Every balance in a modeled steam network closes to round-off, and that is not the same as accuracy
Verification of the steam properties, survey data, boiler and emissions accounting and network balances underlying a modeled steam and cogeneration system, each against an independent reference.
Keywords: steam networks; IAPWS-IF97; model verification; mass and energy balances; MECS; desuperheating
Abstract
Optimization studies of industrial steam systems rest on property calculations, input data, equipment accounting and network balances that are rarely verified independently. We verified each of these layers of a modeled multi-pressure steam and cogeneration network against references the model did not produce: the published IAPWS-IF97 verification tables, the raw workbook of a federal energy survey, independent hand calculations and a separately assembled stream table. The 12 principal continuous checks fall at least 4 orders of magnitude inside their thresholds, most near the round-off limit of double-precision arithmetic; the comparison at steam-table states that the release prints rounded falls a factor of 4.5 inside its threshold, limited by the table's rounding. The worst error at exactly printed steam table states is 3.9 × 10-9, network balances close to 1.4 × 10-15, and 0 of 52 second-law checks fail. Such agreement establishes correct implementation, not physical realism: the low-pressure header, which has no desuperheater, carries about 196 degrees of superheat that a real site would remove, while every balance still closes exactly.
1. Introduction
A dispatch optimization is only as reliable as the models beneath it: steam properties, the data used for context, the accounting of fuel and emissions, and the mass and energy balances of the network. Errors in these layers propagate silently into optimal set-points and savings. Verification against references independent of the model is the standard remedy [1].
This study reports such verification for a modeled mid-size chemical-site steam network and distinguishes what it establishes, correct implementation, from what it cannot, agreement with a physical plant.
2. Methods
2.1 Acceptance criteria
Each check was specified with a numeric threshold before the component it tests was built. The 12 continuous checks are compared with their thresholds on a common logarithmic scale.
2.2 Steam properties
The property layer implements IAPWS-IF97 [2, 3] and was compared with the 18 verification states printed in the release, across 6 tables covering every region. Reference equations were transcribed from the release independently of the production code. A fixture of 57 states spanning the operating envelope and a sweep of 1,666 states across the site's pressures and temperatures were also evaluated.
2.3 Survey data
National context for chemical-sector boiler fuel is taken from MECS 2022 Table 5.2 [4]. An independent reader parsing the workbook's raw XML, without the production parser, re-derived the chemical-industry block.
2.4 Boilers and emissions
Boiler fuel follows a published part-load efficiency form, evaluated on a grid of 151 loads; the boiler energy balance, fuel use and Scope 1 CO2 were compared with independent hand calculations. Scope 2 uses the eGRID2023 CAMX total-output factor [5]. Every thermal quantity carries a label stating whether it is a fuel heating value or a steam enthalpy.
2.5 Network balances
The network comprises two turbines, three letdown valves, three headers, condensate return, makeup, blowdown and a deaerator. It was solved at the design point and three off-design points chosen to exercise the balances (Table 1). Balances were re-summed from a separately assembled stream table and from an independent closed-form solution of the deaerator energy balance, not read from the solver.
3. Results
3.1 Overview
Each of the 12 principal continuous checks falls at least 4 orders of magnitude inside its threshold (Figure 1). The steam-table comparison is plotted at the exactly printed states; at rounded states it falls a factor of 4.5 inside its threshold, as described in Section 3.2. Most measurements lie within a factor of ten of double-precision machine epsilon, where an exact identity computed in floating point is expected to land.
3.2 Steam properties
At states the release prints exactly, the worst relative error is 3.9 × 10-9. At states the release rounds, the worst is 2.2 × 10-5, which reflects the rounding of the printed table rather than an implementation error; the acceptance threshold of 1 × 10-4 accommodates this, and the exact-state comparison is reported separately. The envelope fixture agrees to 6.1 × 10-15, and the sweep found 0 failures, with a worst case of 1.4 × 10-14.
3.3 Survey data
The ingestion reproduces all 144 published cells of the chemical-industry block, and the independent reader
re-derives the boiler-fuel subtotal of 1,477 trillion Btu per year with zero difference. The subtotal is approximate by
construction: residual fuel oil is published only as a value below the reporting unit, carrying the marker
*, and another category is marked not applicable. The figure is used only as
context and never to scale model results.
