# Using orthogonal collocation method solve PDE of adsorption chromatography model

Chromatographic equation with diffusion：

\frac{\partial c}{\partial t} + \frac{1 - \epsilon}{\epsilon} \frac{\partial q}{\partial t} + u \frac{\partial c}{\partial x} = D \frac{\partial^2 c}{\partial x^2}
\frac{\partial q}{\partial t} = - k_f \left( q - \frac{q_m b c}{1 + b c} \right)

Initial conditions:

t = 0, c(x, 0) = q(x, 0) = 0, x > 0


Boundary conditions:

x = 0,

D \frac{\partial c}{\partial x} = u*(c - c0)

x = L,

D \frac{\partial c}{\partial x} = 0

Parameters:

L = 20       # Length of bed ,cm
ϵ = 0.4436   # bed porosity
c0 = 8.8     # Initial concentration ,mg/ml
qm = 76.85   # maximum adsorption capacity ,mg/ml
b = 0.0258   # Langmuir isotherm constant
u = 0.0424   # Velocity ,cm/s
k_f = 0.89   # mass transfer coefficient ,s-1
D = 2.69e-4  # Axial diffusion coefficient ,cm2/s


Discretize the spatial variables using the finite element orthogonal collection method, and divide the column into 30 finite elements, each with 2 configuration points, and solve the pde.

There are some tutorials using Matlab to solve the similar questions:

Solved Multicomponent PDE-ODE System Using MATLAB - Adsorption

The Simple Breakthrough Curve Adsorption using One PDE with Intrinsic Adsorption Coefficient

I’m not sure whether the method they used is orthogonal collection.

Somebody kindly help me out this problem?

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