Our students perform a laboratory experiment todetermine

Consider an opaque horizontal plate that is well insu-lated on its back side. The irradiation on the plate is2500 W/m2, of which 500 W/m2is reflected. The plateis at 227C and has an emissive power of 1200 W/m2.Air at 127C flows over the plate with a heat transferconvection coefficient of 15 W/m2?K. Determine theemissivity, absorptivity, and radiosity of the plate.What is the net heat transfer rate per unit area?

A horizontal, opaque surface at a steady-state tempera-ture of 77C is exposed to an airflow having a freestream temperature of 27C with a convection heattransfer coefficient of 28 W/m2?K. The emissive powerof the surface is 628 W/m2, the irradiation is1380 W/m2, and the reflectivity is 0.40. Determine theabsorptivity of the surface. Determine the net radiationheat transfer rate for this surface. Is this heat transfer tothe surface or from the surface? Determine the com-bined heat transfer rate for the surface. Is this heattransfer to the surface or from the surface?

The top surface of an L?5-mm-thick anodized alu-minum plate is irradiated with G?1000 W/m2whilebeing simultaneously exposed to convection condi-tions characterized by h?40 W/m2?K and T??30C.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 sur-face, the net radiation heat flux at the top surface, andthe rate at which the temperature of the plate is chang-ing with time.

A horizontal semitransparent plate is uniformly irradi-ated from above and below, while air at T??300 Kflows over the top and bottom surfaces, providinga uniform convection heat transfer coefficient ofh?40 W/m2?K. The absorptivity of the plate to theirradiation is 0.40. Under steady-state conditions mea- surements made with a radiation detector above thetop surface indicate a radiosity (which includes trans-mission, as well as reflection and emission) ofJ?5000 W/m2, while the plate is at a uniform tem- perature of T?350 K. Determine the irradiation Gand the emissivity of the plate. Is the plate gray (???) for the prescribedconditions?

What is the irradiation at surfaces A2, A3, and A4ofExample 12.1 due to emission from A1?

Consider a small surface of area A1?10?4m2, whichemits diffusely with a total, hemispherical emissivepower ofE1?5?104W/m2.(a) At what rate is this emission intercepted by a smallsurface of area A2?5?10?4m2, which is ori-ented as shown?(b) What is the irradiation G2on A2?(c) For zenith angles of 2?0, 30, and 60, plot G2as a function of the separation distance for 0.25r21.0 m.

A furnace with an aperture of 20-mm diameter and emis-sive power of 3.72?105W/m2is used to calibrate a heatflux gage having a sensitive area of 1.6?10?5m2.(a) At what distance, measured along a normal fromthe aperture, should the gage be positioned toreceive irradiation of 1000 W/m2?(b) If the gage is tilted off normal by 20, what will beits irradiation?(c) For tilt angles of 0, 20, and 60, plot the gageirradiation as a function of the separation distancefor values ranging from 100 to 300 mm.

A small radiant source A1emits diffusely withan intensity I1?1.2?105W/m2?sr. The radiationdetector A2is aligned normal to the source at adistance of Lo?0.2 m. An opaque screen is posi-tioned midway between A1and A2to prevent radia-tion from the source reaching the detector. The smallsurface Amis a perfectly diffuse mirror that permitsradiation emitted from the source to be reflected intothe detector.(a) Calculate the radiant power incident on Amdue toemission from the source A1, q1lm(W).(b) Assuming that the radiant power, q1lm, is perfectlyand diffusely reflected, calculate the intensity leav-ing Am, Im(W/m2?sr).(c) Calculate the radiant power incident on A2due tothe reflected radiation leavingAm, qml2(?W).(d) Plot the radiant power qml2as a function of the lat-eral separation distance yofor the range 0yo0.2 m. Explain the features of the resulting curve.

According to its directional distribution, solar radiationincident on the earths surface may be divided into twocomponents. The directcomponent consists of parallelrays incident at a fixed zenith angle , while the diffusecomponent consists of radiation that may be approxi-mated as being diffusely distributed with .Consider clear sky conditions for which the direct radi-ation is incident at ?30, with a total flux (based onan area that is normal to the rays) of ,and the total intensity of the diffuse radiation isIdif?70 W/m2?sr. What is the total solar irradiation atthe earths surface?

Solar radiation incident on the earths surface may be divided into the direct and diffuse componentsdescribed in Problem 12.9. Consider conditions for a day in which the intensity of the direct solar radia-tion is Idir?210?107W/m2?sr in the solid anglesubtended by the sun with respect to the earth,s?6.74?10?5sr. The intensity of the diffuseradiation is Idif?70 W/m2?sr.(a) What is the total solar irradiation at the earthssurface when the direct radiation is incident at?30?(b) Verify the prescribed value for s, recognizingthat the diameter of the sun is 1.39?109m andthe distance between the sun and the earth is1.496?1011m (1 astronomical unit).

On an overcast day the directional distribution of thesolar radiation incident on the earths surface may beapproximated by an expression of the form Ii?Incos ,where In?80 W/m2?sr is the total intensity of radiationdirected 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 (asin the case of a rubber balloon). Irradiation on the filmis due to emission from a radiant heater of areaAh?0.0052 m2, which emits diffusely with an inten-sity of Ie,h?169,000 W/m2?sr.(a) Obtain an expression for the irradiation on the filmas a function of the zenith angle .(b) Based on the expressions derived in part (a),which shape provides the more uniform irradiationGand hence provides better quality control for thetreatment process?

To initiate a process operation, an infrared motion sen-sor (radiation detector) is employed to determine theapproach of a hot part on a conveyor system. To setthe sensors amplifier discriminator, the engineerneeds a relationship between the sensor output signal, S, and the position of the part on the conveyor. The sen-sor output signal is proportional to the rate at whichradiation is incident on the sensor.(a) For Ld?1 m, at what location x1will the sensorsignal S1be 75% of the signal corresponding to theposition directly beneath the sensor, So(x?0)?(b) For values of Ld?0.8, 1.0, and 1.2 m, plot the sig-nal ratio, S/So, versus part position, x, for signalratios in the range from 0.2 to 1.0. Compare the x- locations for which S/So?0.75

A small radiant heat source of area A1?2?10?4m2emits diffusely with an intensity I1?1000 W/m2?sr.A second small area, A2?1?10?4m2, is located asshown in the sketch.(a) Determine the irradiation of A2for L20? .5 m.(b) Plot the irradiation of A2over the range 0L210 m

Determine the fraction of the total, hemisphericalemissive power that leaves a diffuse surface in thedirections /4/2 and 0

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

Consider a 5-mm-square, diffuse surface Aohavinga total emissive power of Eo?4000 W/m2. Theradiation field due to emission into the hemispheri-cal space above the surface is diffuse, therebyproviding a uniform intensity I(,). Moreover,if the space is a nonparticipating medium (nonab- sorbing, nonscattering, and nonemitting), the inten-sity is independent of radius for any (, ) direction.Hence intensities at any points P1and P2wouldbe equal.(a) What is the rate at which radiant energy is emittedby Ao, qemit?(b) What is the intensity Io,eof the radiation field emit-ted from the surface Ao?(c) Beginning with Equation 12.13 and presumingknowledge of the intensity Io,e, obtain an expres-sion for qemit.(d) Consider the hemispherical surface located atr?R1?0.5 m. Using the conservation of energyrequirement, determine the rate at which radiantenergy is incident on this surface due to emissionfrom Ao.(e) Using Equation 12.10, determine the rate at whichradiant energy leaving Aois intercepted by thesmall area A2located in the direction (45, ) onthe hemispherical surface. What is the irradiationon A2?(f) Repeat part (e) for the location (0, ). Are theirradiations at the two locations equal?(g) Using Equation 12.18, determine the irradiationG1on the hemispherical surface at r? 1

Assuming blackbody behavior, determine the temper-ature of, and the energy emitted by, areas A1in Exam-ple 12.1 and Problems 12.8 and 12.14, as well as areaAhin Problem 12.12.

The dark surface of a ceramic stove top may beapproximated as a blackbody. The burners, whichare integral with the stove top, are heated from belowby electric resistance heaters.(a) Consider a burner of diameter D?200 mmoperating at a uniform surface temperature ofTs?250C in ambient air at T??20C. With-out a pot or pan on the burner, what are the ratesof heat loss by radiation and convection fromthe burner? If the efficiency associated withenergy transfer from the heaters to the burners is90%, what is the electric power requirement?At what wavelength is the spectral emission amaximum?(b) Compute and plot the effect of the burner tempera-ture on the heat rates for 100Ts350C.

The energy flux associated with solar radiation incidenton the outer surface of the earths atmosphere has beenaccurately measured and is known to be 1368 W/m2.The diameters of the sun and earth are 1.39?109and1.27?107m, respectively, and the distance betweenthe sun and the earth is 1.5?1011m.(a) What is the emissive power of the sun?(b) Approximating the suns surface as black, what isits temperature?(c) At what wavelength is the spectral emissive powerof the sun a maximum?(d) Assuming the earths surface to be black and thesun to be the only source of energy for the earth,estimate the earths surface temperature.

A small flat plate is positioned just beyond the earthsatmosphere and is oriented such that the normal to theplate passes through the center of the sun. Refer toProblem 12.20 for pertinent earthsun dimensions (a) What is the solid angle subtended by the sun abouta point on the surface of the plate?(b) Determine the incident intensity, Ii, on the plateusing the known value of the solar irradiationabove the earths atmosphere (GS? 1368 W/m2).(c) Sketch the incident intensity Iias a function of thezenith angle , where is measured from the nor-mal to the plate.

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

The extremely high temperatures needed to triggernuclear fusion are proposed to be generated by laser-irradiating a spherical pellet of deuterium and tritiumfuel of diameter Dp?1.8 mm.(a) Determine the maximum fuel temperature that canbe achieved by irradiating the pellet with 200lasers, each producing a power of P?500 W. Thepellet 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 ofthe enclosure and have a diameter of DLEH?2 mm.Determine the maximum temperature that can begenerated within the enclosure.

An enclosure has an inside area of 100 m2, and itsinside surface is black and is maintained at a constanttemperature. A small opening in the enclosure has anarea of 0.02 m2. The radiant power emitted from thisopening is 70 W. What is the temperature of the inte-rior enclosure wall? If the interior surface is main-tained at this temperature, but is now polished, whatwill be the value of the radiant power emitted from theopening?

Assuming the earths surface is black, estimate itstemperature if the sun has an equivalent blackbody temperature of 5800 K. The diameters of the sun andearth are 1.39?109and 1.27?107m, respectively,and the distance between the sun and earth is1.51 ? 011m.

