Problem $3(40$ pts$)$ Flow through connective tissues in the body, such as cartilage, the renal

glomerulus, and the interstitium, occurs through complicated flow passages that are difficult

to characterize. One approach is to treat these tissues as porous media, and to apply Darcy's

law, which relates the pressure drop $(P)$ across a porous medium to its permeability $(K)$ :

$P=\frac{QL}{KA}$

Where $$ is the fluid viscosity, $Q$ is the flow rate, $L$ is the length of the porous medium in the

flow$-$wise direction, and A is the cross$-$sectional area of the medium $($solid plus void$)$ facing

the flow. A simple model of a porous medium is to treat it as a number of long tortuous pores

that pass through the medium. Let the medium be characterized as having $n$ pores per unit

cross$-$sectional area. Let the pores have an average radius of a$,$ and an average length of $L,$

where $$ is the tortuosity of the pore path. $($You may ignore all inertial effects.$)$

$($a$)$ Find the porosity $($the void volume per total volume$)$ and the specific surface area $($the

surface area of the pore walls per unit volume$)$ of this medium as functions of $n,$ a$,$

and $$ ONLY.

$($b$)$ Find the permeability as a function of the porosity, the specific surface, and the

tortuosity of the medium ONLY. This relationship is known as the Carman$-$Kozeny

equation.

$($c$)$ Typical extracellular matrices in the body have a permeability of $2{10}^{-14}c{m}^2$ and a

porosity of $80\%($or higher$).$ Calculate a typical pore diameter of such a medium.

Assume a tortuosity of $1\mathrm{.}5.$