### OverviewA common issue in cars during winter is fog accumulation on the glass surfaces, obstructing visibility. A practical solution is to use hot air or electric resistance heaters on the inner surfaces. Consider a car's rear window composed of a 0.4-cm-thick glass with thermal conductivity \( k = 0.84 \, \text{W/m} \cdot \text{K} \) and thermal diffusivity \( \alpha = 0.39 \times 10^{-6} \, \text{m}^2/\text{s} \). Strip heaters, with negligible thickness, are attached 4 cm apart on the inner surface, each producing heat at 25 W/m length. Initially, the whole car, including windows, is at an outdoor temperature of \( T_0 = -3^\circ \text{C} \).The heat transfer coefficients on the inner and outer glass surfaces are \( h_i = 6 \, \text{W/m}^2 \cdot \text{K} \) and \( h_o = 20 \, \text{W/m}^2 \cdot \text{K} \), respectively. The implicit method is used with a time step of 1 minute, a mesh size of \( \Delta x = 0.2 \, \text{cm} \) along the thickness, and \( \Delta y = 1 \, \text{cm} \) in the direction normal to the heater wires.### Diagram ExplanationThe diagram illustrates the setup with these features:- **Glass Layers:** The glass is labeled with nodes 1 to 9, showing the arrangement of the strip heaters and indicating the inner and outer surfaces.- **Heater Wires:** Positioned at specific intervals on the inner surface, generating 25 W/m.- **Mesh Size Indication:** The glass is divided into sections with a thickness of 0.2 cm and intervals of 1 cm between heaters.- **Thermal Symmetry Lines:** Indicate symmetry in the distribution of heat across the window.### TaskDetermine the temperature at node 4, 15 minutes after the strip heaters are activated.

Answered: Image /qna-images/answer/525e6086-2da8-4c5d-ac83-3b9c8ddfff39.jpg

A common annoyance in cars during winter is the formation of fog on the glass surfaces that blocks the view. A practical way of solving this problem is to blow hot air or to attach electric resistance heaters to the inner surfaces. Consider the rear window of a car that consists of a 0.4-cm-thick glass (k = 0.84 W/m-K and a = 0.39 x 10-6 m²/s). Strip heater wires of negligible thickness are attached to the inner surface of the glass, 4 cm apart. Each wire generates heat at a rate of 25 W/m length. Initially, the entire car, including its windows, is at the outdoor temperature of To= -3°C. The heat transfer coefficients at the inner and outer surfaces of the glass can be taken to be h;= 6 and ho= 20 W/m².K, respectively. Use the implicit method with a time step of 1 min with a mesh size of Ax=0.2 cm along the thickness and Ay=1 cm in the direction normal to the heater wires. Inner surface Heater 25 W/m 7 Thermal symmetry line Outer surface Glass 0.2 cm 1 cm Thermal symmetry line Determine the temperature at node 4, 15 min after the strip heaters are turned on.

A common annoyance in cars during winter is the formation of fog on the glass surfaces that blocks the view. A practical way of solving this problem is to blow hot air or to attach electric resistance heaters to the inner surfaces. Consider the rear window of a car that consists of a 0.4-cm-thick glass (k = 0.84 W/m-K and a = 0.39 x 10-6 m²/s). Strip heater wires of negligible thickness are attached to the inner surface of the glass, 4 cm apart. Each wire generates heat at a rate of 25 W/m length. Initially, the entire car, including its windows, is at the outdoor temperature of To= -3°C. The heat transfer coefficients at the inner and outer surfaces of the glass can be taken to be h;= 6 and ho= 20 W/m².K, respectively. Use the implicit method with a time step of 1 min with a mesh size of Ax=0.2 cm along the thickness and Ay=1 cm in the direction normal to the heater wires. Inner surface Heater 25 W/m 7 Thermal symmetry line Outer surface Glass 0.2 cm 1 cm Thermal symmetry line Determine the temperature at node 4, 15 min after the strip heaters are turned on.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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### Overview

A common issue in cars during winter is fog accumulation on the glass surfaces, obstructing visibility. A practical solution is to use hot air or electric resistance heaters on the inner surfaces. Consider a car's rear window composed of a 0.4-cm-thick glass with thermal conductivity \( k = 0.84 \, \text{W/m} \cdot \text{K} \) and thermal diffusivity \( \alpha = 0.39 \times 10^{-6} \, \text{m}^2/\text{s} \). Strip heaters, with negligible thickness, are attached 4 cm apart on the inner surface, each producing heat at 25 W/m length. Initially, the whole car, including windows, is at an outdoor temperature of \( T_0 = -3^\circ \text{C} \).

The heat transfer coefficients on the inner and outer glass surfaces are \( h_i = 6 \, \text{W/m}^2 \cdot \text{K} \) and \( h_o = 20 \, \text{W/m}^2 \cdot \text{K} \), respectively. The implicit method is used with a time step of 1 minute, a mesh size of \( \Delta x = 0.2 \, \text{cm} \) along the thickness, and \( \Delta y = 1 \, \text{cm} \) in the direction normal to the heater wires.

### Diagram Explanation

The diagram illustrates the setup with these features:

- **Glass Layers:** The glass is labeled with nodes 1 to 9, showing the arrangement of the strip heaters and indicating the inner and outer surfaces.
- **Heater Wires:** Positioned at specific intervals on the inner surface, generating 25 W/m.
- **Mesh Size Indication:** The glass is divided into sections with a thickness of 0.2 cm and intervals of 1 cm between heaters.
- **Thermal Symmetry Lines:** Indicate symmetry in the distribution of heat across the window.

### Task

Determine the temperature at node 4, 15 minutes after the strip heaters are activated.

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