A proposed method for generating electricity fromsolar irradiation is to concentrate the irradiation into a cavity that is placed within a large containerof a salt with a high melting temperature. If all heatlosses are neglected, part of the solar irradiationentering the cavity is used to melt the salt whilethe remainder is used to power a Rankine cycle.(The salt is melted during the day and is resolidifiedat night in order to generate electricity around the clock.)Consider conditions for which the solar power enter-ing the cavity is qsol7.50 MW and the time rate ofchange of energy stored in the salt is .For a cavity opening of diameter Ds1 m, determinethe heat transfer to the Rankine cycle, qR. The temper-ature of the salt is maintained at its melting point,Tsalt?Tm?1000C. Neglect heat loss by convectionand irradiation from the surroundings.

Approximations to Plancks law for the spectral emis-sive power are the Wien and Rayleigh-Jeans spectraldistributions, which are useful for the extreme lowand high limits of the product ?T, respectively.(a) Show that the Planck distribution will have theformwhen C2/?T?1 and determine the error (comparedto the exact distribution) for the condition ?T?2898 ?m?K. This form is known as Wiens law. (b) Show that the Planck distribution will have the formwhen C2/?T?1 and determine the error (com-pared to the exact distribution) for the condition?T?100,000?m?K. This form is known as theRayleigh-Jeans law.

Estimate the wavelength corresponding to maximumemission from each of the following surfaces: the sun,a tungsten filament at 2500 K, a heated metal at1500 K, human skin at 305 K, and a cryogenicallycooled metal surface at 60 K. Estimate the fraction ofthe solar emission that is in the following spectralregions: the ultraviolet, the visible, and the infrared.

Thermal imagers have radiation detectors that are sen-sitive to a spectral region and provide white- black orcolor images with shading to indicate relative tempera-ture differences in the scene. The imagers, which haveappearances similar to a video camcorder, have numer-ous applications, such as for equipment maintenance toidentify overheated motors or electrical transformersand for fire-fighting service to determine the directionof 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 particularregion typically depends on the temperature of thescene, although the atmospheric conditions (watervapor, smoke, etc.) may also be important.(a) Determine the band emission fractions for each ofthe spectral regions, 3 to 5?m and 8 to 14?m, fortemperatures of 300 and 900 K.(b) Using the Tools/Radiation/Band Emission Factorfeature within IHT, calculate and plot the bandemission factors for each of the spectral regionsfor the temperature range 300 to 1000 K. Identifythe temperatures at which the fractions are a maxi-mum. What conclusions can you draw from thisgraph concerning the choice of an imager for anapplication?(c) The noise-equivalent temperature (NET) is aspecification of the imager that indicates the mini-mum temperature change that can be resolved in the image scene. Consider imagers operating at the maximum-fraction temperatures identified inpart (b). For each of these conditions, determinethe sensitivity (%) required of the radiation detec-tor in order to provide a NET of 5C. Explain thesignificance of your results. Note:The sensitivity(% units) can be defined as the difference in the band emission fractions for two temperatures dif-fering by the NET, divided by the band emissionfraction at one of the temperatures.

A furnace with a long, isothermal, graphite tube ofdiameter D?12.5 mm is maintained at Tf?2000 Kand is used as a blackbody source to calibrate heat fluxgages. Traditional heat flux gages are constructed asblackened thin films with thermopiles to indicate thetemperature change caused by absorption of the inci-dent radiant power over the entire spectrum. The tradi-tional gage of interest has a sensitive area of 5 mm2and is mounted coaxial with the furnace centerline, butpositioned at a distance of L?60 mm from the begin-ning of the heated section. The cool extension tubeserves to shield the gage from extraneous radiationsources and to contain the inert gas required to preventrapid oxidation of the graphite tube.(a) Calculate the heat flux (W/m2) on the traditionalgage for this condition, assuming that the exten-sion 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 between0.4 and 2.5?m. Calculate the radiant heat fluxincident on the solid-state gage within the pre- scribed spectral region.(c) Calculate and plot the total heat flux and the heatflux in the prescribed spectral region for the solid-state gage as a function of furnace temperature forthe range 2000Tf3000 K. Which gage willhave an output signal that is more sensitive tochanges in the furnace temperature?

Photovoltaic materials convert sunlight directly toelectric power. Some of the photons that are incidentupon the material displace electrons that are in turncollected to create an electric current. The overall effi-ciency of a photovoltaic panel, , is the ratio of elec-trical energy produced to the energy content of theincident radiation. The efficiency depends primarilyon two properties of the photovoltaic material, (i) theband gap, which identifies the energy states of photonshaving 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, ?bgFbgwhere Fbgis the fraction of thephoton energy incident on the surface within the bandgap. Photons that are either outside the materialsband gap or within the band gap but not converted toelectrical energy are either reflected from the panel orabsorbed and converted to thermal energy.Consider a photovoltaic material with a band gapof 1.1B1.8 eV, where Bis the energy state of aphoton. The wavelength is related to the energy stateof a photon by the relationship ??1240 eV?nm/B.The incident solar irradiation approximates that of ablackbody at 5800 K and GS?1000 W/m2.(a) Determine the wavelength range of solar irradia-tion corresponding to the band gap.(b) Determine the overall efficiency of the photo- voltaic material if the interband gap efficiency isbg?0.50.(c) If half of the incident photons that are not con-verted to electricity are absorbed and converted tothermal energy, determine the heat absorption perunit surface area of the panel.

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

Isothermal furnaces with small apertures approximat-ing a blackbody are frequently used to calibrate heat flux gages, radiation thermometers, and other radio-metric devices. In such applications, it is necessary tocontrol power to the furnace such that the variation oftemperature and the spectral intensity of the apertureare within desired limits.(a) By considering the Planck spectral distribution,Equation 12.30, show that the ratio of the frac-tional change in the spectral intensity to the frac-tional change in the temperature of the furnacehas the form(b) Using this relation, determine the allowable vari-ation in temperature of the furnace operating at 2000 K to ensure that the spectral intensity at0.65?m will not vary by more than 0.5%. Whatis the allowable variation at 10?m?

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

A small metal object, initially at Ti?1000 K, iscooled by radiation in a low-temperature vacuumchamber. One of two thin coatings can be applied tothe object so that spectral hemispherical emissivitiesvary with wavelength as shown. For which coatingwill the object most rapidly reach a temperature ofTf?500 K?

The directional total emissivity of nonmetallic materi-als may be approximated as ???ncos, where ?nisthe normal emissivity. Show that the total hemispheri-cal emissivity for such materials is 2/3 of the normal emissivity.

Consider the metallic surface of Example 12.7. Addi-tional measurements of the spectral, hemisphericalemissivity yield a spectral distribution which may beapproximated as follows:(a) Determine corresponding values of the total, hemi-spherical emissivity ?and the total emissive powerEat 2000 K.(b) Plot the emissivity as a function of temperature for500T3000 K. Explain the variation.

The spectral emissivity of unoxidized titanium at roomtemperature is well described by the expression???0.52??0.5for 0.3?m?30?m.(a) Determine the emissive power associated with anunoxidized titanium surface at T?300 K. Assumethe spectral emissivity is ???0.1 for ??30?m.(b) Determine the value of ?maxfor the emissive powerof the surface in part (a).

The spectral, directional emissivity of a diffuse mate-rial at 2000 K has the following distribution:Determine the total, hemispherical emissivity at 2000 K.Determine the emissive power over the spectral range0.8 to 2.5?m andfor the directions 030

A diffuse surface is characterized by the spectralhemispherical emissivity distribution shown. Consider-ing surface temperatures over the range 300Ts 1000 K, at what temperature will the emissive powerbe minimized?

Consider the directionally selective surface having thedirectional emissivity ?, as shown,Assuming that the surface is isotropic in the direc-tion, calculate the ratio of the normal emissivity ?ntothe hemispherical emissivity h ? .

A sphere is suspended in air in a dark room and main-tained at a uniform incandescent temperature. Whenfirst viewed with the naked eye, the sphere appears to bebrighter around the rim. After several hours, however, itappears to be brighter in the center. Of what type mate-rial would you reason the sphere is made? Give plausi-ble reasons for the nonuniformity of brightness of thesphere and for the changing appearance with time.

A proposed proximity meteris based on the physicalarrangement of Problem 12.14. The sensing area of ameter that is installed on a vehicle, A2, is irradiated bya stationary warm object, A1. The sensors electricaloutput signal is proportional to its irradiation.(a) The object temperature and emissivity are 200Cand ??0.85, respectively. Determine the distance,L2,crit, associated with the maximum sensor outputsignal. Assume the object is a diffuse emitter.(b) If the object emits as a nonmetallic material, thetotal directional emissivity may be approximated as???ncos, where ?nis the normal emissivity(Problem 12.36). Determine the distance L2,critasso-ciated with the maximum sensor output signal.

Estimate the total, hemispherical emissivity ?for pol-ished stainless steel at 800 K using Equation 12.43 alongwith information provided in Figure 12.17. Assume thatthe hemispherical emissivity is equal to the normalemissivity. Perform the integration using a band calcu-lation, by splitting the integral into five bands, each ofwhich contains 20% of the blackbody emission at800 K. For each band, assume the average emissivity isthat associated with the median wavelength within theband ?m, for which half of the blackbody radiationwithin the band is above ?m(and half is below ?m). Forexample, the first band runs from ??0 to ?1, such thatF(0l?1)?0.2, and the median wavelength for the firstband is chosen such that F(0l?m)?0.1. Also determinethe surface emissive power

A radiation thermometer is a device that responds to aradiant flux within a prescribed spectral interval and iscalibrated to indicate the temperature of a blackbodythat produces the same flux.(a) When viewing a surface at an elevated tempera-ture Tsand emissivity less than unity, the ther-mometer will indicate an apparent temperaturereferred to as the brightness or spectral radiancetemperature T?. Will T?be greater than, less than,or equal to Ts?(b) Write an expression for the spectral emissivepower of the surface in terms of Wiens spectraldistribution (see Problem 12.27) and the spectralemissivity of the surface. Write the equivalentexpression using the spectral radiance temperatureof the surface and show thatwhere ? represents the wavelength at which thethermometer operates.(c) Consider a radiation thermometer that responds to aspectral flux centered about the wavelength 0.65?m.What temperature will the thermometer indicatewhen viewing a surface with ??(0.65?m)?0.9and Ts?1000 K? Verify that Wiens spectral dis-tribution is a reasonable approximation to Planckslaw for this situation.

