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78 CHAPTER 5 1965 Guide And Data Book Fig. 4 .... Diffusion of Water Vapor Through Stagnant Air both components behave as perfect gases Dalton law is valid, hence that the Gibba- P + pi ^t " constant (11) where - P, -- total pressure, pounds force per square;foot =*'2116 Pt Thus, the partial pressure of the air po must decrease as the partial pressure of the water vapor p, increases, so that the total pressure P, is constant. Dividing.both sides of Equation &c by Af,, the molecular weight of water, Similarly nJA = - (DJR,T)(dp.fdy) . ' (12) ' " A./A - - (D,/R,T){dp./dy) - : ' (13) is constant, and Equation 11 is valid.:The partial pressure gradient of the water vapor causes a partial pressure gradient of the air such that, dpa/dy= --dp,/dy. Air, then, diffuses toward the liquid water interface. Since it cannot be absorbed there, a convective or bulk velocity o of the gas mixture is established in a direction away from the liquid.surface, so that the.net transport of air is zero; i,e.,.it is stagnant. *./A D, dp. $T dy 0 and Dn 1 dp. D, ' 1 dp,': ll/T "dy7; " ~ lt/T X ~dy '\ (15) where A> is the molal density of the air (in lb mols per cu ft). The bulk velocity,-1>, not only gives rise to a transport of air, but also.of water vapor away from the interface. The total rate of water vapor diffusion is therefore */A D, dp. R,T dy Substituting for the velocity v from Equation 15 gives .. n./A ( M p> R.T V + pj dy P dp, R,T A dy But the molal density'ratio /& Pt/p<i from the ideal gas mixture relations, so ,, ,A _ _ _5i. L. ib R,T p. dy D, Pi dp. -- -P R,T p. . dy (16) Integration yields D.P, log(p--/p.) n./A < R,T (y - y<) -- D. Pi iPm -- Prt) (17) R,T p. (y. -- yd where "": ' n, -- molal diffusion rate of water vapor, pound-mob per.hour, R. -- universal gas constant =* 1545 foot-pounds force per (Rankine degree) (pound-mol). } '".i ii.- molal diffusion rate of dry air, pound-mols per. bom. y -- distance measured normal to the transfer surface, feet. D, is the same for diffusion of water -vapor'through'air as -for air through water vapor. Equation' ll requires 'that (dp,/dy)= -?(dpa/dy). By integration between planes l and 2ofFig. 3, r ' i iv; h,/A - iti/A - D.(prt - ptl) R,T(jli - pi) (14) This process is known as equimolal counterdiffusioiu Hie rates of diffusion of air and water vapor are' equal, but in opposite directions. This situation has no counterpart in heat transfer, since heat can be transferred only.-in one direction at a time. . Diffusion of One Gas Through a Second Stag* nant Gas -! A different type of diffusion process b that in which one. gas diffuses through a second stationary or stagnant gas. An important example of this type of diffusion, illustrated in Fig. 4, b where water vapor diffuses from the liquid surface into surrounding stationary air. It b assumed that, equi librium exists through the gas mixture, that the gases are perfect, and that the Gibbs-Dalton law b valid. Ah usual, the diffusion of the water vapor b due to a concentration or par* tial pressure gradient, and b given by Equation 12..:, There b a continuous gas phase, so the total pressure,' Pt, where Pam b the logarithmic mean partial pressure of the stagnant air.-The partial.pressure gradients for this type of diffusion are illustrated in Fig. 5. It should be noted that the.term stagnant refers to the net behavior of the air. 'The air b moving, but the convective flow exactly offsets the diffusion. Comparison of Equation 17 with Equation 14 shows that the fartor Pt/Pm b intro duced when diffusion through a stagnant gas b considered. For a dilute mixture of water vapor in air, pa b approxi mately equal to Pt and Equations 14 and 17 become identical Mass Traitsfer . _ 79 a vcrtics! tube *4 I .AGO in. diameter u partially 2*ter sothat the distance from the water surface to the SSiTtStobeia 2.530 in. (Fig. 4). Perfectly dried air is over the open tube end and the,complete system ts at a SSant temperature of 85 F. The Water evaporated from the ^Siffoundto be 0.0102 lb in 200 far of steady operation. The tntal pressure of tfae system is 14.696-psia. j. a) Using these data, calculate the mass diffuavity of water T#E)Uc5>nmaro.thb experimental result with that calculated by Equation 7. Solution: (a) The molal diffusion rate ri, is I 0 0102 - 0.00000283.. ,200X18 The area of a l in. diameter tube'is 0.00545 .sq ft. Therefore, (*./A), -0.00000283/0.00545 = 0.00052 ib-mols per (hr)(sqft). The'partial pressures are: pt- -- 0; pli -- 0.596 psi p.( - P - P " 14.696 - 0.595 - 14.10 psia The logarithmic mean'partial pressure of the air ia 1 P.