Engineering
Fanno Compressible Pipe Flow Friction Parameter (4fL*/D) & Sonic Choking
Calculate the dimensionless Fanno friction choking parameter 4*f*L*/D = (1 - M^2)/(gamma*M^2) + ((gamma+1)/(2*gamma))*LN(((gamma+1)*M^2)/(2 + (gamma-1)*M^2)) and maximum unchoked gas pipe length L_max.
Determine the exact gas-pipe length where wall friction chokes flow to Mach 1!
In adiabatic constant-area compressible gas pipe flow (Fanno Flow), wall friction ALWAYS drives velocity toward Sonic Mach 1: subsonic gas (M < 1) accelerates as pressure and density drop along the pipe, while supersonic gas (M > 1) decelerates toward M = 1! If pipe length exceeds the Sonic Choking Length L*, a subsonic inlet cannot maintain that Mach number and mass flow rate chokes!
Integrating 1D momentum + adiabatic energy with Darcy friction factor f (where 4 * f_Fanning = f_Darcy; gas tables write 4*f_F*L*/D = f_Darcy*L*/D) yields the exact Fanno Parameter: Phi(M) = ((1 - M^2) / (gamma * M^2)) + ((gamma + 1) / (2 * gamma)) * LN(((gamma + 1) * M^2) / (2 + (gamma - 1) * M^2))! Multiplying Phi(M) by D / (4 * f_Fanning) gives maximum pipe run L_max (m) to sonic choking!
| Inlet Mach Number M (Air gamma = 1.40, Pipe D = 0.05 m, Fanning f = 0.02) | Fanno Parameter 4*f*L*/D | Max Choking Length L* (m) | Compressible Friction Behavior |
|---|---|---|---|
| Low Subsonic (M = 0.30) | 5.2993 | 3.31 m | Accelerates toward M = 1 |
| High Subsonic (M = 0.60) | 0.4908 | 0.31 m | Rapidly chokes to sonic exit |
| Supersonic Duct (M = 2.00) | 0.3050 | 0.19 m | Decelerates toward M = 1 |
Dimensionless Fanno Choking Parameter 4*f*L*/D at Mach M
Evaluates exact Fanno function Phi(M) from Mach number B2 (> 0) and specific heat ratio gamma B3 (> 1).
=IF(OR(B2 <= 0, B3 <= 1), "Invalid Mach or Gamma", ROUND(((1 - (B2 ^ 2)) / (B3 * (B2 ^ 2))) + (((B3 + 1) / (2 * B3)) * LN(((B3 + 1) * (B2 ^ 2)) / (2 + ((B3 - 1) * (B2 ^ 2))))), 4))Maximum Unchoked Pipe Length L* = (4*f*L*/D) * D / (4 * f) (Meters)
Converts dimensionless Fanno parameter into physical pipe length in meters from diameter B4 (m) and Fanning friction factor B5.
=IF(OR(B4 <= 0, B5 <= 0), "Invalid Pipe Diameter or Friction", ROUND((((1 - (B2 ^ 2)) / (B3 * (B2 ^ 2))) + (((B3 + 1) / (2 * B3)) * LN(((B3 + 1) * (B2 ^ 2)) / (2 + ((B3 - 1) * (B2 ^ 2)))))) * B4 / (4 * B5), 2))Try the example
Use the original values or change them to test your understanding.