Parabolic Concentration Profile Research Articles

Flattening of Diluted Species Profile via Passive Geometry in a Microfluidic Device.

In recent years, microfluidic devices have become an important tool for use in lab-on-a-chip processes, including drug screening and delivery, bio-chemical reactions, sample preparation and analysis, chemotaxis, and separations. In many such processes, a flat cross-sectional concentration profile with uniform flow velocity across the channel is desired to achieve controlled and precise solute transport. This is often accommodated by the use of electroosmotic flow, however, it is not an ideal for many applications, particularly biomicrofluidics. Meanwhile, pressure-driven systems generally exhibit a parabolic cross-sectional concentration profile through a channel. We draw inspiration from finite element fluid dynamics simulations to design and fabricate a practical solution to achieving a flat solute concentration profile in a two-dimensional (2D) microfluidic channel. The channel possesses geometric features to passively flatten the solute profile before entering the defined region of interest in the microfluidic channel. An obviously flat solute profile across the channel is demonstrated in both simulation and experiment. This technology readily lends itself to many microfluidic applications which require controlled solute transport in pressure driven systems.

A numerical modelling study of SO2 adsorption on activated carbons with new rate equations

Modelling dynamic adsorption of sulfur dioxide (SO2) on activated carbons (ACs) is significant in guiding practical desulphurization processes and making highly efficient use of adsorbents in terms of the adsorption rate which largely depends on particle size. In this work, models derived from the Vermeulen and an improved linear driving force (LDF) rate equation were studied for the first time on SO2 adsorption over AC particles with different sizes. For larger particles (≥3 mm), breakthrough curves predicted by the Vermeulen equation showed good agreement with experimental data, demonstrating that intraparticle diffusion resistance varied with particle size, feed concentration, adsorption time and location. For smaller particles (1 mm), a correction on the volume-averaged adsorption capacity as a function of adsorption time and saturation in the rate equation was developed to avoid the underestimation of adsorption rate due to the inappropriate parabolic concentration profile inherent in the conventional LDF model. By providing a concentration gradient and adsorption rate closer to actual values, the improved LDF equation was confirmed to provide excellent prediction results on 1-mm particles. Different modelling characteristics of the two models indicates varying effects of intraparticle diffusion on adsorption rate with particle size regarding the specificity of SO2 physisorption on ACs.

New Insights into Computational Fluid Dynamic (CFD) Modeling of Concentration Overpotential

Salinity gradient power (SGP) is a promising source of sustainable energy production involving technologies such as pressure retarded osmosis (PRO), reverse electrodialysis (RED) and capacitive mixing (CAPMIX). Coupling these technologies to their counterparts used for desalination, salinity gradient batteries can in principle be developed [1], as recently demonstrated for RED/ED [2]. Electrode and membrane processes typically involve an accumulation or depletion of specific ions compared the bulk solution, resulting in so called concentration polarization. Concentration polarization is a considerable challenge in RED/ED, effectively reducing the driving force across the membrane, hence reducing the performance of the process [3]. In addition to concentration polarization, the presence of concentration gradients will result in both Ohmic and non-Ohmic contributions to the electrical potential [3], further influencing the characteristics of the system. Different strategies have been proposed in order to mitigate the influence of concentration gradients, specifically increasing the flow velocity and introducing flow promoters – typically by optimizing the spacer geometry [3, 4] – in order to alter momentum and concentration boundary layers. The interaction between ionic species and fluid flow can be achieved by solving the coupled Navier-Stokes (NS) and Nernst-Planck (NP) equations [5,6], allowing for simultaneous prediction of flow and pressure fields, concentration and electrical potential, all of which of interest for the overall performance of the system. The current work describes a simulation framework for simulation of the NS-NP system based on the open source CFD platform OpenFOAM [7], aiming to predict the influence of flow conditions and geometry on concentration and electrical potential. Figure 1 (left) shows constant, linear and parabolic concentration profiles, corresponding to the extreme conditions of plug and Poiseuille flow in a flat channel, with the corresponding results for electrical potential in Figure 1 (right), clearly demonstrating the strong coupling between the two. The suggested framework is applied to spacer filled channels aiming to describe pros and cons of different configurations such as height and pitch of spacers. The presented approach sheds light on future concentration polarization modeling, providing better descriptions of the mass transfer and electric potential and highlighting the role of spacer-filled channel in membrane based processes.

