The Knudsen number, Knmfp/L, is a dimensionlessparameter

Consider the plane wall of Figure 3.1, separating hot andcold fluids at temperatures T?,1and T?,2, respectively.Using surface energy balances as boundary conditions atx?0 and x?L(see Equation 2.34), obtain the tempera-ture distribution within the wall and the heat flux interms of T?,1, T?,2, h1, h2, k, and L

A new building to be located in a cold climate is beingdesigned with a basement that has an L?200- mm-thickwall. Inner and outer basement wall temperatures areTi?20C and To?0C, respectively. The architect canspecify the wall material to be either aerated concreteblock with kac?0.15 W/m?K, or stone mix concrete. Toreduce the conduction heat flux through the stone mixwall to a level equivalent to that of the aerated concretewall, what thickness of extruded polystyrene sheet mustbe applied onto the inner surface of the stone mix concrete wall? Floor dimensions of the basement are20 m30 m, and the expected rental rate is $50/m2/month. What is the yearly cost, in terms of lost rentalincome, if the stone mix concrete wall with polystyreneinsulation is specified?

The rear window of an automobile is defogged by pass-ing warm air over its inner surface.(a) If the warm air is at T?,i?40C and the correspond-ing convection coefficient is hi?30 W/m2?K, whatare the inner and outer surface temperatures of 4-mm-thick window glass, if the outside ambient airtemperature is T?,o??10C and the associated con-vection coefficient is ho?65 W/m2?K?(b) In practice T?,oand hovary according to weatherconditions and car speed. For values of ho?2, 65,and 100 W/m2?K, compute and plot the inner andouter surface temperatures as a function of T?,ofor30T?,o0C

The rear window of an automobile is defogged by attach-ing a thin, transparent, film-type heating element to itsinner surface. By electrically heating this element, a uni-form heat flux may be established at the inner surface.(a) For 4-mm-thick window glass, determine the electricalpower required per unit window area to maintain aninner surface temperature of 15C when the interior airtemperature and convection coefficient are T?,i?25Cand hi?10 W/m2?K, while the exterior (ambient) airtemperature and convection coefficient are T?,o??10C and ho?65 W/m2?K.(b) In practice T?,oand hovary according to weatherconditionsand car speed. For values of ho?2, 20,65, and 100 W/m2?K, determine and plot theelectrical power requirement as a function of T?,ofor?30T?,o0C. From your results, what can youconclude about the need for heater operation at lowvalues of ho? How is this conclusion affected by thevalue of T?,o? If h?Vn, where Vis the vehicle speedand nis a positive exponent, how does the vehiclespeed affect the need for heater operation

A dormitory at a large university, built 50 years ago, hasexterior walls constructed of Ls?25-mm-thick sheath-ing with a thermal conductivity of ks?0.1 W/m?K. Toreduce heat losses in the winter, the university decidesto encapsulate the entire dormitory by applying anLi?25-mm-thick layer of extruded insulation charac-terized by ki?0.029 W/m?K to the exterior of theoriginal sheathing. The extruded insulation is, in turn,covered with an Lg?5-mm-thick architectural glasswith kg?1.4 W/m?K. Determine the heat flux throughthe original and retrofitted walls when the interior andexterior air temperatures are T?,i?22C and T?,o??20C, respectively. The inner and outer convection heattransfer coefficients are hi?5 W/m2K ? and ho?25 W/m2?K, respectively.

In a manufacturing process, a transparent film is beingbonded to a substrate as shown in the sketch. To cure thebond at a temperature T0, a radiant source is used to pro-vide a heat flux q?0(W/m2), all of which is absorbed at the bonded surface. The back of the substrate is main-tained at T1while the free surface of the film is exposed toair at T?and a convection heat transfer coefficient h. (a) Show the thermal circuit representing the steady-stateheat transfer situation. Be sure to label allelements,nodes, and heat rates. Leave in symbolic form.(b) Assume the following conditions: T??20C, h?50 W/m2?K, and T1?30C. Calculate the heat fluxq?0that is required to maintain the bonded surface atT0?60C.(c) Compute and plot the required heat flux as a functionof the film thickness for 0 L1 mm.(d) If the film is not transparent and all of the radiantheat flux is absorbed at its upper surface, determinethe heat flux required to achieve bonding. Plot yourresults as a function of Lfor 0L1 mm.

The walls of a refrigerator are typically constructed bysandwiching a layer of insulation between sheet metalpanels. Consider a wall made from fiberglass insulationof thermal conductivity ki?0.046 W/m?K and thick-ness Li?50 mm and steel panels, each of thermal con-ductivity kp?60 W/m?K and thickness Lp?3 mm. Ifthe wall separates refrigerated air at T?,i?4C fromambient air at T?,o?25C, what is the heat gain perunit surface area? Coefficients associated with naturalconvection at the inner and outer surfaces may beapproximated as hih ? o?5 W/m2?K.

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

A technique for measuring convection heat transfercoefficients involves bonding one surface of a thinmetallic foil to an insulating material and exposing theother surface to the fluid flow conditions of interest. By passing an electric current through the foil, heat isdissipated uniformly within the foil and the correspond-ing flux, P?elec, may be inferred from related voltage andcurrent measurements. If the insulation thickness Landthermal conductivity kare known and the fluid, foil,and insulation temperatures (T?, Ts, Tb) are measured,the convection coefficient may be determined. Considerconditions for which T??Tb?25C, P?elec?2000W/m2, L?10 mm, and k?0.040 W/m?K. (a) With water flow over the surface, the foil tempera-ture measurement yields Ts?27C. Determine theconvection coefficient. What error would beincurred by assuming all of the dissipated power tobe transferred to the water by convection?(b) If, instead, air flows over the surface and the tempera-ture measurement yields Ts?125C, what is the con-vection coefficient? The foil has an emissivity of 0.15and is exposed to large surroundings at 25C. Whaterror would be incurred by assuming all of the dissi-pated power to be transferred to the air by convection?(c) Typically, heat flux gages are operated at a fixedtemperature (Ts), in which case the power dissipa-tion provides a direct measure of the convectioncoefficient. For Ts?27C, plot P?elecas a functionof hofor 10ho1000 W/m2?K. What effectdoes hohave on the error associated with neglectingconduction through the insulation?

The wind chill, which is experienced on a cold, windyday, is related to increased heat transfer from exposedhuman skin to the surrounding atmosphere. Consider alayer of fatty tissue that is 3 mm thick and whose inte-rior surface is maintained at a temperature of 36C. Ona calm day the convection heat transfer coefficient atthe outer surface is 25 W/m2?K, but with 30 km/hwinds it reaches 65 W/m2?K. In both cases the ambientair temperature is ?15C.(a) What is the ratio of the heat loss per unit area fromthe skin for the calm day to that for the windy day?(b) What will be the skin outer surface temperature forthe calm day? For the windy day?(c) What temperature would the air have to assume onthe calm day to produce the same heat loss occurringwith the air temperature at ?15C on the windy day?

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

A thermopane window consists of two pieces of glass7 mm thick that enclose an air space 7 mm thick. Thewindow separates room air at 20C from outside ambi-ent air at ?10C. The convection coefficient associatedwith the inner (room-side) surface is 10 W/m2?K.(a) If the convection coefficient associated with theouter (ambient) air is ho?80 W/m2?K, what is the heat loss through a window that is 0.8 m longby 0.5 m wide? Neglect radiation, and assume theair enclosed between the panes to be stagnant.(b) Compute and plot the effect of hoon the heat loss for10ho100 W/m2?K. Repeat this calculation for a triple-pane construction in which a third pane and asecond air space of equivalent thickness are added.

A house has a composite wall of wood, fiberglass insula-tion, and plaster board, as indicated in the sketch. On acold winter day, the convection heat transfer coefficientsare ho?60 W/m2?K and hi?30 W/m2?K. The totalwall surface area is 350 m2.(a) Determine a symbolic expression for the total thermalresistance of the wall, including inside and outsideconvection effects for the prescribed conditions.(b) Determine the total heat loss through the wall.(c) If the wind were blowing violently, raising hoto300 W/m2?K, determine the percentage increase inthe heat loss.(d) What is the controlling resistance that determines theamount of heat flow through the wall?

Consider the composite wall of Problem 3.13 under con-ditions for which the inside air is still characterized byT?,i?20C and hi?30 W/m2?K. However, use themore realistic conditions for which the outside air ischaracterized by a diurnal (time) varying temperature ofthe formwith ho?60 W/m2?K. Assuming quasi-steady condi-tions for which changes in energy storage within the wallmay be neglected, estimate the daily heat loss throughthe wall if its total surface area is 200 m2.

Consider a composite wall that includes an 8-mm-thickhardwood siding, 40-mm by 130-mm hardwood studson 0.65-m centers with glass fiber insulation (paper faced, 28 kg/m3), and a 12-mm layer of gypsum (vermi-culite) wall board.What is the thermal resistance associated with a wallthat is 2.5 m high by 6.5 m wide (having 10 studs, each2.5 m high)? Assume surfaces normal to the x-directionare isothermal.

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

Consider the oven of Problem 1.54. The walls of theoven consist of L?30-mm-thick layers of insulationcharacterized by kins?0.03 W/m?K that are sand-wiched between two thinlayers of sheet metal. Theexterior surface of the oven is exposed to air at 23Cwith hext?2 W/m2?K. The interior oven air tempera-ture is 180C. Neglecting radiation heat transfer, deter-mine the steady-state heat flux through the oven wallswhen the convection mode is disabled and the free con-vection coefficient at the inner oven surface ishfr?3 W/m2?K. Determine the heat flux through theoven walls when the convection mode is activated, inwhich case the forced convection coefficient at theinner oven surface is hfo?27 W/m2?K. Does operationof the oven in its convection mode result in signifi-cantly increased heat losses from the oven to thekitchen? Would your conclusion change if radiationwere included in your analysis?

The composite wall of an oven consists of three materi-als, two of which are of known thermal conductivity,kA?20 W/m?K and kC?50 W/m?K, and knownthickness, LA?0.30 m and LC?0.15 m. The thirdmaterial, B, which is sandwiched between materials A and C, is of known thickness, LB?0.15 m, butunknown thermal conductivity kB.Under steady-state operating conditions, measurementsreveal an outer surface temperature of Ts,o?20C, aninner surface temperature of Ts,i?600C, and an ovenair temperature of T??800C. The inside convectioncoefficient his known to be 25 W/m2?K. What is thevalue of kB?

The wall of a drying oven is constructed by sandwich-ing an insulation material of thermal conductivity k?0.05 W/m?K between thin metal sheets. The oven air is atT?,i?300C, and the corresponding convection coeffi-cient is hi?30 W/m2?K. The inner wall surface absorbsa radiant flux of q?rad?100 W/m2from hotter objectswithin the oven. The room air is at T?,o?25C, and theoverall coefficient for convection and radiation fromthe outer surface is ho?10 W/m2?K.(a) Draw the thermal circuit for the wall and label alltemperatures, heat rates, and thermal resistances.(b) What insulation thickness Lis required to maintainthe outer wall surface at a safe-to-touchtempera-ture of To?40C?

The t?4-mm-thick glass windows of an automobilehave a surface area of A?2.6 m2. The outside temper- ature is T?,o?32C while the passenger compartmentis maintained at T?,i?22C. The convection heattransfer coefficient on the exterior window surface isho?90 W/m2?K. Determine the heat gain through thewindows when the interior convection heat transfercoefficient is hi?15 W/m2?K. By controlling the air-flow in the passenger compartment the interior heattransfer coefficient can be reduced to hi?5 W/m2?Kwithout sacrificing passenger comfort. Determine the heat gain through the window for the reduced inside heattransfer coefficient.

The thermal characteristics of a small, dormitory refrig-erator are determined by performing two separateexperiments, each with the door closed and the refriger-ator placed in ambient air at T??25C. In one case, anelectric heater is suspended in the refrigerator cavity,while the refrigerator is unplugged. With the heater dis-sipating 20 W, a steady-state temperature of 90C isrecorded within the cavity. With the heater removedand the refrigerator now in operation, the second exper-iment involves maintaining a steady-state cavity tem-perature of 5C for a fixed time interval and recordingthe electrical energy required to operate the refrigera-tor. In such an experiment for which steady operation ismaintained over a 12-hour period, the input electricalenergy is 125,000 J. Determine the refrigerators coeffi-cient of performance (COP).

In the design of buildings, energy conservation require-ments dictate that the exterior surface area, As, be mini-mized. This requirement implies that, for a desired floorOven airT,i,hiRoom airT,o,hoAbsorbed radiation, q"radLToInsulation, k196Chapter 3?One-Dimensional, Steady- State ConductionCH003.qxd 2/24/11 12:26 PM Page 196 space, there may be optimum values associated with the number of floors and horizontal dimensions of thebuilding. Consider a design for which the total floorspace, Af, and the vertical distance between floors, Hf,are prescribed.(a) If the building has a square cross section of width Won a side, obtain an expression for the value of Wthat would minimize heat loss to the surroundings.Heat loss may be assumed to occur from the fourvertical side walls and from a flat roof. Express yourresult in terms of Afand Hf.(b) If Af?32,768 m2and Hf?4 m, for what values ofWand Nf(the number of floors) is the heat lossminimized? If the average overall heat transfercoefficient is U?1 W/m2?K and the differencebetween the inside and ambient air temperatures is25C, what is the corresponding heat loss? What is the percentage reduction in heat loss comparedwith a building for Nf2 ? ?

