EntropyScaling.jl

This is an implementation of the entropy scaling framework introduced in

S. Schmitt, H. Hasse, S. Stephan, Entropy Scaling Framework for Transport Properties using Molecular-based Equations of State, Journal of Molecular Liquids 395 (2024) 123811. [engrxiv]

which enables modelling transport properties (viscosity, thermal conductivity, selff-diffusion coefficient) based on equations of state (EOS).

Installation

To register the module locally, type

Pkg> add https://github.com/se-schmitt/EntropyScaling.jl

in package mode (type ] in REPL to enter Pkg mode).

Then, the module can be loaded by

using EntropyScaling

Module

The module provides two main functions: fit_entropy_scaling and call_entropy_scaling.

Function fit_entropy_scaling

Input:

• T::Vector{Float64}: temperature $T$ vector ${\rm K}$
• ϱ::Vector{Float64}: density $\rho$ vector (${\rm kg/m^{3}}$)
• Y::Vector{Float64}: transport property $Y$ vector, can be
  • viscosity $\eta$ in ${\rm Pa \cdot s}$
  • thermal conductivity $\lambda$ in ${\rm W / (m K)}$
  • self-diffusion coefficient $D$ in ${\rm m^2/s}$
• prop::String: property string, either vis for viscosity, tcn for thermal conductivity, or D for self-diffusion coefficient
• sfun::Function: function to calculate the configurational entropy $ s $ in ${\rm J / (K mol)}$ of the form sfun(T,ϱ,x) (vectorized)
• Bfun::Function: function to calculate the 2nd virial coefficient $B$ in ${\rm m^3 / mol}$ of the form Bfun(T) (vectorized)
• dBdTfun::Function: function to calculate the temperature derivative of the 2nd virial coefficient ${\rm d} B / {\rm d} T$ in ${\rm m^3/(mol K)}$ of the form dBdTfun(T) (vectorized)
• Tc::Float64: critical temperature $T_{\rm c}$ in ${\rm K}$
• pc::Float64: critical pressure $p_{\rm c}$ in ${\rm Pa}$
• M::Float64: molar mass $M$ in ${\rm kg / mol}$
• i_fit: vector determining which parameters of the entropy scaling model should be fitted $(\alpha_{0,i}, \alpha_{{\rm ln},i}, \alpha_{1,i}, \alpha_{2,i}, \alpha_{3,i})$ (if not specified: i_fit=[0,1,1,1,1])
• m_EOS: segment number of the applied EOS (if not specified, m_EOS=1.0)

Output:

• α_par: Fitted component specific parameters $(\alpha_{0,i}, \alpha_{{\rm ln},i}, \alpha_{1,i}, \alpha_{2,i}, \alpha_{3,i})$
• Yˢ: CE-scaled transport property (dimensionless), either $\widehat{\eta}^{+}$, $\widehat{\lambda}^{+}$, or $\widehat{D}^{+}$
• s: reduced configurational entropy $\tilde{s}_{\rm conf}$

Function call_entropy_scaling

Input:

• T::Vector{Float64}: temperature $T$ vector in ${\rm K}$
• ϱ::Vector{Float64}: density $\rho$ vector in ${\rm kg/m^3}$
• α_par: Fitted component specific parameters $(\alpha_{0,i}, \alpha_{{\rm ln},i}, \alpha_{1,i}, \alpha_{2,i}, \alpha_{3,i})$
• prop::String: property string, either vis for Viscosity, tcn for thermal conductivity, or D for self-diffusion coefficient
• x::Matrix{Float64}: mole fraction matrix (rows: state points, columns: components)
• sfun::Function: function to calculate the entropy $s$in ${\rm J / (K mol)}$ of the form sfun(T,ϱ,x) (vectorized)
• Bfun::Vector: array of functions to calculate the 2nd virial coefficient $B$ in ${\rm m^3 / mol}$ of the form Bfun(T) (vectorized)
• dBdTfun::Vector: array of function to calculate the temperature derivative of the 2nd virial coefficient ${\rm d} B / {\rm d} T$ in ${\rm m^3/(mol K)}$ of the form dBdTfun(T) (vectorized)
• Tc::Float64: vector of critical temperatures $T_{{\rm c},i}$ in ${\rm K}$
• pc::Float64: vector of critical pressures $p_{{\rm c},i}$ in ${\rm Pa}$
• M::Float64: vector of molar masses $M_i$ in ${\rm kg/mol}$
• m_EOS: vector of segment numbers of the applied EOS

Output:

• Y: calculated transport property $Y$ vector, can be
  • viscosity $\eta$ in ${\rm Pa \cdot s}$
  • thermal conductivity $\lambda$ in ${\rm W/ (m K)}$
  • self-diffusion coefficient $D$ in ${\rm m^2/s}$

Example

The application of the module to methane modeled by the Peng-Robinson (PR) EOS is shown here. The corresponding files are available in the example folder.

First, the required packages (including EntropyScaling) are loaded and the experimental data are loaded.

