A long, cylindrical, electrical heating element of

Consider the following fluids at a film temperature of300 K in parallel flow over a flat plate with velocityof 1 m/s: atmospheric air, water, engine oil, and mercury.(a) For each fluid, determine the velocity and thermalboundary layer thicknesses at a distance of 40 mmfrom the leading edge.(b) For each of the prescribed fluids and on the samecoordinates, plot the boundary layer thicknesses as a function of distance from the leading edge to aplate length of 40 mm.

Engine oil at 100?C and a velocity of 0.1 m/s flows overboth surfaces of a 1-m-long flat plate maintained at20?C. Determine:(a) The velocity and thermal boundary layer thick-nesses at the trailing edge.(b) The local heat flux and surface shear stress at thetrailing edge.(c) The total drag force and heat transfer per unit widthof the plate.(d) Plot the boundary layer thicknesses and local valuesof the surface shear stress, convection coefficient,and heat flux as a function of xfor 0x1m

Consider steady, parallel flow of atmospheric air over aflat plate. The air has a temperature and free streamvelocity of 300 K and 25 m/s.(a) Evaluate the boundary layer thickness at distancesof x?1, 10, and 100 mm from the leading edge.If a second plate were installed parallel to and at adistance of 3 mm from the first plate, what is thedistance from the leading edge at which boundarylayer merger would occur?(b) Evaluate the surface shear stress and the y-velocitycomponent at the outer edge of the boundary layerfor the single plate at x?1, 10, and 100 mm.(c) Comment on the validity of the boundary layerapproximations

Consider a liquid metal (Pr1), with free stream con-ditions u?and T?, in parallel flow over an isothermalflat plate at Ts. Assuming that u?u?throughout thethermal boundary layer, write the corresponding formof the boundary layer energy equation. Applyingappropriate initial (x?0) and boundary conditions,solve this equation for the boundary layer temperaturefield, T(x, y). Use the result to obtain an expression forthe local Nusselt number Nux. Hint:This problem isanalogous to one- dimensional heat transfer in a semi-infinite medium with a sudden change in surfacetemperature

Consider the velocity boundary layer profile for flowover a flat plate to be of the form u = C1 + C2y. Apply-ing appropriate boundary conditions, obtain an expres-sion for the velocity profile in terms of the boundarylayer thickness ? and the free stream velocity u?. Usingthe integral form of the boundary layer momentumequation (Appendix G), obtain expressions for theboundary layer thickness and the local friction coeffi-cient, expressing your result in terms of the localReynolds number. Compare your results with thoseobtained from the exact solution (Section 7.2.1) and theintegral solution with a cubic profile (Appendix G)

Consider a steady, turbulent boundary layer on anisothermal flat plate of temperature Ts. The boundarylayer is tripped at the leading edge x?0 by a finewire. Assume constant physical properties and velocityand temperature profiles of the formuu???y??1/ 7andT?T?Ts?T??1??y?t? (a) From experiment it is known that the surface shearstress is related to the boundary layer thickness byan expression of the formBeginning with the momentum integral equation(Appendix G), show that?/x?0.376Rex?1/5.Determine the average friction coefficient .(b) Beginning with the energy integral equation, obtainan expression for the local Nusselt number Nuxanduse this result to evaluate the average Nusseltnumber .

Consider flow over a flat plate for which it is desired todetermine the average heat transfer coefficient over theshort span x1to x2, , where (x2?x1) L.Provide three different expressions that can be used toevaluate in terms of (a) the local coefficient at(b) the local coefficients at x1and x2, and(c) the average coefficients at x1and x2. Indicate which ofthe expressions is approximate. Considering whether theflow is laminar, turbulent, or mixed, indicate when it isappropriate or inappropriate to use each of the equations.

A flat plate of width 1 m is maintained at a uniform sur-face temperature of Ts?150?C by using independentlycontrolled, heat-generating rectangular modules ofthickness a?10 mm and length b?50 mm. Eachmodule is insulated from its neighbors, as well as on itsback side. Atmospheric air at 25?C flows over the plateat a velocity of 30 m/s. The thermophysical propertiesof the module are k?5.2 W/m?K, cp?320 J/kg?K,and ??2300 kg/m3.(a) Find the required power generation, (W/m3), in amodule positioned at a distance 700 mm from theleading edge (b) Find the maximum temperature Tmaxin the heat-generating module.

An electric air heater consists of a horizontal array ofthin metal strips that are each 10 mm long in the direction of an airstream that is in parallel flow over the topof the strips. Each strip is 0.2 m wide, and 25 strips arearranged side by side, forming a continuous and smoothsurface over which the air flows at 2 m/s. During operation, each strip is maintained at 500C and the air isat 25C.(a) What is the rate of convection heat transfer fromthe first strip? The fifth strip? The tenth strip? All thestrips?(b) For air velocities of 2, 5, and 10 m/s, determine theconvection heat rates for all the locations of part (a).Represent your results in tabular or bar graph form.(c) Repeat part (b), but under conditions for which theflow is fully turbulent over the entire array of strips.

Consider atmospheric air at 25C and a velocity of 25 m/s flowing over both surfaces of a 1-m-long flatplate that is maintained at 125C. Determine the rate ofheat transfer per unit width from the plate for values ofthe critical Reynolds number corresponding to 105,5105, and 106

Consider laminar, parallel flow past an isothermal flatplate of length L, providing an average heat transfercoefficient of . If the plate is divided into Nsmallerplates, each of length determine an expres-sion for the ratio of the heat transfer coefficient aver-aged over the Nplates to the heat transfer coefficientaveraged over the single plate, .

Repeat Problem 7.11 for the case when the boundarylayer is tripped to a turbulent condition at its leadingedge

Consider a flat plate subject to parallel flow (top andbottom) characterized by u??5 m/s, T??20?C.(a) Determine the average convection heat transfercoefficient, convective heat transfer rate, and dragforce associated with an L?2-m-long, w?2-m-wide flat plate for airflow and surface temperaturesof Ts?50?C and 80?C.(b) Determine the average convection heat transfer coef-ficient, convective heat transfer rate, and drag forceassociated with an L?0.1-m-long, w?0.1-m-wideflat plate for water flow and surface temperatures ofTs?50C ? and 80?C

Consider water at 27?C in parallel flow over an isother-mal, 1-m-long flat plate with a velocity of 2 m/s.(a) Plot the variation of the local heat transfer coeffi-cient, hx(x), with distance along the plate for three flow conditions corresponding to transitionReynolds numbers of (i) 5105, (ii) 3105, and(iii) 0 (the flow is fully turbulent).(b) Plot the variation of the average heat transfer coef-ficient with distance for the three flow condi-tions of part (a).(c) What are the average heat transfer coefficients forthe entire plate for the three flow conditions ofpart (a)?

Explain under what conditions the total rate of heattransfer from an isothermal flat plate of dimensionsL2Lwould be the same, independent of whetherparallel flow over the plate is directed along the sideof length Lor 2L. With a critical Reynolds number of5105, for what values of ReLwould the total heattransfer be independent of orientation?

In fuel cell stacks, it is desirable to operate under condi-tions that promote uniform surface temperatures for theelectrolytic membranes. This is especially true in high-temperature fuel cells where the membrane is con-structed of a brittle ceramic material. Electrochemicalreactions in the electrolytic membranes generate ther-mal energy, while gases flowing above and below themembranes cool it. The stack designer may specify topand bottom flows that are in the same, opposite, ororthogonal directions. A preliminary study of the effectof the relative flow directions is conducted wherebya 150 mm150 mm thinsheet of material, producing auniform heat flux of 100 W/m2, is cooled (top and bot-tom) by air with a free stream temperature and velocityof 25?C and 2 m/s, respectively.(a) Determine the minimum and maximum local mem-brane temperatures for top and bottom flows thatare in the same, opposite, and orthogonal direc-tions. Which flow configuration minimizes themembrane temperature? Hint: For the opposite andorthogonal flow cases, the boundary layers are sub-ject to boundary conditions that are neither uniformtemperature nor uniform heat flux. It is, however,reasonable to expect that the resulting temperatureswould be bracketedby your answers based on theconstant heat flux and constant temperature bound-ary conditions.(b) Plot the surface temperature distribution T(x) forthe cases involving flow in the opposite and samedirections. Thermal stresses are undesirable and arerelated to the spatial temperature gradient alongthe membrane. Which configuration minimizesspatial temperature gradients?

Air at a pressure of 1 atm and a temperature of 50C isin parallel flow over the top surface of a flat plate that is heated to a uniform temperature of 100C. The plate has a length of 0.20 m (in the flow direction) and awidth of 0.10 m. The Reynolds number based on theplate length is 40,000. What is the rate of heat transferfrom the plate to the air? If the free stream velocity ofthe air is doubled and the pressure is increased to10 atm, what is the rate of heat transfer?

Consider the photovoltaic solar panel of Example 3.3.The heat transfer coefficient should no longer be takento be a specified value.(a) Determine the silicon temperature and the electricpower produced by the solar cell for an air velocityof 4 m/s parallel to the long direction, with air andsurroundings temperatures of 20?C. The boundarylayer is tripped to a turbulent condition at the lead-ing edge of the panel.(b) Repeat part (a), except now the panel is orientedwith its short side parallel to the airflow, that is,L?0.1 m and w?1m.(c) Plot the electric power output and the silicon tem- perature versus air velocity over the range 0um10 m /s for the L?0.1 m and w?1 m case

Concentration of sunlight onto photovoltaic cells isdesired since the concentrating mirrors and lenses areless expensive than the photovoltaic material. Con-sider the solar photovoltaic cell of Example 3.3. A100 mm100 mm photovoltaic cell is irradiated withconcentrated solar energy. Since the concentratinglens is glass, it absorbs 10% of the irradiation insteadof the top surface of the solar cell, as in Example 3.3.The remaining irradiation is reflected from the system(7%) or is absorbed in the silicon semiconductormaterial of the photovoltaic cell (83%). The photo-voltaic cell is cooled by air directed parallel to its topand bottom surfaces. The air temperature and velocityare 25?C and 5 m/s, respectively, and the bottomsurface is coated with a high-emissivity paint,?b?0.95.FocusinglensPhotovoltaic cellSee Example 3.3u = 5 m/sL = 100 mmLlensConcentratedirradiation, GcSolar irradiation, GTsur = 25CT = 25CTsur = 25CAir(a) Determine the electric power produced by the photo-voltaic cell and the silicon temperature for a squareconcentrating lens with Llens?400 mm, whichfocuses the irradiation falling on the lens to the smallerarea of the photovoltaic cell. Assume the concentrat-ing lens temperature is 25?C and does not interferewith boundary layer development over the photo-voltaic cells top surface. The top and bottom bound-ary layers are both tripped to turbulent conditions atthe leading edge of the photovoltaic material.(b) Determine the electric power output of the photo-voltaic cell and the silicon temperature over therange 100 mmLlens600 mm.

The roof of a refrigerated truck compartment is of com-posite construction, consisting of a layer of foamedurethane insulation (t2?50 mm, ki?0.026 W/m?K)sandwiched between aluminum alloy panels (t1?5mm,kp?180 W/m?K). The length and width of the roof areL?10 m and W?3.5 m, respectively, and the temper-ature of the inner surface is Ts,i??10?C. Consider con-ditions for which the truck is moving at a speed ofV?105 km/h, the air temperature is T??32?C, and thesolar irradiation is GS?750 W/m2. Turbulent flow maybe assumed over the entire length of the roof.(a) For equivalent values of the solar absorptivity andthe emissivity of the outer surface (?S???0.5),estimate the average temperature Ts,oof the outersurface. What is the corresponding heat load imposedon the refrigeration system?(b) A special finish (?S?0.15, ??0.8) may be appliedto the outer surface. What effect would such anapplication have on the surface temperature and theheat load?(c) If, with ?S???0.5, the roof is not insulated(t2?0), what are the corresponding values of thesurface temperature and the heat load?

The top surface of a heated compartment consists ofvery smooth (A) and highly roughened (B) portions,and the surface is placed in an atmospheric airstream. In the interest of minimizing total convection heattransfer from the surface, which orientation, (1) or (2),is preferred? If Ts?100?C, T??20?C, and u??20 m/s, what is the convection heat transfer from theentire surface for this orientation?

