A thin electrical heating element provides a uniformheat

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

The heat flux that is applied to the left face of a planewall is q" = 20 W/m. The wall is of thickness L = 10mm and of thermal conductivity k = 12 W/mK. If the surface temperatures of the wall are measured to be 50C on the left side and 30C on the right side, do steady-state conditions exist?

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

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

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

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

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

A thermodynamic analysis of a proposed Brayton cyclegas turbine yields P?5 MW of net power production.The compressor, at an average temperature of Tc?400C,is driven by the turbine at an average temperature ofTh?1000C by way of an L?1-m-long, d?70-mm-diameter shaft of thermal conductivity k?40 W/m?K.ProblemsLd PTurbineCompressorShaftminmoutThTcCombustionchamber(a) Compare the steady-state conduction rate throughthe shaft connecting the hot turbine to the warmcompressor to the net power predicted by the ther-modynamics-based analysis.(b) A research team proposes to scale down the gasturbine of part (a), keeping all dimensions in thesame proportions. The team assumes that the samehot and cold temperatures exist as in part (a) andthat the net power output of the gas turbine is pro-portional to the overall volume of the device. Plotthe ratio of the conduction through the shaft to thenet power output of the turbine over the range0.005 m ?? m. Is a scaled-down device withL?0.005 m feasible?

A glass window of width W?1 m and height H?2mis 5 mm thick and has a thermal conductivity of kg?1.4 W/m?K. If the inner and outer surface temperaturesof the glass are 15C and ?20C, respectively, on acold winter day, what is the rate of heat loss through theglass? To reduce heat loss through windows, it is cus-tomary to use a double pane construction in whichadjoining panes are separated by an air space. If thespacing is 10 mm and the glass surfaces in contact withthe air have temperatures of 10C and ?15C, what is the rate of heat loss from a 1 m ?2 m window? Thethermal conductivity of air is ka?0.024 W/m?K

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

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

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

What is the thickness required of a masonry wall havingthermal conductivity 0.75 W/mK if the heat rate is to be 80% of the heat rate through a composite structural wallhaving a thermal conductivity of 0.25 W/mK and athickness of 100 mm? Both walls are subjected to thesame surface temperature difference.

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

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

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

For a boiling process such as shown in Figure 1.5c, theambient temperature T?in Newtons law of cooling isreplaced by the saturation temperature of the fluid Tsat.Consider a situation where the heat flux from the hotplate is q??20?105W/m2. If the fluid is water at atmospheric pressure and the convection heat transfercoefficient is hw?20?103W/m2?K, determine theupper surface temperature of the plate, Ts,w. In an effortto minimize the surface temperature, a technician proposes replacing the water with a dielectric fluidwhose saturation temperature is Tsat,d?52C. If theheat transfer coefficient associated with the dielectricfluid is hd?3?103W/m2?K, will the techniciansplan work?

Youve experienced convection cooling if youve everextended your hand out the window of a moving vehi-cle or into a flowing water stream. With the surface ofyour hand at a temperature of 30C, determine the con-vection heat flux for (a) a vehicle speed of 35 km/h inair at ?5C with a convection coefficient of 40W/m2?K and (b) a velocity of 0.2 m/s in a water streamat 10?C with a convection coefficient of 900 W/m2?K.Which condition would feelcolder? Contrast these results with a heat loss of approximately 30 W/m2under normal room conditions

Air at 40C flows over a long, 25-mm-diameter cylinderwith an embedded electrical heater. In a series of tests,measurements were made of the power per unit length,P, required to maintain the cylinder surface tempera-ture at 300C for different free stream velocities Vof theair. The results are as follows:Air velocity, V(m/s1 2 4 8 12Power, P(W/m) 450 658 983 1507 1963(a) Determine the convection coefficient for each velocity, and display your results graphically.(b) Assuming the dependence of the convection coeffi-cient on the velocity to be of the form h?CVn,determine the parameters Cand nfrom the resultsof part (a)

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

An electric resistance heater is embedded in a longcylinder of diameter 30 mm. When water with a tem-perature of 25C and velocity of 1 m/s flows crosswiseover the cylinder, the power per unit length required tomaintain the surface at a uniform temperature of 90C is 28 kW/m. When air, also at 25C, but with a velocityof 10 m/s is flowing, the power per unit length requiredto maintain the same surface temperature is 400 W/m.Calculate and compare the convection coefficients forthe flows of water and air.

The free convection heat transfer coefficient on a thinhot vertical plate suspended in still air can be deter-mined from observations of the change in plate temper-ature with time as it cools. Assuming the plate isisothermal and radiation exchange with its surround-ings is negligible, evaluate the convection coefficient atthe instant of time when the plate temperature is 225Cand the change in plate temperature with time (dT/dt) is?0.022 K/s. The ambient air temperature is 25C andthe plate measures 0.3? .3 m with a mass of 3.75 kgand a specific heat of 2770 J/kg?K

A transmission case measures W?0.30 m on a sideand receives a power input of Pi?150 hp from theengine.If the transmission efficiency is ??0.93 and airflowover the case corresponds to T??30C and h?200W/m2?K, what is the surface temperature of the transmission?

A cartridge electrical heater is shaped as a cylinder oflength L?200 mm and outer diameter D?20 mm.Under normal operating conditions, the heater dissipates2 kW while submerged in a water flow that is at 20Cand provides a convection heat transfer coefficient ofh?5000 W/m2?K. Neglecting heat transfer from theends of the heater, determine its surface temperature Ts.If the water flow is inadvertently terminated while theheater continues to operate, the heater surface isexposed to air that is also at 20C but for which h ?50W/m2?K. What is the corresponding surface tempera-ture? What are the consequences of such an event?

A common procedure for measuring the velocity of anairstream involves the insertion of an electrically heatedwire (called a hot-wire anemometer) into the airflow,with the axis of the wire oriented perpendicular to theflow direction. The electrical energy dissipated in the wire is assumed to be transferred to the air by forcedconvection. Hence, for a prescribed electrical power, thetemperature of the wire depends on the convection coef-ficient, which, in turn, depends on the velocity of the air.Consider a wire of length L?20 mm and diameterD?0.5 mm, for which a calibration of the formV?6.25?10?5h2has been determined. The velocity Vand the convection coefficient hhave units of m/s andW/m2?K, respectively. In an application involving air ata temperature of T??25C, the surface temperature ofthe anemometer is maintained at Ts?75C with a volt-age drop of 5 V and an electric current of 0.1 A. What isthe velocity of the air

A square isothermal chip is of width w?5 mm on aside and is mounted in a substrate such that its side andback surfaces are well insulated; the front surface is exposed to the flow of a coolant at T??15C. From reliability considerations, the chip temperature must notexceed T ?85C.If the coolant is air and the corresponding convectioncoefficient is h?200 W/m2?K, what is the maximumallowable chip power? If the coolant is a dielectric liquid for which h?3000 W/m2?K, what is the maxi-mum allowable power?

