# Mass balance for diffusion and analogy to heat conduction

• gfd43tg
In summary, the conversation discusses using analogies from heat transfer to solve problems in mass transfer. The process involves using energy balances, Fourier's law, and other equations to derive a solution for mass transfer. The conversation also touches on the issue of finding appropriate boundary conditions for the problem.
gfd43tg
Gold Member
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?

Last edited:
ρ=ρ(0) at x = 0
ρ=ρ(L) at x = L

## What is mass balance for diffusion and how is it related to heat conduction?

Mass balance for diffusion is a fundamental concept in science that describes the movement of particles from an area of high concentration to an area of low concentration. This process is similar to heat conduction, where heat moves from a warmer object to a cooler object until thermal equilibrium is reached. Both diffusion and heat conduction are governed by the laws of thermodynamics.

## What is Fick's law and how does it apply to mass balance for diffusion?

Fick's law is a mathematical equation that describes the rate of diffusion of a substance. It states that the flux of a substance is directly proportional to the concentration gradient and inversely proportional to the distance over which the substance is diffusing. In other words, the greater the concentration gradient, the faster the substance will diffuse.

## How does mass balance for diffusion differ from other types of mass transfer?

Mass balance for diffusion differs from other types of mass transfer, such as convection and advection, in that it does not involve the physical movement of a substance. Diffusion occurs due to the random motion of particles and does not require a bulk flow of fluid. In contrast, convection and advection involve the movement of a substance due to a bulk flow of fluid.

## What factors affect the rate of diffusion?

The rate of diffusion is affected by several factors, including the concentration gradient, the surface area for diffusion, the distance over which diffusion occurs, and the properties of the diffusing substance (such as size and solubility). Temperature can also affect the rate of diffusion, as higher temperatures lead to increased molecular motion and faster diffusion.

## How is mass balance for diffusion applied in real-world scenarios?

Mass balance for diffusion is a fundamental concept that is applied in various fields, including chemistry, biology, and engineering. It is used to understand and predict the movement of substances in different systems, such as the diffusion of gases in the atmosphere, the transfer of nutrients in biological systems, and the diffusion of pollutants in the environment. It is also crucial in the design and optimization of industrial processes, such as in the production of pharmaceuticals or the purification of water.

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