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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

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 \(R e_{x, c}=5 \times 10^{5}\) ?

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).

It is known that on clear nights the air temperature need not drop below \(0^{\circ} \mathrm{C}\) before a thin layer of water on the ground will freeze. Consider such a layer of water on a clear night for which the effective sky temperature is \(-30^{\circ} \mathrm{C}\) and the convection heat transfer coefficient due to wind motion is \(h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The water may be assumed to have an emissivity of \(1.0\) and to be insulated from the ground as far as conduction is concerned. (a) Neglecting evaporation, determine the lowest temperature the air can have without the water freezing. (b) For the conditions given, estimate the mass transfer coefficient for water evaporation \(h_{\mathrm{m}}(\mathrm{m} / \mathrm{s})\). (c) Accounting now for the effect of evaporation, what is the lowest temperature the air can have without the water freezing? Assume the air to be dry.

On a summer day the air temperature is \(27^{\circ} \mathrm{C}\) and the relative humidity is \(30 \%\). Water evaporates from the surface of a lake at a rate of \(0.10 \mathrm{~kg} / \mathrm{h}\) per square meter of water surface area. The temperature of the water is also \(27^{\circ} \mathrm{C}\). Determine the value of the convection mass transfer coefficient. 6.53 It is observed that a 230 -mm-diameter pan of water at \(23^{\circ} \mathrm{C}\) has a mass loss rate of \(1.5 \times 10^{-5} \mathrm{~kg} / \mathrm{s}\) when the ambient air is dry and at \(23^{\circ} \mathrm{C}\). (a) Determine the convection mass transfer coefficient for this situation. (b) Estimate the evaporation mass loss rate when the ambient air has a relative humidity of \(50 \%\). (c) Estimate the evaporation mass loss rate when the water and ambient air temperatures are \(47^{\circ} \mathrm{C}\), assuming that the convection mass transfer coefficient remains unchanged and the ambient air is dry.

Consider the nanofluid of Example 2.2. (a) Calculate the Prandtl numbers of the base fluid and nanofluid, using information provided in the example problem. (b) For a geometry of fixed characteristic dimension \(L\), and a fixed characteristic velocity \(V\), determine the ratio of the Reynolds numbers associated with the two fluids, \(R e_{\text {wf }} / R e_{\mathrm{w}_{\mathrm{d}}-}\) Calculate the ratio of the average Nusselt numbers, \(\overline{N u}_{L, \text {, d }} / \overline{N u}_{\text {L, b }}\), that is associated with identical average heat transfer coefficients for the two fluids, \(\bar{h}_{\mathrm{mf}}=\bar{h}_{\mathrm{bd}}\). (c) The functional dependence of the average Nusselt number on the Reynolds and Prandtl numbers for a broad array of various geometries may be expressed in the general form $$ \overline{N u}_{L}=\bar{h} L / k=C R e^{w N} P r^{1 / 3} $$ where \(C\) and \(m\) are constants whose values depend on the geometry from or to which convection heat transfer occurs. Under most conditions the value of \(m\) is positive. For positive \(m\), is it possible for the base fluid to provide greater convection heat transfer rates than the nanofluid, for conditions involving a fixed geometry, the same characteristic velocities, and identical surface and ambient temperatures?
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