Differential Equations Of Mass Transfer Research Articles

Mathematical Model and Numerical Method of Calculating the Dynamics of High-Temperature Drying of Milled Peat for the Production of Fuel Briquettes

Milled peat must be dried for the production of peat fuel briquettes. The current trend in the creation of drying technologies is the intensification of the dehydration process while obtaining a high-quality final product. An increase in the temperature of the drying agent, above 300 °C, significantly accelerates the reaching of the final moisture content of the peat. In the final stage, it is also accompanied by partial thermal decomposition of the solid phase. Its first stage, which is the decomposition of hemicellulose, contributes to a decrease in weight and an increase in the caloric content of the dry residue. The development of high-temperature drying modes consists of determining the temperature and velocity of the drying agent, wherein the duration of the material reaching the equilibrium moisture content will be minimal and the temperature of the material will not rise above the second-stage decomposition temperature of cellulose. This problem can be solved by the mathematical modeling of the dynamics of peat particles drying in the flow. The article presents a mathematical model of heat and mass transfer, phase transitions, and shrinkage during the dehydration of milled peat particles. The equations of the mathematical model were built based on the differential equation of mass transfer in open deformable systems, which, in the absence of deformations, turns into the known equation of state. A numerical method for implementing a mathematical model has been developed. The adequacy of the mathematical model is confirmed by comparing the results of numerical modeling with known experimental data.

TOWARDS ANALYSIS OF PRIVATE DECISIONS OF DIFFERENTIAL EQUATION OF MASS TRANSFER IN THE MASSIF

On the basis of the analysis of linear system of the equations Onzager is shown, that the equation Luikov is a special case. Therefore private decisions of the differential equation of mass transfer should be analysed on meanings of factors rcm /c and δθ.

Calculation of Pyrolysis Zone in Installation of Activated Carbon Production

Activated carbon can be obtained in a variety of ways. The most promising in terms of resource conservation and economic benefits is the method of producing activated carbon from organic waste. The production of activated carbon by this method is based on the process of pyrogenetic decomposition of pre-crushed and dried waste. As a result of thermochemical processing, activated carbon and pyrolysis gas are formed. Conductive waste pyrolysis is a high-temperature process that runs without oxygen. The temperature regimes of the pyrolysis process are described by the differential heat transfer equation. The calculation of the specific masses of the waste, coal and steam-gas mixture formed during the decomposition is carried out according to the differential equations of mass transfer by complicated ongoing chemical reactions. The hardware design of the pyrolysis process is a set of complex design and technological solutions. When developing equipment designed for the pyrolysis of crushed waste, it is necessary to carry out calculations related to the optimization of the hardware design and operating parameters of the process of thermal decomposition of waste particles, which make it possible to obtain a high-quality product. A pyrolysis zone of crushed waste has been developed for an energy- and resource-saving continuously operating unit for the production of activated carbon. Heating of waste to a temperature of 450-550 ° C is carried out by non-condensing combustible gases obtained during the separation of the steam-gas mixture. Waste flue gases enter a recuperative heat exchanger, and then serve as a heat carrier for drying a new batch of waste. The paper describes a method for calculating the pyrolysis zone in an activated carbon production unit. A mathematical model of the pyrolysis process is presented, by which it is possible to determine the optimal height of the layer of waste particles and the specific mass of the formed coal and gases during the decomposition of organic waste. An algorithm for calculating the pyrolysis zone has been developed. The following characteristics are shown: the dependence of the heating temperature of the layer of wood particles in the pyrolysis chamber on the layer thickness and the kinetic dependence of the specific mass of the solid residue of wood particles during pyrolysis. The calculation of the coal pyrolysis zone for the activated carbon production unit is presented.

Modelling of mass transfer in separation of hydrocarbon mixtures and modernization of industrial columns

A system of differential equations of mass transfer with an interfacial mass source for multicomponent distillation in a column with random and structured packings in the film mode is presented. The interfacial source is associated with the matrix of volumetric mass transfer coefficients as well as a driving force for mass transfer. The diffusion fluxes of mass of components are written in the form of Fick’s law with elements of the matrix of coefficients of molecular diffusion in the gas (vapor) phase. To simplify the mathematical model, one dimensional equations for the diffusion model of the structure of flows in the gas and liquid phases with the interfacial sources and coefficients of back (longitudinal) mixing of gas and liquid flows are used. The Dankwerts boundary conditions are applied to the system of equations. To calculate the back mixing coefficients, experimental data obtained for the selected types of packings are used. When modelling mass transfer in hydrocarbon mixtures consisting of several dozen components, the system of equations for the diffusion models is solved for the key components and fractions of the oil and gas condensate (OGC) mixture stabilization column at an industrial enterprise. A brief description of the technological scheme is given. Scientific and technical solutions are developed to replace obsolete sieve trays in the stabilizer column with a new structured corrugated roll packing having a rough surface. It is found that a packing height of eight meters is sufficient for separating OGC in a given composition range. As a result of implementation of the new packing, the content of isopentane in the stable condensate decreases and its concentration in the broad fraction of light hydrocarbons increases from 9 wt.% to 11.7 wt.%. The implementation of packing in the stabilizer column meets the requirements of the technical specifications.