When raised to very high temperatures, many conven-tional liquid fuels dissociate into hydrogen and othercomponents. Thus the advantage of a solid oxide fuelcellis that such a device can internally reformreadilyavailable liquid fuels into hydrogen that can then beused to produce electrical power in a manner similar toExample 1.5. Consider a portable solid oxide fuel cell,operating at a temperature of Tfc?800C. The fuel cellis housed within a cylindrical canister of diameter D?75 mm and length L?120 mm. The outer surface ofthe canister is insulated with a low-thermal-conductivitymaterial. For a particular application, it is desired thatthe thermal signatureof the canister be small, to avoidits detection by infrared sensors. The degree to whichthe canister can be detected with an infrared sensor maybe estimated by equating the radiation heat flux emittedfrom the exterior surface of the canister (Equation 1.5;Es??s?Ts4) to the heat flux emitted from an equivalentblack surface, (Eb??Tb4). If the equivalent black sur-face temperature Tbis near the surroundings tempera-ture, the thermal signature of the canister is too small tobe detectedthe canister is indistinguishable from thesurroundings.(a) Determine the required thickness of insulation to beapplied to the cylindrical wall of the canister toensure that the canister does not become highly visi-ble to an infrared sensor (i.e., Tb?Tsur?5K).Consider cases where (i) the outer surface is cov-ered with a very thin layer of dirt (?s?0.90) and(ii) the outer surface is comprised of a very thinpolished aluminum sheet (?s?0.08). Calculate therequired thicknesses for two types of insulatingmaterial, calcium silicate (k?0.09 W/m?K) and aerogel (k?0.006 W/m?K). The temperatures ofthe surroundings and the ambient are Tsur?300 Kand T??298 K, respectively. The outer surface ischaracterized by a convective heat transfer coeffi-cient of h?12 W/m2?K.(b) Calculate the outer surface temperature of the can-ister for the four cases (high and low thermal con-ductivity; high and low surface emissivity).(c) Calculate the heat loss from the cylindrical walls ofthe canister for the four cases.

A firefighters protective clothing, referred to as a turnoutcoat, is typically constructed as an ensemble of three lay-ers separated by air gaps, as shown schematically.Representative dimensions and thermal conductivitiesfor the layers are as follows.LayerThickness (mm)k (W/m?K)Shell (s)0.80.047Moisture barrier (mb) 0.55 0.012Thermal liner (tl)3.50.038The air gaps between the layers are 1 mm thick, and heat is transferred by conduction and radiation exchange through the stagnant air. The linearizedradiation coefficient for a gap may be approximated as, , where Tavgrepresents the average temperature of the surfacescomprising the gap, and the radiation flux across thegap may be expressed as .(a) Represent the turnout coat by a thermal circuit,labeling all the thermal resistances. Calculate andtabulate the thermal resistances per unit area (m2?K/W) for each of the layers, as well as for the con-duction and radiation processes in the gaps. Assumethat a value of Tavg?470 K may be used to approx-imate the radiation resistance of both gaps. Com-ment on the relative magnitudes of the resistances.(b) For a pre-flash-ovefire environment in which fire- fighters often work, the typical radiant heat fluxon the fire-side of the turnout coat is 0.25 W/cm2. What is the outer surface temperature of the turnoutcoat if the inner surface temperature is 66C, a con-dition that would result in burn injury?

A particular thermal system involves three objects offixed shape with conduction resistances of R1?1 K/W,R2?2 K/W and R3?4 K/W, respectively. An objec-tive is to minimize the total thermal resistance Rtotasso-ciated with a combination of R1, R2, and R3. The chiefengineer is willing to invest limited funds to specify analternative material for just one of the three objects;the alternative material will have a thermal conductiv-ity that is twice its nominal value. Which object (1, 2,or 3) should be fabricated of the higher thermal con-ductivity material to most significantly decrease Rtot?Hint: Consider two cases, one for which the three ther-mal resistances are arranged in series, and the secondfor which the three resistances are arranged in parallel

A composite wall separates combustion gases at 2600C from a liquid coolant at 100C, with gas- and liquid-side convection coefficients of 50 and 1000W/mK. The wall is composed of a 10-mm-thick layer of beryllium oxide on the gas side and a 20-mm-thickslab of stainless steel (AISI 304) on the liquid side. Thecontact resistance between the oxide and the steel is 0.05 m2K/W. What is the heat loss per unit surface area of the composite? Sketch the temperature distribution from the gas to the liquid.

Approximately 106discrete electrical components canbe placed on a single integrated circuit (chip), withelectrical heat dissipation as high as 30,000 W/m2. Thechip, which is very thin, is exposed to a dielectric liq-uid at its outer surface, with ho?1000 W/m2?K andT?,o?20C, and is joined to a circuit board at its innersurface. The thermal contact resistance between thechip and the board is 10?4m2?K/W, and the boardthickness and thermal conductivity are Lb?5 mm andkb?1 W/m?K, respectively. The other surface of theboard is exposed to ambient air for which hi?40W/m2?K and T?,i?20C. (a) Sketch the equivalent thermal circuit correspondingto steady-state conditions. In variable form, labelappropriate resistances, temperatures, and heat fluxes.(b) Under steady-state conditions for which the chipheat dissipation is q?c?30,000 W/m2, what is thechip temperature?(c) The maximum allowable heat flux, q?c,m, is deter-mined by the constraint that the chip temperaturemust not exceed 85C. Determine q?c,mfor the fore-going conditions. If air is used in lieu of the dielec-tric liquid, the convection coefficient is reduced byapproximately an order of magnitude. What is thevalue of q?c,mfor ho?100 W/m2?K? With air cool-ing, can significant improvements be realized byusing an aluminum oxide circuit board and/or by using a conductive paste at the chip/board inter-face for which R?t,c?1? m2?K/W?

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

Consider a plane composite wall that is composed of two materials of thermal conductivities kA? 0.1 W/m?Kand kB?0.04 W/m?K and thicknesses LA?10 mm andLB?20 mm. The contact resistance at the interfacebetween the two materials is known to be 0.30 m2?K/W.Material A adjoins a fluid at 200C for which h?10W/m2?K, and material B adjoins a fluid at 40C forwhich h?20 W/m2?K.(a) What is the rate of heat transfer through a wall thatis 2 m high by 2.5 m wide?(b) Sketch the temperature distribution.

The performance of gas turbine engines may beimproved by increasing the tolerance of the turbineblades to hot gases emerging from the combustor. Oneapproach to achieving high operating temperaturesinvolves application of a thermal barrier coating(TBC) to the exterior surface of a blade, while passingcooling air through the blade. Typically, the blade ismade from a high-temperature superalloy, such asInconel (k25 W/m?K), while a ceramic, such as zirconia (k1.3 W/m?K), is used as a TBC. Consider conditions for which hot gases at T?,o?1700 K and cooling air at T?,i? 400 K provide outerand inner surface convection coefficients of ho?1000 W/m2?K and hi?500 W/m2?K, respectively. If a0.5-mm-thick zirconia TBC is attached to a 5-mm-thick Inconel blade wall by means of a metallic bond-ing agent, which provides an interfacial thermal resis-tance of R?t,c?10?4m2?K/W, can the Inconel be main-tained at a temperature that is below its maximumallowable value of 1250 K? Radiation effects may beneglected, and the turbine blade may be approximatedas a plane wall. Plot the temperature distribution withand without the TBC. Are there any limits to the thick-ness of the TBC?

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

Physicists have determined the theoretical value of thethermal conductivity of a carbon nanotube to be kcn,T?5000 W/m?K.(a) Assuming the actual thermal conductivity of thecarbon nanotube is the same as its theoretical value,find the thermal contact resistance, Rt,c, that existsbetween the carbon nanotube and the top surfacesof the heated and sensing islands in Example 3.4 .(b) Using the value of the thermal contact resistancecalculated in part (a), plot the fraction of the totalresistance between the heated and sensing islandsthat is due to the thermal contact resistances forisland separation distances of 5 ?ms20 ?m.

Consider a power transistor encapsulated in an alu-minum case that is attached at its base to a squarealuminum plate of thermal conductivity k?240 W/m?K,thickness L?6 mm, and width W?20 mm. The caseis joined to the plate by screws that maintain a contactpressure of 1 bar, and the back surface of the platetransfers heat by natural convection and radiation toambient air and large surroundings at T??Tsur?25C. The surface has an emissivity of ??0.9, and theconvection coefficient is h?4 W/m2?K. The case iscompletely enclosed such that heat transfer may beassumed to occur exclusively through the base plate. (a) If the air-filled aluminum-to-aluminum interfaceis characterized by an area of Ac?210?4m2anda roughness of 10?m, what is the maximum allow-able power dissipation if the surface temperature ofthe case, Ts,c, is not to exceed 85C?(b) The convection coefficient may be increased bysubjecting the plate surface to a forced flow of air.Explore the effect of increasing the coefficient overthe range 4h200 W/m2K ? .

Ring-porouswoods, such as oak, are characterized bygrains. The dark grains consist of very low- densitymaterial that forms early in the springtime. The sur-rounding lighter-colored wood is composed of high-density material that forms slowly throughout most ofthe growing season. Assuming the low- density material is highly porous andthe oak is dry, determine the fraction of the oak cross-section that appears as being grained. Hint: Assume thethermal conductivity parallel to the grains is the sameas the radial conductivity of Table A.3.

A batt of glass fiber insulation is of density ?28 kg/m3. Determine the maximum and minimum pos- sible values of the effective thermal conductivity of theinsulation at T?300 K, and compare with the valuereported in Table A.3.

Air usually constitutes up to half of the volume of commercial ice creams and takes the form of smallspherical bubbles interspersed within a matrix of frozenmatter. The thermal conductivity of ice cream that con-tains no air is kna?1.1 W/m?K at T?20C. Deter-mine the thermal conductivity of commercial ice creamcharacterized by ??0.20, also at T?20C

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

A one-dimensional plane wall of thickness Lis con-structed of a solid material with a linear, nonuniformporosity distribution described by ?(x)??max(x/L). Plot the steady-state temperature distribution, T(x), forks?10 W/m?K, kf?0.1 W/m?K, L?1 m, ?max?0.25, T(x?0)?30C and q?x?100 W/m2 using theexpression for the minimum effective thermal conduc-tivity of a porous medium, the expression for the maxi-mum effective thermal conductivity of a porousmedium, Maxwells expression, and for the case where keff(x)?ks

The diagram shows a conical section fabricated from pure aluminum. It is of circular cross sectionhaving diameter D?ax1/2, where a?0.5 m1/2. The small end is located at x1?25 mm and the large endat x2?125 mm. The end temperatures are T1?600 K and T2?400 K, while the lateral surface iswell insulated.(a) Derive an expression for the temperature distributionT(x) in symbolic form, assuming one-dimensionalconditions. Sketch the temperature distribution.(b) Calculate the heat rate qx.

A truncated solid cone is of circular cross section, andits diameter is related to the axial coordinate by anexpression of the form D?ax3/2, where a?1.0 m?1/2. The sides are well insulated, while the top surface of the cone at x1is maintained at T1and the bottom sur-face at x2is maintained at T2.(a) Obtain an expression for the temperature distributionT(x).(b) What is the rate of heat transfer across the cone if itis constructed of pure aluminum with x1?0.075 m,T1?100C, x2?0.225 m, and T2?20C

From Figure 2.5 it is evident that, over a wide tempera-turerange, the temperature dependence of the thermalconductivity of many solids may be approximated by alinear expression of the form k?ko?aT, where koisa positive constant and ais a coefficient that may bepositive or negative. Obtain an expression for the heatflux across a plane wall whose inner and outer surfacesare maintained at T0and T1, respectively. Sketch theforms of the temperature distribution corresponding toa?0, a?0, and a?0

Consider a tube wall of inner and outer radii riand ro,whose temperatures are maintained at Tiand To, respec-tively. The thermal conductivity of the cylinder is tem-perature dependent and may be represented by anexpression of the form k?ko(1?aT), where koand aare constants. Obtain an expression for the heat transferper unit length of the tube. What is the thermal resis-tance of the tube wall?

Measurements show that steady-state conductionthrough a plane wall without heat generation produceda convex temperature distribution such that the mid-point temperature was ?Tohigher than expected for alinear temperature distribution. Assuming that the thermal conductivity has a lineardependence on temperature, k?ko(1?T), where isa constant, develop a relationship to evaluate in termsof ?To, T1, and T2.

A device used to measure the surface temperature of anobject to within a spatial resolution of approximately50 nm is shown in the schematic. It consists of anextremely sharp-tipped stylus and an extremely smallcantilever that is scanned across the surface. The probetip is of circular cross section and is fabricated of poly-crystalline silicon dioxide. The ambient temperature ismeasured at the pivoted end of the cantilever as T?25C, and the device is equipped with a sensor to mea-sure the temperature at the upper end of the sharp tip,Tsen. The thermal resistance between the sensing probeand the pivoted end is Rt?5 106K/W.(a) Determine the thermal resistance between the sur-face temperature and the sensing temperature.(b) If the sensing temperature is Tsen?28.5C, deter-mine the surface temperature.Hint:Although nanoscale heat transfer effects may beimportant, assume that the conduction occurring in the airadjacent to the probe tip can be described by Fourierslaw and the thermal conductivity found in Table A.4.

A steam pipe of 0.12-m outside diameter is insulatedwith a layer of calcium silicate.(a) If the insulation is 20 mm thick and its inner andouter surfaces are maintained at Ts,1?800 K and Ts,2?490 K, respectively, what is the heat lossper unit length (q?) of the pipe?(b) We wish to explore the effect of insulation thicknesson the heat loss q?and outer surface temperature Ts,2,with the inner surface temperature fixed at Ts,1?800 K. The outer surface is exposed to an airflow (T??25C) that maintains a convection coefficientof h?25 W/m2?K and to large surroundings forwhich Tsur?T??25C. The surface emissivity ofcalcium silicate is approximately 0.8. Compute andplot the temperature distribution in the insulation asa function of the dimensionless radial coordinate,(r?r1)/(r2?r1), where r1?0.06 m and r2is avariable (0.06 ?r20.20 m). Compute and plotthe heat loss as a function of the insulation thicknessfor 0(r2?r1)0.14 m.

