Michael Locht Michelsen

A method for incorporating excess Gibbs energy models in equations of state

A procedure for incorporating information from an excess Gibbs energy model into an equation of state is described. The gE model is used to define mixing rules for an equation of state mixture parameter in such a manner that the equation of state reproduces the vapour—liquid equilibrium predicted by the gE model at low temperatures. The resulting equation of state is fully consistent, performs reasonably at elevated temperatures and is computationally efficient. Typically, the cost of evaluating fugacity coefficients exceeds that of evaluating activity coefficients from the excess Gibbs energy model by 25–50%.

The negative flash

Abstract This paper describes a procedure for calculating vapor-liquid equilibrium for systems that physically exist as a single phase, but still yield non-negative equilibrium compositions that satisfy the material balance and equal fugacity constraints of the P-T flash. Theoretical and practical consequences of the “negative flash” are discussed. For example, it is shown that the negative flash corresponds to a saddle point in the Gibbs energy surface. It is also shown that the limits of pressure and temperature where the negative flash ceases to converge to a non-trivial solution defines the well-known convergence pressure envelope of the mixture. Finally, it is suggested that the continuity of properties across phase boundaries, as calculated by the negative flash, makes the method attractive for algorithms that nest a standard P-T flash inside an outer loop (e.g., an isenthalpic flash or a three-phase dewpoint).

Calculation of phase envelopes and critical points for multicomponent mixtures

Abstract An algorithm for the construction of complete vapour—liquid phase envelopes, capable of accurately determining the critical points and maxima in temperature and pressure is described. Recently developed methods for direct calculation of critical points are critically examined, and a computationally more convenient formulation is suggested.

A Flory–Huggins model based on the Hansen solubility parameters

Abstract The Hansen solubility parameters are typically used, especially in the paint and coatings industry, for selecting suitable solvents for polymer binders. In such applications, the radius of solubility needs to be determined by experiments. Moreover, other calculations, e.g. related to drying of paints, require thermodynamic description over extended concentration ranges. On the other hand, the well-known Flory–Huggins model captures satisfactorily several of the characteristics of polymer solutions, but the interaction parameter of the model usually also has to be determined by experiments. In this work, we present a methodology, based on which the Hansen solubility parameters can be incorporated in the Flory–Huggins model. The resulting model is predictive and yields good predictions of solvent activity coefficients at infinite dilution in several acrylate and acetate polymers. The results are as accurate as other predictive polymer models based on the group-contribution (GC) principle, but in contrast to these models, knowledge of molecular structures is not required in the Hansen/Flory–Huggins model. The new model makes thus use of the extensive literature of Hansen solubility parameters, which are tabulated for very many solvents, pigments and polymers.

The Graetz problem with axial heat conduction

Abstract Using the Graetz problem with axial conduction as an illustrative example a method for solution of an important class of linear partial differential equations is developed. The method is a combination of orthogonal collocation and matrix diagonalization. The reason for the very high accuracy, which is obtained by collocation, is discussed in terms of the eigenvalues of the collocation operator. These are found to increase much faster than the true eigenvalues for k > N 2 where N is the number of collocation points, and this permits a high accuracy also in the “penetration region” of the solution where Fourier Series are slowly convergent. Explicit formulas for the asymptotic Nu-number for large and small Pe-numbers are developed in an appendix. They are based on a perturbation of the eigenfunctions of the simplified model with either infinite or zero Pe-number. A number of variants of the Graetz problem, which can be solved by a repetition of the present computations, are proposed.

Reactive separation systems I. Computation of physical and chemical equilibrium

New algorithms for the computation of the simultaneous chemical and physical equilibrium involved in simulations of reactive flash operations, calculations of phase diagrams, and the determination of reactive azeotropes are presented. These algorithms are based on the use of the elements balance approach where the mass balance equations and the Gibbs energy minimization problem are solved in terms of the elements balance variables and not in terms of component compositions. The total number of elements, which can be atoms, molecules or groups (radicals), is smaller than the number of components in the reactive system. The reduced number of variables allows the visualization of the phase diagrams for many multicomponent reactive systems in two- or three-dimensional figures. Element fractions which are by definition similar to the component mole fractions, are employed to determine the conditions for the existence of element azeotropes. These conditions are identical to those that apply for non-reactive (conventional) determination. Illustrative examples highlighting various features of the proposed algorithms are presented through four reactive systems.

State function based flash specifications

A variety of flash specifications of practical importance can be formulated as minimization of a thermodynamic state function. It is well known that the solution for the PT-flash yields the global minimum of the mixture Gibbs energy, but in addition specifications of PH, PS, TV, UV or SV all permit selections of thermodynamic state functions for which a global minimum must be located. Two important advantages are obtained with the minimization based approach. First, the desired solution is known to be unique, and second, stability analysis can be used to verify its correctness and to determine the number of equilibrium phases. For the PT-flash the solution can be determined by unconstrained minimization, whereas the remaining specifications are accompanied by one or two nonlinear constraints and thus less straightforward to attack. We present here two approaches for dealing with such specifications. The first is a nested optimization approach where T and/or P are the dependent variables in the outer loop, and where a PT-flash is solved in the inner loop. The essential advantage of this formulation is that it is very easy to implement but the drawback is the additional cost of the nested loops. The second approach is based on a modified objective function in which all constraints are removed, but where a saddle point rather than a minimum must be located. The resulting equations are solved by a global Newtons method, and it is shown that a common Jacobian matrix can be used with all the specifications given above.

Calculation of critical points and phase boundaries in the critical region

Abstract A computationally efficient, general method is presented for calculating the critical temperature and pressure of a multicomponent mixture of specified composition. The procedure suggested also provides coefficients αT, αP and u relating changes in temperature, pressure, phase fraction β and composition in the immediate vicinity of the critical point. These variations are shown to be of the general form T − Tc = αT(1 − 2β)δ P − Pc = αP(1 − 2β)δ ln Ki = uizi− 1 2 δ where δ is the distance parameter of the phase-boundary expansion. Finally, an explicit first-order approximation is developed, based on derivatives of the Gibbs-energy surface, givin the phase fractions and equilibrium phase composition for a mixture under near-critical conditions.

Application of the CPA equation of state to glycol/hydrocarbons liquid–liquid equilibria

Abstract The Cubic Plus Association (CPA) equation of state is a thermodynamic model, which combines the well-known cubic SRK (Soave–Redlich–Kwong) equation of state and the association term proposed by Wertheim, typically employed in models like SAFT (statistical associating fluid theory). CPA has been shown in the past to be a successful model for phase equilibria calculations for systems containing water, hydrocarbons and alcohols. In this work, CPA is applied for the first time to liquid–liquid equilibria (LLE) for systems containing glycols and hydrocarbons. It is shown that excellent correlation is achieved with solely a single interaction parameter per binary system. The correlation procedure as well as the nature of the experimental data play a crucial role in the parameter estimation and they are thus extensively discussed.

Hold-up and fluid mixing in gas-liquid fluidised beds

Abstract Axial mixing in the fluid phases of liquid and gas-liquid fluidised beds has been examined by the imperfect pulse method, using radioactive isotopes as tracers. The intensity of mixing is strongly dependent on the particle size of the ballotini and the flow rate of the fluid phases. In beds of 6 mm particles mixing may be neglected except at very high superficial gas velocities, whereas beds of 1 mm particles are characterised by a high degree of mixing in the liquid phase. A new method of data reduction possessing some advantages over the normally used method of moments is applied and discussed.