Derive the nodal finite-difference equations for the

In the method of separation of variables (Section 4.2)for two-dimensional, steady-state conduction, the sepa-ration constant ?2in Equations 4.6 and 4.7 must be apositive constant. Show that a negative or zero value of?2will result in solutions that cannot satisfy the pre-scribed boundary conditions

A two-dimensional rectangular plate is subjected to pre-scribed boundary conditions. Using the results of the exactsolution for the heat equation presented in Section 4.2,calculate the temperature at the midpoint (1, 0.5) by con-sidering the first five nonzero terms of the infinite seriesthat must be evaluated. Assess the error resulting fromusing only the first three terms of the infinite series. Plotthe temperature distributions T(x, 0.5) and T(1.0, y).

Consider the two-dimensional rectangular plate of Problem 4.2 having a thermal conductivity of 50 W/mK. Beginning with the exact solution for the temperature distribution, derive an expression for the heat transfer rate per unit thickness from the plate along the lower surface (0 x 2, y = 0). Evaluate the heat rate considering the first five nonzero terms of the infinite series.

A two-dimensional rectangular plate is subjected to theboundary conditions shown. Derive an expression forthe steady-state temperature distribution T(x, y).

A two-dimensional rectangular plate is subjected toprescribed temperature boundary conditions on threesides and a uniform heat flux into the plate at the top surface. Using the general approach of Section 4.2,derive an expression for the temperature distribution in the plate.

Using the thermal resistance relations developed inChapter 3, determine shape factor expressions for thefollowing geometries:(a) Plane wall, cylindrical shell, and spherical shell.(b) Isothermal sphere of diameter Dburied in an infi-nite medium.

Free convection heat transfer is sometimes quantified bywriting Equation 4.20 as qconv?SkeffT1?2, where keffisan effectivethermal conductivity. The ratio keff/kis greaterthan unity because of fluid motion driven by buoyancyforces, as represented by the dashed streamlines. An experiment for the configuration shown yields aheat transfer rate per unit length of q?conv?110 W/m forsurface temperatures of T1?53C and T2?15C,respectively. For inner and outer cylinders of diametersd?20 mm and D?60 mm, and an eccentricity factorof z?10 mm, determine the value of keff. The actualthermal conductivity of the fluid is k?0.255 W/m?K.

Consider Problem 4.5 for the case where the plate is ofsquare cross section, W?L.(a) Derive an expression for the shape factor, Smax, asso-ciated with the maximumtop surface temperature,such that q?Smaxk(T2,max?T1) where T2,maxis themaximum temperature along y?W.(b) Derive an expression for the shape factor, Savg,associated with the averagetop surface tempera-ture, q?Savgk(T2?T1) where T2is the averagetemperature along y?W.(c) Evaluate the shape factors that can be used to determine the maximum and average temperaturesalong y?W. Evaluate the maximum and averagetemperatures for T1?0C, L?W?10 mm, k?20 W/m?K, and q?s?1000 W/m2.

Radioactive wastes are temporarily stored in a spheri-cal container, the center of which is buried a distanceof 10 m below the earths surface. The outside diame-ter of the container is 2 m, and 500 W of heat arereleased as a result of radioactive decay. If the soil surface temperature is 20C, what is the outside sur-face temperature of the container under steady-stateconditions? On a sketch of the soilcontainer systemdrawn to scale, show representative isotherms and heatflow lines in the soil.

Based on the dimensionless conduction heat rates forcases 1215 in Table 4.1b, find shape factors for the fol-lowing objects having temperature T1, located at the surface of a semi-infinite medium having temperature T2.The surface of the semi-infinite medium is adiabatic.(a) A buried hemisphere, flush with the surface.(b) A disk on the surface. Compare your result to Table4.1a, case 10.(c) A square on the surface.(d) A buried cube, flush with the surface.

Determine the heat transfer rate between two particles ofdiameter D = 100 m and temperatures T1 = 300.1 K and T2 = 299.9 K, respectively. The particles are in contact and are surrounded by air.

A two-dimensional object is subjected to isothermalconditions at its left and right surfaces, as shown in theschematic. Both diagonal surfaces are adiabatic and thedepth of the object is L?100 mm.(a) Determine the two-dimensional shape factor for theobject for a?10 mm, b?12 mm.(b) Determine the two-dimensional shape factor for theobject fora?10 mm, b?15 mm.(c) Use the alternative conduction analysis of Section3.2 to estimate the shape factor for parts (a) and(b). Compare the values of the approximate shapefactors of the alternative conduction analysis to thetwo-dimensional shape factors of parts (a) and (b).(d) For T1?100C and T2?60C, determine the heattransfer rate per unit depth for k?15 W/m?K forparts (a) and (b)

An electrical heater 100 mm long and 5 mm in diameter is inserted into a hole drilled normal to the surfaceof a large block of material having a thermal conductivity of 5 W/mK. Estimate the temperature reachedby the heater when dissipating 50 W with the surface of the block at a temperature of 25C.

Two parallel pipelines spaced 0.5 m apart are buried insoil having a thermal conductivity of 0.5 W/mK. Thepipes have outer diameters of 100 and 75 mm withsurface temperatures of 175C and 5C, respectively.Estimate the heat transfer rate per unit length between the two pipelines.

A small water droplet of diameter D?100 ?m andtemperature Tmp?0C falls on a nonwetting metal sur- face that is at temperature Ts?15C. Determine howlong it will take for the droplet to freeze completely.The latent heat of fusion is hsf?334 kJ/kg

A tube of diameter 50 mm having a surface temperatureof 85C is embedded in the center plane of a concreteslab 0.1 m thick with upper and lower surfaces at 20C.Using the appropriate tabulated relation for this configuration, find the shape factor. Determine the heat transfer rate per unit length of the tube.

Pressurized steam at 450 K flows through a long, thin-walled pipe of 0.5-m diameter. The pipe is enclosed in aconcrete casing that is of square cross section and 1.5 mon a side. The axis of the pipe is centered in the casing,and the outer surfaces of the casing are maintained at300 K. What is the heat loss per unit length of pipe?

The temperature distribution in laser-irradiated materi-als is determined by the power, size, and shape of thelaser beam, along with the properties of the materialbeing irradiated. The beam shape is typically Gaussian,and the local beam irradiation flux (often referred to asthe laser fluenc) isThe x-and y-coordinates determine the location ofinterest on the surface of the irradiated material. Con-sider the case where the center of the beam is located atx?y?r?0. The beam is characterized by a radiusrb, defined as the radial location where the local fluenceis q?(rb)?q?(r?0)/e?0.368q?(r?0).A shape factor for Gaussian heating is S?2?1/2rb,where Sis defined in terms of T1,max?T2[Nissin, Y. I.,A. Lietoila, R. G. Gold, and J. F. Gibbons, J. Appl.Phys.,51, 274, 1980]. Calculate the maximum steady-state surface temperature associated with irradiation ofa material of thermal conductivity k?27 W/m?K andabsorptivity ??0.45 by a Gaussian beam withrb?0.1 mm and power P?1 W. Compare your resultwith the maximum temperature that would occur if theirradiation was from a circular beam of the same diam-eter and power, but characterized by a uniform fluence(aflabeam). Also calculate the average temperature ofthe irradiated surface for the uniform fluence case. Thetemperature far from the irradiated spot is T2?25C.

