Usage Manual

Set up and interpret each of ThermalOne’s 54 calculations.

State & Properties
Refrigeration
- M33 · Liquefaction
- M30 · Cascade and Multistage Refrigeration
Reacting Systems
Thermodynamic Optimisation
Second-Law Design
- M51 · Entropy Generation Minimisation
Moist Air
First Law
Control-Volume Devices
Second Law
Power Cycles
Second Law & Exergy
- M21 · Vapour Plant Exergy Audit
Refrigeration & Heat Pumps

Start a calculation

Choose a calculation from the sidebar. On a narrow window, use the navigation control to return to the list. Select a process or model before filling its active fields: a disabled field is derived or is not used by that choice.

Results update from the current inputs. Read the fluid, phase and model together with the number. A warning can describe an approximation; an invalid or unresolved calculation instead needs corrected inputs before its result can be used. Defaults are a starting case, not a recommendation for a real machine.

Units and numbers

Settings offers SI and US customary units and three to six significant figures. Follow the unit beside each field; changing units converts the number as well as its label. Absolute temperature and a temperature difference are different quantities. Do not enter a Celsius value in a field labelled K.

The significant-figure preference controls reported precision, not solver precision. Slider endpoints are convenient exploration intervals; the property's actual validity domain comes from the selected model and the explanation beside an invalid input.

Readings and graph export

Export Readings saves the published readings as CSV, including each quantity's unit. Copy Readings supplies tab-separated text for pasting into a spreadsheet. These contain the displayed, rounded values rather than every internal solver value. The Settings option Include the inputs in exported readings controls whether the published input rows are included.

A graph's Export graph menu offers PNG image, PDF document, SVG vector and Copy graph. PNG and the copied graph are raster images; PDF and SVG retain vector paths and text. Export uses the plotted model, axes, units, curves and annotations. A graph is not a replacement for the accompanying assumptions or case record.

On macOS choose a destination in the save panel. On iPadOS use the system share sheet, including Save to Files where available. The app exports calculations; it does not import CSV cases or open document files.

Help and settings

Help contains this Usage Manual and the Theory Manual. Choose a book, use its contents or search for the calculation name. Both books are bundled for offline reading. Settings remembers which book Help opens first.

This book explains tasks and active choices. The Theory Manual explains equations, assumptions, numerical methods and model limits. The symbol tables below are reference fields from the specification; a selected task may replace a generic field with a fluid-state pair, a derived value or a model-specific control.

On macOS, use File > Export Readings and Edit > Copy Readings. On iPadOS, the More menu offers export, copy, both manuals and Settings.

State & Properties

M01 · State Explorer

Set up the task

Choose the pair of water properties you know: p–T, p–x, T–x, p–h, p–s, T–v, p–v, h–s or u–v.
Fill the two active fields in the displayed units. Other fields explain why they are derived or inapplicable.
For p–v, use the temperature-root selection when water's density anomaly admits alternatives. Read the phase before using the other properties.

Read the result

The state table gives pressure, temperature, density, specific volume, internal energy, enthalpy and entropy. Quality appears where it is meaningful. The dome diagram places the selected state in its phase region.

Before using the answer

Pressure and temperature on saturation do not determine quality. Choose an independent pair. This entry is water-only, and the IF97 phase and critical-region guards remain applicable.

Input symbols and units

Symbol	Meaning	Unit	Valid range
`substance`	Which working fluid the screen is operating on.	—	Water in 1.0; refrigerants arrive with PE-12.
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`273.15 <= T <= 1073.15` Regions 1–4; Region 5 remains outside the current water solver.
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`0.000611657 <= p <= 100.0`
`v`	Specific volume: the space one kilogram of the substance occupies. The reciprocal of density, and the property that makes a gas different from a liquid.	m^3/kg	`v > 0`
`u`	Specific internal energy: the energy stored in one kilogram of a substance by the motion and arrangement of its molecules, with no reference to where the substance is or how fast it is moving.	kJ/kg	—
`h`	Specific enthalpy: internal energy plus pressure times specific volume. It exists because that combination appears every time a substance flows across a boundary, so it is a bookkeeping convenience that behaves like a property.	kJ/kg	—
`s`	Specific entropy: the property that counts how much of a system's energy is no longer available to do work. It increases in every real process and stays put only in an ideal one.	kJ/(kg K)	—
`x`	Quality: the fraction of a liquid-vapour mixture that is vapour, by mass. Meaningful only between the two saturation lines.	-	`0 <= x <= 1`

Output reference

Symbol	Meaning	Unit	Relation
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	from the chosen pair
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	—
`v`	Specific volume: the space one kilogram of the substance occupies. The reciprocal of density, and the property that makes a gas different from a liquid.	m^3/kg	—
`rho`	Density: mass per unit volume, the reciprocal of specific volume.	kg/m^3	ρ = 1/v
`u`	Specific internal energy: the energy stored in one kilogram of a substance by the motion and arrangement of its molecules, with no reference to where the substance is or how fast it is moving.	kJ/kg	—
`h`	Specific enthalpy: internal energy plus pressure times specific volume. It exists because that combination appears every time a substance flows across a boundary, so it is a bookkeeping convenience that behaves like a property.	kJ/kg	h = u + pv
`s`	Specific entropy: the property that counts how much of a system's energy is no longer available to do work. It increases in every real process and stays put only in an ideal one.	kJ/(kg K)	—
`x`	Quality: the fraction of a liquid-vapour mixture that is vapour, by mass. Meaningful only between the two saturation lines.	-	x=(v-v_f)/v_fg Reported only when strictly between 0 and 1.
`phase`	Which region of the phase diagram the state falls in: compressed liquid, saturated mixture, saturated vapour, superheated vapour, or supercritical.	—	One of the five region labels.

Symbol	Meaning	Unit	Valid range
`substance`	Which working fluid the screen is operating on.	—	Air and the species set carried by the polynomial data.
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`within the selected species source temperature range`
`basis`	Whether specific quantities are reported per kilogram or per kilomole. It changes every number on the screen at once.	—	Mass or molar.
`T_cp_evaluation`	Temperature at which a constant specific heat is evaluated.	K	Active only for the constant-cp model; within the selected species source range.
`gas_cp_model`	Specific-heat law used consistently in the ideal-gas state and inverse calculations.	—	`one declared caloric model` Variable cp, constant cp at an evaluation temperature, or cold-air standard (air only).
`T₁`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`within selected source range` First temperature for entropy change and cp-plot interval.
`p₁`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`p > 0` First pressure for entropy change.
`T₂`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`within selected source range` Second temperature for entropy change and cp-plot interval.
`p₂`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`p > 0` Second pressure for entropy change.

