Solved: The steady-state temperature distribution in a

Assume steady-state, one-dimensional heat conductionthrough the axisymmetric shape shown below.Assuming constant properties and no internal heat generation, sketch the temperature distribution on T?x coordinates. Briefly explain the shape of your curve.

Assume steady-state, one-dimensional conduction in theaxisymmetric object below, which is insulated around itsperimeter. If the properties remain constant and no internal heatgeneration occurs, sketch the heat flux distribution, ,and the temperature distribution, T(x). Explain the shapesof your curves. How do your curves depend on the ther-mal conductivity of the material?

A hot water pipe with outside radius r1 has a temperature T1. A thick insulation, applied to reduce the heat loss, hasan outer radius r2 and temperature T2. On T?rcoordinates, sketch the temperature distribution in the insula-tion for one-dimensional, steady-state heat transfer with constant properties. Give a brief explanation, justifyingthe shape of your curve.

A spherical shell with inner radius r1 and outer radius r2 has surface temperatures T1 and T2, respectively, whereT1?T2. Sketch the temperature distribution on T?rcoordinates assuming steady- state, one-dimensionalconduction with constant properties. Briefly justify theshape of your curve.

Assume steady-state, one-dimensional heat conductionthrough the symmetric shape shown.Assuming that there is no internal heat generation,derive an expression for the thermal conductivity k(x)for these conditions: A(x)=(1?x), T(x)=300(1?2x?x3), and q=6000 W, where A is in squaremeters, T in kelvins, and x in meters.

A composite rod consists of two different materials, A and B, each of length 0.5L.The thermal conductivity of Material A is half that of Material B, that is, kA/kB = 0.5. Sketch the steady- statetemperature and heat flux distributions, T(x) and q"x ,respectively. Assume constant properties and no internal heat generation in either material.

A solid, truncated cone serves as a support for a system that maintains the top (truncated) face of the cone at a temperature T1, while the base of the cone is at atemperature T2T1.The thermal conductivity of the solid depends on tem-perature according to the relation k = k0 ? aT, where a is a positive constant, and the sides of the cone are well insulated. Do the following quantities increase, decrease,or remain the same with increasing x: the heat transfer rate qx, the heat flux q"x , the thermal conductivity k, and the temperature gradient dT/dx?

To determine the effect of the temperature dependenceof the thermal conductivity on the temperature distrib-ution in a solid, consider a material for which thisdependence may be represented aswhere kois a positive constant and ais a coefficient thatmay be positive or negative. Sketch the steady- statetemperature distribution associated with heat transfer ina plane wall for three cases corresponding to a?0,a?0, and a0

A young engineer is asked to design a thermal protec-tion barrier for a sensitive electronic device that mightbe exposed to irradiation from a high-powered infraredlaser. Having learned as a student that a low thermalconductivity material provides good insulating charac-teristics, the engineer specifies use of a nanostructuredaerogel, characterized by a thermal conductivity ofka?0.005 W/m?K, for the protective barrier. The engi-neers boss questions the wisdom of selecting the aero-gel becauseit has a low thermal conductivity. Considerthe sudden laser irradiation of (a) pure aluminum, (b)glass, and (c) aerogel. The laser provides irradiation ofG?10106W/m2. The absorptivities of the materialsare ??0.2, 0.9, and 0.8 for the aluminum, glass, andaerogel, respectively, and the initial temperature of thebarrier is Ti?300 K. Explain why the boss is concerned.Hint:All materials experience thermal expansion (orcontraction), and local stresses that develop within amaterial are, to a first approximation, proportional to thelocal temperature gradient

A one-dimensional plane wall of thickness 2L?100 mm experiences uniform thermal energy generationof and is convectively cooled atx??50 mm by an ambient fluid characterized byT??20C. If the steady-state temperature distribution within the wall is T(x)?a(L2?x2)?bwherea?10C/m2and b?30C, what is the thermal con-ductivity of the wall? What is the value of the convec-tion heat transfer coefficient, h?

Consider steady-state conditions for one-dimensionalconduction in a plane wall having a thermal conductiv-ity k?50 W/m?K and a thickness L?0.25 m, with nointernal heat generation.Determine the heat flux and the unknown quantity foreach case and sketch the temperature distribution, indi-cating the direction of the heat flux.CaseT1(?C)T2(?C)dT/dx(K/m)150?202?30?10370160440?80530200

Consider a plane wall 100 mm thick and of thermal conductivity 100 W/m?K. Steady-state conditions areknown to exist with T1?400 K and T2?600 K. Deter-mine the heat flux and the temperature gradientdT/dxfor the coordinate systems shown

A cylinder of radius ro, length L, and thermal conductivitykis immersed in a fluid of convection coefficient handunknown temperature T?. At a certain instant the temper-ature distribution in the cylinder is T(r)?a?br2, whereaand bare constants. Obtain expressions for the heattransfer rate at roand the fluid temperature

In the two-dimensional body illustrated, the gradient atsurface Ais found to be ?T/?y?30 K/m. What areT? ? and ?T/?xat surface B

onsider the geometry of Problem 2.14 for the casewhere the thermal conductivity varies with temperatureas k?ko?aT, where ko?10 W/m?K, a??10?3W/m?K2, and Tis in kelvins. The gradient at surface B is?T/?x?30 K/m. What is ?? at surface A?

