Question

Define the penetration depth for mass transfer, and explain how it can be determined at a specified time when the diffusion coefficient is known.

Short answer

Answer: To calculate the penetration depth at a specified time, first choose a threshold concentration value (e.g., 95% of the initial concentration). Then, use the error function (erf) solution to the diffusion equation and solve for the distance (x) where the concentration profile reaches the threshold value. The penetration depth (L) can be calculated using the formula: L = 2 * sqrt(D * t) * erf_inv(1 - C_threshold / C0), where D is the diffusion coefficient, t is the specified time, erf_inv is the inverse error function, and C_threshold is the threshold concentration.

Step-by-step solution

Define penetration depth for mass transfer

Penetration depth is a measure of the distance within a medium where an external influence, such as a concentration gradient, leads to significant mass transfer phenomena. In other words, it is the distance over which the concentration of a diffusing species changes significantly due to mass transfer.

Understanding the diffusion process

Mass transfer in a medium depends on the concentration gradient of the diffusing species and its diffusion coefficient. The diffusion coefficient, denoted by D, is a measure of how quickly the species can diffuse through the medium. The higher the diffusion coefficient, the faster the species can diffuse within the medium.

The Fick's Law

Fick's Law describes the relationship between the mass transfer rate and the concentration gradient of the diffusing species. In one-dimensional case, it is given by: Flux (J) = -D * (dC/dx) Where J is the mass flux, D is the diffusion coefficient, dC/dx is the concentration gradient of the species.

Determine the penetration depth

The penetration depth can be determined by analyzing the concentration profile of the diffusing species in the medium. The concentration profile is typically described by diffusion equations, which are derived from Fick's law. For this exercise, we will assume that the concentration profile of the species within the medium can be modeled by the error function (erf) solution to the diffusion equation. The erf solution is given by: C(x, t) = C0 * (1 - erf(x / (2 * sqrt(D * t)))) Where C(x, t) is the concentration at a distance x from the interface at time t, C0 is the initial concentration of the species, and erf is the error function.

Calculate the penetration depth at a specified time

To determine the penetration depth (L) at a specified time (t), we need to identify a threshold concentration value, typically chosen as the concentration level at which the diffusing species has penetrated to a significant level, for example, 95% of the initial concentration (C0 * 0.95). Then, we need to find the value of x (distance from the interface), where the concentration profile C(x, t) reaches the threshold value. We can find the penetration depth by solving the erf equation for x: L = 2 * sqrt(D * t) * erf_inv(1 - C_threshold / C0) Where L is the penetration depth, D is the diffusion coefficient, t is the specified time, erf_inv is the inverse error function, and C_threshold is the threshold concentration (e.g., 0.95 * C0). Now given the diffusion coefficient and time, the penetration depth can be calculated using the above formula.