Calculate the value of the average heat transfer coeffi-cient for the plate of Problem 7.21 when the entire plateis rotated 90 so that half of the leading edge consists ofa very smooth portion (A) and the other half consistsof a highly roughened portion (B)

The proposed design for an anemometer to determinethe velocity of an airstream in a wind tunnel is com-prised of a thin metallic strip whose ends are supportedby stiff rods serving as electrodes for passage of currentused to heat the strip. A fine-wire thermocouple isattached to the trailing edge of the strip and serves asthe sensor for a system that controls the power to main-tain the strip at a constant operating temperature forvariable airstream velocities. Design conditions pertainto an airstream at T??25?C and 1u?50 m/s,with a strip temperature of Ts?35?C.(a) Determine the relationship between the electricalpower dissipation per unit width of the strip in thetransverse direction, P?(mW/mm), and the airstreamvelocity. Show this relationship graphically for thespecified range of u?.(b) If the accuracy with which the temperature of theoperating strip can be measured and maintainedconstant is?0.2?C, what is the uncertainty in theairstream velocity? (c) The proposed design operates in a strip constant-temperature mode for which the airstream velocityis related to the measured power. Consider nowan alternative mode wherein the strip is providedwith a constant power, say, 30 mW/mm, and theairstream velocity is related to the measured striptemperature Ts. For this mode of operation, showthe graphical relationship between the strip temper-ature and airstream velocity. If the temperature canbe measured with an uncertainty of?0.2?C, whatis the uncertainty in the airstream velocity?(d) Compare the features associated with each of theanemometer operating modes.

Steel (AISI 1010) plates of thickness ??6 mm andlength L?1 m on a side are conveyed from a heattreatment process and are concurrently cooled byatmospheric air of velocity u??10 m/s and T??20?Cin parallel flow over the plates.For an initial plate temperature of Ti?300?C, what isthe rate of heat transfer from the plate? What is the cor-responding rate of change of the plate temperature? Thevelocity of the air is much larger than that of the plate.

Consider weather conditions for which the prevailingwind blows past the penthouse tower on a tall building.The tower length in the wind direction is 10 m andthere are 10 window panels. (a) Calculate the average convection coefficient for thefirst, third, and tenth window panels when the windspeed is 5 m/s. Use a film temperature of 300 K toevaluate the thermophysical properties required ofthe correlation. Would this be a suitable value of thefilm temperature for ambient air temperatures in therange?15T?38?C?(b) For the first, third, and tenth windows, on onegraph, plot the variation of the average convectioncoefficient with wind speed for the range5u?100 km/h. Explain the major features ofeach curve and their relative magnitudes.

Consider a rectangular fin that is used to cool a motor-cycle engine. The fin is 0.15 m long and at a tempera-ture of 250C, while the motorcycle is moving at 80 km/h in air at 27C. The air is in parallel flow overboth surfaces of the fin, and turbulent flow conditionsmay be assumed to exist throughout.(a) What is the rate of heat removal per unit width ofthe fin?(b) Generate a plot of the heat removal rate per unit width of the fin for motorcycle speeds ranging from10 to 100 km/h

The Weather Channel reports that it is a hot, muggyday with an air temperature of 90F, a 10 mph breezeout of the southwest, and bright sunshine with a solarinsolation of 400 W/m. Consider the wall of a metalbuilding over which the prevailing wind blows. Thelength of the wall in the wind direction is 10 m, and theemissivity is 0.93. Assume that all the solar irradiationis absorbed, that irradiation from the sky is negligible,and that flow is fully turbulent over the wall. Estimatethe average wall temperature.

In the production of sheet metals or plastics, it is cus-tomary to cool the material before it leaves the produc-tion process for storage or shipment to the customer.Typically, the process is continuous, with a sheet ofthickness ? and width Wcooled as it transits the dis-tance Lbetween two rollers at a velocity V. In thisproblem, we consider cooling of an aluminum alloy(2024-T6) by an airstream moving at a velocity u?incounter flow over the top surface of the sheet. A turbu-lence promoter is used to provide turbulent boundarylayer development over the entire surface.(a) By applying conservation of energy to a differentialcontrol surface of length dx, which either moveswith the sheet oris stationary and through whichthe sheet passes, derive a differential equation thatgoverns the temperature distribution along thesheet. Because of the low emissivity of the alu-minum, radiation effects may be neglected. Express your result in terms of the velocity, thickness, andproperties of the sheet (V,?, ?, cp), the local con-vection coefficient hxassociated with the counterflow, and the air temperature. For a known temper-ature of the sheet (Ti) at the onset of cooling and anegligible effect of the sheet velocity on boundarylayer development, solve the equation to obtain anexpression for the outlet temperature To.(b) For ??2 mm, V?0.10 m/s, L?5 m, W?1m,u??20 m/s, T??20?C, and Ti?300?C, what isthe outlet temperature To?

An array of electronic chips is mounted within a sealedrectangular enclosure, and cooling is implemented byattaching an aluminum heat sink (k?180 W/m?K).The base of the heat sink has dimensions of w1?w2?100 mm, while the 6 fins are of thickness t?10 mmand pitch S?18 mm. The fin length is Lf?50 mm,and the base of the heat sink has a thickness ofLb?10 mm.If cooling is implemented by water flow throughthe heat sink, with u??3 m/s and T??17?C, what isthe base temperature Tbof the heat sink when power dis-sipation by the chips is Pelec?1800 W? The averageconvection coefficient for surfaces of the fins and theexposed base may be estimated by assuming parallelflow over a flat plate. Properties of the water may beapproximated as k?0.62 W/m?K, ??995 kg/m3,cp?4178 J/kg?K, ??7.7310?7m2/s, and Pr5? .2.

Consider the concentrating photovoltaic apparatus ofProblem 7.19. The apparatus is to be installed in adesert environment, so the space between the concen-trating lens and top of the photovoltaic cell isenclosed to protect the cell from sand abrasion inwindy conditions. Since convection cooling from thetop of the cell is reduced by the enclosure, an engi-neer proposes to cool the photovoltaic cell by attach-ing an aluminum heat sink to its bottom surface. Theheat sink dimensions and material are the same asthose of Problem 7.29. A contact resistance of0.510?4m2?K/W exists at the photovoltaiccell/heat sink interface and a dielectric liquid(k?0.064 W/m?K, ??1400 kg/m3, cp?1300 J/kg?K, ??10?6m2/s, Pr?25) flows between the heat sinkfins at u??3 m/s, T??25?C. (a) Determine the electric power produced by the pho-tovoltaic cell and the silicon temperature for asquare concentrating lens with Llens?400 mm.(b) Compare the electric power produced by the photo-voltaic cell with the heat sink in place and with thebottom surface cooled directly by the dielectricfluid (i.e., no heat sink) for Llens?1.5 m.(c) Determine the electric power output and the silicontemperature over the range 100 mmLlens?3000 mm with the aluminum heat sink in place.

In the production of sheet metals or plastics, it iscustomary to cool the material before it leaves the pro-duction process for storage or shipment to the customer.Typically, the process is continuous, with a sheet ofthickness ? and width Wcooled as it transits the distanceLbetween two rollers at a velocity V. In this problem, weconsider cooling of plain carbon steel by an airstreammoving at a velocity u?in cross flow over the top andbottom surfaces of the sheet. A turbulence promoter isused to provide turbulent boundary layer developmentover the entire surface.TurbulencepromoterWTiToPlain carbon steelSurroundings, TsurxLVuTAir(a) By applying conservation of energy to a differentialcontrol surface of length dx, which either moveswith the sheet oris stationary and through whichthe sheet passes, and assuming a uniform sheettemperature in the direction of airflow, derive a dif-ferential equation that governs the temperature dis-tribution, T(x), along the sheet. Consider the effectsof radiation, as well as convection, and express yourresult in terms of the velocity, thickness, and prop- erties of the sheet (V,?, ?, cp, ?), the average con-vection coefficient associated with the crossflow, and the environmental temperatures (T?, Tsur).(b) Neglecting radiation, obtain a closed form solutionto the foregoing equation. For ??3 mm, V?0.10 m/s, L?10 m, W?1 m, u??20 m/s, T??20?C, and a sheet temperature of Ti?500?C at theonset of cooling, what is the outlet temperature To?Assume a negligible effect of the sheet velocityon boundary layer development in the directionof airflow. The density and specific heat of the steelare ??7850 kg/m3and cp?620 J/kg?K, whileproperties of the air may be taken to be k?0.044W/m?K, ??4.510?5m2/s, Pr?0.68.(c) Accounting for the effects of radiation, with ??0.70 and Tsur?20?C, numerically integrate the dif-ferential equation derived in part (a) to determinethe temperature of the sheet at L?10 m. Explore theeffect of Von the temperature distribution alongthe sheet

A steel strip emerges from the hot roll section of a steelmill at a speed of 20 m/s and a temperature of 1200 K.Its length and thickness are L?100 m and ??0.003 m, respectively, and its density and specific heatare 7900 kg/m3and 640 J/kg?K, respectively.Accounting for heat transfer from the top and bottomsurfaces and neglecting radiation and strip conductioneffects, determine the time rate of change of the striptemperature at a distance of 1 m from the leading edgeand at the trailing edge. Determine the distance fromthe leading edge at which the minimum cooling rate is achieved.

In Problem 7.23, an anemometer design wasexplored, and the assumption was made that the striptemperature was uniform. This is a good assumptionwhen the heat transfer coefficient is low or the stripthermal conductivity high, because then conductionwithin the strip redistributes the generated heat andmakes the strip temperature uniform. However, as theheat transfer coefficient increases or strip thermalconductivity decreases, heat generated at a point inthe strip leaves the surface in the vicinity of thatpoint, and the thermal condition is closer to one ofuniform surface heat flux.(a) Develop the calibration equations for both the con-stant surface temperature and constant heat fluxconditions, that is, find the equations that predictthe velocity as a function of the power per unit stripwidth, P?(mW/mm), and the temperature measuredat the trailing edge (as in Problem 7.23). Assumelaminar flow conditions.(b) If the true condition is uniform surface heat flux,but the uniform surface temperature calibration isused, what percentage error will be incurred in thevelocity determination?(c) Where could the thermocouple be placed so thatthe calibration is insensitive to whether the thermalcondition is uniform surface temperature or uni-form surface heat flux?

A flat plate of width 1 m and length 0.2 m is maintainedat a temperature of 32C. Ambient fluid at 22C flowsacross the top of the plate in parallel flow. Determinethe average heat transfer coefficient, the convectionheat transfer rate from the top of the plate, and the dragforce on the plate for the following: (a) The fluid is water flowing at a velocity of 0.5 m/s.(b) The nanofluid of Example 2.2 is flowing at a velocity of 0.5 m/s.(c) Water is flowing at a velocity of 2.5 m/s. (d) The nanofluid of Example 2.2 is flowing at a velocity of 2.5 m/s.

One hundred electrical components, each dissipating25 W, are attached to one surface of a square(0.2 m0.2 m) copper plate, and all the dissipatedenergy is transferred to water in parallel flow over the opposite surface. A protuberance at the leadingedge of the plate acts to tripthe boundary layer, and the plate itself may be assumed to be isothermal. Thewater velocity and temperature are u??2 m/s andT??17?C, and the waters thermophysical propertiesmay be approximated as ??0.9610?6m2/s, k?0.620 W/m?K, and Pr?5.2. (a) What is the temperature of the copper plate?(b) If each component has a plate contact surface areaof 1 cm2and the corresponding contact resistance is210?4m2?K/W, what is the component temper-ature? Neglect the temperature variation across thethickness of the copper plate.

Air at 27?C with a free stream velocity of 10 m/sis used to cool electronic devices mounted on aprinted circuit board. Each device, 4 mm4 mm,dissipates 40 mW, which is removed from the topsurface. A turbulator is located at the leadingedge of the board, causing the boundary layer to beturbulent.(a) Estimate the surface temperature of the fourthdevice located 15 mm from the leading edge of theboard.(b) Generate a plot of the surface temperature of thefirst four devices as a function of the free streamvelocity for 5u?15 m/s.(c) What is the minimum free stream velocity if thesurface temperature of the hottest device is not toexceed 80?C?

The boundary layer associated with parallel flow overan isothermal plate may be tripped at any x- location byusing a fine wire that is stretched across the width of theplate. Determine the value of the critical Reynoldsnumber Rex,c,op that is associated with the optimal loca-tion of the trip wire from the leading edge that willresult in maximum heat transfer from the warm plate tothe cool fluid.

