It is desired to develop a simple model for predicting

The temperature distribution within a laminar thermalboundary layer associated with flow over an isothermalflat plate is shown in the sketch. The temperature distri-bution shown is located at x?x2. (a) Is the plate being heated or cooled by the fluid?(b) Carefully sketch the temperature distributions atx?x1and x?x3. Based on your sketch, at whichof the three x-locations is the local heat flux largest?At which location is the local heat flux smallest?(c) As the free stream velocity increases, the velocityand thermal boundary layers both become thinner.Carefully sketch the temperature distributions atx?x2for (i) a low free stream velocity and (ii) ahigh free stream velocity. Based on your sketch,which velocity condition will induce the largerlocal convective heat flux?

In flow over a surface, velocity and temperature profilesare of the formswhere the coefficients Athrough Gare constants.Obtain expressions for the friction coefficient Cfandthe convection coefficient hin terms of u?, T?, andappropriate profile coefficients and fluid properties.

In a particular application involving airflow over aheated surface, the boundary layer temperature distribu-tion may be approximated aswhere yis the distance normal to the surface and thePrandtl number, Pr?cp?/k?0.7, is a dimensionlessfluid property. If T??400 K, Ts?300 K, andu?/??5000 m?1, what is the surface heat flux?

Water at a temperature of T = 25C flows over one ofthe surfaces of a steel wall (AISI 1010) whose tempera-ture is Ts = 40C. The wall is 0.35 m thick, and its other surface temperature is Ts,2 = 100C. For steady-state conditions what is the convection coefficient associated with the water flow? What is the temperature gradient in the wall and in the water that is in contact with the wall? Sketch the temperature distribution inthe wall and in the adjoining water.

For laminar flow over a flat plate, the local heat transfercoefficient hxis known to vary as x?1/2, where xis thedistance from the leading edge (x?0) of the plate.What is the ratio of the average coefficient betweenthe leading edge and some location xon the plate to thelocal coefficient at x?

A flat plate is of planar dimension 1 m?0.75 m. Forparallel laminar flow over the plate, calculate the ratioof the average heat transfer coefficients over the entireplate, L,1/L,2, for two cases. In Case 1, flow is in theshort direction (L?0.75 m); in Case 2, flow is in the longdirection (L?1 m). Which orientation will result in thelarger heat transfer rate? See Problem 6.5.

Parallel flow of atmospheric air over a flat plate oflength L?3 m is disrupted by an array of stationaryrods placed in the flow path over the plate.Laboratory measurements of the local convection coeffi-cient at the surface of the plate are made for a prescribedvalue of Vand Ts?T?. The results are correlated by anexpression of the form hx?0.7?13.6x?3.4x2, wherehxhas units of W/m2?K and xis in meters. Evaluate theaverage convection coefficient Lfor the entire plate andthe ratio at the trailing edge.

For laminar free convection from a heated vertical sur-face, the local convection coefficient may be expressedas hx?Cx?1/4, where hxis the coefficient at a distance xfrom the leading edge of the surface and the quantity C,which depends on the fluid properties, is independent ofx. Obtain an expression for the ratio x/hx, where xis theaverage coefficient between the leading edge (x?0) andthe x-location. Sketch the variation of hxand xwithx.

A circular, hot gas jet at T?is directed normal to a circu-lar plate that has radius roand is maintained at a uniformtemperature Ts. Gas flow over the plate is axisymmetric,causing the local convection coefficient to have a radialdependence of the form h(r)?a?brn, where a, b, andnare constants. Determine the rate of heat transfer to theplate, expressing your result in terms of T?, Ts, ro, a,b, and n

Experiments have been conducted to determine localheat transfer coefficients for flow perpendicular to along, isothermal bar of rectangular cross section. Thebar is of width cparallel to the flow, and height dnor-mal to the flow. For Reynolds numbers in the range104?Red?5?104, the face- averagedNusselt num-bers are well correlated by an expression of the formThe values of Cand mfor the front face, side faces, andback face of the rectangular rod are found to be the fol- lowing:Facec/dCmFront 0.33 ?c/d?1.33 0.674 1/2Side 0.330.153 2/3Side 1.330.107 2/3Back 0.330.174 2/3Back 1.330.153 2/3Determine the value of the average heat transfer coeffi-cient for the entire exposed surface (that is, averaged overall four faces) of a c?40-mm-wide, d?30-mm-tall rec-tangular rod. The rod is exposed to air in cross flow atV?10 m/s, T??300 K. Provide a plausible explanationof the relative values of the face- averaged heat transfercoefficients on the front, side, and back faces

A concentrating solar collector consists of a parabolicreflector and a collector tube of diameter D, throughwhich flows a working fluid that is heated with concen-trated solar irradiation. Throughout the day, the reflec-tor is slowly repositioned to track the sun. For windconditions characterized by a steady, horizontal flownormal to the tube axis, the local heat transfer coeffi-cient on the tube surface varies, as shown in theschematic for various reflector positions.Case 1V, TCollector tubeParabolic reflectorDNud?hd/k?CRemdPr1/3(a) Estimate the value of the average heat transfercoefficient over the entire collector tube surface foreach of the three cases.(b) Assuming the tube receives the same amount ofsolar irradiation in each case, which case wouldhave the highest collector efficiency?

Air at a free stream temperature of T??20C isin parallel flow over a flat plate of length L?5 m andtemperature Ts?90C. However, obstacles placed inthe flow intensify mixing with increasing distance xfrom the leading edge, and the spatial variation oftemperatures measured in the boundary layer is corre-lated by an expression of the form T(C)?20?70exp (?600xy), where xand yare in meters. Determineand plot the manner in which the local convectioncoefficient hvaries with x. Evaluate the average con-vection coefficient for the plate.

