TransportProperties

TransportProperties lets you calculate the viscosity, thermal conductivity and diffusion coefficients. The code relies on transport.dat for the evaluation of these properties. A sample of the transport.dat file is shown below.

AR                 0   136.500     3.330     0.000     0.000     0.000
CH4                2   141.400     3.746     0.000     2.600    13.000
CO2                1   244.000     3.763     0.000     2.650     2.100
CO                 1    98.100     3.650     0.000     1.950     1.800
H2                 1    38.000     2.920     0.000     0.790   280.000
H2O                2   572.400     2.605     1.844     0.000     4.000
O2                 1   107.400     3.458     0.000     1.600     3.800

The different columns present in the above table is described below.

• Column-1: Name of species
• Column-2: Geometric configuration of the species; 0- single atom, 1- linear molecule, 2-nonlinear molecule
• Column-3: Lennard-Jones potential well depth expressed as $\epsilon/k_B$ and has units K
• Column-4: Lennard-Jones collision diameter ($\sigma$) in Angstroms
• Column-5: Dipole moment ($\mu$) in Debye
• Column-6: Polarizability ($\alpha$) in cubic Angstroms
• Column-7: Rotational relaxation collision number ($Z_\mathrm{rot}$) at 298 K

Documentation for TransportProperties.

Installation

To install the package, use the following commands in the julia REPL

julia> using Pkg
julia> Pkg.add("TransportProperties")

General interfaces

• TransportProperties.D_ii!
• TransportProperties.D_ij
• TransportProperties.D_km!
• TransportProperties.D_km!
• TransportProperties.create_transport_data
• TransportProperties.thermal_coductivity
• TransportProperties.transport_properties
• TransportProperties.viscosity
• TransportProperties.viscosity

D_ii!(D::Array{Float64},sp_trd::Array{TransportData}, T::Float64, p::Float64, molwt::Array{Float64})

D_ij(sp_trd, T, p, molwt)

D_km!(Dkm, sp_trd, T, p, molwt, molefracs)

create_transport_data(gasphase, trans_file)

viscosity(sp_trd, T, molwt, mole_fracs)

viscosity(sp_trd, T, molwt)

Diffusion coefficient calculation

Binary diffusion coefficients

The binary diffusion coefficient is expressed as

\[D_{jk} = \frac{3}{16} \frac{ \sqrt{2\pi N_Ak_B^3T^3/m_{jk}} }{p\pi \sigma_{jk}^2 \Omega^{(1,1)} }\]

Here $N_A$ is the Avogadro's number, $k_B$ is the Boltzmann constant, $p$ is the pressure, and $T$ is the temperature. The reduced molar mass of the species pair (j,k) is defined as

\[m_{jk} = \frac{m_jm_k}{m_j+m_k}\]

The collision integral $\Omega^{(1,1)}$ is determined using the reduced temperature $T^*_{jk}$ and the reduced dipole moment $\delta^*_{jk}$

\[T^*_{jk} = \frac{k_BT}{\epsilon_{jk}}\]

\[\delta^*_{jk} = \frac{1}{2} \frac{\mu^2_{jk}}{\epsilon_{jk}\sigma_{jk}^3}\]

The reduced dipole moment depends on the polarizability of the interacting molecules. In the case where both molecules are either polar or non-polar, then it follows

\[\epsilon_{jk} = \sqrt{\epsilon_j \epsilon_k}\]

\[\mu_{jk}^2 = \mu_j \mu_k\]

and the reduced collision diameter $\sigma_{jk}$ is defined as

\[\sigma_{jk} = \frac{\sigma_k+\sigma_j}{2}\]

In the case of interaction between a polar and non-polar molecule

\[\epsilon_{jk}= \xi^2 \sqrt{\epsilon_j \epsilon_k}\]

\[\sigma_{jk} = \frac{1}{2} (\sigma_j + \sigma_k) \xi^{-1/6}\]

\[\mu_{jk}= 0\]

\[\xi = 1+\frac{1}{4} \alpha_n^* \mu_p^* \sqrt{ \frac{\epsilon_p}{\epsilon_n} }\]

Note that the subscripts $p$ and $n$ represents either $j$ or $k$. If $j$ is polar species, then the subscript $p$ refers to that species

\[\alpha_n^* = \frac{\alpha_n}{\sigma_n^3}\]

\[\mu_p^* = \frac{\mu_p}{\sqrt{\epsilon_p\sigma_p^3}}\]

The estimation of $\Omega^{(1,1)}$ is a table look up procedure using the values of $T^*_{jk}$ and $\delta^*_{jk}$.

