The diffusionequation is a partial differentialequationwhich describes density fluc- tuations in a material undergoing diffusion. The equation can be written as: ∂u(r,t) ∂t =∇·. D(u(r,t),r)∇u(r,t) , (7.1) where u(r,t)is the density of the diffusing material at location r =(x,y,z) and time t.
How do you find the Collective diffusion coefficient?
The equation is usually written as: where ϕ(r, t) is the density of the diffusing material at location r and time t and D(ϕ, r) is the collective diffusion coefficient for density ϕ at location r; and ∇ represents the vector differential operator del.
What is the Maxwell-Stefan description of diffusion?
The necessary equations are formulated as the Maxwell-Stefan description of diffusion; they are often applied to describe gas mixtures, such as syngas in a reactor or the mix of oxygen, nitrogen, and water in a fuel cell cathode.
Is diffusion equation linear or nonlinear?
If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. The equation above applies when the diffusion coefficient is isotropic; in the case of anisotropic diffusion, D is a symmetric positive definite matrix, and the equation is written (for three dimensional diffusion) as:
Is the diffusion equation a nonlinear partial differential?
We found that the diffusivity of diffusion equation depends generally on the concentration of diffusion particles. In that case, the diffusion equation becomes a nonlinear partial differential equation, and the mathematical solution is almost impossible, even if it is a case of the time and one dimension space coordinate.
How do you solve the diffusion equation with zero gradient boundary conditions?
The solution for v(x,t) is the solution to the diffusion equation with zero gradient boundary conditions. This solution is an infinite series in the cosine of n x/L, which was given in equation [63]. [76] The solution for u(x,t) = v(x,t) + w(x) + (t) is then found by combining equations [73] and [76]. [77]
Is the diffusion equation continuous in space and time?
The diffusion equation is continuous in both space and time. One may discretize space, time, or both space and time, which arise in application. Discretizing time alone just corresponds to taking time slices of the continuous system, and no new phenomena arise.