Consider the water heater described in Problem 1.48.We now wish to determine the energy needed to com-pensate for heat losses incurred while the water isstored at the prescribed temperature of 55C. Thecylindrical storage tank (with flat ends) has a capacityof 100 gal, and foamed urethane is used to insulatethe side and end walls from ambient air at an annualaverage temperature of 20C. The resistance to heat transfer is dominated by conduction in the insulationand by free convection in the air, for which h = 2W/mK. If electric resistance heating is used to com-pensate for the losses and the cost of electric power is $0.18/kWh, specify tank and insulation dimensions forwhich the annual cost associated with the heat losses isless than $50.

To maximize production and minimize pumping costs,crude oil is heated to reduce its viscosity during trans-portation from a production field.(a) Consider a pipe-in-pipeconfiguration consisting ofconcentric steel tubes with an intervening insulatingmaterial. The inner tube is used to transport warmcrude oil through cold ocean water. The inner steelpipe (ks?35 W/m?K) has an inside diameter ofDi,1?150 mm and wall thickness ti?10 mm whilethe outer steel pipe has an inside diameter ofDi,2?250 mm and wall thickness to?ti. Determinethe maximum allowable crude oil temperature toensure the polyurethane foam insulation (kp?0.075 W/m?K) between the two pipes does notexceed its maximum service temperature of Tp,max ?70C. The ocean water is at T?,o?5C and providesan external convection heat transfer coefficient ofho?500 W/m2?K. The convection coefficient asso-ciated with the flowing crude oil is hi?450 W/m2?K. (b) It is proposed to enhance the performance of thepipe-in-pipe device by replacing a thin (ta?5 mm)section of polyurethane located at the outside of theinner pipe with an aerogel insulation material(ka?0.012 W/m?K). Determine the maximumallowable crude oil temperature to ensure maximumpolyurethane temperatures are below Tp,max ?70C.

A thin electrical heater is wrapped around the outer sur-face of a long cylindrical tube whose inner surface ismaintained at a temperature of 5C. The tube wall hasinner and outer radii of 25 and 75 mm, respectively, anda thermal conductivity of 10 W/m?K. The thermal con-tact resistance between the heater and the outer surfaceof the tube (per unit length of the tube) is R?t,c?0.01 m?K/W. The outer surface of the heater is exposedto a fluid with T???10C and a convection coefficientof h?100 W/m2?K. Determine the heater power per unit length of tube required to maintain the heater atTo?25C.

In Problem 3.48, the electrical power required to main-tain the heater at To?25C depends on the thermalconductivity of the wall material k, the thermal contactresistance R?t,cand the convection coefficient h. Com-pute and plot the separate effect of changes in k(1k200 W/m?K), R?t,c(0R?t,c0.1 m?K/W),and h(10h1000 W/m2?K) on the total heaterpower requirement, as well as the rate of heat transferto the inner surface of the tube and to the fluid

A stainless steel (AISI 304) tube used to transport a chilled pharmaceutical has an inner diameter of 36 mm and a wall thickness of 2 mm. The pharma-ceutical and ambient air are at temperatures of 6C and23C, respectively, while the corresponding inner and outer convection coefficients are 400 W/m2?K and6 W/m2?K, respectively.(a) What is the heat gain per unit tube length?(b) What is the heat gain per unit length if a 10-mm-thick layer of calcium silicate insulation (kins?0.050 W/m?K) is applied to the tube?

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

A thin electrical heater is inserted between a long circu-lar rod and a concentric tube with inner and outer radii of20 and 40 mm. The rod (A) has a thermal conductivity ofkA?0.15 W/m?K, while the tube (B) has a thermalconductivity of kB?1.5 W/m?K and its outer surfaceis subjected to convection with a fluid of temperatureT???15C and heat transfer coefficient 50 W/m2?K.The thermal contact resistance between the cylindersurfaces and the heater is negligible.(a) Determine the electrical power per unit length ofthe cylinders (W/m) that is required to maintain theouter surface of cylinder B at 5C.(b) What is the temperature at the center of cylinder A?

A wire of diameter D?2 mm and uniform temperatureThas an electrical resistance of 0.01?/m and a currentflow of 20 A.(a) What is the rate at which heat is dissipated per unitlength of wire? What is the heat dissipation per unit volume within the wire?(b) If the wire is not insulated and is in ambient air and large surroundings for which T??Tsur?20C,what is the temperature Tof the wire? The wire has an emissivity of 0.3, and the coefficientassociated with heat transfer by natural convection may be approximated by an expression of theform, h?C[(T?T?)/D]1/4, where C?1.25W/m7/4?K5/4.(c)If the wire is coated with plastic insulation of 2-mmthickness and a thermal conductivity of 0.25 W/m?K,what are the inner and outer surface temperatures ofthe insulation? The insulation has an emissivity of0.9, and the convection coefficient is given by theexpression of part (b). Explore the effect of the insu-lation thickness on the surface temperatures

A 2-mm-diameter electrical wire is insulated by a 2-mm-thick rubberized sheath (k?0.13 W/m?K), andthe wire/sheath interface is characterized by a thermalcontact resistance of . The con-vection heat transfer coefficient at the outer surface ofthe sheath is 10 W/m2?K, and the temperature of the ambient air is 20C. If the temperature of the insula-tion may not exceed 50C, what is the maximum allow-able electrical power that may be dissipated per unitlength of the conductor? What is the critical radius ofthe insulation?

Electric current flows through a long rod generatingthermal energy at a uniform volumetric rate of . The rod is concentric with a hollowceramic cylinder, creating an enclosure that is filledwith air.The thermal resistance per unit length due to radiationbetween the enclosure surfaces is ,and the coefficient associated with free convection inthe enclosure is h?20 W/m2?K.(a) Construct a thermal circuit that can be used to cal-culate the surface temperature of the rod, Tr. Labelall temperatures, heat rates, and thermal resis-tances, and evaluate each thermal resistance.(b) Calculate the surface temperature of the rod for theprescribed conditions.

The evaporator section of a refrigeration unit consistsof thin-walled, 10-mm-diameter tubes through whichrefrigerant passes at a temperature of ?18C. Air iscooled as it flows over the tubes, maintaining a surfaceconvection coefficient of 100 W/m2?K, and is subse-quently routed to the refrigerator compartment.(a) For the foregoing conditions and an air temperatureof ?3C, what is the rate at which heat is extractedfrom the air per unit tube length?(b) If the refrigerators defrost unit malfunctions, frostwill slowly accumulate on the outer tube surface.Assess the effect of frost formation on the coolingcapacity of a tube for frost layer thicknesses in therange 0?4 mm. Frost may be assumed tohave a thermal conductivity of 0.4 W/m?K.(c) The refrigerator is disconnected after the defrostunit malfunctions and a 2-mm-thick layer of frosthas formed. If the tubes are in ambient air forwhich T??20C and natural convection maintainsa convection coefficient of 2 W/m2?K, how longwill it take for the frost to melt? The frost may beassumed to have a mass density of 700 kg/m3and alatent heat of fusion of 334 kJ/kg.

A composite cylindrical wall is composed of twomaterials of thermal conductivity kAand kB, which are separated by a very thin, electric resistance heater for which interfacial contact resistances are negligible.Liquid pumped through the tube is at a temperature T?,iand provides a convection coefficient hiat the inner sur-face of the composite. The outer surface is exposed toambient air, which is at T?,oand provides a convectioncoefficient of ho. Under steady-state conditions, a uni-form heat flux of q?his dissipated by the heater.(a) Sketch the equivalent thermal circuit of the systemand express all resistances in terms of relevantvariables.(b) Obtain an expression that may be used to determinethe heater temperature, Th.(c) Obtain an expression for the ratio of heat flows tothe outer and inner fluids, q?o/q?i. How might thevariables of the problem be adjusted to minimizethis ratio?

An electrical current of 700 A flows through a stainlesssteel cable having a diameter of 5 mm and an electricalresistance of 610?4?/m (i.e., per meter of cablelength). The cable is in an environment having a tem-perature of 30C, and the total coefficient associatedwith convection and radiation between the cable andthe environment is approximately 25 W/m2?K.(a) If the cable is bare, what is its surface temperature?(b) If a very thin coating of electrical insulation isapplied to the cable, with a contact resistance of0.02 m2?K /W, what are the insulation and cablesurface temperatures?(c) There is some concern about the ability of the insula-tion to withstand elevated temperatures. What thick-ness of this insulation (k?0.5 W/m?K) will yieldthe lowest value of the maximum insulation temper-ature? What is the value of the maximum tempera-ture when this thickness is used?

A 0.20-m-diameter, thin-walled steel pipe is used totransport saturated steam at a pressure of 20 bars in aroom for which the air temperature is 25C and the con-vection heat transfer coefficient at the outer surface ofthe pipe is 20 W/mK.(a) What is the heat loss per unit length from the barepipe (no insulation)? Estimate the heat loss per unitlength if a 50-mm-thick layer of insulation (magne-sia, 85%) is added. The steel and magnesia may eachbe assumed to have an emissivity of 0.8, and thesteam-side convection resistance may be neglected.(b) The costs associated with generating the steam andinstalling the insulation are known to be $4/109Jand $100/m of pipe length, respectively. If thesteam line is to operate 7500 h/yr, how many yearsare needed to pay back the initial investment ininsulation?

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

Steam flowing through a long, thin-walled pipe main-tains the pipe wall at a uniform temperature of 500 K.The pipe is covered with an insulation blanket comprisedof two different materials, A and B.The interface between the two materials may beassumed to have an infinite contact resistance, and theentire outer surface is exposed to air for which T??300 K and h?25 W/m2?K.(a) Sketch the thermal circuit of the system. Label(using the preceding symbols) all pertinent nodes andresistances.(b) For the prescribed conditions, what is the total heatloss from the pipe? What are the outer surface tem- peratures Ts,2(A)and Ts,2(B)?

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

A storage tank consists of a cylindrical section that hasa length and inner diameter of L=2 m and Di=1m,respectively, and two hemispherical end sections. The tank is constructed from 20-mm-thick glass (Pyrex) andis exposed to ambient air for which the temperature is 300 K and the convection coefficient is 10 W/mK.The tank is used to store heated oil, which maintains theinner surface at a temperature of 400 K. Determine theelectrical power that must be supplied to a heater sub-merged in the oil if the prescribed conditions are to bemaintained. Radiation effects may be neglected, and thePyrex may be assumed to have a thermal conductivityof 1.4 W/mK.

Consider the liquid oxygen storage system and the lab-oratory environmental conditions of Problem 1.49. Toreduce oxygen loss due to vaporization, an insulatinglayer should be applied to the outer surface of the con-tainer. Consider using a laminated aluminum foil/glassmat insulation, for which the thermal conductivity andsurface emissivity are k=0.00016 W/m?K and ??0.20,respectively.(a) If the container is covered with a 10-mm-thicklayer of insulation, what is the percentage reductionin oxygen loss relative to the uncovered container?(b) Compute and plot the oxygen evaporation rate (kg/s) as a function of the insulation thickness tfor0t50 mm.

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

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

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

A spherical tank for storing liquid oxygen on the spaceshuttle is to be made from stainless steel of 0.80-m outerdiameter and 5-mm wall thickness. The boiling point andlatent heat of vaporization of liquid oxygen are 90 K and213 kJ/kg, respectively. The tank is to be installed in alarge compartment whose temperature is to be maintainedat 240 K. Design a thermal insulation system that willmaintain oxygen losses due to boiling below 1 kg/day.

A spherical, cryosurgical probe may be imbedded indiseased tissue for the purpose of freezing, and therebydestroying, the tissue. Consider a probe of 3-mm diam-eter whose surface is maintained at -30C when imbedded in tissue that is at 37C. A spherical layer offrozen tissue forms around the probe, with a tempera-ture of 0C existing at the phase front (interface)between the frozen and normal tissue. If the thermalconductivity of frozen tissue is approximately1.5 W/mK and heat transfer at the phase front may becharacterized by an effective convection coefficient of 50 W/mK, what is the thickness of the layer of frozentissue (assuming negligible perfusion)?

A spherical vessel used as a reactor for producing phar-maceuticals has a 10-mm-thick stainless steel wall(k?17 W/m?K) and an inner diameter of l m. Theexterior surface of the vessel is exposed to ambient air(T??25C) for which a convection coefficient of6 W/m2?K may be assumed.(a) During steady- state operation, an inner surfacetemperature of 50C is maintained by energy gener-ated within the reactor. What is the heat loss fromthe vessel?(b) If a 20-mm-thick layer of fiberglass insulation(k?0.040 W/m?K) is applied to the exterior of thevessel and the rate of thermal energy generation isunchanged, what is the inner surface temperature ofthe vessel?

The wall of a spherical tank of 1-m diameter containsan exothermic chemical reaction and is at 200C whenthe ambient air temperature is 25C. What thickness ofurethane foam is required to reduce the exterior temper-ature to 40C, assuming the convection coefficient is 20 W/mK for both situations? What is the percentagereduction in heat rate achieved by using the insulation?

