Macroscopic Balance Research Articles

On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients

This study develops a general theory for small-deformation viscoplasticity based on a system of microforces consistent with its own balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that allows for dependences on plastic strain-gradients. The microforce balance and the constitutive equations—suitably restricted by the second law—are shown to be together equivalent to a flow rule that accounts for variations in free energy due to flow. When this energy is the sum of an elastic strain energy and a defect energy quadratic, isotropic, and positive definite in the plastic-strain gradients, the flow rule takes the form of a second-order parabolic PDE for the plastic strain coupled to the usual PDE arising from the standard macroscopic force balance and the elastic stress-strain relation. The classical macroscopic boundary conditions are supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a weak (virtual power) formulation of the nonlocal flow rule is derived.

Implications of Alternative Macroscopic Descriptions Illustrated by General Balance and Continuity Equations

This article demonstrates the outstanding challenges of the application of well-established averaging procedures and rules, and shows that the results are not universal and alternative forms differ by inherent realization, difficulties, and conveniences for implementation in practical analyses of processes in porous media. Two alternative forms of the general macroscopic balance equation are derived by means of the representative elementary volume and mass-weighted-volume averaging rules and illustrated for the equation of continuity. Differences in the resulting mathematical expressions and their implications for practical problems are delineated. The macroscopic equation of continuity derived by combining the volume and mass-weighted-volume averaging rules may be convenient for conforming to the mathematical form of the microscopic equation of continuity. Using only the volume averaging rules yields an additional term compared to the conventional microscopic equation of continuity, which is referred to as transport by hydraulic dispersion due to convective mixing and spreading in porous media and contains a hydraulic dispersion parameter. An equation for empirical determination of the hydraulic dispersion parameter is also derived. Both averaging approaches are shown to be equally applicable when used with proper designations of various quantities as being the volume or mass-weighted-volume averaged. The macroscopic equation of continuity using only the volume-averaged quantities and conforming to the mathematical form of the microscopic equation of continuity is proven to involve an inherent error.

A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations

This study develops a gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations. The theory is based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems. The field equations consist of the yield conditions coupled to the standard macroscopic force balance; these are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. As an aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. To make contact with classical dislocation theory, the microstresses are shown to represent counterparts of the Peach–Koehler force on a single dislocation.

Photocatalytic Conversion of Organic Pollutants Extinction Coefficients and Quantum Efficiencies

This study describes an experimental method for the evaluation of the rate of absorbed photons in an aqueous TiO 2 slurry reactor. This method and the use of a macroscopic radiation balance allow for the estimation of extinction coefficients of the dispersed medium. With this end, the effects of the radiation wavelength on the extinction coefficients of TiO 2 dispersions are established. The photoconversion of phenol as a model pollutant allows the rate of photodegradation and the effect of the catalyst concentration on this rate to be assessed. In addition, the effects of several other operating parameters on both the phenol photodegradation and the quantum efficiency are studied in detail. It is shown that there is good agreement between parameters estimated using the proposed reaction rate model and data from photoconversion and light transmission experiments.

CURRENT AND HEAT FLUX IN A DEGENERATE PLASMA

Looking at the model for a degenerate electron gas in metals at very low temperature, we studied the macroscopic balance equations for current density and heat flux and we analyzed the effects produced on some transport coefficients if the quantum aspects are taken into account. The corrections for diffusion coefficients, electrical conductivity, etc., are explicitly carried out for a quite general collision model and the relations between these transport coefficients are pointed out.

Non‐parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices

AbstractA consistent hydrodynamical model for electron transport in silicon semiconductors, free of any fitting parameter, has been formulated in Anile and Romano (Continuum Mechanics Thermodynamics 1999; 11:307–325) and Romano (Continuum Mechanics Thermodynamics 1999; 12:31–51) on the basis of the maximum entropy principle, by considering the energy band described by the Kane dispersion relation. Explicit constitutive functions for fluxes and production terms in the macroscopic balance equations of density, crystal momentum, energy and energy flux have been obtained. Scatterings of electrons with non‐polar optical phonons (both for intervalley and intravalley interactions), acoustic phonons and impurities have been taken into account.In this article we show the link with other macroscopic models describing the motion of charge carriers. In particular, under suitable scaling assumptions, an energy transport model is recovered. An analysis of the formal properties is given by showing that the evolution equations form a hyperbolic system in the physically relevant region of the space of the dependent variables. At last, by using the numerical method developed in Liotta et al. (International Series of Numerical Mathematics 1999; 130:651–660) and Liotta et al. (SIAM Journal on Numerical Analysis 1999, to appear) simulations for bulk silicon and n+–n–n+ silicon diode are performed. The obtained results are in good agreement with the Monte Carlo data. Copyright © 2001 John Wiley & Sons, Ltd.

