Ville Alopaeus

Simulation of the population balances for liquid–liquid systems in a nonideal stirred tank. Part 2—parameter fitting and the use of the multiblock model for dense dispersions

Abstract In the previous part of this work (Chem. Eng. Sci. 54 (1999) 5887), a multiblock simulation model was developed in order to allow the close examination of different regions of a stirred tank for drop size distribution calculations. In this paper, that model is tested in a parameter fitting procedure. The drop breakage and coalescence parameters are fitted against drop size measurements from dense liquid–liquid dispersions, which were assumed fully turbulent. Since the local turbulence and flow values of a stirred tank are used in the present model, the fundamental breakage and coalescence phenomena can be examined more closely. Furthermore, the present model is capable of predicting inhomogeneities occurring in a stirred tank. It is also to be considered as an improved tool for process scale-up, compared to the simple vessel-averaged population balance approach, or use of correlations of dimensionless numbers only. The present model can use two sources of data for fitting parameters in the drop rate functions. One is to use transient data of the measured drop size distribution as the impeller speed is changed. The other is to use time-averaged data measured at different locations of the stirred tank. It is shown in this paper that the different flow regions can be chosen from the CFD simulations in a straightforward manner. CFD flow simulation results can be used to select the flow regions when no experimentally obtained flow conditions are available. This is especially useful for non-standard vessels, such as reactors containing cooling coils. After fitting the parameters with a multiblock model, the population balance model can be rather easily incorporated into a commercial CFD program to investigate different flow conditions.

Simulation of the population balances for liquid-liquid systems in a nonideal stirred tank.Part 1 Description and qualitative validation of the model

Abstract A simulation model has been developed to model drop populations in a stirred tank. A multiblock stirred tank model has been used with the drop population balance equations developed in the literature. The stirred tank is modeled separately so that local turbulent energy dissipation values and fluid flows are used in the drop breakage and coalescence functions. This model has several attractive features, e.g. it can predict the inhomogenity of dispersions and some scale-up phenomena. Because local conditions can be used in the drop rate functions needed in the population balances, it is possible to take these fundamental processes into closer examination. It seems that the parameter values in the drop breakage and coalescence models depend on flow and turbulence averaging for the vessel. This proposes that for “intrinsic” drop breakage and coalescence rates, a multiblock model for the stirred tank is needed in parameter estimation as well. The stirred tank flow model may be obtained from measurements or from computational fluid dynamics simulations in a straightforward manner.

A dynamic model for plug flow reactor state profiles

A dynamic model for state profiles of a plug flow reactor is developed, including multiple fluid and solid phases. The model is based on conservation of reactor state profile moments along the spatial dimension of the reactor. These moments are transformed analytically into a polynomial approximation at each time step. The method is flexible, and low as well as high order numerical schemes are resulted in by appropriate choice of parameters. A significant advantage of the present method is that boundary conditions of the partial differential equation reactor model are implicitly satisfied via the moment transformation, whereas the polynomial profile in the numerical solution does not have to be forced to satisfy the boundary conditions. The method is tested numerically against analytical solutions in three numerically challenging benchmark cases: prediction of breakthrough curve in packed bed adsorbers; simulation of chromatographic separation; and feeding a step impulse in a plug flow dimerization reactor. It is shown that the high resolution methods result in considerably smaller numerical errors than a simple low-order assumption of piecewise continuous solution.

Approximate high flux corrections for multicomponent mass transfer models and some explicit methods

Abstract New approximate simplifications are made to the high flux correction matrix for the film model and the penetration model, as derived from the Maxwell–Stefan mass transfer theory. Approximations are valid for both predetermined total flux, when explicit method results, and equivalently when total fluxes must be iterated. These presented simplifications are simple enough so as to be always includable in the mass transfer calculations, and the zero total flux assumption (equimolar transfer) is never needed for computational reasons in practical calculations.

A Calculation Method for Multicomponent Mass Transfer Coefficient Correlations

Abstract Mass transfer coefficients are usually based on measurements from binary mixtures. This leads to scalar correlation equation for mass transfer coefficient. These equations are usually obtained from dimensional analysis, and they include fractional powers of the system parameters, like the Reynolds number and the Schmidt number. Generalization according to the linearized Maxwell–Stefan theory (∇ c t and ∇[ D ] are assumed zero along the diffusion path) is then made by replacing the scalar values by corresponding matrix values. This leads to the problem of calculating the matrix fractional powers. This can be done by the similarity transform or by the Sylvester’s expansion, but it is quite a tedious procedure. Krishna and Standart (1976) American Institute of Chemical Engineers Journal 22 , 383–9 proposed that the binary mass transfer correlations could be used for each component pair in multicomponent systems. In this paper, another approximate approach is chosen for the simplification of the calculations. In the Maxwell–Stefan diffusion coefficient matrix, the off-diagonal elements describe diffusional interactions. These elements may be significant, but are usually smaller in magnitude than the diagonal elements. The method presented in this paper is based on that fact. The method gives more accurate results than the practice of using binary mass transfer coefficients. It is applicable to all mass transfer models, such as the film model, penetration model, and models resulting from boundary layer analyses.

