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Radial-Inflow Turbine Geometry for Supercritical CO2: From Pressure Ratio and RPM to Bucket Angles and Diameters

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Radial-Inflow Turbine Geometry for Supercritical CO2: From Pressure Ratio and RPM to Bucket Angles and Diameters

Overview

For compact supercritical CO2 systems, a single radial-inflow turbine stage is common. The goal is to convert a specified total-to-static enthalpy drop into shaft work while keeping inlet Mach numbers and incidence under control. With CO2’s dense, non-ideal behavior, you should size to the actual inlet state (pressure, temperature, cp, k, compressibility Z, density) rather than air-based rules.

This guide gives a first-cut sizing workflow and a FileMaker Pro calculation that returns the key geometry:

Tip speed and tip diameter

Rotor inlet blade height

Inlet absolute and relative metal angles

Sanity checks for Mach number

It assumes a radial-inflow turbine with near-zero exit swirl (a common target), high effectiveness recuperation upstream, and a specified overall pressure ratio across the turbine stage.

Design coefficients to choose

These are typical starting values; adjust during iteration.

Turbine isentropic efficiency, eta_t: 0.80 to 0.90

Stage loading coefficient, psi_t = delta_h0 over U2^2: 1.0 to 1.6

Flow coefficient at rotor inlet, phi_t = Vm2 over U2: 0.15 to 0.25

Slip and exit swirl: target Vw3 approx 0 (zero exit swirl)

Radius ratio, rr = R3 over R2: 0.35 to 0.50 (sets hub exit diameter)

Inlet relative Mach target: Mrel2 less than about 1.2

Inputs and CO2 properties

Provide these as fields or globals in FileMaker. If you do not compute properties in FileMaker, precompute with REFPROP or CoolProp and store them.

T3_K turbine inlet total temperature, K

P3_Pa turbine inlet total pressure, Pa

PR_t turbine pressure ratio P3 over P4 (total to static target)

N_rpm shaft speed, rpm

mdot_kg_s mass flow, kg per s

cp3_J_kgK cp at inlet, J per kg K

k3 ratio of specific heats at inlet

Z3 compressibility at inlet

rho2_kg_m3 CO2 density at rotor inlet plane, kg per m3

rho3_kg_m3 CO2 density at rotor exit plane, kg per m3 (first pass use same as inlet, then refine)

Design choices as fields or constants:

eta_t, psi_t, phi_t, rr

Fixed constant:

R_CO2_J_kgK = 188.9

Sizing logic in words

1. Isentropic head from pressure ratio

Hs = cp3 T3 (1 − PR_t^((k3 − 1) over k3))

Actual specific work to shaft: delta_h0 = eta_t Hs

2. Tip speed and diameter

psi_t = delta_h0 over U2^2 so U2 = sqrt(delta_h0 over psi_t)

D2 = 60 U2 over (pi N_rpm)

3. Inlet meridional velocity and blade height

Vm2 = phi_t U2

mdot = rho2 (2 pi R2 b2) Vm2, with R2 = D2 over 2

Solve for b2

4. Inlet velocity triangles and metal angles

With zero exit swirl, Euler gives Vw2 = delta_h0 over U2

Absolute inlet angle alpha2 = atan(Vm2 over Vw2)

Relative tangential Wt2 = U2 − Vw2

Relative inlet angle beta2 = atan(Vm2 over Wt2)

5. Exit diameter from radius ratio

D3 = rr D2

Use continuity with rho3 to check b3 if you want to size the diffuser and volute later.

6. Mach sanity check

a2 = sqrt(k3 R_CO2 T3 over Z3)

W2 = sqrt( (U2 − Vw2)^2 + Vm2^2 )

Mrel2 = W2 over a2

FileMaker Pro calculation

Paste this into a FileMaker calculation field. Replace field names with your own as needed. It returns a plain text report.

```

Let(

[

R_CO2 = 188.9;

pi = 3.14159265358979;

// Inputs

T3 = T3_K;

P3 = P3_Pa;

PRt = PR_t;

N = N_rpm;

mdot = mdot_kg_s;

cp = cp3_J_kgK;

k = k3;

Z = Z3;

rho_in = rho2_kg_m3;

rho_out = rho3_kg_m3;

// Design coefficients

eta = eta_t;

psi = psi_t;

phi = phi_t;

rr = rr;

// 1) Isentropic head and actual stage work (positive to shaft)

Hs = cp T3 ( 1 • Power( PRt, (k • 1) / k ) );

dh0 = eta Hs; // actual specific work to shaft, J/kg

// 2) Tip speed and tip diameter

U2 = Sqrt( dh0 / psi ); // m/s

D2 = 60 U2 / ( pi N ); // m

// 3) Inlet meridional velocity and blade height

Vm2 = phi U2; // m/s

b2 = mdot / ( rho_in 2 pi ( D2 / 2 ) Vm2 ); // m

// 4) Inlet velocity triangles and metal angles

// Zero exit swirl target => Vw3 approx 0, so Euler: dh0 = U2 Vw2

Vw2 = dh0 / U2; // m/s absolute tangential at inlet

Wt2 = U2 • Vw2; // m/s relative tangential

alpha2_deg = 57.295779513 Atan( Vm2 / Vw2 ); // absolute metal angle at inlet

beta2_deg = 57.295779513 Atan( Vm2 / Wt2 ); // relative metal angle at inlet

// 5) Exit diameter from radius ratio

D3 = rr D2; // m

// Optional: exit blade height from continuity assuming Vm3 approx Vm2

Vm3 = Vm2;

b3 = mdot / ( rho_out 2 pi ( D3 / 2 ) Vm3 ); // m

// 6) Mach check at inlet (relative)

a2 = Sqrt( k R_CO2 T3 / Z ); // m/s

W2 = Sqrt( Wt2^2 + Vm2^2 ); // m/s

Mrel2 = W2 / a2

];

Results & Char(10) &

Tip speed U2 m_s = & Round(U2;3) & Char(10) &

Rotor tip diameter D2 m = & Round(D2;4) & Char(10) &

Rotor inlet blade height b2 m = & Round(b2;4) & Char(10) &

Inlet absolute angle alpha2 deg = & Round(alpha2_deg;1) & Char(10) &

Inlet relative angle beta2 deg = & Round(beta2_deg;1) & Char(10) &

Exit diameter D3 m = & Round(D3;4) & Char(10) &

Exit blade height b3 m = & Round(b3;4) & Char(10) &

Relative inlet Mach Mrel2 = & Round(Mrel2;2)

)

