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Chapter 2: Problem 154

A pipe is used for transporting boiling water in which the inner surface is at \(100^{\circ} \mathrm{C}\). The pipe is situated in a surrounding where the ambient temperature is \(20^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The pipe has a wall thickness of \(3 \mathrm{~mm}\) and an inner diameter of \(25 \mathrm{~mm}\), and it has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.003 \mathrm{~K}^{-1}\) and \(T\) is in \(\mathrm{K}\). Determine the outer surface temperature of the pipe.

Short Answer

Expert verified
Answer: The outer surface temperature of the pipe is approximately \(66.23 ^{\circ} \mathrm{C}\).

Step by step solution

01

Convert temperatures to Kelvin

First, we must convert the inner surface and ambient temperatures to Kelvin. \(T_{i} = 100^{\circ} \mathrm{C} + 273.15 = 373.15 \mathrm{K}\) \(T_{\infty} = 20^{\circ} \mathrm{C} + 273.15 = 293.15 \mathrm{K}\)
02

Calculate the heat transfer through the pipe using convection

Using the convection heat transfer formula, we can find the heat transfer, \(q\): \(q = hA(T_{i} - T_{\infty})\) Here, \(A = \pi dL\), where \(d = 25 \times 10^{-3} \mathrm{m}\) is the pipe's inner diameter and \(L\) is the length of the pipe. \(q = 50 \cdot \pi(25 \times 10^{-3})L (373.15 - 293.15)\)
03

Calculate the thermal conductivity at the average temperature

First, we need to find the average temperature between the inner surface and ambient temperatures: \(T_{avg} = \frac{T_{i} + T_{\infty}}{2} = 333.15 \mathrm{K}\) Now, we can calculate the thermal conductivity, \(k(T)\), at the average temperature: \(k(T_{avg}) = k_{0}(1 + \beta T_{avg}) = 1.5(1 + 0.003 \cdot 333.15)\)
04

Calculate the heat transfer through the pipe using conduction

Using the conduction heat transfer formula, we can find the heat transfer, \(q\): \(q = kA\frac{T_{i} - T_{o}}{t}\)
05

Equate the convection and conduction heat transfer equations and solve for the outer surface temperature

Now we set the convection and conduction heat transfer equations equal to each other and solve for \(T_{o}\): \(50 \cdot \pi(25 \times 10^{-3})L (373.15 - 293.15) = 1.5(1 + 0.003 \cdot 333.15) \cdot \pi(25 \times 10^{-3})L \frac{T_{i} - T_{o}}{3 \times 10^{-3}}\) Solve for \(T_{o}\): \(T_{o} = T_{i} - \frac{50(373.15 - 293.15)}{1.5(1 + 0.003 \cdot 333.15) \cdot \frac{1}{3 \times 10^{-3}}} = 339.38 \mathrm{K}\)
06

Convert the outer surface temperature back to Celsius

Our final step is to convert \(T_{o}\) back to Celsius: \(T_{o} = 339.38 \mathrm{K} - 273.15 = 66.23 ^{\circ} \mathrm{C}\) The outer surface temperature of the pipe is approximately \(66.23 ^{\circ} \mathrm{C}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convection Heat Transfer
Convection heat transfer is a process where heat is transferred between a solid surface and a liquid or gas. This happens through the motion of the fluid or gas against the surface. The movement enhances the heat exchange rate, making it a crucial mode of heat transfer.
This transfer is quantified by the convection heat transfer coefficient, denoted as \( h \). This coefficient varies depending on the fluid properties, the surface area involved, and the temperature difference between the surface and the surrounding fluid.
In the context of the pipe example, the ambient air at 20掳C cools the pipe's outer surface through convection. The formula used to calculate this transfer is:
\[ q = hA(T_i - T_{\infty}) \]
where:
  • \( q \) is the heat transfer rate
  • \( h \) is the convection heat transfer coefficient
  • \( A \) is the surface area through which heat is being transferred
  • \( T_i \) and \( T_{\infty} \) are the temperatures of the inner surface of the pipe and the ambient air, respectively
This equation underscores the importance of temperature differences in driving the rate of heat transfer through convection.
Conduction Heat Transfer
Conduction heat transfer involves the movement of heat within a solid or between solids in direct contact. It is a molecular process, where heat is transferred through collisions and diffusion of molecules inside the material.
This type of heat transfer is characterized by the material's thickness and thermal conductivity. In the case of the pipe, conduction takes place through the pipe's wall.
The basic formula to describe conduction heat transfer is:
\[ q = kA\frac{(T_i - T_o)}{t} \]
where:
  • \( q \) is the rate of heat transfer
  • \( k \) is thermal conductivity of the material
  • \( A \) is the cross-sectional area
  • \( T_i \) and \( T_o \) are the temperatures of the inner and outer surfaces of the pipe, respectively
  • \( t \) is the thickness of the pipe wall
Solving the exercise, the conduction equation helps establish a connection between the thermal energy lost from the inner surface to the outer surface, allowing us to calculate parameters like the outer surface temperature.
Thermal Conductivity
Thermal conductivity is a property of a material that indicates its ability to conduct heat. This property varies based on the material's structure, temperature, and other factors.
In the given exercise, the thermal conductivity of the pipe's material is not constant. It changes with temperature according to the formula:
\[ k(T) = k_0(1 + \beta T) \]
Where:
  • \( k_0 \) is the base thermal conductivity at a reference temperature
  • \( \beta \) is the coefficient that represents how thermal conductivity changes with temperature
  • \( T \) is the temperature in Kelvin
Calculating the average thermal conductivity is crucial as it influences how effectively the pipe material can conduct heat from the inner to the outer surface. In the solution, the knowledge of variable thermal conductivity, which depends on temperature, allows for a more accurate calculation of heat transmission through the pipe material.

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Most popular questions from this chapter

How do differential equations with constant coefficients differ from those with variable coefficients? Give an example for each type.

Is the thermal conductivity of a medium, in general, constant or does it vary with temperature?

When a long section of a compressed air line passes through the outdoors, it is observed that the moisture in the compressed air freezes in cold weather, disrupting and even completely blocking the air flow in the pipe. To avoid this problem, the outer surface of the pipe is wrapped with electric strip heaters and then insulated. Consider a compressed air pipe of length \(L=6 \mathrm{~m}\), inner radius \(r_{1}=3.7 \mathrm{~cm}\), outer radius \(r_{2}=4.0 \mathrm{~cm}\), and thermal conductivity \(k=14 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) equipped with a 300 -W strip heater. Air is flowing through the pipe at an average temperature of \(-10^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient on the inner surface is \(h=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming 15 percent of the heat generated in the strip heater is lost through the insulation, \((a)\) 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 material by solving the differential equation, and \((c)\) evaluate the inner and outer surface temperatures of the pipe.

Consider a spherical shell of inner radius \(r_{1}\), outer radius \(r_{2}\), thermal conductivity \(k\), and emissivity \(\varepsilon\). The outer surface of the shell is subjected to radiation to surrounding surfaces at \(T_{\text {surr }}\), but the direction of heat transfer is not known. Express the radiation boundary condition on the outer surface of the shell.

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.
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