For fully developed laminar flow through a parallel-plate

Fully developed conditions are known to exist for waterflowing through a 25-mm-diameter tube at 0.01 kg/sand 27C. What is the maximum velocity of the waterin the tube? What is the pressure gradient associatedwith the flow?

What is the pressure drop associated with water at 27C flowing with a mean velocity of 0.2 m/s through a 600-m-long cast iron pipe of 0.15-m inside diameter?

Water at 27C flows with a mean velocity of 1 m/sthrough a 1-km-long pipe of 0.25-m inside diameter.(a) Determine the pressure drop over the pipe lengthand the corresponding pump power requirement, ifthe pipe surface is smooth.(b) If the pipe is made of cast iron and its surface isclean, determine the pressure drop and pump powerrequirement.(c) For the smooth pipe condition, generate a plot ofpressure drop and pump power requirement formean velocities in the range from 0.05 to 1.5 m/s.

An engine oil cooler consists of a bundle of 25 smoothtubes, each of length L?2.5 m and diameter D?10 mm.(a) If oil at 300 K and a total flow rate of 24 kg/s is infully developed flow through the tubes, what is thepressure drop and the pump power requirement?(b) Compute and plot the pressure drop and pumppower requirement as a function of flow rate for 10??30 kg/s

For fully developed laminar flow through a parallel-plate channel, the x-momentum equation has the formThe purpose of this problem is to develop expressionsfor the velocity distribution and pressure gradient anal-ogous to those for the circular tube in Section 8.1.(a) Show that the velocity profile, u(y), is parabolicand of the formwhere is the mean velocityum??a212??dpdx? nd?dp/dx??p/L, where ?pis the pressure dropacross the channel of length L.(b) Write an expression defining the friction factor, f,using the hydraulic diameter Dhas the characteris-tic length. What is the hydraulic diameter for theparallel-plate channel?(c) The friction factor is estimated from the expression, where Cdepends upon the flow crosssection, as shown in Table 8.1. What is the coeffi-cient Cfor the parallel-plate channel?(d) Airflow in a parallel-plate channel with a separationof 5 mm and a length of 200 mm experiences a pres-sure drop of ?p?3.75 N/m2. Calculate the meanvelocity and the Reynolds number for air at atmos-pheric pressure and 300 K. Is the assumption of fullydeveloped flow reasonable for this application? Ifnot, what is the effect on the estimate for um?

Consider pressurized water, engine oil (unused), andNaK (22%/78%) flowing in a 20-mm-diameter tube.(a) Determine the mean velocity, the hydrodynamicentry length, and the thermal entry length for eachof the fluids when the fluid temperature is 366 Kand the flow rate is 0.01 kg/s.(b) Determine the mass flow rate, the hydrodynamicentry length, and the thermal entry length for waterand engine oil at 300 and 400 K and a mean velocityof 0.02 m/s.

Velocity and temperature profiles for laminar flow in atube of radius ro?10 mm have the formT(r)?344.8?75.0(r/ro)2?18.8(r/ro)4 with units of m/s and K, respectively. Determine thecorresponding value of the mean (or bulk) temperature,Tm, at this axial position.

At a particular axial station, velocity and temperatureprofiles for laminar flow in a parallel plate channel havethe formwith units of m/s and C, respectively.Determine corresponding values of the mean velocity,um, and mean (or bulk) temperature, Tm. Plot the velocityand temperature distributions. Do your values of umandTmappear reasonable?

In Chapter 1, it was stated that for incompressible liquids,flow work could usually be neglected in the steady-flowenergy equation (Equation 1.12d). In the trans-Alaskapipeline, the high viscosity of the oil and long distancescause significant pressure drops, and it is reasonable to question whether flow work would be significant.Consider an L?100 km length of pipe of diameterD?1.2 m, with oil flow rate ?500 kg/s. The oil prop-erties are ??900 kg/m3, cp?2000 J/kg?K, ??0.765N?s/m2. Calculate the pressure drop, the flow work, andthe temperature rise caused by the flow work.

When viscous dissipation is included, Equation 8.48(multiplied by ?cp) becomesThis problem explores the importance of viscous dissi-pation. The conditions under consideration are laminar,fully developed flow in a circular pipe, with ugiven byEquation 8.15.(a) By integrating the left-hand side over a section of apipe of length Land radius ro, show that this termyields the right-hand side of Equation 8.34.(b) Integrate the viscous dissipation term over the samevolume.(c) Find the temperature rise caused by viscous dissi-pation by equating the two terms calculated above.Use the same conditions as in Problem 8.9.

Consider a circular tube of diameter Dand length L,with a mass flow rate of . (a) For constant heat flux conditions, derive an expres-sion for the ratio of the temperature differencebetween the tube wall at the tube exit and the inlettemperature, Ts(x?L)?Tm,i, to the total heattransfer rate to the fluid q. Express your result interms of , L, the local Nusselt number at the tubeexit NuD(x?L), and relevant fluid properties.(b) Repeat part (a) for constant surface temperatureconditions. Express your result in terms of , L, theaverage Nusselt number from the tube inlet to the tube exit , and relevant fluid properties.

Water enters a tube at 27C with a flow rate of450 kg/h. The heat transfer from the tube wall to thefluid is given as q?s(W/m)?ax, where the coefficient ais 20 W/m2and x(m) is the axial distance from the tubeentrance.(a) Beginning with a properly defined differential con-trol volume in the tube, derive an expression for thetemperature distribution Tm(x) of the water.(b) What is the outlet temperature of the water for aheated section 30 m long?(c) Sketch the mean fluid temperature, Tm(x), and thetube wall temperature, Ts(x), as a function of dis-tance along the tube for fully developed anddevel-oping flow conditions.(d) What value of a uniform wall heat flux, qs(insteadof q?s?ax), would provide the same fluid outlettemperature as that determined in part (b)? For thistype of heating, sketch the temperature distribu-tions requested in part (c).

Consider flow in a circular tube. Within the test sectionlength (between 1 and 2) a constant heat flux qsismaintained.(a) For the following two cases, sketch the surface tem-perature Ts(x) and the fluid mean temperature Tm(x)as a function of distance along the test section x. Incase A, flow is hydrodynamically and thermallyfully developed. In case B, flow is not developed.(b) Assuming that the surface flux qsand the inlet meantemperature Tm,1are identical for both cases, will the exit mean temperature Tm,2for case A be greaterthan, equal to, or less than Tm,2for case B? Brieflyexplain why.

Consider a cylindrical nuclear fuel rod of length Landdiameter Dthat is encased in a concentric tube. Pressur-ized water flows through the annular region between therod and the tube at a rate , and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is knownto vary sinusoidally with distance along the rod. That is, sin(?x/L), where (W/m3) is a constant. Auniform convection coefficient hmay be assumed toexist between the surface of the rod and the water.(a) Obtain expressions for the local heat flux q(x) andthe total heat transfer qfrom the fuel rod to the water.(b) Obtain an expression for the variation of the meantemperature Tm(x) of the water with distance xalong the tube.(c) Obtain an expression for the variation of the rodsurface temperature Ts(x) with distance xalong thetube. Develop an expression for the x-location atwhich this temperature is maximized.

Consider the laminar thermal boundary layer develop-ment near the entrance of the tube shown in Figure 8.4.When the hydrodynamic boundary layer is thin relativeto the tube diameter, the inviscid flow region has a uni-form velocity that is approximately equal to the meanvelocity um. Hence the boundary layer development issimilar to what would occur for a flat plate. (a) Beginning with Equation 7.23, derive an expres-sion for the local Nusselt number NuD, as a func-tion of the Prandtl number Prand the inverseGraetz number GzD?1. Plot the expression using thecoordinates shown in Figure 8.10afor Pr?0.7.(b) Beginning with Equation 7.30, derive an expres-sion for the average Nusselt number , as a func-tion of the Prandtl number Prand the inverseGraetz number GzD?1. Compare your results withthe Nusselt number for the combined entrancelength in the limit of small x.

In a particular application involving fluid flow at a ratethrough a circular tube of length Land diameter D, the surface heat flux is known to have a sinusoidal vari-ation with x, which is of the form qs(x)?qs,msin(?x/L).The maximum flux, qs,m, is a known constant, and thefluid enters the tube at a known temperature, Tm,i.Assuming the convection coefficient to be constant,how do the mean temperature of the fluid and the sur-face temperature vary with x?

flat-plate solar collector is used to heat atmosphericair flowing through a rectangular channel. The bottomsurface of the channel is well insulated, while the topsurface is subjected to a uniform heat flux qo, which isdue to the net effect of solar radiation absorption andheat exchange between the absorber and cover plates.(a) Beginning with an appropriate differential controlvolume, obtain an equation that could be used todetermine the mean air temperature Tm(x) as afunction of distance along the channel. Solve thisequation to obtain an expression for the mean tem-perature of the air leaving the collector.(b) With air inlet conditions of ?0.1 kg/s andTm,i?40C, what is the air outlet temperature ifL?3 m, w?1 m, and qo?700 W/m2? The spe-cific heat of air is cp?1008 J/kg?K

Atmospheric air enters the heated section of a circulartube at a flow rate of 0.005 kg/s and a temperature of20C. The tube is of diameter D?50 mm, and fullydeveloped conditions with h?25 W/m2?K exist overthe entire length of L?3m.(a) For the case of uniform surface heat flux at qs?1000 W/m2, determine the total heat transferrate qand the mean temperature of the air leavingthe tube Tm,o. What is the value of the surface tem-perature at the tube inlet Ts,iand outlet Ts,o? Sketchthe axial variation of Tsand Tm. On the same figure,also sketch (qualitatively) the axial variation of Tsand Tmfor the more realistic case in which the localconvection coefficient varies with x.(b) If the surface heat flux varies linearly with x, suchthat qs(W/m2)?500x(m), what are the values ofq, Tm,o, Ts,i, and Ts,o? Sketch the axial variation ofTsand Tm. On the same figure, also sketch (qualita-tively) the axial variation of Tsand Tmfor the morerealistic case in which the local convection coeffi-cient varies with x. c) For the two heating conditions of parts (a) and (b),plot the mean fluid and surface temperatures, Tm(x)and Ts(x), respectively, as functions of distancealong the tube. What effect will a fourfold increasein the convection coefficient have on the tempera-ture distributions?(d) For each type of heating process, what heat fluxesare required to achieve an air outlet temperature of125C? Plot the temperature distributions.

