Microprocessor Heat Sink Volumetric Thermal Dissipation (Computer & Mechanical Engineering)

A computer systems thermal architect at the UNLV Howard R. Hughes College of Engineering is designing a high-conductivity copper heat-spreader block mounted directly above an enterprise artificial intelligence (AI) accelerator microprocessor.

The heat-spreader block occupies a 3D rectangular box domain in centimeters:

where $0 \le x \le 4\text{ cm}$ and $0 \le y \le 4\text{ cm}$ span the planar die surface footprint, and $0 \le z \le 2\text{ cm}$ represents the vertical thickness toward the finned cooling assembly.

Due to non-uniform computational workload and power gate clustering, steady-state volumetric thermal power generation within the copper substrate is modeled by the spatial scalar density function $q(x, y, z)$ in watts per cubic centimeter ($\text{W/cm}^3$):

To evaluate the required thermal interface material (TIM) conduction capacity and prevent silicon thermal throttling:

1. State the formula for the geometric volume $V(B)$ of the rectangular heat-spreader block and calculate its value.
2. Set up the triple integral over the rectangular domain $B$ using the differential volume element $dV = dz \, dy \, dx$ to calculate the total thermal heat dissipation rate $P_{\text{total}}$ in watts ($\text{W}$).
3. Evaluate the triple integral to find the exact total thermal dissipation rate $P_{\text{total}}$.
4. Calculate the average volumetric thermal generation rate $\overline{q}$ across the heat-spreader block in $\text{W/cm}^3$.
5. Explain how knowing total dissipated thermal wattage and average volumetric heat flux allows computer and mechanical engineers to size forced-air blower CFM requirements or liquid cooling loop flow rates.