# Investigation of nonlinear numerical mathematical model of a multiple shaft gas turbine unit

## Abstract

The development of numerical mathematical model to calculate both the static and dynamic characteristics of a multi-shaft gas turbine consisting of a single combustion chamber, including advanced cycle components such as intercooler and regenerator is presented in this paper. The numerical mathematical model is based on the simplified assumptions that quasi-static characteristic of turbo-machine and injector is used, total pressure loss and heat transfer relation for static calculation neglecting fuel transport time delay can be employed. The supercharger power has a cubical relation to its rotating velocity. The accuracy of each calculation is confirmed by monitoring mass and energy balances with comparative calculations for different time steps of integration. The features of the studied gas turbine scheme are the starting device with compressed air volumes and injector’s supercharging the air directly ahead of the combustion chamber.

## Key Words

Nonlinear Numerical Mathematical Modeling Gas Turbine Performance Multi-shaft Injector Surge Margin## Nomenclature

*A*Heat transfer area, m

_{2}*c*Specific heat, J/(kg K)

*E*Energy, J

*f*Cross sections of an injector, m

^{2}*G*Moment, N-m

*H*Enthalpy, J/kg

*h*Heat transfer coefficient, W/(m

^{2}K)*J*Polar moment of rotor inertia, kg-m

^{2}*k*Thermal conductivity, W/(m K)

*K*Feedback coefficient

*m*Mass flow rate, kg/s

*m*Mass of wall, kg

*n**Exponential order for heat transfer coefficients

*P*Power, W

*p*Pressure, Pa

*Pγ*Compressor and turbine pressure ratio

*Q*Heat flow rate. W

*R*Specific gas constant, J/(kg K)

*S*Relative stability margin coefficient

*T*Total temperature, K.

*t*Temperature, °C

*U*_{s}Control action signal

*U*Overall heat transfer coefficient, W/(m

^{2}K)*u*Internal energy per unit mass, J/kg

*U*_{inj}Injection coefficient

*V*Volume, m

^{3}

## Greek

*γ*Specific heat ratio

*Δp*Pressure loss, Pa

*ΔS*Compressor stability margin ΔSs

^{H}=Ss^{H}-1.0*Δτ*Time step, s

*δ*Channel wall thickness, m

*η*Efficiency

*α*Relative displacement of actuator servo

*v*Kinematic viscosity, m

^{2}/s*ξ*Correlation coefficient

*ρ*Density, kg/m

^{3}*σ*Coefficient for temperature distribution

*τ*Time variable, s

*ω*Rotor angular speed, rad/s

## Subscript

- a
Air

- acc
Accumulated

- av
Average

- bra
Rotor breakaway moment

- cool
Cooling

- df
Direct feedback coefficient

- e
Exit

- f
Fuel

- g
Gas

- ga
Gain feedback coefficient

- i
Index for parameters

- in
Inlet

- inj
Injection coefficient

- int
Intermediate

- mech
Mechanical loss

- n, n+ 1
Old and new time steps

- out
Outlet

- p
Constant pressure

- s
Stability margin

- sm
Servo-motor

- sur
Anti surge valve

- v
Constant volume

- w
Wall

- wor
Working

- set
Pilot signal for regulator

*0*Nominal condition

*1*Outer surface

- 2
Inner surface

- 3
Injector mixing chamber

## Superscript

- *
Injector nozzle throat area

- ^
Reduced parameter

- -
relative parameter

## Abbreviation

- C
Compressor

- CC
Combustion chamber

- DE
Differential equation

- GTU
Gas turbine unit

- HP
High pressure

- HPC
High pressure compressor

- HPT
High pressure turbine

- HPTC
High pressure turbo compressor

- IC
Intercooler

- LHV
Low heating value

- LP
Low pressure

- LPC
Low pressure compressor

- LPT
Low pressure turbine

- LPTC
Low pressure turbo-compressor

- MM
Mathematical model

- PC
Pipeline compressor

- PT
Power turbine

- T
Turbine

- TC
Turbo-compressor

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## References

- Cohen, H., Rogers, G. F. C. and Saravanamuttoo, H. I. H, 1996, “Gas Turbine Theory,” 4th edition. Longman Group Limited... Ch. 8-9.Google Scholar
- Dajneko, V. I., 1984, “Experimental study of a turbine jet starting with the help of aerial boosting and electric starter,” Energetika. 9. pp. 110-111.Google Scholar
- Gittelman, A. I.. 1974, “Dynamic and control of the ship gas turbine units, ” L.: Mashinostroenie. Kim SooYong, Park MooRyong and ChoGoogle Scholar
- SooYong, 1998, “Performance Analysis of a 50kW Turbogenerator Gas Turbine Engine, ” 98-GT-209, ASME TurboExpo 1998.Google Scholar
- Kim SooYong and Soudarev, B., 2000, “Transient Analysis of a Simple Cycle Gas Turbine Engine,” KSAS International Journal. Vol. 1, No. 2, pp. 22–29.Google Scholar
- Kotljar, I. V., 1973, “Transient operations in the gas-turbine units, ” edition of L.: Mashinostroenie.Google Scholar
- Kovalevski, V. P., 1992, “Searching of iterative approaching in calorific, hydraulic, aerodynamic and other complicated computational engineering calculations of heat power machinery, ” The work of VNITIEM.-L.: Nedra. 1992. pp. 186-191.Google Scholar
- Kovalewsky, M. M., 1979, “A Stationary GTU of an open cycle, ” M.: Mashinostroenie.Google Scholar
- Slobodianiuk, L. I. and Dajneko, V. I., 1983, “A calculation of start-up of the gas-turbine unit by boosting”, Energetika. 4. pp. 53–57.Google Scholar
- Sokolov Ja and Singer, N. M., 1970, “Jet vehicles, ” M.: Energija.Google Scholar
- Zhuravlev, V. I. and Kovalevsky, V. P., 1993, “Nonlinear numerical model of a gas-turbine plant for all operation conditions, ” yazheloe Mashinostroenie, 1993, No. 11/12, pp. 2-5.Google Scholar