Adsorbed Solution Theory
Adsorbed Solution Theory
Authors
- Andrés Riedemann, University of Concepción
- Vinicius Santana, Norwegian University of Science and Technology
Adsorbed Solution Theory
Settings
This document was generated with Documenter.jl version 1.7.0 on Monday 30 September 2024. Using Julia version 1.10.5.
Adsorbed Solution Theory
Settings
This document was generated with Documenter.jl version 1.7.0 on Monday 30 September 2024. Using Julia version 1.10.5.
AdsorbedSolutionTheory.LangmuirAdsorbedSolutionTheory.RedlichPetersonAdsorbedSolutionTheory.SipsAdsorbedSolutionTheory.TothAdsorbedSolutionTheory.UnilanAdsorbedSolutionTheory.f∂fAdsorbedSolutionTheory.f∂f∂2fAdsorbedSolutionTheory.henry_coefficientAdsorbedSolutionTheory.iastAdsorbedSolutionTheory.isosteric_heatAdsorbedSolutionTheory.isotherm_lower_boundAdsorbedSolutionTheory.isotherm_upper_boundAdsorbedSolutionTheory.loadingAdsorbedSolutionTheory.nlsolveAdsorbedSolutionTheory.pressureAdsorbedSolutionTheory.saturated_loadingAdsorbedSolutionTheory.sp_resAdsorbedSolutionTheory.x_solAdsorbedSolutionTheory.@MultiSiteAdsorbedSolutionTheory.@with_metadataAdsorbedSolutionTheory.Langmuir — Type`Langmuir(M, K₀, E)`
-Langmuir <: IsothermModelLangmuir(M, K₀, E) represents the Langmuir isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]Description
The Langmuir equation is given by:
n = (M * K₀ * p) / (1 + K₀ * p)
where:
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.RedlichPeterson — Type`RedlichPeterson(M, K₀, E, f)`
+Langmuir <: IsothermModelLangmuir(M, K₀, E) represents the Langmuir isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]Description
The Langmuir equation is given by:
n = (M * K₀ * p) / (1 + K₀ * p)
where:
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.RedlichPeterson — Type`RedlichPeterson(M, K₀, E, f)`
-RedlichPeterson <: IsothermModelRedlichPeterson(M, K₀, E, f) represents the Redlich-Peterson isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]f::T: parameter characterising the heterogeneity of the system (no units)Description
The RedlichPeterson equation is given by:
n = M * p / (1 + (K₀ * p)^f)
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.Sips — Type`Sips(M, K₀, E, f)`
+RedlichPeterson <: IsothermModelRedlichPeterson(M, K₀, E, f) represents the Redlich-Peterson isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]f::T: parameter characterising the heterogeneity of the system (no units)Description
The RedlichPeterson equation is given by:
n = M * p / (1 + (K₀ * p)^f)
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.Sips — Type`Sips(M, K₀, E, f)`
-Sips <: IsothermModelSips(M, K₀, E, f) represents the Sips isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]f::T: parameter characterising the heterogeneity of the system (no units)Description
The Sips equation is given by:
n = M * (K₀ * p)^f / (1 + (K₀ * p)^f)
where:
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.Toth — TypeToth <: IsothermModel
+Sips <: IsothermModelSips(M, K₀, E, f) represents the Sips isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]f::T: parameter characterising the heterogeneity of the system (no units)Description
The Sips equation is given by:
n = M * (K₀ * p)^f / (1 + (K₀ * p)^f)
where:
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.Toth — TypeToth <: IsothermModel
-Toth(M, K₀, E, f₀, β)Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage,[1/Pa]`E::T: adsorption energy, [J/mol]f₀::T: Empirical parameter, -β::T: Empirical parameter, KDescription
Toth isotherm model:
K = K₀exp(-E/(RT)) f = f₀ + β/T nᵢ = MKP/(1 + (KP)ᶠ)¹/ᶠ
AdsorbedSolutionTheory.Unilan — Type`Unilan(M, K₀, E)`
+Toth(M, K₀, E, f₀, β)Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage,[1/Pa]`E::T: adsorption energy, [J/mol]f₀::T: Empirical parameter, -β::T: Empirical parameter, KDescription
Toth isotherm model:
K = K₀exp(-E/(RT)) f = f₀ + β/T nᵢ = MKP/(1 + (KP)ᶠ)¹/ᶠ
AdsorbedSolutionTheory.Unilan — Type`Unilan(M, K₀, E)`
-Unilan <: IsothermModelUnilan(M, K₀, E, f) represents the Unilan (UNIform distribution LANgmuir) isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]f::T: heterogeneity of the adsorbent (no units)Description
The UNILAN equation is given by:
n = M * log((1 + K₀* exp(f) * p)/(1 + K₀ * exp(-f) * p)) / (2 * f)
where:
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.f∂f — Methodf∂f(f,x)returns f and ∂f/∂x evaluated in x, using ForwardDiff.jl, DiffResults.jl and StaticArrays.jl to calculate everything in one pass.
