Adsorption Isotherm Models

NeqSim provides six adsorption isotherm models for calculating gas–solid equilibrium on porous adsorbents. Each model captures different physics — from ideal monolayer coverage to heterogeneous surfaces and micropore filling — and can be selected via the IsothermType enum.

Package: neqsim.physicalproperties.interfaceproperties.solidadsorption

Overview of Available Models

Isotherm Class IsothermType Multi-Component Best For
Langmuir LangmuirAdsorption LANGMUIR Single only Monolayer, homogeneous surfaces
Extended Langmuir LangmuirAdsorption EXTENDED_LANGMUIR Yes Competitive monolayer adsorption
BET BETAdsorption BET Per-component Multilayer, surface area measurement
Freundlich FreundlichAdsorption FREUNDLICH Per-component Heterogeneous surfaces, moderate coverage
Sips SipsAdsorption SIPS Yes (Extended) Heterogeneous + saturation plateau
DRA Potential Theory PotentialTheoryAdsorption DRA Yes Micropore filling, rigorous EOS-based

All models except DRA extend AbstractAdsorptionModel which provides common functionality for selectivity calculations, adsorbed-phase mole fractions, and parameter database loading.

1. Langmuir Isotherm

The Langmuir model assumes monolayer adsorption on a homogeneous surface with a finite number of identical sites. Once a site is occupied, no further adsorption occurs there.

Single-Component Equation

\[q = q_{max} \cdot \frac{K \cdot P}{1 + K \cdot P}\]

where:

Symbol Description Unit
$q$ Equilibrium loading (surface excess) mol/kg
$q_{max}$ Maximum monolayer capacity mol/kg
$K$ Langmuir equilibrium constant (affinity) 1/bar
$P$ Partial pressure of the adsorbate bar

Temperature Dependence (van ‘t Hoff)

The equilibrium constant $K$ varies with temperature according to:

\[K(T) = K(T_{ref}) \cdot \exp\left(\frac{-\Delta H_{ads}}{R} \cdot \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\right)\]

where:

Symbol Description Unit
$T_{ref}$ Reference temperature K
$\Delta H_{ads}$ Isosteric heat of adsorption (negative for exothermic) J/mol
$R$ Universal gas constant ($8.314$) J/(mol K)

Since adsorption is exothermic ($\Delta H_{ads} < 0$), increasing temperature decreases $K$, reducing the equilibrium loading — the thermodynamic basis for Temperature Swing Adsorption (TSA).

Fractional Coverage

The fractional surface coverage (dimensionless, 0 to 1) is:

\[\theta = \frac{q}{q_{max}} = \frac{K \cdot P}{1 + K \cdot P}\]

Limiting Behaviour

Default Parameters

Parameter Default Valid Range
$q_{max}$ 5.0 mol/kg > 0
$K$ 0.1 1/bar > 0
$T_{ref}$ 298.15 K > 0
$\Delta H_{ads}$ −20,000 J/mol typically −5,000 to −50,000

Java API

LangmuirAdsorption model = new LangmuirAdsorption(system);
model.setSolidMaterial("Zeolite 13X");

// Override parameters if needed
model.setQmax(0, 6.5);              // mol/kg for component 0
model.setKLangmuir(0, 0.8);         // 1/bar
model.setHeatOfAdsorption(0, -25000.0); // J/mol

model.calcAdsorption(0);            // phase 0
double loading = model.getSurfaceExcess(0);  // mol/kg
double coverage = model.getCoverage(0);      // 0-1

2. Extended Langmuir (Multi-Component)

When multiple gases compete for the same adsorption sites, the Extended Langmuir model accounts for competitive inhibition:

\[q_i = q_{max,i} \cdot \frac{K_i \cdot P_i}{1 + \displaystyle\sum_{j=1}^{N} K_j \cdot P_j}\]

The denominator sums over all species, so each component’s loading is reduced by the presence of others. This is the standard model for multi-component adsorption in PSA simulation.

Competitive Selectivity

The selectivity of component $i$ over component $j$ is:

\[\alpha_{i/j} = \frac{q_i / y_i}{q_j / y_j} = \frac{q_{max,i} \cdot K_i}{q_{max,j} \cdot K_j}\]

where $y_i$ is the gas-phase mole fraction.