Fluid enters a tube with a flow rate of 0.015 kg/s and aninlet temperature of 20C. The tube, which has a lengthof 6 m and diameter of 15 mm, has a surface tempera-ture of 30C. (a) Determine the heat transfer rate to the fluid if it iswater.(b) Determine the heat transfer rate for the nanofluid of Example 2.2

Water at 300 K and a flow rate of 5 kg/s enters a black,thin-walled tube, which passes through a large furnacewhose walls and air are at a temperature of 700 K. Thediameter and length of the tube are 0.25 m and 8 m,respectively. Convection coefficients associated withwater flow through the tube and airflow over the tubeare 300 W/m2?K and 50 W/m2?K, respectively.(a) Write an expression for the linearized radiationcoefficient corresponding to radiation exchangebetween the outer surface of the pipe and thefurnace walls. Explain how to calculate this coef-ficient if the surface temperature of the tube isrepresented by the arithmetic mean of its inletand outlet values.(b) Determine the outlet temperature of the water, Tm,o.

Slug flow is an idealized tube flow condition for whichthe velocity is assumed to be uniform over the entiretube cross section. For the case of laminar slug flow with a uniform surface heat flux, determine the form of the fully developed temperature distribution T(r) andthe Nusselt number NuD.

Superimposing a control volume that is differential in x on the tube flow conditions of Figure 8.8, derive Equation 8.45a

An experimental nuclear core simulation apparatusconsists of a long thin-walled metallic tube of diameterDand length L, which is electrically heated to producethe sinusoidal heat flux distributionwhere xis the distance measured from the tube inlet.Fluid at an inlet temperature Tm,iflows through the tubeat a rate of . Assuming the flow is turbulent and fullydeveloped over the entire length of the tube, developexpressions for:(a) the total rate of heat transfer, q, from the tube to thefluid;(b) the fluid outlet temperature, Tm,o;(c) the axial distribution of the wall temperature, Ts(x);and(d) the magnitude and position of the highest walltemperature.(e) Consider a 40-mm- diameter tube of 4-m lengthwith a sinusoidal heat flux distribution for whichqo?10,000 W/m2. Fluid passing through the tubehas a flow rate of 0.025 kg/s, a specific heat of4180 J/kg?K, an entrance temperature of 25C, anda convection coefficient of 1000 W/m2?K. Plot themean fluid and surface temperatures as a functionof distance along the tube. Identify important fea-tures of the distributions. Explore the effectof?25% changes in the convection coefficient andthe heat flux on the distributions.

Water at 20C and a flow rate of 0.1 kg/s enters a heated,thin-walled tube with a diameter of 15 mm and length of2 m. The wall heat flux provided by the heating ele-ments depends on the wall temperature according to therelationwhere qs,o?104W/m2, ??0.2 K?1, Tref?20C, andTsis the wall temperature in C. Assume fully devel-oped flow and thermal conditions with a convectioncoefficient of 3000 W/m2?K.(a) Beginning with a properly defined differential con-trol volume in the tube, derive expressions for thevariation of the water, Tm(x), and the wall, Ts(x), temperatures as a function of distance from thetube inlet.(b) Using a numerical integration scheme, calculateand plot the temperature distributions, Tm(x) andTs(x), on the same graph. Identify and comment onthe main features of the distributions. Hint: TheIHTintegral function DER(Tm,x) can be used toperform the integration along the length of the tube.(c) Calculate the total rate of heat transfer to the water.

Engine oil is heated by flowing through a circular tubeof diameter D?50 mm and length L?25 m andwhose surface is maintained at 150C.(a) If the flow rate and inlet temperature of the oil are 0.5 kg/s and 20C, what is the outlet tempera-ture Tm,o? What is the total heat transfer rate qforthe tube?(b) For flow rates in the range 0.5?,compute and plot the variations of Tm,oand qwith. For what flow rate(s) are qand Tm,omaximized?Explain your results

Engine oil flows through a 25-mm-diameter tube at arate of 0.5 kg/s. The oil enters the tube at a temperatureof 25C, while the tube surface temperature is main-tained at 100C.(a) Determine the oil outlet temperature for a 5-m andfor a 100-m long tube. For each case, compare thelog mean temperature difference to the arithmeticmean temperature difference.(b) For 5?L?100 m, compute and plot the averageNusselt number and the oil outlet temperatureas a function of L

In the final stages of production, a pharmaceutical issterilized by heating it from 25 to 75C as it moves at0.2 m/s through a straight thin-walled stainless steeltube of 12.7-mm diameter. A uniform heat flux is main-tained by an electric resistance heater wrapped aroundthe outer surface of the tube. If the tube is 10 m long,what is the required heat flux? If fluid enters the tubewith a fully developed velocity profile and a uniformtemperature profile, what is the surface temperatureat the tube exit and at a distance of 0.5 m from theentrance? Fluid properties may be approximated as ??1000 kg/m3, cp?4000 J/kg?K, m2?? 0?3kg/s?m,? .8 W/m?K, and Pr?10

An oil preheater consists of a single tube of 10-mmdiameter and 5-m length, with its surface maintained at 175C by swirling combustion gases. The engine oil (new) enters at 75C. What flow rate must be suppliedto maintain an oil outlet temperature of 100C? What isthe corresponding heat transfer rate?

Engine oil flows at a rate of 1 kg/s through a 5-mm-diameter straight tube. The oil has an inlet temperatureof 45C and it is desired to heat the oil to a mean tem-perature of 80C at the exit of the tube. The surface ofthe tube is maintained at 150C. Determine the requiredlength of the tube. Hint: Calculate the Reynolds num-bers at the entrance and exit of the tube before proceeding with your analysis.

Air at p?1 atm enters a thin-walled (D?5-mmdiameter) long tube (L?2 m) at an inlet temperature ofTm,i?100C. A constant heat flux is applied to theair from the tube surface. The air mass flow rate is m.?135 ?10?6kg/s.(a) If the tube surface temperature at the exit is Ts,o?160C, determine the heat rate entering thetube. Evaluate properties at T?400 K.(b) If the tube length of part (a) were reduced toL?0.2 m, how would flow conditions at the tubeexit be affected? Would the value of the heat transfercoefficient at the tube exit be greater than, equal to, orsmaller than the heat transfer coefficient for part (a)?(c) If the flow rate of part (a) were increased by a factorof 10, would there be a difference in flow conditionsat the tube exit? Would the value of the heat transfercoefficient at the tube exit be greater than, equal to, or smaller than the heat transfer coefficient forpart (a)?

To cool a summer home without using a vapor-compression refrigeration cycle, air is routed through a plastic pipe (k?0.15 W/m?K, Di?0.15 m, Do?0.17 m) that is submerged in an adjoining body of water.The water temperature is nominally at T??17C, and aconvection coefficient of ho?1500 W/m2?K is main-tained at the outer surface of the pipe.If air from the home enters the pipe at a temperature ofTm,i?29C and a volumetric flow rate ofi?0.025 m3/s, what pipe length Lis needed to provide a discharge tem-perature of Tm,o?21C? What is the fan power requiredto move the air through this length of pipe if its inner sur- face is smooth?

Batch processesare often used in chemical and phar-maceutical operations to achieve a desired chemicalcomposition for the final product. Related heat transferprocesses are typically transient, involving a liquid offixed volume that may be heated from room tempera-ture to a desired process temperature, or cooled fromthe process temperature to room temperature. Considera batch process for which a pharmaceutical (the coldfluid, c) is poured into an insulated, highly agitated ves-sel (a stirred reactor) and heated by passing a hot fluid(h) through a submerged heat exchanger coil of thin- walled tubing and surface area As. The flow rate, ,mean inlet temperature, Th,i, and specific heat, cp,h, ofthe hot fluid are known, as are the initial temperature,Tc,iTh,i, the volume, Vc, mass density, ?c, and specificheat, cv,c, of the pharmaceutical. Heat transfer from thehot fluid to the pharmaceutical is governed by an over-all heat transfer coefficient U.(a) Starting from basic principles, derive expressions thatcan be used to determine the variation of Tcand Th,owith time during the heating process. Hint: Twoequations may be written for the rate of heat transfer,q(t), to the pharmaceutical, one based on the log-mean temperature difference and the other on anenergy balance for flow of the hot fluid through thetube. Equate these expressions to determine Th,o(t) asa function of Tc(t) and prescribed parameters. Use theexpression for Th,o(t) and the energy balance for flowthrough the tube with an energy balance for a controlvolume containing the pharmaceutical to obtain anexpression for Tc(t).(b) Consider a pharmaceutical of volume Vc?1m3, den-sity ?c?1100 kg/m3, specific heat cv,c?2000 J/kg?K,and an initial temperature of Tc,i?25C. A coiled ube of length L?40 m, diameter D?50 mm, andcoil diameter C?500 mm is submerged in the vessel,and hot fluid enters the tubing at Th,i?200C andm.h?2.4 kg/s. The convection coefficient at theouter surface of the tubing may be approximatedasho?1000 W/m2?K, and the fluid propertiesarecp,h?2500 J/kg?K, ?h?0.002 N?s/m2, kh?0.260 W/m?K, and Prh?20. For the foregoingconditions, compute and plot the pharmaceuticaltemperature Tcand the outlet temperature Th,oas afunction of time over the range 0?t?3600 s.How long does it take to reach a batch temperatureof Tc?160C? The process operator may controlthe heating time by varying m.h. For 1?m.h?5kg/s, explore the effect of the flow rate on the timetcrequired to reach a value of Tc?160C

The evaporator section of a heat pump is installed in a large tank of water, which is used as a heat source during the winter. As energy is extracted from the water,it begins to freeze, creating an ice/water bath at 0C,which may be used for air conditioning during the sum-mer. Consider summer cooling conditions for which airis passed through an array of copper tubes, each ofinside diameter D?50 mm, submerged in the bath.(a) If air enters each tube at a mean temperature ofTm,i?24C and a flow rate of m.?0.01 kg/s, whattube length Lis needed to provide an exit tempera-ture of Tm,o?14C? With 10 tubes passing througha tank of total volume V?10 m3, which initiallycontains 80% ice by volume, how long would ittake to completely melt the ice? The density andlatent heat of fusion of ice are 920 kg/m3and3.34?105J/kg, respectively.(b) The air outlet temperature may be regulated byadjusting the tube mass flow rate. For the tubelength determined in part (a), compute and plot Tm,oas a function of m.for 0.005?m.?0.05 kg/s. If thedwelling cooled by this system requires approxi- mately 0.05 kg/s of air at 16C, what design andoperating conditions should be prescribed for thesystem?

