A one-dimensional slab of thickness 2Lis initially at

Consider a thin electrical heater attached to a plate andbacked by insulation. Initially, the heater and plate areat the temperature of the ambient air, T?. Suddenly, thepower to the heater is activated, yielding a constant heatflux at the inner surface of the plate.(a) Sketch and label, on T xcoordinates, the tempera-ture distributions: initial, steady-state, and at twointermediate times.(b) Sketch the heat flux at the outer surface as afunction of time

The inner surface of a plane wall is insulated while theouter surface is exposed to an airstream at T?. The wall is at a uniform temperature corresponding to that of theairstream. Suddenly, a radiation heat source is switchedon, applying a uniform flux to the outer surface.(a) Sketch and label, on T xcoordinates, the tempera-ture distributions: initial, steady-state, and at twointermediate times.(b) Sketch the heat flux at the outer surface as afunction of time.

A microwave oven operates on the principle that application of a high-frequency field causes electrically polarized molecules in food to oscillate. The net effect isa nearly uniform generation of thermal energy within the food. Consider the process of cooking a slab of beef of thickness 2L in a microwave oven and compare it with cooking in a conventional oven, where each side of the slab is heated by radiation.In each case the meat is to be heated from 0C to a minimumtemperature of 90C. Base your comparison on a sketch of the temperature distribution at selected times for each of the cooking processes. In particular, consider the time t0 at which heating is initiated, a time t1 during the heating process,the time t2 corresponding to the conclusion of heating,and a time t3 well into the subsequent cooling process.

A plate of thickness 2L, surface area As, mass M, andspecific heat cp, initially at a uniform temperature Ti, issuddenly heated on both surfaces by a convectionprocess (T?, h) for a period of time to, following whichthe plate is insulated. Assume that the midplane tem-perature does not reach T?within this period of time.(a) Assuming Bi?1 for the heating process, sketch andlabel, on Txcoordinates, the following temperaturedistributions: initial, steady-state (t l?), T(x,to), andat two intermediate times between t?toand t l ?.(b) Sketch and label, on T tcoordinates, the midplaneand exposed surface temperature distributions.(c) Repeat parts (a) and (b) assuming Bi?1 for the plate.(d) Derive an expression for the steady-state tempera-ture T(x,?)?Tf, leaving your result in terms ofplate parameters (M, cp), thermal conditions (Ti, T?,h), the surface temperature T(L, t), and the heatingtime to.

For each of the following cases, determine an appropri-ate characteristic length Lcand the corresponding Biotnumber Bithat is associated with the transient thermalresponse of the solid object. State whether the lumpedcapacitance approximation is valid. If temperature infor-mation is not provided, evaluate properties at T?300 K.(a) A toroidal shape of diameter D?50 mm andcross-sectional area Ac?5mm2is of thermalconductivity k?2.3 W/m?K. The surface of thetorus is exposed to a coolant corresponding to aconvection coefficient of h?50 W/m2?K.(b) A long, hot AISI 304 stainless steel bar of rectan-gular cross section has dimensions w?3 mm,W?5 mm, and L?100 mm. The bar is subjectedto a coolant that provides a heat transfer coefficientof h?15 W/m2?K at all exposed surfaces.(c) A long extruded aluminum (Alloy 2024) tube of innerand outer dimensions w?20 mm and W?24 mm,respectively, is suddenly submerged in water, result-ing in a convection coefficient of h?37 W/m2?K atthe four exterior tube surfaces. The tube is plugged atboth ends, trapping stagnant air inside the tube. (d) An L?300-mm-long solid stainless steel rod of diam-eter D?13 mm and mass M?0.328 kg is exposedto a convection coefficient of h?30 W/m2?K.(e) A solid sphere of diameter D?12 mm andthermal conductivity k?120 W/m?K is sus-pended in a large vacuum oven with internal walltemperatures of Tsur?20C. The initial spheretemperature is Ti?100C, and its emissivity is??0.73.(f) A long cylindrical rod of diameter D?20 mm, den-sity ??2300 kg/m3, specific heat cp?1750 J/kg?K,and thermal conductivity k?16 W/m?K is suddenlyexposed to convective conditions with T??20C.The rod is initially at a uniform temperature ofTi?200C and reaches a spatially averaged tempera-ture of T?100C at t?225 s.(g) Repeat part (f) but now consider a rod diameter ofD?200 mm.Case (e)DTi, ?, kTsurCase (c)wAluminumalloyAirDT, hCase (d)LCases (f,g)

Steel balls 12 mm in diameter are annealed by heating to1150 K and then slowly cooling to 400 K in an air envi-ronment for which T??325 K and h?20 W/m2?K.Assuming the properties of the steel to be k?40 W/m?K,??7800 kg/m3, and c?600 J/kg?K, estimate the timerequired for the cooling process

Consider the steel balls of Problem 5.6, except now theair temperature increases with time as T?(t)?325 Kat, where a?0.1875 K/s.(a) Sketch the ball temperature versus time for 0?t?1 h. Also show the ambient temperature, T?, inyour graph. Explain special features of the balltemperature behavior.(b) Find an expression for the ball temperature as afunction of time T(t), and plot the ball temperaturefor 0?t?1 h. Was your sketch correct?

The heat transfer coefficient for air flowing over a sphere is to be determined by observing the temperaturetime history of a sphere fabricated from pure copper. Thesphere, which is 12.7 mm in diameter, is at 66C before itis inserted into an airstream having a temperature of 27C. A thermocouple on the outer surface of the sphereindicates 55C 69 s after the sphere is inserted into the airstream. Assume and then justify that the sphere behaves as a spacewise isothermal object and calculate the heat transfer coefficient.

A solid steel sphere (AISI 1010), 300 mm in diameter,is coated with a dielectric material layer of thickness2 mm and thermal conductivity 0.04 W/m?K. Thecoated sphere is initially at a uniform temperature of500C and is suddenly quenched in a large oil bath forwhich T??100C and h?3300 W/m2?K. Estimatethe time required for the coated sphere temperature toreach 140C. Hint:Neglect the effect of energy storagein the dielectric material, since its thermal capacitance(?cV) is small compared to that of the steel sphere

A flaked cereal is of thickness 2L?1.2 mm. The density,specific heat, and thermal conductivity of the flake are??700 kg/m3, cp?2400 J/kg?K, and k?0.34 W/m?K,respectively. The product is to be baked by increasing itstemperature from Ti?20C to Tf ?220C in a convec-tion oven, through which the product is carried on a con-veyor. If the oven is Lo?3 m long and the convectionheat transfer coefficient at the product surface and ovenair temperature are h?55 W/m2?K and T??300C,respectively, determine the required conveyor velocity,V. An engineer suggests that if the flake thickness isreduced to 2L?1.0 mm the conveyor velocity can beincreased, resulting in higher productivity. Determinethe required conveyor velocity for the thinner flake.

The base plate of an iron has a thickness of L?7mmand is made from an aluminum alloy (??2800 kg/m3,c?900 J/kg?K, k?180 W/m?K, ??0.80). An electricresistance heater is attached to the inner surface of theplate, while the outer surface is exposed to ambient airand large surroundings at T??Tsur?25C. The areas ofboth the inner and outer surfaces are As?0.040 m2.If an approximately uniform heat flux of qh?1.25104W/m2is applied to the inner surface of the baseplate and the convection coefficient at the outer surfaceis h?10 W/m2?K, estimate the time required for theplate to reach a temperature of 135C. Hint:Numericalintegration is suggested in order to solve the problem.

Thermal energy storage systems commonly involve apacked bedof solid spheres, through which a hot gasflows if the system is being charged, or a cold gas if it isbeing discharged. In a charging process, heat transferfrom the hot gas increases thermal energy stored withinthe colder spheres; during discharge, the stored energydecreases as heat is transferred from the warmerspheres to the cooler gas. Consider a packed bed of 75-mm-diameter alumi-num spheres (??2700 kg/m3, c?950 J/kg?K, k?240 W/m?K) and a charging process for which gasenters the storage unit at a temperature of Tg,i?300C.If the initial temperature of the spheres is Ti?25Cand the convection coefficient is h?75 W/m2?K, howlong does it take a sphere near the inlet of the system toaccumulate 90% of the maximum possible thermalenergy? What is the corresponding temperature at thecenter of the sphere? Is there any advantage to usingcopper instead of aluminum?

A tool used for fabricating semiconductor devicesconsists of a chuck (thick metallic, cylindrical disk)onto which a very thin silicon wafer (??2700 kg/m3,c?875 J/kg?K, k?177 W/m?K) is placed by arobotic arm. Once in position, an electric field in thechuck is energized, creating an electrostatic force thatholds the wafer firmly to the chuck. To ensure a repro-ducible thermal contact resistance between the chuckand the wafer from cycle to cycle, pressurized heliumgas is introduced at the center of the chuck and flows(very slowly) radially outward between the asperities ofthe interface region.An experiment has been performed under conditions forwhich the wafer, initially at a uniform temperatureTw,i?100C, is suddenly placed on the chuck, which is ata uniform and constant temperature Tc?23C. With thewafer in place, the electrostatic force and the helium gasflow are applied. After 15 s, the temperature of the waferis determined to be 33C. What is the thermal contactresistance (m2?K / W) between the wafer and chuck?Will the value of increase, decrease, or remain thesame if air, instead of helium, is used as the purge gas?

A copper sheet of thickness 2L?2 mm has an initialtemperature of Ti?118C. It is suddenly quenched inliquid water, resulting in boiling at its two surfaces. For boiling, Newtons law of cooling is expressed asq?? ?h(Ts?Tsat), where Tsis the solid surface tem-perature and Tsatis the saturation temperature of thefluid (in this case Tsat?100C). The convection heat transfer coefficient may be expressed as h?1010 W/m2?K3(T?Tsat)2. Determine the time needed for the sheet to reach a temperature of T?102C. Plotthe copper temperature versus time for 0 ?t?0.5 s.On the same graph, plot the copper temperature historyassuming the heat transfer coefficient is constant, eval-uated at the average copper temperature .Assume lumped capacitance behavior.

Carbon steel (AISI 1010) shafts of 0.1-m diameter areheat treated in a gas-fired furnace whose gases are at1200 K and provide a convection coefficient of100 W/mK. If the shafts enter the furnace at 300 K,how long must they remain in the furnace to achieve acenterline temperature of 800 K?

A thermal energy storage unit consists of a large rectan-gular channel, which is well insulated on its outer surface and encloses alternating layers of the storagematerial and the flow passage.Each layer of the storage material is an aluminum slabof width W?0.05 m, which is at an initial temperatureof 25C. Consider conditions for which the storage unitis charged by passing a hot gas through the passages,with the gas temperature and the convection coefficientassumed to have constant values of T??600C andh?100 W/m2?K throughout the channel. How longwill it take to achieve 75% of the maximum possibleenergy storage? What is the temperature of the alu-minum at this time?

Small spherical particles of diameter D?50 ?m containa fluorescent material that, when irradiated with whitelight, emits at a wavelength corresponding to the mater-ials temperature. Hence the color of the particle varieswith its temperature. Because the small particles are neu-trally buoyantin liquid water, a researcher wishes to usethem to measure instantaneous local water temperaturesin a turbulent flow by observing their emitted color. If the particles are characterized by a density, specificheat, and thermal conductivity of ??999 kg/m3,k?1.2 W/m?K, and cp?1200 J/kg?K, respectively, determine the time constant of the particles. Hint: Sincethe particles travel with the flow, heat transfer betweenthe particle and the fluid occurs by conduction. Assumelumped capacitance behavior.

A spherical vessel used as a reactor for producingpharmaceuticals has a 5-mm-thick stainless steel wall (k?17 W/m?K) and an inner diameter of Di?1.0 m.During production, the vessel is filled with reactants forwhich ??1100 kg/m3and c?2400 J/kg?K, whileexothermic reactions release energy at a volumetric rateof . As first approximations, the reactantsmay be assumed to be well stirred and the thermalcapacitance of the vessel may be neglected.(a) The exterior surface of the vessel is exposed toambient air (T??25C) for which a convectioncoefficient of h?6 W/m2?K may be assumed. Ifthe initial temperature of the reactants is 25C, what is the temperature of the reactants after 5 h ofprocess time? What is the corresponding tempera-ture at the outer surface of the vessel?(b) Explore the effect of varying the convection coeffi-cient on transient thermal conditions within thereactor

Batch processes are often used in chemical and pharma-ceutical operations to achieve a desired chemical composition for the final product and typically involvea transient heating operation to take the product fromroom temperature to the desired process temperature.Consider a situation for which a chemical of density??l200 kg/m3and specific heat c?2200 J/kg?K occu-pies a volume of V?2.25 m3in an insulated vessel. Thechemical is to be heated from room temperature,Ti?300 K, to a process temperature of T?450 K bypassing saturated steam at Th?500 K through a coiled,thin-walled, 20-mm- diameter tube in the vessel. Steamcondensation within the tube maintains an interior con-vection coefficient of hi?10,000 W/m2?K, while thehighly agitated liquid in the stirred vessel maintains anoutside convection coefficient of ho?2000 W/m2?K. If the chemical is to be heated from 300 to 450 K in60 min, what is the required length Lof the submergedtubing?

