Macroscopic Mass Balance Research Articles

Applying a novel generalized model of sedimentation to the design of settlers and thickeners: Integration and extension of the classical procedures

A new generalized procedure previously introduced and justified by the authors, based on the simultaneous integration of the macroscopic and microscopic differential mass balances by introducing the concept of separation units as an extension of the transfer unit concept was applied here to sedimentation. This similarity leads to a broader analogy with basic transfer and transformation unit operations, at this step focused on the formulation and contrast of a generalized form of the so-called design or sizing equation. This model is comparatively applied under restricted flow Conditions-Vertical Flow Settlers (CSTS) - with the classical methods and extended to the design of horizontal flow settlers (PFS). That opens the door to the sizing of settlers and thickeners in the presence of deviations from these ideal flow models and with other operating configurations. Such a contribution to a desirable generalization and homologation is according to Chemical Engineering methodology specifically in the frequently empirical or casuistic treatment of unit operations with particulate solids.

Dynamic Modeling of Industrial Loop Reactors for Catalytic Ethylene (Co)polymerization: Pressure Safety Valves

AbstractIn industrial loop reactors, for catalytic ethylene (co)polymerization, the heat produced by polymerization can result in temperature and pressure increase, which in turn can lead to reactor breakage, explosion, and consequent release of toxic/flammable fluids in the environment. To predict the reactor temperature and pressure during ethylene (co)polymerization and assess the performance of presure safety valves (PSVs) in response to emergency accidents, an integrated mathematical approach is followed including catalyst kinetics, thermodynamics, and transport properties modeling. A catalytic ethylene (co)polymerization kinetic model is used to predict the catalyst activity variation with respect to reactor temperature and the heat produced by the polymerization reaction during a reaction runaway. The Sanchez–Lacombe equation of state (SL EOS) is utilized in order to predict the reactor pressure for different slurry mixture compositions and the slurry densities at different temperatures, pressures and solids weight fractions. Dynamic macroscopic mass and energy balances are derived to calculate the dynamic evolution of concentrations of the various molecular species as well as the temperature changes in the loop reactor. The model can predict the temperature and pressure increase rates and the maximum flow rate at the PSVs during the runaway.

Nonlinear Model Predictive Control based on multi-scale models: is it worth the complexity?

Nonlinear model predictive control (NMPC) has been used to control a variety of bioprocesses. As NMPC requires full state information, a nonlinear moving horizon estimator (NMHE) is used to reconstruct the states from the available measurements. In context of bioprocesses, both NMPC and NMHE typically use macroscopic mass balance type models. These models ignore all intracellular information that is available via, e.g., Genome-Scale Metabolic networks. However, the use of these models adds significantly to the computational complexity of the controller. In this paper, we aim to assess whether incorporating metabolic networks in an NMPC framework leads to improvement in the tracking performance of a continuous bioreactor. To avoid complex bilevel optimisation problems, our NMPC scheme utilises the dynamic Metabolic Flux Analysis framework. In this framework, the intracellular fluxes are estimated from the measurements of extracellular metabolites. This is achieved by an NMHE scheme. The tracking performance of this metabolic network-based NMPC-NMHE is compared to a simple macroscopic model based NMPC in which all the states, i.e., the extracellular metabolites are assumed to be measured. In our case study, the tracking performance of the metabolic network-based NMPC is marginally better than the performance of the macroscopic model based NMPC. The added advantage of metabolic network based NMPC comes from the insight they offer from the estimated fluxes. Moreover, the performance and the insights obtained depend on the complexity of the metabolic network used.

Designing scalable ultrafiltration/diafiltration process of monoclonal antibodies via mathematical modeling by coupling mass balances and Poisson-Boltzmann equation.

