John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 98
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(11.120) becomesqs = qu + hfg ni,sand eqn. (11.122) takes the formni,s hfg + cp,if (Ts − Tr ) = h(Te − Ts )(11.127)where cp,if is the specific heat of liquid i. Since the latent heat is generallymuch larger than the sensible heat, a comparison of eqn. (11.127) toeqn. (11.123) exposes the greater efficiency per unit mass flow of sweatcooling relative to transpiration cooling.Thermal radiation. When thermal radiation falls on the surface throughwhich mass is transferred, the additional heat flux must enter the energybalances. For example, suppose that thermal radiation were present during transpiration cooling. Radiant heat flux, qrad,e , originating above thee-surface would be absorbed below the u-surface.14 Thus, eqn.
(11.116)becomesni,r ĥi,r = ni,u ĥi,u − qu − αqrad,e(11.128)where α is the radiation absorptance. Equation (11.117) is unchanged.Similarly, thermal radiation emitted by the wall is taken to originate below the u-surface, so eqn. (11.128) is nowni,r ĥi,r = ni,u ĥi,u − qu − αqrad,e + qrad,u(11.129)or, in terms of radiosity and irradiation (see Section 10.4)ni,r ĥi,r = ni,u ĥi,u − qu − (H − B)(11.130)for an opaque surface.14Remember that the s- and u-surfaces are fictitious elements of the enthalpy balances at the phase interface. The apparent space between them need be only a fewmolecules thick. Thermal radiation therefore passes through the u-surface and is absorbed below it.Problems673Chemical Reactions.
The heat and mass transfer analyses in this section and Section 11.8 assume that the transferred species undergo nohomogeneous reactions. If reactions do occur, the mass balances of Section 11.8 are invalid, because the mass flux of a reacting species will varyacross the region of reaction.
Likewise, the energy balance of this sectionwill fail because it does not include the heat of reaction.For heterogeneous reactions, the complications are not so severe. Reactions at the boundaries release the heat of reaction released betweenthe s- and u-surfaces, altering the boundary conditions. The proper stoichiometry of the mole fluxes to and from the surface must be taken intoaccount, and the heat transfer coefficient [eqn.
(11.115)] must be modified to account for the transfer of more than one species [11.30].Problems11.1Derive: (a) eqns. (11.8); (b) eqns. (11.9).11.2A 1000 liter cylinder at 300 K contains a gaseous mixture composed of 0.10 kmol of NH3 , 0.04 kmol of CO2 , and 0.06 kmol ofHe. (a) Find the mass fraction for each species and the pressurein the cylinder. (b) After the cylinder is heated to 600 K, whatare the new mole fractions, mass fractions, and molar concentrations? (c) The cylinder is now compressed isothermally to avolume of 600 liters. What are the molar concentrations, massfractions, and partial densities? (d) If 0.40 kg of gaseous N2is injected into the cylinder while the temperature remains at600 K, find the mole fractions, mass fractions, and molar concentrations.
[(a) mCO2 = 0.475; (c) cCO2 = 0.0667 kmol/m3 ;(d) xCO2 = 0.187.]11.3Planetary atmospheres show significant variations of temperature and pressure in the vertical direction. Observations suggest that the atmosphere of Jupiter has the following composition at the tropopause level:number density of H2= 5.7 × 1021 (molecules/m3 )number density of He= 7.2 × 1020 (molecules/m3 )number density of CH4 = 6.5 × 1018 (molecules/m3 )number density of NH3 = 1.3 × 1018 (molecules/m3 )Chapter 11: An introduction to mass transfer674Find the mole fraction and partial density of each species atthis level if p = 0.1 atm and T = 113 K.
Estimate the number densities at the level where p = 10 atm and T = 400 K,deeper within the Jovian troposphere. (Deeper in the Jupiter’satmosphere, the pressure may exceed 105 atm.)11.4Using the definitions of the fluxes, velocities, and concentrations, derive eqn. (11.34) from eqn. (11.27) for binary diffusion.11.5Show that D12 = D21 in a binary mixture.11.6Fill in the details involved in obtaining eqn. (11.31) from eqn.(11.30).11.7Batteries commonly contain an aqueous solution of sulfuricacid with lead plates as electrodes. Current is generated bythe reaction of the electrolyte with the electrode material. Atthe negative electrode, the reaction is−Pb(s) + SO2−4 PbSO4 (s) + 2ewhere the (s) denotes a solid phase component and the chargeof an electron is −1.609 × 10−19 coulombs.
If the current density at such an electrode is J = 5 milliamperes/cm2 , what isthe mole flux of SO2−4 to the electrode? (1 amp =1 coulomb/s.)What is the mass flux of SO2−4 ? At what mass rate is PbSO4produced? If the electrolyte is to remain electrically neutral,at what rate does H+ flow toward the electrode? Hydrogen= 7.83 ×does not react at the negative electrode. [ṁPbSO4−5210 kg/m ·s.]11.8The salt concentration in the ocean increases with increasingdepth, z. A model for the concentration distribution in theupper ocean isS = 33.25 + 0.75 tanh(0.026z − 3.7)where S is the salinity in grams of salt per kilogram of oceanwater and z is the distance below the surface in meters.