An electronic device, such as a power transistor mountedon a finned heat sink, can be modeled as a spatiallyisothermal object with internal heat generation and anexternal convection resistance.(a) Consider such a system of mass M, specific heat c,and surface area As, which is initially in equilib- rium with the environment at T?. Suddenly, theelectronic device is energized such that a constantheat generation (W) occurs. Show that the tem-perature response of the device iswhere ??T?T(?) and T(?) is the steady-statetemperature corresponding to tl?; ?i?Ti?T(?);Ti?initial temperature of device; R?thermalresistance ; and C?thermal capacitance Mc.(b) An electronic device, which generates 60 W ofheat, is mounted on an aluminum heat sink weigh-ing 0.31 kg and reaches a temperature of 100C inambient air at 20C under steady-state conditions.If the device is initially at 20C, what temperaturewill it reach 5 min after the power is switched on?

Molecular electronicsis an emerging field associated withcomputing and data storage utilizing energy transfer at themolecular scale. At this scale, thermal energy is associ-ated exclusively with the vibration of molecular chains.The primary resistance to energy transfer in these pro-posed devices is the contact resistance at metal-moleculeinterfaces. To measure the contact resistance, individualmolecules are self-assembledin a regular pattern onto avery thin gold substrate. The substrate is suddenly heatedby a short pulse of laser irradiation, simultaneously trans-ferring thermal energy to the molecules. The moleculesvibrate rapidly in their hot state, and their vibrationalintensity can be measured by detecting the randomness ofthe electric field produced by the molecule tips, as indi-cated by the dashed, circular lines in the schematic. Molecules that are of density ??180 kg/m3and spe-cific heat cp?3000 J/kg?K have an initial, relaxedlength of L?2 nm. The intensity of the molecularvibration increases exponentially from an initial value ofIito a steady-state value of Iss?Iiwith the time constantassociated with the exponential response being?I?5 ps. Assuming the intensity of the molecularvibration represents temperature on the molecular scaleand that each molecule can be viewed as a cylinder ofinitial length Land cross-sectional area Ac, determine thethermal contact resistance, , at the metalmoleculeinterface.

A plane wall of a furnace is fabricated from plain carbonsteel (k?60 W/m?K, ??7850 kg/m3, c?430 J/kg?K)and is of thickness L?10 mm. To protect it from thecorrosive effects of the furnace combustion gases, onesurface of the wall is coated with a thin ceramic filmthat, for a unit surface area, has a thermal resistance of?0.01 m2?K/W. The opposite surface is well insu-lated from the surroundings. At furnace start-up the wall is at an initial temperatureof Ti?300 K, and combustion gases at T??1300 Kenter the furnace, providing a convection coefficient ofh?25 W/m2?K at the ceramic film. Assuming thefilm to have negligible thermal capacitance, how longwill it take for the inner surface of the steel to achievea temperature of Ts,i?1200 K? What is the tempera-ture Ts,oof the exposed surface of the ceramic film atthis time?

A steel strip of thickness ?12 mm is annealed bypassing it through a large furnace whose walls aremaintained at a temperature Twcorresponding to that ofcombustion gases flowing through the furnace(Tw?T?). The strip, whose density, specific heat, ther-mal conductivity, and emissivity are ??7900 kg/m3,cp?640 J/kg?K, k?30 W/m?K, and ??0.7, respec-tively, is to be heated from 300C to 600C. (a) For a uniform convection coefficient of h?100 W/m2?K and Tw?T??700C, determine thetime required to heat the strip. If the strip is movingat 0.5 m/s, how long must the furnace be?(b) The annealing process may be accelerated (the stripspeed increased) by increasing the environmentaltemperatures. For the furnace length obtained inpart (a), determine the strip speed for Tw?T??850C and Tw?T??1000C. For each set of envi-ronmental temperatures (700, 850, and 1000C),plot the strip temperature as a function of time overthe range 25C?T?600C. Over this range, alsoplot the radiation heat transfer coefficient, hr, as afunction of time.

In a material processing experiment conducted aboardthe space shuttle, a coated niobium sphere of 10-mmdiameter is removed from a furnace at 900C andcooled to a temperature of 300C. Although propertiesof the niobium vary over this temperature range, con-stant values may be assumed to a reasonable approxi-mation, with ??8600 kg/m3, c?290 J/kg?K, and k?63 W/m?K.(a) If cooling is implemented in a large evacuatedchamber whose walls are at 25C, determine thetime required to reach the final temperature if thecoating is polished and has an emissivity of ??0.1.How long would it take if the coating is oxidizedand ??0.6?(b) To reduce the time required for cooling, considera-tion is given to immersion of the sphere in an inert gas stream for which T??25C and h?200 W/m2?K. Neglecting radiation, what is thetime required for cooling?(c) Considering the effect of both radiation and con- vection, what is the time required for cooling if h?200 W/m2?K and ??0.6? Explore the effecton the cooling time of independently varying hand ?.

Plasma spray-coating processes are often used to pro-vide surface protection for materials exposed to hostileenvironments, which induce degradation through fac-tors such as wear, corrosion, or outright thermal failure.Ceramiccoatings are commonly used for this purpose.By injecting ceramic powder through the nozzle(anode) of a plasma torch, the particles are entrained bythe plasma jet, within which they are then acceleratedand heated.During their time-in-flightthe ceramic particles mustbe heated to their melting point and experience com-plete conversion to the liquid state. The coating isformed as the molten droplets impinge (splat) on thesubstrate material and experience rapid solidification.Consider conditions for which spherical alumina(Al2O3) particles of diameter Dp?50?m, density ?p?3970 kg/m3, thermal conductivity kp?10.5 W/m?K,and specific heat cp?1560 J/kg?K are injected into anarc plasma, which is at T??10,000 K and provides acoefficient of h?30,000 W/m2?K for convective heat-ing of the particles. The melting point and latent heat offusion of alumina are Tmp?2318 K and hsf?3577 kJ/kg,respectively.(a) Neglecting radiation, obtain an expression for thetime-in-flight, ti?f, required to heat a particle fromits initial temperature Tito its melting point Tmp,and, once at the melting point, for the particle toexperience complete melting. Evaluate ti?fforTi?300 K and the prescribed heating conditions.(b) Assuming alumina to have an emissivity of ?p?0.4and the particles to exchange radiation with largesurroundings at Tsur?300 K, assess the validity ofneglecting radiation.

The plasma spray-coating process of Problem 5.25 canbe used to produce nanostructuredceramic coatings.Such coatings are characterized by low thermal conduc-tivity, which is desirable in applications where the coat-ing serves to protect the substrate from hot gases suchas in a gas turbine engine. One method to produce ananostructured coating involves spraying sphericalparticles, each of which is composed of agglomeratedAl2O3nanoscale granules. To form the coating, parti-cles of diameter Dp?50 ?m must be partiallymoltenwhen they strike the surface, with the liquid Al2O3providing a means to adhere the ceramic material tothe surface, and the unmelted Al2O3providing themany grain boundaries that give the coating its lowthermal conductivity. The boundaries between individ-ual granules scatter phonons and reduce the thermalconductivity of the ceramic particle to kp?5W/m?K.The density of the porous particle is reduced to??3800 kg/m3. All other properties and conditions areas specified in Problem 5.25.(a) Determine the time-in-flighcorresponding to 30%of the particle mass being melted.(b) Determine the time-in-flighcorresponding to theparticle being 70% melted.(c) If the particle is traveling at a velocity V?35 m/s,determine the standoff distancesbetween the noz-zle and the substrate associated with your answersin parts (a) and (b).

A chip that is of length L?5 mm on a side and thick-ness t?1 mm is encased in a ceramic substrate, and itsexposed surface is convectively cooled by a dielectricliquid for which h?150 W/m2?K and T??20C.In the off-mode the chip is in thermal equilibrium withthe coolant (Ti?T?). When the chip is energized, how- ever, its temperature increases until a new steady stateis established. For purposes of analysis, the energizedchip is characterized by uniform volumetric heatingwith . Assuming an infinite contactresistance between the chip and substrate and negligibleconduction resistance within the chip, determine thesteady-state chip temperature Tf. Following activationof the chip, how long does it take to come within 1C of this temperature? The chip density and specific heat are??2000 kg/m3and c?700 J/kg?K, respectively.

Consider the conditions of Problem 5.27. In addition totreating heat transfer by convection directly from thechip to the coolant, a more realistic analysis wouldaccount for indirect transfer from the chip to the sub-strate and then from the substrate to the coolant. The total thermal resistance associated with this indirectroute includes contributions due to the chipsubstrateinterface (a contact resistance), multidimensionalconduction in the substrate, and convection from thesurface of the substrate to the coolant. If this total thermal resistance is Rt = 200 K/W, what is the steady-state chip temperature Tf? Following activation of the chip, how long does it take to come within 1C of this temperature?

A long wire of diameter D?1 mm is submerged in anoil bath of temperature T??25C. The wire has anelectrical resistance per unit length of R?e?0.01?/m.If a current of I?100 A flows through the wire and theconvection coefficient is h?500 W/m2?K, what is thesteady-state temperature of the wire? From the timethe current is applied, how long does it take for the wireto reach a temperature that is within 1C of the steady-state value? The properties of the wire are ??8000 kg/m3, c?500 J/kg?K, and k?20 W/m?K

Consider the system of Problem 5.1 where the tempera-ture of the plate is spacewise isothermal during thetransient process.(a) Obtain an expression for the temperature of theplate as a function of time T(t) in terms of , T?, h,L, and the plate properties ?and c.(b) Determine the thermal time constant and thesteady-state temperature for a 12-mm-thick plate ofpure copper when T??27C, h?50 W/m2?K,and ?5000 W/m2. Estimate the time required toreach steady-state conditions.(c) For the conditions of part (b), as well as forh?100 and 200 W/m2?K, compute and plot thecorresponding temperature histories of the plate for0?t?2500 s

Shape memory alloys (SMAs) are metals that undergoa change in crystalline structure within a relatively nar-row temperature range. A phase transformation frommartensiteto austenitecan induce relatively largechanges in the overall dimensions of the SMA. Hence,SMAs can be employed as mechanical actuators. Con-sider an SMA rod that is initially Di?2 mm in diame-ter, Li?40 mm long, and at a uniform temperature ofTi?320 K. The specific heat of the SMA varies signif-icantly with changes in the crystalline phase, hence c varies with the temperature of the material. For aparticular SMA, this relationship is well described byc?500 J/kg?K3630 J/kg?Ke(?0.808 K?1|T?336K|). Thedensity and thermal conductivity of the SMA material are??8900 kg/m3and k?23 W/m?K, respectively.The SMA rod is exposed to a hot gas character-ized by T??350 K, h?250 W/m2?K. Plot the rodtemperature versus time for 0?t?60 s for the cases of variable and constant (c?500 J/kg?K) specific heats.Determine the time needed for the rod temperature toexperience 95% of its maximum temperature change.Hint:Neglect the change in the dimensions of the SMArod when calculating the thermal response of the rod.

Before being injected into a furnace, pulverized coal ispreheated by passing it through a cylindrical tubewhose surface is maintained at Tsur = 1000C. The coal pellets are suspended in an airflow and are known to move with a speed of 3 m/s. If the pellets may be approximated as spheres of 1-mm diameter and it maybe assumed that they are heated by radiation transfer from the tube surface, how long must the tube be to heat coal entering at 25C to a temperature of 600C? Is the use of the lumped capacitance method justified?

As noted in Problem 5.3, microwave ovens operate byrapidly aligning and reversing water molecules withinthe food, resulting in volumetric energy generation and,in turn, cooking of the food. When the food is initiallyfrozen, however, the water molecules do not readilyoscillate in response to the microwaves, and the volu-metric generation rates are between one and two ordersof magnitude lower than if the water were in liquidform. (Microwave power that is not absorbed in thefood is reflected back to the microwave generator,where it must be dissipated in the form of heat to pre-vent damage to the generator.)(a) Consider a frozen, 1-kg spherical piece of groundbeef at an initial temperature of Ti??20C placedin a microwave oven with T??30C and h?15 W/m2?K. Determine how long it will take thebeef to reach a uniform temperature of T?0C,with all the water in the form of ice. Assume theproperties of the beef are the same as ice, andassume 3% of the oven power (P?1 kW total) isabsorbed in the food.(b) After all the ice is converted to liquid, determinehow long it will take to heat the beef to Tf?80Cif 95% of the oven power is absorbed in the food.Assume the properties of the beef are the same asliquid water.(c) When thawing food in microwave ovens, one mayobserve that some of the food may still be frozenwhile other parts of the food are overcooked. Explain why this occurs. Explain why mostmicrowave ovens have thaw cycles that are associ-ated with very low oven powers.

A metal sphere of diameter D, which is at a uniformtemperature Ti, is suddenly removed from a furnace andsuspended from a fine wire in a large room with air at auniform temperature T?and the surrounding walls ata temperature Tsur.(a) Neglecting heat transfer by radiation, obtain anexpression for the time required to cool the sphereto some temperature T.(b) Neglecting heat transfer by convection, obtain anexpression for the time required to cool the sphereto the temperature T.(c) How would you go about determining the timerequired for the sphere to cool to the temperature Tif both convection and radiation are of the sameorder of magnitude?(d) Consider an anodized aluminum sphere (?? 0.75)50 mm in diameter, which is at an initial tempera-ture of Ti?800 K. Both the air and surroundingsare at 300 K, and the convection coefficient is10 W/m2?K. For the conditions of parts (a), (b),and (c), determine the time required for the sphereto cool to 400 K. Plot the corresponding tempera-ture histories. Repeat the calculations for a polishedaluminum sphere (??0.1).

