CFD Model Predictions Research Articles

Analysis of Performance in Homogenous Mixing Systems Using Ansys

Objective: This study aims to understand the mixing phenomenon and the mass transfer occurring simultaneously by simulating the system using Ansys Fluent. Methods: Taking liquid-liquid mixing into account, a simple agitated mixing system can be compared to a CSTR, utilizing the impeller to provide forced convection mixing conditions. The same forced convection can be achieved using high flow rates in smooth vessels instead of mechanical impellers to produce the convection current. The systems are then stimulated using Ansys Fluent software to calculate the mass transfer coefficient and several dimensionless numbers, such as the Reynolds number, Sherwood number, and Schmidt number, in order to understand the underlying mechanism of mass transfer in mixing systems. Results: It has been observed that a 90° pitch impeller tends to have a higher NP value as compared to a 45° pitch impeller. Overall performance can be compared by the time taken to achieve homogeneity at the same angular rotational speed of 100 rpm. Thus, in terms of performance, it can be concluded that 45° pitch is (inclined) > 90° pitch is (in-centre) > 45° pitch (in-centre) is > 90° pitch (inclined). Conclusion: The simulation model used in this study is useful for a combination of CFD model predictions using the sliding mesh approach and the VOF model. It can be applied to study the effect of different designs on the flow pattern and mixing time for a set of axial flow impellers.

On the subsonic and low transonic aerodynamic performance of the land speed record car, Bloodhound LSR

The land speed record vehicle, Bloodhound, undertook testing at subsonic and low transonic speeds (up to Mach 0.8) at Hakskeen Pan, South Africa, during October and November of 2019. A decade of CFD-led aerodynamic design had been undertaken to produce a vehicle with the aim of minimised Mach number aerodynamic dependencies and minimised overall drag. This paper sets out and explains the measured pressure distributions with a focus on the highest speed run of Bloodhound up to a peak speed of 628 mile/h. It compares the measured aerodynamic performance with the various CFD model predictions used throughout the design process showing that, whilst localised discrepancies between CFD model and real behaviour exist, overall the Reynolds-averaged Navier–Stokes (RANS)-based CFD tools used to design the car did result in sufficiently accurate aerodynamic data to predict the overall vehicle performance to a high degree of accuracy. The work outlined in this paper, and the conclusions and recommendations drawn, form the basis for a future record attempt and the understanding of what will be required in principle to extend the World Land Speed Record to 1000 mile/h. It also provides guidance on how to effectively make use of RANS-based CFD modelling predictions for other complex, ground-interacting high-speed applications.

Modelling of a Heated Gas-solid Fluidised Bed using Eulerian Based Models

ABSTRACT An Eulerian-Eulerian granular model was used to simulate the flow and heat transfer through a heatedgassolid fluidised bed. The primary objective of the study was to determine whether the Eulerian-Eulerian granular model adequately predicts the chamber pressure drop, temperature, and bed expansion through the bed. The model predictions were assessed and validated for various flow-regimes, namely the fixed-bed, smooth, bubbling fluidisation, and the maximum fluidisation regimes. This was done on an experimental scale heated gas-solid fluidised bed. However, the results are generalisable for heated gas-solid fluidised beds when the flow is laminar. Numerical models were created using Computational Fluid Dynamics (CFD). The CFD-model predictions were investigated, analysed, and compared to experimental results. Basic experiments were carried out to obtain varying hydrodynamic characteristics. The results showed a slight overprediction of pressure drop and bed expansion, however, the results were still in close agreement with the experiment. In contrast, underprediction of chamber temperatures were obtained. Based on the results of this study, it is recommended that the Eulerian model be used to predict dynamic flow behaviour. Before minimum fluidisation, when in a fixed bed regime, pressure drop in the chamber increases with no increase in bed height. No visible bubbles were present in the fixed bed regime. When fluidisation has been reached, the bed height rises whereas the pressure drop tends to a constant value. Bubble size increases with chamber height and increased superficial velocities. Bubble speed increased with increased chamber height. With increased superficial velocity, the chamber temperatures increase to a maximum temperature of326.65 K with an initial heating element temperature of373.15 K. However, when excessive heat is present in the gas-solid fluidised bed, other methods that sufficiently incorporate particle-particle interactions and bubble-bubble interactions, are recommended. An investigation should be lent to bubble-bubble interactions in the fluidised beds with relation to heat transfer. Additional keywords: Heated fluidised bed, computational fluid dynamics, CFD, Eulerian, granular, fluidisation, gas-solid