The heat transfer rate per unit width (normal to the page)from a longitudinal section, x2?x1, can be expressed asq?12?12(x2?x1)(Ts?T?), where12is the averagecoefficient for the section of length (x2?x1). Considerlaminar flow over a flat plate with a uniform temperatureTs. The spatial variation of the local convection coeffi-cient is of the form hx?Cx?1/2, where Cis a constant.hhh02h(W/m2K)04020Case 1Case 2Case 3Case 3V, TCollector tubeParabolic reflectorCase 2V, TCollector tubeParabolic reflector420Chapter 6?Introduction to ConvectionCH006.qxd 2/24/11 3:32 PM Page 420 (a) Beginning with the convection rate equation in theform dq??hxdx(Ts?T?), derive an expressionfor 12in terms of C, x1, and x2.(b) Derive an expression for 12in terms of x1, x2, andthe average coefficients 1and 2, corresponding tolengths x1and x2, respectively

Experiments to determine the local convection heattransfer coefficient for uniform flow normal to a heatedcircular disk have yielded a radial Nusselt number dis-tribution of the formwhere both nand aare positive. The Nusselt number atthe stagnation point is correlated in terms of theReynolds (ReD?VD/?) and Prandtl numbersObtain an expression for the average Nusselt number,D?D/k,corresponding to heat transfer from anisothermal disk. Typically, boundary layer developmentfrom a stagnation point yields a decaying convectioncoefficient with increasing distance from the stagnationpoint. Provide a plausible explanation for why theopposite trend is observed for the disk.

An experimental procedure for validating results ofProblem 6.14 involves preheating a copper disk to aninitial elevated temperature Tiand recording its temper-ature history T(t) as it is subsequently cooled by the impinging flow to a final temperature Tf. The measuredtemperature decay may then be compared with predic-tions based on the correlation for . Assume thatvalues of a?0.30 and n?2 are associated with thecorrelation.Consider experimental conditions for which a disk ofdiameter D?50 mm and length L?25 mm is preheatedto Ti?1000 K and cooled to Tf?400 K by an impingingairflow at T??300 K. The cooled surface of the disk hasan emissivity of?0.8 and is exposed to large, isother-mal surroundings for which Tsur?T?. The remainingsurfaces of the disk are well insulated, and heat transferthrough the supporting rod may be neglected. Usingresults from Problem 6.14, compute and plot temperaturehistories corresponding to air velocities of V?4, 20, and50 m/s. Constant properties may be assumed for the cop-per (??8933 kg/m3, cp?425 J/kg?K, k?386 W/m?K)and air (??38.8?10?6m2/s, k?0.0407 W/m?K,Pr?0.684).

If laminar flow is induced at the surface of a disk dueto rotation about its axis, the local convection coeffi-cient is known to be a constant, h?C, independent ofradius. Consider conditions for which a disk of radiusro?100 mm is rotating in stagnant air at T??20Cand a value of C?20 W/m2?K is maintained.If an embedded electric heater maintains a surfacetemperature of Ts?50C, what is the local heat flux atthe top surface of the disk? What is the total electricpower requirement? What can you say about the natureof boundary layer development on the disk?

Consider the rotating disk of Problem 6.16. A disk-shaped, stationary plate is placed a short distance awayfrom the rotating disk, forming a gap of width g.The stationary plate and ambient air are at T??20C.If the flow is laminar and the gap-to-radius ratio,G?g/ro, is small, the local radial Nusselt number dis-tribution is of the formNur?h(r)rk?70(1?e?140G) Re?0.456roRe0.478rSupport tuberroConnections toelectric heaterTsAirTNuD?Problems421CH006.qxd 2/24/11 3:32 PM Page 421 where Rer??r2/?[Pelle J., and S. Harmand, Exp.Thermal Fluid Science, 31, 165, 2007]. Determine thevalue of the average Nusselt number, where D?2ro. If the rotating disk temperature isTs?50C, what is the total heat flux from the diskstop surface for g?1 mm, ??150 rad/s? What is thetotal electric power requirement? What can you sayabout the nature of the flow between the disks?

Consider airflow over a flat plate of length L?1munder conditions for which transition occurs atxc?0.5 m based on the critical Reynolds number,Rex,c?5?105.(a) Evaluating the thermophysical properties of air at350 K, determine the air velocity.(b) In the laminar and turbulent regions, the local con-vection coefficients are, respectively,where, at T?350 K, Clam?8.845 W/m3/2?K, Cturb?49.75 W/m1.8?K, and xhas units of m. Develop anexpression for the average convection coefficient, lam(x), as a function of distance from the leadingedge, x, for the laminar region, 0?x?xc.(c) Develop an expression for the average convectioncoefficient, turb(x), as a function of distance from theleading edge, x, for the turbulent region, xc?x?L.(d) On the same coordinates, plot the local and averageconvection coefficients, hxand , respectively, as afunction of xfor 0?x?L

A fan that can provide air speeds up to 50 m/s is to beused in a low-speed wind tunnel with atmospheric air at25C. If one wishes to use the wind tunnel to study flat-plate boundary layer behavior up to Reynolds numbersof Rex?108, what is the minimum plate length thatshould be used? At what distance from the leading edgewould transition occur if the critical Reynolds numberwere Rex,c?5?105?

Consider the flow conditions of Example 6.4 for twosituations, one in which the flow is completely laminar,and the second for flow that is tripped to turbulence atthe leading edge of the plate. Determine whether thereis a plate length L for which the average convectioncoefficient for laminar flow is the same as the averageconvection coefficient for turbulent flow. Assume awater temperature of 300 K.

Assuming a transition Reynolds number of 5?105,determine the distance from the leading edge of a flat plate at which transition will occur for each of thefollowing fluids when u??1 m/s: atmospheric air,engine oil, and mercury. In each case, calculate the tran-sition location for fluid temperatures of 27C and 77C.

To a good approximation, the dynamic viscosity ?, thethermal conductivity k, and the specific heat cpare inde-pendent of pressure. In what manner do the kinematicviscosity ?and thermal diffusivity ?vary with pressurefor an incompressible liquid and an ideal gas? Deter-mine ?of air at 350 K for pressures of 1, 5, and 10 atm.Assuming a transition Reynolds number of 5?105,determine the distance from the leading edge of a flatplate at which transition will occur for air at 350 K atpressures of 1, 5, and 10 atm with u??2 m/s.

