Stefan Tube Research Articles

Analysis of mass transfer in the evaporation-gelation process of polystyrene-benzene solution inside a Stefan tube

The hollow polymer spheres preparation technique of micro-encapsulation requires quantative and precise understandings of the complex evaporation, gelation and solidification processes of the polymer solution. This study focused on the mass transfer during the evaporation-gelation process of a PS-benzene solution inside a Stefan tube. Liquid level reductions of polymer solutions were accurately measured using an image-recognition method with 50μm accuracy. An evaporation model was developed to successfully predict the variation of liquid level, concentration and mass transfer flux, considering solution activity and density as lumped parameters that depend on concentration. This carefully validated evaporation model is expected to be invaluable for the analysis and design of polymer solution evaporation, accommodating various geometries and boundary conditions. Despite the observation of complex processes such as precipitation of polymer molecules and solution-gel phase separation, we characterized and explained them using the effective concentration-density relation. The study showed that reaching a mass fraction of benzene of 0.6 is a significant event in the evaporation process, indicating the end of gelation and the beginning of solidification.

An approach combining the lattice Boltzmann method and Maxwell–Stefan equation for modeling multi-component diffusion

The lattice Boltzmann method is an appropriate mesoscopic-scale tool for investigating the diffusion processes. However, since the state-of-the-art multi-component diffusion lattice Boltzmann (LB) models are based on the kinetic theory and start from the lattice Bhatnagar–Gross–Krook model, some defects cannot be avoided: they are only suitable for steady flow and there are limitations for setting the velocity and viscosity in lattice units. We devise a new incompressible LB model for ideal gases in solid oxide fuel cells (SOFCs), which is based on the advection–diffusion equation and coupled with the Maxwell–Stefan (M–S) equation by relaxation time. The coupled M–S equation is used for correction, considering the driving force in a multi-component diffusion system. Our LB model is implemented to predict the concentration overpotentials of a porous anode in a SOFC. The overpotentials are calculated from an H2–H2O–Ar ternary mass transport simulation and compared to the corresponding experimental results and several published continuum-scale and LB computations, demonstrating that our model offers a better consistency with the experimental measurement. Moreover, a Stefan tube is simulated for benchmarking against the local parameters; this is compared with the related experimental data and demonstrates the accuracy of our LB model.

Diffusion-kinetic model of liquid evaporation from a Stefan tube: A solution to the Stefan diffusion problem

An analytic model that describes the kinetics of the process of liquid evaporation from a vertical tube with an open end (Stefan tube) into the surrounding gas is presented. This model takes into account the intrinsic kinetics of the liquid evaporation process. The case of liquid evaporation from a Stefan tube without consideration of the Stefan flow was examined. In addition, the case of liquid evaporation from a Stefan tube with consideration of the Stefan flow was examined. In the framework of the presented model, it is shown that there are three regimes of liquid evaporation from a Stefan tube: the kinetic, diffusion, and diffusion-kinetic regimes. In the kinetic regime, the rate of liquid evaporation from the Stefan tube is limited only by the intrinsic kinetics of the liquid evaporation process; in the diffusion regime, the rate of liquid evaporation from the Stefan tube is limited only by the intensity of the mass transfer of vapor molecules from the liquid surface to the top of the Stefan tube; and in the diffusion-kinetic regime, the rate of liquid evaporation from the Stefan tube is affected by both the intrinsic kinetics of the liquid evaporation process and the intensity of the mass transfer of vapor molecules from the liquid surface to the top of the Stefan tube. The calculated data obtained in the framework of the presented model are compared with the available experimental data on the kinetics of evaporation of n-pentane, n-hexane, and water from a Stefan tube into nitrogen. As a result of this comparison, the condensation coefficients of n-pentane and water were determined.