Forced air at 25?C and 10 m/s is used to cool elec-tronic elements mounted on a circuit board. Consider a chip of length 4 mm and width 4 mm located120 mm from the leading edge. Because the boardsurface is irregular, the flow is disturbed and theappropriate convection correlation is of the formNux?0.04Rex0.85Pr0.33.Estimate the surface temperature of the chip, Ts, if itsheat dissipation rate is 30 mW.

Air at atmospheric pressure and a temperature of 25?Cis in parallel flow at a velocity of 5 m/s over a 1- m-longflat plate that is heated with a uniform heat flux of1250 W/m2. Assume the flow is fully turbulent over thelength of the plate.(a) Calculate the plate surface temperature, Ts(L), andthe local convection coefficient, hx(L), at the trail-ing edge, x?L.(b) Calculate the average temperature of the platesurface, .(c) Plot the variation of the surface temperature, Ts(x),and the convection coefficient, hx(x), with distanceon the same graph. Explain the key features ofthese distributions

Working in groups of two, our students design andperform experiments on forced convection phenomena using the general arrangement shown schematically. The air box consists of two muffin fans, aplenum chamber, and flow straighteners discharginga nearly uniform airstream over the flat test-plate.The objectives of one experiment were to measure the heat transfer coefficient and to compare the results with standard convection correlations. The velocityof the airstream was measured using a thermistor-based anemometer, and thermocouples were used todetermine the temperatures of the airstream and thetest-plate.With the airstream from the box fully stabilizedat T = 20C, an aluminum plate was preheated ina convection oven and quickly mounted in the test-plate holder. The subsequent temperature historyof the plate was determined from thermocouplemeasurements, and histories obtained for airstreamvelocities of 3 and 9 m/s were fitted by the following polynomial: The temperature Tand time thave units of ?C and s,respectively, and values of the coefficients appropriatefor the time interval of the experiments are tabulated asfollows:Velocity (m/s)39Elapsed Time (s) 300160a(?C)56.8757.00b(?C/s)?0.1472?0.2641c(?C/s2)310?4910?4d(?C/s3)?410?7?210?6e(?C/s4)210?10110?9The plate is square, 133 mm to a side, with a thicknessof 3.2 mm, and is made from a highly polishedaluminum alloy (??2770 kg/m3, c?875 J/kg?K,k?177 W/m?K).(a) Determine the heat transfer coefficients for the twocases, assuming the plate behaves as a spacewiseisothermal object.(b) Evaluate the coefficients Cand mfor a correlationof the formCompare this result with a standard flat- plate corre-lation. Comment on the goodnessof the comparisonand explain any differences.

Consider atmospheric air at u??2 m/s and T??300 Kin parallel flow over an isothermal flat plate of lengthL?1 m and temperature Ts?350 K.(a) Compute the local convection coefficient at theleading and trailing edges of the heated platewith and without an unheated starting length of?1m. (b) Compute the average convection coefficient for theplate for the same conditions as part (a).(c) Plot the variation of the local convection coefficientover the plate with and without an unheated start-ing length.

Consider a thin, 50 mm50 mm fuel cell similar tothat of Example 1.5, with air in parallel flow over itssurfaces. Very small-diameter wires are stretchedacross both sides of the fuel cell at a distance x?xcfrom the leading edge in order to trip the flow into tur-bulent conditions. Using an appropriate correlationfrom Chapter 7, determine the minimum velocityneeded to sustain the fuel cell at Tc?77?C, and theassociated location of the wire. The air and large sur-roundings are at T??Tsur?27?C and the fuel cell dissi-pates . The fuel cell emissivity is ? .85

The cover plate of a flat-plate solar collector is at 15?C,while ambient air at 10?C is in parallel flow over theplate, with u??2 m/s.(a) What is the rate of convective heat loss from theplate?(b) If the plate is installed 2 m from the leading edge ofa roof and flush with the roof surface, what is therate of convective heat loss?

An array of 10 silicon chips, each of length L?10 mmon a side, is insulated on one surface and cooled on theopposite surface by atmospheric air in parallel flow withT??24?C and u??40 m/s. When in use, the sameelectrical power is dissipated in each chip, maintaining auniform heat flux over the entire cooled surface.If the temperature of each chip may not exceed 80?C,what is the maximum allowable power per chip? Whatis the maximum allowable power if a turbulence promoter is used to trip the boundary layer at the lead-ing edge? Would it be preferable to orient the arraynormal, instead of parallel, to the airflow?

A square (10 mm10 mm) silicon chip is insulated onone side and cooled on the opposite side by atmos-pheric air in parallel flow at u??20 m/s and T??24?C. When in use, electrical power dissipation withinthe chip maintains a uniform heat flux at the cooledsurface. If the chip temperature may not exceed 80?C atany point on its surface, what is the maximum allow-able power? What is the maximum allowable power ifthe chip is flush mounted in a substrate that provides foran unheated starting length of 20 mm?

Consider the following fluids, each with a velocity ofV?5 m/s and a temperature of T??20?C, in crossflow over a 10-mm-diameter cylinder maintained at50?C: atmospheric air, saturated water, and engine oil.(a) Calculate the rate of heat transfer per unit length,q?, using the ChurchillBernstein correlation.(b) Generate a plot of q?as a function of fluid velocityfor 0.5V10 m/s

A circular pipe of 25-mm outside diameter is placed inan airstream at 25C and 1-atm pressure. The airmoves in cross flow over the pipe at 15 m/s, while theouter surface of the pipe is maintained at 100C. Whatis the drag force exerted on the pipe per unit length?What is the rate of heat transfer from the pipe per unit length?

An L?1-m-long vertical copper tube of inner diameterDi?20 mm and wall thickness t?2 mm containsliquid water at Tw0?C. On a winter day, air atV?3 m/s, T??20?C is in cross flow over the tube. (a) Determine the heat loss per unit mass from thewater (W/kg) when the tube is full of water.(b) Determine the heat loss from the water (W/kg)when the tube is half full

A long, cylindrical, electrical heating element of diameterD?10 mm, thermal conductivity k?240 W/m?K, den-sity ??2700 kg/m3, and specific heat cp?900 J/kg?K isinstalled in a duct for which air moves in cross flow overthe heater at a temperature and velocity of 27?C and10 m/s, respectively.(a) Neglecting radiation, estimate the steady-state sur-face temperature when, per unit length of theheater, electrical energy is being dissipated at a rateof 1000 W/m. (b) If the heater is activated from an initial temperatureof 27?C, estimate the time required for the surfacetemperature to come within 10?C of its steady-statevalue.

Consider the conditions of Problem 7.49, but nowallow for radiation exchange between the surface of theheating element (??0.8) and the walls of the duct,which form a large enclosure at 27?C.(a) Evaluate the steady-state surface temperature.(b) If the heater is activated from an initial temperatureof 27?C, estimate the time required for the surfacetemperature to come within 10?C of the steady-state value.(c) To guard against overheating due to unanticipatedexcursions in the blower output, the heater con- troller is designed to maintain a fixed surface tem-perature of 275?C. Determine the power dissipationrequired to maintain this temperature for air veloci-ties in the range 5V10 m/s.

Pin fins are to be specified for use in an industrial cool-ing application. The fins will be subjected to a gas incross flow at V?10 m/s. The cylindrical fin has adiameter of D?15 mm, and the cross-sectional area isthe same for each configuration shown in the sketch.For fins of equal length and therefore equal mass, whichfin has the largest heat transfer rate? Assume the gasproperties are those of air at T?350 K. Hint: Assumethe fins can be treated as infinitely long and apply theHilpert correlation to the fin of circular cross section.

A pin fin of 10-mm diameter dissipates 30 W by forcedconvection to air in cross flow with a Reynolds numberof 4000. If the diameter of the fin is doubled and allother conditions remain the same, estimate the fin heatrate. Assume the pin to be infinitely long.

Air at 27?C and a velocity of 5 m/s passes over the smallregion As(20 mm20 mm) on a large surface, which ismaintained at Ts?127?C. For these conditions, 0.5 Wis removed from the surface As. To increase the heatremoval rate, a stainless steel (AISI 304) pin fin ofdiameter 5 mm is affixed to As, which is assumed toremain at Ts?127?C (a) Determine the maximum possible heat removal ratethrough the fin.(b) What fin length would provide a close approxima-tion to the heat rate found in part (a)? Hint: Refer toExample 3.9.(c) Determine the fin effectiveness, ?f.(d) What is the percentage increase in the heat ratefrom Asdue to installation of the fin?

To enhance heat transfer from a silicon chip of widthW?4 mm on a side, a copper pin fin is brazed to thesurface of the chip. The pin length and diameter areL?12 mm and D?2 mm, respectively, and atmos- pheric air at V?10 m/s and T??300 K is in crossflow over the pin. The surface of the chip, and hencethe base of the pin, are maintained at a temperature ofTb?350 K.(a) Assuming the chip to have a negligible effect onflow over the pin, what is the average convectioncoefficient for the surface of the pin?(b) Neglecting radiation and assuming the convectioncoefficient at the pin tip to equal that calculated inpart (a), determine the pin heat transfer rate.(c) Neglecting radiation and assuming the convectioncoefficient at the exposed chip surface to equal thatcalculated in part (a), determine the total rate ofheat transfer from the chip.(d) Independently determine and plot the effect ofincreasing velocity (10V40 m/s) and pin diameter (2D4 mm) on the total rate of heattransfer from the chip. What is the heat rate forV?40 m/s and D?4 mm?

Consider the Nichrome wire (D?1 mm, ?e?10?6??m, k?25 W/m?K, ??0.20) used to fabricate theair heater of Problem 3.86, but now under conditionsfor which the convection heat transfer coefficient mustbe determined.(a) For atmospheric air at 50?C and a cross-flow veloc-ity of 5 m/s, what are the surface and centerlinetemperatures of the wire when it carries a current of25 A and the housing of the heater is also at 50?C?(b) Explore the effect of variations in the flow velocityand electrical current on the surface and centerlinetemperatures of the wire

Hot water at 50?C is routed from one building in whichit is generated to an adjoining building in which it isused for space heating. Transfer between the buildingsoccurs in a steel pipe (k?60 W/m?K) of 100- mmoutside diameter and 8-mm wall thickness. Duringthe winter, representative environmental conditionsinvolve air at T???5?C and V?3 m/s in cross flowover the pipe.(a) If the cost of producing the hot water is $0.10 perkW?h, what is the representative daily cost of heatloss from an uninsulated pipe to the air per meter ofpipe length? The convection resistance associatedwith water flow in the pipe may be neglected.(b) Determine the savings associated with applicationof a 10-mm-thick coating of urethane insulation(k?0.026 W/mK ? ) to the outer surface of the pipe

In a manufacturing process, long aluminum rods ofsquare cross section with d ?25 mm are cooled froman initial temperature of Ti?400?C. Which configura-tion in the sketch should be used to minimize the timeneeded for the rods to reach a safe-to-handletempera-ture of 60?C when exposed to air in cross flow atV?8 m/s, T??30?C? What is the required coolingtime for the preferred configuration? The emissivity ofthe rods is ??0.10 and the surroundings temperatureis Tsur?20?C.

A fine wire of diameter Dis positioned across a passageto determine flow velocity from heat transfer character-istics. Current is passed through the wire to heat it, andthe heat is dissipated to the flowing fluid by convection.The resistance of the wire is determined from electricalmeasurements, and the temperature is known from theresistance.(a) For a fluid of arbitrary Prandtl number, develop anexpression for its velocity in terms of the differencebetween the temperature of the wire and the freestream temperature of the fluid.(b) What is the velocity of an airstream at 1 atm and 25C, if a wire of 0.5-mm diameter achieves a tem-perature of 40C while dissipating 35 W/m?

To determine air velocity changes, it is proposed tomeasure the electric current required to maintain a plat-inum wire of 0.5-mm diameter at a constant tempera-ture of 77?C in a stream of air at 27?C.(a) Assuming Reynolds numbers in the range40?ReD?1000, develop a relationship betweenthe wire current and the velocity of the air that is incross flow over the wire. Use this result to establisha relation between fractional changes in the current,?I/I, and the air velocity, ?V/V.(b) Calculate the current required when the air velocityis 10 m/s and the electrical resistivity of the plat-inum wire is 17.1105 ? ??m.

