Calculating Cooling Time for a Large Building

[mknott]

How do you calculate the time to cool down a 60'X60'X50' building filled with outside air at 96°F to 65°F with 14,250 CFM of supply air at 55°F?

 

[russ_watters]

How do you calculate the time to cool down a 60'X60'X50' building filled with outside air at 96°F to 65°F with 14,250 CFM of supply air at 55°F?

This is a surprisingly difficult question because you have to consider the thermal mass of the building and it's contents and how efficient the airflow is. If you just want a rough estimate of how fast it can cool the air, Newton's law of cooling (or the dilution/mixing equation) can be solved numerically in a spreadsheet in about 30 seconds...

 

[mknott]

So, what is the Newton's Law of Cooling equation and how would you solve this problem with it?

 

[russ_watters]

So, what is the Newton's Law of Cooling equation and how would you solve this problem with it?

Is this homework or a real life situation? Either way, we like to teach here rather than spoonfeed, so you should be able to take partial answers and nudges in the right direction and move them along yourself...

With that in mind, note the late edit to my post. In this situation, Newton's Law of cooling simplifies to a dilution situation. Dilution is just a weighted average. Do you know how to calculate a weighted average?

 

[mknott]

This is a real life project. I've looked up Newton's Law of Cooling, but what I was seeing was it doesn't take into account air flow. I'm not sure what you are talking about in how it simplifies to a dilution situation. I'm not sure how to calculate weighted average either.

 

[mknott]

This is the equation I was looking at: T(t)=T_a + (T_o - T_a)e^(-kt)

 

[russ_watters]

I'm not sure how to calculate weighted average either.

A weighted average is the amount of air at one temperature plus the amount of air at another temperature divided by the total amount of air.
(V1T1+V2T2)/V3=T3

 

[mknott]

So, how does the weighted average pertain to the equation above? I still have two variables that are unknown.

 

[russ_watters]

So, how does the weighted average pertain to the equation above? I still have two variables that are unknown.

Just use the weighted average. You have only one unknown; the temperature after each mixing interval(T3). Do a series of 1 minute increments using that equation and you'll get your answer.

 

[mknott]

So once I calculate all the temperatures after each interval until I reach the desired 65F, how then am I able to calculate the time to get to that temperature? Sorry if I'm not understanding you correctly.

 

[russ_watters]

So once I calculate all the temperatures after each interval until I reach the desired 65F, how then am I able to calculate the time to get to that temperature? Sorry if I'm not understanding you correctly.

Each interval is 1 minute, so you add up the number of intervals (number of calculations). The fact that the same calculation gets repeated several times is why a spreadsheet works so well...

 

[mknott]

Ok, I understand. Thank you for your help.