Mixture diffusion coefficients

The mixture diffusion coefficients are calculated from

\[D_{k,m} = \frac{1-Y_k}{\sum_{j\ne k}^N X_j/D_{jk}}\]

In the above equation, $Y_k$ is the mass fraction of the species $k$, and $X_j$ is the mole fraction of species $j$. $N$ is the total number of species

Viscosity

The viscosity of the mixture is calculated using pure species viscosity. The pure species viscosity is defined as

\[\eta_k = \frac{5}{16} \frac{\sqrt{\pi m_k k_BT/N_A}}{\pi \sigma_k^2 \Omega^{(2,2)}}\]

The Lennard-Jones collision integral $\Omega^{(2,2)}$ is estimated using a table lookup procedure and depends on the reduced temperature.

\[T^*_k = \frac{k_BT}{\epsilon_k}\]

and the reduced dipole moment

\[\delta^*_k = \frac{1}{2} \frac{\mu_k^2}{\epsilon_k \sigma_k^3}\]

The viscosity of the mixture is then defined as

\[\eta = \sum_{k=1}^{N} \frac{X_k\eta_k}{\sum{j=1}^K X_j\Phi_{kj}}\]

\[\Phi_{kj}= \frac{1}{\sqrt{8}} \left(1+\frac{M_k}{M_j}\right)^{-1/2} \left( 1+ \left(\frac{\eta_k}{\eta_j}\right)^{1/2} \left(\frac{M_j}{M_k}\right)^{1/4}\right)^2\]

Thermal conductivity

Similar to calculating viscosity, the thermal conductivity of a mixture is calculated from the pure species thermal conductivity. The pure species thermal conductivity is defined as

\[\lambda_k = \frac{\eta_k}{M_k} \left( f_t C_{v,t} + f_r C_{v,r} + f_v C_{v,v} \right)\]

\[f_t = \frac{5}{2}\left( 1-\frac{2}{\pi} \frac{C_{v,r}}{C_{v,t}} \frac{A}{B} \right)\]

\[f_v = \frac{\rho D_{kk}}{\eta_k}\]

\[f_r = f_v \left( 1+ \frac{2}{\pi} \frac{A}{B}\right)\]

\[A = 2.5-f_v\]

\[B = Z_{rot} + \frac{2}{\pi} \left( \frac{5}{3} \frac{C_{v,r}}{R} + f_v\right)\]

The molar heat capacity $C_v$ for rotational, vibrational or transnational mode depends on the molecule's geometry. For linear molecules

\[\frac{C_{v,t}}{R} = \frac{3}{2}\]

\[\frac{C_{v,r}}{R} = 1\]

\[\frac{C_{v,v}}{R} = C_v - 2.5R\]

\[C_v = C_p -R\]

For non-linear molecules

\[\frac{C_{v,t}}{R} = \frac{3}{2}\]

\[\frac{C_{v,r}}{R} = \frac{3}{2}\]

\[\frac{C_{v,v}}{R} = C_v - 3R\]

\[C_v = C_p -R\]

Executing the code

To see all properties as a screen output

julia>using TransportProperties
julia>transport_properties("transport.xml", "lib_dir")

In the above call, it is assumed that the input file transport.xml is present in the working directory and lib_dir is the path to the lib directory relative to the current working directory. The structure of the transport.xml input file is shown below.

<?xml version="1.0" encoding="ISO-8859-1"?>
<trans>
	<gasphase>CH4 H2O H2 CO CO2 O2 N2</gasphase>
	<molefractions>CH4=0.125, H2O=0.252, CO2=0.084, N2=0.539</molefractions>
	<T>1073.15</T>
	<p>1e5</p>
</trans>

The meaning of the different xml elements are as follows

• <gasphase> : list of gasphase species separated by space
• <molefractions> : mole fractions of the species (instead of <molefractions>, <massfractions> may also be specified)
• <T> : temperature in K
• <p> : pressure in Pa

Input file download

The xml input file and the lib directory containig other required input files may be downloaded from here.

