Question

A beaker contains a fluid of density $\rho kg/m^3$, specific heat $SJ/k{g}^{\circ }C$ and viscosity $\eta .$ The beaker is filled up to height $h$. To estimate the rate of heat transfer per unit area $(\dot{Q}/A)$ by convection when beaker is put on a hot plate, a student proposes that it should depend on $\eta ,\left(\frac{S\Delta \theta }{h}\right)$ and $\left(\frac{1}{\rho g}\right)$ when $\Delta \theta$ (in ${}^{\circ }C)$ is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for $(\dot{Q}/A)$ is

• A. $\eta \frac{S\Delta \theta }{h}$
• B. $\eta \left(\frac{S\Delta \theta }{h}\right)\left(\frac{1}{\rho g}\right)$
• C. $\frac{S\Delta \theta }{\eta h}$
• D. $\left(\frac{S\Delta \theta }{\eta h}\right)\left(\frac{1}{\rho g}\right)$

Answer 1

Answer: (A)

$Let\frac{Q}{A}={\eta }^a{\left(\frac{S\Delta \theta }{h}\right)}^b\left(\rho g\right)$
Using dimensional method
$M{T}^{-3}={\left[M{L}^{-1}{T}^{-1}\right]}^a{\left[L{T}^{-2}\right]}^b{\left[M^{-1}{L}^2{T}^2\right]}^c$
or, $M{T}^{-3}=\left[M^{a-c}{L}^{-a+b+2c}{T}^{-a-2b+2c}\right]$
Equating powers and solving
we get, $a=l,b=c=0$
$\therefore \frac{Q}{A}=\eta \frac{S\Delta \theta }{h}$