TurboExpanderCompressor Model

This document summarizes the mathematical basis of the coupled expander/compressor model, how reference curves are applied (and can be replaced), and provides a usage walkthrough for configuring and running the unit in a process simulation.

Mathematical Basis

The expander and compressor share a shaft speed that is iteratively adjusted until the expander power balances the compressor power plus bearing losses using a Newton-Raphson iteration. The key computational steps are described below.

Expander Calculations

1. Isentropic Enthalpy Drop

The isentropic enthalpy drop across the expander is calculated from an isentropic flash at the target outlet pressure:

\[\Delta h_s = (h_{in} - h_{out,s}) \times 1000 \quad \text{[J/kg]}\]

where $h_{in}$ is the inlet enthalpy and $h_{out,s}$ is the isentropic outlet enthalpy.

2. Velocity Ratio Calculation

The tip speed $U$ and spouting (jet) velocity $C$ are computed as:

\[U = \frac{\pi \cdot D \cdot N}{60}\] \[C = \sqrt{2 \cdot \Delta h_s}\]

The velocity ratio is then:

\[u_c = \frac{U}{C \cdot u_{c,design}}\]

An efficiency correction factor is evaluated from the UC reference curve based on this ratio.

3. Efficiency Corrections

The actual expander isentropic efficiency is calculated by applying correction factors:

\[\eta_s = \eta_{s,design} \cdot f_{UC}(u_c) \cdot f_{Q/N}\left(\frac{Q/N}{(Q/N)_{design}}\right)\]

where:

$f_{UC}$ is the velocity ratio correction factor from the UC curve
$f_{Q/N}$ is the optional flow coefficient correction from the Q/N efficiency spline

4. Expander Shaft Power

The expander shaft power is:

\[W_{expander} = \dot{m} \cdot \Delta h_s \cdot \eta_s\]

where $\dot{m}$ is the mass flow rate.

Compressor Calculations

1. Head and Efficiency Corrections

The compressor polytropic head and efficiency are corrected for off-design operation:

\[H_p = H_{p,design} \cdot f_{head}\left(\frac{Q/N}{(Q/N)_{design}}\right) \cdot \left(\frac{N}{N_{design}}\right)^2\] \[\eta_p = \eta_{p,design} \cdot f_{\eta}\left(\frac{Q/N}{(Q/N)_{design}}\right)\]

where:

$f_{head}$ is the head correction factor from the Q/N head spline
$f_{\eta}$ is the efficiency correction factor from the Q/N efficiency spline
The $(N/N_{design})^2$ term applies the fan law scaling for head

2. Compressor Shaft Power

The compressor shaft power is:

\[W_{comp} = \frac{\dot{m} \cdot H_p}{\eta_p}\]

Power Balance and Speed Iteration

The Newton-Raphson iteration solves for the shaft speed $N$ that satisfies:

\[f(N) = W_{expander} - \left(W_{comp} + W_{bearing}\right) = 0\]

where the bearing losses are modeled as a quadratic function of speed:

\[W_{bearing} = k \cdot N^2\]

The iteration continues until the power mismatch is negligible or iteration limits are reached. The final speed is applied to compute outlet stream properties.

Reference Curves

Three types of reference curves tune performance away from the design point:

1. UC/Efficiency Curve

A constrained parabola through the peak at $(u_c = 1, \eta = 1)$:

\[f_{UC}(u_c) = a \cdot u_c^2 + b \cdot u_c + c\]

The curve can be replaced with setUCcurve(ucValues, efficiencyValues) if alternate test data are available.

2. Q/N Efficiency Curve

A monotonic cubic Hermite spline built from paired Q/N and efficiency arrays via setQNEfficiencycurve:

\[f_{\eta}\left(\frac{Q/N}{(Q/N)_{design}}\right) = \text{spline interpolation}\]

Values are extrapolated linearly outside the provided range, allowing off-map operation while preserving trend continuity.

3. Q/N Head Curve

A similar cubic Hermite spline created with setQNHeadcurve that scales polytropic head at off-design flows:

\[f_{head}\left(\frac{Q/N}{(Q/N)_{design}}\right) = \text{spline interpolation}\]

Like the efficiency spline, it preserves monotonicity and extrapolates linearly beyond the data range.

Note: Curve coefficients are stored on the equipment instance and can be replaced at runtime to test alternative reference maps or updated dynamically from external performance monitoring tools.

Using the Model

Step 1: Construct the Unit and Streams

Clone feeds for the expander and compressor outputs when instantiating the equipment.

Step 2: Set Design Parameters

Provide impeller diameter, design speed, efficiencies, design Q/N, and optional expander design Q/N if expander flow corrections are needed. The defaults mirror the embedded design values but can be overridden through the available setters.

Step 3: Load Reference Curves (Optional)

If site-specific head or efficiency curves exist, call setUCcurve, setQNEfficiencycurve, and setQNHeadcurve with measured points before running the unit.

Step 4: Run the Model

Call run(UUID id) (or the no-argument overload) to iterate speed matching and populate result fields and outlet streams. Retrieve shaft powers with getPowerExpander(unit) and getPowerCompressor(unit) or inspect efficiencies, head, and Q/N ratios through the getters.

