Consider 4.51. (a) The students are each given a flat,

Consider an opaque horizontal plate that is well insulated on its back side. The irradiation on the plate is 2500 W/m2 , of which 500 W/m2 is reflected. The plate is at 227 C and has an emissive power of 1200 W/m2 . Air at 127 C flows over the plate with a heat transfer convection coefficient of 15 W/m2
K. Determine the emissivity, absorptivity, and radiosity of the plate. What is the net heat transfer rate per unit area?

A horizontal, opaque surface at a steady-state temperature of 77 C is exposed to an airflow having a free stream temperature of 27 C with a convection heat transfer coefficient of 28 W/m2
K. The emissive power of the surface is 628 W/m2 , the irradiation is 1380 W/m2 , and the reflectivity is 0.40. Determine the absorptivity of the surface. Determine the net radiation heat transfer rate for this surface. Is this heat transfer to the surface or from the surface? Determine the combined heat transfer rate for the surface. Is this heat transfer to the surface or from the surface?

The top surface of an L 5-mm-thick anodized aluminum plate is irradiated with G 1000 W/m2 while being simultaneously exposed to convection conditions characterized by h 40 W/m2
K and T
30 C. The bottom surface of the plate is insulated. For a plate temperature of 400 K as well as  0.14 and  0.76, determine the radiosity at the top plate surface, the net radiation heat flux at the top surface, and the rate at which the temperature of the plate is changing with time. 12.

A horizontal semitransparent plate is uniformly irradiated from above and below, while air at T
300 K flows over the top and bottom surfaces, providing a uniform convection heat transfer coefficient of h 40 W/m2
K. The absorptivity of the plate to the irradiation is 0.40. Under steady-state conditions measurements made with a radiation detector above the top surface indicate a radiosity (which includes transmission, as well as reflection and emission) of J  5000 W/m2 , while the plate is at a uniform temperature of T 350 K. C Determine the irradiation G and the emissivity of the plate. Is the plate gray ( ) for the prescribed conditions?

What is the irradiation at surfaces A2, A3, and A4 of Example 12.1 due to emission from A1?

Consider a small surface of area A1 104 m2 , which emits diffusely with a total, hemispherical emissive power of E1 5 104 W/m2 . (a) At what rate is this emission intercepted by a small surface of area A2 5 104 m2 , which is oriented as shown? (b) What is the irradiation G2 on A2? (c) For zenith angles of 2 0, 30, and 60 , plot G2 as a function of the separation distance for 0.25 r2 1.0 m. 12.

A furnace with an aperture of 20-mm diameter and emissive power of 3.72 105 W/m2 is used to calibrate a heat flux gage having a sensitive area of 1.6 105 m2 . (a) At what distance, measured along a normal from the aperture, should the gage be positioned to receive irradiation of 1000 W/m2 ? (b) If the gage is tilted off normal by 20 , what will be its irradiation? (c) For tilt angles of 0, 20, and 60 , plot the gage irradiation as a function of the separation distance for values ranging from 100 to 300 mm.

A small radiant source A1 emits diffusely with an intensity I1 1.2 105 W/m2
sr. The radiation detector A2 is aligned normal to the source at a distance of Lo 0.2 m. An opaque screen is positioned midway between A1 and A2 to prevent radiation from the source reaching the detector. The small surface Am is a perfectly diffuse mirror that permits radiation emitted from the source to be reflected into the detector. (a) Calculate the radiant power incident on Am due to emission from the source A1, q1lm(W). (b) Assuming that the radiant power, q1lm, is perfectly and diffusely reflected, calculate the intensity leaving Am, Im (W/m2
sr). (c) Calculate the radiant power incident on A2 due to the reflected radiation leaving Am, qml2(
W). (d) Plot the radiant power qml2 as a function of the lateral separation distance yo for the range 0 yo 0.2 m. Explain the features of the resulting curve. 1

According to its directional distribution, solar radiation incident on the earths surface may be divided into two components. The direct component consists of parallel rays incident at a fixed zenith angle , while the diffuse component consists of radiation that may be approximated as being diffusely distributed with . Consider clear sky conditions for which the direct radiation is incident at 30 , with a total flux (based on an area that is normal to the rays) of , and the total intensity of the diffuse radiation is Idif 70 W/m2
sr. What is the total solar irradiation at the earths surface?

Solar radiation incident on the earths surface may be divided into the direct and diffuse components described in Problem 12.9. Consider conditions for a day in which the intensity of the direct solar radiation is Idir 210 107 W/m2
sr in the solid angle subtended by the sun with respect to the earth, s 6.74 105 sr. The intensity of the diffuse radiation is Idif 70 W/m2
sr. (a) What is the total solar irradiation at the earths surface when the direct radiation is incident at 30 ? (b) Verify the prescribed value for s, recognizing that the diameter of the sun is 1.39 109 m and the distance between the sun and the earth is 1.496 1011 m (1 astronomical unit). 12.11

On an overcast day the directional distribution of the solar radiation incident on the earths surface may be approximated by an expression of the form Ii In cos , where In 80 W/m2
sr is the total intensity of radiation directed normal to the surface and is the zenith angle. What is the solar irradiation at the earths surface?

During radiant heat treatment of a thin-film material, its shape, which may be hemispherical (a) or spherical (b), is maintained by a relatively low air pressure (as in the case of a rubber balloon). Irradiation on the film is due to emission from a radiant heater of area Ah 0.0052 m2 , which emits diffusely with an intensity of Ie,h 169,000 W/m2
sr. (a) Obtain an expression for the irradiation on the film as a function of the zenith angle . (b) Based on the expressions derived in part (a), which shape provides the more uniform irradiation G and hence provides better quality control for the treatment process?

To initiate a process operation, an infrared motion sensor (radiation detector) is employed to determine the approach of a hot part on a conveyor system. To set the sensors amplifier discriminator, the engineer needs a relationship between the sensor output signal, S, and the position of the part on the conveyor. The sensor output signal is proportional to the rate at which radiation is incident on the sensor. (a) For Ld 1 m, at what location x1 will the sensor signal S1 be 75% of the signal corresponding to the position directly beneath the sensor, So (x 0)? (b) For values of Ld 0.8, 1.0, and 1.2 m, plot the signal ratio, S/So, versus part position, x, for signal ratios in the range from 0.2 to 1.0. Compare the xlocations for which S/So 0.75. 12

A small radiant heat source of area A1 2 104 m2 emits diffusely with an intensity I1 1000 W/m2
sr. A second small area, A2 1 104 m2 , is located as shown in the sketch. (a) Determine the irradiation of A2 for L2 0.5 m. (b) Plot the irradiation of A2 over the range 0 L2 10 m. 12.1

Determine the fraction of the total, hemispherical emissive power that leaves a diffuse surface in the directions /4 /2 and 0 .

The spectral distribution of the radiation emitted by a diffuse surface may be approximated as follows (a) What is the total emissive power? (b) What is the total intensity of the radiation emitted in the normal direction and at an angle of 30 from the normal? (c) Determine the fraction of the emissive power leaving the surface in the directions /4 /2.

Consider a 5-mm-square, diffuse surface Ao having a total emissive power of Eo 4000 W/m2 . The radiation field due to emission into the hemispherical space above the surface is diffuse, thereby providing a uniform intensity I( , ). Moreover, if the space is a nonparticipating medium (nonabsorbing, nonscattering, and nonemitting), the intensity is independent of radius for any ( , ) direction. Hence intensities at any points P1 and P2 would be equal. (a) What is the rate at which radiant energy is emitted by Ao, qemit? (b) What is the intensity Io,e of the radiation field emitted from the surface Ao? (c) Beginning with Equation 12.13 and presuming knowledge of the intensity Io,e, obtain an expression for qemit. (d) Consider the hemispherical surface located at r R1 0.5 m. Using the conservation of energy requirement, determine the rate at which radiant energy is incident on this surface due to emission from Ao. (e) Using Equation 12.10, determine the rate at which radiant energy leaving Ao is intercepted by the small area A2 located in the direction (45 , ) on the hemispherical surface. What is the irradiation on A2? (f) Repeat part (e) for the location (0 , ). Are the irradiations at the two locations equal? (g) Using Equation 12.18, determine the irradiation G1 on the hemispherical surface at r R1. Io

Assuming blackbody behavior, determine the temperature of, and the energy emitted by, areas A1 in Example 12.1 and Problems 12.8 and 12.14, as well as area Ah in Problem 12.12.

The dark surface of a ceramic stove top may be approximated as a blackbody. The burners, which are integral with the stove top, are heated from below by electric resistance heaters. (a) Consider a burner of diameter D 200 mm operating at a uniform surface temperature of Ts 250 C in ambient air at T
20 C. Without a pot or pan on the burner, what are the rates of heat loss by radiation and convection from the burner? If the efficiency associated with energy transfer from the heaters to the burners is 90%, what is the electric power requirement? At what wavelength is the spectral emission a maximum? (b) Compute and plot the effect of the burner temperature on the heat rates for 100 Ts 350 C.

The energy flux associated with solar radiation incident on the outer surface of the earths atmosphere has been accurately measured and is known to be 1368 W/m2 . The diameters of the sun and earth are 1.39 109 and 1.27 107 m, respectively, and the distance between the sun and the earth is 1.5 1011 m. (a) What is the emissive power of the sun? (b) Approximating the suns surface as black, what is its temperature? (c) At what wavelength is the spectral emissive power of the sun a maximum? (d) Assuming the earths surface to be black and the sun to be the only source of energy for the earth, estimate the earths surface temperature. 1

A small flat plate is positioned just beyond the earths atmosphere and is oriented such that the normal to the plate passes through the center of the sun. Refer to Problem 12.20 for pertinent earthsun dimensions. (a) What is the solid angle subtended by the sun about a point on the surface of the plate? (b) Determine the incident intensity, Ii , on the plate using the known value of the solar irradiation above the earths atmosphere (GS 1368 W/m2 ). (c) Sketch the incident intensity Ii as a function of the zenith angle , where is measured from the normal to the plate.

A spherical aluminum shell of inside diameter D 2 m is evacuated and is used as a radiation test chamber. If the inner surface is coated with carbon black and maintained at 600 K, what is the irradiation on a small test surface placed in the chamber? If the inner surface were not coated and maintained at 600 K, what would the irradiation be?

The extremely high temperatures needed to trigger nuclear fusion are proposed to be generated by laserirradiating a spherical pellet of deuterium and tritium fuel of diameter Dp 1.8 mm. (a) Determine the maximum fuel temperature that can be achieved by irradiating the pellet with 200 lasers, each producing a power of P 500 W. The pellet has an absorptivity  0.3 and emissivity  0.8. (b) The pellet is placed inside a cylindrical enclosure. Two laser entrance holes are located at either end of the enclosure and have a diameter of DLEH 2 mm. Determine the maximum temperature that can be generated within the enclosure. 12.

An enclosure has an inside area of 100 m2 , and its inside surface is black and is maintained at a constant temperature. A small opening in the enclosure has an area of 0.02 m2 . The radiant power emitted from this opening is 70 W. What is the temperature of the interior enclosure wall? If the interior surface is maintained at this temperature, but is now polished, what will be the value of the radiant power emitted from the opening?

Assuming the earths surface is black, estimate its temperature if the sun has an equivalent blackbody temperature of 5800 K. The diameters of the sun and earth are 1.39 109 and 1.27 107 m, respectively, and the distance between the sun and earth is 1.5 1011 m.

A proposed method for generating electricity from solar irradiation is to concentrate the irradiation into a cavity that is placed within a large container of a salt with a high melting temperature. If all heat losses are neglected, part of the solar irradiation entering the cavity is used to melt the salt while the remainder is used to power a Rankine cycle. (The salt is melted during the day and is resolidified at night in order to generate electricity around the clock.) Consider conditions for which the solar power entering the cavity is qsol 7.50 MW and the time rate of change of energy stored in the salt is . For a cavity opening of diameter Ds 1 m, determine the heat transfer to the Rankine cycle, qR. The temperature of the salt is maintained at its melting point, Tsalt Tm 1000 C. Neglect heat loss by convection and irradiation from the surroundings.

Approximations to Plancks law for the spectral emissive power are the Wien and Rayleigh-Jeans spectral distributions, which are useful for the extreme low and high limits of the product T, respectively. (a) Show that the Planck distribution will have the form when C2 /T  1 and determine the error (compared to the exact distribution) for the condition T 2898
m
K. This form is known as Wiens law. E, b) Show that the Planck distribution will have the form when C2 /T  1 and determine the error (compared to the exact distribution) for the condition T 100,000
m
K. This form is known as the Rayleigh-Jeans law.

Estimate the wavelength corresponding to maximum emission from each of the following surfaces: the sun, a tungsten filament at 2500 K, a heated metal at 1500 K, human skin at 305 K, and a cryogenically cooled metal surface at 60 K. Estimate the fraction of the solar emission that is in the following spectral regions: the ultraviolet, the visible, and the infrared.

Thermal imagers have radiation detectors that are sensitive to a spectral region and provide white-black or color images with shading to indicate relative temperature differences in the scene. The imagers, which have appearances similar to a video camcorder, have numerous applications, such as for equipment maintenance to identify overheated motors or electrical transformers and for fire-fighting service to determine the direction of fire spread and to aid search and rescue for victims. The most common operating spectral regions are 3 to 5
m and 8 to 14
m. The selection of a particular region typically depends on the temperature of the scene, although the atmospheric conditions (water vapor, smoke, etc.) may also be important. (a) Determine the band emission fractions for each of the spectral regions, 3 to 5
m and 8 to 14
m, for temperatures of 300 and 900 K. (b) Using the Tools/Radiation/Band Emission Factor feature within IHT, calculate and plot the band emission factors for each of the spectral regions for the temperature range 300 to 1000 K. Identify the temperatures at which the fractions are a maximum. What conclusions can you draw from this graph concerning the choice of an imager for an application? (c) The noise-equivalent temperature (NET) is a specification of the imager that indicates the minimum temperature change that can be resolved in the image scene. Consider imagers operating at the maximum-fraction temperatures identified in part (b). For each of these conditions, determine the sensitivity (%) required of the radiation detector in order to provide a NET of 5 C. Explain the significance of your results. Note: The sensitivity (% units) can be defined as the difference in the band emission fractions for two temperatures differing by the NET, divided by the band emission fraction at one of the temperatures.

A furnace with a long, isothermal, graphite tube of diameter D 12.5 mm is maintained at Tf 2000 K and is used as a blackbody source to calibrate heat flux gages. Traditional heat flux gages are constructed as blackened thin films with thermopiles to indicate the temperature change caused by absorption of the incident radiant power over the entire spectrum. The traditional gage of interest has a sensitive area of 5 mm2 and is mounted coaxial with the furnace centerline, but positioned at a distance of L 60 mm from the beginning of the heated section. The cool extension tube serves to shield the gage from extraneous radiation sources and to contain the inert gas required to prevent rapid oxidation of the graphite tube. (a) Calculate the heat flux (W/m2 ) on the traditional gage for this condition, assuming that the extension tube is cold relative to the furnace. (b) The traditional gage is replaced by a solid-state (photoconductive) heat flux gage of the same area, but sensitive only to the spectral region between 0.4 and 2.5
m. Calculate the radiant heat flux incident on the solid-state gage within the prescribed spectral region. (c) Calculate and plot the total heat flux and the heat flux in the prescribed spectral region for the solidstate gage as a function of furnace temperature for the range 2000 Tf 3000 K. Which gage will have an output signal that is more sensitive to changes in the furnace temperature? 1

Photovoltaic materials convert sunlight directly to electric power. Some of the photons that are incident upon the material displace electrons that are in turn collected to create an electric current. The overall effi- ciency of a photovoltaic panel, , is the ratio of electrical energy produced to the energy content of the incident radiation. The efficiency depends primarily on two properties of the photovoltaic material, (i) the band gap, which identifies the energy states of photons having the potential to be converted to electric current, and (ii) the interband gap conversion efficiency, bg, which is the fraction of the total energy of photons within the band gap that is converted to electricity. Therefore, bgFbg where Fbg is the fraction of the photon energy incident on the surface within the band gap. Photons that are either outside the materials band gap or within the band gap but not converted to electrical energy are either reflected from the panel or absorbed and converted to thermal energy. Consider a photovoltaic material with a band gap of 1.1 B 1.8 eV, where B is the energy state of a photon. The wavelength is related to the energy state of a photon by the relationship 1240 eV
nm/B. The incident solar irradiation approximates that of a blackbody at 5800 K and GS 1000 W/m2 . (a) Determine the wavelength range of solar irradiation corresponding to the band gap. (b) Determine the overall efficiency of the photovoltaic material if the interband gap efficiency is bg 0.50. (c) If half of the incident photons that are not converted to electricity are absorbed and converted to thermal energy, determine the heat absorption per unit surface area of the panel. 12

An electrically powered, ring-shaped radiant heating element is maintained at a temperature of Th 3000 K and is used in a manufacturing process to heat a small part having a surface area of Ap 0.007 m2 . The surface of the heating element may be assumed to be black. For 1 30 , 2 60 , L 3 m, and W 30 mm, what is the rate at which radiant energy emitted by the heater is incident on the part? 12.3

Isothermal furnaces with small apertures approximating a blackbody are frequently used to calibrate heat flux gages, radiation thermometers, and other radiometric devices. In such applications, it is necessary to control power to the furnace such that the variation of temperature and the spectral intensity of the aperture are within desired limits. (a) By considering the Planck spectral distribution, Equation 12.30, show that the ratio of the fractional change in the spectral intensity to the fractional change in the temperature of the furnace has the form (b) Using this relation, determine the allowable variation in temperature of the furnace operating at 2000 K to ensure that the spectral intensity at 0.65
m will not vary by more than 0.5%. What is the allowable variation at 10
m?

For materials A and B, whose spectral hemispherical emissivities vary with wavelength as shown below, how does the total, hemispherical emissivity vary with temperature? Explain briefly.

A small metal object, initially at Ti 1000 K, is cooled by radiation in a low-temperature vacuum chamber. One of two thin coatings can be applied to the object so that spectral hemispherical emissivities vary with wavelength as shown. For which coating will the object most rapidly reach a temperature of Tf 500 K?

The directional total emissivity of nonmetallic materials may be approximated as  n cos , where n is the normal emissivity. Show that the total hemispherical emissivity for such materials is 2/3 of the normal emissivity.

Consider the metallic surface of Example 12.7. Additional measurements of the spectral, hemispherical emissivity yield a spectral distribution which may be approximated as follows: (a) Determine corresponding values of the total, hemispherical emissivity  and the total emissive power E at 2000 K. (b) Plot the emissivity as a function of temperature for 500 T 3000 K. Explain the variation.

The spectral emissivity of unoxidized titanium at room temperature is well described by the expression  0.520.5 for 0.3
m 30
m. (a) Determine the emissive power associated with an unoxidized titanium surface at T 300 K. Assume the spectral emissivity is  0.1 for
30
m. (b) Determine the value of max for the emissive power of the surface in part (a). 12.39 T

The spectral, directional emissivity of a diffuse material at 2000 K has the following distribution: Determine the total, hemispherical emissivity at 2000 K. Determine the emissive power over the spectral range 0.8 to 2.5
m and for the directions 0 30 .

A diffuse surface is characterized by the spectral hemispherical emissivity distribution shown. Considering surface temperatures over the range 300 Ts 1000 K, at what temperature will the emissive power be minimized?

Consider the directionally selective surface having the directional emissivity  , as shown, Assuming that the surface is isotropic in the direction, calculate the ratio of the normal emissivity n to the hemispherical emissivity h

A sphere is suspended in air in a dark room and maintained at a uniform incandescent temperature. When first viewed with the naked eye, the sphere appears to be brighter around the rim. After several hours, however, it appears to be brighter in the center. Of what type material would you reason the sphere is made? Give plausible reasons for the nonuniformity of brightness of the sphere and for the changing appearance with time.

A proposed proximity meter is based on the physical arrangement of Problem 12.14. The sensing area of a meter that is installed on a vehicle, A2, is irradiated by a stationary warm object, A1. The sensors electrical output signal is proportional to its irradiation. (a) The object temperature and emissivity are 200 C and  0.85, respectively. Determine the distance, L2,crit, associated with the maximum sensor output signal. Assume the object is a diffuse emitter. (b) If the object emits as a nonmetallic material, the total directional emissivity may be approximated as  n cos , where n is the normal emissivity (Problem 12.36). Determine the distance L2,crit associated with the maximum sensor output signal.