Steady-state, one-dimensional conduction occurs in a rodof constant thermal conductivity kand variable cross-sectional areaAx(x)?Aoeax, where Aoand aare con-stants. The lateral surface of the rod is well insulated.(a) Write an expression for the conduction heat rate,qx(x). Use this expression to determine the tempera-ture distribution T(x) and qualitatively sketch thedistribution for T(0)?T(L).(b) Now consider conditions for which thermal energyis generated in the rod at a volumetric ratewhere is a constant. Obtain anexpression for qx(x) when the left face (x?0) iswell insulated

An apparatus for measuring thermal conductivityemploys an electrical heater sandwiched between twoidentical samples of diameter 30 mm and length 60 mm,which are pressed between plates maintained at a uniformtemperature To?77C by a circulating fluid. A conduct-ing grease is placed between all the surfaces to ensuregood thermal contact. Differential thermocouples areimbedded in the samples with a spacing of 15 mm. Thelateral sides of the samples are insulated to ensure one- dimensional heat transfer through the samples a) With two samples of SS316 in the apparatus, theheater draws 0.353 A at 100 V, and the differentialthermocouples indicate ?T1??T2?25.0C. Whatis the thermal conductivity of the stainless steel sam-ple material? What is the average temperature of thesamples? Compare your result with the thermal con-ductivity value reported for this material in Table A.1.(b) By mistake, an Armco iron sample is placed in the lower position of the apparatus with one of theSS316 samples from part (a) in the upper portion. Forthis situation, the heater draws 0.601 A at 100 V, andthe differential thermocouples indicate ?T1??T2?15.0C. What are the thermal conductivity and aver-age temperature of the Armco iron sample?(c) What is the advantage in constructing the apparatuswith two identical samples sandwiching the heaterrather than with a single heatersample combina-tion? When would heat leakage out of the lateralsurfaces of the samples become significant? Underwhat conditions would you expect ?T1??T2?

An engineer desires to measure the thermal conductiv-ity of an aerogel material. It is expected that the aerogelwill have an extremely small thermal conductivity. a) Explain why the apparatus of Problem 2.17 cannotbe used to obtain an accurate measurement of theaerogels thermal conductivity.(b) The engineer designs a new apparatus for which anelectric heater of diameter D?150 mm is sand-wiched between two thin plates of aluminum. Thesteady-state temperatures of the 5-mm- thick alu-minum plates, T1and T2, are measured with ther-mocouples. Aerogel sheets of thickness t?5mmare placed outside the aluminum plates, while acoolant with an inlet temperature of Tc,i?25Cmaintains the exterior surfaces of the aerogel at alow temperature. The circular aerogel sheets areformed so that they encase the heater and alu-minum sheets, providing insulation to minimizeradial heat losses. At steady state, T1?T2?55C,and the heater draws 125 mA at 10 V. Determinethe value of the aerogel thermal conductivity ka.(c) Calculate the temperature difference across thethickness of the 5-mm-thick aluminum plates.Comment on whether it is important to know theaxial locations at which the temperatures of the alu-minum plates are measured.(d) If liquid water is used as the coolant with a totalflow rate of (0.5 kg/min for each ofthe two streams), calculate the outlet temperatureof the water, Tc,o

Consider a 300 mm300 mm window in an aircraft. Fora temperature difference of 80C from the inner to theouter surface of the window, calculate the heat lossthrough L?10-mm-thick polycarbonate, soda limeglass, and aerogel windows, respectively. The thermalconductivities of the aerogel and polycarbonate arekag?0.014 W/m?K and kpc?0.21 W/m?K, respectively.Evaluate the thermal conductivity of the soda lime glassat 300 K. If the aircraft has 130 windows and the cost toheat the cabin air is $1/kW?h, compare the costs associ-ated with the heat loss through the windows for an 8- hourintercontinental flight

Consider a small but known volume of metal that has alarge thermal conductivity. (a) Since the thermal conductivity is large, spatial temperature gradients that develop within the metalin response to mild heating are small. Neglectingspatial temperature gradients, derive a differentialequation that could be solved for the temperature of the metal versus time T(t) if the metal is sub-jected to a fixed surface heat rate qsupplied by anelectric heater.(b) A student proposes to identify the unknown metalby comparing measured and predicted thermal esponses. Once a match is made, relevant thermo-physical properties might be determined, and, inturn, the metal may be identified by comparison topublished property data. Will this approach work?Consider aluminum, gold, and silver as the candi-date metals.

Use IHT to perform the following tasks.(a) Graph the thermal conductivity of pure copper,2024 aluminum, and AISI 302 stainless steel overthe temperature range 300?T?600 K. Includeall data on a single graph, and comment on thetrends you observe. (b) Graph the thermal conductivity of helium and airover the temperature range 300?T?800 K.Include the data on a single graph, and comment onthe trends you observe.(c) Graph the kinematic viscosity of engine oil, ethylene glycol, and liquid water over the tempera-ture range 300?T?360 K. Include all data on a single graph, and comment on the trends youobserve.(d) Graph the thermal conductivity of a water-Al2O3nanofluid at T?300 K over the volume fractionrange 0??? .08. See Example 2.2

Calculate the thermal conductivity of air, hydrogen, and carbon dioxide at 300 K, assuming ideal gas behavior. Compare your calculated values to valuesfrom Table A.4

