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Chapter 6: Problem 13

The heat transfer rate per unit width (normal to the page) from a longitudinal section, \(x_{2}-x_{1}\), can be expressed as \(q_{12}^{\prime}=\bar{h}_{12}\left(x_{2}-x_{1}\right)\left(T_{s}-T_{\infty}\right)\), where \(\bar{h}_{12}\) is the average coefficient for the section of length \(\left(x_{2}-x_{1}\right)\). Consider laminar flow over a flat plate with a uniform temperature \(T_{s}\). The spatial variation of the local convection coefficient is of the form \(h_{x}=C x^{-1 / 2}\), where \(C\) is a constant. (a) Beginning with the convection rate equation in the form \(d q^{\prime}=h_{s} d x\left(T_{s}-T_{x}\right)\), derive an expression for \(\bar{h}_{12}\) in terms of \(C, x_{1}\), and \(x_{2}\). (b) Derive an expression for \(\bar{h}_{12}\) in terms of \(x_{1}, x_{2}\), and the average coefficients \(\bar{h}_{1}\) and \(\bar{h}_{2}\), corresponding to lengths \(x_{1}\) and \(x_{2}\), respectively.

Short Answer

Expert verified
As a short answer: (a) The expression for \(\bar{h}_{12}\) in terms of \(C\), \(x_1\), and \(x_2\) is: \( \bar{h}_{12} = \frac{2C(\sqrt{x_2} - \sqrt{x_1})}{(x_2 - x_1)} \) (b) The expression for \(\bar{h}_{12}\) in terms of \(x_1\), \(x_2\), \(\bar{h}_1\), and \(\bar{h}_2\) is: \( \bar{h}_{12} = \frac{x_1 \frac{C}{\sqrt{x_1}} + x_2 \frac{C}{\sqrt{x_2}}}{x_2 - x_1} \)

Step by step solution

01

(a) Derive the expression for \(\bar{h}_{12}\) in terms of C, \(x_{1}\), and \(x_{2}\)

First, we will use the given convection rate equation: \(dq' = h_s dx(T_s - T_x)\). We know that the spatial variation of the local convection coefficient is given by \(h_x = Cx^{-1/2}\), so we can substitute this expression into the convection rate equation. This will give us: \( dq' = Cx^{-1/2} dx(T_s - T_x) \) Now, we will integrate both sides with respect to x, to find \(q'\): \( \int_{x_1}^{x_2} dq' = \int_{x_1}^{x_2} Cx^{-1/2} dx(T_s - T_x) \) Perform the integration on the right side: \( q' = C(T_s - T_\infty) \int_{x_1}^{x_2} x^{-1/2} dx \) \( q' = 2C(T_s - T_\infty)\left[ x^{1/2} \right]_{x_1}^{x_2} = 2C(T_s - T_\infty)(\sqrt{x_2} - \sqrt{x_1}) \) Now, we can find the average heat transfer coefficient \(\bar{h}_{12}\) over the length \((x_2 - x_1)\) using the equation \(q'_{12} = \bar{h}_{12}(x_2 - x_1)(T_s - T_\infty)\): \( \bar{h}_{12} = \frac{q'_{12}}{(x_2 - x_1)(T_s - T_\infty)} = \frac{2C(\sqrt{x_2} - \sqrt{x_1})}{(x_2 - x_1)} \) So, the expression for \(\bar{h}_{12}\) in terms of \(C\), \(x_1\), and \(x_2\) is: \( \bar{h}_{12} = \frac{2C(\sqrt{x_2} - \sqrt{x_1})}{(x_2 - x_1)} \)
02

(b) Derive the expression for \(\bar{h}_{12}\) in terms of \(x_{1}\), \(x_{2}\), \(\bar{h}_{1}\), and \(\bar{h}_{2}\)

To derive an expression for the average heat transfer coefficient \(\bar{h}_{12}\) in terms of \(x_1\), \(x_2\), \(\bar{h}_1\), and \(\bar{h}_2\), we will use the equation for the average coefficient for the sections of lengths \(x_1\) and \(x_2\): \( \bar{h}_1 = \frac{C}{\sqrt{x_1}} \) and \( \bar{h}_2 = \frac{C}{\sqrt{x_2}} \) Now, we will multiply both sides of each equation by their respective lengths and add them together: \((x_2 - x_1) \bar{h}_{12} = x_1 \bar{h}_1 + x_2 \bar{h}_2 \) Substitute the expressions for \(\bar{h}_1\) and \(\bar{h}_2\): \((x_2 - x_1) \bar{h}_{12} = x_1 \frac{C}{\sqrt{x_1}} + x_2 \frac{C}{\sqrt{x_2}} \) Solve for \(\bar{h}_{12}\): \( \bar{h}_{12} = \frac{x_1 \frac{C}{\sqrt{x_1}} + x_2 \frac{C}{\sqrt{x_2}}}{x_2 - x_1} \) Finally, the expression for \(\bar{h}_{12}\) in terms of \(x_1\), \(x_2\), \(\bar{h}_1\), and \(\bar{h}_2\) is: \( \bar{h}_{12} = \frac{x_1 \frac{C}{\sqrt{x_1}} + x_2 \frac{C}{\sqrt{x_2}}}{x_2 - x_1} \)