For a prescribed wavelength ?, measurement of thespectral intensity I?,e(?, T)???I?,bof radiation emittedby a diffuse surface may be used to determine the sur-face temperature, if the spectral emissivity ??is known,or the spectral emissivity, if the temperature is known. a) Defining the uncertainty of the temperature determination as dT/T, obtain an expression relat-ing this uncertainty to that associated with theintensity measurement, dI?/I?. For a 10% uncer-tainty in the intensity measurement at ??10?m,what is the uncertainty in the temperature forT?500 K? For T?1000 K?(b) Defining the uncertainty of the emissivity determi-nation as d??/??, obtain an expression relating thisuncertainty to that associated with the intensitymeasurement, dI?/I?. For a 10% uncertainty in theintensity measurement, what is the uncertainty inthe emissivity?

Sheet steel emerging from the hot roll section of asteel mill has a temperature of 1200 K, a thickness of??3 mm, and the following distribution for the spec-tral, hemispherical emissivity.The density and specific heat of the steel are7900 kg/m3and 640 J/kg?K, respectively. What is thetotal, hemispherical emissivity? Accounting for emis-sion from both sides of the sheet and neglectingconduction, convection, and radiation from the sur-roundings, determine the initial time rate of change ofthe sheet temperature (dT/dt)i. As the steel cools, it oxi-dizes and its total, hemispherical emissivity increases.If this increase may be correlated by an expression ofthe form ???1200[1200 K/T(K)], how long will it takefor the steel to cool from 1200 to 600 K?

A large body of nonluminous gas at a temperature of1200 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 theemissive power of this gas.

An opaque surface with the prescribed spectral, hemi-spherical reflectivity distribution is subjected to thespectral irradiation shown (a) Sketch the spectral, hemispherical absorptivitydistribution.(b) Determine the total irradiation on the surface.(c) Determine the radiant flux that is absorbed by thesurface.(d) What is the total, hemispherical absorptivity ofthis surface?

A small, opaque, diffuse object at Ts?400 K is sus-pended in a large furnace whose interior walls are atTf?2000 K. The walls are diffuse and gray and havean emissivity of 0.20. The spectral, hemisphericalemissivity for the surface of the small object is givenbelow.(a) Determine the total emissivity and absorptivity ofthe surface.(b) Evaluate the reflected radiant flux and the netradiative flux tothe surface.(c) What is the spectral emissive power at ??2?m?(d) What is the wavelength ?1/2for which one-half ofthe total radiation emitted by the surface is in thespectral region ???1/2?

The spectral reflectivity distribution for white paint(Figure 12.22) can be approximated by the followingstair-step function:?(?m)?0.4 0.43.0?3.0??0.75 0.150.96A small flat plate coated with this paint is suspendedinside a large enclosure, and its temperature is main-tained at 400 K. The surface of the enclosure is maintained at 3000 K and the spectral distribution ofits emissivity has the following characteristics:?(?m)?2.0?2.0??0.2 0.9(a) Determine the total emissivity, ?, of the enclosuresurface.(b) Determine the total emissivity, ?, and absorptivity,?, of the plate.

An opaque surface, 2 m ?2 m, is maintained at 400 Kand is simultaneously exposed to solar irradiation withGS?1200 W/m2. The surface is diffuse and its spec-tral 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 irra-diation, emissive power, radiosity, and net radiationheat transfer from the surface.

Consider Problem 4.51.(a) The students are each given a flat, first-surfacsilver mirror with which they collectively irradi-ate the wooden ship at location B. The reflectionfrom the mirror is specular, and the silvers reflec-tivity is 0.98. The solar irradiation of each mirror,perpendicular to the direction of the suns rays, isGS?1000 W/m2. How many students are neededto conduct the experiment if the solar absorptivityof the wood is ?w?0.80 and the mirror is ori-ented at an angle of 45from the direction of GS?(b) If the students are given second-surfacemirrorsthat consist of a sheet of plain glass that has pol-ished silver on its back side, how many studentsare needed to conduct the experiment? Hint: SeeProblem 12.62.

A diffuse, opaque surface at 700 K has spectral emis-sivities of ???0 for 0?3?m, ???0.5 for3?m??10?m, and ???0.9 for 10?m????.A radiant flux of 1000 W/m2, which is uniformly dis-tributed between 1 and 6?m, is incident on the sur-face at an angle of 30relative to the surface normal. Calculate the total radiant power from a 10?4m2areaof the surface that reaches a radiation detector posi-tioned along the normal to the area. The aperture ofthe detector is 10?5m2, and its distance from the sur-face 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 0.7and is maintained at 900 K. The hemispherical enclo-sure, maintained at 300 K, has a radius of 100 mm andan emissivity of 0.85.Calculate the radiant power leaving an aperture ofdiameter 2 mm located on the enclosure as shown.

The spectral, hemispherical absorptivity of an opaquesurface is as shown.What is the solar absorptivity, ?S? If it is assumed that?????and that the surface is at a temperature of340 K, what is its total, hemispherical emissivity?

The spectral, hemispherical absorptivity of an opaquesurface and the spectral distribution of radiation inci-dent on the surface are as shown. What is the total, hemispherical absorptivity of the sur-face? If it is assumed that ?????and that the surface isat 1000 K, what is its total, hemispherical emissivity?What is the net radiant heat flux to the surface?

Consider an opaque, diffuse surface for which thespectral absorptivity and irradiation are as follows:What is the total absorptivity of the surface for theprescribed irradiation? If the surface is at a tempera-ture of 1250 K, what is its emissive power? How willthe surface temperature vary with time, for the pre-scribed conditions?

The spectral emissivity of an opaque, diffuse surface isas shown.(a) If the surface is maintained at 1000 K, what is thetotal, hemispherical emissivity?(b) What is the total, hemispherical absorptivity of thesurface when irradiated by large surroundings ofemissivity 0.8 and temperature 1500 K?(c) What is the radiosity of the surface when it ismaintained at 1000 K and subjected to the irradia-tion prescribed in part (b)? (d) Determine the net radiation flux into the surfacefor the conditions of part (c).(e) Plot each of the parameters featured in parts (a)(d)as a function of the surface temperature for 750T2000 K.

Radiation leaves a furnace of inside surface temperature1500 K through an aperture 20 mm in diameter. A por-tion of the radiation is intercepted by a detector that is1 m from the aperture, has a surface area of 10?5m2,and is oriented as shown.If the aperture is open, what is the rate at which radia-tion leaving the furnace is intercepted by the detector?If the aperture is covered with a diffuse, semitrans-parent material of spectral transmissivity ???0.8 for?2?m and ???0 for ??2?m, what is the rateat which radiation leaving the furnace is interceptedby the detector?

The spectral transmissivity of a 1-mm-thick layer ofliquid water can be approximated as follows:(a) Liquid water can exist only below its critical tem-perature, Tc?647.3 K. Determine the maximumpossible total transmissivity of a 1-mm-thick layerof liquid water when the water is housed in anopaque container and boiling does not occur.Assume the irradiation is that of a blackbody.(b) Determine the transmissivity of a 1-mm-thicklayer of liquid water associated with melting theplatinum wire used in Nukiyamas boiling experi-ment, as described in Section 10.3.1.(c) Determine the total transmissivity of a 1-mm-thicklayer 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 glasscan be approximated as follows:Plain glass:???0.9 0.3?2.5?mTinted glass:???0.9 0.5?1.5? Outside the specified wavelength ranges, the spectraltransmissivity is zero for both glasses. Compare the solarenergy that could be transmitted through the glasses.With solar irradiation on the glasses, compare the visi-ble radiant energy that could be transmitted

Referring to the distribution of the spectral transmis-sivity of low iron glass (Figure 12.23), describe brieflywhat is meant by the greenhouse effect. That is, howdoes the glass influence energy transfer to and fromthe 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/m2and thespectral distribution of a blackbody at 5800 K isincident on this material, determine the fractionsof the irradiation that are transmitted, reflected,and absorbed by the material.(c) If the temperature of this material is 350 K, deter-mine the emissivity ?.(d) Determine the net heat flux by radiation to thematerial.

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

Four diffuse surfaces having the spectral characteristicsshown are at 300 K and are exposed to solar radiation.Which of the surfaces may be approximated as beinggray

Consider a material that is gray, but directionallyselective with ?(,)?0.5(1?cos). Determinethe hemispherical absorptivity when collimated solarflux irradiates the surface of the material in the direction?45and ?0. Determine the hemispherical emis-sivity ?of the material.

The spectral transmissivity of a 50-?m-thick polymerfilm is measured over the wavelength range 2.5?m?15?m. The spectral distribution may be approxi-mated as ???0.80 for 2.5?m?7?m, ???0.05for 7?m??13?m, and ???0.55 for 13?m??15?m. Transmissivity data outside the range can-not be acquired due to limitations associated with theinstrumentation. An engineer wishes to determine the total transmissivity of the film.(a) Estimate the maximum possibletotal transmissiv-ity of the film associated with irradiation from ablackbody at T?30C.(b) Estimate the minimum possibletotal transmissivityof the film associated with irradiation from ablackbody at T?30C.(c) Repeat parts (a) and (b) for a blackbody at T?600C.

An opaque, horizontal plate has a thickness ofL?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?25C. Air flows above the plate at T?,a?260C with ha?40 W/m2?K. The top of the plate is diffuse and is irradiated withG?1450 W/m2, of which 435 W/m2is reflected. The steady-state top and bottom plate temperatures are Tt?43C and Tb?35C, respectively. Determine the transmissivity, reflectivity, absorptivity, and emis-sivity of the plate. Is the plate gray? What is theradiosity associated with the top of the plate? What isthe convection heat transfer coefficient associatedwith the water flow?

Two small surfaces, A and B, are placed inside anisothermal enclosure at a uniform temperature. Theenclosure provides an irradiation of 6300 W/m2toeach of the surfaces, and surfaces A and B absorbincident radiation at rates of 5600 and 630 W/m2,respectively. Consider conditions after a long timehas 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 charac-teristics is maintained at 500 K when situated in alarge furnace enclosure whose walls are maintained at1500 K:(a) Sketch the spectral distribution of the surfaceemissive power E?and the emissive power E?,bthat the surface would have if it were a blackbody.(b) Neglecting convection effects, what is the netheat flux to the surface for the prescribedconditions?(c) Plot the net heat flux as a function of the surfacetemperature for 500T1000 K. On the samecoordinates, plot the heat flux for a diffuse, graysurface with total emissivities of 0.4 and 0.8.(d) For the prescribed spectral distribution of ??, howdo the total emissivity and absorptivity of the sur-face vary with temperature in the range 500T1000 K?

