KSME International Journal

, Volume 17, Issue 12, pp 2087–2098

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

Article

## 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, m2

c

Specific heat, J/(kg K)

E

Energy, J

f

Cross sections of an injector, m2

G

Moment, N-m

H

Enthalpy, J/kg

h

Heat transfer coefficient, W/(m2K)

J

Polar moment of rotor inertia, kg-m2

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

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

Us

Control action signal

U

Overall heat transfer coefficient, W/(m2 K)

u

Internal energy per unit mass, J/kg

Uinj

Injection coefficient

V

Volume, m3

## Greek

γ

Specific heat ratio

Δp

Pressure loss, Pa

ΔS

Compressor stability margin ΔSsH=SsH-1.0

Δτ

Time step, s

δ

Channel wall thickness, m

η

Efficiency

α

Relative displacement of actuator servo

v

Kinematic viscosity, m2/s

ξ

Correlation coefficient

ρ

Density, kg/m3

σ

Coefficient for temperature distribution

τ

Time variable, s

ω

## 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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