John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 100
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Using eqns. (6.31b) and (6.55)to obtain the dependence of δ and δt on Rex and Pr, showthat stagnant film models gives eqns. (6.33) and (6.58) withina constant on the order of unity. [The constants in these results will differ from the exact results because the effective b.l.thicknesses of the stagnant film model are not the same as theexact values—see eqn. (6.57).]11.38(a) What is the largest value of the mass transfer driving forcewhen one species is transferred? What is the smallest value?(b) Plot the blowing factor as a function of Bm,i for one speciestransferred. Indicate on your graph the regions of blowing,suction, and low-rate mass transfer.
(c) Verify the two limits∗= ρDim /δc .used to show that gm,i11.39Nitrous oxide is bled through the surface of a porous 3/8 in.O.D. tube at 0.025 liter/s per meter of tube length. Air flowsover the tube at 25 ft/s. Both the air and the tube are at 18◦ C,and the ambient pressure is 1 atm. Estimate the mean concentration of N2 O at the tube surface. (Hint : First estimate theconcentration using properties of pure air; then correct theproperties if necessary.)11.40Film absorbtion is a process whereby gases are absorbed intoa falling liquid film. Typically, a thin film of liquid runs downthe inside of a vertical tube through which the gas flows.
Analyze this process under the following assumptions: The filmflow is laminar and of constant thickness, δ0 , with a velocityprofile given by eqn. (8.48); the gas is only slightly soluble inthe liquid, so that it does not penetrate far beyond the liquid surface and so that liquid properties are unaffected by it;and, the gas concentration at the s- and u-surfaces (above andbelow the liquid-vapor interface, respectively) does not varyalong the length of the tube.
The inlet concentration of gas inthe liquid is m1,0 . Show that the mass transfer is given byNum,x =u0 xπ D121/2withu0 =(ρf − ρg )gδ202µfThe mass transfer coefficient here is based on the concentration difference between the u-surface and the bulk liquid atChapter 11: An introduction to mass transfer682m1,0 .
(Hint : The small penetration assumption can be used toreduce the species equation for the film to the diffusion equation, eqn. 11.72.)11.41Benzene vapor flows through a 3 cm I.D. vertical tube. A thinfilm of initially pure water runs down the inside wall of the tubeat a flow rate of 0.3 liter/s. If the tube is 0.5 m long and 40◦ C,estimate the rate (in kg/s) at which benzene is absorbed intowater over the entire length of the tube.
The mass fraction ofbenzene at the u-surface is 0.206. (Hint : Use the result statedin Problem 11.40. Obtain δ0 from the results in Chapter 8.)11.42A mothball consists of a 2.5 cm diameter sphere of naphthalene (C10 H8 ) that is hung by a wire in a closet. The solid naphthalene slowly sublimates to vapor, which drives off the moths.The latent heat of sublimation and evaporation rate are lowenough that the wet-bulb temperature is essentially the ambient temperature.
Estimate the lifetime of this mothball ina closet with a mean temperature of 20◦ C. Use the followingdata:σ = 6.18 Å,ε/kB = 561.5 KforC10 H8 ,and, for solid naphthalene,ρC10 H8 = 1145 kg/m3 at 20◦ CThe vapor pressure (in mmHg) of solid naphthalene near roomtemperature is given approximately by log10 pv = 11.450 −3729.3/(T K). The integral you obtain can be evaluated numerically.11.43In contrast to the napthalene mothball described in Prob. 11.42,other mothballs are made from paradichlorobenzene (PDB). Estimate the lifetime of a 2.5 cm diameter PDB mothball usingthe following room temperature property data:σ = 5.76 Åε/kB = 578.9 KMPDB = 147.0 kg/kmol log10 pv mmHg = 11.985 − 3570/(T K)ρPDB = 1248 kg/m3Problems11.44683Consider the process of catalysis as described in Problem 11.9.The mass transfer process involved is the diffusion of the reactants to the surface and diffusion of products away from it.(a) What is ṁ in catalysis? (b) Reaction rates in catalysis areof the form:Ṙreactant = A e−∆E/R◦T(preactant )n (pproduct )m kmol/m2 ·sfor the rate of consumption of a reactant per unit surfacearea.
The p’s are partial pressures and A, ∆E, n, and m areconstants. Suppose that n = 1 and m = 0 for the reactionB + C → D. Approximate the reaction rate, in terms of mass,as◦ṙB = A e−∆E/R T ρB,s kg/m2 ·sand find the rate of consumption of B in terms of mB,e and themass transfer coefficient for the geometry in question. (c) The◦∗ is called the Damkohler number.ratio Da ≡ ρA e−∆E/R T /gmExplain its significance in catalysis. What features dominatethe process when Da approaches 0 or ∞? What temperaturerange characterizes each?11.45One typical kind of mass exchanger is a fixed-bed catalytic reactor. A flow chamber of length L is packed with a catalystbed.
