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Chapter 14: Problem 26

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

Short Answer

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Q: Write three boundary conditions for mass transfer for species A at x=0 that correspond to specified temperature, specified heat flux, and convection boundary conditions in heat transfer. A: The three boundary conditions for mass transfer for species A at x=0 that correspond to specified temperature, specified heat flux, and convection boundary conditions in heat transfer are: 1. Specified concentration: \(C_A(x=0)=C_{A0}\) 2. Specified mass flux: \(j_A(x=0)=j_{A0}\) 3. Convective mass transfer: \(j_A(x=0) = k_c(C_{Af}-C_{A 0})\)

Step by step solution

01

Understanding the Mass Transfer Analogies for Heat Transfer Boundary Conditions

We will first understand the mass transfer analogies for the different heat transfer boundary conditions: - Specified temperature: This is analogous to a specified concentration condition in mass transfer, where the concentration of species \(A\) at \(x=0\) is specified. - Specified heat flux: This corresponds to a specified mass flux condition in mass transfer, where the mass flux of species \(A\) at \(x=0\) is specified. - Convection boundary condition: This corresponds to a convective mass transfer boundary condition, where the mass transfer between the boundary (\(x=0\)) and the fluid is dictated by a convective mass transfer coefficient and the concentration difference between the surface and the fluid.
02

Boundary condition for a specified concentration

When the boundary condition is given for specified temperature, the mass transfer equivalent is a specified concentration at \(x=0\). The boundary condition can be written as: $$ C_A(x=0)=C_{A0} $$ Where \(C_A\) is the concentration of species \(A\) and \(C_{A0}\) is the specified concentration at \(x=0\).
03

Boundary condition for a specified mass flux

When the boundary condition is given for specified heat flux, the mass transfer equivalent is a specified mass flux at \(x=0\). The boundary condition can be written as: $$ j_A(x=0)=j_{A0} $$ Where \(j_A\) is the mass flux of species \(A\) (in \(\text{kg m}^{-2} \text{s}^{-1}\)) and \(j_{A0}\) is the specified mass flux at \(x=0\).
04

Boundary condition for convective mass transfer

When the boundary condition is given for convection heat transfer, mass transfer equivalent is a convective mass transfer boundary condition. The boundary condition can be written as: $$ j_A(x=0) = k_c(C_{Af}-C_{A 0}) $$ Where \(k_c\) is the mass transfer coefficient at \(x=0\) (in \(\text{kg m}^{-2} \text{s}^{-1}\)), \(C_{Af}\) is the concentration of species \(A\) in the fluid, and \(C_{A0}\) is the concentration of species \(A\) at the surface \(x=0\).

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