Full-Scale CFD Modeling of Multiphase Flow Distribution in a Packed-bed Absorber with Structured Packing Mellapak 250Y

AbstractA full-scale multi-environment Eulerian CFD model for a countercurrent packed-bed absorber with structured packing Mellapak 250Y was built in ANSYS Fluent 2019 R1 in order to model CO2 capture using physical solvents. The objective of the model is to predict the overall absorber gas-liquid internal flow profiles within the complex packing geometry, while accurately predicting the hydrodynamic parameters, such as liquid holdup and pressure drop. The gas-solid and gas-liquid drag coefficients were fitted and validated using the following experimental data by Green et al. (2006. “Hydraulic Characterization of Structured Packing via X-ray Computed Tomography”; 2007. “Novel Application of X-ray Computed Tomography: Determination of Gas/liquid Contact Area and Liquid Holdup in Structured Packing.” Industrial & Engineering Chemistry Research 46: 5734–53.): dry pressure drop, irrigated pressure drop, and liquid holdup. The validated CFD model was used to study the effect of liquid distributor design on the liquid distribution in the absorber using three distributors provided with seven, thirteen, and twenty orifices of 0.2 mm diameter. The CFD model predictions revealed that the distributor with the largest number of orifices resulted in the least liquid maldistribution in the absorber, which led to increasing the overall CO2 absorption efficiency in Selexol as a physical solvent. Also, the overall CO2 absorption efficiency decreased with increasing the superficial liquid velocity due to the shorter contact times between CO2 and Selexol in the absorber at higher superficial liquid velocities.

On the three-dimensional flow of a stable tangential vortex intake

Tangential vortex intakes are compact hydraulic structures commonly used in water supply, drainage and sewerage systems to convey water from high to low elevations efficiently. Tangential vortex intakes can be designed to ensure stable flows for all discharges, with high energy dissipation in the absence of significant air entrainment. However, due to the complex three-dimensional (3D) flow in the tangential vortex intake, current theoretical models are not sufficiently complete to interpret the flow process reliably. This paper presents a 3D computational fluid dynamics study of a steady tangential vortex intake flow using the Volume-of-Fluid method. For the first time, the CFD model predictions are validated against detailed point velocity measurements using laser doppler anemometry (LDA) for a wide range of inflow conditions. For a flow less than the free drainage discharge, the inflow does not interfere with the swirling flow in the drop shaft. The streamwise horizontal x-velocity at the channel-dropshaft junction varies linearly with vertical distance, while the vertical velocity follows a parabolic relation. For larger discharges the pressure and inflow velocity field is notably modified by the dropshaft swirling flow interacting with the inflow issued from the junction. While the swirling flow in the dropshaft is highly asymmetrical, it is found that the local tangential velocity can be well-approximated by a Rankine vortex – with solid rotation (forced vortex) behavior near the air core and a free vortex behavior in the main flow. In the free vortex region the vertical velocity is approximately constant in the radial direction. The predicted head-discharge relations, velocity field, pressure, and air core sizes are in excellent agreement with data. In particular the variation of air core size with flow rate is successfully predicted for the first time. The present study offers comprehensive insights of tangential vortex intake flow, and provides a basis for the hydraulic design of such vortex intake structures.

Two-phase CFD modeling of pulsed disc and doughnut column: Prediction of dispersed phase holdup

CFD simulations of Pulsed Disc and Doughnut Columns (PDDCs), which are used in solvent extraction based separation processes, are preformed to predict dispersed phase holdup in two-phase flow. Flow field is obtained by using Euler-Euler approach embedding continuity and momentum equations along with equations of the standard k-ε mixture model of turbulence. The dispersed phase is assumed to be monodispersed. The momentum equation of the dispersed phase is modified to account for the effect of collisions among drops on dispersed phase holdup. The model is validated with in-house experimental data for different geometric and operating conditions. Predictions of CFD model are found to be better than the predictions of the reported correlations for estimating dispersed phase holdup in PDDCs. The validated computational approach is used for parametric analysis to understand the dependence of dispersed phase holdup and other hydrodynamic variables on operating and geometric parameters and scale-up methodology.

