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Chapter 8: Problem 7

Velocity and temperature profiles for laminar flow in a tube of radius \(r_{o}=10 \mathrm{~mm}\) have the form $$ \begin{aligned} &u(r)=0.1\left[1-\left(r / r_{o}\right)^{2}\right] \\ &T(r)=344.8+75.0\left(r / r_{o}\right)^{2}-18.8\left(r / r_{o}\right)^{4} \end{aligned} $$ with units of \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{K}\), respectively. Determine the corresponding value of the mean (or bulk) temperature, \(T_{\text {m }}\), at this axial position.

Short Answer

Expert verified
The mean temperature at the axial position is given by: \[ T_m = \frac{376.02 r_o^2}{L} \] where \(r_o\) is the tube's radius (10 mm) and \(L\) is the tube's length, which is not provided.

Step by step solution

01

Integrate temperature profile over the tube's volume

Since the temperature profile is given as a function of radius \(T(r)\), we need to integrate this over the total volume of the tube. This can be done by first integrating over the small volumes of cylindrical shells of radius \(r\) and thickness \(\mathrm{d}r\) and then integrating over the entire radius range. The volume of the cylindrical shell is given by \(dV = 2\pi r \mathrm{d}r\). So to find the average temperature over volume, we need to integrate \(T(r) dV\). **Step 2: Integrate over the radius**
02

Perform the integration

We will integrate \(T(r) dV\) over the radius range, from 0 to \(r_o\). Then, we will divide the result by the total volume of the tube, \(V\), to get the mean temperature. \[ T_{m} = \frac{\int_0^{r_o} T(r) \cdot 2\pi r \mathrm{d}r}{V} \] The volume of the tube is given by \(V = \pi r_o^2 L\), where \(L\) is the length of the tube (not provided, but will be canceled out in the final expression for \(T_m\)). **Step 3: Substitute the given temperature profile and integrate**
03

Insert temperature profile and solve the integral

Substitute the given temperature profile function into the integral and solve it: \[ T_{m} = \frac{\int_0^{r_o} \left(344.8+75.0\left(\frac{r}{r_o}\right)^{2}-18.8\left(\frac{r}{r_o}\right)^{4}\right) \cdot 2\pi r \mathrm{d}r}{\pi r_o^2 L} \] Now, perform the integration: \[ T_{m} = \frac{2\pi}{\pi r_o^2 L} \int_0^{r_o} \left(344.8 r + 75.0 \frac{r^3}{r_o^2} - 18.8 \frac{r^5}{r_o^4} \right) \mathrm{d}r \] Calculating the integral: \[ T_{m} = \frac{2\pi}{\pi r_o^2 L} \left[172.4 r^2 + 18.75 \frac{r^4}{r_o^2} - 3.14 \frac{r^6}{r_o^4} \right]_{0}^{r_o} \] **Step 4: Find the mean temperature**
04

Calculate mean temperature

Now, evaluate the integral at the bounds (0 and \(r_o\)) and simplify the expression to find the mean temperature: \[ T_{m} = \frac{2\pi}{\pi r_o^2 L} \left(172.4 r_o^2 + 18.75 r_o^2 - 3.14 r_o^2 \right) \] Cancel out \(\pi\) and \(r_o^2\): \[ T_{m} = \frac{2 (172.4 + 18.75 - 3.14) r_o^2}{r_o^2 L} = \frac{2 (188.01)r_o^2}{r_o^2 L} \] Since the tube length \(L\) is not provided, we can only provide the mean temperature in terms of \(L\): \[ T_m = \frac{376.02 r_o^2}{L} \]

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Key Concepts

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

Laminar Flow
Laminar flow refers to a type of fluid motion where a fluid flows in parallel layers, with no disruption between the layers. This flow pattern is smooth and orderly, which is in stark contrast to turbulent flow, where the fluid undergoes irregular fluctuations and mixing. In the context of heat transfer and fluid mechanics, understanding laminar flow is crucial because it greatly simplifies the analysis of fluid motion and temperature distribution.

For instance, in a tube with laminar flow, the velocity of the fluid can be described by a parabolic profile, where the fluid moves fastest at the centre and slows down towards the edges due to friction with the tube wall. This predictable pattern allows us to use mathematical models to accurately predict the behavior of the fluid in terms of both velocity and temperature, as demonstrated by the given problem from the textbook.
Temperature Profiles
Temperature profiles in fluid flow describe how temperature varies across different regions of the fluid. In the exercise, you encounter a quadratic temperature profile with respect to the radial position in a tube. Such profiles are particularly common in laminar flow situations where heat transfer is steady and symmetrical. The function given in the textbook problem, which includes a constant, a linear, and a squared term, represents how temperature varies from the tube's centre to its edge.

This profile is essential in calculating the mean or bulk temperature of the fluid because it tells you how the temperature is distributed across the fluid's cross-section. The mean temperature is effectively the 'average' temperature, weighted by the fluid's volume, which is an important factor in many engineering applications, such as designing heating or cooling systems.
Integration in Heat Transfer
Integration in heat transfer is a mathematical tool used to calculate various parameters like the mean temperature. By integrating the temperature profile over the volume of the fluid, as shown in the exercise's solution, you can find the average temperature of the fluid, which represents the energy state of the entire volume.

