# Thermal Diffusivity of a Sphere

Thermal Diffusivity of a Sphere

This Demonstration describes a method for determining the thermal diffusivity of a sphere. The sphere is suddenly immersed in a cooling fluid and the temperature of the fluid is observed as a function of time.

Consider a homogeneous solid sphere of radius , initially at uniform temperature that is immersed at time in a volume of fluid in a well-stirred, insulated tank at temperature .

R

T

1

t=0

V

f

T

0

It is convenient to define the following dimensionless variables:

Θ

s

T

s

T

0

T

1

T

0

Θ

f

T

f

T

0

T

1

T

0

ξ=

r

R

τ=t

α

s

2

R

where is the radial coordinate, is the radius of the sphere, is the thermal diffusivity , is the thermal conductivity, and are the solid and fluid densities, and , and are the specific heat capacities of the solid and liquid respectively.

r

R

α

s

k

s

ρ

s

Cp

s

k

s

ρ

s

ρ

f

Cp

s

Cp

f

The heat equation, in terms of these dimensionless variables, can be written [1]:

∂

Θ

s

∂τ

1

2

ξ

∂

∂ξ

2

ξ

∂

Θ

s

∂ξ

with the boundary conditions:

Θ

s

Θ

s

Θ

f

Note that (0,τ) is finite, with

Θ

s

Θ

f

τ

-3

β

∂(1,τ)

Θ

s

∂ξ

and (0)=0, where is the ratio of the heat capacities of the fluid to the solid and is the volume of the solid.

Θ

f

β=

ρ

f

C

p

f

V

f

ρ

s

C

p

s

V

s

V

s

The following solution is obtained in [2]:,

Θ(τ)=-6β

1

1+β

∞

∑

n=1

-τ

2

b

n

e

9(1+β)+

2

β

2

b

n

where the are the nonzero roots of .

b

n

tan=

b

n

3

b

n

3+β

2

b

n

The only place where the thermal diffusivity of the solid appears is in the dimensionless time , so that the temperature rise of the fluid can be used to determine the thermal diffusivity of the solid. Interestingly, the temperature history of the fluid is obtained without requiring the temperature profiles for the solid.

a

s

τ=t

α

s

2

R