A large concrete slab 1 m thick has one-dimensional temperature distribution

T = 4 - 10x + 20x² + 10x³

Where T is temperature and x is the distance from one face towards the other face of the wall. If the slab has a thermal diffusivity of 2 × 10^-3 m²/hr, what is the rate of change of temperature at the other face of the wall? 

Concept:

The general equation for unsteady heat conduction in a slab with no heat generation in one dimension is given by

\(\frac{{{\partial ^2}T}}{{\partial {x^2}}} = \frac{1}{α }\times \frac{{\partial T}}{{\partial \tau }}\)

Calculation:

Given:

T = 4 -10x + 20x2 + 10x3, α = 2 × 10-3 m2/hr

\( \frac{{\partial T}}{{\partial x }}= -10 +40x+30x^2\)

\(\frac {\partial^2T}{\partial x^2} = 40 + 60x\)

Now at the other face of the wall x = 1,

 \(\frac {\partial^2T}{\partial x^2} = 100\)

we have,

\(\frac{{{\partial ^2}T}}{{\partial {x^2}}} = \frac{1}{α }\times \frac{{\partial T}}{{\partial \tau }}\)

\( \frac{{\partial T}}{{\partial \tau }}=α\times( \frac{{{\partial ^2}T}}{{\partial {x^2}}}) =100 × 2 × 10^{-3} = 0.2 °C/hr \)

The rate of change of temperature at the other face of the wall is 0.2°C/hr.