Ageing study of different types of long-term pressure tested PE pipes by IR-microscopy

Although the lifetime of plastic pipes is commonly determined by hydrostatic pressure tests which measure the mechanical resistance, thermo-oxidative degradation plays an important role in practice. It is, therefore, an important question to monitor the elemental processes occurring during ageing of polymers, namely the loss of stabilising additives and the degradation of the polymer. The diffusion of the phenolic antioxidant AO-18, the processing stabiliser, and changes in polymer morphology during hydrostatic pressure testing, have been systematically studied for pipes made of PE 80 and PE 100 using infrared microscopy in a quantitative manner. For both materials a parabolic concentration profile of AO-18 develops over the pipe wall upon hydrostatic pressure testing. Under comparable conditions of testing the loss rate was larger for the PE 80 than for the PE 100. An annealing of the polymer occurs upon ageing with a similar increase in degree of crystallinity for both materials. However, the PE 100 has an overall higher level of crystallinity, and it may be speculated that this is one of the reasons for the slower loss of antioxidant. The results from IR-microscopy are in excellent agreement with analyses of samples taken over the pipe wall by oxidative induction time (OIT) and extraction → HPLC. The conversion of the phosphite additive PS-2 to the corresponding phosphate as a result of oxidation has been quantitatively assessed. The total content of phosphate and phosphite was found to be constant, and almost no extraction is seen before 5000 h. GPC analysis of samples taken proves that no significant reduction in average molar mass of the polymer takes place.

Efficiency of osmotic pipe flows

We present experiments and theory for flows of sugar or salt solutions in cylindrical tubes with semipermeable walls (hollow fiber membranes) immersed in water, quantifying the strength of the osmotic driving force in relation to the dimensionless parameters that specify the system. The pumping efficiency of these flows is limited by the presence of "unstirred" concentration boundary layers near the tube walls, and our primary aim is to understand and quantify these layers and their effect on the flow. We measure the outlet flow rate Q(out) while varying the inlet flow rate Q(*), concentration c(*), and tube length L, and map out the dependence of the flow rate gain γ=Q(out)/Q(*)-1 on these parameters. A theoretical analysis based on (1) the known velocity field for slow flow in cylindrical porous tubes and (2) a parabolic concentration profile allows us to compute analytically how the flow gain depends on the relative magnitude of radial diffusion and advection as well as the ratio of the osmotic velocity to pumping velocity, in very good agreement with experiments and with no adjustable parameters. Our analysis provides criteria that are useful for optimizing osmotic flow processes in, e.g., water purification devices.

Experimental and numerical research for fluidization behaviors in a gas-solid acoustic fluidized bed

The effects of sound assistance on fluidization behaviors were systematically investigated in a gas–solid acoustic fluidized bed. A model modified from Syamlal–O'Brien drag model was established. The original solid momentum equation was developed and an acoustic model was also proposed. The radial particle volume fraction, axial root-mean-square of bed pressure drop, granular temperature, and particle velocity in gas–solid acoustic fluidized bed were simulated using computational fluid dynamics (CFD) code Fluent 6.2. The results showed that radial particle volume fraction increased using modified drag model compared with that using the original one. Radial particle volume fraction was revealed as a parabolic concentration profile. Axial particle volume fraction decreased with the increasing bed height. The granular temperature increased with increasing sound pressure level. It showed that simulation values using CFD code Fluent 6.2 were in agreement with the experimental data. © 2009 American Institute of Chemical Engineers AIChE J, 2010

Stabiliser diffusion in long-term pressure tested polypropylene pipes analysed by IR microscopy

Pipes of random polypropylene was (PP-R) were hydrostatic pressure tested and the distribution of the primary stabiliser, Irganox 1010, was measured in the radial direction over the pipe wall by Infrared (IR) microscopy. Parabolic concentration profiles of the stabiliser develop during testing indicating a loss of stabiliser, at both the inner and outer pipe walls. Raising the temperature of the water bath leads to a uniformly accelerated loss of stabiliser. An increase of the hoop stress accelerates the stabiliser migration at the inner pipe wall. The concentrations of Irganox 1010 averaged over the pipe wall as determined by IR microscopy were in excellent agreement with those obtained from High Performance Liquid Chromatography and Oxidation Induction Time (OIT) analysis. Diffusion constants were calculated from the stabiliser concentration profiles based on the Fickian equations using appropriate initial and boundary conditions. A literature value is compared to these experimental results. The developed IR technique allows monitoring the stabiliser migration faster and more reproducibly than the conventional approach by manual abrasion of layers and measurement of the OIT. Additionally, this brings a tremendous improvement with regard to spatial resolution. As a result the impact of hoop stress and extrusion rate on the migration of stabiliser in PP-R pipes can be shown for the first time.

