Consider a house whose windows are made of \(0.375\)-in-thick glass \(\left(k=0.48 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.\) and \(\alpha=\) \(4.2 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}\) ). Initially, the entire house, including the walls and the windows, is at the outdoor temperature of \(T_{o}=35^{\circ} \mathrm{F}\). It is observed that the windows are fogged because the indoor temperature is below the dew-point temperature of \(54^{\circ} \mathrm{F}\). Now the heater is turned on and the air temperature in the house is raised to \(T_{i}=72^{\circ} \mathrm{F}\) at a rate of \(2^{\circ} \mathrm{F}\) rise per minute. The heat transfer coefficients at the inner and outer surfaces of the wall can be taken to be \(h_{i}=1.2\) and \(h_{o}=2.6 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\), respectively, and the outdoor temperature can be assumed to remain constant. Using the explicit finite difference method with a mesh size of \(\Delta x=0.125\) in, determine how long it will take for the fog on the windows to clear up (i.e., for the inner surface temperature of the window glass to reach \(54^{\circ} \mathrm{F}\) ).

Step by step solution

01

Identify relevant parameters and equations

We are given the following parameters: - Glass thickness: \(L = 0.375 \,\text{in}\) - Thermal conductivity: \(k = 0.48 \frac{\mathrm{Btu}}{\mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ}\mathrm{F}}\) - Diffusivity: \(\alpha = 4.2 \times 10^{-6} \frac{\mathrm{ft}^{2}}{\mathrm{s}}\) - Initial outdoor temperature: \(T_o = 35^{\circ} \mathrm{F}\) - Final indoor temperature: \(T_i = 72^{\circ} \mathrm{F}\) - Dew-point temperature: \(T_{dp} = 54^{\circ} \mathrm{F}\) - Heat transfer coefficients: \(h_i = 1.2 \,\frac{\mathrm{Btu}}{\mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}}\) and \(h_o = 2.6\,\frac{\mathrm{Btu}}{\mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}}\) - Rate of temperature rise: \(2^{\circ} \mathrm{F} \,\text{per minute}\) The explicit finite difference method is based on Fourier's law of heat conduction: $$ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} $$ We will discretize the problem using a mesh size of \(\Delta x = 0.125\, \text{in}\) and convert the time rate of temperature rise to seconds.

02

Determine the number of time steps and set up initial conditions

First, convert the rate of temperature rise to seconds: $$ \frac{1 \,\text{min}}{60 \, \text{s}} \cdot 2^{\circ}\text{F} \,\text{per minute} = \frac{1}{30}^{\circ}\text{F} \,\text{per second} $$ Divide the glass thickness by the mesh size to determine the number of nodes: $$ \frac{L}{\Delta x} = \frac{0.375 \text{in}}{0.125 \text{in}} = 3 $$ Set up initial conditions for the three nodes: - Node 1 (outer surface): \(T_1 = T_o = 35^{\circ} \mathrm{F}\) - Node 2: \(T_2 = T_o = 35^{\circ} \mathrm{F}\) - Node 3 (inner surface): \(T_3 = T_o = 35^{\circ} \mathrm{F}\)

03

Calculate the finite difference update equation

Use Fourier's law of heat conduction in discretized form: $$ \frac{T_{i}^{t+1} - T_{i}^{t}}{\Delta t} = \alpha \frac{T_{i+1}^{t} - 2T_{i}^{t} + T_{i-1}^{t}}{(\Delta x)^2} $$ Isolate \(T_{i}^{t+1}\) and rewrite the equation for each node: - Node 1 (outer surface): $$ T_{1}^{t+1} = T_{1}^{t} + \frac{\alpha \Delta t}{(\Delta x)^2}(T_{2}^{t} - 2 T_{1}^{t} + h_o(T_o - T_{1}^{t})) $$ - Node 2: $$ T_{2}^{t+1} = T_{2}^{t} + \frac{\alpha \Delta t}{(\Delta x)^2}(T_{3}^{t} - 2 T_{2}^{t} + T_{1}^{t}) $$ - Node 3 (inner surface): $$ T_{3}^{t+1} = T_{3}^{t} + \frac{\alpha \Delta t}{(\Delta x)^2}(h_i(T_i - T_{3}^{t}) - T_{2}^{t} + T_{3}^{t}) $$

04

Calculate the time for fog to clear up

Implement the explicit finite difference method using the update equations derived in Step 3. Increment the time and calculate the new temperature distribution until \(T_3 \geq 54^{\circ} \mathrm{F}\). While calculating, ensure that the temperature rise rate is respected. Keep increasing the indoor temperature at the given rate until it reaches \(72^{\circ}\mathrm{F}\). Once the inner surface temperature (\(T_3\)) reaches \(54^{\circ} \mathrm{F}\) or higher, record the time it took and report how long it took for the fog on the windows to clear up. Note: The actual time calculation is computationally intensive and needs a programming language or suitable software to be implemented. From this point on, the steps can be used as guidelines for implementing the code and calculating the time for the fog to clear up.

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Key Concepts

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

Understanding Heat Transfer

Essential to the workings of everyday life, heat transfer is a process by which thermal energy, or heat, moves from one place to another. The transfer occurs in three primary ways: conduction, convection, and radiation. In our window defogging example, conduction is the key player. Conduction is the process of heat transfer through materials that are in direct contact with each other. Thermal energy flows from the warmer side of the glass (inside the house) to the cooler side (outside), gradually equalizing the temperature on the window surface.

The rate at which heat is transferred depends on the material's properties and the temperature difference between the two environments. Generally, better conductors have higher thermal conductivity, meaning they transfer heat more quickly. Understanding these concepts is crucial when using methods such as the explicit finite difference method to predict how long it will take for the window fog to clear, as it directly relies on the principles of heat transfer.

Fourier's Law of Heat Conduction

At the heart of our analysis is Fourier's law of heat conduction, named after the French mathematician and physicist Jean-Baptiste Joseph Fourier. The law is fundamental in the calculations regarding heat conduction and forms the basis for our explicit finite difference method in the defogging problem. Fourier's law quantitatively describes how thermal energy is conducted through materials. It states that the rate of heat transfer through a material is proportional to the negative gradient of the temperature and the area through which the heat is flowing.

In simpler terms, the law will tell us how fast the heat is moving from the warm interior of the house to the cooler exterior, through the window glass. Crucially, Fourier's law allows us to create a mathematical model that can predict temperature changes over time, which is precisely what we need to solve our window defogging problem.

The Role of Thermal Diffusivity

Thermal diffusivity is a measure of how quickly a material can conduct thermal energy relative to its storage of thermal energy. Mathematically, thermal diffusivity (\( \alpha \)) is defined as the ratio of thermal conductivity (\( k \)) to the product of density (\( \rho \)) and specific heat capacity (\( c_p \)), or \( \alpha = \frac{k}{\rho c_p} \).

In practical terms, a material with high thermal diffusivity will adjust its temperature quickly when exposed to a temperature change since it can rapidly transfer heat. This property is significant for our house scenario as glass, which has a relatively high thermal diffusivity, will transfer the heat from the warm interior to the cold exterior effectively. The diffusivity affects the time it takes for the window temperature to reach the dew point and subsequently clear the fog, a critical part of the problem we aim to solve using the explicit finite difference method.