Hot water at 85C flows through a thin-walled coppertube of 30-mm diameter. The tube is enclosed by aneccentric cylindrical shell that is maintained at 35C andhas a diameter of 120 mm. The eccentricity, defined asthe separation between the centers of the tube and shell,is 20 mm. The space between the tube and shell is filledwith an insulating material having a thermal conductiv-ity of 0.05 W/mK. Calculate the heat loss per unit length of the tube, and compare the result with the heatloss for a concentric arrangement.

A furnace of cubical shape, with external dimensions of0.35 m, is constructed from a refractory brick (fireclay).If the wall thickness is 50 mm, the inner surface tem-perature is 600C, and the outer surface temperature is 75C, calculate the heat loss from the furnace.

Laser beams are used to thermally process materials ina wide range of applications. Often, the beam is scannedalong the surface of the material in a desired pattern.Consider the laser heating process of Problem 4.18,except now the laser beam scans the material at a scan-ning velocity of U. A dimensionless maximum surfacetemperature can be well correlated by an expression ofthe form [Nissin, Y. I., A. Lietoila, R. G. Gold, and J. F.Gibbons, J. Appl. Phys.,51, 274, 1980]for the range 0?Pe?10, where Peis a dimensionlessvelocity known as the Peclet number. For this problem,where ?is the thermal diffusivity of thematerial. The maximum material temperature does notoccur directly below the laser beam, but at a lag distance?behind the center of the moving beam. The dimension-less lag distance can be correlated to Peby [Sheng, I. C.,and Y. Chen, J. Thermal Stresses, 14, 129, 1991](a) For the laser beam size and shape and material ofProblem 4.18, determine the laser power required toachieve T1,max?200C for U?2 m/s. The densityand specific heat of the material are ??2000 kg/m3and c?800 J/kg?K, respectively.(b) Determine the lag distance ?associated with U?2 m/s.(c) Plot the required laser power to achieve Tmax,12 ? 00?C for 0 U2 m/s.

A double-glazed window consists of two sheets ofglass separated by an L?0.2-mm-thick gap. The gap is evacuated, eliminating conduction and convectionacross the gap. Small cylindrical pillars, eachL?0.2 mm long and D?0.15 mm in diameter, areinserted between the glass sheets to ensure that theglass does not break due to stresses imposed by the pressure difference across each glass sheet. A con- tact resistance of R?t,c?1.5?10?6m2?K/W existsbetween the pillar and the sheet. For nominal glasstemperatures of T1?20C and T2??10C, deter-mine the conduction heat transfer through an individ- ual stainless steel pillar.

A pipeline, used for the transport of crude oil, is buriedin the earth such that its centerline is a distance of 1.5 mbelow the surface. The pipe has an outer diameter of0.5 m and is insulated with a layer of cellular glass100 mm thick. What is the heat loss per unit length ofpipe when heated oil at 120C flows through the pipeand the surface of the earth is at a temperature of 0C?

A long power transmission cable is buried at a depth(ground-to-cable-centerline distance) of 2 m. The cableis encased in a thin-walled pipe of 0.1-m diameter, and,to render the cable superconducting(with essentiallyzero power dissipation), the space between the cableand pipe is filled with liquid nitrogen at 77 K. If the pipe is covered with a superinsulator (ki?0.005 W/m?K) of 0.05-m thickness and the surface ofthe earth (kg?1.2 W/m?K) is at 300 K, what is thecooling load (W/m) that must be maintained by a cryo-genic refrigerator per unit pipe length?

A small device is used to measure the surface tempera-ture of an object. A thermocouple bead of diameterD?120?m is positioned a distance z?100?m fromthe surface of interest. The two thermocouple wires,each of diameter d?25?m and length L?300?m,are held by a large manipulator that is at a temperatureof Tm?23C.If the thermocouple registers a temperature ofTtc?29C, what is the surface temperature? The thermal conductivities of the chromel and alumel thermocouplewires are kCh?19 W/m?K and kAl?29 W/m?K, respec-tively. You may neglect radiation and convection effects.

A cubical glass melting furnace has exterior dimensionsof width W=5 m on a side and is constructed from refractory brick of thickness L=0.35 m and thermal conductivity k=1.4 W/mK. The sides and top of thefurnace are exposed to ambient air at 25C, with freeconvection characterized by an average coefficient of h=5 W/mK. The bottom of the furnace rests on aframed platform for which much of the surface isexposed to the ambient air, and a convection coefficientof h=5 W/mK may be assumed as a first approximation. Under operating conditions for which combustion gases maintain the inner surfaces of the furnace at 1100C, what is the heat loss from the furnace?

A hot fluid passes through circular channels of a castiron platen (A) of thickness LA?30 mm which is inpoor contact with the cover plates (B) of thicknessLB?7.5 mm. The channels are of diameter D?15 mmwith a centerline spacing of Lo?60 mm. The thermalconductivities of the materials are kA?20 W/m?K andkB?75 W/m?K, while the contact resistance betweenthe two materials is R?t,c?2.0?10?4m2?K/W. The hotfluid is at Ti?150C, and the convection coefficient is1000 W/m2?K. The cover plate is exposed to ambientair at T??25C with a convection coefficient of200 W/m2?K. The shape factor between one channeland the platen top and bottom surfaces is 4.25.(a) Determine the heat rate from a single channel perunit length of the platen normal to the page, q?i.(b) Determine the outer surface temperature of thecover plate, Ts.(c) Comment on the effects that changing the center-line spacing will have on q?iand Ts. How wouldinsulating the lower surface affect q?iand Ts?

An aluminum heat sink (k?240 W/m?K), used to coolan array of electronic chips, consists of a square channelof inner width w?25 mm, through which liquid flow may be assumed to maintain a uniform surfacetemperature of T1?20?C. The outer width and lengthof the channel are W?40 mm and L?160 mm,respectively.If N?120 chips attached to the outer surfaces of theheat sink maintain an approximately uniform surfacetemperature of T2?50?C and all of the heat dissipatedby the chips is assumed to be transferred to the coolant,what is the heat dissipation per chip? If the contact resis- tance between each chip and the heat sink is Rt,c?0.2 K/W, what is the chip temperature?

Hot water is transported from a cogeneration power sta-tion to commercial and industrial users through steelpipes of diameter D?l50 mm, with each pipe centeredin concrete (k?1.4 W/m?K) of square cross section(w?300 mm). The outer surfaces of the concrete areexposed to ambient air for which T??0?C and h?25 W/m2?K.(a) If the inlet temperature of water flowing throughthe pipe is Ti?90?C, what is the heat loss per unitlength of pipe in proximity to the inlet? The tem-perature of the pipe T1may be assumed to be thatof the inlet water.(b) If the difference between the inlet and outlet tem-peratures of water flowing through a 100-m-longpipe is not to exceed 5?C, estimate the minimum allowable flow rate m.. A value of c?4207 J/kg?Kmay be used for the specific heat of the water.

A long constantan wire of 1-mm diameter is butt welded to the surface of a large copper block, forming a thermocouple junction. The wire behaves as a fin, per-mitting heat to flow from the surface, thereby depressingthe sensing junction temperature Tjbelow that of theblock To.(a) If the wire is in air at 25C with a convection coef-ficient of 10 W/m2?K, estimate the measurementerror (Tj?To) for the thermocouple when theblock is at 125C.(b) For convection coefficients of 5, 10, and 25 W/m2?K,plot the measurement error as a function of the ther-mal conductivity of the block material over the range15 to 400 W/m?K. Under what circumstances is itadvantageous to use smaller diameter wire?