Java API

LangmuirAdsorption model = new LangmuirAdsorption(system);
model.setSolidMaterial("Zeolite 13X");
model.calcExtendedLangmuir(0);  // multi-component calculation
double co2Loading = model.getSurfaceExcess("CO2");
double selectivity = model.getSelectivity(co2Index, ch4Index, 0);

3. BET (Brunauer-Emmett-Teller) Isotherm

The BET model extends Langmuir theory to allow multilayer adsorption. It is the standard method for measuring surface areas of porous materials.

Standard BET Equation (Infinite Layers)

\[q = \frac{q_m \cdot C \cdot x}{(1 - x)\left(1 - x + C \cdot x\right)}\]

where:

Symbol Description Unit
$q_m$ Monolayer capacity mol/kg
$C$ BET constant (energy parameter) dimensionless
$x$ Relative pressure $P / P_0$ dimensionless
$P_0$ Saturation (vapour) pressure at temperature $T$ bar

Physical Meaning of the BET Constant

\[C \approx \exp\left(\frac{E_1 - E_L}{RT}\right)\]

where $E_1$ is the heat of adsorption of the first layer and $E_L$ is the heat of liquefaction. Large $C$ ($> 100$) indicates strong adsorbate–surface interaction; small $C$ ($< 20$) indicates weak interaction.

Modified BET for Finite Layers

When adsorption is limited to $n$ layers (e.g., in mesopores):

\[q = q_m \cdot C \cdot x \cdot \frac{1 - (n+1) x^n + n \cdot x^{n+1}}{(1 - x)\left(1 + (C - 1)x - C \cdot x^{n+1}\right)}\]

For $n = 1$, this reduces to the Langmuir isotherm. For $n \to \infty$, it reduces to the standard BET equation.

BET Surface Area Calculation

From the monolayer capacity:

\[A_{BET} = q_m \cdot N_A \cdot \sigma\]

where $N_A = 6.022 \times 10^{23}$ mol$^{-1}$ is Avogadro’s number and $\sigma$ is the cross-sectional area of the adsorbate molecule (typically 0.162 nm$^2$ for N$_2$ at 77 K).

Saturation Pressure Estimation

When experimental $P_0$ values are not available, NeqSim estimates them using the Lee-Kesler correlation (see Fluid Property Estimation section below).

Default Parameters

Parameter Default Notes
$q_m$ $\frac{A_{BET} \times 10^{-3}}{M \times 0.162}$ Derived from BET surface area
$C$ 100 Moderate interaction
Max layers Unlimited Set via setMaxLayers(n)
$A_{BET}$ 1000 m$^2$/g From AbstractAdsorptionModel

Java API

BETAdsorption model = new BETAdsorption(system);
model.setSolidMaterial("AC");
model.setMaxLayers(5);  // Limit to 5 adsorbed layers

model.calcAdsorption(0);
double loading = model.getSurfaceExcess(0);
double relativePressure = model.getRelativePressure(0, 0);
int layers = model.getNumberOfLayers(0);
double surfaceArea = model.calculateBETSurfaceArea(0, 0.162); // nm^2 cross-section

4. Freundlich Isotherm

The Freundlich model is an empirical isotherm for adsorption on heterogeneous surfaces with a distribution of adsorption energies. Unlike Langmuir, it does not predict a saturation plateau.

Freundlich Equation

\[q = K_F \cdot P^{1/n}\]

where:

Symbol Description Unit
$K_F$ Freundlich capacity constant mol/kg/bar$^{1/n}$
$n$ Freundlich intensity parameter dimensionless
$P$ Partial pressure bar

Interpretation of the Exponent $n$

Temperature Dependence

\[K_F(T) = K_F(T_{ref}) \cdot \exp\left(\frac{-\Delta H_{ads}}{R} \cdot \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\right)\]

Limitations

The Freundlich isotherm:

Default Parameters

Parameter Default Valid Range
$K_F$ 2.0 > 0
$n$ 2.0 > 0 (typically 1–10)
$T_{ref}$ 298.15 K > 0
$\Delta H_{ads}$ −20,000 J/mol typically −5,000 to −50,000

Java API

FreundlichAdsorption model = new FreundlichAdsorption(system);
model.setSolidMaterial("AC BPL");

model.setKFreundlich(0, 3.5);
model.setNFreundlich(0, 2.5);

model.calcAdsorption(0);
double loading = model.getSurfaceExcess("CO2");
boolean favorable = model.isFavorableAdsorption(0); // true if n > 1

5. Sips (Langmuir-Freundlich) Isotherm

The Sips isotherm combines Langmuir saturation behaviour with Freundlich heterogeneity. It is the most versatile two-parameter extension and reduces to both Langmuir and Freundlich as special cases.

