PHREEQC version 2 is a computer program written in the C programming language that is designed to perform a wide variety of low-temperature aqueous geochemical calculations. PHREEQC is based on an ion-association aqueous model and has capabilities for (1) speciation and saturation-index calculations; (2) batch-reaction and onedimensional (1D) transport calculations involving reversible reactions, which include aqueous, mineral, gas, solidsolution, surface-complexation, and ion-exchange equilibria, and irreversible reactions, which include specified mole transfers of reactants, kinetically controlled reactions, mixing of solutions, and temperature changes; and (3) inverse modeling, which finds sets of mineral and gas mole transfers that account for differences in composition between waters, within specified compositional uncertainty limits. New features in PHREEQC version 2 relative to version 1 include capabilities to simulate dispersion (or diffusion) and stagnant zones in 1D-transport calculations, to model kinetic reactions with user-defined rate expressions, to model the formation or dissolution of ideal, multicomponent or nonideal, binary solid solutions, to model fixed-volume gas phases in addition to fixed-pressure gas phases, to allow the number of surface or exchange sites to vary with the dissolution or precipitation of minerals or kinetic reactants, to include isotope mole balances in inverse modeling calculations, to automatically use multiple sets of convergence parameters, to print user-defined quantities to the primary output file and (or) to a file suitable for importation into a spreadsheet, and to define solution compositions in a format more compatible with spreadsheet programs. This report presents the equations that are the basis for chemical equilibrium, kinetic, transport, and inverse modeling calculations in PHREEQC; describes the input for the program; and presents examples that demonstrate most of the program's capabilities.

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User's guide to PHREEQC (Version 2)-a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations

Fundamental Principles Laminar Boundary Layer Flow Laminar Duct Flow External Natural Convection Internal Natural Convection Transition to Turbulence Turbulent Boundary Layer Flow Turbulent Duct Flow Free Turbulent Flows Convection with Change of Phase Mass Transfer Convection in Porous Media.

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Convection Heat Transfer

This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.

The MATLAB ODE Suite

Most models for fixed bed adsorbers have used either the homogeneoussolid or pore diffusion model for the pellets. When the adsorption isotherm is linear, the two models can lead to identical breakthrough curves. The conditions for this equivalence are presented here. It is shown that one of the bulk flow factors that was included in the formulation of one pore diffusion model will be significant only for feedstreams containing a relatively high concentration of adsorbate. The prosity factor of the pore model is shown to be very important, especially as the porosity decreases. The importance of the two diffusional models with respect to the predicted breakthrough curves is demonstrated. For comparable beds, it is shown that the breakthrough curve based on the homogeneous model is delayed with respect to that based on the pore model at early times, regardless of the shape of the isotherm. Finally, the various possible solutions for an irreverisble isotherm are reviewed for each of the models, and a solution is presented for the general case of a pore model with an outside film resistance.

Pore and solid diffusion models for fixed-bed adsorbers

Continuous flow systems

Mathematics of Adsorption in Beds. VI. The Effect of Longitudinal Diffusion in Ion Exchange and Chromatographic Columns

An analysis of chemical reactor stability and control—I: The possibility of local control, with perfect or imperfect control mechanisms

A theoretical study of multicomponent chromatography is here presented in which the system is considered to be one-dimensional, isothermal, locally at equilibrium and to have negligible diffusion effects. The discussion starts with constant initial and entry conditions and goes on to stepwise constant data with an arbitrary number of discontinuities. The Langmuir adsorption isotherm is perfectly fitted to the exposition of the mathematical theory of quasilinear equations for it leads to explicit forms for the Riemann invariants and characteristic parameters. This paper develops the theory of simple waves and of shock waves on an independent basis and illustrates this theory by the construction of solutions and the analysis of the interaction of waves. It is shown incidentally that the entropy change across a shock is consistent with the second law of thermodynamics. The separation of solutes is discussed and brief consideration is given to the problems associated with non-uniform geometry and non-isothermal adsorption.

On the theory of multicomponent chromatography

The purpose of this paper is to consider the well agitated continuous reactor from the standpoint of stability of the steady state. It has been shown in the past that chemical-reaction systems may be unstable in the sense that on slight perturbation they tend to move to a more stable state or that they are stable in their steady states, small perturbations being self-correcting so that the system possesses autoregulation. In this paper methods of developing criteria for the quantitative determination of stability or instability or presented and applied to some simple problems. In order that the effect of large perturbations on the system may be determined, complete solutions of the rigorous equations are obtained on the analogue computer (R.E.A.C.). A complete plot of reaction paths in the concentration-temperature plane may be obtained in this manner. Because of the nonlinearity of the system one cannot predict with certainty what steady state will be approached after a given large perturbation, multiple steady states being assumed possible. From the phase plot of reaction paths the regions in the plane which lead to certain steady states are delineated. Also it is shown that the natural behavior of a reactor is not to approach an unstable state. So far as the reactor is concenrned, the unstable state does not exist. The stability of the system is important to the engineer, as control will be easy or difficult and product quality will be satisfactory or not depending upon the relative stability of the steady state. An unstable state would require more elaborate control than a stable state.

Neal R. Amundson

Papers

Mathematics of Adsorption in Beds. VI. The Effect of Longitudinal Diffusion in Ion Exchange and Chromatographic Columns

An analysis of chemical reactor stability and control—I: The possibility of local control, with perfect or imperfect control mechanisms

On the theory of multicomponent chromatography

Chemical reactor stability and sensitivity

Analysis of Breakage in Dispersed Phase Systems