Non-random two-liquid model







VLE of the mixture of chloroform and methanol plus NRTL fit and extrapolation to different pressures


The non-random two-liquid model[1] (short NRTL equation) is an activity coefficient model that correlates the activity coefficients γi{displaystyle gamma _{i}}gamma _{i} of a compound with its mole fractions xi{displaystyle x_{i}}x_{i} in the liquid phase concerned. It is frequently applied in the field of chemical engineering to calculate phase equilibria. The concept of NRTL is based on the hypothesis of Wilson that the local concentration around a molecule is different from the bulk concentration. This difference is due to a difference between the interaction energy of the central molecule with the molecules of its own kind Uii{displaystyle U_{ii}}{displaystyle U_{ii}} and that with the molecules of the other kind Uij{displaystyle U_{ij}}U_{{ij}}. The energy difference also introduces a non-randomness at the local molecular level. The NRTL model belongs to the so-called local-composition models. Other models of this type are the Wilson model, the UNIQUAC model, and the group contribution model UNIFAC. These local-composition models are not thermodynamically consistent for a one-fluid model for a real mixture due to the assumption that the local composition around molecule i is independent of the local composition around molecule j. This assumption is not true, as was shown by Flemr in 1976.[2] However, they are consistent if a hypothetical two-liquid model is used.[3]




Contents






  • 1 Equations for a binary mixture


  • 2 General equations


  • 3 Temperature dependent parameters


  • 4 Parameter determination


  • 5 Literature





Equations for a binary mixture


For a binary mixture the following function[4] are used:


{ln⁡ γ1=x22[τ21(G21x1+x2G21)2+τ12G12(x2+x1G12)2]ln⁡ γ2=x12[τ12(G12x2+x1G12)2+τ21G21(x1+x2G21)2]{displaystyle left{{begin{matrix}ln gamma _{1}=x_{2}^{2}left[tau _{21}left({frac {G_{21}}{x_{1}+x_{2}G_{21}}}right)^{2}+{frac {tau _{12}G_{12}}{(x_{2}+x_{1}G_{12})^{2}}}right]\\ln gamma _{2}=x_{1}^{2}left[tau _{12}left({frac {G_{12}}{x_{2}+x_{1}G_{12}}}right)^{2}+{frac {tau _{21}G_{21}}{(x_{1}+x_{2}G_{21})^{2}}}right]end{matrix}}right.}{displaystyle left{{begin{matrix}ln gamma _{1}=x_{2}^{2}left[tau _{21}left({frac {G_{21}}{x_{1}+x_{2}G_{21}}}right)^{2}+{frac {tau _{12}G_{12}}{(x_{2}+x_{1}G_{12})^{2}}}right]\\ln gamma _{2}=x_{1}^{2}left[tau _{12}left({frac {G_{12}}{x_{2}+x_{1}G_{12}}}right)^{2}+{frac {tau _{21}G_{21}}{(x_{1}+x_{2}G_{21})^{2}}}right]end{matrix}}right.}


with


{ln⁡ G12=−α12 τ12ln⁡ G21=−α21 τ21{displaystyle left{{begin{matrix}ln G_{12}=-alpha _{12} tau _{12}\ln G_{21}=-alpha _{21} tau _{21}end{matrix}}right.}{displaystyle left{{begin{matrix}ln G_{12}=-alpha _{12} tau _{12}\ln G_{21}=-alpha _{21} tau _{21}end{matrix}}right.}


In here τ12{displaystyle tau _{12}}{displaystyle tau _{12}} and τ21{displaystyle tau _{21}}tau_{21} are the dimensionless interaction parameters, which are related to the interaction energy parameters Δg12{displaystyle Delta g_{12}}{displaystyle Delta g_{12}} and Δg21{displaystyle Delta g_{21}}{displaystyle Delta g_{21}} by:


{τ12=Δg12RT=U12−U22RTτ21=Δg21RT=U21−U11RT{displaystyle left{{begin{matrix}tau _{12}={frac {Delta g_{12}}{RT}}={frac {U_{12}-U_{22}}{RT}}\tau _{21}={frac {Delta g_{21}}{RT}}={frac {U_{21}-U_{11}}{RT}}end{matrix}}right.}{displaystyle left{{begin{matrix}tau _{12}={frac {Delta g_{12}}{RT}}={frac {U_{12}-U_{22}}{RT}}\tau _{21}={frac {Delta g_{21}}{RT}}={frac {U_{21}-U_{11}}{RT}}end{matrix}}right.}