For the situation described in Example 6.4, the bound-ary layer can be trippedinto a turbulent state byapplying roughness to the surface of the flat plate at aparticular x-location. Hence the location where transi-tion occurs, xc, can be moved upstream relative to thetransition location associated with the smooth plate ofthe example. Calculate and plot the average convectioncoefficient over the entire plate for roughness appliedover the range 0?xr?L. What values of xrprovidethe minimum and maximum values of ? Assume thewater temperature is 300 K

Consider a laminar boundary layer developing over aflat plate. The flow is incompressible.(a) Substitute Equations 6.31 and 6.32 into Equation6.38 to determine the boundary conditions indimensional form associated with flow over a flatplate of length L. (b) Substitute Equations 6.31, 6.32, as well as the defi-nition of ReLinto Equation 6.35, and compare theresulting expression with Equation 6.28. Note thatfor a flat plate, dp/dx0? and u??V

Consider a laminar boundary layer developing over an isothermal flat plate. The flow is incompressible, and viscous dissipation is negligible.(a) Substitute Equations 6.31 and 6.33 into Equation 6.39 to determine the thermal boundary conditions in dimensional form associated with flow over a flat plate of length L and temperature Ts.(b) Substitute Equations 6.31, 6.32, and 6.33, as wellas the definitions of ReL and Pr, into Equation 6.36,and compare the resulting dimensional expression with Equation 6.29

Experiments have shown that the transition from lami-nar to turbulent conditions for flow normal to the axis of a long cylinder occurs at a critical Reynolds numberof ReD,c2?105, where Dis the cylinder diameter.Moreover, the transition from incompressible to com-pressible flow occurs at a critical Mach number of Mac0.3. For air at a pressure of p?1 atm and tem-perature T?27C, determine the critical cylinderdiameter Dcbelow which, if the flow is turbulent, com-pressibility effects are likely to be important.

An object of irregular shape has a characteristic lengthof L?1 m and is maintained at a uniform surface tem-perature of Ts?400 K. When placed in atmospheric airat a temperature of T??300 K and moving with avelocity of V?100 m/s, the average heat flux from thesurface to the air is 20,000 W/m2. If a second object ofthe same shape, but with a characteristic lengthofL?5 m, is maintained at a surface temperature ofTs?400 K and is placed in atmospheric air atT??300 K, what will the value of the average convec-tion coefficient be if the air velocity is V?20 m/s?

Experiments have shown that, for airflow at T??35Cand V1?100 m/s, the rate of heat transfer from a tur- bine blade of characteristic length L1?0.15 m and sur-face temperature Ts,1?300C is q1?1500 W. Whatwould be the heat transfer rate from a second turbineblade of characteristic length L2?0.3 m operating atTs,2?400C in airflow of T??35C and V2?50 m/s?The surface area of the blade may be assumed to bedirectly proportional to its characteristic length.

Experimental measurements of the convection heattransfer coefficient for a square bar in cross flowyielded the following values:Assume that the functional form of the Nusselt numberis , where C, m, and nare constants.(a) What will be the convection heat transfer coefficientfor a similar bar with L?1 m when V?15 m/s?(b) What will be the convection heat transfer coefficientfor a similar bar with L?1 m when V?30 m/s?(c) Would your results be the same if the side of thebar, rather than its diagonal, were used as the char-acteristic length?

To assess the efficacy of different liquids for cooling anobject of given size and shape by forced convection, it isconvenient to introduce a figureof merit, FF, which com-bines the influence of all pertinent fluid properties on theconvection coefficient. If the Nusselt number is governedby an expression of the form, , obtainthe corresponding relationship between FFand the fluid properties. For representative values of m?0.80and n?0.33, calculate values of FFfor air(k?0.026 W/m?K, ??1.6?10?5m2/s, Pr?0.71),water (k?0.600 W/m?K, ??10?6m2/s, Pr?5.0), anda dielectric liquid (k?0.064 W/m?K, ??10?6m2/s,Pr?25). Which fluid is the most effective cooling agent?

Gases are often used instead of liquids to cool electronicsin avionics applications because of weight considerations.The cooling systems are often closedso that coolants otherthan air may be used. Gases with high figures of merit (seeProblem 6.30) are desired. For representative values ofm?0.85 and n?0.33 in the expression of Problem 6.30,determine the figures of merit for air, pure helium, purexenon (k?0.006 W/m?K, ??24.14?10?6N?s/m2),and an ideal He-Xe mixture containing 0.75 mole fraction ofhelium (k?0.0713 W/m?K, ??25.95?10?6N?s/m2).Evaluate properties at 300 K and atmospheric pressure.For monatomic gases such as helium and xenon and theirmixtures, the specific heat at constant pressure is welldescribed by the relation cp?(5/2)/ ? ?

Experimental results for heat transfer over a flat platewith an extremely rough surface were found to be cor-related by an expression of the formwhere Nuxis the local value of the Nusselt number at aposition xmeasured from the leading edge of the plate.Obtain an expression for the ratio of the average heattransfer coefficient to the local coefficient hx.

Consider conditions for which a fluid with a free streamvelocity of V?1 m/s flows over a surface with a char-acteristic length of L?1 m, providing an averageconvection heat transfer coefficient of .Calculate the dimensionless parameters ,ReL,Pr,and for the following fluids: air, engine oil, mercury,and water. Assume the fluids to be at 300 K.

Consider the nanofluid of Example 2.2. (a) Calculate the Prandtl numbers of the base fluid andnanofluid, using information provided in the exam-ple problem.(b) For a geometry of fixed characteristic dimension L,and a fixed characteristic velocity V, determine the ratio of the Reynolds numbers associated with thetwo fluids, Renf/Rebf. Calculate the ratio of the aver-age Nusselt numbers, , that is associatedwith identical average heat transfer coefficients forthe two fluids, . The functional dependence of the average Nusseltnumber on the Reynolds and Prandtl numbers for abroad array of various geometries may be expressedin the general formwhere Cand mare constants whose values dependon the geometry from or to which convection heattransfer occurs. Under most conditions the value ofmis positive. For positive m, is it possible for thebase fluid to provide greater convection heat trans-fer rates than the nanofluid, for conditions involv-ing a fixed geometry, the same characteristicvelocities, and identical surface and ambienttemperatures?

For flow over a flat plate of length L, the local heattransfer coefficient hxis known to vary as x?1/2, where xis the distance from the leading edge of the plate. What isthe ratio of the average Nusselt number for the entireplate to the local Nusselt number at x?L(NuL)?

