Mixing Cell Approach Research Articles

The Influence of Hyper-Alkaline Leachate on a Generic Host Rock Composition for a Nuclear Waste Repository: Experimental Assessment and Modelling of Novel Variable Porosity and Surface Area

Deep geological disposal is the preferred solution for long-term storage of radioactive waste in many countries. In a deep repository, cementitious materials are widely used in the structure and buffer/backfill of the repository for the stabilisation of the hazardous materials. The cement acts as a physical barrier and also contributes chemically to waste containment by buffering the groundwater to a high pH, limiting the solubility of many radionuclides. This paper describes an experimental and modelling study which evaluates the geochemical interaction between young cement leachate (YCL, pH = 13) and a generic hard rock (in this case Hollington sandstone, representing a ‘hard’ host rock) during permeation with the leachate, as it drives mineralogical changes in the system. One-dimensional reactive transport was modelled using a mixing cell approach within the PHREEQC geochemical code to identify the essential parameters and understand and scale up the effect of variations in these parameters on the observed geochemical processes. This study also focused on the effects of variable porosity, reactive surface area and pore volume on improving the modelling of rock alteration in the system compared to conventional models that assume constant values for these properties. The numerical results showed that the interaction between the injected hyper-alkaline leachate and the sandstone sample results in a series of mineralogical reactions. The main processes were the dissolution of quartz, kaolinite and k-feldspar which was coupled with the precipitation of calcium silicate hydrate gel and tobermorite-14A (C–S–H), prehnite (hydrated silicate), saponite-Mg (smectite clay) and mesolite (Na–Ca zeolite). The simulation showed that the overall porosity of the system increased as primary minerals dissolve and no stable precipitation of the secondary C–S–H /C–A–S–H phases was predicted. The variable porosity scenario provides a better fitting to experimental data and more detailed trends of chemistry change within the column. The time and the number of moles of precipitated secondary phases were also improved which was related to greater exposure surface area of the minerals in the sandstone sample to the YCL.Article HighlightsThe drop in calcium, aluminium and silicate concentrations is mainly due to the formation of calcium silicate hydrate and zeolite minerals as secondary phases. The simulation showed that the overall porosity of the system increased as primary minerals dissolve and no stable precipitation of the secondary C–S–H /C–A–S–H phases was predicted.The dissolution of primary minerals and the precipitation of secondary C–S–H phases had a minimal effect on the pH values, and this was controlled mainly by the initial fluid chemistry.The variable porosity scenario provides a better fitting to experimental data and more detailed trends of chemistry change within the column.

Simulating wheat growth response to potassium availability under field conditions with sandy soils. I. Model development

The uptake of K by crops depends on dynamic interactions between soil and fertiliser K availability, season effects on growth, agronomic management and soil properties. In such complex systems, simulation modelling which accounts for both K supply and demand processes may be the most effective approach to assessing the efficacy of different K fertiliser strategies for crop uptake. We developed a K model for wheat in the crop simulation model APSIM. Reactive solute transport was modelled using the mixing cell approach and root K uptake was modelled based upon the concentration of K in the root system and soil solution and an equilibration between root and shoot concentrations. Photosynthetic assimilation rate and water-use efficiency were modified according to shoot K concentration. The parameter optimisation software PEST was applied to optimise the parameters introduced for this model. Overall the model provided a satisfactory match to the calibration data set for soil surface K, grain yield and shoot K concentration in early growth stages but model predictions were most sensitive to the parameter that describes the shape of the K adsorption isotherm. Evaluation of the calibrated model for an independent data set (n=807) showed reasonable agreement with maturity biomass (r2=0.73, RMSE=1666kgha−1), soil surface K in the year after application (r2=0.72, RMSE=17mgkg−1), and grain yield (r2=0.66, RMSE=637kgha−1). However, it was necessary to modify the grain fill function to include the effect of shoot K concentration on grain filling rate which led to an improvement in grain yield prediction at low K fertiliser rates and in the relative response to K fertiliser application. The current K simulation model provides satisfactory predictions of wheat response to K on sands across a range of seasons. Further improvement could be achieved by evaluation of the solute transport model under field conditions and the grain fill function; both improvements require data sets of wheat K response from experimental sites with well characterised soil water properties.

