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The defroster of an automobile functions by discharging warm air on the inner surface of the windshield. To prevent condensation of water vapor on the surface, the temperature of the air and the surface convection coefficient \(\left(T_{\infty, j}, \overline{h_{i}}\right)\) must be large enough to maintain a surface temperature \(T_{s i}\) that is at least as high as the dewpoint \(\left(T_{s, i} \geq T_{d \mathrm{p}}\right)\). Consider a windshield of length \(L=800 \mathrm{~mm}\) and thickness \(t=6 \mathrm{~mm}\) and driving conditions for which the vehicle moves at a velocity of \(V=70 \mathrm{mph}\) in ambient air at \(T_{\infty \rho}=-15^{\circ} \mathrm{C}\). From laboratory experiments performed on a model of the vehicle, the average convection coefficient on the outer surface of the windshield is known to be correlated by an expression of the form \(\overline{N_{L}}=0.030 \operatorname{Re}_{L}^{0.8} \operatorname{Pr}^{1 / 3}\), where \(R e_{L}=V L \nu\). Air properties may be approximated as \(k=0.023 \mathrm{~W} / \mathrm{m}=\mathrm{K}\), \(v=12.5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\), and \(\operatorname{Pr}=0.71\). If \(T_{d p}=10^{\circ} \mathrm{C}\) and \(T_{m, j}=50^{\circ} \mathrm{C}\), what is the smallest value of \(\bar{h}_{j}\) required to prevent condensation on the inner surface?

Step by step solution

01

1. Convert units

Before we start solving the problem, let's convert the units to SI units. The length L and thickness t are already in SI units. The velocity V is given in miles per hour (mph). We need to convert it into meters per second (m/s). 1 mile = 1609.34 meters 1 hour = 3600 seconds \(V = 70 \text{ mph} = 70 \times \frac{1609.34 \text{ m}}{3600 \text{ s}} = 31.2928 \text{ m/s}\) Now all the given data are in SI units, and we can proceed with the calculations.

02

2. Calculate the Reynolds number

Now, let's calculate the Reynolds number (\(Re_L\)), using the given equation \(Re_L = V L \nu\), where V = 31.2928 m/s, L = 0.8 m, and \(\nu = 12.5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\). \(Re_L = 31.2928 * 0.8 * 12.5 \times 10^{-6}\) \(Re_L = 314.327\)

03

3. Calculate the average Nusselt number

We are given the correlation for the average Nusselt number as \(\overline{N_{L}}=0.030 \operatorname{Re}_{L}^{0.8} \operatorname{Pr}^{1 / 3}\), where \(\operatorname{Re}_{L} = 314.327\), and \(\operatorname{Pr} = 0.71\). \(\overline{N_{L}} = 0.030 \times (314.327)^{0.8} \times (0.71)^{\frac{1}{3}}\) \(\overline{N_{L}} = 17.0542\)

04

4. Calculate the convection heat transfer coefficient on the outer surface of the windshield

Now, let's calculate the convection heat transfer coefficient on the outer surface of the windshield, knowing that for a fluid with a constant property: \(\overline{h_{\rho}} = \frac{\overline{N_{L}}k}{L}\), where \(\overline{N_{L}} = 17.0542\), \(k = 0.023 \mathrm{~W} / \mathrm{m}=\mathrm{K}\), and \(L = 0.8 \mathrm{~m}\). \(\overline{h_{\rho}} = \frac{17.0542 \times 0.023}{0.8}\) \(\overline{h_{\rho}} = 0.4933 \mathrm{~W} / \mathrm{m}^{2}\mathrm{K}\)

