So is it more important to have a good path for airflow at the top or sides? If you have a 37" in a 36" wide opening, how much space is left on the sides?
It definitely is more important to have an airflow path at the top so warm air can exit. In my situation, openings at the sides barely exist. The 37" set fits snugly in the space between the flipper doors when they are open. It's not so tight that the doors can't slide in or out, but there is very little extra room on the sides. LCD TVs generate much less heat than plasma sets, making all this not such a problem.
I didn't mention before that I have a ceiling fan in my family room. Even at its slowest speed, it helps dissipate heat from the TV and the much warmer amp and receiver that are in lower shelves in the same cabinet.
I should probably just decide I'm going with the 46 and cut the dimensions down a bit so I only have 2 inches on both sides and the top. I have the opportunity to make further changes because now they inform me that there is a problem with making the opening 50" wide due to the fact that their sheets of wood are 48" wide. The grain of the wood would have to run horizontally on the back panel when everywhere else it would run vertically.
That would simplify things wouldn't it? Just the same, I think it is worth the effort to try and squeeze in the largest screen possible. I had originally thought I could only fit a 32" set into my space. I visited a few stores with a tape measure in my pocket, and that extra leg work paid off with a larger screen that looks like it was custom made for the space. In your case, if you can fill most of that space with the TV, you may not notice which direction the grain goes on the panel behind it.
The back cut-out is my idea of providing extra ventilation. Currently I have it as 8" x 8", 8" from the bottom, and centered left to right. I arrived at those numbers by looking at the drawings in the manual for the Sharp 46/52D82U manuals and it places the cut-out directly behind the center of the TV.
I was attempting to design it with enough space so I could go either 46 or 52 but that is the big problem. With my current dimensions of 50"w x 36"h it leaves a lot of space at the top if I go 46 and I wonder if that gap will bother your eyes. If I go with the 52, I'd have only 1" space on top and none on the sides.
In my case the TV won't be totally enclosed. The screen will be flush with the sides of the hutch and that leaves a good 6" behind the TV to the back of the hutch (hutch is 13" deep) because the pedestal extends a few inches behind the screen.
As I've said above, instead of centering the opening, you should think about moving the opening closer to the top of the space. You may be better off with two openings, one closer to the bottom for cable access, and another near the top, as wide as possible, for hot air escape.
I hope your cabinet plans work out well. It sounds real good so far. Sometimes it gets difficult to visualize things before they are there in three dimensions. Don't forget to make a space for your center channel speaker

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Whatever way you end up building the cabinet, you are going to love your new Sharp HDTV, whether it is 46" or 52".