A modification of broyden's method for the solution of sparse systems—with application to distillation problems described by nonideal thermodynamic functions

An algorithm that permits Broyden's method to be used for the solution of large systems of algebraic equations with sparse Jacobians is presented. The new procedure is compared to Schubert's modification of Broyden's method and to the Newton-Raphson method by solving an extractive distillation problem. It is demonstrated that the new procedure is competitive with Schubert's method when it is necessary to evaluate Jacobian matrices numerically.     

An analysis of enhanced transverse dispersion on distillation plates

A mathematical model based on the concept of mixing by dispersion is used to analyse the possibility of improving distillation plate efficiency by enhanced transverse dispersion in the liquid, when transverse nonuniformites in liquid cross-flow occur. The intensities of transverse dispersion, which are able to overcome the detrimental effects of typical liquid velocity profiles, are outlined for plates of various diameters. It is concluded that, enhanced plate efficiency is obtained most effectively by combined improvement of liquid velocity profiles and transverse dispersion.     

Dynamic characteristic of binary srv distillation systems

This paper reports a simulation study of SRV (secondary reflux and vaporization) distillation system using a perturbed linear model. The important factors which affect the dynamics are the heat transfer rate between the rectifying and the stripping sections, and the mean temperature differences between the process fluid and the cooling or heating medium at the condenser and the reboiler. The system becomes unstable if the mean temperature differences at the condenser and the reboiler are small, and the heat transfer rate between the rectifying and the stripping sections falls within a certain critical range of values. On either side of this range the system is insensitive to any kind of disturbances because of the inherent regulation occurring between them. Modal analysis shows that the external reflux rate activates the slowest mode which contains mainly the contributions of liquid concentrations on all stages.     

On the dynamics of distillation processes-V: The topology of the boiling temperature surface and its relation to azeotropic distillation

It is shown that ridges and valleys in the boiling temperature surface do not coincide with azeotropic distillation region boundaries. Thus, the prevai and valleys is shown to be false. This allows for a greater flexibility in design and a better understanding of the behavior of azeotropic distillation     

Pole assignment in binary SRV distillation

Closed-loop characteristics of a 10-stage binary SRV distillation system were examined using the single-input pole assignment technique of Crossley & Porter[7]. The results indicated that the external reflux rate was the most preferred input variable for pole shifting, and that top and bottom compositions should be measured for feedback.     

A modular approach to distillation column simulation

THE DYNSYS 2.0 computer program for modeling the dynamic behavior of complex chemical plants is described. The application of this modular program to the simulation of some distillation column control schemes is presented. Tree typical column configurations are studied. The startup of the columns is simulated and the effect of some simple control arrangements is shown.

The advantages of a modular program structure are discussed and the flexibility and utility of DYNSYS is illustrated.     

Heat integration of distillation columns into overall processes

There are many well-known schemes for better energy efficiency in distillation. Examples are thermal coupling, multiple effect, heat pumping, etc. Usually these schemes are discussed for individual columns in isolation, independently from the overall process they are a part of.This paper puts the design of individual distillation columns into context with the heat integration for the overall process. An insight is discussed wGenerally, the paper defines good integration as a column not crossing the heat recovery pinch of the process and either the reboiler or the condenser     

Stepwise periodic distillation—II: Separation of a binary mixture

Stepwise periodic operation of a distillation column in which a binary feed is separated in top and bottoms product is described. A simple but realistic model, both for stepwise periodic and controlled cycling operation and a simulation algorithm are proposed. The results of an extensive parametric study of stepwise periodic operation are presented along those for ideal controlled cycling (no axial mixing).Although in some cases inferior to controlled cycling, it is shown that, for reasonable tray efficiencies and difficult separations, stepwise periodic operation can compare favorably or be superior to ideal controlled cycling.     

Comparison of plate numbers and column lengths in chromatography and distillation

It is sometimes suggested that the number of plates required in chromatography is approximately equal to the square of the number required for distillation. It is shown that this proposition overestimates the number of chromatographic plates by a factor of at least 5 and typically about 30. For a given separation specification, a gas chromatographic column is 3–30 times shorter than the equivalent distillation column. The variation of the two column lengths with relative volatility and desired purity level is shown graphically.     

Simulation of three-phase distillation towers

The approximate models of Boston & Sullivan[1] for estimating K-values and enthalpies are used to solve the MESH equations in the steady-state. The algorithm for phase-splitting uses approximate models in an inner loop and more accurate models in an outer loop. Oscillations between one and two liquid phases are aboided during interative calculations when concentrations approach the two-phase region. Simulation results compare well with those of Block & Hegner[2]. Wide- and narrow-boiling systems are simulated efficiently. For some systems, tray efficiencies determine whether or not a second liquid phase appears in the simulation results.     

A pilot-scale distillation facility for digital computer control research

A new pilot-scale multicomponent distillation facility and associated computer control system are described. The control system design utilizes a hierarchical structure, with a microcomputer for direct digital control of secondary process variables and a general-purpose, real-time computer for high-level control of the primary variables. The facility is used for advanced process control research and real-time system instruction. Some typical results incorporating a new multivariable predictive control algorithm are presented.     

A new approach to distillation column at total reflux

For a distillation column at total reflux, a simulation algorithm is presented, which takes into account the nonideality of both the vapor and liquid phases in addition to the pressure drop at each plate. The solution algorithm based on the Newton-Raphson technique is simple, because this algorithm has only one trial loop and no substitution parts and does not require a large portion of a computer's memory requirements. A numerical example shows the effectiveness of the proposed algorithm.     