Single-Component Sips Equation

\[q = q_{max} \cdot \frac{(K_S \cdot P)^{1/n}}{1 + (K_S \cdot P)^{1/n}}\]

where:

Symbol Description Unit
$q_{max}$ Maximum adsorption capacity mol/kg
$K_S$ Sips affinity constant 1/bar
$n$ Heterogeneity parameter dimensionless

Special Cases

Condition Reduces To
$n = 1$ Langmuir isotherm
Low pressure ($K_S P \ll 1$) Freundlich: $q \approx q_{max} \cdot (K_S P)^{1/n}$
High pressure ($K_S P \gg 1$) Saturation: $q \to q_{max}$

Extended Sips (Multi-Component)

\[q_i = q_{max,i} \cdot \frac{(K_{S,i} \cdot P_i)^{1/n_i}}{1 + \displaystyle\sum_{j=1}^{N} (K_{S,j} \cdot P_j)^{1/n_j}}\]

Temperature Dependence

\[K_S(T) = K_S(T_{ref}) \cdot \exp\left(\frac{-\Delta H_{ads}}{R} \cdot \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\right)\]

Default Parameters

Parameter Default Valid Range
$q_{max}$ 5.0 mol/kg > 0
$K_S$ 0.1 1/bar > 0
$n$ 1.2 > 0 (typically 0.5–5)
$T_{ref}$ 298.15 K > 0
$\Delta H_{ads}$ −20,000 J/mol typically −5,000 to −50,000

Java API

SipsAdsorption model = new SipsAdsorption(system);
model.setSolidMaterial("MOF HKUST-1");

model.setQmax(0, 8.0);
model.setKSips(0, 0.5);
model.setNSips(0, 1.5);

model.calcAdsorption(0);       // single-component
model.calcExtendedSips(0);     // multi-component competitive
double coverage = model.getCoverage(0);

6. Dubinin-Radushkevich-Astakhov (DRA) Potential Theory

The DRA model is NeqSim’s most rigorous adsorption model. It uses Polanyi potential theory with full equation-of-state calculations to determine local composition and density within the adsorbent micropores.

Theory

The adsorbent surface creates a potential field that enhances gas-phase fugacities. The adsorption potential at distance $z$ from the surface is:

\[\varepsilon(z) = \varepsilon_0 \cdot \left[\ln\left(\frac{z_0}{z}\right)\right]^{1/\beta}\]

where:

Symbol Description Unit
$\varepsilon_0$ Characteristic adsorption energy kJ/mol
$z_0$ Limiting adsorption volume (per kg adsorbent) m$^3$/kg $\times 10^{-3}$
$\beta$ Structural heterogeneity exponent dimensionless
$z$ Integration variable (spatial coordinate) same as $z_0$

Fugacity Enhancement

Within the potential field, the fugacity of each component is enhanced:

\[f_i^{field}(z) = f_i^{bulk} \cdot \exp\left(\frac{\varepsilon(z)}{RT}\right)\]

This creates a thermodynamic driving force for molecules to accumulate near the adsorbent surface.

Surface Excess Calculation

The total surface excess of component $i$ is obtained by numerical integration over the pore volume:

\[q_i = \int_0^{z_0} \left(\frac{x_i^{local}(z)}{V_m^{local}(z)} - \frac{y_i^{bulk}}{V_m^{bulk}}\right) dz\]

where $x_i^{local}$ and $V_m^{local}$ are the local mole fraction and molar volume at position $z$, determined by solving the full phase equilibrium at the enhanced fugacity conditions.

Numerical Algorithm

The DRA implementation uses the following algorithm with 500 integration steps:

  1. Clone the thermodynamic system
  2. For each integration step $k$ from 1 to $N_{steps}$:
    • Compute point $z_k = z_0 \cdot k / N_{steps}$
    • Calculate the potential $\varepsilon(z_k)$
    • Compute the correction factor: $\text{corr} = \exp(\varepsilon / RT)$
    • Enhance fugacities: $f_i = \text{corr} \cdot f_i^{bulk}$
    • Iteratively solve for local composition (up to 100 inner iterations):
      • Update mole fractions from enhanced fugacity ratios
      • Adjust local pressure until $\sum x_i = 1$ (tolerance: $10^{-12}$)
    • Accumulate surface excess: $\Delta q_i = \Delta z \cdot (x_i / V_m^{local} - y_i / V_m^{bulk})$
  3. Sum over all steps to get total $q_i$ for each component

