(47h) The Shear Viscosity of Mixtures Revisited – a Novel Way to Define and Calculate Viscosities for Mixtures and Its Relation to Diffusion

Viscosity plays an outstanding role in the design and optimization of reactors, separation units, and pipes in chemical engineering. Together with interfacial properties, diffusion coefficients, densities, and phase equilibria, they form the set of “must-know-data” for complex processes. Generally, viscosity is defined in terms pure components’ viscosities at the system pressure and temperature and an excess viscosity due to mixing. The latter can take on quite complex shapes and therefore is hard to model or even predict. Moreover, connections exist between e.g. diffusive and viscous behavior, which are, however, only known as mathematical relations in very limited circumstances or states. Combining fluid dynamics, multi-component diffusion and equilibrium thermodynamics can be difficult, because e.g. good equations of state need to be known, but also because of technical difficulties and compatibility issues between the different approaches to fluid phenomena. This is especially true, if interfaces are involved. All these challenges are linked.

In contrast to the above theories by themselves, the thermodynamics of irreversible processes (TIP)^{e.g. [1], [2]} can form a connection between these different approaches. Using TIP, a new formulation can be found for the momentum balances of mixtures that directly includes the transport diffusion in a transient manner^[2]. This transient part is, however, often omitted for diffusion processes^{e.g. [3], [4]} and usually not even considered for purely viscous transport^{e.g. for a recent application [5]}. In a formulation consistent with equilibrium thermodynamics and fluid dynamics, the momentum balances in Eulerian frame of reference of the respective species are defined according to Eqs. (1) (Figure), with the mass densities ρ_i of component i, the species velocities vⁱ_α of species i in direction α, the time t, the space coordinates x_α in direction α, the molarities c_i of component i, the chemical potentials μ_i of component i, the total molarity c, Boltzmann’s constant k, the temperature T, the Maxwell-Stefan diffusion coefficient Ð_ij, and the viscous stresses between species i and j namely σ^i,j_αβ in the directions α and β. The latter is especially interesting, since it means that there are viscosities between component pairs, which is also supported by the Green-Kubo relation for viscosity, in which the time-autocorrelation function is included for sums of momenta and pairwise forces^{[6], [7]}.

Because the Eulerian frame of reference is always fixed in space, the appropriate thermodynamic potential from an equilibrium perspective is the free energy density, which has the natural variables c_i and T and for which the volume is always fixed. If these natural variables, a free energy density and the pairwise viscosities are used as a base model, one arrives at a simple viscosity model for mixtures, where the reference viscosities are not the pure component viscosities at the same pressure and temperature, but the pure component viscosities, if the other species were not present in the volume. Thereby a binary viscosity coefficient η₁₂ is left per component pair, which then accounts for mixture viscosity. The approach is applicable in multi-component, multi-phase combined fluid dynamics and mass transport simulations and is thermodynamically consistent.

In this contribution, the application of this model to binary mixtures of simple fluid mixture^[8] is shown. First the “diffusive” part of Eqs. (1), which does not account for viscous momentum dissipation is examined, to exclude errors coming from this part of the equations. Next, pure component viscosities are examined in terms of a correlation with density and temperature, as well as an entropy scaling approach^[9]. In the last part, the dependence of the newly defined binary viscosity coefficient η₁₂ on different variables is examined. The functional form of the binary viscosity coefficient is found to be simpler than that of the excess viscosity coefficient. An entropy scaling approach is found best to account for η₁₂.

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