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 = \(\alpha \cdot \frac{Δt}{(Δx)^2}\) Where α (alpha) is the thermal diffusivity, Δt is the time step, and Δx 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 Δx = 0.2 cm and α = 0.39×10⁻⁶ m²/s, we can determine the maximum Δt that will result in a stable solution.

02

Choose Time Step and Initialize Temperature Grid

Once we determine the maximum Δt 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 Δx = 0.2 cm and Δy = 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°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/m²K at the inner surface and \(h_o = 20\) W/m²K 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: \(T_i^{new} = T_i^{old} + k \cdot \frac{Δt}{(Δx)^2} \cdot (T_{i+1}^{old} - 2T_i^{old} + T_{i-1}^{old})\) 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°C). Once the steady-state temperature distribution has been reached, we can analyze the temperature distribution across the glass.

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