A hierarchy of transport models motivated by studies of the Stefan tube

Motivated by studies of the diffusion of a gas mixture in a Stefan tube, we present a hierarchy of mass transfer models with the species momentum balance and the species continuity equations as the basis. The simplifying conditions and the constitutive equations that lead to the equations of the mixture momentum balance approach are obtained. Further, the form that these equations should take for a multicomponent mixture is derived. These equations are shown to be a generalization of the Meyer–Kostin equations used to study binary diffusion in a Stefan tube. Additional simplifications that result in the so called classic approach to diffusion are identified.

Mixed weak-perturbative solution method for Maxwell's equations of diffusion with Müller's partial stress tensor in the low velocity limit

Maxwell's theory of multicomponent diffusion and subsequent extensions are based on systems of mass and momentum conservation equations. The partial stress tensor, which is involved in these equations, is expressed in terms of the gradients of velocity fields by statistical and continuum mechanical methods. We propose a method for the solution of Maxwell's equations of diffusion coupled with Muller's expression for the partial stress tensor. The proposed method consists in a singular perturbation process, followed by a weak (finite element) analysis of the resulting PDE systems. The singularity involved in the obtained equations was treated by a special technique, by which lower-order systems were supplemented by proper combinations of higher-order equations. The method proved particularly efficient for the solution of the Maxwell-Muller system, eventually reducing the number of unknown fields to that of the classical Navier-Stokes/Fick system. It was applied to the classical Stefan tube problem and the Hagen-Poiseuille flow in a hollow-fiber membrane tube. Numerical results for these problems are presented, and compared with the Navier-Stokes/Fick approximation. It is shown that the 0-th order term of the Maxwell-Muller equations differs from a properly formulated Navier-Stokes/Fick system, by a numerically insignificant amount. Numerical results for 1st-order terms indicate a good agreement of the classical approximation (with properly formulated Navier-Stokes and Fick's equations) with the Maxwell-Muller system, in the studied cases. We propose a method for the solution of Maxwell's equations of diffusion with Muller's expression for the stress tensor.The method is particularly efficient for the solution of the Maxwell-Muller system for low Mach numbers.It reduces the number of unknown fields to that of the classical Navier-Stokes/Fick system.It is applied to Stefan tube and Hagen-Poiseuille flow in a hollow-fiber membrane tube.Numerical results indicate good agreement with classical approximation.

Thermogravimetric measurement of evaporation: Data analysis based on the Stefan tube

Commercial thermogravimetric analyzers (TGA) can precisely measure the evaporation of liquids from an open pot. Analysis of this data to determine vapour pressure has often been based on the Langmuir equation for evaporation in a vacuum. These methods are flawed, since they cannot account correctly for the effects of ambient air. We formulate an improved model for evaporation in a TGA, based on the Stefan tube. It incorporates these effects explicitly. We demonstrate its validity by determining accurate vapour pressures for pure liquids, without using a reference sample. Calculated values typically agree with literature data to within a few per cent, over a range of vapour pressures from 60Pa to 30kPa. A weakness of thermogravimetric determination of vapour pressure has been that its accuracy depends on the end correction. Our data analysis avoids this problem. Also, the air flow rate and the end effect are shown to act separately on evaporation. Accurate results depend on correctly accounting for both. Finally, a simple heat balance is used to account for the effect of evaporative cooling.

Multicomponent diffusion in gas mixtures

The known multicomponent diffusion calculation methods are compared. Main attention is focused on matrix methods of describing diffusion, which are well superior to analytical solutions from the standpoint of universality of description and ease of algorithmization of the computational procedure. Expedients used to calculate the elements of the multicomponent diffusion matrix have been analyzed, and an algorithm has been devised for calculating the dynamics of the formation of a concentration profile in a Stefan tube in the diffusion of a multicomponent mixture through an inert gas layer.

