Recent data from the vortex tube experiment showed a larger temperature differential than expected. Because the vortex tube operates at hydrogen input stream temperatures of about 120K, the goal is to isolate it from external sources of heat by means of a vacuum chamber. This allows any observed temperature differential to be solely attributable to the vortex tube effect as described by the kinetic impinging model or solid core rotational model. The basic experimental setup is depicted in the simple sketch below:
Our initial assessment of the potential sources of the temperature discrepancy included the conductive heat-leak through the copper inlet and outlet pipes that connect to the vortex tube through the bulkhead in the vacuum chamber. We assumed one-dimensional heat transfer through a copper pipe and modeled this heat-leak with the following equations:
Here, Q is heat flow, k is the thermal conductivity (units of W/(m*K), which varies based on temperature), A is the cross-sectional area of the copper pipe, ΔT is the temperature change, and ΔX is the length of the pipe. Because the coefficient of thermal conductivity varies with temperature, we used a vertical lookup function within EES (Engineering Equation Solver, the software we use to build these models) where k is referenced to a table of conductivity in one Kelvin increments. To quantify the rate of heat transfer, we integrated k with respect to T to calculate power (heat flow) per unit area which was expressed in watts on the right side of the equation, as shown below.
However, the approaches shown above neglected the different materials (i.e. brass, copper, aluminum) that comprised the pipe fittings, nor did they account for the varying geometry of said pipe fittings. To further develop the model, we viewed the system as a “thermal circuit”, similar to what is depicted below.
This approach allows for the addition of “thermal resistors” in parallel and series, much like when applying Ohm’s law to an electrical circuit. In taking this approach, we can model each different pipe, fitting, and component as an individual thermal resistor and sum them up to find an equivalent “thermal resistance” for the system. Once the equivalent thermal resistance for the system is found, the computation of the heat transfer rate depends only on the temperature difference.
As the next step, we plan on determining the thermal resistance of the vortex tube piping by constructing a full model of the system in EES based on this approach.