Pipe flow with precipitation at boundary

[Sizhe]

Homework Statement

Consider a plug flow (slippery walls) with a uniform velocity $U$ in a circular pipe of diameter,$d$ and length $L$. The fluid is a supersaturated solution with a concentration $c$. At the pipe wall, precipitation of solute happens at a net reaction rate per unit area of $R'=k(c-c^{sat})$. rate constant $k$ and $c^{sat}$ are independent of time.

Homework Equations

(1) Determine an expression for the solute concentration as a function of axial and radial position. The precipitated solute does not influence the slipperiness of the wall and bulk flow. And the diffusion layer is thin relative to diameter.
(2) determine an expression for the thickness of precipitation layer as a function of axial position $x$, if the solid density is $\rho$.
Hint: the governing equations of this problem bear similarities with a heat transfer problem for which analytical solutions are readily available. And all the transfer coefficients are known.

The Attempt at a Solution

In a cylindrical coordinate system, the mass conservation of solute in bulk at equilibrium state is (neglecting axial diffusion):
$$
U\frac{\partial c}{\partial x} = \frac{D_{salt}}{r}\frac{\partial}{\partial r}(r\frac{\partial c}{\partial r})
$$
with the B.C:
$$
-D_{salt}\frac{\partial c}{\partial r}=k(c-c^{sat}), @ r=d/2
$$
and $\frac{\partial c}{\partial r}=0 @ r=0$. With this governing equation, I think the analytical solution for the forced convection problem can be used because this problem is similar to the convection heat transfer with constant surface heat flux. However, I suspect that the constant surface flux condition might not hold as the solute flux at the surface depends on the precipitation. Any idea of using the similarity between the heat and mass transfer in this case?

 

[Chestermiller]

This is indeed analogous to a heat transfer problem. If you let t = x/U, then the equations reduce to those for transient cooling of a cylinder, with a convective heat transfer coefficient to the surroundings at its surface. The rate constant k is basically the heat transfer coefficient. The analytic solution to this problem is presented in Carslaw and Jaeger, Conduction of Heat in Solids, and in Heat Transmission by McAdams.

The only puzzling part of the problem statement relates to the condition that the diffusion layer is thin compared to the diameter. This suggests that you are supposed to solve for the boundary layer approximation at short times (allowing you to neglect the curvature). However, I don't know of a boundary layer solution to this problem with the given boundary condition (although there are boundary layer solutions both to the constant wall temperature case and the constant wall heat flux case). So I don't see how they can ask you for this thin diffusion layer approximate solution.