558
15 Axial and Radial Turbines for Gases
• Determine the relative inflow Mach numbers of both rotors (A: M w1 = 1.071,
M w3 = 0.805). Remark that the inlet Mach number of the first rotor is much lower
than with impulse blades. This is a big advantage for the 50 % degree of reaction
turbine.
• Determine the deceleration ratio in the outlet guide vanes (A: 0.80). The necessity for a deceleration in the outlet guide vane ring is a small disadvantage of the
reaction turbine (but see next exercise).
15.4.2 Consider once more the axial turbine of the previous exercise, composed
of a stator ring of nozzle vanes, followed by two contra-rotating rotors with large
flow turning and a stator ring of outlet guide vanes. Take the same values for blade
speed, axial through-flow velocity and gas properties. We will adapt the machine in
three steps into a turbine with only rotating components.
• Determine the velocity triangles for ψ 1 = 4 and ψ 2 = 4, such that the couple of the
two contra-rotating rotors forms a repeating system. Remark that these conditions do not determine the system uniquely. Take as particular choice identical
blade shapes, symmetrically positioned, in both rotors (A: obtained for v 2u = 2u).
• Determine the degree of reaction of both rotors (A: 100 %). Remark that the
turbine is equivalent with a Ljungström turbine (see Chap. 6, Sect. 6.2).
• Determine the relative inflow Mach numbers of both rotors (A: M w1 = 0.693,
M w3 = 0.778). Remark that these are lower than the values of the 50 % degree of
reaction turbine of the previous exercise.
• Determine the deceleration ratio in the outlet guide vanes (A: 0.555). Remark
that the deceleration ratio is much too strong for practical realisation of the
vanes.
• Replace the nozzle vane ring with a rotor rotating together with the second rotor of the repeating couple (same speed and same sense) such that the inflow
and outflow velocities remain unchanged (A: this requires ψ = 2). Determine the
degree of reaction of this rotor (A: 200 %: this rotor generates an increase of
kinetic energy in the absolute frame equal to the work done).
• Replace the outlet guide vane ring with a rotor rotating together with the first
rotor of the repeating couple (same speed and same sense) such that the inflow
and outflow velocities remain unchanged (A: this requires ψ = 2 and R = 0).
• The resulting turbine is solely composed of rotors. The inner repeating couple
may be repeated a number of times. A turbine of this type may be used in a propfan or unducted fan aero-engine for driving the contra-rotating propellers.
References
1. AGARD-LS-167 (1989) Blading design for axial turbomachines. AGARD, Paris. ISBN
92-835-0512-3
2. Bertini F, Credi M, Marconcini M, Giovannini M (2013) A path toward the aerodynamic robust
design of low pressure turbines. J Turbomach 135:021018
15 Axial and Radial Turbines for Gases
• Determine the relative inflow Mach numbers of both rotors (A: M w1 = 1.071,
M w3 = 0.805). Remark that the inlet Mach number of the first rotor is much lower
than with impulse blades. This is a big advantage for the 50 % degree of reaction
turbine.
• Determine the deceleration ratio in the outlet guide vanes (A: 0.80). The necessity for a deceleration in the outlet guide vane ring is a small disadvantage of the
reaction turbine (but see next exercise).
15.4.2 Consider once more the axial turbine of the previous exercise, composed
of a stator ring of nozzle vanes, followed by two contra-rotating rotors with large
flow turning and a stator ring of outlet guide vanes. Take the same values for blade
speed, axial through-flow velocity and gas properties. We will adapt the machine in
three steps into a turbine with only rotating components.
• Determine the velocity triangles for ψ 1 = 4 and ψ 2 = 4, such that the couple of the
two contra-rotating rotors forms a repeating system. Remark that these conditions do not determine the system uniquely. Take as particular choice identical
blade shapes, symmetrically positioned, in both rotors (A: obtained for v 2u = 2u).
• Determine the degree of reaction of both rotors (A: 100 %). Remark that the
turbine is equivalent with a Ljungström turbine (see Chap. 6, Sect. 6.2).
• Determine the relative inflow Mach numbers of both rotors (A: M w1 = 0.693,
M w3 = 0.778). Remark that these are lower than the values of the 50 % degree of
reaction turbine of the previous exercise.
• Determine the deceleration ratio in the outlet guide vanes (A: 0.555). Remark
that the deceleration ratio is much too strong for practical realisation of the
vanes.
• Replace the nozzle vane ring with a rotor rotating together with the second rotor of the repeating couple (same speed and same sense) such that the inflow
and outflow velocities remain unchanged (A: this requires ψ = 2). Determine the
degree of reaction of this rotor (A: 200 %: this rotor generates an increase of
kinetic energy in the absolute frame equal to the work done).
• Replace the outlet guide vane ring with a rotor rotating together with the first
rotor of the repeating couple (same speed and same sense) such that the inflow
and outflow velocities remain unchanged (A: this requires ψ = 2 and R = 0).
• The resulting turbine is solely composed of rotors. The inner repeating couple
may be repeated a number of times. A turbine of this type may be used in a propfan or unducted fan aero-engine for driving the contra-rotating propellers.
References
1. AGARD-LS-167 (1989) Blading design for axial turbomachines. AGARD, Paris. ISBN
92-835-0512-3
2. Bertini F, Credi M, Marconcini M, Giovannini M (2013) A path toward the aerodynamic robust
design of low pressure turbines. J Turbomach 135:021018