Question

The temperature distribution, at a certain instant of time in a concrete slab during curing is given by,

\(T = 3{x^2} + 3x + 16\)

Where x is in cm and T is in K. The rate of change of temperature with time is given by (assume diffusivity to be 0.0003 cm²/s)
1. 0.0009 K/s
2. 0.0048 K/s
3. 0.0012 K/s
4. 0.0018 K/s

Answer 1

Correct Answer - Option 4 : 0.0018 K/s

Concept:

General heat conduction equation is,

\(\frac{1}{\alpha }\frac{{\partial T}}{{\partial \tau }} = \left( {\frac{{{\partial ^2}t}}{{\partial {x^2}}} + \frac{{{\partial ^2}t}}{{\partial {y^2}}} + \frac{{{\partial ^2}t}}{{\partial {x^2}}}} \right) + \frac{q}{k}\)

Calculation:

Given:

\(T = 3{x^2} + 3x + 16\)

Without heat generation (q = 0) and in x-direction only,

\(\begin{array}{l} \frac{1}{\alpha }\frac{{\partial T}}{{\partial \tau }} = \frac{{{\partial ^2}t}}{{\partial {x^2}}}\\ \frac{{\partial T}}{{\partial \tau }} = \alpha \frac{{{\partial ^2}t}}{{\partial {x^2}}}\\ \frac{{{\partial ^2}t}}{{\partial {x^2}}} = 6\\ \frac{{\partial T}}{{\partial \tau }} = \left( {0.0003} \right)\left( 6 \right) = 0.0018{\rm{\;K}}/{\rm{s}} \end{array}\)