The RES approach expresses transport properties in terms of thermodynamic properties, which can be obtained directly from an EoS. Various approaches based on RES have been proposed and verified for viscosity of the Lennard-Jones fluid [11] and hundreds of real fluids (e.g., hydrocarbons, [12–14] refrigerants, [6, 15–17], or other commonly used fluids [18, 19]), as well as thermal conductivity of some real fluids [17, 20–24]. Generalized RES approaches for viscosity of more than 100 pure fluids have been developed by Lötgering-Lin et al. [19] and Dehlouz et al. [25]. Here the approach developed in our previous work [6] is extended and the mixing rule is slightly modified.

The fluid viscosity *η* is calculated as the sum of the dilute gas viscosity *η**ρ*◊0(*T*) and the residual part *η*res(*s*r):

$$\eta ={\eta }_{\rho 0}\left(T\right)+{\eta }_{\text{res}}\left({s}^{\text{r}}\right)$$

1

The dilute gas viscosity *η**ρ*◊0(*T*) at temperature *T* of a pure fluid is calculated with the Chapman-Enskog [26] solution of the Boltzmann transport equation, assuming the interactions between molecules can be roughly captured by those of Lennard-Jones (L-J) particles with 12 − 6 potential:

$${\eta }_{\rho 0}\left(T\right)=\frac{5}{16}\sqrt{\frac{m{k}_{\text{B}}T}{\pi }}\frac{1}{{\sigma }^{2}{{\Omega }}^{\left(\text{2,2}\right)*}}$$

2

where *m*, in units of kg is the mass of one molecule; *k*B = 1.380649⋅10− 23 J⋅K− 1 is the Boltzmann constant; *σ* is the collision diameter of the L-J particle; and Ω(2,2)* is the reduced collision integral obtained by integrating the possible approach trajectories of the particles. Neufeld et al.[27] gives an empirical correlation of Ω(2,2)* as a function of temperature as:

$${{\Omega }}^{\left(\text{2,2}\right)*}=1.16145\cdot {\left({T}^{*}\right)}^{-0.14874}+0.52487\cdot \text{e}\text{x}\text{p}(-0.77320\cdot {T}^{*})+2.16178\cdot \text{e}\text{x}\text{p}(-2.43787\cdot {T}^{*})$$

3

where *T**** = *k*B*T*/*ε* is the dimensionless temperature, and *ε*/*k*B is the reduced L-J pair-potential energy. The non-polynomial terms are neglected in this work as REFPROP 10.0 [1] does. The L-J parameters (*σ* and *ε*) in this work were obtained from REFPROP 10.0 as listed in Table S1 in the supporting information (SI).

As introduced by Bell [12, 18], the residual part of viscosity *η*res(*s*r) can be calculated with:

$${\eta }_{\text{res}}\left({s}^{\text{r}}\right)=\frac{{\eta }_{\text{r}\text{e}\text{s}}^{+}{\rho }_{\text{N}}^{2/3}\sqrt{m{k}_{\text{B}}T}}{{\left({s}^{+}\right)}^{2/3}}$$

4

$${s}^{+}= -{s}^{\text{r}}/R$$

5

Here, *ρ*N, in units of m− 3, is the number density; *s*r in units of J·mol− 1·K− 1 is the molar residual entropy, defined as the difference between the real fluid entropy and the ideal gas entropy at the same temperature and density; and *R* = 8.31446261815324 J·mol− 1·K− 1 is the molar gas constant [28]. In this work, the number density *ρ*N and molar residual entropy *s*r were calculated with the reference EoS implemented in REFPROP 10.0 [1] using the python CoolProp package 6.4.1 [3] as an interface. The reference EoS for each pure fluid is listed in Table S2 in the SI. The plus-scaled dimensionless residual viscosity \({\eta }_{\text{r}\text{e}\text{s}}^{+}\) is related to the plus-scaled dimensionless residual entropy *s*+ using the following polynomial equations

$$\text{ln}\left({\eta }_{\text{r}\text{e}\text{s}}^{+}+1\right)={n}_{1}\cdot {(s}^{+})+{n}_{2}\cdot {{(s}^{+})}^{1.5} +{n}_{3}\cdot {{(s}^{+})}^{2} +{n}_{4}\cdot {{(s}^{+})}^{2.5}$$

6

or

$$\text{ln}\left({\eta }_{\text{r}\text{e}\text{s}}^{+}+1\right)={n}_{\text{g}1}\cdot {(s}^{+}/\xi )+{n}_{\text{g}2}\cdot {{(s}^{+}/\xi )}^{1.5} +{n}_{\text{g}3}\cdot {{(s}^{+}/\xi )}^{2} +{n}_{\text{g}4}\cdot {{(s}^{+}/\xi )}^{2.5}$$

7

Eq. (6) is for a pure fluid with fluid-specific fitted parameter *n**k* (*k* = 1, 2, 3, 4), and Eq. (7) is for a group of pure fluids with global fitted parameters *n*g*k* (*k* = 1, 2, 3, 4) and a fluid-specific scaling factor *ξ* for each pure fluid.

To extend the RES model to mixtures, a predictive mixing rule is adopted. The dilute gas viscosity *η**ρ*◊0,mix is calculated with the approximation of Wilke [29]:

$${\eta }_{\rho 0,\text{m}\text{i}\text{x}}=\sum _{i=1}^{N}\frac{{x}_{i}\bullet {\eta }_{\rho 0,i}}{{\sum }_{j=1}^{N}{x}_{j}\bullet {\phi }_{ij}}$$

8

with

$${\phi }_{ij}=\frac{{(1+{({\eta }_{\rho 0,i}/{\eta }_{\rho 0,j})}^{1/2}\bullet {({m}_{j}/{m}_{i})}^{1/4})}^{2}}{{(8\bullet (1+{m}_{i}/{m}_{j}\left)\right)}^{1/2}} ,$$

9

where *x**i* is the mole fraction of component *i* and *m**i* is the mass of one molecule of component *i*. The mole fraction weighted average *m*mix of the components is used to replace the effective mass of one particle *m* in Eq. (4):

$${m}_{\text{m}\text{i}\text{x}}= \sum _{i}{x}_{i}\bullet {m}_{i}$$

10

Attempts to use a mass fraction weighted average result in a negligible statistical difference. Then, in contract to our previous work [6], the mole fraction weighted average coefficient *n**k*,mix is utilised to substitute the parameters *n**k* in Eq. (6), i.e.,

$${n}_{k,\text{m}\text{i}\text{x}}= \sum _{i}{x}_{i}\bullet {n}_{k,i}$$

11

where *n**k,i* (*k* = 1,2,3,4) are fitted *n**k* parameters of component *i*. It is important to note that, only if a pure fluid does not have fluid-specific fitted parameters, the *n**k* (*k* = 1, 2, 3, 4) are replaced by *n*g1/*ξ*, *n*g2/*ξ*1.5, *n*g3/*ξ*2, and *n*g4/*ξ*2.5, respectively.