Hydrodynamic study of two phase flow of column flotation using electrical resistance tomography and pressure probe techniques

Gas dispersion characteristics are very important to design and selection of flotation process for the fine particle separation. Although the separation of flotation process heavily depends on hydrophobicity of given solid particles, but their rate of attachment and detachment depends on local gas dispersion characteristics. In this work, gas dispersion characteristics in a 100 mm laboratory column flotation have been investigated by exploring the flow behavior in terms of the local and mean gas hold-up, bubble rise velocity and bubble size distribution across the column. This experimental data has been used to demonstrate the validation of the two-fluid CFD model predictions in the column flotation. The column hydrodynamics are estimated by using high speed dual planar Electrical Resistance Tomography (ERT) system and pressure transducers. In all the experiments, water is used as the liquid phase and air (bubbles) as the disperse phase. Using the ERT system, measurement of two phase distributions are examined for a wide range of design and operating conditions of the column including different spargers and frother dosage, where the flow changes from homogenous to transition bubbly flow. It is confirmed by ERT that the gas-holdup increases with an increase in the sparger porosity, air superficial velocity, liquid height and liquid feed flow rate. To test the reliability of ERT measurements, simultaneously a set of pressure transducers is used to obtain the sectional average gas holdup and validated. It is observed that bubble density occupancy is more at center region of the column and gas holdup value increases with the increase of gas superficial velocity. Dynamic gas disengagement technique is utilized to measure bubble rise velocity and sauter mean bubble diameter. At end this ERT data has been used to validate two-fluid CFD model (modified with suitable drag and lift forces via user defined functions) predictions in the column and found a close agreement.

Measurements and CFD modeling of a pulverized coal flame with emphasis on ash deposition

Measurements of fly ash deposition in a 15kW pulverized coal jet flame and CFD-based mathematical modeling have been performed. The deposits have been collected at two ports at particle Stokes numbers in the 0.02–0.34 range and particle kinetic energies not larger than 2×10-9J. Inertial impaction and thermophoresis have been identified as main mechanisms of particle transport towards the deposition surfaces. Deposition rates on air-cooled probes (600–700°C surface temperature) have been measured to be 24% (Port 2) and 79.4% (Port 3) larger than those measured on uncooled probes (1150°C surface temperature) due to the enhanced role of thermophoresis. Complex dependencies of the deposition rate on the probe surface temperature and the probe location have been observed. The CFD-model predictions are able to reproduce these dependencies after adjustments to the particle sticking sub-model. The paper contains estimations of both the impaction and sticking efficiencies.

Effects of initial static bed height on fractional conversion and bed pressure drop in tapered-in and tapered-out fluidized bed reactors

In this article, a standard 2D Two-Fluid Model (TFM) closed by the kinetic theory of granular flow (KTGF) has been applied to simulate the behavior of tapered-in and tapered-out fluidized bed reactors. In this regard, two types of chemical reactions with gas volume reduction and increase were considered to investigate the effects of initial static bed height on the fractional conversion and bed pressure drop. To validate the CFD model predictions, the results of hydrodynamic simulations concerning bed pressure drop and bed expansion ratio were compared against experimental data reported in the literature and excellent agreement was observed. The obtained simulation results clearly indicate that there is an appropriate static bed height in a tapered-in reactor in which the fractional conversion becomes maximum at this height; whereas variations of static bed height in a tapered-out reactor have insignificant influences on the fractional conversion. Moreover, it was found that the residence time, temperature, and intensity of turbulence of the gas phase are three important factors affecting the fractional conversion in tapered fluidized bed reactors. In addition, it was observed that increasing the static bed height increases the bed pressure drop for both the tapered-in and tapered-out fluidized bed reactors.

