of Compressors and Turbines (AE 651) Autumn Semester 2009 - PowerPoint PPT Presentation

Aerodynamics of Compressors and Turbines (AE 651) Autumn Semester 2009 Instructor : Bhaskar Roy Professor, Aerospace Engineering Department I.I.T., Bombay e-mail : aeroyia@aero.iitb.ac.in 1

Flow Models for Turbomachinery Three – dimensional flow analysis The assumption that the flow inside the axial flow compressor or turbine annulus is two – dimensional means that any effect of radial movement of the fluid through blade passage is ignored. AE 651 - Prof Bhaskar Roy, IITB 2 Lect - 19

Flow Models for Turbomachinery Radial flow can appear due to the following reasons-- 1. Centrifugal action in the process of transfer of rotational motion is passed on to the fluid 2. Convergence of the annulus passage is substantial for highly loaded compressor stages. 3. Twist and taper (chord-wise and thickness- wise) of the blade from hub to tip; 4. Tip clearance effects; (i.e. effect of tip flow around the open tip of the blades) AE 651 - Prof Bhaskar Roy, IITB 3 Lect - 19

Flow Models for Turbomachinery 5 Double vortex formation in the blade passages; 6. Temperature/ Enthalpy / Entropy gradient in the radial direction (due to above 1 to 4 ); 7. Blade thickness blockage (including the effect of camber and stagger) 8. End wall (casing and hub) boundary layer blockage effects AE 651 - Prof Bhaskar Roy, IITB 4 Lect - 19

Flow Models for Turbomachinery The radial equilibrium theory is based on the assumption that the forces exerted by all these radial gradients contributing to the overall radial movement of the flow must be balanced by forces exerted by the pressure gradient existing in the flow, so that at any instant of time the fluid system is in radial equilibrium . AE 651 - Prof Bhaskar Roy, IITB 5 Lect - 19

Flow Models for Turbomachinery Motion of a particle w.r.t two co – ordinate systems  The two reference systems have relative motion represented by R (motion of position vector R), ω (rotation of the particle r ’ with respect to x,y,z) and r v xyz are the rotational and translational motion of the particle with respect to xyz.  Velocity of particle P with respect to XYZ from fig.   = dr   V XYZ dt   XYZ 6 AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery   d ρ'   v = Velocity of P w.r.to small xyz, xyz   dt        xyz r = R+ ρ ' Vectorially           d ρ' dr dR       = +       dt dt dt Or, XYZ XYZ XyZ  + ω.ρ' V R V = + XYZ xyz AE 651 - Prof Bhaskar Roy, IITB 7 Lect - 19

Flow Models for Turbomachinery Again, acceleration of P w.r.t. space coordinates XYZ,   dV XYZ   = a   XYZ dt XYZ Again, acceleration of P w.r.t. body-fixed coordinates, xyz,   dV xyz   xyz = a   dt xyz AE 651 - Prof Bhaskar Roy, IITB 8 Lect - 19

Flow Models for Turbomachinery Thus, acceleration of P w.r.t. space coordinates XYZ,         d ωρ ' d d V xyz V ** XYZ     a   = = + + R   XYZ     dt dt dt XYZ XYZ xyz     dV dV xyz xyz ω.V     = + Also xyz   XYZ dt   dt xyz     d ω.ρ'     d ρ' dω ρ Using ω       = + '      dt  dt dt XYZ XYZ XYZ     d ρ' dρ' ρ ω   = + '   and     dt dt XYZ xyz AE 651 - Prof Bhaskar Roy, IITB 9 Lect - 19

Flow Models for Turbomachinery Hence,     ρ' ' = + + + + + ρ ω ω ω ω ω a a V R v xyz xyz xyz XYZ      ** ρ' ρ' ω ω ω ω   a R 2 v xyz + + + + = xyz   . . For a frame of reference rotating with constant angular velocity ω , about the z-axis ( axis of the rotating machine )   XYZ = + + ρ'   ω ω ω a a 2 v xyz xyz   AE 651 - Prof Bhaskar Roy, IITB 10 Lect - 19

Flow Models for Turbomachinery For a compressor blade passage, the flow velocities are,     = relative velocity XYZ = absolute velocity V C V V , xyz ω ρ' = = + = V + ω r V + u V V XYZ xyz       Differentiating,   d ω ρ' d V   XYZ   a = - XYZ dt       dt XYZ XYZ   Finally we get , ρ' = + ω + ω ω a a 2 v xyz xyz XYZ AE 651 - Prof Bhaskar Roy, IITB 11 Lect - 19