Estimate the total, hemispherical emissivity  for polished stainless steel at 800 K using Equation 12.43 along with information provided in Figure 12.17. Assume that the hemispherical emissivity is equal to the normal emissivity. Perform the integration using a band calculation, by splitting the integral into five bands, each of which contains 20% of the blackbody emission at 800 K. For each band, assume the average emissivity is that associated with the median wavelength within the band m, for which half of the blackbody radiation within the band is above m (and half is below m). For example, the first band runs from 0 to 1, such that F(0l1) 0.2, and the median wavelength for the first band is chosen such that F(0lm) 0.1. Also determine the surface emissive power. 12.45 A

A radiation thermometer is a device that responds to a radiant flux within a prescribed spectral interval and is calibrated to indicate the temperature of a blackbody that produces the same flux. (a) When viewing a surface at an elevated temperature Ts and emissivity less than unity, the thermometer will indicate an apparent temperature referred to as the brightness or spectral radiance temperature T. Will T be greater than, less than, or equal to Ts? (b) Write an expression for the spectral emissive power of the surface in terms of Wiens spectral distribution (see Problem 12.27) and the spectral emissivity of the surface. Write the equivalent expression using the spectral radiance temperature of the surface and show that where represents the wavelength at which the thermometer operates. (c) Consider a radiation thermometer that responds to a spectral flux centered about the wavelength 0.65
m. What temperature will the thermometer indicate when viewing a surface with (0.65
m) 0.9 and Ts 1000 K? Verify that Wiens spectral distribution is a reasonable approximation to Plancks law for this situation. 12.4

For a prescribed wavelength , measurement of the spectral intensity I,e(, T )  I,b of radiation emitted by a diffuse surface may be used to determine the surface temperature, if the spectral emissivity  is known, or the spectral emissivity, if the temperature is known. 1 Ts (a) Defining the uncertainty of the temperature determination as dT/T, obtain an expression relating this uncertainty to that associated with the intensity measurement, dI/I. For a 10% uncertainty in the intensity measurement at 10
m, what is the uncertainty in the temperature for T 500 K? For T 1000 K? (b) Defining the uncertainty of the emissivity determination as d/, obtain an expression relating this uncertainty to that associated with the intensity measurement, dI/I. For a 10% uncertainty in the intensity measurement, what is the uncertainty in the emissivity? 12.47 Sh

Sheet steel emerging from the hot roll section of a steel mill has a temperature of 1200 K, a thickness of  3 mm, and the following distribution for the spectral, hemispherical emissivity. The density and specific heat of the steel are 7900 kg/m3 and 640 J/kg
K, respectively. What is the total, hemispherical emissivity? Accounting for emission from both sides of the sheet and neglecting conduction, convection, and radiation from the surroundings, determine the initial time rate of change of the sheet temperature (dT/dt)i . As the steel cools, it oxidizes and its total, hemispherical emissivity increases. If this increase may be correlated by an expression of the form  1200[1200 K/T (K)], how long will it take for the steel to cool from 1200 to 600 K?

A large body of nonluminous gas at a temperature of 1200 K has emission bands between 2.5 and 3.5
m and between 5 and 8
m. The effective emissivity in the first band is 0.8 and in the second 0.6. Determine the emissive power of this gas

An opaque surface with the prescribed spectral, hemispherical reflectivity distribution is subjected to the spectral irradiation shown (a) Sketch the spectral, hemispherical absorptivity distribution. (b) Determine the total irradiation on the surface. (c) Determine the radiant flux that is absorbed by the surface. (d) What is the total, hemispherical absorptivity of this surface?

A small, opaque, diffuse object at Ts 400 K is suspended in a large furnace whose interior walls are at Tf 2000 K. The walls are diffuse and gray and have an emissivity of 0.20. The spectral, hemispherical emissivity for the surface of the small object is given below. (a) Determine the total emissivity and absorptivity of the surface. (b) Evaluate the reflected radiant flux and the net radiative flux to the surface. (c) What is the spectral emissive power at 2
m? (d) What is the wavelength 1/2 for which one-half of the total radiation emitted by the surface is in the spectral region  1/2? 12.5

The spectral reflectivity distribution for white paint (Figure 12.22) can be approximated by the following stair-step function: (
m) 0.4 0.43.0
3.0  0.75 0.15 0.96 A small flat plate coated with this paint is suspended inside a large enclosure, and its temperature is maintained at 400 K. The surface of the enclosure is maintained at 3000 K and the spectral distribution of its emissivity has the following characteristics: (
m) 2.0
2.0  0.2 0.9 (a) Determine the total emissivity, , of the enclosure surface. (b) Determine the total emissivity, , and absorptivity, , of the plate.

An opaque surface, 2 m 2 m, is maintained at 400 K and is simultaneously exposed to solar irradiation with GS 1200 W/m2 . The surface is diffuse and its spectral absorptivity is  0, 0.8, 0, and 0.9 for 0 0.5
m, 0.5
m  1
m, 1
m  2
m, and
2
m, respectively. Determine the absorbed irradiation, emissive power, radiosity, and net radiation heat transfer from the surface. 12.53

Consider Problem 4.51. (a) The students are each given a flat, first-surface silver mirror with which they collectively irradiate the wooden ship at location B. The reflection from the mirror is specular, and the silvers reflectivity is 0.98. The solar irradiation of each mirror, perpendicular to the direction of the suns rays, is GS 1000 W/m2 . How many students are needed to conduct the experiment if the solar absorptivity of the wood is w 0.80 and the mirror is oriented at an angle of 45 from the direction of GS? (b) If the students are given second-surface mirrors that consist of a sheet of plain glass that has polished silver on its back side, how many students are needed to conduct the experiment? Hint: See Problem 12.62.

A diffuse, opaque surface at 700 K has spectral emissivities of  0 for 0 3
m,  0.5 for 3
m  10
m, and  0.9 for 10
m  
. A radiant flux of 1000 W/m2 , which is uniformly distributed between 1 and 6
m, is incident on the surface at an angle of 30 relative to the surface normal. GS = 10 Calculate the total radiant power from a 104 m2 area of the surface that reaches a radiation detector positioned along the normal to the area. The aperture of the detector is 105 m2 , and its distance from the surface is 1 m.

A small disk 5 mm in diameter is positioned at the center of an isothermal, hemispherical enclosure. The disk is diffuse and gray with an emissivity of 0.7 and is maintained at 900 K. The hemispherical enclosure, maintained at 300 K, has a radius of 100 mm and an emissivity of 0.85. Calculate the radiant power leaving an aperture of diameter 2 mm located on the enclosure as shown.

The spectral, hemispherical absorptivity of an opaque surface is as shown. What is the solar absorptivity, S? If it is assumed that   and that the surface is at a temperature of 340 K, what is its total, hemispherical emissivity? 1

The spectral, hemispherical absorptivity of an opaque surface and the spectral distribution of radiation incident on the surface are as shown. What is the total, hemispherical absorptivity of the surface? If it is assumed that   and that the surface is at 1000 K, what is its total, hemispherical emissivity? What is the net radiant heat flux to the surface? 1

Consider an opaque, diffuse surface for which the spectral absorptivity and irradiation are as follows: What is the total absorptivity of the surface for the prescribed irradiation? If the surface is at a temperature of 1250 K, what is its emissive power? How will the surface temperature vary with time, for the prescribed conditions?

The spectral emissivity of an opaque, diffuse surface is as shown. (a) If the surface is maintained at 1000 K, what is the total, hemispherical emissivity? (b) What is the total, hemispherical absorptivity of the surface when irradiated by large surroundings of emissivity 0.8 and temperature 1500 K? (c) What is the radiosity of the surface when it is maintained at 1000 K and subjected to the irradiation prescribed in part (b)? (d) Determine the net radiation flux into the surface for the conditions of part (c). (e) Plot each of the parameters featured in parts (a)(d) as a function of the surface temperature for 750 T 2000 K

Radiation leaves a furnace of inside surface temperature 1500 K through an aperture 20 mm in diameter. A portion of the radiation is intercepted by a detector that is 1 m from the aperture, has a surface area of 105 m2 , and is oriented as shown. If the aperture is open, what is the rate at which radiation leaving the furnace is intercepted by the detector? If the aperture is covered with a diffuse, semitransparent material of spectral transmissivity  0.8 for 2
m and  0 for
2
m, what is the rate at which radiation leaving the furnace is intercepted by the detector? 12.6

The spectral transmissivity of a 1-mm-thick layer of liquid water can be approximated as follows: (a) Liquid water can exist only below its critical temperature, Tc 647.3 K. Determine the maximum possible total transmissivity of a 1-mm-thick layer of liquid water when the water is housed in an opaque container and boiling does not occur. Assume the irradiation is that of a blackbody. (b) Determine the transmissivity of a 1-mm-thick layer of liquid water associated with melting the platinum wire used in Nukiyamas boiling experiment, as described in Section 10.3.1. (c) Determine the total transmissivity of a 1-mm-thick layer of liquid water exposed to solar irradiation. Assume the sun emits as a blackbody at Ts 5800 K.

The spectral transmissivity of plain and tinted glass can be approximated as follows: Plain glass:  0.9 0.3 2.5
m Tinted glass:  0.9 0.5 1.5
m 3 Outside the specified wavelength ranges, the spectral transmissivity is zero for both glasses. Compare the solar energy that could be transmitted through the glasses. With solar irradiation on the glasses, compare the visible radiant energy that could be transmitted.

Referring to the distribution of the spectral transmissivity of low iron glass (Figure 12.23), describe briefly what is meant by the greenhouse effect. That is, how does the glass influence energy transfer to and from the contents of a greenhouse?

The spectral absorptivity  and spectral reflectivity  for a spectrally selective, diffuse material are as shown. (a) Sketch the spectral transmissivity . (b) If solar irradiation with GS 750 W/m2 and the spectral distribution of a blackbody at 5800 K is incident on this material, determine the fractions of the irradiation that are transmitted, reflected, and absorbed by the material. (c) If the temperature of this material is 350 K, determine the emissivity . (d) Determine the net heat flux by radiation to the material. 12

Consider a large furnace with opaque, diffuse, gray walls at 3000 K having an emissivity of 0.85. A small, diffuse, spectrally selective object in the furnace is maintained at 300 K. For the specified points on the furnace wall (A) and the object (B), indicate values for , , E, G, and J.

