Question

Consider a steam pipe of length \(L=35 \mathrm{ft}\), inner radius $r_{1}=2 \mathrm{in}\(, outer radius \)r_{2}=24\( in, and thermal conductivity \)k=8 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}{ }^{\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 $h=15 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2},{ }^{\circ} \mathrm{F}$. 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 one-dimensional 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 rate of heat loss from the steam through a pipe. Given information: Length of the pipe (L) = 35 ft Radius of the pipe: inner (r1) = 2 in, outer (r2) = 24 in Thermal conductivity of the pipe material (k) = 15 Btu/(h·ft·°F) Steam temperature (T_steam) = 320°F Heat transfer coefficient (h) = 15 Btu/(h·ft²·°F) Temperature at the outer surface of the pipe (T2) = 160°F Answer: To calculate the rate of heat loss from the steam through the pipe, follow these steps: 1. Express the heat conduction equation in cylindrical coordinates for the radial direction. 2. Set up boundary conditions. 3. Integrate the differential equation. 4. Apply the boundary conditions to find the constants. 5. Find the temperature distribution in the pipe. 6. Calculate the heat loss rate using the convection heat transfer equation and the first boundary condition. After following these steps, you can determine the rate of heat loss from the steam through the pipe.

Step-by-step solution

Determine heat conduction equation

We have a cylindrical pipe, and we need to express the heat conduction equation in cylindrical coordinates for the radial direction: \(\frac{d}{dr}(k \frac{dT}{dr})=0\), where \(T\) is the temperature, \(r\) is the radial distance from the pipe's center, and \(k\) is the thermal conductivity.

Set up boundary conditions

At \(r=r_1 = 2\) in, the inner surface of the pipe, the convection heat transfer is given: \(q_{conv}=h(T_1 - T_{steam})\), where \(T_1\) is the temperature at the inner surface. The heat conduction at the inner surface is: \(q_{cond}=-k \frac{dT}{dr}\big|_{r=r_1}\). At steady state condition, \(q_{conv}=q_{cond}\). At \(r=r_2 = 24\) in, the outer surface of the pipe, the temperature is given: \(T_2=160^{\circ} F\). This is the second boundary condition. #b) Solving the Differential Equation#

Integrate the differential equation

Integrate the heat conduction equation: \(\frac{d}{dr}(k \frac{dT}{dr})=0\). Integration with respect to \(r\) yields: \(k\frac{dT}{dr}=C_1\), where \(C_1\) is a constant. Integrate the second equation with respect to \(r\) to find the temperature distribution in the cylindrical pipe: \(T(r)=\frac{C_1}{k}r+C_2\), where \(C_2\) is another constant.

Apply the boundary conditions

Apply the boundary conditions to find the constants \(C_1\) and \(C_2\). At \(r=r_1 = 2\) in and \(r=r_2 = 24\) in, we have \(T_1\) and \(T_2\), respectively. By plugging the conditions, we get the following system of equations: \(T_1= \frac{C_1}{k} r_1 + C_2\) and \(T_2= \frac{C_1}{k} r_2+C_2\) Now, use the first boundary condition \(q_{conv}=q_{cond}\): \(h(T_1 - T_{steam})=-k \frac{dT}{dr}\big|_{r=r_1} \) By plugging the values of \(h\), \(T_{steam}\), and \(-k \frac{dT}{dr}\big|_{r=r_1} =C_1\), we can solve for \(T_1\). Afterward, we can solve for \(C_1\) and \(C_2\) using the system of equations above.

Find temperature distribution

Using the values of \(C_1\) and \(C_2\), plug them back into the temperature equation: \(T(r)=\frac{C_1}{k}r+C_2\). Now, we have a relation for the temperature distribution in the pipe. #c) Rate of Heat Loss Through the Pipe#

Calculate heat loss rate

Now that we have the temperature distribution, we can calculate the rate of heat loss from the steam through the pipe. Using the convection heat transfer equation and the first boundary condition: \(q_{loss} = 2 \pi r_1 L h (T_1 - T_{steam})\) Here, we can plug in the values for \(r_1 = 2\) in, \(L = 35\) ft, \(h = 15 \frac{Btu}{h\cdot ft^2\cdot^{\circ}F}\), and \(T_1\), which was found in the previous step. The resulting value will give us the rate of heat loss from the steam through the pipe.