Lattice Boltzmann methods for simulation of micro and macrotransport in a packed bed of porous adsorbents under non-isothermal condition

3D simulation of velocity and concentration profiles was carried out for a packed bed of porous adsorbents under non-isothermal conditions using lattice Boltzmann methods. The model developed in this study incorporates microlevel mass transport within the pores of the solid adsorbent particles and macrolevel transport in the bed voids between the particles. In addition, temperature profiles within the packed bed are calculated taking into account temporal rise in the temperature due to the exothermic heat of adsorption. Inherent multi-scaling capability of lattice Boltzmann technique permits simultaneous ascertaining of the magnitude of concentration profiles within the spherical particles due to the Knudsen diffusivity and that in the bulk gas phase due to combined convective and molecular diffusivity effects. The results in this study have significance, since most of the traditional numerical methods are based on 1D or 2D approximations, including averaging of intra-particle concentration gradients by linear driving force approximations or assuming parabolic concentration profiles within the particles. The model predicted breakthrough curves in this study are validated with the experimental data obtained by tomography technique for porous zeolite particles packed in a tubular adsorber.

BOUNDARY LAYER THEORY DESCRIPTION OF SOLUTE TRANSPORT IN SOIL

Solute transport through soil may affect groundwater and surface water quality. The convection-dispersion equation has been widely used to describe soil solute transport processes. Dispersion coefficient (D) and retardation factor (R) are two important transport parameters in the model. Earlier researchers used boundary layer methods to estimate the soil solute transport parameters based on assumptions of parabolic or cubic polynomial concentration profiles. In this article, boundary layer theory is used to calculate solute concentration distributions and the boundary layer distances, based on assumptions of quartic and quintic concentration profiles. Equations describing soil solute concentration profiles are evaluated by comparing them with an exact solution. Equations for estimating D and R are developed, and the effects of instrument sensitivity on determining D and R are analyzed. The results indicate that all of the boundary layer equations may be used to estimate soil solute transport parameters, but the instrument sensitivity required by each equation is different. Therefore, instrument sensitivity should be evaluated before selecting the boundary layer equation used to estimate D and R.

Further Validation of the Quartic Concentration Profile Approximation for Describing Intraparticle Transport in Cyclic Adsorption Processes

Models developed previously by the authors that describe nonlinear adsorption, and simultaneous pore and surface diffusion in a single particle, that are based on intraparticle quartic and parabolic concentration profile approximations, and that utilize the summation of the gas and adsorbed phases approach in the material balance formulations, were further validated under more diverse, yet more realistic, cycling conditions. Periodic square, sinusoidal and triangular wave functions were used to more accurately represent the periodic boundary conditions that the external surface of an adsorbent particle may be exposed to during repeated adsorption and desorption cycles in a fixed bed adsorber. Analytical solutions that describe the periodic uptake and release of the adsorbate by the adsorbent were obtained for all three periodic wave functions, and for both the quartic and parabolic profile approximations. By comparing the predictions obtained from both models with the exact numerical solution, the superiority of the quartic model over the parabolic model was clearly demonstrated for all wave functions, and for a wide range of adsorbate-adsorbent systems and bulk concentrations. Excellent agreement between the quartic and exact models was obtained in most cases. In general, the predictions improved as the wave function changed more gradually with time (triangular more gradual than sinusoidal and sinusoidal more gradual than square), as the degree of mathematical linearity of the adsorbate-adsorbent system increased, and as the maximum external surface concentration decreased (an isotherm nonlinearity effect). Subtle differences in the predictive ability of the new approximate models, stemming from the use of the different wave functions, were exposed. Overall, these results exemplify the importance of comparing the predictive ability of new approximate models that describe intraparticle transport under more diverse cycling conditions than are typically utilized in the literature, which has been dominated by the square wave function.

Dynamics of foam growth: Bubble growth in a limited amount of liquid

AbstractA mathematical model for the growth of a spherical gas bubble in a limited amount of a viscous liquid is presented. The growth of the bubble is assumed to be controlled by both momentum and mass transfer. The simplified approach suggested, by assuming a parabolic concentration profile for the volatile component in the liquid, results in approximated analytical expressions for the final parameters of the process. The numerical results of the model can be used to predict the increase in the bubble size as well as the decrease of the solute concentration in the liquid and the decrease of the cell density with time. The model is able to deal with real processes, such as polymer melt devolatilization and the production of polymeric foams, where many bubbles grow simultaneously. Polym. Eng. Sci. 44:1900–1906, 2004. © 2004 Society of Plastics Engineers.

New Methodology for the Measurement and Analysis of Adsorption Dynamics: Butane on Activated Carbon

The dynamic uptakes of n-butane on Westvaco BAX-1100 activated carbon were measured over the temperature and pressure ranges of 283−383 K and ∼3−1000 Torr, respectively. The experimental uptakes were used to determine the overall linear driving force (LDF) mass-transfer coefficients and their statistically significant dependences on the operating conditions (temperature and loading). Moreover, the effects of temperature on the order of the parabolic concentration profiles inside the adsorbent particle were determined, and the role of these effects in providing the proper description of the energy-activated mechanism of diffusion was established over the studied range of operating conditions. Overall, it was indicated that the energy barrier for diffusion is independent of temperature and that it increases with increasing adsorbent surface coverage with the adsorbate. Nevertheless, a transient state of diffusion was indicated in which an apparent energy barrier for diffusion appears to depend on temperature.

Modeling breakthrough and elution curves in fixed bed of inert core adsorbents: analytical and approximate solutions

A mathematical model is developed for fixed-beds packed with inert core adsorbents. New analytical solutions to predict breakthrough and elution curves are derived for linear adsorption systems coupled with axial dispersion, film mass transfer and intraparticle mass transfer. New approximate solutions are also obtained based on the assumptions of parabolic concentration profile in the adsorbent shell and the quasi-lognormal distribution for the impulse response in order to predict breakthrough and elution curves. The applicability of these approximate solutions is suggested by comparison with the new analytical solutions. The effects of the size of inert core, sample input mode, axial dispersion, film mass transfer resistance and intraparticle diffusion resistance, on the breakthrough and elution curves are discussed. The decrease of the intraparticle mass transfer resistance by using inert core adsorbents is quantitatively analyzed by introducing the parameter 1/ Θ. Furthermore, an analytical expression for the resolution of two components is derived based on the quasi-lognormal distribution approximate solution; the resolution of two components is improved with the inert core adsorbent when compared with the conventional adsorbent, especially for biomacromolecules where the intraparticle diffusion rate is slow.

Alumina-Doped Silica Gradient-Index (GRIN) Lenses by Slurry-Based Three-Dimensional Printing (S-3DP™)

ABSTRACTThe traditional Slurry-based Three-dimensional Printing (S-3DP™) process has been used to fabricate complex structure materials by printing organic binders in selected positions on each printing layer. This process is modified to fabricate functional graded materials, such as gradient index (GRIN) lenses, by depositing different concentrations of dopant at different positions. The modified S-3DP™ process offers advantages over conventional GRIN lens processes, including reduced processing time, improved compositional flexibility, and increased index profile dimensionality. Two different approximately parabolic dopant concentration profiles, which have maximum alumina concentrations of 1.63 mol% and 2.50 mol%, are printed into silica powder beds using S-3DP™. The samples with maximum alumina concentration of 1.63 mol% have been sintered into optical transparency at 1650 0C for 30 minutes in a vacuum furnace (5×10-6 torr) while an additional dehydration process before sintering was required for the samples with maximum alumina concentration of 2.50 mol%. The magnifying effects of GRIN lenses with profiles of 1.63 mol% and 2.50 mol% alumina were observed, yielding effective focal lengths of 10 cm and 6.1 cm, respectively. Light diffraction, which results from the locally inhomogeneous dopant distribution and reduces the optical quality of GRIN lenses, was also observed.

Measurement of shear-induced dispersion in a dilute emulsion

The time-dependent drop distribution of a dilute, polydisperse emulsion is measured in a simple shear flow. The suspending fluid is much more viscous than the dispersed phase (1:1000). Drops are found to drift away from either bounding wall and accumulate near the center of the gap, due to the anisotropy of droplet–plane interactions. An expression for this drift velocity has been derived for single drops by Chan and Leal [J. Fluid Mech. 92, 131 (1979)] and was in agreement with isolated drop migration observed in our work. Eventually the inward drift is balanced by a shear-induced gradient diffusivity, and a steady-state concentration distribution is reached. When the drops are sufficiently far from either wall a self-similar, parabolic concentration profile is predicted at all times. Droplet diffusivities were determined for capillary numbers Ca=γ̇āμ/σ between 0.17 and 0.92, where γ̇ is the shear rate, ā is the mean drop radius, μ is the viscosity of the suspending fluid, and σ the interfacial tension. The values obtained are an order of magnitude lower than theoretical predictions of Loewenberg made in the limit of small deformation.

New Approximate Model for Nonlinear Adsorption and Concentration Dependent Surface Diffusion in a Single Particle

A new approximate model for nonlinear adsorption (Langmuir model) and concentration dependent surface diffusion (HIO model) in a single particle was derived, based on a parabolic concentration profile assumption for the summation of the gas and adsorbed phases. The surface diffusivity was approximated with the adsorbed phase concentration evaluated at the surface of the particle, as the average of the adsorbed phase concentration, and as the average of the first two approximations. Overall, the approximate model based on the average of the first two approximations compared the best with the exact solution for a wide variety of systems and conditions.