A method for determining the thermal conductivity kand the specific heat cpof a material is illustrated in thesketch. Initially the two identical samples of diameterD?60 mm and thickness L?10 mm and the thinheater are at a uniform temperature of Ti?23.00C,while surrounded by an insulating powder. Suddenlythe heater is energized to provide a uniform heat flux on each of the sample interfaces, and the heat flux ismaintained constant for a period of time, ?to. A shorttime after sudden heating is initiated, the temperature atthis interface Tois related to the heat flux asFor a particular test run, the electrical heater dissipates15.0 W for a period of ?to?120 s, and the temperatureat the interface is To(30 s)?24.57C after 30 s of heat-ing. A long time after the heater is deenergized, t??t0, the samples reach the uniform temperature ofTo(?)?33.50C. The density of the sample materials,determined by measurement of volume and mass, is??3965 kg/m3. Determine the specific heat and thermal conductiv-ity of the test material. By looking at values of the ther-mophysical properties in Table A.1 or A.2, identify thetest sample material.

Compare and contrast the heat capacity ?cpof commonbrick, plain carbon steel, engine oil, water, and soil.Which material provides the greatest amount of thermalenergy storage per unit volume? Which material wouldyou expect to have the lowest cost per unit heat capac-ity? Evaluate properties at 300 K

A cylindrical rod of stainless steel is insulated on its exte-rior surface except for the ends. The steady- state tempera-ture distribution is T(x)?a?bx/L, where a?305 Kand b?10 K. The diameter and length of the rod areD?20 mm and L?100 mm, respectively. Determinethe heat flux along the rod, Hint:The mass of the rodis M0? .248 kg.

At a given instant of time, the temperature distributionwithin an infinite homogeneous body is given by thefunction Assuming constant properties and no internal heat generation, determine the regions where the tempera-ture changes with time.

A pan is used to boil water by placing it on a stove, fromwhich heat is transferred at a fixed rate qo. There are twostages to the process. In Stage 1, the water is taken fromits initial (room) temperature Tito the boiling point, asheat is transferred from the pan by natural convection.During this stage, a constant value of the convection coef-ficient hmay be assumed, while the bulk temperature ofthe water increases with time, T??T?(t). In Stage 2, thewater has come to a boil, and its temperature remains at afixed value, T??Tb, as heating continues. Consider apan bottom of thickness Land diameter D, with a coordi-nate system corresponding to x?0 and x?Lfor the sur-faces in contact with the stove and water, respectively.(a) Write the form of the heat equation and the boundary/initial conditions that determine the variation of temperature with position and time, T(x,t), in thepan bottom during Stage 1. Express your result interms of the parameters qo, D, L, h, and T?, as wellas appropriate properties of the pan material.(b) During Stage 2, the surface of the pan in contactwith the water is at a fixed temperature, T(L, t)?TL?Tb. Write the form of the heat equation andboundary conditions that determine the temperaturedistribution T(x) in the pan bottom. Express yourresult in terms of the parameters qo, D, L, and TL, aswell as appropriate properties of the pan material.

Uniform internal heat generation at is occurring in a cylindrical nuclear reactor fuel rod of50-mm diameter, and under steady-state conditions thetemperature distribution is of the form T(r)?a?br2,where Tis in degrees Celsius and ris in meters, whilea?800C and b??4.167105C/m2. The fuel rodproperties are k?30 W/m?K, ??1100 kg/m3, andcp?800 J/kgK.(a) What is the rate of heat transfer per unit length ofthe rod at r?0 (the centerline) and at r?25 mm(the surface)?(b) If the reactor power level is suddenly increased toq.2?108W/m3, what is the initial time rate of tem-perature change at r?0 and r?25 mm?

Consider a one-dimensional plane wall with constantproperties and uniform internal generation . The leftface is insulated, and the right face is held at a uniformtemperature.(a) Using the appropriate form of the heat equation,derive an expression for the x-dependence of thesteady-state heat flux q?(x).(b) Using a finite volume spanning the range 0?x?, derive an expression for q?() and comparethe expression to your result for part (a)

The steady-state temperature distribution in a one-dimensional wall of thermal conductivity 50 W/m?K and hickness 50 mm is observed to be T(C)?a?bx2,where a?200C, b??2000C/m2, and xis in meters.(a) What is the heat generation rate in the wall?(b) Determine the heat fluxes at the two wall faces. Inwhat manner are these heat fluxes related to theheat generation rate?

The temperature distribution across a wall 0.3 m thick ata certain instant of time is T(x)?a?bx?cx2, where Tis in degrees Celsius and xis in meters, a?200C,b??200C/m, and c?30C/m2. The wall has a ther-mal conductivity of 1 W/m?K.(a) On a unit surface area basis, determine the rate ofheat transfer into and out of the wall and the rate of change of energy stored by the wall.(b) If the cold surface is exposed to a fluid at 100C,what is the convection coefficient?

