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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 17.3 Mass Transfer Theories 17.4 Mass Transfer Coefficients 17.1 Theory of Diffusion Flux definitions Velocities If the moles and the mass of component A per unit volume of mixture are cA and ρA, respectively, then the mole fraction of A is xA = cA/c, and mass fraction is wA= ρA/ρ. In a non-uniform fluid mixture with n components that is experiencing bulk motion, the statistical mean velocity of component i in the x direction with respect to stationary coordinates is ui. The mass-average velocity of the mixture in the x direction: ∑= n iiuu 1 1 ρρ 2 The molal-average velocity of the mixture in the x direction: ∑= n iiuccU 1 1 Evidently u = U under the following cases: (a) A binary mixture of very dilute A in B; (b) A non-uniform of components having the same molecular weights MA=MB=······=M; (c) In bulk flow of mixture with uniform composition throughout, regardless of the relative molecular weights of the components: uA = uB = ······ = u ucc uuc c U n i n ii === ∑∑ 11 1 Fluxes Mass fluxes in the x direction for component i Relative to stationary coordinates, nix = ρi ui Relative to the mass-average velocity, iix = ρi（ui − u） Relative to the molal-average velocity, jix = ρi（ui − U） Molal fluxes in the x direction for component i Relative to stationary coordinates, Nix = ci ui 3 Relative to the mass-average velocity, Iix = ci（ui − u） Relative to the molal-average velocity, Jix = ci（ui − U） These expressions enable development of the relationships between the various mass and molal fluxes. Some relationships ∑∑ −=−=−= n ixiix n ii i ixiiix nwnunuui 11 )( ρρ ρρ For the binary system of A and B )( BxAxAAxAx nnwni +−= 0 1 =∑n ixi 0=+ BxAx ii ∑∑ −=−=−= n ixiiix n ii i ixiiix NMxnucc nUuj 11 )( ρρ For the binary system of A and B )()( Bx B A AxAAxBxAxAAAxAx nM MnxnNNMxnj +−=+−= )( 1 Uuj n ix −=∑ ρ )( Uujj BxAx −=+ ρ 4 ∑−=−= n ix i i ixiiix nM wNuucI 1 )( )( 1 uUcI n ix −=∑ ∑−=−= n ixiixiiix NxNUucJ 1 )( 0 1 =∑n ixJ ?=− ixix ij ?=− ixix IJ Steady-state molecular diffusion Fick’s first law Now consider a binary mixture of non-reacting components A and B. Suppose that the total mixture is following steadily with mass- and molal average velocities u and U in x direction. If the composition is nonuniform, molecular diffusion occurs within the mixture in accordance with Fick’s fist law. For steady one-dimensional transfer this diffusive flux may be written as follows: dx dDi AABAx ρ−= which is shown below to require constant density ρ (ρ = ρA + ρB). More generally, dx dwDi AABAx ρ−= which will be shown below not to require constancy of ρ. DAB = DBA is the molecular diffusivity in the 5 binary system. In molal terms, dx dcDJ AABAx −= for which constant total molar concentration c (c = cA + cB) is required. More generally, dx dxcDJ AABAx −= for which variation in c is permissible. Ex.1. Show that dxdDi AABAx /ρ−= requires constant density ρ and dxdwDi AABAx /ρ−= will not to require constancy of ρ. 0)( 1 111 =−=−=−= ∑ ∑∑∑ uuuuuui n n iii n ii n ix ρρρρρ 0=+ BxAx ii BxAx ii −= dx dwDi AABAx ρ−= dx dwDi BBABx ρ−= wA+wB = 1 0=+ dx dw dx dw BA dx dw dx dw BA −= dx dwD dx dwD BBAAAB ρρ =− ∴ DAB = DBA iix = ρi（ui − u） dx dDi AABAx ρ−= 6 ρA uA = ρA u dx dD AAB ρ− ρB uB = ρB u dx dD BBA ρ− ∑= n iiuu 1 1 ρρ BBAA uuu ρρρ += = ρA u dx dD AAB ρ− +ρB u dx dD BBA ρ− ∴ 0==+ dx d dx d dx d BA ρρρ The validity of this result indicates that equation dxdDi AABAx /ρ−= is