On the use of different parabolic concentration profiles for nonlinear adsorption and diffusion in a single particle

Different parabolic concentration profile assumptions have been used in simplifying pore and/or surface diffusion models that describe mass transport inside a single adsorbent particle. A parabolic concentration pofile assumption for the adsorbed phase has been used to describe solid diffusion (Liaw et al., 1979), and also surface diffusion (Yao and Tien, 1993) inside adsorbent particles. A parabolic concentration profile assumption for the gas phase has also been used to account for pore diffusion within adsorbent particles (Yang and Doong, 1985; Do and Rice, 1986; Yao and Tien, 1993; Lai and Tan, 1991) and more recently Zhang and Ritter (1997) used a parabolic concentration profile assumption for the summation of the gas and adsorbed phases to describe pore and/or surface diffusion inside adsorbent particles. These three different parabolic concentration profile assumptions are the same, only for a linear adsorption isotherm, When the adsorption isotherm is nonlinear, the approximate models of the pore-surface diffusion model obtained by using the parabolic concentration profile assumptions of the gas, adsorbed, or summation of the gas and adsorbed phases inside the adsorbent particle are totally different. Nevertheless, the parabolic concentration profile assumptions for the gas and adsorbed phases have been used with essentially no regard for the validity of such an assumption for a given system. Therefore, the objective of this communication is to present three simplified models of the general adsorption and diffusion equation, each based on one of the parabolic distribution concentration profile assumptions. The utility of these models is demonstrated, along with some rules of thumb that a priori indicate which parabolic concentration profile assumption is more appropriate for a given system.

Parabolic intraparticle concentration profile assumption in modeling and simulation of nonlinear simulated moving-bed separation processes

This communication will focus on the application of the LDF (Linear Driving Force) mass transfer model in the mathematical simulation moving-bed separation processes with nonlinear adsorption isotherms to compare with the linear pore diffusion model, in order to determine its validity as well as to establish the equivalence between the two models.

A solute diffusion model for micro-macroscopic analysis of columnar dendritic alloy solidification

A new solute diffusion model for columnar dendritic alloy solidification based on a two-phase expanding volume element is developed to refine the existing multiphase model. In particular, individual effect of dendrite arm coarsening and back diffusion is incorporated in the model equation separately from each other. This formulation not only leads to successful identification of the behind-the-scene cause of the uncertainty associated with the coarsening inherent in the multiphase model, but also yields more accurate prediction compared with available experimental and numerical data. Taking advantage of the analytical solution for the diffusion-controlled limit, the approximation of the parabolic concentration profile commonly adopted in solidification modelings is also justified legitimately. A sample calculation in the absence of back diffusion clarifies the effect of coarsening on the evolutions of the solid fraction along different cooling paths, in that the eutectic fraction with coarsening is always smaller than that without it, and a rapid cooling during the early stage of dendritic solidification is favorable for homogenizing the final composition of alloy microstructures. In addition to the capability of resolving fundamental solute transport mechanisms, the compactness and consistency retained in the formulation make the present model suitable for use in the coupled micro-macroscopic analysis of dendritic alloy solidification as a microscopic component.

IMPLEMENTATION OF THE WEIGHTED SUM OF GRAY GASES MODEL TO A NARROW BAND: APPLICATION AND VALIDITY

The weighted sum of gray gases model (WSGGM) is applied to narrow bands of water vapor for analyzing radiative heat transfer in nongray media. For a given narrow band, the band mean spectral emissivity is represented by a weighted sum of gray gas emissivities expressed in terms of absorption coefficients, weighting factors, absorbing gas density, and path length. Five types of temperature-versus-narrow band absorption coefficient relation are suggested, and comparisons between the modeled emissivity and that of the original narrow band model are made and shown at a few typical bands. Total and low-resolution spectral intensities at the boundary are obtained for uniform, parabolic, boundary-layer-type temperature and parabolic concentration profiles using the WSGGM with five gray gases. Results using the Curtis*Godson approximation are compared with the WSGGM computations for each narrow band. The results using the WSGGM show good agreement with the Curtis-Godson approximation results, and especially, two of the WSGGMs with wavenum-ber-dependent absorption coefficients show highly accurate results with a few percent error in the total wall heat flux. The WSGGMs with wavenumber-independent absorption coefficients are, however, more useful for total spectrum computation, with a slight sacrifice in the accuracy.