397
11.2 Thermodynamic Modelling
Infinitesimal efficiency is defined by
(11.16)
The heat removed from the main flow by cooling may be noted for an entire stage as
(11.17)
where A b is the blade surface, α the heat transfer coefficient (W/m
2
K) and (
)
T T
g
b
−
the average temperature difference between the gas and the blades. The heat
removed may be compared with the stage work according to
where
m g is the gas mass flow rate (kg/s), ψ the work coefficient of the stage and
u the blade speed. The gas flow rate may be noted by g
g
m
v A
r
=
, with ρ and v
representative values of density and through-flow velocity and A g the through-flow
area. The ratio of the heat removed to the work done is
The surface ratio is typically around 8. The Stanton number
pg
St
vc
a r
=
is around
0.005 for convective heat transfer in a flow along a flat plate, with c pg the heat capacity at T g [4]. We define
pg
b
2
g
c
A
1
St
A
u
k
y
=
. With ψ ≈ 2 and blade speed 300 m/s
at the hub, it follows that κ ≈ 0.3 × 10
−3
K
−1
.
By distributing the work and the cooling over the expansion, we may assume the
differential equation:
(11.18)
with − dq and − dW heat removed and work done per mass unit for an infinitesimal
part of the expansion and T g and T b local gas and blade temperatures. To word the
meaning of the heat coefficient, κ may be expressed in J/kJK, so about 0.3 J/kJK.
With state-of-the-art machines the tolerable blade temperature T b ≈ 900 °C. The
value of the heat transfer coefficient κ is lower than estimated up to now. The resistance to heat transfer increases strongly by applying a protective ceramic coating
onto the blades (TBC: Thermal Barrier Coating) and by using film cooling. Due to
these measures, the value of κ may be halved to about 0.15 J/kJK. It is immediately
obvious that the thermal barrier coating constitutes an additional thermal resistance.
Some reasoning is required to understand that a cooling film may be considered as
1
dW
dp .
h
r
∞
−
=
−
,
b
g
b
Q A (T T )
D
a
−
=
−
,
2
g
W m u
D
y
−
=
(
)
pg g
b
b
2
g
pg
c T T
A
Q
1
.
W
A
v c
u
D
a
D
r
y
−
−
=
−
,
g
b
dq
(T T )
dW
k
−
=
−
−
11.2 Thermodynamic Modelling
Infinitesimal efficiency is defined by
(11.16)
The heat removed from the main flow by cooling may be noted for an entire stage as
(11.17)
where A b is the blade surface, α the heat transfer coefficient (W/m
2
K) and (
)
T T
g
b
−
the average temperature difference between the gas and the blades. The heat
removed may be compared with the stage work according to
where
m g is the gas mass flow rate (kg/s), ψ the work coefficient of the stage and
u the blade speed. The gas flow rate may be noted by g
g
m
v A
r
=
, with ρ and v
representative values of density and through-flow velocity and A g the through-flow
area. The ratio of the heat removed to the work done is
The surface ratio is typically around 8. The Stanton number
pg
St
vc
a r
=
is around
0.005 for convective heat transfer in a flow along a flat plate, with c pg the heat capacity at T g [4]. We define
pg
b
2
g
c
A
1
St
A
u
k
y
=
. With ψ ≈ 2 and blade speed 300 m/s
at the hub, it follows that κ ≈ 0.3 × 10
−3
K
−1
.
By distributing the work and the cooling over the expansion, we may assume the
differential equation:
(11.18)
with − dq and − dW heat removed and work done per mass unit for an infinitesimal
part of the expansion and T g and T b local gas and blade temperatures. To word the
meaning of the heat coefficient, κ may be expressed in J/kJK, so about 0.3 J/kJK.
With state-of-the-art machines the tolerable blade temperature T b ≈ 900 °C. The
value of the heat transfer coefficient κ is lower than estimated up to now. The resistance to heat transfer increases strongly by applying a protective ceramic coating
onto the blades (TBC: Thermal Barrier Coating) and by using film cooling. Due to
these measures, the value of κ may be halved to about 0.15 J/kJK. It is immediately
obvious that the thermal barrier coating constitutes an additional thermal resistance.
Some reasoning is required to understand that a cooling film may be considered as
1
dW
dp .
h
r
∞
−
=
−
,
b
g
b
Q A (T T )
D
a
−
=
−
,
2
g
W m u
D
y
−
=
(
)
pg g
b
b
2
g
pg
c T T
A
Q
1
.
W
A
v c
u
D
a
D
r
y
−
−
=
−
,
g
b
dq
(T T )
dW
k
−
=
−
−