A hole of diameter D?0.25 m is drilled through thecenter of a solid block of square cross section withw?1 m on a side. The hole is drilled along thelength, l?2 m, of the block, which has a thermalconductivity of k?150 W/m?K. The four outer sur-faces are exposed to ambient air, with T?,2?25C andh2?4 W/m2?K, while hot oil flowing through the holeis characterized by T?,1?300C and h1?50 W/m2?K.Determine the corresponding heat rate and surfacetemperatures.

In Chapter 3 we assumed that, whenever fins are attachedto a base material, the base temperature is unchanged.What in fact happens is that, if the temperature of the base material exceeds the fluid temperature, attachment of a findepresses the junction temperature Tjbelow the originaltemperature of the base, and heat flow from the basematerial to the fin is two-dimensional.Consider conditions for which a long aluminum pin finof diameter D?5 mm is attached to a base materialwhose temperature far from the junction is maintainedat Tb?100C. Fin convection conditions correspond toh?50 W/m2?K and T??25C.(a) What are the fin heat rate and junction tempera-ture when the base material is (i) aluminum (k?240 W/m?K) and (ii) stainless steel (k?15W/m?K)?(b) Repeat the foregoing calculations if a thermal con-tact resistance of R?t,j?3?10?5m2?K/W is asso-ciated with the method of joining the pin fin to thebase material.(c) Considering the thermal contact resistance, plot theheat rate as a function of the convection coefficientover the range 10 h100 W/m2?K for each ofthe two materials.

An igloo is built in the shape of a hemisphere, with aninner radius of 1.8 m and walls of compacted snow thatare 0.5 m thick. On the inside of the igloo, the surfaceheat transfer coefficient is 6 W/m2?K; on the outside,under normal wind conditions, it is 15 W/m2?K. Thethermal conductivity of compacted snow is 0.15 W/m?K.The temperature of the ice cap on which the igloo sits is?20C and has the same thermal conductivity as thecompacted snow. (a) Assuming that the occupants body heat provides acontinuous source of 320 W within the igloo, cal-culate the inside air temperature when the outsideair temperature is T???40C. Be sure to considerheat losses through the floor of the igloo.(b) Using the thermal circuit of part (a), perform aparameter sensitivity analysis to determine whichvariables have a significant effect on the inside airtemperature. For instance, for very high wind con-ditions, the outside convection coefficient coulddouble or even triple. Does it make sense to con- struct the igloo with walls half or twice as thick?

Consider the thin integrated circuit (chip) of Problem3.150. Instead of attaching the heat sink to the chipsurface, an engineer suggests that sufficient coolingmight be achieved by mounting the top of the chip onto a large copper (k?400 W/m?K) surface that islocated nearby. The metallurgical joint between thechip and the substrate provides a contact resistance ofR?t,c?5?10?6m2?K/W, and the maximum allowablechip temperature is 85C. If the large substrate temper-ature is T2?25C at locations far from the chip, whatis the maximum allowable chip power dissipation qc?

An electronic device, in the form of a disk 20 mm indiameter, dissipates 100 W when mounted flush on a large aluminum alloy (2024) block whose tempera-ture is maintained at 27C. The mounting arrangementis such that a contact resistance of R?t,c?5?10?5m2?K/W exists at the interface between the device andthe block.(a) Calculate the temperature the device will reach,assuming that all the power generated by the devicemust be transferred by conduction to the block.(b) To operate the device at a higher power level, a circuit designer proposes to attach a finned heat sinkto the top of the device. The pin fins and base mate-rial are fabricated from copper (k?400 W/m?K)and are exposed to an airstream at 27C for whichthe convection coefficient is 1000 W/m2?K. For thedevice temperature computed in part (a), what is the permissible operating power?

The elemental unit of an air heater consists of a long circular rod of diameter D, which is encapsulated by afinned sleeve and in which thermal energy is generatedby ohmic heating. The Nfins of thickness tand length Lare integrally fabricated with the square sleeve of widthw. Under steady- state operating conditions, the rate ofthermal energy generation corresponds to the rate of heattransfer to airflow over the sleeve.(a) Under conditions for which a uniform surfacetemperature Tsis maintained around the circum-ference of the heater and the temperature T?and convection coefficient hof the airflow are known, obtain an expression for the rate of heattransfer per unit length to the air. Evaluate theheat rate for Ts?300?C, D?20 mm, an alu-minum sleeve (ks?240 W/m?K), w?40 mm, N?16, t?4 mm, L?20 mm, T??50?C, and h?500 W/m2?K.(b) For the foregoing heat rate and a copper heater ofthermal conductivity kh?400 W/m?K, what is therequired volumetric heat generation within the heaterand its corresponding centerline temperature?(c) With all other quantities unchanged, explore theeffect of variations in the fin parameters (N, L, t) onthe heat rate, subject to the constraint that the finthickness and the spacing between fins cannot beless than 2 mm.

For a small heat source attached to a large substrate, thespreadingresistance associated with multidimensionalconduction in the substrate may be approximated by theexpression [Yovanovich, M. M., and V. W. Antonetti,in Adv. Thermal Modeling Elec. Comp. and Systems, Vol. 1, A. Bar-Cohen and A. D. Kraus, Eds., Hemisphere,NY, 79128, 1988]where Ar?As,h/As,subis the ratio of the heat source areato the substrate area. Consider application of the expres-sion to an in-line array of square chips of width Lh?5 mm on a side and pitch Sh?10 mm. The interface between the chips and a large substrate of thermal con-ductivity ksub?80 W/m?K is characterized by a ther-mal contact resistance of R?t,c?0.5 ?10?4m2?K/W.If a convection heat transfer coefficient of h?100 W/m2?K is associated with airflow (T??15?C)over the chips and substrate, what is the maximumallowable chip power dissipation if the chip tempera-ture is not to exceed Th?85?C?

Consider nodal configuration 2 of Table 4.2. Derive thefinite-difference equations under steady-state conditionsfor the following situations.(a) The horizontal boundary of the internal corner isperfectly insulated and the vertical boundary is sub-jected to the convection process (T, h).(b) Both boundaries of the internal corner are perfectlyinsulated. How does this result compare withEquation 4.41?

Consider nodal configuration 3 of Table 4.2. Derive thefinite-difference equations under steady-state conditionsfor the following situations.(a) The boundary is insulated. Explain how Equation4.42 can be modified to agree with your result.(b) The boundary is subjected to a constant heat flux

Consider nodal configuration 4 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations.(a) The upper boundary of the external corner is per-fectly insulated and the side boundary is subjected to the convection process (T, h).(b) Both boundaries of the external corner are perfectly insulated. How does this result compare with Equation 4.43?

One of the strengths of numerical methods is theirability to handle complex boundary conditions. In the sketch, the boundary condition changes from specified heat flux q"s (into the domain) to convection,at the location of the node (m, n). Write the steady-state, two-dimensional finite difference equation at this node.

Determine expressions for q(m?1,n)(m,n), q(m?1,n)(m,n), q(m,n?1)(m,n)and q(m,n?1)(m,n)for conduction associatedwith a control volume that spans two different materials.There is no contact resistance at the interface between thematerials. The control volumes are Lunits long into thepage. Write the finite difference equation under steady-state conditions for node (m, n).

Consider heat transfer in a one-dimensional (radial)cylindrical coordinate system under steady-state conditions with volumetric heat generation.(a) Derive the finite-difference equation for any inte-rior node m.(b) Derive the finite-difference equation for the node n located at the external boundary subjected to the convection process (T, h).