For laminar boundary layer flow over a flat plate withair at 20C and 1 atm, the thermal boundary layerthickness ?tis approximately 13% larger than thevelocity boundary layer thickness ?. Determinethe ratio ?? if the fluid is ethylene glycol under thesame flow conditions.

Sketch the variation of the velocity and thermal bound-ary layer thicknesses with distance from the leadingedge of a flat plate for the laminar flow of air, water,engine oil, and mercury. For each case assume a meanfluid temperature of 300 K.

Consider parallel flow over a flat plate for air at 300 Kand engine oil at 380 K. The free stream velocity isu??2 m/s. The temperature difference between thesurface and the free stream is the same in both cases,with Ts?T?.(a) Determine the location where transition to turbu-lence occurs, xc, for both fluids. (b) For laminar flow over a flat plate, the velocityboundary layer thickness is given byCalculate and plot the velocity boundary layerthickness ?over the range 0?x?xcfor each fluid. c) Calculate and plot the thermal boundary layerthickness ?tfor the two fluids over the same rangeof xused in part (b). At an x- location where bothfluids experience laminar flow conditions, explainwhich fluid has the largest temperature gradient atthe plate surface,??T/?y?y?0. Which fluid is associ-ated with the largest local Nusselt number Nu?Which fluid is associated with the largest local heattransfer coefficient h?

Forced air at T??25C and V?10 m/s is used to coolelectronic elements on a circuit board. One such ele- ment is a chip, 4 mm?4 mm, located 120 mm fromthe leading edge of the board. Experiments haverevealed that flow over the board is disturbed by theelements and that convection heat transfer is correlatedby an expression of the formEstimate the surface temperature of the chip if it is dis-sipating 30 mW

Consider the electronic elements that are cooled byforced convection in Problem 6.39. The cooling systemis designed and tested at sea level (p1 atm), but thecircuit board is sold to a customer in Mexico City, withan elevation of 2250 m and atmospheric pressure of76.5 kPa.(a) Estimate the surface temperature of the chiplocated 120 mm from the leading edge of the boardwhen the board is operated in Mexico City. Thedependence of various thermophysical propertieson pressure is noted in Problem 6.22.(b) It is desirable for the chip operating temperature tobe independent of the location of the customer.What air velocity is required for operation inMexico City if the chip temperature is to be thesame as at sea level?

Consider the chip on the circuit board of Problem 6.39.To ensure reliable operation over extended periods, thechip temperature should not exceed 85C. Assumingthe availability of forced air at T??25C and applicabilityof the prescribed heat transfer correlation, compute andplot the maximum allowable chip power dissipation Pcas a function of air velocity for 1 ?V?25 m/s. If thechip surface has an emissivity of 0.80 and the board ismounted in a large enclosure whose walls are at 25C,what is the effect of radiation on the Pc Vplot?

A major contributor to product defects in electronicmodules relates to stresses induced during thermalcycling (intermittent heating and cooling). For exam-ple, in circuit cards having active and passive compo-nents with materials of different thermal expansioncoefficients, thermal stresses are the principal source offailure in component joints, such as soldered and wiredconnections. Although concern is generally for fatiguefailure resulting from numerous excursions during thelife of a product, it is possible to identify defective jointsby performing accelerated thermal stress tests beforethe product is released to the customer. In such cases, itis important to achieve rapid thermal cycling to mini-mize disruptions to production schedules.A manufacturer of circuit cards wishes to developan apparatus for imposing rapid thermal transients onthe cards by subjecting them to forced convection char-acterized by a relation of the form ,where m?0.8 and n?0.33. However, he doesnot know whether to use air (k?0.026 W/m?K,??1.6?10?5m2/s, Pr?0.71) or a dielectric liquid(k?0.064 W/m?K, ??10?6m2/s, Pr?25) as theworking fluid. Assuming equivalent air and liquidvelocities and validity of the lumped capacitance modelfor the components, obtain a quantitative estimate ofthe ratio of the thermal time constants for the two flu-ids. What fluid provides the faster thermal response?

The defroster of an automobile functions by discharg-ing warm air on the inner surface of the windshield. Toprevent condensation of water vapor on the surface, thetemperature of the air and the surface convection coeffi-cient ( , ) must be large enough to maintain asurface temperature Ts,ithat is at least as high as thedewpoint (Ts,i ?Tdp).Consider a windshield of length L?800 mm andthickness t?6 mm and driving conditions for whichthe vehicle moves at a velocity of V?70 mph in ambi-ent air at T?,o??15C. From laboratory experimentsperformed on a model of the vehicle, the average con-vection coefficient on the outer surface of the wind-shield is known to be correlated by an expression of theform , where ReL?VL/?. Airproperties may be approximated as k ?0.023 W/m?K,??12.5?10?6m2/s, and Pr?0.71. If Tdp?10C and ?50C, what is the smallest value of required to prevent condensation on the inner surface?

A microscale detector monitors a steady flow(T??27C, V?10 m/s) of air for the possible pres-ence of small, hazardous particulate matter that may besuspended in the room. The sensor is heated to aslightly higher temperature to induce a chemical reac-tion associated with certain substances of interest thatmight impinge on the sensors active surface. Theactive surface produces an electric current if such sur-face reactions occur; the electric current is then sent toan alarm. To maximize the sensor heads surface areaand, in turn, the probability of capturing and detecting aparticle, the sensor head is designed with a very com-plex shape. The value of the average heat transfer coef-ficient associated with the heated sensor must be knownso that the required electrical power to the sensor canbe determined.Consider a sensor with a characteristic dimension ofLs?80?m. A scale model of the sensor is placed in arecirculating (closed) wind tunnel using hydrogen asthe working fluid. If the wind tunnel operates at ahydrogen absolute pressure of 0.5 atm and velocity ofV?0.5 m/s, find the required hydrogen temperatureand characteristic dimension of the scale model, Lm.

A thin, flat plate that is 0.2 m?0.2 m on a side isoriented parallel to an atmospheric airstream having avelocity of 40 m/s. The air is at a temperature ofT??20C, while the plate is maintained at Ts?120C.The airflows over the top and bottom surfaces of theplate, and measurement of the drag force reveals a valueof 0.075 N. What is the rate of heat transfer from bothsides of the plate to the air?