Binary and ternary distillation in a stirred mass transfer cell

A continuous flow stirred cell operating at point conditions has been developed for the study of mass transfer in distillation. Results for the methanol-isopropanol and methanol-ethanol binary systems showed that 85–100% of the resistance to mass transfer resided in the vapour phase, and that the vapour side coefficients correlated well with those obtained in the same cell for gas phase controlled absorption. Flux rates for the ternary system methanol-ethanol-isopropanol were in fair agreement with those predicted from the binary data, using a modified form of the Krishna-Standart theory. It is concluded that the experimental technique shows considerable promise, but requires further development to overcome discrepancies in the interfacial heat balance.     

Local thermodynamic models for high pressure process calculations

This paper discusses the local thermodynamic model concept in the context of high pressure chemical processes where equations of state are usually required for performing thermodynamic property evaluations. Functional forms for local K-value models are developed by simplifying the rigorous equations governing fluid phase equilibria. The local models have imbedded adjustable parameters that are evaluated by regression against either experimental data or thermodynamic properties calculated with more rigorous theoretically based models. During the course of the calculations the parameters in the local models are updated periodically, thereby ensuring that the local models continually provide accurate values for thermodynamic properties. Use of the concepts for process calculations is demonstrated in an example problem.     

Liquid phase mass transfer resistance in a small scale packed distillation column

Binary distillation experiments using the toluene-trichloroethylene system were conducted in a 3-in. diameter column packed with 0.25 in. Raschig rings. In order to determine the liquid phase mass transfer characteristics, gas absorption studies with CO₂-water system were also performed on the same column. It was shown in both cases that end-effects can be ettectively removed by using a “differential height” technique. It was observed in the distillation experiments that the liquid phase was saturated, indicating a negligible resistance to mass transfer in the liquid. Results from the absorption experiments indicate that this resistance is small but not negligible.     

Effectiveness of mass transfer in a packed distillation column in relation to surface tension gradients

In packed columns large differences occur in the wetting of the particles and especially in the refreshing of the liquid on the wetted particles due to gradients in surface tension of the liquid/gas interface. Mass transfer rates may differ with a factor 2. In a column packed with Berl saddles distillation experiments were performed with a mixture n-heptane/cyclohexane. Ceramic Berl saddles of 4, 6 and 10 mm were used as well as aluminum Berl saddles of 4 mm. In some of the experiments the saddles were coated with PTFE (teflon). The driving force for mass transfer was varied over a wide range. Both negative and positive driving forces were realized. The influence of the surface tension driven refreshment of the interface is most pronounced for small particles; for larger liquid flow rates, that may be applied in beds with larger particles, the effect is obscured by the inertia of the downcoming liquid.     

Approximate analytical solutions for systems with axial dispersion. The steady state behaviour of a binary distillation plate

A perturbation method is applied to obtain an approximate analytical solution to the non-linear differential equation which describes the steady state concentration profile of the liquid flowing on a distillation plate. This method proved effective in producing results with reasonable deviations from the exact solution of the differential equation and a considerable reduction in computing time. However, it sometimes requires the solution of a single first order differential equation. Thus, a further approximation was adopted, giving a method which does not involve the solution of any differential equation. Although the deviation in the exit liquid composition is somewhat higher, yet the computing time is less.     

Gear's procedure for the simultaneous solution of differential and algebraic equations with application to unsteady state distillation problems

The dynamic equations modeling a sieve plate at unsteady state are developed. Gear's procedure for the simultaneous solution of systems of stiff differential and algebraic equations is presented and demonstrated for the solution of unsteady state distillation problems. It is shown that the basic stage model can be modified by the addition of one variable and one equation such that Gear's procedures are readily applied. The proposed model and solution procedure is contrasted to recently published procedures. Numerical results are given for the solution of a problem involving an extractive distillation column at unsteady state.     

Thermal effects in non-adiabatic binary distillation effects of partial condensation of mixed vapors on the rates of heat and mass transfer and prediction of H. T. U.

In order to investigate thermal effects in binary distillation, theoretical approaches were made by use of a laminar boundary layer theory with uniform suction or blowing at the interface to give the following correlations for heat and mass fluxes:Nu_Gx = 0.332 Re_Gx^{$\text{1}\text{2}$}Pr_G^{$\text{1}\text{3}$}g(F(0), Pr_G)Sh_Gx(J_Axs/N_Ax) = 0.332 Re_Gx^{$\text{1}\text{2}$}Sc_G^{$\text{1}\text{3}$}g(F(0), Sc_G). From the energy balance at the vapor—liquid interface the factors which affect interfacial velocities or F(0) were discussed.Measurements of heat and mass fluxes in a vertical flat-plate wetted-wall column for the methanol-water system were made and gave a good agreement with the theory.A new method for the prediction of H_OG in binary distillation under partial condensation was proposed. The calculated H_OG's gave a fairly good agreement with the observed results.     

Measurement and differential kinetic analysis of wall temperatures in monolith converters—I kinetic analysis

It is possible to measure axial wall temperature profiles in adiabatic monolith converters under actual reaction conditions. By applying a differential method of kinetic analysis, axial profiles of concentrations at the wall as well as of reaction rates can be derived with proper computation methods. The data thus obtained may serve for further kinetic evaluation, that is for model building and parameter estimation by proven and well-known methods. The whole procedure looks also promising for performing activity tests on fresh and partly deactivated monolith converters. The slightly greater effort of measuring wall temperatures compared with the usual method only to record inlet and outlet temperatures and integral conversions might more than pay off. In the present first part of the paper the differential method of kinetic analysis is developed and tested by some simulation studies. The second part will deal with actual experiments.