Modeling Heat Transfer Through Concentric Cylindrical Layers for Controlled Thermal Regulation of a Commercial Research Cryostat

Abstract

The thermal characteristics of a variable temperature, flowing vapor cryostat are theoretically modeled, accounting for specific geometrical and material constraints, temperature-varying heat transfer coefficients, and thermal conductivities for conductive, convective, and radiative heat transfer. The temperature within the cryostat is controlled by an internal heater and is monitored at both the heater and the sample stage. The system consists of multiple coaxial, cylindrical layers of stainless steel containing various fluids (light vacuum, helium gas, nitrogen gas; the liquid cryogen is nitrogen or helium). Calculated Prandtl and Grashof numbers for the fluid layers suggest that the Churchill-Chu form of the Nusselt equation be used in the analysis of this system. Formulating a model that predicts heat flows throughout the cryostat allows for appropriate articulation of the heater, so the sample quickly reaches the desired temperature without overshooting. Transient and steady-state models were investigated for predictive ability, with both methods reproducing the system's experimentally collected heating and cooling behavior. The established steady-state model replicates temperatures at the heater sensor to 2%, and the transient model replicates temperatures to an average of 1% of experimental values. Functions and values describing the temperature rise at the position of the heater sensor when the heater is turned on, the cooling at the heater sensor when the heater is turned off, the temporal lag of sample heating, the interfacial temperature values, and the heater control parameters have been determined. Methods for refining the explicit finite difference scheme used for solving the diffusion equation are specified.