The Physics of Gas Temperature in the Vortex Tube
To understand the governing physical laws of vortex tubes, we need to first understand gas temperature. Amazingly, this very concept of gas temperature was the first topic of debate when the vortex tube effect was introduced. We read in the article "Ranque's Tube" by C. D. Fulton (Massachusetts Institute of Technology, 1950) the following: "When Georges Joseph Ranque stood before his colleagues of the Societe Francaise de Physique in 1933 and said that hot and cold air came out of a simple piece of pipe, he was received with skepticism. He was a metallurgist associated with a steel company in the mountain town of Montlucon in central France; high-speed gas flow was out of his line. No, said E. Brun, the aerodynamicist, stagnation temperature had been confused with static temperature and adiabatic wall temperature; the two streams weren't really cold and hot. There is no further mention of the subject in the Societe's Journal through 1944." (C. D. Fulton, J. ASRE Refrig. Eng. 58 (1950) 473-479)
It is my personal experience that this very subject of gas temperature was questioned by one of the referees when the new physical theory of vortex tubes was considered for publication in Physical Review Letters. Only after the definitions of temperature were clearly stated in the manuscript text, the objections were withdrawn. These definitions of gas temperatures are not new; they exist in the textbooks, but the reader is to be properly re-introduced to them. Why is that? Because in the general course of physics at university, the only temperature introduced is thermodynamic temperature. This was my experience at the University of Sofia and quite possibly the concepts of static and total (stagnation) gas temperature are not dwelt on in the general course of physics in academia worldwide. And this is understandable, since such concepts are applicable mostly to the narrow field of gas dynamics.
What is static temperature?
In the Physics classroom, we learn that gas temperature is measured with thermometer, when the thermometer is in thermal equilibrium with the gas. Quite importantly, the thermometer is at rest with respect to the gas, when the temperature of the gas is being measured. In physics, this temperature has many names, but they refer to the same quantity: temperature, thermodynamic temperature or static temperature of the gas. We denote this temperature with Ts (static temperature).
Both thermometer and gas are moving with uniform velocity c. The measured temperature is Ts. The velocity c can be zero: then both are at rest. Or it can be nonzero. In the end, the value of c does not matter. The thermometer always shows Ts when both are at rest with respect to each other.
What is total temperature?
Total temperature T of gas is introduced for the more general case of moving gas and stationary thermometer, that is, for the case when both are moving with respect to each other. In this situation, the following happens:
1. Assume the thermometer is at rest, while the gas is moving with uniform velocity c.
2.The moving parcel of gas is forced to stop.
3.The entire process of stopping the gas is done adiabatically: no heat is added to the gas, no heat escapes.
4.Once stopped, the thermometer measures the temperature of the gas. The measured value is T, the total temperature of the previously moving gas: Ts(t1)=T(t0); the static temperature measured at time moment t1 is equal to the total temperature of the gas at time moment t0.
T is known as total temperature or stagnation temperature.
What is the definition of total temperature?
Definition: Total temperature T is defined as the ratio:
where H is the total, or, stagnation enthalpy; m is the mass of the gas; cp is the isobaric heat capacity. The relation between the total temperature T and the static temperature Ts of gas moving with velocity c is defined as
The meaning of this definition is: If a parcel of moving gas has a thermometer moving right next to it, the thermometer measures the static temperature of the gas to be Ts. Now, the gas is forced to stop. This is done adiabatically, no heat escapes, no heat is added. The thermometer also stops. It then measures the temperature of the gas again. It now reads T and equals Ts+c2/(2cp). When moving gas is brought to rest adiabatically, its static temperature rises by c2/(2cp), where c is the velocity of the initially moving gas, before it was forced to stop. This last measurement of the static temperature is called "total" or "stagnation" temperature, because it is a result of the initial static temperature while the gas was still moving plus its rise after the gas was forced to stop adiabatically.
How does one measure total temperature?
There is no thermometer that can measure total temperature. Thermometers are designed to measure static, or thermodynamic temperature only. Total temperature is an effective temperature; it is a sum of entities that have units of temperature. To measure the total temperature of a moving gas parcel, it first needs to be stopped adiabatically. Then its static temperature can be measured, and will equal the total temperature of the previously moving gas parcel. If a parcel of gas is already at rest, its total and static temperatures are equal. If the total temperature of a gas parcel is different than its static temperature, then this gas parcel is moving. Let two parcels of gas have the same static temperature. If one of them has higher total temperature, then it is moving with higher velocity in the reference frame where the total temperature is measured. Because the total temperature depends on the square of the gas velocity c2, it does not depend on the direction of motion, but only on the velocity magnitude.
It is important to understand the difference between static temperature Ts and total temperature T of gas. This is crucial knowledge in the understanding of the vortex tube effect.
How does gas temperature relate to energy?
The total temperature T is related to the total enthalpy of the gas H. Static temperature Ts is related to the internal energy of the gas U. Therefore, Ts reflects the internal energy of the gas, while T reflects the internal energy of the gas plus its kinetic energy.
Is static temperature frame-dependent?
By definition, the static temperature Ts is the same in all frames of reference, because the thermometer is required to move with the gas when Ts is measured. The very definition of Ts is such, that frames of reference do not matter. Unless the thermometer is stationary with respect to the gas, measuring Ts is meaningless.
Is total temperature frame-dependent?
Yes, it is. The total temperature T is not the same in all inertial frames of reference, as it depends on the velocity of the gas c. The term c2/(2cp) is frame-dependent, therefore T is frame-dependent. It is a Galilean invariant, because it retains its mathematical form in any inertial reference frame.
In the stationary frame of reference F, it is
In the inertial frame of reference F', it is
In rotating frames, T no longer has this mathematical form, since a rotating frame is not inertial.
The information contained in this site is based on the following research articles written by Jeliazko G Polihronov and collaborators:
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