$12$B$\mathrm{.}4.$ Heat transfer from a wall to a falling film $($short contact time limit$){?}^2($Fig$.12$B$\mathrm{.}4).$ A cold liq$-$

uid film flowing down a vertical solid wall, as shown in the figure, has a considerable cooling

effect on the solid surface. Estimate the rate of heat transfer from the wall to the fluid for such

short contact times that the fluid temperature changes appreciably only in the immediate

vicinity of the wall.

$($a$)$ Show that the velocity distribution in the falling film, given in $2\mathrm{.}2,$ may be written as

$v_z=v_{z,\text{m}\text{a}\text{x}}[2(\frac{y}{})-(\frac{y}{}{)}^2],$ in which $v_{z,\text{m}\text{a}\text{x}}=g\frac{{}^2}{2}.$ Then show that in the vicinity of the

wall the velocity is a linear function of $y$ given by

$v_z$~~$\frac{g}{}y$

Fig. $12$B$\mathrm{.}4.$ Heat transfer to a film

falling down a vertical wall.$($b$)$ Show that the energy equation for this situation reduces to

$hat(C{)}_p{v}_2\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{z}}=k\frac{de{l}^{2}T}{del{y}^2}$

List all the simplifying assumptions needed to get this result. Combine the preceding two

equations to obtain

$y\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{z}}=\frac{de{l}^{2}T}{del{y}^2}$

in which $=\frac{k}{{}^2}$hat$(C{)}_p.$

$($c$)$ Show that for short contact times we may write as boundary conditions

B$.$C$.1$ :

$T=T_0$ for $z=0$ and $y>0$

$T=T_0$ for $y=$ and $z$ finite

$T=T_1$ for $y=0$ and $z>0$

B$.$C$.2$:

B$.$C$.3$ :

Note that the true boundary condition at $y=$ is replaced by a fictitious boundary condition

at $y=.$ This is possible because the heat is penetrating just a very short distance into the

fluid.

$($d$)$ Use the dimensionless variables $()=\frac{T-T_0}{T_1-T_0}$ and $=\frac{y}{\sqrt[3]{9z}}$ to rewrite

the differential equation as $($see Eq$.$ C$\mathrm{.}1-9)$:

$\frac{d^2}{d{}^2}+3{}^2\frac{d}{d}=0$

Show that the boundary conditions are $=0$ for $=$ and $=1$ at $=0.$

$($e$)$ In Eq$.12$B$\mathrm{.}4-7,$ set $d\frac{}{d}=p$ and obtain an equation for $p().$ Solve that equation to get

$d\frac{}{d}=p()=C_1$exp$(-{}^3).$ Show that a second integration and application of the bound$-$

ary conditions give

$=\frac{{\displaystyle \underset{}{\overset{}{}}}\text{e}\text{x}\text{p}(\frac{-}{b}ar({)}^3)d(\frac{?}{\text{b}\text{a}\text{r}}())}{{\displaystyle \underset{0}{\overset{}{}}}\text{e}\text{x}\text{p}(\frac{-}{b}ar({)}^3)d(\frac{?}{\text{b}\text{a}\text{r}}())}=\frac{1}{(\frac{4}{3})}{\displaystyle \underset{}{\overset{}{}}}$exp$(\frac{-}{b}ar({)}^3)\frac{d}{b}ar()$

$($f$)$ Show that the average heat flux to the fluid is

$q_{\text{a}\text{v}\text{g}}{|}_y=0=\frac{3}{2}\frac{(9L{)}^{-\frac{1}{3}}}{(\frac{4}{3})}k(T_1-T_0)$

where use is made of the Leibniz formula in $C\mathrm{.}3.$Heat transfer in a falling non$-$Newtonian film. Repeat Problem $12$B$\mathrm{.}4$ for a polymeric fluid

that is reasonably well described by the power law model of Eq$.8\mathrm{.}3-3.$