton as it sits on the kitchen counter for durations of t
10 s, 60 s, and 300 s before it is returned to the refrigerator. The convection coefficient associated with natural convection on the sides of the carton is h
10 W/m2
K. The surface emissivity is 0.90. Assume the milk carton temperature remains at 5 C during the process. Your parents have taught you the importance of refrigerating certain foods from the food safety perspective. Comment on the importance of quickly returning the milk carton to the refrigerator from an energy conservation point of view

The energy consumption associated with a home water heater has two components: (i) the energy that must be supplied to bring the temperature of groundwater to the heater storage temperature, as it is introduced to replace hot water that has been used; (ii) the energy needed to compensate for heat losses incurred while the water is stored at the prescribed temperature. In this problem, we will evaluate the first of these components for a family of four, whose daily hot water consumption is approximately 100 gal. If groundwater is available at 15 C, what is the annual energy consumption associated with heating the water to a storage temperature of 55 C? For a unit electrical power cost of $0.18/kW
h, what is the annual cost associated with supplying hot water by means of (a) electric resistance heating or (b) a heat pump having a COP of 3.

Liquid oxygen, which has a boiling point of 90 K and a latent heat of vaporization of 214 kJ/kg, is stored in a spherical container whose outer surface is of 500-mm diameter and at a temperature of 10 C. The container is housed in a laboratory whose air and walls are at 25 C. (a) If the surface emissivity is 0.20 and the heat transfer coefficient associated with free convection at the outer surface of the container is 10 W/m2
K, what is the rate, in kg/s, at which oxygen vapor must be vented from the system? (b) Moisture in the ambient air will result in frost formation on the container, causing the surface emissivity to increase. Assuming the surface temperature and convection coefficient to remain at 10 C and 10 W/m2
K, respectively, compute the oxygen evaporation rate (kg/s) as a function of surface emissivity over the range 0.2   0.94.

The emissivity of galvanized steel sheet, a common roofing material, is
0.13 at temperatures around 300 K, while its absorptivity for solar irradiation is S
0.65. Would the neighborhood cat be comfortable walking on a roof constructed of the material on a day when GS
750 W/m2 , T

16 C, and h
7 W/m2
K? Assume the bottom surface of the steel is insulated.

Three electric resistance heaters of length L
250 mm and diameter D
25 mm are submerged in a 10-gal tank of water, which is initially at 295 K. The water may be assumed to have a density and specific heat of 
990 kg/m3 and c
4180 J/kg
K. (a) If the heaters are activated, each dissipating q1
500 W, estimate the time required to bring the water to a temperature of 335 K. (b) If the natural convection coefficient is given by an expression of the form h
370 (Ts T) 1/3, where Ts and T are temperatures of the heater surface and water, respectively, what is the temperature of each heater shortly after activation and just before deactivation? Units of h and (Ts T) are W/m2  K and K, respectively. (c) If the heaters are inadvertently activated when the tank is empty, the natural convection coefficient associated with heat transfer to the ambient air at T

300 K may be approximated as h
0.70 (Ts T
) 1/3. If the temperature of the tank walls is also 300 K and the emissivity of the heater surface is
0.85, what is the surface temperature of each heater under steady-state conditions? 1.

A hair dryer may be idealized as a circular duct through which a small fan draws ambient air and within which the air is heated as it flows over a coiled electric resistance wire (a) If a dryer is designed to operate with an electric power consumption of Pelec
500 W and to heat air from an ambient temperature of Ti
20 C to a discharge temperature of To
45 C, at what volumetric flow rate should the fan operate? Heat loss from the casing to the ambient air and the surroundings may be neglected. If the duct has a diameter of D
70 mm, what is the discharge velocity Vo of the air? The density and specific heat of the air may be approximated as 
1.10 kg/m3 and cp
1007 J/kg
K, respectively. (b) Consider a dryer duct length of L
150 mm and a surface emissivity of
0.8. If the coefficient associated with heat transfer by natural convection from the casing to the ambient air is h
4 W/m2
K and the temperature of the air and the surroundings is T

Tsur
20 C, confirm that the heat loss from the casing is, in fact, negligible. The casing may be assumed to have an average surface temperature of Ts
40 C.