A composite spherical shell of inner radius r1=0.25 mis constructed from lead of outer radius r2?0.30 m andAISI 302 stainless steel of outer radius r3=0.31 m. Thecavity is filled with radioactive wastes that generate heatat a rate of ?5 105W/m3. It is proposed to submergethe container in oceanic waters that are at a temperature ofT??10C and provide a uniform convection coefficientof h?500 W/m2?K at the outer surface of the container.Are there any problems associated with this proposal?

The energy transferred from the anterior chamber of theeye through the cornea varies considerably dependingon whether a contact lens is worn. Treat the eye as aspherical system and assume the system to be at steadystate. The convection coefficient hois unchanged withand without the contact lens in place. The cornea andthe lens cover one-third of the spherical surface area.Values of the parameters representing this situationare as follows:r1?10.2 mmr2?12.7 mmr3?16.5 mmT?,o?21CT?,i?37Ck2?0.80 W/m?Kk1?0.35 W/m?Kho?6 W/m2?Khi?12 W/m2?K(a) Construct the thermal circuits, labeling all potentialsand flows for the systems excluding the contact lensand including the contact lens. Write resistance ele-ments in terms of appropriate parameters.(b) Determine the heat loss from the anterior chamberwith and without the contact lens in place.(c) Discuss the implication of your results.

The outer surface of a hollow sphere of radius r2is sub-jected to a uniform heat flux q?2. The inner surface at r1is held at a constant temperature Ts,1.(a) Develop an expression for the temperature distribu-tion T(r) in the sphere wall in terms of q?2, Ts,1, r1, r2,and the thermal conductivity of the wall material k.(b) If the inner and outer tube radii are r1?50 mm andr2?100 mm, what heat flux q?2is required to main-tain the outer surface at Ts,2?50C, while the innersurface is at Ts,1?20C? The thermal conductivityof the wall material is k?10 W/m?K.

A spherical shell of inner and outer radii riand ro,respectively, is filled with a heat-generating materialthat provides for a uniform volumetric generation rate(W/m3) of . The outer surface of the shell is exposed toa fluid having a temperature T?and a convection coeffi-cient h. Obtain an expression for the steady-state tem-perature distribution T(r) in the shell, expressing yourresult in terms of ri, ro, , h, T? and the thermal conduc-tivity kof the shell material.

A spherical tank of 3-m diameter contains a liquified-petroleum gas at ?60C. Insulation with a thermal con-ductivity of 0.06 W/m?K and thickness 250 mm isapplied to the tank to reduce the heat gain.(a) Determine the radial position in the insulation layerat which the temperature is 0C when the ambientair temperature is 20C and the convection coeffi-cient on the outer surface is 6 W/m2?K.(b) If the insulation is pervious to moisture from theatmospheric air, what conclusions can you reachabout the formation of ice in the insulation? Whateffect will ice formation have on heat gain to theLP gas? How could this situation be avoided?

A transistor, which may be approximated as a hemi-spherical heat source of radius ro?0.1 mm, is embed-ded in a large silicon substrate (k?125 W/m?K) anddissipates heat at a rate q. All boundaries of the siliconare maintained at an ambient temperature of T??27C,except for the top surface, which is well insulated.Obtain a general expression for the substrate tempera-ture distribution and evaluate the surface temperatureof the heat source for q?4W.

One modality for destroying malignant tissue involvesimbedding a small spherical heat source of radius rowithin the tissue and maintaining local temperaturesabove a critical value Tcfor an extended period. Tissuethat is well removed from the source may be assumedto remain at normal body temperature (Tb?37C).Obtain a general expression for the radial temperaturedistribution in the tissue under steady-state conditionsfor which heat is dissipated at a rate q. If ro?0.5 mm,what heat rate must be supplied to maintain a tissue tem-perature of T?Tc?42C in the domain 0.5r5 mm? The tissue thermal conductivity is approxi-mately 0.5 W/m?K. Assume negligible perfusion.

The air insidea chamber at T?,i?50C is heated con-vectively with hi?20 W/m2?K by a 200-mm-thick wallhaving a thermal conductivity of 4 W/m?K and a uni-form heat generation of 1000 W/m3. To prevent anyheat generated within the wall from being lost to theoutsideof the chamber at T?,o?25C with ho?5W/m2?K, a very thin electrical strip heater is placed onthe outer wall to provide a uniform heat flux, q?o.(a) Sketch the temperature distribution in the wall onT?xcoordinates for the condition where no heatgenerated within the wall is lost to the outsideofthe chamber.(b) What are the temperatures at the wall boundaries,T(0) and T(L), for the conditions of part (a)?(c) Determine the value of q?othat must be supplied by thestrip heater so that all heat generated within the wall istransferred to the insideof the chamber.(d) If the heat generation in the wall were switched offwhile the heat flux to the strip heater remained con-stant, what would be the steady-state temperature,T(0), of the outer wall surface?

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

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

Large, cylindrical bales of hay used to feed livestock inthe winter months are D?2 m in diameter and arestored end-to-end in long rows. Microbial energy gener-ation occurs in the hay and can be excessive if thefarmer bales the hay in a too-wet condition. Assumingthe thermal conductivity of baled hay to bek?0.04 W/m?K, determine the maximum steady-statehay temperature for dry hay (q.?1 W/m3), moist hay (q.?10 W/m3), and wet hay (q.?100 W/m3). Ambientconditions are T??0C and h?25 W/m2?K.

Consider the cylindrical bales of hay in Problem 3.82.It is proposed to utilize the microbial energy generationassociated with wet hay to heat water. Consider a 30-mmdiameter, thin-walled tube inserted lengthwise throughthe middle of a cylindrical bale. The tube carrieswater atT?,i?20C with hi?200 W/m2?K. (a) Determine the steady-state heat transfer to the waterper unit length of tube.(b) Plot the radial temperature distribution in the hay,T(r).(c) Plot the heat transfer to the water per unit length oftube for bale diameters of 0.2 mD2 m

Consider one-dimensional conduction in a plane com-posite wall. The outer surfaces are exposed to a fluid at25C and a convection heat transfer coefficient of1000 W/m2?K. The middle wall B experiences uniformheat generation q.B, while there is no generation in wallsA and C. The temperatures at the interfaces are T1?261C and T2?211C.(a) Assuming negligible contact resistance at the inter-faces, determine the volumetric heat generation q.Band the thermal conductivity kB. (b) Plot the temperature distribution, showing itsimportant features.(c) Consider conditions corresponding to a loss of coolantat the exposed surface of material A (h?0). Determine T1and T2and plot the tempera- ture distribution throughout the system.

Consider a plane composite wall that is composed ofthree materials (materials A, B, and C are arranged leftto right) of thermal conductivities kA?0.24 W/m?K,kB?0.13 W/m?K, and kC?0.50 W/m?K. The thick-nesses of the three sections of the wall are LA?20 mm,LB?13 mm, and LC?20 mm. A contact resistance ofR?t,c?10?2m2?K/W exists at the interface betweenmaterials A and B, as well as at the interface betweenmaterials B and C. The left face of the composite wallis insulated, while the right face is exposed to convec-tive conditions characterized by h?10 W/m2?K, T??20C. For Case 1, thermal energy is generated withinmaterial A at the rate q.A?5000 W/m3. For Case 2,thermal energy is generated within material C at therate q.C?5000 W/m3.(a) Determine the maximum temperature within the com-posite wall under steady-state conditions for Case 1.(b) Sketch the steady-state temperature distribution onT?xcoordinates for Case 1.(c) Sketch the steady-state temperature distribution forCase 2 on the same T?xcoordinates used for Case 1.

An air heater may be fabricated by coiling Nichromewire and passing air in cross flow over the wire.Consider a heater fabricated from wire of diameter D?1 mm, electrical resistivity e?10?6??m, thermalconductivity k?25 W/m?K, and emissivity ??0.20.The heater is designed to deliver air at a temperature ofT??50C under flow conditions that provide a con-vection coefficient of h?250 W/m2?K for the wire.The temperature of the housing that encloses the wireand through which the air flows is Tsur?50C.If the maximum allowable temperature of the wire isTmax?1200C, what is the maximum allowable elec-tric current I? If the maximum available voltage is?E?110 V, what is the corresponding length Lofwire that may be used in the heater and the powerrating of the heater? Hint:In your solution, assume negligible temperature variations within the wire, butafter obtaining the desired results, assess the validity ofthis assumption.

Consider the composite wall of Example 3.7. In theComments section, temperature distributions in the wallwere determined assuming negligible contact resistancebetween materials A and B. Compute and plot the tem-perature distributions if the thermal contact resistance is R"t,c = 10mK/W.

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

A plane wall of thickness 2Land thermal conductivity kexperiences a uniform volumetric generation rate q.. Asshown in the sketch for Case 1, the surface at x??Lisperfectly insulated, while the other surface is main-tained at a uniform, constant temperature To. For Case2, a very thin dielectric strip is inserted at the midpointof the wall (x?0) in order to electrically isolate thetwo sections, A and B. The thermal resistance ofthe strip is R?t?0.0005 m2?K/W. The parametersassociated with the wall are k?50 W/m?K, L?20 mm, and .(a) Sketch the temperature distribution for Case 1 onT?xcoordinates. Describe the key features of thisdistribution. Identify the location of the maximumtemperature in the wall and calculate this temperature.(b) Sketch the temperature distribution for Case 2 onthe same T?xcoordinates. Describe the key fea-tures of this distribution.(c) What is the temperature difference between the twowalls at x?0 for Case 2?(d) What is the location of the maximum temperaturein the composite wall of Case 2? Calculate thistemperature

A nuclear fuel element of thickness 2Lis covered witha steel cladding of thickness b. Heat generated withinthe nuclear fuel at a rate q.is removed by a fluid at T?,which adjoins one surface and is characterized by aconvection coefficient h. The other surface is well insu-lated, and the fuel and steel have thermal conductivitiesof kand ks, respectively.(a) Obtain an equation for the temperature distributionT(x) in the nuclear fuel. Express your results interms of q., k, L, b, ks, h, and T?.(b) Sketch the temperature distribution T(x) for theentire system

Consider the clad fuel element of Problem 3.90.(a) Using appropriate relations from Tables C.1 and C.2,obtain an expression for the temperature distributionT(x) in the fuel element. For kf = 60 W/m?K, L = 15 mm, b = 3 mm, ks = 15 W/m?K, h?10,000W/m2?K, and T??200C, what are the largest andsmallest temperatures in the fuel element if heat isgenerated uniformly at a volumetric rate of?2 107W/m3? What are the corresponding locations?(b) If the insulation is removed and equivalent convec-tion conditions are maintained at each surface,what is the corresponding form of the temperaturedistribution in the fuel element? For the conditionsof part (a), what are the largest and smallest tem-peratures in the fuel? What are the correspondinglocations?(c) For the conditions of parts (a) and (b), plot the tem-perature distributions in the fuel element.

In Problem 3.79 the strip heater acts to guardagainstheat losses from the wall to the outside, and therequired heat flux q?odepends on chamber operatingconditions such as q.and T?,i. As a first step in designing a controller for the guard heater, computeand plot q?oand T(0) as a function of q.for 200q.2000 W/m3and T?,i?30, 50, and 70C

The exposed surface (x?0) of a plane wall of thermalconductivity kis subjected to microwave radiation thatcauses volumetric heating to vary as(x)?o?1?? where q.o(W/m3) is a constant. The boundary at x?Lisperfectly insulated, while the exposed surface is main-tained at a constant temperature To. Determine the tem-perature distribution T(x) in terms of x, L, k, q.o, and To.

A quartz window of thickness Lserves as a viewing portin a furnace used for annealing steel. The inner surface(x?0) of the window is irradiated with a uniform heatflux q?odue to emission from hot gases in the furnace. Afraction, ?, of this radiation may be assumed to beabsorbed at the inner surface, while the remaining radia-tion is partially absorbed as it passes through the quartz.The volumetric heat generation due to this absorptionmay be described by an expression of the formq.(x)?(1??)q?oe?xwhere is the absorption coefficient of the quartz.Convection heat transfer occurs from the outer surface(x?L) of the window to ambient air at T?and is char-acterized by the convection coefficient h. Convectionand radiation emission from the inner surface may beneglected, along with radiation emission from the outersurface. Determine the temperature distribution in thequartz, expressing your result in terms of the foregoingparameters.

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

A cylindrical shell of inner and outer radii, riand ro,respectively, is filled with a heat-generating materialthat provides a uniform volumetric generation rate(W/m3) of q.. The inner surface is insulated, while theouter surface of the shell is exposed to a fluid at T?anda convection coefficient h.(a) Obtain an expression for the steady-state tempera-ture distribution T(r) in the shell, expressing yourresult in terms of ri, ro, q., h, T?, and the thermalconductivity kof the shell material.(b) Determine an expression for the heat rate, q?(ro), atthe outer radius of the shell in terms of q.and shelldimensions

The cross section of a long cylindrical fuel element in anuclear reactor is shown. Energy generation occurs uni-formly in the thorium fuel rod, which is of diameterD?25 mm and is wrapped in a thin aluminum cladding. (a) It is proposed that, under steady-state conditions,the system operates with a generation rate of q.?7108W/m3and cooling system characteristics ofT??95C and h?7000 W/m2?K. Is this pro-posal satisfactory?(b) Explore the effect of variations in q.and hby plot-ting temperature distributions T(r) for a range ofparameter values. Suggest an envelope of accept-able operating conditions.