A liquid food product is processed in a continuous-flow sterilizer. The liquid enters the sterilizer at a tem-perature and flow rate of Tm,i,h?20C, m.?1 kg/s,respectively. A time-at-temperature constraint requiresthat the product be held at a mean temperature ofTm?90C for 10 s to kill bacteria, while a second con-straint is that the local product temperature cannotexceed Tmax?230C in order to preserve a pleasingtaste. The sterilizer consists of an upstream, Lh?5mheating section characterized by a uniform heat flux, an intermediate insulated sterilizing section, and adownstream cooling section of length Lc?10 m. Thecooling section is composed of an uninsulated tubeexposed to a quiescent environment at T??20C. The thin-walled tubing is of diameter D?40 mm.Food properties are similar to those of liquid water atT?330 K. (a) What heat flux is required in the heating section toensure a maximum mean product temperature ofTm?90C? (b) Determine the location and value of the maximumlocal product temperature. Is the second constraintsatisfied?(c) Determine the minimum length of the sterilizingsection needed to satisfy the time-at-temperatureconstraint.(d) Sketch the axial distribution of the mean, surface,and centerline temperatures from the inlet of theheating section to the outlet of the cooling section

Water flowing at 2 kg/s through a 40-mm-diameter tubeis to be heated from 25 to 75C by maintaining the tube surface temperature at 100C.(a) What is the required tube length for these conditions?(b) To design a water heating system, we wish to con-sider using tube diameters in the range from 30 to 50 mm. What are the required tube lengths forwater flow rates of 1, 2, and 3 kg/s? Represent thisdesign information graphically. (c) Plot the pressure gradient as a function of tube diameter for the three flow rates. Assume the tube wall is smooth.

Consider the conditions associated with the hot waterpipe of Problem 7.56, but now account for the convec-tion resistance associated with water flow at a meanvelocity of um?0.5 m/s in the pipe. What is the corre-sponding daily cost of heat loss per meter of the unin-sulated pipe?

A thick-walled, stainless steel (AISI 316) pipe of insideand outside diameters Di?20 mm and Do?40 mm isheated electrically to provide a uniform heat generationrate of . The outer surface of the pipe isinsulated, while water flows through the pipe at a rateof .m ?0.1kg/s (a) If the water inlet temperature is Tm,i?20C and thedesired outlet temperature is Tm,o?40C, what isthe required pipe length?(b) What are the location and value of the maximumpipe temperature?

An air heater for an industrial application consists of aninsulated, concentric tube annulus, for which air flowsthrough a thin-walled inner tube. Saturated steam flowsthrough the outer annulus, and condensation of the steammaintains a uniform temperature Tson the tube surface.Consider conditions for which air enters a 50-mm-diameter tube at a pressure of 5 atm, a temperature ofTm,i?17C, and a flow rate of , whilesaturated steam at 2.455 bars condenses on the outersurface of the tube. If the length of the annulus isL?5 m, what are the outlet temperature Tm,oand pres-sure poof the air? What is the mass rate at which con-densate leaves the annulus?

Consider fully developed conditions in a circular tubewith constant surface temperature TsTm. Determinewhether a small- or large-diameter tube is more effec-tive in minimizing heat loss from the flowing fluidcharacterized by a mass flow rate of . Consider bothlaminar and turbulent conditions

Consider the encased pipe of Problem 4.29, but nowallow for the difference between the mean temperatureof the fluid, which changes along the pipe length, andthat of the pipe.(a) For the prescribed values of k, D, w, h, and T?and apipe of length L?100 m, what is the outlet temper-ature Tm,oof water that enters the pipe at a tempera-ture of Tm,i?90C and a flow rate of ?(b) What is the pressure drop of the water and the cor-responding pump power requirement?(c) Subject to the constraint that the width of the duct isfixed at w?0.30 m, explore the effects of the flowrate and the pipe diameter on the outlet temperature.

Water flows through a thick-walled tube with an innerdiameter of 12 mm and a length of 8 m. The tube isimmersed in a well-stirred, hot reaction tank maintainedat 85C, and the conduction resistance of the tube wall(based on the inner surface area) is Rcd?0.002 m2?K/W.The inlet temperature of the process fluid is Tm,i?20C,and the flow rate is 33 kg/h.(a) Estimate the outlet temperature of the process fluid,Tm,o. Assume, and then justify, fully developed flowand thermal conditions within the tube.(b) Do you expect Tm,oto increase or decrease if com-bined thermal and hydrodynamic entry conditionsexist within the tube? Estimate the outlet tempera-ture of the water for this condition

Atmospheric air enters a 10-m-long, 150-mm-diameteruninsulated heating duct at 60C and 0.04 kg/s. Theduct surface temperature is approximately constant atTs?15C.(a) What are the outlet air temperature, the heat rate q,and pressure drop ?pfor these conditions?(b) To illustrate the tradeoff between heat transfer rateand pressure drop considerations, calculate qand?pfor diameters in the range from 0.1 to 0.2 m. Inyour analysis, maintain the total surface area,As??DL, at the value computed for part (a). Plotq? , and Las a function of the duct diameter.

NaK (45%/55%), which is an alloy of sodium and potas-sium, is used to cool fast neutron nuclear reactors. TheNaK flows at a rate of m.?1 kg/s through a D?50-mm-diameter tube that has a surface temperature ofTs?450 K. The NaK enters the tube at Tm,i?332 Kand exits at an outlet temperature of Tm,o?400 K.Determine the tube length Land the local convectiveheat flux at the tube exit.

The products of combustion from a burner are routed toan industrial application through a thin-walled metallicduct of diameter Di?1 m and length L?100 m. Thegas enters the duct at atmospheric pressure and a mean temperature and velocity of Tm,i?1600 K andum,i?10 m/s, respectively. It must exit the duct at atemperature that is no less than Tm,o?1400 K. What isthe minimum thickness of an alumina-silica insulation(kins?0.125 W/m?K) needed to meet the outletrequirement under worst case conditions for which theduct is exposed to ambient air at T??250 K and across-flow velocity of V?15 m/s? The properties ofthe gas may be approximated as those of air, and as afirst estimate, the effect of the insulation thickness onthe convection coefficient and thermal resistance asso-ciated with the cross flow may be neglected.

Liquid mercury at 0.5 kg/s is to be heated from 300 to400 K by passing it through a 50-mm-diameter tubewhose surface is maintained at 450 K. Calculate therequired tube length by using an appropriate liquidmetal convection heat transfer correlation. Compareyour result with that which would have been obtainedby using a correlation appropriate for Pr0.7.

The surface of a 50-mm-diameter, thin-walled tube ismaintained at 100C. In one case air is in cross flowover the tube with a temperature of 25C and a velocityof 30 m/s. In another case air is in fully developed flowthrough the tube with a temperature of 25C and a meanvelocity of 30 m/s. Compare the heat flux from the tubeto the air for the two cases.

Consider a horizontal, thin-walled circular tube ofdiameter D?0.025 m submerged in a container of n- octadecane (paraffin), which is used to store thermalenergy. As hot water flows through the tube, heat istransferred to the paraffin, converting it from the solidto liquid state at the phase change temperature ofT??27.4C. The latent heat of fusion and densityof paraffin are hsf?244 kJ/kg and ??770 kg/m3,respectively, and thermophysical properties of the watermay be taken as cp?4.185 kJ/kg?K, k?0.653 W/m?K,??467?10?6kg/s?m, and Pr?2.99.(a) Assuming the tube surface to have a uniform tem- perature corresponding to that of the phase change,determine the water outlet temperature and total heattransfer rate for a water flow rate of 0.1 kg/s and aninlet temperature of 60C. If H?W?0.25 m, howlong would it take to completely liquefy the paraffin,from an initial state for which all the paraffin is solidand at 27.4C?(b) The liquefaction process can be accelerated byincreasing the flow rate of the water. Computeand plot the heat rate and outlet temperature as afunction of flow rate for 0.1??0.5 kg/s. Howlong would it take to melt the paraffin form 0 ? .5kg/s?

Consider pressurized liquid water flowing at m.?0.1 kg/sin a circular tube of diameter D?0.1 m and lengthL?6m.(a) If the water enters at Tm,i?500 K and the surfacetemperature of the tube is Ts?510 K, determinethe water outlet temperature Tm,o.(b) If the water enters at Tm,i?300 K and the surfacetemperature of the tube is Ts?310 K, determinethe water outlet temperature Tm,o.(c) If the water enters at Tm,i?300 K and the surfacetemperature of the tube is Ts?647 K, discusswhether the flow is laminar or turbulent.

Cooling water flows through the 25.4-mm-diameterthin-walled tubes of a steam condenser at 1 m/s, and asurface temperature of 350 K is maintained by the con-densing steam. The water inlet temperature is 290 K,and the tubes are 5 m long.(a) What is the water outlet temperature? Evaluatewater properties at an assumed average mean tem-perature, .(b) Was the assumed value for reasonable? If not,repeat the calculation using properties evaluated ata more appropriate temperature.(c) A range of tube lengths from 4 to 7 m is availableto the engineer designing this condenser. Generatea plot to show what coolant mean velocities arepossible if the water outlet temperature is to remainat the value found for part (b). All other conditionsremain the same

he air passage for cooling a gas turbine vane can beapproximated as a tube of 3-mm diameter and 75-mmlength. The operating temperature of the vane is 650C,and air enters the tube at 427C.(a) For an airflow rate of 0.18 kg/h, calculate the airoutlet temperature and the heat removed from thevane.(b) Generate a plot of the air outlet temperature as afunction of flow rate for 0.1??0.6 kg/h. Com-pare this result with those for vanes having 2- and4-mm-diameter tubes, with all other conditionsremaining the same

The core of a high-temperature, gas-cooled nuclear reac-tor has coolant tubes of 20-mm diameter and 780-mmlength. Helium enters at 600 K and exits at 1000 K whenthe flow rate is 8?10?3kg/s per tube.(a) Determine the uniform tube wall surface tempera-ture for these conditions. b) If the coolant gas is air, determine the required flowrate if the heat removal rate and tube wall surfacetemperature remain the same. What is the outlettemperature of the air?

ir at 200 kPa enters a 2-m-long, thin-walled tube of25-mm diameter at 150C and 6 m/s. Steam at 20 barscondenses on the outer surface.(a) Determine the outlet temperature and pressure dropof the air, as well as the rate of heat transfer to the air.(b) Calculate the parameters of part (a) if the pressureof the air is doubled

Heated air required for a food-drying process is generated by passing ambient air at 20C through long,circular tubes (D?50 mm, L?5 m) housed in asteam condenser. Saturated steam at atmospheric pres-sure condenses on the outer surface of the tubes, main-taining a uniform surface temperature of 100C.(a) If an airflow rate of 0.01 kg/s is maintained in eachtube, determine the air outlet temperature Tm,oandthe total heat rate qfor the tube.(b) The air outlet temperature may be controlled byadjusting the tube mass flow rate. Compute andplot Tm,oas a function of for 0.005??0.050 kg/s. If a particular drying process requiresapproximately 1 kg/s of air at 75C, what designand operating conditions should be prescribed forthe air heater, subject to the constraint that the tubediameter and length be fixed at 50 mm and 5 m,respectively?