Here R is the gas constant and T the absolute temperature, and Uij is the energy between molecular surface i and j. Uii is the energy of evaporation. Here Uij has to be equal to Uji, but Δgij{displaystyle Delta g_{ij}}{displaystyle Delta g_{ij}} is not necessary equal to Δgji{displaystyle Delta g_{ji}}{displaystyle Delta g_{ji}}.


The parameters α12{displaystyle alpha _{12}}alpha _{{12}} and α21{displaystyle alpha _{21}}alpha _{{21}} are the so-called non-randomness parameter, for which usually α12{displaystyle alpha _{12}}alpha _{{12}} is set equal to α21{displaystyle alpha _{21}}alpha _{{21}}. For a liquid, in which the local distribution is random around the center molecule, the parameter
α12=0{displaystyle alpha _{12}=0}{displaystyle alpha _{12}=0}. In that case the equations reduce to the one-parameter Margules activity model:


{ln⁡ γ1=x22[τ21+τ12]=Ax22ln⁡ γ2=x12[τ12+τ21]=Ax12{displaystyle left{{begin{matrix}ln gamma _{1}=x_{2}^{2}left[tau _{21}+tau _{12}right]=Ax_{2}^{2}\ln gamma _{2}=x_{1}^{2}left[tau _{12}+tau _{21}right]=Ax_{1}^{2}end{matrix}}right.}{displaystyle left{{begin{matrix}ln gamma _{1}=x_{2}^{2}left[tau _{21}+tau _{12}right]=Ax_{2}^{2}\ln gamma _{2}=x_{1}^{2}left[tau _{12}+tau _{21}right]=Ax_{1}^{2}end{matrix}}right.}


In practice, α12{displaystyle alpha _{12}}alpha _{{12}} is set to 0.2, 0.3 or 0.48. The latter value is frequently used for aqueous systems. The high value reflects the ordered structure caused by hydrogen bonds. However, in the description of liquid-liquid equilibria the non-randomness parameter is set to 0.2 to avoid wrong liquid-liquid description. In some cases a better phase equilibria description is obtained by setting α12=−1{displaystyle alpha _{12}=-1}{displaystyle alpha _{12}=-1}.[5] However this mathematical solution is impossible from a physical point of view, since no system can be more random than random (α12{displaystyle alpha _{12}}alpha _{{12}} =0). In general NRTL offers more flexibility in the description of phase equilibria than other activity models due to the extra non-randomness parameters. However, in practice this flexibility is reduced in order to avoid wrong equilibrium description outside the range of regressed data.


The limiting activity coefficients, aka the activity coefficients at infinite dilution, are calculated by:


{ln⁡ γ1∞=[τ21+τ12exp⁡(−α12 τ12)]ln⁡ γ2∞=[τ12+τ21exp⁡(−α12 τ21)]{displaystyle left{{begin{matrix}ln gamma _{1}^{infty }=left[tau _{21}+tau _{12}exp {(-alpha _{12} tau _{12})}right]\ln gamma _{2}^{infty }=left[tau _{12}+tau _{21}exp {(-alpha _{12} tau _{21})}right]end{matrix}}right.}{displaystyle left{{begin{matrix}ln gamma _{1}^{infty }=left[tau _{21}+tau _{12}exp {(-alpha _{12} tau _{12})}right]\ln gamma _{2}^{infty }=left[tau _{12}+tau _{21}exp {(-alpha _{12} tau _{21})}right]end{matrix}}right.}


The expressions show that at α12=0{displaystyle alpha _{12}=0}{displaystyle alpha _{12}=0} the limiting activity coefficients are equal. This situation that occurs for molecules of equal size, but of different polarities.
It also shows, since three parameters are available, that multiple sets of solutions are possible.



