Consider a fin incorporating facilitated transport only at its base. The reaction occurring there is given...

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Consider a fin incorporating facilitated transport only at its base. The reaction occurring there is given by r=-kc (moles/s). The fin is essentially infinitely long, has a diameter, ro, and species "a" diffuses through it with a diffusivity, Da. It is immersed in a fluid containing species "a" at a concentration of Cape (Ca> Ca throughout the fin). The external fluid provides for a mass transfer coefficient of kc. a) Determine the concentration profile within the fin. (Hint: The fin is infinitely long and so the concentration at the tip should be in equilibrium with the surrounding fluid. Also, for a fin that extends to infinity, sinh and cosh blow up. So, you should use the generic trial solution: c = A exp(-Bx) and solve for A and B) b) What is the flux and concentration at the base of the fin? a) The governing equation is: DA -k.P₁(c₁-c) = 0 d²c dx² b) kP k P cd B²³₁ ) x + ( B²³₁) = 0 D.A DA C ab The boundary conditions state: dx = kx -D. dx Reaction at the base and an effective infinite fin. X=1 d²x dx² -- x=0 x = 00 x-²- goo Consider a fin incorporating facilitated transport only at its base. The reaction occurring there is given by r=-kc (moles/s). The fin is essentially infinitely long, has a diameter, ro, and species "a" diffuses through it with a diffusivity, Da. It is immersed in a fluid containing species "a" at a concentration of Cape (Ca> Ca throughout the fin). The external fluid provides for a mass transfer coefficient of kc. a) Determine the concentration profile within the fin. (Hint: The fin is infinitely long and so the concentration at the tip should be in equilibrium with the surrounding fluid. Also, for a fin that extends to infinity, sinh and cosh blow up. So, you should use the generic trial solution: c = A exp(-Bx) and solve for A and B) b) What is the flux and concentration at the base of the fin? a) The governing equation is: DA -k.P₁(c₁-c) = 0 d²c dx² b) kP k P cd B²³₁ ) x + ( B²³₁) = 0 D.A DA C ab The boundary conditions state: dx = kx -D. dx Reaction at the base and an effective infinite fin. X=1 d²x dx² -- x=0 x = 00 x-²- goo