Mixing Cell Model Research Articles

Application of MCMFIT (Mixing cell model FIT) for phosphorus transport in saturated soil column

An attempt was made to study the effect of model parameters in the prediction of phosphorus (P as orthophosphate) solute transport and the verification of a fitting program, MCMFIT (Mixing Cell Model FIT) to column experimental data. Breakthrough curve pattern for phosphorus solute transport was found to be S-shaped or curvilinear. Simulated results using a dynamic physical non-equilibrium sorption model (DPNSM) and Freundlich isotherm constants K and N (calculated from the column experiments) have satisfactorily fitted the corresponding experimental breakthrough curves. The prediction by optimal parameter values giving the best fit from the MCMFIT was similar to the fitted results by DPNSM although the parameter values were different.

A Conceptual Evaluation of Regional Ground-Water Flow, Southern Nevada-California, USA

We use a mixing-cell model, calibrated with the spatial distribution of corrected 14 C ages of the ground water, to simulate steady-state flow in the regional ground-water flow system underlying the Nevada Test Site and environs in southern Nevada-California. Thirty-five cells covering 12,400 km 2 are used. Total annual inflow to the system averages 5,300 × 10 4 m 3 and is approximately equally divided between subsurface inflow from outside the system and recharge derived from within the system. Almost 23 percent of the total flow originates in the Spring Mountains; subsurface inflow from regions to the north and east of the system accounts for 11 percent and 15 percent of the total flow, respectively. Approximately 10–12 percent of the total flow is derived from recharge in the Fortymile Canyon/Wash area, which corresponds to a range in the average annual areally-distributed recharge rate of 25.6 to 31.7 mm. Rainier Mesa is recharged at an average annual rate of 8.2 mm. The range in long-term average annual recharge to the cell representing Yucca Mountain, the site for a proposed high-level nuclear waste repository, is 15 to 32 × 10 4 m 3 , equivalent to areally-distributed annual rates of approximately 1.9 to 4.2 mm. Approximate age of the ground water beneath Yucca Mountain is 12,000 years old. Minimum mean ground-water ages in the system range from 2,200 years to almost 100,000 years old. Our work suggests that the Ash Meadows subbasin is distinguishable as a somewhat independent unit but that the Oasis Valley and Alkali Flat-Furnace Creek Ranch subbasins can probably be treated as one unit.

Simulation of catalyst fouling at the particle and reactor levels

We present a method for the simulation of catalyst fouling at the particle and reactor levels. At the particle level, diffusion and reaction are described by a continuum model. The accessible pore surface area and the effective diffusivity are obtained from percolation theory. Both these quantities depend on location within the particle, and vary with time. At the reactor level, a mixing-cell model is employed. We demonstrate the method by applying it to the hydrodemetallation of oil. Hydrodemetallation performance depends strongly on the pore structure of the catalyst, and there is a high degree of interplay between events occurring at the particle and reactor levels.

Modeling high‐viscosity, melt‐phase, condensation polymer reactors with a time‐scales analysis

AbstractMathematical models for simultaneous reaction and mass transfer occurring in the manufacture of high‐viscosity, condensation polymers are considered. The specific example of polycondensation of polyethylene terephthalate is examined. Reactor performance is estimated by using a kinetic expression modified by an effectiveness factor. The effectiveness factor is correlated against a ratio of two characteristic times, one identified as the time scale of mixing and the other identified as the time scale of reaction. The time scale of mixing is estimated from experimental mixing data, thereby avoiding the use of potentially inaccurate mixing assumptions. In place of reaction experiments, overall reaction rates are generated using a more detailed mixing‐cell model. The effectiveness factor correlation is compared against previous models. Because the correlation is based on predicted reaction rates rather than experimentally measured reaction rates, the value of the present work lies in the demonstration of the time scales modeling technique. © 1995 John Wiley & Sons, Inc.

Brackish groundwater resources in Bahrain: Current exploitation, numerical evaluation and prospect for utilization