A plane wall of thickness 2L?40 mm and thermal con-ductivity k?5W/m?K experiences uniform volumetricheat generation at a rateq., while convection heat transferoccurs at both of its surfaces (x??L,?L), each ofwhich is exposed to a fluid of temperature T??20C.Under steady-state conditions, the temperature distribu-tion in the wall is of the form T(x)?a?bx?cx2wherea?82.0C, b??210C/m, c??2104C/m2, and xis in meters. The origin of the x-coordinate is at themidplane of the wall.(a) Sketch the temperature distribution and identifysignificant physical features.(b) What is the volumetric rate of heat generation inthe wall?(c) Determine the surface heat fluxes, andHow are these fluxes related to the heatgeneration rate?(d) What are the convection coefficients for the sur-faces at x??Land x??L?(e) Obtain an expression for the heat flux distributionIs the heat flux zero at any location? Explainany significant features of the distribution.(f) If the source of the heat generation is suddenlydeactivated , what is the rate of change ofenergy stored in the wall at this instant?(g) What temperature will the wall eventually reachwith ? How much energy must be removed by the fluid per unit area of the wall (J/m2) to reach this state? The density and specific heat ofthe wall material are 2600 kg/m3and 800 J/kg?K,respectively

Temperature distributions within a series of one-dimensional plane walls at an initial time, at steady state, and at several intermediate times are as shown.For each case, write the appropriate form of the heat dif-fusion equation. Also write the equations for the initialcondition and the boundary conditions that are applied atx?0 and x?L. If volumetric generation occurs, it isuniform throughout the wall. The properties are constant.

One-dimensional, steady-state conduction with uniforminternal energy generation occurs in a plane wall with athickness of 50 mm and a constant thermal conductivity of5W/m?K. For these conditions, the temperature distribu-tion has the form T(x)?a?bx?cx2. The surface atx?0 has a temperature of T(0)?To?120C and experi-ences convection with a fluid for which T??20C andh?500 W/m2?K. The surface at x?Lis well insulated a) Applying an overall energy balance to the wall, cal-culate the volumetric energy generation rate .(b) Determine the coefficients a, b, and cby applyingthe boundary conditions to the prescribed tempera-ture distribution. Use the results to calculate andplot the temperature distribution.(c) Consider conditions for which the convection coef-ficient is halved, but the volumetric energy genera-tion rate remains unchanged. Determine the newvalues of a, b, and c, and use the results to plot thetemperature distribution. Hint:recognize that T(0)is no longer 120C.(d) Under conditions for which the volumetric energygeneration rate is doubled, and the convection coef-ficient remains unchanged (h?500 W/m2?K),determine the new values of a, b, and cand plot thecorresponding temperature distribution. Referringto the results of parts (b), (c), and (d) as Cases 1, 2,and 3, respectively, compare the temperature distri-butions for the three cases and discuss the effects ofhand on the distributions.

Derive the heat diffusion equation, Equation 2.26, forcylindrical coordinates beginning with the differentialcontrol volume shown in Figure 2.12

Derive the heat diffusion equation, Equation 2.29, forspherical coordinates beginning with the differentialcontrol volume shown in Figure 2.13

The steady-state temperature distribution in a semi-transparent material of thermal conductivity kandthickness Lexposed to laser irradiation is of the formwhere A, a, B, and Care known constants. For this situ-ation, radiation absorption in the material is manifestedby a distributed heat generation term, (a) Obtain expressions for the conduction heat fluxes atthe front and rear surfaces.(b) Derive an expression for (c) Derive an expression for the rate at which radiationis absorbed in the entire material, per unit surface area. Express your result in terms of the knownconstants for the temperature distribution, the ther-mal conductivity of the material, and its thickness.

One-dimensional, steady-state conduction with noenergy generation is occurring in a cylindrical shell ofinner radius r1 and outer radius r2. Under what conditionis the linear temperature distribution shown possible

One-dimensional, steady-state conduction with noenergy generation is occurring in a spherical shell ofinner radius r1 and outer radius r2. Under what condi-tion is the linear temperature distribution shown inProblem 2.38 possible?

The steady-state temperature distribution in a one-dimensional wall of thermal conductivity kand thick-ness Lis of the form T?ax3?bx2?cx?d. Deriveexpressions for the heat generation rate per unit volumein the wall and the heat fluxes at the two wall faces (x?0, L)

One-dimensional, steady-state conduction with noenergy generation is occurring in a plane wall of con-stant thermal conductivity.(a) Is the prescribed temperature distribution possible?Briefly explain your reasoning. b) With the temperature at x?0 and the fluid temper-ature fixed at T(0)?0C and T??20C, respec-tively, compute and plot the temperature at x?L,T(L), as a function of hfor 10?h?100 W/m2K ? .Briefly explain your results.

A plane layer of coal of thickness L?1 m experiencesuniform volumetric generation at a rate of due to slow oxidation of the coal particles. Averagedover a daily period, the top surface of the layer trans-fers heat by convection to ambient air for whichh?5 W/m2?K and T??25C, while receiving solarirradiation in the amount GS?400 W/m2. Irradiationfrom the atmosphere may be neglected. The solarabsorptivity and emissivity of the surface are each?S???0.95.a) Write the steady-state form of the heat diffusionequation for the layer of coal. Verify that this equa-tion is satisfied by a temperature distribution of the formFrom this distribution, what can you say about condi-tions at the bottom surface (x?0)? Sketch the temper- ature distribution and label key features.(b) Obtain an expression for the rate of heat transfer byconduction per unit area at x?L. Applying anenergy balance to a control surface about the topsurface of the layer, obtain an expression for Ts.Evaluate Tsand T(0) for the prescribed conditions.(c) Daily average values of GSand hdepend on a num-ber of factors, such as time of year, cloud cover,and wind conditions. For h?5 W/m2?K, computeand plot TSand T(0) as a function of GSfor 50?GS?500 W/m2. For GS?400 W/m2, compute andplot TSand T(0) as a function of hfor 5?h?50 W/m2?K