In one stage of an annealing process, 304 stainless steel sheet is taken from 300 K to 1250 K as it passes through an electrically heated oven at a speed of Vs
10 mm/s. The sheet thickness and width are ts
8 mm and Ws
2 m, respectively, while the height, width, and length of the oven are Ho
2 m, Wo
2.4 m, and Lo
25 m, respectively. The top and four sides of the oven are exposed to ambient air and large surroundings, each at 300 K, and the corresponding surface temperature, convection coefficient, and emissivity are Ts
350 K, h
10 W/m2
K, and s
0.8. The bottom surface of the oven is also at 350 K and rests on a 0.5-m-thick concrete pad whose base is at 300 K. Estimate the required electric power input, Pelec, to the oven.

Convection ovens operate on the principle of inducing forced convection inside the oven chamber with a fan. A small cake is to be baked in an oven when the convection feature is disabled. For this situation, the free convection coefficient associated with the cake and its pan is hfr
3 W/m2
K. The oven air and wall are at temperatures T

Tsur
180 C. Determine the heat flux delivered to the cake pan and cake batter when they are initially inserted into the oven and are at a temperature of Ti
24 C. If the convection feature is activated, the forced convection heat transfer coefficient is hfo
27 W/m2
K. What is the heat flux at the batter or pan surface when the oven is operated in the convection mode? Assume a value of 0.97 for the emissivity of the cake batter and pan.

Annealing, an important step in semiconductor materials processing, can be accomplished by rapidly heating the silicon wafer to a high temperature for a short period of time. The schematic shows a method involving the use of a hot plate operating at an elevated temperature Th. The wafer, initially at a temperature of Tw,i, is suddenly positioned at a gap separation distance L from the hot plate. The purpose of the analysis is to compare the heat fluxes by conduction through the gas within the gap and by radiation exchange between the hot plate and the cool wafer. The initial time rate of change in the temperature of the wafer, (dTw/dt)i , is also of interest. Approximating the surfaces of the hot plate and the wafer as blackbodies and assuming their diameter D to be much larger than the spacing L, the radiative heat flux may be expressed as The silicon wafer has a thickness of d
0.78 mm, a density of 2700 kg/m3 , and a specific heat of 875 J/kg
K. The thermal conductivity of the gas in the gap is 0.0436 W/m
K. (a) For Th
600 C and Tw,i
20 C, calculate the radiative heat flux and the heat flux by conduction across a gap distance of L
0.2 mm. Also determine the value of (dTw /dt)i , resulting from each of the heating modes. (b) For gap distances of 0.2, 0.5, and 1.0 mm, determine the heat fluxes and temperature-time change as a function of the hot plate temperature for 300  Th  1300 C. Display your results graphically. Comment on the relative importance of the two heat

In the thermal processing of semiconductor materials, annealing is accomplished by heating a silicon wafer according to a temperature-time recipe and then maintaining a fixed elevated temperature for a prescribed period of time. For the process tool arrangement shown as follows, the wafer is in an evacuated chamber whose walls are maintained at 27 C and within which heating lamps maintain a radiant flux at its upper surface. The wafer is 0.78 mm thick, has a thermal conductivity of 30 W/m
K, and an emissivity that equals its absorptivity to the radiant flux (
l
0.65). For
3.0  105 W/m2 , the temperature on its lower surface is measured by a radiation thermometer and found to have a value of Tw,l
997 C. To avoid warping the wafer and inducing slip planes in the crystal structure, the temperature difference across the thickness of the wafer must be less than 2 C. Is this condition being met?

A furnace for processing semiconductor materials is formed by a silicon carbide chamber that is zone-heated on the top section and cooled on the lower section. With the elevator in the lowest position, a robot arm inserts the silicon wafer on the mounting pins. In a production operation, the wafer is rapidly moved toward the hot zone to achieve the temperature-time history required for the process recipe. In this position, the top and bottom surfaces of the wafer exchange radiation with the hot and cool zones, respectively, of the chamber. The zone temperatures are Th
1500 K and Tc
330 K, and the emissivity and thickness of the wafer are
0.65 and d
0.78 mm, respectively. With the ambient gas at T

700 K, convection coefficients at the upper and lower surfaces of the wafer are 8 and 4 W/m2
K, respectively. The silicon wafer has a density of 2700 kg/m3 and a specific heat of 875 J/kg
K. (a) For an initial condition corresponding to a wafer temperature of Tw,i
300 K and the position of the wafer shown schematically, determine the corresponding time rate of change of the wafer temperature, (dTw/dt)i . (b) Determine the steady-state temperature reached by the wafer if it remains in this position. How significant is convection heat transfer for this situation? Sketch how you would expect the wafer temperature to vary as a function of vertical distance.

