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The rear window of an automobile is defogged by attaching a thin, transparent, film-type heating element to its inner surface. By electrically heating this element, a uniform heat flux may be established at the inner surface. (a) For 4-mm-thick window glass, determine the electrical power required per unit window area to maintain an inner surface temperature of \(15^{\circ} \mathrm{C}\) when the interior air temperature and convection coefficient are \(T_{\infty, i}=25^{\circ} \mathrm{C}\) and \(h_{i}=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the exterior (ambient) air temperature and convection coefficient are \(T_{\infty, o}=\) \(-10^{\circ} \mathrm{C}\) and \(h_{o}=65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} .\) (b) In practice \(T_{\infty \rho \rho}\) and \(h_{o}\) vary according to weather conditions and car speed. For values of \(h_{o}=2,20\), 65 , and \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine and plot the electrical power requirement as a function of \(T_{\infty \rho}\) for \(-30 \leq T_{\infty, o} \leq 0^{\circ} \mathrm{C}\). From your results, what can you conclude about the need for heater operation at low values of \(h_{o}\) ? How is this conclusion affected by the value of \(T_{\infty, 0}\) ? If \(h \propto V^{n}\), where \(V\) is the vehicle speed and \(n\) is a positive exponent, how does the vehicle speed affect the need for heater operation?

Step by step solution

01

Set up the general equation for heat flux

The general equation for heat flux through a window can be expressed as the sum of conductive and convective heat transfer: \(q = q_{conduction} + q_{convection}\) For both the interior surface and the exterior surface, we'll have the heat transfer by convection and conduction; thus: \(q = q_{cond, int} + q_{conv, int} = q_{cond, ext} + q_{conv, ext}\)

02

Calculate the heat flux for the given conditions

Let's calculate the heat flux for part (a). We are given the temperature for the inner and outer surfaces of the glass (T_inner = 15掳C and T_outter = -10掳C), as well as the convection coefficients (h_i = 10 W/m虏路K and h_o = 65 W/m虏路K). For conductive heat transfer, Fourier's law can be expressed as: \(q_{cond} = k \frac{T_{inner} - T_{outer}}{L}\) For convective heat transfer, \(q_{conv} = h (T - T_{\infty})\) Therefore, \(q_{cond, int} + q_{conv, int} = q_{cond, ext} + q_{conv, ext}\) \(k \frac{T_{inner} - T_{outer}}{L} + h_i(T_{inner} - T_{\infty,i}) = k \frac{T_{outer} - T_{inner}}{L} + h_o(T_{\infty,o} - T_{outer})\) Given values: T_inner = 15掳C, T_outter = -10掳C, h_i = 10 W/m虏路K, h_o = 65 W/m虏路K T_{\infty,i} = 25掳C, T_{\infty,o} = -10掳C, L = 4 mm = 0.004 m Assume the conductivity of window glass, k = 1 W/m路K (this value can change depending on the type of glass being used, but for this exercise, we will use 1 W/m路K as a reasonable approximation). Calculate the heat flux, q, by solving the equation above.

03

Calculate the electrical power required per unit window area

With the heat flux determined, we can now calculate the electrical power required per unit window area for maintaining the inner surface temperature. The formula for electrical power is given by P = q 脳 A, where A represents the unit window area. In this case, since we are looking for electrical power per unit window area, the result will be the heat flux (q) itself.

04

Plotting the electrical power requirement as a function of T_{\infty,o} for different values of h_o

In part (b), we need to determine and plot the electrical power requirement as a function of T_{\infty,o} for different values of h_o: 2, 20, 65, 100 W/m虏路K, and a range of exterior air temperatures from -30掳C to 0掳C. For each value of h_o, use the same method as in Steps 2 and 3 to calculate the heat flux and the electrical power required per unit window area. Then plot the electrical power requirement as a function of T_{\infty,o}.

