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Chapter 6: Problem 50

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 \(T_{s}=5^{\circ} \mathrm{C}\) ?

Short Answer

Expert verified
The average heat flux needed to maintain a surface temperature of \(T_{s}=5^{\circ} \mathrm{C}\) for the small private aircraft is approximately \(14411.18 \mathrm{~W} / \mathrm{m}^{2}\).

Step by step solution

01

Determine the Reynolds number

First, let's find the Reynolds number, which is a dimensionless quantity that helps us describe the flow around the aircraft wing. The Reynolds number can be calculated as: \[Re = \frac{\rho u L}{\mu}\] Where: - \(Re\) is the Reynolds number - \(\rho\) is the air density - \(u\) is the velocity of the aircraft - \(L\) is the characteristic length of the airfoil - \(\mu\) is the air dynamic viscosity We are given the velocity of the plane (\(u = 100 \mathrm{~m} / \mathrm{s}\)), and the characteristic length of the airfoil (\(L = 2 \mathrm{~m}\)). We will assume the values for air density and air dynamic viscosity at a temperature of \(-23^{\circ} \mathrm{C}\): \(\rho = 1.43 \mathrm{~kg} / \mathrm{m}^{3}\) and \(\mu = 1.44 \times 10^{-5} \mathrm{~Pa} \cdot \mathrm{s}\), and calculate the Reynolds number accordingly.
02

Calculate the Reynolds number

Using the provided values, we can calculate the Reynolds number as follows: \[Re = \frac{(1.43 \mathrm{~kg} / \mathrm{m}^{3})(100 \mathrm{~m} / \mathrm{s})(2 \mathrm{~m})}{1.44 \times 10^{-5} \mathrm{~Pa} \cdot \mathrm{s}} \approx 1.99 \times 10^{7}\]
03

Determine the heat transfer coefficient

Now we need to find the heat transfer coefficient (\(h\)) using the Reynolds analogy and the given average friction coefficient (\(\bar{C}_{f}=0.0025\)). The Reynolds analogy states that: \[h = \frac{\rho c_p u \bar{C}_{f}}{Pr}\] Where: - \(c_p\) is the specific heat of air at constant pressure (\(c_p \approx 1005 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\)) - \(Pr\) is the Prandtl number, which is dimensionless and for air usually in the range of 0.7 to 0.8 We will assume a Prandtl number of 0.7 for this problem.
04

Calculate the heat transfer coefficient

Using the given values, we can calculate the heat transfer coefficient (\(h\)): \[h = \frac{(1.43 \mathrm{~kg} / \mathrm{m}^{3})(1005 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K})(100 \mathrm{~m} / \mathrm{s})(0.0025)}{0.7} \approx 514.97 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\]
05

Calculate the required heat flux

Finally, we can find the required heat flux (\(q_{req}\)) to maintain the surface temperature of \(T_{s} = 5^{\circ} \mathrm{C}\) by using the heat transfer coefficient and the temperature difference between the surface and the air: \[q_{req} = h(T_{s} - T_{\infty})\] Where: - \(T_{\infty}\) is the air temperature (\(-23^{\circ} \mathrm{C}\)) - \(T_{s}\) is the surface temperature (\(5^{\circ} \mathrm{C}\))
06

Find the required heat flux

Using the calculated heat transfer coefficient and given temperatures, we can determine the required heat flux: \[q_{req} = (514.97 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K})(5^{\circ} \mathrm{C} - (-23^{\circ} \mathrm{C})) \approx 14411.18 \mathrm{~W} / \mathrm{m}^{2}\] Therefore, the average heat flux needed to maintain a surface temperature of \(T_{s}=5^{\circ} \mathrm{C}\) is approximately \(14411.18 \mathrm{~W} / \mathrm{m}^{2}\).

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Key Concepts

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

Understanding the Reynolds Number
The Reynolds number (Re) is a crucial dimensionless quantity in fluid mechanics that helps differentiate between laminar and turbulent flow patterns around objects, such as aircraft wings. It quantifies the relationship between inertial forces and viscous forces in the flow. The formula for the Reynolds number is:

\[\begin{equation}Re = \dfrac{\rho u L}{\mu}\end{equation}\]

Here
  • \(\rho\) represents the fluid density
  • \(u\) is the velocity of the object through the fluid
  • \(L\) is the characteristic linear dimension (length of the airfoil, in the context of our exercise)
  • \(\mu\) is the dynamic viscosity of the fluid
In practical terms, a high Reynolds number indicates turbulent flow, whereas a low Reynolds number suggests laminar flow. For instance, at high cruising speeds or with larger aircraft, the Reynolds number increases, leading to a turbulent boundary layer, which has implications for the drag experienced by the aircraft and the efficiency of heat transfer. Understanding the Reynolds number is essential for solving problems related to aerodynamics and heat transfer, as it dictates the behavior of the flow around the aircraft, which directly influences the heat flux calculation needed for applications like de-icing.
The Heat Transfer Coefficient Explained
The heat transfer coefficient (\(h\)) is a measure of the convective heat transfer between a surface and a fluid moving past it. It is a parameter that indicates the ease with which heat is transferred from the surface to the fluid, or vice versa. Physically, a high heat transfer coefficient means that the fluid efficiently removes or supplies heat from/to the surface, which is important in systems involving cooling or heating processes.

To compute the heat transfer coefficient, you can use the formula:\[\begin{equation}h = \dfrac{\rho c_p u \bar{C}_{f}}{Pr}\end{equation}\]

In this equation:
  • \(c_p\) represents the specific heat of the fluid at constant pressure
  • \(u\) is the fluid velocity over the surface
  • \(\bar{C}_{f}\) refers to the average friction coefficient, which links the shear stress and kinetic energy of the fluid flow
  • \(Pr\) is the Prandtl number, a dimensionless number that relates the fluid's momentum diffusivity to its thermal diffusivity
In our exercise, the heat transfer coefficient helps us determine the power requirements to prevent ice formation on an aircraft wing. It must be obtained accurately since it directly affects the heat flux needed to maintain the desired surface temperature. Typically, a higher velocity or an increase in the average friction coefficient (\(\bar{C}_{f}\)) will lead to a higher heat transfer coefficient, which influences the effectiveness of systems like electric resistance heating elements installed within the aircraft wings for de-icing.
Electric Resistance Heating
Electric resistance heating is a method of converting electrical energy into heat through the resistance offered by a material to the flow of electric current. This principle can be applied for various purposes, including the de-icing of aircraft wings, as demonstrated in the given exercise.

In resistance heating, an electric current passes through a resistive material, generating heat due to the inherent electrical resistance. The power (\(P\)) dissipated by an electrical heating element is given by Joule's law:\[\begin{equation}P = I^2 R\end{equation}\]

where \(I\) is the current and \(R\) is the electrical resistance of the heating element. To prevent ice from forming on the wings, we calculate the required heat flux (\(q_{req}\)) necessary to maintain a particular surface temperature, taking into consideration the efficiency of the heat transfer to the surrounding air.The solution to our problem involves finding the average heat flux that maintains the surface temperature above the freezing point, thereby preventing the formation of ice. The calculation ties in closely with the principles of heat transfer coefficient and Reynolds number, as the performance and design of the heating element would hinge on the balance between the rate of heat generation and the rate of heat removal by airflow around the wing surfaces.

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Most popular questions from this chapter

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).

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

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?

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.

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