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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

Photosynthesis, as it occurs in the leaves of a green plant, involves the transport of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) from the atmosphere to the chloroplasts of the leaves. The rate of photosynthesis may be quantified in terms of the rate of \(\mathrm{CO}_{2}\) assimilation by the chloroplasts. This assimilation is strongly influenced by \(\mathrm{CO}_{2}\) transfer through the boundary layer that develops on the leaf surface. Under conditions for which the density of \(\mathrm{CO}_{2}\) is \(6 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) in the air and \(5 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\) at the leaf surface and the convection mass transfer coefficient is \(10^{-2} \mathrm{~m} / \mathrm{s}\), what is the rate of photosynthesis in terms of kilograms of \(\mathrm{CO}_{2}\) assimilated per unit time and area of leaf surface?

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 \(V=20 \mathrm{~m} / \mathrm{s}\) ?

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 \(30 \mathrm{~mW}\).

In flow over a surface, velocity and temperature profiles are of the forms $$ \begin{aligned} &u(y)=A y+B y^{2}-C y^{3} \quad \text { and } \\ &T(y)=D+E y+F y^{2}-G y^{3} \end{aligned} $$ where the coefficients \(A\) through \(G\) are constants. Obtain expressions for the friction coefficient \(C_{f}\) and the convection coefficient \(h\) in terms of \(u_{z}, T_{x}\), and appropriate profile coefficients and fluid properties.

A wet-bulb thermometer consists of a mercury-in-glass thermometer covered with a wetted (water) fabric. When suspended in a stream of air, the steady-state thermometer reading indicates the wet-bulb temperature \(T_{\mathrm{ub}}\). Obtain an expression for determining the relative humidity of the air from knowledge of the air temperature \(\left(T_{\infty}\right)\), the wet-bulb temperature, and appropriate air and water vapor properties. If \(T_{\infty}=45^{\circ} \mathrm{C}\) and \(T_{w b}=25^{\circ} \mathrm{C}\), what is the relative humidity of the airstream?
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