Atmospheric air is in parallel flow (u??15 m/s,T??15C) over a flat heater surface that is to be main-tained at a temperature of 140C. The heater surfacearea is 0.25 m2, and the airflow is known to induce adrag force of 0.25 N on the heater. What is the electri-cal power needed to maintain the prescribed surfacetemperature?

Determine the drag force imparted to the top surfaceof the flat plate of Example 6.4 for water temperatures of 300 K and 350 K. Assume the plate dimension in the z-direction is W = 1 m.

For flow over a flat plate with an extremely rough surface, convection heat transfer effects are known to be correlated by the expression of Problem 6.32. For airflow at 50 m/s, what is the surface shear stress at x = 1m from the leading edge of the plate? Assume the air to beat a temperature of 300 K

A thin, flat plate that is 0.2 m?0.2 m on a side withrough top and bottom surfaces is placed in a wind tun-nel so that its surfaces are parallel to an atmosphericairstream having a velocity of 30 m/s. The air is at atemperature of T??20C while the plate is maintainedat Ts?80C. The plate is rotated 45about its centerpoint, as shown in the schematic. Airflows over the topand bottom surfaces of the plate, and measurement ofthe heat transfer rate is 2000 W. What is the drag forceon the plate?

As a means of preventing ice formation on the wings ofa small, private aircraft, it is proposed that electricresistance heating elements be installed within thewings. To determine representative power require-ments, consider nominal flight conditions for which theplane moves at 100 m/s in air that is at a temperatureof?23C. If the characteristic length of the airfoil isL?2 m and wind tunnel measurements indicate anaverage friction coefficient of for the nom-inal conditions, what is the average heat flux needed tomaintain a surface temperature of Ts?5C?

A circuit board with a dense distribution of integratedcircuits (ICs) and dimensions of 120 mm?120 mm ona side is cooled by the parallel flow of atmospheric airwith a velocity of 2 m/s.From wind tunnel tests under the same flow con-ditions, the average frictional shear stress on the upper surface is determined to be 0.0625 N/m2. What is theallowable power dissipation from the upper surface ofthe board if the average surface temperature of the ICsmust not exceed the ambient air temperature by morethan 25C? Evaluate the thermophysical properties ofair at 300 K.

On a summer day the air temperature is 27C and therelative humidity is 30%. Water evaporates fromthe surface of a lake at a rate of 0.10 kg/h per squaremeter of water surface area. The temperature of thewater is also 27C. Determine the value of the convection mass transfer coefficient.

It is observed that a 230-mm-diameter pan of water at23C has a mass loss rate of 1.5?10?5kg/s when theambient air is dry and at 23C.(a) Determine the convection mass transfer coefficientfor this situation.(b) Estimate the evaporation mass loss rate when theambient air has a relative humidity of 50%.(c) Estimate the evaporation mass loss rate when thewater and ambient air temperatures are 47C,assuming that the convection mass transfer coeffi-cient remains unchanged and the ambient air is dry.

The rate at which water is lost because of evaporationfrom the surface of a body of water may be determinedby measuring the surface recession rate. Consider asummer day for which the temperature of both thewater and the ambient air is 305 K and the relativehumidity of the air is 40%. If the surface recessionrate is known to be 0.1 mm/h, what is the rate at whichmass is lost because of evaporation per unit surfacearea? What is the convection mass transfer coefficient?

Photosynthesis, as it occurs in the leaves of a greenplant, involves the transport of carbon dioxide (CO2)from the atmosphere to the chloroplasts of the leaves.The rate of photosynthesis may be quantified in termsof the rate of CO2assimilation by the chloroplasts. Thisassimilation is strongly influenced by CO2transferthrough the boundary layer that develops on the leafsurface. Under conditions for which the density of CO2is 6?10?4kg/m3in the air and 5?10?4kg/m3at theleaf surface and the convection mass transfer coeffi-cient is 10?2m/s, what is the rate of photosynthesis interms of kilograms of CO2assimilated per unit time andarea of leaf surface?

Species A is evaporating from a flat surface into speciesB. Assume that the concentration profile for species Ain the concentration boundary layer is of the form CA(y)?Dy2?Ey?F, where D, E, and Fare con- stants at any x-location and yis measured along a nor-mal from the surface. Develop an expression for themass transfer convection coefficient hmin terms ofthese constants, the concentration of A in the freestream CA,?and the mass diffusivity DAB. Write anexpression for the molar flux of mass transfer by con-vection for species A.

Consider cross flow of gas X over an object having acharacteristic length of L?0.1 m. For a Reynolds num-ber of 1?104, the average heat transfer coefficient is25 W/m2?K. The same object is then impregnated withliquid Y and subjected to the same flow conditions.Given the following thermophysical properties, what isthe average convection mass transfer coefficient??(m2/s)k(W/m?K)?(m2/s)Gas X 21 ?10?60.030 29 ?10?6Liquid Y 3.75 ?10?70.665 1.65 ?10?7Vapor Y 4.25 ?10?50.023 4.55 ?10?5Mixture of gas Xvapor Y:Sc0? .72

Consider conditions for which a fluid with a free streamvelocity of V?1 m/s flows over an evaporating or sub-liming surface with a characteristic length of L?1m,providing an average mass transfer convection coeffi-cient of . Calculate the dimensionlessparameters , ReL, Sc, and for the following combi- nations: airflow over water, airflow over naphthalene,and warm glycerol over ice. Assume a fluid tempera-ture of 300 K and a pressure of 1 atm.

An object of irregular shape has a characteristic lengthof L?1 m and is maintained at a uniform surface tem-perature of Ts?325 K. It is suspended in an airstreamthat is at atmospheric pressure (p?1 atm) and hasa velocity of V?100 m/s and a temperature ofT??275 K. The average heat flux from the surface tothe air is 12,000 W/m2. Referring to the foregoingsituation as case 1, consider the following cases anddetermine whether conditions are analogous to those ofcase 1. Each case involves an object of the same shape,which is suspended in an airstream in the same man-ner. Where analogous behavior does exist, determinethe corresponding value of the average convectioncoefficient.(a) The values of Ts, T?, and premain the same, butL?2 m and V?50 m/s.(b) The values of Tsand T?remain the same, butL?2 m, V?50 m/s, and p?0.2 atm c) The surface is coated with a liquid film that evapo-rates into the air. The entire system is at 300 K,and the diffusion coefficient for the airvapor mix-ture is DAB?1.12?10?4m2/s. Also, L?2m,V?50 m/s, and p?1 atm.(d) The surface is coated with another liquid film forwhich DAB?1.12?10?4m2/s, and the system isat 300 K. In this case L?2 m, V?250 m/s, andp?0.2 atm.