Mathematical Model of Dispersed Phase Gas Separation in a Combined Equipment

A model of turbulent migration of finely dispersed particles (diameter 1–20 μm) in gas streams in structured and random packings of separators and in a separator channel with flow swirl is described. Differential equations of mass transfer of particles in plane-parallel and cylindrical channels, which take account of the source of turbulent-inertial transfer mechanism, are given. Expressions for calculating the parameters of the equations characterizing turbulent migration of particles for a separator packing layer and for a channel with flow swirl are presented. A cellular model of the flow structure is used to reduce the laboriousness of calculation and an expression is derived in explicit form to calculate the efficiency of gas separation of aerosol particles. An example of calculation of combined gas separator for cleaning natural gas from dispersed phase, which is approved in the industry, is given.

Formation of Physic and Mechanical Composition of Dust Emission from the Ventilation Shaft of a Coal Mine as a Factor of Ecological Hazard

During operation of coal mines, polluted air is emitted from the ventilation shafts, it contains large amount of carbon and rock dust with various physical and mechanical composition. Dust contains freshly formed silicon dioxide and other components of the mineral product that are hazardous to human health, which are scattered in the surface layer of the atmosphere. Mainly fine dust comes from the mine workings, and the release of the total dust mass by 80 – 96 % is due to the operation of the loading and unloading complex. It is established that the level of dustiness decreases mainly due to the settling in the mine workings of the coarsely dispersed dust fraction. The environment receives mainly fine dust with a high content of fine particles, which can be intensively dispersed outside the sanitary protection zone of the mine. A mathematical model is obtained in the form of a differential equation of mass transfer, which takes into account the physic and mechanical composition of the dust emission from the ventilation shaft of the coal mine as a factor of ecological hazard. The obtained results determine the directions of effective struggle against dust emissions of coal mines.

Model study of CO2 absorption in aqueous amine solution enhanced by nanoparticles

An investigation for the enhancement of nanoparticles (TiO2, MgO, SiO2) on the mass transfer for CO2 absorption was presented through both the experiment method and a 3-D unsteady model. The effects of nanoparticle loadings, nanoparticle diameters, nanoparticle species, solution species, experimental temperature and the CO2 initial volume fraction on the CO2 absorption were investigated in this paper. Considering the chemical reaction between gas and solution, a 3-D unsteady model was developed to study the mass transfer for CO2 absorption enhanced by nanoparticles in monoethanolamine (MEA) solution. Based on the analysis of zwitterion mechanism in the process of chemical reaction between MEA and CO2, it was shown that the rate of chemical reaction is proportional to the concentration of CO2 due to the large MEA concentration and the zero-order reaction in the liquid membrane. Further, the differential equation of mass transfer in three dimensions was set up, which is related to the rate of chemical reaction and the random distribution of nanoparticles. The random distribution of nanoparticles in a fluid element was determined by Monte Carlo method. In addition, the absorption experiments of CO2 in MEA solution using the bubbling reactor were carried out, which could also verify the correctness of the developed model.

Slow-Equilibration Approximation in Macroscopic Approach to Studying Kinetics at Equilibrium

Macroscopic approach to studying kinetics at equilibrium (MASKE) facilitates measurements of rate constants of formation (k(+)) and dissociation (k(-)) of affinity complexes in the state of chemical equilibrium. MASKE relies on "informational nonequilibrium" created by a nonuniform initial spatial distribution of a label on one of the reactants. In general, finding k(+) and k(-) by MASKE requires fitting experimental label-propagation patterns-dependencies of label concentrations on a coordinate or time-with the simulated label-propagation patterns. Here we introduce a simple fitting-free approach for finding the rate constants in the case of slow equilibration. Slow equilibration means that the characteristic equilibration time of the labeled reactant and labeled complex, t(eq), is much greater than the characteristic separation time of the labeled reactant and labeled complex, t(sep). We developed the mathematics for this approach by solving the differential equations of mass transfer using the assumption of slow equilibration. The approach was then tested and its accuracy was studied by applying it to label-propagation patterns created with the earlier-developed exact solution of the mass-transfer equations. The results proved that the approximate solution was correct. They also showed that k(+) and k(-) can be found with this fitting-free approach with a relative error less than 20% if t(sep) < 0.6t(eq). The practical limitations of our slow-equilibration approximation are discussed.

