A new class of solution methods for multicomponent,multistage separation processes

A new algorithm was developed for the solution of the equations that describe multicomponent, multistage separation processes operating at steady state. The algorithm is based on the use of newly defined energy and volatility parameters as the primary successive approximation variables. A third parameter was defined for each stage as a unique combination of the liquid and vapor phase rates and the temperature, and the quasi-Newton method of Broyden was employed to iterate on these parameters. The exceptional stability of the new algorithm in very difficult cases, as well as its efficiency in easy cases, are demonstrated using a variety of example problems.     

An enhanced pseudo-binary-mixture (PBM) algorithm for multistage separation processes

A scale-shift approach is introduced to further refine the pseudo-binary-mixture (PBM) simultaneous correction procedure for the solution of multicomponent, multistage separation problems [Ishii, Y., & Otto, F. D. (2001). An efficient simultaneous correction procedure for multicomponent, multistage separation calculations for non-ideal systems. Computers and Chemical Engineering, 25, 1285–1298; & Ishii, Y., & Otto, F. D. (2003a). A method to extend the domain of convergence for difficult multicomponent, multistage separation problems. Computers and Chemical Engineering, 27, 855–868]. The scale-shift approach takes advantage of the fact that a solution to a stage-wise separation problem is more easily obtained the fewer the number of stages. In the approach the column specified in a given problem is initially downsized to a configuration having a small number of stages and then scaled up to the original one in a multiple-step manner. A solution is obtained at each scale-shift step and the results are used as a set of initial assumptions to solve the subsequent scaled-up problem. Combining the scale-shift approach along with the gradual non-ideal enhancing method with the PBM algorithm provides a more robust and efficient algorithm for the simulation and solution of difficult multicomponent, multistage separation problems including those involving non-ideal mixtures that are highly sensitive to the quality of initial values. A geometric consideration for the gradual non-ideal enhancing method and details of the unique procedure for the algebraic inversion of matrices employed in the algorithm are also presented.     

Calculation of multistage countercurrent separation processes—II multicomponent multistage separation by relaxation procedure

The relaxation method used for solution of problems arising in design of multicomponent separation processes is discussed and new modifications are proposed. The use of certain techniques for convergence acceleration is studied. The number of iterations is then substantially reduced and effectiveness of relaxation method is thus highly improved.This method is useful also for columns with multiple feeds, intercoolers etc., however, nonideal mixture can be conveniently handled. A modification of the relaxation procedure resulting in a pentadiagonal matrix is presented.     

A general algorithm for multistage multicomponent separation calculations

This paper presents a new general calculational method for equilibrium stage processes. The method employs a multivariate Newton procedure to solve all equations simultaneously for corrections to temperatures, flow rates and compositions. The linearized model equations are selected so that only 2N (1 + NCP) partial derivatives need to be evaluated per iteration. Experience to date indicates that the method is reliable for both distillation and absorption problems and requires less computing time than current procedures. The rigorous solution of a reboiled absorber problem for 16 stages and 15 components required 5 iterations and 14.4 seconds on an IBM 360/67 computer even when the Chao-Seader correlation was employed for the direct calculation of composition dependent equilibrium and enthalpy data.     

Application of the Flory-Huggins theory to the system chloroform-polystyrene-butadiene rubber

According to the Flory-Huggins theory for polymer blends, the polymerpolymer interaction parameter, χ₂₃, must be positive for incompatible polymers. Previous studies of the incompatible ternary system chloroform-polystyrerne-butadiene rubber yielded negative values of χ₂₃ using a solution method based on the Flory-Huggins theory with the chloroform-butadiene rubber interaction parameter used as input. An alternate solution method using the chloroform-polystyrene interaction parameter as input has been used to obtain positive values of χ₂₃ for this incompatible system. This variation in results indicates that the interaction parameters obtained using the Flory-Huggins theory are not physically meaningful. They are, however, useful as correlating parameters. Both sets of results, although quite different, are shown to correlate the experimental data identically well. In addition, the appropriate equations are presented for the case where one of the solvent-polymer interaction parameters is concentration dependent.     

Geometric Aspects of the Theory of Fibre Tension, Storage, and Transport Mechanisms in the Absence of Friction Forces

The mechanisms that perform drawing, storage, or transport of chemical fibres significantly affect their final properties. The possibility of using the small parameter method is examined for calculating the parameters of this effect: if an accurate solution of the problem is known for some initial values of the basic parameters, then its solutions for other values which differ comparatively little from the initial values can be obtained. The method allows gradually increasing the adequacy of the solution by using new or altered values of the basic parameters in the calculation. An accurate solution corresponding to negligibly small friction forces was obtained for the case where the mechanism consists of two rotating elongated cylinders which the fibre is repeatedly wound around. St. Petersburg State University of Technology and Design. Translated fromKhimicheskie Volokna, No. 5, pp. 63–66, September–October, 2000.     