Diffusion and performance of fragranced products: Prediction and validation

The design of fragranced products is a combination of art, technology, and scientific knowledge, although the former still predominates in the industry. For that reason, a theoretical model to assess their performance considering the release and diffusion together with the predicted odor intensity over time and distance from the source was developed and validated. Diffusion profiles were experimentally measured in a diffusion tube, similar to the Stefan tube, and predicted with a model for pure fragrance chemicals, binary, quaternary, and multicomponent (11 chemicals) mixtures. A very good agreement between our purely predictive model and experimental concentration data was observed. Fragrance concentrations in air were then converted into odor intensities using models from psychophysics, making it possible to evaluate the evolution of the odor with time and distance using a performance plot. Accordingly, the performance of these mixtures was modeled and experimentally validated, which constitutes a landmark for fragrance design. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3943–3957, 2013

Two-dimensional diffusion in a Stefan tube: The classical approach

Contrary to claims made in the recent literature, we show that the classical approach to mass transport in gas mixtures can be used to model two-dimensional Stefan tube diffusion. Numerical solutions are obtained for water evaporating into air, using the Kramers–Kistemaker diffusion slip velocity as the boundary condition on the side walls. A comparison is made with the analysis of Salcedo-Diaz et al. (2008 Velocity profiles and circulation in Stefan-diffusion. Chem. Eng. Sci. 63, 4685–4693), in which the classical approach is rejected in favor of the Kerkhof–Geboers equations. Difficulties inherent in this latter approach are discussed in detail.

Drying in porous media with gravity-stabilized fronts: Experimental results

In a recent paper [Yiotis et al., Phys. Rev. E 85, 046308 (2012)] we developed a model for the drying of porous media in the presence of gravity. It incorporated effects of corner film flow, internal and external mass transfer, and the effect of gravity. Analytical results were derived when gravity opposes drying and hence leads to a stable percolation drying front. In this paper, we test the theory using laboratory experiments. A series of isothermal drying experiments in glass bead packings saturated with volatile hydrocarbons is conducted. The transparent glass cells containing the packing allow for the visual monitoring of the phase distribution patterns below the surface, including the formation of liquid films, as the gaseous phase invades the pore space, and for the control of the thickness of the diffusive mass boundary layer over the packing. The experimental results agree very well with theory, provided that the latter is generalized to account for the effects of corner roundness in the film region (which was neglected in the theoretical part). We demonstrate the existence of an early constant rate period (CRP), which lasts as long as the films saturate the surface of the packing, and of a subsequent falling rate period (FRP), which begins practically after the detachment of the film tips from the external surface. During the CRP, the process is controlled by diffusion within the stagnant gaseous phase in the upper part of the cells, yielding a Stefan tube problem solution. During the FRP, the process is controlled by diffusion within the packing, with a drying rate inversely proportional to the observed position of the film tips in the cell. Theoretical and experimental results compare favorably for a specific value of the roundness of the films, which is found to be constant and equal to 0.2 for various conditions, and verify the theoretical dependence on the capillary Ca(f), Bond Bo, and Sherwood Sh numbers.

Diffusion coefficients for some organic and organometallic compounds using quartz crystal microbalance

The diffusion coefficient data for organometallic substances are scarce in the literature. A recently developed quartz crystal microbalance (QCM) system has been used to measure the binary diffusion coefficients. The system consists of a QCM with a driving oscillator circuit, a closed Stefan tube, a temperature controlled unit, a data logger, and a frequency counter. The QCM crystal is placed on the top of a Stefan tube and its active surface is coated with a thin layer of the substance. On the other end of the Stefan tube highly adsorptive charcoal powder is placed. Thus a mass concentration gradient is established in the diffusion tube. The rate of mass loss from the QCM was then used to determine the binary diffusion coefficient in air. Diffusion coefficients of polyaromatic hydrocarbons anthracene and phenanthrene and some organometallic compounds (e.g. ferrocene, aluminium acetylacetonate and chromium acetylacetonate) in air at various temperatures are reported.

