A beaker contains a fluid of density $ρ \frac{k g}{m^{3}},$ specific heat $S \frac{J}{k g^{0} C}$ and viscosity $η$ . The beaker is filled up to height ‘h’. To estimate the heat transferred per unit time per unit area $(\frac{Q}{A})$ by convection when beaker is put on a hot plate, a student proposes that it should depend on $(\frac{S Δ θ}{h}) a n d (\frac{1}{ρ g}) w h e n Δ θ (i n^{0} C)$ is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for $(\frac{Q}{A})$ is

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$\frac{S Δ θ}{r}$

$η (\frac{S Δ θ}{h}) (\frac{1}{ρ g})$

$(\frac{S Δ θ}{η h}) (\frac{1}{ρ g})$

$(\frac{S Δ θ}{h}) η$

answer is C.

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Detailed Solution

$\frac{Q}{A} = η^{x} {(\frac{S Δ θ}{h})}^{γ} (\frac{1}{ρ g}) .... (i) [\frac{Q}{A}] = M T^{- 3}, [η] = M L^{- 1} T^{- 1} [\frac{S Δ θ}{h}] = L T^{- 2}, [\frac{1}{ρ g}] = M^{- 1} L^{2} T^{2} M T^{- 3} = M^{x - z} L^{- x + y + 2 z} T^{- x - 2 y + 2 z} s o 1 = x - z ......... (i i) - x + y + 2 z = 0 ......... (i i i) - x - 2 y + 2 z = - 3 ......... (i v)$
From ii, iii, iv
x = 1, y = 1, z = 0 from (i)
$s o [\frac{Q}{A}] = η \frac{S Δ θ}{h}$