```

Practical guidance

Start with psi_t about 1.2 and phi_t about 0.20 for dense CO2. If Mrel2 is high, reduce pressure ratio per stage, increase D2 slightly, or reduce phi_t.

Keep Mrel2 less than about 1.2 at design to avoid strong shocks and loss.

If b2 becomes too small to manufacture, increase phi_t modestly or reduce delta_h0 by lowering pressure ratio per stage and use two stages.

Recompute rho_in and rho_out using a real gas property model at the rotor planes after you have draft geometry, then iterate.

What this solves

Given T3, P3, pressure ratio, RPM, and mass flow, plus CO2 properties, this first-cut method returns:

The tip diameter and blade height that pass the flow at acceptable velocity

The inlet metal angles that set the velocity triangles and incidence

A Mach check for survivable aerodynamics

From there, you can detail the nozzle, rotor metal camber and thickness, diffuser and volute, and iterate with real-gas CFD and meanline tools.

----

Let(

[

R_CO2 = 188.9;

pi = 3.14159265358979;

// Inputs

T3 = T3_K;

P3 = P3_Pa;

PRt = PR_t;

N = N_rpm;

mdot = mdot_kg_s;

cp = cp3_J_kgK;

k = k3;

Z = Z3;

rho_in = rho2_kg_m3;

rho_out = rho3_kg_m3;

// Design coefficients

eta = eta_t;

psi = psi_t;

phi = phi_t;

rr = rr;

// 1) Isentropic head and actual stage work (positive to shaft)

Hs = cp * T3 * ( 1 - Power( PRt, (k - 1) / k ) );

dh0 = eta * Hs; // actual specific work to shaft, J/kg

// 2) Tip speed and tip diameter

U2 = Sqrt( dh0 / psi ); // m/s

D2 = 60 * U2 / ( pi * N ); // m

// 3) Inlet meridional velocity and blade height

Vm2 = phi * U2; // m/s

b2 = mdot / ( rho_in * 2 * pi * ( D2 / 2 ) * Vm2 ); // m

// 4) Inlet velocity triangles and metal angles

// Zero exit swirl target => Vw3 approx 0, so Euler: dh0 = U2 * Vw2

Vw2 = dh0 / U2; // m/s absolute tangential at inlet

Wt2 = U2 - Vw2; // m/s relative tangential

alpha2_deg = 57.295779513 * Atan( Vm2 / Vw2 ); // absolute metal angle at inlet

beta2_deg = 57.295779513 * Atan( Vm2 / Wt2 ); // relative metal angle at inlet

// 5) Exit diameter from radius ratio

D3 = rr * D2; // m

// Optional: exit blade height from continuity assuming Vm3 approx Vm2

Vm3 = Vm2;

b3 = mdot / ( rho_out * 2 * pi * ( D3 / 2 ) * Vm3 ); // m

// 6) Mach check at inlet (relative)

a2 = Sqrt( k * R_CO2 * T3 / Z ); // m/s

W2 = Sqrt( Wt2^2 + Vm2^2 ); // m/s

Mrel2 = W2 / a2

];

"Results" & Char(10) &

"Tip speed U2 m_s = " & Round(U2;3) & Char(10) &

"Rotor tip diameter D2 m = " & Round(D2;4) & Char(10) &

"Rotor inlet blade height b2 m = " & Round(b2;4) & Char(10) &

"Inlet absolute angle alpha2 deg = " & Round(alpha2_deg;1) & Char(10) &

"Inlet relative angle beta2 deg = " & Round(beta2_deg;1) & Char(10) &

"Exit diameter D3 m = " & Round(D3;4) & Char(10) &

"Exit blade height b3 m = " & Round(b3;4) & Char(10) &

"Relative inlet Mach Mrel2 = " & Round(Mrel2;2)

)

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CONTACT TEL: +1-608-238-6001 (Chicago Time Zone USA) Email: greg@infinityturbine.com | AMP | PDF