In Bahrain, where water resources available for direct use are finite and the best of its quality has a salinity of over 2.5 g L−1, utilization of brackish groundwater is an essential part in the management of the country's water resources. Bahrain's brackish water occurs in the Rus-Umm Er Radhuma formations in the form of a lens of a finite lateral extent, with a salinity ranges between 8 and 15 g L−1. Planning for utilization of brackish groundwater for desalination purposes in Bahrain was based on simulation modeling of the aquifer system using a mixing cell model developed originally in 1983. The model was used to predict the aquifer response to pumping from the proposed wellfield in terms of changes of TDS over a period of 20 years. Construction and operation of the wellfield in 1984 was based on the predicted salinity changes. Over the past 9 uears of wellfield operation (1984–1993), and through continuous monitoring of the aquifer response to pumping, the collected data is used to post-audit the original model by history matching. The calibration process adopted has resulted in a statisfactory agreement between the model output and the observed data. The model is then used to predict the wellfield salinity changes and the aquifer potentiometric levels. The expected life span for the brackish groundwater utilization by the wellfield is redefined through constrained utilization that takes into account salinity deterioration coupled with the effect of head decline on hydraulic interaction between the brackish water and the upper fresh water aquifer. The results suggest that the operation of the wellfield should cease by the year 2007. Construction of a new model that enables testing and evaluating different development scenarios is recommended to aid future management decisions regarding the utilization of brackish groundwater.

Analysis of one-dimensional multispecies transport experiments in laboratory soil columns

A numerical scheme based on the mixing cell concept is used to analyse multispecies solute transport under the condition of steady flow. Explicit examples are presented for the case of two chemical species that are reacting during transport. Also, unlike the standard mixing cell approach, the improved method used here maintains second-order accuracy. The mixing cell scheme is applied to various data sets from multispecies transport experiments. First, the scheme is used to analyse the snow-plow effect as observed in a laboratory experiment. This phenomenon results from large concentration differences in the solution and solid phases within the soil profile. The analysis revealed that the solute exchange process is time dependent, rather than being in equilibrium. In another case, the scheme was used to analyse the observed snow-plow peaks of Ca resulting from ion exchange reactions between Ca and Mg in a soil profile. The analysis revealed that the selectivity coefficient changes with total cation concentration in the system. The results demonstrate that nonlinear reactions in multispecies transport can be analysed efficiently, accurately, and quickly using the improved mixing cell model.

Simulation of in situ uraninite leaching. part I: A partial equilibrium model of the NH4HCO3-(NH4)2CO3-H2O2 leaching system

In situ leaching of uraninite and calcite by H2O2-NH4HCO2-(NH4)2CO3 solutions has been simulated using a partial equilibrium model which incorporates a one-parameter mixing cell model of solution flow. Rate laws for UO2 dissolution and for CaCO2 dissolution/precipitation were taken from the literature, as were equilibrium constants for solution phase reactions. Parameters of the model include the UO2 and CaCO3 ore grades, the concentrations of the H2O2, NH4HCO3, and (NH4)2CO3 components, porosity, exit solution flow rate, ore and mineral densities, and mineral rate constants and surface areas. Mineral conversions, component and species concentrations, and porosity are among the time-dependent quantities calculated using the model. For the conditions simulated, calcite dissolved somewhat faster than uraninite. The results emphasize the importance of the coupling between the mineral reactions and solution flow. Changes in the concentrations of the CO32- and HCO3- species are particularly complicated and not predictable from the calcite kinetics alone or from a purely equilibrium model; although the simulations did not reveal any conditions under which the solution would become saturated with CaCO3, the pH continued to change throughout the calcite dissolution and is buffered only after calcite has been consumed.

Note on common mixing cell models

Different types of mixing cell models are analysed in detail. One type is found to be second-order accurate when reactions are not considered but reduces to first-order accuracy when a sorption reaction is taken into account. The second type is only first-order accurate even when a reaction is not considered. The most commonly used mixing cell model is extended to account for different surface boundary conditions, including that for a landfill. The exit conditions for a finite spatial domain were taken into consideration by making use of boundary layer corrections. It is found that the results obtained using the mixing cell model coupled with a non-linear isotherm are very accurate when compared with results obtained from a Crank-Nicolson scheme. It was observed that the accuracy of the results is dependent on the spatial step size. Best results are obtained when the spatial step size is close to twice the dispersivity. Application of the mixing cell model with a boundary layer correction was demonstrated by comparison with experimental data from a laboratory Na-Ca exchange experiment in which a solution containing Na was passed through a column filled with Ca-saturated loamy sand. The mixing cell model was also applied to breakthrough curves of two organic compounds (carbontetrachloride and tetrachloroethylene) observed in a field experiment conducted at Borden, Ontario, Canada. The mixing cell model described satisfactorily the Na transport in loamy sand and also the transport of organics in a Borden aquifer.