On a cool day in April a scantily clothed runner isknown to lose heat at a rate of 500 W because of con-vection to the surrounding air at T??10C. The run-ners skin remains dry and at a temperature ofTs?30C. Three months later, the runner is moving atthe same speed, but the day is warm and humid with atemperature of T??30C and a relative humidity of???60%. The runner is now drenched in sweat andhas a uniform surface temperature of 35C. Under bothconditions constant air properties may be assumed with??1.6?10?5m2/s, k?0.026 W/m?K, Pr?0.70, andDAB(water vaporair)?2.3?10?5m2/s.(a) What is the rate of water loss due to evaporation onthe summer day?(b) What is the total convective heat loss on the sum-mer day?

An object of irregular shape 1 m long maintained at a constant temperature of 100C is suspended in anairstream having a free stream temperature of 0C, a pres-sure of 1 atm, and a velocity of 120 m/s. The air tempera-ture measured at a point near the object in the airstream is 80C. A second object having the same shape is 2 mlong and is suspended in an airstream in the same manner.The air free stream velocity is 60 m/s. Both the air and the object are at 50C, and the total pressure is 1 atm. Aplastic coating on the surface of the object is being driedby this process. The molecular weight of the vapor is 82,and the saturation pressure at 50C for the plastic materialis 0.0323 atm. The mass diffusivity for the vapor in air at50C is 2.60?10?5m2/s.(a) For the second object, at a location corresponding tothe point of measurement on the first object, deter-mine the vapor concentration and partial pressure.(b) If the average heat flux q?is 2000 W/m2for the firstobject, determine the average mass flux nA? (kg/s?m2)for the second object.

An industrial process involves the evaporation of waterfrom a liquid film that forms on a contoured surface.Dry air is passed over the surface, and from laboratorymeasurements the convection heat transfer correlationis of the form (a) For an air temperature and velocity of 27C and10 m/s, respectively, what is the rate of evaporationfrom a surface of 1-m2area and characteristiclength L?1 m? Approximate the density of satu-rated vapor as ?A,sat?0.0077 kg/m3.(b) What is the steady-state temperature of the liquidfilm?

The naphthalene sublimation techniqueinvolves theuse of a mass transfer experiment coupled with ananalysis based on the heat and mass transfer analogy toobtain local or average convection heat transfer coeffi-cients for complex surface geometries. A coating ofnaphthalene, which is a volatile solid at room tempera-ture, is applied to the surface and is then subjected toairflow in a wind tunnel. Alternatively, solid objectsmay be cast from liquid naphthalene. Over a designatedtime interval, t, there is a discernible loss of naphtha-lene due to sublimation, and by measuring the surfacerecession at locations of interest or the mass loss of thesample, local or average mass transfer coefficients maybe determined.Consider a rectangular rod of naphthalene exposedto air in cross flow at V?10 m/s, T??300 K, as inProblem 6.10, except now c?10 mm and d?30 mm.Determine the change in mass of the L?500-mm-longrod over a time period of t?30 min. Naphthalenehas a molecular weight of ?A?128.16 kg/kmol, andits solidvapor saturation pressure at 27C and 1 atm ispA,sat?1.33?10?4bar

Consider application of the naphthalene sublimationtechnique (Problem 6.63) to a gas turbine blade that iscoated with naphthalene and has a surface area ofAs?0.05 m2.To determine the average convection heat transfer coef-ficient for a representative operating condition, anexperiment is performed in which the coated blade isexposed for 30 min to atmospheric air at the desiredvelocity and a temperature of T??27C. During theexperiment the surface temperature is Ts?27C, and atits conclusion the mass of the blade is reduced bym?8 g. What is the average convection heat transfercoefficient associated with the operating condition?

A manufacturer of ski equipment wishes to developheadgear that will offer enhanced thermal protectionfor skiers on cold days at the slopes. Headgear can bemade with good thermal insulating characteristics,but it tends to be bulky and cumbersome. Skiersprefer comfortable, lighter gear that offers good visi-bility, but such gear tends to have poor thermal insu-lating characteristics. The manufacturer decides totake a new approach to headgear design by concen-trating the insulation in areas about the head that areprone to the highest heat losses from the skier andminimizing use of insulation in other locations.Hence, the manufacturer must determine the localheat transfer coefficients associated with the humanhead with a velocity of V?10 m/s directed normal tothe face and an air temperature of?13C. A youngengineerdecides to make use of the heat and masstransfer analogy and the naphthalene sublimationtechnique (see Problem 6.63) and casts head shapesof solid naphthalene with characteristic dimensionsthat are half-scale (that is, the models are half as largeas the full-scale head).(a) What wind tunnel velocity (T??300 K) is neededto apply the experimental results to the human headassociated with V?10 m/s?(b) A wind tunnel experiment is performed fort?120 min, T??27C. The engineer finds thatthe naphthalene has receded by ?1?0.1 mm at theback of the head, ?2?0.32 mm in the middle ofthe forehead, and ?3?0.64 mm on the ear. Deter-mine the heat transfer coefficients at these locationsfor the full-scale head at?13C. The density ofsolid naphthalene is ?A,sol?1025 kg/m3.(c) After the new headgear is designed, the models arefitted with the new gear (half-scale) and the experi-ments are repeated. Some areas of the model thatwere found to have small local heat transfer coeffi-cients are left uncovered since insulating theseareas would have little benefit in reducing overallheat losses during skiing. Would you expect thelocal heat transfer coefficients for these exposedareas to remain the same as prior to fitting themodel with the headgear? Explain why

A streamlined strut supporting a bearing housingis exposed to a hot airflow from an engine exhaust. It isnecessary to run experiments to determine the averageconvection heat transfer coefficient from the air to thestrut in order to be able to cool the strut to the desiredsurface temperature Ts. It is decided to run mass trans-fer experiments on an object of the same shape and toobtain the desired heat transfer results by using the heatand mass transfer analogy. The mass transfer experiments were conducted using ahalf-size model strut constructed from naphthaleneexposed to an airstream at 27C. Mass transfer mea-surements yielded these results:ReLShL60,000 282120,000 491144,000 568288,000 989(a) Using the mass transfer experimental results, deter- mine the coefficients Cand mfor a correlation ofthe form .(b) Determine the average convection heat transfer coef-ficient for the full-sized strut, LH?60 mm, whenexposed to a free stream airflow with V?60 m/s,T??184C, and p??1 atm when Ts?70C.(c) The surface area of the strut can be expressed asAs?2.2LH?l, where lis the length normal to the page.For the conditions of part (b), what is the change inthe rate of heat transfer to the strut if the characteristiclength LHis doubled?

