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I would like to warn you that the reason for corrosion in the coper-iron pipe union is not the Oxygen but the difference in their electronegativity, therfore I warmly recommend using the dielectric unions.

In order to understand what determines the design temeperature, for any space, that is acceptable to human occupants I would recommend you to read ASHRAE Standard 55 (latest edition)"Thermal Enviromental Conditions for Human Occupancy". Or some of the books writen by Fanger.Good luck.

Usually in food processing industry they use plate heat exchangers, for that kind applications. Personally i don't think that cooling tower is good solution for something like that, because you will have very little control over your procees.

You shoud pay attention to word adiabatic = no heat exchage and therefore dQ = 0, but if the process is non quasistatic than you will have dS > 0 or dS < 0.

You should be more specific, and you should pay attention to phase change, because due to the differnce in temperatures, and partial pressures of vater vapor in hot air and bondary layer of the cold plate you can have condensation (which will drasticly change the way of calculting the heat ransfer).

Be aware that in Aerobic studio you have much higher heat realise from ocuppants usually Sensible heat = 790 Btu/hr per occuapant and Latent Heat = 1090 Btu/hr per occupant. I would also recomend providing the ceiling fans in order to increase comfort of occupants and to avoid stratification of...

You are suppose to get capture velocity when you divide hood area with air flow through hood. Actually you are suppose to determine how much air you want to exhaust. Then you have to divide exhaust air flow with the hood capture velocity and that will give you the hood required area.

You should be also careful with using volumetric flow instead of mass flow, because when air passes through cooling coil it changes density.