Experimental study of a capillary pumped loop in comparison with the prediction of a 3D CFD

A comparison is presented between experimental and numerical results regarding the operation of a capillary pumped loop evaporator. Two cylindrical evaporators were tested, with different heated porous lengths, 20 and 40 mm, respectively. Both have 22 mm external diameter, 9 mm porous thickness and 80 mm porous length. The working liquid was water. The loop was made from copper tubes and the evaporator from copper porous wick covered with aluminum with grooves formed in the inner surface. All tests took place on a horizontal level using heat load applied to the evaporator surface from an 85-W electric resistance. The experimental measurements were compared with the predictions of a three-dimensional CFD model of the evaporator and were found to be in satisfactory agreement. For the 20-mm wick heated length evaporator CFD model with water initial temperature of 20°C the divergence with the experimental pressure drop mean value was 0 Pa for volume flow rates between 0.4 and 0.6 l/min and 50 Pa for the rest of the values. For 30°C the divergence was 0 Pa <0.4 l/min and 50 Pa for larger flow rates. Moreover, for 40°C the difference was up to 50 Pa from 0 to 0.9 l/min. In every case predictions were below the wick capillary limit. The computed outflow temperature presented a maximum difference of 1.5% compared with the experimental data, which is very satisfying. On the other hand, the predictions of the evaporator CFD model with a 40-mm wick heated length were even better.

Quantitative Analysis and Design of a Spray Aerosol Inhaler. Part 2: Improvements in Mouthpiece Performance

The objective of this study was to utilize previously identified critical design attributes for the capillary aerosol generator as a model spray inhaler in order to develop a second-generation device that minimized aerosol drug deposition in the mouthpiece. Computational fluid dynamics (CFD) predictive analysis of the critical design attributes indicated that turbulence intensity should be reduced and the effective mouthpiece diameter should be increased. Two second-generation inhaler mouthpieces meeting these specifications were manufactured and tested. The first device (Design 1) implemented a larger cross-sectional area in the mouthpiece and streamlined flow, whereas the second device (Design 2) used a perforated mouthpiece wall. An in vitro deposition study was performed to quantify the deposition of drug mass in the mouthpieces and connected induction ports, and the results were compared with the CFD predictions. The two second-generation mouthpieces reduced in vitro aerosol deposition from the original value of 7.8% to values of 2.1% (Device 1) and 4.3% (Device 2), without largely altering the induction port deposition. This was achieved by design alterations aimed at reducing turbulence intensity and increasing the effective mouthpiece diameter. CFD model predictions were in good agreement with the in vitro experimental data. A second-generation spray inhaler mouthpiece with low drug deposition was developed using a predictive CFD model and in vitro experiments. Applying this quantitative analysis and design methodology to medical devices, which is similar to the Quality by Design paradigm, could provide significant advantages compared with traditional approaches.

Analysis of swirling flow in hydrocyclones operating under dense regime

There are many circumstances where hydrocyclone performance and dense flow are intertwined, such as for example when feed solids flow exceeds hydrocyclone capacity during continuous operations. The work reported here, which is part of an ongoing research effort to develop a robust CFD model for prediction of hydrocyclone performance, focuses on hydrocyclone operation under high solids concentration. The paper presents the basic physics framework that accounts for solid–liquid and solid–solid interactions under hydrocyclone’s swirling flow. Operating conditions that are past the transition from spray to rope regime are deliberately chosen for this purpose. Model predictions are validated by comparison with solids split and separation curves measured on a 100mm diameter hydrocyclone. CFD model predictions permit taking an insightful look at the inside of a hydrocyclone under extreme operating conditions, which would be difficult to achieve experimentally. Velocity profiles, G-force distribution and distribution of solids predicted by CFD are bound to lead to a better understanding of the separation that takes place inside a hydrocyclone, which may eventually help improve hydrocyclone design and performance.

Validation of numerical modeling of conditions in an atrium space with a hybrid ventilation system

Quite extensive measurements undertaken in a three-story atrium space (floors 14th–16th) with a hybrid solar-assisted natural ventilation system in the Engineering building of the Concordia University, Montreal, Canada have recently become available. The thermal conditions of the atrium space have been studied numerically using the Reynolds Averaged Navier-Stokes (RANS) modeling approach. The RANS turbulence models that were tested include the standard k-ε, RNG k-ε, ‘realizable’ k-ε, and SST k-ω turbulence models. The radiation exchange between the surfaces of the atrium space was considered using the Discrete Transfer Radiation Model (DTRM). The resultant steady state governing equations were solved using a commercial CFD solver FLUENT. Numerical results were obtained for the conditions existing when measurements were taken in the Concordia atrium on typical clear days with the blinds fully open or fully closed and with the natural ventilation system (NV) ON or OFF. The CFD model predictions were validated by the comparison against the experimental measurements available and it was found that the numerical predictions implying CFD model agree well with the measurements.

