We use liquid nitrogen to mantain the temperature at 77K for at least 24h. The bench and the nitrogen is heated by radiative and conductive thermal exchanges. Radiative thermal exchange is decreased by using a passive thermal shield between the active shield at cryogenic temperature and the vacuum jacket at room temperature. The equilibrium temperature of the passive shield is given by its surface and the one of the passive shield, ie the radius of the passive and active shield. Heating by radiation depends on that temperature assuming shields as grey bodies with emissivity epsilon~0.01.
The conductive heating comes from the front of the instrument. The bulkhead at cryogenic temperature is attached to the telescope at room temperature through a G10 fiber ring with low thermal conductivity. Heating will depend on the thickness and length of this ring. In opposition to the tilt of the instrument heating will be higher for high cross-section and short length of the cylinder. Finally, the time of the instrument will remain cold is proportional to the volume of nitrogen on the cane. The effect is also opposite to the mechanical behavior since increasing the volume of nitrogen will increase the distributed weight and the bending of the bench. The thermo-mechanical analysis requires iterations in both calculations in order to optimize the bending while keeping the autonomy of the instrument at least 24 hours.
Heating of Dewar:
Conduction from front = 21.9834 W
Conduction from Passive Shield = 0.0880333 W
Radiation from Passive Shield = 2.63343 W
Radiation of front = 4.96393 W
Radiation through Window = 25.7159 W
Conduction by Electronics = 2.84458 W
Electronics = 0.00000 W
Total Heating of the Dewar = 58.2292 W
Since the heating of the liquid is ~58W and there is 0.022L/Wh of liquid nitrogen consumed the time of total consumption for a total volume of 143L is 111h. The total weight of the bench (20% lightened) + dewar + mirrors + liquid is 552kg. This does not include the weight of the instrument ring, shields and flange. In order to decrease the weight of the system, we could limit the volume of liquid nitrogen to 30% whilst verifying the specifications.