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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 fluid moving smoothly along in layers without disruption between them. This type of flow is often observed when a fluid flows over a flat surface at low speed. In the context of heat transfer, laminar flow plays a key role in understanding how thermal energy is conveyed in fluids.
  • Particles travel in parallel paths, leading to minimal mixing and momentum exchange between the layers.
  • Laminar flow usually occurs at lower Reynolds numbers, typically less than 2000.
  • This stable flow type is characterized by streamlined motion, which allows for easier calculation of variables such as the heat transfer coefficient.
Looking at this particular exercise, the flow of air or fluid over a flat plate would be considered laminar, meaning calculations can assume a consistent transfer layer is maintained throughout the flow's path on the plate.
Heat Transfer Coefficient
The heat transfer coefficient is a measure of how easily heat moves from a surface into or out of a fluid. It's important when designing systems for effectively transferring energy, such as radiant heat barriers or liquid cooling systems.
  • Represented as \( h \), the heat transfer coefficient is used to quantify convection occurring at a specific point or over an average distance.
  • In the laminar flow over a flat plate scenario, the local heat transfer coefficient \( h_x \) is defined as a function of position: \( h_x = C x^{-1/2} \).
  • The average heat transfer coefficient across a distance \([x_1, x_2] \) offers a simplified expression for system design.
In exercises like this, understanding the spatial variables and how they affect \( h \) becomes crucial, as opposed to turbulent flow scenarios where behavior might be less predictable.
Flat Plate
A flat plate in convection heat transfer studies often serves as a reference scenario to analyze and simplify flow dynamics. When fluid approaches, flows over, and leaves the plate, engineers can measure and predict heat transfer rates effectively.
  • The geometric simplicity allows for detailed analysis of boundary layers, which are thin regions near surfaces where flow velocity gradients and temperature gradients occur due to viscosity.
  • The flat plate is used in many engineering applications ranging from wind tunnels to thermal analysis, making its study crucial for foundational heat transfer knowledge.
  • In laminar flow context, it's assumed that the fluid successfully wets the plate, ensuring consistent measurement of variables like the heat transfer coefficient \( \bar{h}_{12} \).
Through exercises utilizing a flat plate, students can grasp core concepts of convection, one of the three methods of heat transfer alongside conduction and radiation.

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Most popular questions from this chapter

Dry air at \(32^{\circ} \mathrm{C}\) flows over a wetted (water) plate of \(0.2 \mathrm{~m}^{2}\) area. The average convection coefficient is \(\bar{h}=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the heater power required to maintain the plate at a temperature of \(27^{\circ} \mathrm{C}\) is \(432 \mathrm{~W}\). Estimate the power required to maintain the wetted plate at a temperature of \(37^{\circ} \mathrm{C}\) in dry air at \(32^{\circ} \mathrm{C}\) if the convection coefficients remain unchanged.

Consider cross flow of gas \(\mathrm{X}\) over an object having a characteristic length of \(L=0.1 \mathrm{~m}\). For a Reynolds number of \(1 \times 10^{4}\), the average heat transfer coefficient is \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The same object is then impregnated with liquid \(Y\) and subjected to the same flow conditions. Given the following thermophysical properties, what is the average convection mass transfer coefficient?

Consider airflow over a flat plate of length \(L=1 \mathrm{~m}\) under conditions for which transition occurs at \(x_{c}=0.5 \mathrm{~m}\) based on the critical Reynolds number, \(R e_{x, c}=5 \times 10^{5}\). (a) Evaluating the thermophysical properties of air at \(350 \mathrm{~K}\), determine the air velocity. (b) In the laminar and turbulent regions, the local convection coefficients are, respectively, \(h_{\text {lam }}(x)=C_{\text {lam }} x^{-05}\) and \(h_{\text {marb }}=C_{\text {marb }} x^{-0.2}\) where, at \(T=350 \mathrm{~K}, C_{\text {lum }}=8.845 \mathrm{~W} / \mathrm{m}^{3 / 2} \cdot \mathrm{K}, C_{\text {tub }}=\) \(49.75 \mathrm{~W} / \mathrm{m}^{1.8} \cdot \mathrm{K}\), and \(x\) has units of \(\mathrm{m}\). Develop an expression for the average convection coefficient, \(\bar{h}_{\mathrm{hm}}(x)\), as a function of distance from the leading edge, \(x\), for the laminar region, \(0 \leq x \leq x_{x}\). (c) Develop an expression for the average convection coefficient, \(\bar{h}_{\text {art }}(x)\), as a function of distance from the leading edge, \(x\), for the turbulent region, \(x_{c} \leq x \leq L\). (d) On the same coordinates, plot the local and average convection coefficients, \(h_{x}\) and \(\bar{h}_{x}\), respectively, as a function of \(x\) for \(0 \leq x \leq L\).

As a means of preventing ice formation on the wings of a small, private aircraft, it is proposed that electric resistance heating elements be installed within the wings. To determine representative power requirements, consider nominal flight conditions for which the plane moves at \(100 \mathrm{~m} / \mathrm{s}\) in air that is at a temperature of \(-23^{\circ} \mathrm{C}\). If the characteristic length of the airfoil is \(L=2 \mathrm{~m}\) and wind tunnel measurements indicate an average friction coefficient of \(\bar{C}_{f}=0.0025\) for the nominal conditions, what is the average heat flux needed to maintain a surface temperature of \(T_{s}=5^{\circ} \mathrm{C}\) ?

Experiments have shown that the transition from laminar to turbulent conditions for flow normal to the axis of a long cylinder occurs at a critical Reynolds number of \(R e_{D,} \approx 2 \times 10^{5}\), where \(D\) is the cylinder diameter. Moreover, the transition from incompressible to compressible flow occurs at a critical Mach number of \(M a_{e}=0.3\). For air at a pressure of \(p=1 \mathrm{~atm}\) and temperature \(T=27^{\circ} \mathrm{C}\), determine the critical cylinder diameter \(D_{c}\) below which, if the flow is turbulent, compressibility effects are likely to be important.
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