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

The correct option is Logarithmic.

Analysis of the Given System:
From the provided image, we see a solid of varying cross-section along the direction of heat transfer (the horizontal x-axis). The image explicitly specifies the linear variation of the cross-sectional area with the label:

$$A_x=A_0+A_{c}x$$

where:
• $A_0$ is the cross-sectional area at $x=0$.
• $A_c$ is the constant rate of change of area along the direction of heat transfer.
• $Q$ is the rate of heat transfer through the solid.

Step-by-Step Mathematical Derivation:
According to Fourier's law of heat conduction, the rate of one-dimensional heat transfer is expressed as:

$$Q=-k{A}_x\frac{dT}{dx}$$

where:
• $k$ is the constant thermal conductivity of the material.
• $\frac{dT}{dx}$ is the temperature gradient.

Under steady-state conditions with no internal heat generation, the heat flow rate $Q$ remains constant along the length of the solid. Substituting the expression for the linear area variation into Fourier's equation yields:

$$Q=-k(A_0+A_{c}x)\frac{dT}{dx}$$

To find the temperature distribution $T\left(x\right)$, we separate the variables $T$ and $x$:

$$dT=-\frac{Q}{k}\left(\frac{dx}{A_0+A_{c}x}\right)$$

Integrating both sides of the equation with respect to their corresponding variables:

$$\int dT=-\frac{Q}{k}\int \frac{dx}{A_0+A_{c}x}$$

Performing the integration, we get:

$$T\left(x\right)=-\frac{Q}{k{A}_c}\ln (A_0+A_{c}x)+C$$

where $C$ is the constant of integration determined by the boundary conditions.

Conclusion:
The derived equation shows that the temperature distribution $T\left(x\right)$ is a function of the natural logarithm of the position coordinate $x$. Therefore, the temperature distribution in the solid is Logarithmic.