Ultrafiltration/diafiltration (UF/DF) operations are employed for achieving the desired therapeutic monoclonal antibody (mAb) formulations. Due to electrostatic interactions between the charged proteins, solute ions, and uncharged excipients, the final pH and concentration values are not always equal to those in the DF buffer. At high protein concentrations, typical for industrial formulations, this effect becomes predominant. To account for challenges occurring in industrial environments, a robust mathematical framework enabling the prediction of pH and concentration profiles throughout the UF/DF process is provided. The proposed mechanistic model combines a macroscopic mass balance approach with a molecular approach based on a Poisson-Boltzmann equation dealing with electrostatic interactions and accounting for protein exclusion volume effect. The mathematical model was validated with experimental data of two commercially relevant mAbs obtained from an industrial UF/DF process using scalable laboratory equipment. The robustness and flexibility of the model were tested by using proteins with different isoelectric points and net charges. The latter was determined via a titration curve, enabling realistic protein charge-pH evaluation. In addition, the model was tested for different DF buffer types containing both monovalent and polyvalent ions, with various types of uncharged excipients. The model generality enables its implementation for the UF/DF processes of other protein varieties.

Modeling and simulation of an industrial slurry phase ethylene polymerization reactor: effect of reactor operating variables

A hierarchical and computationally efficient mathematical model was developed to explain the polymerization of high-density polyethylene (HDPE) in an isothermal, industrial, continuous stirred tank slurry reactor (CSTR). A modified polymeric multi-grain model (PMGM) was used. Steady-state macroscopic mass balance equations were derived for all species (namely, monomer, solvent, catalyst and polymer) to obtain the final particle size and the required monomer and solvent input rates for a given catalyst input and the reactor residence time. The interphase mass transfer coefficients were calculated for the industrial CSTR using the operating data on the reactor. The present model was tuned with some data on an isothermal industrial reactor and the simulation results were compared with data on another set of industrial reactor. The comparison revealed that the present tuned model is capable of predicting the productivity and the polymer yield at various catalyst feed rates and the mean residence times. The effects of variation of two operating variables (catalyst feed rate and mean residence time) on the productivity, the polymer yield, the polydispersity index (PDI) and the operational safety were analyzed. The present study indicated that an optimal value of the reactor residence time (for maximum productivity per catalyst particle) exists at any catalyst feed rate.

An implicit joint-continuum model for the hydro-mechanical analysis of fractured rock masses

Fractures control the hydro-mechanical behavior of rock masses. Explicit numerical analyses require detailed information on fracture properties, spacing, and orientation. This paper advances an implicit joint-continuum model for the coupled hydro-mechanical analysis of regularly spaced-persistent fractured rock masses. The stiffness tensor combines the compliance of each fracture set and the intact rock; similarly, the permeability tensor adds the fluid transport through fractures and the matrix. The fully coupled hydro-mechanical analysis incorporates the rock mass stiffness and permeability tensors, and satisfies force equilibrium and macroscopic fluid mass balance. We implement the implicit joint-continuum model within a finite element framework and verify the numerical simulator against closed-form solutions for simple boundary conditions. The application of the code to the hydraulic stimulation of a fractured rock mass shows the effect of stress anisotropy and fracture orientation on the development of open-mode discontinuities (i.e., hydraulic fracture) and hydro-shearing. The implicit joint-continuum model can be readily extended to more complex coupled processes, including thermal and chemical phenomena.

Fba-Based Prediction of Biomass and Ethanol Concentration Time Profiles in Saccharomyces cerevisiae Fed-Bath Cultures

Abstract A Flux Balance Analysis (FBA)-based dynamical model is proposed for predicting biomass and ethanol concentration time profiles in Saccharomyces cerevisiae fed-batch cultures, with glucose and ammonium uptake rates as inputs. It is based on a metabolic network compiling the necessary internal fluxes for reproducing respiratory as well as respiro-fermentative metabolisms. While the objective cost function classically accounts for biomass growth maximization, additional linear constraints are used to reproduce overflow metabolism phenomena. New conditional equality constraints are introduced in the FBA linear programs to account for variable biomass composition (based on protein mass fraction identification with or without ammonium feeding). The proposed FBA-based model contains only 7 parameters which are estimated with the experimental data. A two-step procedure for separately solving the FBA linear programs and the macroscopic mass balance ODEs significantly lowers the computational load for parameter identification. Direct and leave-one-out cross-validation results are provided.