The cylindrical system illustrated has negligible varia-tion of temperature in the r-and z-directions. Assume that ?r?ro?riis small compared to ri, and denotethe length in the z-direction, normal to the page, as L.(a) Beginning with a properly defined control volumeand considering energy generation and storageeffects, derive the differential equation that prescribesthe variation in temperature with the angular coordi-nate . Compare your result with Equation 2.26.(b) For steady-state conditions with no internal heat gen-eration and constant properties, determine the tem-perature distribution T() in terms of the constantsT1, T2, ri, and ro. Is this distribution linear in ?(c) For the conditions of part (b) write the expressionfor the heat rate .

Beginning with a differential control volume in theform of a cylindrical shell, derive the heat diffusionequation for a one-dimensional, cylindrical, radial coor-dinate system with internal heat generation. Compareyour result with Equation 2.26

Beginning with a differential control volume in theform of a spherical shell, derive the heat diffusion equa-tion for a one-dimensional, spherical, radial coordinatesystem with internal heat generation. Compare yourresult with Equation 2.29

A steam pipe is wrapped with insulation of inner andouter radii riand ro, respectively. At a particular instantthe temperature distribution in the insulation is knownto be of the form Are conditions steady- state or transient? How do theheat flux and heat rate vary with radius?

For a long circular tube of inner and outer radii r1andr2, respectively, uniform temperatures T1and T2aremaintained at the inner and outer surfaces, while ther-mal energy generation is occurring within the tube wall(r1rr2). Consider steady-state conditions forwhich T1T2. Is it possible to maintain a linearradialtemperature distribution in the wall? If so, what specialconditions must exist?

Passage of an electric current through a long conduct-ing rod of radius riand thermal conductivity krresultsin uniform volumetric heating at a rate of . The con-ducting rod is wrapped in an electrically nonconductingcladding material of outer radius roand thermal con-ductivity kc, and convection cooling is provided by anadjoining fluid.For steady-state conditions, write appropriate forms ofthe heat equations for the rod and cladding. Expressappropriate boundary conditions for the solution ofthese equations.

Two-dimensional, steady-state conduction occurs in a hollow cylindrical solid of thermal conductivityk?16 W/m?K, outer radiusro?1 m and overalllength 2zo?5 m, where the origin of the coordinate system is located at the midpoint of the center line. The inner surface of the cylinder is insulated, and thetemperature distribution within the cylinder has the form T(r,z)?a?br2?clnr?dz2, where a??20C, b?150C/m2, c??12C, d??300C/m2and rand zare in meters.(a) Determine the inner radius riof the cylinder.(b) Obtain an expression for the volumetric rate of heatgeneration, (c) Determine the axial distribution of the heat flux at theouter surface, What is the heat rate at the outer surface? Is it into or out of the cylinder?(d) Determine the radial distribution of the heat flux atthe end faces of the cylinder, andWhat are the corresponding heat rates?Are they into or out of the cylinder?(e) Verify that your results are consistent with an over-all energy balance on the cylinder.

An electric cable of radius r1and thermal conductivitykcis enclosed by an insulating sleeve whose outer sur-face is of radius r2and experiences convection heattransfer and radiation exchange with the adjoining airand large surroundings, respectively. When electric current passes through the cable, thermal energy is gen-erated within the cable at a volumetric rate .(a) Write the steady-state forms of the heat diffusionequation for the insulation and the cable. Verifythat these equations are satisfied by the followingtemperature distributions:Insulation: Cable: Sketch the temperature distribution, T(r), in thecable and the sleeve, labeling key features.(b) Applying Fouriers law, show that the rate of con- duction heat transfer per unit length through thesleeve may be expressed asApplying an energy balance to a control surfaceplaced around the cable, obtain an alternativeexpression for q?r, expressing your result in terms ofand r1.(c) Applying an energy balance to a control surfaceplaced around the outer surface of the sleeve, obtainan expression from which Ts,2may be determined asa function of , r1, h, T?, ?, and Tsur.(d) Consider conditions for which 250 A are passingthrough a cable having an electric resistance per unitlength of R?e?0.005 ?/m, a radius of r1?15 mm,and a thermal conductivity of kc?200 W/m?K.For ks?15 W/m?K, r2?15.5 mm, h?25W/m2K, ??0.9, T??25C, and Tsur?35C, evaluate the surface temperatures, Ts,1and Ts,2, as well as the temperature Toat the centerline of the cable.(e) With all other conditions remaining the same, com-pute and plot To, Ts,1, and Ts,2as a function of r2for15.5?r2?20 mm.

A spherical shell of inner and outer radii riand ro,respectively, contains heat-dissipating components, andat a particular instant the temperature distribution in theshell is known to be of the formAre conditions steady-state or transient? How do theheat flux and heat rate vary with radius?

