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Total Pressure vs Total Temperature

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Old   October 20, 2020, 16:01
Default Total Pressure vs Total Temperature
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Why one is defined adiabatically and the other is defined reversibly?



Acarding to Anderson,

Definition of Total Temperature

Imagine that you grab hold of the fluid element and adiabatically slow it down to zero velocity. Clearly, you would expect (correctly) that the values of p, T , and ρ would change as the fluid element is brought to rest. In particular, the value of the temperature of the fluid element after it has been brought to rest adiabatically is defined as the total temperature, denoted by T0.

Definition of Total Pressure

Imagine that you grab hold of the fluid element and slow it down to zero velocity, but this time, let us slow it down both adiabatically and reversibly. That is, let us slow the fluid element down to zero velocity isentropically. When the fluid element is brought to rest isentropically, the resulting pressure and density are defined as the total pressure p0 and total density ρ0.
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Old   October 20, 2020, 16:27
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Quote:
Originally Posted by mardar572 View Post
Why one is defined adiabatically and the other is defined reversibly?



Acarding to Anderson,

Definition of Total Temperature

Imagine that you grab hold of the fluid element and adiabatically slow it down to zero velocity. Clearly, you would expect (correctly) that the values of p, T , and ρ would change as the fluid element is brought to rest. In particular, the value of the temperature of the fluid element after it has been brought to rest adiabatically is defined as the total temperature, denoted by T0.

Definition of Total Pressure

Imagine that you grab hold of the fluid element and slow it down to zero velocity, but this time, let us slow it down both adiabatically and reversibly. That is, let us slow the fluid element down to zero velocity isentropically. When the fluid element is brought to rest isentropically, the resulting pressure and density are defined as the total pressure p0 and total density ρ0.



I suggest to understand these definitions by starting from the Crocco's theorem and then how the total pressure is defined from it driving to the Bernoulli integral.
Remember that T0 defines also the total enthalpy.
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Old   October 21, 2020, 11:53
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It goes back to the question of how much useful work you can get out of a system. If you consider the stagnation temperature first, the equivalent thermodynamic system would be a heat reservoir at temperature T0. Slowing the flow adiabatically is all that is required to reach the max temperature, since the lack of reversibility only shows up as an increase in the internal energy of the gas, and thus leads back to the temperature. For the total pressure, the equivalent thermodynamic system is a work reservoir, and the reversibilility condition means that the molecules are slowed down in a way that minimizes the appearance of energy in the gas in a mode that can't be extracted as work.
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