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Centrifugal Compressor Bucket Geometry for Supercritical CO2 Turbines OverviewFor an sCO2 turbomachine using a centrifugal compressor, the bucket geometry (impeller blade metal angles and key diameters) is set by the head you must add, the rotational speed, and the flow you must pass at acceptable Mach and incidence. Because CO2 behaves as a dense real gas near the critical region, use CO2 properties at the actual inlet state when sizing.Below is a compact method that ties pressure ratio, inlet temperature and pressure, and RPM to the main geometry:Tip speed from required head.Tip diameter from speed.Exit meridional velocity from a chosen flow coefficient.Exit blade height from mass flow and density.Exit metal angle from the velocity triangle with slip.Inlet eye diameter from allowable inlet Mach.Inlet metal angle from the inlet velocity triangle.Where possible, pass in CO2 property values at the design point (density and k, Z). If you cannot compute them inside FileMaker, precompute with REFPROP/CoolProp and store them as fields.Design coefficients to chooseCompressor isentropic efficiency, eta_c: 0.70 to 0.85Head coefficient, psi: 0.45 to 0.70 (total to shaft)Flow coefficient, phi: 0.06 to 0.12 for dense sCO2Slip factor, sigma: 0.85 to 0.92 for backswept bladesInlet Mach limit, M1_max: 0.60 to 0.80 at the eye tipHub to tip ratio at eye, htr: 0.35 to 0.50Variables and constantsRequired inputs as FileMaker fields or globals:T1_K inlet temperature, KP1_Pa inlet pressure, PaPR pressure ratio, P2/P1N_rpm rotational speed, rpmmdot_kg_s mass flow, kg s^-1rho1_kg_m3 CO2 density at inlet, kg m^-3k1 ratio of specific heats at inletZ1 compressibility at inletcp1_J_kgK cp at inlet, J kg^-1 K^-1rho2_kg_m3 CO2 density at impeller exit, kg m^-3 (estimate or property)Design choices as fields or constants:eta_c, psi, phi, sigma, M1_max, htrFixed constant:R_CO2_J_kgK = 188.9NotesIf rho2_kg_m3 is not known, a first pass can take rho2 approximately equal to rho1 for dense sCO2, then refine later with a property model.FileMaker-ready sizing formulaPaste the following into a FileMaker calculation (returns a readable block of results). Replace field names with your own if needed.```Let([R_CO2 = 188.9;pi = 3.14159265358979;// InputsT1 = T1_K;P1 = P1_Pa;PRr = PR;N = N_rpm;mdot = mdot_kg_s;rho1 = rho1_kg_m3;k = k1;Z = Z1;cp = cp1_J_kgK;// Design coefficientseta = eta_c;psi = psi;phi = phi;sigma = sigma;M1 = M1_max;htr = htr;// 1) Isentropic head required for the pressure ratio// Hs = cp T1 ( PR^((k-1)/k) • 1 )Hs = cp T1 ( Power(PRr,(k-1)/k) • 1 );// Shaft head allowing for compressor efficiencyH = Hs / eta;// 2) Tip speed from head coefficient: psi = H / U2^2 => U2 = sqrt( H / psi )U2 = Sqrt( H / psi );// 3) Impeller tip diameter: U2 = pi D2 N / 60 => D2 = 60 U2 / ( pi N )D2 = 60 U2 / ( pi N );// 4) Exit meridional velocity from flow coefficient: Vm2 = phi U2Vm2 = phi U2;// 5) Exit blade height from continuity: mdot = rho2 (2piR2b2) Vm2// R2 = D2/2 => b2 = mdot / ( rho2 2 pi (D2/2) Vm2 )rho2 = rho2_kg_m3;b2 = mdot / ( rho2 2 pi ( D2 / 2 ) Vm2 );// 6) Exit blade metal angle beta2 from velocity triangle with slip// Ideal tangential absolute at exit: Vw2_ideal = U2 • H / U2// With slip: Vw2 = sigma Vw2_ideal// Relative tangential at exit: Wt2 = U2 • Vw2// beta2 = arctan( Vm2 / Wt2 ) in degreesVw2_ideal = U2 • ( H / U2 );Vw2 = sigma Vw2_ideal;Wt2 = U2 • Vw2;beta2_deg = 57.295779513 Atan( Vm2 / Wt2 );// 7) Inlet eye sizing from allowable Mach at the eye tip// Speed of sound a1 = sqrt( k R_CO2 T1 / Z )a1 = Sqrt( k R_CO2 T1 / Z );// Inlet meridional velocity target: Vm1 = M1 a1Vm1 = M1 a1;// Inlet density = rho1 (input). Required eye flow area: A1 = mdot / ( rho1 Vm1 )A1 = mdot / ( rho1 Vm1 );// Annulus area with hub to tip ratio htr: A1 = pi/4 D1_tip^2 ( 1 • htr^2 )// Solve for D1_tipD1_tip = Sqrt( 4 A1 / ( pi ( 1 • Power(htr,2) ) ) );// Hub diameterD1_hub = htr D1_tip;// 8) Inlet metal angle beta1 at the tip for zero prewhirl// U1 = pi D1_tip N / 60; beta1 = arctan( Vm1 / U1 )U1 = pi D1_tip N / 60;beta1_deg = 57.295779513 Atan( Vm1 / U1 )];Results: & Char(10) &Tip speed U2 m_s = & Round(U2;3) & Char(10) &Impeller tip diameter D2 m = & Round(D2;4) & Char(10) &Exit