Problems on radial-flow gas turbines.

PROBLEMS

1. A small inward radial-flow gas turbine, comprising a ring of nozzle blades, a radial-vaned rotor and an axial diffuser, operates at the nominal design point with a total-to-total efficiency of 0.90. At turbine entry, the stagnation pressure and temperature of the gas are 400 kPa and 1140 K. The flow leaving the turbine is diffused to a pressure of 100 kPa and has negligible final velocity. Given that the flow is just choked at nozzle exit, determine the impeller peripheral speed and the flow outlet angle from the nozzles. For the gas assume

γ 5 1.333 and R 5 287 J/(kg oC).

2. The mass flow rate of gas through the turbine given in Problem 1 is 3.1 kg/s, the ratio of the rotor axial width-rotor tip radius (b2/r2) is 0.1, and the nozzle isentropic velocity ratio (φ2) is 0.96. Assuming that the space between nozzle exit and rotor entry is negligible and ignoring the effects of blade blockage, determine

a. the static pressure and static temperature at nozzle exit;

b. the rotor tip diameter and rotational speed;

c. the power transmitted assuming a mechanical efficiency of 93.5%.

3. A radial turbine is proposed as the gas expansion element of a nuclear powered Brayton cycle space power system. The pressure and temperature conditions through the stage at the design point are to be as follows:

upstream of nozzles, p01 5 699 kPa, T01 5 1145 K;

nozzle exit, p2 5 527.2 kPa, T2 5 1029 K;

rotor exit, p3 5 384.7 kPa, T3 5 914.5 K; T03 5 924.7 K.

The ratio of rotor exit mean diameter to rotor inlet tip diameter is chosen as 0.49 and the required rotational speed as 24,000 rev/min. Assuming the relative flow at rotor inlet is radial and the absolute flow at rotor exit is axial, determine

a. the total-to-static efficiency of the turbine;

b. the rotor diameter;

c. the implied enthalpy loss coefficients for the nozzles and rotor row.

The gas employed in this cycle is a mixture of helium and xenon with a molecular weight of 39.94 and a ratio of specific heats of 5/3. The universal gas constant is R0 5 8.314 kJ/(kg- mol K).

4. A film-cooled radial-inflow turbine is to be used in a high-performance open Brayton cycle gas turbine. The rotor is made of a material able to withstand a temperature of 1145 K at a tip speed of 600 m/s for short periods of operation. Cooling air is supplied by the compressor that operates at a stagnation pressure ratio of 4-1, with an isentropic efficiency of 80%, when air is admitted to the compressor at a stagnation temperature of 288 K. Assuming that the effectiveness of the film cooling is 0.30 and the cooling air temperature at turbine entry is the same as that at compressor exit, determine the maximum permissible gas temperature at entry to the turbine. Take γ 5 1.4 for the air. Take γ 5 1.333 for the gas entering the turbine. Assume R 5 287 J/(kg K) in both cases.

5. The radial-inflow turbine in Problem 3 is designed for a specific speed Ωs of 0.55 rad.

Determine

a. the volume flow rate and the turbine power output;

b. the rotor exit hub and tip diameters;

c. the nozzle exit flow angle and the rotor inlet passage width-diameter ratio, b2/D2.

6. An IFR gas turbine with a rotor diameter of 23.76 cm is designed to operate with a gas mass flow of 1.0 kg/s at a rotational speed of 38,140 rev/min. At the design condition, the inlet stagnation pressure and temperature are to be 300 kPa and 727oC. The turbine is to be “cold” tested in a laboratory where an air supply is available only at the stagnation conditions of 200 kPa and 102oC.

a. Assuming dynamically similar conditions between those of the laboratory and the projected design determine, for the “cold” test, the equivalent mass flow rate and the speed of rotation. Assume the gas properties are the same as for air.

b. Using property tables for air, determine the Reynolds numbers for both the hot and cold running conditions. The Reynolds number is defined in this context as where ρ01 and μ1 are the stagnation density and viscosity of the air, Ω is the rotational speed (rev/s), and D is the rotor diameter.

7. For the radial-flow turbine described in the previous problem and operating at the prescribed “hot” design point condition, the gas leaves the exducer directly to the atmosphere at a

pressure of 100 kPa and without swirl. The absolute flow angle at rotor inlet is 72o to the radial direction. The relative velocity w3 at the mean radius of the exducer (which is one-half of the rotor inlet radius r2) is twice the rotor inlet relative velocity w2. The nozzle enthalpy loss coefficient, ζN 5 0.06. Assuming the gas has the properties of air with an average value of γ 5 1.34 (this temperature range) and R 5 287 J/kg K, determine

a. the total-to-static efficiency of the turbine;

b. the static temperature and pressure at the rotor inlet;

c. the axial width of the passage at inlet to the rotor;

d. the absolute velocity of the flow at exit from the exducer;

e. the rotor enthalpy loss coefficient;

f. the radii of the exducer exit given that the radius ratio at that location is 0.4.

8. One of the early space power systems built and tested for NASA was based on the Brayton cycle and incorporated an IFR turbine as the gas expander. Some of the data available concerning the turbine are as follows:

total-to-total pressure ratio (turbine inlet to turbine exit), p01/p03 5 1.560; total-to-static pressure ratio, p01/p3 5 1.613;

total temperature at turbine entry, T01 5 1083 K; total pressure at inlet to turbine, T01 5 91 kPa; shaft power output (measured on a dynamometer), Pnet 5 22.03 kW; bearing and seal friction torque (a separate test), τf 5 0.0794 Nm; rotor diameter, D2 5 15.29 cm;

absolute flow angle at rotor inlet, α2 5 72o;

absolute flow angle at rotor exit, α3 5 0o;

the hub-to-shroud radius ratio at rotor exit, r3h/r3s 5 0.35;

ratio of blade speed to jet speed, ν 5 U2/c0 5 0.6958, c0 based on total-to-static pressure ratio.