A nuclear reactor fuel element consists of a solidcylindrical pin of radius r1and thermal conductivity kf.The fuel pin is in good contact with a cladding materialof outer radius r2and thermal conductivity kc. Considersteady-state conditions for which uniform heat genera-tion occurs within the fuel at a volumetric rate q.and theouter surface of the cladding is exposed to a coolantthat is characterized by a temperature T?and a convec-tion coefficient h.(a) Obtain equations for the temperature distributionsTf(r) and Tc(r) in the fuel and cladding, respec-tively. Express your results exclusively in terms ofthe foregoing variables.(b) Consider a uranium oxide fuel pin for which k?2W/m?K and r1?6 mm and cladding for whichkc?25 W/m?K and r2?9 mm. If q.?2 108W/m3, h?2000 W/m2?K, and T??300 K, whatis the maximum temperature in the fuel element?(c) Compute and plot the temperature distribution,T(r), for values of h?2000, 5000, and 10,000W/m2?K. If the operator wishes to maintain thecenterline temperature of the fuel element below1000 K, can she do so by adjusting the coolant flowand hence the value of h?

Consider the configuration of Example 3.8, where uni-form volumetric heating within a stainless steel tube isinduced by an electric current and heat is transferred byconvection to air flowing through the tube. The tubewall has inner and outer radii of r1?25 mm and r2?35 mm, a thermal conductivity of k?15 W/m?K, anelectrical resistivity of e?0.7 10?6??m, and amaximum allowable operating temperature of 1400 K.(a) Assuming the outer tube surface to be perfectlyinsulated and the airflow to be characterized by atemperature and convection coefficient of T?,1?400 K and h1?100 W/m2?K, determine the maxi-mum allowable electric current I.(b) Compute and plot the radial temperature distribu-tion in the tube wall for the electric current of part (a)and three values of h1(100, 500, and 1000 W/m2?K).For each value of h1, determine the rate of heattransfer to the air per unit length of tube. In practice, even the best of insulating materialswould be unable to maintain adiabatic conditions atthe outer tube surface. Consider use of a refractoryinsulating material of thermal conductivity k?1.0W/m?K and neglect radiation exchange at its outersurface. For h1? 100 W/m2?K and the maximumallowable current determined in part (a), computeand plot the temperature distribution in the compos-itewall for two values of the insulation thickness(??25 and 50 mm). The outer surface of the insula-tion is exposed to room air for which T?,2?300 Kand h2?25 W/m2?K. For each insulation thickness,determine the rate of heat transfer per unit tubelength to the inner airflow and the ambient air

A high-temperature, gas-cooled nuclear reactor consistsof a composite cylindrical wall for which a thorium fuelelement (k57 W/m?K) is encased in graphite (k3W/m?K) and gaseous helium flows through an annularcoolant channel. Consider conditions for which the heliumtemperature is T??600 K and the convection coefficientat the outer surface of the graphite is h?2000 W/m2?K.(a) If thermal energy is uniformly generated in the fuelelement at a rate q.?108W/m3, what are the tem-peratures T1and T2at the inner and outer surfaces,respectively, of the fuel element?(b) Compute and plot the temperature distribution inthe composite wall for selected values of q.. Whatis the maximum allowable value of q.?

A long cylindrical rod of diameter 200 mm with ther-mal conductivity of 0.5 W/m?K experiences uniformvolumetric heat generation of 24,000 W/m3. The rod isencapsulated by a circular sleeve having an outerdiameter of 400 mm and a thermal conductivity of4 W/m?K. The outer surface of the sleeve is exposedto cross flow of air at 27C with a convection coeffi-cient of 25 W/m2?K.(a) Find the temperature at the interface between therod and sleeve and on the outer surface.(b) What is the temperature at the center of the rod?

A radioactive material of thermal conductivity kis castas a solid sphere of radius ro and placed in a liquid bath for which the temperature T and convection coeffi-cient hare known. Heat is uniformly generated withinthe solid at a volumetric rate of q.. Obtain the steady-state radial temperature distribution in the solid,expressing your result in terms of r, q., k, h, and T.

Radioactive wastes are packed in a thin-walled sphericalcontainer. The wastes generate thermal energy nonuni-formly according to the relation q.?q.o[1?(r/ro)2] whereq.is the local rate of energy generation per unit volume, q.is a constant, and rois the radius of the container. Steady-state conditions are maintained by submerging the con-tainer in a liquid that is at T?and provides a uniform convection coefficient h.Determine the temperature distribution, T(r), in the con-tainer. Express your result in terms of q.o, ro, T?, h, andthe thermal conductivity kof the radioactive wastes.

Radioactive wastes (krw?20 W/m?K) are stored in aspherical, stainless steel (kss?15 W/m?K) container ofinner and outer radii equal to ri?0.5 m and ro?0.6 m.Heat is generated volumetrically within the wastes at auniform rate of q.?105W/m3, and the outer surface ofthe container is exposed to a water flow for which h?1000 W/m2?K and T??25C.(a) Evaluate the steady-state outer surface tempera-ture, Ts,o.(b) Evaluate the steady-state inner surface tempera-ture, Ts,i.(c) Obtain an expression for the temperature distribu-tion, T(r), in the radioactive wastes. Express yourresult in terms of ri, Ts,i, krw, and q.. Evaluate thetemperature at r?0. (d) A proposed extension of the foregoing designinvolves storing waste materials having the samethermal conductivity but twice the heat generation(q.?2 105W/m3) in a stainless steel containerof equivalent inner radius (ri?0.5 m). Safetyconsiderations dictate that the maximum systemtemperature not exceed 475C and that the con-tainer wall thickness be no less than t?0.04 mand preferably at or close to the original design (t?0.1 m). Assess the effect of varying the outsideconvection coefficient to a maximum achievablevalue of h?5000 W/m2?K (by increasing thewatervelocity) and the container wall thickness. Isthe proposed extension feasible? If so, recommendsuitable operating and design conditions for hand t,respectively.

Unique characteristics of biologically active materi-als such as fruits, vegetables, and other productsrequire special care in handling. Following harvestand separation from producing plants, glucose iscatabolized to produce carbon dioxide, water vapor,and heat, with attendant internal energy generation.Consider a carton of apples, each of 80-mm diameter,which is ventilated with air at 5C and a velocity of 0.5 m/s. The corresponding value of the heat trans-fer coefficient is 7.5 W/m2?K. Within each applethermal energy is uniformly generated at a total rateof 4000 J/kg?day. The density and thermal conduc-tivity of the apple are 840 kg/m3and 0.5 W/m?K,respectively.(a) Determine the apple center and surfacetemperatures.(b) For the stacked arrangement of apples within thecrate, the convection coefficient depends on the velocity as h?C1V0.425, where C1?10.1W/m2?K?(m/s)0.425. Compute and plot the centerand surface temperatures as a function of the airvelocity for 0.1 V1 m/s.

Consider the plane wall, long cylinder, and sphereshown schematically, each with the same characteris-tic length a, thermal conductivity k, and uniform volu-metric energy generation rate q..(a) On the same graph, plot the steady-state dimen-sionless temperature, [T(xor r)?T(a)]/[(q.a2)/2k],versus the dimensionless characteristic length, x/aor r/a, for each shape.(b) Which shape has the smallest temperature differ-ence between the center and the surface? Explainthis behavior by comparing the ratio of the volume-to-surface area.(c) Which shape would be preferred for use as anuclear fuel element? Explain why.

The radiation heat gage shown in the diagram ismade from constantan metal foil, which is coated blackand is in the form of a circular disk of radius Randthickness t. The gage is located in an evacuated enclo-sure. The incident radiation flux absorbed by the foil, q?i,diffuses toward the outer circumference and into thelarger copper ring, which acts as a heat sink at the con-stant temperature T(R). Two copper lead wires areattached to the center of the foil and to the ring to com-plete a thermocouple circuit that allows for measure-ment of the temperature difference between the foilcenter and the foil edge, ?T?T(0) ?T(R). Obtain the differential equation that determines T(r), thetemperature distribution in the foil, under steady-stateconditions. Solve this equation to obtain an expressionrelating ?Tto q?i. You may neglect radiation exchangebetween the foil and its surroundings

Copper tubing is joined to the absorber of a flat-plate solar collector as shown.The aluminum alloy (2024-T6) absorber plate is 6 mm thick and well insulated on its bottom. The top surfaceof the plate is separated from a transparent cover plateby an evacuated space. The tubes are spaced a distanceLof 0.20 m from each other, and water is circulated through the tubes to remove the collected energy. The water may be assumed to be at a uniform temperature of Tw = 60C. Under steady-state operating conditions for which the netradiation heat flux to the surface is q"rad = 800 W/m, what is the maximum temperature on theplate and the heat transfer rate per unit length of tube?Note that q"rad represents the net effect of solar radiationabsorption by the absorber plate and radiation exchangebetween the absorber and cover plates. You may assume the temperature of the absorber plate directlyabove a tube to be equal to that of the water.

One method that is used to grow nanowires (nanotubeswith solid cores) is to initially deposit a small dropletof a liquid catalyst onto a flat surface. The surface andcatalyst are heated and simultaneously exposed to ahigher-temperature, low-pressure gas that contains amixture of chemical species from which the nanowireis to be formed. The catalytic liquid slowlyabsorbs thespecies from the gas through its top surface and con-verts these to a solid material that is deposited onto theunderlying liquid-solid interface, resulting in construc-tion of the nanowire. The liquid catalyst remains sus- pended at the tip of the nanowire.Consider the growth of a 15-nm-diameter siliconcarbide nanowire onto a silicon carbide surface. Thesurface is maintained at a temperature of Ts?2400 K,and the particular liquid catalyst that is used must bemaintained in the range 2400 KTc3000 K to per-form its function. Determine the maximum length of a nanowire that may be grown for conditions character- ized by h?105W/m2?K and T??8000 K. Assumeproperties of the nanowire are the same as for bulk sil- icon carbide.

Consider the manufacture of photovoltaic silicon, asdescribed in Problem 1.42. The thin sheet of siliconis pulled from the pool of molten material veryslowlyand is subjected to an ambient temperature ofT??527C within the growth chamber. A convec-tion coefficient of h?7.5 W/m2?K is associatedwith the exposed surfaces of the silicon sheet when itis inside the growth chamber. Calculate the maxi-mum allowable velocity of the silicon sheet Vsi. Thelatent heat of fusion for silicon is hsf?1.8106J/kg. It can be assumed that the thermal energyreleased due to solidification is removed by conduc-tion along the sheet.

Copper tubing is joined to a solar collector plate ofthickness t, and the working fluid maintains the tem-perature of the plate above the tubes at To. There is auniform net radiation heat flux q?radto the top surfaceof the plate, while the bottom surface is well insulated.The top surface is also exposed to a fluid at T?thatprovides for a uniform convection coefficient h.(a) Derive the differential equation that governs thetemperature distribution T(x) in the plate. (b) Obtain a solution to the differential equation forappropriate boundary conditions.

A thin flat plate of length L, thickness t, and widthW?Lis thermally joined to two large heat sinks thatare maintained at a temperature To. The bottom of theplate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniformvalue of q?o.(a) Derive the differential equation that determinesthe steady-state temperature distribution T(x) inthe plate.(b) Solve the foregoing equation for the temperaturedistribution, and obtain an expression for the rateof heat transfer from the plate to the heat sinks

Consider the flat plate of Problem 3.112, but with theheat sinks at different temperatures, T(0)?ToandT(L)?TL, and with the bottom surface no longer insu-lated. Convection heat transfer is now allowed tooccur between this surface and a fluid at T?, with aconvection coefficient h.(a) Derive the differential equation that determines thesteady-state temperature distribution T(x) in the plate.(b) Solve the foregoing equation for the temperaturedistribution, and obtain an expression for the rateof heat transfer from the plate to the heat sinks.(c) For q?o?20,000 W/m2, To?100C, TL?35C,T??25C, k?25 W/m?K, h?50 W/m2?K, L?100 mm, t?5 mm, and a plate width of W?30 mm, plot the temperature distribution anddetermine the sink heat rates, qx(0) and qx(L). Onthe same graph, plot three additional temperaturedistributions corresponding to changes in the fol-lowing parameters, with the remaining parametersunchanged: (i) q?o?30,000 W/m2, (ii) h?200W/m2?K, and (iii) the value of q?ofor which qx(0)?0 when h?200 W/m2?K.

The temperature of a flowing gas is to be measured witha thermocouple junction and wire stretched betweentwo legs of a sting, a wind tunnel test fixture. The junc-tion is formed by butt-welding two wires of differentmaterial, as shown in the schematic. For wires of diame-ter D?125 ?m and a convection coefficient ofh?700 W/m2?K, determine the minimum separation distance between the two legs of the sting, L?L1?L2,to ensure that the sting temperature does not influencethe junction temperature and, in turn, invalidate the gastemperature measurement. Consider two different typesof thermocouple junctions consisting of (i) copper andconstantan wires and (ii) chromel and alumel wires.Evaluate the thermal conductivity of copper and con-stantan at T?300 K. Use kCh?19 W/m?K andkAl?29 W/m?K for the thermal conductivities of thechromel and alumel wires, respectively.

A bonding operation utilizes a laser to provide a constantheat flux, q?o, across the top surface of a thin adhesive-backed, plastic film to be affixed to a metal strip asshown in the sketch. The metal strip has a thicknessd?1.25 mm, and its width is large relative to that of thefilm. The thermophysical properties of the strip are?7850 kg/m3, cp?435 J/kg?K, and k?60 W/m?K.The thermal resistance of the plastic film of widthw1?40 mm is negligible. The upper and lower surfacesof the strip (including the plastic film) experience con-vection with air at 25C and a convection coefficient of10 W/m2?K. The strip and film are very long in thedirection normal to the page. Assume the edges of themetal strip are at the air temperature (T?).(a) Derive an expression for the temperature distribu-tion in the portion of the steel strip with the plasticfilm (?w1/2x??w1/2).(b) If the heat flux provided by the laser is 10,000W/m2, determine the temperature of the plastic filmat the center (x?0) and its edges (x??w1/2).(c) Plot the temperature distribution for the entire stripand point out its special features.

