John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 99
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Note that the solution must be assumed to be very dilute.11.17A dilute aqueous solution at 300 K contains potassium ions,K+ . If the velocity of aqueous K+ ions is 6.61 × 10−4 cm2 /s·Vper unit electric field (1 V/cm), estimate the effective radius ofK+ ions in an aqueous solution. Criticize this estimate. (Thecharge of an electron is −1.609 × 10−19 coulomb and a volt =1J/coulomb.)11.18(a) Obtain diffusion coefficients for: (1) dilute CCl4 diffusingthrough liquid methanol at 340 K; (2) dilute benzene diffusing through water at 290 K; (3) dilute ethyl alcohol diffusing through water at 350 K; and (4) dilute acetone diffusingthrough methanol at 370 K.
(b) Estimate the effective radius ofa methanol molecule in a dilute aqueous solution.[(a) Dacetone,methanol = 6.8 × 10−9 m2 /s.]11.19If possible, calculate values of the viscosity, µ, for methane,hydrogen sulfide, and nitrous oxide, under the following conditions: 250 K and 1 atm, 500 K and 1 atm, 250 K and 2 atm,250 K and 12 atm, 500 K and 12 atm.11.20(a) Show that k = (5/2)µcv for a monatomic gas. (b) ObtainEucken’s formula for the Prandtl number of a dilute gas:Pr = 4γ (9γ − 5)(c) Recall that for an ideal gas, γ (D + 2)/D, where D is thenumber of modes of energy storage of its molecules.
Obtainan expression for Pr as a function of D and describe what itmeans. (d) Use Eucken’s formula to compute Pr for gaseousAr, N2 , and H2 O. Compare the result to data in Appendix Aover the range of temperatures. Explain the results obtainedfor steam as opposed to Ar and N2 . (Note that for each modeof vibration, there are two modes of energy storage but thatvibration is normally inactive until T is very high.)11.21A student is studying the combustion of a premixed gaseousfuel with the following molar composition: 10.3% methane,15.4% ethane, and 74.3% oxygen. She passes 0.006 ft3/s of theChapter 11: An introduction to mass transfer678mixture (at 70◦ F and 18 psia) through a smooth 3/8 inch I.D.tube, 47 inches long. (a) What is the pressure drop? (b) Thestudent’s advisor recommends preheating the fuel mixture, using a Nichrome strip heater wrapped around the last 5 inchesof the duct.
If the heater produces 0.8 W/inch, what is the walltemperature at the outlet of the duct? Let cp,CH4 = 2280 J/kg·K,γCH4 = 1.3, cp,C2 H6 = 1730 J/kg·K, and γC2 H6 = 1.2, and evaluate the properties at the inlet conditions.11.22(a) Work Problem 6.36. (b) A fluid is said to be incompressible ifthe density of a fluid particle does not change as it moves aboutin the flow (i.e., if Dρ/Dt = 0). Show that an incompressible = 0. (c) How does the condition of incomflow satisfies ∇ · upressibility differ from that of “constant density”? Describe aflow that is incompressible but that does not have “constantdensity.”11.23Carefully derive eqns. (11.62) and (11.63).
Note that ρ is notassumed constant in eqn. (11.62).11.24Derive the equation of species conservation on a molar basis,using ci rather than ρi . Also obtain an equation in ci alone,similar to eqn. (11.63) but without the assumption of incompressibility. What assumptions must be made to obtain thelatter result?11.25Find the following concentrations: (a) the mole fraction of airin solution with water at 5◦ C and 1 atm, exposed to air at thesame conditions, H = 4.88 × 104 atm; (b) the mole fractionof ammonia in air above an aqueous solution, with xNH3 =0.05 at 0.9 atm and 40◦ C and H = 1522 mm Hg; (c) the molefraction of SO2 in an aqueous solution at 15◦ C and 1 atm, ifpSO2 = 28.0 mm Hg and H = 1.42 × 104 mm Hg; and (d) thepartial pressure of ethylene over an aqueous solution at 25◦ Cand 1 atm, with xC2 H4 = 1.75 × 10−5 and H = 11.4 × 103 atm.11.26Use a steam table to estimate (a) the mass fraction of watervapor in air over water at 1 atm and 20◦ C, 50◦ C, 70◦ C, and90◦ C; (b) the partial pressure of water over a 3 percent-byweight aqueous solution of HCl at 50◦ C; (c) the boiling pointat 1 atm of salt water with a mass fraction mNaCl = 0.18.[(c) TB.P .
