The conservation of energy is a fundamental concept of physics along with the conservation of mass and the conservation of momentum. Within some problem domain, the amount of energy remains constant and energy is neither created nor destroyed. Energy can be converted from one form to another (potential energy can be converted to kinetic energy) but the total energy within the domain remains fixed. On some separate pages, we have discussed the state of a static gas, the properties which define the state, and the first law of thermodynamics as applied to any system, in general. On this page we derive a useful form of the energy conservation equation for a gas beginning with the first law of thermodynamics.
Thermodynamics is a branch of physics which deals with the energy and work of a system. As mentioned on the gas properties slide, thermodynamics deals only with the large scale response of a system which we can observe and measure in experiments. As aerodynamicists, we are most interested in thermodynamics in the study of propulsion systems and understanding high speed flows.
Aeronautical engineers usually simplify a thermodynamic analysis by using intensive variables; variables that do not depend on the mass of the gas. We call these variables specific variables, and many of the state properties listed on this page, such as the work, internal energy, and volume are specific quantities. Engineers usually use the lower case letter for the "specific" version of a variable.
Let us begin with the specific work (w) equation.
Equation 1: w = (p * v) sub2 - (p * v) sub1 + wsh
There are two parts to the specific work equation for a moving gas. Some of the work, called the shaft work (wsh) is used to move the fluid or turn a shaft, while the rest of the work goes into changing the state of the gas.
The first law of thermodynamics defines another state variable called the specific internal energy (e). The change in internal energy is equal to the heat flow (q) minus the work done by the fluid. For a moving gas, we add an additional term which involves the specific kinetic energy of the gas (k). (If we were dealing with a large amount of gas, we would also have to include some terms for potential energy, but we are neglecting those effects here.) The first law then becomes:
Equation 2: e sub2 - e sub1 + k sub2 - k sub1 = q - wsh - (p * v) sub2 + (p * v) sub1
If we perform a little algebra on the first law of thermodynamics, we can begin to group some terms of the equations.
Equation 3: e sub2 + (p * v) sub2 - e sub 1 - (p * v) sub1 + (.5 * u ^2) sub2 - (.5 * u ^2) sub1 = q - wsh
A useful additional state variable for a gas is the specific enthalpy (h) of the gas.
Equation 4: h = e + p * v
Use Equation 4 to simplify Equation 3:
Equation 5: h sub 2 + (.5 * u ^2) sub2 - h sub1 - (.5 * u ^2) sub 1 = q - wsh
By combining the velocity terms with the enthalpy terms to form the total specific enthalpy we can further simplify the equation.
Equation 6: h + .5 * u ^2 = ht = total enthalpy
The total specific enthalpy is analogous to the total pressure in Bernoulli's equation; both expressions involve a "static" value plus one half the square of the velocity.
Use Equation 6 to simplify Equation 5
Equation 7: ht sub 2 - ht sub1 = q - wsh
For a compressor or power turbine, there is no external heat flow into the gas and the "q" term is set equal to zero. In the burner, no work is performed and the "wsh" term is set to zero.
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