Consider a laterally insulated rod of length L and constant thermal conductivity. Assuming one-dimensional heat conduction in the rod, which of the following steady-state temperature profile(s) can occur without internal heat generation?

The correct temperature profiles are the linear profiles shown in the first and second images (straight lines with positive and negative slopes, respectively).

1. Governing Physical Principles
For one-dimensional, steady-state heat conduction along the length ($x$-direction) of a laterally insulated rod of length $L$ and constant thermal conductivity $k$, the heat conduction equation is:

$$\frac{d^{2}T}{d{x}^2}+\frac{{\dot{q}}_g}{k}=0$$

where:
• $T$ is the temperature as a function of position $x$.
• $k$ is the constant thermal conductivity of the rod material.
• ${\dot{q}}_g$ is the rate of internal heat generation per unit volume.

2. Applying the Constraints
The problem states that there is no internal heat generation, meaning:

$${\dot{q}}_g=0$$

Substituting this value back into the heat conduction equation, we obtain:

$$\frac{d^{2}T}{d{x}^2}=0$$

3. Mathematical Derivation of the Temperature Profile
To find the temperature distribution $T\left(x\right)$, we integrate the simplified governing equation twice with respect to $x$:
Integrating once gives:

$$\frac{dT}{dx}=C_1$$

where $C_1$ is the first integration constant representing the constant temperature gradient.
Integrating a second time gives the temperature profile equation:

$$T\left(x\right)=C_{1}x+C_2$$

where $C_2$ is the second constant of integration.

4. Analysis of the Profiles Shown in the Images
The equation $T\left(x\right)=C_{1}x+C_2$ is mathematically a straight line. Therefore, any steady-state temperature profile in a rod without internal heat generation must be linear:
• Image 0 (Increasing Linear Profile): Shows a linear temperature profile starting at a lower temperature at $x=0$ and rising to a higher temperature at $x=L$ (positive slope $C_1>0$). This occurs when the right boundary is maintained at a higher temperature than the left boundary.
• Image 1 (Decreasing Linear Profile): Shows a linear temperature profile starting at a higher temperature at $x=0$ and dropping to a lower temperature at $x=L$ (negative slope $C_1<0$). This occurs when heat flows from left to right due to a higher temperature at the left boundary.
• Image 2 (Parabolic Profile) & Image 3 (Oscillatory Profile): These non-linear curves require either internal heat sources/sinks or transient (unsteady) conditions to occur, as they have non-zero second derivatives ($d^{2}T/d{x}^2\ne 0$).

Thus, the linear profiles (such as those depicted in the first two options) are the only steady-state profiles that can exist under the given conditions.