Consider laminar flow of a fluid with Pr?4 thatundergoes a combined entrance process within a con- stant surface temperature tube of length Lxfd,twith aflow rate of . An engineer suggests that the total heattransfer rate can be improved if the tube is divided intoNshorter tubes, each of length LN?L/Nwith a flowrate of . Determine an expression for the ratio ofthe heat transfer coefficient averaged over the Ntubes,each experiencing a combined entrance process, to theheat transfer coefficient averaged over the single tube,

A common procedure for cooling a high-performancecomputer chip involves joining the chip to a heat sinkwithin which circular microchannels are machined.During operation, the chip produces a uniform heatflux at its interface with the heat sink, while a liquidcoolant (water) is routed through the channels. Con-sider a square chip and heat sink, each Lon a side,with microchannels of diameter Dand pitch S?C1D,where the constant C1is greater than unity. Water is supplied at an inlet temperature Tm,iand a total massflow rate (for the entire heat sink).(a) Assuming that is dispersed in the heat sink suchthat a uniform heat flux is maintained at the sur-face of each channel, obtain expressions for thelongitudinal distributions of the mean fluid, Tm(x),and surface, Ts(x), temperatures in each channel.Assume laminar, fully developed flow throughouteach channel, and express your results in terms ofm., qc, C1, D, and/or L, as well as appropriate ther-mophysical properties.(b) For L?12 mm, D?1 mm, C1?2, qc?20 W/cm2,m.?0.010 kg/s, and Tm,i?290 K, compute and plotthe temperature distributions Tm(x) and Ts(x).(c) A common objective in designing such heat sinksis to maximize while maintaining the heat sink atan acceptable temperature. Subject to prescribedvalues of L?12 mm and Tm,i?290 K and theconstraint that Ts,max?50C, explore the effect onof variations in heat sink design and operatingconditions

One way to cool chips mounted on the circuit boards ofa computer is to encapsulate the boards in metal framesthat provide efficient pathways for conduction to sup-porting cold plates. Heat generated by the chips is thendissipated by transfer to water flowing through pas-sages drilled in the plates. Because the plates are madefrom a metal of large thermal conductivity (typicallyaluminium or copper), they may be assumed to be at atemperature, Ts,cp (a) Consider circuit boards attached to cold plates ofheight H?750 mm and width L?600 mm, eachwith N?10 holes of diameter D?10 mm. Ifoperating conditions maintain plate temperatures ofTs,cp?32C with water flow at perpassage and Tm,i?7C, how much heat may be dis-sipated by the circuit boards?(b) To enhance cooling, thereby allowing increasedpower generation without an attendant increase insystem temperatures, a hybrid cooling scheme maybe used. The scheme involves forced airflow over theencapsulated circuit boards, as well as water flowthrough the cold plates. Consider conditions forwhich Ncb?10 circuit boards of width W?350 mmare attached to the cold plates and their average sur-face temperature is Ts,cb?47C when Ts,cp?32C. Ifair is in parallel flow over the plates with u??10 m/sand T??7C, how much of the heat generated bythe circuit boards is transferred to the air?

Refrigerant-134a is being transported at 0.1 kg/sthrough a Teflon tube of inside diameter Di?25 mmand outside diameter Do?28 mm, while atmosphericair at V?25 m/s and 300 K is in cross flow over thetube. What is the heat transfer per unit length of tube toRefrigerant-134a at 240 K?

Oil at 150C flows slowlythrough a long, thin-walledpipe of 30-mm inner diameter. The pipe is suspended in a room for which the air temperature is 20C and the convection coefficient at the outer tube surface is 11 W/m2K ? . Estimate the heat loss per unit length of tube.

Exhaust gases from a wire processing oven are dis-charged into a tall stack, and the gas and stack surfacetemperatures at the outlet of the stack must be estimated.Knowledge of the outlet gas temperature Tm,ois useful for predicting the dispersion of effluents in the thermalplume, while knowledge of the outlet stack surface tem-perature Ts,oindicates whether condensation of the gasproducts will occur. The thin-walled, cylindrical stack is0.5 m in diameter and 6.0 m high. The exhaust gas flowrate is 0.5 kg/s, and the inlet temperature is 600C.(a) Consider conditions for which the ambient air tem-perature and wind velocity are 4C and 5 m/s,respectively. Approximating the thermophysicalproperties of the gas as those of atmospheric air,estimate the outlet gas and stack surface tempera-tures for the given conditions.(b) The gas outlet temperature is sensitive to variationsin the ambient air temperature and wind velocity.For T???25C, 5C, and 35C, compute and plotthe gas outlet temperature as a function of windvelocity for 2?V?10 m/s.

A hot fluid passes through a thin-walled tube of 10-mmdiameter and 1-m length, and a coolant at T??25C is in cross flow over the tube. When the flow rate ism??18 kg/h and the inlet temperature is Tm,i?85C,the outlet temperature is Tm,o?78C.m = 18 kg/hTm,i= 85CCoolantHot fluidT= 25CTube, D = 10 mm,L = 1 mTm,o = 78COven exhaust gasesOutlet Diameter,0.5 mInletStack baseStackHeight, 6 mOvenBuildingThermal plumeAssuming fully developed flow and thermal conditionsin the tube, determine the outlet temperature, Tm,o, if the flow rate is increased by a factor of 2. That is,m??36 kg/h, with all other conditions the same. The thermophysical properties of the hot fluid are ??1079 kg/m3, cp?2637 J/kgK? , ??0.0034 N?s/m2, an? .261 W/m?K

Consider a thin-walled tube of 10-mm diameter and 2-m length. Water enters the tube from a large reservoirat m??0.2 kg/s and Tm,i?47C.(a) If the tube surface is maintained at a uniform temper-ature of 27C, what is the outlet temperature of thewater, Tm,o? To obtain the properties of water, assumean average mean temperature of T_m?300 K.(b) What is the exit temperature of the water if it isheated by passing air at T??100C and V?10 m/sin cross flow over the tube? The properties of airmay be evaluated at an assumed film temperature ofTf?350 K.(c) In the foregoing calculations, were the assumedvalues of T_mand Tfappropriate? If not, use prop-erly evaluated properties and recompute Tm,ofor theconditions of part (b).

Water at a flow rate of m??0.215 kg/s is cooled from70C to 30C by passing it through a thin-walled tubeof diameter D?50 mm and maintaining a coolant atT??15C in cross flow over the tube.(a) What is the required tube length if the coolant is airand its velocity is V?20 m/s?(b) What is the tube length if the coolant is water andV?2 m/s

The problem of heat losses from a fluid moving througha buried pipeline has received considerable attention.Practical applications include the trans-Alaska pipeline,as well as power plant steam and water distribution lines.Consider a steel pipe of diameter Dthat is used to trans-port oil flowing at a rate m?othrough a cold region. Thepipe is covered with a layer of insulation of thickness tand thermal conductivity kiand is buried in soil to adepth z(distance from the soil surface to the pipe center-line). Each section of pipe is of length Land extendsbetween pumping stations in which the oil is heated toensure low viscosity and hence low pump power require-ments. The temperature of the oil entering the pipe froma pumping station and the temperature of the groundabove the pipe are designated as Tm,iand Ts, respectively,and are known.Consider conditions for which the oil (o) propertiesmay be approximated as ?o?900 kg/m3, cp,o?2000J/kg?K, ?o?8.5?10?4m2/s, ko?0.140 W/m?K, Pro?104; the oil flow rate is m?o?500 kg/s; and thepipe diameter is 1.2 m.(a) Expressing your results in terms of D, L, z, t, m?o,Tm,i, and Ts, as well as the appropriate oil (o), insula-tion (i), and soil (s) properties, obtain all the expres-sions needed to estimate the temperature Tm,oof theoil leaving the pipe.(b) If Ts??40C, Tm,i?120C, t?0.15 m, ki?0.05W/m?K, ks?0.5 W/m?K, z?3 m, and L?100 km,what is the value of Tm,o? What is the total rate ofheat transfer qfrom a section of the pipeline?(c) The operations manager wants to know the tradeoffbetween the burial depth of the pipe and insulationthickness on the heat loss from the pipe. Develop agraphical representation of this design information.

To maintain pump power requirements per unit flow ratebelow an acceptable level, operation of the oil pipeline ofProblem 8.63 is subject to the constraint that the oil exittemperature Tm,oexceed 110C. For the values of Tm,i, Ts,D, ti, z, L, and kiprescribed in Problem 8.63, operatingparameters that are variable and affect Tm,oare the ther-mal conductivity of the soil and the flow rate of the oil.Depending on soil composition and moisture and thedemand for oil, representative variations are 0.25?ks?1.0 W/m?K and 250?m?o?500 kg/s. Using the proper-ties prescribed in Problem 8.63, determine the effect ofthe foregoing variations on Tm,oand the total heat rate q.What is the worst case operating condition? If necessary,what adjustments could be made to ensure that Tm,o?110C for the worst case conditions?

Consider a thin-walled, metallic tube of length L?1mand inside diameter Di?3 mm. Water enters the tubeat m??0.015 kg/s and Tm,i?97C.(a) What is the outlet temperature of the water if thetube surface temperature is maintained at 27C?(b) If a 0.5-mm-thick layer of insulation of k?0.05W/m?K is applied to the tube and its outer surfaceis maintained at 27C, what is the outlet tempera-ture of the water?(c) If the outer surface of the insulation is no longermaintained at 27C but is allowed to exchange heatby free convection with ambient air at 27C, what isthe outlet temperature of the water? The free convec- tion heat transfer coefficient is 5 W/m2?K

A circular tube of diameter D?0.2 mm and length L?100 mm imposes a constant heat flux of q ?20?103W/m2on a fluid with a mass flow rate of m??0.1 g/s.For an inlet temperature of Tm,i?29C, determine thetube wall temperature at x?Lfor pure water. Evaluate luid properties at T_?300 K. For the same conditions,determine the tube wall temperature at x?Lfor thenanofluid of Example 2.2.

Repeat Problem 8.66 for a circular tube of diameterD?2 mm, an applied heat flux of q ?200,000 W/m2,and a mass flow rate of m??10 g/s.

