T-S Diagrams for Parametric Cycle Analysis of an Ideal Ramjet Engine

The text is available for reading at http://3dimensionaldesigningandmanufacturing.blogspot.com/2014/09/parametric-cycle-analysis-of-ideal.html, somehow, the diagrams are showing there.

1,600 K, Mach 1 

1,900 K Mach 1 

2,200 K Mach 1 

 1,900 K, Mach 2

 2,200 K, Mach 2

1,600 K, Mach 2

Parametric Cycle Analysis of an Ideal Ramjet Engine

Extract from my rejected thesis from my university days. Optimization of Thrust Per Unit Mass Flow of a Jet Engines by Optimizing the Overall Compression Ratio, to design Multi-diameter Inlets, because an inlet works best for a set specific of conditions. Hope it helps you in your scholarly work.

Ramjet engine

A ramjet engine is the simplest of all aircraft engines. It consists of an inlet or a diffuser, a combustor and a nozzle. Air is fed into the diffuser, it increases the static pressure and temperature of air by slowing it down. Then the air is fed into the combustor, where it is mixed with fuel and is ignited, resulting in increased energy. This corresponds to an increased temperature. The combustion occurs at a constant pressure for sub sonic flows. The nozzle then expand the gas to ambient pressure, with a decrease in temperature. This process results in increased in KE for gas.

Formulations

The formulations for the cycle analysis are as follows.

General Gas Constant, R=((g-1)/ g)*C_p

Velocity of Sound, a_o=(1000*g*R*T_o).^(1/2)
Total to Static Temperature Ratio, t_r=1+(((g-1)/2)*(M_o.^2))
Burner Exit Enthalpy to Ambient Enthalpy Ratio, t_l=T_t4/T_o
Velocity Ratio, v₉/a_o=M_o*((t_l/t_r).^(1/2))
Thrust per Air Flow, F/m_o=a_o*((v₉/a_o)-M_o)
Fuel Air Ratio, f=((C_p*T_o)/h_PR)*( t_l-t_r)
Thrust Specific Fuel Consumption, S=f/(F/m_o)
Thermal Efficiency, h_T=1-(1/t_r)
Propulsive Efficiency, h_P=2/(((t_l/t_r).^(1/2))+1)
Overall Efficiency h_O=h_T*h_P
Total to Static Temperature Ratio for Maximum Thrust per Air Flow, t_{r (maxFmo)}=t _l.^(1/3)
Mach Number of Flight for Maximum Thrust per Air Flow , M_{o (maxFmo)}=((2/(g-1))*( t_{r (maxFmo)}-1)).^(1/2)
Temperature at Station T_to (after Compression), T_to=t_r*T_o
Nozzle Exhaust Temperature, T₉=T_o*(t_l/t_r)

Calculations

By using these equations, and by entering the following inputs:

For Mach Number of Flight= 1

T_o= 216.7 K, g= 1.4, C_p= 1.004 KJ/(Kg.K), T_t4= 1,600 K, M_o= 1, h_PR= 42,800 KJ/Kg

We get the Following Results:

Station Temperatures for T-S Diagram

Ambient Temperature, Station T₀, 216.7 K
Temperature after Compression, Station T_to and T_t2, 260.04 K
Temperature at Nozzle Entry, Station T_t4, 1,600 K
Temperature at Nozzle Exit, Station T₉, 1,333.33 K

T-S Diagram generated from MATLAB

Results Based on the Data Entered

General Gas Constant= 0.286857 KJ/(Kg.K)

Velocity of Sound= 295.003 m/s

Total to Static Temperature Ratio= 1.2

Burner Exit Temperature to Ambient Temperature Ratio= 7.38348

Gas Exit Velocity to Velocity of Sound Ratio= 2.4805

Thrust per Unit Airflow= 436.753 N/(Kg/s)

Fuel to Air Ratio= 0.0314327

Thrust Specific Fuel Consumption= 71.9691 (mg/s)/N

Thermal Efficiency= 16.6667 %

Propulsive Efficiency= 57.4629 %

Overall Efficiency= 9.57716 %

For Optimum Mach Number of Flight

Total to Static Temperature Ratio for Maximum Thrust per Air Flow= 1.94724
Optimum Mach Number of Flight for Maximum Thrust per Air Flow= 2.17629

For all the same values as above, except T_t4= 1,900 K, we get these results:

Station Temperatures for T-S Diagram

Ambient Temperature, Station T_o, 216.7 K
Temperature after Compression, Station T_t0 and T_t2, 260.04 K
Temperature at Nozzle Entry, Station T_t4, 1,900 K
Temperature at Nozzle Exit, Station T₉, 1,583.33 K

T-S Diagram from MATLAB

Results Based on the Data Entered
General Gas Constant= 0.286857 KJ/(Kg.K)
Velocity of Sound= 295.003 m/s
Total to Static Temperature Ratio= 1.2
Burner Exit Temperature to Ambient Temperature Ratio= 8.76788
Gas Exit Velocity to Velocity of Sound Ratio= 2.70307
Thrust per Unit Airflow= 502.41 N/(Kg/s)
Fuel to Air Ratio= 0.0384701
Thrust Specific Fuel Consumption= 76.5712 (mg/s)/N
Thermal Efficiency= 16.6667 %
Propulsive Efficiency= 54.0093 %
Overall Efficiency= 9.00155 %

For Optimum Mach Number of Flight

Total to Static Temperature Ratio for Maximum Thrust per Air Flow= 2.06205
Optimum Mach Number of Flight for Maximum Thrust per Air Flow= 2.30439

For all the same values as above, except T_t4= 2,200 K, we get these results:

