Question

In a case of one dimensional heat conduction in a medium with constant properties, T is the temperature at position x, at time t. Then $\frac{\partial T}{\partial t}$ is proportional to

• A. $\frac{T}{x}$
• B. $\frac{\partial T}{\partial x}$
• C. $\frac{{\partial }^{2}T}{\partial x\partial t}$
• D. $\frac{{\partial }^{2}T}{\partial x^2}$

Answer 1

The correct option is D $\frac{{\partial }^{2}T}{\partial x^2}$

The equation of heat transfer in cartesian coordinates without heat generation is
$\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial Z^2}=\frac{\rho c_p}{k}\frac{\partial T}{\partial t}$

For one dimensional heat transfer,
$\frac{{\partial }^{2}T}{\partial x^2}=\frac{\rho c_p}{k}\frac{\partial T}{\partial t}$

i.e., $\frac{\partial T}{\partial t}\propto \frac{{\partial }^{2}T}{\partial x^2}$