How long will it take for the temperature in the engine room to reach the maximum allowable value?

Question

Data center

Gentlemen, the designers of data centers and their sympathizers, I am trying to figure out how to determine (in theory, not in practice) the time during which, with the air conditioning turned off, the average temperature in the machine room will rise to a critical maximum. It is assumed that I have the following initial data: the dimensions of the machine room; temperature at the time of turning off the air conditioners; maximum allowable temperature; heat dissipation of equipment in the hall. It is not supposed to take into account heat losses from walls, air leakage and other non-quantifiable parameters. Maybe there are some formulas from thermodynamics suitable for such calculations, maybe there are formulas that perfectly correlate with practice? I do not hope to get the exact value, but at least understand whether this time will be calculated in seconds, minutes or tens of minutes.
Thank you!
UPD: I made the following calculation attempt. I used the formula Q=cm(t2-t1), where the
specific heat capacity of air at 20 deg. Celsius c=1005 J/(kg deg);
air mass in the turbine hall with dimensions, for example, 5x5x4 m and at an air density of 1.2 kg / m3 (at 20 degrees Celsius) m = 1.2 kg / m3 x 100 m3 = 120 kg;
t1 - let's say 20 degrees;
t2- 35 degrees.
Then Q = 1809000 J.
So, if I have equipment heat dissipation (30 servers), for example, 20000 BTU / hour = 21100000 J / hour, then the temperature in the server room will rise by 15 degrees in 1809000/21100000 = 0.086 hours = 5 minutes.
But such calculations do not fit in with practice; in reality, the numbers will be at least an order of magnitude higher.
UPD2: Some commentators noticed that after all, in vain I neglected the walls, equipment and other things. I agree, it sounds reasonable. But I have no idea how to put it all together. For example, walls and equipment do take some of the heat, but other formulas are needed here, not the one I used for air, at least because we are not going to heat them (walls and equipment) to 35 degrees. At the same time, heat still flows out through the walls, and the equipment simply accumulates heat. Complicated! Nevertheless, I still have hope that someone has already gone down this path, all the more confirmed their calculations in practice.

Answer 1

Eh ... I remember back in my school years I loved and knew physics very well.
As you have already noticed, there are at least three components in your task: walls (also ceiling and floor), air and equipment (I will consider them because they consume the most heat). So for the air you have already calculated everything. If I understood you correctly, then you took 35 as the critical temperature. But this is only the air temperature (yes, I'm a cap, but it's easier for me to reason this way). Question: what temperature will the other components have, because they even have different heat capacities. The easiest way out, in my opinion, is simply to take "by eye" the final temperatures of these objects. For example, the final temperature of the walls will change by a maximum of several degrees ( t2 - (~ 22-25)). However, the equipment will heat up much more, because it is the server equipment that gives off heat to the air (cap again). Total about 50-60 degrees. Again, this is all so to speak "by eye". Thus, you have three Q formulas. They are added and the result is found. What is there in terms of masses and specific heat capacities - you already know better. I took the walls, ceiling and floor of concrete 20cm thick. As for the equipment, it’s hard to say: I would just take iron so as not to rack my brains for a long time, if you really find fault, then you can also take plastic into account. The volume of equipment can be roughly estimated by subtracting a third or a quarter from the volume of the room (as you like). You can also take plastic into account. The volume of equipment can be roughly estimated by subtracting a third or a quarter from the volume of the room (as you like). You can also take plastic into account. The volume of equipment can be roughly estimated by subtracting a third or a quarter from the volume of the room (as you like).
Having done all this, you can get, albeit very rounded, but the result. More precisely it will turn out only in practice.

Answer 2

I purely practically did it - turned off the air conditioning in the server room, and followed the temperature growth graph according to the monitoring data in Zabbix.
Having reached a certain control value, the air conditioning was turned on again, and the graph was extrapolated.
In my case, the critical threshold was reached in about 40 minutes from the moment the air conditioning was turned off.

Answer 3

But such calculations do not fit in with practice.

Because the heat from the server will go not only to heat the air (with a small specific heat), but also to heat metal racks, radiators, cases, concrete walls / ceilings, and other things (with a high specific heat). That's just the concrete will not warm up entirely (this takes many days), but only superficially. It would be very difficult to calculate.
Conduct practical experiments in the desired data center or at least a comparable server room: turn off the air conditioners, put thermometers at several heights and note the time. It is not necessary to bring it to the limit - it is enough to stop the experiment (turn on the air conditioners) after 10 degrees.