Fluid velocities can be measured using hot-film sen-sors, and a common design is one for which the sens-ing element forms a thin film about the circumferenceof a quartz rod. The film is typically comprised of athin (100 nm) layer of platinum, whose electricalresistance is proportional to its temperature. Hence,when submerged in a fluid stream, an electric currentmay be passed through the film to maintain itstemperature above that of the fluid. The temperatureof the film is controlled by monitoring its electricresistance, and with concurrent measurement of theelectric current, the power dissipated in the film maybe determined. Proper operation is assured only if the heat generatedin the film is transferred to the fluid, rather than con-ducted from the film into the quartz rod. Thermally,the film should therefore be strongly coupled to thefluid and weakly coupled to the quartz rod. This condi- tion is satisfied if the Biot number is very large, Bi?D/2k?1, where is the convection coefficientbetween the fluid and the film and kis the thermal con-ductivity of the rod.(a) For the following fluids and velocities, calculateand plot the convection coefficient as a functionof velocity: (i) water, 0.5V5 m/s; (ii) air,1V20 m/s.(b) Comment on the suitability of using this hot-filmsensor for the foregoing conditions.

Consider use of the hot-film sensor described in Prob-lem 7.60 to determine the velocity of water entering thecooling system of an electric power plant from anadjoining lake. The sensor is mounted within an intakepipe, and its controls are set to maintain an average hot-film temperature that is 5?C larger than the fluid tem-perature (Ts,hf?T??5?C).(a) If an independent measurement of the water tem- perature yields a value of T??17?C, use theChurchillBernstein correlation to estimate thevelocity of the water under conditions for whichthe power input to the sensor maintains a heatflux of from the film to thewater.(b) If the sensor is exposed to the water for anextended period, its surface will be fouledby anaccumulation of deposits from the water. Considerconditions for which the deposits form a 0.l-mm- thick shell around the sensor and have a thermalconductivity of kd?2W/m?K. For T??17?C andthe flow velocity determined in part (a), what heatflux must be supplied to the sensor to maintain itstemperature at Ts,hf?22?C? What is the corre-sponding error in the velocity measurement? Note:Conduction across the deposit may be approxi-mated as that across a plane wall

Determine the convection heat loss from both the topand the bottom of a flat plate at Ts?80?C with air inparallel flow at T??25?C, u??3 m/s. The plate ist?1 mm thick, L?25 mm long, and of depthw?50 mm. Neglect the heat loss from the edges of theplate. Compare the convection heat loss from the plateto the convection heat loss from an Lc?50-mm-longcylinder of the same volume as that of the plate. Theconvective conditions associated with the cylinder arethe same as those associated with the plate.

Consider two very long, straight fins of uniform crosssection, as shown in Figure 3.17. The rectangular finhas dimensions t?1 mm and w?20 mm. The circularpin fin has the same cross-sectional area as the rectan-gular fin. Both fins are constructed of aluminum withk?237 W/m?K. In both cases, the base temperature isTb?85?C. Airflow is directed as shown in the figure,with T??20?C and u??5 m/s.(a) Calculate the heat loss from each fin. Assume thatthe heat transfer coefficient on the edges of the rec-tangular fin is equal to the average value on theupper and lower surfaces. (b) What diameter cylindrical fin would be needed toprovide the same fin heat transfer rate as the rectan-gular cross- section fin?

A computer code is being developed to analyze a tem-perature sensor of 12.5-mm diameter experiencing crossflow of water with a free stream temperature of 80?Cand variable velocity. Derive an expression for the con-vection heat transfer coefficient as a function of the sen-sor surface temperature Tsfor the range 20?Ts?80?Cand for velocities Vin the range 0.005?V?0.20 m/s.Use the Zukauskas correlation for the range40?ReD?1000 and assume that the Prandtl numberof water has a linear temperature dependence

A 25-mm-diameter, high-tension line has an electricalresistance of 10?4?/m and is transmitting a current of1000 A.(a) If ambient air at 10?C and 5 m/s is in cross flowover the line, what is its surface temperature?(b) If the line may be approximated as a solid copperrod, what is its centerline temperature?(c) Generate a plot that depicts the variation of thesurface temperature with air velocity for 1V10 m/s.

An aluminum transmission line with a diameter of20 mm has an electrical resistance of and carries a current of 700 A. The line issubjected to frequent and severe cross winds, increas-ing the probability of contact between adjacent lines,thereby causing sparks and creating a potential fire haz-ard for nearby vegetation. The remedy is to insulate theline, but with the adverse effect of increasing the con- ductor operating temperature.(a) Calculate the conductor temperature when the airtemperature is 20?C and the line is subjected tocross flow with a velocity of 10 m/s.(b) Calculate the conductor temperature for the sameconditions, but with a 2-mm-thick insulation havinga thermal conductivity of 0.15 W/m?K. (c) Calculate and plot the temperatures of the bare andinsulated conductors for wind velocities in therange from 2 to 20 m/s. Comment on features ofthe curves and the effect of the wind velocity on theconductor temperatures

To augment heat transfer between two flowing fluids, itis proposed to insert a 100-mm-long, 5-mm- diameter2024 aluminum pin fin through the wall separating thetwo fluids. The pin is inserted to a depth of dinto fluid 1.Fluid 1 is air with a mean temperature of 10?C andvelocity of 10 m/s. Fluid 2 is air with a mean tempera-ture of 40?C and velocity of 3 m/s.(a) Determine the rate of heat transfer from the warmair to the cool air through the pin fin for d?50 mm.(b) Plot the variation of the heat transfer rate with theinsertion distance, d. Does an optimal insertiondistance exist?

An uninsulated steam pipe is used to transport high-temperature steam from one building to another. Thepipe is of 0.5-m diameter, has a surface temperature of150?C, and is exposed to ambient air at?10?C. Theair moves in cross flow over the pipe with a velocityof 5 m/s.(a) What is the heat loss per unit length of pipe?(b) Consider the effect of insulating the pipe with arigid urethane foam (k?0.026 W/m?K). Evaluateand plot the heat loss as a function of the thickness? of the insulation layer for 0?50 mm

A thermocouple is inserted into a hot air duct to measurethe air temperature. The thermocouple (T1) is soldered to the tip of a steel thermocouple wellof lengthL?0.15 m and inner and outer diameters of Di?5mm and Do?10 mm. A second thermocouple (T2) is used tomeasure the duct wall temperature.Consider conditions for which the air velocity in theduct is V?3 m/s and the two thermocouples registertemperatures of T1?450 K and T2?375 K. Neglectingradiation, determine the air temperature T?. Assume that,for steel, k?35 W/m?K, and, for air, ??0.774 kg/m3,??25110?7N?s/m2, k?0.0373 W/m?K, andPr?0.686.

Consider conditions for which a mercury-in-glassthermometer of 4-mm diameter is inserted to alengthLthrough the wall of a duct in which air at77?C is flowing. If the stem of the thermometer atthe duct wall is at the wall temperatureTw?15?C,conduction heat transfer through the glass causes thebulb temperature to be lower than that of theairstream.(a) Develop a relationship for the immersionerror,?Ti?T(L)?T?, as a function of air velocity,thermometer diameter, and insertion length L.(b) To what length Lmust the thermometer be insertedif the immersion error is not to exceed 0.25?C whenthe air velocity is 10 m/s?(c) Using the insertion length determined in part (b),calculate and plot the immersion error as a functionof air velocity for the range 2 to 20 m/s.(d) For a given insertion length, will the immersionerror increase or decrease if the diameter ofthe thermometer is increased? Is the immersionerror more sensitive to the diameter or airvelocity?

In a manufacturing process, a long, coated plastic rod(??2200 kg/m3, c?800 J/kg?K, k?1W/m?K) ofdiameter D?20 mm is initially at a uniform tempera-ture of 25?C and is suddenly exposed to a cross flow ofair at T??350?C and V?50 m/s.(a) How long will it take for the surface of the rod toreach 175?C, the temperature above which thespecial coating will cure?(b) Generate a plot of the time to reach 175?C as afunction of air velocity for 5V50 m/s

In an extrusion process, copper wire emerges from theextruder at a velocity Veand is cooled by convectionheat transfer to air in cross flow over the wire, as wellas by radiation to the surroundings.(a) By applying conservation of energy to a differentialcontrol surface of length dx, which either moveswith the wire oris stationary and through which thewire passes, derive a differential equation that gov-erns the temperature distribution, T(x), along thewire. In your derivation, the effect of axial conduc-tion along the wire may be neglected. Express yourresult in terms of the velocity, diameter, and prop-erties of the wire (Ve, D, ?, cp, ?), the convectioncoefficient associated with the cross flow , andthe environmental temperatures (T?, Tsur).(b) Neglecting radiation, obtain a closed form solutionto the foregoing equation. For Ve?0.2 m/s,D?5 mm, V?5 m/s, T??25?C, and an initialwire temperature of Ti?600?C, compute the tem-perature Toof the wire at x?L?5 m. The densityand specific heat of the copper are ??8900 kg/m3and cp?400 J/kg?K, while properties of the airmay be taken to be k?0.037 W/m?K, ??310?5m2/s, and Pr?0.69.(c) Accounting for the effects of radiation, with??0.55 and Tsur?25?C, numerically integrate thedifferential equation derived in part (a) to determine the temperature of the wire at L?5 m. Explore theeffects of Veand ?on the temperature distributionalong the wire.

The objective of an experiment performed by ourstudents is to determine the effect of pin fins on thethermal resistance between a flat plate and an airstream.A 25.9-mm-square polished aluminum plate is sub-jected to an airstream in parallel flow at T??20?C andu??6 m/s. An electrical heating patch is attached tothe backside of the plate and dissipates 15.5 W underall conditions. Pin fins of diameter D?4.8 mm andlength L?25.4 mm are fabricated from brass and canbe firmly attached to the plate at various locations overits surface. Thermocouples are attached to the platesurface and the tip of one of the fins.Measured temperatures for five pin-fin configurationsare tabulated.Temperature (?C)Numberof Pin FinsFin Tip Plate Base070.2140.667.4239.564.7536.457.4834.252.1(a) Using the experimental observations and neglect-ing the effect of flow interactions between pins,determine the thermal resistance between the plateand the airstream for the five configurations.(b) Develop a model of the platepin fin system andusing appropriate convection correlations, predictthe thermal resistances for the five configurations.Compare your predictions with the observationsand explain any differences.(c) Use your model to predict the thermal resistanceswhen the airstream velocity is doubled.

Air at 25C flows over a 10-mm-diameter sphere with avelocity of 25 m/s, while the surface of the sphere ismaintained at 75C.(a) What is the drag force on the sphere?(b) What is the rate of heat transfer from the sphere?(c) Generate a plot of the heat transfer from the sphereas a function of the air velocity for the range 1 to 25 m/s

Consider a sphere with a diameter of 20 mm and a sur-face temperature of 60C that is immersed in a fluid at atemperature of 30C and a velocity of 2.5 m/s. Calcu-late the drag force and the heat rate when the fluid is(a) water and (b) air at atmospheric pressure. Explainwhy the results for the two fluids are so different.

Consider the material processing experiment of Prob-lem 5.24, with atmospheric nitrogen used to implementcooling by convection. However, instead of using aprescribed value of the convection coefficient, computethe coefficient from an appropriate correlation.(a) Neglecting radiation, determine the time requiredto cool the sphere from 900C to 300C if thevelocity and temperature of the nitrogen areV = 5 m/s and T = 25C (b) Accounting for the effects of both convection and radiation, with = 0.6 and Tsur = 25C, determine the time required to cool the sphere. Explore theeffects of the flow velocity on your result.

A spherical, underwater instrument pod used to makesoundings and to measure conditions in the water has adiameter of 85 mm and dissipates 300 W.(a) Estimate the surface temperature of the pod whensuspended in a bay where the current is 1 m/s andthe water temperature is 15C.(b) Inadvertently, the pod is hauled out of the waterand suspended in ambient air without deactivatingthe power. Estimate the surface temperature of thepod if the air temperature is 15C and the windspeed is 3 m/s.