A horizontal structure consists of an LA?10-mm-thicklayer of copper and an LB?10-mm-thick layer of alu-minum. The bottom surface of the composite structurereceives a heat flux of q?? ?100 kW/m2, while the topsurface is exposed to convective conditions character-ized by h?40 W/m2?K, T??25C. The initial tem-perature of both materials is Ti,A?Ti,B?25C, and acontact resistance of ?40010?6m2?K/W existsat the interface between the two materials.(a) Determine the times at which the copper and alu-minum each reach a temperature of Tf?90C. Thecopper layer is on the bottom.(b) Repeat part (a) with the copper layer on the top.Hint: Modify Equation 5.15 to include a term associ-ated with heat transfer across the contact resistance.Apply the modified form of Equation 5.15 to each ofthe two slabs. See Comment 3 of Example 5.2

As permanent space stations increase in size, there is anattendant increase in the amount of electrical powerthey dissipate. To keep station compartment tempera-tures from exceeding prescribed limits, it is necessaryto transfer the dissipated heat to space. A novel heatrejection scheme that has been proposed for this purpose is termed a Liquid Droplet Radiator (LDR).The heat is first transferred to a high vacuum oil, whichis then injected into outer space as a stream of smalldroplets. The stream is allowed to traverse a distance L,over which it cools by radiating energy to outer space atabsolute zero temperature. The droplets are then col-lected and routed back to the space station.Consider conditions for which droplets of emissivity??0.95 and diameter D?0.5 mm are injected at a tem-perature of Ti?500 K and a velocity of V?0.1 m/s.Properties of the oil are ??885 kg/m3, c?1900 J/ kg?K,and k?0.145 W/m?K. Assuming each drop to radiate todeep space at Tsur?0 K, determine the distance Lrequired for the droplets to impact the collector at a finaltemperature of Tf?300 K. What is the amount of thermalenergy rejected by each droplet?

Thin film coatings characterized by high resistance toabrasion and fracture may be formed by using microscalecomposite particles in a plasma spraying process. Aspherical particle typically consists of a ceramic core,such as tungsten carbide (WC), and a metallic shell, suchas cobalt (Co). The ceramic provides the thin film coat-ing with its desired hardness at elevated temperatures,while the metal serves to coalesce the particles on thecoated surface and to inhibit crack formation. In theplasma spraying process, the particles are injected into aplasma gas jet that heats them to a temperature above themelting point of the metallic casing and melts the casingbefore the particles impact the surface.Consider spherical particles comprised of a WCcore of diameter Di?16?m, which is encased in a Coshell of outer diameter Do?20 ?m. If the particlesflow in a plasma gas at T??10,000 K and the coeffi- cient associated with convection from the gas to theparticles is h?20,000 W/m2?K, how long does ittake to heat the particles from an initial temperatureofTi?300 K to the melting point of cobalt,Tmp?1770 K? The density and specific heat of WC(the core of the particle) are ?c?16,000 kg/m3and cc?300 J/kg?K, while the corresponding values forCo (the outer shell) are ?s?8900 kg/m3andcs?750 J/kg?K. Once having reached the meltingpoint, how much additional time is required to com-pletely melt the cobalt if its latent heat of fusion ishsf?2.59105J/kg? You may use the lumped capac-itance method of analysis and neglect radiationexchange between the particle and its surroundings.

A long, highly polished aluminum rod of diameterD?35 mm is hung horizontally in a large room. Theinitial rod temperature is Ti?90C, and the room airisT??20C. At time t1?1250 s, the rod temperature isT1?65C, and, at time t2?6700 s, the rod tempera-ture is T2?30C. Determine the values of the con- stants Cand nthat appear in Equation 5.26. Plot the rodtemperature versus time for 0?t?10,000 s. On thesame graph, plot the rod temperature versus time for aconstant value of the convection heat transfer coeffi-cient, evaluated at a rod temperature of .For all cases, evaluate properties at .

Thermal stress testing is a common procedure used toassess the reliability of an electronic package. Typi-cally, thermal stresses are induced in soldered or wiredconnections to reveal mechanisms that could causefailure and must therefore be corrected before the prod-uct is released. As an example of the procedure,consider an array of silicon chips (?ch?2300 kg/m3, cch?710 J/kg?K) joined to an alumina substrate(?sb?4000 kg/m3, csb?770 J/kg?K) by solder balls(?sd?11,000 kg/m3, csd?130 J/kg?K). Each chip ofwidth Lchand thickness tchis joined to a unit substratesection of width Lsband thickness tsbby solder balls ofdiameter D.A thermal stress test begins by subjecting the multichipmodule, which is initially at room temperature, to a hotfluid stream and subsequently cooling the module byexposing it to a cold fluid stream. The process isrepeated for a prescribed number of cycles to assess theintegrity of the soldered connections.(a) As a first approximation, assume that there isnegligible heat transfer between the components(chip/solder/substrate) of the module and thatthe thermal response of each component may bedetermined from a lumped capacitance analysis involving the same convection coefficient h.Assuming no reduction in surface area due to con-tact between a solder ball and the chip or substrate,obtain expressions for the thermal time constant ofeach component. Heat transfer is to all surfaces of achip, but to only the top surface of the substrate.Evaluate the three time constants for Lch?15 mm,tch?2 mm, Lsb?25 mm, tsb?10 mm, D?2 mm,and a value of h?50 W/m2?K, which is character- istic of an airstream. Compute and plot the temper-ature histories of the three components for theheating portion of a cycle, with Ti?20C andT??80C. At what time does each component expe-rience 99% of its maximum possible temperature rise,that is, (T?Ti)/(T??Ti)?0.99? If the maximumstress on a solder ball corresponds to the maximumdifference between its temperature and that of thechip or substrate, when will this maximum occur?(b) To reduce the time required to complete a stresstest, a dielectric liquid could be used in lieu of airto provide a larger convection coefficient ofh?200 W/m2?K. What is the corresponding sav-ings in time for each component to achieve 99% ofits maximum possible temperature rise?

The objective of this problem is to develop thermalmodels for estimating the steady-state temperature andthe transient temperature history of the electrical trans-former shown.The external transformer geometry is approximatelycubical, with a length of 32 mm to a side. The com-bined mass of the iron and copper in the transformer is0.28 kg, and its weighted-average specific heat is400 J/kg?K. The transformer dissipates 4.0 W and isoperating in ambient air at T??20C, with a convec-tion coefficient of 10 W/m2?K. List and justify theassumptions made in your analysis, and discuss limita-tions of the models.(a) Beginning with a properly defined control volume,develop a model for estimating the steady-statetemperature of the transformer, T(?). EvaluateT(?) for the prescribed operating conditions.(b) Develop a model for estimating the thermalresponse (temperature history) of the transformer ifit is initially at a temperature of Ti?T?and poweris suddenly applied. Determine the time requiredfor the transformer to come within 5C of itssteady-state operating temperature.

In thermomechanical data storage, a processing head,consisting of Mheated cantilevers, is used to write dataonto an underlying polymer storage medium. Electricalresistance heaters are microfabricated onto each can-tilever, which continually travel over the surface of themedium. The resistance heaters are turned on and offby controlling electrical current to each cantilever. As acantilever goes through a complete heating and coolingcycle, the underlying polymer is softened, and one bitof data is written in the form of a surface pitin thepolymer. A track of individual data bits (pits), each sep-arated by approximately 50 nm, can be fabricated. Mul-tiple tracks of bits, also separated by approximately50 nm, are subsequently fabricated into the surface ofthe storage medium. Consider a single cantilever that isfabricated primarily of silicon with a mass of5010?18kg and a surface area of 60010?15m2. Thecantilever is initially at Ti?T??300 K, and the heattransfer coefficient between the cantilever and theambient is 200103W/m2?K.(a) Determine the ohmic heating required to raise thecantilever temperature to T?1000 K within aheating time of th?1?s. Hint: See Problem 5.20.(b) Find the time required to cool the cantilever from1000 K to 400 K (tc) and the thermal processingtime required for one complete heating and coolingcycle, tp?thtc.(c) Determine how many bits (N) can be written onto a1mm1 mm polymer storage medium. If M?100cantilevers are ganged onto a single processinghead, determine the total thermal processing timeneeded to write the data.

The melting of water initially at the fusion temperature,Tf?0C, was considered in Example 1.6. Freezing ofwater often occurs at 0C. However, pure liquids that undergo a cooling process can remain in a supercooledliquid state well below their equilibrium freezing tem-perature, Tf, particularly when the liquid is not in con-tact with any solid material. Droplets of liquid water inthe atmosphere have a supercooled freezing tempera-ture, Tf,sc, that can be well correlated to the dropletdiameter by the expression Tf,sc??280.87 ln(Dp)in the diameter range 10?7?Dp?10?2m, where Tf,schas units of degrees Celsius and Dpis expressed in unitsof meters. For a droplet of diameter D?50 ?m andinitial temperature Ti?10C subject to ambient condi-tions of T???40C and h?900 W/m2?K, comparethe time needed to completely solidify the droplet forcase A, when the droplet solidifies at Tf?0C, andcase B, when the droplet starts to freeze at Tf,sc. Sketchthe temperature histories from the initial time to thetime when the droplets are completely solid. Hint:When the droplet reaches Tf,scin case B, rapid solidifi-cation occurs during which the latent energy releasedby the freezing water is absorbed by the remaining liq- uid in the drop. As soon as any ice is formed within thedroplet, the remaining liquid is in contact with a solid(the ice) and the freezing temperature immediatelyshifts from Tf,scto Tf?0C.

Consider the series solution, Equation 5.42, for theplane wall with convection. Calculate midplane(x*?0) and surface (x*?1) temperatures ?* forFo?0.1 and 1, using Bi?0.1, 1, and 10. Consideronly the first four eigenvalues. Based on these results,discuss the validity of the approximate solutions, Equa-tions 5.43 and 5.44

Consider the one-dimensional wall shown in the sketch,which is initially at a uniform temperature Tiand is sud-denly subjected to the convection boundary conditionwith a fluid at T?.For a particular wall, case 1, the temperature at x?L1after t1?100 s is T1(L1, t1)?315C. Another wall,case 2, has different thickness and thermal conditionsas shown.Wall, T(x, 0) = Ti,k,InsulationLxT, h356Chapter 5?Transient ConductionCH005.qxd 2/24/11 12:37 PM Page 356 L?kTiT?hCase (m) (m2/s) (W/m?K) (C)(C) (W/m2?K)1 0.10 15 10?650 300 400 2002 0.40 25 10?6100 30 20 100How long will it take for the second wall to reach 28.5Cat the position x?L2? Use as the basis for analysis, thedimensionless functional dependence for the transienttemperature distribution expressed in Equation 5.41

Copper-coated, epoxy-filled fiberglass circuit boardsare treated by heating a stack of them under highpressure, as shown in the sketch. The purpose of thepressingheating operation is to cure the epoxy thatbonds the fiberglass sheets, imparting stiffness to theboards. The stack, referred to as a book,is comprised of10 boardsand 11 pressing plates,which prevent epoxyfrom flowing between the boards and impart a smoothfinish to the cured boards. In order to perform simpli-fied thermal analyses, it is reasonable to approximatethe book as having an effective thermal conductivity (k)and an effective thermal capacitance (?cp). Calculatethe effective properties if each of the boards andplates has a thickness of 2.36 mm and the followingthermophysical properties: board (b) ?b?1000 kg/m3,cp,b?1500 J/kg?K, kb?0.30 W/m?K; plate (p) ?p?8000 kg/m3, cp,p?480 J/kg?K, kp?12 W/m?K.

Circuit boards are treated by heating a stack of themunder high pressure, as illustrated in Problem 5.45. Theplatens at the top and bottom of the stack are main-tained at a uniform temperature by a circulating fluid.The purpose of the pressingheating operation is tocure the epoxy, which bonds the fiberglass sheets, andimpart stiffness to the boards. The cure condition isachieved when the epoxy has been maintained at orabove 170C for at least 5 min. The effective thermo-physical properties of the stack or book(boardsand metal pressing plates) are k?0.613 W/m?K and?cp?2.73106J/m3?K.(a) If the book is initially at 15C and, following appli-cation of pressure, the platens are suddenly brought to a uniform temperature of 190C, calculate theelapsed time terequired for the midplane of thebook to reach the cure temperature of 170C.(b) If, at this instant of time, t?te, the platen tempera-ture were reduced suddenly to 15C, how muchenergy would have to be removed from the book bythe coolant circulating in the platen, in order toreturn the stack to its initial uniform temperature?

A constant-property, one-dimensional plane slab ofwidth 2L, initially at a uniform temperature, is heatedconvectively with Bi?1.(a) At a dimensionless time of Fo1, heating is suddenlystopped, and the slab of material is quickly coveredwith insulation. Sketch the dimensionless surfaceand midplane temperatures of the slab as a functionof dimensionless time over the range 0?Fo??.By changing the duration of heating to Fo2, thesteady-statemidplane temperature can be set equalto the midplane temperature at Fo1. Is the value ofFo2equal to, greater than, or less than Fo1? Hint:Assume both Fo1and Fo2are greater than 0.2.(b) Letting Fo2?Fo1 ?Fo, derive an analyticalexpression for ?Fo, and evaluate ?Fofor the con-ditions of part (a).(c) Evaluate ?Fofor Bi?0.01, 0.1, 10, 100, and ?when Fo1and Fo2are both greater than 0.2

Referring to the semiconductor processing tool ofProblem 5.13, it is desired at some point in the manu-facturing cycle to cool the chuck, which is made ofaluminum alloy 2024. The proposed cooling schemepasses air at 20C between the air-supply head and thechuck surface.(a) If the chuck is initially at a uniform temperature of100C, calculate the time required for its lowersurface to reach 25C, assuming a uniform convec-tion coefficient of 50 W/m2?K at the headchuckinterface.(b) Generate a plot of the time-to-cool as a function of the convection coefficient for the range 10?h?2000 W/m2?K. If the lower limit repre-sents a free convection condition without any headpresent, comment on the effectiveness of the headdesign as a method for cooling the chuck.