In a two-dimensional cylindrical configuration, the radial(r) and angular () spacings of the nodes are uniform.The boundary at r?riis of uniform temperature Ti. Theboundaries in the radial direction are adiabatic (insulated)and exposed to surface convection (T?,h), as illustrated.Derive the finite-difference equations for (i) node 2,(ii) node 3, and (iii) node 1.

Upper and lower surfaces of a bus bar are convectivelycooled by air at T?, with hu?hl. The sides are cooledby maintaining contact with heat sinks at To, through athermal contact resistance of R?t,c. The bar is of thermalconductivity k, and its width is twice its thickness L. Consider steady-state conditions for which heat is uni-formly generated at a volumetric rate q.due to passageof an electric current. Using the energy balance method,derive finite-difference equations for nodes 1 and 13.

Derive the nodal finite-difference equations for the fol-lowing configurations.(a) Node (m,n) on a diagonal boundary subjected toconvection with a fluid at T?and a heat transfercoefficient h. Assume x??y. (b) Node (m,n) at the tip of a cutting tool with the uppersurface exposed to a constant heat flux q?o, and thediagonal surface exposed to a convection coolingprocess with the fluid at T?and a heat transfercoefficient h. Assume xy ?? .

Consider the nodal point 0 located on the boundarybetween materials of thermal conductivity kA and kB. Derive the finite-difference equation, assuming no internal generation.

onsider the two-dimensional grid (x??y) represent-ing steady-state conditions with no internal volumetricgeneration for a system with thermal conductivity k. Oneof the boundaries is maintained at a constant temperatureTswhile the others are adiabatic.Derive an expression for the heat rate per unit length nor-mal to the page crossing the isothermal boundary (Ts).

Consider a one-dimensional fin of uniform cross-sectional area, insulated at its tip, x=L. (See Table 3.4,case B). The temperature at the base of the fin Tb and of the adjoining fluid T, as well as the heat transfer coefficient hand the thermal conductivity k, are known.(a) Derive the finite-difference equation for any interior node m.(b) Derive the finite-difference equation for a node n located at the insulated tip.

Consider the network for a two-dimensional system with-out internal volumetric generation having nodal tempera-tures shown below. If the grid spacing is 125 mm and thethermal conductivity of the material is 50 W/mK, calculate the heat rate per unit length normal to the page fromthe isothermal surface (Ts).

An ancient myth describes how a wooden ship wasdestroyed by soldiers who reflected sunlight from theirpolished bronze shields onto its hull, setting the shipablaze. To test the validity of the myth, a group of col-lege students are given mirrors and they reflect sunlightonto a 100 mm?100 mm area of a t?10-mm-thickplywood mockup characterized by k?0.8 W/m?K. Thebottom of the mockup is in water at Tw?20C, whilethe air temperature is T??25C. The surroundings areat Tsur?23C. The wood has an emissivity of ??0.90;both the front and back surfaces of the plywood are char-acterized by h?5 W/m2?K. The absorbed irradiationfrom the Nstudents mirrors is GS,N?70,000 W/m2onthe front surface of the mockup.(a) A debate ensues concerning where the beam shouldbe focused, location A or location B. Using a finite difference method with x??y?100 mm andtreating the wood as a two-dimensional extendedsurface (Figure 3.17a), enlighten the students as towhether location A or location B will be moreeffective in igniting the wood by determining themaximum local steady-state temperature.(b) Some students wonder whether the same techniquecan be used to melt a stainless steel hull. Repeatpart (a) considering a stainless steel mockup of thesame dimensions with k?15 W/m?K and ??0.2.The value of the absorbed irradiation is the same asin part (a).

Consider the square channel shown in the sketch oper-ating under steady-state conditions. The inner surfaceof the channel is at a uniform temperature of 600 K,while the outer surface is exposed to convection with afluid at 300 K and a convection coefficient of 50 W/m2?K.From a symmetrical element of the channel, a two-dimensional grid has been constructed and the nodeslabeled. The temperatures for nodes 1, 3, 6, 8, and 9 areidentified.(a) Beginning with properly defined control volumes,derive the finite-difference equations for nodes 2, 4, and 7 and determine the temperatures T2, T4, andT7(K).(b) Calculate the heat loss per unit length from thechannel.

A long conducting rod of rectangular cross section(20 mm?30 mm) and thermal conductivity k?20 W/m?K experiences uniform heat generation at arate q.?5?107W/m3, while its surfaces are main-tained at 300 K.(a) Using a finite-difference method with a grid spac-ing of 5 mm, determine the temperature distributionin the rod.(b) With the boundary conditions unchanged, whatheat generation rate will cause the midpoint tem-perature to reach 600 K?

A flue passing hot exhaust gases has a square crosssection, 300 mm to a side. The walls are constructed of refractory brick 150 mm thick with a thermal conductivity of 0.85 W/mK. Calculate the heat loss from theflue per unit length when the interior and exterior surfaces are maintained at 350 and 25C, respectively. Use a grid spacing of 75 mm.

Steady-state temperatures (K) at three nodal points of along rectangular rod are as shown. The rod experiencesa uniform volumetric generation rate of 5?107W/m3and has a thermal conductivity of 20 W/m?K. Two of itssides are maintained at a constant temperature of 300 K,while the others are insulated.(a) Determine the temperatures at nodes 1, 2, and 3.(b) Calculate the heat transfer rate per unit length(W/m) from the rod using the nodal temperatures.Compare this result with the heat rate calculatedfrom knowledge of the volumetric generation rateand the rod dimensions.

Functionally graded materials are intentionally fabri-cated to establish a spatial distribution of properties inthe final product. Consider an L?Ltwo-dimensionalobject with L?20 mm. The thermal conductivity dis-tribution of the functionally graded material is k(x)?20 W/m?K?(7070 W/m5/2?K) x3/2. Two sets of boundaryconditions, denoted as cases 1 and 2, are applied.Case Surface Boundary Condition11T?100C2T?50C3Adiabatic4Adiabatic21Adiabatic2Adiabatic3T?50C4T?100C(a) Determine the spatially averaged value of the ther-mal conductivity . Use this value to estimate theheat rate per unit length for cases 1 and 2.(b) Using a grid spacing of 2 mm, determine the heatrate per unit depth for case 1. Compare your resultto the estimated value calculated in part (a).(c) Using a grid spacing of 2 mm, determine the heatrate per unit depth for case 2. Compare your resultto the estimated value calculated in part (a).

Steady-state temperatures at selected nodal points ofthe symmetrical section of a flow channel are knownto be T2?95.47?C, T3?117.3?C, T5?79.79?C, T6?77.29?C, T8?87.28?C, and T10?77.65?C. Thewall experiences uniform volumetric heat generation ofq.?106W/m3and has a thermal conductivity of k?10 W/m?K. The inner and outer surfaces of the chan-nel experience convection with fluid temperatures ofT?,i?50?C and T?,o?25?C and convection coeffi-cients of hi?500 W/m2?K and ho?250 W/m2?K.(a) Determine the temperatures at nodes 1, 4, 7, and 9.(b) Calculate the heat rate per unit length (W/m) fromthe outer surface A to the adjacent fluid.(c) Calculate the heat rate per unit length from theinner fluid to surface B.(d) Verify that your results are consistent with an over-all energy balance on the channel section.