A gas mixture containing some species i to be consumedby the catalytic reaction flows through the bed at a rate ṁ. Theeffectiveness of such a exchanger (cf. Chapter 3) isε = 1 − e−NTU ,where NTU = gm,oa P L/ṁwhere gm,oa is the overall mass transfer coefficient for the catalytic packing, P is the surface area per unit length, and ε isdefined in terms of mass fractions. In testing a 0.5 m catalyticreactor for the removal of ethane, it is found that the ethaneconcentration drops from a mass fraction of 0.36 to 0.05 at aflow rate of 0.05 kg/s. The packing is known to have a surfacearea of 11 m2 .
What is the exchanger effectiveness? What isthe overall mass transfer coefficient in this bed?11.46(a) Perform the integration to obtain eqn. (11.112). Then takethe derivative and the limit needed to get eqns. (11.113) and(11.114). (b) What is the general form of eqn. (11.115) whenmore than one species is transferred?Chapter 11: An introduction to mass transfer68411.47(a) Derive eqn. (11.125) from eqn. (11.124).
(b) Suppose that1.5 m2 of the wing of a spacecraft re-entering the earth’s atmosphere is to be cooled by transpiration; 900 kg of the vehicle’s weight is allocated for this purpose. The low-rate heattransfer coefficient is 1800 W/m2 ·K, and the hottest period ofre-entry is expected to last 3 minutes. If the air behind theshock wave ahead of the wing is at 2500◦ C and the reservioris at 5◦ C , which of these gases—H2 , He, and N2 —keeps thesurface coolest? (Of course, the result for H2 is invalidated bythe fact that H2 would burn under these conditions.)11.48Dry ice (solid CO2 ) is used to cool medical supplies transportedby a small plane to a remote village in Alaska.
A roughly spherical chunk of dry ice, 5 cm in diameter, falls from the planethrough air at 5◦ C with a terminal velocity of 15 m/s. If steadystate is reached quickly, what are the temperature and sublimation rate of the dry ice? The latent heat of sublimationis 574 kJ/kg and log10 (pv mmHg) = 9.9082 − 1367.3/(T K).The temperature will be well below the “sublimation point” ofCO2 (solid-to-vapor saturation temperature), which is −78.6◦ Cat 1 atm. Use the heat transfer relation for spheres in a lam1/3. (Hint : first estimate theinar flow, NuD = 2 + 0.3 Re0.6D Prsurface temperature using properties for pure air; then correctthe properties if necessary.)11.49The following data were taken at a weather station over a period of several months:DateTdry-bulbTwet-bulb3/154/215/135/317/415.5◦ C22.027.332.739.011.0◦ C16.825.820.031.2Use eqn.
(11.109) to find the mass fraction of water in the airat each date. Compare to values from a psychrometric chart.11.50Biff Harwell has taken Deb sailing. Deb, and Biff’s towel, fallinto the harbor. Biff rescues them both from a passing dolphinand then spreads his wet towel out to dry on the fiberglas foredeck of the boat. The incident solar radiation is 1050 W/m2 ;the ambient air is at 31◦ C, with mH2 O = 0.017; the wind speedProblems685is 8 knots relative to the boat (1 knot = 1.151 mph); εtowel αtowel 1; and the sky has the properties of a black body at280 K. The towel is 3 ft long in the windward direction and 2 ftwide. Help Biff figure out how rapidly (in kg/s) water evaporates from the towel.11.51Steam condenses on a 25 cm high, cold vertical wall in a lowpressure condenser unit. The wall is isothermal at 25◦ C, andthe ambient pressure is 8000 Pa.
Air has leaked into the unitand has reached a mass fraction of 0.04. The steam–air mixture is at 45◦ C and is blown downward past the wall at 8 m/s.(a) Estimate the rate of condensation on the wall. (Hint : Thesurface of the condensate film is not at the mixture or wall temperature.) (b) Compare the result of part (a) to condensationwithout air in the steam. What do you conclude?11.52During a coating process, a thin film of ethanol is wiped ontoa thick flat plate, 0.1 m by 0.1 m. The initial thickness of theliquid film is 0.1 mm, and the initial temperature of both theplate and the film is 303 K. The air above the film is at 303 Kand moves at 10 m/s.
(a) Assume that the plate is a poorconductor, so that heat transfer from it is negligible. After ashort initial transient, the liquid film reaches a steady temperature. Find this temperature and calculate the time requiredfor the film to evaporate. (b) Discuss what happens whenthe plate is a very good conductor of heat, and estimate theshortest time required for evaporation.
Properties of ethanolare as follow: log10 (pv mmHg) = 9.4432 − 2287.8/(T K);hfg = 9.3 × 105 J/kg; liquid density, ρeth = 789 kg/m3 ; Sc= 1.30 for ethanol vapor in air; vapor specific heat capacity,cpeth = 1420 J/kg·K.11.53Ice cubes left in a freezer will slowly sublimate into the air.Suppose that a tray of ice cubes is left in a freezer with air at−10◦ C and a relative humidity of 50%. The air in the freezeris circulated by a small fan, creating a heat transfer coefficientfrom the top of the ice cube tray of 5 W/m2 K.
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