Transient gas–liquid flow in horizontal T-junctions

A model is developed to calculate the transient behavior of co-current gas–liquid flow with small liquid holdup values ( ε L ⩽0.1) through a horizontal regular T-junction. The present model combines transient two-phase pipe flow and gas–liquid flow splitting in a T-junction. The transient model has been derived from the mass and momentum balances for the gas and liquid phase. The splitting model has been derived from the steady-state macroscopic mechanical energy balance (extended Bernoulli equation) applied to the “inlet-to-run” streamline and “inlet-to-branch” streamline of both the gas and the liquid phase. The calculated results with the introduced transient model are compared with experimental data of the transient flow of the system air/water through a horizontal regular (D 1=D 2=D 3=0.051 m) T-junction. The calculated steady-state values as well as the time-dependent dynamic changes of gas–liquid flow through a horizontal regular T-junction are in good agreement with the on-line registered experimental data.

Modeling the vertical motion of a soft contact lens

Purpose. The motion of a soft contact lens in the up-down or vertical (inferior-superior) and in-out (anterior-posterior) directions drives mixing in the post lens tear film. Thus, it is important to obtain an accurate assessment of lens motion. The commonly used experimental technique to measure the vertical motion, video microscopy, only gauges motion during the interblink period. Since most of the eyelid force, which drives the motion, is exerted during the blink, it is reasonable to assume that the majority of the lens motion in the up-down direction takes place during the blink and is, therefore, hidden from view. Thus, experimentally measured values of vertical lens travel are currently underpredicted. Methods. In this paper, we use a simple mechanical force balance on the lens to predict its motion in the vertical direction. The forces included in our model are due to the upper and the lower eyelids, gravity, elasticity, and viscous stresses. We input the lens physical parameters, the various tear film thicknesses, and the upper eyelid velocity. Then we integrate a macroscopic force balance to obtain the lens vertical position as a function of time. Results. The proposed model predicts that the downward lens motion during a blink is about 2–3 times the downward motion during the interblink (centration). Conclusions. The up-down motion observable during the interblink period is only a small fraction of the total lens vertical travel.

Equilibrium Force Isotherms of a Deformable Bubble/Drop Interacting with a Solid Particle across a Thin Liquid Film

Comparison of experimental force−distance isotherms obtained for a deformable drop/bubble and a solid particle interacting across a thin liquid film with colloidal theories, for example, the DLVO theory, currently relies on two incompatible and nonphysical assumptions. These are approximating the bubble/drop as a purely elastic solid and, contradictorily, as a nondeformable solid. To avoid these aphysical assumptions, we have developed a more rigorous method for interpreting the experimental force−distance isotherms. Equilibrium shapes of the bubbles/drops are calculated from the augmented Young−Laplace equation pertinent to the relevant geometry of the experiments. The overall bubble/drop−solid interaction force is then calculated using a macroscopic force balance. We find that the nondeformable and the elastic-deformation models for the bubble/drop are rather inaccurate. The developed methodology is of particular relevance to interparticle force measurements using an atomic force microscope (AFM). To th...

Mesoscopic dynamics of microcracks

The mesoscopic concept is applied to the description of microcracks. The balance equations of the cracked continuum result in mesoscopic directional balances of mass, momentum, angular momentum, and energy. Averaging over the length of the cracks gives the corresponding orientational balances. A further averaging process leads to the macroscopic balance equations of the microcracked continua. Dynamic equations for the fabric tensors of different order are derived using a multipole moment expansion of the orientational crack distribution function. The simple example of Griffith cracks is treated. The role of physical assumptions in the microcrack representations and the different macroscopic internal variable representations of microcracks are discussed.

Microscopic discussions of macroscopic balance equations for solids based on atomic configurations

Macroscopic behavior of solids is widely studied from microscopic viewpoints such as molecular dynamics and lattice dynamics. However, there are some difficulties with the averaging methods in the microscopic expressions of stress, higher-order stress and heat flux. In this study, we discuss the microscopic expressions of macroscopic variables and macroscopic balance equations of a solid that is modeled as an assembly of atoms, on the basis of generalized Cosserat continuum theories. The concept of mesodomain is introduced to relate microscopic quantities of atoms to macroscopic quantities of continua. Microscopic expressions of stresses and heat flux are described as area averages of microscopic quantities such as velocities of atoms, interatomic potential forces. Balance equations for stress and higher-order stresses are derived from the equations of atomic motion. The energy equation, represented by the averages for the values in the mesodomain, is obtained by dividing the velocity of atoms into macroscopic motion and thermal motion. Since the generalized polar effects are taken into account in this model, derived macroscopic balance equations have the same form as the equations of generalized Cosserat continua. Moreover, microscopic descriptions obtained here show that the higher-order mechanical power of the generalized Cosserat continua is equivalent to the thermodynamic quantities of simple materials. The values of velocities of atoms calculated by molecular dynamics simulation are substituted into the newly obtained microscopic expressions of stress and higher-order stresses. The obtained values of stresses and the theoretical values derived from Eringen's formula correspond well, which demonstrates the usefulness of the present microscopic expressions for the practical application in computer simulations.

