In class, we derived the following unsteady-state differential mass balance on component a where the flux of a (na) was in the b direction. a. starting with a balance on component a within a spherical shell having an incremental thickness ar, derive the corresponding unsteady-state differential mass balance for spherical geometery. a hint is provided on the following page. b. explain the analogy between diffusional mass transfer and heat conduction. include in your discussion the analogy between the biot number for heat tranfer discussed in chapter 10 and the biot number for mass transfer defined in chapter 17 (p. 559) c. describe how figures 10.5 and 10.8 could be used to solve a problem involving diffusion of component a in the r direction from a porous, sphere into the fluid surrounding the sphere
hint on problem 2 of hw #9: a similar shell balance derivation is shown in the topic 9, lesson 2 slides for one-dimensional diffusion in the b direction. in that derivation the cross-sectional area (a) remains the constant with b. a is constant in the direction of diffusion for rectangular geometry and for cylindrical geometry when mass transfer is in the z direction (parallel to the cylinder's axis). however, when diffusion is in the radial (r) direction, a varies in the direction of diffusion. specifically, for cylindrical coordinates when mass transfer occurs is in the r direction, a 27arl, and for spherical coordinates, a = 47r2. the derivation of a shell balance in which a varies withr was demonstrated for the unsteady- state heat conduction (topic 5, lesson 3, parts 2 and 3). because fourier's law of conduction and fick's law of diffusion have similar mathematical forms, the final differential mass balance for spherical coordinates (neglecting convection) should have the same mathematical form as the differential energy balance when n = 2. however, wherever the energy balance has a temperature, the mass balance will have a concentration, and wherever the energy balance has a thermal diffusivity, the mass balance will have a mass diffusivity