The temperature controller for a clothes dryer consistsof a bimetallic switch mounted on an electrical heaterattached to a wall-mounted insulation pad. The switch is set to open at 70C, the maximum dryer air temperature. To operate the dryer at a lower air tem-perature, sufficient power is supplied to the heater such thatthe switch reaches 70C (Tset) when the air temperature Tis less than Tset. If the convection heat transfer coefficientbetween the air and the exposed switch surface of 30 mm2is 25 W/m2?K, how much heater power Peis requiredwhen the desired dryer air temperature is T??50C?

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

Under conditions for which the same room temperatureis maintained by a heating or cooling system, it is notuncommon for a person to feel chilled in the winter butcomfortable in the summer. Provide a plausible expla-nation for this situation (with supporting calculations)by considering a room whose air temperature is main-tained at 20C throughout the year, while the walls ofthe room are nominally at 27C and 14C in the sum-mer and winter, respectively. The exposed surface of aperson in the room may be assumed to be at a tempera-ture of 32C throughout the year and to have an emis-sivity of 0.90. The coefficient associated with heattransfer by natural convection between the person andthe room air is approximately 2 W/m K

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

An instrumentation package has a spherical outer surfaceof diameter D?100 mm and emissivity ??0.25. Thepackage is placed in a large space simulation chamberwhose walls are maintained at 77 K. If operation of theelectronic components is restricted to the temperature range 40 ?T ?85C, what is the range of acceptablepower dissipation for the package? Display your resultsgraphically, showing also the effect of variations in theemissivity by considering values of 0.20 and 0.30

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

If in Equation 1.9, the radiation heat transfercoefficient may be approximated aswhere . We wish to assess the validityof this approximation by comparing values of hrandhr,afor the following conditions. In each case, representyour results graphically and comment on the validity ofthe approximation.(a) Consider a surface of either polished aluminum (??0.05) or black paint (??0.9), whose temperaturemay exceed that of the surroundings (Tsur?25C)by 10 to 100C. Also compare your results with val-ues of the coefficient associated with free convectionin air , where h(W/m2?K)?0.98?T1/3.(b) Consider initial conditions associated with placing aworkpiece at in a large furnace whosewall temperature may be varied over the range . According to the surface finish orcoating, its emissivity may assume values of 0.05,0.2, and 0.9. For each emissivity, plot the relativeerror, (hr?hr,a)/hr, as a function of the furnacetemperature.

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

An electrical resistor is connected to a battery, asshown schematically. After a brief transient, the resistorassumes a nearly uniform, steady-state temperature of95C, while the battery and lead wires remain at theambient temperature of 25C. Neglect the electricalresistance of the lead wires.(a) Consider the resistor as a system about which a control surface is placed and Equation 1.12c is applied. Determine the corresponding values of in(W), g(W), out(W), and st(W). If a controlsurface is placed about the entire system, what arethe values of in, g, out, and st?(b) If electrical energy is dissipated uniformly within theresistor, which is a cylinder of diameter D?60 mmand length L?250 mm, what is the volumetric heatgeneration rate, (W/m3)?(c) Neglecting radiation from the resistor, what is theconvection coefficient?

Pressurized water (pin?10 bar, Tin?110C) enters thebottom of an L?10-m-long vertical tube of diameterD?100 mm at a mass flow rate of ?1.5 kg/s. Thetube is located inside a combustion chamber, resultingin heat transfer to the tube. Superheated steam exits thetop of the tube at pout?7 bar, Tout?600C. Determinethe change in the rate at which the following quantitiesenter and exit the tube: (a) the combined thermal andflow work, (b) the mechanical energy, and (c) the totalenergy of the water. Also, (d) determine the heat trans-fer rate, q. Hint: Relevant properties may be obtainedfrom a thermodynamics text.

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

An internally reversible refrigerator has a modifiedcoefficient of performance accounting for realistic heattransfer processes ofwhere qinis the refrigerator cooling rate, qoutis the heatrejection rate, and is the power input. Show that COPmcan be expressed in terms of the reservoir temperaturesTcand Th, the cold and hot thermal resistances Rt,candRt,h, and qin, aswhere Rtot?Rt,cRt,h. Also, show that the power inputmay be expressed as

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

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

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

One method for growing thin silicon sheets for photo-voltaic solar panels is to pass two thin strings of highmelting temperature material upward through a bath ofmolten silicon. The silicon solidifies on the strings nearthe surface of the molten pool, and the solid silicon sheetis pulled slowly upward out of the pool. The silicon isreplenished by supplying the molten pool with solid sili-con powder. Consider a silicon sheet that is Wsi?85 mmwide and tsi?150m thick that is pulled at a velocity ofVsi?20 mm/min. The silicon is melted by supplyingelectric power to the cylindrical growth chamber ofheight H?350 mm and diameter D?300 mm. Theexposed surfaces of the growth chamber are at Ts?320 K, the corresponding convection coefficient at the exposed surface is h?8 W/m2?K, and the surface ischaracterized by an emissivity of ?s?0.9. The solid sili-con powder is at Tsi,i?298 K, and the solid silicon sheetexits the chamber at Tsi,o?420 K. Both the surroundingsand ambient temperatures are T??Tsur?298 K.54Chapter 1?IntroductionVsiSolid silicon powderVsiSolidsiliconsheetMolten siliconPelecTsurTs, sAirT, htsiSolidsiliconsheetMolten siliconStringCrucibleHTs,oD(a) Determine the electric power, Pelec, needed to oper-ate the system at steady state.(b) If the photovoltaic panel absorbs a time- averagedsolar flux of ?180 W/m2and the panel has aconversion efficiency (the ratio of solar power absorbed to electric power produced) of??0.20,how long must the solar panel be operated to pro-duce enough electric energy to offset the electric energy that was consumed in its manufacture?

