Question

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

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

Answer 1

Answer: (A)

$\eta \frac{S\Delta \theta }{h}$

The dimensions of $\frac{\dot{Q}}{A}$ is $\frac{J/s}{m^2}=\frac{\left[M{L}^2{T}^{-2}\right]}{L^2{T}^{-1}}=\left[M{T}^{-3}\right]$

Checking units of options

A)$\eta \frac{S\Delta \theta }{h}=\left[M{L}^{-1}{T}^{-1}\right]\frac{\left[M{L}^2{T}^{-2}\right]}{\left[M\right]\left[K\right]}\frac{\left[K\right]}{\left[L\right]}=\left[M{T}^{-3}\right]$ option A is correct

B) The dimensions of option B can be derived from option A by multiplying dimension of $\frac{1}{\rho g}$

$\left[M{T}^{-3}\right]\times \frac{1}{\left[M{L}^{-3}\right]\left[L{T}^{-2}\right]}=\left[L^2{T}^{-1}\right]$

C) The dimensions of option C can be derived from option B by multiplying dimension of ${\eta }^2$

$\left[L^2{T}^{-1}\right]\times \frac{1}{{\left[M{L}^{-1}{T}^{-1}\right]}^2}=\left[M^{-2}{L}^{4}T\right]$

D) The dimensions of option D can be derived from option C by multiplying dimension of $\rho g$

$\left[M^{-2}{L}^{4}T\right]\times \left[M{L}^{-3}\right]\left[L{T}^{-2}\right]=\left[M^{-1}{L}^2{T}^{-1}\right]$

Hence correct answer is option A.