User:Tohline/PGE/FirstLawOfThermodynamics
First Law of Thermodynamics
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Standard Presentation
Following the detailed discussion of the laws of thermodynamics that can be found, for example, in Chapter I of [C67] we know that, "for an infinitesimal quasi-statical change of state," the change <math>~dQ</math> in the total heat content <math>~Q</math> of a fluid element is given by, what we will label as,
Form A
of the First Law of Thermodyamics
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<math>~dQ</math> |
<math>~=</math> |
<math>~ d\epsilon + PdV \, , </math> |
[C67], Chapter II, Eq. (2)
where, <math>~\epsilon</math> is the specific internal energy, <math>~P</math> is the pressure, and <math>~V</math><math>~= 1/</math><math>~\rho</math> is the specific volume of the fluid element. Generally, the change in the total heat content can be rewritten in terms of the gas temperature, <math>~T</math>, and the specific entropy of the fluid, <math>~s</math>, via the expression,
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<math>~dQ</math> |
<math>~=</math> |
<math>~T ds \, .</math> |
[C67], Chapter II, Eq. (44)
If, in addition, it is understood that the specified changes are occurring over a certain interval of time <math>~dt</math>, then from this pair of expressions we derive what will henceforth be referred to as the,
Standard Form
of the First Law of Thermodyamics
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<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math> |
[C67], Chapter II, Eq. (2)
If the state changes occur in such a way that no heat seeps into or leaks out of the fluid element, then <math>~ds/dt = 0</math> and the changes are said to have been made adiabatically. For an adiabatically evolving system, therefore, the First Law assumes who henceforth will be referred to as the,
Adiabatic Form
of the First Law of Thermodyamics
<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math>
[C67], Chapter II, Eq. (2)
Clearly this form of the First Law also may be viewed as a statement of specific entropy conservation.
Entropy Tracer
Multiplying the Adiabatic Form of the First Law of Thermodynamics through by <math>~\rho</math> and rearranging terms, we find that,
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<math>~0</math> |
<math>~=</math> |
<math>~ \rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr) </math> |
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<math>~=</math> |
<math>~ \frac{d(\rho\epsilon)}{dt} - \epsilon \frac{d\rho}{dt} - \frac{P}{\rho} \frac{d\rho}{dt} </math> |
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<math>~=</math> |
<math>~ \frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon) \frac{1}{\rho}\frac{d\rho}{dt} </math> |
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<math>~=</math> |
<math>~ \frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon)\frac{d\ln\rho}{dt} </math> |
is an equally valid statement of the conservation of specific entropy in an adiabatic flow. In combination with Form B of the ideal gas equation of state, namely,
<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>
we may furthermore write,
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<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d(\rho\epsilon)}{dt} - \gamma_g (\rho\epsilon)\frac{d\ln\rho}{dt} </math> |
Lagrangian Form of the Equation of Continuity, we may furthermore write,
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