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Chapter 3: Problem 87

Consider the composite wall of Example 3.7. In the Comments section, temperature distributions in the wall were determined assuming negligible contact resistance between materials A and B. Compute and plot the temperature distributions if the thermal contact resistance is \(R_{t, c}^{\prime \prime}=10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\).

Short Answer

Expert verified
We are given a composite wall with materials A and B and a thermal contact resistance, \(R_{t, c}^{\prime \prime}=10^{-4} \: m^2 \cdot K / W\). From the temperature and thermal conductivity data given in Example 3.7, we compute the temperature resistances of materials A and B. Next, we calculate the heat flux through the wall, and find the temperature at the contact point between the materials. Using these values, we determine the temperature distributions for each material, \(T_A(x)\) and \(T_B(x)\). Finally, we plot these temperature distributions as a function of distance x along the composite wall, which allows us to analyze the effect of the thermal contact resistance on the overall temperature distribution.

Step by step solution

01

Given data and variables

We are given that the thermal contact resistance between materials A and B is \(R_{t, c}^{\prime \prime}=10^{-4} \: m^2 \cdot K / W\). The temperatures at the two end points of the composite wall are given in Example 3.7. Let's denote these temperatures as \(T_1\), the temperature at the surface in contact with material A, and \(T_3\), the temperature at the surface in contact with material B. The thicknesses of the materials and their coefficients of thermal conductivity are also given from Example 3.7. Let the thickness of A be \(L_A\), the thickness of B be \(L_B\), the thermal conductivity of A be \(k_A\), and the thermal conductivity of B be \(k_B\).
02

Determine the heat flux through the composite wall

Since the heat transfer through the composite wall is steady-state, the heat flux through it must be constant. To find the heat flux through the wall, we can consider material A and B separately and find the overall temperature resistance. The equations that link the heat flux with the resistance and temperature difference are \( q = \frac{T_1 - T_2}{R_{t, A} + R_{t, c}^{\prime\prime}} = \frac{T_2 - T_3}{R_{t, B}} \), where \(R_{t, A}\) and \(R_{t, B}\) are the temperature resistances of materials A and B, respectively, and \(T_2\) is the temperature at the point where materials A and B are in contact. Compute the temperature resistance of materials A and B using \(R_{t, A} = \frac{L_A}{k_A}\) and \(R_{t, B} = \frac{L_B}{k_B}\), then solve for the temperature at the contact point between the materials (i.e., \(T_2\)).
03

Compute temperature distribution in each material

Now we have the temperatures at the two ends of the composite wall, and the temperature at the point of contact between the materials (i.e., \(T_1\), \(T_2\), and \(T_3\)). We can compute the temperature distribution in each material using the equation: \(T(x) = -q R_t + T_c\) For material A, where \(x\) is the distance from the surface in contact with A, set the appropriate \(R_t\) and \(T_c\), and compute the temperature distribution \(T_A(x)\). Repeat the process for the material and obtain the temperature distribution \(T_B(x)\).
04

Plot the temperature distributions

Now that we have the temperature distributions for materials A and B, plot \(T_A(x)\) and \(T_B(x)\) as functions of distance \(x\) from the surface in contact with material A. The x-axis should range from 0 to \(L_A + L_B\), and the y-axis should range from the \(T_3\) to \(T_1\). The plot will show the temperature distribution of the composite wall with the thermal contact resistance between the materials.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Transfer
Heat transfer deals with the movement of heat from one area or substance to another. This process occurs via three main mechanisms: conduction, convection, and radiation. Most problems, including those related to composite walls, are concerned with conduction, which involves the transfer of heat through materials without the movement of the material itself.

In the context of our exercise, the concept is applied to understand how heat flows through two different materials, A and B, that make up a composite wall. The heat transfer is influenced by several factors, including the thermal conductivity of the materials, the temperature difference across the wall, and the wall's thickness.

Specifically, it's important to understand that the thermal contact resistance at the interface of materials A and B can significantly affect the overall heat transfer rate. This resistance to heat flow, denoted as \(R_{t, c}^{''}\), is a result of the microscopic imperfections and air gaps at the interface, which impede the heat flow and can create temperature discontinuities in the system.
Temperature Distribution
Temperature distribution in a material indicates how the temperature varies within the substance at steady state, meaning that the temperature at any given point does not change with time. To find the temperature distribution in the composite wall, one must consider both the rate of heat transfer and the resistances to heat flow, which include both the materials' resistances and any contact resistance present.

In our exercise, we use the concept of thermal resistance to calculate the temperature distribution. The resistance is inversely proportional to the thermal conductivity of the material and directly proportional to its thickness. By including thermal contact resistance, \(R_{t, c}^{''}\), we can mathematically model how temperature changes across the materials at the interface and within each material of the composite wall.