Heat is transferred by radiation and convection betweenthe inner surface of the nacelle of the wind turbine of Example 1.3 and the outer surfaces of the gearbox andgenerator. The convection heat flux associated with the gearbox and the generator may be described by q?conv,gb?h(Tgb?T?) and q?conv,gen?h(Tgen?T?),respectively, where the ambient temperature T??Ts(which is the nacelle temperature) and h?40 W/m2?K.The outer surfaces of both the gearbox and the generatorare characterized by an emissivity of ??0.9. If the sur-face areas of the gearbox and generator are Agb?6m2and Agen?4m2, respectively, determine their surfacetemperatures.

Radioactive wastes are packed in a long, thin-walledcylindrical container. The wastes generate thermal energynonuniformly according to the relation ?[1?(r/ro)2], where is the local rate of energy generation perunit volume, is a constant, and rois the radius of thecontainer. Steady-state conditions are maintained by sub-merging the container in a liquid that is at T?and pro-vides a uniform convection coefficient h. Obtain an expression for the total rate at which energy is generated in a unit length of the container. Use thisresult to obtain an expression for the temperature Tsof thecontainer wall.

An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface is well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density \(\rho\) and specific heat c of aluminum are known to be 2700 \(\mathrm{kg} / \mathrm{m}^{3}\) and 900 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\), respectively. (a) Consider conditions for which the plate is at a temperature of 25°C and its top surface is suddenly exposed to ambient air at \(T_{\infty}=20^{\circ} \mathrm{C}\) and to solar radiation that provides an incident flux of 900 \(\mathrm{W} / \mathrm{m}^{2}\).The convection heat transfer coefficient between the surface and the air is h = 20 \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\). What is the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? (c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for \(0.05 \leq \varepsilon \leq 1\), with all other conditions remaining as prescribed. Repeat your calculations for values of \(\alpha_{S}\) = 0.5 and 1.0, and plot the results with those obtained for \(\alpha_{S}\) = 0.8. If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation?

A blood warmer is to be used during the transfusion ofblood to a patient. This device is to heat blood takenfrom the blood bank at 10C to 37C at a flow rate of200 ml/min. The blood passes through tubing of length2 m, with a rectangular cross section 6.4 mm?1.6 mmAt what rate must heat be added to the blood to accom-plish the required temperature increase? If the fluidoriginates from a large tank with nearly zero velocityand flows vertically downward for its 2-m length Obtain an expression for the total rate at which energy is generated in a unit length of the container. Use thisresult to obtain an expression for the temperature Tsof thecontainer wall.1.45An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface is well insu-lated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solarradiation, while having an emissivity of 0.25. The density ?and specific heat cof aluminum are known tobe 2700 kg/m3and 900 J/kg?K, respectively.(a) Consider conditions for which the plate is at a tem-perature of 25C and its top surface is suddenly exposed to ambient air at T??20C and to solar radiation that provides an incident flux of 900 W/m2.The convection heat transfer coefficient between thesurface and the air is h?20 W/m2?K. What is theinitial rate of change of the plate temperature?(b) What will be the equilibrium temperature of theplate when steady-state conditions are reached?(c) The surface radiative properties depend on the spe-cific nature of the applied coating. Compute and plotthe steady-state temperature as a function of theemissivity for 0.05???1, with all other conditionsremaining as prescribed. Repeat your calculations forvalues of ?S?0.5 and 1.0, and plot the results withthose obtained for ?S?0.8. If the intent is to maxi-mize the plate temperature, what is the mostdesirable combination of the plate emissivity and itsabsorptivity to solar radiation?1.46A blood warmer is to be used during the transfusion ofblood to a patient. This device is to heat blood takenfrom the blood bank at 10C to 37C at a flow rate of200 ml/min. The blood passes through tubing of length2 m, with a rectangular cross section 6.4 mm?1.6 mmAt what rate must heat be added to the blood to accom-plish the required temperature increase? If the fluidoriginates from a large tank with nearly zero velocityand flows vertically downward for its 2- m length, T, hroq = qo [1 (r/ro)2]estimate the magnitudes of kinetic and potential energychanges. Assume the bloods properties are similar tothose of water.

Consider a carton of milk that is refrigerated at a tem-perature of Tm?5C. The kitchen temperature on a hotsummer day is T??30C. If the four sides of the cartonare of height and width L?200 mm and w?100 mm,respectively, determine the heat transferred to the milkcarton as it sits on the kitchen counter for durations oft?10 s, 60 s, and 300 s before it is returned to the refrigerator. The convection coefficient associated withnatural convection on the sides of the carton is h?10W/m2?K. The surface emissivity is 0.90. Assume themilk carton temperature remains at 5C during theprocess. Your parents have taught you the importance ofrefrigerating certain foods from the food safety perspec-tive. Comment on the importance of quickly returningthe milk carton to the refrigerator from an energy con-servation point of view

The energy consumption associated with a home waterheater has two components: (i) the energy that must besupplied to bring the temperature of groundwater to theheater storage temperature, as it is introduced to replacehot water that has been used; (ii) the energy needed tocompensate for heat losses incurred while the water isstored at the prescribed temperature. In this problem,we will evaluate the first of these components for afamily of four, whose daily hot water consumption isapproximately 100 gal. If groundwater is available at15C, what is the annual energy consumption associ-ated with heating the water to a storage temperature of55C? For a unit electrical power cost of $0.18/kW?h,what is the annual cost associated with supplying hotwater by means of (a) electric resistance heating or (b) a heat pump having a COP of 3.

Liquid oxygen, which has a boiling point of 90 K and alatent heat of vaporization of 214 kJ/kg, is stored in a spherical container whose outer surface is of 500-mmdiameter and at a temperature of ?10C. The container ishoused in a laboratory whose air and walls are at 25C.(a) If the surface emissivity is 0.20 and the heat transfercoefficient associated with free convection at theouter surface of the container is 10 W/m2?K, what isthe rate, in kg/s, at which oxygen vapor must bevented from the system?(b) Moisture in the ambient air will result in frost forma-tion on the container, causing the surface emissivityto increase. Assuming the surface temperature andconvection coefficient to remain at ?10C and 10 W/m2?K, respectively, compute the oxygen evap-oration rate (kg/s) as a function of surface emissivityover the range 0.2 ???0.94.

