Two thin metallic spherical shells of radii 20 cm and 30 cm, respectively are placed with their centres coinciding. A material of thermal conductivity α is filled in the space between the shells. The inner shell is maintained at 300 K and the outer shell at 310 K . If the rate at which heat flows radially through the material is 40 W, find the value of α (in units of .J s -1 m -1 K -1).

Question

Two thin metallic spherical shells of radii 20 cm and 30 cm, respectively are placed with their centres coinciding. A material of thermal conductivity α is filled in the space between the shells. The inner shell is maintained at 300 K and the outer shell at 310 K . If the rate at which heat flows radially through the material is 40 W, find the value of α (in units of .J s -1 m -1 K -1). (A) (3/π) (B) (4 π/3) (C) (5/3 π) (D) (π/2)

Tardigrade Admin · Accepted Answer

Given, radius of two thin metallic spherical shells

r_{1} = 20 c m = 0.2 m

r_{2} = 30 c m = 0.3 m

temperature of inner shell,

T_{1} = 300 K

,
temperature of outer shell,

T_{2} = 310 K

,
and rate of heat flow,

H = 40 W

.
Radial rate of flow of heat through the shell in the steady state,

H = \frac{d Q}{d t} = α A \frac{d T}{d r} = α (4 π r^{2}) \frac{d T}{d r}

\frac{d r}{r ^{2}} = \frac{4 π α}{H} \cdot d T

Integrate on both sides, we get

\Rightarrow r_{1} \int r_{2} \frac{d r}{r ^{2}} = \frac{4 π α}{H} T_{1} \int T_{2} d T

\frac{r _{2} - r _{1}}{r _{1} r _{2}} = \frac{4 π α}{H} (T_{2} - T_{1})

∴ α = \frac{H ( r _{2} - r _{1} )}{4 π r _{1} r _{2} ( T _{2} - T _{1} )}

Putting the given values, we get

α = \frac{40 ( 0.3 - 0.2 )}{4 π ( 0.3 ) ( 0.2 ) ( 310 - 300 )} = \frac{4}{2.4 π} = \frac{5}{3 π}

Hence, the value of

α = \frac{5}{3 π} J s^{- 1} m^{- 1} K^{- 1}

.

$\frac{3}{π}$

$\frac{4 π}{3}$

$\frac{5}{3 π}$

$\frac{π}{2}$

π3​

34π​

3π5​

2π​

$\frac{3}{π}$

$\frac{4 π}{3}$

$\frac{5}{3 π}$

$\frac{π}{2}$