Question

A rod of length L is made of a material with thermal conductivity k, initially with a tem- perature distribution along its length (the x-direction) that can be described by a function (x). Find the temperature distribution in the rod T(x,t) at time t>0 if all the surfaces and the two ends of the rod are thermally insulated.

Answer 1

Answer:

Temperature distribution = ( T( λ , T )  - Ta ) / (T₉ - Ta) =  erf ( λ / 2 √∝T )

Explanation:

Determine the temperature distribution in the Rod T( x,t )

Given data :

Length: L

Thermal conductivity ; K

t > 0

Taking a 2nd order derivative

d^2 T / dx^2 = 1 /∝ * dT/dS  

considering boundary conditions

T( λ , 0 ) = Ti

T ( 0,T ) = Ta given that   t>0

T ( ∝ , 1 ) = T₉ for ₉ >0

Finally the General equation for the temperature distribution in the Rod

= ( T( λ , T )  - Ta ) / (T₉ - Ta) =  erf ( λ / 2 √∝T )