Question

Consider a steam pipe of length \(L=30 \mathrm{ft}\), inner radius $r_{1}=2 \mathrm{in}\(, outer radius \)r_{2}=2.4 \mathrm{in}$, and thermal conductivity $k=7.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}$. Steam is flowing through the pipe at an average temperature of \(250^{\circ} \mathrm{F}\), and the average convection heat transfer coefficient on the inner surface is given to be $h=12 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}$. If the average temperature on the outer surfaces of the pipe is \(T_{2}=160^{\circ} \mathrm{F},(a)\) express the differential equation and the boundary conditions for steady onedimensional heat conduction through the pipe, \((b)\) obtain a relation for the variation of temperature in the pipe by solving the differential equation, and (c) evaluate the rate of heat loss from the steam through the pipe.

Short answer

Question: Determine the relation for the variation of temperature in a steam pipe and evaluate the rate of heat loss from the steam through the pipe. Answer: The relation for the variation of temperature in a steam pipe is given by \(T(r) = C_1 \ln(r) + C_2\), where \(C_1\) and \(C_2\) are constants determined by the boundary conditions. The rate of heat loss from the steam through the pipe can be calculated using the formula: \(q_{loss} = h \cdot 2 \pi r_1 L \cdot (T(r_1) - 250)\).

Step-by-step solution

Express the differential equation

The differential equation for steady one-dimensional heat conduction through a cylindrical pipe is given by the following formula: \( \frac{d}{dr} \left( r \frac{d T}{d r} \right) = 0 \) This equation represents the change in temperature (\(T\)) with respect to the radial distance (\(r\)).

Determine the boundary conditions

There are two boundary conditions for this problem: 1. At the inner surface of the pipe (\(r = r_1\)), the convection heat transfer is given by: \( h (T - 250) = -k \frac{d T}{d r}|_{r = r_1} \) 2. At the outer surface of the pipe (\(r = r_2\)), the temperature is given: \( T|_{r = r_2} = T_2 = 160 \)

Solve the differential equation

Now, we need to solve the differential equation to obtain the temperature distribution along the radial distance. We can integrate the differential equation twice to find \(T(r)\). First, integrate with respect to \(r\): \( r \frac{dT}{dr} = C_1 \) Divide by \(r\) and integrate again: \( \frac{dT}{dr} = \frac{C_1}{r} \) \( T(r) = C_1 \ln(r) + C_2 \)

Apply boundary conditions

Now, we need to determine the constants of integration \(C_1\) and \(C_2\) by applying the boundary conditions. Boundary condition 1: \( h (T(r_1) - 250) = -k \frac{d T}{d r}|_{r = r_1} \) \( 12 (C_1 \ln(r_1) + C_2 - 250) = -7.2 \cdot \frac{C_1}{r_1} \) Boundary condition 2: \( T(r_2) = 160 \) \( C_1 \ln(r_2) + C_2 = 160 \) Solve these two equations simultaneously to find \(C_1\) and \(C_2\).

Find the temperature distribution

After solving the system of equations for \(C_1\) and \(C_2\), substitute their values back into the temperature distribution equation: \( T(r) = C_1 \ln(r) + C_2 \) Now, we have the relation for the variation of temperature in the pipe.

Evaluate the rate of heat loss

Finally, we can evaluate the rate of heat loss from the steam through the pipe. The heat loss can be determined by using the convection heat transfer equation on the inner surface of the pipe: \( q_{loss} = h \cdot A_1 \cdot (T_1 - T_{steam}) \) Where \(A_1\) is the surface area of the inner surface of the pipe and \(T_1 = T(r_1)\) is the temperature at the inner surface. \( q_{loss} = h \cdot 2 \pi r_1 L \cdot (T(r_1) - 250) \) Substitute the values of \(L\), \(r_1\), \(h\), and \(T(r_1)\) to calculate the rate of heat loss from the steam through the pipe.