Single fuel cells such as the one of Example 1.5 can be scaled up by arranging them into a fuel cell stack. A stack consists of multiple electrolytic membranes that are sandwiched between electrically conducting bipolar plates. Air and hydrogen are fed to each membrane through flow channels within each bipolar plate, as shown in the sketch. With this stack arrangement, the individual fuel cells are connected in series, electrically, producing a stack voltage of Estack
N  Ec, where Ec is the voltage produced across each membrane and N is the number of membranes in the stack. The electrical current is the same for each membrane. The cell voltage, Ec, as well as the cell efficiency, increases with temperature (the air and hydrogen fed to the stack are humidified to allow operation at temperatures greater than in Example 1.5), but the membranes will fail at temperatures exceeding T 85 C. Consider L  w membranes, where L
w
100 mm, of thickness tm
0.43 mm, that each produce Ec
0.6 V at I
60 A, and
45 W of thermal energy when operating at T
80 C. The external surfaces of the stack are exposed to air at T

25 C and surroundings at Tsur
30 C, with
0.88 and h
150 W/m2
K. (a) Find the electrical power produced by a stack that is Lstack
200 mm long, for bipolar plate thickness in the range 1 mm  tbp  10 mm. Determine the total thermal energy generated by the stack. (b) Calculate the surface temperature and explain whether the stack needs to be internally heated or cooled to operate at the optimal internal temperature of 80 C for various bipolar plate thicknesses. (c) Identify how the internal stack operating temperature might be lowered or raised for a given bipolar plate thickness, and discuss design changes that would promote a more uniform temperature distribution within the stack. How would changes in the external air and surroundings temperature affect your answer? Which membrane in the stack is most likely to fail due to high operating temperature?

Consider the wind turbine of Example 1.3. To reduce the nacelle temperature to Ts
30 C, the nacelle is vented and a fan is installed to force ambient air into and out of the nacelle enclosure. What is the minimum mass flow rate of air required if the air temperature increases to the nacelle surface temperature before exiting the nacelle? The specific heat of air is 1007 J/kg
K. 1.60 Consider the conducting rod of Exa

Consider the conducting rod of Example 1.4 under steady-state conditions. As suggested in Comment 3, the temperature of the rod may be controlled by varying the speed of airflow over the rod, which, in turn, alters the convection heat transfer coefficient. To consider the effect of the convection coefficient, generate plots of T versus I for values of h
50, 100, and 250 W/m2
K. Would variations in the surface emissivity have a significant effect on the rod temperature?

A long bus bar (cylindrical rod used for making electrical connections) of diameter D is installed in a large conduit having a surface temperature of 30 C and in which the ambient air temperature is T

30 C. The electrical resistivity, e(
m), of the bar material is a function of temperature, e,o
e [1 (T To)], where e,o
0.0171
m, To
25 C, and
0.00396 K1 . The bar experiences free convection in the ambient air, and the convection coefficient depends on the bar diameter, as well as on the difference between the surface and ambient temperatures. The governing relation is of the form, h
CD0.25 (T T
) 0.25, where C
1.21 W
m1.75
K1.25. The emissivity of the bar surface is
0.85. (a) Recognizing that the electrical resistance per unit length of the bar is
e /Ac, where Ac is its cross-sectional area, calculate the current-carrying capacity of a 20-mm-diameter bus bar if its temperature is not to exceed 65 C. Compare the relative importance of heat transfer by free convection and radiation exchange. (b) To assess the trade-off between current-carrying capacity, operating temperature, and bar diameter, for diameters of 10, 20, and 40 mm, plot the bar temperature T as a function of current for the range 100  I  5000 A. Also plot the ratio of the heat transfer by convection to the total heat transfer. 1.62 A

A small sphere of reference-grade iron with a specific heat of 447 J/kg
K and a mass of 0.515 kg is suddenly immersed in a waterice mixture. Fine thermocouple wires suspend the sphere, and the temperature is observed to change from 15 to 14 C in 6.35 s. The experiment is repeated with a metallic sphere of the same diameter, but of unknown composition with a mass of 1.263 kg. If the same observed temperature change occurs in 4.59 s, what is the specific heat of the unknown material?