Annealing is a process by which steel is reheated andthen cooled to make it less brittle. Consider the reheatstage for a 100-mm-thick steel plate (??7830 kg/m3,c?550 J/kg?K, k?48 W/m?K), which is initially at auniform temperature of Ti?200C and is to be heatedto a minimum temperature of 550C. Heating iseffected in a gas-fired furnace, where products of com-bustion at T??800C maintain a convection coeffi-cient of h?250 W/m2?K on both surfaces of the plate.How long should the plate be left in the furnace?

Consider an acrylic sheet of thickness L?5 mm that isused to coat a hot, isothermal metal substrate at Th?300C. The properties of the acrylic are ??1990 kg/m3, c?1470 J/kg?K, and k?0.21 W/m?K.Neglecting the thermal contact resistance between theacrylic and the metal substrate, determine how long itwill take for the insulated back side of the acrylic toreach its softening temperature, Tsoft?90C. The initialacrylic temperature is Ti2? 0C.

The 150-mm-thick wall of a gas-fired furnace is con-structed of fireclay brick (k?1.5 W/m?K, ??2600kg/m3, cp?1000 J/kg?K) and is well insulated at itsouter surface. The wall is at a uniform initial tempera-ture of 20C, when the burners are fired and the innersurface is exposed to products of combustion for whichT??950C and h?100 W/m2?K.(a) How long does it take for the outer surface of thewall to reach a temperature of 750C?(b) Plot the temperature distribution in the wall at theforegoing time, as well as at several intermediatetimes.

Steel is sequentially heated and cooled (annealed) torelieve stresses and to make it less brittle. Consider a100-mm-thick plate (k?45 W/m?K, ??7800 kg/m3,cp?500 J/kg?K) that is initially at a uniform tempera-ture of 300C and is heated (on both sides) in a gas-fired furnace for which T??700C andh?500 W/m2?K. How long will it take for a minimumtemperature of 550C to be reached in the plate?

Stone mix concrete slabs are used to absorb thermalenergy from flowing air that is carried from a large con-centrating solar collector. The slabs are heated duringthe day and release their heat to cooler air at night. If thedaytime airflow is characterized by a temperature andconvection heat transfer coefficient of T??200C and h?35 W/m2?K, respectively, determine the slab thickness 2Lrequired to transfer a total amount ofenergy such that Q/Qo?0.90 over a t?8-h period.The initial concrete temperature is Ti4? 0C.

A plate of thickness 2L?25 mm at a temperature of600C is removed from a hot pressing operation andmust be cooled rapidly to achieve the required physicalproperties. The process engineer plans to use air jets tocontrol the rate of cooling, but she is uncertain whetherit is necessary to cool both sides (case 1) or only oneside (case 2) of the plate. The concern is not just for thetime-to-cool, but also for the maximum temperaturedifference within the plate. If this temperature differ-ence is too large, the plate can experience significantwarping.The air supply is at 25C, and the convection coefficienton the surface is 400 W/m2?K. The thermophysical prop-erties of the plate are ??3000 kg/m3, c?750 J/kg?K,and k?15 W/m?K.(a) Using the IHTsoftware, calculate and plot on onegraph the temperature histories for cases 1 and 2for a 500-s cooling period. Compare the timesrequired for the maximum temperature in the plateto reach 100C. Assume no heat loss from theunexposed surface of case 2.(b) For both cases, calculate and plot on one graph thevariation with time of the maximum temperaturedifference in the plate. Comment on the relativemagnitudes of the temperature gradients within theplate as a function of time.

During transient operation, the steel nozzle of a rocketengine must not exceed a maximum allowable operating temperature of 1500 K when exposed to combustiongases characterized by a temperature of 2300 K and aconvection coefficient of 5000 W/m2?K. To extend theduration of engine operation, it is proposed that aceramic thermal barrier coating(k?10 W/m?K,??610?6m2/s) be applied to the interior surface ofthe nozzle.(a) If the ceramic coating is 10 mm thick and at an ini-tial temperature of 300 K, obtain a conservativeestimate of the maximum allowable duration ofengine operation. The nozzle radius is much largerthan the combined wall and coating thickness.(b) Compute and plot the inner and outer surface tem-peratures of the coating as a function of time for0?t?150 s. Repeat the calculations for a coatingthickness of 40 mm

Two plates of the same material and thickness Lare atdifferent initial temperatures Ti,1and Ti,2, where Ti,2?Ti,1.Their faces are suddenly brought into contact. Theexternal surfaces of the two plates are insulated.(a) Let a dimensionless temperature be defined as T* (Fo)?(T Ti,1)/(Ti,2 Ti,1). Neglecting the ther-mal contact resistance at the interface between theplates, what are the steady-state dimensionless tem-peratures of each of the two plates, and ?What is the dimensionless interface temperatureat any time?(b) An effective overall heat transfer coefficientbetween the two plates can be defined based on theinstantaneous, spatially averaged dimensionlessplate temperatures, . Notingthat a dimensionless heat transfer rate to or fromeither of the two plates may be expressed as, determine an expression for for Fo?0.2

In a tempering process, glass plate, which is initially ata uniform temperature Ti, is cooled by suddenly reduc-ing the temperature of both surfaces to Ts. The plate is20 mm thick, and the glass has a thermal diffusivity of610?7m2/s.(a) How long will it take for the midplane temperatureto achieve 50% of its maximum possible tempera-ture reduction?(b) If (Ti?Ts)?300C, what is the maximum tem- perature gradient in the glass at the time calculatedin part (a)?

The strength and stability of tires may be enhanced byheating both sides of the rubber (k?0.14 W/m?K,??6.3510?8m2/s) in a steam chamber for whichT??200C. In the heating process, a 20-mm-thick rubber wall (assumed to be untreaded) is taken from aninitial temperature of 25C to a midplane temperatureof 150C.(a) If steam flow over the tire surfaces maintains aconvection coefficient of h?200 W/m2?K, howlong will it take to achieve the desired midplanetemperature?(b) To accelerate the heating process, it is recom-mended that the steam flow be made sufficientlyvigorous to maintain the tire surfaces at 200Cthroughout the process. Compute and plot the mid-plane and surface temperatures for this case, aswell as for the conditions of part (a)

A plastic coating is applied to wood panels by firstdepositing molten polymer on a panel and then cool-ing the surface of the polymer by subjecting it to airflow at 25C. As first approximations, the heat ofreaction associated with solidification of the polymermay be neglected and the polymer/wood interfacemay be assumed to be adiabatic.If the thickness of the coating is L?2 mm and it has aninitial uniform temperature of Ti?200C, how longwill it take for the surface to achieve a safe-to- touchtemperature of 42C if the convection coefficient ish?200 W/m2?K? What is the corresponding value ofthe interface temperature? The thermal conductivityand diffusivity of the plastic are k?0.25 W/m?K and1 ?? .2010?7m2/s, respectively

A long rod of 60-mm diameter and thermophysicalproperties ??8000 kg/m3, c?500 J/kg?K, andk?50 W/m?K is initially at a uniform temperature andis heated in a forced convection furnace maintained at750 K. The convection coefficient is estimated to be1000 W/m2?K.(a) What is the centerline temperature of the rod whenthe surface temperature is 550 K? (b) In a heat-treating process, the centerline tempera-ture of the rod must be increased from Ti?300 Kto T?500 K. Compute and plot the centerline temperature histories for h?100, 500, and1000 W/m2?K. In each case the calculation may beterminated when T5 ? 00 K.

A long cylinder of 30-mm diameter, initially at a uni-form temperature of 1000 K, is suddenly quenched in alarge, constant-temperature oil bath at 350 K. The cylin-der properties are k?1.7 W/m?K, c?1600 J/kg?K,and ??400 kg/m3, while the convection coefficient is50 W/m2?K.(a) Calculate the time required for the surface of thecylinder to reach 500 K.(b) Compute and plot the surface temperature historyfor 0?t?300 s. If the oil were agitated, provid-ing a convection coefficient of 250 W/m2?K, howwould the temperature history change?

Work Problem 5.47 for a cylinder of radius ro and length L = 20 ro.

A long pyroceram rod of diameter 20 mm is clad with avery thin metallic tube for mechanical protection. Thebonding between the rod and the tube has a thermalcontact resistance of .(a) If the rod is initially at a uniform temperature of900 K and is suddenly cooled by exposure toan airstream for which T??300 K and h?100 W/m2?K, at what time will the centerline reach600 K?(b) Cooling may be accelerated by increasing the air-speed and hence the convection coefficient. Forvalues of h?100, 500, and 1000 W/m2?K, com-pute and plot the centerline and surface tempera-tures of the pyroceram as a function of time for0?t?300 s. Comment on the implications ofachieving enhanced cooling solely by increasing h.

A long rod 40 mm in diameter, fabricated fromsapphire (aluminum oxide) and initially at a uniformtemperature of 800 K, is suddenly cooled by a fluid at 300 K having a heat transfer coefficient of1600 W/m2?K. After 35 s, the rod is wrapped in insula-tion and experiences no heat losses. What will be thetemperature of the rod after a long period of time?

A cylindrical stone mix concrete beam of diameter D?0.5 m initially at Ti?20C is exposed to hot gases at T??500C. The convection coefficient is h?10 W/m2?K.(a) Determine the centerline temperature of the beamafter an exposure time of t?6h.(b) Determine the centerline temperature of a secondbeam that is of the same size and exposed to the sameconditions as in part (a) but fabricated of lightweightaggregate concrete with density ??1495 kg/m3,thermal conductivity k?0.789 W/m?K, and specificheat cp?880 J/kg?K

A long plastic rod of 30-mm diameter (k?0.3 W/m?Kand ?cp?1040 kJ/m3?K) is uniformly heated in anoven as preparation for a pressing operation. For bestresults, the temperature in the rod should not be lessthan 200C. To what uniform temperature should therod be heated in the oven if, for the worst case, the rodsits on a conveyor for 3 min while exposed to convec-tion cooling with ambient air at 25C and with a con-vection coefficient of 8 W/m2?K? A further conditionfor good results is a maximumminimum temperaturedifference of less than 10C. Is this condition satisfied?If not, what could you do to satisfy it?

As part of a heat treatment process, cylindrical, 304 stainless steel rods of 100-mm diameter are cooled froman initial temperature of 500C by suspending them inan oil bath at 30C. If a convection coefficient of 500 W/mK is maintained by circulation of the oil,how long does it take for the centerline of a rod to reacha temperature of 50C, at which point it is withdrawnfrom the bath? If 10 rods of length L = 1 m are processed per hour, what is the nominal rate at whichenergy must be extracted from the bath (the coolingload)?

In a manufacturing process, long rods of differentdiameters are at a uniform temperature of 400C in acuring oven, from which they are removed and cooledby forced convection in air at 25C. One of the lineoperators has observed that it takes 280 s for a 40-mm-diameter rod to cool to a safe-to- handletemperature of60C. For an equivalent convection coefficient, howlong will it take for an 80-mm- diameter rod to cool tothe same temperature? The thermophysicalpropertiesof the rod are ??2500 kg/m3, c?900 J/kg?K, andk?15 W/m?K. Comment on your result. Did youanticipate this outcome?

The density and specific heat of a particular materialare known (??l200 kg/m3, cp?1250 J/kg?K), but itsthermal conductivity is unknown. To determine thethermal conductivity, a long cylindrical specimen ofdiameter D?40 mm is machined, and a thermocouple is inserted through a small hole drilled along thecenterline.The thermal conductivity is determined by performingan experiment in which the specimen is heated to a uni-form temperature of Ti?100C and then cooled bypassing air at T??25C in cross flow over the cylin-der. For the prescribed air velocity, the convectioncoefficient is h?55 W/m2?K.(a) If a centerline temperature of T(0, t)?40C isrecorded after t?1136 s of cooling, verify thatthe material has a thermal conductivity ofk?0.30 W/m?K.(b) For air in cross flow over the cylinder, the pre- scribed value of h?55 W/m2?K corresponds to avelocity of V?6.8 m/s. If h?CV0.618, where theconstant Chas units of W?s0.618/m2.618?K, how doesthe centerline temperature at t?1136 s vary withvelocity for 3?V?20 m/s? Determine the center-line temperature histories for 0?t?1500 s andvelocities of 3, 10, and 20 m/s.

In Section 5.2 we noted that the value of the Biot number significantly influences the nature of the temper-ature distribution in a solid during a transient conductionprocess. Reinforce your understanding of this importantconcept by using the IHTmodel for one-dimensionaltransient conduction to determine radial temperaturedistributions in a 30-mm-diameter, stainless steel rod(k?15 W/m?K, ??8000 kg/m3, cp?475 J/kg?K), asit is cooled from an initial uniform temperature of 325Cby a fluid at 25C. For the following values of the con-vection coefficient and the designated times, determinethe radial temperature distribution: h?100 W/m2?K(t?0, 100, 500 s); h?1000 W/m2?K(t?0, 10, 50 s);h?5000 W/m2?K(t?0, 1, 5, 25 s). Prepare a separategraph for each convection coefficient, on which tempera- ture is plotted as a function of dimensionless radius at thedesignated times.

