As a process engineer you have been asked to design the agitation system for a culture of Aspergillus niger within a stirred tank bioreactor. It has been observed that in filamentous fungal cultures the rheological parameters of the medium, such as the consistency index $($K$)$ and the flow behavior index $($n$),$ are usually dependent on the biomass concentration according to one of the following expressions: $K=a{X}^b$

$n=d{X}^e$

$K=\frac{K_0{e}^{cx}}{1-\frac{K_0}{K_f}(1-e^{cx})}$

$n=n_f+\frac{1-n_f}{1+d{x}^e}$ where a$,$ c and d are empirical parameters, K$0$ and Kf are the initial and final values $$of the consistency index and nf is the final value of flow behavior.

Considering that the density of the medium is $1000$ kg$/$m$3,$ at a cell density of $20$ g$/$L and the information in tables $1$ and $2,$ answer the following: The tables are the following:

a$)$ Fit the experimental data to equations $(1-4)$ and select the one that best describes the rheological behavior based on its regression coefficient $($R$2)$ or the sum of least squares.