A 50 mm  45 mm  20 mm cell phone charger has a surface temperature of Ts
33 C when plugged into an electrical wall outlet but not in use. The surface of the charger is of emissivity
0.92 and is subject to a free convection heat transfer coefficient of h
4.5 W/m2
K. The room air and wall temperatures are T

22 C and Tsur
20 C, respectively. If electricity costs C
$0.18/kW
h, determine the daily cost of leaving the charger plugged in when not in use.

A spherical, stainless steel (AISI 302) canister is used to store reacting chemicals that provide for a uniform heat flux to its inner surface. The canister is suddenly submerged in a liquid bath of temperature T
 Ti , where Ti is the initial temperature of the canister wall. (a) Assuming negligible temperature gradients in the canister wall and a constant heat flux , develop an equation that governs the variation of the wall temperature with time during the transient process. What is the initial rate of change of the wall temperature if
105 W/m2 ? (b) What is the steady-state temperature of the wall? (c) The convection coefficient depends on the velocity associated with fluid flow over the canister and whether the wall temperature is large enough to induce boiling in the liquid. Compute and plot the steady-state temperature as a function of h for the range 100  h  10,000 W/m2
K. Is there a value of h below which operation would be unacceptable?

A freezer compartment is covered with a 2-mm-thick layer of frost at the time it malfunctions. If the compartment is in ambient air at 20 C and a coefficient of h
2 W/m2
K characterizes heat transfer by natural convection from the exposed surface of the layer, estimate the time required to completely melt the frost. 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 vertical slab of Woods metal is joined to a substrate on one surface and is melted as it is uniformly irradiated by a laser source on the opposite surface. The metal is initially at its fusion temperature of Tf
72 C, and the melt runs off by gravity as soon as it is formed. The absorptivity of the metal to the laser radiation is 1
0.4, and its latent heat of fusion is hsf
33 kJ/kg. (a) Neglecting heat transfer from the irradiated surface by convection or radiation exchange with the surroundings, determine the instantaneous rate of melting in kg/s
m2 if the laser irradiation is 5 kW/m2 . How much material is removed if irradiation is maintained for a period of 2 s? (b) Allowing for convection to ambient air, with T

20 C and h
15 W/m2
K, and radiation exchange with large surroundings (
0.4, Tsur
20 C), determine the instantaneous rate of melting during irradiation.

A photovoltaic panel of dimension 2 m  4 m is installed on the roof of a home. The panel is irradiated with a solar flux of GS
700 W/m2 , oriented normal to the top panel surface. The absorptivity of the panel to the solar irradiation is S
0.83, and the efficiency of conversion of the absorbed flux to electrical power is 
P/SGSA
0.553 0.001 K1 Tp, where Tp is the panel temperature expressed in kelvins and A is the solar panel area. Determine the electrical power generated for (a) a still summer day, in which Tsur
T

35 C, h
10 W/m2
K, and (b) a breezy winter day, for which Tsur
T

15 C, h
30 W/m2
K. The panel emissivity is
0.90. 1.6

Following the hot vacuum forming of a paper-pulp mixture, the product, an egg carton, is transported on a conveyor for 18 s toward the entrance of a gas-fired oven where it is dried to a desired final water content. Very little water evaporates during the travel time. So, to increase the productivity of the line, it is proposed that a bank of infrared radiation heaters, which provide a uniform radiant flux of 5000 W/m2 , be installed over the conveyor. The carton has an exposed area of 0.0625 m2 and a mass of 0.220 kg, 75% of which is water after the forming process. The chief engineer of your plant will approve the purchase of the heaters if they can reduce the water content by 10% of the total mass. Would you recommend the purchase? Assume the heat of vaporization of water is hfg
2400 kJ/kg.