Mixing cell models for nonlinear nonequilebrium single species adsorption and transport : Bajracharya, K; Barry, D A Water Resour ResV29, N5, May 1993, P1405–1413

Mixing cell models for nonlinear nonequilebrium single species adsorption and transport : Bajracharya, K; Barry, D A Water Resour ResV29, N5, May 1993, P1405–1413

Application of VLEACH to Vadose Zone Transport of VOCs at an Arizona Superfund Site

AbstractCleanup standards for volatile organic compounds in thick vadose zones can be based on indirect risk (transport to ground water) when contamination is below depths of significant direct risk. At one Arizona Superfund site, a one‐dimensional vadose zone transport model (VLE‐ACH) was used to estimate the continued transport of VOCs from the vadose zone to ground water. VLEACH is a relatively simple and readily available model that proved useful for estimating indirect risk from VOCs in the vadose zone at this site. The estimates of total soil concentrations used as initial conditions for VLF.ACH incorporated a variety of data from the site. Soil gas concentrations were found to be more useful than soil matrix data for estimating total soil concentrations at this arid‐zone site. A simple mixing cell model was used with the VLEACH‐derived mass loading estimates from the vadose zone over time to estimate the resulting changes in ground water concentrations. For this site, the results of the linked VLEACH/mixing cell simulations indicate it is likely that the federal MCI. for TCE will be exceeded in underlying ground water if remedial action on I he vadose zone is not pursued.

Hydrologic Controls in Nitrate, Sulfate, and Chloride Concentrations

AbstractNitrate‐nitrogen, SO4, and Cl concentrations and discharge rates from a small, upland watershed in east central Pennsylvania were analyzed in the context of a layered aquifer. A pattern of greater NO3‐N concentrations during periods of greater discharge was generally followed by declining concentrations during hydrograph recession. The coincident temporal pattern of NO3‐N and discharge appears hydrologically controlled because both NO3‐N and SO4 exhibited similar patterns even though there are important differences in their behavior and reactivity in biologically controlled systems. Chemical concentration patterns in stream flow arose from flow through a two‐layer geologic system in which NO3‐N and SO4 concentrations were different between layers. The substantially lower concentrations observed in the deeper water were attributed to recharge from areas with forest and mixed land uses. Discharge from this deeper layer supported the lowest flows and the lowest chemical concentrations observed in stream flow. Increases in stream flow resulted from greater discharge from the shallower groundwater layer. Adding discharge from the shallower groundwater resulted in higher chemical concentrations in stream flow. Nitrate‐nitrogen concentration estimated with a mixing cell model, in which concentration changes result from dilution only, closely matched measured concentration.

Mixing cell models for nonlinear nonequilibrium single species adsorption and transport

The oft‐used “standard” mixing cell model describing reactive solute transport is extended to cater for nonequilibrium reactions. For tracer transport, the standard model is second‐order accurate in the space and time discretization used. However, an error analysis of the standard model reveals that its accuracy is degraded when reactions are included. An “improved” model which maintains second‐order accuracy is developed. The improved mixing cell model is demonstrated to be more accurate than the standard model. The solution obtained from the improved model is found to agree very well with the results of a numerical Crank‐Nicolson solution for various isotherms. The improved mixing cell predictions are also shown to agree very well with the exact solution for the two‐site adsorption model, where a linear isotherm is considered. Application of the improved model was illustrated by simulating the experimental data from a laboratory Ca‐K exchange experiment in which a solution containing Ca was passed through a column filled with K‐saturated soil. The model was used in conjunction with the experimentally determined nonlinear adsorption isotherm to predict the experimental breakthrough data. By changing the value of kinetic coefficient, it was demonstrated that the predictions varied markedly. However, the simulations suggest that the equilibrium model is more appropriate than the assumption of nonequilibrium.

Mixing cell models for nonlinear equilibrium single species adsorption and transport

A simple improved mixing cell model is shown to be more accurate than the standard mixing cell model when used to solve solute transport problems with a nonlinear adsorption isotherm included. The solution obtained from the improved model is found to agree closely with that calculated using a Crank-Nicolson scheme, and an exact analytical solution for solute transport subject to a nonlinear solute adsorption isotherm. Application of the improved mixing cell model was demonstrated by application to experimental data from a laboratory CaK exchange experiment in which a solution containing Ca was passed through a column filled with K-saturated soil. The model was used in conjunction with the experimentally determined nonlinear adsorption isotherm to predict the experimental breakthrough data. The data and the predictions agreed reasonably well, indicating that the equilibrium assumption is reasonable. However, it is suggested that an appropriate isotherm is necessary to describe well the concentration history curve.