Consider the conditions of Problem 6.3, but with a thin film of water on the surface. If the air is dry and theSchmidt number Scis 0.6, what is the evaporative massflux? Is there net energy transfer to or from the water?

Consider the conditions of Problem 6.7, for which aheat transfer experiment yielded the prescribed distrib-ution of the local convection coefficient, hx(x). Theexperiment was performed for surface and free streamtemperatures of 310 and 290 K, respectively. Now con-sider repeating the experiment under conditions forwhich the surface is coated with a thin layer of naphtha-lene and both the surface and air are at 300 K. What isthe corresponding value of the average convection masstransfer coefficient, ?

Using the naphthalene sublimation technique, the radialdistribution of the local convection mass transfer coeffi-cient for uniform flow normal to a circular disk hasbeen correlated by an expression of the formShD?hm(r)DDAB?Sho?1?a?rro?n? The stagnation point Sherwood number (Sho) depends onthe Reynolds (ReD?VD/?) and Schmidt (Sc??/DAB)numbers, and data have been correlated by the follow-ing expression:Obtain an expression for the average Nusselt numbercorresponding to heat transfer froman isothermal disk exposed to the foregoing flowconditions. If a?1.2 and n?5.5, what is the rate ofheat transfer from a disk of diameter D?20 mm and sur-face temperature Ts?125C to an airstream for whichReD?5?104and T??25C? Typically, boundarylayer development from a stagnation point yields adecaying convection coefficient with increasing distancefrom the stagnation point. Provide a plausible explana-tion for why the opposite trend is observed for the disk.

To reduce the threat of predators, the sand grouse, abird of Kenya, will lay its eggs in locations wellremoved from sources of groundwater. To bring waterto its chicks, the grouse will then fly to the nearestsource and, by submerging the lower part of its body,will entrain water within its plumage. The grouse willthen return to its nest, and the chicks will imbibe waterfrom the plumage. Of course, if the time of flight is toolong, evaporative losses could cause a significantreduction in the water content of the plumage, and thechicks could succumb to dehydration.To gain a better understanding of convective transferduring flight, wind tunnel studies were performed usingmolded models of the grouse. By heating the portion ofthe model that corresponds to the water-encapsulatingplumage, an average convection heat transfer coeffi-cient was determined. Results for different air speedsand model sizes were then used to develop an empiricalcorrelation of the formNuL?0.034 ReL4/5Pr1/ The effective surface area of the water-encapsulatingportion of the plumage is designated as As, and thecharacteristic length is defined as L?(As)1/2.Consider conditions for which a grouse has entrained0.05 kg of water within plumage of As?0.04 m2and isreturning to its nest at a constant speed of V?30 m/s.The ambient air is stagnant and at a temperature and rela-tive humidity of T??37C and ???25%, respectively.If, throughout the flight, the surface Asis covered with aliquid water film at Ts?32C, what is the maximumallowable distance of the nest from the water source,if the bird must return with at least 50% of its initialwater supply?

A laboratory experiment involves simultaneous heatand mass transfer from a water-soaked towel experienc-ing irradiation from a bank of radiant lamps and paral-lel flow of air over its surface. Using a convectioncorrelation to be introduced in Chapter 7, the averageheat transfer convection coefficient is estimated to be. Assume that the radiative propertiesof the towel are those of water, for which ???0.96, and that the surroundings are at 300 K.(a) Determine the rate at which water evaporates fromthe towel, nA(kg/s).(b) Perform an energy balance on the towel to deter-mine the net rate of radiation transfer, qrad(W), tothe towel. Determine the irradiation G(W/m2).

In the spring, concrete surfaces such as sidewalks anddriveways are sometimes very wet in the morning, evenwhen it has not rained during the night. Typical night-time conditions are shown in the sketch.BreezeT= 290 K,?= 0.7h_= 53 W/m2KTsky = 240 KConcreteTs = 275 K, = ?= 0.96Thin layer of liquid water (a) Determine the heat fluxes associated with convec-tion, , evaporation, , and radiation exchangewith the sky, .(b) Do your calculations suggest why the concrete iswet instead of dry? Explain briefly.(c) Is heat flowing from the liquid layer to the con-crete? Or from the concrete to the liquid layer?Determine the heat flux by conduction into or outof the concrete.

Dry air at 32C flows over a wetted (water) plate of0.2 m2area. The average convection coefficient is, and the heater power required tomaintain the plate at a temperature of 27C is 432 W.Estimate the power required to maintain the wettedplate at a temperature of 37C in dry air at 32C if theconvection coefficients remain unchanged.

Dry air at 32C flows over a wetted plate of length200 mm and width 1 m (case A). An embedded electri-cal heater supplies 432 W and the surface temperatureis 27C.(a) What is the evaporation rate of water from the plate(kg/h)?(b) After a long period of operation, all the water isevaporated from the plate and its surface is dry(case B). For the same free stream conditions andthe same heater power as case A, estimate the tem-perature of the plate, Ts.

A 20-mm-diameter sphere is suspended in a dry airstream with a temperature of 22C. The power sup-plied to an embedded electrical heater within the sphere is 2.51 W when the surface temperature is 32C. Howmuch power is required to maintain the sphere at 32C ifits outer surface has a thin porous covering saturated with water? Evaluate the properties of air and the diffusion coefficient of the airwater vapor mixture at 300 K.

