Equation of State

Equation of State and Gas Properties

The gas phase is treated as an ideal-gas mixture. This page lists exactly how each gas property in the model is obtained, following core/src/objects/fluid.cpp, core/include/adsim/objects/fluid.h and the property updates in core/src/equations/adsorbent_equations.cpp.

Ideal gas law

In every cell the total molar concentration \(C\), the pressure \(P\) and the gas temperature \(T\) are tied by

\[ C = \frac{P}{R\thinspace T} \]

This is one of the residuals the solver satisfies at every time step (the Concentration equation), and the same relation is used wherever a concentration is needed at a cell face or a boundary, with the face pressure and temperature taken as the mean of the two neighbouring cells. Component concentrations follow from the mole fractions, \(c_i = y_i\thinspace C\). There is no compressibility factor, so the model is meant for pressures and temperatures where the gas is close to ideal; at high pressure, such as the 15 bar of the Casas (2012) decks, check that the ideal-gas error is acceptable for your gas.

Symbol	Description	Unit
\(C\)	Total gas molar concentration	mol m⁻³
\(c_i\)	Molar concentration of component \(i\)	mol m⁻³
\(y_i\)	Mole fraction of component \(i\)	–
\(P\)	Pressure	Pa
\(T\)	Gas temperature	K
\(R\)	Gas constant, 8.314462618 J mol⁻¹ K⁻¹	J mol⁻¹ K⁻¹

Where CoolProp is used

Components are chosen by their CoolProp fluid names in the deck's fluid package. CoolProp is called only when a run starts (Fluid::Initialise), never inside the time integration:

• Molar mass \(M_i\) of each component, from CoolProp.
• Heat capacity \(C_{p,i}\) [J kg⁻¹ K⁻¹] of each pure component, evaluated once, at the pressure and temperature of the initial conditions, and then held constant for the whole run. A TSA cycle that heats the gas by 100 K therefore uses the gas heat capacity of the initial temperature throughout.
• The heat capacity is always the gas-phase value. A component that would be condensed in its pure state at the initial pressure and temperature (water, ethanol, hexane or toluene at 1 bar and 298 K) takes its ideal-gas heat capacity \(C_{p0,i}(T)\) instead of CoolProp's liquid value: about 1865 J kg⁻¹ K⁻¹ (33.6 J mol⁻¹ K⁻¹) for water vapour at 298 K, not the liquid's 4181. In the bed such a component is a vapour at a partial pressure no higher than its saturation pressure, where it is close to ideal, and \(C_{p0}\) depends on temperature only, so it exists at every state. Components in a gas or supercritical state, which includes the permanent gases (N₂, O₂, CH₄, H₂, Ar, He, and CO₂ below its saturation pressure, about 64 bar at 298 K), keep the real-gas \(C_{p,i}(P, T)\).

If CoolProp cannot evaluate a property, the error is printed to the console and the value is set to 0.

Constant gas heat capacity and viscosity

Models in the literature often use fixed gas properties; for example, Yancy-Caballero et al. (2020) and Haghpanah et al. (2013) use \(C_{p,g} = 30.7\) J mol⁻¹ K⁻¹ and \(\mu = 1.72\times10^{-5}\) Pa s. Two fluid switches reproduce them:

• Gas heat capacity mode (fluid.gas_heat_capacity_mode): coolprop (the default, as above) or constant. With constant, the molar heat capacity of each component is the value given in fluid.gas_heat_capacity[<component>] [J mol⁻¹ K⁻¹], and \(C_{p,i}\) is that value divided by \(M_i\). The mixing rule below is unchanged, so equal values give that molar heat capacity at every composition. A new component starts at 29.1 J mol⁻¹ K⁻¹ (7R/2).
• Viscosity mode (fluid.viscosity_mode): polynomial (the default, the mole-fraction average below) or constant. With constant, every cell has the mixture viscosity fluid.viscosity [Pa s], whatever its temperature and composition.

The Gas phase adsorbed-phase heat capacity takes the same \(C_{p,i}\thinspace M_i\), so with a constant gas heat capacity it is the constant too.