Four diffuse surfaces having the spectral characteristics shown are at 300 K and are exposed to solar radiation. Which of the surfaces may be approximated as being gray?

Consider a material that is gray, but directionally selective with  ( , ) 0.5(1  cos ). Determine the hemispherical absorptivity when collimated solar flux irradiates the surface of the material in the direction 45 and 0 . Determine the hemispherical emissivity  of the material. 1

The spectral transmissivity of a 50-
m-thick polymer film is measured over the wavelength range 2.5
m 15
m. The spectral distribution may be approximated as  0.80 for 2.5
m 7
m,  0.05 for 7
m  13
m, and  0.55 for 13
m  15
m. Transmissivity data outside the range cannot be acquired due to limitations associated with the instrumentation. An engineer wishes to determine the total transmissivity of the film. (a) Estimate the maximum possible total transmissivity of the film associated with irradiation from a blackbody at T 30 C. (b) Estimate the minimum possible total transmissivity of the film associated with irradiation from a blackbody at T 30 C. (c) Repeat parts (a) and (b) for a blackbody at T 600 C. Energy Bala

An opaque, horizontal plate has a thickness of L 21 mm and thermal conductivity k 25 W/m
K. Water flows adjacent to the bottom of the plate and is at a temperature of T
,w 25 C. Air flows above the plate at T
,a 260 C with ha 40 W/m2
K. The top of the plate is diffuse and is irradiated with G 1450 W/m2 , of which 435 W/m2 is reflected. The steady-state top and bottom plate temperatures are Tt 43 C and Tb 35 C, respectively. Determine the transmissivity, reflectivity, absorptivity, and emissivity of the plate. Is the plate gray? What is the radiosity associated with the top of the plate? What is the convection heat transfer coefficient associated with the water flow? 12.

Two small surfaces, A and B, are placed inside an isothermal enclosure at a uniform temperature. The enclosure provides an irradiation of 6300 W/m2 to each of the surfaces, and surfaces A and B absorb incident radiation at rates of 5600 and 630 W/m2 , respectively. Consider conditions after a long time has elapsed. (a) What are the net heat fluxes for each surface? What are their temperatures? (b) Determine the absorptivity of each surface. (c) What are the emissive powers of each surface? (d) Determine the emissivity of each surface.

A diffuse surface having the following spectral characteristics is maintained at 500 K when situated in a large furnace enclosure whose walls are maintained at 1500 K: (a) Sketch the spectral distribution of the surface emissive power E and the emissive power E,b that the surface would have if it were a blackbody. (b) Neglecting convection effects, what is the net heat flux to the surface for the prescribed conditions? (c) Plot the net heat flux as a function of the surface temperature for 500 T 1000 K. On the same coordinates, plot the heat flux for a diffuse, gray surface with total emissivities of 0.4 and 0.8. (d) For the prescribed spectral distribution of , how do the total emissivity and absorptivity of the surface vary with temperature in the range 500 T 1000 K? 0

Consider an opaque, diffuse surface whose spectral reflectivity varies with wavelength as shown. The surface is at 750 K, and irradiation on one side varies with wavelength as shown. The other side of the surface is insulated. What are the total absorptivity and emissivity of the surface? What is the net radiative heat flux to the surface?

A special diffuse glass with prescribed spectral radiative properties is heated in a large oven. The walls of the oven are lined with a diffuse, gray refractory brick having an emissivity of 0.75 and are maintained at Tw 1800 K. Consider conditions for which the glass temperature is Tg 750 K. (a) What are the total transmissivity , the total reflectivity , and the total emissivity  of the glass? (b) What is the net radiative heat flux, , to the glass? (c) For oven wall temperatures of 1500, 1800, and 2000 K, plot as a function of glass temperature for 500 Tg 800 K.

The 50-mm peephole of a large furnace operating at 450 C is covered with a material having  0.8 and 0 for irradiation originating from the furnace. The material has an emissivity of 0.8 and is opaque to irradiation from a source at room temperature. The outer surface of the cover is exposed to surroundings and ambient air at 27 C with a convection heat transfer coefficient of 50 W/m2
K. Assuming that convection effects on the inner surface of the cover are negligible, calculate the heat loss by the furnace and the temperature of the cover.

The window of a large vacuum chamber is fabricated from a material of prescribed spectral characteristics. A collimated beam of radiant energy from a solar simulator is incident on the window and has a flux of 3000 W/m2 . The inside walls of the chamber, which are large compared to the window area, are maintained at 77 K. The outer surface of the window is subjected to surroundings and room air at 25 C, with a convection heat transfer coefficient of 15 W/m2
K. (a) Determine the transmissivity of the window material to radiation from the solar simulator, which approximates the solar spectral distribution. (b) Assuming that the window is insulated from its chamber mounting arrangement, what steady-state temperature does the window reach? (c) Calculate the net radiation transfer per unit area of the window to the vacuum chamber wall, excluding the transmitted simulated solar flux.

A thermocouple whose surface is diffuse and gray with an emissivity of 0.6 indicates a temperature of 180 C when used to measure the temperature of a gas flowing through a large duct whose walls have an emissivity of 0.85 and a uniform temperature of 450 C. (a) If the convection heat transfer coefficient between the thermocouple and the gas stream is and there are negligible conduction losses from the thermocouple, determine the temperature of the gas. (b) Consider a gas temperature of 125 C. Compute and plot the thermocouple measurement error as a function of the convection coefficient for 10 . What are the implications of your results? h 1000 W/m2

A thermocouple inserted in a 4-mm-diameter stainless steel tube having a diffuse, gray surface with an emissivity of 0.4 is positioned horizontally in a large airconditioned room whose walls and air temperature are 30 and 20 C, respectively. (a) What temperature will the thermocouple indicate if the air is quiescent? (b) Compute and plot the thermocouple measurement error as a function of the surface emissivity for 0.1  1.0

A temperature sensor embedded in the tip of a small tube having a diffuse, gray surface with an emissivity of 0.8 is centrally positioned within a large airconditioned room whose walls and air temperature are 30 and 20 C, respectively. (a) What temperature will the sensor indicate if the convection coefficient between the sensor tube and the air is 5 W/m2
K? (b) What would be the effect of using a fan to induce airflow over the tube? Plot the sensor temperature as a function of the convection coefficient for 2 h 25 W/m2
K and values of  0.2, 0.5, and 0.8.

A sphere (k 185 W/m
K,  7.25 105 m2 /s) of 30-mm diameter whose surface is diffuse and gray with an emissivity of 0.8 is placed in a large oven whose walls are of uniform temperature at 600 K. The temperature of the air in the oven is 400 K, and the convection heat transfer coefficient between the sphere and the oven air is 15 W/m2
K. (a) Determine the net heat transfer to the sphere when its temperature is 300 K. (b) What will be the steady-state temperature of the sphere? (c) How long will it take for the sphere, initially at 300 K, to come within 20 K of the steady-state temperature? (d) For emissivities of 0.2, 0.4, and 0.8, plot the elapsed time of part (c) as a function of the convection coefficient for 10 h 25 W/m2
K. R

A thermograph is a device responding to the radiant power from the scene, which reaches its radiation detector within the spectral region 912
m. The thermograph provides an image of the scene, such as the side of a furnace, from which the surface temperature can be determined. (a) For a black surface at 60 C, determine the emissive power for the spectral region 912
m. (b) Calculate the radiant power (W) received by the thermograph in the same range (912
m) when viewing, in a normal direction, a small black wall area, 200 mm2 , at Ts 60 C. The solid angle subtended by the aperture of the thermograph when viewed from the target is 0.001 sr. (c) Determine the radiant power (W) received by the thermograph for the same wall area (200 mm2 ) and solid angle (0.001 sr) when the wall is a gray, opaque, diffuse material at Ts 60 C with emissivity 0.7 and the surroundings are black at Tsur 23 C. 1

A radiation thermometer is a radiometer calibrated to indicate the temperature of a blackbody. A steel billet having a diffuse, gray surface of emissivity 0.8 is heated in a furnace whose walls are at 1500 K. Estimate the temperature of the billet when the radiation thermometer viewing the billet through a small hole in the furnace indicates 1160 K.

A radiation detector has an aperture of area Ad 106 m2 and is positioned at a distance of r 1 m from a surface of area As 104 m2 . The angle formed by the normal to the detector and the surface normal is 30 . The surface is at 500 K and is opaque, diffuse, and gray with an emissivity of 0.7. If the surface irradiation is 1500 W/m2 , what is the rate at which the detector intercepts radiation from the surface? 12

A small anodized aluminum block at 35 C is heated in a large oven whose walls are diffuse and gray with  0.85 and maintained at a uniform temperature of 175 C. The anodized coating is also diffuse and gray with  0.92. A radiation detector views the block through a small opening in the oven and receives the radiant energy from a small area, referred to as the target, At , on the block. The target has a diameter of 3 mm, and the detector receives radiation within a solid angle 0.001 sr centered about the normal from the block. (a) If the radiation detector views a small, but deep, hole drilled into the block, what is the total power (W) received by the detector? (b) If the radiation detector now views an area on the block surface, what is the total power (W) received by the detector?

Consider the diffuse, gray opaque disk A1, which has a diameter of 10 mm, an emissivity of 0.3, and is at a temperature of 400 K. Coaxial to the disk A1, there is a black, ring-shaped disk A2 at 1000 K having the dimensions shown in the sketch. The backside of A2 is insulated and does not directly irradiate the cryogenically cooled detector disk A3, which is of diameter 10 mm and is located 2 m from A1.

