Application of Newton's law of cooling to biophysics experiment

[herpetology]

Hi All,

I am hoping to create an equation which I can use to describe the thermodynamic properties of microhabitats used by Chuckwalla lizards. Basically,the habitat in question is a crevice that is shaped like a rectangular prism within an igneous rock. I am trying to develop an equation that can predict the air temperature within the crevice, if the ambient temperature is known. This will help us understand the chuckwalla's energy budget and behavior, etc.

I was wondering if you guys could look over my math to see if it looks like I'm on the right track. If not, I'd certainly appreciate your help!

So, first, I wanted to create an equation to model the head flow across an igneous rock.
So, H= k*A(T_out-T_in)

Where k is the thermal conductivity of igneous rock, A is the total cross sectional area of conducting surface, Tout is the ambient temperature and Tin is the temperature within the crevice, and x is the thickness of the rock.

The air in the crevice will heat up a certain number of degrees for every Joule that flows through the rock:

ΔQ/(v*C)= ΔT_in, where C is the volumetric specific heat of air and v is the volume of air within the crevice.

The amount of heat transferred at time T can be found by multiplying heat flow by time:

H*Δt= ΔQ

Plug H*Δt in for ΔQ and you get:

HΔt/(v*C)= ΔT_in

divide by Δt:

H/(v*C) = ΔT_in/Δt

or, plugging in for H:

(k*A(T_out-T_in)/(v*C) = ΔT_in/Δt

k*A/(v*C) is constant = K, so

K(T_out-T_in) = ΔT_in/Δt

Finally, solving the differential equation using e^Kt as the integrating factor, I ended up with:

T_in(t) = C*e^-Kt - T_out

how does this look? obv, i just ended up with Newton's cooling equation, but I have some idea how to figure out how to estimate K since i did it this way.

 

[herpetology]

Looks like I forgot the include the thickness of the rock, X, in the equation.

should look like this:

H= k*A(Tout-Tin)/x

 

[herpetology]

anyone ?

 

[timthereaper]

Well, it really depends on what your assumptions are and how accurate you want to be. You seem to be assuming that the source of the heat and the outside temperature of the wall are constant, thus you have a constant heat flow from the source through the rock to the air. You might need to verify this, but depending on how little the variation is you probably can safely assume this. If you do, then you can use your equations to find out how much heat is being transferred into the air.

One problem I see is that you are not taking into consideration convection in your equation. Your equations are all Fourier's law of conduction, but convection is another thing entirely. The temperature at the wall might be Tout, but the actual free stream temperature of the air might be less than the wall temperature. You might need to take into consideration air speed and bulk temperatures of the air in order to estimate h better. Since you know the heat being transferred in, once you have h and A, you can find the free-stream temperature better. In retrospect, you probably could use conduction as a good model for enclosed spaces, but you have to make sure you can or else your results will be wrong.

Another thing might be that you have to see if the temperatures are fairly constant over the time interval you are considering. For example, the air temperature in a desert will fluctuate wildly depending on time of day. If you need to take this into consideration, you might want to do a transient nodal analysis of the system.

I hope that made sense to you. If anyone here knows more than I do about this, please correct me.