Question

Consider a solid cylindrical rod whose ends are maintained at constant but different temperatures while the side surface is perfectly insulated. There is no heat generation. It is claimed that the temperature along the axis of the rod varies linearly during steady heat conduction. Do you agree with this claim? Why?

Short answer

Based on the given conditions, the temperature distribution along the axis of the rod follows a linear equation: T(x) = (T2 - T1)/L)x + T1. This conclusion is in agreement with the claim that the temperature along the axis varies linearly during steady heat conduction (assuming perfectly insulated side surfaces and no heat generation).

Step-by-step solution

Define the problem variables

Let's consider a few variables related to the cylindrical rod: - L: Length of the rod - A: Cross-sectional area of the rod - k: Thermal conductivity of the rod - T1: Temperature at one end of the rod - T2: Temperature at the other end of the rod - x: Distance along the axis of the rod (from one end to another) - T(x): Temperature distribution along the axis of the rod

Write the heat conduction equation for steady state

The heat conduction equation for one-dimensional steady-state heat conduction in the rod, with no heat generation is given by: \begin{equation} \frac{d^2T}{dx^2} = 0 \end{equation}

Solve the differential equation

Let's solve the differential equation by integrating it twice: 1. Integrate once: \begin{equation} \frac{dT}{dx} = C_1 \end{equation} Where C_1 is a constant. 2. Integrate a second time: \begin{equation} T(x) = C_1x + C_2 \end{equation} Where C_2 is another constant. From the equation, we can see that the temperature T(x) along the axis of the rod is linearly dependent on the distance x.

Verify the boundary conditions

Now let's verify that this solution works for the boundary conditions given: 1. At x = 0, T(x) = T1: \begin{equation} T1 = C_1 \cdot 0 + C_2 \Rightarrow C_2 = T1 \end{equation} 2. At x = L, T(x) = T2: \begin{equation} T2 = C_1 \cdot L + T1 \Rightarrow C_1 = \frac{T2 - T1}{L} \end{equation} With these constant values, we can write the temperature distribution equation along the axis of the rod as: \begin{equation} T(x) = (\frac{T2 - T1}{L})x + T1 \end{equation}

Conclusion

From the analysis and our temperature distribution equation, we can confirm that we agree with the claim that the temperature along the axis of the rod varies linearly during steady heat conduction, for the given problem constraints (perfectly insulated side surface and no heat generation within the rod).

Key concepts

Cylindrical Rod Temperature Distribution

Understanding the temperature distribution within a cylindrical rod during steady-state heat conduction is essential for various engineering applications. For instance, consider a scenario where a cylindrical rod has different temperatures imposed upon its two ends and is perfectly insulated along its surface. This leads to heat transfer solely along the length of the rod, from the hotter end to the cooler one. The essence of a steady-state condition is that the temperature at each point in the material remains constant over time, suggesting that the incoming and outgoing heat flux at any cross-section is balanced.

To determine if the temperature distribution along the axis is indeed linear, as claimed, we must assess the differential equation governing heat conduction under the given conditions. This approach relies on mathematical principles to describe how temperature changes within the rod and helps engineers predict the rod's thermal behavior under steady-state conditions.

Thermal Conductivity

Thermal conductivity, denoted by the symbol 'k', is a property of materials that indicates their ability to conduct heat. It plays a pivotal role in the analysis of heat transfer problems, including steady-state heat conduction. A higher 'k' value means that the material can transport heat more efficiently, while a lower 'k' signifies that the material is a better insulator. In the context of the cylindrical rod exercise, 'k' would determine how rapidly the heat is transferred from the hot end to the cold end of the rod.

In the analysis, 'k' remains a silent factor in the differential equation as the equation itself simplifies to a statement indicating uniformity in temperature irrespective of 'k'. However, one must be aware that 'k' greatly influences the rate at which the rod achieves steady-state and should be considered when evaluating materials for heat transfer applications.

Differential Equation in Heat Transfer

The assessment of heat transfer within a body often requires setting up and solving differential equations. These equations mathematically represent the rate of temperature change within the material in relation to space and time. The basic form of the steady-state heat conduction equation is \[\begin{equation}\frac{d^2T}{dx^2} = 0\r\end{equation}\], which assumes no internal heat generation and that the temperature does not change with time.

By solving this equation through integration, we can relate the temperature distribution to the positional coordinate within the rod. For a cylindrical rod with linear boundary conditions, the temperature indeed shows a linear relation, confirming the initial claim. This resulting equation, \[\begin{equation}\T(x) = C_1x + C_2\r\end{equation}\], where \[\begin{equation}\C_1 = \frac{T2 - T1}{L}\r\end{equation}\], provides a clear and direct way to ascertain the temperature at any point along the rod’s axis.

Boundary Conditions in Heat Transfer

Boundary conditions are crucial in solving heat transfer problems because they define the parameters at the boundaries of the system. In the case of the cylindrical rod, we have specific temperatures fixed at both ends, which are our boundary conditions. These conditions allow us to solve the differential equation by giving precise values to the integration constants.

For the rod at x = 0, T(x) must equal T1, and similarly, at x = L, T(x) must equal T2. Once these conditions are applied, the resulting temperature distribution can be accurately defined, which in this scenario confirms a linear distribution along the rod. Ensuring that the boundary conditions accurately reflect the physical scenario is paramount as they anchor the theoretical model to the real system's behavior, allowing for meaningful predictions and analyses.