1. Tardigrade
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  3. Physics
  4. A beaker contains a fluid of density ρ kg / m 3, specific heat S J / kg ° C and viscosity η . The beaker is filled up to height h. To estimate the rate of heat transfer per unit area ( dotQ / A) by convection when beaker is put on a hot plate, a student proposes that it should depend on η,((S Δ θ/h)) and ((1/ρ g)) when Δ θ (in . ° C ) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for ( dotQ / A) is

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Q. A beaker contains a fluid of density $\rho\, kg / m ^{3}$, specific heat $S\, J / kg ^{\circ} C$ 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 $\left.{ }^{\circ} C \right)$ 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

JEE MainJEE Main 2015Physical World, Units and Measurements

Solution:

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