- - P-i ' 0-596 _ 14 38 P" " log. (p../p.i) " log. (14.696/14.10) The diffuarity is now computed from Equation 17 as (.M),gfrp(p. - y.) D, = Pi(p - P--) (0.00052) (1545K545) (14.38) (2.53/12) (14.696) (0.596) (144) .050 sq ft per hr b) By Equation 7 noting that Pr atmospheres, 0.000146 T*. Z). Pr , (T+441) -Pt/14.696 D. * 0.006l46(545)*-V986 - 1.025 sq ft per hr Note that neglecting' the correction factor PJpM for this ex ample would cause only a 2 percent ehange in. the calculated experimental value of D,. . . CONVECTIVE OR EDDY DIFFUSION The concept of.turbulent flow b described by the character istic phenomenon of random yelocity.fluctuations superim posed on the time-average velocity. These velocity~fluctua tions result in snail particles of the fluid .moving, at.one instant, faster than the average velocity; at another instant, slower. The fluctuations occur not only in the direction of flow but also normal to the flow direction. As a.result, small mixing actions or eddy currents are established within the turbulent flow held; If a'small particle of fluid b carried by an eddy current from a region of high velocity to one of low sjocity, it will'give up momentum to -the slower moving fund. Hence, these eddies result in an exchange of momentum between different layers of the moving fluid. If there b a tem perature gradient in the fluid, energy will be exchanged by this same mixing action of the eddies in exactly the same way. TMibrly, if a mass concentration gradient exists; -there..will be a.mass exchange by the same mechanism. -Thb mass exr b known as eddy diffusion.. The ^difference between molecular and eddy diffusion b that of microscopic,or molecu 4 Hffing action versus macroscopic or particle mixing action. ^ exPecteI eddy:diffusion is relatively fast,as coraPWW to molecular diffusion. Furthermore, .the rate of eddy usion will depend on the intensify of,.the.velocity, fluctua. os,.that b, on the intensity of tiie.turbulence. Since,-the intensity of turbulence b determined by the Reynolds number o the flow, the rate of eddy diffusion wilTdepend-on- the _ tiqmqlda number..It b posable .to, define;an ,fdy diffusinty fn hy the same form of equation as that in molecular diffusion (le., Equation 8), so that m,/A = -- tb{dp,/dy) (18) where to " eddy diffusivity, square feet per hour. Experiments have shown that to b a strong function of Reynolds number.* Because data on eddy diffusivities are rare and difficult to obtain common practice b to define a masa transfer coefficient, analogous to the heat transfer coefficient in convective heat transfer, and to'determine it experimen tally. Mass Transfer Coefficient Consider an air stream in steady turbulent flow over a wetted surface as illustrated in Fig. 6. It b assumed that the liquid-vapor interface b stationary and, therefore, the ve locity b zero at thb surface. Thb results in a slow-moving layer of fluid next to the surface which b in laminar flow. Be tween thia laminar sublayer and the main body of the turbu lent stream there b a transition region in which the fluid may be alternately in laminar flow and in turbulent flow. Thb flow regime b referred to as the. buffer layer.:Within the laminar sublayer, only molecular diffusion can occur; whereas in tiie buffer layer both molecular and eddy diffusion are important contributors to the mass transfer process. In the turbulent region eddy diffusion predominates and is so rapid as to soon equalize the concentration gradient. Because of the presence of the laminar sublayer, the rate of molecular mass diffusion from the wetted surface to the air stream, from Equation 5c b Diffusion rate = -- (Dw/R,T)(dp,/dy)t where (dp,/dy)i b the partial pressure gradient at the inter face. Making the usual assumptions that the gases are ideal, that they obey the Gibbs-Dalton law, and that the total pre& sure b constant, then a partial pressure gradient must also' exist in the air. The wetted surface b impermeable to air. Therefore, a convective or bulk velocity must be established to counter the air diffusion rate. The total mass transfer from the wetted surface to the air stream must then be given by m,/A - - {0./R.T){dp,/dy)i + (19) where t\ b the convective velocity at the interface (Fig. 6), and p.i - p'i/RtT is the partial mass density of the water vapor at the interface. It was shown previously that for the diffusion of one gas through a second stagnant gas, the simple molecular diffusion equation could be properly corrected by the factor. (Pi/p), thus accounting for the mass transfer contribution of the con vective velocity. Furthermore, in the case of dilute mixtures, as in Example 1 above, thb contribution b small. It might be assumed, therefore, that the same correction factor could be Fig. 6 .... Turbulent Diffusion Boundary Layer on a. . . Rat Surface. .