A chemically reacting mixture is stored in a thin-walledspherical container of radius r1?200 mm, and the exother-mic reaction generates heat at a uniform, but temperature-dependent volumetric rate of ?oexp(?A/To), whereo?5000 W/m3, A?75 K, and Tois the mixture temper-ature in kelvins. The vessel is enclosed by an insulatingmaterial of outer radius r2, thermal conductivity k, andemissivity ?. The outer surface of the insulation experi-ences convection heat transfer and net radiation exchangewith the adjoining air and large surroundings, respectively.(a) Write the steady-state form of the heat diffusionequation for the insulation. Verify that this equa-tion is satisfied by the temperature distributionSketch the temperature distribution, T(r), labelingkey features.(b) Applying Fouriers law, show that the rate of heattransfer by conduction through the insulation maybe expressed asApplying an energy balance to a control surfaceabout the container, obtain an alternative expressionfor qr, expressing your result in terms of and r1 (c) Applying an energy balance to a control surfaceplaced around the outer surface of the insulation,obtain an expression from which Ts,2may be deter-mined as a function of , r1, h, T?, ?, and Tsur.(d) The process engineer wishes to maintain a reactortemperature of To?T(r1)?95C under conditionsfor which k?0.05 W/m?K, r2?208 mm, h?5W/m2?K, ??0.9, T??25C, and Tsur?35C.What is the actual reactor temperature and the outersurface temperature Ts,2of the insulation?(e) Compute and plot the variation of Ts,2with r2for201?r2?210 mm. The engineer is concernedabout potential burn injuries to personnel who maycome into contact with the exposed surface of theinsulation. Is increasing the insulation thickness apractical solution to maintaining Ts,2?45C? Whatother parameter could be varied to reduce Ts,2?

A thin electrical heater dissipating 4000 W/m2is sand-wiched between two 25-mm-thick plates whoseexposed surfaces experience convection with a fluid forwhich and . The thermo- physical properties of the plate material are ??2500kg/m3, , and . (a) On T?xcoordinates, sketch the steady-state tem-perature distribution for?L?x??L. Calculatevalues of the temperatures at the surfaces, x??L,and the midpoint, x?0. Label this distribution asCase 1, and explain its salient features.(b) Consider conditions for which there is a loss ofcoolant and existence of a nearly adiabatic con-dition on the x??Lsurface. On the T?x coordi-nates used for part (a), sketch the correspondingsteady-state temperature distribution and indicatethe temperatures at x?0,?L. Label the distribu-tion as Case 2, and explain its key features c) With the system operating as described in part (b),the surface x??Lalso experiences a sudden loss ofcoolant. This dangerous situation goes undetected for15 min, at which time the power to the heater isdeactivated. Assuming no heat losses from the sur-faces of the plates, what is the eventual (tl?),uniform, steady-state temperature distribution in theplates? Show this distribution as Case 3 on yoursketch, and explain its key features. Hint: Apply theconservation of energy requirement on a time-intervalbasis, Eq. 1.12b, for the initial and final conditionscorresponding to Case 2 and Case 3, respectively.(d) On T?tcoordinates, sketch the temperature his-tory at the plate locations x?0,?Lduring thetransient period between the distributions for Cases2 and 3. Where and when will the temperature inthe system achieve a maximum value?

The one-dimensional system of mass Mwith constantproperties and no internal heat generation shown in thefigure is initially at a uniform temperature Ti. The elec-trical heater is suddenly energized, providing a uniformheat flux at the surface x?0. The boundaries at x?Land elsewhere are perfectly insulated.(a) Write the differential equation, and identify theboundary and initial conditions that could be usedto determine the temperature as a function of posi-tion and time in the system.(b) On T?xcoordinates, sketch the temperature distri-butions for the initial condition (t?0) and for severaltimes after the heater is energized. Will a steady-statetemperature distribution ever be reached?(c) On q?x?tcoordinates, sketch the heat flux q?x(x, t) atthe planes x?0, x?L/2, and x?Las a functionof time.(d) After a period of time tehas elapsed, the heaterpower is switched off. Assuming that the insulationis perfect, the system will eventually reach a finaluniform temperature Tf. Derive an expression thatcan be used to determine Tfas a function of theparameters , te, Ti, and the system characteristicsM, cp, and As(the heater surface area).

Consider a one-dimensional plane wall of thickness 2L.The surface at x??Lis subjected to convectiveconditions characterized by T?,1, h1, while the surface atx??Lis subjected to conditions T?,2,h2. The initialtemperature of the wall is To?(T?,1?T?,2)/2 whereT?,1?T?,2.(a) Write the differential equation, and identify theboundary and initial conditions that could be usedto determine the temperature distribution T(x, t) asa function of position and time.(b) On T?xcoordinates, sketch the temperature dis-tributions for the initial condition, the steady-statecondition, and for two intermediate times for thecase h1?h2. (c) On ?tcoordinates, sketch the heat flux at the planes x?0,?L, and?L.(d) The value of h1is now doubled with all other con-ditions being identical as in parts (a) through (c).On T?xcoordinates drawn to the same scale asused in part (b), sketch the temperature distribu-tions for the initial condition, the steady-state con-dition, and for two intermediate times. Compare thesketch to that of part (b).(e) Using the doubled value ofh1, sketch the heat fluxat the planes x?0,?L, and?Lon the sameplot you prepared for part (c). Compare the two responses.