The emissivity of galvanized steel sheet, a commonroofing material, is ??0.13 at temperatures around300 K, while its absorptivity for solar irradiation is?S?0.65. Would the neighborhood cat be comfortablewalking on a roof constructed of the material on a day when GS?750 W/m2, T??16C, and h?7W/m2?K? Assume the bottom surface of the steel is insulated

Three electric resistance heaters of length L?250 mmand diameter D?25 mm are submerged in a 10- galtank of water, which is initially at 295 K. The watermay be assumed to have a density and specific heat of ??990 kg/m3and c?4180 J/kg?K.(a) If the heaters are activated, each dissipatingq1?500 W, estimate the time required to bring thewater to a temperature of 335 K.(b) If the natural convection coefficient is given by anexpression of the form h?370 (Ts?T)1/3, whereTsand Tare temperatures of the heater surface and water, respectively, what is the temperature ofeach heater shortly after activation and just beforedeactivation? Units of hand (Ts?T) are W/m2?Kand K, respectively.(c) If the heaters are inadvertently activated when thetank is empty, the natural convection coefficientassociated with heat transfer to the ambient air atT??300 K may be approximated as h?0.70(Ts?T?)1/3. If the temperature of the tank walls isalso 300 K and the emissivity of the heater surfaceis ??0.85, what is the surface temperature of eachheater under steady-state conditions?

A hair dryer may be idealized as a circular duct throughwhich a small fan draws ambient air and within whichthe air is heated as it flows over a coiled electric resis-tance wire. (a) If a dryer is designed to operate with an electricpower consumption of Pelec?500 W and to heatair from an ambient temperature of Ti?20C to adischarge temperature of To?45C, at what volu-metric flow rate should the fan operate? Heat lossfrom the casing to the ambient air and the surround-ings may be neglected. If the duct has a diameter ofD?70 mm, what is the discharge velocity Voofthe air? The density and specific heat of the air maybe approximated as ??1.10 kg/m3and cp?1007J/kg?K, respectively.(b) Consider a dryer duct length of L?150 mm and asurface emissivity of ??0.8. If the coefficientassociated with heat transfer by natural convectionfrom the casing to the ambient air is h?4 W/m2?K and the temperature of the air and the surroundings is T??Tsur?20C, confirm that the heat loss from the casing is, in fact, negligible.The casing may be assumed to have an average surface temperature of Ts?40C

n one stage of an annealing process, 304 stainless steelsheet is taken from 300 K to 1250 K as it passesthrough an electrically heated oven at a speed of Vs?10 mm/s. The sheet thickness and width arets?8 mm and Ws?2 m, respectively, while theheight, width, and length of the oven are Ho?2m,Wo?2.4 m, and Lo?25 m, respectively. The top andfour sides of the oven are exposed to ambient air and large surroundings, each at 300 K, and the corre-sponding surface temperature, convection coefficient,and emissivity are Ts?350 K, h?10 W/m2?K, and?s?0.8. The bottom surface of the oven is also at350 K and rests on a 0.5-m-thick concrete pad whosebase is at 300 K. Estimate the required electric powerinput, Pelec, to the oven.

Convection ovens operate on the principle of inducingforced convection inside the oven chamber with a fan.A smallcake is to be baked in an oven when the con-vection feature is disabled. For this situation, the freeconvection coefficient associated with the cake and its pan is hfr?3 W/m2?K. The oven air and wall are attemperatures T??Tsur?180C. Determine the heatflux delivered to the cake pan and cake batter whenthey are initially inserted into the oven and are at a tem-perature of Ti?24C. If the convection feature is acti-vated, the forced convection heat transfer coefficient ishfo?27 W/m2?K. What is the heat flux at the batter orpan surface when the oven is operated in the convectionmode? Assume a value of 0.97 for the emissivity of thecake batter and pan.

Annealing, an important step in semiconductor materi-als processing, can be accomplished by rapidly heatingthe silicon wafer to a high temperature for a short period of time. The schematic shows a method involv-ing the use of a hot plate operating at an elevated tem-perature Th. The wafer, initially at a temperature of Tw,i,is suddenly positioned at a gap separation distance Lfrom the hot plate. The purpose of the analysis is tocompare the heat fluxes by conduction through the gaswithin the gap and by radiation exchange between thehot plate and the cool wafer. The initial time rate ofchange in the temperature of the wafer, (dTw/dt)i, is alsoof interest. Approximating the surfaces of the hot plateand the wafer as blackbodies and assuming their diame-ter Dto be much larger than the spacing L, the radiativeheat flux may be expressed as The silicon wafer has a thickness of d?0.78 mm, adensity of 2700 kg/m3, and a specific heat of 875J/kg?K. The thermal conductivity of the gas in the gapis 0.0436 W/m?K.q?rad??(T4h?T4w).transfer modes and the effect of the gap distance onthe heating process. Under what conditions could awafer be heated to 900C in less than 10 s?

n the thermal processing of semiconductor materials,annealing is accomplished by heating a silicon waferaccording to a temperature-time recipe and then main-taining a fixed elevated temperature for a prescribedperiod of time. For the process tool arrangement shownas follows, the wafer is in an evacuated chamber whose walls are maintained at 27C and within whichheating lamps maintain a radiant flux at its upper surface. The wafer is 0.78 mm thick, has a thermal con-ductivity of 30 W/m?K, and an emissivity that equalsits absorptivity to the radiant flux (???l?0.65). For?3.0?105W/m2, the temperature on its lower sur-face is measured by a radiation thermometer and foundto have a value of Tw,l?997C. To avoid warping the wafer and inducing slip planes inthe crystal structure, the temperature difference acrossthe thickness of the wafer must be less than 2C. Is thiscondition being met?

A furnace for processing semiconductor materials isformed by a silicon carbide chamber that is zone- heatedon the top section and cooled on the lower section. Withthe elevator in the lowest position, a robot arm inserts thesilicon wafer on the mounting pins. In a production oper-ation, the wafer is rapidly moved toward the hot zone toachieve the temperature-time history required for theprocess recipe. In this position, the top and bottom sur-faces of the wafer exchange radiation with the hot andcool zones, respectively, of the chamber. The zone temperatures are Th?1500 K and Tc?330 K, and theemissivity and thickness of the wafer are ??0.65 andd?0.78 mm, respectively. With the ambient gas atT??700 K, convection coefficients at the upper andlower surfaces of the wafer are 8 and 4 W/m2?K, respec-tively. The silicon wafer has a density of 2700 kg/m3anda specific heat of 875 J/kg?K. (a) For an initial condition corresponding to a wafertemperature of Tw,i?300 K and the position of thewafer shown schematically, determine the corre-sponding time rate of change of the wafer temper-ature, (dTw/dt)i.(b) Determine the steady-state temperature reachedby the wafer if it remains in this position. Howsignificant is convection heat transfer for thissituation? Sketch how you would expect thewafer temperature to vary as a function of verti-cal distance.