Example Code

A realistic setup that mirrors common plant data collection and map-updating workflows:

TurboExpanderCompressor turboExpanderComp = new TurboExpanderCompressor(
    "TurboExpanderCompressor", jt_tex_splitter.getSplitStream(0));
turboExpanderComp.setUCcurve(
    new double[] {0.9964751359624449, 0.7590835113213541, 0.984295619176559, 0.8827799803397821,
        0.9552460269880922, 1.0},
    new double[] {0.984090909090909, 0.796590909090909, 0.9931818181818183, 0.9363636363636364,
        0.9943181818181818, 1.0});
turboExpanderComp.setQNEfficiencycurve(
    new double[] {0.5, 0.7, 0.85, 1.0, 1.2, 1.4, 1.6},
    new double[] {0.88, 0.91, 0.95, 1.0, 0.97, 0.85, 0.6});
turboExpanderComp.setQNHeadcurve(
    new double[] {0.5, 0.8, 1.0, 1.2, 1.4, 1.6},
    new double[] {1.1, 1.05, 1.0, 0.9, 0.7, 0.4});
turboExpanderComp.setImpellerDiameter(0.424);
turboExpanderComp.setDesignSpeed(6850.0);
turboExpanderComp.setExpanderDesignIsentropicEfficiency(0.88);
turboExpanderComp.setDesignUC(0.7);
turboExpanderComp.setDesignQn(0.03328);
turboExpanderComp.setExpanderOutPressure(inp.expander_out_pressure);
turboExpanderComp.setCompressorDesignPolytropicEfficiency(0.81);
turboExpanderComp.setCompressorDesignPolytropicHead(20.47);
turboExpanderComp.setMaximumIGVArea(1.637e4);

// Run the coupled model and retrieve power with unit conversion
turboExpanderComp.run();
double expanderPowerMW = turboExpanderComp.getPowerExpander("MW");
double compressorPowerMW = turboExpanderComp.getPowerCompressor("MW");

Parameter Reference

Velocity Ratio (UC) Curve

Parameter	Description
`setUCcurve(ucValues, effValues)`	Normalizes the velocity ratio $u_c = \frac{U}{C \cdot u_{c,design}}$ to an efficiency multiplier via a constrained parabola fitted to the supplied points

Q/N Curves

Parameter	Description
`setQNEfficiencycurve(qnValues, effValues)`	Cubic Hermite spline that scales expander and compressor efficiencies against flow coefficient deviations $Q/N$
`setQNHeadcurve(qnValues, headValues)`	Spline used to scale the compressor polytropic head for off-design flows before applying the $(N/N_{design})^2$ speed law

Geometry and Design Point

Parameter	Description
`setImpellerDiameter(D)`	Impeller diameter [m] — sets the peripheral velocity $U$ at design, anchoring UC corrections
`setDesignSpeed(N)`	Design rotational speed [rpm] — anchor for Newton iteration speed matching
`setExpanderDesignIsentropicEfficiency(η)`	Base isentropic efficiency multiplied by curve correction factors
`setCompressorDesignPolytropicEfficiency(η)`	Base polytropic efficiency for the compressor
`setCompressorDesignPolytropicHead(Hp)`	Design polytropic head [kJ/kg]
`setDesignUC(uc)`	Design velocity ratio for the expander
`setDesignQn(qn)`	Reference flow coefficient $(Q/N)_{design}$ for the compressor
`setDesignExpanderQn(qn)`	Reference flow coefficient for the expander (optional)

Operating Conditions

Parameter	Description
`setExpanderOutPressure(P)`	Target outlet pressure for the isentropic flash that produces $\Delta h_s$

IGV Geometry

Parameter	Description
`setMaximumIGVArea(A)`	Maximum inlet guide vane throat area [mm²]
`setIgvAreaIncreaseFactor(f)`	Optional factor to expand available IGV area

Tip: The same update paths can be invoked during runtime if monitoring identifies drift in the reference maps; supplying new curve points and re-running will propagate the new performance predictions.

IGV Handling

The Inlet Guide Vane (IGV) opening is computed from the last stage enthalpy drop, mass flow, and volumetric flow each time run() completes.

IGV Calculation Method

The helper evaluateIGV performs the following:

Infer density from the fluid properties
Estimate nozzle velocity from half the stage enthalpy drop:

\[v_{nozzle} = \sqrt{\Delta h_{stage}}\]

Derive required area to pass the flow:

\[A_{required} = \frac{\dot{V}}{v_{nozzle}}\]

Calculate IGV opening as the area ratio:

\[\text{IGV}_{opening} = \min\left(\frac{A_{required}}{A_{throat}}, 1.0\right)\]

IGV Area Expansion

If the required area exceeds the installed IGV area, an optional enlargement factor (setIgvAreaIncreaseFactor) increases the available area:

\[A_{available} = A_{max} \cdot f_{increase}\]

IGV Output Methods

Method	Description
`calcIGVOpening()`	Returns the calculated IGV opening fraction (0–1)
`calcIGVOpenArea()`	Returns the actual open area [mm²]
`getCurrentIGVArea()`	Returns the current IGV throat area [mm²]