Question

Computer memory chips are mounted on a finned metallic mount to protect them from overheating. A \(152-\mathrm{MB}\) memory chip dissipates \(5 \mathrm{~W}\) of heat to air at \(25^{\circ} \mathrm{C}\). If the temperature of this chip is not to exceed \(60^{\circ} \mathrm{C}\), the overall heat transfer coefficient- area product of the finned metal mount must be at least (a) \(0.14 \mathrm{~W} /{ }^{\circ} \mathrm{C}\) (b) \(0.20 \mathrm{~W} /{ }^{\circ} \mathrm{C}\) (c) \(0.32 \mathrm{~W} /{ }^{\circ} \mathrm{C}\) (d) \(0.48 \mathrm{~W} /{ }^{\circ} \mathrm{C}\) (e) \(0.76 \mathrm{~W} /{ }^{\circ} \mathrm{C}\)

Short answer

Answer: (a) 0.14 W/℃

Step-by-step solution

We know the following values: Q = 5 W (heat dissipation of the memory chip) \(T_{chip}\) = 60℃ (maximum chip temperature) \(T_{air}\) = 25℃ (air temperature) Let's find the temperature difference ΔT and the minimum value for U·A. #Step 2: Calculate the temperature difference ΔT#

To find the temperature difference, we use the following equation: $$\Delta T = T_{chip} - T_{air}$$ Substitute the given values: $$\Delta T = 60^{\circ} \mathrm{C} - 25^{\circ} \mathrm{C} = 35^{\circ} \mathrm{C}$$ #Step 3: Use the heat transfer equation to find the minimum value for U·A#

In order to maintain the memory chip temperature below \(60^{\circ} \mathrm{C}\), the heat transfer equation must be satisfied with the given ΔT and the unknown U·A: $$Q = U\cdot A\cdot \Delta T$$ Now, we need to solve for U·A: $$U\cdot A = \frac{Q}{\Delta T}$$ Substitute the known values: $$U\cdot A = \frac{5 \mathrm{~W}}{35^{\circ} \mathrm{C}}$$ #Step 4: Calculate the minimum overall heat transfer coefficient-area product U·A#

Finally, we calculate the numerical value for U·A: $$U\cdot A = \frac{5 \mathrm{~W}}{35^{\circ} \mathrm{C}} = 0.142857 \mathrm{~W} /{ }^{\circ} \mathrm{C}$$ Upon analyzing the given choices, we can see that option (a) 0.14 W/℃ most closely matches our calculated result. Therefore, the overall heat transfer coefficient-area product of the finned metal mount must be at least 0.14 W/℃ to prevent the memory chip from exceeding 60℃. The correct answer is (a).