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Chapter 6: Problem 5

For laminar flow over a flat plate, the local heat transfer coefficient \(h_{x}\) is known to vary as \(x^{-1 / 2}\), where \(x\) is the distance from the leading edge \((x=0)\) of the plate. What is the ratio of the average coefficient between the leading edge and some location \(x\) on the plate to the local coefficient at \(x\) ?

Short Answer

Expert verified
The ratio of the average coefficient between the leading edge and some location \(x\) on the plate to the local coefficient at \(x\) is 2.

Step by step solution

01

Write the given relation for local heat transfer coefficient \(h_{x}\)

The local heat transfer coefficient \(h_{x}\) is given to be proportional to \(x^{-1/2}\): \[h_{x} = Cx^{-1/2}\] Where \(C\) is the proportionality constant.
02

Calculate the average coefficient over the distance between leading edge and some location \(x\)

To find the average coefficient, we need to integrate \(h_{x}\) over the distance \(x\) from the leading edge and then divide by the total distance \(x\). The average heat transfer coefficient \(h_{avg}\) is given by: \[h_{avg} = \frac{1}{x}\int_{0}^{x} h_{x'} dx'\]
03

Substitute the expression of \(h_{x}\) in the integral for \(h_{avg}\)

Substituting the expression for \(h_{x}\) in the expression for \(h_{avg}\): \[h_{avg} = \frac{1}{x}\int_{0}^{x} Cx'^{-1/2} dx'\]
04

Evaluate the integral for \(h_{avg}\)

Now let's evaluate the integral: \[h_{avg} = \frac{C}{x}\left[\int_{0}^{x} x'^{-1/2} dx'\right]\] \[h_{avg} = \frac{C}{x} \left[2x'^{1/2}\right]_{0}^{x}\]
05

Solve for \(h_{avg}\)

Solve for \(h_{avg}\): \[h_{avg} = \frac{C}{x} \left[2x^{1/2} - 0\right]\] \[h_{avg} = 2Cx^{-1/2}\]
06

Find the ratio of \(h_{avg}\) to \(h_{x}\)

Now we can calculate the desired ratio by dividing \(h_{avg}\) by \(h_{x}\): \[\frac{h_{avg}}{h_{x}} = \frac{2Cx^{-1/2}}{Cx^{-1/2}}\]
07

Simplify the ratio and find the answer

Simplify the ratio: \[\frac{h_{avg}}{h_{x}} = \frac{2Cx^{-1/2}}{Cx^{-1/2}} = 2\] Therefore, the ratio of the average coefficient between the leading edge and some location \(x\) on the plate to the local coefficient at \(x\) is 2.

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Key Concepts

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

Local Heat Transfer Coefficient
Understanding the local heat transfer coefficient is crucial for predicting how heat will transfer at specific points along a surface. This coefficient, denoted as \( h_x \), represents the rate at which heat is transferred per unit surface area per unit temperature difference between the surface and the surrounding fluid at a particular location \( x \).

In the context of laminar flow over a flat plate, the local heat transfer coefficient varies inversely with the square root of the distance from the leading edge of the plate. Mathematically, this relationship is expressed as \( h_x = Cx^{-1/2} \), where \( C \) is a constant that encapsulates the properties of the fluid and flow conditions.

What this tells us is that as one moves away from the leading edge, the local heat transfer coefficient decreases. This is because the boundary layer, which acts as a thermal resistance, grows thicker further away from the leading edge, thus reducing the rate of heat transfer.
Average Heat Transfer Coefficient
The average heat transfer coefficient, denoted by \( h_{avg} \), is a way of assessing the overall heat transfer performance over a certain length of the plate, rather than at a specific point. It is derived by integrating the local heat transfer coefficient along the length of the plate from the leading edge to a point \( x \) and then normalizing by the length.

