10.73. One method to assess axial dispersion in packed beds is to inject a small amount...

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10.73. One method to assess axial dispersion in packed beds is to inject a small amount of solute upstream, No, and measure its distribution down- stream C(x, t). Thus, for plug flow with interstitial velocity u, the material balance can be shown to be ax + at Da a CA x2 where D, denotes axial dispersion coefficient. A simpler equation re- sults by introducing a coordinate moving with the fluid: z = x - vt. (a) Show that the change of coordinate gives a form of Fick's second law at = Da a CA az (b) A new particular solution (to some unspecified problem) can be obtained by differentiating the error function solution for step input; show that this gives K CA = exp - 4Dat 4Dat We shall use this solution to describe response to a pulse of solute. (c) Find the constant K by noting that No must be conserved at any time, t0 to get' CA(z, t) = A No 1 4Dat exp 4Dat where A, is flow area, defined as total area times porosity. 10.73. One method to assess axial dispersion in packed beds is to inject a small amount of solute upstream, No, and measure its distribution down- stream C(x, t). Thus, for plug flow with interstitial velocity u, the material balance can be shown to be ax + at Da a CA x2 where D, denotes axial dispersion coefficient. A simpler equation re- sults by introducing a coordinate moving with the fluid: z = x - vt. (a) Show that the change of coordinate gives a form of Fick's second law at = Da a CA az (b) A new particular solution (to some unspecified problem) can be obtained by differentiating the error function solution for step input; show that this gives K CA = exp - 4Dat 4Dat We shall use this solution to describe response to a pulse of solute. (c) Find the constant K by noting that No must be conserved at any time, t0 to get' CA(z, t) = A No 1 4Dat exp 4Dat where A, is flow area, defined as total area times porosity.