Question

A group of 25 power transistors, dissipating \(1.5 \mathrm{~W}\) each, are to be cooled by attaching them to a black-anodized square aluminum plate and mounting the plate on the wall of a room at \(30^{\circ} \mathrm{C}\). The emissivity of the transistor and the plate surfaces is 0.9. Assuming the heat transfer from the back side of the plate to be negligible and the temperature of the surrounding surfaces to be the same as the air temperature of the room, determine the size of the plate if the average surface temperature of the plate is not to exceed \(50^{\circ} \mathrm{C}\). Answer: $43 \mathrm{~cm} \times 43 \mathrm{~cm}$

Short answer

Answer: The size of the aluminum plate should be 43 cm × 43 cm.

Step-by-step solution

Calculate total power dissipated

First, we need to calculate the total power dissipated by the group of 25 power transistors. Total power dissipated = Number of transistors × Power dissipated per transistor Total power dissipated = \(25 \times 1.5 \mathrm{~W}\)

Calculate temperature difference

Next, we need to find the difference in temperature between the plate's average surface temperature and the air temperature of the room. Temperature difference = Plate's average surface temperature - Room air temperature Temperature difference = \(50^{\circ} \mathrm{C} - 30^{\circ} \mathrm{C}\)

Calculate heat transfer rate

Now, we need to calculate the heat transfer rate from the plate using the Stefan-Boltzmann Law. The Stefan-Boltzmann Law states that the heat transfer rate (Q) is proportional to the emissivity of the surfaces, the Stefan-Boltzmann constant, the surface area, and the difference in the temperatures raised to the power of 4. Q = emissivity × Stefan-Boltzmann constant × Area × (Temperature difference)^4

Determine the area of the plate

We can solve for the area by rearranging the formula from Step 3 and entering the given values. Area = Q / (emissivity × Stefan-Boltzmann constant × (Temperature difference)^4) Area = (Total power dissipated) / (0.9 × (5.67 x \(10^{-8} \mathrm{W/m^2 K^4})\) × \((20^{\circ} \mathrm{C})^4\)) Plug in the values calculated in Steps 1 and 2: Area = \((25 \times 1.5 \mathrm{W}) / (0.9 × (5.67 x 10^{-8} \mathrm{W/m^2 K^4}) × (20^{\circ} \mathrm{C})^4)\) Area = \(0.1875 \mathrm{m^2}\)

Determine the size of the plate

Since the aluminum plate is square, we can take the square root of the area calculated in Step 4 to find the length of each side. Side length = \(\sqrt{0.1875 \mathrm{m^2}}\) Side length = 0.433 \mathrm{m} Finally, convert the side_length to centimeters to get the final answer. Side length = \(0.433 \mathrm{m} \times 100 \mathrm{cm/m}\) = \(43 \mathrm{cm}\) Therefore, the size of the aluminum plate should be \(43 \mathrm{cm} \times 43 \mathrm{cm}\).