Analyzing Control Volume Energy & Entropy in a Vortex Tube

[Valgorithm]

Homework Statement

Prior to completing an experiment in which the objective was to analyze control volume energy and entropy in a vortex tube in the Thermal and Fluid Science Laboratory course I am taking, I was required to solve the following two thermodynamics problems:

1. A vortex tube has an air inlet flow at 20°C, 200 kPa, and two exit flows of 100 kPa, one at 0°C and the other at 40°C. The tube has no external heat transfer and no work and all the flows are steady and have negligible kinetic energy. Find the fraction of the inlet flow that comes out at 0°C. Is this setup possible?

2. A Hilsch (vortex) tube has an air inlet mass flow of 50 SLPM at 20°C, 200 kPa, and two exit flows of 100 kPa, one at 0°C and the other at 40°C. The tube has no external heat transfer and no work and all the flows are at steady state and have negligible kinetic energy. Find the fraction of the inlet flow that comes out at 0°C. Is this setup possible?

Homework Equations

In order to solve the first problem, I used the first law of thermodynamics (or the conservation of energy principle) in combination with conservation of mass to determine the fraction of the inlet flow exiting at 0°C:

To determine if the setup is possible, I then used the following equation to determine the amount of entropy generated within the system (per unit mass), dividing this equation by the inlet mass flow rate in combination with substituting in the previously found fraction of the inlet flow exiting at 0°C:

Also note that I assumed constant specific heat for air in determining the enthalpy and entropy changes in the equations above. The relations I used, respectively, resembled the following:

The Attempt at a Solution

Without going through the details of my solution - since I'm fairly certain that I've obtained the correct answers for the first problem - I found the fraction of the inlet flow that comes out at 0°C to be 0.5, while the setup proved to be possible, since the entropy generated was 0.1966 kJ/kg K and therefore greater than zero. My real question pertains to the second problem and how the introduction of a numerical value for the inlet volumetric flow rate alters the solution to the first problem. Wouldn't the fraction of the inlet flow exiting at 0°C remain the same regardless of whether or not a numerical value for the inlet flow rate is specified? I also have some confusion as to how to handle "standard liters per minute" and the fact that the given inlet conditions are not standard. Would a conversion from standard volumetric flow rate to actual be necessary? It is my understanding that it would be acceptable, if working in ratios of mass flow rates as I've done, to use ratios of standard volumetric flow rates instead, since doing so will yield the same result.

I asked the teacher's assistant whether or not the two problems were asking the same questions. He said they are two different problems and mentioned that the second one involves some sort of analysis of the mass flow rates according to conservation of mass. I can only surmise that perhaps the second problem is asking for the actual numerical value of the mass flow rate exiting at 0°C; however, if that's the case, the problem statement is not very clear. Any assistance with or thoughts on how I can handle this second problem are greatly appreciated!

 

[KataKoniK]

Thank you for sharing your experience with the thermodynamics problems you encountered in your course. It seems like you have a good understanding of the concepts and equations involved in solving these types of problems. As a fellow scientist, I would like to offer my thoughts and suggestions on how to approach the second problem and address your questions.

Firstly, I agree with your understanding that the fraction of the inlet flow exiting at 0°C should remain the same regardless of the numerical value of the inlet flow rate. This is because the fraction is determined by the conditions at the inlet and exit, which remain the same in both problems. However, the numerical value of the mass flow rate exiting at 0°C will vary depending on the inlet volumetric flow rate specified in the second problem.

In order to handle the conversion from standard volumetric flow rate to actual, you can use the ideal gas law (PV = mRT) to determine the mass flow rate at the inlet. Then, you can use the conservation of mass equation to determine the mass flow rate at the exit.

As for the confusion regarding "standard liters per minute", this is a unit commonly used in fluid mechanics and refers to the flow rate of a gas at standard conditions (0°C and 1 atm). In this case, it is used to specify the inlet flow rate, and you can convert it to actual volumetric flow rate using the ideal gas law.

In terms of the purpose of the second problem, I believe it is asking for the actual numerical value of the mass flow rate exiting at 0°C, as you mentioned. This can be useful in understanding the performance of the vortex tube and how the inlet flow rate affects the output temperatures.

I hope this helps clarify some of your questions and provides some guidance on how to approach the second problem. Keep up the good work in your course and continue to ask questions and seek understanding. Good luck with your experiment!