How to obtain 3 parameters from a linearized model?

Question

I have the following model:

$$ q_e=\dfrac{K_s\ {C_e}^\beta}{1+a_s\ {C_e}^\beta} $$

Its linearized form is:

$$ \beta\ln(C_e)=-\ln\left(\dfrac{K_s}{q_e}\right)+\ln a_s $$

The known values are $C_e$ and $q_e$. According to the literature, I can find $\beta$, $K_s$ and $a_s$ plotting $\ln\left(\dfrac{K_s}{q_e}\right)\ \rm{vs.}\ ln(C_e)$. However, I have no idea how to find these parameters because $K_s$ is unknown.

The model that I'm trying to use is called "Sips isotherm"

Answer 1

$$ q_e=\dfrac{K_s\ {C_e}^\beta}{1+a_s\ {C_e}^\beta} $$ The fitting of the equation to given data is usually done thanks to non-linear regression. This requires an initial guessing of the parameters and iterative numerical calculus.

The principle of a non-conventional method which doesn't require guessing and no iteration is explained in the paper : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales .

The application to the present function is shown below.

In interest of simpler writing the notations are changed : $$\begin{cases} y=q_e \\ x=C_e \\ K=K_s \\ a=a_s \\ b=\beta \end{cases}$$ $$ y=\frac{K\,x^b}{1+a\,x^b} \tag 1 $$

NUMEICAL EXAMPLE :

NOTE :

The linearization is obtained thanks to an integral equation. The least mean squares regression is carried out with respect to the integral equation which leads to slightly different values of the parameters than a classical non-linear regression with respect to the initial function $(1)$. If least mean squares with respect to $(1)$ is absolutely required, a non-linear regression is necessary. Then one can use the above results instead of initial guess which is much more robust.

FOR INFORMATION :

In the present case the integral equation used is : $$\int \frac{1}{y(x)}dx = \frac{1}{1-b}\frac{x}{y}+\frac{ab}{K(1-b)}x+\text{constant}$$ The above $S_i$ are the approximates of $\int \frac{1}{y}dx$ by numerical integration. Note that for the numerical integration the data has to be ranked in increasing order of $x_i$.

Answer 2

Your model can be rewritten in the form $$y=ax^c+b,$$ where $a,b,c$ are the unknown parameters.

You can solve such a nonlinear model by

• fixing $c$ and solving the corresponding linear problem $y=a(x^c)+b$ by least-squares,

• computing the error residue $e(c)$,

• minimizing the nonlinear univariate function $e(c)$.

Starting from a search interval of length $L$ and stopping the search at absolute accuracy $\epsilon$, the number of iterations using the Golden Section method will be about

$$\log_\phi\frac{L}{\epsilon}$$

(about $33$ iterations for single-precision accuracy).