General equations


The general equation for ln⁡(γi){displaystyle ln(gamma _{i})}{displaystyle ln(gamma _{i})} for species i{displaystyle i}i in a mixture of n{displaystyle n}n components is:[6]


ln⁡(γi)=∑j=1nxjτjiGji∑k=1nxkGki+∑j=1nxjGij∑k=1nxkGkj(τij−∑m=1nxmτmjGmj∑k=1nxkGkj){displaystyle ln(gamma _{i})={frac {displaystyle sum _{j=1}^{n}{x_{j}tau _{ji}G_{ji}}}{displaystyle sum _{k=1}^{n}{x_{k}G_{ki}}}}+sum _{j=1}^{n}{frac {x_{j}G_{ij}}{displaystyle sum _{k=1}^{n}{x_{k}G_{kj}}}}{left({tau _{ij}-{frac {displaystyle sum _{m=1}^{n}{x_{m}tau _{mj}G_{mj}}}{displaystyle sum _{k=1}^{n}{x_{k}G_{kj}}}}}right)}}{displaystyle ln(gamma _{i})={frac {displaystyle sum _{j=1}^{n}{x_{j}tau _{ji}G_{ji}}}{displaystyle sum _{k=1}^{n}{x_{k}G_{ki}}}}+sum _{j=1}^{n}{frac {x_{j}G_{ij}}{displaystyle sum _{k=1}^{n}{x_{k}G_{kj}}}}{left({tau _{ij}-{frac {displaystyle sum _{m=1}^{n}{x_{m}tau _{mj}G_{mj}}}{displaystyle sum _{k=1}^{n}{x_{k}G_{kj}}}}}right)}}


with


Gij=exp⁡(−αijτij){displaystyle G_{ij}=exp left({-alpha _{ij}tau _{ij}}right)}{displaystyle G_{ij}=exp left({-alpha _{ij}tau _{ij}}right)}


αij=αij0+αij1T{displaystyle alpha _{ij}=alpha _{ij_{0}}+alpha _{ij_{1}}T}{displaystyle alpha _{ij}=alpha _{ij_{0}}+alpha _{ij_{1}}T}


τi,j=Aij+BijT+CijT2+Dijln⁡(T)+EijTFij{displaystyle tau _{i,j}=A_{ij}+{frac {B_{ij}}{T}}+{frac {C_{ij}}{T^{2}}}+D_{ij}ln {left({T}right)}+E_{ij}T^{F_{ij}}}{displaystyle tau _{i,j}=A_{ij}+{frac {B_{ij}}{T}}+{frac {C_{ij}}{T^{2}}}+D_{ij}ln {left({T}right)}+E_{ij}T^{F_{ij}}}


There are several different equation forms for αij{displaystyle alpha _{ij}}alpha_{ij} and τij{displaystyle tau _{ij}}tau _{ij}, the most general of which are shown above.



Temperature dependent parameters


To describe phase equilibria over a large temperature regime, i.e. larger than 50 K, the interaction parameter has to be made temperature dependent.
Two formats are frequently used. The extended Antoine equation format:


τij=f(T)=aij+bijT+cij ln⁡ T+dijT{displaystyle tau _{ij}=f(T)=a_{ij}+{frac {b_{ij}}{T}}+c_{ij} ln T+d_{ij}T}{displaystyle tau _{ij}=f(T)=a_{ij}+{frac {b_{ij}}{T}}+c_{ij} ln T+d_{ij}T}

Here the logarithmic and linear terms are mainly used in the description of liquid-liquid equilibria (miscibility gap).


The other format is a second-order polynomial format:


Δgij=f(T)=aij+bij⋅T+cijT2{displaystyle Delta g_{ij}=f(T)=a_{ij}+b_{ij}cdot T+c_{ij}T^{2}}{displaystyle Delta g_{ij}=f(T)=a_{ij}+b_{ij}cdot T+c_{ij}T^{2}}


Parameter determination


The NRTL parameters are fitted to activity coefficients that have been derived from experimentally determined phase equilibrium data (vapor–liquid, liquid–liquid, solid–liquid) as well as from heats of mixing. The source of the experimental data are often factual data banks like the Dortmund Data Bank. Other options are direct experimental work and predicted activity coefficients with UNIFAC and similar models.
Noteworthy is that for the same liquid mixture several NRTL parameter sets might exist. The NRTL parameter set to use depends on the kind of phase equilibrium (i.e. solid–liquid (SL), liquid–liquid (LL), vapor–liquid (VL)). In the case of the description of a vapor–liquid equilibria it is necessary to know which saturated vapor pressure of the pure components was used and whether the gas phase was treated as an ideal or a real gas. Accurate saturated vapor pressure values are important in the determination or the description of an azeotrope. The gas fugacity coefficients are mostly set to unity (ideal gas assumption), but for vapor-liquid equilibria at high pressures (i.e. > 10 bar) an equation of state is needed to calculate the gas fugacity coefficient for a real gas description.