meridional velocity Vm2 m_s = & Round(Vm2;3) & Char(10) &Exit blade height b2 m = & Round(b2;4) & Char(10) &Exit metal angle beta2 deg = & Round(beta2_deg;1) & Char(10) &Inlet speed of sound a1 m_s = & Round(a1;1) & Char(10) &Inlet meridional velocity Vm1 m_s = & Round(Vm1;1) & Char(10) &Eye tip diameter D1_tip m = & Round(D1_tip;4) & Char(10) &Eye hub diameter D1_hub m = & Round(D1_hub;4) & Char(10) &Inlet metal angle beta1 deg = & Round(beta1_deg;1))```How this maps to CO2 and pressure ratioThe pressure ratio enters via the isentropic head Hs.The turbine system preference for moderate pressure ratios in sCO2 is reflected here because higher PR quickly raises H, which increases U2 and thus D2 for a fixed RPM.CO2 specificity appears through R_CO2, k, Z, and densities, which affect the eye sizing, Mach check, and blade height.For dense sCO2 near the critical region, rho1 is high and the eye can be compact, but Mach and incidence must still be controlled.Practical tips1. Start with psi 0.55 and phi 0.08 for dense sCO2, sigma 0.90, M1_max 0.7, htr 0.45.2. Keep tip relative Mach at exit below about 1.2; if high, reduce PR per stage or raise RPM and split into two stages.3. If b2 becomes too small for manufacturability, increase phi slightly or reduce PR per stage and add a second stage.4. After a first pass, refine rho2, k, and Z with a real gas property call at the estimated exit state and rerun.This workflow gives you a consistent way to go from target pressure ratio, RPM, and inlet state to a first cut of centrifugal impeller bucket geometry that is specific to CO2.-----Let([R_CO2 = 188.9;pi = 3.14159265358979;// InputsT1 = T1_K;P1 = P1_Pa;PRr = PR;N = N_rpm;mdot = mdot_kg_s;rho1 = rho1_kg_m3;k = k1;Z = Z1;cp = cp1_J_kgK;// Design coefficientseta = eta_c;psi = psi;phi = phi;sigma = sigma;M1 = M1_max;htr = htr;// 1) Isentropic head required for the pressure ratio// Hs = cp * T1 * ( PR^((k-1)/k) - 1 )Hs = cp * T1 * ( Power(PRr,(k-1)/k) - 1 );// Shaft head allowing for compressor efficiencyH = Hs / eta;// 2) Tip speed from head coefficient: psi = H / U2^2 => U2 = sqrt( H / psi )U2 = Sqrt( H / psi );// 3) Impeller tip diameter: U2 = pi * D2 * N / 60 => D2 = 60 * U2 / ( pi * N )D2 = 60 * U2 / ( pi * N );// 4) Exit meridional velocity from flow coefficient: Vm2 = phi * U2Vm2 = phi * U2;// 5) Exit blade height from continuity: mdot = rho2 * (2*pi*R2*b2) * Vm2// R2 = D2/2 => b2 = mdot / ( rho2 * 2 * pi * (D2/2) * Vm2 )rho2 = rho2_kg_m3;b2 = mdot / ( rho2 * 2 * pi * ( D2 / 2 ) * Vm2 );// 6) Exit blade metal angle beta2 from velocity triangle with slip// Ideal tangential absolute at exit: Vw2_ideal = U2 - H / U2// With slip: Vw2 = sigma * Vw2_ideal// Relative tangential at exit: Wt2 = U2 - Vw2// beta2 = arctan( Vm2 / Wt2 ) in degreesVw2_ideal = U2 - ( H / U2 );Vw2 = sigma * Vw2_ideal;Wt2 = U2 - Vw2;beta2_deg = 57.295779513 * Atan( Vm2 / Wt2 );// 7) Inlet eye sizing from allowable Mach at the eye tip// Speed of sound a1 = sqrt( k * R_CO2 * T1 / Z )a1 = Sqrt( k * R_CO2 * T1 / Z );// Inlet meridional velocity target: Vm1 = M1 * a1Vm1 = M1 * a1;// Inlet density = rho1 (input). Required eye flow area: A1 = mdot / ( rho1 * Vm1 )A1 = mdot / ( rho1 * Vm1 );// Annulus area with hub to tip ratio htr: A1 = pi/4 * D1_tip^2 * ( 1 - htr^2 )// Solve for D1_tipD1_tip = Sqrt( 4 * A1 / ( pi * ( 1 - Power(htr,2) ) ) );// Hub diameterD1_hub = htr * D1_tip;// 8) Inlet metal angle beta1 at the tip for zero prewhirl// U1 = pi * D1_tip * N / 60; beta1 = arctan( Vm1 / U1 )U1 = pi * D1_tip * N / 60;beta1_deg = 57.295779513 * Atan( Vm1 / U1 )];"Results:" & Char(10) &"Tip speed U2 m_s = " & Round(U2;3) & Char(10) &"Impeller tip diameter D2 m = " & Round(D2;4) & Char(10) &"Exit meridional velocity Vm2 m_s = " & Round(Vm2;3) & Char(10) &"Exit blade height b2 m = " & Round(b2;4) & Char(10) &"Exit metal angle beta2 deg = " & Round(beta2_deg;1) & Char(10) &"Inlet speed of sound a1 m_s = " & Round(a1;1) & Char(10) &"Inlet meridional velocity Vm1 m_s = " & Round(Vm1;1) & Char(10) &"Eye tip diameter D1_tip m = " & Round(D1_tip;4) & Char(10) &"Eye hub diameter D1_hub m = " & Round(D1_hub;4) & Char(10) &"Inlet metal angle beta1 deg = " & Round(beta1_deg;1)) |
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