Heat is to be removed from a reaction vessel operatingat 75C by supplying water at 27C and 0.12 kg/sthrough a thin-walled tube of 15-mm diameter. Theconvection coefficient between the tube outer surfaceand the fluid in the vessel is 3000 W/m2?K.(a) If the outlet water temperature cannot exceed 47C,what is the maximum rate of heat transfer from thevessel?(b) What tube length is required to accomplish the heattransfer rate of part (a)?

A heating contractor must heat 0.2 kg/s of water from15C to 35C using hot gases in cross flow over a thin-walled tube.Your assignment is to develop a series of design graphsthat can be used to demonstrate acceptable combinationsof tube dimensions (Dand L) and of hot gas conditions(T?and V) that satisfy this requirement. In your analysis,consider the following parameter ranges: D?20, 30, or40 mm; L?3, 4, or 6 m; T??250, 375, or 500C; and20?V?40 m/s.

A thin-walled tube with a diameter of 6 mm and lengthof 20 m is used to carry exhaust gas from a smoke stackto the laboratory in a nearby building for analysis. Thegas enters the tube at 200C and with a mass flow rateof 0.003 kg/s. Autumn winds at a temperature of 15Cblow directly across the tube at a velocity of 5 m/s.Assume the thermophysical properties of the exhaustgas are those of air. (a) Estimate the average heat transfer coefficient forthe exhaust gas flowing inside the tube.(b) Estimate the heat transfer coefficient for the airflowing across the outside of the tube.(c) Estimate the overall heat transfer coefficient Uand the temperature of the exhaust gas when itreaches the laboratory.

A 50-mm-diameter, thin-walled metal pipe covered witha 25-mm-thick layer of insulation (0.085 W/m?K) andcarrying superheated steam at atmospheric pressure issuspended from the ceiling of a large room. The steamtemperature entering the pipe is 120C, and the air tem-perature is 20C. The convection heat transfer coefficienton the outer surface of the covered pipe is 10 W/m2?K. If the velocity of the steam is 10 m/s, at what point alongthe pipe will the steam begin condensing?

A thin-walled, uninsulated 0.3-m-diameter duct is usedto route chilled air at 0.05 kg/s through the attic of alarge commercial building. The attic air is at 37C, andnatural circulation provides a convection coefficient of2 W/m2?K at the outer surface of the duct. If chilled airenters a 15-m-long duct at 7C, what is its exit tempera-ture and the rate of heat gain? Properties of the chilledair may be evaluated at an assumed average tempera-ture of 300 K.

Pressurized water at Tm,i?200C is pumped atm??2 kg/s from a power plant to a nearby industrialuser through a thin-walled, round pipe of inside dia-meter D?1 m. The pipe is covered with a layer ofinsulation of thickness t?0.15 m and thermal con-ductivity k?0.05 W/m?K. The pipe, which is oflength L?500 m, is exposed to a cross flow of air atT???10C and V?4 m/s. Obtain a differentialequation that could be used to solve for the variationof the mixed mean temperature of the water Tm(x) withthe axial coordinate. As a first approximation, theinternal flow may be assumed to be fully developedthroughout the pipe. Express your results in terms ofm?, V, T?, D, t, k, and appropriate water (w) and air (a)properties. Evaluate the heat loss per unit length of the pipe at the inlet. What is the mean temperature of the water at the outlet?

Water at 290 K and 0.2 kg/s flows through a Teflontube (k?0.35 W/m?K) of inner and outer radii equalto 10 and 13 mm, respectively. A thin electrical heatingtape wrapped around the outer surface of the tube deliv-ers a uniform surface heat flux of 2000 W/m2, while aconvection coefficient of 25 W/m2?K is maintained onthe outer surface of the tape by ambient air at 300 K.What is the fraction of the power dissipated by the tape, which is transferred to the water? What is the outer sur-face temperature of the Teflon tube?

The temperature of flue gases flowing through the largestack of a boiler is measured by means of a thermocoupleenclosed within a cylindrical tube as shown. The tubeaxis is oriented normal to the gas flow, and the thermo-couple senses a temperature Ttcorresponding to that ofthe tube surface. The gas flow rate and temperature aredesignated as m?gand Tg, respectively, and the gas flowmay be assumed to be fully developed. The stack is fab-ricated from sheet metal that is at a uniform temperatureTsand is exposed to ambient air at T?and large sur-roundings at Tsur. The convection coefficient associatedwith the outer surface of the duct is designated as ho,while those associated with the inner surface of the ductand the tube surface are designated as hiand ht, respec-tively. The tube and duct surface emissivities are desig-nated as?tand?s, respectively.(a) Neglecting conduction losses along the thermocou-ple tube, develop an analysis that could be used topredict the error (Tg?Tt) in the temperature mea-surement.(b) Assuming the flue gas to have the properties ofatmospheric air, evaluate the error for Tt?300C,Ds?0.6 m, Dt?10 mm, m?g?1 kg/s, T??Tsur?27C, ?t??s?0.8, and ho?25 W/m2K ? .

In a biomedical supplies manufacturing process, arequirement exists for a large platen that is to be main-tained at 45?0.25C. The proposed design featuresthe attachment of heating tubes to the platen at a relative spacing S. The thick-walled, copper tubes havean inner diameter of Di?8 mm and are attached to the platen with a high thermal conductivity solder,which provides a contact width of 2Di. The heatingfluid (ethylene glycol) flows through each tube at afixed rate of m??0.06 kg/s. The platen has a thickness f w?25 mm and is fabricated from a stainless steelwith a thermal conductivity of 15 W/m?K.Considering the two-dimensional cross section of theplaten shown in the inset, perform an analysis to determinethe heating fluid temperature Tmand the tube spacing Srequired to maintain the surface temperature of the platen,T(x, w), at 45?0.25C, when the ambient temperature is25C and the convection coefficient is 100 W/m2?K.

Consider the ground source heat pump of Problem5.100 under winter conditions for which the liquid isdischarged from the heat pump into high-density poly-ethylene tubing of thickness t?8 mm and thermal con-ductivity k?0.47 W/m?K. The tubing is routed throughsoil that maintains a uniform temperature of approxi-mately 10C at the tube outer surface. The properties ofthe fluid may be approximated as those of water.(a) For a tube inner diameter and flow rate of Di?25 mm and m??0.03 kg/s and a fluid inlet temper-ature of Tm,i?0C, determine the tube outlet tem-perature (heat pump inlet temperature), Tm,o, as afunction of the tube length Lfor 10?L?50 m.(b) Recommend an appropriate length for the system.How would your recommendation be affected byvariations in the liquid flow rate?

For a sharp-edged inlet and a combined entry region,the average Nusselt number may be computed fromEquation 8.63, with C?24ReD?0.23and m?0.815?2.08?10?6ReD[23]. Determine at x/D?10and 60 for ReD?104and 105

Fluid enters a thin-walled tube of 5-mm diameter and2-m length with a flow rate of 0.04 kg/s and a tempera-ture of Tm,i?85C. The tube surface is maintained at a temperature of Ts?25C, and for this operating con-dition, the outlet temperature is Tm,o?31.1C. What isthe outlet temperature if the flow rate is doubled? Fullydeveloped, turbulent flow may be assumed to exist inboth cases, and the fluid properties may be assumed tobe independent of temperature

Air at 3?10?4kg/s and 27C enters a rectangular ductthat is 1 m long and 4 mm?16 mm on a side. A uniformheat flux of 600 W/m2is imposed on the duct surface.What is the temperature of the air and of the duct surfaceat the outlet?

Air at 25C flows at 30?10?6kg/s within 100-mm-long channels used to cool a high thermal conductivitymetal mold. Assume the flow is hydrodynamically andthermally fully developed.(a) Determine the heat transferred to the air for a circu-lar channel (D?10 mm) when the mold tempera- ture is 50C (case A).(b) Using new manufacturing methods (see Problem8.105), channels of complex cross section can bereadily fabricated within metal objects, such asmolds. Consider air flowing under the same condi-tions as in case A, except now the channel is seg-mented into six smaller triangular sections. Theflow area of case A is equal to the total flow area ofcase B. Determine the heat transferred to the air forthe segmented channel.(c) Compare the pressure drops for cases A and B.

A cold plateis an active cooling device that is attachedto a heat-generating system in order to dissipate theheat while maintaining the system at an acceptable tem-perature. It is typically fabricated from a material ofhigh thermal conductivity, kcp, within which channelsare machined and a coolant is passed. Consider a cop-per cold plate of height Hand width Won a side, withinwhich water passes through square channels of widthw?h. The transverse spacing between channels ?istwice the spacing between the sidewall of an outerchannel and the sidewall of the cold plate.Consider conditions for which equivalentheat-generatingsystems are attached to the top and bottom of the coldplate, maintaining the corresponding surfaces at the sametemperature Ts. The mean velocity and inlet temperatureof the coolant are umand Tm,i, respectively.(a) Assuming fully developed turbulent flow throughouteach channel, obtain a system of equations that maybe used to evaluate the total rate of heat transfer tothe cold plate, q, and the outlet temperature of thewater, Tm,o, in terms of the specified parameters.(b) Consider a cold plate of width W?100 mm andheight H?10 mm, with 10 square channels of widthw?6 mm and a spacing of ??4 mm betweenchannels. Water enters the channels at a tempera-ture of Tm,i?300 K and a velocity of um?2 m/s. If the top and bottom cold plate surfaces are atTs?360 K, what is the outlet water temperature andthe total rate of heat transfer to the cold plate? Thethermal conductivity of the copper is 400 W/m?K,while average properties of the water may be takento be ??984 kg/m3, cp?4184 J/kg?K, ??489?10?6N?s/m2, k?0.65 W/m?K, and Pr?3.15. Isthis a good cold plate design? How could its perfor-mance be improved?

The cold plate design of Problem 8.82 has not been opti-mized with respect to selection of the channel width, andwe wish to explore conditions for which the rate of heattransfer may be enhanced. Assume that the width andheight of the copper cold plate are fixed at W?100 mm and H?10 mm, while the channel height and spacingbetween channels are fixed at h?6 mm and ??4mm.The mean velocity and inlet temperature of the water aremaintained at um?2 m/s and Tm,i?300 K, while equiv-alent heat- generating systems attached to the top and bot-tom of the cold plate maintain the corresponding surfacesat 360 K. Evaluate the effect of changing the channelwidth, and hence the number of channels, on the rate ofheat transfer to the cold plate. Include consideration of thelimiting case for which w9? 6 mm (one channel).