Consider an opaque, diffuse surface whose spectralreflectivity varies with wavelength as shown. The sur-face is at 750 K, and irradiation on one side varies withwavelength as shown. The other side of the surface isinsulated.What are the total absorptivity and emissivity ofthe surface? What is the net radiative heat flux to thesurface?

A special diffuse glass with prescribed spectral radia-tive properties is heated in a large oven. The walls ofthe oven are lined with a diffuse, gray refractory brickhaving an emissivity of 0.75 and are maintained atTw?1800 K. Consider conditions for which the glasstemperature is Tg?750 K.(a) What are the total transmissivity ?, the total reflec-tivity ?, 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, and2000 K, plot as a function of glass tempera-ture for 500Tg800 K.

The 50-mm peephole of a large furnace operating at450C is covered with a material having ??0.8 and ??0 for irradiation originating from the furnace. Thematerial has an emissivity of 0.8 and is opaque to irradi-ation from a source at room temperature. The outer sur-face of the cover is exposed to surroundings and ambientair at 27C with a convection heat transfer coefficient of50 W/m2?K. Assuming that convection effects on theinner surface of the cover are negligible, calculate theheat loss by the furnace and the temperature of the cover.

The window of a large vacuum chamber is fabricatedfrom a material of prescribed spectral characteristics.A collimated beam of radiant energy from a solar sim-ulator is incident on the window and has a flux of3000 W/m2. The inside walls of the chamber, whichare large compared to the window area, are maintainedat 77 K. The outer surface of the window is subjectedto surroundings and room air at 25C, with a convec-tion heat transfer coefficient of 15 W/m2?K.(a) Determine the transmissivity of the window mate-rial to radiation from the solar simulator, whichapproximates the solar spectral distribution.(b) Assuming that the window is insulated from itschamber mounting arrangement, what steady-statetemperature does the window reach?(c) Calculate the net radiation transfer per unit area ofthe window to the vacuum chamber wall, exclud-ing the transmitted simulated solar flux.

A thermocouple whose surface is diffuse and gray withan emissivity of 0.6 indicates a temperature of 180Cwhen used to measure the temperature of a gas flowingthrough a large duct whose walls have an emissivity of0.85 and a uniform temperature of 450C.(a) If the convection heat transfer coefficient betweenthe thermocouple and the gas stream isand there are negligible conduc-tion losses from the thermocouple, determine thetemperature of the gas.(b) Consider a gas temperature of 125C. Computeand plot the thermocouple measurement erroras afunction of the convection coefficient for 10. What are the implications ofyour results?

A thermocouple inserted in a 4-mm-diameter stainlesssteel tube having a diffuse, gray surface with an emis-sivity of 0.4 is positioned horizontally in a large air-conditioned room whose walls and air temperature are30 and 20C, respectively.(a) What temperature will the thermocouple indicateif the air is quiescent?(b) Compute and plot the thermocouple measurementerroras a function of the surface emissivity for0.1?1.0

A temperature sensor embedded in the tip of a smalltube having a diffuse, gray surface with an emissivityof 0.8 is centrally positioned within a large air-conditioned room whose walls and air temperature are30 and 20C, respectively.(a) What temperature will the sensor indicate if theconvection coefficient between the sensor tube andthe air is 5 W/m2?K?(b) What would be the effect of using a fan to induceairflow over the tube? Plot the sensor temperatureas a function of the convection coefficient for 2h25 W/m2?K and values of ?0 ? .2, 0.5, and 0.8

A sphere (k?185 W/m?K, ??7.25?10?5m2/s) of30-mm diameter whose surface is diffuse and graywith an emissivity of 0.8 is placed in a large ovenwhose walls are of uniform temperature at 600 K. Thetemperature of the air in the oven is 400 K, and theconvection heat transfer coefficient between the sphereand the oven air is 15 W/m2?K.(a) Determine the net heat transfer to the sphere whenits temperature is 300 K.(b) What will be the steady-state temperature of thesphere?(c) How long will it take for the sphere, initially at300 K, to come within 20 K of the steady-statetemperature?(d) For emissivities of 0.2, 0.4, and 0.8, plot theelapsed time of part (c) as a function of the con-vection coefficient for 10h25 W/m2K ? .

A thermograph is a device responding to the radiantpower from the scene, which reaches its radiationdetector within the spectral region 912?m. The ther-mograph provides an image of the scene, such as theside of a furnace, from which the surface temperaturecan be determined. (a) For a black surface at 60C, determine the emis-sive power for the spectral region 912?m.(b) Calculate the radiant power (W) received by thethermograph in the same range (912 ?m) whenviewing, in a normal direction, a small black wallarea, 200 mm2, at Ts?60C. The solid angle subtended by the aperture of the thermographwhen viewed from the target is 0.001 sr.(c) Determine the radiant power (W) received by thethermograph for the same wall area (200 mm2) andsolid angle (0.001 sr) when the wall is a gray,opaque, diffuse material at Ts?60C with emissiv-ity 0.7 and the surroundings are black at Tsur2? 3C

A radiation thermometer is a radiometer calibrated toindicate the temperature of a blackbody. A steel billethaving a diffuse, gray surface of emissivity 0.8 isheated in a furnace whose walls are at 1500 K. Esti-mate the temperature of the billet when the radiationthermometer viewing the billet through a small hole inthe furnace indicates 1160 K.

A radiation detector has an aperture of area Ad?10?6m2and is positioned at a distance of r?1 m from a sur-face of area As?10?4m2. The angle formed by the nor-mal to the detector and the surface normal is ?30.The surface is at 500 K and is opaque, diffuse, andgray with an emissivity of 0.7. If the surface irradia-tion is 1500 W/m2, what is the rate at which the detec-tor intercepts radiation from the surface?

A small anodized aluminum block at 35C is heated ina large oven whose walls are diffuse and gray with??0.85 and maintained at a uniform temperature of 175C. The anodized coating is also diffuse and graywith ??0.92. A radiation detector views the blockthrough a small opening in the oven and receives theradiant energy from a small area, referred to as the tar-get, At, on the block. The target has a diameter of3 mm, and the detector receives radiation within asolid angle 0.001 sr centered about the normal fromthe 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 theblock 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 atemperature of 400 K. Coaxial to the disk A1, there is a black, ring-shaped disk A2at 1000 K having thedimensions shown in the sketch. The backside of A2isinsulated and does not directly irradiate the cryogeni-cally cooled detector disk A3, which is of diameter10 mm and is located 2 m from A1.Calculate the rate at which radiation is incident on A3due to emission and reflection from A.

An infrared (IR) thermograph is a radiometer that pro-vides an image of the target scene, indicating theapparent temperature of elements in the scene by ablackwhite brightness or bluered color scale. Radia-tion originating from an element in the target scene isincident on the radiation detector, which provides a sig-nal proportional to the incident radiant power. The signalsets the image brightness or color scale for the imagepixel associated with that element. A scheme is proposedfor field calibration of an infrared thermograph having aradiation detector with a 3- to 5-?m spectral bandpass.A heated metal plate, which is maintained at 327C andhas four diffuse, gray coatings with different emissivi- ties, is viewed by the IR thermograph in surroundingsfor which Tsur?87C. a) Consider the thermograph output when viewingthe black coating, ?o?1. The radiation reaching thedetector is proportional to the product of the black-body emissive power (or emitted intensity) at thetemperature of the surface and the band emissionfraction corresponding to the IR thermograph spec-tral bandpass. The proportionality constant isreferred to as the responsivity, R(?V?m2/W). Writean expression for the thermograph output signal, So,in terms of R, the coating blackbody emissivepower, and the appropriate band emission fraction.Assuming R?1?V?m2/W, evaluate So(?V).(b) Consider the thermograph output when viewing oneof the coatings for which the emissivity ?cis lessthan unity. Radiation from the coating reaches thedetector due to emission and the reflection of irradia-tion from the surroundings. Write an expression forthe signal, Sc, in terms of R, the coating blackbodyemissive power, the blackbody emissive power ofthe surroundings, the coating emissivity, and theappropriate band emission fractions. For the diffuse,gray coatings, the reflectivity is ?c?1??c.(c) Assuming R?1?V?m2/W, evaluate the thermo-graph signals, Sc(?V), when viewing panels withemissivities 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 correctscale indication of Ts?327C. The signals fromthe other three coatings, Sc, are less than So.Hence the thermograph will indicate an apparent(blackbody) temperature less than Ts. Estimate thetemperatures indicated by the thermograph for thethree panels of part (c).

A charge-coupled device (CCD) infrared imaging sys-tem (see Problem 12.85) operates in a manner similarto a digital video camera. Instead of being sensitive toirradiation in the visible part of the spectrum, how-ever, each small sensor in the infrared systems CCDarray is sensitive in the spectral region 912?m. Notethat the system is designed to only view radiationcoming from directly in front of it. An experimenter wishes to use the infrared imaging system to map thesurface temperature distribution of a heated object in a wind tunnel experiment. The air temperature in thewind tunnel, as well as the surroundings temperaturein the laboratory, is 23C.(a) In a preliminary test of the concept, the experi-menter views a small aluminum billet locatedin the wind tunnel that is at a billet temperatureof 50C. The aluminum is coated with a high-emissivity paint, ?? 0.96. If the infrared imagingsystem is calibrated to indicate the temperature ofa blackbody, what temperature will be indicatedby the infrared imaging system as it is used toview the aluminum billet through a 6-mm- thickfused quartz window?(b) In a subsequent experiment, the experimenterreplaces the quartz window with a thin (130-?m-thick) household polyethylene film with ??0.78within the spectral range of the imaging system.What temperature will be indicated by the infraredimaging system when it is used to view the alu-minum billet through the polyethylene film?

12.87A diffuse, spherical object of diameter and temperature9 mm and 600 K, respectively, has an emissivity of 0.95.Two very sensitive radiation detectors, each with anaperture area of 300?10?6m2, detect the object as itpasses over at high velocity from left to right as shownin the schematic. The detectors capture hemisphericalirradiation and are equipped with filters characterized by???0.9 for ??2.5 ?m and ???0 for ??2.5?m. Attime t1?0, detectors A and B indicate irradiations ofGA,1?5.060 mW/m2and GB,1?5.000 mW/m2, respec-tively. At time t2?4 ms, detectors A and B indicateirradiations of GA,2?5.010 mW/m2and GB,2?5.050mW/m2, respectively. The environment is at 300 K.Determine the velocity components of the particle, vxand vy. Determine when and where the particle willstrike a horizontal plane located at y?0. Hint:Theobject is located at an elevation above y?2 m when itis detected. Assume the objects trajectory is a straightline in the plane of the page. Recall that the projectedarea of a sphere is a circle.