Electronic power devices are mounted to a heat sink having an exposed surface area of 0.045 m2 and an emissivity of 0.80. When the devices dissipate a total power of 20 W and the air and surroundings are at 27 C, the average sink temperature is 42 C. What average temperature will the heat sink reach when the devices dissipate 30 W for the same environmental condition?

A computer consists of an array of five printed circuit boards (PCBs), each dissipating Pb
20 W of power. Cooling of the electronic components on a board is provided by the forced flow of air, equally distributed in passages formed by adjoining boards, and the convection coefficient associated with heat transfer from the components to the air is approximately h
200 W/m2
K. Air enters the computer console at a temperature of Ti
20 C, and flow is driven by a fan whose power consumption is Pf
25 W.(a) If the temperature rise of the airflow, (To Ti ), is not to exceed 15 C, what is the minimum allowable volumetric flow rate of the air? The density and specific heat of the air may be approximated as 
1.161 kg/m3 and cp
1007 J/kg
K, respectively. (b) The component that is most susceptible to thermal failure dissipates 1 W/cm2 of surface area. To minimize the potential for thermal failure, where should the component be installed on a PCB? What is its surface temperature at this location?

Consider a surface-mount type transistor on a circuit board whose temperature is maintained at 35 C. Air at 20 C flows over the upper surface of dimensions 4 mm  8 mm with a convection coefficient of 50 W/m2
K. Three wire leads, each of cross section 1 mm  0.25 mm and length 4 mm, conduct heat from the case to the circuit board. The gap between the case and the board is 0.2 mm (a) Assuming the case is isothermal and neglecting radiation, estimate the case temperature when 150 mW is dissipated by the transistor and (i) stagnant air or (ii) a conductive paste fills the gap. The thermal conductivities of the wire leads, air, and conductive paste are 25, 0.0263, and 0.12 W/m
K, respectively. (b) Using the conductive paste to fill the gap, we wish to determine the extent to which increased heat dissipation may be accommodated, subject to the constraint that the case temperature not exceed 40 C. Options include increasing the air speed to achieve a larger convection coefficient h and/or changing the lead wire material to one of larger thermal conductivity. Independently considering leads fabricated from materials with thermal conductivities of 200 and 400 W/m
K, compute and plot the maximum allowable heat dissipation for variations in h over the range 50  h  250 W/m2
K.

The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m2 , and the underside is perfectly insulated. The convection coefficient between the roof and the ambient air is 12 W/m2
K. (a) Neglecting radiation exchange with the surroundings, calculate the temperature of the roof under steadystate conditions if the ambient air temperature is 20 C. (b) For the same ambient air temperature, calculate the temperature of the roof if its surface emissivity is 0.8. (c) The convection coefficient depends on airflow conditions over the roof, increasing with increasing air speed. Compute and plot the roof temperature as a function of h for 2  h  200 W/m2
K.

Consider the conditions of Problem 1.22, but the surroundings temperature is 25 C and radiation exchange with the surroundings is not negligible. If the convection coefficient is 6.4 W/m2
K and the emissivity of the plate is
0.42, determine the time rate of change of the plate temperature, dT/dt, when the plate temperature is 225 C. Evaluate the heat loss by convection and the heat loss by radiation.

Most of the energy we consume as food is converted to thermal energy in the process of performing all our bodily functions and is ultimately lost as heat from our bodies. Consider a person who consumes 2100 kcal per day (note that what are commonly referred to as food calories are actually kilocalories), of which 2000 kcal is converted to thermal energy. (The remaining 100 kcal is used to do work on the environment.) The person has a surface area of 1.8 m2 and is dressed in a bathing suit. (a) The person is in a room at 20 C, with a convection heat transfer coefficient of 3 W/m2
K. At this air temperature, the person is not perspiring much. Estimate the persons average skin temperature. (b) If the temperature of the environment were 33 C, what rate of perspiration would be needed to maintain a comfortable skin temperature of 33 C?