Symbol	Meaning	Unit	Relation
`R`	Specific gas constant: the universal gas constant divided by the molar mass of this particular gas.	kJ/(kg K)	R=R̄/M Core value is mass specific; molar basis uses ui.gas_properties_view.value and changes kg to kmol in the unit.
`M`	Molar mass: the mass of one kilomole of the substance. For a mixture, the mole-fraction-weighted average.	kg/kmol	—
`cp`	Specific heat at constant pressure: how much energy raises one kilogram by one degree while the pressure is held fixed.	kJ/(kg K)	c̄ₚ(T)=R̄(a₁T⁻²+a₂T⁻¹+a₃+a₄T+a₅T²+a₆T³+a₇T⁴) Core value is mass specific; molar basis uses ui.gas_properties_view.value and changes kg to kmol in the unit.
`cv`	Specific heat at constant volume: the same question with the volume held fixed instead, which is a smaller number because none of the energy goes into pushing the surroundings back.	kJ/(kg K)	cᵥ=cₚ-R Core value is mass specific; molar basis uses ui.gas_properties_view.value and changes kg to kmol in the unit.
`k`	Specific heat ratio: cp divided by cv. It governs how steeply temperature changes when a gas is compressed without heat transfer.	-	k=cₚ/cᵥ
`h`	Specific enthalpy: internal energy plus pressure times specific volume. It exists because that combination appears every time a substance flows across a boundary, so it is a bookkeeping convenience that behaves like a property.	kJ/kg	h(T)=∫ cₚ dT Core value is mass specific; molar basis uses ui.gas_properties_view.value and changes kg to kmol in the unit.
`u`	Specific internal energy: the energy stored in one kilogram of a substance by the motion and arrangement of its molecules, with no reference to where the substance is or how fast it is moving.	kJ/kg	u=h-RT Core value is mass specific; molar basis uses ui.gas_properties_view.value and changes kg to kmol in the unit.
`s0`	The standard-state entropy function: the part of an ideal gas's entropy that depends only on temperature, with the pressure dependence stripped out and handled separately.	kJ/(kg K)	s°(T)=∫ cₚ/T dT Core value is mass specific; molar basis uses ui.gas_properties_view.value and changes kg to kmol in the unit.
`pr`	Relative pressure: a tabulated function of temperature alone whose ratio between two states equals the pressure ratio of an isentropic process between them. A device for avoiding an integral, not a physical pressure.	-	p_r(T)= exp ((s°(T)-s°(T_ref))/R) Normalised to 1 at the reference temperature. Defined only up to a multiplicative constant; every use is a ratio.
`vr`	Relative volume: the companion to relative pressure, whose ratio gives the volume ratio of an isentropic process.	-	v_r(T)=(T/T_ref)/(p_r(T)) Same normalisation, same reason.
`specific_entropy_change`	Entropy change per unit mass, or per unit amount when molar basis is selected.	kJ/(kg K)	Δ s=s⁰(T₂)-s⁰(T₁)-R ln (p₂/p₁) Specific entropy change, not total entropy. Molar basis multiplies the mass-specific value by molar mass and displays kJ/(kmol K). Core value is mass specific; molar basis uses ui.gas_properties_view.value and changes kg to kmol in the unit.

Symbol	Meaning	Unit	Valid range
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	—
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	—
`p_plot_min`	Lower pressure of the displayed isotherm interval.	MPa	`0.000611213 <= p_plot_min < p_plot_max <= 100` Lower pressure of the displayed isotherm interval.
`p_plot_max`	Upper pressure of the displayed isotherm interval.	MPa	`0.000611213 <= p_plot_min < p_plot_max <= 100` Upper pressure of the displayed isotherm interval.
`root`	Candidate outer branch selected independently for each EOS.	-	`largest root: vapour, smallest root: liquid` Candidate outer branch selected independently for each EOS.
`plot_quantity`	Quantity on the common isotherm chart.	-	`compressibility, relative volume difference` Quantity on the common isotherm chart.

Symbol	Meaning	Unit	Relation
`Z`	Compressibility factor: how far the substance departs from ideal-gas behaviour, expressed as the ratio of its actual specific volume to the ideal-gas value at the same temperature and pressure.	-	Z=pv/RT
`v`	Specific volume: the space one kilogram of the substance occupies. The reciprocal of density, and the property that makes a gas different from a liquid.	m^3/kmol	Molar basis, matching the universal gas constant used by the cubic relations.
`TR`	Reduced temperature: temperature as a fraction of the critical temperature.	-	T_R=T/T_c
`pR`	Reduced pressure: pressure as a fraction of the critical pressure.	-	p_R=p/p_c
`vR_pseudo`	Pseudo-reduced specific volume: a dimensionless volume built from the critical constants, used so a volume can be entered on a chart drawn in reduced coordinates.	-	v_R'=v p_c/(RT_c)
`error_vs_reference`	How far the chosen model sits from the reference formulation at this state, so the model's accuracy is a number rather than a claim.	-	Relative molar-volume difference from IF97: table values are fractions; the relative-difference chart uses percent.
`phase`	Which region of the phase diagram the state falls in: compressed liquid, saturated mixture, saturated vapour, superheated vapour, or supercritical.	-	IF97 reference phase.
`root_count`	Per-model root counts or Lee–Kesler fitted-fluid crossing count; counts do not establish phase coexistence.	-	Per-model root counts or Lee–Kesler fitted-fluid crossing count; counts do not establish phase coexistence.
`branch_status`	Resolved per-model candidate branch and reasons for unavailable comparisons.	-	Resolved per-model candidate branch and reasons for unavailable comparisons.

Symbol	Meaning	Unit	Valid range
`substance`	Which working fluid the screen is operating on.	—	Selected species supplies its critical constants and acentric factor; these are not editable inputs.
`T₁`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	—
`p₁`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	—
`T₂`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	—
`p₂`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	—
`model`	Which equation of state the screen is evaluating.	—	—

Symbol	Meaning	Unit	Relation
`dh_departure`	How much the real enthalpy differs from the ideal-gas enthalpy at the same temperature.	kJ/kg	—
`ds_departure`	How much the real entropy differs from the ideal-gas entropy at the same temperature and pressure.	kJ/(kg K)	—
`DeltaH`	Change in total enthalpy between the two states.	kJ/kg	—
`DeltaS`	Change in total entropy between the two states.	kJ/(kg K)	—
`phi_fug`	Fugacity coefficient: the factor by which a real gas's escaping tendency differs from its pressure. One for an ideal gas.	-	ln φ=∫₀^p(Z-1)/p dp