In heat treating to harden steel ball bearings(c?500 J/kg?K, ??7800 kg/m3, k?50 W/m?K), it is desirable to increase the surface temperature for ashort time without significantly warming the interior ofthe ball. This type of heating can be accomplished bysudden immersion of the ball in a molten salt bath withT??1300 K and h?5000 W/m2?K. Assume that anylocation within the ball whose temperature exceeds1000 K will be hardened. Estimate the time required toharden the outer millimeter of a ball of diameter20 mm, if its initial temperature is 300 K.

A cold air chamber is proposed for quenching steel ballbearings of diameter D?0.2 m and initial temperatureTi?400C. Air in the chamber is maintained at?15Cby a refrigeration system, and the steel balls passthrough the chamber on a conveyor belt. Optimumbearing production requires that 70% of the initial ther-mal energy content of the ball above?15C beremoved. Radiation effects may be neglected, and theconvection heat transfer coefficient within the chamberis 1000 W/m2?K. Estimate the residence time of theballs within the chamber, and recommend a drive veloc-ity of the conveyor. The following properties may beused for the steel: k?50 W/m?K, ??210?5m2/s,and c?450 J/kg?K

A soda lime glass sphere of diameter D1 = 25 mm isencased in a bakelite spherical shell of thickness L = 10 mm. The composite sphere is initially at a uniformtemperature, Ti = 40C, and is exposed to a fluid at T = 10C with h = 30 W/mK. Determine the center temperature of the glass at t = 200 s. Neglect the thermal contact resistance at the interface between the two materials.

Stainless steel (AISI 304) ball bearings, which have uni-formly been heated to 850C, are hardened by quenching them in an oil bath that is maintained at 40C. Theball diameter is 20 mm, and the convection coefficientassociated with the oil bath is 1000 W/mK. (a) If quenching is to occur until the surface temperature of the balls reaches 100C, how long must theballs be kept in the oil? What is the center temperature at the conclusion of the cooling period?(b) If 10,000 balls are to be quenched per hour, what isthe rate at which energy must be removed by the oilbath cooling system in order to maintain its temper-ature at 40C?

A sphere 30 mm in diameter initially at 800 K isquenched in a large bath having a constant temperatureof 320 K with a convection heat transfer coefficient of75 W/m2?K. The thermophysical properties of thesphere material are: ??400 kg/m3, c?1600 J/kg?K,and k?1.7 W/m?K.(a) Show, in a qualitative manner on T tcoordinates,the temperatures at the center and at the surface of thesphere as a function of time.(b) Calculate the time required for the surface of thesphere to reach 415 K.(c) Determine the heat flux (W/m2) at the outer surfaceof the sphere at the time determined in part (b).(d) Determine the energy (J) that has been lost by thesphere during the process of cooling to the surfacetemperature of 415 K.(e) At the time determined by part (b), the sphere isquickly removed from the bath and covered withperfect insulation, such that there is no heat lossfrom the surface of the sphere. What will be thetemperature of the sphere after a long period oftime has elapsed?(f ) Compute and plot the center and surface tempera-ture histories over the period 0?t?150 s. Whateffect does an increase in the convection coefficientto h?200 W/m2?K have on the foregoing temper-ature histories? For h?75 and 200 W/m2?K, com-pute and plot the surface heat flux as a function oftime for 0?t?150 s

Work Problem 5.47 for the case of a sphere of radius ro.

Spheres A and B are initially at 800 K, and they aresimultaneously quenched in large constant temperaturebaths, each having a temperature of 320 K. The follow-ing parameters are associated with each of the spheresand their cooling processes.Sphere A Sphere BDiameter (mm)300 30Density (kg/m3)1600 400Specific heat (kJ/kg?K)0.400 1.60Thermal conductivity (W/m?K) 170 1.70Convection coefficient (W/m2?K) 5 50(a) Show in a qualitative manner, on T tcoordinates,the temperatures at the center and at the surface foreach sphere as a function of time. Briefly explainthe reasoning by which you determine the relativepositions of the curves.(b) Calculate the time required for the surface of eachsphere to reach 415 K (c) Determine the energy that has been gained by eachof the baths during the process of the spheres cool-ing to 415 K

Spheres of 40-mm diameter heated to a uniformtemperature of 400C are suddenly removed from theoven and placed in a forced-air bath operating at 25Cwith a convection coefficient of 300 W/m2?K on thesphere surfaces. The thermophysical properties of thesphere material are ??3000 kg/m3, c?850 J/kg?K,and k?15 W/m?K.(a) How long must the spheres remain in the air bathfor 80% of the thermal energy to be removed?(b) The spheres are then placed in a packing carton thatprevents further heat transfer to the environment.What uniform temperature will the spheres eventu-ally reach?

To determine which parts of a spiders brain are trig-gered into neural activity in response to various opticalstimuli, researchers at the University of MassachusettsAmherst desire to examine the brain as it is shown images that might evoke emotions such as fear or hunger. Consider a spider at Ti = 20C that is shown afrightful scene and is then immediately immersed inliquid nitrogen at T = 77 K. The brain is subsequently dissected in its frozen state and analyzed to determine which parts of the brain reacted to the stimulus. Using your knowledge of heat transfer, determine how much time elapses before the spiders brain begins to freeze. Assume the brain is a sphere of diameter Db = 1 mm,centrally located in the spiders cephalothorax, whichmay be approximated as a spherical shell of diameter Dc = 3 mm. The brain and cephalothorax propertiescorrespond to those of liquid water. Neglect the effectsof the latent heat of fusion and assume the heat transfer coefficient is h = 100 W/mK.

Consider the packed bed operating conditions ofProblem 5.12, but with Pyrex (??2225 kg/m3,c?835 J/kg?K, k?1.4 W/m?K) used instead of alu-minum. How long does it take a sphere near the inlet ofthe system to accumulate 90% of the maximum possi-ble thermal energy? What is the corresponding temper-ature at the center of the sphere?

The convection coefficient for flow over a solid spheremay be determined by submerging the sphere, which isinitially at 25C, into the flow, which is at 75C, andmeasuring its surface temperature at some time duringthe transient heating process.(a) If the sphere has a diameter of 0.1 m, a thermal con- ductivity of 15 W/m?K, and a thermal diffusivity of10?5m2/s, at what time will a surface temperature of 60C be recorded if the convection coefficient is300 W/m2?K?(b) Assess the effect of thermal diffusivity on the ther-mal response of the material by computing centerand surface temperature histories for ??10?6,10?5, and 10?4m2/s. Plot your results for the period0?t?300 s. In a similar manner, assess theeffect of thermal conductivity by considering val-ues of k1 ? .5, 15, and 150 W/m?K

Consider the sphere of Example 5.6, which is initiallyat a uniform temperature when it is suddenly removedfrom the furnace and subjected to a two-step coolingprocess. Use the Transient Conduction, Spheremodelof IHTto obtain the following solutions.(a) For step 1, calculate the time required for the centertemperature to reach T(0, t)?335C, while cool-ing in air at 20C with a convection coefficient of10 W/m2?K. What is the Biot number for this cool-ing process? Do you expect radial temperature gra- dients to be appreciable? Compare your results tothose of the example.(b) For step 2, calculate the time required for the centertemperature to reach T(0, t)?50C, while coolingin a water bath at 20C with a convection coeffi-cient of 6000 W/m2?K.(c) For the step 2 cooling process, calculate and plotthe temperature histories, T(r, t), for the center andsurface of the sphere. Identify and explain key fea- tures of the histories. When do you expect the tem-perature gradients in the sphere to be the largest?

Two large blocks of different materials, such as copper and concrete, have been sitting in a room (23C) for avery long time. Which of the two blocks, if either, will feel colder to the touch? Assume the blocks to be semi-infinite solids and your hand to be at a temperature of 37C.

A plane wall of thickness 0.6 m (L?0.3 m) is made ofsteel (k?30 W/m?K, ??7900 kg/m3, c?640 J/kg?K).It is initially at a uniform temperature and is thenexposed to air on both surfaces. Consider two differentconvectionconditions: natural convection, characterizedby h?10 W/m2?K, and forced convection, withh?100 W/m2?K. You are to calculate the surface tem-perature at three different timest?2.5 min, 25 min,and 250 minfor a total of six different cases.(a) For each of these six cases, calculate the nondimen-sional surface temperature, ?*s?(Ts?T?)/(Ti?T?), using four different methods: exact solution, first-term-of-the-series solution, lumped capacitance, andsemi-infinite solid. Present your results in a table.(b) Briefly explain the conditions for which (i) thefirst-term solution is a good approximation tothe exact solution, (ii) the lumped capacitance solu-tion is a good approximation, (iii) the semi-infinitesolid solution is a good approximation.

Asphalt pavement may achieve temperatures as high as 50C on a hot summer day. Assume that such a temperature exists throughout the pavement, when suddenly arainstorm reduces the surface temperature to 20C. Cal-culate the total amount of energy (J/m2) that will betransferred from the asphalt over a 30-min period in which the surface is maintained at 20C

A thick steel slab (??7800 kg/m3, c?480 J/kg?K,k?50 W/m?K) is initially at 300C and is cooled bywater jets impinging on one of its surfaces. The temper-ature of the water is 25C, and the jets maintain anextremely large, approximately uniform convectioncoefficient at the surface. Assuming that the surface ismaintained at the temperature of the water throughoutthe cooling, how long will it take for the temperature toreach 50C at a distance of 25 mm from the surface?

A tile-iron consists of a massive plate maintained at150C by an embedded electrical heater. The iron isplaced in contact with a tile to soften the adhesive,allowing the tile to be easily lifted from the subflooring.The adhesive will soften sufficiently if heated above50C for at least 2 min, but its temperature should notexceed 120C to avoid deterioration of the adhesive.Assume the tile and subfloor to have an initial tempera-ture of 25C and to have equivalent thermophysicalproperties of k?0.15 W/m?K and ?cp?1.5106J/m3?K.(a) How long will it take a worker using the tile-iron tolift a tile? Will the adhesive temperature exceed120C?(b) If the tile-iron has a square surface area 254 mm tothe side, how much energy has been removed fromit during the time it has taken to lift the tile?

A simple procedure for measuring surface convectionheat transfer coefficients involves coating the surfacewith a thin layer of material having a precise meltingpoint temperature. The surface is then heated and, bydetermining the time required for melting to occur, the convection coefficient is determined. The followingexperimental arrangement uses the procedure to deter-mine the convection coefficient for gas flow normal to asurface. Specifically, a long copper rod is encased ina super insulator of very low thermal conductivity, and avery thin coating is applied to its exposed surface.If the rod is initially at 25C and gas flow forwhich h?200 W/m2?K and T??300C is initiated,what is the melting point temperature of the coating ifmelting is observed to occur at t?400 s?

An insurance company has hired you as a consultant toimprove their understanding of burn injuries. They areespecially interested in injuries induced when a portionof a workers body comes into contact with machinerythat is at elevated temperatures in the range of 50 to100C. Their medical consultant informs them that irre-versible thermal injury (cell death) will occur in anyliving tissue that is maintained at T?48C for a dura-tion ?t?10 s. They want information concerning theextent of irreversible tissue damage (as measured bydistance from the skin surface) as a function of themachinery temperature and the time during which con-tact is made between the skin and the machinery.Assume that living tissue has a normal temperature of37C, is isotropic, and has constant properties equiva- lent to those of liquid water.(a) To assess the seriousness of the problem, computelocations in the tissue at which the temperature willreach 48C after 10 s of exposure to machinery at50C and 100C.(b) For a machinery temperature of 100C and0?t?30 s, compute and plot temperature historiesat tissue locations of 0.5, 1, and 2 mm from the skin.

A procedure for determining the thermal conductivityof a solid material involves embedding a thermocouple in a thick slab of the solid and measuring the responseto a prescribed change in temperature at one surface.Consider an arrangement for which the thermocoupleis embedded 10 mm from a surface that is suddenly brought to a temperature of 100C by exposure to boil-ing water. If the initial temperature of the slab was30C and the thermocouple measures a temperature of 65C, 2 min after the surface is brought to 100C, whatis its thermal conductivity? The density and specificheat of the solid are known to be 2200 kg/m and 700 J/kgK.

A very thick slab with thermal diffusivity 5.610?6m2/s and thermal conductivity 20 W/m?K is ini-tially at a uniform temperature of 325C. Suddenly, thesurface is exposed to a coolant at 15C for which theconvection heat transfer coefficient is 100 W/m2?K.(a) Determine temperatures at the surface and at adepth of 45 mm after 3 min have elapsed.(b) Compute and plot temperature histories (0 ?t?300 s) at x?0 and x?45 mm for the followingparametric variations: (i) ??5.6 10?7, 5.6 10?6, and 5.610?5m2/s; and (ii) k?2, 20, and200 W/m?K.

A thick oak wall, initially at 25C, is suddenly exposedto combustion products for which T??800C andh?20 W/m2?K.(a) Determine the time of exposure required for thesurface to reach the ignition temperature of 400C.(b) Plot the temperature distribution T(x) in themedium at t?325 s. The distribution shouldextend to a location for which T?25C.

Standards for firewalls may be based on their thermalresponse to a prescribed radiant heat flux. Consider a0.25-m-thick concrete wall (??2300 kg/m3,c?880 J/kg?K, k?1.4 W/m?K), which is at an initialtemperature of Ti?25C and irradiated at one surfaceby lamps that provide a uniform heat flux of. The absorptivity of the surface to theirradiation is ?s?1.0. If building code requirementsdictate that the temperatures of the irradiated and backsurfaces must not exceed 325C and 25C, respec-tively, after 30 min of heating, will the requirementsbe met?