05

5. Determine the smallest value of \(\bar{h}_{j}\)

Since we need to maintain a surface temperature at least equal to the dewpoint, we can use the following energy balance equation for the surface temperature: \(T_{d p} - T_{\infty \rho} = \left( \frac{\overline{h_{j}}t}{k} + \frac{1}{\overline{h_{\rho}}} \right)(T_{m, j} - T_{\infty \rho}) \). Let's first calculate the left part of the equation, \(T_{d p} - T_{\infty \rho}\): \(T_{d p} - T_{\infty \rho} = 10 - (-15) = 25 \mathrm{^\circ C}\) Now, let's find the smallest value of \(\bar{h}_{j}\) by solving the equation: \(25 = \left( \frac{\overline{h_{j}} \times 0.006}{0.023} + \frac{1}{0.4933} \right)(50 - (-15))\) \(25 = \left( \frac{\overline{h_{j}} \times 0.006}{0.023} + \frac{1}{0.4933} \right)(65)\) \(\frac{\overline{h_{j}} \times 0.006}{0.023} = \frac{25}{65} - \frac{1}{0.4933}\) \(\overline{h_{j}} = \frac{0.023}{0.006}\left( \frac{25}{65} - \frac{1}{0.4933} \right)\) \(\overline{h_{j}} = 253.45 \mathrm{~W} / \mathrm{m}^{2}\mathrm{K}\) The smallest value of \(\overline{h_{j}}\) required to prevent condensation on the inner surface of the windshield is approximately \(253.45 \mathrm{~W} / \mathrm{m}^{2}\mathrm{K}\).

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Key Concepts

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

Convection Heat Transfer Coefficient

Understanding the convection heat transfer coefficient is crucial for various engineering applications, including the design of vehicle windshields that require defrosting. Convection is one of the three modes of heat transfer, the others being conduction and radiation. In the context of a windshield, convection occurs when the warm air from the defroster comes into contact with the cooler glass surface.

The convection heat transfer coefficient, denoted as \( \bar{h} \), quantifies the ability of convective flow to transfer heat between a surface and a fluid moving over it. It's measured in watts per square meter per Kelvin \( \mathrm{W/m^2K} \). A higher \( \bar{h} \) suggests more efficient heat transfer. In our scenario, preventing condensation involves maintaining a surface temperature above the dew point, necessitating an adequate convection heat transfer coefficient \( \bar{h}_j \). The exact value required can be derived from balancing the energy equation and considering both the internal and external convection coefficients, as shown in the problem's solution.

To facilitate students' understanding:

• Provide a clear explanation of how convection involves the movement of heat due to the movement of fluid.
• Use analogies, such as comparing the convection heat transfer process to a fan blowing away the heat from hot metal.
• Emphasize the importance of \( \bar{h} \) in determining how effectively the defroster warms the windshield to prevent condensation.

Reynolds Number

The Reynolds number, often denoted as \( Re \), is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It compares the relative importance of inertial forces and viscous forces and is calculated by the formula \( Re = \frac{\rho V L}{\mu} \) or \( Re = \frac{V L}{u} \) where:\

• \(\rho\) is the fluid density
• \(V\) is the velocity of the flow
• \(L\) is the characteristic length (like the windshield's length)
• \(\mu\) is the dynamic viscosity of the fluid
• \(u\) is the kinematic viscosity of the fluid

In layman terms, if you think about it as a race between the forces within the fluid, the Reynolds number tells us whether inertia (suggesting turbulent flow) or viscosity (suggesting laminar flow) is winning. The unique conditions of windshield defrosting require knowledge of this number to design an effective system.

For educational improvement, it's effective to:

• Describe the context of the problem and how the Reynolds number provides insights into the flow over a windshield.
• Utilize diagrams or visuals to demonstrate the difference between laminar and turbulent flows.
• Relate the concept to everyday experiences, such as comparing the airflow around a moving car to water flowing around rocks in a stream.

By understanding the Reynolds number in the windshield defrosting context, students can appreciate its role in predicting the effectiveness of the defrosting process.

Nusselt Number

The Nusselt number is a dimensionless parameter in heat transfer that relates the convective heat transfer to conduction heat transfer across a boundary. Denoted as \( Nu \), it's given by the formula \( Nu = \frac{h L}{k} \), where:\

• \(h\) is the convection heat transfer coefficient
• \(L\) is the characteristic length
• \(k\) is the thermal conductivity of the fluid

For a windshield in a car, an ideal defrosting system depends significantly on a well-calculated Nusselt number, which allows engineers to understand and optimize the heat transfer process on the windshield's surface.

Explaining the Nusselt number with clarity can include:

• Breaking down its calculation into understandable steps, and relating it to the real-world application of defrosting a windshield.
• Using simple language to explain how it represents the efficiency of heat transfer from the warmer air due to the defroster to the cooler windshield.
• Discussing its importance in ensuring the car's visibility during cold weather by effectively preventing condensation and freezing on the windshield.

The Nusselt number in our exercise helped us calculate the required convection heat transfer coefficient to prevent condensation, showcasing its value in practical engineering problems.