An infrared (IR) thermograph is a radiometer that provides an image of the target scene, indicating the apparent temperature of elements in the scene by a blackwhite brightness or bluered color scale. Radiation originating from an element in the target scene is incident on the radiation detector, which provides a signal proportional to the incident radiant power. The signal sets the image brightness or color scale for the image pixel associated with that element. A scheme is proposed for field calibration of an infrared thermograph having a radiation detector with a 3- to 5-
m spectral bandpass. A heated metal plate, which is maintained at 327 C and has four diffuse, gray coatings with different emissivities, is viewed by the IR thermograph in surroundings for which Tsur 87 C. (a) Consider the thermograph output when viewing the black coating, o 1. The radiation reaching the detector is proportional to the product of the blackbody emissive power (or emitted intensity) at the temperature of the surface and the band emission fraction corresponding to the IR thermograph spectral bandpass. The proportionality constant is referred to as the responsivity, R(
V
m2 /W). Write an expression for the thermograph output signal, So, in terms of R, the coating blackbody emissive power, and the appropriate band emission fraction. Assuming R 1
V
m2 /W, evaluate So (
V). (b) Consider the thermograph output when viewing one of the coatings for which the emissivity c is less than unity. Radiation from the coating reaches the detector due to emission and the reflection of irradiation from the surroundings. Write an expression for the signal, Sc, in terms of R, the coating blackbody emissive power, the blackbody emissive power of the surroundings, the coating emissivity, and the appropriate band emission fractions. For the diffuse, gray coatings, the reflectivity is c 1  c. (c) Assuming R 1
V
m2 /W, evaluate the thermograph signals, Sc (
V), when viewing panels with emissivities of 0.8, 0.5, and 0.2. (d) The thermograph is calibrated so that the signal So (with the black coating) will give a correct scale indication of Ts 327 C. The signals from the other three coatings, Sc, are less than So. Hence the thermograph will indicate an apparent (blackbody) temperature less than Ts. Estimate the temperatures indicated by the thermograph for the three panels of part (c). 12.

A charge-coupled device (CCD) infrared imaging system (see Problem 12.85) operates in a manner similar to a digital video camera. Instead of being sensitive to irradiation in the visible part of the spectrum, however, each small sensor in the infrared systems CCD array is sensitive in the spectral region 912
m. Note that the system is designed to only view radiation coming from directly in front of it. An experimenter wishes to use the infrared imaging system to map the surface temperature distribution of a heated object in a wind tunnel experiment. The air temperature in the wind tunnel, as well as the surroundings temperature in the laboratory, is 23 C. (a) In a preliminary test of the concept, the experimenter views a small aluminum billet located in the wind tunnel that is at a billet temperature of 50 C. The aluminum is coated with a highemissivity paint,  0.96. If the infrared imaging system is calibrated to indicate the temperature of a blackbody, what temperature will be indicated by the infrared imaging system as it is used to view the aluminum billet through a 6-mm-thick fused quartz window? (b) In a subsequent experiment, the experimenter replaces the quartz window with a thin (130-
mthick) household polyethylene film with  0.78 within the spectral range of the imaging system. What temperature will be indicated by the infrared imaging system when it is used to view the aluminum billet through the polyethylene film?

A diffuse, spherical object of diameter and temperature 9 mm and 600 K, respectively, has an emissivity of 0.95. Two very sensitive radiation detectors, each with an aperture area of 300 106 m2 , detect the object as it passes over at high velocity from left to right as shown in the schematic. The detectors capture hemispherical irradiation and are equipped with filters characterized by  0.9 for  2.5
m and  0 for  2.5
m. At time t1 0, detectors A and B indicate irradiations of GA,1 5.060 mW/m2 and GB,1 5.000 mW/m2 , respectively. At time t2 4 ms, detectors A and B indicate irradiations of GA,2 5.010 mW/m2 and GB,2 5.050 mW/m2 , respectively. The environment is at 300 K. Determine the velocity components of the particle, vx and vy. Determine when and where the particle will strike a horizontal plane located at y 0. Hint: The object is located at an elevation above y 2 m when it is detected. Assume the objects trajectory is a straight line in the plane of the page. Recall that the projected area of a sphere is a circle. Detector A

A radiation detector having a sensitive area of Ad 4 106 m2 is configured to receive radiation from a target area of diameter Dt 40 mm when located a distance of Lt 1 m from the target. For the experimental apparatus shown in the sketch, we wish to determine the emitted radiation from a hot sample of diameter Ds 20 mm. The temperature of the aluminum sample is Ts 700 K and its emissivity is s 0.1. A ring-shaped cold shield is provided to minimize the effect of radiation from outside the sample area, but within the target area. The sample and the shield are diffuse emitters. (a) A (a) Assuming the shield is black, at what temperature, Tsh, should the shield be maintained so that its emitted radiation is 1% of the total radiant power received by the detector? (b) Subject to the parametric constraint that radiation emitted from the cold shield is 0.05, 1, or 1.5% of the total radiation received by the detector, plot the required cold shield temperature, Tsh, as a function of the sample emissivity for 0.05 s 0.35.

A two-color pyrometer is a device that is used to measure the temperature of a diffuse surface, Ts. The device measures the spectral, directional intensity emitted by the surface at two distinct wavelengths separated by . Calculate and plot the ratio of the intensities I ,e(  , , , Ts) and I,e(, , , Ts) as a function of the surface temperature over the range 500 K Ts 1000 K for 5
m and 0.1, 0.5, and 1
m. Comment on the sensitivity to temperature and on whether the ratio depends on the emissivity of the surface. Discuss the tradeoffs associated with speci- fication of the various values of . Hint: The change in the emissivity over small wavelength intervals is modest for most solids, as evident in Figure 12.17. Lt = 1 m

Consider a two-color pyrometer such as in Problem 12.89 that operates at 1 0.65
m and 2 0.63
m. Using Wiens law (see Problem 12.27) determine the temperature of a sheet of stainless steel if the ratio of radiation detected is . Ap

Square plates freshly sprayed with an epoxy paint must be cured at for an extended period of time. The plates are located in a large enclosure and heated by a bank of infrared lamps. The top surface of each plate has an emissivity of  0.8 and experiences convection with a ventilation airstream that is at T
27 C and provides a convection coefficient of h 20 W/m2
K. The irradiation from the enclosure walls is estimated to be Gwall 450 W/m2 , for which the plate absorptivity is wall 0.7. (a) Determine the irradiation that must be provided by the lamps, Glamp. The absorptivity of the plate surface for this irradiation is lamp 0.6. (b) For convection coefficients of h 15, 20, and 30 W/m2
K, plot the lamp irradiation, Glamp, as a function of the plate temperature, Ts, for 100 Ts 300 C. (c) For convection coefficients in the range from 10 to 30 W/m2
K and a lamp irradiation of Glamp 3000 W/m2 , plot the airstream temperature T
required to maintain the plate at Ts 140 C. 12.92 A

An apparatus commonly used for measuring the reflectivity of materials is shown below. A water-cooled sample, of 30-mm diameter and temperature Ts 300 K, is mounted flush with the inner surface of a large enclosure. The walls of the enclosure are gray and diffuse with an emissivity of 0.8 and a uniform temperature Tf 1000 K. A small aperture is located at the bottom of the enclosure to permit sighting of the sample or the enclosure wall. The spectral reflectivity of an opaque, diffuse sample material is as shown. The heat transfer coefficient for convection between the sample and the air within the cavity, which is also at 1000 K, is h 10 W/m2
K. a) Calculate the absorptivity of the sample. (b) Calculate the emissivity of the sample. (c) Determine the heat removal rate (W) by the coolant. (d) The ratio of the radiation in the A direction to that in the B direction will give the reflectivity of the sample. Briefly explain why this is so.

A very small sample of an opaque surface is initially at 1200 K and has the spectral, hemispherical absorptivity shown. The sample is placed inside a very large enclosure whose walls have an emissivity of 0.2 and are maintained at 2400 K. (a) What is the total, hemispherical absorptivity of the sample surface? (b) What is its total, hemispherical emissivity? (c) What are the values of the absorptivity and emissivity after the sample has been in the enclosure a long time? (d) For a 10-mm-diameter spherical sample in an evacuated enclosure, compute and plot the variation of the sample temperature with time, as it is heated from its initial temperature of 1200 K

A manufacturing process involves heating long copper rods, which are coated with a thin film, in a large furnace whose walls are maintained at an elevated temperature Tw. The furnace contains quiescent nitrogen gas at 1-atm pressure and a temperature of T
Tw. The film is a diffuse surface with a spectral emissivity of  0.9 for 2
m and  0.4 for
2
m. (a) Consider conditions for which a rod of diameter D and initial temperature Ti is inserted in the furnace, such that its axis is horizontal. Assuming validity of the lumped capacitance approximation, derive an equation that could be used to determine the rate of change of the rod temperature at the time of insertion. Express your result in terms of appropriate variables. (b) If Tw T
1500 K, Ti 300 K, and D 10 mm, what is the initial rate of change of the rod temperature? Confirm the validity of the lumped capacitance approximation. (c) Compute and plot the variation of the rod temperature with time during the heating process. 12.95 A p

A procedure for measuring the thermal conductivity of solids at elevated temperatures involves placement of a sample at the bottom of a large furnace. The sample is of thickness L and is placed in a square container of width W on a side. The sides are well insulated. The walls of the cavity are maintained at Tw, while the bottom surface of the sample is maintained at a much lower temperature Tc by circulating coolant through the sample container. The sample surface is diffuse and gray with an emissivity s. Its temperature Ts is measured optically. (a) Neglecting convection effects, obtain an expression from which the sample thermal conductivity may be evaluated in terms of measured and known quantities (Tw, Ts, Tc, s, L). The measurements are made under steady-state conditions. If Tw 1400 K, Ts 1000 K, s 0.85, L 0.015 m, and Tc 300 K, what is the sample thermal conductivity? Coo (b) If W 0.10 m and the coolant is water with a flow rate of , is it reasonable to assume a uniform bottom surface temperature Tc?

One scheme for extending the operation of gas turbine blades to higher temperatures involves applying a ceramic coating to the surfaces of blades fabricated from a superalloy such as inconel. To assess the reliability of such coatings, an apparatus has been developed for testing samples under laboratory conditions. The sample is placed at the bottom of a large vacuum chamber whose walls are cryogenically cooled and which is equipped with a radiation detector at the top surface. The detector has a surface area of Ad 105 m2 , is located at a distance of Lsd 1 m from the sample, and views radiation originating from a portion of the ceramic surface having an area of Ac 104 m2 . An electric heater attached to the bottom of the sample dissipates a uniform heat flux, , which is transferred upward through the sample. The bottom of the heater and sides of the sample are well insulated. Consider conditions for which a ceramic coating of thickness Lc 0.5 mm and thermal conductivity kc 6 W/m
K has been sprayed on a metal substrate of thickness Ls 8 mm and thermal conductivity ks 25 W/m
K. The opaque surface of the ceramic may be approximated as diffuse and gray, with a total, hemispherical emissivity of c 0.8. (a) Consider steady-state conditions for which the bottom surface of the substrate is maintained at T1 1500 K, while the chamber walls (including the surface of the radiation detector) are maintained at Tw 90 K. Assuming negligible thermal contact resistance at the ceramicsubstrate interface, determine the ceramic top surface temperature T2 and the heat flux . (b) For the prescribed conditions, what is the rate at which radiation emitted by the ceramic is intercepted by the detector? q h R (c) After repeated experiments, numerous cracks develop at the ceramicsubstrate interface, creating an interfacial thermal contact resistance. If Tw and are maintained at the conditions associated with part (a), will T1 increase, decrease, or remain the same? Similarly, will T2 increase, decrease, or remain the same? In each case, justify your answer.