The synthesis of the optimal spatial pseudostructures of imitations of the porous structure of a catalyst grain based on the concept of fractals

The problem of the synthesis of fractal structures in the form of dendrite and a Menger sponge as the statistical equivalents of natural adsorbents and catalysts during the identification of their models by real porometric data is defined. Differential equations of mass transfer by diffusion, adsorption, and chemical reaction, which include the fractal parameters of the mentioned models, are given.

Mathematical modeling of the uptake of curing salts in pork meat

A mathematical model was developed in order to represent the uptake of curing salts (NaNO 2, KNO 3, and NaCl) in pork meat pieces with the purpose of determining immersion times and suitable salt concentrations in the brine of wet curing processes. The partial differential equations of mass transfer under unsteady-state conditions were solved numerically, in three-dimensional geometry (finite cylinders), considering the diffusive and also the convective contributions of the different solutes, due to water uptake by the tissue. The variation of the diffusion coefficients of the curing salts with NaCl concentration was also considered. The numerical model was validated using results obtained from experiments of salt diffusion in pork meat cylinders of different diameters immersed in curing brines. The model was applied to predict the time necessary for an industrial piece of meat to be immersed in brine without exceeding the maximum permitted nitrite value of 200 ppm and the recommendable sodium chloride concentration.

A Critical Review for Proper Use of Water/Oil/Gas Transfer Functions in Dual-Porosity Naturally Fractured Reservoirs: Part II

Summary This paper continues the work presented in Ramirez et al. (2009). In Part I, we discussed the viability of the use of simple transfer functions to accurately account for fluid exchange as the result of capillary, gravity, and diffusion mass transfer for immiscible flow between fracture and matrix in dual-porosity numerical models. Here, we show additional information on several relevant topics, which include (1) flow of a low-concentration water-soluble surfactant in the fracture and the extent to which the surfactant is transported into the matrix; (2) an adjustment to the transfer function to account for the early slow mass transfer into the matrix before the invading fluid establishes full connectivity with the matrix; and (3) an analytical approximation to the differential equation of mass transfer from the fracture to the matrix and a method of solution to predict oil-drainage performance. Numerical experiments were performed involving single-porosity, fine-grid simulation of immiscible oil recovery from a typical matrix block by water, gas, or surfactant-augmented water in an adjacent fracture. Results emphasize the viability of the transfer-function formulations and their accuracy in quantifying the interaction of capillary and gravity forces to produce oil depending on the wettability of the matrix. For miscible flow, the fracture/matrix mass transfer is less complicated because the in-terfacial tension (IFT) between solvent and oil is zero; nevertheless, the gravity contrast between solvent in the fracture and oil in the matrix creates convective mass transfer and drainage of the oil.

Mathematical modelling of flue-gas desulphurisation using lime slurry in a turbulent contact absorber

A mathematical model has been developed for absorption of SO2 from flue gases using lime slurry in Turbulent Contact Absorbers (TCA). In this model, the TCA was simulated by considering it as a falling film absorption system. Turbulent effects were taken into account by using eddy viscosity and eddy diffusivity. The system of partial differential equations of mass transfer with moving boundaries was solved numerically in order to obtain the concentration profiles of the reactants, the thickness of the reaction zone and the outlet gas concentration. A bench-scale TCA was built up, and the mathematical model was evaluated against experimental data obtained from the built device. The model predictions showed good agreement with the experimental data and literature data.

A Computational Transport Model for Wall-Cooled Catalytic Reactor

A computational transport model is proposed for simulating the concentration, temperature, and velocity distributions in a wall-cooled fixed-bed catalytic reactor producing vinyl acetate from acetic acid and acetylene without assuming empirical Sct and Prt numbers or using the experimental coefficient of dispersion. The model consists of the basic differential equation of mass transfer with the recently developed− εc model for its closure and the accompanied computational fluid dynamics and heat transfer models. With the present model, the radial and axial profiles of turbulent mass-transfer diffusivity Dt, turbulent thermal diffusivity αt, and turbulent kinematic viscosity νt, as well as their distributions in the reactor are obtained. It is found that the local ratios of νt/Dt and νt/αt, or the local Prandtl (Prt) and Schmidt (Sct) numbers, are varying in the reactor, although the shapes of the νt, αt, and Dt profiles show somewhat similarity. The simulated results are found to be in agreement with the experimental measurements reported by Valstar et al.33

