Intersection of Fanno and a Rayleigh Line

Intersection of Fanno and a Rayleigh Line 

          Fanno and Rayleigh line, when plotted on h-s plane, for same mass velocity G, intersect at 1 and 2.as shown in fig. All states of Fanno line have same stagnation temperature or stagnation enthalpy, and all states of Rayleigh line have same stream thrust F / A. Therefore, 1 and 2 have identical values of G, h0 and F / A. from 1 to 2 possible by a compression shock wave without violating Second Law Thermodynamics. A shock is a sudden compression which increases the pressure and entropy of the fluid but the velocity is decrease from supersonic to subsonic.

A change from states 2 to 1 from subsonic to supersonic flow is not possible in view Second Law Thermodynamics. (Entropy can not decrease in a flow process)

3.The pressure temperature and Mach no. of air in combustion chamber are 4 bar,100°C and 0.2 respectively. The stagnation temperature of air in combustion chamber is increased 3 times the initial value. Calculate:

1.   The mach no., pressure and temperature at exit.

2.   Stagnation pressure

3.   Heat supplied per Kg of air

Solution

Refer isentropic flow table for γ=1.4and m1=.2

Tutorial Problems of Flow Through Ducts:

1.A circular duct passes 8.25Kg/s of air at an exit Mach number of 0.5. The entry pressure temperature are 3.45 bar and 38°C respectively and the coefficient of friction 0.005.If the Mach number at entry is 0.15, determine : I. The diameter of the duct,

II. Length of the duct, III. Pressure and temperature at the exit, IV. Stagnation pressure loss, and V. Verify the exit Mach number through exit velocity and temperature.

2) A gas (γ =1.3,R=0.287 KJ/KgK) at p1 =1bar, T1 =400 k enters a 30cm diameter duct at a Mach number of 2.0.A normal shock occurs at a Mach number of 1.5 and the exit Mach number is1.0, If the mean value of the friction factor is 0.003 determine:

1)Lengths of the duct upstream and downstream of the shock wave, 2)Mass flow rate of the gas and downstream of the shock.

3) Air enters a long circular duct ( d =12.5cm,f=0.0045) at a Mach number 0.5, pressure 3.0 bar and temperature 312 K.If the flow is isothermal throughout the duct determine (a) the length of the duct required to change the Mach number to 0.7,(b) pressure and temperature of air at M =0.7 (c) the lengthof the duct required to attain limiting Mach number, and (d) state of air at the limiting Mach umber.compare these values with those obtained in adiabatic flow.

4. Show that the upper and lower branches of a Fanno curve represent subsonic and supersonic flows respectively . prove that at the maximum entropy point Mach number is unity and all processes approach this point .How would the state of a gas in a flow change from the supersonic to subsonic branch ? Flow in constant area ducts with heat transfer(Rayleigh flow)

5) The Mach number at the exit of a combustion chamber is 0.9. The ratio of stagnation temperature at exit and entry is 3.74. If the pressure and temperature of the gas at exit are 2.5 bar and 1000°C respectively determine (a) Mach number, pressure and temperature of the gas at entry, (b) the heat supplied per kg of the gas and (c) the maximum heat that can be supplied. Take γ= 1.3, Cp= 1.218 KJ/KgK

6) The conditions of a gas in a combuster at entry are: P1=0.343 bar ,T1 = 310K ,C1=60m/s.Detemine the Mach number ,pressure ,temperature and velocity at the exit if the increase in stagnation enthalpy of the gas between entry and exit is 1172.5KJ/Kg. Take Cp=1.005KJ/KgK, γ =1.4

7) A combustion chamber in a gas turbine plant receives air at 350 K ,0.55bar and 75 m/s

.The air –fuel ratio is 29 and the calorific value of the fuel is 41.87 MJ/Kg .Taking γ=1.4 and R =0.287 KJ/kg K for the gas determine.

a)           The initial and final Mach numbers

b)Final pressure, temperature and velocity of the gas

c)Percent stagnation pressure loss in the combustion chamber, and

d)The maximum stagnation temperature attainable.