A storage tank contains an oil whose density varies with temperature according to the…

PHYSICS · Subjective · CBSE · Physics · Mechanical Properties of Fluids · Mechanical Properties of Fluids

Problem

A storage tank contains an oil whose density varies with temperature according to the relation

\rho_T = \rho_{20}\,[1 - \beta\,(T - 20)]

, where

\rho_{20} = 870\,\text{kg}\,\text{m}^{-3}

is the density at

20^{\circ}\text{C}

T

is the temperature in

^{\circ}\text{C}

and

\beta = 9.0\times10^{-4}\;^{\circ}\text{C}^{-1}

. A pressure gauge located

12\,\text{m}

below the free surface of the oil shows an absolute pressure of

2.01\times10^{5}\,\text{Pa}

. The atmospheric pressure at the surface is

1.01\times10^{5}\,\text{Pa}

. (a) Derive an expression linking the temperature

T

of the oil to the measured gauge pressure, the depth and the given parameters. (b) Calculate the temperature of the oil in the tank.