A thin metallic wire of thermal conductivity k, diame-ter D, and length 2Lis annealed by passing an electri-cal current through the wire to induce a uniform volu-metric heat generation q.. The ambient air around the wire is at a temperature T?, while the ends of thewire at x??Lare also maintained at T?. Heat trans-fer from the wire to the air is characterized by the convection coefficient h. Obtain expressions for thefollowing:(a) The steady-state temperature distribution T(x) alongthe wire,(b) The maximum wire temperature.(c) The average wire temperature.

A motor draws electric power Pelecfrom a supply lineand delivers mechanical power Pmechto a pumpthrough a rotating copper shaft of thermal conductiv-ity ks, length L, and diameter D. The motor ismounted on a square pad of width W, thickness t, andthermal conductivity kp. The surface of the housingexposed to ambient air at T?is of area Ah, and the cor-responding convection coefficient is hh. Opposite ends of the shaft are at temperatures of Thand T?, andheat transfer from the shaft to the ambient air is char-acterized by the convection coefficient hs. The base ofthe pad is at T?.(a) Expressing your result in terms of Pelec, Pmech, ks,L, D, W, t, kp, Ah, hh, and hs, obtain an expressionfor (Th?T?).(b) What is the value of Thif Pelec?25 kW, Pmech?15 kW, ks?400 W/m?K, L?0.5 m, D?0.05 m,W?0.7 m, t?0.05 m, kp?0.5 W/m?K, Ah?2m2, hh?10 W/m2?K, hs?300 W/m2?K, andT??25C?

Consider the fuel cell stack of Problem 1.58. Thet?0.42-mm-thick membranes have a nominal thermalconductivity of k?0.79 W/m?K that can be increasedto keff,x?15.1 W/m?K by loading 10%, by volume, car-bon nanotubes into the catalyst layers. The membrane experiences uniform volumetric energy generation at arate of q.?10 106W/m3. Air at Ta?80C provides a convection coefficient of ha?35 W/m2?K on one sideof the membrane, while hydrogen at Th?80C,hh?235 W/m2?K flows on the opposite side of themembrane. The flow channels are 2L?3 mm wide. Themembrane is clamped between bipolar plates, each ofwhich is at a temperature Tbp?80C.(a) Derive the differential equation that governs thetemperature distribution T(x) in the membrane.(b) Obtain a solution to the differential equation,assuming the membrane is at the bipolar platetemperature at x?0 and x?2L.(c) Plot the temperature distribution T(x) from x?0to x?Lfor carbon nanotube loadings of 0% and10% by volume. Comment on the ability of thecarbon nanotubes to keep the membrane below itssoftening temperature of 85C

Consider a rod of diameter D, thermal conductivity k,and length 2Lthat is perfectly insulated over one por-tion of its length, ?Lx0, and experiences con-vection with a fluid (T?, h) over the other portion,0x??L. One end is maintained at T1, while theother is separated from a heat sink at T3by an interfa-cial thermal contact resistance a) Sketch the temperature distribution on T?xcoor-dinates and identify its key features. Assume thatT1?T3?T?.(b) Derive an expression for the midpoint temperatureT2in terms of the thermal and geometric parame-ters of the system.(c) For T1?200C, T3?100C, and the conditions provided in the schematic, calculate T2and plot the tem-perature distribution. Describe key features of thedistribution and compare it to your sketch of part (a)

A carbon nanotube is suspended across a trench of widths?5 ?m that separates two islands, each at T??300 K.A focused laser beam irradiates the nanotube at a dis-tance ?from the left island, delivering q?10?W of energy to the nanotube. The nanotube temperatureis measured at the midpoint of the trench using a point probe. The measured nanotube temperature isT1?324.5 K for ?1?1.5 ?m and T2?326.4 K for?2?3.5 ?m.Determine the two contact resistances, Rt,c,Land Rt,c,Rat the left and right ends of the nanotube, respec-tively. Theexperiment is performed in a vacuum withTsur?300 K. The nanotube thermal conductivity anddiameter are kcn?3100 W/m?K and D?14 nm,respectively.

A probe of overall length L?200 mm and diameter D?12.5 mm is inserted through a duct wall such that a por-tion of its length, referred to as the immersion length Li,is in contact with the water stream whose temperature,T?,i, is to be determined. The convection coefficientsover the immersion and ambient- exposed lengths are hi?1100 W/m2?K and ho?10 W/m2?K, respectively.The probe has a thermal conductivity of 177 W/m?K andis in poor thermal contact with the duct wall. a) Derive an expression for evaluating the measure-ment error, ?Terr?Ttip?T?,i, which is the differ-ence between the tip temperature, Ttip, and thewater temperature, T?,i. Hint:Define a coordinatesystem with the origin at the duct wall and treatthe probe as two fins extending inward and out-ward from the duct, but having the same base tem-perature. Use Case A results from Table 3.4.(b) With the water and ambient air temperatures at 80and 20C, respectively, calculate the measurementerror, ?Terr, as a function of immersion length forthe conditions Li/L?0.225, 0.425, and 0.625.(c) Compute and plot the effects of probe thermalconductivity and water velocity (hi) on the mea-surement error.

A rod of diameter D?25 mm and thermal conductiv-ity k?60 W/m?K protrudes normally from a furnacewall that is at Tw?200C and is covered by insulationof thickness Lins?200 mm. The rod is welded to thefurnace wall and is used as a hangerfor supportinginstrumentation cables. To avoid damaging the cables,the temperature of the rod at its exposed surface, To,must be maintained below a specified operating limit ofTmax?100C. The ambient air temperature is T??25C, and the convection coefficient is h?15 W/m2?K.(a) Derive an expression for the exposed surface temper-ature Toas a function of the prescribed thermal and geometrical parameters. The rod has an exposedlength Lo, and its tip is well insulated.(b) Will a rod with Lo?200 mm meet the specifiedoperating limit? If not, what design parameterswould you change? Consider another material,increasing the thickness of the insulation, andincreasing the rod length. Also, consider how youmight attach the base of the rod to the furnace wallas a means to reduce To.

A metal rod of length 2L, diameter D, and thermalconductivity kis inserted into a perfectly insulatingwall, exposing one-half of its length to an airstreamthat is of temperature T?and provides a convectioncoefficient hat the surface of the rod. An electro-magnetic field induces volumetric energy generation ata uniform rate q.within the embeddedportion of the rod.(a) Derive an expression for the steady-state tempera-ture Tbat the base of the exposed half of the rod.The exposed region may be approximated as avery long fin.(b) Derive an expression for the steady-state tempera-ture Toat the end of the embedded half of the rod.(c) Using numerical values provided in the schematic,plot the temperature distribution in the rod anddescribe key features of the distribution. Does therod behave as a very long fin?

A very long rod of 5-mm diameter and uniform thermalconductivity k?25 W/m?K is subjected to a heat treat-ment process. The center, 30-mm-long portion of therod within the induction heating coil experiences uni-form volumetric heat generation of 7.5106W/m3.The unheated portions of the rod, which protrude fromthe heating coil on either side, experience convectionwith the ambient air at T??20C and h?10 W/m2?K.Assume that there is no convection from the surface ofthe rod within the coil. (a) Calculate the steady-state temperature Toof the rodat the midpoint of the heated portion in the coil.(b) Calculate the temperature of the rod Tbat the edgeof the heated portion.

From Problem 1.71, consider the wire leads connectingthe transistor to the circuit board. The leads are of ther-mal conductivity k, thickness t, width w, and length L.One end of a lead is maintained at a temperature Tccor-responding to the transistor case, while the other endassumes the temperature Tbof the circuit board. Duringsteady-state operation, current flow through the leadsprovides for uniform volumetric heating in the amountq., while there is convection cooling to air that is at T?and maintains a convection coefficient h.(a) Derive an equation from which the temperaturedistribution in a wire lead may be determined. Listall pertinent assumptions.(b) Determine the temperature distribution in a wirelead, expressing your results in terms of the pre-scribed variables.

Turbine blades mounted to a rotating disc in a gasturbine engine are exposed to a gas stream that is atT??1200C and maintains a convection coefficientofh?250 W/m2?K over the blade. The blades, which are fabricated from Inconel,k20 W/m?K, have a length of L?50 mm. Theblade profile has a uniform cross- sectional area ofAc?6 10?4m2and a perimeter of P?110 mm. Aproposed blade-cooling scheme, which involves routing air through the supporting disc, is able tomaintain the base of each blade at a temperature of Tb?300C.(a) If the maximum allowable blade temperature is1050C and the blade tip may be assumed to be adi-abatic, is the proposed cooling scheme satisfactory?(b) For the proposed cooling scheme, what is the rateat which heat is transferred from each blade to thecoolant?

In a test to determine the friction coefficient ?associ-ated with a disk brake, one disk and its shaft arerotated at a constant angular velocity , while anequivalent disk/shaft assembly is stationary. Each diskhas an outer radius of r2?180 mm, a shaft radius ofr1?20 mm, a thickness of t?12 mm, and a thermalconductivity of k?15 W/m?K. A known force Fisapplied to the system, and the corresponding torque ?required to maintain rotation is measured. The diskcontact pressure may be assumed to be uniform (i.e.,independent of location on the interface), and the disksmay be assumed to be well insulated from the sur-roundings.(a) Obtain an expression that may be used to evaluate?from known quantities.(b) For the region r1rr2, determine the radialtemperature distribution T(r) in the disk, whereT(r1)?T1is presumed to be known.(c) Consider test conditions for which F?200 N,?40 rad/s, ??8 N?m, and T1?80C. Evalu-ate the friction coefficient and the maximum disk temperature.

Consider an extended surface of rectangular cross sec-tion with heat flow in the longitudinal direction. The blades, which are fabricated from Inconel,k20 W/m?K, have a length of L?50 mm. Theblade profile has a uniform cross-sectional area ofAc?6 10?4m2and a perimeter of P?110 mm. Aproposed blade- cooling scheme, which involves routing air through the supporting disc, is able tomaintain the base of each blade at a temperature of Tb?300C.(a) If the maximum allowable blade temperature is1050C and the blade tip may be assumed to be adi-abatic, is the proposed cooling scheme satisfactory?(b) For the proposed cooling scheme, what is the rateat which heat is transferred from each blade to thecoolant?3.127In a test to determine the friction coefficient ?associ-ated with a disk brake, one disk and its shaft arerotated at a constant angular velocity , while anequivalent disk/shaft assembly is stationary. Each diskhas an outer radius of r2?180 mm, a shaft radius ofr1?20 mm, a thickness of t?12 mm, and a thermalconductivity of k?15 W/m?K. A known force Fisapplied to the system, and the corresponding torque ?required to maintain rotation is measured. The diskcontact pressure may be assumed to be uniform (i.e.,independent of location on the interface), and the disksmay be assumed to be well insulated from the sur-roundings.(a) Obtain an expression that may be used to evaluate?from known quantities.(b) For the region r1rr2, determine the radialtemperature distribution T(r) in the disk, whereT(r1)?T1is presumed to be known.(c) Consider test conditions for which F?200 N,?40 rad/s, ??8 N?m, and T1?80C. Evalu-ate the friction coefficient and the maximum disktemperature.3.128Consider an extended surface of rectangular cross sec-tion with heat flow in the longitudinal direction.Fr2r1tDisk interface,friction coefficient, T1

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

A brass rod 100 mm long and 5 mm in diameterextends horizontally from a casting at 200C. The rodis in an air environment with T = 20C and h = 30W/mK. What is the temperature of the rod 25, 50,and 100 mm from the casting?

The extent to which the tip condition affects the ther-mal performance of a fin depends on the fin geometryand thermal conductivity, as well as the convectioncoefficient. Consider an alloyed aluminum (k?180 W/m?K) rectangular fin of length L?10 mm,thickness t?1 mm, and width w?t. The base tem- perature of the fin is Tb?l00C, and the fin is exposedto a fluid of temperature T??25C.(a) Assuming a uniform convection coefficient of h?100 W/m2?K over the entire fin surface, deter-mine the fin heat transfer rate per unit width q?f,efficiency ?f, effectiveness ?f, thermal resistanceper unit width R?t,f, and the tip temperature T(L) forCases A and B of Table 3.4. Contrast your resultswith those based on an infinite fapproximation.(b) Explore the effect of variations in the convectioncoefficient on the heat rate for 10?h?1000W/m2?K. Also consider the effect of such varia-tions for a stainless steel fin (k?15 W/m?K).

A pin fin of uniform, cross-sectional area is fabricatedof an aluminum alloy (k?160 W/m?K). The findiameter is D?4 mm, and the fin is exposed to con-vective conditions characterized by h?220 W/m2?K.It is reported that the fin efficiency is ?f?0.65. Deter-mine the fin length Land the fin effectiveness ?f.Account for tip convection.

The extent to which the tip condition affects the thermalperformance of a fin depends on the fin geometry andthermal conductivity, as well as the convection coeffi-cient. Consider an alloyed aluminum (k?180 W/m?K)rectangular fin whose base temperature is Tb?100C.The fin is exposed to a fluid of temperature T??25C,and a uniform convection coefficient of h?100W/m2?K may be assumed for the fin surface.(a) For a fin of length L?10 mm, thicknesst?1 mm, and width w?t, determine the fin heattransfer rate per unit width q?f, efficiency ?f, effec-tiveness ?f, thermal resistance per unit width R?t,f,and tip temperature T(L) for Cases A and B ofTable 3.4. Contrast your results with those basedon an infinite fapproximation.(b) Explore the effect of variations in Lon the heat ratefor 3?L?50 mm. Also consider the effect of suchvariations for a stainless steel fin (k?15 W/m?K).