Characteristics of liquid and tracer dispersion in trickle-bed reactors: Effect on CFD modeling and experimental analyses

The characteristics of mechanical dispersion of tracer and liquid are analyzed using CFD modeling and experimental results from the literature. The most significant differences are underlined and their impact is discussed further. When compared to uniform liquid distribution, the more complicated flow conditions in liquid source measurements are considered to have a significant effect on result analysis and should be paid more attention to. Modeling of mechanical dispersion of liquid using CFD is discussed. Finally, liquid source dispersion cases are simulated and the results are compared to the experimental liquid as well as tracer dispersion results.

Solution of bivariate population balance equations with high-order moment-conserving method of classes

Abstract In this work the high-order moment-conserving method of classes (HMMC) ( Alopaeus et al., 2006 ) is extended to solve the bivariate Population Balance Equation (PBE). The method is capable of guaranteeing the internal consistency of the discretized equations for a generic moment set, including mixed-order moments of the distribution. The construction of the product tables in the case of aggregation, breakage and convection in internal coordinate space are discussed. Eventually, several test cases are considered to assess the accuracy of the method. The application to a realistic mass transfer problems in a liquid–liquid system is preliminarily discussed. The comparison with analytical solutions of pure aggregation problems shows that the proposed method is accurate with only a limited number of categories.

A model for chlorine dioxide delignification of chemical pulp

Abstract A phenomenon based model for chlorine dioxide delignification of chemical pulp is introduced. The pulp suspension environment is modeled using the concept of two liquid phases, one inside and the other external to the fiber wall. Physico-chemical processes taking place during delignification are implemented with thermodynamic, mass transfer and reaction kinetic models. A broad library of chemical reactions is introduced. Inclusion of each reaction is justified. The model response is tested against experimental laboratory delignification results (o-delignified birch pulp). The experimental data consists of kappa number, hexenuronic acid, inorganic oxy-chlorine compound, and organochlorine (AOX, OX) measurements at several time points during five delignification experiments. The model predictions are mainly in good agreement with the experimental results. The predictions regarding hypochlorous acid driven processes (HexA removal, organochlorine formation, chlorite and chlorate concentration) are somewhat incoherent, indicating that knowledge regarding the intermediately formed hypochlorous acid is presently insufficient.

Utilization of Population Balances in Simulation of Liquid-Liquid Systems in Mixed Tanks

A simulation model has been developed to model drop populations in a mixed tank. A multiblock mixed tank model has been used with the drop population balance equations developed in the literature. The drop breakage and coalescence functions used in the population balance model take into account the local turbulent energy dissipation values. The drop breakage and coalescence function parameters are fitted against drop size measurements from dense liquid-liquid dispersions, which were assumed fully turbulent. Since the local turbulence and flow values of a mixed tank are used in the present model, the fundamental breakage and coalescence phenomena can be taken into closer examination. Furthermore, the present model is capable of predicting inhomogeneities occurring in a mixed tank. It is also considered as an improved tool for process scale-up, compared to the simple vessel-averaged population balance approach, or use of correlations of dimensionless numbers only. The present model can use two sources of data for fitting parameters in the drop rate functions. One is the transient data of the measured drop size distribution as the impeller speed is changed. The other is the time-averaged data measured at different locations of the mixed tank. Different flow regions can be chosen from direct measurements or from the CFD simulations in a straightforward manner. CFD flow simulation results can be used when no experimentally obtained flow conditions are available. This is especially useful for nonstandard vessels, such as reactors containing cooling coils.

Solubility of Organosolv Lignin in γ-Valerolactone/Water Binary Mixtures.

Abstract The solubility of lignin in a mixture of γ‐valerolactone (GVL) and water at different weight ratios was measured using the Hildebrand solubility parameters. Based on the molecular structure of lignin, its solubility parameter (δ‐value) was calculated as 25.5 MPa1/2. The δ‐value for aqueous GVL solvent increased from 23.1 MPa1/2 for pure GVL to 45.6 MPa1/2 for pure water. Therefore, the lignin solubility was predicted to increase with increasing GVL concentration in the aqueous mixture up to approximately 92–96 wt % of GVL. A ternary diagram describing the phase behavior of water–GVL–lignin mixtures at room temperature was constructed based on the experimental results. The three‐component system exhibited a complex behavior with a liquid–liquid and solid–liquid–liquid phase split. The efficiency of the selected fractionation trials in a previous work was validated using the ternary solubility diagram. A promising recovery pathway and lignin isolation method were deduced from the results of this work.