Determination of NRTL parameters from LLE data is more complicated than parameter regression from VLE data as it involves solving isoactivity equations which are highly non-linear. In addition, parameters obtained from LLE may not always represent the real activity of components due to lack of knowledge on the activity values of components in the data regression.[7][8][9] For this reason it is necessary to confirm the consistency of the obtained parameters in the whole range of compositions (including binary subsystems, experimental and calculated lie-lines, Hessian matrix, etc.).[10][11][12]



Literature





  1. ^ Renon H., Prausnitz J. M., "Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures", AIChE J., 14(1), S.135–144, 1968


  2. ^ McDermott (Fluid Phase Equilibrium 1(1977)33) and Flemr (Coll. Czech. Chem.Comm., 41 (1976) 3347)


  3. ^ Hu, Y.; Azevedo, E.G.; Prausnitz, J.M. (1983). "The molecular basis for local compositions in liquid mixture models". Fluid Phase Equilibria. 13: 351–360. doi:10.1016/0378-3812(83)80106-X..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  4. ^ Reid R. C., Prausnitz J. M., Poling B. E., The Properties of Gases & Liquids, 4th Edition, McGraw-Hill, 1988


  5. ^ Effective Local Compositions in Phase Equilibrium Correlations, J. M. Marina, D. P. Tassios Ind. Eng. Chem. Process Des. Dev., 1973, 12 (1), pp 67–71


  6. ^ http://users.rowan.edu/~hesketh/0906-316/Handouts/Pages%20from%20SimBasis%20appendix%20A%20property%20packages.pdf


  7. ^ Reyes-Labarta, J.A.; Olaya, M.M.; Velasco, R.; Serrano, M.D.; Marcilla, A. (2009). "Correlation of the Liquid-Liquid Equilibrium Data for Specific Ternary Systems with One or Two Partially Miscible Binary Subsystems". Fluid Phase Equilibria. 278 (1–2): 9–14. doi:10.1016/j.fluid.2008.12.002.


  8. ^ Marcilla, A.; Reyes-Labarta, J.A.; Serrano, M.D.; Olaya, M.M. (2011). "GE Models and Algorithms for Condensed Phase Equilibrium Data Regression in Ternary Systems: Limitations and Proposals". The Open Thermodynamics Journal. 5: 48–62. doi:10.2174/1874396X01105010048.


  9. ^ Marcilla, A.; Serrano, M.D.; Reyes-Labarta, J.A.; Olaya, M.M. (2012). "Checking Liquid-Liquid Critical Point Conditions and their Application in Ternary Systems". Industrial & Engineering Chemistry Research. 51 (13): 5098–5102. doi:10.1021/ie202793r.


  10. ^ Li, Z.; Smith, K. H.; Mumford, K. A.; Wang, Y.; Stevens, G. W., Regression of NRTL parameters from ternary liquid–liquid equilibria using particle swarm optimization and discussions. Fluid Phase Equilib. 2015, 398, 36-45.


  11. ^ Marcilla, Antonio; Reyes-Labarta, Juan A.; Olaya, M.Mar (2017). "Should we trust all the published LLE correlation parameters in phase equilibria? Necessity of their Assessment Prior to Publication". Fluid Phase Equilibria. 433: 243–252. doi:10.1016/j.fluid.2016.11.009.


  12. ^ Reyes-Labarta, Juan A.; Olaya, Maria del Mar; Marcilla, Antonio (2015-11-27). "Graphical User Interface (GUI) for the analysis of Gibbs Energy surfaces, including LL tie-lines and Hessian matrix". University of Alicante. hdl:10045/51725.









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