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Chapter 2: Problem 60

Consider a 20-cm-thick large concrete plane wall \((k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) subjected to convection on both sides with \(T_{\infty 1}=27^{\circ} \mathrm{C}\) and \(h_{1}=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inside, and \(T_{\infty 2}=8^{\circ} \mathrm{C}\) and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperatures at the inner and outer surfaces of the wall.

Short Answer

Expert verified
Question: Determine the temperatures at the inner and outer surfaces of a concrete wall, given the following information: - The wall has a thickness of 20 cm. - The thermal conductivity (k) of the concrete is 1.4 W/mK. - The wall is exposed to convective heat transfer with coefficients h1 and h2 on the inner and outer surfaces, respectively, with h1 = 20 W/m²K and h2 = 10 W/m²K. - The ambient temperatures T∞1 and T∞2 on the inner and outer surfaces are 25°C and 0°C, respectively. Answer: To determine the temperatures at the inner and outer surfaces of the concrete wall, follow these steps: 1. Derive the differential equation for steady one-dimensional heat conduction: \(\frac{d^2 T}{dx^2} = 0\) 2. Apply the boundary conditions and solve for temperature distribution: At x = 0 (inner surface): \(h_1(T(0) - T_\infty 1) = -k C_1\) At x = L (outer surface): \(h_2(T(20 \text{ cm}) - T_\infty 2) = k C_1\) Solve for C1 and C2 using these equations. 3. Calculate the temperature at the inner and outer surfaces of the wall: \(T_{inner} = T(0) = C_2\) \(T_{outer} = T(20 \text{ cm}) = C_1 \cdot (20 \text{ cm}) + C_2\) By using these steps and solving for C1 and C2, you can determine the temperatures at the inner and outer surfaces of the concrete wall.

Step by step solution

01

Derive the Differential Equation for Steady One-dimensional Heat Conduction

We know that, for temperature to be steady, there should be no heat generation and our problem must be one-dimensional. Fourier's Law states that: \(q_x = -k \frac{dT}{dx}\) where \(q_x\) is the heat flux, \(k\) is the thermal conductivity, \(T\) is the temperature, and \(x\) is the distance in the x-axis. Since there is no heat generation and we are considering one-dimensional heat conduction, the heat flux (\(q_x\)) must be constant. \(\frac{d q_x}{dx} = 0\) Differentiating Fourier's Law with respect to \(x\) gives: \(\frac{d}{dx}(-k \frac{dT}{dx}) = k \frac{d^2 T}{dx^2} = 0\) Therefore, the differential equation for steady one-dimensional heat conduction is: \(\frac{d^2 T}{dx^2} = 0\)
02

Apply the Boundary Conditions and Solve for Temperature Distribution

At x = 0 (inner surface), we have a convective boundary condition given as: \(h_1(T(0) - T_\infty 1) = -k \frac{dT}{dx}(0)\) At x = L (outer surface), we have another convective boundary condition given as: \(h_2(T(L) - T_\infty 2) = k \frac{dT}{dx}(L)\) Since the differential equation is a second-order ODE, we need two boundary conditions. Integrate the differential equation once to obtain: \(\frac{dT}{dx} = C_1\) Integrating again, we get: \(T(x) = C_1 x + C_2\) Now, we apply the boundary conditions: \(h_1(T(0) - T_\infty 1) = -k C_1\) \(h_2(T(20\text{ cm}) - T_\infty 2) = k C_1\) From these two equations, solve for \(C_1\) and \(C_2\).
03

Calculate the Temperature at the Inner and Outer Surfaces of the Wall

With the constant values \(C_1\) and \(C_2\) determined, we now substitute these values back into our temperature equation and find the temperature at the inner and outer surfaces of the wall: \(T_{inner} = T(0) = C_1 \cdot 0 + C_2 = C_2\) \(T_{outer} = T(20\text{ cm}) = C_1 \cdot (20\text{ cm}) + C_2\) The values of \(T_{inner}\) and \(T_{outer}\) are the temperatures at the inner and outer surfaces of the concrete wall.

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Most popular questions from this chapter

Consider a plane wall of thickness \(L\) whose thermal conductivity varies in a specified temperature range as \(k(T)=\) \(k_{0}\left(1+\beta T^{2}\right)\) where \(k_{0}\) and \(\beta\) are two specified constants. The wall surface at \(x=0\) is maintained at a constant temperature of \(T_{1}\), while the surface at \(x=L\) is maintained at \(T_{2}\). Assuming steady one-dimensional heat transfer, obtain a relation for the heat transfer rate through the wall.

Exhaust gases from a manufacturing plant are being discharged through a 10 - \(\mathrm{m}\) tall exhaust stack with outer diameter of \(1 \mathrm{~m}\), wall thickness of \(10 \mathrm{~cm}\), and thermal conductivity of \(40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The exhaust gases are discharged at a rate of \(1.2 \mathrm{~kg} / \mathrm{s}\), while temperature drop between inlet and exit of the exhaust stack is \(30^{\circ} \mathrm{C}\), and the constant pressure specific heat of the exhaust gasses is \(1600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). On a particular day, the outer surface of the exhaust stack experiences radiation with the surrounding at \(27^{\circ} \mathrm{C}\), and convection with the ambient air at \(27^{\circ} \mathrm{C}\) also, with an average convection heat transfer coefficient of \(8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Solar radiation is incident on the exhaust stack outer surface at a rate of \(150 \mathrm{~W} / \mathrm{m}^{2}\), and both the emissivity and solar absorptivity of the outer surface are 0.9. Assuming steady one-dimensional heat transfer, (a) obtain the variation of temperature in the exhaust stack wall and (b) determine the inner surface temperature of the exhaust stack.

In a food processing facility, a spherical container of inner radius \(r_{1}=40 \mathrm{~cm}\), outer radius \(r_{2}=41 \mathrm{~cm}\), and thermal conductivity \(k=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is used to store hot water and to keep it at \(100^{\circ} \mathrm{C}\) at all times. To accomplish this, the outer surface of the container is wrapped with a 800 -W electric strip heater and then insulated. The temperature of the inner surface of the container is observed to be nearly \(120^{\circ} \mathrm{C}\) at all times. Assuming 10 percent of the heat generated in the heater is lost through the insulation, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the container, \((b)\) obtain a relation for the variation of temperature in the container material by solving the differential equation, and \((c)\) evaluate the outer surface temperature of the container. Also determine how much water at \(100^{\circ} \mathrm{C}\) this tank can supply steadily if the cold water enters at \(20^{\circ} \mathrm{C}\).

A spherical metal ball of radius \(r_{o}\) is heated in an oven to a temperature of \(T_{i}\) throughout and is then taken out of the oven and allowed to cool in ambient air at \(T_{\infty}\) by convection and radiation. The emissivity of the outer surface of the cylinder is \(\varepsilon\), and the temperature of the surrounding surfaces is \(T_{\text {surr }}\). The average convection heat transfer coefficient is estimated to be \(h\). Assuming variable thermal conductivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

The variation of temperature in a plane wall is determined to be \(T(x)=110-60 x\) where \(x\) is in \(\mathrm{m}\) and \(T\) is in \({ }^{\circ} \mathrm{C}\). If the thickness of the wall is \(0.75 \mathrm{~m}\), the temperature difference between the inner and outer surfaces of the wall is (a) \(30^{\circ} \mathrm{C}\) (b) \(45^{\circ} \mathrm{C}\) (c) \(60^{\circ} \mathrm{C}\) (d) \(75^{\circ} \mathrm{C}\) (e) \(84^{\circ} \mathrm{C}\)
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