Hydrodynamics and Reaction Performances of Multiphase Reactors for Marine Applications – A Review

AbstractHydrodynamics and (catalytic & non-catalytic) reaction [high-pressure hydrodesulfurization; Fischer-Tropsch synthesis; CO2-monoethanolamine (MEA) absorption; simultaneous CO2and H2S absorption in MEA; CO2thermal desorption, enzymatic CO2hydration] performances of trickle-bed and packed-bed column reactors subjected to static inclination, rolling (symmetrical/asymmetrical externally-generated reactor oscillations) and heaving motions were analyzed via detailed dynamic 3-D models which couple the macroscopic volume-averaged momentum, mass, energy and species balance equations in the liquid/gas phases with diffusion/chemical reaction inside the catalyst particles, enzyme washcoat or liquid film near the gas-liquid interface. Axial symmetry breakdown once the packed bed systems become inclined inflicts noticeable reductions of the reactor (or scrubber) performances. Only the performance of Fischer-Tropsch synthesis in the presence of water-gas shift reaction increases slightly with the increase of trickle-bed reactor inclination because of facile uptake of reactants in the key reactions from the gas phase while a fraction of valuable CO can be forwarded in water gas-shift reaction. For most of the reactions examined, the reactor performance is negatively impacted in asymmetric oscillating multiphase reactors with Fischer-Tropsch synthesis as an exception owing to the presence of water-gas shift reaction the performance of which is slightly improved. This performance deterioration/enhancement is maximal for the reactor moving between vertical and an inclined position when the time-dependent performance waves develop around the steady-state solution of the mid-inclination angle. The oscillatory reactor performance moves towards the steady-state solution of the vertical state when the asymmetry between the two inclined positions dwindles. Symmetric oscillating and heaving trickle-bed and packed-bed column reactors generate an oscillatory performance around the steady-state solution of vertical state which is affected by the amplitude and period of the angular and heaving motions of the vessel.

Study of model application for drying of pulp fruit in spouted bed with intermittent feeding and accumulation

ABSTRACTIn the present study, we applied and improved a model to describe the behavior of a spouted bed dryer with intermittent feeding for suspension drying, considering suspension accumulation inside the device. This model is important in macroscopic heat and mass balances, represented by a system of ordinary differential balances, implemented and resolved numerically using Fortran routines, in which the influence of process variables on drying dynamics were assessed. The results obtained in modeling and mathematical simulations were compared with experimental data, indicating that the model fits the process well, based on outlet air temperature and moisture.

Work conjugate principle-constrained volume averaging technique for multiphase porous media

Volume averaging is a standard method for the development of macroscopic balance equations for modelling the thermodynamic behaviors of multiphase porous media. However, work conjugate principle which is a common practice in continuum mechanics is not emphasized by the volume averaging technique resulting in the macroscopic balance equations are not capable of comprehensively describing the kinematic behaviors of multiphase porous media due to the loss of essential macroscopic variables. This study derives the macroscopic mass and momentum balance equations for the pore fluid of a fluid-solid porous medium by use of the volume averaging technique. We show (1) if the procedure of the volume averaging is implemented in its traditional manner, only the average flux of the pore fluid described by its mass average velocity is captured; (2) if the work conjugate principle is employed to define a work-conjugate velocity for the pore fluid at the macroscale, both the average flux (described by the mass average velocity) and the dispersive flux (described by the deviation of the mass average velocity from the work-conjugate one) are reproduced. This theoretical analysis demonstrates that the work conjugate principle is an essential thermodynamic constraint to improve the volume averaging technique, in the sense that the macroscopic balance equations are required to be capable of comprehensively describing the macroscopic kinematic behaviors of multiphase porous media.

