Macroscopic Mass Balance Research Articles

Macroscopic Balance Model for Wave Rotors

A mathematical model for multi-port wave rotors is described. The wave processes that effect energy exchange within the rotor passage are modeled using one-dimensional gas dynamics. Macroscopic mass and energy balances relate volume-averaged thermodynamic properties in the rotor passage control volume to the mass, momentum, and energy fluxes at the ports. Loss models account for entropy production in boundary layers and in separating flows caused by blade-blockage, incidence, and gradual opening and closing of rotor passages. The mathematical model provides a basis for predicting design-point wave rotor performance, port timing, and machine size. Model predictions are evaluated through comparisons with CFD calculations and three-port wave rotor experimental data. A four-port wave rotor design example is provided to demonstrate model applicability. The modeling approach is amenable to wave rotor optimization studies and rapid assessment of the trade-offs associated with integrating wave rotors into gas turbine engine systems.

Wave Propagation in Poroelastic Media Saturated by Two Fluids

A theory of wave propagation in isotropic poroelastic media saturated by two immiscible Newtonian fluids is presented. The macroscopic constitutive relations, and mass and momentum balance equations are obtained by volume averaging the microscale balance and constitutive equations and assuming small deformations. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy’s law. The coefficients of macroscopic constitutive relations are expressed in terms of measurable quantities in a novel way. The theory demonstrates the existence of three compressional and one rotational wave. The third compressional wave is associated with the pressure difference between the fluid phase and dependent on the slope of the capillary pressure-saturation relation.

Kinetic Analysis of the Simultaneous Nondispersive Extraction and Back-Extraction of Chromium(VI)

This work has been focused on the study of the viability and kinetics of Cr(VI) removal from industrial waste waters and simultaneous concentration for its reuse in electroplating processes, working with liquid−liquid systems in two hollow-fiber (HF) modules, and using Aliquat 336 as an organic carrier. For the kinetic analysis of the non-steady-state system the macroscopic mass balances of the chromium in the fluid phases into HF modules were developed and solved simultaneously with the mass balances in the homogenization stirred tanks. An optimization procedure to calculate the mass transport parameters was developed, leading to the value Km= 1.92 × 10-8 m/s (membrane mass-transport coefficient). In the extraction process a kinetic mass-transfer parameter depending on the experimental conditions was obtained.

Modulated Drug Release Using Iontophoresis Through Heterogeneous Cation-Exchange Membranes. 2. Influence of Cation-Exchanger Content on Membrane Resistance and Characteristic Times

An implantable drug delivery method using iontophoresis through cation-selective membranes was further developed. Heterogeneous cation-exchange membranes (HCMs) were prepared by mixing conductive sulfonated polystyrene beads into a nonconductive silicone rubber matrix. The membrane resistivity and lag time to steady-state transport of two salts, (±)-phenylpropanolamine hydrochloride (PPA) and NaCl, were evaluated during constant current iontophoresis at 37 °C as a function of the resin content in the HCMs. A continuous decline in membrane resistivity was observed as fractional resin content (l) was increased over the entire usable region (l= 0.29–0.52), a characteristic that could be described by a percolation scaling law (for an infinite lattice, 3-D geometry). Morphological analysis of the membranes before and after swelling strongly suggested that the conducting clusters of resin beads form during the swelling period prior to use. The response time to steady-state transport of PPA into NaCl during a 40 μA constant current (0.27 cm2) was found to increase with increasingl, but not without decreasing the permselectivity of the HCMs for the drug cation. The lag time effect could be explained in terms of an increasing number of fixed charge groups in the membrane available for transport (mfcA), which was derived from a macroscopic mass balance model. The values ofmfcAwere also found to be related to the characteristic time of diffusion in a homogenous transport projection of the HCM (or an effective medium), an essential parameter for future non-steady-state simulations. The characteristic time of diffusion was found to be invariant with changing resin content, suggesting that the membranes are fairly nontortuous (ca. seven beads thick). By assuming that the thickness of the HCM approaches the thickness of its homogeneous projection, an expression was derived to predict lag time to steady-state PPA transport requiring resistance measurements only (provided that the resin capacity is known). There was excellent agreement between the theoretical and experimental lag time to steady-state transport of PPA (r= 0.96,p< 0.001), further implicating the role of membrane resistance in the bi-ionic system. These modeling approaches have already found utility in iontophoretic implant design for prevention of cardiac arrhythmias and may be valuable in future non-steady-state analysis to further develop on-line detection-implant response technology.