A radiation detector having a sensitive area of Ad?4?10?6m2is configured to receive radiation from a tar- get area of diameter Dt?40 mm when located a dis-tance of Lt?1 m from the target. For the experimentalapparatus shown in the sketch, we wish to determine theemitted radiation from a hot sample of diameter Ds?20 mm. The temperature of the aluminum sample isTs?700 K and its emissivity is ?s?0.1. A ring-shapedcold shield is provided to minimize the effect of radia-tion from outside the sample area, but within the targetarea. The sample and the shield are diffuse emitters.(a) Assuming the shield is black, at what temperature,Tsh, should the shield be maintained so that itsemitted radiation is 1% of the total radiant powerreceived by the detector?(b) Subject to the parametric constraint that radiationemitted from the cold shield is 0.05, 1, or 1.5% ofthe total radiation received by the detector, plot therequired cold shield temperature, Tsh, as a functionof the sample emissivity for 0.05? 0.35.

A two-color pyrometeris a device that is used to mea-sure the temperature of a diffuse surface, Ts. The devicemeasures the spectral, directional intensity emitted bythe surface at two distinct wavelengths separated by ?.Calculate and plot the ratio of the intensitiesI?? ?,e(?? ?, , , Ts) and I?,e(?, , , Ts) as a func-tion of the surface temperature over the range500 KTs1000 K for ??5?m and ??0.1, 0.5,and 1?m. Comment on the sensitivity to temperatureand on whether the ratio depends on the emissivity ofthe surface. Discuss the tradeoffs associated with speci-fication of the various values of ?. Hint: The changein the emissivity over small wavelength intervals ismodest for most solids, as evident in Figure 12.17.

Consider a two-color pyrometer such as in Problem12.89 that operates at ?1?0.65?m and ?2?0.63?m.Using Wiens law (see Problem 12.27) determine thetemperature of a sheet of stainless steel if the ratio ofradiation detected is .

Square plates freshly sprayed with an epoxy paintmust be cured at for an extended period of time.The plates are located in a large enclosure and heatedby a bank of infrared lamps. The top surface of eachplate has an emissivity of ??0.8 and experiencesconvection with a ventilation airstream that is atT??27C and provides a convection coefficient ofh?20 W/m2?K. The irradiation from the enclosurewalls is estimated to be Gwall?450 W/m2, for whichthe plate absorptivity is ?wall?0.7.(a) Determine the irradiation that must be provided bythe lamps, Glamp. The absorptivity of the plate sur- face for this irradiation is ?lamp?0.6.(b) For convection coefficients of h?15, 20, and30 W/m2?K, plot the lamp irradiation, Glamp, as afunction of the plate temperature, Ts, for 100Ts300C.(c) For convection coefficients in the range from 10 to30 W/m2?K and a lamp irradiation of Glamp?3000 W/m2, plot the airstream temperature T?required to maintain the plate at Ts?140C.

An apparatus commonly used for measuring the reflec-tivity of materials is shown below. A water- cooledsample, of 30-mm diameter and temperature Ts?300 K, is mounted flush with the inner surface of alarge enclosure. The walls of the enclosure are grayand diffuse with an emissivity of 0.8 and a uniformtemperature Tf?1000 K. A small aperture is locatedat the bottom of the enclosure to permit sighting of thesample or the enclosure wall. The spectral reflectivityof an opaque, diffuse sample material is as shown.The heat transfer coefficient for convection betweenthe sample and the air within the cavity, which is alsoat 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 thecoolant.(d) The ratio of the radiation in the A direction to thatin the B direction will give the reflectivity of thesample. Briefly explain why this is so

A very small sample of an opaque surface is initially at1200 K and has the spectral, hemispherical absorptiv-ity shown.The sample is placed inside a very large enclosurewhose walls have an emissivity of 0.2 and are main-tained at 2400 K.(a) What is the total, hemispherical absorptivity of thesample surface?(b) What is its total, hemispherical emissivity?(c) What are the values of the absorptivity and emis-sivity after the sample has been in the enclosure along time?(d) For a 10-mm- diameter spherical sample in an evacu-ated enclosure, compute and plot the variation of thesample temperature with time, as it is heated from itsinitial temperature of 1200 K

A manufacturing process involves heating long copperrods, which are coated with a thin film, in a large fur-nace whose walls are maintained at an elevated tem-perature 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 emissivityof ???0.9 for ?2?m and ???0.4 for ??2 ?m.(a) Consider conditions for which a rod of diameter Dand initial temperature Tiis inserted in the furnace,such that its axis is horizontal. Assuming validityof the lumped capacitance approximation, derivean equation that could be used to determine therate of change of the rod temperature at the time ofinsertion. Express your result in terms of appropri-ate variables.(b) If Tw?T??1500 K, Ti?300 K, and D?10 mm,what is the initial rate of change of the rod temper-ature? Confirm the validity of the lumped capaci-tance approximation.(c) Compute and plot the variation of the rod tempera-ture with time during the heating process.

A procedure for measuring the thermal conductivity of solids at elevated temperatures involves placement ofa sample at the bottom of a large furnace. The sampleis of thickness Land is placed in a square container ofwidth Won a side. The sides are well insulated. Thewalls of the cavity are maintained at Tw, while the bot-tom surface of the sample is maintained at a muchlower temperature Tcby circulating coolant throughthe sample container. The sample surface is diffuseand gray with an emissivity ?s. Its temperature Tsismeasured optically.(a) Neglecting convection effects, obtain an expres-sion from which the sample thermal conductivitymay be evaluated in terms of measured andknown quantities (Tw, Ts, Tc, ?s, L). The measure-ments are made under steady-state conditions. If Tw?1400 K, Ts?1000 K, ?s?0.85, L?0.015 m, andTc?300 K, what is the samplethermal conductivity? (b) If W?0.10 m and the coolant is water with a flowrate of , is it reasonable to assume auniform bottom surface temperature Tc?

One scheme for extending the operation of gas turbineblades to higher temperatures involves applying aceramic coating to the surfaces of blades fabricatedfrom a superalloy such as inconel. To assess the relia-bility of such coatings, an apparatus has been devel-oped for testing samples under laboratory conditions.The sample is placed at the bottom of a large vacuumchamber whose walls are cryogenically cooled andwhich is equipped with a radiation detector at thetop surface. The detector has a surface area ofAd?10?5m2, is located at a distance of Lsd?1mfrom the sample, and views radiation originating froma portion of the ceramic surface having an area ofAc?10?4m2. An electric heater attached to the bot-tom of the sample dissipates a uniform heat flux, ,which is transferred upward through the sample. Thebottom of the heater and sides of the sample are wellinsulated.Consider conditions for which a ceramic coating ofthickness Lc?0.5 mm and thermal conductivity kc?6W/m?K has been sprayed on a metal substrate ofthickness Ls?8 mm and thermal conductivity ks?25 W/m?K. The opaque surface of the ceramic may beapproximated as diffuse and gray, with a total, hemi-spherical emissivity of ?c?0.8.(a) Consider steady-state conditions for which thebottom surface of the substrate is maintained atT1?1500 K, while the chamber walls (includingthe surface of the radiation detector) are main-tained at Tw?90 K. Assuming negligible thermalcontact resistance at the ceramicsubstrate inter-face, determine the ceramic top surface tempera-ture T2and the heat flux .(b) For the prescribed conditions, what is the rate atwhich radiation emitted by the ceramic is inter-cepted by the detector? c) After repeated experiments, numerous cracksdevelop at the ceramicsubstrate interface, creat-ing an interfacial thermal contact resistance. If Twand are maintained at the conditions associatedwith part (a), will T1increase, decrease, or remainthe same? Similarly, will T2increase, decrease, orremain the same? In each case, justify youranswer

The equipment for heating a wafer during a semicon-ductor manufacturing process is shown schematically.The wafer is heated by an ion beam source (not shown)to a uniform, steady-state temperature. The large chambercontains the process gas, and its walls are at a uniformtemperature of Tch?400 K. A 5 mm?5 mm target areaon the wafer is viewed by a radiometer, whose objec-tive lens has a diameter of 25 mm and is located500 mm from the wafer. The line-of-sight of theradiometer is off the wafer normal.(a) In a preproduction test of the equipment, a blackpanel (??1.0) is mounted in place of the wafer.Calculate the radiant power (W) received by theradiometer if the temperature of the panel is 800 K.(b) The wafer, which is opaque, diffuse-gray with anemissivity of 0.7, is now placed in the equipment,and the ion beam is adjusted so that the powerreceived by the radiometer is the same as thatfound for part (a). Calculate the temperature of thewafer for this heating condition.

The fire brick of Example 12.10 is used to constructthe walls of a brick oven. The irradiation on the inte-rior surface of the wall is G?50,000 W/m2and has aspectral distribution proportional to that of a black-body at 2000 K. The temperature of the gases adjacentto the inner wall of the oven is 500 K, and the convec-tion heat transfer coefficient is 25 W/m2. Find the wallsurface temperature if the heat loss through the wall isnegligible. If the brick wall is 0.1 m thick and of ther-mal conductivity kb?1.0 W/m?K, and is insulatedwith a 0.1-m-thick layer of thermal conductivityki?0.05 W/m?K, what is the steady-state interior wall surface temperature if the temperature of theexternal surface of the insulation is 300 K?

A laser-materials-processing apparatus encloses a sam-ple in the form of a disk of diameter D?25 mm andthickness w?1 mm. The sample has a diffuse surfacefor which the spectral distribution of the emissivity,??(?), is prescribed. To reduce oxidation, an inert gasstream of temperature T??500 K and convectioncoefficient h?50 W/m2?K flows over the sampleupper and lower surfaces. The apparatus enclosure islarge, with isothermal walls at Tenc?300 K. To main-tain the sample at a suitable operating temperature ofTs?2000 K, a collimated laser beam with an operatingwavelength of ??0.5?m irradiates its upper surface.(a) Determine the total emissivity ?of the sample.(b) Determine the total absorptivity ? of the samplefor irradiation from the enclosure walls.(c) Perform an energy balance on the sample anddetermine the laser irradiation, Glaser, required tomaintain the sample at Ts?2000 K.(d) Consider a cool-downprocess, when the laser andthe inert gas flow are deactivated. Sketch the totalemissivity as a function of the sample temperature,Ts(t), during the process. Identify key features,including the emissivity for the final condition(tl?).(e) Estimate the time to cool a sample from its operat-ing condition at Ts(0)?2000 K to a safe-to-touchtemperature of Ts(t)?40C. Use the lumpedcapacitance method and include the effect of con-vection to the inert gas with h?50 W/m2?K andT??Tenc?300 K. The thermophysical propertiesof the sample material are ??3900 kg/m3,cp?760 J/kg?K, and k?45 W/m?K.

