A Mathematical Solution To A Two-Phase Mass Query

I was recently exposed to a complex technical problem that has thus far been resolved with a simple overestimated assumption. Calculating the mass of refrigerant in an evaporator or condenser. How is this complex? A refrigerant is typically undergoing a phase change across the component turning from a liquid to a gas and visa versa. (I said "typically" because super-critical carbon dioxide remains a gas across a condenser!)

The phase change results in a dramatic change in fluid density across the component meaning, at each unit of length (mm, inches, furlongs, what ever you want) from the entrance of the evaporator or condenser the density and, therefore mass of refrigerant is different.

Taking the evaporator example forward, the density reduces as the fluid becomes a gas. To maintain the conservation of mass, the velocity of the fluid must increase. When viewed on a system scale, it has been typical practice to take the inlet and outlet densities, assume a linear change, calculate an average density across the component, and determine the refrigerant mass by the product of average density and known volume. This assumption overstates the refrigerant mass as the density change is exponential, not linear.

Using a fluid properties software such as REFPROP, I produced a graph of carbon dioxide density for a constant pressure (31.3 bar) for every unit of specific enthalpy between 220 and 440 kJ/kg. A simple visual review of the graph validates the exponential trend rather than linear. Not everyone has access to REFPROP, nor the computing power to calculate the density per enthalpy unit, so I strived to derive a mathematical equation that can be applied to any scenario.

By taking the established function for an exponential curve, I was able to derive an equation that closely matched the REFPROP iterations (a full proof is in the works!). An exact match is impossible due to the imperfect nature of the data. The equation requires an understanding of the inlet, outlet and one intermediate point density and specific enthalpy value combination. The intermediate point is the intersection point between the REFPROP and Exponential curves.

By calculating the integral of the curve, the area under the graph is calculated, representing a true average density across the component. It is clear that the area of cross section above the REFPROP curve is counter balanced with the cross section area under the REFPROP curve. The product of the true average density with the known volume produces a mass value closer to the true value (within 2% of the cases I've sampled). In the example illustrated, the linear assumption overstate the refrigerant mass by 90%, a major deviation!

I am hoping to run practical experiments to validate the mathematical theory but considering how close the model matches the REFPROP iterations (which is already established as a leading authority on fluid properties), I am confident it is a stronger calculation method than the linear assumption. While this will not dramatically change the way refrigeration systems are designed, it will enable us to produce more accurate calculations and further understand the natural world.