Pressure-driven flow of suspensions: simulation and theory

Dynamic simulations of the pressure-driven flow in a channel of a non-Brownian suspension at zero Reynolds number were conducted using Stokesian Dynamics. The simulations are for a monolayer of identical particles as a function of the dimensionless channel width and the bulk particle concentration. Starting from a homogeneous dispersion, the particles gradually migrate towards the centre of the channel, resulting in an homogeneous concentration profile and a blunting of the particle velocity profile. The time for achieving steady state scales as (H/a)3a/〈u〉, where H is the channel width, a the radii of the particles, and 〈u〉 the average suspension velocity in the channel. The concentration and velocity profiles determined from the simulations are in qualitative agreement with experiment.A model for suspension flow has been proposed in which macroscopic mass, momentum and energy balances are constructed and solved simultaneously. It is shown that the requirement that the suspension pressure be constant in directions perpendicular to the mean motion leads to particle migration and concentration variations in inhomogeneous flow. The concept of the suspension ‘temperature’ – a measure of the particle velocity fluctuations – is introduced in order to provide a nonlocal description of suspension behaviour. The results of this model for channel flow are in good agreement with the simulations.

Model studies of slag carry-over during drainage of metallurgical vessels

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Physical models of fumarolic flow

Abstract Transport of gas from subsurface magma to the atmosphere is modelled as flow through a narrow, rough pipe, and through a wider conduit filled with spheres, representing flow in a porous pipe. Macroscopic mass, momentum and energy balances are written for a control volume representing a short vertical section of the pipe. The balances are solved for a perfect gas with the approximate thermodynamic properties of steam, losing heat by steady-state radial conduction through the pipe walls, with boundary conditions of magmatic temperature at the base of the pipe, atmospheric pressure at the surface, and an ambient geotherm at a specified distance from the conduit. Results yield the variation of exit temperature with magmatic source depth, conduit radius, mass flow rate per unit area, distance to ambient temperatures, vent porosity, and porous particle size. Two ambient geotherms are used: (1) a linear gradient from atmospheric to magmatic temperatures; and (2) a ‘hydrothermal’ geotherm, based on water saturation temperatures assuming a hydrostatic pressure gradient. Because individual vents commonly cluster, the porous pipe model is considered to be more realistic than the rough pipe model, mainly because the conductive cooling approximation is more likely to remain valid. Exit temperatures fall as conductive losses increase, i.e. as the source depth increases, the conduit narrows, the mass flow decreases, the distance to ambient temperatures reduces, or the ambient geotherm cools. The porous pipe model is applied to data from two fumarole fields, and indicates magmatic source depths of

A general method of designing gas and gas‐liquid injectors using laws of turbulent jet mixing

AbstractBased on the equation for entrainment of a turbulent free jet, a simple method of designing gas and gas‐liquid injectors is presented. Using the formula for mass flow ratio of an injector consisting of a nozzle and a short cylindrical mixing tube open at both ends, a general equation for designing any injector with variable geometry and flow resistances is given. The optimization procedure is outlined. Compared to the conventional design of injectors, starting from macroscopic mass and momentum balances, the present method is much simpler, easily applicable and involves only one empirical entrainment coefficient. It is particularly advantageous in the case of variable density of the motive and entrained fluids since the density ratio appears explicitly in the relevant equations.

