Question

An electronic package in the shape of a sphere with an outer diameter of $100 \mathrm{~mm}$ is placed in a large laboratory room. The surface emissivity of the package can assume three different values \((0.2,0.25\), and \(0.3)\). The walls of the room are maintained at a constant temperature of $77 \mathrm{~K}$. The electronics in this package can only operate in the surface temperature range of $40^{\circ} \mathrm{C} \leq T_{s} \leq 85^{\circ} \mathrm{C}\(. Determine the range of power dissipation \)(\dot{W})$ for the electronic package over this temperature range for the three surface emissivity values \((\varepsilon)\). Plot the results in terms of \(\dot{W}(\mathrm{~W})\) vs. \(T_{s}\left({ }^{\circ} \mathrm{C}\right)\) for the three different values of emissivity over a surface temperature range of 40 to \(85^{\circ} \mathrm{C}\) with temperature increments of \(5^{\circ} \mathrm{C}\) (total of 10 data points for each \(\varepsilon\) value). Provide a computer- generated graph for the display of your results, and tabulate the data used for the graph. Comment on the results obtained.

Short answer

Question: Plot the power dissipation (𝑊̇) vs. surface temperature (Ts) for three different emissivity values (0.2, 0.25, and 0.3) of an electronic package in the shape of a sphere with an outer diameter of 100mm, placed in a laboratory room with a constant wall temperature of 77 K. The package can operate in a surface temperature range of 40 to 85 °C. Comment on the results obtained. Answer: Using the Stefan-Boltzmann law, we calculated the power dissipation for each emissivity value (0.2, 0.25, and 0.3) and temperature increment (every 5 °C from 40 °C to 85 °C). After plotting the graph, we observed that the power dissipation increases as the surface temperature increases and decreases as the emissivity value decreases. This indicates that an electronic package with a higher surface emissivity requires more power dissipation to maintain its temperature in the given range, making it less efficient compared to packages with lower emissivity values. Therefore, it is crucial to consider the emissivity values while designing electronic packages to improve their performance and efficiency.

Step-by-step solution

Convert temperatures to Kelvin

First, convert the given surface temperature range from Celsius to Kelvin. For this, add 273.15 to each temperature value. For example: 40 °C = 40 + 273.15 = 313.15 K 85 °C = 85 + 273.15 = 358.15 K Step 2: Calculate the surface area of the sphere

Calculate surface area of the sphere

Given that the electronic package is a sphere with an outer diameter of 100mm, we can calculate the surface area (A) using the formula: A = 4πr^2, where r is the radius of the sphere. r = diameter / 2 = 100 / 2 = 50mm = 0.05m A = 4π(0.05)^2 = 0.0314 m^2 Step 3: Use the Stefan-Boltzmann law to find the power dissipation (𝑊̇)

Calculate power dissipation using Stefan-Boltzmann law

The Stefan-Boltzmann law states that the power dissipation (𝑊̇) is proportional to the difference in the fourth power of the temperatures (Ts and Tw), the surface area (A), and the emissivity (𝜀). Let Tw = 77 K (constant temperature of the walls). Formula: 𝑊̇ = 𝜀 * A * σ * (Ts^4 - Tw^4) Here, σ is the Stefan-Boltzmann constant (5.67 × 10^(-8) W/m^2K^4). Step 4: Compute the power dissipation for emissivity values and temperature increments

Compute power dissipation for various cases

For each emissivity value (0.2, 0.25, and 0.3) and temperature increment (every 5 °C from 40 °C to 85 °C), calculate the power dissipation (𝑊̇) using the Stefan-Boltzmann law. Once you have calculated the power dissipation for each of the 10 data points for each emissivity value, tabulate the data and create a computer-generated graph to display the results. Step 5: Comment on the results obtained

Analyze and comment on the results

Once you have plotted and analyzed the graph, comment on the results obtained. Observe any trends or patterns in power dissipation with respect to surface temperature and emissivity values.