$4-5.$ Transient Diffusion in a Permeable Tube with Open Ends

The objective is to examine in more detail the effects of the capillary ends in the permeabil$-$

ity experiment described in Example $3\mathrm{.}5-2.$ In that experiment $($Fig$.3-16)$ an open$-$ended cylindrical

tube $($isolated kidney capillary$)$ of radius a and length $2L$ was immersed initially in a fluid with solute

concentration $C_0.$ At $t=0$ the concentration in the surrounding fluid was changed rapidly to a new

constant value, $k_{m}C(z,t)\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{t}}=D\frac{de{l}^{2}C}{del{z}^2}-\frac{2k_m}{R}(C-C_1).t\frac{L^2}{D}z(t)=\frac{C-C_1}{C_0-C_1}=e^{-2k_m\frac{t}{R}}.t\frac{L^2}{D}z=0C(0,t)=C_{1}t>0C_1,$ causing outward diffusion both across the wall and through the open ends.

The membrane permeability $is{k}_m.$

$(a)In$ Example $3\mathrm{.}5-2it$ was found that radial concentration variations within the tube can $be$

neglected. Show that the concentration $in$ the lumen, $C(z,t),is$ governed then $by$

$\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{t}}=D\frac{de{l}^{2}C}{del{z}^2}-\frac{2k_m}{R}(C-C_1).$

$(b)It$ was also pointed out $in$ Example $3\mathrm{.}5-2$ that, for $t\frac{L^2}{D},$ the concentration $in$ the cen$-$

tral part $of$ the tube $(near$ the observation windows$)is$ independent $ofz.$ Show that the dimen$-$

sionless concentration there $is$ given $by$

$(t)=\frac{C-C_1}{C_0-C_1}=e^{-2k_m\frac{t}{R}}.$

Sketch qualitatively the complete concentration profiles for several values $oft,$ including

times both less than and greater than$\frac{L^2}{D}.$

$(c)$ For the early times discussed $in$ part $(b),$ use similarity $to$ obtain a solution that includes posi$-$

tions near the ends. Choose $z=0at$ one end and assume that $C(0,t)=C_1$ for all $t>0.$ For

the similarity method $to$ work, how must the concentration variable $be$ modified?