Question

The wall of a constant diameter pipe of length $1m$ is heated uniformly with flux $q^{\prime \prime }$by wrapping a heater coil around it. The flow at the inlet to the pipe is hydrodynamically fully developed. The fluid is incompressible and the flow is assumed to be laminar and steady all through the pipe. The bulk temperature of the fluid is equal to $0^{\circ }C$ at the inlet and ${50}^{\circ }C$ at the exit. The wall temperatures are measured at three locations, $P,Q$ and $R,$ as shown in the figure .The flow thermally develops after some distance from the inlet. The following measurements are made:

Among the locations $P,Q$ and $R$ the flow is thermally developed at

• A.
  $P,Q$ and $R$
• B.
  $Q$ and $R$ only
• C.
  $R$ only
• D.
  $P$ and $Q$ only

Answer 1

The correct option is B

$Q$ and $R$ only
Internal forced convection
Boundary condition: Constant Heat Flux Case
$T_m$(Bulk mean temperature of fluid)
At inlet, $T_{m.i}=0^{\circ }C$
At outlet, $T_{m.\theta }={50}^{\circ }C$
Variation of Bulk Mean Temperature $T_m$
$T_m=T_{m.i}+\frac{q^{\prime \prime }P}{\dot{m}{c}_P}x...$ linear with ${}^{\prime }{x}^{\prime }$
$x=1m;T_{m.\theta }={50}^{\circ }C$
${50}^{\circ }C=0^{\circ }C+\frac{q^{\prime \prime }P}{\dot{m}{C}_p}\times 1$
$\Rightarrow \frac{q^{\prime \prime }P}{\dot{m}{C}_p}=50\Rightarrow T_m=0+50\times x\Rightarrow T_m=50x$
at $x=0.2m,T_m={10}^{\circ }Cx=0.4m,T_m={20}^{\circ }C\left(P\right)$
$x=0.6m,T_m={30}^{\circ }C\left(Q\right)$
$x=0.8m,T_m={40}^{\circ }C\left(R\right)$
$x=1.0m,T_m={50}^{\circ }$

When flow is thermally developed ${}^{\prime }{h}^{\prime }$ becomes constant.
From Newton's law of cooling,
$q^{\prime \prime }=h(T_s-T_m)$
$[T_s\to Surfacetemperatureofpipe]$
$\Rightarrow$ $T_S-T_m=Constant$
Difference between pipe surface and fluid temperature is constant when flow is thermally developed. Also, when flow is thermally developed, pipe surface temperature ${}^{\prime }{T}_s^{\prime }$varies linearly.
Now, given that, $T_s$ at location $P,Q,R,$
$T_{S.P}={50}^{\circ }C$
$T_{S.Q}={80}^{\circ }C$
$T_{S.R}={90}^{\circ }C$

Temperature difference between pipe surface and fluid,

at $P,T_s-T_m=50-20={30}^{\circ }C$
at $Q,T_s-T_m=80-30={50}^{\circ }C$
at $R,T_s-T_m=90-40={50}^{\circ }C$

Now, it is clear that, from $Q$ to $R,$ $T_s-T_m$ is constant and $T_s$ varies linearly. So flow is thermally developed at $"Q$ and $R"$