Symbol	Meaning	Unit	Relation
`beta_v`	Volume expansivity: the fractional change in volume per degree of temperature rise at constant pressure.	1/K	β=(1/v)(∂ v/(∂ T))ₚ
`kappa_T`	Isothermal compressibility: the fractional reduction in volume per unit pressure rise at constant temperature.	1/MPa	κ=-(1/v)(∂ v/(∂ p))_T
`cp_minus_cv`	The gap between the two specific heats, which for a real substance depends on how much it expands when heated and how much it resists being squeezed.	kJ/(kg K)	cₚ-cᵥ=vTβ²/κ
`mu_JT`	Joule-Thomson coefficient: how much the temperature changes per unit pressure drop in a throttling process. Its sign decides whether throttling cools or warms, and refrigeration only works where it is positive.	K/MPa	Kernel returns K/kPa; presentation multiplies by 1000.
`dpdT_sat`	The slope of the saturation line on the pressure-temperature plane.	MPa/K	Kernel Clapeyron slope with kJ/kg is kPa/K; presentation divides by 1000.
`hfg`	Latent enthalpy: the energy one kilogram absorbs turning from saturated liquid into saturated vapour at fixed temperature.	kJ/kg	h_fg=T v_fg(dp/(dT))_sat
`c_sound`	Speed of sound in the substance at this state.	m/s	—

Symbol	Meaning	Unit	Valid range
`composition₁`	The make-up of the mixture, as fractions of its components.	-	First stream or property-state composition; proportional nonnegative amounts are normalised.
`composition₂`	The make-up of the mixture, as fractions of its components.	-	Second stream composition in mixing mode.
`basis`	Whether specific quantities are reported per kilogram or per kilomole. It changes every number on the screen at once.	-	`mole fractions, mass fractions` Controls composition fractions only; both entered stream flows remain molar.
`T₁`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	First stream or property-state temperature.
`p₁`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	First stream or property-state pressure.
`ndot₁`	Molar flow: amount of substance passing a section per unit time.	kmol/s	`positive finite` First inlet molar flow; active in mixing mode.
`T₂`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	Second inlet temperature.
`p₂`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	Second inlet pressure.
`ndot₂`	Molar flow: amount of substance passing a section per unit time.	kmol/s	`positive finite` Second inlet molar flow; active in mixing mode.
`p₃`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	Outlet mixing pressure; momentum is not solved.

Symbol	Meaning	Unit	Relation
`M`	Molar mass: the mass of one kilomole of the substance. For a mixture, the mole-fraction-weighted average.	kg/kmol	M=Σ yᵢMᵢ
`R`	Specific gas constant: the universal gas constant divided by the molar mass of this particular gas.	kJ/(kg K)	—
`cp`	Specific heat at constant pressure: how much energy raises one kilogram by one degree while the pressure is held fixed.	kJ/(kg K)	—
`cv`	Specific heat at constant volume: the same question with the volume held fixed instead, which is a smaller number because none of the energy goes into pushing the surroundings back.	kJ/(kg K)	—
`k`	Specific heat ratio: cp divided by cv. It governs how steeply temperature changes when a gas is compressed without heat transfer.	-	—
`p_partial`	Partial pressure: the pressure one component of a mixture would exert if it alone occupied the whole volume.	MPa	pᵢ=yᵢp Same pressure unit as the input, before the display-unit conversion.
`DeltaS_mixing`	The entropy created purely by letting distinct species share a volume, with no temperature or pressure change involved. Forgetting it is the classic mixture error.	kJ/(kg K)	-(R̄/M)Σᵢ yᵢ ln yᵢ Displayed mass-specific entropy of mixing; the raw kernel identity is per kmol and is divided by apparent molar mass.
`T₃`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	—
`sigmadot`	Rate of entropy production.	kW/K	Total = thermal + composition + pressure contributions, each in kW/K. Pressure changes can contribute negatively; the total must satisfy the second law. The mixer does not solve momentum.
`ndot`	Molar flow: amount of substance passing a section per unit time.	kmol/s	Total outlet molar flow; sum of the inlet molar flows.
`sigma_thermal`	Thermal contribution to the entropy-production rate.	kW/K	Thermal contribution to the entropy-production rate.
`sigma_composition`	Composition contribution to the entropy-production rate.	kW/K	Composition contribution to the entropy-production rate.
`sigma_pressure`	Pressure contribution to entropy rate; it may be negative while the total remains nonnegative.	kW/K	Pressure contribution to entropy rate; it may be negative while the total remains nonnegative.

Symbol	Meaning	Unit	Valid range
`substance`	Which working fluid the screen is operating on.	—	`CO2, N2, CH4` CO2 uses the cubic/NASA model; N2 and CH4 use their cryogenic Helmholtz models.
`T₁`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	Fresh-feed and aftercooler temperature; actual compressor suction follows return mixing.
`p₁`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`0 < p_low < p_high` Common low pressure of feed, separator and return.
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`0 < p_low < p_high` Compressor discharge and high-pressure heat-exchanger pressure; not station 2 pressure.
`eta_reg`	Regenerator effectiveness: how much of the available temperature rise the regenerator actually delivers.	-	`0 < eta_reg <= 1` Linde uses one recuperator; Claude adds a positive expander split and a second recuperator. Warm-end enthalpy approach applies to the entire Linde recuperator but only the warm Claude recuperator. At a fixed approach below one, zero split is not a continuous cross-arrangement limit.
`eta_c`	Isentropic compressor efficiency: the work an ideal compression would have taken divided by the work actually taken.	-	`0 < eta_c <= 1` Adiabatic isentropic efficiency from the actual mixed suction state; not the legacy ideal isothermal work estimate.
`alpha_split`	Fraction of compressed feed sent through the work-producing expander.	-	`0 < alpha_split < 1 for Claude; zero for Linde` Fraction of feed through the expander.
`T_split`	Temperature where the high-pressure flow divides between the expander and cold recuperator.	K	`T_sat_low < T_split < T_supply` High-pressure split-point temperature for Claude.
`eta_t`	Isentropic turbine efficiency: actual work out divided by the work an ideal expansion to the same pressure would have produced.	-	`0 < eta_t <= 1` Dry-expander isentropic efficiency.

Symbol	Meaning	Unit	Valid range
`fuel`	Which fuel is being burned.	—	—
`a_C`	Number of carbon atoms in one molecule of the fuel.	-	—
`b_H`	Number of hydrogen atoms in one molecule of the fuel.	-	—
`c_O`	Number of oxygen atoms already in one molecule of the fuel.	-	—
`d_N`	Number of nitrogen atoms in one molecule of the fuel.	-	—
`e_S`	Number of sulphur atoms in one molecule of the fuel.	-	—
`pct_theo_air`	How much air was supplied as a percentage of exactly enough. A hundred per cent means exactly enough.	-	—
`phi_eq`	Equivalence ratio: how much fuel is present relative to exactly enough. One means exactly enough, above one means rich.	-	`0 < phi_eq <= 1`
`AF`	Air-fuel ratio on a mass basis: kilograms of air supplied per kilogram of fuel.	-	—
`omega`	Humidity ratio: kilograms of water vapour carried per kilogram of dry air. The quantity that stays fixed when moist air is merely heated or cooled.	-	Humidity ratio of the combustion air, when moist air is used.
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`positive product pressure` Pressure for the product-water dew-point calculation; does not change stoichiometric coefficients.

