237
6.10 Exercises
drop with formula (4.32). The polytropic exponent follows from fitting a polytropic
pressure-temperature relation through the states at start and end of the process. A
similar procedure can be used with a rotor or a stage when the work is determined
by the Euler formula and measurement of velocity components.
6.10.2. In basic analysis of turbomachines, adiabatic flow is supposed, but a real
flow is seldom perfectly adiabatic. The definitions of isentropic efficiency and loss
coefficients are all based on adiabatic flow. Use of these notions is therefore often
problematic in practise. Study the influence of heat transfer on the example of nozzle flow of an ideal gas with p 00 and T 00 as inlet conditions and p 1 as backpressure
(the h-s diagram for adiabatic flow is Fig. 4.5). Assume distributed heat transfer
proportional to the enthalpy drop according to
(
),
dq
dh
k
= −
with constant ,
k positive for heat addition to the nozzle and negative for heat removal. The equations for
work and energy are (work is zero):
2
1
1
irr
2
dW d v
dp dq ,
=
+
+
r
or
2
1
1
irr
2
d v
dp dq ,
r
= −
−
2
1
2
dW dq dh d v ,
+ = +
or
2
1
2
d v
dh dq ( 1
)( dh ).
= − + = +
−
k
The work equation allows the introduction of the infinitesimal efficiency in the
same way as with adiabatic flow:
2
1
1
2
d v
(
dp ).
∞
=
− r
h
With this relation substituted
in the energy equation, it follows that, for ideal gas and constant values of heat capacity, h ∞ and ,
k the expansion process is polytropic with coefficient:
Derive the formulae for generated kinetic energy, converted pressure energy, −∆h r ,
and enthalpy drop −∆h, analogous to the formulae (4.30) and (4.32) for adiabatic
flow. With constant coefficients, the following relations hold:
Thus, in an experiment, kinetic energy can be measured and h ∞ and k can be determined in average sense. Verify by comparison with adiabatic flow for k = 0.20 and
k = −0.20, together with h ∞ = 0.90, that the generated kinetic energy is not much
affected by heat transfer. Verify for pressure ratios p
p
00
0
/ = 1.20, 1.50 and 2 (A:
the difference with adiabatic flow for pressure ratio 2 is + 1.45 % for k = 0.20 and
−2.13 % for k = −0.20). So, in an experiment, the measured kinetic energy may be
taken as the value for adiabatic flow with only little error. But the measured value
can be corrected to adiabatic flow with the determined values of h ∞ and .
k A similar
correction can be done for rotor work when the work is determined by the Euler
formula and measurement of velocity components.
n
1
.
n 1
1
∞
+
=
−
−
k g
h g
2
2
1
1
1
r
1
2
2
v
( h ),
v
( 1
)( h ).
∞
=
−
= +
−
h
D
k
D
6.10 Exercises
drop with formula (4.32). The polytropic exponent follows from fitting a polytropic
pressure-temperature relation through the states at start and end of the process. A
similar procedure can be used with a rotor or a stage when the work is determined
by the Euler formula and measurement of velocity components.
6.10.2. In basic analysis of turbomachines, adiabatic flow is supposed, but a real
flow is seldom perfectly adiabatic. The definitions of isentropic efficiency and loss
coefficients are all based on adiabatic flow. Use of these notions is therefore often
problematic in practise. Study the influence of heat transfer on the example of nozzle flow of an ideal gas with p 00 and T 00 as inlet conditions and p 1 as backpressure
(the h-s diagram for adiabatic flow is Fig. 4.5). Assume distributed heat transfer
proportional to the enthalpy drop according to
(
),
dq
dh
k
= −
with constant ,
k positive for heat addition to the nozzle and negative for heat removal. The equations for
work and energy are (work is zero):
2
1
1
irr
2
dW d v
dp dq ,
=
+
+
r
or
2
1
1
irr
2
d v
dp dq ,
r
= −
−
2
1
2
dW dq dh d v ,
+ = +
or
2
1
2
d v
dh dq ( 1
)( dh ).
= − + = +
−
k
The work equation allows the introduction of the infinitesimal efficiency in the
same way as with adiabatic flow:
2
1
1
2
d v
(
dp ).
∞
=
− r
h
With this relation substituted
in the energy equation, it follows that, for ideal gas and constant values of heat capacity, h ∞ and ,
k the expansion process is polytropic with coefficient:
Derive the formulae for generated kinetic energy, converted pressure energy, −∆h r ,
and enthalpy drop −∆h, analogous to the formulae (4.30) and (4.32) for adiabatic
flow. With constant coefficients, the following relations hold:
Thus, in an experiment, kinetic energy can be measured and h ∞ and k can be determined in average sense. Verify by comparison with adiabatic flow for k = 0.20 and
k = −0.20, together with h ∞ = 0.90, that the generated kinetic energy is not much
affected by heat transfer. Verify for pressure ratios p
p
00
0
/ = 1.20, 1.50 and 2 (A:
the difference with adiabatic flow for pressure ratio 2 is + 1.45 % for k = 0.20 and
−2.13 % for k = −0.20). So, in an experiment, the measured kinetic energy may be
taken as the value for adiabatic flow with only little error. But the measured value
can be corrected to adiabatic flow with the determined values of h ∞ and .
k A similar
correction can be done for rotor work when the work is determined by the Euler
formula and measurement of velocity components.
n
1
.
n 1
1
∞
+
=
−
−
k g
h g
2
2
1
1
1
r
1
2
2
v
( h ),
v
( 1
)( h ).
∞
=
−
= +
−
h
D
k
D