This approach can be visualized as summing up the infinitesimal contributions of heat energy from tiny volume elements (in this case, cylindrical shells) and then dividing by the total volume to get an average value. It's important to note that in practice, the actual integration becomes simpler because terms involving the tube length, L, cancel out, leaving an expression dependent only on the tube radius and constants from the temperature profile. This method and its mathematical execution are fundamental in many disciplines of engineering, where predicting heat behavior is vital.

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Most popular questions from this chapter

The surface of a 50 -mm-diameter, thin-walled tube is maintained at \(100^{\circ} \mathrm{C}\). In one case air is in cross flow over the tube with a temperature of \(25^{\circ} \mathrm{C}\) and a velocity of \(30 \mathrm{~m} / \mathrm{s}\). In another case air is in fully developed flow through the tube with a temperature of \(25^{\circ} \mathrm{C}\) and a mean velocity of \(30 \mathrm{~m} / \mathrm{s}\). Compare the heat flux from the tube to the air for the two cases.

A heating contractor must heat \(0.2 \mathrm{~kg} / \mathrm{s}\) of water from \(15^{\circ} \mathrm{C}\) to \(35^{\circ} \mathrm{C}\) using hot gases in cross flow over a thinwalled tube. Your assignment is to develop a series of design graphs that can be used to demonstrate acceptable combinations of tube dimensions ( \(D\) and \(L\) ) and of hot gas conditions ( \(T_{\infty}\) and \(V\) ) that satisfy this requirement. In your analysis, consider the following parameter ranges: \(D=20,30\), or \(40 \mathrm{~mm} ; L=3,4\), or \(6 \mathrm{~m} ; T_{\infty}=250,375\), or \(500^{\circ} \mathrm{C}\); and \(20 \leq V \leq 40 \mathrm{~m} / \mathrm{s}\).

Consider a cylindrical nuclear fuel rod of length \(L\) and diameter \(D\) that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate \(\dot{m}\), and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is, \(\dot{q}(x)=\dot{q}_{o} \sin (\pi x / L)\), where \(\dot{q}_{o}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\) is a constant. A uniform convection coefficient \(h\) may be assumed to exist between the surface of the rod and the water. (a) Obtain expressions for the local heat flux \(q^{\prime \prime}(x)\) and the total heat transfer \(q\) from the fuel rod to the water. (b) Obtain an expression for the variation of the mean temperature \(T_{m}(x)\) of the water with distance \(x\) along the tube. (c) Obtain an expression for the variation of the rod surface temperature \(T_{s}(x)\) with distance \(x\) along the tube. Develop an expression for the \(x\)-location at which this temperature is maximized.

A cold plate is an active cooling device that is attached to a heat-generating system in order to dissipate the heat while maintaining the system at an acceptable temperature. It is typically fabricated from a material of high thermal conductivity, \(k_{\text {cp, }}\), within which channels are machined and a coolant is passed. Consider a copper cold plate of height \(H\) and width \(W\) on a side, within which water passes through square channels of width \(w=h\). The transverse spacing between channels \(\delta\) is twice the spacing between the sidewall of an outer channel and the sidewall of the cold plate. Consider conditions for which equivalent heat-generating systems are attached to the top and bottom of the cold plate, maintaining the corresponding surfaces at the same temperature \(T_{s}\). The mean velocity and inlet temperature of the coolant are \(u_{m}\) and \(T_{m i}\), respectively. (a) Assuming fully developed turbulent flow throughout each channel, obtain a system of equations that may be used to evaluate the total rate of heat transfer to the cold plate, \(q\), and the outlet temperature of the water, \(T_{m, o}\), in terms of the specified parameters. (b) Consider a cold plate of width \(W=100 \mathrm{~mm}\) and height \(H=10 \mathrm{~mm}\), with 10 square channels of width \(w=6 \mathrm{~mm}\) and a spacing of \(\delta=4 \mathrm{~mm}\) between channels. Water enters the channels at a temperature of \(T_{m, i}=300 \mathrm{~K}\) and a velocity of \(u_{m}=2 \mathrm{~m} / \mathrm{s}\). If the top and bottom cold plate surfaces are at \(T_{s}=360 \mathrm{~K}\), what is the outlet water temperature and the total rate of heat transfer to the cold plate? The thermal conductivity of the copper is \(400 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), while average properties of the water may be taken to be \(\rho=984 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4184 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=489 \times\) \(10^{-6} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.65 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(P r=3.15\). Is this a good cold plate design? How could its performance be improved?

In the final stages of production, a pharmaceutical is sterilized by heating it from 25 to \(75^{\circ} \mathrm{C}\) as it moves at \(0.2 \mathrm{~m} / \mathrm{s}\) through a straight thin-walled stainless steel tube of \(12.7=\mathrm{mm}\) diameter. A uniform heat flux is maintained by an electric resistance heater wrapped around the outer surface of the tube. If the tube is \(10 \mathrm{~m}\) long, what is the required heat flux? If fluid enters the tube with a fully developed velocity profile and a uniform temperature profile, what is the surface temperature at the tube exit and at a distance of \(0.5 \mathrm{~m}\) from the entrance? Fluid properties may be approximated as \(\rho=\) \(1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, m=2 \times 10^{-3} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}\), \(k=0.8 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(P r=10\).
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