384
11 Power Gas Turbines
11.2 Thermodynamic Modelling
At the inlet and the outlet of a compressor or expander, flow is mostly homogeneous
and steady with a good approximation. This is not the case at the inlet or the outlet
of a stage. For a stage we assume averages over the inlet area and outlet area as representative states. As unsteadiness originates from rotor revolution, such averages
are steady at constant rotor speed. We wish to assess stage efficiency only using the
average inlet and outlet states.
11.2.1 Isentropic Efficiency with Adiabatic Compression
or Expansion
For assessing the efficiency of a compressor or a turbine stage, we consider steady
flow between the average states at inlet and outlet. To a streamline in the absolute
frame applies:
Energy equation:
(11.1)
Work equation:
(11.2)
where dW is the elementary work done on the fluid and dq the elementary heat
supplied to it. The gravitational potential energy is denoted by U. Further, dq irr is
the heat by irreversibility (dissipation). Firstly, we consider adiabatic processes, i.e.
dq = 0.
The energy equation is a total differential and thus can be integrated. The work
equation forms a total differential if ρ is only a function of p. A special case meeting this condition is ρ = constant. The reversible part of the work done on the fluid
is then
2
1 2 v
p
U
D
D r D
+
+
and the quantity
2
1
m
2
E
v
p
U
r
=
+
+ is called the
mechanical energy, the sum of kinetic energy, pressure potential energy and gravitational potential energy (see Chap. 1). Historically, the term head is used for the
mechanical energy divided by the gravity acceleration g, so expressed in metres.
But with machines receiving or delivering work, it is more convenient to use the
term head for the difference of mechanical energy across a component or across the
machine, so expressed in J/kg. We have used the term head in this sense in previous
chapters and we will continue this terminology. From the difference between the
work equation and the energy equation it follows dq irr = dh − 1/ρ dp. For constant
density, the dissipated part of the work is then Δe, with e the internal energy. Since
internal energy cannot be a source of work in a constant density fluid, the efficiency
,
2
1 2
dW dq d v dh dU
+ =
+ +
,
2
1
irr
2
1
dW d v
dp dU dq
r
=
+
+
+
11 Power Gas Turbines
11.2 Thermodynamic Modelling
At the inlet and the outlet of a compressor or expander, flow is mostly homogeneous
and steady with a good approximation. This is not the case at the inlet or the outlet
of a stage. For a stage we assume averages over the inlet area and outlet area as representative states. As unsteadiness originates from rotor revolution, such averages
are steady at constant rotor speed. We wish to assess stage efficiency only using the
average inlet and outlet states.
11.2.1 Isentropic Efficiency with Adiabatic Compression
or Expansion
For assessing the efficiency of a compressor or a turbine stage, we consider steady
flow between the average states at inlet and outlet. To a streamline in the absolute
frame applies:
Energy equation:
(11.1)
Work equation:
(11.2)
where dW is the elementary work done on the fluid and dq the elementary heat
supplied to it. The gravitational potential energy is denoted by U. Further, dq irr is
the heat by irreversibility (dissipation). Firstly, we consider adiabatic processes, i.e.
dq = 0.
The energy equation is a total differential and thus can be integrated. The work
equation forms a total differential if ρ is only a function of p. A special case meeting this condition is ρ = constant. The reversible part of the work done on the fluid
is then
2
1 2 v
p
U
D
D r D
+
+
and the quantity
2
1
m
2
E
v
p
U
r
=
+
+ is called the
mechanical energy, the sum of kinetic energy, pressure potential energy and gravitational potential energy (see Chap. 1). Historically, the term head is used for the
mechanical energy divided by the gravity acceleration g, so expressed in metres.
But with machines receiving or delivering work, it is more convenient to use the
term head for the difference of mechanical energy across a component or across the
machine, so expressed in J/kg. We have used the term head in this sense in previous
chapters and we will continue this terminology. From the difference between the
work equation and the energy equation it follows dq irr = dh − 1/ρ dp. For constant
density, the dissipated part of the work is then Δe, with e the internal energy. Since
internal energy cannot be a source of work in a constant density fluid, the efficiency
,
2
1 2
dW dq d v dh dU
+ =
+ +
,
2
1
irr
2
1
dW d v
dp dU dq
r
=
+
+
+