Specific features of multicomponent diffusion

Using a computational experiment, the existence of the abnormal regimes of nonequimolar diffusion in three-component gas-vapor mixtures such as acetone-methanol-air, hydrogen-water-carbon dioxide, and hydrogen-ammonia-nitrogen is shown. Due to osmotic and reverse diffusion, the molar fraction of water vapors and ammonia can be increased by more than a factor of two in the individual segments of a Stefan tube in comparison with their concentration at the inlet.

Removal of Lead from Fe–Pb Mixtures by Evaporation in Stagnant Argon at 1 448 K and the Determination of the Lead Vapor-Argon Interdiffusivity

Lead, considered as a tramp element in steel, was successfully removed from a 50 % Fe-50 % Pb mixture by evaporation in argon at 1 448 K. The progress of the removal shows that it takes 5 h to completely remove the lead from the mixture, the nonvolatile iron (solid) being left behind in the crucible (alumina). Results point toward a diffusion-controlled evaporation, and the system, which closely simulates Stefan's tube for studying evaporation of a liquid through a stagnant gas film, allowed the measurement of the interdiffusivity of lead and argon (D Pb-Ar ). Since this study dealt with a pseudosteady state diffusion, rather than the steady state one for which the diffusion flux equation is well established, the required pseudosteady state equation was first developed, and, subsequently, utilized with the help of the rate data to find the D Pb-Ar at 1 448 K. The measured value, which is 1.766(+0.378) cm 2 /s, compares very well with the estimated value of 1.774 cm 2 /s.

Velocity profiles and circulation in Stefan-diffusion

Recently, a new multicomponent fluid transport theory was presented by Kerkhof and Geboers [2005a. Towards a unified theory of isotropic molecular transport phenomena. A.I.Ch.E. Journal 51, 79–121], in which the transport is described by simultaneous equations of motion for each species. The theory is applied to water vapor and nitrogen in the Stefan tube as an example of two-dimensional fluid transport. Numerical solution results in axial and radial velocity profiles for both species and clearly indicates circulation of nitrogen. Comparison with the classic Stefan-diffusion equation shows that for wide tubes the latter approach is still reasonable, but for narrow tubes shear effects dominate. The nitrogen circulation causes radial concentration profiles. For small diameter tubes, the axial velocity profiles from the present model are close to those from the velocity profile model of Kerkhof et al. [2001. On the isothermal binary transport in a single pore. Chemical Engineering Journal 83, 107–121], in which radial transport was assumed to be infinitely fast.

Removal of Chlorinated Organic Compounds from Porous Materials by Using Ultrasound

The removal performance of chlorinated organic compound from the porous material under ultrasonic irradiation was experimentally investigated. As the model PCB, o-dichlorobenzene was used. The solvent was hexane or dodecane. Wood chips impregnated with o-dichlorobenzene were used as the sample. The ultrasonic and solvent conditions were changed, and the removal ratio of o-dichlorobenzene from sample was measured. In order to examine the removal behavior of chlorinated organic compound from pore, the Stefan tube injected with o-dichlorobenzene was used as the sample. The pore and solvent conditions were changed, the interface position between o-dichlorobenzene and solvent in Stefan tube was measured and the penetration depth of solvent in Stefan tube was determined.By the ultrasonic irradiation, the removal ratio rapidly increased and became almost unity after 120 minutes for the sample with 10 mm in length. The removal rate for the ultrasonic irradiation was much higher than that for the shaking. The difference in removal ratio between the ultrasonic irradiation and the shaking became larger as the sample was bigger. The removal ratio increased with increasing sound pressure amplitude and became constant beyond a certain sound pressure amplitude. At the fixed sound pressure amplitude, the removal ratio at 28 kHz was nearly equal to that at 45 kHz. The removal ratio hardly depended on the solvent temperature. The removal ratio for degassed solvent was larger than that for non-degassed solvent.The penetration depth in Stefan tube had a maximum value against location of Stefan tube in solvent, but had no significant influence on size and depth of pore in Stefan tube. When the saturation degree of dissolved oxygen in solvent was 0.65, the penetration depth in Stefan tube was largest.