Single fuel cells such as the one of Example 1.5 can bescaled up by arranging them into a fuel cell stack.A stackconsists of multiple electrolytic membranes that aresandwiched between electrically conducting bipolarplates.Air and hydrogen are fed to each membranethrough flowchannelswithin each bipolar plate, asshown in the sketch. With this stack arrangement, theindividual fuel cells are connected in series, electrically,producing a stack voltage of Estack?N?Ec, where Ecisthe voltage produced across each membrane and Nis thenumber of membranes in the stack. The electrical currentis the same for each membrane. The cell voltage, Ec, aswell as the cell efficiency, increases with temperature(the air and hydrogen fed to the stack are humidified toallow operation at temperatures greater than in Example1.5), but the membranes will fail at temperatures exceed-ing T85C. Consider L?wmembranes, whereL?w?100 mm, of thickness tm?0.43 mm, that eachproduce Ec?0.6 V at I?60 A, and ?45 W of thermal energy when operating at T?80C. The exter-nal surfaces of the stack are exposed to air at T??25Cand surroundings at Tsur?30C, with ??0.88 andh?150 W/m2?K. (a) Find the electrical power produced by a stack thatis Lstack?200 mm long, for bipolar plate thicknessin the range 1 mm ?tbp ?10 mm. Determine thetotal thermal energy generated by the stack.(b) Calculate the surface temperature and explainwhether the stack needs to be internally heated orcooled to operate at the optimal internal tempera-ture of 80C for various bipolar plate thicknesses.(c) Identify how the internal stack operating tempera-ture might be lowered or raised for a given bipolarplate thickness, and discuss design changes thatwould promote a more uniform temperature distrib-ution within the stack. How would changes in theexternal air and surroundings temperature affectyour answer? Which membrane in the stack is mostlikely to fail due to high operating temperature?

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

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

A long bus bar (cylindrical rod used for making electrical connections) of diameter Dis installed in alarge conduit having a surface temperature of 30Cand in which the ambient air temperature is T??30C. The electrical resistivity, ?e(?m), of the bar material is a function of temperature, ?e,o??e[1?(T?To)], where ?e,o?0.0171?m,To?25C, and ??0.00396 K?1. The bar experiences freeconvection in the ambient air, and the convectioncoefficient depends on the bar diameter, as well as on the difference between the surface and ambienttemperatures. The governing relation is of the form, h?CD?0.25(T?T?)0.25, where C?1.21W?m?1.75?K?1.25. The emissivity of the bar surface is??0.85.(a) Recognizing that the electrical resistance per unitlength of the bar is ??e/Ac, where Acis itscross-sectional area, calculate the current- carryingcapacity of a 20-mm-diameter bus bar if its tem-perature is not to exceed 65C. Compare the rela-tive importance of heat transfer by free convectionand radiation exchange.(b) To assess the trade-off between current-carrying capacity, operating temperature, and bar diameter,for diameters of 10, 20, and 40 mm, plot the bar temperature Tas a function of current for the range 100?I?5000 A. Also plot the ratio of the heat transfer by convection to the total heattransfer

A small sphere of reference-grade iron with a specificheat of 447 J/kg?K and a mass of 0.515 kg is suddenlyimmersed in a waterice mixture. Fine thermocouplewires suspend the sphere, and the temperature isobserved to change from 15 to 14C in 6.35 s. Theexperiment is repeated with a metallic sphere of thesame diameter, but of unknown composition with amass of 1.263 kg. If the same observed temperaturechange occurs in 4.59 s, what is the specific heat of theunknown material?

A 50 mm?45 mm?20 mm cell phone charger has a surface temperature of Ts?33C when pluggedinto an electrical wall outlet but not in use. The surface of the charger is of emissivity ??0.92 and issubject to a free convection heat transfer coefficient of h?4.5 W/m2?K. The room air and wall tempera-tures are T??22C and Tsur?20C, respectively. If electricity costs C?$0.18/kW?h, determine thedaily cost of leaving the charger plugged in when notin use

A spherical, stainless steel (AISI 302) canister is used tostore reacting chemicals that provide for a uniform heatflux to its inner surface. The canister is suddenly sub-merged in a liquid bath of temperature T??Ti, where Tiis the initial temperature of the canister wall.(a) Assuming negligible temperature gradients in thecanister wall and a constant heat flux , develop anequation that governs the variation of the wall tem-perature with time during the transient process.What is the initial rate of change of the wall tem-perature if ?105W/m2?(b) What is the steady-state temperature of the wall?(c) The convection coefficient depends on the velocityassociated with fluid flow over the canister andwhether the wall temperature is large enough toinduce boiling in the liquid. Compute and plot thesteady-state temperature as a function of hfor the range 100?h?10,000 W/m2?K. Is there avalue of hbelow which operation would be unacceptable

A freezer compartment is covered with a 2-mm-thicklayer of frost at the time it malfunctions. If the compartment is in ambient air at 20C and a coefficientof h ?2 W/m2?K characterizes heat transfer by naturalconvection from the exposed surface of the layer, esti-mate the time required to completely melt the frost.The frost may be assumed to have a mass density of700 kg/m3and a latent heat of fusion of 334 kJ/kg.