Ungauged catchment contributions to Lake Tana's water balance

AbstractLake Tana's flow system is governed by four main components: the inflow from surrounding river catchments into the lake, the outflow at Bahir Dar through the Blue Nile, the direct rainfall on the lake and the direct evaporation from the lake. While recent studies applied simple pragmatic approaches to estimate runoff from ungauged catchments, here emphasis is placed on more advanced approaches based on regionalization and spatial proximity principles. In the regionalization approach, model parameters of the conceptual HBV (Hydrologiska Byråns Vattenbalansavdelning) rainfall‐runoff modelling of gauged catchments are transferred to ungauged catchments to allow runoff simulation. Parameter transfer was attempted through regression, proximity procedures and catchment size. This yielded 42, 47 and 46%, respectively, of the total river inflow for the three procedures. Lake areal rainfall is estimated by interpolation of the rain gauges around the lake, open water evaporation is estimated by the Penman‐combination equation while observed inflows and outflow data are directly used in the lake water balance. The water balance closure term was established by comparing the measured lake levels with the calculated levels. Results show that runoff from ungauged catchments is around 880 mm per year for the simulation period 1995–2001 with a water balance closure error of 5%. In addition, use is made of river and lake water chemistry to arrive at an estimate of the unknown inflow and outflow components through the mixing cell approach. The results obtained with this method also provide independent information with regard to the errors in the individual water balance components. Copyright © 2009 John Wiley & Sons, Ltd.

Evaluating the recharge mechanism of the Lower Kuiseb Dune area using mixing cell modeling and residence time data

Summary A mixing cell approach was extended by a method incorporating mean cell residence times derived from 14C to further constrain and validate the modeling results. This extended approach was used to model the groundwater system of the Lower Kuiseb Dune area in Namibia. The Kuiseb river is a 560 km long ephemeral river that crosses the Namib Desert from east to west. Transmission losses from the riverbed during flood events are an important source of groundwater recharge to the underlying aquifer system. The Lower Kuiseb area is located in a hyper arid region with annual precipitation less than 25 mm/yr. Hydrochemical data from 13 wells in the area were used in the mixing cell model (MCM). End members were identified as sources for the groundwater found in the Lower Kuiseb, including inflow from the crystalline basement plateau north of the Kuiseb as well as floodwater from the Kuiseb river. A conceptual groundwater recharge and flow model was developed, and then inverse mixing cell modeling was carried out using hydrochemical tracers. This approach generally allows for several possible solutions. After completing the inverse modeling, a forward mixing cell model was developed by varying the mean residence time of each cell to fit calculated 14C data to the measured 14C data. This new approach joins previously developed methods solely based on conservative mixing or residence time optimization. Based on the results of the model the fraction of floodwater in different sections of the Lower Kuiseb groundwater systems was calculated, ranging from 61% to 98.2%. In addition to floodwater, groundwater inflow from the crystalline basement north of the Kuiseb was shown to contribute to the Lower Kuiseb aquifer, accounting for salinization during periods without flooding.

Modeling of bubble column slurry reactor for reductive alkylation of p-phenylenediamine

A bubble column slurry reactor (BCSR) model has been developed for the reductive alkylation of p-phenylenediamine (PPDA) with methyl ethyl ketone (MEK) to N , N ′ -di-secondary-alkyl- p-phenylenediamine (Di-amine). This particular reaction system is commercially relevant and involves a combination of parallel and consecutive reactions comprising equilibrium non-catalytic (homogeneous) and catalytic (heterogeneous) steps. The proposed model is based on the ‘mixing cell approach’. In this work the mixing cell approach has been extended by including a liquid backflow stream from all but the bottommost mixing cell. The model incorporates the contributions of gas–liquid and liquid–solid mass transfer, heat effects, and complex multistep reaction kinetics. CFD model is used to estimate the extent of backflow among mixing cells and its dependence on operating parameters. The effect of gas and liquid velocities, catalyst loading, inlet PPDA concentration, and temperature on the conversion, selectivity, global rate of hydrogenation, and temperature rise is discussed. The comparison of the current approach with the traditional mixing cell model is discussed. The BCSR model presented here will be useful to provide guidelines for designing and improving overall performance of bubble column reactors.