It is well known that, although two materials are at thesame temperature, one may feel cooler to the touch thanthe other. Consider thick plates of copper and glass,each at an initial temperature of 300 K. Assuming yourfinger to be at an initial temperature of 310 K and tohave thermophysical properties of ??1000 kg/m3,c?4180 J/kg?K, and k?0.625 W/m?K, determinewhether the copper or the glass will feel cooler tothe touch

Two stainless steel plates (??8000 kg/m3, c?500J/kg?K, k?15 W/m?K), each 20 mm thick and insulated on one surface, are initially at 400 and 300 K when they are pressed together at their uninsulated sur- faces. What is the temperature of the insulated surfaceof the hot plate after 1 min has elapsed?

Special coatings are often formed by depositing thinlayers of a molten material on a solid substrate. Solidifi-cation begins at the substrate surface and proceeds untilthe thickness Sof the solid layer becomes equal to thethickness of the deposit.(a) Consider conditions for which molten material atits fusion temperature Tfis deposited on a largesubstrate that is at an initial uniform temperatureTi. With S?0 at t?0, develop an expression forestimating the time tdrequired to completelysolidify the deposit if it remains at Tfthroughoutthe solidification process. Express your result interms of the substrate thermal conductivityand thermal diffusivity (ks, ?s), the density andlatent heat of fusion of the deposit (?, hsf), thedeposit thickness , and the relevant temperatures(Tf, Ti).(b) The plasma spray deposition process of Problem5.25 is used to apply a thin (?2 mm) aluminacoating on a thick tungsten substrate. The substratehas a uniform initial temperature of Ti?300 K,and its thermal conductivity and thermal diffusivitymay be approximated as ks?120 W/m?K and?s?4.010?5m2/s, respectively. The densityand latent heat of fusion of the alumina are??3970 kg/m3and hsf?3577 kJ/kg, respectively,and the alumina solidifies at its fusion temperature(Tf?2318 K). Assuming that the molten layer isinstantaneously deposited on the substrate, estimatethe time required for the deposit to solidify

When a molten metal is cast in a mold that is apoor conductor, the dominant resistance to heat flowis within the mold wall. Consider conditions forwhich a liquid metal is solidifying in a thick-walledmold of thermal conductivity kwand thermal diffusiv-ity ?w. The density and latent heat of fusion of themetal are designated as ?and hsf, respectively, and inboth its molten and solid states, the thermal conduc-tivity of the metal is very much larger than that ofthe mold. Just before the start of solidification (S?0), the moldwall is everywhere at an initial uniform temperature Tiand the molten metal is everywhere at its fusion (melt-ing point) temperature of Tf. Following the start ofsolidification, there is conduction heat transfer into themold wall and the thickness of the solidified metal Sincreases with time t.(a) Sketch the one-dimensional temperature distribu-tion, T(x), in the mold wall and the metal at t?0 and at two subsequent times during thesolidification. Clearly indicate any underlyingassumptions.(b) Obtain a relation for the variation of the solid layerthickness Swith time t, expressing your result interms of appropriate parameters of the system.

Joints of high quality can be formed by friction welding.Consider the friction welding of two 40-mm- diameterInconel rods. The bottom rod is stationary, while thetop rod is forced into a back-and-forth linear motioncharacterized by an instantaneous horizontal dis-placement, d(t)?acos(?t) where a?2 mm and??1000 rad/s. The coefficient of sliding frictionbetween the two pieces is ??0.3. Determine thecompressive force that must be applied to heat the jointto the Inconel melting point within t?3 s, startingfrom an initial temperature of 20C. Hint: The fre-quency of the motion and resulting heat rate are veryhigh. The temperature response can be approximatedas if the heating rate were constant in time, equal to itsaverage value.

rewritable optical disc (DVD) is formed by sand-wiching a 15-nm-thick binary compound storage mater-ial between two 1-mm-thick polycarbonate sheets. Dataare writtento the opaque storage medium by irradiatingit from below with a relatively high-powered laserbeam of diameter 0.4 ?m and power 1 mW, resulting inrapid heating of the compound material (the polycar-bonate is transparent to the laser irradiation). If thetemperature of the storage medium exceeds 900 K, anoncrystalline, amorphous material forms at the heatedspot when the laser irradiation is curtailed and the spotis allowed to cool rapidly. The resulting spots of amor-phous material have a different reflectivity from the sur-rounding crystalline material, so they can subsequentlybe readby irradiating them with a second, low-powerlaser and detecting the changes in laser radiation trans-mitted through the entire DVD thickness. Determine theirradiation (write) time needed to raise the storagemedium temperature from an initial value of 300 K to1000 K. The absorptivity of the storage medium is 0.8.The polycarbonate properties are ??1200 kg/m3,k?0.21 W/m?K, and cp?1260 J/kg?K

Ground source heat pumps operate by using the soil,rather than ambient air, as the heat source (or sink) forheating (or cooling) a building. A liquid transfersenergy from (to) the soil by way of buried plastic tub-ing. The tubing is at a depth for which annual varia-tions in the temperature of the soil are much less thanthose of the ambient air. For example, at a locationsuch as South Bend, Indiana, deep-ground tempera-tures may remain at approximately 11C, while annualexcursions in the ambient air temperature may rangefrom 25C to37C. Consider the tubing to be laidout in a closely spacedserpentine arrangement. To what depth should the tubing be buried so that thesoil can be viewed as an infinite medium at constanttemperature over a 12-month period? Account for theperiodic cooling (heating) of the soil due to bothannual changes in ambient conditions and variations inheat pump operation from the winter heating to thesummer cooling mode.

To enable cooking a wider range of foods in microwaveovens, thin, metallic packaging materials have beendeveloped that will readily absorb microwave energy.As the packaging material is heated by the microwaves,conduction simultaneously occurs from the hot packag-ing material to the cold food. Consider the sphericalpiece of frozen ground beef of Problem 5.33 that is nowwrapped in the thin microwave-absorbing packagingmaterial. Determine the time needed for the beef that isimmediately adjacent to the packaging material to reachT?0C if 50% of the oven power (P?1 kW total) isabsorbed in the packaging material.

Derive an expression for the ratio of the total energytransferred from the isothermal surface of an infinitecylinder to the interior of the cylinder, Q/Qo, that isvalid for Fo?0.2. Express your results in terms ofthe Fourier number Fo.

The structural components of modern aircraft are com-monly fabricated of high-performance composite mate-rials. These materials are fabricated by impregnatingmats of extremely strong fibers that are held within aform with an epoxy or thermoplastic liquid. After theliquid cures or cools, the resulting component is ofextremely high strength and low weight. Periodically,these components must be inspected to ensure that thefiber mats and bonding material do not become delami-nated and, in turn, the component loses its airworthiness.One inspection method involves application of a uniform, constant radiation heat flux to the surface beinginspected. The thermal response of the surface is mea-sured with an infrared imaging system, which capturesthe emission from the surface and converts it to a color-coded map of the surface temperature distribution. Con-sider the case where a uniform flux of 5 kW/m2isapplied to the top skin of an airplane wing initially at20C. The opposite side of the 15-mm-thick skin is adja-cent to stagnant air and can be treated as well insulated.The density and specific heat of the skin material are1200 kg/m3and 1200 J/kg?K, respectively. The effec-tive thermal conductivity of the intact skin material isk1?1.6 W/m?K. Contact resistances develop internal tothe structure as a result of delamination between thefiber mats and the bonding material, leading to a reducedeffective thermal conductivity of k2?1.1 W/m?K.Determine the surface temperature of the componentafter 10 and 100 s of irradiation for (i) an area where thematerial is structurally intact and (ii) an adjacent areawhere delamination has occurred within the wing.

Consider the plane wall of thickness 2L, the infinitecylinder of radius ro, and the sphere of radius ro. Eachconfiguration is subjected to a constant surface heatflux . Using the approximate solutions of Table 5.2bfor Fo?0.2, derive expressions for each of the threegeometries for the quantity (Ts,act Ti)/(Ts,lc Ti). Inthis expression, Ts,actis the actual surface temperatureas determined by the relations of Table 5.2b, and Ts,lcis the temperature associated with lumped capaci-tance behavior. Determine criteria associated with(Ts,act Ti)/ (Ts,lc Ti)?1.1, that is, determine when the lumped capacitance approximation is accurate towithin 10%.

Problem 4.9 addressed radioactive wastes stored under-ground in a spherical container. Because of uncertaintyin the thermal properties of the soil, it is desired to mea-sure the steady-state temperature using a test container(identical to the real container) that is equipped withinternal electrical heaters. Estimate how long it will takethe test container to come within 10C of its steady-statevalue, assuming it is buried very far underground. Usethe soil properties from Table A.3 in your analysis.

Derive an expression for the ratio of the total energytransferred from the isothermal surface of a sphere to theinterior of the sphere Q/Qo that is valid for Fo < 0.2. Express your result in terms of the Fourier number, Fo.

Consider the experimental measurement of Example5.10. It is desired to measure the thermal conductivity of an extremely thin sample of the same nanostruc-tured material having the same length and width. To minimize experimental uncertainty, the experimenterwishes to keep the amplitude of the temperatureresponse, T, above a value of 0.1C. What is the min-imum sample thickness that can be measured? Assumethe properties of the thin sample and the magnitude ofthe applied heating rate are the same as those measured and used in Example 5.10

The stability criterion for the explicit method requiresthat the coefficient of the term of the one- dimen-sional, finite-difference equation be zero or positive.Consider the situation for which the temperatures at thetwo neighboring nodes (Tpm?1, Tpm1) are 100C whilethe center node (Tpm) is at 50C. Show that for values ofthe finite-difference equation will predict a valueof that violates the second law of thermodynamics.

A thin rod of diameter Dis initially in equilibriumwith its surroundings, a large vacuum enclosure attemperature Tsur. Suddenly an electrical current I(A) ispassed through the rod having an electrical resistivity?eand emissivity ?. Other pertinent thermophysicalproperties are identified in the sketch. Derive the tran-sient, finite-difference equation for node m.

A one-dimensional slab of thickness 2Lis initially at auniform temperature Ti. Suddenly, electric current ispassed through the slab causing uniform volumetricheating q.(W/m3). At the same time, both outer sur-faces (x??L) are subjected to a convection processat T?with a heat transfer coefficient h.Write the finite-difference equation expressing conser-vation of energy for node 0 located on the outer sur-face at x??L. Rearrange your equation and identifyany important dimensionless coefficients

Consider Problem 5.9 except now the combined vol-ume of the oil bath and the sphere is Vtot?1m3. Theoil bath is well mixed and well insulated.(a) Assuming the quenching liquids properties arethat of engine oil at 380 K, determine the steady-state temperature of the sphere.(b) Derive explicit finite difference expressions forthe sphere and oil bath temperatures as a func-tion of time using a single node each for thesphere and oil bath. Determine any stabilityrequirements that might limit the size of the timestep ?t.(c) Evaluate the sphere and oil bath temperaturesafter one time step using the explicit expressionsof part (b) and time steps of 1000, 10,000, and20,000 s.(d) Using an implicit formulation with ?t?100 s,determine the time needed for the coated sphere toreach 140C. Compare your answer to the timeassociated with a large, well-insulated oil bath.Plot the sphere and oil temperatures as a functionof time over the interval 0 h?t?15 h. Hint:SeeComment 3 of Example 5.2.

A plane wall (??4000 kg/m3, cp?500 J/kg?K,k?10 W/m?K) of thickness L?20 mm initially hasa linear, steady- state temperature distribution withboundaries maintained at T1?0C and T2?100C.Suddenly, an electric current is passed through thewall, causing uniform energy generation at a rate. The boundary conditions T1and T2remain fixed.(a) On T xcoordinates, sketch temperature distrib-utions for the following cases: (i) initial condi-tion (t?0); (ii) steady-state conditions (tl?),assuming that the maximum temperature inthe wall exceeds T2; and (iii) for two intermedi-ate times. Label all important features of thedistributions.(b) For the system of three nodal points shownschematically (1, m, 2), define an appropriatecontrol volume for node mand, identifying all rel-evant processes, derive the corresponding finite-difference equation using either the explicitorimplicitmethod.(c) With a time increment of ?t?5 s, use the finite-difference method to obtain values of Tmfor thefirst 45 s of elapsed time. Determine the corre-sponding heat fluxes at the boundaries, that is, (0, 45 s) and qx(20 mm, 45 s).(d) To determine the effect of mesh size, repeat youranalysis using grids of 5 and 11 nodal points(?x? .0 and 2.0 mm, respectively).

A round solid cylinder made of a plastic material(??610?7m2/s) is initially at a uniform tempera-ture of 20C and is well insulated along its lateral sur-face and at one end. At time t?0, heat is applied tothe left boundary causing T0to increase linearly withtime at a rate of 1C/s.(a) Using the explicit method with , derive thefinite-difference equations for nodes 1, 2, 3, and 4.(b) Format a table with headings of p, t(s), and thenodal temperatures T0to T4. Determine the surfacetemperature T0when T4?35C

Derive the explicit finite-difference equation foran interior node for three-dimensional transient conduction. Also determine the stability criterion.Assume constant properties and equal grid spacing inall three directions.

Derive the transient, two-dimensional finite-differenceequation for the temperature at nodal point 0 locatedon the boundary between two different materials.

