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1 Chapter 17 Principles of Diffusion and Mass Transfer Between Phases 17.1 Theory of Diffusion 17.2 Prediction of Diffusivities Diffusivities are best estimated by experimental measurements, and where such information is available for the system of interest, it should be used directly. Often the desired values are not available, however, and they must be estimated from published correlations. Sometimes a value is available for one set of conditions of temperature and pressure; the correlations are then useful in predicting, from the known value, the desired values for other conditions. In gases ⎟⎠ ⎞⎜⎝ ⎛⎟⎟⎠ ⎞⎜⎜⎝ ⎛= P P T TDD 0 81.1 0 0 In liquids ⎟⎠ ⎞⎜⎝ ⎛⎟⎟⎠ ⎞⎜⎜⎝ ⎛= μ μo T TDD 0 0 17.3 Mass Transfer Theories Convective-transfer and its rate 2 In a turbulent stream the moving eddies transport matter from one location to another. The total molal flux, relative to entire phase, becomes dz dcDJ AA )( ε+−= The eddy diffusivity ε depends on the fluid properties but also on the velocity and position in the flowing stream. Equation cannot be directly integrated to determine the flux. In most mass-transfer operations turbulent flow is desired to increase the rate of transfer per unit area or to help disperse one fluid in another and create more interfacial area. Mass transfer in most cases is treated using the same type of equations, which feature a mass-transfer coefficient k. This coefficient is defined as rate of mass transfer per area per unit concentration difference. )( AiAGA ppkN −= )( iyA yykN −= )( AAiLA cckN −= )( xxkN ixA −= Gy Pkk = , Lx ckk = 3 Two-film theory and mass transfer between phases Two-film theory This theory supposes that motion in the two phases dies near the interface and the entire resistance to transfer is considered as being contained in two fictitious films on either side of the interface, in which transfer occurs by purely molecular diffusion. It is postulated that local equilibrium prevails at the interface (yi = f (xi) = mxi ) and that the concentration gradients are established so rapidly in the films compared to the total time of contact that steady-state diffusion may be assumed. Individual- and overall mass transfer coefficients In the two-film theory, the rate of transfer to the interface is set equal to the rate of the transfer from the interface. yi = f (xi) = mxi esistance/1/1 R forceDriving k xx k yyN x i y i A =−=−= y e xy e xy ii A K yy k m k yy k m k mxxyyN 111 )( −= + −= + −+−= 4 x e xy e xy ii A K xx kmk xx kmk xxmyyN 11111 /)( −= + −= + −+−= )( eyA yyKN −= )( xxKN exA −= Ky and Kx are the overall coefficients based on the overall driving forces. Gas film control and liquid film control xyy k m kK += 11 xyx kmkK 111 += The term yK/1 can be considered an overall resistance to mass transfer, and the terms xkm / and yk/1 are the resistances in the liquid and gas films. xy kmk >>/1 yy kK ≈ Gas film control, NH3, HCl absorbed by water; xy kmk 1)/(1 << xx kK ≈ Liquid film control, CO2, O2 absorbed by water. Ex. Comparison with heat transfer yk , xk differ from 1α , 2α , and may not be determined, because the iy , ix may not be measured. 