Station Temperatures for T-S Diagram

Ambient Temperature, Station T_o, 216.7 K
Temperature after Compression, Station T_t0 and T_t2, 260.04 K
Temperature at Nozzle Entry, Station T_t4, 2,200 K
Temperature at Nozzle Exit, Station T₉, 1,833.33 K

T-S Diagram from MATLAB

Results Based on the Data Entered


General Gas Constant= 0.286857 KJ/(Kg.K)
Velocity of Sound= 295.003 m/s
Total to Static Temperature Ratio= 1.2
Burner Exit Temperature to Ambient Temperature Ratio= 10.1523
Gas Exit Velocity to Velocity of Sound Ratio= 2.90865
Thrust per Unit Airflow= 563.057 N/(Kg/s)
Fuel to Air Ratio= 0.0455075
Thrust Specific Fuel Consumption= 80.8222 (mg/s)/N
Thermal Efficiency= 16.6667 %
Propulsive Efficiency= 51.1686 %
Overall Efficiency= 8.5281 %

For Optimum Mach Number of Flight

Total to Static Temperature Ratio for Maximum Thrust per Air Flow= 2.16532
Optimum Mach Number of Flight for Maximum Thrust per Air Flow= 2.41383

For Mach Number of Flight= 2

By using these equations, and by entering the following inputs:

T_o= 216.7 K, g= 1.4, C_p= 1.004 KJ/(Kg.K), T_t4= 1,600 K, M_o= 2, h_PR= 42,800 KJ/Kg

Station Temperatures for T-S Diagram

Ambient Temperature, Station T_o, 216.7 K
Temperature after Compression, Station T_t0 and T_t2, 390.06 K
Temperature at Nozzle Entry, Station T_t4, 1600 K
Temperature at Nozzle Exit, Station T₉, 888.889 K

T-S Diagram from MATLAB

Results Based on the Data Entered

General Gas Constant= 0.286857 KJ/(Kg.K)
Velocity of Sound= 295.003 m/s
Total to Static Temperature Ratio= 1.8
Burner Exit Temperature to Ambient Temperature Ratio= 7.38348
Gas Exit Velocity to Velocity of Sound Ratio= 4.05065
Thrust per Unit Airflow= 604.947 N/(Kg/s)
Fuel to Air Ratio= 0.0283827
Thrust Specific Fuel Consumption= 46.9177 (mg/s)/N
Thermal Efficiency= 44.4444 %
Propulsive Efficiency= 66.1086 %
Overall Efficiency= 29.3816 %

For Optimum Mach Number of Flight

Total to Static Temperature Ratio for Maximum Thrust per Air Flow= 1.94724
Optimum Mach Number of Flight for Maximum Thrust per Air Flow= 2.17629


For all the same values as above, except T_t4= 1,900 K, we get these results:

Station Temperatures for T-S Diagram

Ambient Temperature, Station T_o, 216.7 K
Temperature after Compression, Station T_t0 and T_t2, 390.06 K
Temperature at Nozzle Entry, Station T_t4, 1900 K
Temperature at Nozzle Exit, Station T₉, 1055.56 K

T-S Diagram from MATLAB

Results Based on the Data Entered

General Gas Constant= 0.286857 KJ/(Kg.K)
Velocity of Sound= 295.003 m/s
Total to Static Temperature Ratio= 1.8
Burner Exit Temperature to Ambient Temperature Ratio= 8.76788
Gas Exit Velocity to Velocity of Sound Ratio= 4.41409
Thrust per Unit Airflow= 712.163 N/(Kg/s)
Fuel to Air Ratio= 0.0354201
Thrust Specific Fuel Consumption= 49.7359 (mg/s)/N
Thermal Efficiency= 44.4444 %
Propulsive Efficiency= 62.3627 %
Overall Efficiency= 27.7168 %

For Optimum Mach Number of Flight

Total to Static Temperature Ratio for Maximum Thrust per Air Flow= 2.06205
Optimum Mach Number of Flight for Maximum Thrust per Air Flow= 2.30439

For all the same values as above, except T_t4= 2,200 K, we get these results:

Station Temperatures for T-S Diagram

Ambient Temperature, Station T_o, 216.7 K
Temperature after Compression, Station T_t0 and T_t2, 390.06 K
Temperature at Nozzle Entry, Station T_t4, 2200 K
Temperature at Nozzle Exit, Station T₉, 1222.22 K

T-S Diagram from MATLAB

Results Based on the Data Entered

General Gas Constant= 0.286857 KJ/(Kg.K)
Velocity of Sound= 295.003 m/s
Total to Static Temperature Ratio= 1.8
Burner Exit Temperature to Ambient Temperature Ratio= 10.1523
Gas Exit Velocity to Velocity of Sound Ratio= 4.7498
Thrust per Unit Airflow= 811.2 N/(Kg/s)
Fuel to Air Ratio= 0.0424575
Thrust Specific Fuel Consumption= 52.3391 (mg/s)/N
Thermal Efficiency= 44.4444 %
Propulsive Efficiency= 59.261 %
Overall Efficiency= 26.3382 %

For Optimum Mach Number of Flight

Total to Static Temperature Ratio for Maximum Thrust per Air Flow= 2.16532
Optimum Mach Number of Flight for Maximum Thrust per Air Flow= 2.41383

Conclusions

By looking at the results, we find the following results.

1.       The performance of any ramjet engine relies heavily on the stagnation temperature increase across the burner.
2.       To have efficient compression of the air, the ramjet requires high flight speeds.
3.       For ramjets, the static thrust is zero, they must be moving to develop thrust.

T-s diagrams available here, some how not showing here.
http://3dimensionaldesigningandmanufacturing.blogspot.com/2014/09/t-s-diagrams-for-parametric-cycle.html