Method of Asymptotic Interpolation in Problems of Chemical Hydrodynamics and Mass Transfer

A new modification of the asymptotic interpolation method is put forward, which uses the known two-sided asymptotics of the solution of a problem and the values of the function in an intermediate region. Special switching functions are suggested for joining the asymptotic solutions. Their constants are determined from their values at specified reference points. By way of an example, this method is applied to some problems of chemical hydrodynamics and mass transfer that have no exact analytical solution. Approximate solutions are obtained for the entire variation range of the characteristic dimensionless parameter. Their accuracy was estimated by collating them with the results of numerical computation.     

Fourier series solution for multistage countercurrent cake washing and segregated wash effluent circulation

A Fourier series solution method was developed for solving a multistage countercurrent cake washing problem, so that the solute concentration gradient inside a cake between washing stages was preserved. The cake washing process was described by the advection-dispersion equation. The solution method was employed to explore the multistage countercurrent cake washing in a segregated and a non-segregated wash effluent circulation. The segregated wash effluent circulation gave better solute recovery than the non-segregated wash effluent circulation especially when the wash ratio was around one.     

Stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition

The problem of a steady laminar two‐dimensional stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition is studied. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically using the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter and the convective parameter on the fluid flow and heat transfer characteristics are disscussed. It is found that the skin friction coefficient and the heat transfer rate at the surface decrease with increasing values of the material parameter. Moreover, dual solutions are found to exist for the shrinking case, while for the stretching case, the solution is unique. © 2011 Canadian Society for Chemical Engineering     

Residence time distribution of solids in multistage fluidisation

The residence time distribution of the solids in multistage fluidisation, wherein downcomers for transferring the solids from stage to stage are provided to the horizontal perforated plates, is experimentally investigated covering a wide range in process variables using flat, baffle and spiral plates developed in the study. The RTD data is modelled using (i) the multiple parameter model, (ii) the diffusional mixing model and (iii) the fractional tank extension model when it is noted that over certain conditions of operation both spiral and baffle plates indicate near piston-flow for the solids without hampering the fluidisation characteristics.     

Use of Hildebrand and Lamoreaux's theory to predict solubility and related properties of gas-liquid systems

There is a need in many fields such as chemical engineering, process chemistry and pharmacology for reliable data on the solubility, entropy of solution and partial molar volume of gases in liquids. Experimental data are not available for many systems of practical and academic importance and consequently a reliable theory to predict values for these properties would be of great use. Recent refinements of solubility parameter theory have been successful in predicting such values for several systems and in this paper results obtained from this theory are compared with experimental measurements on a large range of gas-liquid systems. The agreement found is satisfactory in the most part, but for helium, neon, xenon, hydrogen and deuterium, the predicted results deviate consistently from the experimental values. This is to be expected on the basis of some of the earlier results of solubility parameter theory and an empirical modification is proposed that yields significant improvements in the predicted values. This has also been used to predict values for carbon dioxide, tetrafluoromethane and sulphur hexafluoride to which the theory is not directly applicable. Values predicted by solubility parameter theory are compared with those obtained by Battino and co-workers from scaled particle theory. In general, the two theories are equally successful in predicting solubilities, but refined solubility parameter theory yields better values of entropy of solution. It is concluded that the theory provides a useful means of predicting solubility and entropy of solution but that prediction of partial molar volume results is limited because of the paucity of data for the thermal pressure coefficients of the solvents. Attention is drawn to apparent inaccuracies in certain experimental measurements. The interrelationship between the various forms of the entropy of solution used by different workers is clarified. A value of the energy of vaporisation of carbon dioxide, δE₂^v=2.64±0.15 kcal/mol is calculated.     

Prediction of Flory–Huggins interaction parameters from intrinsic viscosities

Previous articles from this laboratory have described a model for predicting the second virial coefficient of a polymer solution given the polymer molecular weight and its intrinsic viscosity in the particular solvent. The same theory is used in this report to calculate values of the Flory–Huggins interaction parameter χ. The method is different in that χ is obtained through manipulation of simulated exprimental data in exactly the same way as in an actual experiment. Agreement between estimated χ values and those obtained at infinite dilutions from membrane osmometry or light scattering is within 2%, on the average. The model accounts for the molecular weight dependence of χ.     