For reasons of crew safety, an inert gas argon (R 5 208.2 J/(kg K), ratio of specific heats, γ 5 1.667) was used in the cycle. The turbine design scheme was based on the concept of optimum efficiency. Determine, for the design point

a. the rotor vane tip speed;

b. the static pressure and temperature at rotor exit;

c. the gas exit velocity and mass flow rate;

d. the shroud radius at rotor exit;

e. the relative flow angle at rotor inlet;

f. the specific speed.

Note: The volume flow rate to be used in the definition of the specific speed is based on the rotor exit conditions.

9. What is meant by the term nominal design in connection with a radial-flow gas turbine rotor?

Sketch the velocity diagrams for a 90o IFR turbine operating at the nominal design point.

At entry to a 90o IFR turbine, the gas leaves the nozzle vanes at an absolute flow angle, α2, of 73o. The rotor-blade tip speed is 460 m/s and the relative velocity of the gas at rotor exit is twice the relative velocity at rotor inlet. The rotor mean exit diameter is 45% of the rotor inlet diameter. Determine

a. the exit velocity from the rotor;

b. the static temperature difference, T2 2 T3, of the flow between nozzle exit and rotor exit.

Assume the turbine operates at the nominal design condition and that Cp 5 1.33 kJ/kg K.

10. The initial design of an IFR turbine is to be based upon Whitfield’s procedure for optimum efficiency. The turbine is to be supplied with 2.2 kg/s of air, a stagnation pressure of 250 kPa,

a stagnation temperature of 800oC, and have an output power of 450 kW. At turbine exit the static pressure is 105 kPA. Assuming for air that γ 5 1.33 and R 5 287 J/kg K, determine the value of Whitfield’s power ratio, S, and the total-to-static efficiency of the turbine.

11. By considering the theoretical details of Whitfield’s design problem for obtaining the optimum efficiency of an IFR turbine show that the correct choice for the relationship of the rotor inlet flow angles is obtained from the following equation:

12. An IFR turbine rotor is designed with 13 vanes and is expected to produce 400 kW from a supply of gas heated to a stagnation temperature of 1100 K at a flow rate of 1.2 kg/s. Using Whitfield’s optimum efficiency design method and assuming ηts 5 0.85, determine

a. the overall stagnation pressure to static pressure ratio and

b. the rotor tip speed and inlet Mach number, M2, of the flow.

Assume Cp 5 1.187 kJ/kg K and γ 5 1.33.

13. Another IFR turbine is to be built to develop 250 kW of shaft power from a gas flow of 1.1 kg/s. The inlet stagnation temperature, T01, is 1050 K, the number of rotor blades is 13, and the outlet static pressure, p3, is 102 kPa. At rotor exit the area ratio, ν 5 r3h/r3s 5 0.4, and the velocity ratio, cm3/U2 5 0.25. The shroud-to-rotor inlet radius, r3s/r2, is 0.4. Using the optimum efficiency design method, determine

a. the power ratio, S, and the relative and absolute flow angles at rotor inlet;

b. the rotor-blade tip speed;

c. the static temperature at rotor exit;

d. the rotor speed of rotation and rotor diameter.

Evaluate the specific speed, Ωs. How does this value compare with the optimum value of specific speed determined in Figure 8.15?

14. Using the same input design data for the IFR turbine given in Problem 5 and given that the total-to-static efficiency is 0.8, determine

a. the stagnation pressure of the gas at inlet and

b. the total-to-total efficiency of the turbine.

15. An IFR turbine is required with a power output of 300 kW driven by a supply of gas at a stagnation pressure of 222 kPa, at a stagnation temperature of 1100 K, and at a flow rate of 1.5 kg/s. The turbine selected by the engineer has 13 vanes and preliminary tests indicate it should have a total-to-static efficiency of 0.86. Based upon the optimum efficiency design method, sketch the appropriate velocity diagrams for the turbine and determine

a. the absolute and relative flow angles at rotor inlet;

b. the overall pressure ratio;

c. the rotor tip speed.

16. For the IFR turbine of the previous problem, the following additional information is made available:

Again, based upon the optimum efficiency design criterion, determine

a. the rotor diameter and speed of rotation;

b. the enthalpy loss coefficients of the rotor and the nozzles given that the nozzle loss coefficient is (estimated) to be one quarter of the rotor loss coefficient.

Incoming search terms:

• A 90 IFR gas turbine has the following data total to staticpressure ratio 3 5 and exite pressure is 1bar inlet total temprature 923K blade isentropic speed ratio 0 66 rotor diamter ratio 0 45 speed 16000 rpm nozzle exite angle 20 nozzle effecency 0 95 and
• An inward flow radial gas turbine with a rotor diameter of 23 76 cm is designed to operatewith a gas mass flow of 1 0 kg/s at a rotational speed of 38 140 rev/min At the design conditionthe inlet stagnation pressure and temperature are to be 300 kPa and 7
• problem on radial flow
• The performance data for an inward flow exhaust gas turbine are as follows: Stagnation pressure at inlet to nozzles Stagnation temperature at inlet to nozzles Static pressure at exit from nozzles Static temperature at exit from nozzles Static pressure at
• the performance data for inward exhaust gas turbine are as follows;stagnation pressure at inlet to nozzles p01=705kpa