Consider Problem 1.1. (a) If the exposed cold surface of the insulation is at T2
20 C, what is the value of the convection heat transfer coefficient on the cold side of the insulation if the surroundings temperature is Tsur
320 K, the ambient temperature is T

5 C, and the emissivity is
0.95? Express your results in units of W/m2
K and W/m2
C. (b) Using the convective heat transfer coefficient you calculated in part (a), determine the surface temperature, T2, as the emissivity of the surface is varied over the range 0.05   0.95. The hot wall temperature of the insulation remains fixed at T1
30 C. Display your results graphically.

The wall of an oven used to cure plastic parts is of thickness L
0.05 m and is exposed to large surroundings and air at its outer surface. The air and the surroundings are at 300 K. (a) If the temperature of the outer surface is 400 K and its convection coefficient and emissivity are h
20 W/m2
K and
0.8, respectively, what is the temperature of the inner surface if the wall has a thermal conductivity of k
0.7 W/m2
K? (b) Consider conditions for which the temperature of the inner surface is maintained at 600 K, while the air and large surroundings to which the outer surface is exposed are maintained at 300 K. Explore the effects of variations in k, h, and on (i) the temperature of the outer surface, (ii) the heat flux through the wall, and (iii) the heat fluxes associated with convection and radiation heat transfer from the outer surface. Specifically, compute and plot the foregoing dependent variables for parametric variations about baseline conditions of k
10 W/m
K, h
20 W/m2
K, and
0.5. The suggested ranges of the independent variables are 0.1  k  400 W/m
K, 2  h  200 W/m2
K, and 0.05   1. Discuss the physical implications of your results. Under what conditions will the temperature of the outer surface be less than 45 C, which is a reasonable upper limit to avoid burn injuries if contact is made? 1.

An experiment to determine the convection coefficient associated with airflow over the surface of a thick stainless steel casting involves the insertion of thermocouples into the casting at distances of 10 and 20 mm from the surface along a hypothetical line normal to the surface. The steel has a thermal conductivity of 15 W/m
K. If the thermocouples measure temperatures of 50 and 40 C in the steel when the air temperature is 100 C, what is the convection coefficient?

A thin electrical heating element provides a uniform heat flux q o to the outer surface of a duct through which airflows. The duct wall has a thickness of 10 mm and a thermal conductivity of 20 W/m
K. (a) At a particular location, the air temperature is 30 C and the convection heat transfer coefficient between the air and inner surface of the duct is 100 W/m2
K. What heat flux q o is required to maintain the inner surface of the duct at Ti
85 C? (b) For the conditions of part (a), what is the temperature (To) of the duct surface next to the heater? (c) With Ti
85 C, compute and plot q o and To as a function of the air-side convection coefficient h for the range 10  h  200 W/m2
K. Briefly discuss your results.

A rectangular forced air heating duct is suspended from the ceiling of a basement whose air and walls are at a temperature of T

Tsur
5 C. The duct is 15 m long, and its cross section is 350 mm  200 mm. (a) For an uninsulated duct whose average surface temperature is 50 C, estimate the rate of heat loss from the duct. The surface emissivity and convection coefficient are approximately 0.5 and 4 W/m2
K, respectively. (b) If heated air enters the duct at 58 C and a velocity of 4 m/s and the heat loss corresponds to the result of part (a), what is the outlet temperature? The density and specific heat of the air may be assumed to be 
1.10 kg/m3 and c
1008 J/kg
K, respectively.

Consider the steam pipe of Example 1.2. The facilities manager wants you to recommend methods for reducing the heat loss to the room, and two options are proposed. The first option would restrict air movement around the outer surface of the pipe and thereby reduce the convection coefficient by a factor of two. The second option would coat the outer surface of the pipe with a low emissivity (
0.4) paint. (a) Which of the foregoing options would you recommend? (b) To prepare for a presentation of your recommendation to management, generate a graph of the heat loss q as a function of the convection coefficient for 2  h  20 W/m2
K and emissivities of 0.2, 0.4, and 0.8. Comment on the relative efficacy of reducing heat losses associated with convection and radiation.

During its manufacture, plate glass at 600 C is cooled by passing air over its surface such that the convection heat transfer coefficient is h
5 W/m2
K. To prevent cracking, it is known that the temperature gradient must not exceed 15 C/mm at any point in the glass during the cooling process. If the thermal conductivity of the glass is 1.4 W/m
K and its surface emissivity is 0.8, what is the lowest temperature of the air that can initially be used for the cooling? Assume that the temperature of the air equals that of the surroundings.