Reverse osmosis separation of inorganic solutes in aqueous solutions using porous cellulose acetate membranes

Using a modified form of the Born expression for the free energy of ion-solvent interaction, to both the bulk solution phase and the membrane–solution interface, a parameter is obtained to express the repulsion of the ion at the interface. This parameter, called the free energy parameter for ions, is then related to solute transport parameter obtained from reverse osmosis experiments. Numerical values of this free energy parameter have been obtained for six monovalent and four divalent cations and for 12 monovalent anions. Using the experimental data for the reverse osmosis separation of sodium chloride as reference, the utility of the above parameter for predicting solute separation in reverse osmosis is illustrated for 32 other inorganic salts.     

An approximate solution to the model of a sparged packed bed electrode reactor

An approximate analytical solution to the mathematical model of a sparged packed bed electrochemical reactor (SPBER) in which a gaseous reactant undergoes direct anodic oxidation is presented. The model is based on two strongly nonlinear partial differential equations and is solved by the inverse operator method (IOM). The solution is used to study the direct anodic oxidation of propylene. The approximate IOM solution is useful in that it enables the direct investigation of the effect of model parameters on the operation of the reactor. Current density against electrode potential, polarization data, for the anodic oxidation of propylene in the SPBER is presented. Nonlinear parameter estimate methods are used to fit the model to experimental data to obtain physically meaningful values of kinetic parameters.     

The dissolution of a stationary bubble enhanced by chemical reaction

The problem of the dissolution of a gas bubble aided by a first order chemical reaction is treated theoretically. Two new approximate solutions are developed. The first one is based on a time-expansion introduced by us recently, is useful for small values of the reaction rate parameter. The second utilizes the assumption of a thin boundary layer, and is applicable for relatively large values of the reaction rate parameter. The accuracy of these and other approximations is assessed by comparison with a numerical solution of the complete model equations.     

Inverse problems of biological systems using multi-objective optimization

Mathematical modeling for dynamic biological systems is a central theme in systems biology. There are still many challenges in using time-course data to obtain an inverse problem of nonlinear dynamic biological systems. In this study, a multi-objective optimization technique is introduced to determine kinetic parameter values of biochemical reaction systems. The multi-objective parameter estimation was converted into the minimax problem through the satisfying trade-off method. The aspiration value was assigned as the minimum solution to the corresponding single objective estimation. The aim of this trade-off estimation was to obtain a compromised result by simultaneously minimizing both concentration and slope error criteria. Hybrid differential evolution was applied to solve the minimax problem and to yield a global estimation.     

Modeling the thermal and physical properties of liquid and gas mixtures of Fischer–Tropsch synthesis products

The method of pseudocritical thermodynamic parameters and the Lee-Kesler equation of state are used for calculating the thermal and physical properties of a solution and a mixture of gaseous paraffins that are major constituents of Fischer-Tropsch synthesis products. The dynamic viscosities of a solution and a vapor are modeled using an original procedure proposed by the authors. A surface tension is calculated by the parachor method. The composition of liquid and gaseous products is found using a stable iteration scheme proposed by the authors. The model is tested by a comparison with experimental data from the literature for model mixtures of hydrocarbons. The results of calculating the composition and thermophysical properties of a mixture of paraffins that model the composition of synthesis products at different values of the chain propagation parameter, temperatures, and pressures are presented.     

Finite element collocation solution of tubular and packed-bed reactor two point boundary value problems

Nonlinear two point boundary value problems (TPBVP's) are frequently encountered in chemical engineering. TPBVP's describing tubular and packed-bed reactors are important and have received much attention in the literature. For certain ranges of parameter values of the reactors, the TPBVP's become stiff and pose problems in numerical solution. These problems have been efficiently solved by the finite element collocation method using Hermite and B-spline functions in conjunction with quasilinearization. Numerical results have been presented and discussed. The problems have been solved for a wide range of parameter values without encountering any numerical difficulty.     

2—十一烷基—N—羟甲基—N—羟乙基咪唑啉与阴离子表面活性剂的协同效应

用最大气泡法测定咪唑啉的表面张力和临界胶束浓度,并与N-月桂酸-N-甲基牛磺酸钠（AMT）复配后测其表面张力及其分子间的相互作用参数。利用正规溶液理论对它们的协同效应进行了讨论,表明实验值与计算结果相符。