You want to model interstitial fluid flow and drug delivery in the brain using a cylindrical tissue culture insert system $($see below$).$ You encapsulate neural cells in a porous $3$D hydrogel and place the gels inside the culture inserts. The inserts have a porous membrane at the bottom, allowing you to apply a pressure head on top of the gel and drive flow through the hydrogel. The pressure at the top and bottom of the gel are both assumed to be atmospheric, and you were able to look up the fluid properties, permeability, and diffusion coefficient of your drug of interest. The insert has a diameter of $4cm,$ the gel is $1cm$ thick $($L$),$ and the fluid height $($h$)$ is $9cm.$

Media

Gel

$T=37$ Celsius

$=993k\frac{g}{m^{{?}^{{?}^{?}}}}3$

$=6\mathrm{.}91{10}^{{?}^{{?}^{?}}}(-4)P{a}^{***}s$

$D=6{10}^{{?}^{{?}^{?}}}(-11)m^{{?}^{{?}^{?}}}\frac{2}{s}$

$k=3\mathrm{.}19{10}^{{?}^{{?}^{?}}}(-14)m^{{?}^{{?}^{?}}}2$

a$)$ Determine the pressure at the top of the gel $(P_1).$

b$)$ Determine the volumetric flow rate through the gel.

c$)$ Using the mass transfer coefficient for a packed bed, calculate the flux through the gel if the drug concentration at the top of the gel is $10M$ and the concentration at the bottom of the gel is initially zero. Assume a particle diameter of $1$ micron.