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A common annoyance in cars in winter months 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 \(\left(k=0.84 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\) and \(\left.\alpha=0.39 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\). Strip heater wires of negligible thickness are attached to the inner surface of the glass, \(4 \mathrm{~cm}\) apart. Each wire generates heat at a rate of \(25 \mathrm{~W} / \mathrm{m}\) length. Initially the entire car, including its windows, is at the outdoor temperature of \(T_{o}=-3^{\circ} \mathrm{C}\). The heat transfer coefficients at the inner and outer surfaces of the glass can be taken to be \(h_{i}=6\) and \(h_{o}=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Using the explicit finite difference method with a mesh size of \(\Delta x=\) \(0.2 \mathrm{~cm}\) along the thickness and \(\Delta y=1 \mathrm{~cm}\) in the direction normal to the heater wires, determine the temperature distribution throughout the glass \(15 \mathrm{~min}\) after the strip heaters are turned on. Also, determine the temperature distribution when steady conditions are reached.

Step by step solution

01

Calculate Explicit Stability Criterion

We need to determine if the explicit finite difference method is stable for this problem. This is done by calculating the Fourier number (Fo) using the explicit stability criterion formula: Fo = \(\alpha \cdot \frac{螖t}{(螖x)^2}\) Where 伪 (alpha) is the thermal diffusivity, 螖t is the time step, and 螖x is the mesh size in the x-direction. The criterion for stability is that Fo must be less than or equal to 1/2. Using the given 螖x = 0.2 cm and 伪 = 0.39脳10鈦烩伓 m虏/s, we can determine the maximum 螖t that will result in a stable solution.

02

Choose Time Step and Initialize Temperature Grid

Once we determine the maximum 螖t for stability, we can choose a time step for our analysis. It is usually a good idea to choose a time step slightly smaller than the maximum to ensure stability. Next, we'll create a grid to represent the temperature values at each mesh point. Since our mesh size is 螖x = 0.2 cm and 螖y = 1 cm, we can determine the dimensions of the grid. The initial temperature of the entire grid will be set equal to the outdoor temperature (\(T_o\)) of -3掳C.

03

Apply Boundary Conditions

Now, we need to apply boundary conditions at the inner and outer surfaces of the glass. The heat transfer coefficients are provided for both surfaces: \(h_i = 6\) W/m虏K at the inner surface and \(h_o = 20\) W/m虏K at the outer surface. The heat flux at each surface can be calculated using the heat transfer coefficient at that surface, and the temperature difference between the surface and the adjacent nodes will be calculated. We can then update the corresponding boundary nodes using the calculated heat fluxes.

04

Update Interior Nodes Using Explicit Finite Difference Method

With the boundary conditions applied, we can now update the temperatures at the interior nodes using the explicit finite difference method. The temperature at each interior node can be updated using the neighboring temperatures and the thermal conductivity (k) of the glass: \(T_i^{new} = T_i^{old} + k \cdot \frac{螖t}{(螖x)^2} \cdot (T_{i+1}^{old} - 2T_i^{old} + T_{i-1}^{old})\) We'll perform these calculations iteratively for each time step until the total simulation time (in our case, 15 minutes) is reached.

05

Calculate Steady-State Temperature Distribution

After 15 minutes have passed, we can calculate the steady-state temperature distribution by running the simulation until the temperature change between time steps for each mesh point is much smaller than a given tolerance (for example, 0.01掳C). Once the steady-state temperature distribution has been reached, we can analyze the temperature distribution across the glass.

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Key Concepts

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

Heat Transfer

Heat transfer is a fundamental concept in physics that describes how thermal energy moves from one point to another. In our scenario, the focus is on the process of preventing fogging on a car's rear window by heating it.
This heating process involves transferring heat from heater wires to the glass surface. Understanding how heat moves is key to figuring out how quickly and evenly the temperature will change across the window.
There are three main modes of heat transfer:

• **Conduction**: Direct transfer of heat through a material, like the heat flowing through the glass.
• **Convection**: Transfer of heat by the movement of fluids or gases, such as the hot air blown against the window surfaces.
• **Radiation**: Transfer of heat in the form of electromagnetic waves that doesn't require a medium.

In this specific problem, conduction through glass and convective heat transfer from the air to the glass surface are the primary mechanisms at work. Knowing these processes helps facilitate designing systems that provide efficient heating, avoiding fog with minimal energy use.

Thermal Conductivity

Thermal conductivity is a material property that quantifies how well it can conduct heat. It's represented by the symbol \(k\) and measured in units of \( ext{W/m} \, \cdot \text{K}\) (watts per meter per Kelvin). In simpler terms, it describes how easily heat can travel through a material.
In the case of the glass window, the given thermal conductivity is \(0.84 \, \text{W/m} \, \cdot \text{K}\), indicating that glass is moderately conductive. This affects how quickly the glass can heat up and distribute the heat generated by the heater wires.
Higher thermal conductivity implies faster heat transfer through the material. For instance, metals typically have high thermal conductivity. In contrast, materials like wood or rubber are poor conductors, being better insulators. Understanding these properties is crucial for designing effective heating elements and systems.

Stability Criterion

When using numerical methods like the finite difference method to solve heat transfer problems, ensuring stability is crucial. Stability ensures that the numerical solution behaves and converges correctly, providing accurate results.
In this problem, the stability criterion is addressed by calculating the Fourier number \(\text{Fo}\). It is defined in the finite difference method as:\[\text{Fo} = \alpha \cdot \frac{\Delta t}{(\Delta x)^2}\]where:

• \(\alpha\) is the thermal diffusivity of the material.
• \(\Delta t\) is the time step size.
• \(\Delta x\) is the size of the spatial step.

For stability in an explicit finite difference method, \(\text{Fo}\) must be less than or equal to \(\frac{1}{2}\). This ensures that the temperature updates at each step do not "explode" and lead to unrealistic values.
This calculation helps determine appropriate time steps for simulations, ensuring accurate modeling of the heat transfer over time.

Temperature Distribution

Temperature distribution refers to how temperatures vary throughout the material, in this case, the glass window over time and space.
By simulating with the finite difference method, we can understand how heat moves from the heater wires through the glass. The initial state has the glass at a uniform temperature of \(-3^{\circ} \text{C}\). Over time, as the heater wires transfer heat, the temperature within the glass changes.
The goal is to determine two main temperature distributions:

• **Transient Distribution**: The temperature profile at a specific time, such as 15 minutes after the heater activation.
• **Steady-State Distribution**: The temperature profile once equilibrium has been reached, with no further changes over time.

Accurate knowledge of these distributions is essential for assessing how efficiently the window defogs and ensures a clear view. It also informs designs to optimize heating and energy usage.