3.4 Boilers and emissions
Efficiency rises monotonically from 0.83125 of its nominal value at minimum stable load to the nominal value, with 0 violations; requests below minimum stable load are refused, with 0 accepted. The boiler energy balance agrees with the hand calculation across 32 points to 2.3 × 10-16, fuel at three loads to 1.8 × 10-16, and Scope 1 CO2 to 1.5 × 10-16. The CAMX factor of 195.04 kg CO2e per MWh, grossed up for 4.1 percent grid loss, gives a loss-inclusive 203.38 kg CO2e per delivered MWh; the loss share is an end user's Scope 3 under the GHG Protocol [6], so the model reports purchased-electricity emissions as grid emissions, Scope 2 plus grid losses. All 15 thermal quantities carry a heating-value or enthalpy label (0 unlabelled); mixing higher and lower heating values for natural gas would introduce an error of about ten percent without any arithmetic fault.
3.5 Network balances
Table 1. Network operating points.
| Operating point | Boiler steam, t/h | Makeup, t/h | Generation, MW | LP temperature, deg C |
|---|---|---|---|---|
| Design point | 156.1 | 53.7 | 13.30 | 343.6 |
| Low load | 102.5 | 32.8 | 10.01 | 346.2 |
| High load | 182.9 | 64.2 | 14.03 | 352.9 |
| Low return | 159.7 | 85.2 | 13.30 | 346.3 |
Across all 4 operating points, the worst turbine isentropic-drop error is 2.0 × 10-15 and the worst letdown enthalpy error 4.0 × 10-16; header mass and energy balances close to 6.6 × 10-16 and 7.2 × 10-16, the site mass balance to 2.2 × 10-16, and the deaerator and return balances to 1.4 × 10-15. Of 52 second-law checks, covering the isentropic work bound of each turbine section and the sign of entropy generation in every valve, mixer and the deaerator, 0 failed. One discrepancy remains unexplained: an independent integration of volume against pressure along the condensing tail's isentrope reproduces the enthalpy drop to a relative 1.2 × 10-6, a difference of 0.0011 kJ/kg against an estimated quadrature error of 5.9 × 10-6 kJ/kg. It is well within the threshold, but its cause was not isolated.
3.6 Code and source checks
A scan of the model's source code found 0 numeric literals outside the single inputs module across 31 files, and all 8 cited input sources were checked against their primary publications.
4. Discussion
Agreement to sixteen digits shows that the model computes what it was specified to compute. It does not show that the specification describes a plant. The low-pressure header illustrates the difference. Letdown valves conserve enthalpy, so steam let down from the medium-pressure header arrives hot: 343.6 degrees C at the design point, against a saturation temperature of 147.9 degrees C at 0.45 MPa, about 196 degrees of superheat. At the dispatch cost optimum it is still 318.4 degrees C, 170 degrees above saturation (Figure 2). A real site would install a desuperheater, which sprays water into the steam and changes the deaerator's water balance. The model has none, and every balance still closes exactly.
Verification of this kind is necessary but not sufficient. Realism checks, such as bounds on header superheat, would need to be specified as acceptance criteria in their own right.
5. Limitations
- The site is a plausible mid-size chemical site consistent with published steam-system practice, not measured data from any facility.
- Property consistency is not empirical accuracy: IF97 differs from the more accurate IAPWS-95 formulation [7] by a relative 8.5 × 10-5 on the high-pressure turbine section's enthalpy drop.
- Feedwater and condensate pump work, the drum-to-header pressure drop and deaerator venting are omitted.
- Part-load forms are published correlations, not vendor curves.
- The survey subtotal is context only; no model result is scaled to it.
6. Conclusions
The property layer, data ingestion, boiler accounting and network balances of the modeled steam network are correctly implemented to the limit of double precision. The model nonetheless retains about 196 degrees of low-pressure superheat that a real site would remove, showing that exact verification and physical realism must be assessed separately.
References
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