Briefly outline the principles underpinning the penetration theory of mass trans

Question

Briefly outline the principles underpinning the penetration theory of mass transfer. The mathematical model for penetration theory relies on the solution of Kick's second law: partialderivative c/partialderivative t = D partialderivative^2 c/partialderivative x^2 subject to the following boundary conditions (note that c_1 is a constant value, and represents an interfacial concentration). c(x, t = 0) = 0 x (x = 0, t > 0) = c, c (x rightarrow infinity, t > 0) = 0 Define all of the symbols in the equation for Fick's second law. Explain whether the boundary conditions correspond to the case of semi-infinite diffusion. Using the method of Laplace transformation, show that the diffusion problem solves to give: c = c [1 - erf (x/2 squareroot Dot)] You will need the inverse transformation: L^-1 {1/s exp (-k squareroot s)} = 1 - erf (k/w squareroot t) where k is a constant (independent of time), s is the Laplace variable and erf(z) represents the error function of the variable z. Given that the interfacial flux to the surface (j) is: j = -00D (partialderivative c/partialderivative x)_x =0 and using the standard result: d/dx erf (x)] = 2/squareroot pi exp (-x^2) show that the instantaneous mass transfer coefficient (k_1) is given by: k_L = squareroot D/pi t

Explanation / Answer

In most industrial processes of mass tranfe is unsready state processthat means the contact time between phases is too short and very difficult to achive stationary state. But this theory is not taken by the film model. In the absorption of gas from bubbles or by wetted wall columns, the mass transfer surface id formed instantaneously and transient diffusion of material takes place.
There are some assumption in the penetration theory:-
1) Unsteady state masss transfer occurs to liquid only when it incontact with the surface or other pase
2) Equilibrium exists at gas liquid interface.
3) each of the element stays incontact with the gas for sme period of time.

Under these circumstances, the convective terms in the diffusion can be
neglected and the unsteady state mass transfer of gas (penetration) to the liquid
element can be written as :-

c/t = D(a->b) ^2/Z^2

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