Unified modeling technique for high‐viscosity, melt‐phase, condensation polymer reactors

AbstractMathematical models for chemical reaction and mass transfer occurring in the manufacture of high‐viscosity condensation polymers are considered. A preliminary study indicates that several diverse models can be represented using a single formula based upon an effectiveness factor. The effectiveness factor is shown to depend upon a ratio of time scales of mixing and reaction. The formula giving the effectiveness factor in terms of the time scales ratio is shown to depend upon mixing assumptions only. Starting with a mixing‐cell model, a new modeling framework is developed and shown to include the previous models as special cases. The framework is free of inherent mixing assumptions and can be applied to a wide variety of situations once the mixing characteristics are specified. © 1993 John Wiley & Sons, Inc.

Quantitative assessment of the flow pattern in the southern Arava Valley (Israel) by environmental tracers and a mixing cell model

This paper demonstrates the implementation of a novel mathematical model to quantify subsurface inflows from various sources into the arid alluvial basin of the southern Arava Valley divided between Israel and Jordan. The model is based on spatial distribution of environmental tracers and is aimed for use on basins with complex hydrogeological structure and/or with scarce physical hydrologic information. However, a sufficient qualified number of wells and springs are required to allow water sampling for chemical and isotopic analyses. Environmental tracers are used in a multivariable cluster analysis to define potential sources of recharge, and also to delimit homogeneous mixing compartments within the modeled aquifer. Six mixing cells were identified based on 13 constituents. A quantitative assessment of 11 significant subsurface inflows was obtained. Results revealed that the total recharge into the southern Arava basin is around 12.52 × 10 6 m 3 year −1 . The major source of inflow into the alluvial aquifer is from the Nubian sandstone aquifer which comprises 65–75% of the total recharge. Only 19–24% of the recharge, but the most important source of fresh water, originates over the eastern Jordanian mountains and alluvial fans.

Interwell Tracer Test To Determine Residual Oil Saturation In A Gas-Saturated Reservoir. Part I: Theory And Design

Abstract The residual oil saturation to gas-flood is an important parameter to evaluate the potential of the Golden Spike D3 ‘A’ pool as a candidate for enhanced oil recovery and gas storage. Conventional methods, including logging, sponge coring and single well tracer testing, are not applicable to this low-pressure, low-porosity, gas filled carbonate reservoir. An interwell tracer method which works on the chromatographic separation of tracers with different Henry's law constants was therefore proposed and tested in the lab. Stringent selection criteria based on the detection limit and Henrys law constant were established to screen chemicals for application. According to the criteria, sulphur hexafluoride and halocarbons (Freon), which can be detected in the sub ppm range using a gas chromatograph equipped with an electron capture detector, were selected for this study. Henry's law constants were determined experimentally using a static equilibrium method and a dynamic slim tube displacement method. In the preliminary screening of chemicals for lab testing, the Henry's law constant could be estimated from vapor pressure after correcting for the non-ideal behavior using the regular solution theory. It was demonstrated in the slim tube tests that residual oil saturation could be determined within I pore volume % accuracy from tracer separation using the simple chromagraphic theory. A mixing cell model was also developed to simulate the slim tube test results. This model was also capable of handling dual porosities that are common to carbonates. Introduction The Golden Spike D3 ‘A’ pool(1,2), located 29 km southwest of Edmonton in central Alberta, is a carbonate reservoir. With the on-coming of the Beaufort Sea gas development, Golden Spike is being considered as a candidate for Beaufort gas storage and enhanced oil recovery. In as much as both projects have substantial economic incentive and are somewhat adversely affected by each other, the two projects need to be carefully evaluated using the best possible data. Residual oil saturation to gas-flood is the key parameter for the evaluation of either process. Various conventional methods for Sor determination, such as production history, sponge coring, laboratory gas floods, logging and single-well tracer testing, have been extensively reported and compared in the literature(3,4). Unfortunately, these methods are not satisfactory for low-pressure, low-porosity, gas-filled carbonate reservoirs such as Golden Spike. Therefore, an interwell method(5,6) using Freon's with different vapor pressures was proposed and studied in the lab. The interwell method works on the chromatographic separation of Freon in the reservoir. According to the chromatographic theory, the Freon with the highest Henry's law constant (or K value) or the lowest solubility in oil is produced first. Thus, by comparing the production profiles of various Freon's, the average residual oil saturation between wells can be determined. Although the principle of the interwell method was disclosed by Cooke(6) in 1971, not a single test has been reported in the literature due to a lack of suitable chemicals and the potential interpretation problems(3). The Golden Spike test is the first attempt ever in the industry to apply the theory to the field.