The equipment for heating a wafer during a semiconductor manufacturing process is shown schematically. The wafer is heated by an ion beam source (not shown) to a uniform, steady-state temperature. The large chamber contains the process gas, and its walls are at a uniform temperature of Tch 400 K. A 5 mm 5 mm target area on the wafer is viewed by a radiometer, whose objective lens has a diameter of 25 mm and is located 500 mm from the wafer. The line-of-sight of the radiometer is off the wafer normal. (a) In a preproduction test of the equipment, a black panel ( 1.0) is mounted in place of the wafer. Calculate the radiant power (W) received by the radiometer if the temperature of the panel is 800 K. (b) The wafer, which is opaque, diffuse-gray with an emissivity of 0.7, is now placed in the equipment, and the ion beam is adjusted so that the power received by the radiometer is the same as that found for part (a). Calculate the temperature of the wafer for this heating condition.

The fire brick of Example 12.10 is used to construct the walls of a brick oven. The irradiation on the interior surface of the wall is G 50,000 W/m2 and has a spectral distribution proportional to that of a blackbody at 2000 K. The temperature of the gases adjacent to the inner wall of the oven is 500 K, and the convection heat transfer coefficient is 25 W/m2 . Find the wall surface temperature if the heat loss through the wall is negligible. If the brick wall is 0.1 m thick and of thermal conductivity kb 1.0 W/m
K, and is insulated with a 0.1-m-thick layer of thermal conductivity ki 0.05 W/m
K, what is the steady-state interior W wall surface temperature if the temperature of the external surface of the insulation is 300 K?

A laser-materials-processing apparatus encloses a sample in the form of a disk of diameter D 25 mm and thickness w 1 mm. The sample has a diffuse surface for which the spectral distribution of the emissivity, (), is prescribed. To reduce oxidation, an inert gas stream of temperature T
500 K and convection coefficient h 50 W/m2
K flows over the sample upper and lower surfaces. The apparatus enclosure is large, with isothermal walls at Tenc 300 K. To maintain the sample at a suitable operating temperature of Ts 2000 K, a collimated laser beam with an operating wavelength of 0.5
m irradiates its upper surface. (a) Determine the total emissivity  of the sample. (b) Determine the total absorptivity  of the sample for irradiation from the enclosure walls. (c) Perform an energy balance on the sample and determine the laser irradiation, Glaser, required to maintain the sample at Ts 2000 K. (d) Consider a cool-down process, when the laser and the inert gas flow are deactivated. Sketch the total emissivity as a function of the sample temperature, Ts(t), during the process. Identify key features, including the emissivity for the final condition (t l
). (e) Estimate the time to cool a sample from its operating condition at Ts(0) 2000 K to a safe-to-touch temperature of Ts(t) 40 C. Use the lumped capacitance method and include the effect of convection to the inert gas with h 50 W/m2
K and T
Tenc 300 K. The thermophysical properties of the sample material are  3900 kg/m3 , cp 760 J/kg
K, and k 45 W/m
K. Laser beam Windo

A cylinder of 30-mm diameter and 150-mm length is heated in a large furnace having walls at 1000 K, while air at 400 K is circulating at 3 m/s. Estimate the steady-state cylinder temperature under the following specified conditions. (a) The cylinder is in cross flow, and its surface is diffuse and gray with an emissivity of 0.5. (b) The cylinder is in cross flow, but its surface is spectrally selective with  0.1 for 3
m and  0.5 for
3
m. (c) The cylinder surface is positioned such that the airflow is longitudinal and its surface is diffuse and gray. (d) For the conditions of part (a), compute and plot the cylinder temperature as a function of the air velocity for 1 V 20 m/s. 12.1

An instrumentation transmitter pod is a box containing electronic circuitry and a power supply for sending sensor signals to a base receiver for recording. Such a pod is placed on a conveyor system, which passes through a large vacuum brazing furnace as shown in the sketch. The exposed surfaces of the pod have a special diffuse, opaque coating with spectral emissivity as shown. To stabilize the temperature of the pod and prevent overheating of the electronics, the inner surface of the pod is surrounded by a layer of a phase-change material (PCM) having a fusion temperature of 87 C and a heat of fusion of 25 kJ/kg. The pod has an exposed surface area of 0.040 m2 and the mass of the PCM is 1.6 kg. Furthermore, it is known that the power dissipated by the electronics is 50 W. Consider the situation when the pod enters the furnace at a uniform temperature of 87 C and all the PCM is in the solid state. How long will it take before all the PCM changes to the liquid state?

A thin-walled plate separates the interior of a large furnace from surroundings at 300 K. The plate is fabricated from a ceramic material for which diffuse surface behavior may be assumed and the exterior surface is air cooled. With the furnace operating at 2400 K, convection at the interior surface may be neglected. (a) If the temperature of the ceramic plate is not to exceed 1800 K, what is the minimum value of the outside convection coefficient, ho, that must be maintained by the air-cooling system? (b) Compute and plot the plate temperature as a function of ho for 50 ho 250 W/m2
K.

A thin coating, which is applied to long, cylindrical copper rods of 10-mm diameter, is cured by placing the rods horizontally in a large furnace whose walls are maintained at 1300 K. The furnace is filled with nitrogen gas, which is also at 1300 K and at a pressure of 1 atm. The coating is diffuse, and its spectral emissivity has the distribution shown. (a) What are the emissivity and absorptivity of the coated rods when their temperature is 300 K? (b) What is the initial rate of change of their temperature? (c) What are the emissivity and absorptivity of the coated rods when they reach a steady-state temperature? (d) Estimate the time required for the rods to reach 1000 K.

A large combination convectionradiation oven is used to heat-treat a small cylindrical product of diameter 25 mm and length 0.2 m. The oven walls are at a uniform temperature of 1000 K, and hot air at 750 K is in cross flow over the cylinder with a velocity of 5 m/s. The cylinder surface is opaque and diffuse with the spectral emissivity shown. (a) Determine the rate of heat transfer to the cylinder when it is first placed in the oven at 300 K. (b) What is the steady-state temperature of the cylinder? (c) How long will it take for the cylinder to reach a temperature that is within 50 C of its steady-state value?

A 10-mm-thick workpiece, initially at 25 C, is to be annealed at a temperature above 725 C for a period of at least 5 minutes and then cooled. The workpiece is opaque and diffuse, and the spectral distribution of its emissivity is shown schematically. Heating is effected in a large furnace with walls and circulating air at 750 C and a convection coefficient of 100 W/m2
K. The thermophysical properties of the workpiece are  2700 kg/m3 , c 885 J/kg
K, and k 165 W/m
K. (a) Calculate the emissivity and the absorptivity of the workpiece when it is placed in the furnace at its initial temperature of 25 C. (b) Determine the net heat flux into the workpiece for this initial condition. What is the corresponding rate of change in temperature, dT/dt, for the workpiece? 1 (c) Calculate the time for the workpiece to cool from 750 C to a safe-to-touch temperature of 40 C, if the surroundings and cooling air temperature are 25 C and the convection coefficient is 100 W/m2
K.

After being cut from a large single-crystal boule and polished, silicon wafers undergo a high-temperature annealing process. One technique for heating the wafer is to irradiate its top surface using highintensity, tungsten-halogen lamps having a spectral distribution approximating that of a blackbody at 2800 K. To determine the lamp power and the rate at which radiation is absorbed by the wafer, the equipment designer needs to know its absorptivity as a function of temperature. Silicon is a semiconductor material that exhibits a characteristic band edge, and its spectral absorptivity may be idealized as shown schematically. At low and moderate temperatures, silicon is semitransparent at wavelengths larger than that of the band edge, but becomes nearly opaque above 600 C. (a) What are the 1% limits of the spectral band that includes 98% of the blackbody radiation corresponding to the spectral distribution of the lamps? Over what spectral region do you need to know the spectral absorptivity? (b) How do you expect the total absorptivity of silicon to vary as a function of its temperature? Sketch the variation and explain its key features. (c) Calculate the total absorptivity of the silicon wafer for the lamp irradiation and each of the five temperatures shown schematically. From the data, calculate the emissivity of` the wafer at 600 and 900 C. Explain your results and why the emissivity changes with temperature. Hint: Within IHT, create a look-up table to specify values of the spectral properties and the LOOKUPVAL and INTEGRAL functions to perform the necessary integrations. (d) If the wafer is in a vacuum and radiation exchange only occurs at one face, what is the irradiation needed to maintain a wafer temperature of 600 C?

Solar irradiation of 1100 W/m2 is incident on a large, flat, horizontal metal roof on a day when the wind blowing over the roof causes a convection heat transfer coefficient of 25 W/m2
K. The outside air temperature is 27 C, the metal surface absorptivity for incident solar radiation is 0.60, the metal surface emissivity is 0.20, and the roof is well insulated from below. (a) Estimate the roof temperature under steady-state conditions. (b) Explore the effect of changes in the absorptivity, emissivity, and convection coefficient on the steady-state temperature

Neglecting the effects of radiation absorption, emission, and scattering within their atmospheres, calculate the average temperature of Earth, Venus, and Mars assuming diffuse, gray behavior. The average distance from the sun of each of the three planets, Lsp, along with their measured average temperatures, , are shown in the table below. Based upon a comparison of the calculated and measured average temperatures, which planet is most affected by radiation transfer in its atmosphere?

A deep cavity of 50-mm diameter approximates a blackbody and is maintained at 250 C while exposed to solar irradiation of 800 W/m2 and surroundings and ambient air at 25 C. A thin window of spectral transmissivity and reflectivity 0.9 and 0, respectively, for the spectral range 0.2 to 4
m is placed over the cavity opening. In the spectral range beyond 4
m, the window behaves as an opaque, diffuse, gray body of emissivity 0.95. Assuming that the convection coefficient on the upper surface of the window is 10 W/m2
K, determine the temperature of the window and the power required to maintain the cavity at 250 C.

