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Thermodynamics is a branch of physics that deals with the energy and work of a system. As aerodynamicists, we are most interested in thermodynamics in the study of propulsion systems and understanding high speed flows. The First Law of Thermodynamics indicates that the total energy of a system is conserved. This includes the potential and kinetic energy, the work done by the system, and the transfer of heat through the system. The Second Law of Thermodynamics indicates that, while many physical processes that satisfy the first law are possible, the only processes that occur in nature are those for which the entropy of the system either remains constant or increases. Entropy, like temperature and pressure, can be explained on both a macro scale and a micro scale. Since thermodynamics deals only with the macro scale, entropy is defined here to be the heat transfer into the system divided by the temperature.
In the text only version of the equations, * denotes multiplication, / denotes division, and ln is the natural logarithm. S is the entropy, p is the pressure, T is the temperature and V is the volume of the gas. H is the enthalpy, E is the internal energy, Q is the heat transfer. R is the gas constant, Cp is the heat capacity at constant pressure, Cv the heat capacity at constant volume. d is a differential change of a variable, delta is a large change.
Equation 1: 2nd law of thermodynamics: S2 - S1 = delta Q / T
During a thermodynamic process, the temperature of an object changes as heat is applied or extracted. A more correct definition of the entropy is the differential form that accounts for this variation.
Equation 2: : dS = dQ / T
The change in entropy is then the inverse of the temperature integrated over the change in heat transfer. For gases, there are two possible ways to evaluate the change in entropy. The equations can be formulated in terms of the internal energy and the definition of work for a gas. Or the equations can be formulated in terms of the enthalpy of the gas. Let's perform the first derivation in terms of the internal energy. The first law of thermodynamics is then written as:
Equation 3: dQ = dE + p * dV
The equation of state gives:
Equation 4: p * V = R * T
Substituting for p in Equation 3, we obtain:
Equation 5: dQ = dE + R * T * dV / V
The heat transfer of a gas is equal to the specific heat constant times the change in temperature; in differential form, dQ = constant * dT. If we have a constant volume process, then, from Equation 5:
Equation 6: dE = dQ = Cv * dT
Equation 7: dQ = Cv * dT + R * T * dV / V
Substituting the value of dQ into the differential form of the second law, Equation 2, we arrive at differential equations that we can easily integrate.
Equation 8: dS = Cv * dT / T + R * dV / V
Equation 9: S2 - S1 = Cv * ln ( T2 / T1) + R * ln ( V2 / V1)
The derivation in terms of the enthalpy proceeds as follows: The form of the first law is now:
Equation 10: dQ = dH - V * dp
Substituting for V from Equation 4, we obtain:
Equation 11: dQ = dH - R * T * dp / p
Again, the heat transfer of a gas is equal to the specific heat constant times the change in temperature; in differential form, dQ = constant * dT. If the pressure is constant the second term of Equation 11 is zero and
Equation 12: dH = dQ = Cp * dT
Equation 13: dQ = Cp * dT - R * T * dp / p
Substituting the value of dQ into the differential form of the second law, we arrive at differential equations that we can easily integrate.
Equation 14: dS = Cp * dT / T - R * dp / p
Equation 15: S2 - S1 = Cp * ln ( T2 / T1) - R * ln ( p2 / p1)
Specific forms of Equation 15 and Equation 9 can be developed by dividing by the mass to obtain the specific heat capacities and the specific volumes.
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