Universelle Methode zur Berechnung von Injektormischern

Chemie Ingenieur TechnikVolume 65, Issue 4 p. 455-457 Wissenschaftliche Kurzbeiträge Universelle Methode zur Berechnung von Injektormischern† Dr.-Ing. Natti S. Rao, Dr.-Ing. Natti S. Rao Lehrstuhl für Energieanlagentechnik, Ruhr-Universität Bochum, Postfach 10 21 48, 4630 Bochum 1Search for more papers by this authorProf. Dr.-Ing. Hans Kremer, Prof. Dr.-Ing. Hans Kremer Lehrstuhl für Energieanlagentechnik, Ruhr-Universität Bochum, Postfach 10 21 48, 4630 Bochum 1Search for more papers by this author Dr.-Ing. Natti S. Rao, Dr.-Ing. Natti S. Rao Lehrstuhl für Energieanlagentechnik, Ruhr-Universität Bochum, Postfach 10 21 48, 4630 Bochum 1Search for more papers by this authorProf. Dr.-Ing. Hans Kremer, Prof. Dr.-Ing. Hans Kremer Lehrstuhl für Energieanlagentechnik, Ruhr-Universität Bochum, Postfach 10 21 48, 4630 Bochum 1Search for more papers by this author First published: April 1993 https://doi.org/10.1002/cite.330650419 † Vortrag von N. S. Rao auf dem Jahrestreffen der Verfahrens-Ingenieure, 25. bis 27. Sept. 1991 in Köln. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Volume65, Issue4April 1993Pages 455-457 RelatedInformation

Mass balance approaches for estimating the intestinal absorption and metabolism of peptides and analogues: theoretical development and applications.

A theoretical analysis for estimating the extent of intestinal peptide and peptide analogue absorption was developed on the basis of a mass balance approach that incorporates convection, permeability, and reaction. The macroscopic mass balance analysis (MMBA) was extended to include chemical and enzymatic degradation. A microscopic mass balance analysis, a numerical approach, was also developed and the results compared to the MMBA. The mass balance equations for the fraction of a drug absorbed and reacted in the tube were derived from the general steady state mass balance in a tube: [formula: see text] where M is mass, z is the length of the tube, R is the tube radius, Pw is the intestinal wall permeability, kr is the reaction rate constant, C is the concentration of drug in the volume element over which the mass balance is taken, VL is the volume of the tube, and vz is the axial velocity of drug. The theory was first applied to the oral absorption of two tripeptide analogues, cefaclor (CCL) and cefatrizine (CZN), which degrade and dimerize in the intestine. Simulations using the mass balance equations, the experimental absorption parameters, and the literature stability rate constants yielded a mean estimated extent of CCL (250-mg dose) and CZN (1000-mg dose) absorption of 89 and 51%, respectively, which was similar to the mean extent of absorption reported in humans (90 and 50%). It was proposed previously that 15% of the CCL dose spontaneously degraded systematically; however, our simulations suggest that significant CCL degradation occurs (8 to 17%) presystemically in the intestinal lumen.(ABSTRACT TRUNCATED AT 250 WORDS)

On the motion of a fluid-saturated porous solid

Two-parameter structure model of a porous solid is proposed as an approximation of a real porous structure and the macroscopic mass and momentum balance equations are derived for such a medium filled with liquid. The approach presented leads to the equations of motion for a fluid-saturated porous medium with coupling terms via cross-mass couplings. The linear form of these equations is equivalent to the well-known Biot equations.

Entrance Region Flow of a Casson Fluid in a Straight Channel

Abstract The entrance region flow of a Casson fluid in a straight channel has been investigated numerically without making prior assumptions on the form of velocity profile within the boundary layer region, which is determined by a cross-sectional integration of the momentum differential equation for a given distance from the channel entrance. Using the macroscopic mass and momentum balance equations, the thickness of the core, the entrance length, the plug core velocity, and the pressure drop have been obtained at each cross section of the entrance region of the channel for specific values of Casson number.