A vertical slab of Woods metal is joined to a substrate onone surface and is melted as it is uniformly irradiated by alaser source on the opposite surface. The metal is initiallyat its fusion temperature of Tf?72C, and the melt runsoff by gravity as soon as it is formed. The absorptivity ofthe metal to the laser radiation is ?1?0.4, and its latentheat of fusion is hsf ?33 kJ/kg.(a) Neglecting heat transfer from the irradiated surfaceby convection or radiation exchange with the surroundings, determine the instantaneous rate ofmelting in kg/s?m2if the laser irradiation is 5 kW/m2.How much material is removed if irradiation is main-tained for a period of 2 s?(b) Allowing for convection to ambient air, withT??20C and h?15 W/m2?K, and radiationexchange with large surroundings (??0.4,Tsur?20C), determine the instantaneous rate ofmelting during irradiation

A photovoltaic panel of dimension 2 m?4m isinstalled on the roof of a home. The panel is irradiatedwith a solar flux of GS?700 W/m2, oriented normal tothe top panel surface. The absorptivity of the panel to thesolar irradiation is ?S?0.83, and the efficiency of con-version of the absorbed flux to electrical power is??P/?SGSA?0.553?0.001 K?1Tp, where Tpis thepanel temperature expressed in kelvins and Ais the solarpanel area. Determine the electrical power generated for(a) a still summer day, in which Tsur?T??35C,h?10 W/m2?K, and (b) a breezy winter day, for whichTsur?T???15C, h?30 W/m2?K. The panel emis- sivity is ??0.90

Following the hot vacuum forming of a paper-pulpmixture, the product, an egg carton, is transported on aconveyor for 18 s toward the entrance of a gas-firedoven where it is dried to a desired final water content.Very little water evaporates during the travel time. So, to increase the productivity of the line, it is pro-posed that a bank of infrared radiation heaters, whichprovide a uniform radiant flux of 5000 W/m2, beinstalled over the conveyor. The carton has an exposedarea of 0.0625 m2and a mass of 0.220 kg, 75% ofwhich is water after the forming process. The chief engineer of your plant will approve the pur-chase of the heaters if they can reduce the water contentby 10% of the total mass. Would you recommend thepurchase? Assume the heat of vaporization of water is hfg?2400 kJ/kg.

Electronic power devices are mounted to a heat sink hav-ing an exposed surface area of 0.045 m2 and an emissiv-ity of 0.80. When the devices dissipate a total power of20 W and the air and surroundings are at 27C, the aver-age sink temperature is 42C. What average temperaturewill the heat sink reach when the devices dissipate 30 Wfor the same environmental condition?

A computer consists of an array of five printed circuitboards (PCBs), each dissipating Pb ?20 W of power.Cooling of the electronic components on a board is pro-vided by the forced flow of air, equally distributed inpassages formed by adjoining boards, and the convec-tion coefficient associated with heat transfer from thecomponents to the air is approximately h?200 W/m2?K.Air enters the computer console at a temperature of Ti?20C, and flow is driven by a fan whose powerconsumption is Pf?25 W. (a) If the temperature rise of the airflow, (To?Ti), is notto exceed 15C, what is the minimum allowable volu-metric flow rate of the air? The density and specificheat of the air may be approximated as ??1.161kg/m3and cp?1007 J/kg?K, respectively.(b) The component that is most susceptible to thermalfailure dissipates 1 W/cm2of surface area. To mini-mize the potential for thermal failure, where shouldthe component be installed on a PCB? What is itssurface temperature at this location

Consider a surface-mount type transistor on a circuitboard whose temperature is maintained at 35C. Air at 20C flows over the upper surface of dimensions 4 mm x 8 mm with a convection coefficient of 50 W/m2K. Three wire leads, each of cross section 1 mm x 0.25 mm andlength 4 mm, conduct heat from the case to the circuitboard. The gap between the case and the board is 0.2 mm.(a) Assuming the case is isothermal and neglecting radia-tion, estimate the case temperature when 150 mW isdissipated by the transistor and (i) stagnant air or (ii) aconductive paste fills the gap. The thermal conductiv-ities of the wire leads, air, and conductive paste are25, 0.0263, and 0.12 W/mK, respectively. (b) Using the conductive paste to fill the gap, we wish todetermine the extent to which increased heat dissipa-tion may be accommodated, subject to the constraintthat the case temperature not exceed 40C. Options include increasing the air speed to achieve a largerconvection coefficient hand/or changing the leadwire material to one of larger thermal conductivity.Independently considering leads fabricated frommaterials with thermal conductivities of 200 and 400 W/mK, compute and plot the maximum allow-able heat dissipation for variations in hover the range 50 h 250 W/m2K

The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m2, and the underside is perfectly insulated. The convection coefficient between the roof and the ambient air is 12 W/m2K.(a) Neglecting radiation exchange with the surroundings,calculate the temperature of the roof under steady-state conditions if the ambient air temperature is 20C. (b) For the same ambient air temperature, calculate the temperature of the roof if its surface emissivityis 0.8.(c) The convection coefficient depends on airflow condi-tions over the roof, increasing with increasing airspeed. Compute and plot the roof temperature as afunction of h for 2 h 200 W/m2K.

Consider the conditions of Problem 1.22, but the sur-roundings temperature is 25C and radiation exchange with the surroundings is not negligible. If the convection coefficient is 6.4 W/m2K and the emissivity of the plate is = 0.42, determine the time rate of change ofthe plate temperature, dT/dt, when the plate temperatureis 225C. Evaluate the heat loss by convection and the heat loss by radiation.

Most of the energy we consume as food is converted tothermal energy in the process of performing all our bodilyfunctions and is ultimately lost as heat from our bodies.Consider a person who consumes 2100 kcal per day (notethat what are commonly referred to as food calories areactually kilocalories), of which 2000 kcal is converted tothermal energy. (The remaining 100 kcal is used to dowork on the environment.) The person has a surface areaof 1.8 m2and is dressed in a bathing suit.(a) The person is in a room at 20C, with a convectionheat transfer coefficient of 3 W/m2K. At this airtemperature, the person is not perspiring much.Estimate the persons average skin temperature.(b) If the temperature of the environment were 33C,what rate of perspiration would be needed to main- tain a comfortable skin temperature of 33C?