A device that recovers heat from high-temperaturecombustion products involves passing the combustiongas between parallel plates, each of which is main-tained at 350 K by water flow on the opposite surface.The plate separation is 40 mm, and the gas flow is fullydeveloped. The gas may be assumed to have the proper-ties of atmospheric air, and its mean temperature andvelocity are 1000 K and 60 m/s, respectively.(a) What is the heat flux at the plate surface?(b) If a third plate, 20 mm thick, is suspended midwaybetween the original plates, what is the surface heatflux for the original plates? Assume the tempera-ture and flowrateof the gas to be unchanged andradiation effects to be negligible.

Air at 1 atm and 285 K enters a 2-m-long rectangularduct with cross section 75 mm?150 mm. The duct ismaintained at a constant surface temperature of 400 K,and the air mass flow rate is 0.10 kg/s. Determine theheat transfer rate from the duct to the air and the airoutlet temperature.

A double-wall heat exchanger is used to transfer heatbetween liquids flowing through semicircular coppertubes. Each tube has a wall thickness of t?3 mm andan inner radius of ri?20 mm, and good contact ismaintained at the plane surfaces by tightly woundstraps. The tube outer surfaces are well insulated.(a) If hot and cold water at mean temperatures ofTh,m?330 K and Tc,m?290 K flow through the adjoining tubes at m?h?m?c?0.2 kg/s, what is therate of heat transfer per unit length of tube? The wallcontact resistance is 10?5m2?K/W. Approximate theproperties of both the hot and cold water as??800?10?6kg/s?m, k?0.625 W/m?K, andPr?5.35. Hint: Heat transfer is enhanced by con-duction through the semicircular portions of the tubewalls, and each portion may be subdivided into twostraight fins with adiabatic tips.(b) Using the thermal model developed for part (a),determine the heat transfer rate per unit length whenthe fluids are ethylene glycol. Also, what effect willfabricating the exchanger from an aluminum alloyhave on the heat rate? Will increasing the thicknessof the tube walls have a beneficial effect?

Consider laminar, fully developed flow in a channel ofconstant surface temperature Ts. For a given mass flowrate and channel length, determine which rectangularchannel, b/a?1.0, 1.43, or 2.0, will provide the high-est heat transfer rate. Is this heat transfer rate greaterthan, equal to, or less than the heat transfer rate associ-ated with a circular tube?

ou have been asked to perform a feasibility study on the design of a blood warmer to be used during the transfusion of blood to a patient. This exchanger is to heat blood taken from the bank at 10C to 37C at a flow rate of 200 ml/min. The blood passes througha rectangular cross-section tube, 6.4 mm?1.6 mm,which is sandwiched between two plates held at a con-stant temperature of 40C.(a) Compute the length of the tubing required toachieve the desired outlet conditions at the speci-fied flow rate. Assume the flow is fully developedand the blood has the same properties as water.(b) Assess your assumptions and indicate whether youranalysis over- or underestimates the necessarylength.

A coolant flows through a rectangular channel (gallery)within the body of a mold used to form metal injection parts. The gallery dimensions are a?90 mm andb?9.5 mm, and the fluid flow rate is 1.3?10?3m3/s.The coolant temperature is 15C, and the mold wall isat an approximately uniform temperature of 140C.To minimize corrosion damage to the expensive mold, itis customary to use a heat transfer fluid such as ethyleneglycol, rather than process water. Compare the convec-tion coefficients of water and ethylene glycol for thisapplication. What is the tradeoff between thermal perfor-mance and minimizing corrosion?

An electronic circuit board dissipating 50 W is sand-wiched between two ducted, forced-air-cooled heatsinks. The sinks are 150 mm in length and have 20 rectangular passages 6 mm?25 mm. Atmospheric airat a volumetric flow rate of 0.060 m3/s and 27C isdrawn through the sinks by a blower. Estimate theoperating temperature of the board and the pressuredrop across the sinks.

To slow down large prime movers like locomotives, aprocess termed dynamic electric braking is used toswitch the traction motor to a generator mode inwhich mechanical power from the drive wheels isabsorbed and used to generate electrical current. Asshown in the schematic, the electric power is passedthrough a resistor grid (a), which consists of an arrayof metallic blades electrically connected in series (b).The blade material is a high-temperature, high electri-cal resistivity alloy, and the electrical power is dissi-pated as heat by internal volumetric generation. Tocool the blades, a motor- fan moves high-velocity airthrough the grid.(a) Treating the space between the blades as a rectan- gular channel of 220-mm?4-mm cross sectionand 70-mm length, estimate the heat removal rateper blade if the airstream has an inlet temperatureand velocity of 25C and 50 m/s, respectively,while the blade has an operating temperature of600C.(b) On a locomotive pulling a 10-car train, there maybe 2000 of these blades. Based on your result frompart (a), how long will it take to slow a train whosetotal mass is 106kg from a speed of 120 km/h to50 km/h using dynamic electric braking?

A printed circuit board (PCB) is cooled by laminar,fully developed airflow in adjoining, parallel- platechannels of length Land separation distance a. Thechannels may be assumed to be of infinite extent in thetransverse direction, and the upper and lower surfacesare insulated. The temperature Tsof the PCB board isuniform, and airflow with an inlet temperature of Tm,iis driven by a pressure difference ?p.Calculate the average heat removal rate per unitarea (W/m2) from the PCB.

Water at m??0.02 kg/s and Tm,i?20C enters an annularregion formed by an inner tube of diameter Di?25 mmand an outer tube of diameter Do?100 mm. Saturatedsteam flows through the inner tube, maintaining its sur-face at a uniform temperature of Ts,i?100C, while theouter surface of the outer tube is well insulated. If fullydeveloped conditions may be assumed throughout theannulus, how long must the system be to provide an out-let water temperature of 75C? What is the heat flux fromthe inner tube at the outlet?

For the conditions of Problem 8.93, how long must theannulus be if the water flow rate is 0.30 kg/s instead of0.02 kg/s?

Referring to Figure 8.11, consider conditions in anannulus having an outer surface that is insulated(qo?0) and a uniform heat flux qiat the inner surface.Fully developed, laminar flow may be assumed to exist.(a) Determine the velocity profile u(r) in the annularregion.(b) Determine the temperature profile T(r) and obtainan expression for the Nusselt number Nuiassoci-ated with the inner surface.

Consider the air heater of Problem 8.38, but now withairflow through the annulus and steam flow through theinner tube. For the prescribed conditions and an outertube diameter of Do?65 mm, determine the outlettemperature and pressure of the air, as well as the massrate of steam condensation.

Consider a concentric tube annulus for which the innerand outer diameters are 25 and 50 mm. Water enters theannular region at 0.04 kg/s and 25C. If the inner tubewall is heated electrically at a rate (per unit length) ofq??4000 W/m, while the outer tube wall is insulated,how long must the tubes be for the water to achieve anoutlet temperature of 85C? What is the inner tube sur-face temperature at the outlet, where fully developedconditions may be assumed?

It is common practice to recover waste heat from an oil-or gas-fired furnace by using the exhaust gases to preheatthe combustion air. A device commonly used for thispurpose consists of a concentric pipe arrangement forwhich the exhaust gases are passed through the inner pipe, while the cooler combustion air flows through anannular passage around the pipe.Consider conditions for which there is a uniform heat transfer rate per unit length, q?i?1.25?105W/m,from the exhaust gases to the pipe inner surface, while air flows through the annular passage at a rate of m?a?2.1 kg/s. The thin-walled inner pipe is ofdiameter Di?2 m, while the outer pipe, which is well insulated from the surroundings, is of diameterDo?2.05 m. The air properties may be taken to becp?1030 J/kg?K, ??270?10?7N?s/m2, k?0.041W/m?K, and Pr?0.68.(a) If air enters at Ta,1?300 K and L?7 m, what isthe air outlet temperature Ta,2?(b) If the airflow is fully developed throughout theannular region, what is the temperature of the innerpipe at the inlet (Ts,i,1) and outlet (Ts,i,2) sections ofthe device? What is the outer surface temperatureTs,o,1at the inlet?

A concentric tube arrangement, for which the inner andouter diameters are 80 mm and 100 mm, respectively, isused to remove heat from a biochemical reaction occur-ring in a 1-m-long settling tank. Heat is generated uni-formly within the tank at a rate of 105W/m3, and wateris supplied to the annular region at a rate of 0.2 kg/s.Reaction tank,q = 1 105 W/m3,L = 1 mm = 0.2 kg/sWaterDi = 80 mmDo = 100 mmInsulation wrap a) Determine the inlet temperature of the supply waterthat will maintain an average tank surface tempera-ture of 37C. Assume fully developed flow andthermal conditions. Is this assumption reasonable? (b) It is desired to have a slight, axial temperature gra-dient on the tank surface, since the rate of the bio-chemical reaction is highly temperature dependent.Sketch the axial variation of the water and surfacetemperatures along the flow direction for the fol-lowing two cases: (i) the fully developed conditionsof part (a), and (ii) conditions for which entranceeffects are important. Comment on features of thetemperature distributions. What change to the sys-tem or operating conditions would you make toreduce the surface temperature gradient?

Consider the air cooling system and conditions of Problem 8.31, but with a prescribed pipe length ofL?15 m.(a) What is the air outlet temperature, Tm,o? What isthe fan power requirement?(b) The convection coefficient associated with airflowin the pipe may be increased twofold by insertinga coiled spring along the length of the pipe to disrupt flow conditions near the inner surface. Ifsuch a heat transfer enhancement scheme isadopted, what is the attendant value of Tm,o? Useof the insert would not come without a corre-sponding increase in the fan power requirement.What is the power requirement if the friction factor is increased by 50%?(c) After extended exposure to the water, a thin coat-ing of organic matter forms on the outer surface ofthe pipe, and its thermal resistance (for a unit areaof the outer surface) is Rt,o?0.050 m2?K/W. Whatis the corresponding value of Tm,owithout the insertof part (b)?

Consider sterilization of the pharmaceutical product ofProblem 8.27. To avoid any possibility of heating theproduct to an unacceptably high temperature, atmos-pheric steam is condensed on the exterior of the tubeinstead of using the resistance heater, providing a uni-form surface temperature, Ts?100C.(a) For the conditions of Problem 8.27, determine therequired length of straight tube, Ls, that would be needed to increase the mean temperature of thepharmaceutical product from 25C to 75C.(b) Consider replacing the straight tube with a coiledtube characterized by a coil diameter C?100 mmand a coil pitch S?25 mm. Determine the overall length of the coiled tube, Lcl(i.e., the product ofthe tube pitch and the number of coils), necessaryto increase the mean temperature of the pharma- ceutical to the desired value.(c) Calculate the pressure drop through the straighttube and through the coiled tube.(d) Calculate the steam condensation rate.

