This page shows the mathematical derivation of the isentropic flow equations beginning with the definition of the gas constant from the equation of state and using information from the entropy slide. A graphical version of this slide is also available. In the text only version presented here, * denotes multiplication, / denotes division, ^ denotes exponentiation, ^2 means quantity squared, sqrt means square root. r is the density, p is the pressure, T is the temperature, e is the entropy, and g is the gas constant, cp is the specific heat coefficient at constant pressure, cv is the specific heat coefficient at constant volume, and gam is the ratio of specific heats. u is the velocity, s is the speed of sound, A is the area, M is the Mach number, q is the dynamic pressure. pt, Tt, and rt are total values of the variable. d anything is a differential change of that variable.
Equation 1. Definitions: gam = cp / cv, cp - cv = g
Equation 2. Divide by cp: 1 - 1 / gam = g / cp
Equation 3. Regroup: cp / g = gam / (gam - 1)
Equation 4. Equation of State: p = r * g * T
Equation 5. Entropy of a gas: de = cp * dT / T - g * dp / p
Equation 6. Isentropic (de = 0): cp * dT / t = g * dp / p
Equation 7. Use equation 4 to simplify: cp * dT = dp / r
Equation 8. Substitute equation 4 into equation 7: (cp / g) * d(p / r) = dp / r
Equation 9. Differentiate: (cp / g) * (dp / r - (p / r ^2) * dr) = dp /r
Equation 10. Regroup: ((cp / g) - 1) * dp / p = (cp / g) * dr / r
Equation 11. Substitute equation 3 into equation 10: (1 / (gam -1)) * dp /p = (gam / (gam -1)) * dr /r
Equation 12. Simplify: dp / p = gam * dr / r
Equation 13. Integrate: p / r ^ gam = constant
We can determine the value of the constant in Equation 13, by defining the total conditions to be the pressure and density when the flow is brought to rest isentropically. We then obtain the other forms of this relation, which are shown on the isentropic flow equation slide.
The second derivation involves determining the relation between the static and total conditions for an isentropic flow in terms of the Mach number. In this derivation, we use the information from the enthalpy equation. For this text only version, the variables are the same as defined above with the addition of the gas enthalpy = h.
Equation 1. Definitions: gam = cp / cv, cp - cv = g
Equation 2. Divide by cp: 1 - 1 / gam = g / cp
Equation 3. Regroup: cp / g = gam / (gam - 1)
Equation 4. Definitions: u = M * s, s ^2 = gam * g * T, h = cp * T
Equation 5. Total enthalpy of a gas: ht = h + u ^2 / 2
Equation 6. Substitute equation 4 into equation 5: cp * Tt = cp * T + M ^2 * s ^2 / 2
Equation 7. Substitute equation 4 into equation 6: cp * Tt = cp * T + M ^2 * gam * g * T / 2
Equation 8. Divide by cp * T : Tt / T = 1 + M ^2 * gam * g / (2 * cp)
Equation 9. Substitute equation 3 into equation 8: Tt / T = 1 + (gam - 1) / 2 * M ^2
Equation 9 can be inverted to give the form on the isentropic flow slide. Using the previously derived results for pressure and density for an isentropic flow and the equation of state, all other forms can be easily derived.
Compressible Aerodynamics:
Interactive Isentropic Flow:
Interactive Nozzle Flow:
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byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov