Mathematical Model Of Mass Transfer Research Articles

Model of Mass Transfer in Liquid–Liquid Extraction in a Turbulent Forward Flow

A mathematical description of the process of mass transfer in a continuous phase in the turbulent motion of two mutually insoluble liquids has been obtained on the basis of an assigned law of damping of turbulent pulsations in the boundary layer. The basic parameters of the model have been found in terms of the hydraulic resistance of a droplet in motion. Consideration has been given to the use of the mathematical mass-transfer model in apparatuses with mixing devices. Agreement has been shown between results of calculation of mass-transfer coefficients and experimental data.

Turbulent Mass Transfer Optimization Control Technology in Coalmine

Gas control in coalmine was the critical issue since gas accidents give rise to a lot of harm. This paper put forward the optimization control strategy based on the turbulent mass transfer mechanism on the basis of previous research. First of all, gas turbulent mass transfer mathematical model was established in heading laneway. Three aspects were describing in detail, gas diffusion law and quality entransy conservation when no wind, gas fluid flow process and momentum entransy conservation when no wind, ventilation airflow turbulent mass transfer process and mathematics when ventilation exits. Turbulent mass transfer optimization control strategy on the basis of the method of Lagrange multipliers was adopted. Moreover, turbulent mass transfer Euler equation which satisfy mass entransy dissipation extremum in turbulent mass transfer process. Finally, simulation experiments were carried out in the experimental platform; the data shows that turbulent mass transfer optimization control strategy was effective.

Predictive Microbiology Coupled with Gas (O2 /CO2 ) Transfer in Food/Packaging Systems: How to Develop an Efficient Decision Support Tool for Food Packaging Dimensioning.

Coupling gas transfer with predictive microbiology is essential to rationally design modified atmosphere packaging (MAP) strategies to ensure and guarantee food safety. Nowadays, these strategies are generally empirically built and over-sized since packaging material with high barrier properties is often chosen by default even if such a high level of protection is not systematically required. Protection strategies could be improved using rational sizing based on quantitative analysis and mathematical modeling of mass transfer. This paper aims at reviewing the current knowledge available for developing such a tool and the further research needed. First there is a special focus on oxygen (O2 ) and carbon dioxide (CO2 ) solubility and diffusivity parameters, which are absolutely indispensable to accurately model mass transfer in MAP systems. Next, the current knowledge of the effect of O2 /CO2 on the growth of microorganisms is explored with an emphasis on predictive microbiology. The last part points out the main bottlenecks and further research needed to be carried out in order to develop an efficient MAP modeling tool for food safety coupling O2 /CO2 transfer and predictive microbiology.

Modeling of extraction process of crude polysaccharides from Basil seeds (Ocimum basilicum l.) as affected by process variables.

Basil seed (Ocimum basilicum L.) has practical amounts of gum with good functional properties. In this work, extraction of gum from Basil seed was studied. Effect of pH, temperature and water/seed ratio on the kinetic and thermodynamic parameters; entropy, enthalpy and free energy of extraction were investigated. The maximum gum yield was 17.95% at 50°C for pH=7 and water/seed ratio 30:1. In this study, the experimental data were fitted to a mathematical model of mass transfer and equations constants were obtained. The kinetic of Basil seed gum extraction was found to be a first order mass transfer model. Statistical results indicated that the model used in this study will be able to predict the gum extraction from Basil seed adequately. It also found that ΔH and ΔS were positive and ΔG was negative indicating that the extraction process was spontaneous, irreversible and endothermic. The ΔH, ΔS and ΔG values were 0.26-7.87kJ/mol, 8.12-33.2J/molK and 1.62-4.42kJ/mol, respectively.

Development of a mathematical model and simulation of mass transfer of solar ethanol distillation in modified brewery tank

The objective of this study was to develop a mathematical model of the mass transfer in a modified brewery tank for producing fuel ethanol. To reduce fossil fuel consumption, 50 flat-plate solar collectors were used as the heat source for the two stages of a distillation process for increasing the ethanol concentration. A 350-L distillation tank with 10%v/v (Stage 1) and a 70-L distillation tank with 40%v/v (Stage 2) were employed in the experiment used to develop the mathematical model of the mass transfer. A difference of approximately 10% was observed between the model predictions and the experimental results of the distillation product of Stage 1, whereas the predicted concentration was approximately 30% higher than that of the experiment, although this was reduced to approximately 5% by homogeneous mixing of the solution. Regarding the distillation process of Stage 2, there was approximately 10% difference between the predicted and experimental products, and approximately 3% difference between the predicted and experimental concentrations. The differences are attributed to errors in the heat transfer rate prediction of the model, which varies directly with the solar radiation values.

Efficiency assessment of air stripping packed towers for removal of VOCs (volatile organic compounds) from industrial and drinking waters

Packed column strippers are being used for removing VOCs (volatile organic compounds) contaminants from drinking and industrial waste waters. In order to evaluate the capability of packed towers for reducing the contaminant concentration to its permissible limits, a mathematical mass transfer model is developed. The model, which is based on resistances-in-series approach and considers liquid-vapor equilibria in terms of temperature-dependent Henry's constants, is successfully validated against several experimental data. The effects of key parameters, such as column pressure, temperature, gas-to-liquid flow rate and VOC inlet concentration, on the removal of different VOCs from waste waters and also on the required tower height are thoroughly considered. It is found that these parameters can have different impacts on the removal of different VOCs, and the required tower height is highly dependent on the VOC solubility and volatility. The optimum design and operating conditions of a packed tower are also determined for the maximum VOC removal and the minimum packing height.

Molecular-diffusion mass transfer of substance in soil medium

The process of the diffusion transfer of substances in soil medium, which causes the contamination of soil and groundwater, is considered. Four theoretical mathematical models of mass transfer are proposed. A dimensionless complex is proposed that acts as a measure of ratio of flow of biologically digestible component to molecular-diffusion flow. Experimental results are provided for one case, and the value of molecular diffusion coefficient in sandy soil medium is obtained.

Mathematical modeling of mass transfer in supercritical fluid extraction of oleoresin from red pepper

Supercritical fluid extraction (SFE) from red pepper (Capsicum frutescens) using CO2 was performed in order to evaluate mass transfer. Kinetic extractions were done at 15MPa and 40°C, varying solvent flow rate, particle diameter and extraction bed volume. The highest extraction rates were obtained for high solvent flow rates, low particle diameters and low extraction bed volume. These results can be explained by the high importance of the convective phenomenon under these conditions. A classical model based on the concept of intact and broken cells was applied to experimental SFE curves, and model parameters were obtained. Two modeling strategies were used: (1) simultaneous fitting creating a set of parameters for pairs of duplicates, (2) multiple fitting that adjusts a single value for the solute concentration in unbroken cell for curves with equal particle diameter. These fits allowed estimating the convective mass transfer coefficient for each condition, and the respective values of the experimental Sherwood number. Experimental data was used to calculate dimensionless numbers of Reynolds and Schmidt for each condition. A second series of mathematical modeling was performed using these dimensionless numbers, which allowed proposing new mass transfer correlations. These new equations were based on the existence of forced and free convection, although the importance of the latter was small. The convective coefficients calculated with the correlations were compared to those obtained from the model, and a good coherence was found between both.

Models of mass transfer in gas transmission systems

The models of gas movement in pipelines and gas filtration processes in complex porous media are considered in entire and fractional derivatives. The method for linearization of equations, which are included in the mathematical model of mass transfer, is suggested as well as an iterative scheme for solving initial systems of nonlinear differential equations is constructed. The finite-element model of the problem with the use of the Petrov-Galerkin method and Grunwald-Letnikov scheme concerning derivatives of the fractional order are implemented. The research of the models is carried out as well as comparative analysis of the numerical results is done.