AdsorbedSolutionTheory.f∂f∂2f — Methodf∂f∂2f(f,x)returns f,∂f/∂x,and ∂²f/∂²x and evaluated in x, using ForwardDiff.jl, DiffResults.jl and StaticArrays.jl to calculate everything in one pass.
AdsorbedSolutionTheory.henry_coefficient — Methodhenry_coefficient(model::IsothermModel, T) -> HCalculate the Henry's coefficient for a single component system using the specified isotherm model and temperature T.
Arguments
model::IsothermModel: An instance of IsothermModel, representing the isotherm model to be used for the calculation.T: The temperature at which the Henry's coefficient is to be calculated.Returns
H: The Henry's coefficient in the default units of [mol/kg].Description
This function returns the Henry's coefficient, which is a measure of the initial slope of the adsorption isotherm at low pressures. It is defined as the derivative of the loading q with respect to pressure p at p = 0:
H = (∂q/∂p) at p = 0 at a given T.
AdsorbedSolutionTheory.iast — Functioniast(models,p,T,y,method = FastIAS(),gas_model = nothing;x0 = nothing,maxiters = 100,reltol = 1e-12, abstol = 1e-10)TODO: docs
returns qtot,x,convergencesymbol (:success, or :maxiters_exceeded)
AdsorbedSolutionTheory.isosteric_heat — Methodisosteric_heat(model::IsothermModel, Vᵍ, p, T; Vᵃ = zero(eltype(model))) -> QₛₜCalculate the isosteric heat of adsorption for a given isotherm model.
Arguments
model::IsothermModel: The isotherm model used to describe the adsorption process.Vᵍ: The molar volume of the gas phase.Vᵃ: The molar volume of the adsorbed phase (typically Vᵃ << Vᵍ; default is zero).p: Pressure at which the isosteric heat is evaluated.T: Temperature at which the isosteric heat is evaluated.Returns
Qₛₜ: The estimated isosteric heat of adsorption.Description
The function estimates the isosteric heat of adsorption Qₛₜ for a single component using its isotherm and the Clausius-Clapeyron equation:
Qₛₜ = -T * (Vᵍ - Vᵃ) * (∂n/∂T)ₚ / (∂n/∂p)ₜ
where:
This equation is derived based on the Clausius-Clapeyron relation, which relates the temperature dependence of the loading to the isosteric heat.
References:
AdsorbedSolutionTheory.isotherm_lower_bound — Methodisotherm_lower_bound(model::IsothermModel)
+Unilan <: IsothermModelUnilan(M, K₀, E, f) represents the Unilan (UNIform distribution LANgmuir) isotherm model, which describes the adsorption of a gas on a solid surface.
Inputs
M::T: maximum loading capacity of the adsorbent, [mol/kg]K₀::T: equilibrium constant at zero coverage, [1/Pa]E::T: adsorption energy, [J/mol]f::T: heterogeneity of the adsorbent (no units)Description
The UNILAN equation is given by:
n = M * log((1 + K₀* exp(f) * p)/(1 + K₀ * exp(-f) * p)) / (2 * f)
where:
The adsorption energy E is related to the equilibrium constant K₀ by the equation:
K₀ = exp(-E / (R * T))
where:
AdsorbedSolutionTheory.f∂f — Methodf∂f(f,x)returns f and ∂f/∂x evaluated in x, using ForwardDiff.jl, DiffResults.jl and StaticArrays.jl to calculate everything in one pass.