restricted to constant density ρ. dx dD dx dD dx dD dx dwDi A AB A AB A AB A ABAx ρ ρ ρρ ρρρρ +−= −=−= )/( dx dD dx dDi BBABBABx ρ ρ ρρ +−= 0=+ BxAx ii DAB = DBA 0=+−− dx d dx d dx d BA ρρρ dx d dx d dx d BA ρρρ =+ The validity of this result indicates that equation dxdwDi AABAx /ρ−= does not require constant 7 density ρ, but ρ = f (x) is required. Steady-state molecular diffusion Under steady-state conditions the concentration at a given point is constant with time. Attention is here confined to non-reacting of components A and B for steady one-dimensional transfer. Molal flux relative to stationary coordinates, NA = cA uA = cA U + cA (uA − U) A steady total or bulk flow is imposed upon the fluid mixture in the direction in which component A is diffusing (NA = convective flux + diffusive flux). ABBAAAA Jucucc cN ++⋅= )(1 ABAAA JNNxN ++= )( Equimolal counter-diffusion In this case, the total molal flux with respect to stationary coordinates is zero, so NA = − NB, dz dcDJN AAA −== At steady state NA is constant, and integrating for constant D to give 8 )( 21 AAA cc DN −= δ δ is z2−z1; cA1 and cA2 are the concentrations of A at z1 and z2, respectively. For ideal gas, RT p V nc AAA == )( 21 AAA ppRT DN −= δ pA1 and pA2 are the partial pressures of A at z1 and z2, respectively. The partial pressure distribution is linear. Unimolal unidirectional (one-way) diffusion In this case the total of component B in one direction because of the bulk flow is equal to the flux of B in the opposite direction because of molecular diffusion. Component B is therefore motionless in relation to stationary coordinates and NB = 0. ABAAA JNNxN ++= )( dz dcD c cN AAA −=− )1( At steady state NA is constant, and integrating for constant D to give 9 ))(( 21 AA Bm A ccc cDN −= δ 1 2 12 ln B B BB Bm c c ccc −= ))(( 21 AA Bm A ppp P RT DN −= δ 1 2 12 ln B B BB Bm p p ppp −= The increase in transfer — by the factor (P/pBm) — due to bulk flow in the direction of diffusion of A is indicated. The partial pressure distribution is non-linear. Unsteady-state molecular diffusion In unsteady state diffusion processes, the concentration at a given point varies with time. Continuity Consider the volume element dxdydz in an isotropic medium that is free from convection. The concentration of diffusing solute A at the center of the 10 element (x, y, z) is cA. The rate at which solute A enters through the left face of the element by molecular diffusion is ] 2 [ dz z JJdxdy AZAZ ∂ ∂− where JAZ is the flux through the plane at (x, y, z) parallel to the left face. The rate at which solute A leaves by diffusion through the right face is ] 2 [ dz z JJdxdy AZAZ ∂ ∂+ The rate of accumulation of A within the volume element from these two sources is z Jdxdydz AZ∂ ∂− . Similarly, from the top and bottom faces the accumulation rate is y J dxdydz Ay∂ ∂− and from the front and rear faces it is x Jdxdydz Ax∂ ∂− . Another expression for the rate of accumulation of A in the volume element is τ∂ ∂ Acdxdydz 11 ∴ z J y J x Jc AzAyAxA ∂ ∂−∂ ∂−∂ ∂−=∂ ∂ τ Combine with Fick’s first law for constant D, A AAAA cD z c y c x cDc 22 2 2 2 2 2 )( ∇=∂ ∂+∂ ∂+∂ ∂=∂ ∂ τ For the reacting systems, if the reaction rate of component A is nAiA ckr = , AA A cDrc 2∇=+∂ ∂ τ Fick’s second law For one-dimensional diffusion without reaction, as in a direction normal to the two large surfaces of a slab, 22 / xcA ∂∂ and 22 / ycA ∂∂ are zero and the equation reduces to 2 2 z cDc AA ∂ ∂=∂ ∂ τ which is Fick’s second law of molecular diffusion.