Calculation of properties

The following methods may be used to calculate the properties of pure species or mixtures

Pure species voscosity

julia>using TransportProperties, IdealGas
julia>gasphase = ["CH4", "CO2", "H2O", "H2", "CO"]
julia>sp_tr_data = create_transport_data(gasphase,"transport.dat")
julia>thermo_all = IdealGas.create_thermo(gasphase,"therm.dat")
julia>mu = viscosity(sp_tr_data,T,thermo_all.molwt)

Mixture viscosity

julia>using TransportProperties, IdealGas
julia>gasphase = ["CH4", "CO2", "H2O", "H2", "CO"]
julia>sp_tr_data = create_transport_data(gasphase,"transport.dat")
julia>thermo_all = IdealGas.create_thermo(gasphase,"therm.dat")
julia>mu = viscosity(sp_tr_data,T,thermo_all.molwt,mole_fracs)    

Binary diffusion coefficients

julia>using TransportProperties, IdealGas
julia>gasphase = ["CH4", "CO2", "H2O", "H2", "CO"]
julia>sp_tr_data = create_transport_data(gasphase,"transport.dat")
julia>thermo_all = IdealGas.create_thermo(gasphase,"therm.dat")
julia>Dij = D_ij(sp_tr_data,T,p,thermo_all.molwt)

Mixture diffusion coefficients

julia>using TransportProperties, IdealGas
julia>gasphase = ["CH4", "CO2", "H2O", "H2", "CO"]
julia>sp_tr_data = create_transport_data(gasphase,"transport.dat")
julia>thermo_all = IdealGas.create_thermo(gasphase,"therm.dat")
julia>Dkm = zeros(length(gasphase))
julia>D_km!(Dkm,sp_tr_data,T,p,thermo_all.molwt,mole_fracs)        

or

julia>using TransportProperties, IdealGas
julia>gasphase = ["CH4", "CO2", "H2O", "H2", "CO"]
julia>sp_tr_data = create_transport_data(gasphase,"transport.dat")
julia>thermo_all = IdealGas.create_thermo(gasphase,"therm.dat")
julia>Dkm = zeros(length(gasphase))
julia>Dij = D_ij(sp_tr_data,T,p,thermo_all.molwt)
julia>D_km!(Dkm, bdc, mole_fracs, thermo_all.molwt)

Thermal conductivity

julia>using TransportProperties, IdealGas
julia>gasphase = ["CH4", "CO2", "H2O", "H2", "CO"]
julia>sp_tr_data = create_transport_data(gasphase,"transport.dat")
julia>thermo_all = IdealGas.create_thermo(gasphase,"therm.dat")
julia>tc = thermal_coductivity(sp_tr_data,T,p,thermo_all,mole_fracs)

In the above calls mole_fracs is an array of mole fractions; must be of same size as the number of gasphase species. T and p are respectively the temperature (K) and pressure (Pa)

Output

The method transport_properties creates a screen output

Pure species viscosity:
     Species 	 viscosity(Kg/m-s)
-----------------------------------
         CH4 	      2.8992e-05
         H2O 	      3.8879e-05
          H2 	      2.0596e-05
          CO 	      4.2735e-05
         CO2 	      4.3121e-05
          O2 	      5.0182e-05
          N2 	      4.3463e-05
Mixture viscosity: 4.1079e-05 Kg/m-s

Mixture diffusion coefficients:
     Species 	 Diff.Coeff(m^2/s)
-----------------------------------
         CH4 	      2.2091e-04
         H2O 	      2.5665e-04
          H2 	      6.8730e-04
          CO 	      1.9235e-04
         CO2 	      1.4955e-04
          O2 	      1.9554e-04
          N2 	      1.8077e-04

Binary diffusion coefficients:
Note: self diffusion coefficients are not printed below:
-----------------------------------
	        CH4	        H2O	         H2	         CO	        CO2	         O2	         N2
         CH4	0.0000e+00	2.4993e-04	6.3858e-04	2.0108e-04	1.6734e-04	2.0549e-04	2.0277e-04	
         H2O	2.4993e-04	0.0000e+00	8.2047e-04	2.3675e-04	1.8888e-04	2.4327e-04	2.3919e-04	
          H2	6.3858e-04	8.2047e-04	0.0000e+00	6.6030e-04	5.8600e-04	6.9276e-04	6.6647e-04	
          CO	2.0108e-04	2.3675e-04	6.6030e-04	0.0000e+00	1.4726e-04	1.8413e-04	1.8318e-04	
         CO2	1.6734e-04	1.8888e-04	5.8600e-04	1.4726e-04	0.0000e+00	1.4794e-04	1.4849e-04	
          O2	2.0549e-04	2.4327e-04	6.9276e-04	1.8413e-04	1.4794e-04	0.0000e+00	1.8572e-04	
          N2	2.0277e-04	2.3919e-04	6.6647e-04	1.8318e-04	1.4849e-04	1.8572e-04	0.0000e+00	

Thermal conductivity of mixture: 8.1707e-02 (W/m-K)