(a) Plotthe mass fraction of salt as a function of z. (The region of rapidtransition of msalt (z) is called the halocline.) (b) Ignoring theeffects of waves or currents, compute jsalt (z). Use a value ofProblems675Dsalt,water = 1.5 × 10−5 cm2 /s. Indicate the position of maximum diffusion on your plot of the salt concentration. (c) Theupper region of the ocean is well mixed by wind-driven wavesand turbulence, while the lower region and halocline tend tobe calmer.
Using jsalt (z) from part (b), make a simple estimateof the amount of salt carried upward in one week in a 5 km2horizontal area of the sea.11.9In catalysis, one gaseous species reacts with another on a passive surface (the catalyst) to form a gaseous product. For example, butane reacts with hydrogen on the surface of a nickelcatalyst to form methane and propane.
This heterogeneousreaction, referred to as hydrogenolysis, isNiC4 H10 + H2 → C3 H8 + CH4The molar rate of consumption of C4 H10 per unit area in the◦−2.4, where A = 6.3 ×reaction is ṘC4 H10 = A(e−∆E/R T )pC4 H10 pH2102810 kmol/m ·s, ∆E = 1.9 × 10 J/kmol, and p is in atm.(a) If pC4 H10 ,s = pC3 H8 ,s = 0.2 atm, pCH4 ,s = 0.17 atm, andpH2 ,s = 0.3 atm at a nickel surface with conditions of 440◦ Cand 0.87 atm total pressure, what is the rate of consumption ofbutane? (b) What are the mole fluxes of butane and hydrogento the surface? What are the mass fluxes of propane and ethaneaway from the surface? (c) What is ṁ ? What are v, v ∗ , andvC4 H10 ? (d) What is the diffusional mole flux of butane? Whatis the diffusional mass flux of propane? What is the flux of Ni?[(b) nCH4 ,s = 0.0441 kg/m2 ·s; (d) jC3 H8 = 0.121 kg/m2 ·s.]11.10Consider two chambers held at temperatures T1 and T2 , respectively, and joined by a small insulated tube.
The chambersare filled with a binary gas mixture, with the tube open, andallowed to come to steady state. If the Soret effect is takeninto account, what is the concentration difference between thetwo chambers? Assume that an effective mean value of thethermal diffusion ratio is known.11.11Compute D12 for oxygen gas diffusing through nitrogen gasat p = 1 atm, using eqns. (11.39) and (11.42), for T = 200 K,500 K, and 1000 K. Observe that eqn.
(11.39) shows large deviations from eqn. (11.42), even for such simple and similarmolecules.Chapter 11: An introduction to mass transfer67611.12(a) Compute the binary diffusivity of each of the noble gaseswhen they are individually mixed with nitrogen gas at 1 atmand 300 K. Plot the results as a function of the molecularweight of the noble gas. What do you conclude? (b) Considerthe addition of a small amount of helium (xHe = 0.04) to a mixture of nitrogen (xN2 = 0.48) and argon (xAr = 0.48). Compute DHe,m and compare it with DAr,m . Note that the higherconcentration of argon does not improve its ability to diffusethrough the mixture.11.13(a) One particular correlation shows that gas phase diffusioncoefficients vary as T 1.81 and p −1 .
If an experimental value ofD12 is known at T1 and p1 , develop an equation to predict D12at T2 and p2 . (b) The diffusivity of water vapor (1) in air (2) wasmeasured to be 2.39 × 10−5 m2 /s at 8◦ C and 1 atm. Provide aformula for D12 (T , p).11.14Kinetic arguments lead to the Stefan-Maxwell equation for adilute-gas mixture:⎛⎞n$Jj∗Ji∗ci cj⎝⎠−∇xi =c 2 Dij cjcij=1(a) Derive eqn. (11.44) from this, making the appropriate assumptions. (b) Show that if Dij has the same value for eachpair of species, then Dim = Dij .11.15Compute the diffusivity of methane in air using (a) eqn. (11.42)and (b) Blanc’s law. For part (b), treat air as a mixture of oxygenand nitrogen, ignoring argon. Let xmethane = 0.05, T = 420◦ F,and p = 10 psia.
[(a) DCH4 ,air = 7.66×10−5 m2 /s; (b) DCH4 ,air =8.13 × 10−5 m2 /s.]11.16Diffusion of solutes in liquids is driven by the chemical potential, µ. Work is required to move a mole of solute A from aregion of low chemical potential to a region of high chemicalpotential; that is,dµAdxdxunder isothermal, isobaric conditions. For an ideal (very dilute)solute, µA is given bydW = dµA =µA = µ0 + R ◦ T ln(cA )Problems677where µ0 is a constant. Using an elementary principle of mechanics, derive the Nernst-Einstein equation.
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