Worldwide, over a billion solder balls must be manu-factured daily for assembling electronics packages. Theuniform droplet spraymethod uses a piezoelectricdevice to vibrate a shaft in a pot of molten solder that,in turn, ejects small droplets of solder through a preci-sion-machined nozzle. As they traverse a collectionchamber, the droplets cool and solidify. The collectionchamber is flooded with an inert gas such as nitrogen toprevent oxidation of the solder ball surfaces. (a) Molten solder droplets of diameter 130?m areejected at a velocity of 2 m/s at an initial temperatureof 225?C into gaseous nitrogen that is at 30C andslightly above atmospheric pressure. Determine theterminal velocity of the particles and the distance theparticles have traveled when they become completelysolidified. The solder properties are ??8230 kg/m3,c?240 J/kg?K, k?38 W/m?K, hsf?42 kJ/kg. Thesolders melting temperature is 183?C.(b) The piezoelectric device oscillates at 1.8 kHz, pro- ducing 1800 particles per second. Determine the sep-aration distance between the particles as they traversethe nitrogen gas and the pot volume needed in orderto produce the solder balls continuously for one week.

A spherical workpiece of pure copper with a diameterof 15 mm and an emissivity of 0.5 is suspended in alarge furnace with walls at a uniform temperature of 600C. Air flows over the workpiece at a temperature of 900C and a velocity of 7.5 m/s.(a) Determine the steady-state temperature of theworkpiece.(b) Estimate the time required for the workpiece to comewithin 5C of the steady-state temperature if it is at aninitial, uniform temperature of 25C.(c) To decrease the time to heat the workpiece, the airvelocity is doubled, with all other conditions remain-ing the same. Determine the steady-state temperatureof the workpiece and the time required for it to comewithin 5C of this value. Plot on the same graph the workpiece temperature histories for the two velocities

Copper spheres of 20-mm diameter are quenched bybeing dropped into a tank of water that is maintainedat 280 K. The spheres may be assumed to reach the terminal velocity on impact and to drop freely through the water. Estimate the terminal velocity by equating thedrag and gravitational forces acting on the sphere. What is the approximate height of the water tank needed to cool the spheres from an initial temperature of 360 K to a center temperature of 320 K?

For the conditions of Problem 7.80, what are the termi-nal velocity and the tank height if engine oil at 300 K,rather than water, is used as the coolant?

Consider the plasma spray coating process of Problem5.25. In addition to the prescribed conditions, the argonplasma jet is known to have a mean velocity ofV?400 m/s, while the initialvelocity of the injectedalumina particles may be approximated as zero. Thenozzle exit and the substrate are separated by a distanceof L?100 mm, and pertinent properties of the argonplasma may be approximated as k?0.671 W/m?K,cp?1480 J/kg?K, ??2.7010?4kg/s?m, and ??5.610?3m2/s.(a) Assuming the motion of particles entrained by theplasma jet to be governed by Stokes law, deriveexpressions for the particle velocity, Vp(t), and itsdistance of travel from the nozzle exit, xp(t), as afunction of time, t, where t?0 corresponds to par-ticle injection. Evaluate the time-in-flight requiredfor a particle to traverse the separation distance,xp?L, and the velocity Vpat this time.(b) Assuming an average relative velocity of during the time-of-flight, estimate the con-vection coefficient associated with heat transferfrom the plasma to the particle. Using this coeffi-cient and assuming an initial particle temperatureof Ti?300 K, estimate the time-in-flight requiredto heat a particle to its melting point, Tmp, and, onceat Tmp, for the particle to experience complete melt-ing. Is the prescribed value of Lsufficient to ensurecomplete particle melting before surface impact?

Highly reflective aluminum coatings may be formed onthe surface of a substrate by impacting the surface withmolten drops of aluminum. The droplets are dischargedfrom an injector, proceed through an inert gas (helium),and must still be in a molten state at the time of impact.Consider conditions for which droplets with a diameter,velocity, and initial temperature of D?500?m, V?3 m/s, and Ti?1100 K, respectively, traverse astagnant layer of atmospheric helium that is at a tem-perature of T??300 K. What is the maximum allow-able thickness of the helium layer needed to ensure thatthe temperature of droplets impacting the substrate isgreater than or equal to the melting point of aluminum(Tf?Tmp?933 K)? Properties of the molten alu-minum may be approximated as ??2500 kg/m3, c?1200 J/kg?K, and k?200 W/mK ? .

issue engineering involves the development of biologi-cal substitutes that restore or improve tissue function.Once manufactured, engineered organs can be implantedand grow within the patient, obviating chronic shortagesof natural organs that arise when traditional organ trans-plant procedures are used. Artificial organ manufactureinvolves two major steps. First, a porous scaffoldis fab-ricated with a specific pore size and pore distribution, aswell as overall shape and size. Second, the top surfaceof the scaffold is seeded with human cells that grow intothe pores of the scaffold. The scaffold material isbiodegradable and is eventually replaced with healthytissue. The artificial organ is then ready to be implantedin the patient.The complex pore shapes, small pore sizes, andunusual organ shapes preclude use of traditional manu-facturing methods to fabricate the scaffolds. A methodthat has been used with success is a solid freeform fab-ricationtechnique whereby small spherical drops aredirected to a substrate. The drops are initially moltenand solidify when they impact the room- temperaturesubstrate. By controlling the location of the dropletdeposition, complex scaffolds can be built up, one dropat a time. A device similar to that of Problem 7.78 isused to generate uniform, 75-?m- diameter drops at aninitial temperature of Ti?150?C. The particles are sentthrough quiescent air at T??25?C. The droplet proper-ties are ??2200 kg/m3, c?700 J/kg?K.LTi = 150CAirT= 25C, VExit nozzleDroplet generatorTissue scaffold (a) It is desirable for the droplets to exit the nozzle attheir terminal velocity. Determine the terminalvelocity of the drops.(b) It is desirable for the droplets to impact the structureat a temperature of T2?120?C. What is the requireddistance between the exit nozzle and the structure, L

A spherical thermocouple junction 1.0 mm in diameteris inserted in a combustion chamber to measure thetemperature T?of the products of combustion. The hotgases have a velocity of V?5 m/s.(a) If the thermocouple is at room temperature, Ti, whenit is inserted in the chamber, estimate the timerequired for the temperature difference, T??T, toreach 2% of the initial temperature difference,T??Ti. Neglect radiation and conduction throughthe leads. Properties of the thermocouple junction areapproximated as k?100 W/m?K, c?385 J/kg?K,and ??8920 kg/m3, while those of the combustiongases may be approximated as k?0.05 W/m?K,??5010?6m2/s, and Pr?0.69.(b) If the thermocouple junction has an emissivity of0.5 and the cooled walls of the combustor are atTc?400 K, what is the steady-state temperature ofthe thermocouple junction if the combustion gasesare at 1000 K? Conduction through the lead wiresmay be neglected.(c) To determine the influence of the gas velocity onthe thermocouple measurement error, compute thesteady-state temperature of the thermocouple junc- tion for velocities in the range 1V25 m/s. Theemissivity of the junction can be controlled throughapplication of a thin coating. To reduce the mea-surement error, should the emissivity be increasedor decreased? For V?5 m/s, compute the steady-state junction temperature for emissivities in therange 0.1?1.0.

A thermocouple junction is inserted in a large duct tomeasure the temperature of hot gases flowing throughthe duct. (a) If the duct surface temperature Tsis less than thegas temperature Tg, will the thermocouple sense atemperature that is less than, equal to, or greaterthan Tg? Justify your answer on the basis of a sim-ple analysis.(b) A thermocouple junction in the shape of a 2-mm-diameter sphere with a surface emissivity of 0.60is placed in a gas stream moving at 3 m/s. If thethermocouple senses a temperature of 320?Cwhen the duct surface temperature is 175?C, whatis the actual gas temperature? The gas may beassumed to have the properties of air at atmos-pheric pressure.(c) How would changes in velocity and emissivityaffect the temperature measurement error? Deter-mine the measurement error for velocities in therange 1V25 m/s (??0.6) and for emissivi-ties in the range 0.1?1.0 (V?3 m/s).

Consider temperature measurement in a gas streamusing the thermocouple junction described in Problem7.86 (D?2 mm, ??0.60). If the gas velocity andtemperature are 3 m/s and 500?C, respectively, whattemperature will be indicated by the thermocouple ifthe duct surface temperature is 200?C? The gas may beassumed to have the properties of atmospheric air.What temperature will be indicated by the thermocou-ple if the gas pressure is doubled and all other condi-tions remain the same?

A silicon chip (k ?150 W/m?K, ??2300 kg/m3, cp?700 J/kg?K), 10 mm on a side and 1 mm thick, is con- nected to a substrate by solder balls (k?40 W/m?K,??10,000 kg/m3, cp?150 J/kg?K) of 1-mm diame-ter, and during an accelerated thermal stress test, thesystem is exposed to the flow of a dielectric liquid(k?0.064 W/m?K, ??10?6m2/s, Pr?25). As firstapproximations, treat the top and bottom surfaces of thechip as flat plates in turbulent, parallel flow and assumethe substrate and lower chip surfaces to have a negligi-ble effect on flow over the solder balls. Also assumepoint contact between the chip and the solder, therebyneglecting heat transfer by conduction between thecomponents.(a) The stress test begins with the components at ambi-ent temperature (Ti?20?C) and proceeds withheating by the fluid at T??80?C. If the fluidvelocity is V?0.2 m/s, estimate the ratio of the time constant of the chip to that of a solder ball.Which component responds more rapidly to theheating process?(b) The thermal stress acting on the solder joint is pro-portional to the chip-to-solder temperature differ-ence. What is this temperature difference 0.25 safter the start of heating?

Repeat Example 7.7 for a more compact tube bank inwhich the longitudinal and transverse pitches are SL = ST = 20.5 mm. All other conditions remain the same.

A preheater involves the use of condensing steam at100C on the inside of a bank of tubes to heat air thatenters at 1 atm and 25C. The air moves at 5 m/s incross flow over the tubes. Each tube is 1 m long andhas an outside diameter of 10 mm. The bank consists of 196 tubes in a square, aligned array for which ST = SL = 15 mm. What is the total rate of heat trans-fer to the air? What is the pressure drop associatedwith the airflow?

Consider the in-line tube bank of Problem 7.90(D?10 mm, L?1 m, and ST?SL?15 mm), withcondensing steam used to heat atmospheric air enteringthe tube bank at Ti?25?C and V?5 m/s. In this case,however, the desired outlet temperature, not the numberof tube rows, is known. What is the minimum value ofNLneeded to achieve an outlet temperature of To?75?C? What is the corresponding pressure drop acrossthe tube bank?

A tube bank uses an aligned arrangement of 10-mm-diameter tubes with ST = SL = 20 mm. There are 10rows of tubes with 50 tubes in each row. Consider an application for which cold water flows through thetubes, maintaining the outer surface temperature at 27C, while flue gases at 427C and a velocity of 5 m/sare in cross flow over the tubes. The properties of theflue gas may be approximated as those of atmosphericair at 427C. What is the total rate of heat transfer perunit length of the tubes in the bank?

An air duct heater consists of an aligned array of elec-trical heating elements in which the longitudinal andtransverse pitches are SL?ST?24 mm. There are 3rows of elements in the flow direction (NL?3) and 4elements per row (NT?4). Atmospheric air with anupstream velocity of 12 m/s and a temperature of 25?Cmoves in cross flow over the elements, which have adiameter of 12 mm, a length of 250 mm, and are main-tained at a surface temperature of 350?C.(a) Determine the total heat transfer to the air and thetemperature of the air leaving the duct heater.(b) Determine the pressure drop across the elementbank and the fan power requirement.(c) Compare the average convection coefficient obtainedin your analysis with the value for an isolated (single) element. Explain the difference betweenthe results.(d) What effect would increasing the longitudinal andtransverse pitches to 30 mm have on the exit tem-perature of the air, the total heat rate, and the pres-sure drop?

A tube bank uses an aligned arrangement of 30-mm-diameter tubes with ST?SL?60 mm and a tube lengthof 1 m. There are 10 tube rows in the flow direction(NL?10) and 7 tubes per row (NT?7). Air withupstream conditions of T??27?C and V?15 m/s isin cross flow over the tubes, while a tube wall tempera-ture of 100?C is maintained by steam condensationinside the tubes. Determine the temperature of air leav-ing the tube bank, the pressure drop across the bank,and the fan power requirement.

Repeat Problem 7.94, but with NL?7, NT?10, andV1 ? 0.5 m/s.