Microforces and the theory of solute transport

A generalized continuum framework for the theory of solute transport in fluids is proposed and systematically developed. This framework rests on the introduction of a generic force balance for the solute, a balance distinct from the macroscopic momentum balance associated with the mixture. Special forms of such a force balance have been proposed and used going back at least as far as Nernst's 1888 theory of diffusion. Under certain circumstances, this force balance yields a Fickian constitutive relation for the diffusive solute flux, and, in conjunction with the solute mass balance, provides a generalized Smoluchowski equation for the mass fraction. Our format furnishes a systematic procedure for generalizing convection-diffusion models of solute transport, allowing for constitutive nonlinearities, external forces acting on the diffusing constituents, and coupling between convection and diffusion.

Solid substrate fermentation of Monascus purpureus: growth, carbon balance, and consistency analysis.

Solid substrate fermentation (SSF) of Monascus purpureus on rice is a promising new technology for obtaining natural pigments. However, before attempts can be made at maximizing pigment yield, all significant macroscopic compounds should be assayed. Here, Monascus purpureus has been grown on rice in batch mode, and the evolution of the main components, biomass, residual rice, O(2), CO(2), ethanol, acetic acid, and pigments, have been followed. This set of data, never previously studied for Monascus SSF, allowed both the performance of a macroscopic elemental balance, which accounted for 83-94% of the initial substrate carbon, and a check of data consistency. Standard consistency analysis showed a significant underestimation of the nitrogen fraction of biomass, but it was unable to discriminate the errors in the carbon balance as a result of the simultaneous presence of two gross errors in the system. A simple stoichiometric model in tandem with consistency analysis explained unaccounted carbon as an underestimation of CO(2) and ethanol. Using the simplified method to estimate ethanol, the macroscopic balance accounted for 87-99% of the initial carbon.

Solid Substrate Fermentation of Monascus purpureus: Growth, Carbon Balance, and Consistency Analysis

AbstractSolid Substrate fermentation (SSF) of Monascus purpureus on rice is a promising new technology for obtaining natural pigments. However, before attempts can be made at maximizing pigment yield, all significant macroscopic compounds should be assayed. Here, Monascus purpureus has been grown on rice in batch mode, and the evolution of the main components, biomass, residual rice, O2, CO2, ethanol, acetic acid, and pigments, have been followed. This set of data, never previously studied for Monascus SSF, allowed both the performance of a macroscopic elemental balance, which accounted for 83–94% of the initial substrate carbon, and a check of data consistency. Standard consistency analysis showed a significant underestimation of the nitrogen fraction of biomass, but it was unable to discriminate the errors in the carbon balance as a result of the simultaneous presence of two gross errors in the system. A simple stoichiometric model in tandem with consistency analysis explained unaccounted carbon as an underestimation of CO2 and ethanol. Using the simplified method to estimate ethanol, the macroscopic balance accounted for 87–99% of the initial carbon.

Nonlinear control of an evaporator using an error trajectory technique

This paper outlines the successful application of an adaptive control method in an industrial environment. The control strategy incorporated MRAC (Model Reference Adaptive Control) using an adaptive PID where the adaptive parameter was estimated using a macroscopic energy balance over the control zone based on the concept of a tracking error trajectory. Success is demonstrated in a plant for sugar production where the formation of candy, due to incorrect management, the steam flow to the evaporators prevented the sugar syrups leaving the system to pass to the centrifugal screens. This stopped production three or four times a day for 30min, leading to serious problems in relation to cost and quality, the evaporators being very sensitive to temperature changes.