5 )( tTK A q −= )( eyA yyKN −= )( xxKN exA −= mxye = , xy KmK =⋅ Penetration theory The penetration theory supposes that turbulence transports eddies from the bulk of the phase to the interface, where they remain for a short but constant time before being displaced back into the interior of the phase to be mixed with the bulk fluid. Solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion, in accordance with Fick's second law: 6 2 2 z cDc AA ∂ ∂=∂ ∂ τ The boundary conditions are 0)0,( AA czc = AiA cc =),0( τ The average flux over the time interval 0 to τ0 is )()(2 0 AAicAAiA cckcc DN −=−= πτ Surface renewal theory The surface renewal theory proposes an “infinite” range of ages for element of the interface. The probability of an element of surface being replaced by a fresh eddy is considered to be independent of the age of that element. Define a surface-age- distribution function, )(τφ , such that the fraction of surface with ages between τ and τ+dτ is ττφ d)( . If the probability of replacement of a surface element is independent of its age, )exp()( ττφ ss −= where s is the fractional rate of surface renewal. )()( AAicAAiA cckccDsN −=−= 17.4 Mass Transfer Coefficients 7 Mass transfer coefficients Dimensional analysis From the mechanism of mass transfer, it can be expected that the coefficient k would depend on the diffusivity D and on the variables that control the character of the fluid flow, namely, the velocity u, the viscosity μ, the density ρ, and some linear dimension d. ),,,,( ρμudDfkc = Dimensional analysis gives ),( D DG D dkc ρ μ μϕ= )(Re, ScSh ϕ= Sh and Sc are the Sherwood and Schmidt numbers. Correlations for mass transfer Turbulent flow mass transfer to pipe walls, 3/18.0Re023.0 ScSh = 3/18.0 )()(023.0 D du D dkc ρ μ μ ρ= Analogy between transfer of momentum, heat and mass Newton’s law, Fourier’s law and Fick’s first law are respectively 8 dz ud dz ud dz du )()( ρνρρ μμτ −=−=−= dz Tcd dz Tcd c k dz dTk A q pp p )()( ραρρ −=−=−= dz dcDJ AABAz −= The diffusivity of momentum ν, the thermal diffusivity α and the molecular diffusivity DAB are true fluid properties; their values depend on the temperature and pressure. The Fanning friction factor for turbulent flow in the smooth tube, 2.0Re046.0 −=f Heat transfer by forced convection for turbulent flow in the long tube, 3/18.0 )()(023.0 k cdu k dh pi μ μ ρ= 3/18.0 PrRe023.0=Nu 2 Re023.0 PrRe 2.0 3/1 fNujH === − Turbulent flow mass transfer to pipe walls, 9 3/18.0 )()(023.0 D du D dkc ρ μ μ ρ= 3/18.0Re023.0 ScSh = 2 Re023.0 Re 2.0 3/1 f Sc ShjD === − 2.0Re023.0 2 −=== fjj HD The analogy shown in this equation is general for heat and mass transfer in the same equipment. 3/23/2 3/1 )(Re Sc u k Du k Sc Shj ccD === ρ μ RT Py RT p V nc AAA === RT PM=ρ )()( icAiAcA yyRT PkcckN −=−= RT Pkk cy = )()( iGAiAGA yyPkppkN −=−= Pkk Gy = 32323232 /G/y/y/c D ScG PMkSc G Mk Sc Pu RTk Sc u kj ==== Ex.1 Estimate the time required for complete evaporation of the water on the way. Assuming δ = const. OHA MALAN 2/ρτ ⋅⋅=⋅⋅ 10 Bm AAA p Ppp RT DN )( 21 −= δ 1 2 12 ln B B BB Bm p p ppp −= Ex.2 The effect of pressure on mass transfer coefficients in wetted-wall tower. PP 2' = G = const. ' yk = yk ' yK > yK Theoretical analysis: )( )1( 1))(( i mg AiA Bmg A yyyRT DPpp p P RT DN −−=−= δδ Low concentration, 1 )1( 1 ≈− my , g y RT DPk δ= P D 1∝ yk is independent of P. Experimental: 33.083.0 023.0 ⎟⎠ ⎞⎜⎝ ⎛⎟⎠ ⎞⎜⎝ ⎛= D dG D dkc ρ μ μ 11 Gas: P∝ρ P D 1∝ P kc 1∝ )( iAAcA cckN −= )( RT Py RT Pyk ic −= )( ic yyRT Pk −= RT Pkk cy = yk is independent of P. ∴ yy kk =' xyy k m kK += 11 P↑, m↓, PP Em 1∝= ∴ P↑, yK ↑, yy KK >'