A cylinder of 30-mm diameter and 150-mm length isheated in a large furnace having walls at 1000 K,while air at 400 K is circulating at 3 m/s. Estimate thesteady-state cylinder temperature under the followingspecified conditions.(a) The cylinder is in cross flow, and its surface isdiffuse and gray with an emissivity of 0.5.(b) The cylinder is in cross flow, but its surface isspectrally selective with ???0.1 for ?3?mand ???0.5 for ??3?m.(c) The cylinder surface is positioned such that theairflow is longitudinal and its surface is diffuseand gray.(d) For the conditions of part (a), compute and plotthe cylinder temperature as a function of the airvelocity for 1V20 m/s

An instrumentation transmitter pod is a box contain-ing electronic circuitry and a power supply for send-ing sensor signals to a base receiver for recording.Such a pod is placed on a conveyor system, whichpasses through a large vacuum brazing furnace asshown in the sketch. The exposed surfaces of the podhave a special diffuse, opaque coating with spectralemissivity as shown.To stabilize the temperature of the pod and preventoverheating of the electronics, the inner surface of thepod is surrounded by a layer of a phase-change mate-rial (PCM) having a fusion temperature of 87C and aheat of fusion of 25 kJ/kg. The pod has an exposedsurface area of 0.040 m2and the mass of the PCM is 1.6 kg. Furthermore, it is known that the powerdissipated by the electronics is 50 W. Consider thesituation when the pod enters the furnace at a uni-form temperature of 87C and all the PCM is in thesolid state. How long will it take before all the PCMchanges to the liquid state?

A thin-walled plate separates the interior of a largefurnace from surroundings at 300 K. The plate isfabricated from a ceramic material for which diffusesurface behavior may be assumed and the exteriorsurface is air cooled. With the furnace operating at2400 K, convection at the interior surface may beneglected.(a) If the temperature of the ceramic plate is not toexceed 1800 K, what is the minimum value of theoutside convection coefficient, ho, that must bemaintained by the air-cooling system?(b) Compute and plot the plate temperature as a func-tion of hofor 50ho250 W/m2?K

A thin coating, which is applied to long, cylindricalcopper rods of 10-mm diameter, is cured by placingthe rods horizontally in a large furnace whose wallsare maintained at 1300 K. The furnace is filled withnitrogen gas, which is also at 1300 K and at a pressureof 1 atm. The coating is diffuse, and its spectral emis-sivity has the distribution shown.(a) What are the emissivity and absorptivity of thecoated rods when their temperature is 300 K?(b) What is the initial rate of change of theirtemperature?(c) What are the emissivity and absorptivity of thecoated rods when they reach a steady-statetemperature?(d) Estimate the time required for the rods to reach1000 K

A large combination convectionradiation oven isused to heat-treat a small cylindrical product of dia- meter 25 mm and length 0.2 m. The oven walls are ata uniform temperature of 1000 K, and hot air at 750 Kis in cross flow over the cylinder with a velocity of5 m/s. The cylinder surface is opaque and diffuse withthe spectral emissivity shown.(a) Determine the rate of heat transfer to the cylinderwhen it is first placed in the oven at 300 K.(b) What is the steady-state temperature of thecylinder?(c) How long will it take for the cylinder to reach atemperature that is within 50C of its steady-statevalue?

A 10-mm-thick workpiece, initially at 25C, is to beannealed at a temperature above 725C for a periodof at least 5 minutes and then cooled. The work-piece is opaque and diffuse, and the spectral distribu-tion of its emissivity is shown schematically. Heatingis effected in a large furnace with walls and circu-lating air at 750C and a convection coefficient of100 W/m2?K. The thermophysical properties of theworkpiece are ??2700 kg/m3, c?885 J/kg?K, andk?165 W/m?K.(a) Calculate the emissivity and the absorptivity ofthe workpiece when it is placed in the furnace atits initial temperature of 25C.(b) Determine the net heat flux into the workpiecefor this initial condition. What is the correspond-ing rate of change in temperature, dT/dt, for theworkpiece? c) Calculate the time for the workpiece to cool from750C to a safe-to-touch temperature of 40C, ifthe surroundings and cooling air temperatureare 25C and the convection coefficient is100 W/m2?K.

After being cut from a large single-crystal boule andpolished, silicon wafers undergo a high- temperatureannealing process. One technique for heating thewafer is to irradiate its top surface using high-intensity, tungsten-halogen lamps having a spectraldistribution approximating that of a blackbody at2800 K. To determine the lamp power and the rate atwhich radiation is absorbed by the wafer, the equip-ment designer needs to know its absorptivity as afunction of temperature. Silicon is a semiconductormaterial that exhibits a characteristic band edge, andits spectral absorptivity may be idealized as shownschematically. At low and moderate temperatures,silicon is semitransparent at wavelengths larger thanthat of the band edge, but becomes nearly opaqueabove 600C.(a) What are the 1% limits of the spectral band thatincludes 98% of the blackbody radiation corre-sponding to the spectral distribution of thelamps? Over what spectral region do you need toknow the spectral absorptivity?(b) How do you expect the total absorptivity of siliconto vary as a function of its temperature? Sketch thevariation and explain its key features.(c) Calculate the total absorptivity of the siliconwafer for the lamp irradiation and each of the fivetemperatures shown schematically. From the data,calculate the emissivity of` the wafer at 600 and900C. Explain your results and why the emissiv-ity changes with temperature. Hint: Within IHT,create a look-uptable to specify values of the spectral properties and the LOOKUPVAL andINTEGRAL functions to perform the necessaryintegrations.(d) If the wafer is in a vacuum and radiation exchangeonly occurs at one face, what is the irradiationneeded to maintain a wafer temperature of 600C

Solar irradiation of 1100 W/m2is incident on a large,flat, horizontal metal roof on a day when the windblowing over the roof causes a convection heattransfer coefficient of 25 W/m2?K. The outside airtemperature is 27C, the metal surface absorptivityfor incident solar radiation is 0.60, the metal surfaceemissivity is 0.20, and the roof is well insulatedfrom below.(a) Estimate the roof temperature under steady-stateconditions.(b) Explore the effect of changes in the absorptivity,emissivity, and convection coefficient on thesteady-state temperature.

Neglecting the effects of radiation absorption, emis-sion, and scattering within their atmospheres, calcu-late the average temperature of Earth, Venus, andMars assuming diffuse, gray behavior. The averagedistance from the sun of each of the three planets, Lsp, along with their measuredaverage tempera-tures, , are shown in the table below. Based upon acomparison of the calculated and measured averagetemperatures, which planet is most affected by radia-tion transfer in its atmosphere?PlanetLsp(m)Venus1.08?1011735Earth1.50?1011287Mars2.30 ?101122

A deep cavity of 50-mm diameter approximates ablackbody and is maintained at 250C while exposedto solar irradiation of 800 W/m2and surroundings andambient air at 25C. A thin window of spectral trans-missivity and reflectivity 0.9 and 0, respectively, forthe spectral range 0.2 to 4?m is placed over the cavityopening. In the spectral range beyond 4?m, the win-dow behaves as an opaque, diffuse, gray body of emis-sivity 0.95. Assuming that the convection coefficienton the upper surface of the window is 10 W/m2?K, determine the temperature of the window and thepower required to maintain the cavity at 250C.

Consider the evacuated tube solar collector describedin part (d) of Problem 1.87 of Chapter 1. In the inter-est of maximizing collector efficiency, what spectralradiative characteristics are desired for the outer tubeand for the inner tube

Solar flux of 900 W/m2is incident on the top sideof a plate whose surface has a solar absorptivity of0.9 and an emissivity of 0.1. The air and surround-ings are at 17C and the convection heat transfercoefficient between the plate and air is 20 W/m2?K.Assuming that the bottom side of the plate isinsulated, determine the steady-state temperature ofthe plate.

Consider an opaque, gray surface whose directionalabsorptivity is 0.8 for 060and 0.1 for ?60.The surface is horizontal and exposed to solar irradia-tion comprised of direct and diffuse components.(a) What is the surface absorptivity to direct solarradiation that is incident at an angle of 45fromthe normal? What is the absorptivity to diffuseirradiation?(b) Neglecting convection heat transfer between thesurface and the surrounding air, what wouldbe the equilibrium temperature of the surface ifthe direct and diffuse components of the irradia-tion were 600 and 100 W/m2, respectively? Theback side of the surface is insulated.

The absorber plate of a solar collector may be coatedwith an opaque material for which the spectral,directional absorptivity is characterized by relationsof the form??,(?, )??2??? The zenith angle is formed by the suns rays andthe plate normal, and ?1and ?2are constants.(a) Obtain an expression for the total, hemisphericalabsorptivity, ?S, of the plate to solar radiation inci-dent at ?45. Evaluate ?Sfor ?1?0.93, ?2?0.25, and a cut-off wavelength of ?c?2?m.(b) Obtain an expression for the total, hemisphericalemissivity ?of the plate. Evaluate ?for a platetemperature of Tp?60C and the prescribedvalues of ?1, ?2, and ?c.(c) For a solar flux of incident atand the prescribed values of and Tp, what is the net radiant heat flux, , tothe plate?(d) Using the prescribed conditions and the Radiation/Band Emission Factoroption in the Toolssectionof IHTto evaluate F(0l?c), explore the effect of ?con ? , ?, and? etfor the wavelength range0.7c ? 5?m.

A contractor must select a roof covering materialfrom the two diffuse, opaque coatings with ??(?) asshown. Which of the two coatings would result in alower roof temperature? Which is preferred for sum-mer use? For winter use? Sketch the spectral distribu-tion of ??that would be ideal for summer use. Forwinter use.

It is not uncommon for the night sky temperature indesert regions to drop to?40C. If the ambient airtemperature is 20C and the convection coefficientfor still air conditions is approximately 5 W/m2K ? ,can a shallow pan of water freeze?

