A beaker contains a fluid of density ρ kg m ^ 3 , specific heat S J...

A beaker contains a fluid of density

ρ\dfrac{kg}{{m}^{3}}

, specific heat

S\dfrac{J}{k{g}^{o}C}

and viscosity

η

. The beaker is filled up to height h. To estimate the rate of heat transfer per unit area

\left(\dfrac{\overset{˙}{Q}}{A}\right)

by convection when beaker is put on a hot plate, a student proposes that it should depend on

η

, \left(\dfrac{S\Delta\theta}{h}\right)

and \left(\dfrac{1}{ρg}\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(\dfrac{\overset{˙}{Q}}{A}\right)

is:

Options

A. $\left(\dfrac{S\Delta\theta}{h}\right)η$

\dfrac{Q}{A} = {η}^{x}{\left(\dfrac{s ∆ \theta}{h}\right)}^{y}{\left(\dfrac{1}{ρg}\right)}

^{z}

. . . (1)<br /><br />\left[\dfrac{Q}{A}\right] = M{T}^{- 3}, \left[η\right] = M{L}^{- 1}{T}^{- 1}

\left[\dfrac{s ∆ \theta}{h}\right] = L{T}^{- 2}, \left[\dfrac{1}{ρg}\right] = {M}^{- 1}{L}^{2}{T}^{2}

<br /><br />

M{T}^{- 3} = \left[{M}^{x}{L}^{- x}{T}^{- x}\right]\left[{L}^{y}{T}^{- 2y}\right]\left[{M}^{- 2}{L}^{2z}{T}^{2z}\right]

<br /><br />

M{T}^{- 3} = {M}^{x - z}{L}^{- x + y + 2z}{T}^{- x - 2y + 2z}

<br /><br />So

1 = x - z

. . . (2)<br /><br />

- x + y + 2z = 0

. . . (3)<br /><br />

- x - 2y + 2z = - 3

. . . (4)<br /><br />from 2, 3, 4<br /><br />

x = 1, y = 1, z = 0,

from (1)<br /><br />so \left[\dfrac{Q}{A}\right] = η\dfrac{s ∆ \theta}{h}