Question

Consider a steam pipe of length \(L\), inner radius \(r_{1}\), outer radius \(r_{2}\), and constant thermal conductivity \(k\). Steam flows inside the pipe at an average temperature of \(T_{i}\) with a convection heat transfer coefficient of \(h_{i}\). The outer surface of the pipe is exposed to convection to the surrounding air at a temperature of \(T_{0}\) with a heat transfer coefficient of \(h_{o^{*}}\) Assuming steady one-dimensional heat conduction through the pipe, \((a)\) express the differential equation and the boundary conditions for heat conduction through the pipe material, \((b)\) obtain a relation for the variation of temperature in the pipe material by solving the differential equation, and (c) obtain a relation for the temperature of the outer surface of the pipe.

Short answer

Q: Express the heat conduction through the material of a steam pipe under steady-state conditions in terms of boundary conditions and temperature distribution. A: For steady-state heat conduction in a steam pipe, the governing equation in cylindrical coordinates is: \(\frac{1}{r} \frac{d}{d r}(r k \frac{dT}{dr}) = 0\) The boundary conditions are given at the inner surface and outer surface of the pipe as: \(h_i A (T_i - T(r_1)) = 2 \pi r_1 L (T_i - T(r_1))\) \(h_o^* A (T(r_2) - T_0) = 2 \pi r_2 L (T(r_2) - T_0)\) The temperature distribution in the pipe wall is given by the expression: \(T(r) = \frac{C_1}{k} \ln\vert r\vert + C_2\) The temperature at the outer surface of the pipe is: \(T(r_2) = \frac{-h_i k^2 (T_i - C_2)}{r_1 r_2} + C_2\)

Step-by-step solution

(a) Governing equation and boundary conditions

To find the governing equation for steady-state heat conduction through the pipe's material, we will use the heat equation in cylindrical coordinates, assuming one-dimensional heat conduction in the radial direction, which is given by: \(\frac{1}{r} \frac{d}{d r}(r k \frac{dT}{dr}) = 0\) To solve the problem, we need boundary conditions at the inner and outer surfaces of the pipe. At the inner surface of the pipe, the heat transfer rate is given by convection between the steam and the pipe wall: \(h_i A (T_i - T(r_1)) = 2 \pi r_1 L (T_i - T(r_1))\) At the outer surface of the pipe, the heat transfer rate is given as: \(h_o^* A (T(r_2) - T_0) = 2 \pi r_2 L (T(r_2) - T_0)\)

(b) Solve the governing equation for temperature distribution

Now we'll solve the differential equation for the temperature distribution in the pipe wall: Step 1: Separating variables \(\frac{1}{r} \frac{d}{dr}(r k \frac{dT}{dr}) = 0\) \(\frac{d}{dr}(r k \frac{dT}{dr}) = 0\) Step 2: Integrate the equation \(\int\frac{d}{dr}(r k \frac{dT}{dr}) dr = \int 0 dr\) \(r k \frac{dT}{dr} = C_1\) Step 3: Integrate once more \(k \frac{dT}{dr} = \frac{C_1}{r}\) \(\int\frac{dT}{ dr = \int\frac{C_1}{r k}dr\) \(T(r) = \frac{C_1}{k} \ln\vert r\vert + C_2\)