Ion chromatography of fresh- and salt-water displacement: Laboratory experiments and multicomponent transport modelling

Column experiments with fresh- and salt-water displacements were performed to obtain information on multicomponent ion exchange at varying concentration levels as well as to validate a multicomponent transport model. During salt-water displacement by fresh groundwater, a three times larger dispersivity was observed in the interface than for the reverse situation. Experiments with synthetic solutions with varying Ca/Na ratios indicate that solution density differences and chemical interaction between displacing solution and exchange complex determine the change in dispersivity. Using calculated selectivity coefficients, covering a broad composition range, and dispersivity values as input, a geochemical mixing cell model was applied to simulate the displacement experiments. The results show that fresh-water displacement by once diluted seawater can only be modelled when the Gapon selectivity of Ca over Mg decreases by a factor of two during the breakthrough.

A deuterium-calibrated groundwater flow model of a regional carbonate-alluvial system

The White River Flow System (WRFS), a regional carbonate-alluvial groundwater system in southeastern Nevada, U.S.A., contains large amounts of water in storage, especially in the underlying carbonate reservoir. As the population of Nevada grows, it may become necessary to tap the resources of this and other regional carbonate systems. Because of the depth to the carbonate reservoir and, until now, lack of motivation to collect detailed hydrogeological data on it, the state of knowledge of flow in the carbonate system is poor. However, a simple mixing-cell flow model of the WRFS can be constructed and calibrated with the spatial distribution of the stable isotope deuterium. This type of model subdivides the system into carbonate and alluvial cells and routes water and deuterium through the entire cell network. It provides estimates of recharge rates, groundwater ages and volumes of water in storage. Transience in recharge rates and their deuterium signatures are unaccounted for by the model. The lack of constraints on the system mandates the calibration of three different flow scenarios, each of which differs slightly from the other. Despite these differences, some consistent quantitative results are obtained. Foremost among these are: (1) the carbonate aquifer may contain as much as 752 km 3 of water in storage; (2) recharge from the Sheep Range to Coyote Spring Valley is at least 90% greater than previously believed; (3) Lower Meadow Valley is part of the WRFS and contributes underflow to Upper Moapa Valley; (4) underflow with an average value of 0.163 m 3 s −1 flows westward out of the system along the Pahranagat Shear Zone; (5) recharge to the alluvial system is greater than that to the carbonate system; (6) groundwater mean ages range from 1600 to 34 000 years, with the oldest waters exceeding 100 000 years old. The results also demonstrate that deuterium can be used to calibrate simple flow models and provide groundwater ages. Despite the uncertainties and lack of constraints in mixing-cell models, they provide first approximations to information which, until now, has been difficult, if not impossible, to obtain. These models are especially useful for analyzing sparse-data systems, testing different flow hypotheses with minimal effort, providing ranges in parameter estimates, guiding future data collection and serving as precursors for the development of more sophisticated models.

Comparison of axial dispersion and mixing cell models for design and simulation of fischer-tropsch slurry bubble column reactors

Two different general approaches for simulating the performance of Fischer-Tropsch slurry bubble column reactors based upon the axial dispersion model and mixing-cell models are described. Comparisons between semi-batch operation and continuous operation with either cocurrent or countercurrent flow of gas and liquid are performed to examine the effect of overall gas-liquid contacting pattern on various measures of reactor performance. For the ranges of operating conditions investigated here, the catalyst distribution is the primary intrinsic variable that is most affected by the use of cocurrent versus countercurrent gas-liquid contacting pattern for the case of continuous operation. If overall synthesis gas conversion is used as a common measure of performance, then cocurrent flow is preferred over countercurrent flow. In terms of describing the liquid-phase flow pattern, a mixing cell model represents a more realistic approach and has computational advantages when compared to the axial dispersion model. A correlation which relates the liquid-phase Peclet number to the number of mixing cells for a particular case is also given and discussed.

FCC Regenerator flow model

Catalyst flow patterns in FCC cross-flow and swirl-type dense-bed regenerators affect regenerator efficiency. A two-dimensional, pseudo-homogeneous flow model describing such flow patterns is developed. The predicted catalyst flow patterns for swirl regenerators indicate catalyst bypassing and recirculating zones. More severe bypassing is predicted for cross-flow regenerators, the severity of bypassing increases with increasing Reynolds number. The calculated flow patterns are combined with a radial mixing-cell model to account for radial mixing in fluidized beds. Radial mixing rates in commercial size regenerators counteract somewhat the effect of bypassing. Model calculations coupling coke burning kinetics with the flow model, however, show that bypassing leads to non-uniform coke levels and oxygen breakthrough. This modeling technology is used to optimize FCC regenerator operation.