You last two posters missed two key points. It's not the coolant. That passage was set to coolant temp before the test was run in my model, and unaccounted for in the older model. I did not allow the coolant temp to rise above normal in the test. That heat came from elsewhere.
Neither my model (shown) nor the older model accounted for the EGR at all.
The old model shown (black and white) was made in CAD, and various known temperatures were constrained before the test.
2: Constraints
The bottom surface was constrained to 120° Celsius (250° Fahrenheit), the intake (the hole in the trapezoid) to 36° C/ 97° F, and the other surfaces exposed directly to the engine bay were set at 76° C/169° F. Surfaces exposed to the engine bay that shared an edge with the bottom mounting plate were left unconstrained.
I did the same, but with some differences. I made my own 3D model based on an empty housing, and constrained the bottom surface of the supercharger and the coolant passage to known temperatures off my cars when warmed up, then began the test with a typical CFM of air through the inlet.
My model was created in CoCreate Solid Modeling (hp designed it originally under UNIX as ME30, then sold it to CoCreate who now license it back to us as a discount under windows). I used the finite element analysis module for my thermal testing.
One possibility for the odd rise in temperature is that the lower area of the inlet isn't exposed to engine bay air. Their model was more simplified, causing the entire inlet to rise in termperature. My model was far more advanced, and showed a specific rise in one area.
I did my modelling quite some time ago, and don't recall the results of every test. I varied alot of the parameters including coolant on/off, coolant temp, LIM surface temp, air temp, etc. Every time, whether the EGR or Coolant was on or off, the lower surface of the inlet was always hot. In some tests, the outlet did rise in temp, until I smoothed off the edges in the model (which is why I port/smooth my inlet and outlet the way I do now).