Question

One-dimensional steady state heat conduction takes place through a solid whose cross-sectional area varies linearly in the direction of heat transfer. Assume there is no heat generation in the solid and thermal conductivity of the material is constant and independent of temperature. The temperature distribution in the solid is

• A. Quadratic
• B. Logarithmic
• C. Linear
• D. Exponential

Answer 1

The correct option is B Logarithmic


A=f(x)
Let, A = a + bx

$\dot{E_{in}}+\dot{E_{out}}+\text{/}{E}_g=\text{/}{E}_{st}$

($\because$ steady state and no heat generation)
$\dot{E_{in}}=\dot{E_{out}}$
$q\ne f\left(x\right)$ i.e., is constant
$q=-k{A}_{cs}\frac{dT}{dx}=constant$
$A_{cs}\frac{dT}{dx}=constant$
$\because$ k will be constant for a material)

$\int dT=\int \frac{C.dx}{(a+bx)}$
$T=\frac{C}{b}ln(a+bx)+C_1$

Hence, temperature distribution in the solid is logarithmic.