Mixture properties

All mixture properties use linear mole-fraction mixing, evaluated in every cell at every residual evaluation:

\[ M = \sum_i y_i\thinspace M_i,\qquad \rho = \sum_i c_i\thinspace M_i = C\thinspace M,\qquad C_{p,g} = \frac{\sum_i y_i\thinspace M_i\thinspace C_{p,i}}{\sum_i y_i\thinspace M_i} \]

\[ \mu = \sum_i y_i\thinspace \mu_i(T),\qquad k_g = \sum_i y_i\thinspace k_i(T) \]

The heat capacity is mixed on the molar basis: mole fractions average the pure-component molar heat capacities \(M_i C_{p,i}\) (exact for an ideal-gas mixture), and dividing by the mixture molar mass \(M\) gives the per-kilogram \(C_{p,g}\) [J kg⁻¹ K⁻¹] that the energy balance multiplies by \(\rho\). The volumetric heat capacity is therefore \(\rho\thinspace C_{p,g} = C \sum_i y_i\thinspace M_i\thinspace C_{p,i}\). Every user of the gas heat capacity (the energy balances, the convective enthalpy terms, the thermal dispersion and heat-transfer correlations and the logged heat capacity) takes this one value. For 50/50 H₂/CO₂ it is about 33 J mol⁻¹ K⁻¹, where averaging the per-kilogram \(C_{p,i}\) by mole fraction would give about 174.

The linear rules for viscosity and conductivity are simpler than the Wilke or Wassiljewa mixing rules; they are adequate for the dilute, chemically similar mixtures typical of PSA, and are what the model uses.

Viscosity and thermal conductivity polynomials

Calling CoolProp in every cell at every residual evaluation would make a run minutes slower, so the pure-component viscosity \(\mu_i\) and conductivity \(k_i\) are degree-4 polynomials in temperature with coefficients fitted offline and compiled into the program (viscosity_coefficients.h, thermal_conductivity_coefficients.h):

\[ \mu_i(T) = \sum_{n=0}^{4} a_{n,i}\thinspace T_c^{\thinspace n},\qquad k_i(T) = \sum_{n=0}^{4} b_{n,i}\thinspace T_c^{\thinspace n},\qquad T_c = \min\negthinspace \big(\max(T,\thinspace 150\ \text{K}),\thinspace 650\ \text{K}\big) \]

• The temperature is clamped to 150–650 K before evaluation: outside that window the property is held at its value at the nearest end, never extrapolated.
• The polynomials are least-squares fits of CoolProp's dilute-gas values, sampled at low pressure so that condensable components (water, alcohols, hydrocarbons) get their vapour values, not the liquid's (water at 298 K: 9.7 × 10⁻⁶ Pa s and 0.0184 W m⁻¹ K⁻¹). For the components CoolProp has no viscosity model for (CarbonMonoxide, Ethylene, Fluorine, Krypton, Neon, NitrousOxide, SulfurDioxide, Xenon), the viscosity fit is to Chapman–Enskog kinetic theory with the component's Lennard-Jones parameters; for the components without a conductivity model, the conductivity fit is to the Eucken relation \(k = \mu\thinspace (c_p + 1.25\thinspace R/M)\). The headers state a maximum fit error of 1 % (viscosity) and 1.8 % (conductivity) over the window.
• Pressure dependence is neglected; gas viscosity changes by less than about 1 % over 0.1–10 bar.
• A floor of 10⁻⁶ Pa s (viscosity) and 10⁻⁴ W m⁻¹ K⁻¹ (conductivity) guards against a fit undershooting at the window edges. A component with no coefficients at all falls back to 1 × 10⁻⁵ Pa s and 0.024 W m⁻¹ K⁻¹.

Other gas properties

• Molecular diffusivities come from the Chapman–Enskog binary-diffusion model with the Lennard-Jones energy and length of each component (Poling, Prausnitz and O'Connell, The Properties of Gases and Liquids, 5th ed.). They feed the dispersion and film mass-transfer correlations; see Transport Correlations.
• Adsorbed-phase heat capacity is either a constant per component or, with the Gas phase option of the energy balance, \(C_{p,i}\thinspace M_i\) from the same initial-condition heat capacity; see Energy Balance.
• Isothermal runs evaluate all of the above at the fixed temperature \(T_{\text{iso}}\); see Isothermal mode.