Plant leaves possess small channels that connect theinterior moist region of the leaf to the environment.The channels, called stomata, pose the primary resis-tance to moisture transport through the entire plant,and the diameter of an individual stoma is sensitiveto the level of CO2in the atmosphere. Consider a leafof corn (maize) whose top surface is exposed to solarirradiation of GS?600 W/m2and an effective sky temperature of Tsky?0C. The bottom side of theleaf is irradiated from the ground which is at atemperature of Tg?20C. Both the top and bottomof the leaf are subjected to convective conditionscharacterized by h?35 W/m2?K, T??25C andalso experience evaporation through the stomata.Assuming the evaporative flux of water vapor is50?10?6kg/m2?s under rural atmospheric CO2concentrations and is reduced to 5?10?6kg/m2?swhen ambient CO2concentrations are doubled nearan urban area, calculate the leaf temperature in therural and urban locations. The heat of vaporization ofwater is hfg?2400 kJ/kg and assume ????0.97for radiation exchange with the sky and the ground,and ?S?0.76 for solar irradiation.

In the central receiver concept of solar energy collec-tion, a large number of heliostats (reflectors) providea concentrated solar flux of to thereceiver, which is positioned at the top of a tower.The receiver wall is exposed to the solar flux at itsouter surface and to atmospheric air for whichT?,o?300 K and ho?25 W/m2?K. The outer sur-face is opaque and diffuse, with a spectral absorp-tivity of ???0.9 for ??3?m and ???0.2 for ??3?m. The inner surface is exposed to a workingfluid (a pressurized liquid) for which T?,i?700 K andhi?1000 W/m2?K. The outer surface is also exposedto surroundings for which Tsur?300 K. If the wall isfabricated from a high-temperature material for whichk?15 W/m?K, what is the minimum thickness Lneeded to ensure that the outer surface temperaturedoes not exceed Ts,o?1000 K? What is the collectionefficiency associated with this thickness?

Consider the central receiver of Problem 12.117 to be acylindrical shell of outer diameter D?7 m and lengthL?12 m. The outer surface is opaque and diffuse,with a spectral absorptivity of ???0.9 for ??3?mand ???0.2 for ??3?m. The surface is exposed toquiescentambient air for which T??300 K. a) Consider representative operating conditions forwhich solar irradiation at GS?80,000 W/m2isuniformly distributed over the receiver surfaceand the surface temperature is Ts?800 K.Determine the rate at which energy is collectedby the receiver and the corresponding collectorefficiency.(b) The surface temperature is affected by conditionsinternal to the receiver. For GS?80,000 W/m2,compute and plot the rate of energy collection andthe collector efficiency for 600Ts1000 K

Radiation from the atmosphere or sky can be esti-mated as a fraction of the blackbody radiation corre-sponding to the air temperature near the ground, Tair.That is, irradiation from the sky can be expressed asand for a clear night sky, the emis-sivity is correlated by an expression of the form?sky?0.741?0.0062Tdp, where Tdpis the dew pointtemperature (C). Consider a flat plate exposed to the night sky and in ambient air at 15C with a rela-tive humidity of 70%. Assume the back side of theplate is insulated, and that the convection coefficienton the front side can be estimated by the correlation, where Tis the absolutevalue of the plate-to-air temperature difference. Willdew form on the plate if the surface is (a) clean andmetallic with ?? 0.23, and (b) painted with ??0.85?

A thin sheet of glass is used on the roof of a green-house and is irradiated as shown. The irradiation comprises the total solar flux GS, theflux Gatmdue to atmospheric emission (sky radiation),and the flux Gidue to emission from interior surfaces.The fluxes Gatmand Giare concentrated in the far IRregion (??8?m). The glass may also exchangeenergy by convection with the outside and insideatmospheres. The glass may be assumed to be totallytransparent for ??1 ?m (???1.0 for ??1?m) andopaque, with ???1.0 for ??1?m.(a) Assuming steady-state conditions, with all radia-tive fluxes uniformly distributed over the sur-faces and the glass characterized by a uniformtemperature Tg, write an appropriate energy bal-ance for a unit area of the glass.(b) For Tg?27C, hi?10 W/m2?K, GS?1100W/m2, T?,o?24C, ho?55 W/m2?K, Gatm?250 W/m2, and Gi?440 W/m2, calculate thetemperature of the greenhouse ambient air, T?,i.

A solar furnace consists of an evacuated chamberwith transparent windows, through which concen- trated solar radiation is passed. Concentration may beachieved by mounting the furnace at the focal pointof a large curved reflector that tracks radiation inci-dent directly from the sun. The furnace may be usedto evaluate the behavior of materials at elevated tem-peratures, and we wish to design an experiment toassess the durability of a diffuse, spectrally selectivecoating for which ???0.95 in the range ?4.5?mand ???0.03 for ??4.5?m. The coating isapplied to a plate that is suspended in the furnace.(a) If the experiment is to be operated at a steady-state plate temperature of T?2000 K, how muchsolar irradiation GSmust be provided? The irradiation may be assumed to be uniformly distributedover the plate surface, and other sources of inci-dent radiation may be neglected.(b) The solar irradiation may be tunedto allow opera-tion over a range of plate temperatures. Computeand plot GSas a function of temperature for500T3000 K. Plot the corresponding values of ? and ?as a function of Tfor the designated range.

The flat roof on the refrigeration compartment of afood delivery truck is of length L?5 m and widthW?2 m. It is fabricated from thin sheet metal towhich a fiberboard insulating material of thicknesst?25 mm and thermal conductivity k?0.05 W/m?Kis bonded. During normal operation, the truck movesat a velocity of V?30 m/s in air at T??27C, with a rooftop solar irradiation of GS?900 W/m2andwith the interior surface temperature maintained atTs,i??13C.(a) The owner has the option of selecting a roofcoating from one of the three paints listed inTable A.12 (Parsons Black, Acrylic White, orZinc Oxide White). Which should be chosenand why?(b) For the preferred paint of part (a), determine the steady-state value of the outer surface tem-perature Ts,o. The boundary layer is tripped at the leading edge of the roof, and turbulentflow may be assumed to exist over the entireroof. Properties of the air may be taken to be ??15?10?6m2/s, k?0.026 W/m?K, andPr?0.71.(c) What is the load (W) imposed on the refrigera-tion system by heat transfer through the roof?(d) Explore the effect of the truck velocity on theouter surface temperature and the heat load

Growers use giant fans to prevent grapes from freez-ing when the effective sky temperature is low. Thegrape, which may be viewed as a thin skin of negligi-ble thermal resistance enclosing a volume of sugarwater, is exposed to ambient air and is irradiatedfrom the sky above and ground below. Assume thegrape to be an isothermal sphere of 15-mm diameter,and assume uniform blackbody irradiation over its top and bottom hemispheres due to emission from thesky and the earth, respectively.(a) Derive an expression for the rate of change of thegrape temperature. Express your result in termsof a convection coefficient and appropriate tem-peratures and radiative quantities.(b) Under conditions for which Tsky?235 K, T??273 K, and the fan is off (V?0), determinewhether the grapes will freeze. To a goodapproximation, the skin emissivity is 1 and thegrape thermophysical properties are those of sug- arless water. However, because of the sugar con-tent, the grape freezes at?5C.(c) With all conditions remaining the same, exceptthat the fans are now operating with V?1 m/s,will the grapes freeze?

A circular metal disk having a diameter of 0.4 m isplaced firmly against the ground in a barren horizon-tal region where the earth is at a temperature of280 K. The effective sky temperature is also 280 K.The disk is exposed to quiescent ambient air at 300 Kand direct solar irradiation of 745 W/m2. The surfaceof the disk is diffuse with ???0.9 for 0???1?mand ???0.2 for ??1?m. After some time haselapsed, the disk achieves a uniform, steady-statetemperature. The thermal conductivity of the soil is0.52 W/m?K.(a) Determine the fraction of the incident solar irra-diation 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 theaverage free convection heat transfer coefficientat the upper surface of the disk.(d) Show that a disk temperature of 340 K doesindeed yield a steady-state condition for the disk.

The neighborhood cat likes to sleep on the roof of ourshed in the backyard. The roofing surface is weath-ered galvanized sheet metal (??0.65, ?S?0.8).Consider a cool spring day when the ambient air tem- perature is 10C and the convection coefficient canbe estimated from an empirical correlation of theform , where Tis the differencebetween the surface and ambient temperatures.Assume the sky temperature is?40C.(a) Assuming the backside of the roof is well insu-lated, calculate the roof temperature when thesolar irradiation is 600 W/m2. Will the cat enjoysleeping under these conditions?(b) Consider the case when the backside of the roofis not insulated, but is exposed to ambient airwith the same convection coefficient relationand experiences radiation exchange with theground, also at the ambient air temperature. Cal-culate the roof temperature and comment onwhether the roof will be a comfortable place forthe cat to snooze.

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

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

The oxidized-aluminum wing of an aircraft has achord length of Lc?4 m and a spectral, hemispher-ical emissivity characterized by the followingdistribution.(a) Consider conditions for which the plane is on the ground where the air temperature is 27C, thesolar irradiation is 800 W/m2, and the effectivesky temperature is 270 K. If the air is quiescent,what is the temperature of the top surface of thewing? The wing may be approximated as a hori-zontal, flat plate.(b) When the aircraft is flying at an elevation ofapproximately 9000 m and a speed of 200 m/s, theair temperature, solar irradiation, and effective skytemperature are?40C, 1100 W/m2, and 235 K,respectively. What is the temperature of thewings top surface? The properties of the air maybe approximated as ??0.470 kg/m3, ??1.50?10?5N?s/m2, k?0.021 W/m?K, and Pr0 ? .72.

Two plates, one with a black painted surface and theother with a special coating (chemically oxidizedcopper) are in earth orbit and are exposed to solarradiation. The solar rays make an angle of 30 with the normal to the plate. Estimate the equilibrium tem-perature of each plate assuming they are diffuse andthat the solar flux is 1368 W/m2. The spectral absorp-tivity of the black painted surface can be approxi-mated by ???0.95 for 0??and that of thespecial coating by ???0.95 for 0??3?m and???0.05 for ??3 ?m.

A spherical satellite of diameter Dis in orbit aboutthe earth and is coated with a diffuse material forwhich the spectral absorptivity is ???0.6 for ?3 ?m and ???0.3 for ??3?m. When it is on thedark side of the earth, the satellite sees irradiationfrom the earths surface only. The irradiation may beassumed to be incident as parallel rays, and its mag-nitude is GE?340 W/m2. On the bright side of theearth the satellite sees the earth irradiation GEplusthe solar irradiation GS?1368 W/m2. The spectraldistribution of radiation from the earth may beapproximated as that of a blackbody at 280 K, andthe temperature of the satellite may be assumed toremain below 500 K.What is the steady-state temperature of the satellitewhen it is on the dark side of the earth and when it ison the bright side?

