91Ó°ÊÓ

The rear window of an automobile is defogged by attaching a thin, transparent, film-type heating clement 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 cocfficient are \(T_{=i}=25^{\circ} \mathrm{C}\) and \(h_{i}=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the exterior (ambient) air temperature and convection coefficient are \(T_{\text {we }}=\) \(-10^{\circ} \mathrm{C}\) and \(h_{e}=65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (b) In practice \(T_{w e}\) and \(h_{y}\) vary according to weather conditions and car speed. For values of \(h_{\mathrm{m}}=2,20,65\), and \(100 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\), determine and plot the electrical power requirement as a function of \(T_{z,}\) for \(-30 \leq T_{x p} \leq\) \(0^{\circ} \mathrm{C}\). From your results, what can you conclude about the need for heater operation at low values of \(h_{e}\) ? How is this conclusion affected by the value of \(T_{\text {w, }, 0}\) ? If \(h \propto\) \(V^{*}\), where \(V\) is the vehicle speed and \(n\) is a positive exponent, how does the vehicle speed affect the need for heater eperation?

Step by step solution

01

Understand the Heat Transfer Equation

In order to determine the electrical power required per unit window area, we need to consider the heat transfer through the window. The heat transfer involves convection on both the interior and exterior surfaces, and conduction through the window glass. The equation relating these is: \[ q = h_i(T_i - T_{si}) = k\left(\frac{T_{si} - T_{se}}{L}\right) = h_e(T_{se} - T_{we}) \]where:- \( h_i \) is the interior heat transfer coefficient,- \( h_e \) is the exterior heat transfer coefficient,- \( k \) is the thermal conductivity of glass,- \( T_i \) is the interior air temperature,- \( T_{si} \) is the inner surface temperature,- \( T_{se} \) is the outer surface temperature,- \( T_{we} \) is the exterior air temperature, and- \( L \) is the thickness of the glass.

02

Calculate the Power Requirement (Part a)

First, calculate the rate of heat transfer (\( q \)) needed to maintain the inner surface temperature \( T_{si} = 15^{\circ}C \) using both sides of the window:1. Heat transfer through the interior: \[ q = h_i (T_i - T_{si}) = 10 (25 - 15) = 100 \, \text{W/m}^2 \]2. Consider heat transfer through the glass and exterior convection: - Assuming properties: Thermal conductivity \( k = 1.4 \, \text{W/m-K} \), and glass thickness \( L = 0.004 \, \text{m} \). - Balance equation becomes:\[ q = k \left(\frac{T_{si} - T_{se}}{L}\right) + h_e(T_{se} + 10) \]We calculate the final \( q \, \text{W/m}^2 \) ensuring both equations equal, solving for the power requirement on the electrical heating element.

03

Calculate Power Requirement Across Variable Conditions (Part b)

For this part, we need to determine how the power requirement changes with the exterior conditions:1. Vary \( h_e \) and compute \( q \) using the relationship defined in Step 1, modifying \( T_{we} \) from \(-30^{\circ}C\) to \(0^{\circ}C\) as per each specified \( h_m \).2. Plot \( q \) against \( T_{we} \) for each \( h_m \) value to visualize how the electrical power requirement changes with conditions.

04

Conclusion Interpretation and Vehicle Speed Effects

From the plots, determine that higher power is required in extreme conditions (lower \( T_{we} \) and higher \( h_m \)). At low \( h_e \) values, heating is less crucial because natural convection is weaker and the heat loss is smaller. Finally, since \( h \propto V^n \), as speed increases, \( h_e \) increases, suggesting more heating would be necessary to compensate for the increased convective heat loss.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Convection

Convection is one of the primary modes of heat transfer, crucial to understanding how heat moves between surfaces and fluids. It plays a significant role in the heating of the rear window of a vehicle, as described in our exercise. When a fluid, such as air, moves over a surface, it can enhance the transfer of heat between the surface and the moving fluid.
This movement can be natural, due to temperature-induced fluid movement, or forced, like when driving a car. The heat transfer coefficient, denoted as \(h\), quantifies the efficiency of convective heat transfer. This coefficient depends on both the nature of the fluid and its velocity over the surface. In the context of the vehicle's rear window, the interior and exterior coefficients are different due to varying airflow conditions.

• The interior coefficient \(h_i\) represents heat transferring from the cabin's air to the window surface.
• The exterior coefficient \(h_e\) captures heat exchange between the outside air and the window.

Understanding these coefficients is essential to calculating the necessary electrical power for heating. Variations in vehicle speed or weather will change these coefficients, thereby affecting heat transfer rates across the vehicle window.

Thermal Conductivity

Thermal conductivity is a material property indicating how well a material conducts heat. In this exercise, it relates to the car window's glass, and is a vital parameter in calculating heat transfer through the window. It is denoted by \( k \) and is expressed in units of \( ext{W/m-K} \).
A material with higher thermal conductivity efficiently transfers heat, which could lead to significant heat loss if not managed properly. In the equation \[ q = krac{T_{si} - T_{se}}{L} \], we see how thermal conductivity works alongside temperature differences across the window glass and its thickness:

• \(T_{si}\) and \(T_{se}\) are the inner and outer surface temperatures of the glass.
• \(L\) is the thickness of the window glass.

The balance achieved by thermal conduction through the glass, together with convection internally and externally, helps maintain the target surface temperature within the vehicle. Adjusting the internal heating element ensures this balance remains effective, illustrating the importance of selecting appropriate materials with the right thermal characteristics for automotive applications.

Electrical Power Calculation

Electrical power calculation is essential to determining the energy needed to maintain desired temperatures within the rear window's surface. In our problem, the heating element provides the required power to combat the effects of convection and conduction. Applying the heat transfer relationship, we find the appropriate power use.
This involves equating the convective heat loss at the interior surface with the conductive loss through the glass, as the heating element must offset this total loss. The formula \( q = h_i(T_i - T_{si}) = k\left(\frac{T_{si} - T_{se}}{L}\right) + h_e(T_{se} - T_{we}) \) links the electrical power to the rates of heat loss and the thermal characteristics of the window.

• This means the initially calculated \( q \) value on the inner surface is balanced out by electrical power input, ensuring consistent temperature maintenance.
• Accurate calculations ensure efficiency and prevent overheating or energy waste.

Adapting the electrical power to different conditions as explored in part (b) of the exercise guarantees effectiveness in various external environments.

Vehicle Speed Effects on Heat Transfer

Vehicle speed greatly affects heat transfer, particularly through its impact on the external convective heat transfer coefficient, \(h_e\). As speed increases, the airflow over the vehicle intensifies, generally increasing \(h_e\), which can influence convective heat loss.
The equation \( h \propto V^n \), with \(V\) being the vehicle's speed and \(n\) a positive exponent, captures this relationship. Faster-moving air enhances heat dissipation, implying that under increasing speeds, greater electrical heating may be necessary to maintain the interior surface temperature.

• At higher speeds, the greater \(h_e\) means more heat is carried away by external air, necessitating more energy input.
• Conversely, lower speeds result in lesser convective losses, sometimes reducing the need for active heating.

Thus, understanding speed effects helps in dynamically adjusting the heating system of the window, ensuring it operates efficiently across different driving conditions and minimizes energy consumption during lesser convection scenarios.