Mathematical modeling and simulation for gas–liquid reactors

Gas–liquid reactors are widely used in many industrial processes such as oxidation, hydroformylation, chlorination, etc. The paper develops comprehensive model for reactors using the mixing cell approach. It incorporates heat and mass transfer effects in the film and uses a boundary element method to solve the film model equations. The fluxes obtained at the interface are then directly used as the link to the reactor model. Simple isothermal and non-isothermal reactions were numerically tested. Application to two industrially important case studies, chlorination of butanoic acid and oxidation of cyclohexane are briefly illustrated. For the autocatalytic chlorination of butanoic acid, the yield of desired product, monochlorobutanoic acid, is favored by the high degree of mixing in the liquid phase. Therefore, this reaction should be carried out in a CSTR. A series of five bubble tanks with parallel gas reactant feed for cyclohexane oxidation was also simulated. It was found that the cyclohexane conversion is low while the oxygen conversion is relatively high and almost constant in each tank. Due to the complex multistep nature of this reaction scheme, oxygen is consumed in many steps of oxidation and selectivity of main products (which are intermediate products in the reaction scheme) depends on the critical control of over-oxidation in the kinetic mechanism.

A computational model for predicting particle size distribution and performance of fluidized bed polypropylene reactor

Gas phase polymerization of propylene is one of the most widely accepted and commercially used processes for manufacturing of polypropylene. The present work describes a comprehensive mathematical model for simulating fluidized bed polypropylene reactors. Unlike previously published models, the present model simultaneously predicts both polymer properties as well as particle size distribution. A generalized framework of a dynamic model based on mixing cell approach and detailed polymerization kinetics coupled with population balance model for particle size distribution (PSD) is developed. Need for coupling the reaction engineering model with population balance models is demonstrated. The coupled model was then used to understand influence of operating parameters on polymer properties and particle size distribution. The model is also used to understand the effects of multiple active sites and reaction kinetics on macroscopic variables. The developed framework is useful for simulating multi-monomer, multi-site Ziegler-Natta type olefin fluidized bed polymerization reactors.

Modeling of hydrogenation of maleic acid in a bubble‐column slurry reactor

AbstractA bubble‐column slurry‐reactor model has been developed for the hydrogenation of aqueous maleic acid (MAC) to tetrahydrofuran (THF). This particular reaction system has recent commercial relevance and represents a case where complex multistep catalytic hydrogenation reactions are conducted at high pressure (>15 MPa) and high temperature (>240°C). It also has additional complexities associated with the reaction chemistry, since the THF reaction product is volatile and the reaction is highly exothermic. The proposed model is derived using the mixing cell approach and incorporates the contributions of gas–liquid and liquid–solid mass transfer, intraparticle diffusion effects, product volatility, heat effects, and complex multistep reaction kinetics. The effect of gas and liquid velocities, catalyst loading, inlet maleic acid concentration, and temperature on the conversion, selectivity, temperature rise, and productivity of the desired products (THF and γ‐butyrolactone (GBL)) is also discussed. Since the reaction step involving the hydrogenation of GBL to THF is relatively slow, severe operating conditions are necessary to achieve high THF selectivity. The distribution pattern of THF in the gas and liquid phases is also discussed. The model proposed could be useful for simulation of existing pilot‐ or industrial‐scale reactors, as well as the design and scale‐up of new reactors for this particular reaction or one that has similar characteristics.

MCMFIT: Efficient optimal fitting of a generalized nonlinear advection-dispersion model to experimental data

The use of standard numerical schemes to solve nonlinear advective-dispersive equations for the estimation of parameters is CPU-time consuming and hence not desirable for routine use. An efficient scheme using a novel mixing cell approach has been used to estimate parameter values by nonlinear least-squares fitting for nonlinear adsorption of a single solute species coupled with one-dimensional transport. A problem with gradient methods of nonlinear least-squares fitting is that they are prone to determine best-fit parameters corresponding to local minima rather than the global minimum. As is well known, this problem can be avoided by judicious selection of the starting values. The present code, MCMFIT, includes a random search of the parameter space in order to determine a suitable set of initial parameter values. The program also includes the option of selecting user-defined initial parameter values because of possible physical considerations. These values then are passed to the nonlinear least-squares fitting program to obtain the optimal parameter values. Penalty functions have been employed to maintain user-imposed constraints on the parameter values. MCMFIT is capable of handling linear, Freundlich, Langmuir, and S-curve adsorption isotherms. The use of MCMFIT is demonstrated with the use of synthetic as well as laboratory and field data.

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.