A wall 0.12 m thick having a thermal diffusivity of1.510?6m2/s is initially at a uniform temperature of85C. Suddenly one face is lowered to a temperatureof 20C, while the other face is perfectly insulated.(a) Using the explicit finite-difference technique withspace and time increments of 30 mm and 300 s,respectively, determine the temperature distribu-tion at t?45 min.(b) With ?x?30 mm and ?t?300 s, compute T(x, t)for 0?t?tss, where tssis the time required forthe temperature at each nodal point to reach avalue that is within 1C of the steady-state temper-ature. Repeat the foregoing calculations for?t?75 s. For each value of ?t, plot temperaturehistories for each face and the midplane.

A molded plastic product (??1200 kg/m3, c?1500 J/kg?K, k?0.30 W/m?K) is cooled by exposingone surface to an array of air jets, while the oppositesurface is well insulated. The product may be approxi-mated as a slab of thickness L?60 mm, which is ini-tially at a uniform temperature of Ti?80C. The airjets are at a temperature of T??20C and provide auniform convection coefficient of h?100 W/m2?K atthe cooled surface.Using a finite-difference solution with a space incre-ment of ?x?6 mm, determine temperatures at thecooled and insulated surfaces after 1 h of exposure tothe gas jets

Consider a one-dimensional plane wall at a uniforminitial temperature Ti. The wall is 10 mm thick, andhas a thermal diffusivity of ??610?7m2/s. Theleft face is insulated, and suddenly the right face islowered to a temperature Ts,r.(a) Using the implicit finite-difference technique with?x?2 mm and ?t?2 s, determine how long itwill take for the temperature at the left face Ts,ltoachieve 50% of its maximum possible temperaturereduction.(b) At the time determined in part (a), the right face issuddenly returned to the initial temperature. Deter-mine how long it will take for the temperature atthe left face to recover to a 20% temperaturereduction, that is, Ti Ts,l?0.2(Ti Ts,r).

The plane wall of Problem 2.60 (k?50 W/m?K,??1.510?6m2/s) has a thickness of L?40 mmand an initial uniform temperature of To?25C. Sud-denly, the boundary at x?Lexperiences heating by afluid for which T??50C and h?1000 W/m2?K,while heat is uniformly generated within the wall at. The boundary at x?0 remains at To.(a) With ?x?4 mm and ?t?1 s, plot temperaturedistributions in the wall for (i) the initial condi- tion, (ii) the steady-state condition, and (iii) twointermediate times.(b) On qx?tcoordinates, plot the heat flux at x? and x?L. At what elapsed time is there zero heatflux at xL ? ?

Consider the fuel element of Example 5.11. Initially, theelement is at a uniform temperature of 250C with noheat generation. Suddenly, the element is inserted intothe reactor core, causing a uniform volumetric heat gen-eration rate of q.?108W/m3. The surfaces are convec-tively cooled with T??250C and h?1100 W/m2?K.Using the explicit method with a space increment of2 mm, determine the temperature distribution 1.5 s afterthe element is inserted into the core

Consider two plates, A and B, that are each initiallyisothermal and each of thickness L?5 mm. The facesof the plates are suddenly brought into contact in ajoining process. Material A is acrylic, initially at Ti,A ?20C with ?A?1990 kg/m3, cA?1470 J/kg?K, andkA?0.21 W/m?K. Material B is steel initially at Ti,B ?300C with ?B?7800 kg/m3, cB?500 J/kg?K, andkB?45 W/m?K. The external (back) surfaces of theacrylic and steel are insulated. Neglecting the thermalcontact resistance between the plates, determine howlong it will take for the external surface of the acrylicto reach its softening temperature, Tsoft?90C. Plotthe acrylics external surface temperature as well as theq ?1107W/m3?Problems369CH005.qxd 2/24/11 12:37 PM Page 369 average temperatures of both materials over the timespan 0?t?300 s. Use 20 equally spaced nodal points

Consider the fuel element of Example 5.11, which oper-ates at a uniform volumetric generation rate ofq.?107W/m3, until the generation rate suddenlychanges to q.?2107W/m3. Use the Finite- DifferenceEquations, One-Dimensional, Transientconductionmodel builder of IHTto obtain the implicit form of thefinite-difference equations for the 6 nodes, with?x?2 mm, as shown in the example.(a) Calculate the temperature distribution 1.5 s afterthe change in operating power, and compare yourresults with those tabulated in the example.(b) Use the Exploreand Graphoptions of IHTto calcu- late and plot temperature histories at the midplane(00) and surface (05) nodes for 0?t?400 s. Whatare the steady-state temperatures, and approxi-mately how long does it take to reach the new equi- librium condition after the step change in operatingpower?

In a thin-slab, continuous casting process, molten steelleaves a mold with a thin solid shell, and the moltenmaterial solidifies as the slab is quenched by water jetsen route to a section of rollers. Once fully solidified,the slab continues to cool as it is brought to an accept-able handling temperature. It is this portion of theprocess that is of interest. Consider a 200-mm-thick solid slab of steel(??7800 kg/m3, c?700 J/kg?K, k?30 W/m?K),initially at a uniform temperature of Ti?1400C. The slab is cooled at its top and bottom surfaces by water jets(T??50C), which maintain an approximately uniformconvection coefficient of h?5000 W/m2?K at both sur-faces. Using a finite-difference solution with a spaceincrement of ?x?1 mm, determine the time required tocool the surface of the slab to 200C. What is the corre-sponding temperature at the midplane of the slab? If theslab moves at a speed of V?15 mm/s, what is therequired length of the cooling section?

Determine the temperature distribution at t = 30 min for the conditions of Problem 5.116.(a) Use an explicit finite-difference technique with atime increment of 600 s and a space increment of30 mm.(b) Use the implicit method of the IHT Finite-Difference Equation Tool Pad for One-Dimensional Transient Conduction

A very thick plate with thermal diffusivity5.610?6m2/s and thermal conductivity 20 W/m?Kis initially at a uniform temperature of 325C. Sud-denly, the surface is exposed to a coolant at 15C forwhich the convection heat transfer coefficient is100 W/m2?K. Using the finite-difference method witha space increment of ?x?15 mm and a time incre-ment of 18 s, determine temperatures at the surfaceand at a depth of 45 mm after 3 min have elapsed

Referring to Example 5.12, Comment 4, consider asudden exposure of the surface to large surroundingsat an elevated temperature (Tsur) and to convection (T?, h).(a) Derive the explicit, finite- difference equation forthe surface node in terms of Fo, Bi, and Bir.(b) Obtain the stability criterion for the surface node.Does this criterion change with time? Is the crite-rion more restrictive than that for an interior node?(c) A thick slab of material (k?1.5 W/m?K,??710?7m2/s,??0.9), initially at a uniformtemperature of 27C, is suddenly exposed to largesurroundings at 1000 K. Neglecting convectionand using a space increment of 10 mm, determinetemperatures at the surface and 30 mm from thesurface after an elapsed time of 1 min.

A constant-property, one-dimensional plane wall ofwidth 2L, at an initial uniform temperature Ti, isheated convectively (both surfaces) with an ambientfluid at T??T?,1, h?h1. At a later instant in time,t?t1, heating is curtailed, and convective cooling isinitiated. Cooling conditions are characterized byT??T?,2?Ti, h?h2. (a) Write the heat equation as well as the initial andboundary conditions in their dimensionless formfor the heating phase (Phase 1). Express theequations in terms of the dimensionless quanti-ties ?*, x*, Bi1, and Fo, where Bi1is expressed interms of h1.(b) Write the heat equation as well as the initial andboundary conditions in their dimensionless form forthe cooling phase (Phase 2). Express the equationsin terms of the dimensionless quantities ?*, x*, Bi2,Fo1, and FowhereFo1 is the dimensionless timeassociated with t1, and Bi2is expressed in terms ofh2. To be consistent with part (a), express thedimensionless temperature in terms of T??T?,1.(c) Consider a case for which Bi1?10, Bi2?1, andFo1?0.1. Using a finite-difference method with?x*?0.1 and ?Fo?0.001, determine the tran-sient thermal response of the surface (x*?1),midplane (x*?0), and quarter-plane (x*?0.5)of the slab. Plot these three dimensionless temper-atures as a function of dimensionless time over therange 0?Fo?0.5.(d) Determine the minimum dimensionless tempera-ture at the midplane of the wall, and the dimen- sionless time at which this minimum temperatureis achieved.

Consider the thick slab of copper in Example 5.12,which is initially at a uniform temperature of 20Cand is suddenly exposed to a net radiant flux of3105W/m2. Use the Finite-Difference Equations/One-Dimensional/Transientconduction model builderof IHTto obtain the implicit form of the finite-differenceequations for the interior nodes. In your analysis, use aspace increment of ?x?37.5 mm with a total of 17nodes (0016), and a time increment of ?t?1.2 s.For the surface node 00, use the finite- difference equa-tion derived in Section 2 of the Example.(a) Calculate the 00 and 04 nodal temperatures att?120 s, that is, T(0, 120 s) and T(0.15 m, 120 s),and compare the results with those given in Com-ment 1 for the exact solution. Will a time incrementof 0.12 s provide more accurate results?(b) Plot temperature histories for x?0, 150, and600 mm, and explain key features of your results.

In Section 5.5, the one-term approximation to theseries solution for the temperature distribution wasdeveloped for a plane wall of thickness 2Lthat is ini-tially at a uniform temperature and suddenly sub-jected to convection heat transfer. If Bi?0.1, the wall can be approximated as isothermal and repre- sented as a lumped capacitance (Equation 5.7). Forthe conditions shown schematically, we wish tocompare predictions based on the one-term approxi-mation, the lumped capacitance method, and a finite-difference solution.(a) Determine the midplane, T(0, t), and surface, T(L,t), temperatures at t?100, 200, and 500 s usingthe one-term approximation to the series solu-tion, Equation 5.43, What is the Biot number forthe system?(b) Treating the wall as a lumped capacitance, calcu-late the temperatures at t?50, 100, 200, and500 s. Did you expect these results to comparefavorably with those from part (a)? Why are thetemperatures considerably higher?(c) Consider the 2- and 5-node networks shownschematically. Write the implicit form of the finite-difference equations for each network, and deter-mine the temperature distributions for t?50, 100,200, and 500 s using a time increment of ?t?1s.You may use IHTto solve the finite-differenceequations by representing the rate of change of thenodal temperatures by the intrinsic function, Der(T, t). Prepare a table summarizing the resultsof parts (a), (b), and (c). Comment on the relativedifferences of the predicted temperatures. Hint:See the Solver/Intrinsic Functionssection ofIHT/Helpor the IHT Examplesmenu (Example5.2) for guidance on using the Der(T, t) function.

Steel-reinforced concrete pillars are used in the con-struction of large buildings. Structural failure can occurat high temperatures due to a fire because of softeningof the metal core. Consider a 200-mm- thick compositepillar consisting of a central steel core (50 mm thick)sandwiched between two 75-mm- thick concrete walls.The pillar is at a uniform initial temperature of Ti?27C and is suddenly exposed to combustion productsat T??900C, h?40 W/m2?K on both exposed sur-faces. The surroundings temperature is also 900C.(a) Using an implicit finite difference method with?x?10 mm and ?t?100 s, determine the tem-perature of the exposed concrete surface and thecenter of the steel plate at t?10,000 s. Steelproperties are: ks?55 W/m?K, ?s?7850 kg/m3,and cs?450 J/kg?K. Concrete properties are:kc?1.4 W/m?K, ?c?2300 kg/m3, cc?880 J/kg?K,and ??0.90. Plot the maximum and minimumconcrete temperatures along with the maximumand minimum steel temperatures over the duration0?t?10,000 s.(b) Repeat part (a) but account for a thermal contactresistance of ?0.20 m2?K/W at the concrete-steel interface.(c) At t?10,000 s, the fire is extinguished, and thesurroundings and ambient temperatures return toT??Tsur?27C. Using the same convection heattransfer coefficient and emissivity as in parts (a)and (b), determine the maximum steel temperatureand the critical timeat which the maximum steeltemperature occurs for cases with and without thecontact resistance. Plot the concrete surface tem-perature, the concrete temperature adjacent to thesteel, and the steel temperatures over the duration10,000?t? 0,000 s.

Consider the bonding operation described in Problem3.115, which was analyzed under steady-state condi-tions. In this case, however, the laser will be used toheat the film for a prescribed period of time, creatingthe transient heating situation shown in the sketch. The strip is initially at 25C and the laser provides auniform flux of 85,000 W/m2over a time interval of?ton?10 s. The system dimensions and thermo-physical properties remain the same, but the convec-tion coefficient to the ambient air at 25C is now 100 W/m2?K and w1?44 mm.Using an implicit finite-difference method with?x?4 mm and ?t?1 s, obtain temperature historiesfor 0?t?30 s at the center and film edge, T(0, t) andT(w1/2, t), respectively, to determine if the adhesive issatisfactorily cured above 90C for 10 s and if itsdegradation temperature of 200C is exceeded.

One end of a stainless steel (AISI 316) rod of diameter10 mm and length 0.16 m is inserted into a fixturemaintained at 200C. The rod, covered with an insulat-ing sleeve, reaches a uniform temperature throughoutits length. When the sleeve is removed, the rod is sub-jected to ambient air at 25C such that the convectionheat transfer coefficient is 30 W/m2?K.(a) Using the explicit finite- difference technique witha space increment of ?x?0.016 m, estimate thetime required for the midlength of the rod to reach100C.(b) With ?x?0.016 m and ?t?10 s, compute T(x, t) for 0?t?t1, where t1is the time requiredfor the midlength of the rod to reach 50C. Plot thetemperature distribution for t?0, 200 s, 400 s,and t1.

