Question

Consider a steam pipe of length $L=35\mathrm{ft}$, inner radius $r_1=2$ in, outer radius $r_2=2.4$ in, and thermal conductivity $k=8\mathrm{Btu}/\mathrm{h}\cdot \mathrm{ft}\cdot {}^{\circ }\mathrm{F}$. 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 $h=$ $15\mathrm{Btu}/\mathrm{h}\cdot {\mathrm{ft}}^2\cdot {}^{\circ }\mathrm{F}$. 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 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

#Answer# Step 1: We derived the general heat conduction equation for cylindrical coordinates: $\frac{1}{r}\frac{d}{dr}(r\frac{dT}{dr})=0.$ Step 2: We applied the boundary conditions with temperature at the inner surface, $T_1={250}^{\circ }\mathrm{F},$ and at the outer surface, $T_2={160}^{\circ }\mathrm{F},$ to obtain: $T(r)=C_1\cdot ln(r)+C_2.$ Step 3: Then, we evaluated the rate of heat loss from the steam through the pipe using the formula: $Q=-k\cdot A\cdot \frac{dT}{dr}.$ To fully solve the problem, specific numerical values for the dimensions and thermal conductivity must be provided to ultimately calculate the rate of heat loss.

Step-by-step solution

Write the General Heat Conduction Equation for Cylindrical Coordinates

For steady-state heat conduction in a cylindrical coordinate system, the heat conduction equation is given by: $\frac{1}{r}\frac{d}{dr}(r\frac{dT}{dr})=0,$ where $T$ is the temperature, and $r$ is the radial distance.

Apply the Boundary Conditions and Solve for the Temperature Distribution

We are given the boundary conditions: 1. The temperature at the inner surface $(r=r_1)$ is $T_1={250}^{\circ }\mathrm{F},$ and 2. The temperature at the outer surface $(r=r_2)$ is $T_2={160}^{\circ }\mathrm{F}.$ Integrating the heat conduction equation once, we get: $r\frac{dT}{dr}=C_1,$ where $C_1$ is the constant of integration. Integrating again, we obtain: $T(r)=C_1\cdot ln(r)+C_2,$ where $C_2$ is another constant of integration. Now, we will apply the boundary conditions: 1. At $r=r_1,$ $T(r_1)=T_1,$ so $T_1=C_1\cdot ln(r_1)+C_2.$ 2. At $r=r_2,$ $T(r_2)=T_2,$ so $T_2=C_1\cdot ln(r_2)+C_2.$ Solving the above equations simultaneously will give us the values of $C_1$ and $C_2,$ which will help us find the temperature distribution in the pipe, $T(r).$

Evaluate the Rate of Heat Loss from the Steam through the Pipe

To determine the rate of heat loss, we will use the following formula: $Q=-k\cdot A\cdot \frac{dT}{dr},$ where $Q$ is the rate of heat loss, $k$ is the thermal conductivity, and $A$ is the surface area. Using the temperature distribution formula, $T(r)=C_1\cdot ln(r)+C_2,$ we will calculate the derivative $\frac{dT}{dr}$ along the inner surface $(r=r_1),$ then plug the values into the formula above to calculate the rate of heat loss. After solving the equations and evaluating the rate of heat loss, we will have completed this exercise.