Electrical components mounted to each of two isother-mal plates are cooled by passing atmospheric airbetween the plates, and an in-line array of aluminumpin fins is used to enhance heat transfer to the air.The pins are of diameter D?2 mm, length L?100 mm, and thermal conductivity k?240 W/m?K.The longitudinal and transverse pitches areSL?ST?4 mm, with a square array of 625 pins(NT?NL?25) mounted to square plates that are eachof width W?100 mm on a side. Air enters the pinarray with a velocity of 10 m/s and a temperatureof 300 K (a) Evaluating air properties at 300 K, estimate the aver-age convection coefficient for the array of pin fins.(b) Assuming a uniform convection coefficient overall heat transfer surfaces (plates and pins), usethe result of part (a) to determine the air outlet tem- perature and total heat rate when the plates aremaintained at 350 K. Hint:The air outlet tempera-ture is governed by an exponential relation of theform, whereis the mass flow rate of air passingthrough the array, Atis the total heat transfer sur-face area (plates and fins), and ?ois the overallsurface efficiency defined by Equation 3.107.

Consider the chip cooling scheme of Problem 3.146,but with an insulated top wall placed at the pin tips toforce airflow across the pin array. Air enters the array at20?C and with a velocity Vthat may be varied but can-not exceed 10 m/s due to pressure drop considerations.The pin fin geometry, which includes the number ofpins in the NNsquare array, as well as the pin diam-eter Dpand length Lp, may also be varied, subject to theconstraint that the product NDpnot exceed 9 mm.Neglecting heat transfer through the board, assess theeffect of changes in air velocity, and hence ho, as wellas pin fin geometry, on the air outlet temperature andthe chip heat rate, if the remaining conditions of Prob-lems 3.146 and 3.27, including a maximum allowablechip temperature of 75?C, remain in effect. Recom- mend design and operating conditions for which chipcooling is enhanced. Hint:The air outlet temperatureis governed by a relation of the form [(Ts?To)/(Ts?Ti)]?exp[?(At?o)/cp], where is the massflow rate of air passing through the array, Atis the totalheat transfer surface area (chip and pins), and o? is theoverall surface efficiency defined by Equation 3.107.

An air-cooled steam condenser is operated with air incross flow over a square, in-line array of 400 tubes(NL?NT?20), with an outside tube diameter of 20 mmand longitudinal and transverse pitches of SL?60 mmand ST?30 mm, respectively. Saturated steam at apressure of 2.455 bars enters the tubes, and a uniform tube outer surface temperature of Ts?390 K may beassumed to be maintained as condensation occurs withinthe tubes.(a) If the temperature and velocity of the air upstreamof the array are Ti?300 K and V?4 m/s, what isthe temperature Toof the air that leaves the array?As a first approximation, evaluate the properties ofair at 300 K.(b) If the tubes are 2 m long, what is the total heattransfer rate for the array? What is the rate atwhich steam is condensed in kg/s?(c) Assess the effect of increasing NLby a factor of 2,while reducing SLto 30 mm. For this configuration,explore the effect of changes in the air velocity.

Heating and cooling with miniatureimpinging jets hasbeen proposed for numerous applications. For a singleround jet, determine the minimum jet diameter forwhich Equation 7.71 may be applied for air at atmos-pheric pressure (a) at Te0 ? ? and (b) at Te? 00?C

A circular transistor of 10-mm diameter is cooled byimpingement of an air jet exiting a 2-mm- diameterround nozzle with a velocity of 20 m/s and a tempera-ture of 15?C. The jet exit and the exposed surface ofthe transistor are separated by a distance of 10 mm.If the transistor is well insulated at all but its exposedsurface and the surface temperature is not to exceed85?C, what is the transistors maximum allowableoperating power?

A long rectangular plate of AISI 304 stainless steel is ini-tially at 1200 K and is cooled by an array of slot jets (seeFigure 7.17). The nozzle width and pitch areW?10 mmandS?100 mm, respectively, and the nozzle-to-plateseparation isH?200 mm. The plate thickness andwidth aret?8 mm andL?1 m, respectively. If air exits the nozzles at a temperature of 400 K and a velocityof 30 m/s, what is the initial cooling rate of the plate?

A cryogenic probe is used to treat cancerous skintissue. The probe consists of a single round jet ofdiameter De?2 mm that issues from a nozzle concen-trically situated within a larger, enclosed cylindricaltube of outer diameter Do?15 mm. The wall thick-ness of the AISI 302 stainless steel probe is t?2 mm,and the separation distance between the nozzle and theinner surface of the probe is H?5 mm.Assuming the cancerous skin tissue to be a semi-infinite medium with kc?0.20 W/m?K and Tc?37?Cfar from the probe location, determine the surface tem-perature Ts. Neglect the contact resistance between theprobe and the tissue. Cold nitrogen exits the jet atTe?100 K, Ve?20 m/s. Hint:Due to the probewalls, the jet is confined and behaves as if it were onein an array such as in Figure 7.18c

Air at 10 m/s and 15?C is used to cool a square hotmolded plastic plate 0.5 m to a side having a surfacetemperature of 140?C. To increase the throughput ofthe production process, it is proposed to cool the plateusing an array of slotted nozzles with width and pitchof 4 mm and 56 mm, respectively, and a nozzle-to-plate separation of 40 mm. The air exits the nozzle at atemperature of 15?C and a velocity of 10 m/s.(a) Determine the improvement in cooling rate thatcan be achieved using the slotted nozzle arrange-ment in lieu of turbulated air at 10 m/s and 15?C inparallel flow over the plate.(b) Would the heat rates for both arrangements changesignificantly if the air velocities were increased bya factor of 2?(c) What is the air mass rate requirement for the slot-ted nozzle arrangement?

Consider Problem 7.103, in which the improvement inperformance of slot-jet cooling over parallel- flowcooling was demonstrated. Design an optimal roundnozzle array, using the same air jet velocity and temperature, 10 m/s and 15C, respectively, and comparethe cooling rates and supply air requirements. Discussthe features associated with each of the three methodsrelevant to selecting one for this application of coolingthe plastic part

Consider the plasma spraying process of Problems5.25 and 7.82. For a nozzle exit diameter of D?10 mm and a substrate radius of r?25 mm, estimatethe rate of heat transfer by convection qconvfrom theargon plasma to the substrate, if the substrate tempera-ture is maintained at 300 K. Energy transfer to the sub-strate is also associated with the release of latent heatqlat, which occurs during solidification of the impactedmolten droplets. If the mass rate of droplet impinge-ment is p?0.02 kg/s?m2, estimate the rate of latentheat release

You have been asked to determine the feasibility ofusing an impinging jet in a soldering operation forelectronic assemblies. The schematic illustrates the useof a single, round nozzle to direct high- velocity, hotair to a location where a surface mountjoint is to beformed.For your study, consider a round nozzle with a diame-ter of 1 mm located a distance of 2 mm from the regionof the surface mount, which has a diameter of 2.5 mm.(a) For an air jet velocity of 70 m/s and a temperatureof 500?C, estimate the average convection coeffi-cient over the area of the surface mount.(b) Assume that the surface mount region on theprinted circuit board (PCB) can be modeled asa semi-infinite medium, which is initially at auniform temperature of 25?C and suddenly expe-riences convective heating by the jet. Estimatethe time required for the surface to reach 183?C.The thermophysical properties of a typical solderare ??8333 kg/m3, cp?188 J/kg?K, and k?51 W/m?K (c) For each of three air jet temperatures of 500, 600,and 700?C, calculate and plot the surface tempera-ture as a function of time for 0t 150 s. Onthis plot, identify important temperature limits forthe soldering process: the lower limit correspond-ing to the solders eutectic temperature, Tsol?183?C, and the upper limit corresponding to theglass transition temperature, Tgl?250?C, at whichthe PCB becomes plastic. Comment on the out- come of your study, the appropriateness of theassumptions, and the feasibility of using the jet fora soldering application

Consider the packed bed of aluminum spheresdescribed in Problem 5.12 under conditions for whichthe bed is charged by hot air with an inlet velocity ofV?1 m/s and temperature of Tg,i?300?C, but forwhich the convection coefficient is not prescribed. Ifthe porosity of the bed is ??0.40 and the initial tem-perature of the spheres is Ti?25?C, how long does ittake a sphere near the inlet of the bed to accumulate90% of its maximum possible energy?

The use of rock pile thermal energy storage systemshas been considered for solar energy and industrialprocess heat applications. A particular system involvesa cylindrical container, 2 m long by 1 m in diameter, inwhich nearly spherical rocks of 0.03-m diameter arepacked. The bed has a void space of 0.42, and the den-sity and specific heat of the rock are ??2300 kg/m3and cp?879 J/kg?K, respectively. Consider condi-tions for which atmospheric air is supplied to the rockpile at a steady flow rate of 1 kg/s and a temperatureof 90?C. The air flows in the axial direction throughthe container. If the rock is at a temperature of 25?C,what is the total rate of heat transfer from the air to therock pile

The cylindrical chamber of a pebble bed nuclear reac-toris of length L?10 m, and diameter D?3 m. The chamber is filled with spherical uranium oxide pel-lets of core diameter Dp?50 mm. Each pellet gener- ates thermal energy in its core at a rate of and iscoated with a layer of non-heat-generating graphite,which is of uniform thickness ??5 mm, to form apebble. The uranium oxide and graphite each have a thermal conductivity of 2 W/m?K. The packed bedhas a porosity of ??0.4. Pressurized helium at 40 barsis used to absorb the thermal energy from the pebbles.The helium enters the packed bed at Ti?450?Cwith a velocity of 3.2 m/s. The properties of the helium may be assumed to be cp?5193 J/kg?K,k?0.3355 W/m?K, ??2.1676 kg/m3, ??4.21410?5kg/s?m, Pr?0.654.(a) For a desired overall thermal energy transfer rateof q?125 MW, determine the mean outlet tem-perature of the helium leaving the bed, To, and theamount of thermal energy generated by eachpellet, .(b) The amount of energy generated by the fueldecreases if a maximum operating temperature ofapproximately 2100?C is exceeded. Determine themaximum internal temperature of the hottest pelletin the packed bed. For Reynolds numbers in therange 4000ReD10,000, Equation 7.81 maybe replaced by ?jH?2.876 ReD1 ?? .3023 ReD0? .35

Latent heat capsulesconsist of a thin-walled sphericalshell within which a solid-liquid, phase-change mate-rial (PCM) of melting point Tmpand latent heat offusion hsfis enclosed. As shown schematically, thecapsules may be packed in a cylindrical vessel throughwhich there is fluid flow. If the PCM is in its solidstate and Tmp?Ti, heat is transferred from the fluid tothe capsules and latent energy is stored in the PCM as itmelts. Conversely, if the PCM is a liquid and Tmp?Ti,energy is released from the PCM as it freezes and heatis transferred to the fluid. In either situation, all of thecapsules within the packed bed would remain at Tmpthrough much of the phase change process, in whichcase the fluid outlet temperature would remain at afixed value To.Phase-changematerial , Tmp, hsfCapsuleShell, DcLvV, Ti Consider an application for which air at atmosphericpressure is chilled by passing it through a packed bed(??0.5) of capsules (Dc?50 mm) containing anorganic compound with a melting point of Tmp?4?C.The air enters a cylindrical vessel (Lv?Dv?0.40 m)at Ti?25?C and V?1.0 m/s.(a) If the PCM in each capsule is in the solid state atTmpas melting occurs within the capsule, what isthe outlet temperature of the air? If the density andlatent heat of fusion of the PCM are ??1200 kg/m3and hsf?165 kJ/kg, what is the massrate (kg/s) at which the PCM is converted fromsolid to liquid in the vessel? (b) Explore the effect of the inlet air velocity and cap-sule diameter on the outlet temperature.(c) At what location in the vessel will complete melt-ing of the PCM in a capsule first occur? Oncecomplete melting begins to occur, how will theoutlet temperature vary with time and what is itsasymptotic value?

The porosity of a packed bed can be decreased by vibrating the containment vessel as the vessel is filled with theparticles. The vibration promotes particle settling.(a) Consider the air chilling process of Problem7.110a. Determine the outlet air temperature To and mass rate at which the PCM is melted for = 0.30. Assume the total mass of PCM and themass flow rate of air are unchanged. The length ofthe containment vessel Lvis decreased to compensate for the reduced porosity. (b) Determine To and the PCM melting rate for thecase where the diameter of the containment vessel Dv is decreased to compensate for the reducedporosity. Which containment vessel configurationis preferred?

