Fanno Flow

Fanno Curves
Fanno Flow Equations
Solution of Fanno Flow Equations
Integration of Fanno Flow Equations

1. FANNO CURVE & FANNO EQUATION Dhaval Chauhan Mechanical Engineer

2. Fanno Curve Consider a flow of a fluid in a perfectly insulated (Q=0) constant area duct. Continuity equation: ṁ …(1) Where G = Mass flow density. Energy equation: …(2) Where = Stagnation enthalpy  

3. But from Equation (1), C= …(3) If the properties of and G are known on upstream side, the values of h and ρ can be obtained at any section of the duct. Fig. 1 shows the relation defined by Equation (1) for a single value of and various values of G on the plot of h Vs V or . These curves are called Fanno lines. According to second law of thermodynamics, the entropy during the adiabatic process with friction always increases.  

4. Therefore, the path of states for portion of curves must be from left to right. It follows that during subsonic flow, the effect of friction will be to increase the velocity and Mach number with reduction in enthalpy and pressure. While, during supersonic flow, the effect of friction will be to decrease the velocity and Mach number with increase in enthalpy and pressure.

5. Fig. (1) Fanno lines on ( h -s) diagram

6. Fanno flow equation: Continuity equation: Perfect gas equation: By definition of Mach number: Energy equation:  

7. Momentum equation Stagnation pressure – Mach number relationship Impulse function  

8. Solution of Fanno Lines Equations and Effect of Wall Friction on Fluid Properties We have discussed above simultaneous Equations which relate to following eight different variables: , , , , , , and Out of the above, variable is independent variable which is responsible for changes in flow properties during the flow in a duct. Now we can solve the simultaneous equation discussed in terms of removing seven variables as follows: = + …(1)  

9. and, + = 0 …(2) On substituting the value of from Eq.(2) in Eq.(1), = - …(3) But, from Continuity equation, hence, …(4) On substituting the value of Eq.(4) in Momentum equation we get, …(5)  

10. On substituting the value of Eq.(5) in (4) we get, …(6) Since, from Continuity equation, therefore substituting the value of from Eq.(6) in Continuity equation we get, …(7) From Perfect gas equation we have, …(8) On substituting the value of Eq.(5) & (7) we get, …(9)  

11. From Equation by definition of Mach number, On substituting the values from Eq.(6) & (9) we get, …(10) From Momentum equation, …(11) On substituting the values from Eq.(5) & (10) we get, …(12)  

12. From equation of Impulse Function, …(13) On substituting the values from Eq.(5) & (10) we get, …(14)  

13. Integration of F anno equations Variation of Mach number with duct length Temperature ratio Density ratio  

14. Velocity ratio Pressure ratio Stagnation pressure ratio  

15. Impulse function ratio Change in entropy