A radiator on a proposed satellite solar power stationmust dissipate heat being generated within the satel-lite by radiating it into space. The radiator surface hasa solar absorptivity of 0.5 and an emissivity of 0.95.What is the equilibrium surface temperature when thesolar irradiation is 1000 W/m2and the required heatdissipation is 1500 W/m2?

A spherical satellite in near-earth orbit is exposed tosolar irradiation of 1368 W/m2. To maintain a desiredoperating temperature, the thermal control engineer intends to use a checker pattern for which a fractionFof the satellite surface is coated with an evaporatedaluminum film (??0.03, ?S?0.09), and the frac-tion (1?F) is coated with a white, zinc-oxide paint(??0.85, ?S?0.22). Assume the satellite isisothermal and has no internal power dissipation.Determine the fraction Fof the checker patternrequired to maintain the satellite at 300 K.

An annular fin of thickness tis used as a radiator todissipate heat for a space power system. The fin isinsulated on the bottom and may be exposed to solarirradiation GS. The fin is coated with a diffuse, spec-trally selective material whose spectral reflectivity isspecified.Heat is conducted to the fin through a solid rod ofradius ri, and the exposed upper surface of the finradiates to free space, which is essentially at absolutezero temperature.(a) If conduction through the rod maintains a finbase temperature of T(ri)?Tb?400 K and thefin efficiency is 100%, what is the rate of heatdissipation for a fin of radius ro?0.5 m? Considertwo cases, one for which the radiator is exposed tothe sun with GS?1000 W/m2and the other withno exposure (GS?0).(b) In practice, the fin efficiency will be less than100% and its temperature will decrease withincreasing radius. Beginning with an appropriatecontrol volume, derive the differential equationthat determines the steady-state, radial tempera-ture distribution in the fin. Specify appropriateboundary conditions.

A rectangular plate of thickness t, length L, andwidth Wis proposed for use as a radiator in a space- craft application. The plate material has a thermalconductivity of 300 W/m?K, a solar absorptivity of0.45, and an emissivity of 0.9. The radiator isexposed to solar radiation only on its top surface,while both surfaces are exposed to deep space at atemperature of 4 K (a) If the base of the radiator is maintained atTb?80C, what is its tip temperature and therate of heat rejection? Use a computer- based,finite-difference method with a space incrementof 0.1 m to obtain your solution.(b) Repeat the calculation of part (a) for the casewhen the space ship is on the dark side of theearth and is not exposed to the sun.(c) Use your computer code to calculate the heat rateand tip temperature for GS?0 and an extremelylarge value of the thermal conductivity. Compareyour results to those obtained from a hand calcu-lation that assumes the radiator to be at a uniformtemperature Tb. What other approach might youuse to validate your code

The directional absorptivity of a gray surface varieswith as follows.(a) What is the ratio of the normal absorptivity ?ntothe hemispherical emissivity of the surface?(b) Consider a plate with these surface characteris-tics on both sides in earth orbit. If the solarflux incident on one side of the plate is, what equilibrium temperaturewill the plate assume if it is oriented normal tothe suns rays? What temperature will it assumeif it is oriented at 75to the suns rays?

Two special coatings are available for application toan absorber plate installed below the cover glassdescribed in Example 12.9. Each coating is diffuse and is characterized by the spectral distributionsshown.Which coating would you select for the absorberplate? Explain briefly. For the selected coating, whatis the rate at which radiation is absorbed per unit areaof the absorber plate if the total solar irradiation atthe cover glass is GS?1000 W/m2?

Consider the spherical satellite of Problem 12.130.Instead of the entire satellite being coated with amaterial that is spectrally selective, half of the satel-lite is covered with a diffuse gray coating character-ized by ?1?0.6. The other half of the satellite iscoated with a diffuse gray material with ?2?0.3.(a) Determine the steady-state satellite temperaturewhen the satellite is on the bright side of the earthwith the high-absorptivity coating facing the sun.Determine the steady-state satellite temperaturewhen the low-absorptivity coating faces the sun.Hint: Assume one hemisphere of the satellite isirradiated by the sun and the opposite hemisphereis irradiated by the earth.(b) Determine the steady-state satellite temperaturewhen the satellite is on the dark side of the earthwith the high- absorptivity coating facing theearth. Determine the steady-state satellite tem-perature when the low- absorptivity coating facesthe earth.(c) Identify a scheme to minimize the temperaturevariations of the satellite as it travels between thebright and dark sides of the earth

A spherical capsule of 3-m radius is fired from aspace platform in earth orbit, such that it travelstoward the center of the sun at 16,000 km/s. Assumethat the capsule is a lumped capacitance body with adensityspecific heat product of 4?106J/m3?K andthat its surface is black.(a) Derive a differential equation for predicting thecapsule temperature as a function of time. Solvethis equation to obtain the temperature as a func-tion of time in terms of capsule parameters andits initial temperature Ti (b) If the capsule begins its journey at 20C, predict theposition of the capsule relative to the sun at whichits destruction temperature, 150C, is reached

The spectral absorptivity of aluminum coated with athin layer of silicon dioxide may be approximated as??,1?0.98 for ???cand ??,2?0.05 for ???cwhere the cutoff wavelengthis ?c?0.15?m undernormal circumstances.(a) Determine the equilibrium temperature of a flatpiece of the coated aluminum that is exposed tosolar irradiation, GS?1368 W/m2on its uppersurface. The opposite surface is insulated.(b) The cutoff wavelength can be modified by vary-ing the coating thickness. Determine the value of?cthat will maximize the equilibrium tempera-ture of the surface.

Consider the spherical satellite of Problem 12.130.By changing the thickness of the diffuse materialused for the coating, engineers can control the cut-off wavelengththat marks the boundary between???0.6 and ???0.3.(a) What cutoff wavelength will minimize thesteady-state temperature of the satellite when it ison the bright side of the earth? Using this coat-ing, what will the steady-state temperature on thedark side of the earth be?(b) What cutoff wavelength will maximize the steady- state temperature of the satellite when it is on thedark side of the earth? What will the correspond-ing steady-state temperature be on the bright side?

A solar panel mounted on a spacecraft has an area of1m2and a solar-to-electrical power conversion effi-ciency of 12%. The side of the panel with the photo-voltaic array has an emissivity of 0.8 and a solarabsorptivity of 0.8. The back side of the panel has anemissivity of 0.7. The array is oriented normal tosolar irradiation of 1500 W/m2.(a) Determine the steady-state temperature of thepanel and the electrical power (W) produced forthe prescribed conditions.(b) If the panel were a thin plate without the solar cells,but with the same radiative properties, determinethe temperature of the plate for the prescribed con-ditions. Compare this result with that from part (a).Are they the same or different? Explain why.(c) Determine the temperature of the solar panel1500 s after the spacecraft is eclipsed by a planet.The thermal capacity of the panel per unit area is.9000 J/m2?K.

It is known that on clear nights a thin layer of water onthe ground will freeze before the air temperature dropsbelow 0C. Consider such a layer of water on a clearnight for which the effective sky temperature is?30Cand the convection heat transfer coefficient due to windmotion is h?25 W/m2?K. The water may be assumedto have an emissivity of 1.0 and to be insulated fromthe ground as far as conduction is concerned. Neglect-ing evaporation, determine the lowest temperature thatthe air can have without the water freezing. Accountingnow for the effect of evaporation, what is the lowesttemperature that the air can have without the waterfreezing? Assume the air to be dry.

A shallow layer of water is exposed to the naturalenvironment as shown.Consider conditions for which the solar and atmos-pheric irradiations are GS?600 W/m2and Gatm?300 W/m2, respectively, and the air temperature andrelative humidity are T??27C and ??0.50,respectively. The reflectivities of the water surfaceto the solar and atmospheric irradiation are ?S?0.3and ?atm?0, respectively, while the surface emissiv-ity is ??0.97. The convection heat transfer coeffi-cient at the airwater interface is h?25 W/m2?K.If the water is at 27C, will this temperatureincrease or decrease with time?

A roof-cooling system, which operates by maintain-ing a thin film of water on the roof surface, may beused to reduce air-conditioning costs or to maintain acooler environment in nonconditioned buildings. Todetermine the effectiveness of such a system, con-sider a sheet metal roof for which the solar absorptiv-ity ?Sis 0.50 and the hemispherical emissivity ?is0.3. Representative conditions correspond to a sur-face convection coefficient hof 20 W/m2?K, a solarirradiation GSof 700 W/m2, a sky temperature of?10C, an atmospheric temperature of 30C, and arelative humidity of 65%. The roof may be assumedto be well insulated from below. Determine the roofsurface temperature without the water film. Assum-ing the film and roof surface temperatures to beequal, determine the surface temperature with the ilm. The solar absorptivity and the hemisphericalemissivity of the filmsurface combination are ?S?0.8 and ??0.9, respectively

A wet towel hangs on a clothes line under conditionsfor which one surface receives solar irradiation ofGS?900 W/m2and both surfaces are exposed toatmospheric (sky) and ground radiation of Gatm?200 W/m2and Gg?250 W/m2, respectively. Undermoderately windy conditions, airflow at a temperatureof 27C and a relative humidity of 60% maintains aconvection heat transfer coefficient of 20 W/m2?K atboth surfaces. The wet towel has an emissivity of 0.96and a solar absorptivity of 0.65. As a first approxima-tion the properties of the atmospheric air may be eval-uated at a temperature of 300 K.Towel (Ts, , S)Ground AirDetermine the temperature Tsof the towel. What isthe corresponding evaporation rate for a towel that is0.75 m wide by 1.50 m long?

Our students perform a laboratory experiment todetermine mass transfer from a wet paper towel expe-riencing forced convection and irradiation from radi-ant lamps. For the values of T?and Twbprescribedon the sketch, the towel temperature was found to be Ts?310 K. In addition, flat-plate correlationsyielded average heat and mass transfer convectioncoefficients of and respectively. The towel has dimensions of92.5 mm?92.5 mm and is diffuse and gray with anemissivity of 0.96.(a) From the foregoing results, determine the vapordensities, ?A,sand ?A,?, the evaporation rate, nA(kg/s), and the net rate of radiation transfer to thetowel, qrad(W).(b) Using results from part (a) and assuming that theirradiation Gis uniform over the towel, deter-mine the emissive power E, the irradiation G, andthe radiosity J.