The curing process of Example 1.9 involves exposure of the plate to irradiation from an infrared lamp and attendant cooling by convection and radiation exchange with the surroundings. Alternatively, in lieu of the lamp, heating may be achieved by inserting the plate in an oven whose walls (the surroundings) are maintained at an elevated temperature. (a) Consider conditions for which the oven walls are at 200C, airflow over the plate is characterized by T

20 C and h
15 W/m2
K, and the coating has an emissivity of
0.5. What is the temperature of the plate? (b) For ambient air temperatures of 20, 40, and 60 C, determine the plate temperature as a function of the oven wall temperature over the range from 150 to 250 C. Plot your results, and identify conditions for which acceptable curing temperatures between 100 and 110 C may be maintained.

The diameter and surface emissivity of an electrically heated plate are D
300 mm and
0.80, respectively. (a) Estimate the power needed to maintain a surface temperature of 200 C in a room for which the air and the walls are at 25 C. The coefficient characterizing heat transfer by natural convection depends on the surface temperature and, in units of W/m2
K, may be approximated by an expression of the form h
0.80(Ts T
) 1/3. (b) Assess the effect of surface temperature on the power requirement, as well as on the relative contributions of convection and radiation to heat transfer from the surface.

Bus bars proposed for use in a power transmission station have a rectangular cross section of height H
600 mm and width W
200 mm. The electrical resistivity, e(
m), of the bar material is a function of temperature, e
e,o[1 (T To)], where e,o
0.0828
m, To
25 C, and
0.0040 K1 . The emissivity of the bars painted surface is 0.8, and the temperature of the surroundings is 30 C. The convection coefficient between the bar and the ambient air at 30 C is 10 W/m2
K. (a) Assuming the bar has a uniform temperature T, calculate the steady-state temperature when a current of 60,000 A passes through the bar. (b) Compute and plot the steady-state temperature of the bar as a function of the convection coefficient for 10  h  100 W/m2
K. What minimum convection coefficient is required to maintain a safe-operating temperature below 120 C? Will increasing the emissivity significantly affect this result? 1.

A solar flux of 700 W/m2 is incident on a flat-plate solar collector used to heat water. The area of the collector is 3 m2 , and 90% of the solar radiation passes through the cover glass and is absorbed by the absorber plate. The remaining 10% is reflected away from the collector. Water flows through the tube passages on the back side of the absorber plate and is heated from an inlet temperature Ti to an outlet temperature To. The cover glass, operating at a temperature of 30 C, has an emissivity of 0.94 and experiences radiation exchange with the sky at 10 C. The convection coef- ficient between the cover glass and the ambient air at 25 C is 10 W/m2
K. (a) Perform an overall energy balance on the collector to obtain an expression for the rate at which useful heat is collected per unit area of the collector, q u. Determine the value of q u. (b) Calculate the temperature rise of the water, To Ti , if the flow rate is 0.01 kg/s. Assume the specific heat of the water to be 4179 J/kg
K. (c) The collector efficiency  is defined as the ratio of the useful heat collected to the rate at which solar energy is incident on the collector. What is the value of ?

In analyzing the performance of a thermal system, the engineer must be able to identify the relevant heat transfer processes. Only then can the system behavior be properly quantified. For the following systems, identify the pertinent processes, designating them by appropriately labeled arrows on a sketch of the system. Answer additional questions that appear in the problem statement. (a) Identify the heat transfer processes that determine the temperature of an asphalt pavement on a summer day. Write an energy balance for the surface of the pavement. (b) Microwave radiation is known to be transmitted by plastics, glass, and ceramics but to be absorbed by materials having polar molecules such as water. Water molecules exposed to microwave radiation align and reverse alignment with the microwave radiation at frequencies up to 109 s 1 , causing heat to be generated. Contrast cooking in a microwave oven with cooking in a conventional radiant or convection oven. In each case, what is the physical mechanism responsible for heating the food? Which oven has the greater energy utilization efficiency? Why? Microwave heating is being considered for drying clothes. How would the operation of a microwave clothes dryer differ from a conventional dryer? Which is likely to have the greater energy utilization efficiency? Why? (c) To prevent freezing of the liquid water inside the fuel cell of an automobile, the water is drained to an onboard storage tank when the automobile is not in use. (The water is transferred from the tank back to the fuel cell when the automobile is turned on.) Consider a fuel cellpowered automobile that is parked outside on a very cold evening with T

