eksponential kurve (thermodynamic heattransportation) (mathmatics)

[Kartoffelmos]

Hello everyone!

This is my situation:

I have a room, where i know the temperature at this exact moment. (20C).

i have a freezer in the room, this freezer trasports a low amount of cold temperture through the walls, and a high amount of temperture from the compresser (lets just say engine).
what i am trying to do is make a graph of the temperature in the room, when there is no ventilation which means the freezer will increase the rooms temperature.

Now the problem:

The formula for the freezers cold temperatur transportation is dependant on the temperature in the room. (when the room temp rises the amount of warmth transported into the freezer rises)

The compressor on the freezer will turn on every time a set temperature is reached, but this should also increase overtime until its constantly turned on. (at which point it will be throwing ALOT of wartm into the room)

lets assume the room is 100% isolated.

the problem is i dont know how to make a formula that actually depends on another formula, (which then depends on the first)
I.E the freezer temperature is dependant on the room tempeature, but the room temperature is dependant on the freezer compresser, and the compressor is dependant on the temperature in the freezer.

ANY IDEAS HERE WOULD BE AMAZING! dont need a solution just some brainstorming/input on what i could do to get one step closer to a solution, thanks

[teylyn]

Hello,

What you describe are iterative calculations. If you go to Excel's Options > Formulas, you can enable iterative calculations and define the number of iterations to allow. If you don't do that and have formulas where B1 depends on A1 and A1 depends on C1 and C1 depends on B1, Excel will throw a warning for circular references.

cheers, teylyn

[MrShorty]

the problem is i dont know how to make a formula that actually depends on another formula, (which then depends on the first)
I.E the freezer temperature is dependant on the room tempeature, but the room temperature is dependant on the freezer compresser, and the compressor is dependant on the temperature in the freezer.

My first response to something like this is to consider if this is a math problem or an Excel problem. When you say that you don't know how to "make a formula that depends on another formula which depends on the first", do you mean that you know how to do it on paper, but cannot translate the problem into Excel, or are having trouble formulating the problem in any format?

I find that I first have to understand the math behind a problem like this. How are the equations expressed? Are they expressed as differential equations? Would the problem make more sense expressed as differential equations.

[Kartoffelmos]

Well its a bit of both tbh.

the equation on pen an papir, would be quite easy (i think), i will try to explain the mathematics behind each part.

The equation behind the freezer: the higher the room temperature, the faster the cold temperature in the freezer will rise, so this is not linear.

i analysed that the compresser it ran 21 times over 12 hours, (10min avg on each start), at a 20C room temperature,
problem here is the higher the room temperature is the more it will run, since the temperture in the freezer is rising to a specifc setpoint at which point the compresser wil start,
so what i do is say when the compressers run, they generate XkW which means the temperature will rise X over Y time)

so i dont realy know how to combine the two first parts, which i can actually define mathematically, with
the heat generated from the compresser, i use the kW to anylise how much energy is pored into One m3. (with this i know how much the temperature will rise over time)
but im trying to figure out a formula with the compressor atm.

[Kartoffelmos]

freezer temperature.jpg

Here is the equation for the freezer, (general heat excange)

[MrShorty]

I am not real familiar with these kind of problems. It seems to me that you need to focus on the math before you worry about programming in Excel.

You have a differential equation for the heat flow from the room into the freezer. In order to keep the temperature inside the freezer constant, we have to "pump" that heat from the freezer back into the room. It seems to me that, if we could assume the system is perfectly ideal, Q(room to freezer)=Q(freezer to room) and there is no net change in the total heat (and therefore temperature) of the system. Of course, it requires additional energy/heat to pump heat from the cold freezer into the warm room. What equations are we using to describe how much heat enters the system as you pump the heat from the freezer to the room? How are you modeling the "heat capacity" of the room (how much does the temperature of the room change as you bring heat into it)?

It seems to me that there should be three differential equations to describe this:
1) heat flow from room to freezer (Fourier's law). This one seems fairly straightforward, and it will occur continuously.
2) heat flow from freezer to room (some kind of heat pump equation).
3) heat flow from "outside" into the room to "drive" equation 2.
4) an additional equation to relate d(heat) to d(temperature) for both the room and the freezer.

With equations for those quantities of heat, it seems like it should be fairly straightforward to derive an expression for dTemp/dtime. There might need to be two equations (one for when the compressor is not running and one for when the compressor is running), then figure out when to switch from one equation to another.

Does that help you see where the problem seems to be headed? Is there something I am misunderstanding?

[Kartoffelmos]

Hey MrShorty,

Since you are allready getting your hands dirty, i might aswell give you a detailed explanation of the situation.

but yes, you definetly got the idea of the problem.

i will make a post when i have figured something out with one or two more equations aswell as give you a better explanation, since i left some things out just to try to keep the complexity down a notch.

[Kartoffelmos]

Just an update, the thread is most definetly not dead, i am having some trouble making the correct assumtions/equation of how the freezers' kompressers actually effekt the room temperature, the reason is the freezer effeciency, im trying to analyze how much of the Energy (kW) the freezer rechievies goes "cooling", and how much goes to "heating". but i will keep in touch, as soon as i find a "hole" in this.

[Kartoffelmos]

Okay time for an update, this is how the system is actually setup

The freezer and kompressor:
When setting the freezer to its setpoint of -80decree C. it cycles at -78(start kompressor) to -82 stop kompressor.
the freezer has a heat rejection of 595W, at a room temperature of 25C (this is pr hour).
the kompressor runs 31minutes (53% of an hour) at 25C room temperature.

the compressor has a stamp with U=230V I=12A cosphi=0,8
so the power = U*I*cosphi => 230*12*0,8 =2208W

Everything above is fact

From the -80C room, there is a constant flow of heat from the room(25C) into freezer of: (2208*53,1%) 1170W (not 100% sure about this asumption)

there are 9 of these types of freezers in the room

The room
the size of the room is 349,14 m^3

[Kartoffelmos]

So this means the maximum power of the kompressor is 2208W.
and i know the power rejection of the kompressor at 53,1 %runtime is 595. this means the heat rejection is (595/53,1*100)=1120W when the kompressor runs all the time.

this is where im in a bit of a pickle, im thinking when the kompressor runs, the energy it uses is equal to the energy it removes from the freezer right, (1170W).
but it it runs 100% of the time it removes 2208W which dosent realy make sence if it ALSO has a heat rejection of 1120W (that would mean it is actually using 2208+1120W=3328W)

Another solution could be that if i know the heat rejection from when it runs 100% (595/53,1*100)=1120W
then the rest of the 2208 could be what is available to cool the freezer (2208 - 1120) =1087.W

meaning when the compressor is on 53% of the time it cools the freezer with 53% 1087 = 576W.

[Kartoffelmos]

okay im making a new thread, becouse there are some mistakes in this thread and i dont want people to mix up the numbers hope its okay.