# Heat transfer inside a rectangular cross-section box

I am required to design a water cooling system to a motor controller (unfortunately our application doesn’t support the use of ambient air as a coolant), the controller usually as an aluminum plate where the heat is usually transmitted through, the following figure shows how water will flow above this plate cooling it. These walls will direct the flow of water. I am now required to analyze the heat transfer through this set up. The first step is to work out h and I had a couple of questions.

$1$-I needed to work out Reynolds numbers, so I chose the characteristic length to be the hydraulic diameter of a rectangular duct, but when then I realized since heat transfer comes only from the bottom plate then, this case is closer to the flat plate case, so I assumed it’s a rectangular duct when it comes to fluid dynamics but a flat plate when it comes to heat transfer. Are those assumptions correct? If not, what’s the correct thing to do ?

$2$-If I am going to consider it a flat plate should I assume that the length of this plate is the total length of the path of the water inside the box or the maximum straight length?

$3$-Water goes in cold and comes out hot, so in the equation $Q=hA(T_w-T_p)$ should $T_w$ be the logarithmic mean or the arithmetic mean of $T_h$ and $T_c$? $1$ - Definitely use the hydraulic diameter of a rectangular duct. In the spirit of the Reynolds Number, water is flowing through that duct.
$2$ - Use the path length as you drew it (green line).
$3$ - If you want effective cooling of the plate, the difference $T_h-T_c$ should be small. That means that volumetric throughput should be sufficiently large. In that case which average you choose will not matter that much.