A straight fin fabricated from 2024 aluminum alloy(k?185 W/m?K) has a base thickness of t?3mmand a length of L?15 mm. Its base temperature isTb?100C, and it is exposed to a fluid for whichT??20C and h?50 W/m2?K. For the foregoingconditions and a fin of unit width, compare the fin heat rate, efficiency, and volume for rectangular, triangular,and parabolic profiles.

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

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

Circular copper rods of diameter D?1 mm andlength L?25 mm are used to enhance heat transferfrom a surface that is maintained at Ts,1?100C. Oneend of the rod is attached to this surface (at x?0),while the other end (x?25 mm) is joined to a secondsurface, which is maintained at Ts,2?0C. Air flowingbetween the surfaces (and over the rods) is also at atemperature of T??0C, and a convection coefficientof h?100 W/m2?K is maintained.(a) What is the rate of heat transfer by convectionfrom a single copper rod to the air?(b) What is the total rate of heat transfer from a1m1 m section of the surface at 100C, if abundle of the rods is installed on 4-mm centers?

During the initial stages of the growth of the nanowireof Problem 3.109, a slight perturbation of the liquidcatalyst droplet can cause it to be suspended on the topof the nanowire in an off-center position. The resultingnonuniform deposition of solid at the solid-liquidinterface can be manipulated to form engineered shapessuch as a nanospring, that is characterized by a springradius r, spring pitch s, overall chord length Lc(lengthrunning along the spring), and end-to-end length L, asshown in the sketch. Consider a silicon carbidenanospring of diameter D?15 nm, r?30 nm, s?25 nm, and Lc?425 nm. From experiments, it is knownthat the average spring pitch svaries with average tem-perature Tby the relation ds/dT?0.1 nm/K. Usingthis information, a student suggests that a nanoactuatorcan be constructed by connecting one end of thenanospring to a small heater and raising the tempera-ture of that end of the nano spring above its initial value.Calculate the actuation distance ?Lfor conditionswhere h?106W/m2?K, T??Ti?25C, with a base temperature of Tb?50C. If the base temperature canbe controlled to within 1C, calculate the accuracy towhich the actuation distance can be controlled. Hint:Assume the spring radius does not change when thespring is heated. The overall spring length may beapproximated by the formula,

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

A 40-mm-long, 2-mm-diameter pin fin is fabricated ofan aluminum alloy (k?140 W/m?K).(a) Determine the fin heat transfer rate for Tb?50C,T??25C, h?1000 W/m2?K, and an adiabatictip condition.(b) An engineer suggests that by holding the fin tip ata low temperature, the fin heat transfer rate can beincreased. For T(x?L)?0C, determine the newfin heat transfer rate. Other conditions are as inpart (a).(c) Plot the temperature distribution, T(x), over therange 0xLfor the adiabatic tip case and the prescribed tip temperature case. Also show theambient temperature in your graph. Discuss relevantfeatures of the temperature distribution.(d) Plot the fin heat transfer rate over the range0h1000 W/m2?K for the adiabatic tip caseand the prescribed tip temperature case. For theprescribed tip temperature case, what would the calculated fin heat transfer rate be if Equation3.78 were used to determine qfrather than Equa-tion 3.76?

An experimental arrangement for measuring the ther-mal conductivity of solid materials involves the use of two long rods that are equivalent in every respect,except that one is fabricated from a standard materialof known thermal conductivity kAwhile the other isfabricated from the material whose thermal conductivity kB is desired. Both rods are attached at one end to a heat source of fixed temperature Tb, are exposed to a fluid of temperature T, and are instrumented with thermocouples to measure the temperature at a fixed distance x1from the heat source. If the standard material is aluminum, with kA = 200 W/mK, and measurements reveal values of TA = 75C and TB = 60C at x1 for Tb = 100C and T = 25C, what is the therma lconductivity kB of the test material?

Finned passages are frequently formed between paral-lel plates to enhance convection heat transfer in com-pact heat exchanger cores. An important application isin electronic equipment cooling, where one or moreair-cooled stacks are placed between heat-dissipatingelectrical components. Consider a single stack of rec-tangular fins of length Land thickness t, with convec-tion conditions corresponding to hand T?.(a) Obtain expressions for the fin heat transfer rates,qf,oand qf,L, in terms of the base temperatures, Toand TL.(b) In a specific application, a stack that is 200 mmwide and 100 mm deep contains 50 fins, each oflength L?12 mm. The entire stack is made fromaluminum, which is everywhere 1.0 mm thick. Iftemperature limitations associated with electricalcomponents joined to opposite plates dictate maxi-mum allowable plate temperatures of To?400 K and TL?350 K, what are the corresponding maxi-mum power dissipations if h1 ? 50 W/m2?K andT?3 ? 00 K?

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

An isothermal silicon chip of width W?20 mm on aside is soldered to an aluminum heat sink (k?180 W/m?K) of equivalent width. The heat sink has abase thickness of Lb?3 mm and an array of rectangu- lar fins, each of length Lf?15 mm. Airflow at T??20C is maintained through channels formed by thefins and a cover plate, and for a convection coefficientof h?100 W/m2?K, a minimum fin spacing of1.8 mm is dictated by limitations on the flow pressuredrop. The solder joint has a thermal resistance of.AirT,hWLbLftSChip, Tc, qcSolder, Rt",cHeat sink, kCover plateR?t,c?210?6m2?K/W. (a) Consider limitations for which the array has N?11 fins, in which case values of the fin thicknesst?0.182 mm and pitch S?1.982 mm are obtainedfrom the requirements that W?(N?1)S?tandS?t?1.8 mm. If the maximum allowable chiptemperature is Tc?85C, what is the correspondingvalue of the chip power qc? An adiabatic fin tip con-dition may be assumed, and airflow along the outersurfaces of the heat sink may be assumed to providea convection coefficient equivalent to that associatedwith airflow through the channels. (b) With (S?t) and hfixed at 1.8 mm and 100W/m2?K, respectively, explore the effect of increas-ing the fin thickness by reducing the number of fins.With N?11 and S?tfixed at 1.8 mm, butrelaxation of the constraint on the pressure drop,explore the effect of increasing the airflow, andhence the convection coefficient.

As seen in Problem 3.109, silicon carbide nanowires ofdiameter D?15 nm can be grown onto a solid siliconcarbide surface by carefully depositing droplets of cata-lyst liquid onto a flat silicon carbide substrate. Siliconcarbide nanowires grow upward from the depositeddrops, and if the drops are deposited in a pattern, anarray of nanowire fins can be grown, forming a siliconcarbide nano-heat sink. Consider finned and unfinnedelectronics packages in which an extremely small, 10?m10?m electronics device is sandwichedbetween two d?100-nm-thick silicon carbide sheets. Inboth cases, the coolant is a dielectric liquid at 20C. Aheat transfer coefficient of h?1 105W/m2?K existson the top and bottom of the unfinned package and on allsurfaces of the exposed silicon carbide fins, which areeach L?300 nm long. Each nano-heat sink includes a200200 array of nanofins. Determine the maximumallowable heat rate that can be generated by the elec-tronic device so that its temperature is maintained at Tt?85C for the unfinned and finned packages.

As more and more components are placed on asingle integrated circuit (chip), the amount of heatthat is dissipated continues to increase. However,this increase is limited by the maximum allowablechip operating temperature, which is approximately75C. To maximize heat dissipation, it is proposedthat a 44 array of copper pin fins be metallurgi-cally joined to the outer surface of a square chip thatis 12.7 mm on a side.(a) Sketch the equivalent thermal circuit for the pinchipboard assembly, assuming one- dimensional,steady-state conditions and negligible contactresistance between the pins and the chip. In vari-able form, label appropriate resistances, tempera-tures, and heat rates.(b) For the conditions prescribed in Problem 3.27,what is the maximum rate at which heat can bedissipated in the chip when the pins are in place?That is, what is the value of qcfor Tc?75C? Thepin diameter and length are Dp?1.5 mm andLp?15 mm.

A homeowners wood stove is equipped with a topburner for cooking. The D?200-mm-diameter burneris fabricated of cast iron (k?65 W/m?K). The bottom(combustion) side of the burner has 8 straight fins of uni-form cross section, arranged as shown in the sketch. Avery thin ceramic coating (??0.95) is applied to all sur-faces of the burner. The top of the burner is exposed toroom conditions (Tsur,t?T?,t?20C, ht?40 W/m2?K),while the bottom of the burner is exposed to combus-tion conditions (Tsur,b?T?.b?450C, hb?50 W/m2?K).Compare the top surface temperature of the finnedburner to that which would exist for a burner withoutfins. Hint: Use the same expression for radiation heattransfer to the bottom of the finned burner as for theburner with no fins.

In Problem 3.146, the prescribed value of ho?1000W/m2?K is large and characteristic of liquid cooling. Inpractice it would be far more preferable to use air cool-ing, for which a reasonable upper limit to the convec-tion coefficient would be ho?250 W/m2?K. Assess theeffect of changes in the pin fin geometry on the chipheat rate if the remaining conditions of Problem 3.146,including a maximum allowable chip temperature of75C, remain in effect. Parametric variations that maybe considered include the total number of pins Nin thesquare array, the pin diameter Dp, and the pin length Lp.However, the product N1/2Dpshould not exceed 9 mmto ensure adequate airflow passage through the array.Recommend a design that enhances chip cooling.

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

As a means of enhancing heat transfer from high-performance logic chips, it is common to attach a heat sinkto the chip surface in order to increase thesurface area available for convection heat transfer.Because of the ease with which it may be manufac-tured (by taking orthogonal sawcuts in a block ofmaterial), an attractive option is to use a heat sinkconsisting of an array of square fins of width won aside. The spacing between adjoining fins would bedetermined by the width of the sawblade, with thesum of this spacing and the fin width designated asthe fin pitch S. The method by which the heat sink isjoined to the chip would determine the interfacialcontact resistance, R?t,c Consider a square chip of width Wc?16 mm andconditions for which cooling is provided by a dielec-tric liquid with T??25C and h?1500 W/m2?K. Theheat sink is fabricated from copper (k?400 W/m?K),and its characteristic dimensions are w?0.25 mm, S?0.50 mm, L?6 mm, and Lb?3 mm. The pre-scribed values of wand Srepresent minima imposedby manufacturing constraints and the need to maintainadequate flow in the passages between fins.(a) If a metallurgical joint provides a contact resis-tance of R?t,c?5 10?6m2?K/W and the maxi-mum allowable chip temperature is 85C, what isthe maximum allowable chip power dissipationqc? Assume all of the heat to be transferredthrough the heat sink.(b) It may be possible to increase the heat dissipationby increasing w, subject to the constraint that (S?w) ?0.25 mm, and/or increasing L(subjectto manufacturing constraints that L10 mm).Assess the effect of such changes.

Because of the large number of devices in todays PCchips, finned heat sinks are often used to maintain thechip at an acceptable operating temperature. Two findesigns are to be evaluated, both of which have base(unfinned) area dimensions of 53 mm 57 mm. Thefins are of square cross section and fabricated from anextruded aluminum alloy with a thermal conductivityof 175 W/m?K. Cooling air may be supplied at 25C,and the maximum allowable chip temperature is 75C.Other features of the design and operating conditionsare tabulated.Fin DimensionsConvectionCross Section Length Number of CoefficienDesignw?w (mm)L (mm) Fins in Array (W/m2?K)A3 3306 9 125B1 1 7 14 17 375Determine which fin arrangement is superior. In youranalysis, calculate the heat rate, efficiency, and effec-tiveness of a single fin, as well as the total heat rateand overall efficiency of the array. Since real estateinside the computer enclosure is important, comparethe total heat rate per unit volume for the two designs.

Consider design B of Problem 3.151. Over time, dustcan collect in the fine grooves that separate the fins. Consider the buildup of a dust layer of thicknessLd, as shown in the sketch. Calculate and plot the totalheat rate for design B for dust layers in the range 0 Ld5 mm. The thermal conductivity of the dust canbe taken as kd= 0.032 W/m?K. Include the effects ofconvection from the fin tip.

A long rod of 20-mm diameter and a thermal conduc-tivity of 1.5 W/m?K has a uniform internal volumetricthermal energy generation of 106W/m3. The rod is cov-ered with an electrically insulating sleeve of 2-mmthickness and thermal conductivity of 0.5 W/m?K. A spi-der with 12 ribs and dimensions as shown in the sketchhas a thermal conductivity of 175 W/m?K, and is used tosupport the rod and to maintain concentricity with an 80-mm-diameter tube. Air at T??25C passes over the spi-der surface, and the convection coefficient is 20 W/m2?K.The outer surface of the tube is well insulated.We wish to increase volumetric heating within therod, while not allowing its centerline temperature toexceed 100C. Determine the impact of the followingchanges, which may be effected independently or con- currently: (i) increasing the air speed and hence the con-vection coefficient; (ii) changing the number and/orthickness of the ribs; and (iii) using an electrically non-conducting sleeve material of larger thermal conductivity(e.g., amorphous carbon or quartz). Recommend a realis-tic configuration that yields a significant increase in q..

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

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

Heat is uniformly generated at the rate of 2 x 10W/m in a wall of thermal conductivity 25 W/mK and thickness 60 mm. The wall is exposed to convection on both sides, with different heat transfer coefficients and temperatures as shown. There are straight rectangular fins on the right-hand side of the wall, with dimensions as shown and thermal conductivity of 250 W/mK. What is the maximum temperature that will occur in the wall?