# Implementation of the Peng-Robinson (PR) equation of state for methane
using DelimitedFiles, Roots
using PyCall, PyPlot
using EntropyScaling

# Read experiemntal data
header = split(readline("example/methane_vis.csv"),',')
dat = readdlm("example/methane_vis.csv",',',skipstart=1)
T_exp = Float64.(dat[:,findfirst(header.=="T")])                # K
p_exp = Float64.(dat[:,findfirst(header.=="p")])                # MPa
η_exp = Float64.(dat[:,findfirst(header.=="eta")])              # Pa s
state = dat[:,findfirst(header.=="state")]                      # -
ref = dat[:,findfirst(header.=="shortref")]                     # -

Next, the PR EOS is provided with the parameters for methane. No EOS are implemented in this repository. The EOS calculations have to be done externally.

# Parameters for methane
Tc = 190.4                                                      # K
pc = 4.6e6                                                      # Pa
ω = 0.011                                                       # -
M = 0.016043                                                    # kg mol⁻¹

# Load PR equation of state functions
include("PR.jl")
(p_PR, s_PR, B_PR, dBdT_PR) = PR(Tc, pc, ω, M)

# Implicit calculation of density
ϱ_exp = Float64[]
for (i,state_i) in enumerate(state)
    rts = find_zeros(x -> p_PR(T_exp[i],x,1.0) - p_exp[i]*1e6,0,M/(0.07780*EntropyScaling.R*Tc/pc ))
    if state_i in ["G","SCR"]
        push!(ϱ_exp,rts[1])
    else
        push!(ϱ_exp,rts[end])
    end
end

Now, the fit_entropy_scaling function is executed and the result is plotted in a $\tilde{s}-\widehat{\eta}^{+}$ plot using the PyPlot package.

# Fit entropy scaling
α_CH₄, ηˢ, s = fit_entropy_scaling(T_exp, ϱ_exp, η_exp, "vis"; sfun=s_PR, Bfun=B_PR, dBdTfun=dBdT_PR, Tc=Tc, pc=pc, M=M)

# Plot results
figure()
xs = [0.:0.01:1.1*maximum(s);]
plot(xs,EntropyScaling.fun_es(xs,α_CH₄;prop="vis"),"k-",zorder=0)
scat=scatter(s,log.(ηˢ),c=T_exp,marker="o",s=15,zorder=10)
xlabel(L"\tilde{s}_{\rm conf}",fontsize=10)
ylabel(L"\ln\left(\widehat{\eta}^{+}\right)",fontsize=10)
xlim(minimum(xs),maximum(xs))
cb = colorbar(scat)
cb.ax.set_title(L"T\,/\,{\rm K}",fontsize=10)
tight_layout()

At the end, the application of the adjusted parameters is demonstrated by computing muliple isotherms and comparing them to experimental data.

# Calculation of the viscosity along isotherms
what_ref = ref .== "Hellemans et al. (1970) [A]" .&& T_exp .< 140.0
T_iso = unique(T_exp[what_ref])

# Plot experimental data
figure()
scatter(p_exp[what_ref],η_exp[what_ref].*1e3,marker="o",c=T_exp[what_ref],facecolor="white",s=15.0)
cb = colorbar()
cb.ax.set_title(L"T\,/\,{\rm K}",fontsize=10)

for Ti in T_iso
    what_i = what_ref .&& T_exp .== Ti
    pi = LinRange(0.01,11.0,20)
    coli = get_cmap()((Ti-minimum(T_iso))/(maximum(T_iso)-minimum(T_iso)))
    
    # Implicit calculation of density
    ϱi = Float64[]
    for pj in pi push!(ϱi,find_zeros(x -> p_PR(Ti,x,1.0) - pj*1e6,0,M/(0.07780*EntropyScaling.R*Tc/pc ))[end]) end
    
    # Calculate and plot isotherm
    η_i = call_entropy_scaling(repeat([Ti], length(pi)), ϱi, [α_CH₄], "vis"; sfun=s_PR, Bfun=[B_PR], dBdTfun=[dBdT_PR], Tc=[Tc], pc=[pc], M=[M])
    plot(pi,η_i.*1e3,"-",c=coli,linewidth=1.0)
end
plot(NaN,NaN,"-k",label="Entropy scaling")
scatter(NaN,NaN,marker="o",s=15,c="k",label="Hellemans et al. (1970)")
legend(loc="upper left",fontsize=10,frameon=false)
xlim(0,11)
ylim(0.06,0.22)
xlabel(L"p\;/\;{\rm MPa}",fontsize=10)
ylabel(L"\eta\;/\;{\rm mPa\,s}",fontsize=10)
tight_layout()

Cite

@article{entropy_scaling_framework_2023,
    title={Entropy Scaling Framework for Transport Properties using Molecular-based Equations of State},
    author={Sebastian Schmitt, Hans Hasse, and Simon Stephan},
    journal={Journal of Molecular Liquids},
    volume={395},
    pages={123811},
    year={2024},
    doi={10.1016/j.molliq.2023.123811},
    url={https://doi.org/10.1016/j.molliq.2023.123811}
}