Question

The right end of a bar of length \(a\) with thermal conductivity \(\kappa_{1}\) and cross-sectional area \(A_{1}\) is joined to the left end of a bar of thermal conductivity \(\kappa,\) and cross-sectional area \(A_{2}\) The composite bar has a total length \(L\). Suppose that the end \(x=0\) is held at temperature zero, while the end \(x=L\) is held at temperature \(T\). Find the steady-state temperature in the composite bar, assuming that the temperature and rate of heat flow are continuous at \(x=a .\) Hint: See Eq. ( 2 ) of Appendix A.

Short answer

Answer: To find the steady-state temperature distribution for a composite bar, we can (1) obtain the temperature functions for the individual bars, (2) apply the boundary conditions for these bars, (3) apply the continuity conditions for temperature and rate of heat flow at the interface, and (4) solve for the constants in the temperature functions to obtain the steady-state temperature distribution.

Step-by-step solution

Temperature Functions for Individual Bars

We will find the temperature function \(T_1(x)\) for the bar with thermal conductivity \(\kappa_1\) and area \(A_1\), and temperature function \(T_2(x)\) for the bar with thermal conductivity \(\kappa and\) area \(A_2\). According to Eq. 2 of Appendix A, these functions will follow the form \(T(x) = C_1 x + C_2\), where \(C_1\) and \(C_2\) are constants to be determined from boundary conditions.

Apply Boundary Conditions for Individual Bars

To determine the constants, we will apply the boundary conditions: For the bar with thermal conductivity \(\kappa_1\) and area \(A_1\): - At the end \(x=0\), the temperature \(T_1=0\) (given in the problem) - At the end \(x=a\), let the temperature be T_1(a)=T_a (unknown, but continuous with the second bar) For the bar with thermal conductivity \(\kappa\) and area \(A_2\): - At the end \(x=a\), let the temperature be \(T_2(a)=T_a\) - At the end \(x=L\), the temperature \(T_2=L\) (given in the problem) By applying these boundary conditions, we can determine the constants \(C_1\) and \(C_2\) for both temperature functions.

Apply Continuity Conditions at \(x=a\)

According to the problem, the temperature and rate of heat flow are continuous at \(x=a\). As both \(T_1\) and \(T_2\) are continuous, we have \(T_1(a)=T_2(a)=T_a\). The heat flow for both bars is given by \(q_i= -\kappa_i A_i \frac{d T_i}{d x}\), where \(i = 1, 2\). From the continuity of heat flow, we have: $$ q_1 = q_2 \Rightarrow -\kappa_1 A_1 \frac{d T_1}{d x} = - \kappa_2 A_2 \frac{d T_2}{d x} \text{ at } x=a $$

Determine Steady-State Temperature Distribution

Now we will use the expressions for \(T_1(x)\) and \(T_2(x)\) obtained in Step 2, plug in x=a, and apply the continuity conditions derived in Step 3. From this, we can find the remaining constant, which will give us the steady-state temperature distribution for the composite bar.

Key concepts

Thermal Conductivity

Thermal conductivity is a fundamental concept in the study of heat transfer within materials. It refers to a material's ability to conduct heat and is denoted by the Greek letter \( \kappa \). Specifically, it quantifies the amount of heat energy that passes per unit time through a unit area with a temperature gradient of one degree per unit distance. In the context of the textbook exercise, two bars with different thermal conductivities, \( \kappa_1 \) and \( \kappa \), are considered.

Understanding thermal conductivity is critical as it dictates how heat spreads throughout a medium. A high thermal conductivity indicates that the material is able to easily transfer heat, while a lower thermal conductivity means that the material insulates better and heat transfer is not as effective. For instance, metals typically have high thermal conductivity, while materials like wood, plastic, or glass have lower thermal conductivity.

When studying the steady-state temperature distribution in a composite bar, as proposed in the exercise, thermal conductivity affects how the temperature varies across the material. Students should note that at steady state, heat entering a section of the bar must equal the heat leaving it, meaning that temperature changes are no longer occurring with time and the system is in equilibrium.

Boundary Conditions

Boundary conditions are a set of constraints that provide solutions to differential equations used in various physical scenarios, like our textbook exercise on temperature distribution across a composite bar. They are necessary to uniquely determine the solution to the differential equation that models physical phenomena such as temperature variation. In essence, without proper boundary conditions, a problem involving differential equations could have infinitely many solutions.

There are three common types of boundary conditions:

• Dirichlet boundary conditions, where the value of the function is specified at the boundary.
• Neumann boundary conditions, which give the derivative (rate of change) of the function at the boundary.
• Robin boundary conditions which are a weighted combination of Dirichlet and Neumann conditions.

In our exercise, the boundaries are defined at the ends of the composite bar (\(x=0\) and \(x=L\)). At \(x=0\), the temperature is fixed at zero, while at \(x=L\), it is fixed at \(T\). These are examples of Dirichlet boundary conditions. The continuity of \(T_1(a)\) and \(T_2(a)\) at \(x=a\) is crucial for solving the differential equations governing heat flow and determining the steady-state temperature distribution.

Differential Equations

Differential equations play a central role in formulating physical problems where quantities vary continuously with respect to some variable, frequently time or space. In thermal physics, for example, they describe how temperature changes within an object over time or distance. Differential equations can either be ordinary (ODEs), involving derivatives with respect to only one variable, or partial (PDEs), involving derivatives with respect to multiple variables.

In our textbook exercise, the differential equation describes how temperature varies within the bars along the \(x\)-axis. This equation relates the rate of change of temperature with respect to position, known as the temperature gradient, to the heat flow in the material. Solving this equation requires understanding both thermal conductivity and boundary conditions.

The general solution to the temperature distribution problem typically takes the form \(T(x) = C_1 x + C_2\), which is derived from an ODE. \(C_1\) and \(C_2\) are constants specific to the particular system in question, and their values are determined by applying the given boundary conditions and ensuring the continuity conditions at the junction of the two bars. Eventually, this process provides the complete distribution of temperature along the bar at steady state, showing us exactly how temperature varies from one end of the bar to the other in equilibrium.