Acoustics of multiscale sorptive porous materials

This paper investigates sound propagation in multiscale rigid-frame porous materials that support mass transfer processes, such as sorption and different types of diffusion, in addition to the usual visco-thermo-inertial interactions. The two-scale asymptotic method of homogenization for periodic media is successively used to derive the macroscopic equations describing sound propagation through the material. This allowed us to conclude that the macroscopic mass balance is significantly modified by sorption, inter-scale (micro- to/from nanopore scales) mass diffusion, and inter-scale (pore to/from micro- and nanopore scales) pressure diffusion. This modification is accounted for by the dynamic compressibility of the effective saturating fluid that presents atypical properties that lead to slower speed of sound and higher sound attenuation, particularly at low frequencies. In contrast, it is shown that the physical processes occurring at the micro-nano-scale do not affect the macroscopic fluid flow through the material. The developed theory is exemplified by introducing an analytical model for multiscale sorptive granular materials, which is experimentally validated by comparing its predictions with acoustic measurements on granular activated carbons. Furthermore, we provide empirical evidence supporting an alternative method for measuring sorption and mass diffusion properties of multiscale sorptive materials using sound waves.

Hydrocarbon hydrodesulfurization in vertical, inclined and oscillating trickle beds – Hydrodynamics & reactor performance for offshore petroleum marine applications

Two-phase flow dynamics and hydrodesulfurization (HDS) performance were numerically analyzed for vertical, inclined, and oscillating trickle beds with a prospect of assessing the potential of implementing HDS operations on marine floating platforms. An unsteady-state three-dimensional model based on the macroscopic volume-averaged mass, momentum, energy and species balance equations coupled with simultaneous diffusion and chemical reaction within sulfide CoMo/alumina catalyst was developed for the purpose of comparative analyses using H2/dibenzothiophene/n-hexadecane model system. Angular uniform and sinusoidal oscillatory motion of the reactor between two angled symmetrical positions and between vertical and an inclined position was considered. As in the case of cold-flow passive conditions, in chemical reaction environment two-phase downflow deviates considerably from axial symmetry at higher reactor inclinations with noticeable liquid accumulation in the bottommost reactor cross-sectional area on the tilting side. Also, externally-induced reactor periodic oscillation generates complex reverse secondary flow in radial and circumferential directions and oscillatory patterns of flow field with the amplitude and propagation frequency affected by the type (uniform or sinusoidal), amplitude and period of angular motion of reactor. Inclined, symmetric and asymmetric oscillating trickle-bed reactors underperform the vertical configuration and the decline of symmetric oscillating trickle-bed reactor hydrodesulfurization performance is lower under uniform oscillatory motion conditions. Oscillating trickle-bed reactors generate an oscillatory hydrodesulfurization performance which is affected by the parameters of angular motion of reactor. The inhibiting effect induced by H2S on the hydrogenolysis is considerable at higher packed bed inclinations and in symmetric oscillating trickle-bed reactors.

Modelling the evolution of microstructure and transport properties of cement pastes under conditions of accelerated leaching

Calcium leaching might be a significant degradation process in concrete and reinforced structures with an anticipated long-term service life such as nuclear waste disposal systems or large hydro structures (dams, bridges, water tanks). The leaching process is extremely slow under environmental conditions, which fosters the use of accelerated experimental approaches such as leaching in an ammonium nitrate (NH4NO3) solution. In this paper, we develop a one-dimensional diffusion-based transport model with the purpose to predict the changes in microstructure and transport properties of saturated cement pastes in contact with a NH4NO3 solution. The model helps to better understand the transient state of leaching which is difficult to observe by experimental work.The main new elements in this model are change in model configuration with extended solution domains; ability to predict the spatial profiles of diffusivity and permeability; including the effect on solubility of the spatial-temporal evolution of nitrate concentration; and including the effect of limestone addition to the cement paste of leaching kinetics. This model is based on macroscopic mass balances for Ca in aqueous and solid phases which are linked together by applying a variable solid-liquid Ca equilibrium curve. Besides the prediction of the leached depth, porosity increase, portlandite and C-S-H contents, and the amount of leached Ca, the model also enables to estimate the variation of permeability and diffusivity over the domain at different immersion periods in NH4NO3 solution. The model is verified by accelerated leaching experiments in 6mol/l NH4NO3 solution on CEM I cement pastes with/without limestone fillers. Verification with experimental results shows a good agreement.

