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Diffusion and Mass Transfer Btw Molecules

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Mass Transfer Coefficients
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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 c  A  and  ρ  A ,   respectively, then the mole fraction of A is  x  A  = c  A / c , and mass fraction is w  A =  ρ  A /  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 u i . The mass-average velocity of the mixture in the  x  direction: ∑ = nii uu 1 1  ρ  ρ      2 The molal-average velocity of the mixture in the  x  direction: ∑ = nii uccU  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  M   A =  M   B =······=  M  ; (c) In bulk flow of mixture with uniform composition throughout, regardless of the relative molecular weights of the components: u  A   = u  B   = ······ = u uccuuccU  ninii  ===  ∑∑ 11 1   Fluxes  Mass fluxes in the x direction for component i Relative to stationary coordinates, n ix  =  ρ i u i  Relative to the mass-average velocity, i ix  =  ρ i ( u i   −   u )  Relative to the molal-average velocity,  j ix  =  ρ i ( u i   −   U  )    Molal fluxes in the x direction for component i Relative to stationary coordinates,  N  ix  = c i u i   3 Relative to the mass-average velocity,  I  ix  = c i ( u i   −   u )  Relative to the molal-average velocity,  J  ix  = c i ( u i   −   U  )  These expressions enable development of the relationships between the various mass and molal fluxes. Some relationships ∑∑  −=−=−= nixiixniiiixiiix  nwnunuui 11 )(  ρ  ρ  ρ  ρ   For the binary system of A and B )(  Bx x x x  nnwni  +−=  0 1 = ∑ nix i  0 =+  Bx x  ii   ∑∑  −=−=−= nixiiix niiiixiiix  N  M  xnuc cnU u j 11 )(  ρ  ρ   For the binary system of A and B )()(  Bx B A Ax A Ax Bx Ax A A Ax Ax  n M n xn N  N  M  xn j  +−=+−=  )( 1 U u j nix  −= ∑  ρ   )(  U u j j  Bx x  −=+  ρ      4 ∑ −=−= nixiiixiiix  nw N uuc I  1 )( )( 1 uU c I  nix  −= ∑   ∑ −=−= nixiixiiix  N  x N U uc J  1 )( 0 1 = ∑ nix  J   ? =− ixix  i j  ? =− ixix  I    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: dxd  Di  A AB Ax  ρ  −=  which is shown below to require constant density  ρ   (  ρ  =  ρ  A  +  ρ  B ). More generally, dxdw Di  A AB Ax  ρ  −=  which will be shown below not to require constancy of  ρ .  D  AB  =  D  BA  is the molecular diffusivity in the
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