Symbol	Meaning	Unit	Relation
`alpha_stoich`	Moles of oxygen needed to burn one mole of the fuel completely with none left over.	-	—
`AF`	Air-fuel ratio on a mass basis: kilograms of air supplied per kilogram of fuel.	-	AF=AF̄ M_air/M_fuel
`AF_molar`	The same ratio counted in moles instead of kilograms.	-	—
`phi_eq`	Equivalence ratio: how much fuel is present relative to exactly enough. One means exactly enough, above one means rich.	-	—
`pct_theo_air`	How much air was supplied as a percentage of exactly enough. A hundred per cent means exactly enough.	-	—
`products`	The species and amounts the reaction produces.	—	—
`Tdp`	Dew point: the temperature at which the air being cooled would start to condense.	K	Below this the water in the exhaust condenses, which is what limits stack temperature in a real plant.

Symbol	Meaning	Unit	Valid range
`y_CO2`	Mole fraction of carbon dioxide in the dry product gas.	-	`0 <= y_CO2 <= 1` Mole-fraction mode requires sum 1 within 0.002. Relative amounts have a separate explicit input mode. Nitrogen may be the remainder only when the four dry species are exhaustive.
`y_CO`	Mole fraction of carbon monoxide in the dry product gas, which is what incomplete combustion leaves behind.	-	`0 <= y_CO <= 1` Mole-fraction mode requires sum 1 within 0.002. Relative amounts have a separate explicit input mode. Nitrogen may be the remainder only when the four dry species are exhaustive.
`y_O2`	Mole fraction of oxygen in the dry product gas, which is what excess air leaves behind.	-	`0 <= y_O2 <= 1` Mole-fraction mode requires sum 1 within 0.002. Relative amounts have a separate explicit input mode. Nitrogen may be the remainder only when the four dry species are exhaustive.
`y_N2`	Mole fraction of nitrogen in the dry product gas.	-	`0 <= y_N2 <= 1` Mole-fraction mode requires sum 1 within 0.002. Relative amounts have a separate explicit input mode. Nitrogen may be the remainder only when the four dry species are exhaustive.
`fuel`	Which fuel is being burned.	—	—
`AF`	Air-fuel ratio on a mass basis: kilograms of air supplied per kilogram of fuel.	-	`AF > 0` Known mode: molar dry-air amount per mole of fuel, not a mass ratio.
`a_C`	Number of carbon atoms in one molecule of the fuel.	-	`a_C > 0` Known-fuel carbon count.
`b_H`	Number of hydrogen atoms in one molecule of the fuel.	-	`b_H >= 0` Known-fuel hydrogen count.

Symbol	Meaning	Unit	Relation
`AF`	Air-fuel ratio on a mass basis: kilograms of air supplied per kilogram of fuel.	-	—
`phi_eq`	Equivalence ratio: how much fuel is present relative to exactly enough. One means exactly enough, above one means rich.	-	—
`a_C`	Number of carbon atoms in one molecule of the fuel.	-	—
`b_H`	Number of hydrogen atoms in one molecule of the fuel.	-	—
`pct_theo_air`	How much air was supplied as a percentage of exactly enough. A hundred per cent means exactly enough.	-	—

Symbol	Meaning	Unit	Relation
`y_liquid`	Liquid yield: the fraction of the gas processed that comes out as liquid. Small, and seeing how small is the point.	-	Liquid production divided by compressor circulation; the fresh make-up fraction equals this yield at steady state. Single-species molar and mass fractions agree.
`w_specific`	Work required per unit mass of product, which for liquefaction is per kilogram of liquid actually made rather than per kilogram processed.	kJ/kg	Net input work per kg liquid product. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`w_min_liquid`	Reversible minimum work per kg liquid product. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/kg	Reversible minimum work per kg liquid product. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`beta`	Coefficient of performance of a refrigerator: heat removed from the cold space divided by the work it cost. Routinely greater than one, which is why it is not called an efficiency.	-	Minimum work divided by actual net work on the same liquid basis.
`w_feed`	Net input work per kg feed. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/kg	Net input work per kg compressor circulation. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`w_compressor`	Compressor work per kg feed. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/kg	Compressor work per kg compressor circulation. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`w_expander`	Recovered expander work per kg feed; the flow-split factor is already included. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/kg	Recovered expander work per kg compressor circulation; the flow-split factor is already included. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`q_aftercooler`	Aftercooler heat rejection per kg feed. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/kg	Aftercooler heat rejection per kg compressor circulation. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`closure_feed`	Whole-process energy residual per kg feed. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/kg	Whole-process energy residual per kg compressor circulation. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`cold_box_closure`	Cold-box energy residual per kg feed. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/kg	Cold-box energy residual per kg compressor circulation. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`s_gen_feed`	Whole-process entropy generation per kg feed and kelvin. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/(kg K)	Whole-process entropy generation per kg compressor circulation and kelvin. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`s_mix_feed`	Return-mixing entropy generation per kg feed and kelvin. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.	kJ/(kg K)	Return-mixing entropy generation per kg compressor circulation and kelvin. Raw result is molar; ui.liquefaction_view.per_mass divides by the selected molar mass before display.
`T_inversion`	The temperature above which throttling warms the gas instead of cooling it, so simple liquefaction stops working.	K	Independent warm-gas Peng–Robinson/NASA diagnostic; not the N2/CH4 Helmholtz cycle inversion locus.
`flow_fraction`	Each station flow divided by fresh compressor-feed basis.	-	Each station flow divided by compressor circulation (station 2 flow = 1); fresh make-up at station 1 equals liquid yield, not unity.
`sampled_approach`	Minimum computed temperature difference at sampled heat fractions.	K	Minimum of 65 recuperator heat-load samples; temperature difference, not absolute temperature or a continuous certificate.

Symbol	Meaning	Unit	Relation
`beta`	Coefficient of performance of a refrigerator: heat removed from the cold space divided by the work it cost. Routinely greater than one, which is why it is not called an efficiency.	-	—
`Wdot`	Rate of work transfer, that is, power.	kW	—
`pi_opt`	The intermediate pressure that minimises total work, which for two ideal stages is the geometric mean of the end pressures.	MPa	Flash staging optimises intermediate pressure. The separate-loop route instead optimises the exchanger midpoint temperature, with the chosen temperature approach held fixed.
`mdot`	Mass flow rate.	kg/s	Flash core flows are kmol/s and are multiplied by the selected molar mass at the display boundary. Separate-loop outputs are kg/s for each fluid independently.

Symbol	Meaning	Unit	Relation
`Qdot_per_fuel`	Heat transferred per kilomole of fuel burned.	kJ/kmol	—
`hRP`	Enthalpy of combustion: the difference between product and reactant enthalpies at the same temperature and pressure.	kJ/kmol	h̄_RP=Σ_P n_eh̄_e-Σ_R n_ih̄_i Standard gas-phase reference, including water as vapor, independent of the selected product phase.
`HHV`	Higher heating value: energy released per kilogram of fuel with the product water counted as liquid.	kJ/kg	Water in the products taken as liquid.
`LHV`	Lower heating value: the same with the water counted as vapour. Smaller, and quoting the wrong one is a common way to overstate an efficiency.	kJ/kg	Water in the products taken as vapour.
`hf0`	Enthalpy of formation: the energy bookkeeping entry that lets substances of different chemical identity be added together. Zero by convention for a stable element in its reference form.	kJ/kmol	Standard gas-phase reference, including water as vapor, independent of the selected product phase.

