Question

Write a computer program to evaluate the variation of temperature with time of thin square metal plates that are removed from an oven at a specified temperature and placed vertically in a large room. The thickness, the size, the initial temperature, the emissivity, and the thermophysical properties of the plate as well as the room temperature are to be specified by the user. The program should evaluate the temperature of the plate at specified intervals and tabulate the results against time. The computer should list the assumptions made during calculations before printing the results. For each step or time interval, assume the surface temperature to be constant and evaluate the heat loss during that time interval and the temperature drop of the plate as a result of this heat loss. This gives the temperature of the plate at the end of a time interval, which is to serve as the initial temperature of the plate for the beginning of the next time interval. Try your program for \(0.2-\mathrm{cm}\)-thick vertical copper plates of $40 \mathrm{~cm} \times 40 \mathrm{~cm}\( in size initially at \)300^{\circ} \mathrm{C}\( cooled in a room at \)25^{\circ} \mathrm{C}$. Take the surface emissivity to be \(0.9\). Use a time interval of \(1 \mathrm{~s}\) in calculations, but print the results at \(10-\mathrm{s}\) intervals for a total cooling period of \(15 \mathrm{~min}\).

Short answer

Question: Write a short answer discussing how to create a computer program that calculates the temperature variation of a thin square metal plate over time as it's cooled in a room. Answer: To create a computer program to calculate the temperature variation of a thin square metal plate, follow these steps: 1) Define input parameters and constants, including thickness, size, initial temperature, emissivity, thermophysical properties, and room temperature. 2) Calculate heat loss and temperature drop using Fourier's Law and Newton's Law of Cooling. 3) Update the initial temperature for the next time interval. 4) Output results and assumptions at specified time intervals (e.g., every 10 seconds for 15 minutes). The program should account for assumptions such as constant surface temperature during each time interval and plate cooling solely due to heat transfer through conduction.

Step-by-step solution

Define Input Parameters and Constants

The input parameters from the user include thickness, size, initial temperature, emissivity, thermophysical properties of the plate, and the room temperature. In our example, we are given the following specific values: - Thickness = 0.2 cm - Size = 40 cm x 40 cm - Initial Temperature = \(300^{\circ} \mathrm{C}\) - Emissivity = 0.9 - Room Temperature = \(25^{\circ} \mathrm{C}\) - Time interval = 1s In addition to these input parameters, are constants for Copper, as follows: - Density = 8,960 \(kg/m^3\) - Specific heat capacity = 385 \(J/kg\cdot K\) - Thermal conductivity = 400 \(W/m\cdot K\)

Calculate Heat Loss and Temperature Drop

For each time interval, we assume constant surface temperature and calculate the heat loss during that time interval. This is then used to evaluate the temperature drop of the plate as a result of the heat loss. We can use Fourier's Law combined with Newton's Law of Cooling to calculate the heat loss and the temperature drop as follows: Heat Loss: \(q = kA\frac{dT}{dx}\) Temperature Drop: \(\Delta T = \frac{q \cdot \Delta t}{m \cdot c_p}\), Where: - \(q\) represents heat loss - \(k\) is the thermal conductivity of the metal (400 \(W/m\cdot K\) for copper) - \(A\) is the surface area of the metal plate - \(dT\) is the temperature difference between plate and room temperature - \(dx\) is the thickness of the metal plate - \(\Delta t\) is the time interval - \(m\) is the mass of the metal plate - \(c_p\) is the specific heat capacity of the metal (385 \(J/kg\cdot K\) for copper)

Update Initial Temperature for Next Time Interval

After calculating the temperature drop, we can update the initial temperature (\(T_{initial}\)) for the next time interval using the following equation: \(T_{initial} = T_{initial} - \Delta T\)

Output Results and Assumptions

The program should print the results at 10-second intervals for a total cooling period of 15 minutes and include any assumptions made during calculations. Some of the assumptions include: 1. Constant surface temperature during each time interval. 2. Plate cooling solely due to heat transfer through conduction. By following these steps, the program will provide the required solution for the variation of temperature with time for thin square metal plates.