AdsorbedSolutionTheory.f∂f∂2f — Methodf∂f∂2f(f,x)returns f,∂f/∂x,and ∂²f/∂²x and evaluated in x, using ForwardDiff.jl, DiffResults.jl and StaticArrays.jl to calculate everything in one pass.
AdsorbedSolutionTheory.henry_coefficient — Methodhenry_coefficient(model::IsothermModel, T) -> HCalculate the Henry's coefficient for a single component system using the specified isotherm model and temperature T.
Arguments
model::IsothermModel: An instance of IsothermModel, representing the isotherm model to be used for the calculation.T: The temperature at which the Henry's coefficient is to be calculated.Returns
H: The Henry's coefficient in the default units of [mol/kg].Description
This function returns the Henry's coefficient, which is a measure of the initial slope of the adsorption isotherm at low pressures. It is defined as the derivative of the loading q with respect to pressure p at p = 0:
H = (∂q/∂p) at p = 0 at a given T.
AdsorbedSolutionTheory.iast — Functioniast(models,p,T,y,method = FastIAS(),gas_model = nothing;x0 = nothing,maxiters = 100,reltol = 1e-12, abstol = 1e-10)TODO: docs
returns qtot,x,convergencesymbol (:success, or :maxiters_exceeded)
AdsorbedSolutionTheory.isosteric_heat — Methodisosteric_heat(model::IsothermModel, Vᵍ, p, T; Vᵃ = zero(eltype(model))) -> QₛₜCalculate the isosteric heat of adsorption for a given isotherm model.
Arguments
model::IsothermModel: The isotherm model used to describe the adsorption process.Vᵍ: The molar volume of the gas phase.Vᵃ: The molar volume of the adsorbed phase (typically Vᵃ << Vᵍ; default is zero).p: Pressure at which the isosteric heat is evaluated.T: Temperature at which the isosteric heat is evaluated.Returns
Qₛₜ: The estimated isosteric heat of adsorption.Description
The function estimates the isosteric heat of adsorption Qₛₜ for a single component using its isotherm and the Clausius-Clapeyron equation:
Qₛₜ = -T * (Vᵍ - Vᵃ) * (∂n/∂T)ₚ / (∂n/∂p)ₜ
where:
This equation is derived based on the Clausius-Clapeyron relation, which relates the temperature dependence of the loading to the isosteric heat.
References:
AdsorbedSolutionTheory.isotherm_lower_bound — Methodisotherm_lower_bound(model::IsothermModel)
isotherm_lower_bound(T,model::IsothermModel)
-isotherm_lower_bound(T,::Type{M}) where M <:IsothermModelReturns the lower bound for the parameters of the isotherm model model of type M. with number type T, as a Ntuple{model_length(M),T}. The default assumes that all parameters are nonnegative.
AdsorbedSolutionTheory.isotherm_upper_bound — Methodisotherm_upper_bound(model::IsothermModel)
+isotherm_lower_bound(T,::Type{M}) where M <:IsothermModelReturns the lower bound for the parameters of the isotherm model model of type M. with number type T, as a Ntuple{model_length(M),T}. The default assumes that all parameters are nonnegative.
AdsorbedSolutionTheory.isotherm_upper_bound — Methodisotherm_upper_bound(model::IsothermModel)
isotherm_upper_bound(T,model::IsothermModel)
-isotherm_upper_bound(T,::Type{M}) where M <:IsothermModelReturns the upper bound for the parameters of the isotherm model model of type M. with number type T, as a Ntuple{model_length(M),T}. The default assumes no upper bound for the parameters.
AdsorbedSolutionTheory.loading — Methodloading(model::IsothermModel, p, T) -> qCalculate the loading q based on the provided isotherm model, pressure p, and temperature T.
Arguments
model::IsothermModel: An instance of IsothermModel, representing the isotherm model to be used for the calculation.p: The pressure at which the loading is to be calculated.T: The temperature at which the loading is to be calculated.Returns
q: The calculated loading based on the isotherm model, pressure, and temperature.Description
This function computes the loading q based on the given isotherm model, pressure p, and temperature T.
AdsorbedSolutionTheory.nlsolve — Functionfunction nlsolve(f!,x0,method=TrustRegion(Newton(), Dogleg()), options=NEqOptions(),chunk = ForwardDiff.Chunk{2}())Given a function f!(result,x) that returns a system of equations, nlsolve(f!,x0) returns a NLSolvers.ConvergenceInfo struct that contains the results of the non-linear solving procedure.