Symbol	Meaning	Unit	Relation
`T_ad`	Adiabatic flame temperature: how hot the products get when none of the released energy is allowed to leave.	K	—
`products`	The species and amounts the reaction produces.	—	—
`residual`	How far from zero the energy balance still is at the returned root. Reported rather than assumed, so convergence is visible.	kJ/kmol	The residual at the returned root, reported so convergence is visible rather than asserted.

Symbol	Meaning	Unit	Valid range
`components`	Which species are present.	—	—
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`T > 0 within the selected property model` Binary saturation uses mechanical PR roots, independent of caloric polynomial bounds. Pure water uses IF97; four refrigerants use Helmholtz; CO2 uses PR.
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`p > 0` Pressure is input for binary bubble/dew modes; pure coexistence computes pressure from temperature.
`composition`	The make-up of the mixture, as fractions of its components.	—	`0 <= composition <= 1` First-component liquid mole fraction, or vapor fraction in dew mode. Pure endpoints ignore the absent component.
`nphase`	How many phases coexist.	-	`nphase = 2` Two coexisting bulk phases in pure and binary modes; the membrane constraint is not counted by the free bulk phase rule.
`solvent_fraction`	Mole fraction of solvent in the ideal solution.	—	`0 < solvent_fraction <= 1` Solvent mole fraction in the ideal solution.
`solvent_molar_volume`	Constant partial molar volume of solvent used to convert a chemical-potential difference to osmotic pressure.	m³/kmol	`solvent_molar_volume > 0` Constant solvent partial molar volume in the membrane model.

Symbol	Meaning	Unit	Relation
`F_dof`	Degrees of freedom: how many properties can still be chosen freely once the phases and components are fixed.	-	F=2+N-P One for pure coexistence, two for a binary with two independent components; the freely exchanging bulk phase rule does not apply to the selective membrane.
`composition`	The make-up of the mixture, as fractions of its components.	—	—
`T_bubble`	The temperature at which the first bubble of vapour appears in a liquid mixture being heated.	K	—
`T_dew`	The temperature at which the first drop of liquid appears in a vapour mixture being cooled.	K	—
`pi_osmotic`	Osmotic pressure: the pressure that has to be applied to a solution to stop pure solvent flowing into it through a membrane.	MPa	—

Symbol	Meaning	Unit	Valid range
`reaction`	Which reaction the equilibrium is being computed for.	—	Choose a library or declare species and an arbitrary independent signed reaction matrix. Decimal/fraction coefficients are preserved exactly; every element and charge must balance.
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`T>0 in given-lnK mode; thermochemical mode uses the valid ranges of participating records, including the pure-water IF97 pressure restriction.`
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`p > 0; declared pure liquid water with nonzero inventory or participation also requires IF97 region 1 at T,p` User-defined pressure slider explores 0.001 to 10 MPa logarithmically. This is an exploration interval, not a validity limit; typed values remain subject to phase and model checks.
`n_initial`	How much of each species is present before the reaction proceeds.	kmol	`non-negative finite amounts with nonzero total` CO2, CO, H2O vapor, H2, O2, N2 and pure H2O liquid; omitted species mean zero. Atomic mode accepts H, N, O, Ar, Na, Cs with each singly charged ion and electrons; initial total charge must be zero. User-defined mode accepts declared species with explicit integer element counts, charge and phase; total initial charge must be zero.
`inert`	Species present that take no part in the reaction but dilute the mixture, which shifts the equilibrium anyway.	—	Any species excluded from all supplied reactions remains chemically frozen, even if its elements occur in reactive species.
`lnK_given`	User-supplied natural logarithm of the equilibrium constant at the current temperature and 1 atm standard state.	—	`finite natural-log constants, one per active atom or written user-defined reaction` Given at the entered temperature and 1 atm standard state; does not specify reaction enthalpy or temperature dependence.
`nu_matrix`	Exact signed reaction coefficients; zero entries exclude a species from a reaction.	—	`explicit user-defined system` Exact signed reaction coefficients; zero entries exclude a species from a reaction.
`species_atoms`	Positive integer count of each declared element in one species.	—	`explicit user-defined system` Positive integer count of each declared element in one species.
`species_charge`	Signed integer electric charge of one species.	—	`explicit user-defined system` Signed integer electric charge of one species.
`species_phase`	Ideal gas or a separate pure condensed phase with unit activity.	—	`explicit user-defined system` Ideal gas or a separate pure condensed phase with unit activity.

Symbol	Meaning	Unit	Relation
`lnK`	The logarithm of the equilibrium constant. The app works in this throughout because the constant itself spans about forty decades.	-	ln K=-Δ G°/(R̄T) Gas standard-state K depends on T; effective condensation K includes actual-pressure liquid Gibbs and therefore depends on T and p.
`K_eq`	Equilibrium constant: how far the reaction goes at this temperature. Shown for reading only, never used as an intermediate.	-	Display only; unavailable when exp(lnK) cannot retain six significant digits. Never used in the solver.
`epsilon_ext`	Extent of reaction: how far the reaction has actually proceeded, from none of it to all of it.	kmol	Signed extent in each written reaction coordinate; inverse rows favor smaller inventory scales. An unresolved near-zero extent is unavailable, not a false exact zero.
`composition`	The make-up of the mixture, as fractions of its components.	—	—
`dH_reaction`	Energy absorbed or released by the reaction, which is what decides whether heating pushes it forwards or backwards.	kJ/kmol	Computed only from matching thermochemical records; given-lnK mode does not invent reaction heat.
`n_e`	Amount of free electrons, included as a gas species in charge and mole balances.	kmol	Common electron inventory, included in total gas moles and partial pressures.
`z_ion`	Fraction of a conserved atom inventory present as singly charged ions.	—	Ionized fraction for each conserved atom inventory.

Symbol	Meaning	Unit	Valid range
`fuel`	Which fuel is being burned.	—	Hydrogen for PEM; H2, CO or CH4 net-reaction bounds for solid oxide. Pure oxygen supply; no reforming or polarization model.
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`200 <= T <= 6000 for gas products; liquid-water model requires IF97 R1` Mathematical property range, not an electrode or membrane operating specification.
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	`p > 0` Common pure fuel / pure oxygen / mixed-product stream pressure; liquid product feasibility checked with T.
`n_electrons`	How many electrons move per mole of fuel in the cell reaction.	-	Derived from the selected oxidation reaction: 2 for H2/CO, 8 for CH4; displayed, not arbitrarily editable.
`V_operating`	The voltage the cell is actually running at, which is always below the reversible value.	V	`0 <= V_operating <= Ecell` Four ULPs only accommodate representational roundoff at the reversible boundary.
`current`	Current drawn from the cell.	A	`current >= 0` Total cell current. Unity fuel utilization and Faradaic efficiency; zero current returns zero power and consumed flow.