Consider Problem 1.1.(a) If the exposed cold surface of the insulation is at T2?20C, what is the value of the convection heattransfer coefficient on the cold side of the insulationif the surroundings temperature is Tsur?320 K, theambient temperature is T??5C, and the emissiv-ity is ??0.95? Express your results in units ofW/m2?K and W/m2?C.(b) Using the convective heat transfer coefficient youcalculated in part (a), determine the surface tempera-ture, T2, as the emissivity of the surface is variedover the range 0.05 ???0.95. The hot wall tem-perature of the insulation remains fixed at T1?30C.Display your results graphically

The wall of an oven used to cure plastic parts is ofthickness L?0.05 m and is exposed to large surround-ings and air at its outer surface. The air and the sur-roundings are at 300 K.(a) If the temperature of the outer surface is 400 K and its convection coefficient and emissivity are h?20 W/m2?K and ??0.8, respectively, what isthe temperature of the inner surface if the wall has athermal conductivity of k?0.7 W/m2?K?(b) Consider conditions for which the temperature of theinner surface is maintained at 600 K, while the airand large surroundings to which the outer surface isexposed are maintained at 300 K. Explore the effectsof variations in k, h, and ?on (i) the temperature ofthe outer surface, (ii) the heat flux through the wall,and (iii) the heat fluxes associated with convectionand radiation heat transfer from the outer surface.Specifically, compute and plot the foregoing depen-dent variables for parametric variations about base-line conditions of k?10 W/m?K, h?20 W/m2?K,and ??0.5. The suggested ranges of the indepen-dent variables are 0.1 ?k?400 W/m?K, 2 ?h?200 W/m2?K, and 0.05 ???1. Discuss the physi-cal implications of your results. Under what condi-tions will the temperature of the outer surface be lessthan 45C, which is a reasonable upper limit to avoidburn injuries if contact is made

An experiment to determine the convection coefficientassociated with airflow over the surface of a thick stainless steel casting involves the insertion of thermo-couples into the casting at distances of 10 and 20 mm from the surface along a hypothetical line normal to thesurface. The steel has a thermal conductivity of 15 W/mK. If the thermocouples measure temperatures of 50 and 40C in the steel when the air temperature is100C, what is the convection coefficient?

A thin electrical heating element provides a uniformheat flux q?oto the outer surface of a duct through whichairflows. The duct wall has a thickness of 10 mm and athermal conductivity of 20 W/m?K.(a) At a particular location, the air temperature is 30Cand the convection heat transfer coefficientbetween the air and inner surface of the duct is100 W/m2?K. What heat flux qois required tomaintain the inner surface of the duct at Ti?85C? b) For the conditions of part (a), what is the tempera-ture (To) of the duct surface next to the heater?(c) With Ti?85C, compute and plot qoand Toas afunction of the air-side convection coefficient hforthe range 10?h?200 W/m2?K. Briefly discussyour results.

A rectangular forced air heating duct is suspended fromthe ceiling of a basement whose air and walls are at atemperature of T??Tsur?5C. The duct is 15 m long,and its cross section is 350 mm ?200 mm.(a) For an uninsulated duct whose average surface tem-perature is 50C, estimate the rate of heat loss fromthe duct. The surface emissivity and convectioncoefficient are approximately 0.5 and 4 W/m2?K,respectively.(b) If heated air enters the duct at 58C and a velocity of4 m /s and the heat loss corresponds to the result ofpart (a), what is the outlet temperature? The densityand specific heat of the air may be assumed to be ??1.10 kg/m3and c??1008 J/kg?K, respectively.

Consider the steam pipe of Example 1.2. The facilitiesmanager wants you to recommend methods for reduc-ing the heat loss to the room, and two options are pro-posed. The first option would restrict air movementaround the outer surface of the pipe and thereby reducethe convection coefficient by a factor of two. The sec-ond option would coat the outer surface of the pipe witha low emissivity (??0.4) paint.(a) Which of the foregoing options would you recommend?(b) To prepare for a presentation of your recommenda-tion to management, generate a graph of the heatloss qas a function of the convection coefficientfor 2?h?20 W/m2?K and emissivities of 0.2,0.4, and 0.8. Comment on the relative efficacy ofreducing heat losses associated with convection andradiation.

During its manufacture, plate glass at 600C is cooled by passing air over its surface such that the convectionheat transfer coefficient is h = 5 W/m2K. To prevent cracking, it is known that the temperature gradient must not exceed 15C/mm at any point in the glass during thecooling process. If the thermal conductivity of the glassis 1.4 W/mK and its surface emissivity is 0.8, what isthe lowest temperature of the air that can initially beused for the cooling? Assume that the temperature ofthe air equals that of the surroundings.

The curing process of Example 1.9 involves exposureof the plate to irradiation from an infrared lamp andattendant cooling by convection and radiation exchange with the surroundings. Alternatively, in lieu of thelamp, heating may be achieved by inserting the plate inan oven whose walls (the surroundings) are maintainedat an elevated temperature.(a) Consider conditions for which the oven walls areat 200?C, airflow over the plate is characterized byT??20C and h?15 W/m2?K, and the coatinghas an emissivity of ??0.5. What is the tempera-ture of the plate?(b) For ambient air temperatures of 20, 40, and 60C,determine the plate temperature as a function of theoven wall temperature over the range from 150 to250C. Plot your results, and identify conditions forwhich acceptable curing temperatures between 100and 110C may be maintained.

The diameter and surface emissivity of an electricallyheated plate are D?300 mm and ??0.80, respectively.(a) Estimate the power needed to maintain a surfacetemperature of 200C in a room for which the airand the walls are at 25C. The coefficient character-izing heat transfer by natural convection dependson the surface temperature and, in units ofW/m2?K, may be approximated by an expression ofthe form h?0.80(Ts?T?)1/3.(b) Assess the effect of surface temperature on thepower requirement, as well as on the relative con-tributions of convection and radiation to heat trans-fer from the surface.

Bus bars proposed for use in a power transmission station have a rectangular cross section of height H?600 mm and width W?200 mm. The electricalresistivity, ?e(?m), of the bar material is a functionof temperature, ?e??e,o[1?(T?To)], where ?e,o?0.0828 ?m, To?25C, and ??0.0040 K?1. Theemissivity of the bars painted surface is 0.8, and the temperature of the surroundings is 30C. The con-vection coefficient between the bar and the ambient airat 30C is 10 W/m2?K.(a) Assuming the bar has a uniform temperature T, calculate the steady-state temperature when a cur-rent of 60,000 A passes through the bar.(b) Compute and plot the steady-state temperature ofthe bar as a function of the convection coefficientfor 10?h?100 W/m2?K. What minimum convection coefficient is required to maintain asafe- operating temperature below 120C? Willincreasing the emissivity significantly affect thisresult?