System Model of Heat and Mass Transfer Process for Mobile Solvent Vapor Phase Drying Equipment

The solvent vapor phase drying process is one of the most important processes during the production and maintenance for large oil-immersed power transformer. In this paper, the working principle, system composition, and technological process of mobile solvent vapor phase drying (MVPD) equipment for transformer are introduced in detail. On the basis of necessary simplification and assumption for MVPD equipment and process, a heat and mass transfer mathematical model including 40 mathematical equations is established, which represents completely thermodynamics laws of phase change and transport process of solvent, water, and air in MVPD technological processes and describes in detail the quantitative relationship among important physical quantities such as temperature, pressure, and flux in key equipment units and process. Taking a practical field drying process of 500 KV/750 MVA power transformer as an example, the simulation calculation of a complete technological process is carried out by programming with MATLAB software and some relation curves of key process parameters changing with time are obtained such as body temperature, tank pressure, and water yield. The change trend of theoretical simulation results is very consistent with the actual production record data which verifies the correctness of mathematical model established.

Mathematical modeling of polluting migration in soils

The paper gives the analysis of mathematical tools in the sphere of hydro-ecological research in the study and prediction of the state of aquatic ecosystems. The specific of mathematical description of industrial effluents distribution was revealed to study the influence of anthropogenic impact on soils. It was found that the mathematical models of the influence of waste water on soil condition should be formed taking into account the type of waste waters, that is impurities containing in them and interacting with soil particles; soil properties and composition, that is type, structure, shape of pores and cracks in the soil, as well as the presence of water in its composition; interaction processes between them. The mathematical model of mass transfer of solutes in seepage flows of non-conservative pollutants was proposed, which describes the interaction between soils and waste waters. It was established that the difficulty occurs in solving mass transfer equations, which describe fluid motion in a porous media, and depends mainly on the structure and properties of soils, shape of pores and cracks that must be taken into account in the preparation of boundary conditions

Mass Transfer Modeling during Osmotic Dehydration of Chub Mackerel (Scomber japonicus ) Slices in Salt and Glycerol Solution at Different Temperatures

The use of osmotic dehydration to preserve fish products could be an interesting option in order to maintain the characteristics of food. The objective of this research was to study the mathematical modeling of water loss and solute gain during the cooking-infusion of chub mackerel (Scomber japonicus) slices at different temperatures. Osmotic dehydration was performed by cooking-infusion in solutions containing 54% (w/w) glycerol and 7% (w/w) salt (solution aw = 0.64) at different temperatures (50, 70 and 90C). Experimental results were adjusted to the Crank, Weibull, and Zugarramurdi & Lupín models. Mass transfer kinetics of water, salt and glycerol during osmotic dehydration of chub mackerel slices were adequately adjusted, with similar accuracy, by the three models. However, the Zugarramurdi & Lupín model was preferred because it also allowed the prediction of water and solids contents at equilibrium, which is relevant for the development of this kind of products. Practical Applications This article deals with the mathematical modeling of mass transfer during osmotic dehydration (OD) of chub mackerel (Scomber japonicus) slices. Based on the importance of developing new preservation technologies of this species, we consider that OD by immersion in osmotic solutions represents a good alternative for the industrialized production of chub mackerel. The models proposed in this work allowed the accurate prediction of water and soluble solid transfer kinetics during processing at different temperatures. Moreover, Zugarramurdi & Lupín model satisfactorily predicted the equilibrium values. The development of new products and the optimization of OD process depend on the adequate understanding of the moisture and solids contents and on the knowledge of the diffusion coefficients during OD. The results of this work are of sum importance for the fish processing industry to develop intermediate moisture fish products, which, in turn, can be ready-to-eat products, as well as input material of further processes.

Operational and quality characteristics of elastoplastic coatings of rubberized objects

A mathematical model of heat transfer in heat treatment of a two-layer rubber-metal object is developed. A mathematical model of mass transfer of a vulcanizing agent in heat treatment of an elastoplastic coating is developed. A comparative analysis of theoretical data is carried out and the results of an experiment on heat treatment of rubberized objects are presented. The influence of heat and mass transfer on chemical stability of rubberized objects in corrosive media is studied.

Study of iodine removal efficiency in self-priming venturi scrubber

Venturi scrubber is used in filtered vented containment system of nuclear power plants to remove the gaseous pollutants from contaminated gas during severe accidents. In this research, an experimental and theoretical investigation has been carried out to study the iodine removal efficiency in a self-priming venturi scrubber. The aqueous solution is prepared by adding weight percentage of sodium hydroxide 0.5% and sodium thiosulphate 0.2% in scrubbing water to increase the absorbance of inorganic iodine (I2) from the contaminated gas during emission. The iodine removal efficiency is investigated at various gas and liquid flow rates, and iodine inlet concentrations. The iodine removal efficiency is measured experimentally by measuring the inlet and outlet concentration of iodine at sampling ports. The petite droplets are formed in a venturi scrubber to absorb the iodine through the mass transfer phenomenon. A mathematical model for mass transfer based on a gas liquid interface is employed for the verification of experimental results. The contact time between iodine and scrubbing solution depends on the total volumetric flow of gas and liquid, and volume of throat and diffuser of the venturi scrubber. Sauter mean diameter is calculated from the Nukiyama and Tanasawa correlation. Steinberger and Treybal’s correlation is used to measure the mass transfer coefficient for the gas phase. The results calculated from the model under predict the experimental data.

Self-organization of copper nanosystems under Volmer–Weber conditions during quasi-equilibrium condensation

Copper nanosystems were obtained by means of magnetron sputtering and further deposition of extremely weak vapour fluxes onto glass ceramic substrates and KCl cleaved facets. The increased pressure (∼10Pa) of highly purified argon, low discharge powers (∼24W) of the magnetron sputterer, as well as relatively high growth surface temperature (∼653K) have been used during the technological process. The mechanisms of structure formation on different substrates were analysed by SEM, TEM and electron and x-ray diffraction. To explain the self-organization of low-dimensional systems we developed a mathematical model of mass transfer of sputtered substance within the space between the magnetron sputterer and substrate. On the basis of experimental and modelling studies it was found that Ostwald ripening of Cu nanocrystal systems is possible to realize on KCl (001) cleaved facets under conditions of critically low supersaturations and at increased concentration of nucleation sites.

Mathematical modeling of mass transfer in multicomponent gas mixture across the synthesized composite polymeric membrane

This study presents a new mathematical model to investigate the ternary gas mixture permeation across a synthesized composite PDMS/PA membrane. A novel algorithm is introduced for direct determination of diffusion coefficients. It pertains to study gas permeation through concentration dependent systems and comparing with traditional time lag method confirms the precision of this approach. Feature is that this method does not require physical properties of the membrane. Accordingly, it can be used as a general comprehensive model. In addition, molecular pair and molecular trio interactions were taken into account and in order to investigate the deviation of gas mixture from ideality, fugacities were calculated. The results showed that permeabilites of H2 and CH4 increase with increasing feed temperature and fugacity, while that of C3H8 decreases. Moreover, increasing C3H8 concentration improved permeation properties of all components. The results demonstrated that considering the concentration dependent system (CDS) leads to the small deviation of about less than 10%, while the deviation of 50–100% by the concentration independent system (CIS) was acquired. Additionally the results indicated that permeability of the lighter gases is specially affected by diffusivity, while solubility is dominant on permeability of the heavier gases.

Osmotic Dehydration of Nectarines: Influence of the Operating Conditions and Determination of the Effective Diffusion Coefficients

The aim of the present work is to study the kinetics of osmotic dehydration of Caldesi nectarines (Prunus persica var. nectarina) evaluating the effect of osmotic solution concentration, type of solute, temperature, fruit/solute ratio and process time on moisture content, water loss, soluble solids content and solids gain. The process analysis was carried out experimentally and numerically through the mathematical modelling of mass transfer. Hypertonic solutions of glucose syrup and sorbitol (40 and 60 % w/w) were used for dehydration, during 2 h of process at temperatures of 25 and 40 °C, with fruit/osmotic agent ratio of 1:4 and 1:10. Water loss and solids gain showed significant differences depending on the type and concentration of the osmotic agent, process time and fruit/solution ratio. The concentration interacted significantly with all variables; in addition, there was an interaction between the type of osmotic agent and the relationship between fruit and the osmotic agent. The effective diffusion coefficients were obtained from the analytical solution of Fick’s second law applied to flat-plate geometry and by solving the mass transfer microscopic balances by finite element method, taking into account the real geometry of the nectarine pieces. The values obtained from Fick’s law varied between 1.27 × 10−10 and 1.37 × 10−08 m2 s−1 for water and from 1.14 × 10−10 to 1.08 × 10−08 m2 s−1 for soluble solids, while the values calculated by finite elements method ranges were between 0.70 × 10−09 and 4.80 × 10−09 m2 s−1 for water and between 0.26 × 10−09 and 1.70 × 10−09 m2 s−1 for soluble solids. The diffusion coefficients values obtained from the numerical solution are consistent with those published in literature.

A mathematical model for mass transfer in hydrophobic pervaporation for organic compounds separation from aqueous solutions

In this study, a numerical predictive model was developed based on the solution–diffusion theory to describe the mass transfer of the pervaporation process through the PDMS membrane taking into account fluxes coupling. The Flory–Huggins model was used to predict equilibrium sorption of ethanol and water into the PDMS membrane. The transport of penetrants through the membrane was described with generalized Fick’s law which is able to calculate the mass fluxes and selectivities as well as the concentration profile of components inside the membrane. Three modeling cases were considered: (I) constant diffusion coefficient inside the membrane, (II) concentration dependent diffusion coefficient and (III) concentration and temperature dependent diffusion coefficient. For estimating the diffusion coefficients of penetrants through the membrane, the Duda’s free volume theory was applied. The resulting nonlinear transport equations were solved simultaneously by formulating the weak form Galerkin finite element method. The model was then validated using the experimental data obtained from the pervaporative separation of ethanol from aqueous solutions. Finally, the effect of operating parameters such as feed concentration and temperature on the permeation flux and penetrant selectivities were investigated via both the model prediction and experiments. The results showed that the penetrant diffusion coefficient dependencies on the component concentrations must be taken into account.

Optimization and mathematical modeling of mass transfer between Zostera marina residues and supercritical CO2 modified with ethanol

Abstract Supercritical CO 2 extraction of phenolic compounds from Zostera marina residues was optimized by developing a mathematical model based on mass transfer balances. A linear driving force model was applied considering model parameters such as solute concentration on the surface of the solid ( C s ) and in the supercritical fluid phase ( C f ), film mass transfer coefficient ( k f ) and molecular diffusivity ( D AB ) and axial dispersion ( D ax ) coefficients. Henry's law was used to describe the equilibrium state of solid and fluid phases. The results of the proposed model were compared to that of the experimental data in terms of transport properties and extraction yield at various temperatures (303.15, 323.15, 353.15 K), pressures (15, 25, 35 MPa) and co-solvent mass ratios (0, 10, 20%). The optimum parameters were elicited as 25 MPa, 353.15 K and a co-solvent ratio of 20% yielding 77.22 μg g −1 dry feed. The model satisfactorily described the extraction yield which can be used for scale-up purposes.

Study of CO<sub>2</sub> Hydrate Growth Process Based on Diffusion-Controlled Reaction

A one-dimensional mathematical model of mass transfer during CO2 gas hydrate film formation at the boundary between the gas and water is established by analyzing the process of the static system. According to the model, we can get the concentration distribution of free CO2 in the hydrate film along with the time, and the conversion rate of hydrate with time can also be obtained. The results calculated at specific experiment condition in the system are analyzed. It indicates that the conversion rate can reach about 85% after 10 hours under static condition with 3.5MPa pressure and 0.5°C temperature by pure water when the initial reaction liquid depth is 20 mm