Consider an aluminum heat sink (k?240 W/m?K),such as that shown schematically in Problem 4.28. The inner and outer widths of the square channel arew?20 mm and W?40 mm, respectively, and an outersurface temperature of Ts?50?C is maintained by thearray of electronic chips. In this case, it is not the innersurface temperature that is known, but conditions(T?,h) associated with coolant flow through the chan-nel, and we wish to determine the rate of heat transferto the coolant per unit length of channel. For this pur-pose, consider a symmetrical section of the channel anda two-dimensional grid with x??y?5 mm.(a) For T??20?C and h?5000 W/m2?K, determinethe unknown temperatures, T1, ...,T7, and the rateof heat transfer per unit length of channel, q?. (b) Assess the effect of variations in hon the unknowntemperatures and the heat rate.

Conduction within relatively complex geometries cansometimes be evaluated using the finite- differencemethods of this text that are applied to subdomainsandpatchedtogether. Consider the two- dimensionaldomain formed by rectangular and cylindrical subdo-mains patched at the common, dashed control surface.Note that, along the dashed control surface, tempera-tures in the two subdomains are identical and localconduction heat fluxes to the cylindrical subdomain are identical to local conduction heat fluxes from therectangular subdomain. Calculate the heat transfer per unit depth into the page,q?, using x??y??r?10 mm and ??/8. Thebase of the rectangular subdomain is held at Th?20C,while the vertical surface of the cylindrical subdomainand the surface at outer radius roare at Tc?0C. Theremaining surfaces are adiabatic, and the thermal con-ductivity is k?10 W/m?K.

Consider the two-dimensional tube of a noncircularcross section formed by rectangular and semicylindricalsubdomains patched at the common dashed control sur-faces in a manner similar to that described in Problem4.59. Note that, along the dashed control surfaces, tem-peratures in the two subdomains are identical and localconduction heat fluxes to the semicylindrical subdo-main are identical to local conduction heat fluxes fromthe rectangular subdomain. The bottom of the domainis held at Ts?100C by condensing steam, while theflowing fluid is characterized by the temperature andconvection coefficient shown in the sketch. Theremaining surfaces are insulated, and the thermal con-ductivity is k?15 W/m?K.Find the heat transfer rate per unit length of tube, q?,using x??y??r?10 mm and ??/8. Hint:Take advantage of the symmetry of the problem byconsidering only half of the entire domain.

The steady-state temperatures (C) associated with selec-ted nodal points of a two-dimensional system having athermal conductivity of 1.5 W/m?K are shown on theaccompanying grid.Insulatedboundary0.1 m0.1 m129.4T245.8137.0 103.5T3172.9T1132.8 67.0Isothermal boundaryT0 = 200CT = 30Ch = 50 W/m2KDi= 40 mmL = Do= 80 mmT,i= 20Chi= 240 W/m2KTs= 100Crxyk = 15 W/mKt = 10 mm268Chapter 4?Two-Dimensional, Steady-State ConductionCH004.qxd 2/21/11 8:53 AM Page 268 (a) Determine the temperatures at nodes 1, 2, and 3.(b) Calculate the heat transfer rate per unit thicknessnormal to the page from the system to the fluid.

A steady-state, finite-difference analysis has been per-formed on a cylindrical fin with a diameter of 12 mmand a thermal conductivity of 15 W/m?K. The convec-tion process is characterized by a fluid temperature of25C and a heat transfer coefficient of 25 W/m2?K.(a) The temperatures for the first three nodes, sepa-rated by a spatial increment of x?10 mm, aregiven in the sketch. Determine the fin heat rate.(b) Determine the temperature at node 3, T3.

Consider the two-dimensional domain shown. All sur-faces are insulated except for the isothermal surfaces atx?0 and L.(a) Use a one-dimensional analysis to estimate theshape factor S.(b) Estimate the shape factor using a finite differenceanalysis with x??y?0.05L. Compare youranswer with that of part (a), and explain the differ-ence between the two solutions.

Consider two-dimensional, steady-state conduction in asquare cross section with prescribed surface temperatures. (a) Determine the temperatures at nodes 1, 2, 3, and 4.Estimate the midpoint temperature.(b) Reducing the mesh size by a factor of 2, determinethe corresponding nodal temperatures. Compareyour results with those from the coarser grid.(c) From the results for the finer grid, plot the 75, 150,and 250C isotherms.

Consider a long bar of square cross section (0.8 m to theside) and of thermal conductivity 2 W/mK. Three of thesesides are maintained at a uniform temperature of 300C.The fourth side is exposed to a fluid at 100C for whichthe convection heat transfer coefficient is 10 W/mK.(a) Using an appropriate numerical technique with agrid spacing of 0.2 m, determine the midpoint tem-perature and heat transfer rate between the bar andthe fluid per unit length of the bar.(b) Reducing the grid spacing by a factor of 2,determine the midpoint temperature and heattransfer rate. Plot the corresponding temperaturedistribution across the surface exposed to the fluid.Also, plot the 200 and 250C isotherms.

Consider a two-dimensional, straight triangular fin oflength L?50 mm and base thickness t?20 mm. Thethermal conductivity of the fin is k?25 W/m?K. Thebase temperature is Tb?50C, and the fin is exposed toconvection conditions characterized by h?50 W/m2?K,T??20C. Using a finite difference mesh with x?10 mm and y?2 mm, and taking advantage of symme-try, determine the fin efficiency, f. Compare your value ofthe fin efficiency with that reported in Figure 3.19.

A common arrangement for heating a large surface areais to move warm air through rectangular ducts belowthe surface. The ducts are square and located midwaybetween the top and bottom surfaces that are exposed toroom air and insulated, respectively.1.5LConcreteT1 = 80CT2 = 30CAir ductLLLxyT= 20Ch = 50 W/m2Kt = 20 mmL = 50 mm171216192128131720391418410155116?Problems269CH004.qxd 2/21/11 8:53 AM Page 269 For the condition when the floor and duct temperaturesare 30 and 80C, respectively, and the thermal conductiv-ity of concrete is 1.4 W/m?K, calculate the heat rate fromeach duct, per unit length of duct. Use a grid spacing withx?2y, where ? .125Land L?150 mm.

Consider the gas turbine cooling scheme of Example 4.3.In Problem 3.23, advantages associated with applying athermal barrier coating(TBC) to the exterior surface ofa turbine blade are described. If a 0.5-mm-thick zirco-nia coating (k?1.3 W/m?K, R?t,c?10?4m2?K/W) isapplied to the outer surface of the air- cooled blade,determine the temperature field in the blade for theoperating conditions of Example 4.3.

A long, solid cylinder of diameter D?25 mm is formedof an insulating core that is covered with a very thin(t?50 ?m), highly polished metal sheathing of thermalconductivity k?25 W/m?K. Electric current flowsthrough the stainless steel from one end of the cylinder tothe other, inducing uniform volumetric heating within thesheathing of q.?5 ?106W/m3. As will become evidentin Chapter 6, values of the convection coefficientbetween the surface and air for this situation are spa-tially nonuniform, and for the airstream conditions ofthe experiment, the convection heat transfer coefficientvaries with the angle ?as h(?)?26?0.637??8.92?2for 0???/2 and h(?)?5 for ?/2??.(a) Neglecting conduction in the ?-direction within thestainless steel, plot the temperature distributionT(?) for 0??for T??25C. (b) Accounting for ?-direction conduction in the stain-less steel, determine temperatures in the stainlesssteel at increments of ???/20 for 0??.Compare the temperature distribution with that ofpart (a).Hint: The temperature distribution is symmetricalabout the horizontal centerline of the cylinder.

