VELOCITY DIAGRAMS FOR IMPULSE-REACTION TURBINE

The steam passes to the fixed blades with velocity v_a1. Fixed blades also act as nozzles with pressure drop occurring, while steam passes through them so that there is gain in kinetic energy and steam leaves the fixed blades with velocity v_a2. The expansion of steam and the enthalpy drop occur in fixed and moving blades. The velocity diagrams for an impulse-reaction turbine are shown in Fig. 7.11.

1. Degree of Reaction: The degree of reaction R_d of a reaction turbine is defined as the ratio of enthalpy drop over moving blades to the total enthalpy drop in the stage. Thus Enthalpy drop through fixed blades, Enthalpy drop along the moving blades, ∆h_m = Kinetic energy supplied to moving blades, K.E.Neglecting friction of blade surface,Axial thrust on rotor = (v_f 1 − v_f 2) + (∆p)_m × area of blade discNow total enthalpy drop in the stage = Work done by steam in the stage Δh_f + Δh_m = u (v_w1 + v_w2)Figure 7.11 Velocity diagrams for impulse-reaction turbine From Fig. 7.11, we havev_r2 = v_f2 cosec β₂ and v_r1 = v_f1 cosec β₁and v_w1 + v_w2 = v_f 1 cot β₁ + v_f 2 cot β₂The velocity of flow generally remains constant through the blades.∴ v_f 1 = v_f 2 = v_f If R_d = 0.5, then From Fig. 7.11, we have u = v_f (cot β₂ − cot α₂) = v_f (cot α₁ − cot β₁)This means that the moving and fixed blades must have the same shape if the degree of reaction is 50%. This condition gives symmetrical velocity diagrams. This type of turbine is known as Parson’s Reaction Turbine.
2. Efficiency: We assume that the degree of reaction is 50%, that is, ∆h_f = ∆h_m, the moving and fixed blades are of the same shape, and the velocity of steam at exit from the preceding stage is the same as the velocity of steam at the entrance to the succeeding stage.The work done per kg of steam, w = u(v_w1 + v_w2) = u [v_a1 cos α₁ + (v_r2 cos β₂ − u)]Now, β₂ = α₁ and v_r2 = v_a1 where  is the speed ratio.K.E. supplied to fixed blade = K.E. supplied to moving blade Total energy supplied to stage,For symmetrical blades, v_r2 = v_a1From Fig. 7.11, we have   For η_b to be maximum,  (1 + 2ρ cos α₁ − ρ²) (4 cos α₁ − 4ρ) − 2ρ (2 cos α₁ − ρ) (2 cos α₁ − ρ) = 04 (cos α₁ − ρ) (1 + 2ρ cos α₁ − ρ²) − 4ρ (cos α₁ − ρ) (2 cos α₁ − ρ) = 0(cos α₁ − ρ) [(1 + 2ρ cos α₁ − ρ²) − ρ (2 cos α₁ − ρ)] = 0∴ cos α₁ − ρ² = 0 
3. Height of Turbine Blading:The height and thickness of blading are shown in Fig. 7.12.Volume flow = Area × flow velocitywhere d = mean diameter of turbine wheel   h_1, h₂ = blade heights at inlet and outlet respectively   t_1, t₂ = thickness of blades at inlet and outlet respectively.      n = number of bladesFigure 7.12 Height of turbine bladingFor most of the turbines, v_f1 = v_f2 = v_f, h₁ = h₂ = h and t₁ = t₂ = tEquation (7.31) gives height of blades.The pitch of blades, p is given by: ṁv_s = [n (p – t)h] v_fGenerally t << p, ∴ ṁv_s = n ph v_f = π dhv_f