Key Characteristics

Parameters and Typical Values

Parameter Typical Range Physical Basis
$\varepsilon_0$ 0.05 – 16.0 kJ/mol Interaction strength (surface energy)
$z_0$ 0.1 – 0.45 m$^3$/kg $\times 10^{-3}$ Micropore volume per unit mass
$\beta$ 2.0 – 3.0 $\beta = 2$: Gaussian energy distribution (DR); $\beta \neq 2$: general DA

Java API

PotentialTheoryAdsorption model = new PotentialTheoryAdsorption(system);
model.setSolidMaterial("AC Calgon F400");
model.calcAdsorption(0);

double co2Loading = model.getSurfaceExcess("CO2");
double totalLoading = model.getTotalSurfaceExcess();

Fluid Property Estimation

Several isotherm models (BET, capillary condensation) require pure-component physical properties that may not be directly available from the thermodynamic system. The FluidPropertyEstimator utility class provides correlations for these:

Lee-Kesler Vapour Pressure

\[\ln P_r^{sat} = f^{(0)}(T_r) + \omega \cdot f^{(1)}(T_r)\] \[f^{(0)} = 5.92714 - \frac{6.09648}{T_r} - 1.28862 \ln T_r + 0.169347\, T_r^6\] \[f^{(1)} = 15.2518 - \frac{15.6875}{T_r} - 13.4721 \ln T_r + 0.43577\, T_r^6\]

where $T_r = T / T_c$ and $P_r = P / P_c$.

Rackett Liquid Molar Volume

\[V_m^{sat} = V_c \cdot Z_{RA}^{(1 - T_r)^{2/7}} \times 10^{-3}\] \[Z_{RA} = 0.29056 - 0.08775 \omega\]

Macleod-Sugden Surface Tension

\[\sigma = \left(\frac{[P] \cdot \rho_L}{10^6}\right)^4\]

with a fallback correlation:

\[\sigma = 0.02 \cdot (1 - T_r)^{1.26}\]

where $[P]$ is the parachor.

Parameter Database

All isotherm parameters are stored in AdsorptionParameters.csv and loaded automatically when calcAdsorption() is called.

Database Schema

Column Description Unit
ID Row identifier —
Name Component name (NeqSim convention) —
Solid Adsorbent material code —
eps DRA characteristic energy kJ/mol
z0 DRA limiting adsorption volume m$^3$/kg $\times 10^{-3}$
beta DRA structural heterogeneity dimensionless
qmax Langmuir/Sips maximum capacity mol/kg
K_langmuir Langmuir equilibrium constant 1/bar
C_BET BET constant dimensionless
TempRef Reference temperature K
dH_ads Heat of adsorption J/mol
K_freundlich Freundlich capacity constant mol/kg/bar$^{1/n}$
n_freundlich Freundlich intensity parameter dimensionless
K_sips Sips affinity constant 1/bar
n_sips Sips heterogeneity parameter dimensionless

Supported Adsorbent Materials (14)

Material Code Type Typical Application
AC Activated Carbon (generic) General gas separation
AC Calgon F400 Activated Carbon VOC removal, gas purification
AC Norit R1 Activated Carbon Air/gas purification
AC BPL Activated Carbon Gas-phase adsorption
Zeolite 13X Faujasite zeolite CO$_2$ removal, PSA air separation
Zeolite 5A LTA zeolite N$_2$/O$_2$ separation, gas drying
Zeolite 4A LTA zeolite Dehydration
Zeolite ZSM-5 MFI zeolite Hydrocarbon separation
Silica Gel Amorphous silica Dehydration
MOF HKUST-1 Metal-Organic Framework CO$_2$ capture, gas storage
MOF-5 Metal-Organic Framework H$_2$ storage
CMS Carbon Molecular Sieve N$_2$/O$_2$ kinetic separation
Alumina Activated alumina Dehydration, fluoride removal
MCM-41 Mesoporous silica Catalysis, controlled release

Supported Components (10)

methane, CO2, nitrogen, argon, ethane, propane, H2S, water, hydrogen, n-butane

Fallback Parameters

When specific isotherm parameters are not available in the database, models fall back to DRA parameters with the following conversions:

Isotherm Fallback Expression
Langmuir $q_{max}$ $z_0 \times 1000 / M$
Langmuir $K$ $\exp(\varepsilon_0 / (R \times 298.15))$
Langmuir $\Delta H$ $-\varepsilon_0 \times 1000$
BET $q_m$ $z_0 \times 1000 / M \times 0.4$
BET $C$ $\exp(\varepsilon_0 \times 1000 / (R \times T))$
Freundlich $K_F$ $z_0 \times 1000 / M \times 0.5$
Freundlich $n$ $1 + \varepsilon_0 / 5$
Sips $q_{max}$ $z_0 \times 1000 / M$
Sips $K_S$ $\exp(\varepsilon_0 / (R \times 298.15))$
Sips $n$ 1.2

Capillary Condensation

For mesoporous materials (pore radius 2–50 nm), adsorption can trigger capillary condensation — the liquid fills the pore below the bulk saturation pressure. This is modelled separately via CapillaryCondensationModel.

Kelvin Equation

The critical pore radius at which condensation occurs is:

\[r_K = -\frac{g_f \cdot \gamma \cdot V_m^{liq} \cdot \cos\theta}{R \cdot T \cdot \ln(P / P_0)} \times 10^9 \quad (\text{nm})\]

where:

Symbol Description Unit
$g_f$ Geometry factor (2 for cylindrical/spherical, 1 for slit) dimensionless
$\gamma$ Surface tension of adsorbate N/m
$V_m^{liq}$ Liquid molar volume m$^3$/mol
$\theta$ Contact angle radians
$P/P_0$ Relative pressure dimensionless

Condensation Onset Pressure

\[\frac{P}{P_0} = \exp\left(-\frac{g_f \cdot \gamma \cdot V_m^{liq} \cdot \cos\theta}{R \cdot T \cdot r_{eff} \times 10^{-9}}\right)\]

where $r_{eff} = r_{pore} - t_{ads}$ accounts for the pre-adsorbed layer thickness $t_{ads}$.

Pore Size Distribution (Log-Normal)

\[f(r) = \frac{1}{r \cdot \sigma_{ln} \cdot \sqrt{2\pi}} \cdot \exp\left(-\frac{(\ln r - \ln \bar{r})^2}{2\sigma_{ln}^2}\right)\]

with $\sigma_{ln} = \ln(1 + \sigma_r / \bar{r})$.

The total condensed amount is integrated over pores where $P > P_{condensation}(r)$:

\[n_{condensed} = \int_{r_{min}}^{r_{max}} \frac{V_{pore}(r)}{V_m^{liq}} \cdot f(r) \, dr \quad \text{(mol/g)}\]

Pore Geometry Types

Pore Type Geometry Factor $g_f$ Description
CYLINDRICAL 2.0 Most common, capillary-tube model
SLIT 1.0 Layered materials (clays, graphite)
SPHERICAL 2.0 Ink-bottle pores
INK_BOTTLE 2.0 Constricted-neck pores (hysteresis)

Default Parameters

Parameter Default Unit
Mean pore radius 5.0 nm
Pore radius std. dev. 2.0 nm
Min pore radius 1.0 nm
Max pore radius 25.0 nm
Total pore volume 0.5 cm$^3$/g
Contact angle 0.0 radians
Adsorbed layer thickness 0.35 nm
Integration steps 100 —

Java API

CapillaryCondensationModel ccModel = new CapillaryCondensationModel(system);
ccModel.setMeanPoreRadius(4.0);
ccModel.setPoreRadiusStdDev(1.5);
ccModel.setTotalPoreVolume(0.6);
ccModel.setPoreType(CapillaryCondensationModel.PoreType.CYLINDRICAL);

ccModel.calcCapillaryCondensation(0);
double kelvinRadius = ccModel.getKelvinRadius("CO2");
double condensate = ccModel.getCondensateAmount("CO2");

Model Selection Guide

By Application

Application Recommended Model Rationale
PSA process design Extended Langmuir or Sips Multi-component, fast computation
TSA process design Langmuir + van ‘t Hoff Temperature dependence built in
Material screening Langmuir or Freundlich Simple, fits most data
Surface area measurement BET Industry standard (ISO 9277)
Rigorous thermodynamic study DRA Full EOS treatment
Mesoporous materials BET + Capillary Condensation Multilayer + pore filling
Carbon molecular sieves DRA Micropore filling physics
MOFs and novel materials Sips Heterogeneity + saturation

By Data Availability

Available Data Recommended Model
Only $q_{max}$ and $K$ Langmuir
Heterogeneous isotherm shape Sips or Freundlich
BET surface area measurements BET
Full thermodynamic properties DRA
No data (estimation needed) Langmuir with database fallback