Investigations on hydrodynamics and mass transfer in gas–liquid stirred reactor using computational fluid dynamics

In this work, the hydrodynamics and mass transfer in a gas–liquid dual turbine stirred tank reactor are investigated using multiphase computational fluid dynamics coupled with population balance method (CFD–PBM). A steady state method of multiple frame of reference (MFR) approach is used to model the impeller and tank regions. The population balance for bubbles is considered using both homogeneous and inhomogeneous polydispersed flow (MUSIG) equations to account for bubble size distribution due to breakup and coalescence of bubbles. The gas–liquid mass transfer is implemented simultaneously along with the hydrodynamic simulation and the mass transfer coefficient is obtained theoretically using the equation based on the various approaches like penetration theory, slip velocity, eddy cell model and rigid based model. The CFD model predictions of local hydrodynamic parameters such as gas holdup, Sauter mean bubble diameter and interfacial area as well as averaged quantities of hydrodynamic and mass transfer parameters for different mass transfer theoretical models are compared with the reported experimental data of Alves et al. (2002a, b, 2004). The predicted hydrodynamic and mass transfer parameters are in reasonable agreement with the experimental data.

Evaluation of CFD model predictions of local dispersion from an area source on a complex industrial site

The CFD model Fluidyn-Panache was configured to model atmospheric transport from an area source. Modelled flow and turbulence were evaluated by comparison with on-site meteorological measurements, whilst atmospheric dispersion was compared with wind tunnel measurements. The results showed that higher rates of vertical and lateral dispersion were modelled than were determined in the wind tunnel, though modelled and measured ground-level centreline concentration data were within a factor of two. Uncertainties in wind tunnel and numerical modelling were highest close to the source. Consideration of fine-scale features was only necessary for receptors in the immediate near-field.

CFD modeling of gas–liquid–solid mechanically agitated contactor

In the present work, CFD simulations have been carried out to study solid suspension in gas–liquid–solid mechanically agitated contactor using Eulerian–Eulerian multifluid approach along with standard k−ɛ turbulence model. A multiple frame of reference (MFR) has been used to model the impeller and tank region. The CFD model predictions are compared qualitatively with the literature experimental data and quantitatively with our experimental data. The present study also involves the effects of impeller design, particle size and gas flow rate on the critical impeller speed for solid suspension in gas–liquid–solid mechanically agitated contactor. The values predicted by CFD simulation for critical impeller speed agrees well with experimental data for various operating conditions.

Comparison of ambient and spray aerosol deposition in a standard induction port and more realistic mouth–throat geometry

For inhalation aerosols, it is well known that spray momentum and geometry characteristics can significantly influence deposition in the mouth–throat (MT) region. However, little is know about the quantitative influence of spray momentum on aerosol transport and deposition. The objective of this study was to evaluate the effect of spray momentum on deposition in a standard induction port (IP) and a representative MT geometry using capillary-generated aerosols. Capillary aerosol generation (CAG) was selected as a model spray aerosol system that has not been previously tested in a realistic throat geometry. To evaluate the effects of spray momentum, the transport and deposition characteristics of transient capillary-generated aerosols were compared with ambient particles of the same size inhaled at a steady flow rate of 30 L/min. To evaluate the influence of geometry, aerosols were considered in a standard IP and a more realistic MT model. A previously tested CFD model was employed to simulate aerosol transport and deposition for ambient and CAG spray aerosols in both the IP and MT geometries. Considering the capillary-generated spray, good agreement was observed for the deposition of drug mass between the in vitro experiments (IP—15.3%, MT—19.4%) and CFD model predictions without droplet evaporation (IP—14.7%, MT—20.8%). In all cases considered, deposition was increased for spray vs. ambient aerosols and in the MT geometry vs. the IP model. Based on CFD results for a representative polydisperse aerosol distribution, the deposition of ambient particles was highly sensitive to the geometry considered, with 2.9 times more deposition in the MT compared to the IP model. In contrast, deposition was less influenced by the geometry for a CAG spray aerosol, with only a 25–40% deposition increase in the MT. As a result, use of the simple IP model may provide a reasonable approximation of total MT deposition for systems with high spray momentum. However, the IP model may be less useful for evaluating the total MT deposition in systems with reduced spray momentum effects.