Symbol	Meaning	Unit	Relation
`Ecell`	Reversible cell voltage: the most a fuel cell can produce before any losses, set by the Gibbs function change and the charge moved.	V	E=-Δḡ/(nF)
`Wmax`	The most work obtainable from a reaction, which is set by the Gibbs function change rather than by the energy released.	kJ/kmol	Wₘₐₓ=-Δḡ
`eta`	Thermal efficiency: net work out divided by heat in. The fraction of what was paid for that came back as work.	-	—
`epsilon`	Exergetic efficiency: what the component actually delivered as a fraction of the exergy it was given. It asks a harder question than thermal efficiency and usually gets a worse answer.	-	Electrical work divided by all incoming exergy resources: positive fuel/oxygen/heat terms plus negative outlet-stream exergy. Distinct from voltage efficiency V/E.
`Qdot`	Rate of heat transfer.	kW	Positive heat rejection; negative when heat is absorbed. Current converts per-fuel heat to a rate.

Contents

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Units and numbers

Readings and graph export

Help and settings

State & Properties

M01 · State Explorer

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Output reference

M02 · Ideal Gas Properties

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M34 · Real Gas: Compressibility and Cubic Equations of State

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M35 · Departure Functions

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M36 · Thermodynamic Relations

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M37 · Ideal Gas Mixtures

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Refrigeration

M33 · Liquefaction

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M30 · Cascade and Multistage Refrigeration

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Reacting Systems

M41 · Combustion Stoichiometry

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M42 · Products Analysis

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M43 · Reacting Energy Balance

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M44 · Adiabatic Flame Temperature

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Output reference

Symbol	Meaning	Unit	Valid range
`T₁`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`T > T0 > 0` Source stream inlet.
`T0`	Dead-state temperature: the temperature of the environment the system is eventually going to equilibrate with. Exergy is meaningless without it.	K	`0 < T0 < T`
`mdot`	Mass flow rate.	kg/s	`mdot > 0` Mass flow of charging stream.
`cp`	Specific heat at constant pressure: how much energy raises one kilogram by one degree while the pressure is held fixed.	kJ/(kg K)	`cp > 0` Constant specific heat of charging stream.
`UA_store`	Conductance between the charging stream and the store.	kW/K	`UA_store > 0`
`t_charge`	How long the store is charged for.	s	`t_charge > 0` Finite entered charging duration; latent charge must fit the available latent heat.
`T_store`	Temperature the store is held at.	K	`T0 < T_store < T` Latent melting temperature. Sensible store begins at ambient and its temperature evolves.
`storage_kind`	Whether the store works by rising in temperature or by melting at a fixed one.	—	`sensible or latent`
`M_store`	Sensible store total heat capacity.	kJ/K	`M_store > 0` Sensible store total heat capacity.
`Q_latent`	Available latent heat; no warming outside the phase-change plateau is modeled.	kJ	`Q_latent > 0` Available latent heat; no warming outside the phase-change plateau is modeled.
`storage_flow`	Latent stream transfer factor: 1-exp(-N) or N/(1+N). Sensible model uses plug flow.	—	`plug flow or mixed flow` Latent stream transfer factor: 1-exp(-N) or N/(1+N). Sensible model uses plug flow.

Symbol	Meaning	Unit	Relation
`sigma`	Entropy production: entropy that was created inside the boundary rather than carried across it. Zero for an ideal process, positive for every real one, and never negative.	kJ/K	Integrated heat-exchanger entropy generation, excluding outlet disposal.
`Ed`	Exergy destroyed: work potential that was permanently lost, equal to the dead-state temperature times the entropy produced.	kJ	Integrated heat-exchanger exergy destruction, T0*sigma.
`T_store_opt`	The storage temperature that destroys the least exergy, which is neither the highest available nor the source temperature.	K	Capacity-constrained latent melting temperature maximizing captured exergy at fixed duration.
`eta_storage`	How much of the exergy put into a store is still there to take out.	-	Stored / incoming exergy over the full duration.
`t_charge_opt`	Sensible charging duration maximizing stored / incoming exergy.	s	Sensible charging duration maximizing stored / incoming exergy.
`Q_stored`	Integrated stored heat.	kJ	Integrated stored heat.
`B_stored`	Integrated captured exergy.	kJ	Integrated captured exergy.
`B_in`	Incoming stream exergy over the entered time.	kJ	Incoming stream exergy over the entered time.
`B_out`	Unused outgoing stream exergy over the entered time.	kJ	Unused outgoing stream exergy over the entered time.
`S_dump`	Additional entropy if the outlet is discarded to ambient.	kJ/K	Additional entropy if the outlet is discarded to ambient.
`S_total`	Heat-exchanger plus outlet-disposal entropy.	kJ/K	Heat-exchanger plus outlet-disposal entropy.
`T_final_store`	Store temperature at the end of the charge.	K	Store temperature at the end of the charge.
`T_final_out`	Final outlet temperature.	K	Final outlet temperature.

Symbol	Meaning	Unit	Valid range
`T`	Temperature: the property two bodies share when nothing flows between them on contact. Always absolute inside the app.	K	`(0, infinity)` Blackbody source temperature, at or above T0; collector requires strictly above.
`T0`	Dead-state temperature: the temperature of the environment the system is eventually going to equilibrate with. Exergy is meaningless without it.	K	`0 < T0 <= T` Environment temperature bounded by the entered source; zero is an excluded endpoint.
`A`	Cross-sectional area the stream passes through.	m^2	`(0, infinity)` Emitting area in reference models, receiver area in collector model.
`flux`	Radiant energy arriving per unit area per unit time.	kW/m^2	`(0, infinity)` Incident irradiance before concentration; collector only.
`concentration`	How many times the incoming radiation is concentrated before it reaches the collector.	-	`(0, infinity)` Collector irradiance multiplier; concentration*flux cannot exceed the source blackbody emissive power.