20 C. The storage tank is initially empty at Ti,t
20 C, when liquid water, at atmospheric pressure and temperature Ti,w
50 C, is introduced into the tank. The tank has a wall thickness tt and is blanketed with insulation of thickness tins. Identify the heat transfer processes that will promote freezing of the water. Will the likelihood of freezing change as the insulation thickness is modified? Will the likelihood of freezing depend on the tank walls thickness and material? Would freezing of the water be more likely if plastic (low thermal conductivity) or stainless steel (moderate thermal conductivity) tubing is used to transfer the water to and from the tank? Is there an optimal tank shape that would minimize the probability of the water freezing? Would freezing be more likely or less likely to occur if a thin sheet of aluminum foil (high thermal conductivity, low emissivity) is applied to the outside of the insulation? W (d) Your grandmother is concerned about reducing her winter heating bills. Her strategy is to loosely fit rigid polystyrene sheets of insulation over her double-pane windows right after the first freezing weather arrives in the autumn. Identify the relevant heat transfer processes on a cold winter night when the foamed insulation sheet is placed (i) on the inner surface and (ii) on the outer surface of her window. To avoid condensation damage, which configuration is preferred? Condensation on the window pane does not occur when the foamed insulation is not in place. (e) There is considerable interest in developing building materials with improved insulating qualities. The development of such materials would do much to enhance energy conservation by reducing space heating requirements. It has been suggested that superior structural and insulating qualities could be obtained by using the composite shown. The material consists of a honeycomb, with cells of square cross section, sandwiched between solid slabs. The cells are filled with air, and the slabs, as well as the honeycomb matrix, are fabricated from plastics of low thermal conductivity. For heat transfer normal to the slabs, identify all heat transfer processes pertinent to the performance of the composite. Suggest ways in which this performance could be enhanced.(g) A double-glazed, glass fire screen is inserted between a wood-burning fireplace and the interior of a room. The screen consists of two vertical glass plates that are separated by a space through which room air may flow (the space is open at the top and bottom). Identify the heat transfer processes associated with the fire screen.

In considering the following problems involving heat transfer in the natural environment (outdoors), recognize that solar radiation is comprised of long and short wavelength components. If this radiation is incident on a semitransparent medium, such as water or glass, two things will happen to the nonreflected portion of the radiation. The long wavelength component will be absorbed at the surface of the medium, whereas the short wavelength component will be transmitted by the surface. (a) The number of panes in a window can strongly influence the heat loss from a heated room to the outside ambient air. Compare the single- and double-paned units shown by identifying relevant heat transfer processes for each case. (b) In a typical flat-plate solar collector, energy is collected by a working fluid that is circulated through tubes that are in good contact with the back face of an absorber plate. The back face is insulated from the surroundings, and the absorber plate receives solar radiation on its front face, which is typically covered by one or more transparent plates. Identify the relevant heat transfer processes, first for the absorber plate with no cover plate and then for the absorber plate with a single cover plate. (c) The solar energy collector design shown in the schematic has been used for agricultural applications. Air is blown through a long duct whose cross section is in the form of an equilateral triangle. One side of the triangle is comprised of a double-paned, semitransparent cover; the other two sides are constructed from aluminum sheets painted flat black on the inside and covered on the outside with a layer of styrofoam insulation. During sunny periods, air entering the system is heated for delivery to either a greenhouse, grain drying unit, or storage system. Identify all heat transfer processes associated with the cover plates, the absorber plate(s), and the air. (d) Evacuated-tube solar collectors are capable of improved performance relative to flat-plate collectors. The design consists of an inner tube enclosed in an outer tube that is transparent to solar radiation. The annular space between the tubes is evacuated. The outer, opaque surface of the inner tube absorbs solar radiation, and a working fluid is passed through the tube to collect the solar energy. The collector design generally consists of a row of such tubes arranged in front of a reflecting panel. Identify all heat transfer processes relevant to the performance of this device.