An engineer proposes to insert a solid rod of diameterDiinto a circular tube of diameter Doto enhance heattransfer from the flowing fluid of temperature Tmto theouter tube wall of temperature Ts,o. Assuming laminarflow, calculate the ratio of the heat flux from the fluidto the outer tube wall with the rod to the heat fluxwithout the rod, qo/qo,wo, for Di/Do?0, 0.10, 0.25 and0.50. The rod is placed concentrically within the tube

An electrical power transformer of diameter 230 mmand height 500 mm dissipates 1000 W. It is desired tomaintain its surface temperature at 47C by supplyingethylene glycol at 24C through thin- walled tubing of20-mm diameter welded to the lateral surface of thetransformer. All the heat dissipated by the transformeris assumed to be transferred to the ethylene glycol.Assuming the maximum allowable temperature rise ofthe coolant to be 6C, determine the required coolantflow rate, the total length of tubing, and the coil pitchSbetween turns of the tubing

A bayonet cooleris used to reduce the temperature ofa pharmaceutical fluid. The pharmaceutical fluid flowsthrough the cooler, which is fabricated of 10-mm-diameter, thin-walled tubing with two 250-mm- longstraight sections and a coil with six and a half turnsand a coil diameter of 75 mm. A coolant flows outsidethe cooler, with a convection coefficient at the outsidesurface of ho?500 W/m2?K and a coolant tempera-ture of 20C. Consider the situation where the pharma-ceutical fluid enters at 90C with a mass flow rate of 0.005 kg/s. The pharmaceutical has the followingproperties: ??1200 kg/m3, ??4?10?3N?s/m2,cp?2000 J/kg?K, and k?0.5 W/m?K. (a) Determine the outlet temperature of the pharma- ceutical fluid.(b) It is desired to further reduce the outlet tempera-ture of the pharmaceutical. However, because thecooling process is just one part of an intricate pro-cessing operation, flow rates cannot be changed. Ayoung engineer suggests that the outlet temperaturemight be reduced by inserting stainless steel coiledsprings into the straight sections of the cooler withthe notion that the springs will disturb the flow adja-cent to the inner tube wall and, in turn, increase theheat transfer coefficient at the inner tube wall. Asenior engineer asserts that insertion of the springsshould double the heat transfer coefficient at thestraight inner tube walls. Determine the outlet tem-perature of the pharmaceutical fluid with the springsinserted into the tubes, assuming the senior engineeris correct in his assertion.(c) Would you expect the outlet temperature of thepharmaceutical to depend on whether the springshave a left-hand or right-hand spiral? Why?

The mold used in an injection molding process con-sists of a top half and a bottom half. Each half is60 mm?60 mm?20 mm and is constructed of metal(??7800 kg/m3, c?450 J/kg?K). The cold mold(100C) is to be heated to 200C with pressurizedwater (available at 275C and a total flow rate of0.02 kg/s) prior to injecting the thermoplastic material.The injection takes only a fraction of a second, and thehot mold (200C) is subsequently cooled with coldwater (available at 25C and a total flow rate of0.02 kg/s) prior to ejecting the molded part. After partejection, which also takes a fraction of a second, theprocess is repeated. a) In conventional mold design, straight cooling(heating) passages are bored through the mold in alocation where the passages will not interfere withthe molded part. Determine the initial heating rateand the initial cooling rate of the mold when five5-mm-diameter, 60-mm-long passages are boredin each half of the mold (10 passages total). Thevelocity distribution of the water is fully devel-oped at the entrance of each passage in the hot (orcold) mold.(b) New additive manufacturing processes, known as selective freeform fabrication, or SFF, are usedto construct molds that are configured with con-formal cooling passages.Consider the same moldas before, but now a 5-mm- diameter, coiled, con-formal cooling passage is designed within eachhalf of the SFF-manufactured mold. Each of thetwo coiled passages has N?2 turns. The coiledpassage does not interfere with the molded part.The conformal channels have a coil diameterC?50 mm. The total water flow remains thesame as in part (a) (0.01 kg/s per coil). Determinethe initial heating rate and the initial cooling rateof the mold.(c) Compare the surface areas of the conventional andconformal cooling passages. Compare the rate atwhich the mold temperature changes for moldsconfigured with the conventional and conformalheating and cooling passages. Which cooling pas-sage, conventional or conformal, will enable pro-duction of more parts per day? Neglect the pres-ence of the thermoplastic material

Consider the pharmaceutical product of Problem 8.27.Prior to finalizing the manufacturing process, test trialsare performed to experimentally determine the depen-dence of the shelf life of the drug as a function of thesterilization temperature. Hence, the sterilization tem-perature must be carefully controlled in the trials. Topromote good mixing of the pharmaceutical and, inturn, relatively uniform outlet temperatures across theexit tube area, experiments are performed using adevice that is constructed of two interwoven coiledtubes, each of 10-mm diameter. The thin-walled tubingis welded to a solid high thermal conductivity rod ofdiameter Dr?40 mm. One tube carries the pharma-ceutical product at a mean velocity of up?0.1 m/s andinlet temperature of 25C, while the second tube carriespressurized liquid water at uw?0.12 m/s with an inlettemperature of 127C. The tubes do not contact eachother but are each welded to the solid metal rod, witheach tube making 20 turns around the rod. The exteriorof the apparatus is well insulated a) Determine the outlet temperature of the pharmaceu-tical product. Evaluate the liquid water properties at 380 K.(b) Investigate the sensitivity of the pharmaceuticalsoutlet temperature to the velocity of the pressurizedwater over the range 0.10uw0.25 m/s.

An extremely effective method of cooling high-power-density silicon chips involves etching microchannelsin the back (noncircuit) surface of the chip. The chan-nels are covered with a silicon cap, and cooling ismaintained by passing water through the channels.Consider a chip that is 10 mm?10 mm on a side andin which fifty 10-mm-long rectangular microchannels,each of width W?50?m and height H?200?m,have been etched. Consider operating conditions forwhich water enters each microchannel at a tempera-ture of 290 K and a flow rate of 10?4kg/s, while thechip and cap are at a uniform temperature of 350 K.Assuming fully developed flow in the channel and thatall the heat dissipated by the circuits is transferred to the water, determine the water outlet temperatureand the chip power dissipation. Water properties maybe evaluated at 300 K.

An ideal gas flows within a small diameter tube.Derive an expression for the transition density of thegas ?cbelow which microscale effects must beaccounted for. Express your result in terms of the gas molecule diameter, universal gas constant, Boltz-manns constant, and the tube diameter. Evaluate thetransition density for a D?10-?m-diameter tube forhydrogen, air, and carbon dioxide. Compare the cal-culated transition densities with the gas density atatmospheric pressure and T?23C

Consider the microchannel cooling arrangement ofProblem 8.107. However, instead of assuming theentire chip and cap to be at a uniform temperature,adopt a more conservative (and realistic) approach thatprescribes a temperature of Ts?350 K at the base ofthe channels (x?0) and allows for a decrease in tem-perature with increasing xalong the side walls of eachchannel.(a) For the operating conditions prescribed in Problem8.107 and a chip thermal conductivity of kch?140 W/m?K, determine the water outlet tempera-ture and the chip power dissipation. Heat transferfrom the sides of the chip to the surroundings andfrom the side walls of a channel to the cap may beneglected. Note that the spacing between channels,??S?W, is twice the spacing between the sidewall of an outer channel and the outer surface ofthe chip. The channel pitch is S?L/N, whereL?10 mm is the chip width and N?50 is thenumber of channels.(b) The channel geometry prescribed in Problem8.107 and considered in part (a) is not optimized,and larger heat rates may be dissipated by adjust-ing related dimensions. Consider the effect ofreducing the pitch to a value of S?100?m,while retaining a width of W?50 ?m and a flowrate per channel of m?1?10?4kg/s

The onset of turbulence in a gas flowing within a cir-cular tube occurs at ReD,c?2300, while a transitionfrom incompressible to compressible flow occurs at a critical Mach number of Mac?0.3. Determine thecritical tube diameter Dc, below which incompressibleturbulent flow and heat transfer cannot exist for (i) air,(ii) CO2, (iii) He. Evaluate properties at atmosphericpressure and a temperature of T?300 K

Due to its comparatively large thermal conductivity,water is a preferred fluid for convection cooling. How-ever, in applications involving electronic devices, watermust not come into contact with the devices, whichwould therefore have to be hermetically sealed. To cir-cumvent related design and operational complexities andto ensure that the devices are not rendered inoperable bycontact with the coolant, a dielectric fluid is commonlyused in lieu of water. Many gases have excellent dielec-tric characteristics, and despite its poor heat transferproperties, air is the common choice for electronic cool-ing. However, there is an alternative, which involves aclass of perfluorinated liquidsthat are excellentdielectrics and have heat transfer properties superior tothose of gases.Consider the microchannel chip cooling applica-tion of Problem 8.109 but now for a perfluorinatedliquid with properties of cp?1050 J/kg?K, k?0.065W/m?K, ??0.0012 N?s/m2, and Pr?15.(a) For channel dimensions of H?200?m, W?50?m, and S?20?m, a chip thermal conductivity ofkch?140 W/m?K and width L?10 mm, a chan- nel base temperature (x?0) of Ts?350 K, achannel inlet temperature of Tm,i?290 K, and aflow rate of m?1?10?4kg/s per channel, determinethe outlet temperature and the chip power dissipa-tion for the dielectric liquid.(b) Consider the foregoing conditions, but with air ata flow rate of m?1?10?6kg/s used as the coolant.Using properties of cp?1007 J/kg?K, k?0.0263W/m?K, and ??185?10?7N?s/m2, determinethe air outlet temperature and the chip power dissipation.

Many of the solid surfaces for which values of thethermal and momentum accommodation coefficientshave been measured are quite different from thoseused in micro- and nanodevices. Plot the Nusselt num-ber NuDassociated with fully developed laminar flowwith constant surface heat flux versus tube diameterfor 1?m?D?1 mm and (i) ?t?1, ?p?1, (ii)?t?0.1, ?p?0.1, (iii) ?t?1, ?p?0.1, and (iv)?t?0.1, ?p?1. For tubes of what diameter do theaccommodation coefficients begin to influence con-vection heat transfer? For which combination of ?tand?pdoes the Nusselt number exhibit the least sensitivityto changes in the diameter of the tube? Which combi-nation results in Nusselt numbers greater than the con-ventional fully developed laminar value for constantheat flux conditions, NuD?4.36? Which combinationis associated with the smallest Nusselt numbers? Whatcan you say about the ability to predict convection heattransfer coefficients in a small-scale device if the accommodation coefficients are not known for mate-rial from which the device is fabricated? Use proper-ties of air at atmospheric pressure and T?300 K.

