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Chapter 6: Problem 7

Parallel flow of atmospheric air over a flat plate of length \(L=3 \mathrm{~m}\) is disrupted by an array of stationary rods placed in the flow path over the plate. Laboratory measurements of the local convection coefficient at the surface of the plate are made for a prescribed value of \(V\) and \(T_{x}>T_{x}\). The results are correlated by an expression of the form \(h_{x}=0.7+13.6 x-3.4 x^{2}\), where \(h_{x}\) has units of \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(x\) is in meters. Evaluate the average convection coefficient \(\bar{h}_{L}\) for the entire plate and the ratio \(\bar{h}_{L} / h_{L}\) at the trailing edge.

Short Answer

Expert verified
The average convection coefficient \(\bar{h}_L\) for the entire plate is \(4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the ratio \(\bar{h}_L / h_L\) at the trailing edge is approximately \(0.5526\).

Step by step solution

01

Calculate average convection coefficient \(\bar{h}_L\)

First, we need to find the average convection coefficient \(\bar{h}_L\) for the entire plate using the given equation for the local convection coefficient \(h_x = 0.7 + 13.6x - 3.4x^2\). To do this, we integrate \(h_x\) over the length of the plate (\(0\) to \(L\), where \(L = 3\,\text{m}\)) and divide by the length of the plate: \[\bar{h}_L = \frac{1}{L} \int_{0}^{L} h_x \,\mathrm{d}x\] Substitute the given equation for \(h_x\): \[\bar{h}_L = \frac{1}{3} \int_{0}^{3} (0.7 + 13.6x - 3.4x^2) \,\mathrm{d}x\]
02

Integrate the given expression of \(h_x\)

Now integrate the expression of \(h_x\) with respect to \(x\): \[\int (0.7 + 13.6x - 3.4x^2) \,\mathrm{d}x = 0.7x + \frac{13.6}{2}x^2 - \frac{3.4}{3}x^3 + C\] Since the integration is definite, we don't need to include the constant \(C\). Now we can evaluate the integral at the limits: \begin{align*} \int_{0}^{3} (0.7 + 13.6x - 3.4x^2) \,\mathrm{d}x &= \left[ 0.7x + \frac{13.6}{2}x^2 - \frac{3.4}{3}x^3 \right]_{0}^{3} \\ &= \left(0.7(3) + \frac{13.6}{2}(3)^2 - \frac{3.4}{3}(3)^3\right) - \left(0.7(0) + \frac{13.6}{2}(0)^2 - \frac{3.4}{3}(0)^3\right) \\ &= 12.6 \end{align*}
03

Calculate the average convection coefficient \(\bar{h}_L\)

Now that we have the integral value, we can calculate the average convection coefficient \(\bar{h}_L\): \[\bar{h}_L = \frac{1}{3}(12.6) = 4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\] The average convection coefficient \(\bar{h}_L\) for the entire plate is \(4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\).
04

Calculate \(h_L\) at the trailing edge

To find the local convection coefficient at the trailing edge, evaluate \(h_x\) at \(x = L = 3\,\text{m}\): \[h_L = 0.7 + 13.6(3) - 3.4(3)^2 = 7.6\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\]
05

Calculate the ratio \(\bar{h}_L / h_L\) at the trailing edge

Now, we can calculate the ratio \(\bar{h}_L / h_L\) at the trailing edge: \[\frac{\bar{h}_L}{h_L} = \frac{4.2}{7.6} = 0.5526\] The ratio \(\bar{h}_L / h_L\) at the trailing edge is approximately \(0.5526\). In summary, the average convection coefficient \(\bar{h}_L\) for the entire plate is \(4.2\, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the ratio \(\bar{h}_L / h_L\) at the trailing edge is approximately \(0.5526\).

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Key Concepts

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

Heat Transfer
Heat transfer is a fundamental concept in engineering and physics, describing the movement of thermal energy from one object to another.
This process is essential for understanding how objects warm up or cool down.
In heat transfer, thermal energy can move through three main methods: conduction, convection, and radiation. - **Conduction** occurs when heat moves through a solid material from a high temperature area to a low temperature region.
This process relies on the direct contact of molecules. - **Convection** involves the transfer of heat through fluids, which can be liquids or gases.
When a fluid flows over a surface, it can carry away or supply heat, depending on its temperature relative to the surface.
The convection coefficient is a measure of how effectively this transfer takes place. - **Radiation** involves the transfer of heat through electromagnetic waves without requiring a medium. Understanding these processes helps in calculating the convection coefficient, which is central to the problem.
The convection coefficient describes how well a fluid can absorb or disperse heat while flowing over a surface like a flat plate.
Parallel Flow
In fluid mechanics, parallel flow refers to when a fluid moves over a surface with streamline patterns remaining relatively uniform and consistent.
This type of flow is common in heat transfer problems, especially when analyzing air or liquid moving over flat surfaces.
Parallel flow ensures that all points along the flow path experience relatively similar conditions. With parallel flow, evaluating the heat transfer becomes more straightforward.
One can model the heat transfer using known equations that depend on factors like fluid velocity, surface characteristics, and temperature differences.
In exercises like the one provided, laboratory measurements of convection coefficients help illustrate how parallel flow affects heat distribution across the surface. Parallel flow in particular allows us to easily integrate heat transfer equations over a specified surface length.
This enables the calculation of properties like the average convection coefficient, which offers insight into the overall heat transfer effects throughout the surface area.
Understanding parallel flow is essential for predicting how efficient heat transfer will be during different scenarios.
Flat Plate
The flat plate is a simple yet widely used model in heat transfer analysis.
Researchers and engineers employ flat plates to understand fundamental concepts of thermal energy distribution and transfer. - A flat plate provides a consistent surface over which controlled experiments can take place.
This includes analyzing air or fluid flow and measuring temperature changes as heat moves through the material. - The flat plate assumption simplifies many mathematical models. This simplification is vital because it allows for basic, accurate predictions about how convection and other forms of heat transfer occur. In the exercise, the flat plate is crucial for the analysis of the convection coefficient.
Using the formula provided, the convection coefficient changes along the length of the plate as defined by the length parameter, illustrating how local conditions can vary. Understanding the role of a flat plate makes it easier to grasp more complex heat transfer scenarios.
It provides an essential foundational model for assessing the effectiveness and dynamics of heat transfer mechanisms in engineering applications.