Consider Problem 4.69. An engineer desires to measure the surface temperature of the thin sheathing by paintingit black (=0.98) and using an infrared measurement device to nonintrusively determine the surface tempera-ture distribution. Predict the temperature distribution ofthe painted surface, accounting for radiation heat transfer with large surroundings at Tsur = 25C.

Consider using the experimental methodology of Prob-lem 4.70 to determine the convection coefficient distri-bution about an airfoil of complex shape.Accounting for conduction in the metal sheathing andradiation losses to the large surroundings, determine theconvection heat transfer coefficients at the locationsshown. The surface locations at which the temperaturesare measured are spaced 2 mm apart. The thickness ofthe metal sheathing is t?20 ?m, the volumetric gener-ation rate is q.?20?106W/m3, the sheathings ther-mal conductivity is k?25 W/m?K, and the emissivityof the painted surface is ??0.98. Compare yourresults to cases where (i) both conduction along thesheathing and radiation are neglected, and (ii) whenonly radiation is neglected.TemperatureTemperature Temperature Location(C)Location(C)Location(C)1 27.77 11 34.29 21 31.132 27.67 12 36.78 22 30.643 27.71 13 39.29 23 30.604 27.83 14 41.51 24 30.775 28.06 15 42.68 25 31.166 28.47 16 42.84 26 31.527 28.98 17 41.29 27 31.858 29.67 18 37.89 28 31.519 30.66 19 34.51 29 29.9110 32.18 20 32.36 30 28.42

A thin metallic foil of thickness 0.25 mm with a patternof extremely small holes serves as an acceleration gridto control the electrical potential of an ion beam. Such agrid is used in a chemical vapor deposition (CVD)process for the fabrication of semiconductors. The topsurface of the grid is exposed to a uniform heat flux caused by absorption of the ion beam, q"s = 600 W/m.The edges of the foil are thermally coupled to water-cooled sinks maintained at 300 K. The upper and lowersurfaces of the foil experience radiation exchange withthe vacuum enclosure walls maintained at 300 K. Theeffective thermal conductivity of the foil material is40 W/mK, and its emissivity is 0.45. Assuming one-dimensional conduction and using afinite-difference method representing the grid by 10nodes in the x-direction, estimate the temperature distribution for the grid. Hint:For each node requiring anenergy balance, use the linearized form of the radiationrate equation, Equation 1.8, with the radiation coefficienthr, Equation 1.9, evaluated for each node.

A long bar of rectangular cross section, 0.4 m?0.6 m ona side and having a thermal conductivity of 1.5 W/m?K,is subjected to the boundary conditions shown.Two of the sides are maintained at a uniform tempera-ture of 200C. One of the sides is adiabatic, and theremaining side is subjected to a convection processwith T??30C and h?50 W/m2?K. Using an appro-priate numerical technique with a grid spacing of 0.1 m,determine the temperature distribution in the bar andthe heat transfer rate between the bar and the fluid perunit length of the bar.

The top surface of a plate, including its grooves, is main-tained at a uniform temperature of T1=200C. The lowersurface is at T2=20C, the thermal conductivity is15 W/mK, and the groove spacing is 0.16 m. (a) Using a finite-difference method with a mesh sizeof x??y?40 mm, calculate the unknown nodaltemperatures and the heat transfer rate per width ofgroove spacing (w) and per unit length normal tothe page.(b) With a mesh size of x??y?10 mm, repeat theforegoing calculations, determining the temperaturefield and the heat rate. Also, consider conditions forwhich the bottom surface is not at a uniform tempera-ture T2but is exposed to a fluid at T??20C. Withx??y?10 mm, determine the temperature fieldand heat rate for values of h?5, 200, and 1000W/m2?K, as well as for h?.

Refer to the two-dimensional rectangular plate of Prob-lem 4.2. Using an appropriate numerical method withx??y?0.25 m, determine the temperature at themidpoint (1, 0.5).

The shape factor for conduction through the edge ofadjoining walls for which D?L/5, where Dand Larethe wall depth and thickness, respectively, is shown inTable 4.1. The two-dimensional symmetrical elementof the edge, which is represented by inset (a), isbounded by the diagonal symmetry adiabat and a sec-tion of the wall thickness over which the temperaturedistribution is assumed to be linear between T1and T2.(a) Using the nodal network of inset (a) with L?40 mm,determine the temperature distribution in the elementfor T1?100C and T2?0C. Evaluate the heat rate per unit depth (D?1 m) if k?1W/m?K. Determinethe corresponding shape factor for the edge, andcompare your result with that from Table 4.1.(b) Choosing a value of n?1 or n?1.5, establish anodal network for the trapezoid of inset (b) anddetermine the corresponding temperature field.Assess the validity of assuming linear temperaturedistributions across sections aaand bb.

The diagonal of a long triangular bar is well insulated,while sides of equivalent length are maintained at uni-form temperatures Taand Tb.(a) Establish a nodal network consisting of five nodesalong each of the sides. For one of the nodes on thediagonal surface, define a suitable control volume andderive the corresponding finite-difference equation.Using this form for the diagonal nodes and appropri-ate equations for the interior nodes, find the temper-ature distribution for the bar. On a scale drawing ofthe shape, show the 25, 50, and 75C isotherms.(b) An alternate and simpler procedure to obtain thefinite-difference equations for the diagonal nodes fol-lows from recognizing that the insulated diagonal sur-face is a symmetry plane. Consider a square 5 ?5nodal network, and represent its diagonal as a symme-try line. Recognize which nodes on either side of thediagonal have identical temperatures. Show that youcan treat the diagonal nodes as interior nodes andwrite the finite- difference equations by inspection.

A straight fin of uniform cross section is fabricated froma material of thermal conductivity 50 W/m?K, thicknessw?6 mm, and length L?48 mm, and it is very long inthe direction normal to the page. The convection heattransfer coefficient is 500 W/m2?K with an ambient airtemperature of T??30C. The base of the fin is main-tained at Tb?100C, while the fin tip is well insulated. (a) Using a finite-difference method with a spaceincrement of 4 mm, estimate the temperature distri-bution within the fin. Is the assumption of one-dimensional heat transfer reasonable for this fin?(b) Estimate the fin heat transfer rate per unit lengthnormal to the page. Compare your result with theone-dimensional, analytical solution, Equation 3.81.(c) Using the finite-difference mesh of part (a), computeand plot the fin temperature distribution for valuesof h?10, 100, 500, and 1000 W/m2?K. Determineand plot the fin heat transfer rate as a function of h.

A rod of 10-mm diameter and 250-mm length has oneend maintained at 100C. The surface of the rod expe-riences free convection with the ambient air at 25Cand a convection coefficient that depends on the differ-ence between the temperature of the surface and theambient air. Specifically, the coefficient is prescribedby a correlation of the form, hfc?2.89[0.6?0.624(T?T?)1/6]2, where the units are hfc(W/m2?K) and T(K).The surface of the rod has an emissivity ??0.2 andexperiences radiation exchange with the surroundingsat Tsur?25C. The fin tip also experiences free convec-tion and radiation exchange.Assuming one-dimensional conduction and using afinite-difference method representing the fin by fivenodes, estimate the temperature distribution for the fin.Determine also the fin heat rate and the relative contri-butions of free convection and radiation exchange.Hint:For each node requiring an energy balance, usethe linearized form of the radiation rate equation, Equa-tion 1.8, with the radiation coefficient hr, Equation 1.9,evaluated for each node. Similarly, for the convectionrate equation associated with each node, the free con-vection coefficient hfcmust be evaluated for each node.