We calculate it using the integral \( h_{avg} = \frac{1}{x}\int_{0}^{x} h_{x'} dx' \). The steps outlined in the given solution walk through the calculation process and demonstrate that the average heat transfer coefficient along the plate is twice the local value at the point \( x \), i.e., \( h_{avg} = 2Cx^{-1/2} \). This value provides a very useful comparison, indicating that although the local coefficient decreases along the plate, the average can give us a more practical understanding of the heat transfer over the length of the plate.
Flat Plate Boundary Layer
The boundary layer on a flat plate is a fundamental concept in fluid dynamics and heat transfer that denotes the thin layer of fluid near the plate where frictional forces are significant. This region is characterized by a velocity gradient from the no-slip condition at the plate surface, where the fluid velocity is zero, to the free stream velocity away from the surface.

In heat transfer analysis, the thermal boundary layer is also pertinent, which is often similar but not identical to the velocity boundary layer. It is within this boundary layer that the temperature gradient exists, and hence, this is where heat is being transferred from the plate to the fluid, or vice versa.

For laminar flow, which is smooth and orderly, the boundary layer starts out very thin at the leading edge and grows with distance along the plate. This change in thickness affects the local heat transfer coefficient, as heat transfer is more efficient where the boundary layer is thinnest. Understanding how the boundary layer develops and influences heat transfer is essential for designing systems that cool or heat surfaces efficiently.

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Most popular questions from this chapter

The naphthalene sublimation technique involves the use of a mass transfer experiment coupled with an analysis based on the heat and mass transfer analogy to obtain local or average convection heat transfer coefficients for complex surface geometries. A coating of naphthalene, which is a volatile solid at room temperature, is applied to the surface and is then subjected to airflow in a wind tunnel. Alternatively, solid objects may be cast from liquid naphthalene. Over a designated time interval, \(\Delta t\), there is a discernible loss of naphthalene due to sublimation, and by measuring the surface recession at locations of interest or the mass loss of the sample, local or average mass transfer coefficients may be determined. Consider a rectangular rod of naphthalene exposed to air in cross flow at \(V=10 \mathrm{~m} / \mathrm{s}, T_{\mathrm{s}}=300 \mathrm{~K}\), as in Problem 6.10, except now \(c=10 \mathrm{~mm}\) and \(d=30 \mathrm{~mm}\). Determine the change in mass of the \(L=500\)-mm-long rod over a time period of \(\Delta t=30 \mathrm{~min}\). Naphthalene has a molecular weight of \(M_{\mathrm{A}}=128.16 \mathrm{~kg} / \mathrm{kmol}\), and its solid-vapor saturation pressure at \(27^{\circ} \mathrm{C}\) and \(1 \mathrm{ltm}\) is \(p_{\text {A, } a t}=1.33 \times 10^{-4}\) bar.

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.

An expression for the actual water vapor partial pressure in terms of wet-bulb and dry-bulb temperatures, referred to as the Carrier equation, is given as $$ p_{v}=p_{g w}-\frac{\left(p-p_{g w}\right)\left(T_{d b}-T_{\mathrm{wb}}\right)}{1810-T_{\mathrm{wb}}} $$ where \(p_{v}, p_{g w}\) and \(p\) are the actual partial pressure, the saturation pressure at the wet-bulb temperature, and the total pressure (all in bars), while \(T_{\mathrm{db}}\) and \(T_{\mathrm{wb}}\) are the dry- and wet-bulb temperatures in kelvins. Consider air at \(1 \mathrm{~atm}\) and \(37.8^{\circ} \mathrm{C}\) flowing over a wet-bulb thermometer that indicates \(21.1^{\circ} \mathrm{C}\). (a) Using Carrier's equation, calculate the partial pressure of the water vapor in the free stream. What is the relative humidity? (b) Refer to a psychrometric chart and obtain the relative humidity directly for the conditions indicated. Compare the result with part (a). (c) Use Equation \(6.65\) to determine the relative humidity. Compare the result to parts (a) and (b).

A fan that can provide air speeds up to \(50 \mathrm{~m} / \mathrm{s}\) is to be used in a low-speed wind tunnel with atmospheric air at \(25^{\circ} \mathrm{C}\). If one wishes to use the wind tunnel to study flatplate boundary layer behavior up to Reynolds numbers of \(R e_{x}=10^{8}\), what is the minimum plate length that should be used? At what distance from the leading edge would transition occur if the critical Reynolds number were \(R e_{x, c}=5 \times 10^{5}\) ?

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