= 101.8◦ C.]Problems11.27679Suppose that a steel fitting with a carbon mass fraction of 0.2%is put into contact with carburizing gases at 940◦ C, and thatthese gases produce a steady mass fraction, mC,u , of 1.0% carbon just within the surface of the metal. The diffusion coefficient of carbon in this steel is DC,Fe = 1.50 × 10−5 m2 s exp −(1.42 × 108 J/kmol) (R ◦ T )for T in kelvin. How long does it take to produce a carbonconcentration of 0.6% by mass at a depth of 0.5 mm? Howmuch less time would it take if the temperature were 980◦ C?11.28(a) Write eqn. (11.62) in its boundary layer form.
(b) Write thisconcentration boundary layer equation and its b.c.’s in termsof a nondimensional mass fraction, ψ, analogous to the dimensionless temperature in eqn. (6.42). (c) For ν = Dim , relate ψto the Blasius function, f , for flow over a flat plate. (d) Note thesimilar roles of Pr and Sc in the two boundary layer transportprocesses. Infer the mass concentration analog of eqn.
(6.55)and sketch the concentration and momentum b.l. profiles forSc = 1 and Sc 1.11.29When Sc is large, momentum diffuses more easily than mass,and the concentration b.l. thickness, δc , is much less than themomentum b.l. thickness, δ. On a flat plate, the small partof the velocity profile within the concentration b.l. is approximately u/Ue = 3y/2δ. Compute Num,x based on this velocityprofile, assuming a constant wall concentration.
(Hint : Use themass transfer analogs of eqn. (6.47) and (6.50) and note thatqw /ρcp becomes ji,s /ρ.).11.30Consider a one-dimensional, binary gaseous diffusion processin which species 1 and 2 travel in opposite directions along thez-axis at equal molar rates. (The gas mixture will be at rest,with v = 0 if the species have identical molecular weights).This process is known as equimolar counter-diffusion.
(a) Whatare the relations between N1 , N2 , J1∗ , and J2∗ ? (b) If steady stateprevails and conditions are isothermal and isobaric, what isthe concentration of species 1 as a function of z? (c) Writethe mole flux in terms of the difference in partial pressure ofspecies 1 between locations z1 and z2 .Chapter 11: An introduction to mass transfer68011.31Consider steady mass diffusion from a small sphere.
Whenconvection is negligible, the mass flux in the radial direction isnr ,i = jr ,i = −ρDim dmi /dr . If the concentration is mi,∞ farfrom the sphere and mi,s at its surface, use a mass balance toobtain the surface mass flux in terms of the overall concentration difference (assuming that ρDim is constant). Then applythe definition eqns. (11.94) and (11.78) to show that Num,D = 2for this situation.11.32An experimental Stefan tube is 1 cm in diameter and 10 cmfrom the liquid surface to the top. It is held at 10◦ C and 8.0 ×104 Pa.
Pure argon flows over the top and liquid CCl4 is atthe bottom. The pool level is maintained while 0.086 ml ofliquid CCl4 evaporates during a period of 12 hours. What is thediffusivity of carbon tetrachloride in argon measured underthese conditions? The specific gravity of liquid CCl4 is 1.59and its vapor pressure is log10 pv = 8.004 − 1771/T , where pvis expressed in mm Hg and T in K.11.33Repeat the analysis given in Section 11.7 on the basis of massfluxes, assuming that ρDim is constant and neglecting anybuoyancy-driven convection. Obtain the analog of eqn.
(11.88).11.34In Sections 11.5 and 11.7, it was assumed at points that cD12or ρD12 was independent of position. (a) If the mixture composition (e.g., x1 ) varies in space, this assumption may be poor.Using eqn. (11.42) and the definitions from Section 11.2, examine the composition dependence of these two groups. Forwhat type of mixture is ρD12 most sensitive to composition?What does this indicate about molar versus mass-based analysis? (b) How do each of these groups depend on pressure andtemperature? Is the analysis of Section 11.7 really limited toisobaric conditions? (c) Do the Prandtl and Schmidt numbersdepend on composition, temperature, or pressure?11.35A Stefan tube contains liquid bromine at 320 K and 1.2 atm.Carbon dioxide flows over the top and is also bubbled up throughthe liquid at the rate of 4.4 ml/hr. If the distance from the liquid surface to the top is 16 cm and the diameter is 1 cm, whatis the evaporation rate of Br2 ? (psat,Br2 = 0.680 bar at 320 K.)[NBr2 ,s = 1.90 × 10−6 kmol/m2 ·s.]11.36Show that gm,1 = gm,2 and Bm,1 = Bm,2 in a binary mixture.Problems68111.37Demonstrate that stagnant film models of the momentum andthermal boundary layers reproduce the proper dependence ofCf ,x and Nux on Rex and Pr.
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