Consider the packed bed (??0.5) of latent heat cap-sules (Dc?50 mm) described in Problem 7.110, butnow for an application in which ambient air is to beheated by passing it through the bed. In this case thecapsules contain an organic compound with a meltingpoint of Tmp?50?C, and the air enters the vessel(Lv?Dv?0.40 m) at Ti?20?C and V?1.0 m/s.(a) If the PCM in each capsule is in the liquid state atTmpas solidification occurs within the capsule, whatis the outlet temperature of the air? If the density andlatent heat of fusion of the PCM are ??900 kg/m3and hsf?200 kJ/kg, what is the mass rate (kg/s) atwhich the PCM is converted from liquid to solid inthe vessel?(b) Explore the effect of the inlet air velocity and cap-sule diameter on the outlet temperature.

Packed beds of spherical particles can besinteredat hightemperature to form permeable, rigid foams. A foamsheet of thicknesst?10 mm is comprised of sinteredbronze spheres, each of diameterD?0.6 mm. The metalfoam has a porosity of??0.25, and the foam sheet fillsthe cross section of anL?40 mmW?40 mm windtunnel. The upper and lower surfaces of the foam are attemperaturesTs?80?C, and the two other foam edges(the front edge shown in the schematic and the corre-sponding back edge) are insulated. Air flows in the windtunnel at an upstream temperature and velocity ofTi?20?C andV?10 m/s, respectively.(a) Assuming the foam is at a uniform temperature Ts,estimatethe convection heat transfer rate to theair. Do you expect the actual heat transfer rate tobe equal to, less than, or greater than your esti-mated value?(b) Assuming one-dimensional conduction in the x- direction, use an extended surface analysis to esti-matethe heat transfer rate to the air. To do so,show that the effectiveperimeter associated withEquation 3.70 is Peff?Ap,t/L. Determine the effec-tive thermal conductivity of the foam keffby usingEquation 3.25. Do you expect the actual heattransfer rate to be equal to, less than, or greaterthan your estimated value?

Consider mass loss from a smooth wet flat plate due toforced convection at atmospheric pressure. The plateis 0.5 m long and 3 m wide. Dry air at 300 K and a freestream velocity of 35 m/s flows over the surface,which is also at a temperature of 300 K. Estimate theaverage mass transfer coefficient and determinethe water vapor mass loss rate (kg/s) from the plate.

Consider dry, atmospheric air in parallel flow over a0.5-m-long plate whose surface is wetted. The airvelocity is 35 m/s, and the air and water are each at atemperature of 300 K.(a) Estimate the heat loss and evaporation rate perunit width of the plate, q?and n?A, respectively.(b) Assuming the air temperature remains at 300 K,generate plots of q?and n?Afor a range of watertemperatures from 300 to 350 K, with air veloci-ties of 10, 20, and 35 m/s.(c) For the air velocities and air temperature of part(b), determine the water temperatures for whichthe heat loss will be zero.

A flat plate coated with a volatile substance (species A)is exposed to dry, atmospheric air in parallel flow withT??20?C and u??8 m/s. The plate is maintained at aconstant temperature of 134?C by an electrical heatingelement, and the substance evaporates from the surface.The plate has a width of 0.25 m (normal to the plane ofthe sketch) and is well insulated on the bottom.The molecular weight and the latent heat of vaporiza-tion of species A are ?A?150 kg/kmol andhfg?5.44106J/kg, respectively, and the mass dif-fusivity is DAB?7.7510?7m2/s. If the saturatedvapor pressure of the substance is 0.12 atm at 134?C,what is the electrical power required to maintainsteady-state conditions?

Dry air at atmospheric pressure and 350 K, with a freestream velocity of 25 m/s, flows over a smooth, porousplate 1 m long.(a) Assuming the plate to be saturated with liquidwater at 350 K, estimate the mass rate of evapora-tion per unit width of the plate, (kg/s?m).(b) For air and liquid water temperatures of 300, 325,and 350 K, generate plots of as a function ofvelocity for the range from 1 to 25 m/s

A scheme for dissipating heat from an array of N = 100 integrated circuits involves joining the cir-cuits to the bottom of a plate and exposing the top ofthe plate to a water bath. The water container is of length L?100 mm on a side and is exposed to airflowat its top surface. The flow is turbulated by the pro-truding lip of the side wall.If the sides and bottom of the container are well insu-lated from the surroundings and heat is uniformlydissipated in each circuit, at what rate may heat be dissipated from each circuit when the water temperatureis maintained at Tb?350 K?

A series of water-filled trays, each 222 mm long, expe-riences an evaporative drying process. Dry air atT??300 K flows over the trays with a velocity of15 m/s, while radiant heaters maintain the surface tem- perature at Ts?330 K.(a) What is the evaporative flux (kg/s?m2) at a dis-tance 1 m from the leading edge?(b) What is the irradiation (W/m2) that should be sup-plied to the tray surface at this location to maintainthe water temperature at 330 K?(c) Assuming the water temperature is uniform overthe tray at this location, what is the evaporation rate(kg/s?m) from the tray per unit width of the tray?(d) What irradiation should be applied to each of thefirst four trays such that the corresponding evapo-ration rates are identical to that found in part (c)?

Consider the physical system of Problem 7.119 (aseries of water-filled trays heated radiatively), butunder operating conditions for which each tray is0.25 m long by 1 m wide and is uniformly irradiated,with G?104W/m2. Dry air at T??300 K continuesto flow over the trays at a velocity of 15 m/s. (a) What is the rate of water loss (kg/s) from the first,third, and fourth trays?(b) Estimate the temperature of the water in each ofthe designated trays

The apparatus described in Problem 7.40 is used by ourstudents to experimentally determine convection heatand mass transfer coefficients, to confirm the heatmassanalogy, and to compare measured results with predic-tions based on standard correlations. The velocity, V, ofthe airstream is measured using a thermistor-basedanemometer, and its relative humidity is determinedfrom measurements of the wet and dry bulb tempera-tures, Twband Tdb, respectively. Thermocouples areattached to the test-plate, which is covered with a sheetof wet paper in the mass transfer experiments.(a)Convection heat transfer coefficientUsing thedata provided in Problem 7.40, determine the heattransfer coefficients for the two velocities, assum-ing the plate to behave as a spacewise isothermalobject. Evaluate the coefficients Cand mfor a cor-relation of the form ?CRemPr1/3. Comparethis result with a standard flat-plate correlation.Comment on the goodnessof the comparison andprovide reasons for any differences.(b)Convection mass transfer coefficientA sheet ofwater-saturated paper, 133 mm to a side, was usedas the test surface and its mass measured at two dif-ferent times, m(t) and m(t??t). Thermocoupleswere used to monitor the paper temperature as afunction of time, from which the average tempera-ture, T_s, was determined. The wet and dry bulbtemperatures were Twb?13?C and Tdb?27?C,and data recorded for two airstream velocities areas follows:Water Mass Loss ObservationsVm(t)m(t??t)?t(m/s) (?C) (g) (g) (s)3 15.3 55.62 54.45 4759 16.0 55.60 54.50 240Determine the convection mass transfer coeffi-cients for the two flow conditions. Evaluate thecoefficients Cand mfor a correlation of the form?CRemSc1/3.(c) Using the heatmass analogy, compare the experi-mental results with each other and against standardcorrelations. Comment on the goodnessof the com- parison and provide reasons for any differences

Dry air at 35C and a velocity of 20 m/s flows over awetted plate of length 500 mm and width 150 mm. Anembedded electrical heater supplies power to maintainthe plate surface temperature at 20C.(a) What is the evaporation rate (kg/h) of water fromthe plate? What electrical power is required tomaintain steady-state conditions?(b) After a long period of operation, all the water isevaporated from the plate and its surface is dry.For the same free stream conditions and heaterpower of part (a), estimate the temperature of the plate.

A minivan traveling 90 km/h has just passed through athunderstorm that left a film of water 0.1 mm thick onthe top of the van. The top of the van can be assumedto be a flat plate 6 m long. Assume isothermal condi-tions at 27C, an ambient air relative humidity of 80%,and turbulent flow over the entire surface. What location on the van top will be the last to dry? What isthe water evaporation rate per unit area (kg/sm2) at thetrailing edge of the van top?

Benzene, a known carcinogen, has been spilled on thelaboratory floor and has spread to a length of 2 m. If afilm 1 mm deep is formed, how long will it take forthe benzene to completely evaporate? Ventilation in thelaboratory provides for airflow parallel to the surfaceat 1 m/s, and the benzene and air are both at 25C. The mass densities of benzene in the saturated vapor andliquid states are known to be 0.417 and 900 kg/m3,respectively.

Atmospheric air of 40% relative humidity and temper-ature T??300 K is in parallel flow over a series ofwater-filled trays, with u??12 m/s.What is the rate at which energy must be supplied toeach of the first three trays to maintain the water at300 K?

A stream of atmospheric air is used to dry a series ofbiological samples on plates that are each of lengthLi?0.25 m in the direction of the airflow. The air isdry and at a temperature equal to that of the plates(T??Ts?50?C). The air speed is u??9.1 m/s. (a) Sketch the variation of the local convection masstransfer coefficient hm,xwith distance xfrom theleading edge. Indicate the specific nature of the xdependence.(b) Which of the plates will dry the fastest? Calculatethe drying rate per meter of width for this plate(kg/s?m).(c) At what rate would heat have to be supplied to thefastest drying plate to maintain it at Ts?50?Cduring the drying process?

Condenser cooling water for a power plant is stored ina cooling pond that is 1000 m long by 500 m wide.However, because of evaporative losses, it is necessary to periodically add makeup water to the pond inorder to maintain a suitable water level. Assumingisothermal conditions at 27C for the water and the air,that the free stream air is dry and moving at a velocityof 2 m/s in the direction of the 1000- m pond length,and that the boundary layer on the water surface iseverywhere turbulent, determine the amount of makeupwater that should be added to the pond daily.

Consider the plate conveyor system of Problem 7.24,but now under conditions for which the plates arebeing transported from a liquid bath used for surfacecleaning. The initial plate temperature is Ti?40?C,and the surfaces are covered with a thin liquid film. Ifthe air velocity and temperature are u??1 m/s andT??20?C, respectively, what is the initial rate of heattransfer from the plate? What is the corresponding rateof change of the plate temperature? The latent heat ofvaporization of the solvent, the diffusion coefficientassociated with transport of its vapor in air, and its sat-urated vapor density at 40?C are hfg?900 kJ/kg,DAB?10?5m2/s, and ?A,sat?0.75 kg/m3, respec-tively. The velocity of the conveyor can be neglectedrelative to that of the air

In a paper-drying process, the paper moves on a con-veyor belt at 0.2 m/s, while dry air from an in- linearray of round jets (Figure 7.18b) impinges normal toits surface. The nozzle diameter and pitch areD?20 mm and S?100 mm, respectively, and thenozzle-to-paper separation is H?200 mm. Air exitsthe nozzle at a velocity and temperature of 20 m/s and 300 K, while the wet paper is maintained at 300 K. Inkg/s?m2, what is the average drying rate of the paper?

In a paper mill drying process, a sheet of paper slurry(waterfiber mixture) has a linear velocity of 5 m/s as itis rolled. Radiant heaters maintain a sheet temperatureof Ts?330 K, as evaporation occurs to dry, ambientair at 300 K above and below the sheet.(a) What is the evaporative flux at a distance ofx?1 m from the leading edge of the roll? What isthe corresponding value of the radiant flux (irradi- ation, G) that must be supplied to the sheet tomaintain its temperature at 330 K? The sheet hasan absorptivity of ??1.(b) To accelerate the drying and paper productionprocesses, the velocity and temperature of the stripare increased to 10 m/s and 340 K, respectively.To maintain a uniform strip temperature, the irra-diation Gmust be varied with xalong the strip.For 0x1 m, compute and plot the variationshm,x(x), N?A(x), and G(x).

