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gfd43tg

Gold Member

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Hello,

I just began learning mass transfer, and I am trying to use analogies from heat transfer to help me solve problems. For example, if you have one dimensional heat transfer through a plane wall, I would start with a general energy balance.

$$\frac {dE}{dt} = \dot Q_{x} - \dot Q_{x + \Delta x} + \dot e_{gen}A \Delta x$$

$$ \rho A \Delta x \hat c_{p} \frac {dT}{dt} = \dot Q_{x} - \dot Q_{x + \Delta x} + \dot e_{gen}A \Delta x$$

dividing through and taking the limit as x approaches 0,

$$\rho \hat c_{p} \frac {dT}{dt} = - \frac {1}{A} \frac {d \dot Q}{dx} + \dot e_{gen}$$

Then using Fourier's law ##\dot Q = -kA \frac {dT}{dx}## I substitute and get

$$ \rho \hat c_{p} \frac {dT}{dt} = \frac {d}{dx} (k \frac {dT}{dx}) + \dot e_{gen}$$

Now I want to extend this to mass transfer, and I find my textbook to be sorely lacking in even properly showing a derivation from the beginning, so I try myself

$$ \frac {dm}{dt} = \dot m_{x} - \dot m_{x + \Delta x} + \dot m_{gen} A \Delta x$$

$$ A \Delta x \frac {d \rho}{dt} = \dot m_{x} - \dot m_{x + \Delta x} + \dot m_{gen} A \Delta x$$

Diving through and the limit as delta x approaches 0

$$ \frac {d \rho}{dt} = - \frac {1}{A} \frac {d}{dx} (-D_{AB}A \frac {d \rho}{dx}) + \dot m_{gen} $$

Now I can remove the generation term to get

$$ \frac {d \rho}{dt} = \frac {d}{dx} (D_{AB} \frac {d \rho}{dx})$$

Assume steady state, so then I integrate once

$$ D_{AB} \frac {d \rho}{dx} = C_{1} $$

$$ D_{AB} \rho = C_{1}x + C_{2} $$

Now what boundary conditions are appropriate to use?

I just began learning mass transfer, and I am trying to use analogies from heat transfer to help me solve problems. For example, if you have one dimensional heat transfer through a plane wall, I would start with a general energy balance.

$$\frac {dE}{dt} = \dot Q_{x} - \dot Q_{x + \Delta x} + \dot e_{gen}A \Delta x$$

$$ \rho A \Delta x \hat c_{p} \frac {dT}{dt} = \dot Q_{x} - \dot Q_{x + \Delta x} + \dot e_{gen}A \Delta x$$

dividing through and taking the limit as x approaches 0,

$$\rho \hat c_{p} \frac {dT}{dt} = - \frac {1}{A} \frac {d \dot Q}{dx} + \dot e_{gen}$$

Then using Fourier's law ##\dot Q = -kA \frac {dT}{dx}## I substitute and get

$$ \rho \hat c_{p} \frac {dT}{dt} = \frac {d}{dx} (k \frac {dT}{dx}) + \dot e_{gen}$$

Now I want to extend this to mass transfer, and I find my textbook to be sorely lacking in even properly showing a derivation from the beginning, so I try myself

$$ \frac {dm}{dt} = \dot m_{x} - \dot m_{x + \Delta x} + \dot m_{gen} A \Delta x$$

$$ A \Delta x \frac {d \rho}{dt} = \dot m_{x} - \dot m_{x + \Delta x} + \dot m_{gen} A \Delta x$$

Diving through and the limit as delta x approaches 0

$$ \frac {d \rho}{dt} = - \frac {1}{A} \frac {d}{dx} (-D_{AB}A \frac {d \rho}{dx}) + \dot m_{gen} $$

Now I can remove the generation term to get

$$ \frac {d \rho}{dt} = \frac {d}{dx} (D_{AB} \frac {d \rho}{dx})$$

Assume steady state, so then I integrate once

$$ D_{AB} \frac {d \rho}{dx} = C_{1} $$

$$ D_{AB} \rho = C_{1}x + C_{2} $$

Now what boundary conditions are appropriate to use?

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