Kinetic analysis of the vacuum membrane distillation of chloroform from aqueous solutions

In this work, the vacuum membrane distillation (VMD) technology has been applied to the removal of chloroform from a dilute aqueous stream. Microporous polypropylene hollow fiber membranes were used to immobilize the liquid–vapor interface. The influence of the following operational variables was experimentally studied: (i) initial chloroform concentration in the feed (212–2012 mg/l), (ii) feed flow rate in the laminar flow regime (0.23–0.98 l/min) and in the transition to the turbulent flow regime (2.7–8 l/min), (iii) temperature (5–47.5°C) and vacuum pressure in the permeate side (7 and 14 mm Hg). The mathematical model proposed to describe the kinetic results includes the mass balance to the feed tank and the mass balance to the VMD module. In the laminar regime the solution of the continuity mass conservation equation makes the diffusion coefficient of chloroform in the aqueous phase the only parameter needed to describe the separation rates. In the turbulent flow regime, a macroscopic mass balance including an overall mass transfer coefficient that accounts both for liquid film diffusion and membrane mass transport was developed. It was found that only at higher Reynolds numbers within the turbulent flow regime the resistance to mass transfer in the membrane had influence on the overall mass transfer coefficient.

Reaction set simplification using variable selection techniques

A novel approach to reaction set reduction is presented which utilizes variable selection (VS) techniques from the statistics literature to identify a family of reduced reaction sets. By taking advantage of the structure of elementary reaction sets and the nature of macroscopic reactor balances, a linear regression model of the complete reaction set is developed from a sampling of data points along the reactor concentration and reaction-rate profiles. Utilizing a modification of the Leaps and Bounds algorithm (Furnival and Wilson, 1974, Techometrics, 16 (4), 499–511), the best reduced linear models for each parameter subset size are identified, and these solutions are used as approximations of the global solutions of the reaction set reduction problem that would be found by complete enumeration of all reduced reaction sets. With the use of weighted least squares, the procedure can be directed to preferentially select reduced reaction sets that accurately represent the full reaction set kinetics for specific species while downplaying the importance of other species. The proposed VS procedure offers significant computational savings over complete enumeration by avoiding the repeated integration of differential equations for reduced reaction sets. The VS procedure is tested on an 18 reaction 10 species reaction set for a batch reactor, and results compare favorably with those found using complete enumeration.

Analysis of bioreactor experimental data by the application of metabolic pathway stoichiometry to polyhydroxyalkanoate production by Alcaligenes Eutrophus

In biochemical processes, the stoichiometry can be the result of macroscopic balances (where the microorganism is specified by the elementary composition and only the chemical reactions of the conversion process are considered) or the balances of the metabolic pathway, where biochemical knowledge available for metabolic reactions, in addition to the chemical characteristics of the system, are considered. It is possible to identify several linear relationships among the conversion rates in these processes. While several rates are measured, others can be calculated. If a calculated conversion rate is also measured, measurement errors or errors of the described model can be detected or even diagnosed by comparing these values, and accurate estimates can be obtained by combining them.

Derivation of Forchheimer terms and their verification by application to waves propagation in porous media

Forchheimer terms are developed for the macroscopic momentum and energy balance equations considering saturated thermoelastic porous media, and for the macroscopic momentum balance equations in the case of multiphase porous media. It is shown that these terms represent the exchange between the interacting phases at their common interface. Using these Forchheimer terms, a very good agreement was evident when the 1-D numerical simulation of the propagation of compaction waves in a saturated thermoelastic porous medium was compared with shock tube experiments.

Controlled Firing of Reaction‐Bonded Aluminum Oxide (RBAO) Ceramics: Part I, Continuum‐Model Predictions

The reaction‐bonded aluminum oxide (RBAO) process is a novel, reaction‐forming technique for producing monolithic, alumina‐based ceramics. Although there has been extensive work on the RBAO process, it is often difficult to reproduce the process and avoid sample cracking. To solve the problems that are associated with the RBAO process, it is necessary to have a fundamental understanding of the reaction‐bonding process and the effects of various processing parameters on the reaction behavior. To gain some insight into the process, a continuum model has been developed. The model, which considers the interaction between the macroscopic material and energy balances, is used to predict conditions under which RBAO bodies may be fired in a controlled manner, i.e., avoiding the runaway reaction. In particular, the effects of the oxygen content of the atmosphere, the heat loss by convection and radiation, the heating cycle, and scale (sample size) have been investigated. For small sample sizes, model predictions indicate that the reaction may be controlled by reducing the oxygen content of the atmosphere, increasing the heat loss, and/or incorporating an isothermal hold into the heating cycle at a temperature just below the ignition temperature. For larger sample sizes, model predictions indicate the need for multiple low‐temperature holds at increasing temperatures. It is believed that firing RBAO bodies in a controlled manner will allow one to avoid sample cracking. Part II of this work presents a complementary experimental study that investigates the reaction behavior and structural integrity of samples that have been fired under the predicted conditions.