Phase separation of dispersed mist and dispersed annular (rivulet or thin film) flow in a tee—II: Analysis

An experimental and analytical investigation of dispersed mist and dispersed annular (rivulet or thin film) flow phase separation in a tee was performed. The analysis portion consisted of developing physically based models and incorporating the models into a computer program that calculates liquid drop and liquid pathline trajectories for these two flow regimes. Macroscopic mass balances were then calculated and compared with data. In addition, a correlation for dispersed mist phase separation was developed from computer calculations and is presented.

Metabolic modeling of fumaric acid production byrhizopus arrhizus

A metabolic model is developed for fumaric acid production byRhizopus arrhizus. The model describes the reaction network and the extents of reaction in terms of the concentrations of the measurable species. The proposed pathway consists of the Embden-Meyerhof pathway and two pathways to FA production, both of which require CO2 fixation (the forward and the reverse TCA cycles). Relationships among the measurable quantities, in addition to those obtainable by a macroscopic mass balance, are found by invoking a pseudo-steady-state assumption on the nonaccumulating species in the pathway. Applications of the metabolic model, such as verifying the proposed pathway, obtaining the theoretical yield and selectivity, and detecting experimental errors, are discussed.

Predicting fraction dose absorbed in humans using a macroscopic mass balance approach.

A theoretical approach for estimating fraction dose absorbed in humans has been developed based on a macroscopic mass balance that incorporates membrane permeability and solubility considerations. The macroscopic mass balance approach (MMBA) is a flow model approach that utilizes fundamental mass transfer theory for estimating the extent of absorption for passively as well as nonpassively absorbed drugs. The mass balance on a tube with steady input and a wall flux of Jw = PwCb results in the following expression for fraction dose absorbed, F: [formula; see text] where the absorption number, An = L/R.Pw/(vz), L and R are the intestinal length and radius, Pw is the unbiased drug wall permeability, (vz) is the axial fluid velocity, C*b = Cb/Co and is the dimensionless bulk or lumen drug concentration, Cb and Co are the bulk and initial drug concentrations, respectively, and z* is the fractional intestinal length and is equal to z/L. Three theoretical cases are considered: (I) Co less than or equal to S, Cm less than or equal to S, (II) Co greater than S, Cm less than or equal to S, and (III) Co greater than S, Cm greater than S, where S is the drug solubility and Cm is the outlet drug concentration. Solving the general steady-state mass balance result for fraction dose absorbed using the mixing tank (MT) and complete radial mixing (CRM) models results in the expressions for the fraction dose absorbed in humans. Two previously published empirical correlations for estimating fraction dose absorbed in humans are discussed and shown to follow as special cases of this theoretical approach. The MMBA is also applied to amoxicillin, a commonly prescribed orally absorbed beta-lactam antibiotic for several doses. The parameters used in the correlation were determined from in situ or in vitro experiments along with a calculated system scaling parameter. The fraction dose absorbed calculated using the MMBA is compared to human amoxicillin pharmacokinetic results from the literature with initial doses approximated to be both above and below its solubility. The results of the MMBA correlation are discussed with respect to the nonpassive absorption mechanism and solubility limitation of amoxicillin. The MMBA is shown to be a fundamental, theoretically based model for estimating fraction dose absorbed in humans from in situ and in vitro parameters from which previously published empirical correlations follow as special cases.

Transport and deposition of indoor radon decay products—I. Model development and validation

