Getting Started with Langmuir.jl

This is an introductory tutorial for Langmuir.jl. We will demonstrate the basics of the package by building an isotherm model and estimating properties with it.

Installing Langmuir.jl

To install Langmuir, use the Julia package manager.

using Pkg; Pkg.add("Langmuir")

Initializing an IsothermModel and estimating properties for single component adsorption

In this package, we support several isotherm models. You can refer to the list of supported models here. Here is how you construct a LangmuirS1 model.

M = 1.913 # mol.kg⁻¹
K₀ = 6.82e-10 # Pa⁻¹
E  = -21_976.40 # J.mol⁻¹
isotherm = LangmuirS1(M, K₀, E)

LangmuirS1{Float64}(1.913, 6.82e-10, -21976.4)

You can use a instantiated model to estimate the equilibrium properties of the adsorption system. To estimate the loading (amount of adsorbate per mass of adsorbent) in the adsorbent, given the temperature T and pressure p, you can do as follows:

p = 101325.0
T = 298.15
l = loading(isotherm, p, T)

0.6285277134382115

When estimating loading with a model, it is common to plot isotherms, i.e., pressure vs loading for a fixed temperature. To do it, you can use the loading_at_T(isotherm, P, T) function.

P = 0.0:5_000.0:100_000.0 |> collect
l_at_300 = loading_at_T(isotherm, P, 300.)
l_at_350 = loading_at_T(isotherm, P, 350.)
plot(P, l_at_300, size = (500, 250), label = "300K")
plot!(P, l_at_350, label = "350K")
xlabel!("P (Pa)")
ylabel!("l (mol/kg)")

You can also estimate other properties from the isotherm such as the henry coefficient at a given temperature by calling henry_coefficient(model::IsothermModel, T). The henry coefficient should correspond to the slope of the isotherm when $P \rightarrow 0.0$. In Langmuir.jl, this is obtained using automatic differentiation and introduces no numerical error in the estimate. You can see in the example below how to visualize the tangent line built from the henry coefficient at $300K$.

P_ = P[1:3]
plot(P_, l_at_300[1:3], size = (500, 250), label = "300K")
H = henry_coefficient(isotherm, 300.0)
plot!(P_, H*P_, label = "Tangent line")

To finish this section for single component adsorption, one can also estimate the isosteric heat of adsorption by calling isosteric_heat(model, Vg, p, T) where Vg is the molar volume of the gas phase, p the pressure in Pascal and T the temperature in Kelvin. For the Langmuir model, the isosteric heat should be constant and equal to the energy parameter E. You can plot the isosteric heat either as a function of the pressure or loading.

Below it is assumed that the ideal gas law is a good approximation to describe the molar volume of the gas phase.

import Langmuir: Rgas
ΔH = map(P -> isosteric_heat(isotherm, P, 300.), P[2:end]) |> x -> round.(x, digits = 7)
scatter(l_at_300[2:end], ΔH, size = (500, 250),  ylabel = "Isosteric heat (J/mol)", xlabel = "loading (mol/kg)", label = "Estimated isosteric heat with AD")
plot!([first(l_at_300), last(l_at_300)], [E, E], label = "Expected value")

Estimating properties in multicomponent adsorption.

When it comes to estimating properties in multicomponent adsorption, the Ideal Adsorption Solution Theory (IAST) has been proven accurate for a number of systems. It allows one to estimate multicomponent adsorption behavior from single component isotherms.

When formulated, estimating the loading with IAST becomes a nonlinear solve problem which can be solved in different ways. Here, we support the Nested Loop and FastIAS methods. To know more about the two and which one to choose, refer to this paper: 10.1002/aic.14684.

It can be shown analytically that IAST estimation of multicomponent loading is the same as the extendend Langmuir method when the parameter $M_i$ (saturation loading) are the same for all components, i.e., $M_1 = M_2 = ... = M_{N_c}$. The extended langmuir has the form $n_i = \frac{M_i \times K_{i,0} \exp{\frac{\Delta H}{RT}}}{1 + \sum_i K_i \times P_i}$. Below you can see a numerical verification of IAST for that condition.

using Langmuir
import Langmuir: Rgas
isotherm_1 = LangmuirS1(1.913, 6.82e-10, -21_976.40)
isotherm_2 = LangmuirS1(1.913, 1.801e-9, -16_925.01)
models = (isotherm_1, isotherm_2)
(n_total, x, is_success) = iast(models, 101325.0, 300., [0.5, 0.5], FastIAS())
loading_1 =  n_total*x[1]
loading_2 = n_total*x[2]

K_1 = 6.82e-10*exp(21_976.40/Rgas(isotherm_1)/300.)
K_2 = 1.801e-9*exp(16_925.01/Rgas(isotherm_2)/300.)
p_1 = 0.5*101325.0
p_2 = 0.5*101325.0
loading_1_expected = 1.913*K_1*p_1/(1.0 + K_1*p_1 + K_2*p_2)
loading_2_expected = 1.913*K_2*p_2/(1.0 + K_1*p_1 + K_2*p_2)

println("IAST estimated loading for component 1 is: ", round(loading_1, digits = 4))
println("Extende langmuir estimated loading for component 1 is: ", round(loading_1_expected, digits = 4))

IAST estimated loading for component 1 is: 0.3377
Extende langmuir estimated loading for component 1 is: 0.3377