The inherent unsteadiness of the flow within turbomachines
It is a less well-known fact often ignored by designers of turbomachinery that turbomachines can only work the way they do because of flow unsteadiness. This subject was discussed by Dean (1959), Horlock and Daneshyar (1970), and Greitzer (1986). Here, only a brief introduction to an extensive subject is given.
In the absence of viscosity, the equation for the stagnation enthalpy change of a fluid particle moving through a turbomachine is
where D/Dt is the rate of change following the fluid particle. Eq. (1.57) shows us that any change in stagnation enthalpy of the fluid is a result of unsteady variations in static pressure. In fact, with- out unsteadiness, no change in stagnation enthalpy is possible and thus no work can be done by the fluid. This is the so-called “Unsteadiness Paradox.” Steady approaches can be used to determine the work transfer in a turbomachine, yet the underlying mechanism is fundamentally unsteady.
A physical situation considered by Greitzer (1986) is the axial compressor rotor as depicted in Figure 1.16a. The pressure field associated with the blades is such that the pressure increases from the suction surface (S) to the pressure surface (P). This pressure field moves with the blades and is therefore steady in the relative frame of reference. However, for an observer situated at the point* (in the absolute frame of reference), a pressure that varies with time would be recorded, as shown
PROBLEMS
1. a. Air flows adiabatically through a long straight horizontal duct, 0.25 m diameter, at a measured mass flow rate of 40 kg/s. At a particular section along the duct the measured values of static temperature T 5 150oC and static pressure p 5 550 kPa. Determine the average velocity of the airflow and its stagnation temperature.
b. At another station further along the duct, measurements reveal that the static temperature has dropped to 147oC as a consequence of wall friction. Determine the average velocity and the static pressure of the airflow at this station.
Also determine the change in entropy per unit of mass flow between the two stations. For air assume that R 5 287 J/(kg K) and γ 5 1:4.
2. Nitrogen gas at a stagnation temperature of 300 K and a static pressure of 2 bar flows adiabatically through a pipe duct of 0.3 m diameter. At a particular station along the duct length the Mach number is 0.6. Assuming the flow is frictionless, determine a. the static temperature and stagnation pressure of the flow;
b. the mass flow of gas if the duct diameter is 0.3 m. For nitrogen gas take R 5 297 J=ðkg KÞ and γ 5 1:4.
3. Air flows adiabatically through a horizontal duct and at a section numbered (1) the static pressure p1 5 150 kPa, the static temperature T1 5 200oC and the velocity c1 5 100 m/s. At a station further downstream the static pressure p2 5 50 kPa and the static temperature T2 5 150oC. Determine the velocity c2 and the change in entropy per unit mass of air. For air take R 5 287 J/(kg K) and γ 5 1:4.
4. For the adiabatic expansion of a perfect gas through a turbine, show that the overall efficiency ηt and small stage efficiency ηp are related by
where ε 5 r(12γ)/γ, and r is the expansion pressure ratio, γ is the ratio of specific heats. An axial flow turbine has a small stage efficiency of 86%, an overall pressure ratio of 4.5 to 1 and a mean value of γ equal to 1.333. Calculate the overall turbine efficiency.
5. Air is expanded in a multistage axial flow turbine, the pressure drop across each stage being very small. Assuming that air behaves as a perfect gas with ratio of specific heats γ, derive pressure-temperature relationships for the following processes:
a. reversible adiabatic expansion;
b. irreversible adiabatic expansion, with small stage efficiency ηp;
c. reversible expansion in which the heat loss in each stage is a constant fraction k of the enthalpy drop in that stage;
d. reversible expansion in which the heat loss is proportional to the absolute temperature T. Sketch the first three processes on a T, s diagram. If the entry temperature is 1100 K and the pressure ratio across the turbine is 6 to 1, calculate the exhaust temperatures in each of the first three cases. Assume that γ is 1.333, that ηp 5 0.85, and that k 5 0.1.
6. Steam at a pressure of 80 bar and a temperature of 500oC is admitted to a turbine where it expands to a pressure of 0.15 bar. The expansion through the turbine takes place adiabatically with an isentropic efficiency of 0.9 and the power output from the turbine is 40 MW. Using a Mollier chart and/or steam tables determine the enthalpy of the steam at exit from the turbine and the flow rate of the steam.
7. A multistage high-pressure steam turbine is supplied with steam at a stagnation pressure of 7 MPa and a stagnation temperature of 500oC. The corresponding specific enthalpy is 3410 kJ/kg. The steam exhausts from the turbine at a stagnation pressure of 0.7 MPa, the steam having been in a superheated condition throughout the expansion. It can be assumed that the steam behaves like a perfect gas over the range of the expansion and that γ 5 1.3. Given that the turbine flow process has a small-stage efficiency of 0.82, determine
a. the temperature and specific volume at the end of the expansion;
b. the reheat factor.
The specific volume of superheated steam is represented by pv 5 0.231(h 2 1943), where p is in kPa, v is in m3/kg, and h is in kJ/kg.
8. A 20 MW back-pressure turbine receives steam at 4 MPa and 300oC, exhausting from the last stage at 0.35 MPa. The stage efficiency is 0.85, reheat factor 1.04, and external losses 2% of the actual isentropic enthalpy drop. Determine the rate of steam flow. At the exit from the first stage nozzles, the steam velocity is 244 m/s, specific volume 68.6 dm3/kg, mean diameter 762 mm, and steam exit angle 76o measured from the axial direction. Determine the nozzle exit height of this stage.
9. Steam is supplied to the first stage of a five-stage pressure-compounded steam turbine at a stagnation pressure of 1.5 MPa and a stagnation temperature of 350oC. The steam leaves the last stage at a stagnation pressure of 7.0 kPa with a corresponding dryness fraction of 0.95. By using a Mollier chart for steam and assuming that the stagnation state point locus is a straight line joining the initial and final states, determine
a. the stagnation conditions between each stage assuming that each stage does the same amount of work;
b. the total-to-total efficiency of each stage;
c. the overall total-to-total efficiency and total-to-static efficiency assuming the steam enters the condenser with a velocity of 200 m/s;
d. the reheat factor based upon stagnation conditions.
10. Carbon dioxide gas (CO2) flows adiabatically along a duct. At station 1 the static pressure p1 5 120 kPa and the static temperature T1 5 120oC. At station 2 further along the duct the static pressure p2 5 75 kPa and the velocity c2 5 150 m/s.
Determine
a. the Mach number M2;
b. the stagnation pressure p02;
c. stagnation temperature T02;
d. the Mach number M1.
For CO2 take R 5 188 J/(kg K) and γ 5 1.30.
11. Air enters the first stage of an axial flow compressor at a stagnation temperature of 20oC and at a stagnation pressure of 1.05 bar and leaves the compressor at a stagnation pressure of 11 bar. The total-to-total efficiency of the compressor is 83%. Determine, the exit stagnation temperature of the air and the polytropic efficiency of the compressor. Assume for air that γ 5 1:4.