A large plate of thickness 2Lis at a uniform tempera-ture of Ti?200C, when it is suddenly quenched bydipping it in a liquid bath of temperature T??20C.Heat transfer to the liquid is characterized by the con-vection coefficient h.(a) If x?0 corresponds to the midplane of the wall, onT?xcoordinates, sketch the temperature distribu-tions for the following conditions: initial condition(t?0), steady-state condition (tl?), and twointermediate times.(b) On ?tcoordinates, sketch the variation withtime of the heat flux at x?L. (c) If h?100 W/m2?K, what is the heat flux at x?Land t?0? If the wall has a thermal conductivity ofk?50 W/m?K what is the corresponding tempera-ture gradient at x?L?(d) Consider a plate of thickness 2L?20 mm with adensity of ??2770 kg/m3and a specific heatcp?875 J/kg?K. By performing an energy balanceon the plate, determine the amount of energy perunit surface area of the plate (J/m2) that is trans-ferred to the bath over the time required to reachsteady-state conditions.(e) From other considerations, it is known that, duringthe quenching process, the heat flux at x??Landx??Ldecays exponentially with time according tothe relation, ?Aexp(?Bt), where tis in seconds,A?1.80104W/m2, and B?4.12610?3s?1.Use this information to determine the energy perunit surface area of the plate that is transferred tothe fluid during the quenching process.

The plane wall with constant properties and no internalheat generation shown in the figure is initially at a uniformtemperature Ti. Suddenly the surface at x?Lis heated bya fluid at T?having a convection heat transfer coefficient h.The boundary at x?0 is perfectly insulated.(a) Write the differential equation, and identify theboundary and initial conditions that could be usedto determine the temperature as a function of posi-tion and time in the wall.(b) On T?xcoordinates, sketch the temperature dis-tributions for the following conditions: initial con-dition (t?0), steady-state condition (tl?), andtwo intermediate times.(c) On q?x?tcoordinates, sketch the heat flux at thelocations x?0, x?L. That is, show qualitativelyhow (0, t) and (L,t) vary with time.(d) Write an expression for the total energy transferredto the wall per unit volume of the wall (J/m3).

Consider the steady-state temperature distributionswithin a composite wall composed of Material A andMaterial B for the two cases shown. There is no nternal generation, and the conduction process is one-dimensional. Answer the following questions for each case. Whichmaterial has the higher thermal conductivity? Does thethermal conductivity vary significantly with tempera-ture? If so, how? Describe the heat flux distributionthrough the composite wall. If the thickness andthermal conductivity of each material were both dou-bled and the boundary temperatures remained the same,what would be the effect on the heat flux distribution?Case 1. Linear temperature distributions exist in bothmaterials, as shown. Case 2. Nonlinear temperature distributions exist inboth materials, as shown.

A plane wall has constant properties, no internal heatgeneration, and is initially at a uniform temperature Ti.Suddenly, the surface at x?Lis heated by a fluid at T?having a convection coefficient h. At the same instant,the electrical heater is energized, providing a constantheat flux at x?0.(a) On T?xcoordinates, sketch the temperaturedistributions for the following conditions: initialcondition (t?0), steady-state condition (tl?),and for two intermediate times (b) On coordinates, sketch the heat flux corre-sponding to the four temperature distributions ofpart (a).(c) On q?x?tcoordinates, sketch the heat flux at thelocations x?0 and x?L. That is, show qualita-tively how (0, t) and (L, t) vary with time.(d) Derive an expression for the steady-state tempera-ture at the heater surface, T(0,?), in terms of ,T?, k, h, a

A plane wall with constant properties is initially at a uni-form temperature To. Suddenly, the surface at x?Lisexposed to a convection process with a fluid at T?(?To)having a convection coefficient h. Also, suddenly thewall experiences a uniform internal volumetric heating that is sufficiently large to induce a maximum steady-state temperature within the wall, which exceeds that ofthe fluid. The boundary at x?0 remains at To.(a) On T?xcoordinates, sketch the temperature dis-tributions for the following conditions: initial con-dition (t?0), steady-state condition (tl?), andfor two intermediate times. Show also the distribu-tion for the special condition when there is no heatflow at the x?Lboundary.(b) On q?x?tcoordinates, sketch the heat flux for thelocations x?0 and x?L, that is, and, respectively.

Consider the conditions associated with Problem 2.60,but now with a convection process for which T?To.(a) On T?xcoordinates, sketch the temperature dis-tributions for the following conditions: initial con-dition (t?0), steady-state condition (tl?), andfor two intermediate times. Identify key features ofthe distributions, especially the location of themaximum temperature and the temperature gradi-ent at x?L.(b) On q?x?tcoordinates, sketch the heat flux for the locations x?0 and x?L, that is, and, respectively. Identify key features of the fluxhistories.