A novel scheme for dissipating heat from the chips ofa multichip array involves machining coolant channelsin the ceramic substrate to which the chips areattached. The square chips (Lc?5 mm) are alignedabove each of the channels, with longitudinal andtransverse pitches of SL?ST?20 mm. Water flowsthrough the square cross section (W?5 mm) of eachchannel with a mean velocity of um?1 m/s, and itsproperties may be approximated as ??1000 kg/m3,cp?4180 J/kg?K, ??855?10?6kg/s?m, k?0.610W/m?K, and Pr?5.8. Symmetry in the transversedirection dictates the existence of equivalent conditionsfor each substrate section of length Lsand width ST.(a) Consider a substrate whose length in the flowdirection is Ls?200 mm, thereby providing atotal of NL?10 chips attached in-line above eachflow channel. To a good approximation, all theheat dissipated by the chips above a channel maybe assumed to be transferred to the water flowingthrough the channel. If each chip dissipates 5 W,what is the temperature rise of the water passingthrough the channel?(b) The chip-substrate contact resistance is Rt,c?0.5?10?4m2?K/W, and the three-dimensionalconduction resistance for the Ls?STsubstratesection is Rcond?0.120 K/W. If water enters thesubstrate at 25C and is in fully developed flow,estimate the temperature Tcof the chips and thetemperature Tsof the substrate channel surface.

Consider air flowing in a small-diameter steel tube.Graph the Nusselt number associated with fully devel-oped laminar flow with constant surface heat flux for tube diameters ranging from 1?m?D?1 mm.Evaluate air properties at T?350 K and atmosphericpressure. The thermal and momentum accommodation coefficients are ?t?0.92 and ?p?0.87, respectively.Compare the Nusselt number you calculate to thevalue provided in Equation 8.53, NuD4 ? .36.

An experiment is designed to study microscale forcedconvection. Water at Tm,i?300 K is to be heated in a straight, circular glass tube with a 50-?m innerdiameter and a wall thickness of 1 mm. Warm water atT??350 K, V?2 m/s is in cross flow over the exte-rior tube surface. The experiment is to be designed to cover the operating range 1?ReD?2000, whereReDis the Reynolds number associated with the inter-nal flow.(a) Determine the tube length Lthat meets a designrequirement that the tube be twice as long as thethermal entrance length associated with the high-est Reynolds number of interest. Evaluate waterproperties at 305 K.(b) Determine the water outlet temperature, Tm,o, thatis expected to be associated with ReD?2000.Evaluate the heating water (water in cross flowover the tube) properties at 330 K.(c) Calculate the pressure drop from the entrance tothe exit of the tube for ReD?2000.(d) Based on the calculated flow rate and pressuredrop in the tube, estimate the height of a columnof water (at 300 K) needed to supply the neces-sary pressure at the tube entrance and the timeneeded to collect 0.1 liter of water. Discuss how the outlet temperature of the water flowingfrom the tube, Tm,o, might be measured.

Determine the tube diameter that corresponds to a 10% reduction in the convection heat transfer coeffi-cient for thermal and momentum accommodationcoefficients of ?t?0.92 and ?p?0.89, respectively.Determine the channel spacing, a, that is associatedwith a 10% reduction in husing the same accommoda-tion coefficients. The gas is air at T?350 K andatmospheric pressure for both the tube and the parallelplate configurations. The flow is laminar and fullydeveloped with constant surface heat flux.

An experiment is devised to measure liquid flow andconvective heat transfer rates in microscale channels.The mass flow rate through a channel is determined bymeasuring the amount of liquid that has flowedthrough the channel and dividing by the duration ofthe experiment. The mean temperature of the outletfluid is also measured. To minimize the time needed toperform the experiment (that is, to collect a significantamount of liquid so that its mass and temperature canbe accurately measured), arraysof microchannels aretypically used. Consider an array of microchannels of circular cross section, each with a nominal diameter of50?m, fabricated into a copper block. The channelsare 20 mm long, and the block is held at 310 K. Waterat an inlet temperature of 300 K is forced into thechannels from a pressurized plenum, so that a pressuredifference of 2.5?106Pa exists from the entrance tothe exit of each channel.In many microscale systems, the characteristicdimensions are similar to the tolerances that can becontrolled during the manufacture of the experimentalapparatus. Hence, careful consideration of the effect ofmachining tolerances must be made when interpretingthe experimental results.(a) Consider the case in which three microchannelsare machined in the copper block. The channeldiameters exhibit some deviation due to manufac-turing constraints and are of actual diameter45?m, 50?m, and 55?m, respectively. Calculatethe mass flow rate through each of the three chan-nels, along with the mean outlet temperature ofeach channel.(b) If the water exiting each of the three channels iscollected and mixed in a single container, calculatethe average flow rate through each of the threechannels and the average mixed temperature of thewater that is collected from all three channels.(c) The enthusiastic experimentalist uses the averageflow rate and the average mixed outlet tempera-ture to analyze the performance of the average(50?m) diameter channel and concludes that flow rates and heat transfer coefficients areincreased and decreased, respectively, by about5% when forced convection occurs in microchan-nels. Comment on the validity of the experimen-talists conclusion.

In the processing of very long plastic tubes of 2-mminside diameter, air flows inside the tubing with aReynolds number of 1000. The interior layer of theplastic material evaporates into the air under fullydeveloped conditions. Both plastic and air are at400 K, and the Schmidt number for the mixture of theplastic vapor and air is 2.0. Determine the convectionmass transfer coefficient

Air at 300 K and a flow rate of 3 kg/h passes upwardthrough a 30-mm tube, as shown in the sketch. A thinfilm of water, also at 300 K, slowly falls downward onthe inner surface of the tube. Determine the convec-tion mass transfer coefficient for this situation.

What is the convection mass transfer coefficient asso-ciated with fully developed atmospheric airflow at 27C and 0.04 kg/s through a 50-mm-diameter tubewhose surface has been coated with a thin layer ofnaphthalene? Determine the velocity and concentra-tion entry lengths

Air flowing through a tube of 75-mm diameter passesover a 150-mm-long roughened section that is con-structed from naphthalene having the properties??128.16 kg/kmol and psat(300 K)?1.31?10?4bar.The air is at 1 atm and 300 K, and the Reynolds numberis ReD?35,000. In an experiment for which flow wasmaintained for 3 h, mass loss due to sublimation from theroughened surface was determined to be 0.01 kg. What isthe associated convection mass transfer coefficient? Whatwould be the corresponding convection heat transfercoefficient? Contrast these results with those predicted by conventional smooth tube correlations

Dry air at 35C and a velocity of 10 m/s flows over athin-walled tube of 20-mm diameter and 200- mmlength, having a fibrous coating that is water-saturated.AirT= 35CV= 10 m/sWaterm,Tm,iSaturated surface conditionTs = 27C o maintain an approximately uniform surface tem-perature of 27C, water at a prescribed flow rate andtemperature passes through the tube.(a) Considering the heat and mass transfer processeson the external surface of the tube, determine theheat rate from the tube.(b) For a flow rate of 0.025 kg/s, determine the inlettemperature, Tm,i, at which water must be suppliedto the tube.

Consider gas flow of mass density ? and rate m?through atube whose inner surface is coated with a liquid or a sub-limable solid of uniform vapor density ?A,s. Derive Equa-tion 8.86 for variation of the mean vapor density, ?A,m,with distance xfrom the tube entrance and Equation 8.83for the total rate of vapor transfer from a tube of length L

Atmospheric air at 25C and 3?10?4kg/s flowsthrough a 10-mm-diameter, 1-m-long circular tubewhose inner surface is wetted with a water film. Deter-mine the water vapor density at the tube outlet, assum-ing the inlet air to be dry. What is the rate at whichvapor is added to the air?

Air at 25C and 1 atm is in fully developed flow atm??10?3kg/s through a 10-mm-diameter circulartube whose inner surface is wetted with water. Deter-mine the tube length required for the water vapor inthe air to reach 99% of saturation. The inlet air is dry

A humidifier consists of a bundle of vertical tubes, eachof 20-mm diameter, through which dry atmospheric airis in fully developed flow at 10?3kg/s and 298 K. Theinner tube surface is wetted with a water film. Deter-mine the tube length required for the water vapor toreach 99% of saturation. What is the rate at whichenergy must be supplied to each tube to maintain itstemperature at 298 K

he final step of a manufacturing process in which aprotective coating is applied to the inner surface of a circular tube involves passage of dry, atmosphere airthrough the tube to remove a residual liquid associatedwith the process. Consider a coated 5-m-long tubewith an inner diameter of 50 mm. The tube is main-tained at a temperature of 300 K, and the residual liquidexists as a thin film whose corresponding vapor pres-sure is 15 mm Hg. The molecular weight and diffusioncoefficient of the vapor are ?A?70 kg/kmol andDAB?10?5m2/s, respectively. Air enters the tube at amean velocity of 0.5 m/s and a temperature of 300 K.(a) Estimate the partial pressure and mass density ofvapor in the air exiting the tube.(b) What is the rate of liquid removal from the tube inkg/s?

Dry air is inhaled at a rate of 10 liter/min through atrachea with a diameter of 20 mm and a length of125 mm. The inner surface of the trachea is at a normal body temperature of 37C and may beassumed to be saturated with water.(a) Assuming steady, fully developed flow in thetrachea, estimate the mass transfer convectioncoefficient.(b) Estimate the daily water loss (liter/day) associatedwith evaporation in the trachea.

A mass transfer operation is preceded by laminar flowof a gaseous species B through a circular tube that issufficiently long to achieve a fully developed velocityprofile. Once the fully developed condition is reached,the gas enters a section of the tube that is wetted with aliquid film (A). The film maintains a uniform vapordensity ?A,salong the tube surface.(a) Write the differential equation and boundary condi-tions that govern the species A mass density distri-bution, ?A(x, r), for x?0.(b) What is the heat transfer analog to this problem?From this analog, write an expression for theaverage Sherwood number associated with massexchange over the region 0?x?L.(c) Beginning with application of conservation ofspecies to a differential control volume of extent, derive an expression (Equation 8.86) thatmay be used to determine the mean vapor density?A,m,oat x?L.(d) Consider conditions for which species B is air at25C and 1 atm and the liquid film consists of water,also at 25C. The flow rate is m??2.5?10?4kg/s,and the tube diameter is D?10 mm. What is themean vapor density at the tube outlet if L?1m?