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

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

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

It is proposed to air-cool the cylinders of a combustionchamber by joining an aluminum casing with annu-lar fins (k = 240 W/mK) to the cylinder wall (k = 50 W/mK).The air is at 320 K and the corresponding convection coefficient is 100 W/mK. Although heating at the inner surface is periodic, it is reasonable to assume steady-state conditions with a time-averaged heat flux of q?i?105W/m2. Assuming negligible contact resistancebetween the wall and the casing, determine the wall innertemperature Ti, the interface temperature T1, and the finbase temperature Tb. Determine these temperatures if theinterface contact resistance is R?t,c?10?4m2?K/W.

Consider the air-cooled combustion cylinder of Problem3.160, but instead of imposing a uniform heat flux atthe inner surface, consider conditions for which thetime-averaged temperature of the combustion gases isTg?1100 K and the corresponding convection coeffi-cient is hg?150 W/m2?K. All other conditions,including the cylinder/casing contact resistance,remain the same. Determine the heat rate per unitlength of cylinder (W/m), as well as the cylinder innertemperature Ti, the interface temperatures T1,iand T1,o,and the fin base temperature Tb. Subject to the con-straint that the fin gap is fixed at ??2 mm, assess theeffect of increasing the fin thickness at the expense ofreducing the number of fins.

Heat transfer from a transistor may be enhanced byinserting it in an aluminum sleeve (k?200 W/m?K)having 12 integrally machined longitudinal fins on itsouter surface. The transistor radius and height are r1?2.5 mm and H?4 mm, respectively, while the fins are of length L?r3?r2?8 mm and uniform thickness t?0.8 mm. The thickness of the sleeve base is r2?r1?1 mm, and the contact resistance of the sleeve-transistorinterface is R?t,c?0.6 10?3m2?K/W. Air at T??20Cflows over the fin surface, providing an approximatelyuniform convection coeffficient of h?30 W/m2?K.(a) When the transistor case temperature is 80C, whatis the rate of heat transfer from the sleeve?(b) Identify all of the measures that could be taken toimprove design and/or operating conditions, suchthat heat dissipation may be increased while stillmaintaining a case temperature of 80C. In words,assess the relative merits of each measure. Choose what you believe to be the three most promisingmeasures, and numerically assess the effect of cor-responding changes in design and/or operating con-ditions on thermal performance.

Consider the conditions of Problem 3.149 but now allowfor a tube wall thickness of 5 mm (inner and outer diam-eters of 50 and 60 mm), a fin-to-tube thermal contactresistance of 10?4m2?K/W, and the fact that the watertemperature, Tw?350 K, is known, not the tube surfacetemperature. The water-side convection coefficient ishw?2000 W/m2?K. Determine the rate of heat transferper unit tube length (W/m) to the water. What would bethe separate effect of each of the following designchanges on the heat rate: (i) elimination of the contactresistance; (ii) increasing the number of fins from four toeight; and (iii) changing the tube wall and fin materialfrom copper to AISI 304 stainless steel (k?20W/m?K)?

A scheme for concurrently heating separate water andair streams involves passing them through and over an array of tubes, respectively, while the tube wall isheated electrically. To enhance gas-side heat transfer,annular fins of rectangular profile are attached to theouter tube surface. Attachment is facilitated with adielectric adhesive that electrically isolates the fins fromthe current-carrying tube wall. (a) Assuming uniform volumetric heat generationwithin the tube wall, obtain expressions for theheat rate per unit tube length (W/m) at the inner(ri) and outer (ro) surfaces of the wall. Expressyour results in terms of the tube inner and outersurface temperatures, Ts,iand Ts,o, and other perti-nent parameters.(b) Obtain expressions that could be used to determineTs,iand Ts,oin terms of parameters associated withthe water- and air-side conditions.(c) Consider conditions for which the water and airare at T?,i?T?,o?300 K, with correspondingconvection coefficients of hi?2000 W/m2?K andho?100 W/m2?K. Heat is uniformly dissipated ina stainless steel tube (kw?15 W/m?K), havinginner and outer radii of ri?25 mm and ro?30mm, and aluminum fins (t???2 mm, rt?55mm) are attached to the outer surface, with R?t,c?10?4m2?K/W. Determine the heat rates and tem-peratures at the inner and outer surfaces as a func-tion of the rate of volumetric heating q.. The upperlimit to q.will be determined by the constraints thatTs,inot exceed the boiling point of water (100C)and Ts,onot exceed the decomposition temperatureof the adhesive (250C).

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

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

Consider heat transfer in a forearm, which can beapproximated as a cylinder of muscle of radius 50 mm(neglecting the presence of bones), with an outer layerof skin and fat of thickness 3 mm. There is metabolicheat generation and perfusion within the muscle. Themetabolic heat generation rate, perfusion rate, arterialtemperature, and properties of blood, muscle, andskin/fat layer are identical to those in Example 3.12. The environment and surroundings are the same as forthe air environment in Example 3.12.(a) Write the bioheat transfer equation in radial coor-dinates. Write the boundary conditions thatexpress symmetry at the centerline of the forearmand specified temperature at the outer surface ofthe muscle. Solve the differential equation andapply the boundary conditions to find an expres-sion for the temperature distribution. Note that thederivatives of the modified Bessel functions aregiven in Section 3.6.4.(b) Equate the heat flux at the outer surface of themuscle to the heat flux through the skin/fat layerand into the environment to determine the temper-ature at the outer surface of the muscle.(c) Find the maximum forearm temperature.

For one of the M?48 modules of Example 3.13, deter-mine a variety of different efficiency values concerningthe conversion of waste heat to electrical energy.(a) Determine the thermodynamic efficiency, ?therm?PM?1/q1. (b) Determine the figure of merit for one module,and the thermoelectric efficiency, ?TEusing Equa-tion 3.128.(c) Determine the Carnot efficiency, ?Carnot?1 T2/T1.(d) Determine both the thermoelectric efficiency and theCarnot efficiency for the case where h1?h2l?.(e) The energy conversion efficiency of thermoelec-tric devices is commonly reported by evaluatingEquation 3.128, but with T?,1and T?,2 usedinstead of T1and T2, respectively. Determine thevalue of ?TEbased on the inappropriate use of T?,1and T?,2, and compare with your answers for parts(b) and (d).

One of the thermoelectric modules of Example 3.13 isinstalled between a hot gas at T?,1?450C and a coldgas at T?,2?20C. The convection coefficient associated with the flowing gases is h?h1?h2?80 W/m2?Kwhile the electrical resistance of the load is Re,load?4 ?.(a) Sketch the equivalent thermal circuit and deter-mine the electric power generated by the modulefor the situation where the hot and cold gases pro-vide convective heating and cooling directly to themodule (no heat sinks).(b) Two heat sinks (k?180 W/m?K; see sketch),each with a base thickness of Lb?4 mm and finlength Lf?20 mm, are soldered to the upper andlower sides of the module. The fin spacing is3 mm, while the solder joints each have a thermalresistance of R?t,c?2.510?6m2?K/W. Eachheat sink has N?11 fins, so that t?2.182 mmand S?5.182 mm, as determined from therequirements that W?(N?1)S?tandS?t?3 mm. Sketch the equivalent thermal cir-cuit and determine the electric power generated by the module. Compare the electric power gener-ated to your answer for part (a). Assume adiabaticfin tips and convection coefficients that are thesame as in part (a).

Thermoelectric modules have been used to generateelectric power by tapping the heat generated by woodstoves. Consider the installation of the thermoelectricmodule of Example 3.13 on a vertical surface of awood stove that has a surface temperature ofTs?375C. A thermal contact resistance of R?t,c?510?6m2?K/W exists at the interface between thestove and the thermoelectric module, while the roomair and walls are at T??Tsur?25C. The exposedsurface of the thermoelectric module has an emissiv-ity of ??0.90 and is subjected to a convection coef-ficient of h?15 W/m2?K. Sketch the equivalent thermal circuit and determine the electric power generated by the module. The load electrical resistance is Re,load ?3?.

The electric power generator for an orbiting satelliteis composed of a long, cylindrical uranium heatsource that is housed within an enclosure of squarecross section. The only way for heat that is gener-ated by the uranium to leave the enclosure isthrough four rows of the thermoelectric modules ofExample 3.13. The thermoelectric modules generateelectric power and also radiate heat into deep spacecharacterized by Tsur?4 K. Consider the situationfor which there are 20 modules in each row for a total of M?420?80 modules. The modulesare wired in series with an electrical load of Re,load?250 ?, and have an emissivity of ??0.93.Determine the electric power generated for and 100 kW. Also determine the surface tempera-tures of the modules for the three thermal energygeneration rates.

Rows of the thermoelectric modules of Example 3.13are attached to the flat absorber plate of Problem 3.108.The rows of modules are separated by Lsep?0.5 m andthe backs of the modules are cooled by water at a tem-perature of Tw?40C, with h?45 W/m2?K.Determine the electric power produced by one rowof thermoelectric modules connected in series electri-cally with a load resistance of 60 ?. Calculate the heat transfer rate to the flowing water. Assume rows of 20 immediately adjacent modules, with the lengths ofboth the module rows and water tubing to beLrow?20Wwhere W?54 mm is the module dimen-sion taken from Example 3.13. Neglect thermal con-tact resistances and the temperature drop across thetube wall, and assume that the high thermal conductiv-ity tube wall creates a uniform temperature around the tube perimeter. Because of the thermal resistanceprovided by the thermoelectric modules, it is nolonger appropriate to assume that the temperature ofthe absorber plate directly above a tube is equal to thatof the water.

Determine the conduction heat transfer through an air layer held between two 10 mm10 mm parallelaluminum plates. The plates are at temperatures Ts,1?305 K and Ts,2?295 K, respectively, and the air is at atmospheric pressure. Determine the con-duction heat rate for plate spacings of L?1 mm,L?1?m, and L?10 nm. Assume a thermal accom-modation coefficient of t?0.92.

Determine the parallel plate separation distance L,above which the thermal resistance associated withmolecule-surface collisions Rt,m?sis less than 1% ofthe resistance associated with moleculemolecule col-lisions, Rt,m?mfor (i) air between steel plates witht?0.92 and (ii) helium between clean aluminumplates with t?0.02. The gases are at atmosphericpressure, and the temperature is T?300 K.

Determine the conduction heat flux through various planelayers that are subjected to boundary temperatures of Ts,1?301 K and Ts,2?299 K at atmospheric pressure.Hint: Do not account for micro- or nanoscale effectswithin the solid, and assume the thermal accommodationcoefficient for an aluminumair interface is t?0.92.(a) Case A: The plane layer is aluminum. Determinethe heat fluxq?xfor Ltot?600 ?m and Ltot?600 nm.(b) Case B: Conduction occurs through an air layer.Determine the heat fluxq?xfor Ltot?600 ?m andLtot?600 nm. (c) Case C: The composite wall is composed of air heldbetween two aluminum sheets. Determine the heatfluxq?xfor Ltot?600?m (with aluminum sheetthicknesses of ??40?m) and Ltot?600 nm (withaluminum sheet thicknesses of ??40 nm).(d) Case D: The composite wall is composed of 7 airlayers interspersed between 8 aluminum sheets. Determine the heat fluxq?xfor Ltot?600 ?m (withaluminum sheet and air layer thicknesses of??40?m) and Ltot?600 nm (with aluminumsheet and air layer thicknesses of ??40 nm).

The Knudsen number, Kn??mfp/L, is a dimensionlessparameter used to describe potential micro- or nanoscaleeffects. Derive an expression for the ratio of the thermalresistance due to moleculesurface collisions to the ther-mal resistance associated with moleculemolecule colli-sions, Rt,m?s/Rt,m?m, in terms of the Knudsen number,the thermal accommodation coefficient t, and theratio of specific heats , for an ideal gas. Plot the criti-cal Knudsen number, Kncrit, that is associated withRt,m?s/Rt,m?m?0.01 versus t, for ?1.4 and 1.67(corresponding to air and helium, respectively).

A nanolaminated material is fabricated with an atomiclayer deposition process, resulting in a series of stacked, alternating layers of tungsten and aluminumoxide, each layer being ??0.5 nm thick. Each tung- stenaluminum oxide interface is associated with athermal resistance of R?t,i?3.85 10?9m2?K/W. Thetheoretical values of the thermal conductivities of thethinaluminum oxide and tungsten layers arekA?1.65 W/m?K and kT?6.10 W/m?K, respec-tively. The properties are evaluated at T?300 K.(a) Determine the effective thermal conductivity ofthe nanolaminated material. Compare the value ofthe effective thermal conductivity to the bulk ther-mal conductivities of aluminum oxide and tung-sten, given in Tables A.1 and A.2.(b) Determine the effective thermal conductivity ofthe nanolaminated material assuming that the ther-mal conductivities of the tungsten and aluminumoxide layers are equal to their bulk values.

Gold is commonly used in semiconductor packagingto form interconnections (also known as interconnects)that carry electrical signals between different devicesin the package. In addition to being a good electricalconductor, gold interconnects are also effective atprotecting the heat- generating devices to which theyare attached by conducting thermal energy away fromthe devices to surrounding, cooler regions. Consider a thin film of gold that has a cross section of60 nm250 nm.(a) For an applied temperature difference of 20C,determine the energy conducted along a 1-?m-long, thin- film interconnect. Evaluate properties at300 K.(b) Plot the lengthwise (in the 1-?m direction) andspanwise (in the thinnest direction) thermal conductivities of the gold film as a function of the filmthickness Lfor 30L140 nm.