(c) Obtain the temperature at the outer surface of the pipe

Now we'll use the boundary conditions to find the temperature at the outer surface of the pipe: Step 1: Use the first boundary condition at \(r = r_1\) \(h_i (T_i - T(r_1)) = \frac{k}{r_1}(T(r_1) - T(r_2))\) Using the temperature distribution function we derived: \(h_i (T_i - (\frac{C_1}{k} \ln\vert r_1\vert + C_2)) = \frac{k}{r_1}(T(r_1) - T(r_2))\) Now, we can solve for \(C_1\) and \(C_2\): \(C_1 = -\frac{h_i k^2 (T_i - C_2)}{r_1}\) Step 2: Use the second boundary condition at \(r = r_2\) \(h_o^* (T(r_2) - T_0) = \frac{k}{r_2}(T(r_1) - T(r_2))\) Using the temperature distribution function and \(C_1\) expression: \(h_o^* (\frac{-h_i k^2 (T_i - C_2)}{r_1 r_2} + C_2 - T_0) = \frac{k}{r_2}(T(r_1) - T(r_2))\) Finally, solving for the temperature at the outer surface of the pipe: \(T(r_2) = \frac{-h_i k^2 (T_i - C_2)}{r_1 r_2} + C_2\) In conclusion, we obtained a relation for the temperature distribution in the pipe wall under steady-state heat conduction conditions. The temperature at the outer surface of the pipe is given by \(T(r_2) = \frac{-h_i k^2 (T_i - C_2)}{r_1 r_2} + C_2\).

Key concepts

Differential Equation for Heat Transfer

Heat conduction in pipes is a common scenario in engineering, often analyzed using cylindrical coordinates due to the geometry of pipes. The differential equation that describes heat transfer in a pipe assumes one-dimensional, radial heat conduction. This is expressed as:\[ \frac{1}{r} \frac{d}{dr}(r k \frac{dT}{dr}) = 0 \]Where:

• \( r \) is the radial coordinate.
• \( k \) is the thermal conductivity of the pipe's material, assumed constant.
• \( T \) is the temperature distribution within the pipe's wall.

This equation arises from the principle of energy conservation, asserting that heat flow into any radial segment equals the heat flow out, given a steady state with no heat accumulation. The assumption of one-dimensional heat flow simplifies the problem, allowing us to focus solely on radial changes without considering changes along the pipe’s length.
Recognizing this fundamental equation is crucial for analyzing the temperature changes and predicting where and how heat will spread through the pipe material. It lays the groundwork for determining how thermal conduction interacts with the pipe's boundary conditions to define the temperature profile.

Boundary Conditions in Heat Transfer

Boundary conditions are essential to completing the heat transfer analysis. They define how the pipe exchanges heat with its surroundings at the inner and outer surfaces. For the inner boundary (at the inner radius \(r_1\)), the heat transfer rate is due to convection, expressed by:\[ h_i A (T_i - T(r_1)) = 2 \pi r_1 L (T_i - T(r_1)) \] Where:

• \(h_i\) is the convection heat transfer coefficient between the steam and the pipe.
• \(A\) is the surface area, and \(L\) is the pipe's length.
• \(T_i\) is the steam temperature.

At the outer boundary (at the outer radius \(r_2\)), the heat transfer rate to the surrounding air, which is also convective, is described by: \[ h_o^* A (T(r_2) - T_0) = 2 \pi r_2 L (T(r_2) - T_0) \]Where:

• \(h_o^*\) is the convection heat transfer coefficient with the surrounding air.
• \(T_0\) is the ambient air temperature.

These boundary conditions are necessary to solve the differential equations. They account for the interaction between the pipe and its environment, allowing the determination of unknowns in the temperature distribution equation.

Temperature Distribution in Cylindrical Coordinates

Once the differential equation is established with proper boundary conditions, we move on to solving for the **temperature distribution** inside the pipe. The process involves integrating the main heat conduction equation:1. Start with the separated form: \(\frac{1}{r} \frac{d}{dr}(r k \frac{dT}{dr}) = 0\)2. Integrate once: \(rk \frac{dT}{dr} = C_1\)
Integrating a second time gives you the general solution:\[ T(r) = \frac{C_1}{k} \ln|r| + C_2 \]Here, \(C_1\) and \(C_2\) are constants determined using boundary conditions from previous calculations.
The final expression indicates how temperature varies from the inner to the outer radius. The logarithmic relation signifies that as one moves radially outward from the inner surface to the outer surface, the temperature changes in a specific logarithmic pattern. Understanding this pattern ensures the prediction of thermal behavior and assists engineers in managing heat loss or gain within pipe systems. Solving the temperature distribution thus requires applying the derived constants into the general solution, giving insight into precise thermal gradients and conditions in real-world situations.