Symbol	Meaning	Unit	Relation
`psi_radiation`	The fraction of a radiation stream's energy that is available as work. Lower than the reversible factor at the source temperature, and the two are routinely confused.	-	—
`E_radiation`	Rate at which radiation delivers exergy.	kW	Total available/reference power includes area; density is separately labeled kW/m². The selected model defines the energy reference.
`eta_max`	The largest thermal efficiency any device can have between the two given reservoir temperatures.	-	Shown beside it, since the two are routinely confused.
`T_opt`	The collector temperature that maximises work output, balancing conversion quality against loss.	K	—

Symbol	Meaning	Unit	Valid range
`TH`	Temperature of the hot reservoir.	K	`(0, infinity)` Hot reservoir or hot-stream inlet, greater than TC.
`TC`	Temperature of the cold reservoir.	K	`(0, infinity)` Cold reservoir, also the exergy reference temperature.
`UA_hot`	Conductance between the hot reservoir and the engine.	kW/K	`(0, infinity)` Hot-side conductance; derived from total and fraction in fixed-reservoir mode.
`UA_cold`	Conductance between the engine and the cold reservoir.	kW/K	`(0, infinity)` Derived in fixed-reservoir mode; does not apply to the ideal cold sink of the continuous hot-stream mode.
`mdot`	Mass flow rate.	kg/s	`(0, infinity)` Hot-stream mass flow.
`cp`	Specific heat at constant pressure: how much energy raises one kilogram by one degree while the pressure is held fixed.	kJ/(kg K)	`(0, infinity)` Constant stream specific heat.
`UA_total`	The fixed sum of hot and cold exchanger conductances.	kW/K	`(0, infinity)` Fixed-reservoir total conductance budget.
`hot_conductance_fraction`	The share of the fixed conductance budget assigned to the hot side.	-	`(0, 1)` Fraction of total conductance assigned to the hot exchanger; zero endpoints cannot run the engine.

Symbol	Meaning	Unit	Relation
`eta_maxpower`	The efficiency a device actually runs at when it is producing as much power as it can, which is lower than the reversible bound and much closer to what real plants achieve.	-	η=1-√(T_C/T_H) Model-dependent result; see explicitly selected source constraint.
`Wdot_max`	The largest power the arrangement can produce under the stated constraint, as opposed to the largest efficiency.	kW	Model-dependent result; see explicitly selected source constraint.
`eta_max`	The largest thermal efficiency any device can have between the two given reservoir temperatures.	-	Shown beside it, for the comparison. Model-dependent result; see explicitly selected source constraint.
`T_internal`	The internal working temperatures of an endoreversible engine, which sit between the reservoir temperatures because the heat has to flow across a finite gap.	K	Model-dependent result; see explicitly selected source constraint.

Symbol	Meaning	Unit	Valid range
`geometry`	Arrangement: heated duct, balanced counterflow exchanger, or fixed-length pin fin in crossflow.	—	`one of three declared arrangements` Arrangement: heated duct, balanced counterflow exchanger, or fixed-length pin fin in crossflow.
`temperature`	Fluid temperature for duct and pin fin.	K	`finite positive; active only in named arrangement` Fluid temperature for duct and pin fin.
`density`	Constant fluid density for duct and pin fin.	kg/m^3	`finite positive; active only in named arrangement` Constant fluid density for duct and pin fin.
`viscosity`	Constant dynamic viscosity for duct and pin fin.	Pa s	`finite positive; active only in named arrangement` Constant dynamic viscosity for duct and pin fin.
`conductivity`	Constant fluid thermal conductivity for duct and pin fin.	W/(m K)	`finite positive; active only in named arrangement` Constant fluid thermal conductivity for duct and pin fin.
`mass_flow`	Duct fluid mass flow.	kg/s	`finite positive; active only in named arrangement` Duct fluid mass flow.
`heat_per_length`	Heat transferred to the duct fluid per unit length.	W/m	`finite positive; active only in named arrangement` Heat transferred to the duct fluid per unit length.
`duct_length`	Duct segment length over which heat and pressure losses are evaluated.	m	`finite positive; active only in named arrangement` Duct segment length over which heat and pressure losses are evaluated.
`cp`	Specific heat at constant pressure: how much energy raises one kilogram by one degree while the pressure is held fixed.	kJ/(kg K)	`finite positive; active only in named arrangement` Duct mass-specific heat; the display adapter alone converts to J/(kg K) for the composition calculation.
`prandtl`	Duct heat-transfer correlation Prandtl number. Pin-fin Prandtl number is fixed at 0.71, not this input.	-	`0.7 < prandtl < 160` Duct heat-transfer correlation Prandtl number. Pin-fin Prandtl number is fixed at 0.71, not this input.
`hot`	Balanced exchanger hot-side inlet temperature.	K	`hot > cold > 0` Balanced exchanger hot-side inlet temperature.
`cold`	Balanced exchanger cold-side inlet temperature.	K	`hot > cold > 0` Balanced exchanger cold-side inlet temperature.
`capacity`	Equal heat-capacity rate of either single side of the balanced exchanger; not the sum.	kW/K	`finite positive` Equal heat-capacity rate of either single side of the balanced exchanger; not the sum.
`hot_drop`	Hot fractional inlet-pressure loss per unit NTU.	-	`non-negative; at least one loss slope positive` Hot fractional inlet-pressure loss per unit NTU.
`cold_drop`	Cold fractional inlet-pressure loss per unit NTU.	-	`non-negative; at least one loss slope positive` Cold fractional inlet-pressure loss per unit NTU.
`hot_ratio`	Hot-side gas constant divided by specific heat.	-	`0 < hot_ratio < 1` Hot-side gas constant divided by specific heat.
`cold_ratio`	Cold-side gas constant divided by specific heat.	-	`0 < cold_ratio < 1` Cold-side gas constant divided by specific heat.
`minimum_effectiveness`	Minimum fraction of the balanced exchanger maximum heat duty required by the design task.	-	`0 < minimum_effectiveness < 1` Useful-duty lower bound on balanced-exchanger effectiveness.
`maximum_drop`	Upper bound on either side fractional pressure drop.	-	`0 < maximum_drop < 1` Upper bound on either side fractional pressure drop.
`heat`	Fixed pin-fin heat duty.	W	`finite positive; pin-fin arrangement` Fixed pin-fin heat duty.
`speed`	Fluid speed across the pin fin.	m/s	`finite positive; pin-fin arrangement` Fluid speed across the pin fin.
`fin_length`	Fixed pin-fin length; only diameter is optimized.	m	`finite positive; pin-fin arrangement` Fixed pin-fin length; only diameter is optimized.
`fin_conductivity`	Solid pin-fin thermal conductivity.	W/(m K)	`finite positive; pin-fin arrangement` Solid pin-fin thermal conductivity.
`reference_capacity`	Fixed pin-fin reference capacity used solely to normalize the entropy-generation number.	kW/K	`finite positive; pin-fin arrangement` Fixed pin-fin reference capacity used solely to normalize the entropy-generation number.

