To develop an accurate physical model to be used as a testing ground for the machine learning controls to be implemented in the system, all parameters (material and dimensional) must be known as accurately as possible. The parameters A, k as in https://nodus.ligo.caltech.edu:8081/CTN/2191 are not known very accurately so a measurement of the time constant (time taken for the can to naturally cool to 1/e of its initial temperature) was attempted in order to gauge how well the model matches with experiment.
Several previous measurements of the time constant were undertaken, but results varying from 2-5 hours were obtained https://nodus.ligo.caltech.edu:8081/CTN/1728 and therefore, to further investigate what was going on another such experiment was undertaken.
__Experiment:__
On 23rd May, Andrew and I surrounded the assembly of the vacuum can with aluminium foil (as seen in the picture). A setpoint of 45 C was chosen on 23rd May at 20:24, then on 24th May at 15:40 after the set point was reached the heater and PID were turned off. The evolution of the system was recorded.
The steady state temperature obtained was 45.792 0.005 C, where the uncertainty is calculated from a fit of steady state data as seen in the attached figure. A small region before cooldown was used for an estimate of the rms value for the noise obtained by subtracting a polynomial fit (10th order) to detrend the data.
The region of the data corresponding to the cooldown was fitted with an exponential decay using scipy.optimize.curve_fit() with:
The following are the parameters from the fit:
Fit parameters
Parameter |
Estimate |
Uncertainty |
a |
2.4e2 |
1.16e5 |
b |
5.610 |
5e-3 |
c |
2.1e-1 |
2.69e3 |
d |
23.4 |
3.3e-3 |
The uncertainty here is taken from the square root of the correspoding covariance matrix for the fit parameters. This seems very unreliable given the unreasonably large uncertainties in a,c and relatively tiny uncertainties in b,d, even though visually the fit seems good.
According to the fit, the time constant should be 5.610 0.005 hours. But there seems to be many issues with the model including the large uncertainty and the very large value that was calculated from the fit.
It seems like this model is an inaccurate description of the system at this level of sensitivity of measurement. The exponential decay curve does not even visually appear to fit the data to the level of the calculated rms noise value. This can be seen even in previous such experiments (as seen in https://nodus.ligo.caltech.edu:8081/CTN/1728). The possible reasons for this may be that the simple conduction model of the vacuum can may be leaving out conduction or radiation through other significant channels, the gradient across the foam is not linear at all time steps (as is assumed in the conduction equation), the geometrical effects of the foam and can may be more significant than is assumed, or the inner components of the can may be responsible for significant heat transfer. The answer to this may be evident from performing more complex models to fit the data.
The jupyter notebook can be found at https://github.com/CaltechExperimentalGravity/NonlinearControl/tree/master/TemperatureControl/Data/20180518_CoolDownTestVacCan |