Using the equation \(T(x) = -q R_t + T_c\), we identify the temperature at any point \(x\) within each material, considering the known temperatures at the boundaries and the heat flux, \(q\), that is constant throughout the system.
Composite Wall
A composite wall is a structure made up of multiple layers of different materials that are in contact with each other, leading to a composite system through which heat must transfer. The key property in analyzing such a system is how well each material conducts heat, termed as its thermal conductivity.

In our exercise, we deal with a composite wall made up of materials A and B. When we introduce thermal contact resistance between these materials, it adds an additional layer of complexity to the system. This resistance must be accounted for in the computations to accurately predict how heat travels through the wall.

Understanding the behavior of composite walls is essential in practical applications such as building construction, where walls often consist of layers of different materials designed to provide insulation and structural stability while also managing how heat enters and exits the building.

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Most popular questions from this chapter

A bonding operation utilizes a laser to provide a constant heat flux, \(q_{o}^{\prime \prime}\), across the top surface of a thin adhesivebacked, plastic film to be affixed to a metal strip as shown in the sketch. The metal strip has a thickness \(d=1.25 \mathrm{~mm}\), and its width is large relative to that of the film. The thermophysical properties of the strip are \(\rho=7850 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=435 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The thermal resistance of the plastic film of width \(w_{1}=40 \mathrm{~mm}\) is negligible. The upper and lower surfaces of the strip (including the plastic film) experience convection with air at \(25^{\circ} \mathrm{C}\) and a convection coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The strip and film are very long in the direction normal to the page. Assume the edges of the metal strip are at the air temperature \(\left(T_{\infty}\right)\). (a) Derive an expression for the temperature distribution in the portion of the steel strip with the plastic film \(\left(-w_{1} / 2 \leq x \leq+w_{1} / 2\right)\). (b) If the heat flux provided by the laser is 10,000 \(\mathrm{W} / \mathrm{m}^{2}\), determine the temperature of the plastic film at the center \((x=0)\) and its edges \(\left(x=\pm w_{1} / 2\right)\). (c) Plot the temperature distribution for the entire strip and point out its special features.

The fin array of Problem \(3.142\) is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider conditions for which the second fluid maintains equivalent temperatures at the parallel plates, \(T_{o}=T_{L}\), thereby establishing symmetry about the midplane of the fin array. The heat exchanger is \(1 \mathrm{~m}\) long in the direction of the flow of air (first fluid) and \(1 \mathrm{~m}\) wide in a direction normal to both the airflow and the fin surfaces. The length of the fin passages between adjoining parallel plates is \(L=8 \mathrm{~mm}\), whereas the fin thermal conductivity and convection coefficient are \(k=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (aluminum) and \(h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. (a) If the fin thickness and pitch are \(t=1 \mathrm{~mm}\) and \(S=4 \mathrm{~mm}\), respectively, what is the value of the thermal resistance \(R_{t, o}\) for a one-half section of the fin array? (b) Subject to the constraints that the fin thickness and pitch may not be less than \(0.5\) and \(3 \mathrm{~mm}\), respectively, assess the effect of changes in \(t\) and \(S\).

A thin flat plate of length \(L\), thickness \(t\), and width \(W \geqslant L\) is thermally joined to two large heat sinks that are maintained at a temperature \(T_{o}\). The bottom of the plate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniform value of \(q_{o}^{\prime \prime}\) (a) Derive the differential equation that determines the steady-state temperature distribution \(T(x)\) in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks.

A nuclear fuel element of thickness \(2 L\) is covered with a steel cladding of thickness \(b\). Heat generated within the nuclear fuel at a rate \(\dot{q}\) is removed by a fluid at \(T_{\infty}\), which adjoins one surface and is characterized by a convection coefficient \(h\). The other surface is well insulated, and the fuel and steel have thermal conductivities of \(k_{f}\) and \(k_{s}\), respectively. (a) Obtain an equation for the temperature distribution \(T(x)\) in the nuclear fuel. Express your results in terms of \(\dot{q}, k_{f}, L, b, k_{s}, h\), and \(T_{\infty}\). (b) Sketch the temperature distribution \(T(x)\) for the entire system.

Circular copper rods of diameter \(D=1 \mathrm{~mm}\) and length \(L=25 \mathrm{~mm}\) are used to enhance heat transfer from a surface that is maintained at \(T_{s, 1}=100^{\circ} \mathrm{C}\). One end of the rod is attached to this surface (at \(x=0\) ), while the other end \((x=25 \mathrm{~mm})\) is joined to a second surface, which is maintained at \(T_{s, 2}=0^{\circ} \mathrm{C}\). Air flowing between the surfaces (and over the rods) is also at a temperature of \(T_{\infty}=0^{\circ} \mathrm{C}\), and a convection coefficient of \(h=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) is maintained. (a) What is the rate of heat transfer by convection from a single copper rod to the air? (b) What is the total rate of heat transfer from a \(1 \mathrm{~m} \times 1 \mathrm{~m}\) section of the surface at \(100^{\circ} \mathrm{C}\), if a bundle of the rods is installed on 4 -mm centers?
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