Modelling the carbonation of cement pastes under a CO2 pressure gradient considering both diffusive and convective transport

Underground concrete structures in radioactive waste disposal have the potential to be subjected to a high hydrostatic pressure and the surrounding environment may contain a high dissolved CO2 concentration. Therefore, a combination of diffusion and advection should be taken into account when one considers the carbonation mechanism. This study aims at developing a model to predict the evolution of the microstructure and transport properties of hardened cement pastes due to carbonation under accelerated conditions in which a pressure gradient of pure CO2 is applied. The current model is improved from the preliminary model in terms of extension to limestone cement paste and accounting for the transport of moisture. The proposed model is based on a macroscopic mass balance for CO2 and moisture in both gaseous and aqueous phases. A simplified solid-liquid equilibrium curve is used to relate the Ca content in aqueous and solid phases. Besides the prediction of the changes in porosity, diffusivity, permeability, and saturation degree, the model also enables prediction of the carbonation degree, portlandite content, and CO2 uptake. Verification with experimental results from accelerated carbonation tests shows a good agreement.

Hybrid Multiscale Finite Volume Method for Advection-Diffusion Equations Subject to Heterogeneous Reactive Boundary Conditions

We present a hybrid scheme for the coupling of macro- and microscale continuum models for reactive contaminant transport in fractured and porous media. The transport model considered is the advection-diffusion equation, subject to linear heterogeneous reactive boundary conditions. The multiscale finite volume method (MsFV) is employed to define an approximation to the microscale concentration field defined in terms of macroscopic or global degrees of freedom, together with local interpolator and corrector functions capturing microscopic spatial variability. The macroscopic mass balance relations for the MsFV global degrees of freedom are coupled with the macroscopic model, resulting in a global problem for the simultaneous time-stepping of all macroscopic degrees of freedom throughout the domain. In order to perform the hybrid coupling, the micro- and macroscale models are applied over overlapping subdomains of the simulation domain, with the overlap denoted as the handshake subdomain $\Omega^{hs}$, over which continuity of concentration and transport fluxes between models is enforced. Continuity of concentration is enforced by posing a restriction relation between models over $\Omega^{hs}$. Continuity of fluxes is enforced by prolongating to microscopic resolution the macroscopic model fluxes across the boundary of $\Omega^{hs}$. The microscopic interpolator and corrector functions are solutions to local microscopic advection-diffusion problems decoupled from the global degrees of freedom and from each other by virtue of the MsFV decoupling ansatz. The error introduced by the decoupling ansatz is reduced iteratively by the preconditioned GMRES algorithm, with the hybrid MsFV operator serving as the preconditioner.

Design of an integrated dryer and conveyor belt for woody biofuels

Combustion or gasification of high-moisture content biomass is associated with a number of drawbacks, such as operational instabilities and lowered total efficiency. The present work proposes an integrated dryer and conveyor belt for woody biofuels with steam as the heat transfer medium. The use of low-temperature steam is favorable from a heat management point of view, but also helps to minimize the risk of fire, self-ignition and dust explosions. Furthermore, the presented dryer design represents an efficient combination of fuel transport, drying equipment and fuel feeding system.The proposed design is developed from a macroscopic energy and mass balance model that uses results from computational fluid dynamics (CFD) fuel bed modeling and experiments as its input. This CFD simulation setup can be further used to optimize the design with respect to bed height, steam injection temperatures and fuel type. The macroscopic model can be used to investigate the integration of the dryer within a larger biomass plant. Such a case study is also presented, where the dryer is tailored for integration within an indirect steam gasification system. It is found that the exergy efficiency of this dryer is 52.9%, which is considerably higher than those of other dryers using air or steam, making the proposed drying technology a very competitive choice for operation with indirect steam gasification units.