Uses NLSolvers.jl as backend, the jacobian is calculated with ForwardDiff.jl, with the specified chunk size
To obtain the underlying solution vector, use x_sol
To see available solvers and options, check NLSolvers.jl
AdsorbedSolutionTheory.pressure — Methodpressure(model::IsothermModel, x, T, f; approx = :exact)
given an isotherm::IsothermModel and x = f(model,p,T), find p such that f(model,p,T) = x. There are two options for f and x:
f is sp_res, then x = sp_res(model, p, T)f is loading, then x = loading(model, p, T)By default, it performs a root-finding over the isotherm. but custom implementations can be done by overloading pressure_impl(model::MyModel,x,T,f::typeof(f),approx) The approx::Symbol argument indicates if the procedure is exact or approximate. by default a henry coefficient aproximation is used when approx =:henry is used.
AdsorbedSolutionTheory.saturated_loading — Methodsaturated_loading(model::IsothermModel, T)Returns the loading of of an adsorption isotherm when the pressure tends to infinity. by default it is evaluated at 1/√eps(eltype(model)) (6.7108864e7 for Float64 inputs.).
Inputs
AdsorbedSolutionTheory.sp_res — Methodsp_res(model::IsothermModel, p, T) -> ΠCalculate the reduced spreading pressure for a given isotherm model at a specific pressure p and temperature T.
Arguments
model::IsothermModel: An instance of IsothermModel, representing the isotherm model used for the calculation.p: The pressure at which the reduced spreading pressure is to be calculated.T: The temperature at which the reduced spreading pressure is to be calculated.Returns
Π: The reduced spreading pressure Description
The reduced spreading pressure is a key quantity in Ideal Adsorbed Solution Theory (IAST), used to describe the adsorption behavior of mixtures. This function calculates the reduced spreading pressure Π by integrating the isotherm equation over the pressure range from 0 to p.
The reduced spreading pressure is often calculated numerically as:
Π = ∫ (q(p') / p') dp' from 0 to p
where:
q(p') is the loading at pressure p'.AdsorbedSolutionTheory.x_sol — Methodx_sol(res::NLSolvers.ConvergenceInfo)Returns the scalar or vector x that solves the system of equations or is the minimizer of an optimization procedure.
AdsorbedSolutionTheory.@MultiSite — Macro@MultiSite(isotherms)Utility macro to build MultiSite types.
Example:
julia> v1 = @MultiSite{Langmuir,Langmuir} #abstract eltype
+isotherm_upper_bound(T,::Type{M}) where M <:IsothermModelReturns the upper bound for the parameters of the isotherm model model of type M. with number type T, as a Ntuple{model_length(M),T}. The default assumes no upper bound for the parameters.
AdsorbedSolutionTheory.loading — Methodloading(model::IsothermModel, p, T) -> qCalculate the loading q based on the provided isotherm model, pressure p, and temperature T.
Arguments
model::IsothermModel: An instance of IsothermModel, representing the isotherm model to be used for the calculation.p: The pressure at which the loading is to be calculated.T: The temperature at which the loading is to be calculated.Returns
q: The calculated loading based on the isotherm model, pressure, and temperature.Description
This function computes the loading q based on the given isotherm model, pressure p, and temperature T.
AdsorbedSolutionTheory.nlsolve — Functionfunction nlsolve(f!,x0,method=TrustRegion(Newton(), Dogleg()), options=NEqOptions(),chunk = ForwardDiff.Chunk{2}())Given a function f!(result,x) that returns a system of equations, nlsolve(f!,x0) returns a NLSolvers.ConvergenceInfo struct that contains the results of the non-linear solving procedure.
Uses NLSolvers.jl as backend, the jacobian is calculated with ForwardDiff.jl, with the specified chunk size
To obtain the underlying solution vector, use x_sol
To see available solvers and options, check NLSolvers.jl
AdsorbedSolutionTheory.pressure — Methodpressure(model::IsothermModel, x, T, f; approx = :exact)
given an isotherm::IsothermModel and x = f(model,p,T), find p such that f(model,p,T) = x. There are two options for f and x:
f is sp_res, then x = sp_res(model, p, T)f is loading, then x = loading(model, p, T)By default, it performs a root-finding over the isotherm. but custom implementations can be done by overloading pressure_impl(model::MyModel,x,T,f::typeof(f),approx) The approx::Symbol argument indicates if the procedure is exact or approximate. by default a henry coefficient aproximation is used when approx =:henry is used.