Consider the evacuated tube solar collector described in part (d) of Problem 1.87 of Chapter 1. In the interest of maximizing collector efficiency, what spectral radiative characteristics are desired for the outer tube and for the inner tube?

Solar flux of 900 W/m2 is incident on the top side of a plate whose surface has a solar absorptivity of 0.9 and an emissivity of 0.1. The air and surroundings are at 17 C and the convection heat transfer coefficient between the plate and air is 20 W/m2
K. Assuming that the bottom side of the plate is insulated, determine the steady-state temperature of the plate.

Consider an opaque, gray surface whose directional absorptivity is 0.8 for 0 60 and 0.1 for
60 . The surface is horizontal and exposed to solar irradiation comprised of direct and diffuse components. (a) What is the surface absorptivity to direct solar radiation that is incident at an angle of 45 from the normal? What is the absorptivity to diffuse irradiation? (b) Neglecting convection heat transfer between the surface and the surrounding air, what would be the equilibrium temperature of the surface if the direct and diffuse components of the irradiation were 600 and 100 W/m2 , respectively? The back side of the surface is insulated.

The absorber plate of a solar collector may be coated with an opaque material for which the spectral, directional absorptivity is characterized by relations of the form , (, ) 2
c , (, ) 1 cos  c Tsur = The zenith angle is formed by the suns rays and the plate normal, and 1 and 2 are constants. (a) Obtain an expression for the total, hemispherical absorptivity, S, of the plate to solar radiation incident at 45 . Evaluate S for 1 0.93, 2 0.25, and a cut-off wavelength of c 2
m. (b) Obtain an expression for the total, hemispherical emissivity  of the plate. Evaluate  for a plate temperature of Tp 60 C and the prescribed values of 1, 2, and c. (c) For a solar flux of incident at and the prescribed values of and Tp, what is the net radiant heat flux, , to the plate? (d) Using the prescribed conditions and the Radiation/ Band Emission Factor option in the Tools section of IHT to evaluate F(0lc), explore the effect of c on S, , and q net for the wavelength range 0.7 c 5
m. 12.114 A

A contractor must select a roof covering material from the two diffuse, opaque coatings with () as shown. Which of the two coatings would result in a lower roof temperature? Which is preferred for summer use? For winter use? Sketch the spectral distribution of  that would be ideal for summer use. For winter use. 1

It is not uncommon for the night sky temperature in desert regions to drop to 40 C. If the ambient air temperature is 20 C and the convection coefficient for still air conditions is approximately 5 W/m2
K, can a shallow pan of water freeze?

Plant leaves possess small channels that connect the interior moist region of the leaf to the environment. The channels, called stomata, pose the primary resistance to moisture transport through the entire plant, and the diameter of an individual stoma is sensitive to the level of CO2 in the atmosphere. Consider a leaf of corn (maize) whose top surface is exposed to solar irradiation of GS 600 W/m2 and an effective sky temperature of Tsky 0 C. The bottom side of the leaf is irradiated from the ground which is at a temperature of Tg 20 C. Both the top and bottom of the leaf are subjected to convective conditions characterized by h 35 W/m2
K, T
25 C and also experience evaporation through the stomata. Assuming the evaporative flux of water vapor is 50 106 kg/m2
s under rural atmospheric CO2 concentrations and is reduced to 5 106 kg/m2
s when ambient CO2 concentrations are doubled near an urban area, calculate the leaf temperature in the rural and urban locations. The heat of vaporization of water is hfg 2400 kJ/kg and assume   0.97 for radiation exchange with the sky and the ground, and S 0.76 for solar irradiation. 12.117 I

In the central receiver concept of solar energy collection, a large number of heliostats (reflectors) provide a concentrated solar flux of to the receiver, which is positioned at the top of a tower. The receiver wall is exposed to the solar flux at its outer surface and to atmospheric air for which T
,o 300 K and ho 25 W/m2
K. The outer surface is opaque and diffuse, with a spectral absorptivity of  0.9 for  3
m and  0.2 for
3
m. The inner surface is exposed to a working fluid (a pressurized liquid) for which T
,i 700 K and hi 1000 W/m2
K. The outer surface is also exposed to surroundings for which Tsur 300 K. If the wall is fabricated from a high-temperature material for which k 15 W/m
K, what is the minimum thickness L needed to ensure that the outer surface temperature does not exceed Ts,o 1000 K? What is the collection efficiency associated with this thickness? 12.118 Cons

Consider the central receiver of Problem 12.117 to be a cylindrical shell of outer diameter D 7 m and length L 12 m. The outer surface is opaque and diffuse, with a spectral absorptivity of  0.9 for  3
m and  0.2 for
3
m. The surface is exposed to quiescent ambient air for which T
300 K. Atmosph (a) Consider representative operating conditions for which solar irradiation at GS 80,000 W/m2 is uniformly distributed over the receiver surface and the surface temperature is Ts 800 K. Determine the rate at which energy is collected by the receiver and the corresponding collector efficiency. (b) The surface temperature is affected by conditions internal to the receiver. For GS 80,000 W/m2 , compute and plot the rate of energy collection and the collector efficiency for 600 Ts 1000 K. 1

Radiation from the atmosphere or sky can be estimated as a fraction of the blackbody radiation corresponding to the air temperature near the ground, Tair. That is, irradiation from the sky can be expressed as and for a clear night sky, the emissivity is correlated by an expression of the form sky 0.741  0.0062Tdp, where Tdp is the dew point temperature ( C). Consider a flat plate exposed to the night sky and in ambient air at 15 C with a relative humidity of 70%. Assume the back side of the plate is insulated, and that the convection coefficient on the front side can be estimated by the correlation , where T is the absolute value of the plate-to-air temperature difference. Will dew form on the plate if the surface is (a) clean and metallic with  0.23, and (b) painted with  0.85?

A thin sheet of glass is used on the roof of a greenhouse and is irradiated as shown. The irradiation comprises the total solar flux GS, the flux Gatm due to atmospheric emission (sky radiation), and the flux Gi due to emission from interior surfaces. The fluxes Gatm and Gi are concentrated in the far IR region (  8
m). The glass may also exchange energy by convection with the outside and inside atmospheres. The glass may be assumed to be totally transparent for  1
m ( 1.0 for  1
m) and opaque, with  1.0 for  1
m. (a) Assuming steady-state conditions, with all radiative fluxes uniformly distributed over the surfaces and the glass characterized by a uniform temperature Tg, write an appropriate energy balance for a unit area of the glass. (b) For Tg 27 C, hi 10 W/m2
K, GS 1100 W/m2 , T
,o 24 C, ho 55 W/m2
K, Gatm 250 W/m2 , and Gi 440 W/m2 , calculate the temperature of the greenhouse ambient air, T
,i . 12.121 A sola

A solar furnace consists of an evacuated chamber with transparent windows, through which concentrated solar radiation is passed. Concentration may be achieved by mounting the furnace at the focal point of a large curved reflector that tracks radiation incident directly from the sun. The furnace may be used to evaluate the behavior of materials at elevated temperatures, and we wish to design an experiment to assess the durability of a diffuse, spectrally selective coating for which  0.95 in the range 4.5
m and  0.03 for
4.5
m. The coating is applied to a plate that is suspended in the furnace. (a) If the experiment is to be operated at a steadystate plate temperature of T 2000 K, how much solar irradiation GS must be provided? The irradiation may be assumed to be uniformly distributed over the plate surface, and other sources of incident radiation may be neglected. (b) The solar irradiation may be tuned to allow operation over a range of plate temperatures. Compute and plot GS as a function of temperature for 500 T 3000 K. Plot the corresponding values Trans of  and  as a function of T for the designated range

The flat roof on the refrigeration compartment of a food delivery truck is of length L 5 m and width W 2 m. It is fabricated from thin sheet metal to which a fiberboard insulating material of thickness t 25 mm and thermal conductivity k 0.05 W/m
K is bonded. During normal operation, the truck moves at a velocity of V 30 m/s in air at T
27 C, with a rooftop solar irradiation of GS 900 W/m2 and with the interior surface temperature maintained at Ts,i 13 C. (a) The owner has the option of selecting a roof coating from one of the three paints listed in Table A.12 (Parsons Black, Acrylic White, or Zinc Oxide White). Which should be chosen and why? (b) For the preferred paint of part (a), determine the steady-state value of the outer surface temperature Ts,o. The boundary layer is tripped at the leading edge of the roof, and turbulent flow may be assumed to exist over the entire roof. Properties of the air may be taken to be 15 106 m2 /s, k 0.026 W/m
K, and Pr 0.71. (c) What is the load (W) imposed on the refrigeration system by heat transfer through the roof? (d) Explore the effect of the truck velocity on the outer surface temperature and the heat load. 12.123 Gro

Growers use giant fans to prevent grapes from freezing when the effective sky temperature is low. The grape, which may be viewed as a thin skin of negligible thermal resistance enclosing a volume of sugar water, is exposed to ambient air and is irradiated from the sky above and ground below. Assume the grape to be an isothermal sphere of 15-mm diameter, and assume uniform blackbody irradiation over its top and bottom hemispheres due to emission from the sky and the earth, respectively. (a) Derive an expression for the rate of change of the grape temperature. Express your result in terms of a convection coefficient and appropriate temperatures and radiative quantities. (b) Under conditions for which Tsky 235 K, T
273 K, and the fan is off (V 0), determine whether the grapes will freeze. To a good approximation, the skin emissivity is 1 and the grape thermophysical properties are those of sugarless water. However, because of the sugar content, the grape freezes at 5 C. (c) With all conditions remaining the same, except that the fans are now operating with V 1 m/s, will the grapes freeze? 12

A circular metal disk having a diameter of 0.4 m is placed firmly against the ground in a barren horizontal region where the earth is at a temperature of 280 K. The effective sky temperature is also 280 K. The disk is exposed to quiescent ambient air at 300 K and direct solar irradiation of 745 W/m2 . The surface of the disk is diffuse with  0.9 for 0   1
m and  0.2 for
1
m. After some time has elapsed, the disk achieves a uniform, steady-state temperature. The thermal conductivity of the soil is 0.52 W/m
K. (a) Determine the fraction of the incident solar irradiation that is absorbed. (b) What is the emissivity of the disk surface? (c) For a steady-state disk temperature of 340 K, employ a suitable correlation to determine the average free convection heat transfer coefficient at the upper surface of the disk. (d) Show that a disk temperature of 340 K does indeed yield a steady-state condition for the disk. Air

The neighborhood cat likes to sleep on the roof of our shed in the backyard. The roofing surface is weathered galvanized sheet metal ( 0.65, S 0.8). Consider a cool spring day when the ambient air temperature is 10 C and the convection coefficient can be estimated from an empirical correlation of the form , where T is the difference between the surface and ambient temperatures. Assume the sky temperature is 40 C. (a) Assuming the backside of the roof is well insulated, calculate the roof temperature when the solar irradiation is 600 W/m2 . Will the cat enjoy sleeping under these conditions? (b) Consider the case when the backside of the roof is not insulated, but is exposed to ambient air with the same convection coefficient relation and experiences radiation exchange with the ground, also at the ambient air temperature. Calculate the roof temperature and comment on whether the roof will be a comfortable place for the cat to snooze.

