Question

How do I calculate the heat being produced by a bunch of servers enclosed in a room. I have measurements for the air flow in and air out of the room through fans. I have the temperature at both fans, too. The diameter of both fans is 6inches and they are circular. Any ideas?

Answer 1

We can model the room as an open thermodynamic system. From the first law, we know that\[Q - W = \dot m \Delta h\]where \(Q\) is the heat being added to the system, \(W\) is the work being done by the system, \(\dot m\) is the mass flow rate of the air, and \(\Delta h\) is the change in enthalpy.
We can simply our analysis considerably if the make a couple of assumptions:
1) Air as having a constant specific heat.
2) The fans simply move the air and don't do any measurable amount of work on the air. (Obviously the fan do work on the air to move it, but will consider this work as not adding any thermal energy to the system.)
3) Both fans have the same mass flow rate.
4) The room is adiabatic. Therefore, the only heat addition is from the servers.
Let's observe the impact of these assumptions to our first law equations.
From (1), \(\Delta h = c_p \Delta T\) where \(\Delta T\) is the change in temperature of the air at the fans and \(c_p\) is the constant pressure specific heat of air (Can be found here: http://www.engineeringtoolbox.com/air-properties-d_156.html) (Use the specific heat corresponding to the average temperature at the inlet and outlet fans.)
From (2), \(W = 0\)
From (3), \(\dot m_{in} = \dot m_{out}\) (Refer to the diagram below.)
From (4), \(Q=\) the heat generated by the servers.

Now, let's draw our system boundary.

Now, we are able to come up with a manageable first law expression, as follows:\[Q_{server} = \dot m c_p(T_{out} - T_{i n})\]\(Q\) will have units Watts.

If we don't known the mass flow rate data, we can still solve this problem, as follows:\[q_{server} = c_p (T_{out} - T_{ i n})\]Here \(q\) has units \(\rm J \over kg\)