AdsorbedSolutionTheory.saturated_loading — Methodsaturated_loading(model::IsothermModel, T)Returns the loading of of an adsorption isotherm when the pressure tends to infinity. by default it is evaluated at 1/√eps(eltype(model)) (6.7108864e7 for Float64 inputs.).
Inputs
AdsorbedSolutionTheory.sp_res — Methodsp_res(model::IsothermModel, p, T) -> ΠCalculate the reduced spreading pressure for a given isotherm model at a specific pressure p and temperature T.
Arguments
model::IsothermModel: An instance of IsothermModel, representing the isotherm model used for the calculation.p: The pressure at which the reduced spreading pressure is to be calculated.T: The temperature at which the reduced spreading pressure is to be calculated.Returns
Π: The reduced spreading pressure Description
The reduced spreading pressure is a key quantity in Ideal Adsorbed Solution Theory (IAST), used to describe the adsorption behavior of mixtures. This function calculates the reduced spreading pressure Π by integrating the isotherm equation over the pressure range from 0 to p.
The reduced spreading pressure is often calculated numerically as:
Π = ∫ (q(p') / p') dp' from 0 to p
where:
q(p') is the loading at pressure p'.AdsorbedSolutionTheory.x_sol — Methodx_sol(res::NLSolvers.ConvergenceInfo)Returns the scalar or vector x that solves the system of equations or is the minimizer of an optimization procedure.
AdsorbedSolutionTheory.@MultiSite — Macro@MultiSite(isotherms)Utility macro to build MultiSite types.
Example:
julia> v1 = @MultiSite{Langmuir,Langmuir} #abstract eltype
MultiSite{T, Tuple{Langmuir{T}, Langmuir{T}}} where T
julia> AdsorbedSolutionTheory.from_vec(v,[1,2,0,3,4,0])
@@ -23,7 +23,7 @@
MultiSite{Float64, Tuple{Langmuir{Float64}, Langmuir{Float64}}}
julia> AdsorbedSolutionTheory.from_vec(v2,[1,2,0,3,4,0])
-MultiSite{Float64, Tuple{Langmuir{Float64}, Langmuir{Float64}}}((Langmuir{Float64}(1.0, 2.0, 0.0), Langmuir{Float64}(3.0, 4.0, 0.0)))AdsorbedSolutionTheory.@with_metadata — Macro@with_metadata(struct_expr)macro that allows to define an isotherm model with additional metadata, about parameter bounds and descriptions of parameters:
Usage:
AdsorbedSolutionTheory.@with_metadata struct MyIsotherm{T} <: IsothermModel{T}
+MultiSite{Float64, Tuple{Langmuir{Float64}, Langmuir{Float64}}}((Langmuir{Float64}(1.0, 2.0, 0.0), Langmuir{Float64}(3.0, 4.0, 0.0)))AdsorbedSolutionTheory.@with_metadata — Macro@with_metadata(struct_expr)macro that allows to define an isotherm model with additional metadata, about parameter bounds and descriptions of parameters:
Usage:
AdsorbedSolutionTheory.@with_metadata struct MyIsotherm{T} <: IsothermModel{T}
A::T,(0,1),"field A" #bounds and description provided
B::T #nothing provided
C::T,(1,10) #only bounds provided
@@ -32,4 +32,4 @@
from_vec(MyIsotherm,(1,2,3,4)) #ok
from_vec(MyIsotherm,(-1,2,3,4)) #ArgumentError: MyIsotherm: value for the field `A` (field A) is out of the parameter bounds: (0.0 <= -1 <= 1.0) == false
-from_vec(MyIsotherm,(1,2,-3,4)) #ArgumentError: MyIsotherm: value for the field `C` is out of the parameter bounds: (1.0 <= -3 <= 10.0) == falseSettings
This document was generated with Documenter.jl version 1.7.0 on Monday 30 September 2024. Using Julia version 1.10.5.
Settings
This document was generated with Documenter.jl version 1.7.0 on Monday 30 September 2024. Using Julia version 1.10.5.