The exposed surface of a power amplifier for an earth satellite receiver of area 130 mm 130 mm has a diffuse, gray, opaque coating with an emissivity of 0.5. For typical amplifier operating conditions, the surface temperature is 58 C under the following environmental conditions: air temperature, T
27 C; sky temperature, Tsky 20 C; convection coefficient, h 15 W/m2
K; and solar irradiation, GS 800 W/m2 . (a) For the above conditions, determine the electrical power being generated within the amplifier. (b) It is desired to reduce the surface temperature by applying one of the diffuse coatings (A, B, C) shown as follows. Which coating will result in the coolest surface temperature for the same amplifier operating and environmental conditions? 12.

Consider a thin opaque, horizontal plate with an electrical heater on its backside. The front side is exposed to ambient air that is at 20 C and provides a convection heat transfer coefficient of 10 W/m2
K, solar irradiation of 600 W/m2 , and an effective sky temperature of 40 C. What is the electrical power (W/m2 ) required to maintain the plate surface temperature at Ts 60 C if the plate is diffuse and has the designated spectral, hemispherical reflectivity?

The oxidized-aluminum wing of an aircraft has a chord length of Lc 4 m and a spectral, hemispherical emissivity characterized by the following distribution. (a) Consider conditions for which the plane is on the ground where the air temperature is 27 C, the solar irradiation is 800 W/m2 , and the effective sky temperature is 270 K. If the air is quiescent, what is the temperature of the top surface of the wing? The wing may be approximated as a horizontal, flat plate. (b) When the aircraft is flying at an elevation of approximately 9000 m and a speed of 200 m/s, the air temperature, solar irradiation, and effective sky temperature are 40 C, 1100 W/m2 , and 235 K, respectively. What is the temperature of the wings top surface? The properties of the air may be approximated as  0.470 kg/m3 ,
1.50 105 N
s/m2 , k 0.021 W/m
K, and Pr 0.72. Spac

Two plates, one with a black painted surface and the other with a special coating (chemically oxidized copper) are in earth orbit and are exposed to solar radiation. The solar rays make an angle of 30 with the normal to the plate. Estimate the equilibrium temperature of each plate assuming they are diffuse and that the solar flux is 1368 W/m2 . The spectral absorptivity of the black painted surface can be approximated by  0.95 for 0
and that of the special coating by  0.95 for 0  3
m and  0.05 for  3
m. 12.130

A spherical satellite of diameter D is in orbit about the earth and is coated with a diffuse material for which the spectral absorptivity is  0.6 for 3
m and  0.3 for
3
m. When it is on the dark side of the earth, the satellite sees irradiation from the earths surface only. The irradiation may be assumed to be incident as parallel rays, and its magnitude is GE 340 W/m2 . On the bright side of the earth the satellite sees the earth irradiation GE plus the solar irradiation GS 1368 W/m2 . The spectral distribution of radiation from the earth may be approximated as that of a blackbody at 280 K, and the temperature of the satellite may be assumed to remain below 500 K. What is the steady-state temperature of the satellite when it is on the dark side of the earth and when it is on the bright side? 12.131

A radiator on a proposed satellite solar power station must dissipate heat being generated within the satellite by radiating it into space. The radiator surface has a solar absorptivity of 0.5 and an emissivity of 0.95. What is the equilibrium surface temperature when the solar irradiation is 1000 W/m2 and the required heat dissipation is 1500 W/m2 ?

A spherical satellite in near-earth orbit is exposed to solar irradiation of 1368 W/m2 . To maintain a desired operating temperature, the thermal control engineer intends to use a checker pattern for which a fraction F of the satellite surface is coated with an evaporated aluminum film ( 0.03, S 0.09), and the fraction (1F) is coated with a white, zinc-oxide paint ( 0.85, S 0.22). Assume the satellite is isothermal and has no internal power dissipation. Determine the fraction F of the checker pattern required to maintain the satellite at 300 K. 12

An annular fin of thickness t is used as a radiator to dissipate heat for a space power system. The fin is insulated on the bottom and may be exposed to solar irradiation GS. The fin is coated with a diffuse, spectrally selective material whose spectral reflectivity is specified. Heat is conducted to the fin through a solid rod of radius ri , and the exposed upper surface of the fin radiates to free space, which is essentially at absolute zero temperature. (a) If conduction through the rod maintains a fin base temperature of T(ri ) Tb 400 K and the fin efficiency is 100%, what is the rate of heat dissipation for a fin of radius ro 0.5 m? Consider two cases, one for which the radiator is exposed to the sun with GS 1000 W/m2 and the other with no exposure (GS 0). (b) In practice, the fin efficiency will be less than 100% and its temperature will decrease with increasing radius. Beginning with an appropriate control volume, derive the differential equation that determines the steady-state, radial temperature distribution in the fin. Specify appropriate boundary conditions. 12.

A rectangular plate of thickness t, length L, and width W is proposed for use as a radiator in a spacecraft application. The plate material has a thermal conductivity of 300 W/m
K, a solar absorptivity of 0.45, and an emissivity of 0.9. The radiator is exposed to solar radiation only on its top surface, while both surfaces are exposed to deep space at a temperature of 4 K. (a) If the base of the radiator is maintained at Tb 80 C, what is its tip temperature and the rate of heat rejection? Use a computer-based, finite-difference method with a space increment of 0.1 m to obtain your solution. (b) Repeat the calculation of part (a) for the case when the space ship is on the dark side of the earth and is not exposed to the sun. (c) Use your computer code to calculate the heat rate and tip temperature for GS 0 and an extremely large value of the thermal conductivity. Compare your results to those obtained from a hand calculation that assumes the radiator to be at a uniform temperature Tb. What other approach might you use to validate your code?

The directional absorptivity of a gray surface varies with as follows. (a) What is the ratio of the normal absorptivity n to the hemispherical emissivity of the surface? (b) Consider a plate with these surface characteristics on both sides in earth orbit. If the solar flux incident on one side of the plate is , what equilibrium temperature will the plate assume if it is oriented normal to the suns rays? What temperature will it assume if it is oriented at 75 to the suns rays?

Two special coatings are available for application to an absorber plate installed below the cover glass described in Example 12.9. Each coating is diffuse Which coating would you select for the absorber plate? Explain briefly. For the selected coating, what is the rate at which radiation is absorbed per unit area of the absorber plate if the total solar irradiation at the cover glass is GS 1000 W/m2 ?

Consider the spherical satellite of Problem 12.130. Instead of the entire satellite being coated with a material that is spectrally selective, half of the satellite is covered with a diffuse gray coating characterized by 1 0.6. The other half of the satellite is coated with a diffuse gray material with 2 0.3. (a) Determine the steady-state satellite temperature when the satellite is on the bright side of the earth with the high-absorptivity coating facing the sun. Determine the steady-state satellite temperature when the low-absorptivity coating faces the sun. Hint: Assume one hemisphere of the satellite is irradiated by the sun and the opposite hemisphere is irradiated by the earth. (b) Determine the steady-state satellite temperature when the satellite is on the dark side of the earth with the high-absorptivity coating facing the earth. Determine the steady-state satellite temperature when the low-absorptivity coating faces the earth. (c) Identify a scheme to minimize the temperature variations of the satellite as it travels between the bright and dark sides of the earth.

A spherical capsule of 3-m radius is fired from a space platform in earth orbit, such that it travels toward the center of the sun at 16,000 km/s. Assume that the capsule is a lumped capacitance body with a densityspecific heat product of 4 106 J/m3
K and that its surface is black. (a) Derive a differential equation for predicting the capsule temperature as a function of time. Solve this equation to obtain the temperature as a function of time in terms of capsule parameters and its initial temperature Ti (b) If the capsule begins its journey at 20 C, predict the position of the capsule relative to the sun at which its destruction temperature, 150 C, is reached.

The spectral absorptivity of aluminum coated with a thin layer of silicon dioxide may be approximated as ,1 0.98 for  c and ,2 0.05 for  c where the cutoff wavelength is c 0.15
m under normal circumstances. (a) Determine the equilibrium temperature of a flat piece of the coated aluminum that is exposed to solar irradiation, GS 1368 W/m2 on its upper surface. The opposite surface is insulated. (b) The cutoff wavelength can be modified by varying the coating thickness. Determine the value of c that will maximize the equilibrium temperature of the surface. 12.140 Co

Consider the spherical satellite of Problem 12.130. By changing the thickness of the diffuse material used for the coating, engineers can control the cutoff wavelength that marks the boundary between  0.6 and  0.3. (a) What cutoff wavelength will minimize the steadystate temperature of the satellite when it is on the bright side of the earth? Using this coating, what wi

A solar panel mounted on a spacecraft has an area of 1 m2 and a solar-to-electrical power conversion effi- ciency of 12%. The side of the panel with the photovoltaic array has an emissivity of 0.8 and a solar absorptivity of 0.8. The back side of the panel has an emissivity of 0.7. The array is oriented normal to solar irradiation of 1500 W/m2 . (a) Determine the steady-state temperature of the panel and the electrical power (W) produced for the prescribed conditions. (b) If the panel were a thin plate without the solar cells, but with the same radiative properties, determine the temperature of the plate for the prescribed conditions. Compare this result with that from part (a). Are they the same or different? Explain why. (c) Determine the temperature of the solar panel 1500 s after the spacecraft is eclipsed by a planet. The thermal capacity of the panel per unit area is 9000 J/m . 2
K