Derive the expression for the temperature reached by a solid at a given time 10 't' in case of transient heat conduction

Answer 1

Consider a solid of thickness L and thermal conductivity k. Let the initial temperature of the solid be T1 and the temperature of the surrounding fluid be T2. At t=0, the entire solid is at temperature T1.

The heat conduction equation for the transient case is given by:

ρc ∂T/∂t = k ∂2T/∂x2

where ρ is the density of the solid, c is the specific heat capacity, T is the temperature distribution as a function of time 't' and position 'x'.

Assuming the heat flow to be one-dimensional, we can write the temperature distribution as:

T(x,t) = T2 + (T1 - T2) [1 - erf(x / 2√(k.t / ρ.c))]

Here, erf is the error function defined as:

erf(z) = 2/√π ∫0z e^(-t^2) dt

Substituting t=10t in the above expression, we get:

T(x,10t) = T2 + (T1 - T2) [1 - erf(x / 2√(k.10t / ρ.c))]

Thus, we have derived the expression for the temperature reached by a solid at a given time 10 't' in case of transient heat conduction.