Ehsan Heidaryan, Ph.D.Chemical Engineering · Process & Thermophysical Modelingheidaryan.ehsan@gmail.com
Offline single-file tool. Pick a fluid, then calculate single-state properties or an adiabatic turbine expansion.
The residual Helmholtz terms and the ideal-gas heat capacity terms come from the supplied REFPROP-format fluid files,
and the fluid data page carries the published reference for each individual equation. Below the critical temperature the page
solves the saturation line first, so a single-phase state is identified as liquid or as vapor before its density root is
taken; the ancillary curves in the same file and the Peng-Robinson density serve only as starting values for those two solves.
The selected fluid drives both calculator pages. Changing the fluid resets the state and turbine inputs to
values inside that equation’s validity range.
Subpage 1
Thermodynamic calculator, carbon dioxide
Single-state Helmholtz property calculation from pressure and temperature. The solver identifies the phase from the equation’s own saturation line before it picks the density root.
Enter pressure and temperature, then press Calculate.
Residual term types read from the fluid file: polynomial and exponential, Gaussian bell, non-analytic critical, and association terms.
Phase equilibrium: two-variable Newton on the saturated densities with equal pressure and equal chemical potential, started from the ancillary curves in the same file.
Density solver convergence
The density at your temperature and pressure is the root of F(ρ) = P(T, ρ) − P. The solver starts
from the Peng-Robinson seed and uses a safeguarded Newton iteration inside the bracket for the identified phase.
The magenta dot sits where F crosses zero.
Calculate a state to show the density solver convergence.
Subpage 2
Turbine calculator, carbon dioxide
Adiabatic turbine expansion from the active fluid’s equation of state. The tool solves the isentropic outlet at fixed outlet pressure, then the real outlet from turbine efficiency. An outlet inside the two-phase dome is reported with its vapor quality.
Enter turbine conditions, then press Calculate turbine.
Property
Value
Unit
Definitions: state 1 is turbine inlet, state 2s is the isentropic outlet at P2 and s1, and state 2 is the real outlet from turbine efficiency. Kinetic and potential energy changes are not included.
Turbine performance sweep
Inlet state, turbine efficiency, mechanical efficiency, and mass flow stay fixed while the outlet pressure varies.
Left axis: specific work. Right axis: second-law efficiency. The dashed vertical line marks your current pressure ratio.
Calculate the turbine to build the sweep.
Subpage 3
Fluid data and equation-of-state parameters
Everything the calculator uses for the active fluid, read from its REFPROP-format file. Use the download button to export the full parameter set for the selected fluid.
Carbon dioxide
State diagrams: the reference chart dataset stored in this file covers carbon dioxide only.
Select carbon dioxide to view the T–s, P–h, h–s, and Z–P diagrams. Diagram data for the
other fluids comes in a later version.
Thermodynamic state diagrams
Reference CO₂ chart data · saturation dome, isotherms, isobars
The thermodynamic page marks only one solved state on each chart. The turbine page shows two paths: ideal isentropic 1 to 2s and real 1 to 2. Axes and colorbars follow the active unit selectors,
so changing temperature, pressure, or energy units relabels every chart. Pressures below 1 atm are not plotted on pressure-axis charts.
The T–s chart explicitly labels the minimum and maximum isobars. Hover a chart to enlarge it.
T – s Temperature vs Entropy
Isobars colored by pressure. The minimum and maximum isobars are labeled.
P – h Pressure vs Enthalpy
Log pressure axis. Isotherms colored by temperature (300 to 1000 K).
h – s Enthalpy vs Entropy
Mollier chart. Isotherms colored by temperature (300 to 1000 K).
P – Z Compressibility factor
Z versus pressure on a log axis. Isotherms colored by temperature (220 to 1000 K).
Methods
This section explains how the reported numbers are produced and lists the sources behind the models.
