A model of vacuum can was constructed taking into account the heat conduction across the foam, neglecting the geometry of the foam and approximating an effective cross-sectional area and thickness.
The analytically calculated time for the decay of temperature to 1/e of the initial value (time constant) = d*m*C/k/A = 2.99 hours, with constant ambient temperature and no heating. The parameters used in this calculation :
Material and dimensional parameters of vac can
| Mass of assembly |
m |
15.76 kg |
| Specific heat of steel can |
C |
505 J/kg-K |
| Heat conduction cross sec area |
A |
1.3 m^2 |
| Thickness |
d |
5.08e-2 m |
| Thermal conductivity constant |
k |
1.136*25.4e-3 W/m-K |
'A' is not known very accurately and could even be as low as 0.48 m^2, and since the time constant is very sensitive to A a better estimate is required for a more accurate model. The units for 'k' in the spec sheet was confusingly W/m^2-K, nonetheless an approximation was made assuming it scales by thickness. An experiment to measure the time constant is now underway. This will give us a better idea of the values.
In order to study the evolution of temperature, scipy.odeint was used to perform numerical integration of the following equation:
 ,
where 'H' is the heating power.
This can be done even in the presence of fluctuating ambient temperatures and variable heating power (taken as a lists, piecewise functions).
In the two trial plots shown, the ambient temperature was stepped up from 300K to 315K at 10000 s (~2.77hours). One plot shows the evolution without heating and the other with a heating power of 18.46 W. |