A channel of triangular cross section, which is 25 mlong and 1 m deep, is used for the storage of water.The water and the surrounding air are each at a tem-perature of 25?C, and the relative humidity of the airis 50%.(a) If the air moves at a velocity of 5 m/s along thelength of the channel, what is the rate of water lossdue to evaporation from the surface?(b) Obtain an expression for the rate at which thewater depth would decrease with time due to evap-oration. For the above conditions, how long wouldit take for all the water to evaporate

Mass transfer experiments have been conducted on anaphthalene cylinder of 18.4-mm diameter and 88.9-mm length subjected to a cross flow of air in a low-speedwind tunnel. After exposure for 39 min to theairstream at a temperature of 26?C and a velocity of12 m/s, it was determined that the cylinder massdecreased by 0.35 g. The barometric pressure wasrecorded at 750.6 mm Hg. The saturation pressure psatof naphthalene vapor in equilibrium with solid naph-thalene is given by the relation psat?p10E, whereE?8.67?(3766/T), with T(K) and p(bar) being thetemperature and pressure of air. Naphthalene has amolecular weight of 128.16 kg/kmol.(a) Determine the convection mass transfer coefficientfrom the experimental observations.(b) Compare this result with an estimate from anappropriate correlation for the prescribed flowconditions

Dry air at 1-atm pressure and a velocity of 15 m/s is tobe humidified by passing it in cross flow over a porouscylinder of diameter D?40 mm, which is saturatedwith water.(a) Assuming the water and air to be at 300 K, calculatethe mass rate of water evaporated under steady-stateconditions from the cylindrical medium per unitlength.(b) How will the evaporation rate change if the air andwater are maintained at a higher temperature?Generate a plot for the temperature range 300 to350 K to illustrate the effect of temperature on the evaporation rate.

Dry air at 35?C and a velocity of 15 m/s flows over along cylinder of 20-mm diameter. The cylinder iscovered with a thin porous coating saturated withwater, and an embedded electrical heater suppliespower to maintain the coating surface temperatureat 20?C.(a) What is the evaporation rate of water from thecylinder per unit length (kg/h?m)? What electricalpower per unit length of the cylinder (W/m) isrequired to maintain steady-state conditions?(b) After a long period of operation, all the water isevaporated from the coating and its surface is dry.For the same free stream conditions and heaterpower of part (a), estimate the temperature of thesurface.

Dry air at 20?C and a velocity of 15 m/s flows over a20-mm-diameter rod covered with a thin porous coatingthat is saturated with water. The rod (k?175 W/m?K)is 250 mm long and its ends are attached to heat sinksmaintained at 35?C.Perform a steady-state, finite-difference analysis of therodporous coating system, considering conduction inthe rod as well as energy transfer from the surface byconvection heat and mass transfer. Use the analysis toestimate the temperature at the midspan of the rod andthe evaporation rate from the surface. (Suggestions:Use 10 nodes to represent the half-length of the sys- tem. Estimate the overall average convection heattransfer coefficient based on an average film tempera-ture for the system, and use the heatmass transferanalogy to determine the average convection masstransfer coefficient. Validate your code by using it topredict a temperature distribution that agrees with theanalytical solution for a fin without evaporation.)

Approximate the human form as an unclothed verticalcylinder of 0.3-m diameter and 1.75-m length with asurface temperature of 30C.(a) Calculate the heat loss in a 10-m/s wind at 20C.(b) What is the heat loss if the skin is covered with athin layer of water at 30C and the relative humid-ity of the air is 60%?

t has been suggested that heat transfer from a surfacecan be augmented by wetting it with water. As a spe-cific example, consider a horizontal tube that isexposed to a transverse stream of dry air. You mayassume that the tube, which is maintained at a temper-ature Ts?T?, is completely wetted on the outsidewith a thin film of water. Derive an equation to deter-mine the extent of heat transfer enhancement due towetting. Evaluate the enhancement for V?10 m/s,D?10 mm, Ts?320 K, and T??300 K.

In the first stage of a paper-drying process, a cylinderof diameter 0.15 m is covered by moisture- soakedpaper. The temperature of the paper is maintained at70C by embedded electrical heaters. Dry air at avelocity of 10 m/s and temperature of 20C flows overthe cylinder. (a) Calculate the required electrical power and theevaporation rate per unit length of the cylinder, q?and n?A, respectively.(b) Generate plots of q?and n?Aas a function of the dryair velocity for 5V20 m/s and for paper tem- peratures of 65, 70, and 75C ? .

Cylindrical dry-bulb and wet-bulb thermometers areinstalled in a large-diameter duct to obtain the temper-ature T?and the relative humidity ?of moist airflowing through the duct at a velocity V. The dry- bulbthermometer has a bare glass surface of diameter Ddband emissivity ?g. The wet-bulb thermometer is cov-ered with a thin wick that is saturated with water flow-ing continuously by capillary action from a bottomreservoir. Its diameter and emissivity are designated asDwband ?w. The duct inside surface is at a known tem-perature Ts, which is less than T?. Develop expressionsthat may be used to obtain T?and ?from knowledgeof the dry-bulb and wet-bulb temperatures Tdband Twband the foregoing parameters. Determine T?and ?when Tdb?45?C, Twb?25?C, Ts?35?C, p?1 atm,V?5 m/s, Ddb?3 mm, Dwb?4 mm, and ?g??w?0.95. As a first approximation, evaluate the dry- andwet-bulb air properties at 45 and 25C? , respectively

The thermal pollution problem is associated with dis-charging warm water from an electrical power plant orfrom an industrial source to a natural body of water.Methods for alleviating this problem involve coolingthe warm water before allowing the discharge tooccur. Two such methods, involving wet cooling tow-ers or spray ponds, rely on heat transfer from the warmwater in droplet form to the surrounding atmosphere.To develop an understanding of the mechanisms thatcontribute to this cooling, consider a spherical dropletof diameter Dand temperature T, which is moving at avelocity Vrelative to air at a temperature T?andrelative humidity ?. The surroundings are character-ized by the temperature Tsur. Develop expressionsfor the droplet evaporation and cooling rates. Calcu-late the evaporation rate (kg/s) and cooling rate (K/s)whenD?3 mm, V?7 m/s, T?40?C, T??25?C,Tsur?15?C, and ??0.60. The emissivity of wateris w? ?0.96

Cranberries are harvested by flooding the bogs inwhich they are grown and raking them into troughs fortransport. At the processing plant, the surface moistureon the berries is removed as they roll over a fine screenthrough which warm air is blown. The berries have anaverage diameter of 15 mm, and the thickness of thewater layer is 0.2 mm. f the velocity and temperature of the heated air are2 m/s and 30?C, respectively, estimate the timerequired to dry the berries. Assume that the water filmon the berries is also at 30?C.

A spherical drop of water, 0.5 mm in diameter, isfalling at a velocity of 2.15 m/s through dry, still air at1-atm pressure. Estimate the instantaneous rate ofevaporation from the drop if the drop surface is at 60C and the air is at 100C.

A spherical droplet of alcohol, 0.5 mm in diameter, isfalling freely through quiescent air at a velocity of1.8 m/s. The concentration of alcohol vapor at thesurface of the droplet is 0.0573 kg/m3, and the diffu-sion coefficient for alcohol in air is 10?5m2/s. Neglect-ing radiation and assuming steady-state conditions,calculate the surface temperature of the droplet if theambient air temperature is 300 K. The latent heat ofvaporization is 8.42105J/kg.

As described in Problem 7.84, the second step in tissueengineering is to seed the top surface of the scaffoldwith human cells that subsequently grow into thepores of the scaffold. A seeding method that has beenproposed is to use a droplet generator similar to that ofProblem 7.84 to generate Dp?50 ?m diameter drops.The material in the droplet generator is a slurry con-sisting of a mixture of a host liquid and human livercells. The host liquid has properties similar to water,and the liver cells are spherical with a diameter ofDlc?20 ?m and density ?lc?2400 kg/m3. Dropletsare injected into atmospheric air with a relative humid-ity and temperature of ??0.50 and T??25?C,respectively. The particles are injected with an initialtemperature of Ti?25?C.(a) It is desirable for each drop to contain one livercell. Determine the volume fraction, f, of livercells in the slurry and the terminal velocity for adrop containing one liver cell.(b) The droplet containing one liver cell is injected atits terminal velocity. Determine the time of flightfor a distance between the ejector nozzle and thescaffold of L?4 mm. c) Determine the initial evaporation rate from thedroplet.(d) The tissue engineer is concerned that evaporationwill change the mass of the droplet and, in turn, willaffect its time of flight and the precision withwhich the seeds can be placed on the scaffold.Estimate the maximum change in mass due toevaporation during the time of flight. Compare thevariation of mass due to evaporation to the varia-tion associated with there being one to three livercells per droplet. Does evaporation or the liver cellpopulation per droplet influence the variability ofthe droplet mass most significantly?

Motile bacteria are equipped with flagella that arerotated by tiny, biological electrochemical engineswhich, in turn, propel the bacteria through a hostliquid. Consider a nominally spherical Escherichiacolibacterium that is of diameter D?2 ?m. The bac-terium is in a water-based solution at 37?C containinga nutrient which is characterized by a binary diffusioncoefficient of DAB?0.710?9m2/s and a foodenergy value of ??16,000 kJ/kg. There is a nutrientdensity difference between the fluid and the shell ofthe bacterium of ??A?86010?12kg/m3. Assuminga propulsion efficiency of ??0.5, determine the max-imum speed of the E. coli. Report your answer in bodydiameters per second.

In a home furnace humidification system, water dropletsof diameter Dare discharged in a direction opposing themotion of warm air emerging from the heater. The air ishumidified by evaporation from the droplets, and theexcess water is collected on a splash plate,from whichit is routed to a drain. Consider conditions for which air enters the heater ata temperature and relative humidity of 17?C and70%, respectively, and leaves the heater at a temper-ature of 47?C. The droplet diameter is 1 mm, andthe relative velocity between the droplets and theheated air is 15 m/s. During the time-of-flight,the change in droplet diameter may be neglectedand the droplet temperature may be assumed to remainat 47?C. What is the rate of evaporation from a single droplet?

Evaporation of liquid fuel droplets is often studied inthe laboratory by using a porous sphere techniquein which the fuel is supplied at a rate just sufficient tomaintain a completely wetted surface on the sphere.Consider the use of kerosene at 300 K with a poroussphere of 1-mm diameter. At this temperature thekerosene has a saturated vapor density of 0.015 kg/m3and a latent heat of vaporization of 300 kJ/kg. Themass diffusivity for the vaporair mixture is 10?5m2/s.If dry, atmospheric air at V?15 m/s and T??300 Kflows over the sphere, what is the minimum mass rateat which kerosene must be supplied to maintain a wettedsurface? For this condition, by how much mustT?actually exceed Tsto maintain the wetted surfaceat 300 K?

Consider an air-conditioning system composed of abank of tubes arranged normal to air flowing in a ductat a mass rate of (kg/s). A coolant flowing throughthe tubes is able to maintain the surface temperatureof the tubes at a constant value of Ts?Ta,i, where Ta,iis the inlet air temperature (upstream of the tubebank). It has been suggested that air cooling may beenhanced if a thin, uniform film of water is main-tained on the outer surface of each of the tubes.(a) Assuming the water film to be at the temperatureTs, develop an expression for the ratio of theamount of cooling that occurs with the water filmto the amount of cooling that occurs without thefilm. The amount of cooling may be defined asTa,i?Ta,o, where Ta,ois the outlet air temperature(downstream of the tube bank). The upstream air may be assumed to be dry, and the driving poten-tials for convection heat and mass transfer may beapproximated as (Ta,i?Ts) and ?A,sat(Ts), respec-tively. Note:The total rate of heat loss from theair may be expressed as (Ta,i?Ta,o).Estimate the value of this ratio under conditionsfor which Ta,i?35?C and Ts?10?C.(b) Consider a tube bank that is 5 rows deep, with 12tubes in a row. Each tube is 0.5 m long, with anoutside diameter of 8 mm, and a staggered arrange-ment is used for which ST?SL?24 mm. Underconditions for which m.a?0.5 kg/s,V?3 m/s,Ta,i?35?C, and Ts?10?C, what is the value of Ta,oif the tubes are wetted? What is the specifichumidity of the air leaving the tube bank?

In a paper-drying process, the paper moves on a con-veyor belt at 0.2 m/s, while dry air from an array ofslot jets (Figure 7.17) impinges normal to its surface.The nozzle width and pitch are W?10 mm andS?100 mm, respectively, and the nozzle-to-plate sep-aration is H?200 mm. The wet paper is of widthL?1 m and is maintained at 300 K, while the air exitsthe nozzles at a temperature of 300 K and a velocity of20 m/s. In kg/s?m2, what is the average drying rate perunit surface area of the paper?