A simplified representation for cooling in very large-scaleintegration (VLSI) of microelectronics is shown in thesketch. A silicon chip is mounted in a dielectric substrate,and one surface of the system is convectively cooled,while the remaining surfaces are well insulated from thesurroundings. The problem is rendered two-dimensional by assuming the system to be very long in the directionperpendicular to the paper. Under steady-state operation,electric power dissipation in the chip provides for uni-form volumetric heating at a rate of q.. However, theheating rate is limited by restrictions on the maximumtemperature that the chip is allowed to achieve.For the conditions shown on the sketch, will the maxi-mum temperature in the chip exceed 85C, the maximumallowable operating temperature set by industry stan-dards? A grid spacing of 3 mm is suggested.

A heat sink for cooling computer chips is fabricated fromcopper (ks?400 W/m?K), with machined microchan-nels passing a cooling fluid for which T?25C and h?30,000 W/m2?K. The lower side of the sink experi-ences no heat removal, and a preliminary heat sink designcalls for dimensions of a?b?ws?wf?200?m. Asymmetrical element of the heat path from the chip to thefluid is shown in the inset.(a) Using the symmetrical element with a square nodalnetwork of x??y?100?m, determine the corre-sponding temperature field and the heat rate q?to thecoolant per unit channel length (W/m) for a maximumallowable chip temperature Tc,max?75C. Estimatethe corresponding thermal resistance between thechip surface and the fluid, R?t,c?(m?K/W). What isthe maximum allowable heat dissipation for a chipthat measures 10 mm?10 mm on a side? (b) The grid spacing used in the foregoing finite- differencesolution is coarse, resulting in poor precision forthe temperature distribution and heat removal rate.Investigate the effect of grid spacing by consider-ing spatial increments of 50 and 25?m.(c) Consistent with the requirement that a?b?400?m,can the heat sink dimensions be altered in a mannerthat reduces the overall thermal resistance?

A plate (k?10 W/m?K) is stiffened by a series of lon-gitudinal ribs having a rectangular cross section withlength L?8 mm and width w?4 mm. The base of theplate is maintained at a uniform temperature Tb?45C, while the rib surfaces are exposed to air at a tem-perature of T??25C and a convection coefficient of h?600 W/m2?K.(a) Using a finite-difference method with x??y?2 mm and a total of 5 ?3 nodal points and regions,estimate the temperature distribution and the heatrate from the base. Compare these results with thoseobtained by assuming that heat transfer in the rib isone-dimensional, thereby approximating the behav-ior of a fin.(b) The grid spacing used in the foregoing finite-difference solution is coarse, resulting in poor pre-cision for estimates of temperatures and the heatrate. Investigate the effect of grid refinement byreducing the nodal spacing to x??y?1 mm (a9?3 grid) considering symmetry of the center line.(c) Investigate the nature of two-dimensional conduc-tion in the rib and determine a criterion for which theone-dimensional approximation is reasonable. Do soby extending your finite-difference analysis to deter-mine the heat rate from the base as a function of thelength of the rib for the range 1.5 L/w10, keep-ing the length Lconstant. Compare your results withthose determined by approximating the rib as a fin.

The bottom half of an I-beam providing support for afurnace roof extends into the heating zone. The web iswell insulated, while the flange surfaces experience convection with hot gases at T??400C and a convec-tioncoefficient of h?150 W/m2?K. Consider thesymmetrical element of the flange region (inset a),assuming that the temperature distribution across theweb is uniform at Tw?100C. The beam thermal con-ductivity is 10 W/m?K, and its dimensions arew?80 mm, ww?30 mm, and L?30 mm.(a) Calculate the heat transfer rate per unit length tothe beam using a 5 ?4 nodal network.(b) Is it reasonable to assume that the temperature dis-tribution across the webflange interface is uni-form? Consider the L-shaped domain of inset (b)and use a fine grid to obtain the temperature distri-bution across the webflange interface. Make thedistance wo?ww/2.

A long bar of rectangular cross section is 60 mm?90 mm on a side and has a thermal conductivity of 1W/mK. One surface is exposed to a convection process with air at 100C and a convection coefficient of 100 W/mK, while the remaining surfaces are maintained at 50C. (a) Using a grid spacing of 30 mm and the Gauss-Seideliteration method, determine the nodal temperaturesand the heat rate per unit length normal to the pageinto the bar from the air.(b) Determine the effect of grid spacing on the temper-ature field and heat rate. Specifically, consider agrid spacing of 15 mm. For this grid, explore theeffect of changes in hon the temperature field andthe isotherms.

A long trapezoidal bar is subjected to uniform tempera-tures on two surfaces, while the remaining surfaces arewell insulated. If the thermal conductivity of the mate-rial is 20 W/mK, estimate the heat transfer rate per unitlength of the bar using a finite-difference method. Usethe GaussSeidel method of solution with a space increment of 10 mm.

Small-diameter electrical heating elements dissipating50 W/m (length normal to the sketch) are used to heat aceramic plate of thermal conductivity 2 W/m?K. Theupper surface of the plate is exposed to ambient air at30C with a convection coefficient of 100 W/m2?K,while the lower surface is well insulated.(a) Using the GaussSeidel method with a grid spac-ing of x?6 mm and y?2 mm, obtain the tem-perature distribution within the plate.(b) Using the calculated nodal temperatures, sketchfour isotherms to illustrate the temperature distri-bution in the plate.(c) Calculate the heat loss by convection from theplate to the fluid. Compare this value with the ele-ment dissipation rate. (d) What advantage, if any, is there in not makingx??yfor this situation?(e) With x??y?2 mm, calculate the temperaturefield within the plate and the rate of heat transferfrom the plate. Under no circumstances may thetemperature at any location in the plate exceed400C. Would this limit be exceeded if the airflowwere terminated and heat transfer to the air were bynatural convection with h?10 W/m2?K?

A straight fin of uniform cross section is fabricatedfrom a material of thermal conductivity k?5W/m?K,thickness w?20 mm, and length L?200 mm. The finis very long in the direction normal to the page. Thebase of the fin is maintained at Tb?200C, and the tipcondition allows for convection (case A of Table 3.4),with h?500 W/m2?K and T??25C.(a) Assuming one-dimensional heat transfer in the fin,calculate the fin heat rate, , and the tiptemperature TL. Calculate the Biot number for thefin to determine whether the one-dimensionalassumption is valid.(b) Using the finite-element method of FEHT, perform atwo-dimensional analysis on the fin to determine thefin heat rate and tip temperature. Compare yourresults with those from the one-dimensional, analyti-cal solution of part (a). Use the View/TemperatureContoursoption to display isotherms, and discusskey features of the corresponding temperature fieldand heat flow pattern. Hint: In drawing the outline ofthe fin, take advantage of symmetry. Use a fine meshnear the base and a coarser mesh near the tip. Why?(c) Validate your FEHTmodel by comparing predic-tions with the analytical solution for a fin withthermal conductivities of k?50 W/m?K and500 W/m?K. Is the one-dimensional heat transferassumption valid for these conditions?

