Question

Write three boundary conditions for mass transfer (on a mass basis) for species \(\mathrm{A}\) at \(x=0\) that correspond to specified temperature, specified heat flux, and convection boundary conditions in heat transfer.

Short answer

Answer: The three boundary conditions are as follows: 1. Specified Temperature: \(C_A(x=0) = C_{A_0}\) 2. Specified Heat Flux: \(\cfrac{d C_A}{d x}(x=0) = -\cfrac{q_0}{D}\) 3. Convection Boundary Condition: \(\cfrac{d C_A}{d x}(x=0) = -\cfrac{k_c}{D}(C_{A_{\infty}} - C_A(x=0))\)

Step-by-step solution

Boundary Condition 1: Specified Temperature

For this boundary condition, the temperature at x=0 is specified, so we will need to relate this temperature to the concentration of species A. We can assume that the concentration of species A is directly proportional to the specified temperature. At x=0, let the specified temperature be \(T_0\) and the concentration of species A be \(C_{A_0}\). The boundary condition can be written as: $$C_A(x=0) = C_{A_0}$$

Boundary Condition 2: Specified Heat Flux

For a specified heat flux boundary condition, the heat flux at x=0 is given, which can be related to mass flux using Fick's law of diffusion. Let the specified heat flux at x=0 be \(q_0\). Fick's law of diffusion states that the mass flux \(\mathrm{J_A}\) is related to the concentration gradient of species A: $$\mathrm{J_A} = -D\cfrac{d C_A}{d x}$$ where D is the diffusion coefficient. Since the heat and mass fluxes are assumed to be proportional, at x=0, we can write the boundary condition as: $$\cfrac{d C_A}{d x}(x=0) = -\cfrac{q_0}{D}$$

Boundary Condition 3: Convection Boundary Condition

For a convection boundary condition, a convective mass transfer is taking place at x=0. The boundary condition can be formulated using Newton's law of cooling, stating that the convective mass transfer flux is related to the concentration difference between the boundary concentration and the concentration of species \(\mathrm{A}\) at x=0. Let the surrounding concentration be \(C_{A_\infty}\), the mass transfer coefficient be \(k_c\), and the convective mass transfer flux be \(\mathrm{J_{conv}}\). Newton's law states that: $$\mathrm{J_{conv}} = k_c(C_{A_{\infty}} - C_A(x=0))$$ Assuming the diffusive and convective mass transfer processes are coupled, we can again use Fick's law to write the convection boundary condition as: $$\cfrac{d C_A}{d x}(x=0) = -\cfrac{k_c}{D}(C_{A_{\infty}} - C_A(x=0))$$