Adsorption from Micellar Surfactant Solutions: Nonlinear Theory and Experiment

A theoretical approach to the kinetics of adsorption on the expanding interface of a micellar surfactant solution is proposed. The nonlinear partial differential equations of mass transfer of the micelles and monomers are reduced to ordinary differential equations, which are solved numerically along with the nonlinearized adsorption isotherm as a boundary condition. Being more general, this theoretical approach is consistent also with the special case of small deviations from equilibrium widely used in the kinetics of micellization. The theory is applied further to fit the experimental data for dynamic surface tension of micellar solutions of sodium dodecyl sulfate and sodium polyoxyethylene-2 sulfate measured by the maximum bubble pressure (MBP) method. The expansion of the bubble surface is accounted for by using the experimental dependence of the bubble size on time. The calculated effective rate constant of micelle decay for sodium dodecyl sulfate turns out to be sensitive to the mechanism of the micelle's disintegration. This provides a possibility for one to determine the rate constants of the fast and slow processes in the relaxation kinetics of micellization by using a relatively simple experimental technique for dynamic surface tension measurement like the MBP method.

Calculation of Intercrystalline Solution Composition during in Vitro Subsurface Lesion Formation in Dental Minerals†

Applications of a novel technique to calculate intercrystalline solution composition during enamel demineralization are presented. Bovine tooth enamel blocks and carbonated apatite (CAP) compressed disks were demineralized in an in vitro subsurface lesion system. The demineralization medium was a 0.1 M acetate buffer at pH 4.5, containing calcium, phosphate, and fluoride (0.5 ppm). Mineral samples were demineralized for various times, and fluoride profiles and mineral density profiles of these samples were determined by electron microprobe and X-ray microradiography, respectively. A model independent data analysis (MIDA) technique uses these data along with the differential equations for mass transfer and permits calculation of the local intercrystalline solution composition profiles inside the porous mineral matrix as functions of time and position. The invariance in diffusivity with time as calculated in the analysis was taken as an indicator of the physical reasonableness of the method. Current outcomes suggest that it is the sharp gradient of fluoride concentration in the intercrystalline solution which causes the formation of subsurface lesions. Since the driving force for mineral dissolution is a function of solution composition, a gradient of this driving force is consequently formed. Using a compressed disk of carbonated apatite powder as a model for block enamel excluded the possibility of the existence of a gradient of mineral composition which could also cause a gradient of the driving force for mineral dissolution. An FAP surface complex hypothesis is consistent with the current view that fluoride in the intercrystalline solution has a stronger inhibition effect on the dissolution of mineral than does fluoride in the mineral phase. With the help of the MIDA technique, calculated results indicate that the mechanism of the formation of subsurface lesions is dynamically controlled by the intercrystalline solution composition.

An analysis of the elution curve in preparative chromatography with moving feed ports

AbstractThe present work deals with the mathematical analysis of the elution curve in preparative chromatography with moving feed ports. To predict the elution curve for this moving feed mode, simple differential equations for mass transfer were employed and solved by using algorithms based on the Plate Theory and iteration methods. As a result, the least chromatographic band spreading was found on the moving feed mode rather than the non‐moving feed mode only if mass transfer resistance between the phases does not participate in the adsorption process. On the other hand, when the mass transfer is not very rapid, the chromatographic performance of the moving feed mode was found to be compatible with that of the non‐moving feed mode.

Mass transfer in laminar flow

Starting from the differential equation of mass transfer in laminar flow and the appropriate boundary condition, expressions are derived for the rate of mass transfer from (a) a flat plate in a longitudinal fluid stream, (b) a vertical flat plate by natural convection, (c) the forward stagnation point of a sphere in a fluid stream. Only outward mass transfer is considered; this corresponds to blowing outwards from the plate at a rate inversely proportional to the boundary-layer thickness. The Kármán-Pohlhausen-Kroujiline method is used. Where appropriate the Prandtl or Schmidt number has been taken as 0⋅71. The calculations are valid for all mass-transfer processes for which a single diffusion coefficient can be ascribed to the diffusing property, but are particularly relevant to the combustion of liquid fuels, for which the outward mass-transfer rates are so high that important deviations occur from boundary-layer profiles without mass transfer. Despite the great temperature variations present in boundary layers with combustion, mean values for the fluid properties are assumed. In the case of natural convection, it is assumed that the body forces on the fluid in the boundary layer are everywhere zero; this leads to a less serious over-estimate of the buoyancy than the usual assumptions which are valid only for small temperature differences.