A tantalum rod of diameter 3 mm and length 120 mmis supported by two electrodes within a large vacuumenclosure. Initially the rod is in equilibrium with theelectrodes and its surroundings, which are main-tained at 300 K. Suddenly, an electrical current,I?80 A, is passed through the rod. Assume theemissivity of the rod is 0.1 and the electrical resistiv-ity is 9510?8??m. Use Table A.1 to obtain theother thermophysical properties required in yoursolution. Use a finite-difference method with a spaceincrement of 10 mm (a) Estimate the time required for the midlength of therod to reach 1000 K.(b) Determine the steady-state temperature distribu-tion and estimate approximately how long it willtake to reach this condition

A support rod (k?15 W/m?K, ??4.010?6m2/s)of diameter D?15 mm and length L?100 mm spansa channel whose walls are maintained at a temperatureof Tb?300 K. Suddenly, the rod is exposed to a crossflow of hot gases for which T??600 K andh?75 W/m2?K. The channel walls are cooled andremain at 300 K.(a) Using an appropriate numerical technique, deter-mine the thermal response of the rod to the con- vective heating. Plot the midspan temperature as afunction of elapsed time. Using an appropriateanalytical model of the rod, determine the steady-state temperature distribution, and compare theresult with that obtained numerically for very longelapsed times.(b) After the rod has reached steady-state conditions,the flow of hot gases is suddenly terminated, andthe rod cools by free convection to ambient air atT??300 K and by radiation exchange with largesurroundings at Tsur?300 K. The free convectioncoefficient can be expressed as h(W/m2?K)?C?Tn, where C?4.4 W/m2?K1.188and n?0.188.The emissivity of the rod is 0.5. Determine thesubsequent thermal response of the rod. Plot themidspan temperature as a function of coolingtime, and determine the time required for the rodto reach a safe-to-touchtemperature of 315 K.

Consider the acceleration-grid foil (k?40 W/m?K,??310?5m2/s, ??0.45) of Problem 4.72. Developan implicit, finite-difference model of the foil, whichcan be used for the following purposes.(a) Assuming the foil to be at a uniform temperatureof 300 K when the ion beam source is activated,obtain a plot of the midspan temperaturetime his-tory. At what elapsed time does this point on thefoil reach a temperature within 1 K of the steady-state value? (b) The foil is operating under steady-state conditionswhen, suddenly, the ion beam is deactivated. Obtaina plot of the subsequent midspan temperaturetimehistory. How long does it take for the hottest point on the foil to cool to 315 K, a safe- to-touchcondition?

Circuit boards are treated by heating a stack of themunder high pressure as illustrated in Problem 5.45 anddescribed further in Problem 5.46. A finite-differencemethod of solution is sought with two additional con-siderations. First, the book is to be treated as havingdistributed, rather than lumped, characteristics, byusing a grid spacing of ?x?2.36 mm with nodes atthe center of the individual circuit board or plate. Sec-ond, rather than bringing the platens to 190C in onesudden change, the heating schedule Tp(t) shown inthe sketch is to be used to minimize excessive thermalstresses induced by rapidly changing thermal gradientsin the vicinity of the platens.(a) Using a time increment of ?t?60 s and theimplicit method, find the temperature history ofthe midplane of the book and determine whethercuring will occur (170C for 5 min).(b) Following the reduction of the platen temperaturesto 15C (t?50 min), how long will it take for themidplane of the book to reach 37C, a safe temper-ature at which the operator can begin unloadingthe press?(c) Validate your program code by using the heatingschedule of a sudden change of platen tempera-ture from 15 to 190C and compare results withthose from an appropriate analytical solution(see Problem 5.46).

A thin circular disk is subjected to induction heatingfrom a coil, the effect of which is to provide a uniformheat generation within a ring section as shown. Convection occurs at the upper surface, while thelower surface is well insulated.(a) Derive the transient, finite-difference equation fornode m, which is within the region subjected toinduction heating.(b) On T rcoordinates sketch, in qualitative manner,the steady-state temperature distribution, identify-ing important features.

An electrical cable, experiencing uniform volumetricgeneration , is half buried in an insulating materialwhile the upper surface is exposed to a convectionprocess (T?, h).(a) Derive the explicit, finite- difference equations foran interior node (m, n), the center node (m?0),and the outer surface nodes (M, n) for the convec-tion and insulated boundaries.(b) Obtain the stability criterion for each of the finite- difference equations. Identify the most restrictivecriterion

Two very long (in the direction normal to the page)bars having the prescribed initial temperature distribu-tions are to be soldered together. At time t = 0, the m = 3 face of the copper (pure) bar contacts the m = 4 face of the steel (AISI 1010) bar. The solder and flux act as an interfacial layer of negligible thickness andeffective contact resistance .Initial Temperatures (K)n/m1237001 700 700 1000 900 8002 700 700 700 1000 900 8003 700 700 700 1000 900 800(a) Derive the explicit, finite-difference equation interms of Foand for T4,2and deter-mine the corresponding stability criterion.(b) Using Fo?0.01, determine T4,2one time stepafter contact is made. What is ?t? Is the stabilitycriterion satisfied?

Consider the system of Problem 4.92. Initially with noflue gases flowing, the walls (??5.510?7m2/s) areat a uniform temperature of 25C. Using the implicit,finite-difference method with a time increment of 1 h,find the temperature distribution in the wall 5, 10, 50,and 100 h after introduction of the flue gases.

Consider the system of Problem 4.86. Initially, theceramic plate (??1.510?6m2/s) is at a uniformtemperature of 30C, and suddenly the electricalheating elements are energized. Using the implicit,finite-difference method, estimate the time requiredfor the difference between the surface and initial tem-peratures to reach 95% of the difference for steady-state conditions. Use a time increment of 2 s.

Consider the fuel element of Example 5.11, whichoperates at a uniform volumetric generation rate ofuntil the generation rate suddenly changes to . Use the finite-elementsoftware FEHTto obtain the following solutions.(a) Calculate the temperature distribution 1.5 s afterthe change in operating power and compare yourresults with those tabulated in the example. Hint:First determine the steady-state temperature distri-bution for , which represents the initial conditionfor the transient temperature distribution after thestep change in power to . Next, in the Setupmenu, click on Transient: in the Specify/InternalGenerationbox, change the value to ; and in theRuncommand, click on Continue(not Calculate).See the Runmenu in the FEHTHelp section forbackground information on the Continueoption.(b) Use your FEHTmodel to plot temperature histo-ries at the midplane and surface for 0?t?400 s.What are the steady-state temperatures, andapproximately how long does it take to reach thenew equilibrium condition after the step change inoperating power?

Consider the thick slab of copper in Example 5.12,which is initially at a uniform temperature of 20C andis suddenly exposed to large surroundings at 1000C(instead of a prescribed heat flux).(a) For a surface emissivity of 0.94, calculate the tem-peratures T(0, 120 s) and T(0.15 m, 120 s) using thefinite-element software FEHT. Hint: In the Convec-tion Coefficienbox of the Specify/Boundary Condi-tionsmenu of FEHT, enter the linearized radiationcoefficient (see Equation 1.9) for the surface(x?0). Enter the temperature of the surroundingsin the Fluid Temperaturebox. See also the Helpsec-tion on Entering Equations. Click on Setup/Temper-atures in Kto enter all temperatures in kelvins.(b) Plot the temperature histories for x?0, 150, and600 mm, and explain key features of your results.

Consider the composite wall of Problem 2.53. In part(d), you are asked to sketch the temperature histories atx?0, Lduring the transient period between cases 2 and3. Calculate and plot these histories using the finite-ele-ment method of FEHT, the finite-difference method ofIHT(with ?x?5mm and ?t?1.2 s), and/or an alter-native procedure of your choice. If you use more thanone method, compare the respective results. Note that,in using FEHTor IHT, a look-up table must be createdfor prescribing the variation of the heater flux with time(see the appropriate Helpsection for guidance)

Common transmission failures result from the glazingof clutch surfaces by deposition of oil oxidation and decomposition products. Both the oxidation anddecomposition processes depend on temperature histo-ries of the surfaces. Because it is difficult to measure these surface temperatures during operation, it is useful to develop models to predict clutch-interfacethermal behavior. The relative velocity between mat-ing clutch plates, from the initial engagement to thezero-sliding (lock-up) condition, generates heat that is transferred to the plates. The relative velocitydecreases at a constant rate during this period, produc-ing a heat flux that is initially very large and decreaseslinearly with time, until lock-up occurs. Accordingly,qf?qo?[1?(t/tlu)], where qo?1.6107W/m2and tlu?100 ms is the lock-up time. The plates havean initial uniform temperature of Ti?40C, when theprescribed frictional heat flux is suddenly applied tothe surfaces. The reaction plate is fabricated fromsteel, while the composite plate has a thinner steel cen-ter section bonded to low-conductivity friction mater-ial layers. The thermophysical properties are ?s?7800 kg/m3, cs?500 J/kg?K, and ks?40 W/m?Kfor the steel and ?fm?1150 kg/m3, cfm?1650 J/kg?K,and kfm?4W/m?K for the friction material.(a) On T?tcoordinates, sketch the temperature his-tory at the midplane of the reaction plate, at theinterface between the clutch pair, and at the mid-plane of the composite plate. Identify key features.(b) Perform an energy balance on the clutch pair overthe time interval ?t?tluto determine the steady-state temperature resulting from clutch engage-ment. Assume negligible heat transfer from theplates to the surroundings.(c) Compute and plot the three temperature historiesof interest using the finite-element method ofFEHTor the finite- difference method of IHT(with?x?0.1 mm and ?t?1 ms). Calculate and plotthe frictional heat fluxes to the reaction andcomposite plates, and , respectively, as afunction of time. Comment on features of the tem-perature and heat flux histories. Validate yourmodel by comparing predictions with the results from part (b). Note: Use of both FEHTand IHTrequires creation of a look-up data table for pre-scribing the heat flux as a function of time.

A process mixture at 200C flows at a rate of 207 kg/minonto a conveyor belt of 3-mm thickness, 1-m width, and30-m length traveling with a velocity of 36 m/min. Theunderside of the belt is cooled by a water spray at a tem-perature of 30C, and the convection coefficient is3000 W/m2?K. The thermophysical properties of theprocess mixture are ?m?960 kg/m3, cm?1700 J/kg?K,and km?1.5 W/m?K, while the properties for theconveyor (metallic) belt are ?b?8000 kg/m3, cb?460 J/kg?K, and kb?15 W/m?K.Using the finite-difference method of IHT(?x?0.5 mm, ?t?0.05 s), the finite-elementmethod of FEHT, or a numerical procedure of yourchoice, calculate the surface temperature of the mix-ture at the end of the conveyor belt To,s. Assume negli-gible heat transfer to the ambient air by convection orby radiation to the surroundings

In a manufacturing process, stainless steel cylinders(AISI 304) initially at 600 K are quenched bysubmersion in an oil bath maintained at 300 Kwithh?500 W/m2?K. Each cylinder is of length2L?60 mm and diameter D?80 mm. Use theready-to-solvemodel in the Examplesmenu of FEHTto obtain the following solutions (a) Calculate the temperatures, T(r, x, t), after 3 min atthe cylinder center, T(0, 0, 3 min), at the center ofa circular face, T(0, L, 3 min), and the midheightof the side, T(ro, 0, 3 min).(b) Plot the temperature history at the center, T(0, 0, t),and at the midheight of the side, T(ro, 0, t), for0?t?10 min using the View/Temperatures vs.Timecommand. Comment on the gradients occur-ring at these locations and what effect they mighthave on phase transformations and thermal stresses.(c) Having solved the model for a total integrationtime of 10 min in part (b), now use the View/Tem-perature Contourscommand with the shaded bandoption for the isotherm contours. Select the FromStart to Stoptime option, and view the tempera-ture contours as the cylinder cools during thequench process. Describe the major features of thecooling process revealed by this display. Use otheroptions of this command to create a 10-isothermtemperature distribution for t?3 min.(d) For the location of part (a), calculate the tempera-tures after 3 min if the convection coefficient isdoubled (h?1000 W/m2?K). Also, for convec-tion coefficients of 500 and 1000 W/m2?K, deter-mine how long the cylinder needs to remain in theoil bath to achieve a safe-to-touchsurface temper-ature of 316 K. Tabulate and comment on theresults of parts (a) and (d).

The operations manager for a metals processingplant anticipates the need to repair a large furnaceand has come to you for an estimate of the timerequired for the furnace interior to cool to a safeworking temperature. The furnace is cubical with a16-m interior dimension and 1-m thick walls forwhich ??2600 kg/m3, c?960 J/kg?K, and k?1W/m?K. The operating temperature of the furnaceis 900C, and the outer surface experiences convec-tion with ambient air at 25C and a convection coef-ficient of 20 W/m2?K.(a) Use a numerical procedure to estimate the timerequired for the inner surface of the furnace tocool to a safe working temperature of 35C. Hint:Consider a two-dimensional cross section of thefurnace, and perform your analysis on the smallestsymmetrical section.(b) Anxious to reduce the furnace downtime, theoperations manager also wants to know whateffect circulating ambient air through the furnacewould have on the cool-down period. Assumeequivalent convection conditions for the innerand outer surfaces