Thread: Problem finding explicit formulation of series coefficient

1. Problem finding explicit formulation of series coefficient

Hi,

I am currently working on gas diffusion through membranes and I am having issues finding a coefficient for the ODE's solution, based on the method presented here.

My problem concerns the coefficient lambda, appearing after developping Fick's law, given by eq.16 :

tan(λn) = (k*SR − δRλnn + SRβn)) / (SRδRλn + βn(k*SR − δRλn2)

where k = 1,
SR remains a symbolic variable,
δR is a known constant,
βn is a function of λnR,SR and k

To perform a curve fit with my experimental data, I need to find an explicit formulation of λn. In practice, I can consider only the first term of the summation (λ1) for sufficient accuracy.

How can I express explicitly λ1? The tan() function is giving me a headache

Thank you,

edit : I would like to do something similar to the transformation in eqs 11-12 in this paper

2. Re: Problem finding explicit formulation of series coefficient

If $\beta_1$ is an unspecified function of $\lambda_1$ I don't see how you have any hope of even approximately solving for $\lambda_1$

Do you have bounds on $\lambda_1$ ?

On $\left(-\dfrac {\pi}{4}, \dfrac {\pi}{4}\right), \tan(x) \approx x$

3. Re: Problem finding explicit formulation of series coefficient

Hi,

β is a known function, sorry for the misunderstanding.

β = (δR * λn2 - k SR) / (SR * λn * δR)

Could you give me hints on the method applied in the second paper (Aachib et al.) to transform the alpha parameter from eq. 11 to eq. 12? It would work for my purposes, but I don't see what the intermediate steps are ...

Thank you,