A solar flux of 700 W/m2is incident on a flat-plate solarcollector used to heat water. The area of the collector is 3 m2, and 90% of the solar radiation passes throughthe cover glass and is absorbed by the absorber plate. The remaining 10% is reflected away from thecollector. Water flows through the tube passages on the back side of the absorber plate and is heated froman inlet temperature Tito an outlet temperature To. Thecover glass, operating at a temperature of 30C, has an emissivity of 0.94 and experiences radiationexchange with the sky at ?10C. The convection coef-ficient between the cover glass and the ambient air at25C is 10 W/m2?K.(a) Perform an overall energy balance on the collectorto obtain an expression for the rate at which usefulheat is collected per unit area of the collector, qu.Determine the value of qu.(b) Calculate the temperature rise of the water, To?Ti,if the flow rate is 0.01 kg/s. Assume the specificheat of the water to be 4179 J/kg?K.(c) The collector efficiency ?is defined as the ratio ofthe useful heat collected to the rate at which solarenergy is incident on the collector. What is thevalue of ??

An analyzing the performance of a thermal system, the engineer must be able to identify the relevant heattransfer processes. Only then can the system behaviorbe properly quantified. For the following systems, iden-tify the pertinent processes, designating them by appropriately labeled arrows on a sketch of the system.Answer additional questions that appear in the problemstatement.(a) Identify the heat transfer processes that determinethe temperature of an asphalt pavement on a summer day. Write an energy balance for the surface ofthe pavement (b) Microwave radiation is known to be transmitted byplastics, glass, and ceramics but to be absorbed by materials having polar molecules such as water.Water molecules exposed to microwave radiation align and reverse alignment with the microwave radiation at frequencies up to 10 s 1, causing heat to be generated. Contrast cooking in a microwaveoven with cooking in a conventional radiant orconvection oven. In each case, what is the physicalmechanism responsible for heating the food?Which oven has the greater energy utilization efficiency? Why? Microwave heating is being considered for drying clothes. How would the operation of a microwave clothes dryer differ from aconventional dryer? Which is likely to have thegreater energy utilization efficiency? Why?(c) To prevent freezing of the liquid water inside thefuel cell of an automobile, the water is drained toan onboard storage tank when the automobile is notin use. (The water is transferred from the tank back tothe fuel cell when the automobile is turned on.) Consider a fuel cellpowered automobile that is parked outside on a very cold evening with T = -20C.The storage tank is initially empty at Ti,t??20C,when liquid water, at atmospheric pressure and tem-perature Ti,w?50C, is introduced into the tank. Thetank has a wall thickness ttand is blanketed withinsulation of thickness tins. Identify the heat transferprocesses that will promote freezing of the water.Will the likelihood of freezing change as the insula-tion thickness is modified? Will the likelihood offreezing depend on the tank walls thickness andmaterial? Would freezing of the water be more likelyif plastic (low thermal conductivity) or stainless steel(moderate thermal conductivity) tubing is used totransfer the water to and from the tank? Is there anoptimal tank shape that would minimize the probabil-ity of the water freezing? Would freezing be morelikely or less likely to occur if a thin sheet of alu-minum foil (high thermal conductivity, low emissiv-ity) is applied to the outside of the insulation? (d) Your grandmother is concerned about reducing her winter heating bills. Her strategy is to looselyfit rigid polystyrene sheets of insulation over herdouble-pane windows right after the first freezingweather arrives in the autumn. Identify the relevantheat transfer processes on a cold winter night whenthe foamed insulation sheet is placed (i) on theinner surface and (ii) on the outer surface of herwindow. To avoid condensation damage, whichconfiguration is preferred? Condensation on thewindow pane does not occur when the foamedinsulation is not in place.(e) There is considerable interest in developing buildingmaterials with improved insulating qualities. Thedevelopment of such materials would do much toenhance energy conservation by reducing space heat-ing requirements. It has been suggested that superiorstructural and insulating qualities could be obtainedby using the composite shown. The material consistsof a honeycomb, with cells of square cross section,sandwiched between solid slabs. The cells are filledwith air, and the slabs, as well as the honeycombmatrix, are fabricated from plastics of low thermalconductivity. For heat transfer normal to the slabs,identify all heat transfer processes pertinent to theperformance of the composite. Suggest ways inwhich this performance could be enhanced. (f) A thermocouple junction (bead) is used to measurethe temperature of a hot gas stream flowing through achannel by inserting the junction into the mainstreamof the gas. The surface of the channel is cooled suchthat its temperature is well below that of the gas.Identify the heat transfer processes associated withthe junction surface. Will the junction sense a tem-perature that is less than, equal to, or greater than thegas temperature? A radiation shield is a small, open-ended tube that encloses the thermocouple junction,yet allows for passage of the gas through the tube.How does use of such a shield improve the accuracyof the temperature measurement? (g) A double- glazed, glass fire screen is insertedbetween a wood-burning fireplace and the interiorof a room. The screen consists of two vertical glassplates that are separated by a space through whichroom air may flow (the space is open at the top andbottom). Identify the heat transfer processes associ-ated with the fire screen. h) A thermocouple junction is used to measure the tem-perature of a solid material. The junction is insertedinto a small circular hole and is held in place byepoxy. Identify the heat transfer processes associatedwith the junction. Will the junction sense a tempera-ture less than, equal to, or greater than the solid tem-perature? How will the thermal conductivity of theepoxy affect the junction temperature?

In considering the following problems involving heattransfer in the natural environment (outdoors), recognizethat solar radiation is comprised of long and short wave-length components. If this radiation is incident on a semi-transparent medium, such as water or glass, two thingswill happen to the nonreflected portion of the radiation.The long wavelength component will be absorbed at thesurface of the medium, whereas the short wavelengthcomponent will be transmitted by the surface.(a) The number of panes in a window can strongly influ-ence the heat loss from a heated room to the outsideambient air. Compare the single- and double-panedunits shown by identifying relevant heat transferprocesses for each case. b) In a typical flat-plate solar collector, energy is col-lected by a working fluid that is circulated throughtubes that are in good contact with the back face ofan absorber plate. The back face is insulated from Identify all heat transfer processes associated with thecover plates, the absorber plate(s), and the air.(d) Evacuated-tube solar collectors are capable ofimproved performance relative to flat-plate collec-tors. The design consists of an inner tube enclosedin an outer tube that is transparent to solar radia-tion. The annular space between the tubes is evacu-ated. The outer, opaque surface of the inner tubeabsorbs solar radiation, and a working fluid ispassed through the tube to collect the solar energy.The collector design generally consists of a row ofsuch tubes arranged in front of a reflecting panel.Identify all heat transfer processes relevant to theperformance of this device .