05

Evaluate and interpret the results

After plotting the curves for different values of h_o, analyze the results and answer the questions: 1. What can you conclude about the need for heater operation at low values of h_o? 2. How is this conclusion affected by the value of T_{\infty,o}? 3. If h 鈭� V^n, where V is the vehicle speed and n is a positive exponent, how does the vehicle speed affect the need for heater operation?

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Key Concepts

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

Conduction

Conduction is a fundamental process of heat transfer that occurs within a material or between materials in direct contact with each other. It is the transfer of heat through a material without any movement of the material itself. Conduction happens due to the transfer of kinetic energy from hotter molecules to cooler ones. In simpler terms, when you touch a warm object, you experience conduction.

Key points about conduction include:

• It is driven by a temperature difference.
• No mass movement occurs within the substance.
• The rate of conduction depends on the thermal conductivity of the material.

In the context of the rear window defogger, conduction occurs through the glass, with the heat moving from the inner surface (with the heating element) to the outer surface exposed to colder outside air. Fourier's Law quantifies conduction as follows:
\[ q_{cond} = k \frac{T_{inner} - T_{outer}}{L} \]
where:

• \( k \) is the thermal conductivity of the glass.
• \( T_{inner} \) and \( T_{outer} \) are the temperatures at the inner and outer surfaces.
• \( L \) is the thickness of the glass.

The more thermally conductive the material, the easier it is for heat to transfer through it via conduction.

Convection

Convection is another mode of heat transfer and occurs when heat is transferred by the movement of fluids鈥攅ither gases or liquids. In convection, the fluid motion is the primary driver for heat transfer, caused either by external forces (like a fan or a pump) or through natural buoyancy forces arising from temperature differences.

Key characteristics of convection include:

• Involves fluid movement, enhancing heat transfer.
• Can be classified as either natural (due to buoyancy) or forced (due to mechanical means).
• Is a key factor in designing heating and cooling systems.

In our example of the automobile defogger, convection occurs at both the inner and outer surfaces. The interior air warms the glass through the heating element's inner surface, while the exterior air cools the outer surface. The heat transfer by convection can be represented by:
\[ q_{conv} = h(T - T_{\infty}) \]
where:

• \( h \) represents the convection heat transfer coefficient.
• \( T \) is the temperature of the surface.
• \( T_{\infty} \) is the ambient temperature.

Understanding convection helps in evaluating how efficiently a surface can transfer heat to its surroundings or how fast it can cool down.

Heat Flux

Heat flux is an important concept in thermal dynamics, quantifying the rate of heat transfer per unit area. It is a vector quantity, having both a magnitude and a direction. In simpler terms, heat flux tells us how much heat is being transferred through a given area over a specific period of time.

Important aspects of heat flux are:

• Measured in watts per square meter (\( W/m^2 \)).
• Combines conduction and convection contributions.
• Determines the energy transfer efficiency.

In the automobile window problem, the heat flux through the glass is a crucial factor, as it combines both the conduction through the glass material and the convection at the surfaces. It is crucial for calculating the necessary power for the defogger, as the formula for electrical power required is \( P = q \times A \), where \( A \) is the area through which heat flows.

By establishing a uniform heat flux, we ensure even heating across the surface, preventing hot or cold spots in the glass.

Thermal Conductivity

Thermal conductivity is a critical property of materials, indicating how well a material can conduct heat. Materials with high thermal conductivity quickly transfer heat, making them good conductors, while materials with low thermal conductivity are insulators. This property is integral when designing systems that need effective heat management or insulation.

Key points about thermal conductivity include:

• Denoted as \( k \), measured in watts per meter per Kelvin (\( W/m\cdot K \)).
• Varies depending on the material's structure and composition.
• Plays a vital role in determining the rate of heat transfer via conduction in materials.

In the context of the automobile window, thermal conductivity (\( k \)) of the glass determines how efficiently heat flows from the heated inner surface to the colder outer surface. It is essential for calculating conduction heat flux and thereby other parameters like heat flux and electrical power required for heating. A balance is needed to ensure the glass effectively transfers enough heat to keep the surface free of fog without excessive power consumption.