I suppose you are looking for an expression for Ta in closed form. What are the independent variables?
Why are you trying to solve this in a closed form as opposed to a numerical solution?
The independant variable is time. I would like an equation (or algorithm) for Ta =F(Tt,Tt') to implement in software to allow a better
estimate of the value minus effects of the acquisition system.
As an example I will illustrate a first order approximation:
Tt is the temperature reading
Ta is the estimate of that actual temperature
Rat is the thermal resistance factor between the subject and the measuring device
Tt' = (Ta-Tt)/Rat
This simply states the the rate of change in the measuring device is the difference in temperatures between
the subject and the measuring device divided by the thermal resistance between them.
the answer would be simply:
Ta = Tt + Rat*Tt'
This is easy to implement in software but it does not work well enough due to other system effects not accounted for.
If it did work it would give a good approximation of the subject's temperature without waiting for equilibrium.
The problem is that in the acquisition system there is coupling between the measuring device and the body of the system.
The body of the system is also coupled to the subject. The temperature of the body is Rm.
The thermal resistance from the subject to the body is Ram. The thermal resistance from the body to the measuring device is Rmt.
I see now that there is an error in the problem I wrote yesterday. The correct problem is:
Tt' = (Ta-Tt)/Rat + (Tm-Tt)/Rmt
Tm' = (Ta-Tm)/Ram
This states that the temperature of the measuring device changes at rate influenced both by the body and the subject
at rates depending on their temperature differences and thermal resistances.
It becomes second order because the measuring device is influenced indirectly through the body.
One numerical method to solve this would be to continually integrate Tm but I would much prefer a solution of the form
Ta =F(Tt,Tt') if at all possible because the in the system that I am using Tm would have to be visible and that would lead to too many questions.
I have modeled the system as described here and it matches the response observed very well whereas the first order approximation did not.
The independent variable of time as used here is simplified because the data are regularly sampled. Tt' is Tt(sample number)-Tt(sample number-1)
The resistance factors are catch-all to account for mass and time also.
Thanks for your consideration of this!