Investigation of liquid maldistribution in trickle-bed reactors using porous media concept in CFD

A three-dimensional CFD model for simulating two-phase flow in trickle-bed reactors (TBRs) is presented. Based on porous media concept, a two-phase Eulerian model (rather than computationally demanding traditional three-phase Eulerian model) describing the flow domain as porous region is presented to understand and forecast the liquid maldistribution in TBRs under cold-flow conditions. The drag forces between phases have been accounted by employing the relative permeability concept [Sàez, A. E., Carbonell, R. G., 1985. Hydrodynamic parameters for gas–liquid cocurrent flow in packed beds. A.I.Ch.E. Journal 31, 52–62]. The model predictions are validated against experimental data reported in literature, notably using the liquid distribution studies of Marcendelli [1999. Hydrodynamique, Transfert de Chaleur Particule-Fluide et Distribution des phases dans les Reacteurs a lit Fixe a Ecoulement a Co-courant Descendant de Gaz et de Liquide. Doctoral Thesis. INPL, Nancy, France]. Various distributor configurations reported therein have been recreated in the CFD model and sensitivity studies have been performed. Good agreement is obtained between the reported experimental results and this proposed first-principle based CFD model. Finally, the concept of distribution uniformity is discussed and applied to the CFD model predictions. The CFD model is subjected to a systematic sensitivity study in order to explore better liquid distribution alternatives.

CFD based investigations on hydrodynamics and energy dissipation due to solid motion in liquid fluidised bed

Abstract CFD simulations are carried out for the prediction of flow patterns in a liquid–solid fluidised bed using Eulerian–Eulerian framework. The CFD model predictions are compared with the experimental findings reported by Limtrakul et al. [S. Limtrakul, J. Chen, P.A. Ramachandran, M.P. Dudukovic, Solids motion and holdup profiles in liquid fluidised beds, Chem. Eng. Sci. 60 (2005) 1909–1920] and the comparison shows good agreement. The CFD model has been further extended to compute solid mass balance in the core and annular regions for verifying conservation of mass and energy flows due to various dissipation mechanisms. Energy required for solid expansion in liquid fluidised bed is also compared with energy required for solid suspension in an equivalent stirred tank contactor at similar operating conditions. The influence of various interphase drag models proposed by Gidaspow [D. Gidaspow, Multiphase Flow and Fluidisation, 1st ed., Academic Press, San Diego, 1994], Di Felice et al. [R. Di Felice, The voidage functions for fluid–particle interaction system, Int. J. Multiphase Flow 20 (1994) 153–159] and Syamlal and O’Brien [M. Syamlal, T.J. O’Brien, Simulation of granular layer inversion in liquid fluidised beds, Int. J. Multiphase Flow 14 (1988) 473–481] on solid motion in liquid fluidised bed have been investigated. Even though these models predict the flow pattern of solid motion inside the fluidised bed with reasonable accuracy, the model proposed by Gidaspow showed the better quantitative agreement with experimental data. For ensuring accuracy of numerical simulation prediction, comparisons between 2D and 3D simulation, the effect of grid sensitivity, time step sensitivity and effect of inlet feed conditions have been carried out and a comprehensive CFD methodology is proposed to model the hydrodynamics of liquid–solid fluidised bed.

Eddy contact model: CFD simulations of liquid reactions in nearly homogeneous turbulence

An eddy contact model is proposed that describes the mixing rate between two miscible liquids in nearly homogeneous turbulence. The model is patterned after the eddy-break-up concept of Spalding (1971, 13th International Symposium on Combustion, The Combustion Institute (p. 649)) but replaces the eddy dissipation with the Kolmolgoroff time scale and fluid mass with volume fractions. The concept is used to describe the aqueous distribution of several pure fluids engaged in mixing sensitive parallel reactions. Numerical results for a free jet in an axisymmetric tubular reactor are compared with the experimental yield from both azo-coupled and acid-base-ester parallel reactions. It was found by direct comparison with other CFD-based calculations that the eddy contact model is nearly as accurate as Monte-Carlo/PDF methods and is comparable to a Beta function/mixture fraction technique. Moreover, CFD model predictions of selectivity within a scaled down plug flow reactor are shown to be comparable to direct numerical simulations. The eddy contact model is demonstrated to be conceptually simple, easy to implement and computationally efficient. Limitations on the use of the model are also discussed.