Where the parameters come from
Each fluid file in the supplied set holds a Helmholtz equation of state written as a dimensionless energy,
φ = φ⁰ + φʳ, with δ = ρ/ρred and τ = Tred/T. The page reads the full
coefficient set for the recommended equation in each file: polynomial and exponential terms
nδdτtexp(−gδl), Gaussian bell terms, the non-analytic
critical terms used by carbon dioxide and water, and the association terms used by the newer ammonia equation.
The ideal-gas part is built from the ideal-gas heat capacity block in the same file, integrated analytically
term by term, with the two integration constants fitted so that enthalpy and entropy reproduce the reference
state declared in that file. Four fluids in the set store only a modified Benedict-Webb-Rubin equation; three
of them also carry a Helmholtz form, which is what this page uses, and nitrogen trifluoride is therefore not
in the list.
How the thermodynamic properties are calculated
For a given temperature and pressure the density solves P = ρRT(1 + δφʳδ). Below the
critical temperature the page first solves the saturation line for that temperature, then brackets the density
root on the liquid side or the vapor side according to the pressure. The saturation solve is a two-variable
Newton iteration on the saturated liquid and vapor densities that enforces equal pressure and equal chemical
potential, started from the ancillary vapor-pressure and saturated-density curves in the same fluid file.
All other properties are exact derivatives of the Helmholtz energy. The compressibility factor is
Z = 1 + δφʳδ. Internal energy, enthalpy, entropy, and the Gibbs and Helmholtz energies follow
the standard Helmholtz relations. The isochoric heat capacity is cv = −Rτ²(φ⁰ττ +
φʳττ), and the isobaric heat capacity adds the volumetric term. Speed of sound and the fugacity
coefficient use the same first and second derivatives. The derivatives come from exact automatic differentiation
of every term, so accuracy holds close to the critical point.
How the turbine parameters are calculated
The turbine page treats the expander as adiabatic, steady flow, with kinetic and potential energy neglected.
State 1 is the inlet. The isentropic outlet, state 2s, sits at the outlet pressure with the same entropy as the
inlet, s2s = s1. The real outlet, state 2, uses the isentropic efficiency through h2 = h1 − ηt(h1 − h2s).
Each outlet state is found at the outlet pressure. If the target enthalpy or entropy falls between the saturated
liquid and saturated vapor values at that pressure, the outlet is a two-phase mixture and the page reports its
vapor quality instead of a superheated temperature.
Specific work is w = h1 − h2. Fluid power is the mass flow times the specific work, and shaft power applies
the mechanical efficiency, ηm · ṁ · w. Entropy generation for the adiabatic expansion is
s2 − s1. Exergy destruction follows the Gouy-Stodola relation, X_dest = T0 · sgen, with T0 the
dead-state temperature. Flow exergy at a state is ψ = (h − h0) − T0(s − s0), evaluated against
the chosen dead state. The second-law efficiency is the real work divided by the flow-exergy drop,
η_II = w / (ψ1 − ψ2).
References
Lemmon, E. W., Bell, I. H., Huber, M. L., and McLinden, M. O. NIST Reference Fluid Thermodynamic and Transport
Properties Database, REFPROP, Version 10.0. The fluid files supplied to this page carry the source citation for
each individual equation of state in their own header.
Span, R. and Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the
Triple-Point Temperature to 1100 K at Pressures up to 800 MPa. Journal of Physical and Chemical Reference Data,
25(6), 1509-1596, 1996.
Wagner, W. and Pruss, A. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water
Substance for General and Scientific Use. Journal of Physical and Chemical Reference Data, 31(2), 387-535, 2002.
Peng, D. Y. and Robinson, D. B. A New Two-Constant Equation of State. Industrial and Engineering Chemistry
Fundamentals, 15(1), 59-64, 1976. Used here only for the density seed.
Akasaka, R. A Reliable and Useful Method to Determine the Saturation State from Helmholtz Energy Equations of
State. Journal of Thermal Science and Technology, 3(3), 442-451, 2008. Basis of the two-variable saturation solve.
Çengel, Y. A. and Boles, M. A. Thermodynamics: An Engineering Approach. McGraw-Hill Education. Source for the
control-volume energy balance, isentropic turbine efficiency, exergy, and second-law efficiency definitions.