Gleaning More Data From Newton's Law of Cooling

[teilhardo]

Hi,

I am currently trying to expand upon the data set that I recently recorded from a heat exchanger that I am trying to better understand. I'd like to use Newton's law of cooling (the differential version) to try and find a way of determining the time it will take for the cold side of my heat exchanger to come to steady state temperature when different powers are applied to the hot side. The surface area, mass and materials are not easily accesible. What I do have is a very accurate thermistor and a variable power supply that I can use to adjust input power to the heater on the hot side.

Thus far I have applied 2,4,6 and 8 watts of power to the heated side and waited for the cold side to come to steady state. I have recorded all these temperature values. I have also recorded temperature data every 200ms when I applied 12W of power until steady state temperature was achieved. I plotted this data in JMP and came up with a really nice model in the form of Newton's Law (the model determined the value of "r" as seen in the equation here:

http://upload.wikimedia.org/math/2/5/6/256d51eb7ebb24952c4744af00c443e7.png

However, I'd like to determine the value of "r" pre-emptively so that I will know in advance the time it will take my system to equilibriate. In effect, it seems like I need to calculate the heat capacity and heat transfer coefficient without knowing any physical parameters (like surface area, mass, diffusive flux, etc). Is there any way that I can do this with the data I have?

Thanks,
Tei

 

[boyze]

I've had situations similar to this where I didn't know some of the thermal system parameters but had T(t) data and could make a reasonable assumption on the form of the thermal model. I used the "Solver" function in Excel to back out the unknown parameters. It's essentially a regression analysis.

Here's one example.

Have Fun!

James A. Pike

http://www.xl4sim.com

http://www.erieztechnologies.com

 

[IRstuff]

?? It seems to me that you are mixing apples and oranges in your question.

Your heat exchanger's time constant is generally negligible, since its cooling efficiency tends to minimize its net thermal mass. Therefore, your transient response is essentially dependent on your load parameters, i.e., its thermal mass, thermal conductivity, and load.

So, I don't get how you think you can calculate the transient performance without knowing the thermal mass and conductivity of the load. You can measure the results and derive the parameters, but with a priori knowledge, you cannot calculate it ahead of time.

You seem to asking, "what is the 0 to 60 time of this truck, but I'm not going to tell you how big a load it's towing." The answer is going to be different if it's pulling nothing compared to if it's pulling 40 tons. Newton's law is only describing the steady state speed of the truck for a given rpm, say.

TTFN
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[teilhardo]

Hi James,
Thanks for the response. I am indeed finding a solution to my data which looks very similar to this graph:

The statistical tool that I am using has determined that my value of k (as shown in the picture) is 5.66E-5 (s^-1). However, this data was determined empirically using a fixed input power (as determined by input voltage and current).

According to the equations that I have found on the internet, namely:

1) k=hA/C (known from model)
where
2) h=heat transfer coefficient (unknown) in [W/m^2*K]
3) A=surface area (unknown) [m^2]
4) C=heat capacity (unknown)[J/K]

I'm assuming that the graph referenced above changes according to my input power. Assuming a constant ambient temperature, it seems that the final, steady state temperature as well as the time it takes to get to the final resting temperature will change according tot he input power.

Based upon the equations above, it seems like A and C should be constant (based on the physical characteristics of the system) but that h might change according to the temperature extremes. I'm just trying to see if my unknown parameters can be solved using the data that I have already recovered.

Thanks,
Tei

 

[teilhardo]

IRSTUFF,

Thanks for your response. Based upon the equations that I have listed, it seems like all the unknown constants can be lumped together as they are the only things staying the same between power levels. I'm just wondering how I can group them all together and come up with a method of making k a function of input power. Is this not possible?

Thanks,
Tei

 

[IRstuff]

k is NOT a function of input power in an electrical analog, since it's analogous to the RC time constant. The only reason it would change, in your case, is because you are changing R or C when you change the input.

TTFN
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[teilhardo]

IRSTUFF,

Thanks for your help. So, I am assuming that we are incapable of empirically determining the value of this exponential value as a function of input power?

Thanks,
Tei

 

[IRstuff]

Because it's not a function of input power. Whatever you think is the relationship, it's not.

TTFN
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