A successful California engineer has installed a circularhot tub in his backyard and finds that, for the typicaloperating conditions shown in the sketch, water mustbe added at a rate of 0.001 kg/s to maintain a fixedwater level in the tub. If the tub is well insulated on its sides and bottom and ifthe temperature of the makeup water is equal to thatof the tub water, at what rate must electrical heaterssupply energy to maintain the tub water at 310 K?

It is known that on clear nights the air temperature neednot drop below 0C before a thin layer of water on theground will freeze. Consider such a layer of water on aclear night for which the effective sky temperatureis?30C and the convection heat transfer coefficientdue to wind motion is h?25 W/m2?K. The water maybe assumed to have an emissivity of 1.0 and to be insu-lated from the ground as far as conduction is concerned.(a) Neglecting evaporation, determine the lowest tem-perature the air can have without the water freezing.(b) For the conditions given, estimate the mass transfercoefficient for water evaporation hm(m/s).(c) Accounting now for the effect of evaporation, whatis the lowest temperature the air can have withoutthe water freezing? Assume the air to be dry.

An expression for the actual water vapor partial pres-sure in terms of wet-bulb and dry-bulb temperatures,referred to as the Carrier equation, is given aswhere pv, pgw, and pare the actual partial pressure, thesaturation pressure at the wet-bulb temperature, andthe total pressure (all in bars), while Tdband Twbare thedry- and wet-bulb temperatures in kelvins. Consider airat 1 atm and 37.8C flowing over a wet-bulb thermome-ter that indicates 21.1C.(a) Using Carriers equation, calculate the partial pres-sure of the water vapor in the free stream. What isthe relative humidity?(b) Refer to a psychrometric chart and obtain the rela-tive humidity directly for the conditions indicated.Compare the result with part (a).(c) Use Equation 6.65 to determine the relative humid-ity. Compare the result to parts (a) and (b).

A mist cooler is used to provide relief for a fatiguedathlete. Water at Ti?10C is injected as a mist into afan airstream with ambient temperature of T??32C.The droplet diameters are 100 ?m. For small dropletsthe average Nusselt number is correlated by an expres-sion of the form(a) At the initial time, calculate the rate of convectionheat transfer to the droplet, the rate of evaporativeheat loss, and the rate of change of temperature ofthe droplet for two values of the relative humidityof the fan airstream, ???0.20 and 0.95. Explainwhat is happening to the droplet in each case. (b) Calculate the steady-state droplet temperature foreach of the two relative humidity values in part (a).

A wet-bulb thermometer consists of a mercury-in-glassthermometer covered with a wetted (water) fabric.When suspended in a stream of air, the steady-statethermometer reading indicates the wet- bulb tempera-ture Twb. Obtain an expression for determining therelative humidity of the air from knowledge of the airtemperature (T?), the wet-bulb temperature, and appro-priate air and water vapor properties. If T??45C andTwb?25C, what is the relative humidity of theairstream?

An industrial process involves evaporation of a thinwater film from a contoured surface by heating it frombelow and forcing air across it. Laboratory measure-ments for this surface have provided the following heattransfer correlation:The air flowing over the surface has a temperature of290 K, a velocity of 10 m/s, and is completely dry(???0). The surface has a length of 1 m and a surfacearea of 1 m2. Just enough energy is supplied to maintainits steady-state temperature at 310 K.(a) Determine the heat transfer coefficient and the rateat which the surface loses heat by convection.(b) Determine the mass transfer coefficient and theevaporation rate (kg/h) of the water on the surface.(c) Determine the rate at which heat must be suppliedto the surface for these conditions

A 2-mm-thick layer of water on an electrically heatedplate is maintained at a temperature of Tw?340 K, asdry air at T??300 K flows over the surface of thewater (case A). The arrangement is in large surround- ings that are also at 300 K. (a) If the evaporative flux from the surface of the waterto the air is , what is the corre-sponding value of the convection mass transfercoefficient? How long will it take for the water tocompletely evaporate?(b) What is the corresponding value of the convectionheat transfer coefficient and the rate at whichelectrical power must be supplied per unit area ofthe plate to maintain the prescribed temperature of the water? The emissivity of water is w?0.95.(c) If the electrical power determined in part (b) ismaintained after complete evaporation of the water(case B), what is the resulting temperature of theplate, whose emissivity is ?

A disk of 20-mm diameter is covered with a water film.Under steady-state conditions, a heater power of200 mW is required to maintain the diskwater film at305 K in dry air at 295 K and the observed evaporationrate is 2.55?10?4kg/h.(a) Calculate the average mass transfer convectioncoefficient for the evaporation process.(b) Calculate the average heat transfer convectioncoefficient .(c) Do the values of mand satisfy the heatmassanalogy?(d) If the relative humidity of the ambient air at295 K were increased from 0 (dry) to 0.50, but he power supplied to the heater was maintainedat 200 mW, would the evaporation rate increase ordecrease? Would the disk temperature increaseor decrease?

An experiment is conducted to determine the averagemass transfer convection coefficient of a small dropletusing a heater controlled to operate at a constant tem-perature. The power history required to completelyevaporate the droplet at a temperature of 37C is shownin the sketch. It was observed that, as the droplet dried,its wetted diameter on the heater surface remainednearly constant at a value of 4 mm.(a) Calculate the average mass transfer convectioncoefficient based on the wetted area during theevaporation process when the droplet, heater, andthe dryambient air are at 37C.(b) How much energy will be required to evaporatethe droplet if the dryambient air temperature is27C, while the dropletheater temperature remainsat 37C?

It is desired to develop a simple model for predicting thetemperaturetime history of a plate during the dryingcycle in a dishwasher. Following the wash cycle the plateis at Tp(t)?Tp(0)?65C and the air in the dishwasheris completely saturated (???1.0) at T??55C.The values of the plate surface area As, mass M, and spe-cific heat care such that Mc/As?1600 J/m2?K.(a) Assuming the plate is completely covered by a thinfilm of water and neglecting the thermal resistancesof the film and plate, derive a differential equationfor predicting the plate temperature as a functionof time.(b) For the initial conditions (t?0) estimate thechange in plate temperature with time, dT/dt(C/s),assuming that the average heat transfer coefficienton the plate is 3.5 W/m2?K