Symbol	Meaning	Unit	Relation
`Ns`	Entropy generation number: entropy produced, made dimensionless by the stream's capacity rate, so arrangements of different size can be compared.	-	Total entropy generation divided by duct mass-flow heat capacity, either balanced-exchanger side capacity, or fixed pin-fin reference capacity.
`sigmadot_dT`	The share of entropy production caused by heat crossing a finite temperature difference.	kW/K	Thermal part at the selected constrained design.
`sigmadot_dp`	The share caused by fluid friction.	kW/K	Fluid-friction part at the selected constrained design.
`D_opt`	Duct or pin-fin diameter at the constrained minimum; only these two arrangements.	m	Duct or pin-fin diameter at the constrained minimum; only these two arrangements.
`NTU_opt`	Balanced-exchanger NTU at the constrained minimum; only this arrangement.	-	Balanced-exchanger NTU at the constrained minimum; only this arrangement.
`design_boundary`	Whether the selected minimum is on a constraint boundary or is an interior stationary point.	—	Whether the selected minimum is on a constraint boundary or is an interior stationary point.
`Re`	Reynolds number: the ratio of inertial to viscous effects in the flow, which is what fixes the friction.	-	duct arrangement: metric named Reynolds number. Derived at the selected design, not a direct input.
`Nu`	Nusselt number.	-	duct arrangement: metric named Nusselt number. Derived at the selected design, not a direct input.
`f_fanning`	Fanning friction factor.	-	duct arrangement: metric named Fanning friction factor. Derived at the selected design, not a direct input.
`wall_bulk_difference`	Wall minus bulk temperature.	ΔK	duct arrangement: metric named Wall minus bulk temperature. Derived at the selected design, not a direct input.
`segment_pressure_drop`	Segment pressure drop.	MPa	duct arrangement: metric named Segment pressure drop. Derived at the selected design, not a direct input.
`bulk_rise_fraction`	Bulk temperature rise fraction.	-	duct arrangement: metric named Bulk temperature rise fraction. Derived at the selected design, not a direct input.
`property_prandtl`	Property-derived Prandtl number.	-	duct arrangement: metric named Property-derived Prandtl number. Derived at the selected design, not a direct input.
`epsilon`	Exergetic efficiency: what the component actually delivered as a fraction of the exergy it was given. It asks a harder question than thermal efficiency and usually gets a worse answer.	-	balanced arrangement: metric named Effectiveness. Derived at the selected design, not a direct input.
`Qdot`	Rate of heat transfer.	kW	balanced arrangement: metric named Heat transferred. Derived at the selected design, not a direct input.
`hot_outlet_temperature`	Hot outlet temperature.	K	balanced arrangement: metric named Hot outlet temperature. Derived at the selected design, not a direct input.
`cold_outlet_temperature`	Cold outlet temperature.	K	balanced arrangement: metric named Cold outlet temperature. Derived at the selected design, not a direct input.
`hot_drop_fraction`	Hot pressure drop / inlet pressure.	-	balanced arrangement: metric named Hot pressure drop / inlet pressure. Derived at the selected design, not a direct input.
`cold_drop_fraction`	Cold pressure drop / inlet pressure.	-	balanced arrangement: metric named Cold pressure drop / inlet pressure. Derived at the selected design, not a direct input.
`NTU_asymptotic`	Asymptotic unconstrained NTU.	-	balanced arrangement: metric named Asymptotic unconstrained NTU. Derived at the selected design, not a direct input.
`thermal_asymptotic`	Asymptotic thermal entropy at selected NTU.	kW/K	balanced arrangement: metric named Asymptotic thermal entropy at selected NTU. Derived at the selected design, not a direct input.
`friction_asymptotic`	Asymptotic friction entropy at selected NTU.	kW/K	balanced arrangement: metric named Asymptotic friction entropy at selected NTU. Derived at the selected design, not a direct input.
`Re`	Reynolds number: the ratio of inertial to viscous effects in the flow, which is what fixes the friction.	-	pin fin arrangement: metric named Reynolds number. Derived at the selected design, not a direct input.
`Nu`	Nusselt number.	-	pin fin arrangement: metric named Nusselt number. Derived at the selected design, not a direct input.
`C_drag`	Drag coefficient.	-	pin fin arrangement: metric named Drag coefficient. Derived at the selected design, not a direct input.
`fin_base_temperature`	Fin base temperature.	K	pin fin arrangement: metric named Fin base temperature. Derived at the selected design, not a direct input.
`base_ambient_difference`	Base minus ambient temperature.	ΔK	pin fin arrangement: metric named Base minus ambient temperature. Derived at the selected design, not a direct input.
`Bi_transverse`	Transverse Biot number.	-	pin fin arrangement: metric named Transverse Biot number. Derived at the selected design, not a direct input.
`aspect`	The geometric design variable being optimised over.	-	pin fin arrangement: metric named Length / diameter. Derived at the selected design, not a direct input.
`drag_force`	Drag force.	N	pin fin arrangement: metric named Drag force. Derived at the selected design, not a direct input.
`fin_conductance`	Fin thermal conductance.	kW/K	pin fin arrangement: metric named Fin thermal conductance. Derived at the selected design, not a direct input.

Symbol	Meaning	Unit	Valid range
`p`	Pressure: the normal force a fluid exerts per unit of the area it pushes on. Always absolute inside the app; gauge is a display mode.	MPa	Barometric pressure; altitude-corrected.
`Tdb`	Dry-bulb temperature: what an ordinary thermometer in the air stream reads.	K	—
`Twb`	Wet-bulb temperature: what a thermometer with a wet wick reads, which is lower because evaporation cools it. How much lower is a measure of how dry the air is.	K	—
`Tdp`	Dew point: the temperature at which the air being cooled would start to condense.	K	—
`omega`	Humidity ratio: kilograms of water vapour carried per kilogram of dry air. The quantity that stays fixed when moist air is merely heated or cooled.	-	—
`phi`	Relative humidity: how much water vapour the air holds as a fraction of the most it could hold at that temperature.	-	`0 <= phi <= 1`
`h_da`	Enthalpy of moist air per kilogram of dry air, rather than per kilogram of mixture, because the dry air is what stays constant through the processes.	kJ/kg	Enthalpy per unit mass of dry air.
`altitude`	Height above sea level, used to correct the barometric pressure, which shifts the whole chart.	m	`-500 <= altitude <= 11000`

Symbol	Meaning	Unit	Relation
`omega`	Humidity ratio: kilograms of water vapour carried per kilogram of dry air. The quantity that stays fixed when moist air is merely heated or cooled.	-	ω=0.622pᵥ/(p-pᵥ)
`phi`	Relative humidity: how much water vapour the air holds as a fraction of the most it could hold at that temperature.	-	—
`Tdp`	Dew point: the temperature at which the air being cooled would start to condense.	K	—
`Twb`	Wet-bulb temperature: what a thermometer with a wet wick reads, which is lower because evaporation cools it. How much lower is a measure of how dry the air is.	K	—
`h_da`	Enthalpy of moist air per kilogram of dry air, rather than per kilogram of mixture, because the dry air is what stays constant through the processes.	kJ/kg	h=h_a+ω h_g(T)
`v_da`	Volume occupied per kilogram of dry air.	m^3/kg	—
`mu_sat`	Degree of saturation: the humidity ratio as a fraction of the saturated value at the same temperature and pressure.	-	—
`p_partial`	Partial pressure: the pressure one component of a mixture would exert if it alone occupied the whole volume.	MPa	—