Commonly used mathematical models of indoor radon decay product behavior are based on macroscopic mass-balances, often referred to as ‘uniformly-mixed models’. The uniformly-mixed model's applicability is limited by its inability to track the movement of pollutants from their sources to other areas within the enclosure, to permit spatial- or time-dependent sources, or to take proper account of interactions with macroscopic surfaces. Although the uniformly-mixed model parameterizes the deposition process as a constant volumetric removal rate, in reality the deposition process is actually a surface phenomenon and is strongly affected by environmental conditions. This paper describes the development of RADTRAN, a two-dimensional radon progeny transport model that begins with the differential conservation equations describing the motion of air and the transport of reactive pollutants, introduces appropriate boundary conditions to represent surface deposition, and then calculates the concentration distribution of radon progeny throughout the entire region of interest. Knowing the concentration gradient near the surface, a local mass-transfer coefficient (the deposition velocity) can be determined as a function of environmental conditions. RADTRAN simulations have been based on several flow conditions: buoyancy-driven recirculating enclosure flows, free and forced-convection boundary layer flows, and one-dimensional diffusion. Free progeny diffusivity, D f, and attachment rate, X, were varied over representative ranges. For these conditions, RADTRAN calculated free deposition velocities of u f = 0.014–0.079 cm s −1, for 218Po. RADTRAN predictions are compared to a range of experimental measurements. It was found that the predicted range of deposition velocities is in rough agreement with findings from experiments conducted in flow conditions similar to the simplified flows used in RADTRAN.

Entrance flow of Herschel-Bulkley fluids in a duct

The entrance region flow of Herschel-Bulkley fluids in a duct has been investigated numerically without pre-assuming the form of velocity profile within the boundary layer region which is determined by a cross-sectional integration of the momentum differential equation for a given distance from the duct entrance. Using the macroscopic mass and momentum balance equations, the thickness of the core, the entrance length, the plug core velocity and the pressure drop have been obtained at each cross section of the entrance region of the duct for specific values of yield number and flow behavior index.

Phase separation of dispersed mist and dispersed annular (rivulet or thin film) flow in a tee—I: Experiments

An experimental and analytical investigation of dispersed and dispersed annular (rivulet or thin film) flow phase separation in a tee was performed. The primary objective of the experimental portion of the reseach was to obtain data and observations to help formulate and test mechanistically based analytical models of phase separation for these two flow regimes. A variety of measurements were obtained, including: gas velocity profiles; pressure; macroscopic mass balances; streamline and eddy boundary maps; and rivulet, drop and solid particle trajectories.

Evolution of governing mass and momentum balances following an abrupt pressure impact in a porous medium

A mathematical model is developed of an abrupt pressure impact applied to a compressible fluid flowing through a porous medium domain. Nondimensional forms of the macroscopic fluid mass and momentum balance equations yield two new scalar numbers relating storage change to pressure rise. A sequence of four reduced forms of mass and momentum balance equations are shown to be associated with a sequence of four time periods following the onset of a pressure change. At the very first time period, pressure is proven to be distributed uniformly within the affected domain. During the second time interval, the momentum balance equation conforms to a wave form. The behavior during the third time period is governed by the averaged Navier-Stokes equation. After a long time, the fourth period is dominated by a momentum balance similar to Brinkman's equation which may convert to Darcy's equation when friction at the solid-fluid interface dominates.

Use of Referential Coordinates in Deforming Soils

AbstractAnalysis of hydrological processes in deforming soils generally involves use of some form of a referential, or material, coordinate transformation. In the most commonly used of these relationships, there appear two macroscopic soil mass density factors, that associated with the configuration of the soil at a reference time t = 0, and that associated with its configuration at a later (arbitrary) time. In some previous work, the assumption was made that these bulk densities could be evaluated at the same space‐fixed point. In an illustrative application to dual‐energy gamma‐ray data on the swelling of a clay loam during infiltration, this approximation is shown here to predict vertical displacement values that are as much as 7 mm in excess of values obtained without the approximation. Moreover, the approximation fails to satisfy macroscopic mass balance. It is also shown that the direction, downward or upward, of the vertical displacement of macroscopic body‐points depends on the boundary condition used in the integration of the referential coordinate transformation; i.e., depends on whether the integration begins at the soil surface or at a point below the wetting front. Since these two integration pathways are mathematically equivalent, it is argued that a systematic comparison between results obtained using them can serve a useful purpose in detecting occurrences of lateral expansion or systematic errors in the experimental data.