Flow Models for Turbomachinery Considering the equilibrium of forces along the arbitrary flow direction, s – direction, we get, between any two axial stations, separated by a small distances ∆ s , where area, A i is constant. 1 Δp = ∆ p.A i = A i .ρ.∆s. a XYZ , a ρ Δs XYZ 1 Dv  2 + ω r + 2ωV . p = ρ Dt The flow in compressor blade is diffusing, DV is negative, , Dt 1 Dv   2 - ω r + 2 ω V . p = x x ρ Dt AE 651 - Prof Bhaskar Roy, IITB 12 Lect - 19

Flow Models for Turbomachinery Change of axis notations where r , t and a are the radial, tangential (peripheral) and axial directions respectively. Generalized flow path direction AE 651 - Prof Bhaskar Roy, IITB 13 Lect - 19

Flow Models for Turbomachinery Assumptions made are:- • The fluid is frictionless • The rotor is rigid and rotates with constant angular velocity ω • The flow is steady relative to the rotor • The radial variation of density is neglected • This still leaves enough scope for formation of i) vorticity, ii) entropy gradients, and iii) stagnation enthalpy gradients in the flow field. AE 651 - Prof Bhaskar Roy, IITB 14 Lect - 19

Flow Models for Turbomachinery Then from the definition of unit vectors, ∧ ∧ i V i i V V D D ∧ ∧ ∧ ∧ r ω i i i i ω i i i i t = = = = = - = - t t t t And t t t t Dt r Dt r r r r r ∧ ∧ ∧ V = V i +V i +V i Again, r r t t a a       ds = 0 ds Steady D D And   = V + As State dt   Dt Ds dt Ο 0 DV DV DV DV DV d θ dθ DV DV   ∧ ∧ ∧   i i i t t a a = r - + + + V V V V     r w r a Dt Dt dt Dt dt Dt t     AE 651 - Prof Bhaskar Roy, IITB 15 Lect - 19

Flow Models for Turbomachinery Now, the equation for flow inside the compressor blade passage may be resolved in its three components, using r, q and z coordinate system and modified the above,   2 + + ω r V 1 ∂ p D V   t r - = - V   ---------- (a) ρ ∂ r D s   r 1 ∂ p D V .   V V V V t a - . = + + 2. ω.V r V   r ρ r. ∂ θ D s r -------- (b)   1 ∂ p D V   a - = V -------------- (c)   ρ ∂ z D s   AE 651 - Prof Bhaskar Roy, IITB 16 Lect - 19

Flow Models for Turbomachinery V t = V w AE 651 - Prof Bhaskar Roy, IITB 17 Lect - 19

Flow Models for Turbomachinery Assuming some arbitrary velocity triangle for the flow at the station under consideration, C ω r V t = = + + t Then equation (a) and (b) can be rewritten as   2  C   1 p DV t r - =   - V ---------- (d)  ρ r  Ds  r   D r.C   1 1 ∂ p p V t - =   ----------- (e) ρ r. θ ∂ D s r   AE 651 - Prof Bhaskar Roy, IITB 18 Lect - 19

Flow Models for Turbomachinery Now, we can write the kinematic relation as,     D D D D = V V V V Ds Da a Where V a and a are axial components of V and s respectively Now, if we define a new meridional direction by ∧ ∧ ∧ D i = D i + D i m m r r a a   2 Hence from equation (d) we can write , C 1 ∂p DV DV   t - = r - V   ρ ∂r Dm m r   AE 651 - Prof Bhaskar Roy, IITB 19 Lect - 19

Flow Models for Turbomachinery By the definition of meridional direction tan. φ = V r and V = V sin  V Z Hence the flow equation in radial direction can be written down as 2 D 1 ∂p Dsin.φ C V 2 t = - - si sin. φ m V V V V m m m m ρ ∂ρ r Dm Dm     D D D D Now, by = V V V V definition Ds Dm m AE 651 - Prof Bhaskar Roy, IITB 20 Lect - 19

Flow Models for Turbomachinery D D 1 1  D sin. D   = - = - = cos.  r Now, and Dm Dm Dm D m m r m is the radius of curvature of the meridional plane, The negative sign is arbitrary. But, for axial flow compressor the flow track inside generally moves towards lesser φ or higher rm . Hence, 2 2 D 1 1 ∂ p p = C V V V V t m m m m + cos. φ - V ρ ∂ ρ r Dm r r m This is the full radial – equilibrium Equation for circumferentially averaged (blade to blade) flow properties inside of a turbo machine blade row 21 21 AE 651 - Prof Bhaskar Roy, IITB Lect - 19

Flow Models for Turbomachinery 22 22 AE 651 - Prof Bhaskar Roy, IITB Lect - 19