Consider the diffusion of solute A into the single cylindrical pore shown in the figure below. The

Question:

Consider the diffusion of solute A into the single cylindrical pore shown in the figure below. The end of the pore at z = L is sealed. The pore space is initially filled with inert fluid B. As solute A diffuses into the quiescent fluid space inside the pore space, it adsorbs onto the inner walls of the pore. The €œadsorption isotherm€ of solute A onto the solid surface of the pore is described by the Langmuir equation, given by

where q_A is the amount adsorbed on the surface (moles A/cm² surface area), c_A is the local concentration of solute A right above the surface (moles A/cm³), and K is the equilibrium constant (moles/cm³), and q_A,max is the maximum amount of solute A, which can be adsorbed on the surface (moles A/cm² surface area). At high concentrations where c_A >> K; q_A ‰ˆ q_A;max, and at low concentrations where K >> c_A, the adsorption isotherm becomes linear, so that

q_A ‰ˆ q_A,max/K C_A

a. Think of a specific physical system€”i.e., propose specific materials for solute A, the fluid B, and the solid surface from an outside literature reference. What does the plot of the Langmuir isotherm (q_A vs. c_A) look like for this specific physical system? Develop an algebraic expression that describes the maximum amount of solute A, which can be adsorbed within a single pore.

b. You may now consider that the concentration profiles of solute A is only in the axial direction, not in the radial direction. You may also assume that the process is dilute with respect to solute A, the linear adsorption isotherm is valid, and the rate processes of adsorption are extremely fast. Using the €œshell balance€ approach, develop the differential forms of the general differential equation for mass transfer and Fick€™s flux equation, taking into account the adsorption of solute A onto the surface of pore in the differential mass balance. Then combine the simplified forms of the general differential equation of mass transfer and Fick€™s flux equation to arrive at a single differential equation for the transfer of solute A within the pore in terms of concentration c_A. State all assumptions and boundary/ initial conditions as part of the analysis.

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