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Most popular questions from this chapter

An object of irregular shape has a characteristic length of \(L=1 \mathrm{~m}\) and is maintained at a uniform surface temperature of \(T_{s}=400 \mathrm{~K}\). When placed in atmospheric air at a temperature of \(T_{x}=300 \mathrm{~K}\) and moving with a velocity of \(V=100 \mathrm{~m} / \mathrm{s}\), the average heat flux from the surface to the air is \(20,000 \mathrm{~W} / \mathrm{m}^{2}\). If a second object of the same shape, but with a characteristic length of \(L=5 \mathrm{~m}\), is maintained at a surface temperature of \(T_{s}=400 \mathrm{~K}\) and is placed in atmospheric air at \(T_{\infty}=300 \mathrm{~K}\), what will the value of the average convection coefficient be if the air velocity is \(V=20 \mathrm{~m} / \mathrm{s}\) ?

Species A is evaporating from a flat surface into species B. Assume that the concentration profile for species A in the concentration boundary layer is of the form \(C_{\mathrm{A}}(y)=D y^{2}+E y+F\), where \(D, E\), and \(F\) are constants at any \(x\)-location and \(y\) is measured along a normal from the surface. Develop an expression for the mass transfer convection coefficient \(h_{w}\) in terms of these constants, the concentration of \(A\) in the free stream \(C_{\mathrm{A}, \infty}\) and the mass diffusivity \(D_{\mathrm{AB}}\). Write an expression for the molar flux of mass transfer by convection for species \(A\).

Forced air at \(T_{\infty}=25^{\circ} \mathrm{C}\) and \(V=10 \mathrm{~m} / \mathrm{s}\) is used to cool electronic elements on a circuit board. One such element is a chip, \(4 \mathrm{~mm} \times 4 \mathrm{~mm}\), located \(120 \mathrm{~mm}\) from the leading edge of the board. Experiments have revealed that flow over the board is disturbed by the elements and that convection heat transfer is correlated by an expression of the form Estimate the surface temperature of the chip if it is dissipating \(30 \mathrm{~mW}\).

Experiments to determine the local convection heat transfer coefficient for uniform flow normal to a heated circular disk have yielded a radial Nusselt number distribution of the form $$ N u_{D}=\frac{h(r) D}{k}=N u_{o}\left[1+a\left(\frac{r}{r_{o}}\right)^{n}\right] $$ where both \(n\) and \(a\) are positive. The Nusselt number at the stagnation point is correlated in terms of the Reynolds \(\left(R e_{D}=V D / v\right)\) and Prandtl numbers $$ N u_{o}=\frac{h(r=0) D}{k}=0.814 \operatorname{Re}_{D}^{1 / 2} \mathrm{Pr}^{0.36} $$ Obtain an expression for the average Nusselt number, \(\overline{N u}_{D}=\bar{h} D / k\), corresponding to heat transfer from an isothermal disk. Typically, boundary layer development from a stagnation point yields a decaying convection coefficient with increasing distance from the stagnation point. Provide a plausible explanation for why the opposite trend is observed for the disk.

A 2-mm-thick layer of water on an electrically heated plate is maintained at a temperature of \(T_{w}=340 \mathrm{~K}\), as dry air at \(T_{\infty}=300 \mathrm{~K}\) flows over the surface of the water (case A). The arrangement is in large surroundings that are also at \(300 \mathrm{~K}\). (a) If the evaporative flux from the surface of the water to the air is \(n_{\mathrm{A}}^{\prime \prime}=0.030 \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2}\), what is the corresponding value of the convection mass transfer coefficient? How long will it take for the water to completely evaporate? (b) What is the corresponding value of the convection heat transfer coefficient and the rate at which electrical power must be supplied per unit area of the plate to maintain the prescribed temperature of the water? The emissivity of water is \(\varepsilon_{w}=0.95\). (c) If the electrical power determined in part (b) is maintained after complete evaporation of the water (case B), what is the resulting temperature of the plate, whose emissivity is \(\varepsilon_{p}=0.60\) ?
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