Consider the steady-state temperature distribution withina composite wall composed of Materials A and B.The conduction process is one-dimensional. Withinwhich material does uniform volumetric generationoccur? What is the boundary condition at x??LA?How would the temperature distribution change if thethermal conductivity of Material A were doubled? How would the temperature distribution change if thethermal conductivity of Material B were doubled? Doesa contact resistance exist at the interface between thetwo materials? Sketch the heat flux distribution through the composite wall

A spherical particle of radius r1experiences uniform ther-mal generation at a rate of q.. The particle is encapsulatedby a spherical shell of outside radius r2that is cooled byambient air. The thermal conductivities of the particle andshell are k1and k2, respectively, where k1?2k2.(a) By applying the conservation of energy principle tospherical control volume A, which is placed at anarbitrary location within the sphere, determine arelationship between the temperature gradientdT/drand the local radius r, for 0?r?r1.(b) By applying the conservation of energy principle to spherical control volume B, which is placed at an arbitrary location within the spherical shell, determine a relationship between the temperaturegradient dT/drand the local radius r, for r1?r?r2.(c) On T?rcoordinates, sketch the temperature dis-tribution over the range 0?r?r2.

A longcylindrical rod, initially at a uniform tempera-ture Ti, is suddenly immersed in a largecontainer ofliquid at T?Ti. Sketch the temperature distributionwithin the rod, T(r), at the initial time, at steady state,and at two intermediate times. On the same graph,carefully sketch the temperature distributions thatwould occur at the same times within a second rod thatis the same size as the first rod. The densities and spe-cific heats of the two rods are identical, but the thermalconductivity of the second rod is very large. Which rodwill approach steady-state conditions sooner? Write theappropriate boundary conditions that would be appliedat r?0 and r?D/2 for either rod.

A plane wall of thickness L?0.1 m experiences uniformvolumetric heating at a rate q.. One surface of the wall(x?0) is insulated, and the other surface is exposed to afluid at T??20C, with convection heat transfer charac-terized by h?1000 W/m2?K. Initially, the temperaturedistribution in the wall is T(x,0)?a?bx2, wherea?300C, b??1.0104C/m2, and xis in meters.Suddenly, the volumetric heat generation is deactivated(q.?0 for t0), while convection heat transfer contin-ues to occur at x?L. The properties of the wall are??7000 kg/m3, cp?450 J/kg?K, and k?90 W/m?K.(a) Determine the magnitude of the volumetric energygeneration rate q.associated with the initial condi-tion (t0).(b) On T?xcoordinates, sketch the temperature dis-tribution for the following conditions: initial condi-tion (t0), steady-state condition (tl?), andtwo intermediate conditions.(c) On ?tcoordinates, sketch the variation withtime of the heat flux at the boundary exposed to theconvection process, . Calculate the corre-sponding value of the heat flux at t?0, .(d) Calculate the amount of energy removed from thewall per unit area (J/m2) by the fluid stream q?x(L, 0)q?x(L, t)q?xk, , cp, q(t< 0) xLT,hr1r2Ambient airT, hqControl volume BControl volume AChemical reactionq?x(x)xLALBkAkBT(x)108Chapter 2?Introduction to ConductionCH002.qxd 2/24/11 12:22 PM Page 108 as the wall cools from its initial to steady-state condition

A plane wall that is insulated on one side (x?0) isinitially at a uniform temperature Ti, when its exposedsurface at x?Lis suddenly raised to a temperature Ts.(a) Verify that the following equation satisfies the heatequation and boundary conditions:where C1is a constant and ?is the thermal diffusivity.(b) Obtain expressions for the heat flux at x?0 andx?L.(c) Sketch the temperature distribution T(x) at t?0, attl?, and at an intermediate time. Sketch the vari-ation with time of the heat flux at x?L, .(d) What effect does ?have on the thermal response ofthe material to a change in surface temperature

A composite one-dimensional plane wall is of overallthickness 2L. Material A spans the domain ?L?x0and experiences an exothermicchemical reaction leadingto a uniform volumetric generation rate of q.A. Material Bspans the domain 0?x?Land undergoes an endo-thermicchemical reaction corresponding to a uniformvolumetric generation rate of q.B??q.A. The surfaces at x??Lare insulated. Sketch the steady- statetemperature and heat flux distributions T(x) and q?x(x), respectively, over the domain?L?x?LforkA?kB, kA?0.5kB, and kA?2kB. Point out theimportant features of the distributions you have drawn.If , can you sketch the steady-state tempera-ture distribution?

Typically, air is heated in a hair dryer by blowing itacross a coiled wire through which an electric current ispassed. Thermal energy is generated by electric resis-tance heating within the wire and is transferred by con-vection from the surface of the wire to the air. Considerconditions for which the wire is initially at roomtemperature, Ti, and resistance heating is concurrentlyinitiated with airflow at t?0. (a) For a wire radius ro, an air temperature T?, and aconvection coefficient h, write the form of theheat equation and the boundary/initial conditionsthat govern the transient thermal response, T(r, t),of the wire.(b) If the length and radius of the wire are 500 mm and1 mm, respectively, what is the volumetric rate ofthermal energy generation for a power consump-tion of Pelec?500 W? What is the convection heatflux under steady-state conditions?(c) On T?rcoordinates, sketch the temperaturedistributions for the following conditions: initialcondition (t?0), steady-state condition (tl?),and for two intermediate times.(d) On q?r?tcoordinates, sketch the variation of theheat flux with time for locations at r?0 and r?ro.

The steady-state temperature distribution in a compos-ite plane wall of three different materials, each of con-stant thermal conductivity, is shown.(a) Comment on the relative magnitudes of and ,and of and .(b) Comment on the relative magnitudes of kAand kB,and of kBand kC.(c) Sketch the heat flux as a function of x.