A spherical particle of coal is slowly being oxidized in a stream of pure oxygen (O2) gas. The solid carbon within the coal particle oxidizes to carbon dioxide (CO2) gas as shown: C(s) + O2(g)-C02(9) The coal particle contains a significant amount of noncombustible ash that leaves behind a highly porous layer. The entire diameter of the particle remains constant, but the unreacted inner core of the solid, nonporous coal shrinks slowly with time. The oxidation reaction is very rapid so the transport of Oz and CO2 gas is controlled by diffusion through the porous layer. Convective mass transfer resistances associated with the flow of O, gas around the particle can be neglected so the concentration of Oz gas at the outer surface of the particle is essentially 100% O2. The temperature and pressure of the gas inside and outside the porous particle are constant.
A) Draw a picture of the physical system and state at least five reasonable assumptions for the mass-transfer aspects of the process. What coordinate system should be used?
B) What is the simplified differential form of Fick's equation for Oz gas (species A)?
C) What is the simplified form of the general differential equation for mass transfer in terms of flux NA? What term is "constant" over the diffusion path?
D) What is the general differential equation for mass transfer in terms of concentration C/? Propose two boundary conditions that may be used to solve the resulting differential equation
E) Parts (b) and (d) consider the gas space in the porous ash layer as the "system" for mass transfer. Now, consider the unsteady-state material balance on the shrinking non-porous carbon core itself, and relate the rate of shrinkage of the flux N . Leave the equation in differential form with respect to time t, but propose reasonable initial and final conditions. Processes of this time are called "pseudo steady state" because the flux is considered steady state whereas the depletion of the source for mass transfer is an unsteady-state process.