Modelling of Hg0 Removal from Gaseous Streams and its Fixation in Hydroxyapatite-Based Adsorbents Modified with Copper Sulphide

In this work, a mathematical model of a fixed-bed process for elemental mercury removal from gaseous streams is proposed. Advective and diffusive processes were taken into account in the macroscopic mass balance along the bed, and diffusive process was considered in the microscopic mass balance inside the adsorbent particle in which the adsorption process occurs. In addition, a chemical reaction term was used to represent the mercury fixation by the adsorbent. The proposed model was implemented in the Environment for Modelling, Simulation and Optimization dynamic simulator. Model parameters were estimated based on experimental breakthrough curves, by minimizing the least-square function. The model satisfactorily described two experimental breakthrough curves of mercury removal using adsorbents with different preparation methods, both with high mercury fixation capacity. The proposed mathematical model was also able to consistently describe the mercury concentrations in the bed and in the particle.

Prediction of metabolic flux distribution from gene expression data based on the flux minimization principle.

Prediction of possible flux distributions in a metabolic network provides detailed phenotypic information that links metabolism to cellular physiology. To estimate metabolic steady-state fluxes, the most common approach is to solve a set of macroscopic mass balance equations subjected to stoichiometric constraints while attempting to optimize an assumed optimal objective function. This assumption is justifiable in specific cases but may be invalid when tested across different conditions, cell populations, or other organisms. With an aim to providing a more consistent and reliable prediction of flux distributions over a wide range of conditions, in this article we propose a framework that uses the flux minimization principle to predict active metabolic pathways from mRNA expression data. The proposed algorithm minimizes a weighted sum of flux magnitudes, while biomass production can be bounded to fit an ample range from very low to very high values according to the analyzed context. We have formulated the flux weights as a function of the corresponding enzyme reaction's gene expression value, enabling the creation of context-specific fluxes based on a generic metabolic network. In case studies of wild-type Saccharomyces cerevisiae, and wild-type and mutant Escherichia coli strains, our method achieved high prediction accuracy, as gauged by correlation coefficients and sums of squared error, with respect to the experimentally measured values. In contrast to other approaches, our method was able to provide quantitative predictions for both model organisms under a variety of conditions. Our approach requires no prior knowledge or assumption of a context-specific metabolic functionality and does not require trial-and-error parameter adjustments. Thus, our framework is of general applicability for modeling the transcription-dependent metabolism of bacteria and yeasts.

One-flow feed divided electrochemical reactor for indirect electrolytic production of hypochlorite from brine for swimming pool treatment-experimental and theoretical optimization

• Optimization of an electrochemical chlorinator for swimming pool. • Divided asymmetric electrochemical continuously stirred reactor. • Convective transfer of the anolyte across the separator to the cathodic compartment. • The brine (5 M NaCl) conversion reaches 50% under 3 kA m −2 with a faradic yield of 80%. • Correlation of current density versus chloride concentration for concentrated brine. A ‘two-compartment’ asymmetric electrochemical reactor, operating without electrodes polarity inversion, was designed and optimized for the chlorination of swimming pools. Gaseous chlorine, produced at the anode and absorbed in the alkalinized catholyte, provides the hypochlorite solution. Empiric equations providing the chloride concentration dependence on the initial current density magnitude were established. Experimental optimization of the effect of the various operating parameters allows a chloride conversion close to 50% with faradic yields higher than 80%, and a chlorine production of 1 kmol/day/m 2 to be achieved. Macroscopic mass balance, was performed and the obtained theoretical results correlate with the experimental ones ( X theor . Cl - - X exper . Cl - < 10%).

Entrance Region Flow of Herschel-Bulkley Fluid in an Annular Cylinder

The entrance region flow of a Herschel-Bulkley fluid in an annular cylinder has been investigated numerically without making prior assumptions on the form of velocity profile within the boundary layer region. This velocity distribution is determined as part of the procedure by cross sectional integration of the momentum differential equation for a given distance z from the channel entrance. Using the macroscopic mass and momentum balance equation, the entrance length at each cross section of the entrance region of the annuli and pressure distribution have been calculated for specific values of Herschel-Bulkley number and various values of aspect ratio and flow behavior index. The effects of non-Newtonian characteristics and channel width on the velocity profile, pressure distribution and the entrance length have been discussed.