Cation and proton exchange, pH variations, and carbonate reactions in a freshening aquifer

Freshening of aquifers is accompanied by sequential elution of the saltwater (seawater) cations from the sediment's exchange complex. The resulting Chromatographic patterns are modeled with a one‐dimensional geochemical transport model that can handle the complex interplay of transport and mineral and ion exchange equilibria. The transport part is based on the mixing cell approach, with different time steps for advective and diffusive transport used when required by small grid size. The chemical reactions are calculated explicitly after each time step with the geochemical model PHREEQE. Ion exchange is included in the form of association half reactions, which allows simulation of the dynamic nature of the exchange process. The variation in the constant for proton association is obviated with an activity coefficient for H‐X that is derived from the constant capacitance model. All coefficients for the exchange model are obtained by fitting to literature data to be able to perform the modeling as realistically as possible. The code is applied to a laboratory column experiment, and subsequently used to demonstrate Chromatographic development of solute profiles in a freshening aquifer. Sequential peaks of Mg2+, K+, and Na+ along a flow path in the Aquia aquifer in Maryland are modeled, and the results confirm that the variation of water qualities in this aquifer has basically a Chromatographic origin. Proton exchange acts here as a source of acid in NaHCO3 water in which calcite dissolves. This explains the Na+ to HCO3− ratio and high δ13C observed in these waters.

Intrusion of Seawater into a Porous Fracture zone a Coupled Geohydrochemical Modelling Exercise

ABSTRACTA coupled geochemical transport program PHREEQM2D has been tested in order to examine intrusion of seawater into fracture zones which can be connected to tunnels, such as nuclear-waste repositories on the sea shores. The code consists of a flow and transport program HST2D and a chemical equilibrium program PHREEQE. It uses a mixing cell approach, where chemistry and transport are solved separately. Ion exchange is calculated by means of a simple surface complexation model.In the present artificial exercise, replacement of groundwater by seawater in a connecting porous fracture zone was modelled without comparing the results to observed data. Dilution, mixing, ion exchange, and dissolution/precipitation reactions were examined.The results of modeling showed that the sulphate front of seawater moved some distance towards the tunnel during the simulations. The characteristic solutes of dilute groundwater were assumed to enter the tunnel in a short period of time. Amounts of sulphates decreased due to dilution and mixing. The chemical contents of waters were also affected by dissolution, precipitation, and ion exchange. Calcite dissolved and oxyhydroxides precipitated in oxidizing water. The strongest exchangeable ions in water, Ca and Na, were retarded due to ion exchange in the mixing zones. This caused an increase in dissolution of calcite in the mixing zones. Calcite would, however, start to precipitate in the mixing zone when the ion exchange capacity of rock would be consumed.Some important factors, such as retardation on oxide surfaces (adsorption models) and microbial activity (organic complexation models) have not been tested in the present exercise. Otherwise, the PHREEQM2D showed its applicability to modelling of chemical equilibrium, dilution, mixing, ion exchange, and elution/precipitation in discrete porous zones in transient conditions. It could be used to model temperature-enchanced, coupled geohyd-Γochemical, processes in porous zones consisting of densely fractured or crushed crystalline rock.

A Review of Chemical Solute Transport Models

This paper presents a review of groundwater models capable of simulating solute transport (i.e. advection-dispersion) and equilibrium chemistry processes. Introductory, basic model components are discussed including a physical and a chemical model component and the coupling between the two components. However, emphasis is placed on the chemical model component, since this represents the recent addition to solute transport modeling. Several ways exist of formulating either model component. For the physical model component this may include a continous description based on the governing mass balance equations or a discontinous description by a mixing cell approach. For the chemical model component, i.e. processes such as ion exchange, adsorption, complexation, precipitation/dissolution and open/closed carbonate system reactions, local equilibrium is usually assumed. The assumption of local equilibrium is investigated. The relation between the continuous formulation and a coupling/decoupling of the physical and chemical components and the resulting nature of the mathematical problem and its numerical strategy is briefly discussed. In all, 27 models have been surveyed in terms of their physical and chemical model components, their verification and applicability to field scale problems. Due to a lack of actual field applications it has not been possible to evaluate all models in natural settings and hence, each model will, in theory, be valid when the constraints given by the assumptions and limitations taken in developing the model are met. The models show great diversity with respect to complexity and general applicability to field scale problems. Most of the models were developed for specific purposes and it is difficult to point out one or more »general« models. However, it may be concluded that there exist many ways of formulating a model both with respect to transport and chemical calculations, but also to numerical techniques. Many of the models are very complex and future research should be directed towards validation studies of these models on a field scale, rather than developing still more complex models.