This problem is about viscous heating. A Newtonian fluid of constant physical properties $($ and $k)$ flows down a circular pipe. Taking the $z-$axis downwardly along the center of the pipe $($and parallel to the gravity $g)$ and assuming that the flow is incompressible and laminar, we have

$v_z(r)=v_{\text{m}\text{a}\text{x}}[1-(\frac{r}{R}{)}^2],$

where $R$ is the inner radius of the pipe and

$v_{\text{m}\text{a}\text{x}}=\frac{R^2}{4},$

in which

$=g-\frac{dp}{(d)z}$

is a constant. The other components of the velocity are zero.

Suppose that the pipe is insulated and let $T_0$ denote the temperature of the fluid for $z0.$ You are asked to find the temperature profile in the flind for $z>0.$

$($a$)$ List assumptions you would impose on the thermal energy balance equation.

$($b$)$ Set up the partial differential equation for the temperature field.

$($c$)$ Assume that the conductive heat transfer in the $z-$direction is negligible compared to the convective heat transfer in the same direction and let $T(r,z)=f(r)+g(z).$ Find two ordinary differential equations, one for $f(r)$ and the other for $g(z).$

$($d$)$ Find the temperature field involving arbitrary constants.

$($e$)$ Impose boundary conditions at $r=0$ and $R$ and determine some of the constants in $T(r,z).$

$($f$)$ Show that the other boundary condition at $z=0$ cannot be satisfied.

$($g$)$ The macroscopic energy balance written for a control volume $V_f($i$.$e$.,$ volume fixed in space$)$ takes the form

$\frac{d}{dt}{\displaystyle \underset{V_f}{\overset{\phantom{}}{}}}(u+\frac{1}{2}{v}^2+)dV+{\displaystyle \underset{S_f}{\overset{\phantom{}}{}}}(h+\frac{1}{2}{v}^2+)v_{*}{n}_{d}S$

$=-{\displaystyle \underset{S_f}{\overset{\phantom{}}{}}}{n}_{*}{q}_{d}S+{\displaystyle \underset{V_f}{\overset{\phantom{}}{}}}RdV$

where $h(x_{,}t)$ is the enthalpy per unit mass of the fluid at point $x_{?}$ at time $t.$ Applying this equation to the section of the pipe between $z=0$ and $z,$ show that

${\displaystyle \underset{0}{\overset{R}{}}}\{hat(c{)}_p[T(r,z)-T_0]-z\}{v}_{z}rdr=0.$

Hint: From thermodynamics, the specific enthalpy hat$(h)(T,)$ is known to satisfy$*$

dhat$(h)=$hat$(c{)}_{p}dT+\frac{1}{}dp$

for a fluid of fixed composition.

$($h$)$ Substituting the temperature profile from part $($e$)$ into Eq$.(2),$ find two equations that can be used to determine the constants in $T(r,z).$ Are they consistent with your result from part $($e$)?$

$($i$)$ Express the temperature gradient del$\frac{T}{d}$elz in terms of $,,$hat$(c{)}_p,R,$ and $Re,$ where $Re$ is defined by

$Re$:$=\frac{2R(\text{:}{v}_z\text{:})}{}$

with

$($:$v_z$:$)$:$=\frac{1}{R^2}{\displaystyle \underset{0}{\overset{R}{}}}{v}_z(r)2rdr.$

$($j$)$ Evaluate the temperature rise over a distance of $10m$ under the following conditions. ${?}^{}$

i$.$ Water at room temperature

$Re={10}^3$

$\frac{}{}={10}^{-2}c\frac{m^2}{s}$

$R=0\mathrm{.}5cm$

hat$(c{)}_p=1ca\frac{l}{g}*K$

ii$.$ Petroleum oil at room temperature

$Re={10}^3$

$\frac{}{}=10c\frac{m^2}{s}$

$R=0\mathrm{.}5cm$

hat$(c{)}_p=0\mathrm{.}5ca\frac{l}{g}*K$

iii. The same as ii above, but with $R=1\mathrm{.}5cm.$