I replicated Koji's recent cooldown analysis on data prior to painting the test mass with black coating. The model first considers conductive cooling of the coldplate + inner shield from the cold head, via copper braid. Then it considers radiative cooling of the test mass from the coldplate + inner shield.
To model the conductive cooling of the coldplate + inner shield, I used:
dT_coldplate/dt = C * k_Cu(T) * A_Cu/l_Cu * (T_coldhead - T_coldplate) / (Cp_Al(T) * m_coldplate)
where A_Cu and l_Cu are the cross-sectional area and length of the copper braid. I used a temperature-varying heat capacity of Cu and specific heat of Al. In order to align this model with the data, I found that the constant C=0.085. I am not sure what extra factors should be contributing to this scaling, but once it is added, the model aligns well with the two time constants apparent in the data [Attachment 1]. I will connect with Koji to determine what he considered here / if there is something I am missing.
I then took the aligned coldplate + inner shield cooling model to consider the radiative cooling of the test mass. I used:
dT_tm/dt = Fe * sigma * A_tm * (T_coldplate^4 - T_tm^4) / (Cp_Si * m_tm)
where we assume T_coldplate is the temperature of the coldplate + inner shield. Fe is the emissivity coefficient Koji considers in his analysis, which I found to be consistent with his result: 0.15. As he stated in a previous entry , Fe can be broken down as:
1/Fe = 1/e_tm + (1/e_surr - 1)*A_tm/A_surr,
where e_surr and A_surr are the emissivity and area, respectively, of the surrounding coldplate and inner shield. Using e_surr=0.07 (rough Al), we get that an Fe value of 0.15 corresponds to a test mass emissivity of 0.18. This is slightly lower than Koji's value, due to differences in our calculated surface areas, but otherwise consistent.
A key point is that once the conductive cooling of the coldplate is modeled accurately (with a fudge factor of 0.085), the radiative cooling model of the test mass lines up well with the data without the need for another fudge factor. Note that the radiative cooling model above does not use a temperature-varying specific heat of Si [Attachment 1]. If a temperature-dependent value is used, we end up with the test mass cooldown model seen in Attachment 2. This causes the model to diverge from the data, so another factor might be missing in the model. Perhaps using temperature-dependent emissivities will correct for the deviation and cause even better agreement. This is a future step for the model.
Lastly, the painting of the test mass will increase its emissivity value, strengthening the radiative link between the test mass and its surroundings. (Koji has already posted updates on this cooling trend, and I will use this data once I obtain a copy.) Based on Koji's entry , we can consider a new e_paint of 0.5 and 1. Attachment 3 compares radiative cooling models of the test mass using different emissivity values. We can expect that if the coating performs as expected, the test mass can reach 123K between ~87-110 hours. A next step is to plot the cooldown data for the painted test mass to see how accurate this prediction is.
I will next aim to understand the 0.085 fudge factor needed to align the conductive cooling model with the coldplate cooling data. I will also add a fitting feature to directly spit out the optimal factors needed in both conductive and radiative cooling.