Consider the long rectangular bar of Problem 4.84 withthe prescribed boundary conditions.(a) Using the finite-element method of FEHT, determinethe temperature distribution. Use the View/Tempera-ture Contourscommand to represent the isotherms.Identify significant features of the distribution.(b) Using the View/Heat Flowscommand, calculatethe heat rate per unit length (W/m) from the bar tothe airstream.(c) Explore the effect on the heat rate of increasing theconvection coefficient by factors of two and three.Explain why the change in the heat rate is not pro-portional to the change in the convection coefficient.

Consider the long rectangular rod of Problem 4.53,which experiences uniform heat generation while itssurfaces are maintained at a fixed temperature.(a) Using the finite-element method of FEHT, determinethe temperature distribution. Use the View/Tempera-ture Contourscommand to represent the isotherms.Identify significant features of the distribution.(b) With the boundary conditions unchanged, whatheat generation rate will cause the midpoint tem-perature to reach 600 K?

Consider the symmetrical section of the flow channel ofProblem 4.57, with the prescribed values of q., k, T?,i,T?,o, hi, and ho. Use the finite-element method of FEHTto obtain the following results.(a) Determine the temperature distribution in the sym-metrical section, and use the View/TemperatureContourscommand to represent the isotherms.Identify significant features of the temperature dis-tribution, including the hottest and coolest regionsand the region with the steepest gradients. Describethe heat flow field.(b) Using the View/Heat Flowscommand, calculatethe heat rate per unit length (W/m) from the outersurface A to the adjacent fluid.(c) Calculate the heat rate per unit length from theinner fluid to surface B.(d) Verify that your results are consistent with an over-all energy balance on the channel section.

The hot-film heat flux gage shown schematically maybe used to determine the convection coefficient of anadjoining fluid stream by measuring the electric powerdissipation per unit area, P?e (W/m2), and the averagesurface temperature, Ts,f, of the film. The power dissi-pated in the film is transferred directly to the fluid byconvection, as well as by conduction into the substrate. If substrate conduction is negligible, the gage measure-ments can be used to determine the convection coeffi-cient without application of a correction factor. Yourassignment is to perform a two-dimensional, steady-state conduction analysis to estimate the fraction of thepower dissipation that is conducted into a 2-mm- thickquartz substrate of width W?40 mm and thermal con-ductivity k?1.4 W/m?K. The thin, hot-film gage has awidth of w?4 mm and operates at a uniform powerdissipation of 5000 W/m2. Consider cases for which thefluid temperature is 25C and the convection coefficientis 500, 1000, and 2000 W/m2?K.Use the finite-element method of FEHTto analyze asymmetrical half-section of the gage and the quartzsubstrate. Assume that the lower and end surfaces ofthe substrate are perfectly insulated, while the uppersurface experiences convection with the fluid.(a) Determine the temperature distribution and the con-duction heat rate into the region below the hot filmfor the three values of h. Calculate the fractions ofelectric power dissipation represented by theserates. Hint: Use the View/Heat Flowcommand tofind the heat rate across the boundary elements.(b) Use the View/Temperature Contourscommand toview the isotherms and heat flow patterns. Describethe heat flow paths, and comment on features of thegage design that influence the paths. What limita-tions on applicability of the gage have beenrevealed by your analysis?

Consider the system of Problem 4.54. The interior sur-face is exposed to hot gases at 350C with a convectioncoefficient of 100 W/mK, while the exterior surface experiences convection with air at 25C and a convection coefficient of 5 W/mK.(a) Using a grid spacing of 75 mm, calculate the temper- ature field within the system and determine the heatloss per unit length by convection from the outer sur-face of the flue to the air. Compare this result withthe heat gained by convection from the hot gases tothe air. (b) Determine the effect of grid spacing on the tem-perature field and heat loss per unit length to theair. Specifically, consider a grid spacing of 25 mmand plot appropriately spaced isotherms on aschematic of the system. Explore the effect ofchanges in the convection coefficients on the tem-perature field and heat loss.

Electronic devices dissipating electrical power can becooled by conduction to a heat sink. The lower surfaceof the sink is cooled, and the spacing of the devices ws,the width of the device wd, and the thickness Land ther-mal conductivity kof the heat sink material each affectthe thermal resistance between the device and thecooled surface. The function of the heat sink is tospreadthe heat dissipated in the device throughout thesink material.(a) Beginning with the shaded symmetrical element,use a coarse (5?5) nodal network to estimate thethermal resistance per unit depth between the deviceand lower surface of the sink, R?t,d?s(m?K/W). Howdoes this value compare with thermal resistancesbased on the assumption of one-dimensional con-duction in rectangular domains of (i) width wdandlength Land (ii) width wsand length L?(b) Using nodal networks with grid spacings three andfive times smaller than that in part (a), determinethe effect of grid size on the precision of the ther- mal resistance calculation.(c) Using the finer nodal network developed for part (b), determine the effect of device width on thethermal resistance. Specifically, keeping wsand Lfixed, find the thermal resistance for values ofwd/ws?0.175, 0.275, 0.375, and 0.475.

Consider one-dimensional conduction in a planecomposite wall. The exposed surfaces of materials Aand B are maintained at T1?600 K and T2?300 K,respectively. Material A, of thickness La?20 mm,has a temperature-dependent thermal conductivity ofka?ko[1??(T?To)], where ko?4.4 W/m?K, ??0.008 K?1, To?300 K, and T is in kelvins.Material B is of thickness Lb?5 mm and has a ther-mal conductivity of kb?1W/m?K. (a) Calculate the heat flux through the composite wallby assuming material A to have a uniform thermalconductivity evaluated at the average temperatureof the section.(b) Using a space increment of 1 mm, obtain the finite-difference equations for the internal nodes andcalculate the heat flux considering the temperature-dependent thermal conductivity for Material A. Ifthe IHTsoftware is employed, call-up functionsfrom Tools/Finite-Difference Equationsmay be usedto obtain the nodal equations. Compare your resultwith that obtained in part (a).(c) As an alternative to the finite- difference method ofpart (b), use the finite-element method of FEHTtocalculate the heat flux, and compare the result withthat from part (a). Hint: In the Specify/MaterialPropertiesbox, properties may be entered as a func-tion of temperature (T), the space coordinates (x,y),or time (t). See the Helpsection for more details.

A platen of thermal conductivity k?15 W/m?K isheated by flow of a hot fluid through channels of widthL?20 mm, with T?,i?200?C and hi?500 W/m2?K.The upper surface of the platen is used to heat a processfluid at T?,o?25?C with a convection coefficient ofho?250 W/m2?K. The lower surface of the platen isinsulated. To heat the process fluid uniformly, the tem-perature of the platens upper surface must be uniformto within 5?C. Use a finite-difference method, such as that of IHT, or the finite-element method of FEHTtoobtain the following results.(a) Determine the maximum allowable spacing Wbetween the channel centerlines that will satisfy thespecified temperature uniformity requirement.(b) What is the corresponding heat rate per unit lengthfrom a flow channel?

Consider the cooling arrangement for the very large-scaleintegration (VLSI) chip of Problem 4.93. Use the finite-element method of FEHTto obtain the following results.(a) Determine the temperature distribution in the chip-substrate system. Will the maximum temperatureexceed 85C?(b) Using the FEHTmodel developed for part (a),determine the volumetric heating rate that yields amaximum temperature of 85C.(c) What effect would reducing the substrate thicknesshave on the maximum operating temperature? For avolumetric generation rate of q.?107W/m3, reducethe thickness of the substrate from 12 to 6 mm, keep-ing all other dimensions unchanged. What is the max-imum system temperature for these conditions? Whatfraction of the chip power generation is removed byconvection directly from the chip surface?