Analysis of the Stefan Tube at Supercritical Conditions and Diffusion Coefficient Measurements

An analysis of the Stefan tube for binary diffusion coefficient measurement of a system comprised of a solid solute and a supercritical fluid has been carried out. The prototype system used to evaluate the approach is naphthalene−supercritical carbon dioxide. An initial theoretical calculation with the available data shows that the diffusion process would be in the unsteady-state region for a practical duration of the experiment. The development of a theoretical model was carried out for systems where the transport of the species is dominated only by diffusion as well as for systems involving convective transport along with diffusion. Tractable analytical solutions are developed for finite control volume, relaxing the semi-unbounded boundary assumption often used. The binary diffusion coefficients were determined at temperatures of 35, 50, and 65 °C for the pressures 80, 100, and 120 atm. The quantification of the convective term showed little effect on the diffusion coefficient for the supercritical naph...

Vapor transport within the thermal diffusion cloud chamber

A review of two different, one-dimensional models of the vapor transport within the thermal diffusion cloud chamber (TDCC) is presented. In one case the assumption is made that there are no convective fluxes within the chamber and that heat and mass transport occur by diffusion only. Although in this model there are no restrictions on the transport of the two components within the chamber, the assumption of no velocities within the chamber results in an incorrect flux boundary condition for the background, carrier gas. The second model is based on the typical, stagnant background gas assumption and the equations of this model closely follow those of the classical Stefan tube problem in which there is transport of a volatile species through a noncondensible, carrier gas. Unfortunately, this model of the TDCC also suffers from the same inconsistencies as noted by several researchers for the Stefan tube. When the convective contributions to the flux are low in the stagnant background gas model, the two models give reasonably close results. For more convective situations, the supersaturation results can differ by more than 50%. One interesting feature of the zero velocity model is that it predicts a change in the supersaturation profile with pressure, whereas no pressure dependence is predicted with the stagnant background gas model. Unfortunately, the direction of this pressure change is opposite to that seen in experimental observations.

New Light on Some Old Problems: Revisiting the Stefan Tube, Graham's Law, and the Bosan-quet Equation.

ADVERTISEMENT RETURN TO ISSUEPREVAddition/CorrectionNEXTORIGINAL ARTICLEThis notice is a correctionNew Light on Some Old Problems: Revisiting the Stefan Tube, Graham's Law, and the Bosan-quet Equation.Piet J. A. M. KerkhofCite this: Ind. Eng. Chem. Res. 2000, 39, 1, 243Publication Date (Web):December 3, 1999Publication History Published online3 December 1999Published inissue 1 January 2000https://doi.org/10.1021/ie9910625Copyright © 2000 American Chemical SocietyRIGHTS & PERMISSIONSArticle Views264Altmetric-Citations-LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (20 KB) Get e-Alerts Get e-Alerts

New Light on Some Old Problems: Revisiting the Stefan Tube, Graham's Law, and the Bosanquet Equation

The Stefan diffusion tube is modeled with the recently developed binary friction model (BFM), and it is shown that this model can describe all kinds of vapor flow regimes from Stefan diffusion to pure pressure-driven flow but also Knudsen and (diffusive) slip flow. From the BFM simple criteria are derived for the transition between various regions, the criteria of which are also relevant to the fields of gas adsorption, heterogeneous catalysis, and drying of porous materials. For isobaric counterdiffusion the BFM shows where the limits of Graham's law are. For equimolar counterdiffusion analogously the area of applicability of the Bosanquet equation is derived from the BFM and an extended equation is given.

Novel technique for investigating the temperature effect on the diffusion coefficient of naphthalene into air

This article describes a technique that uses a piezoelectric quartz crystal microbalance (QCM) for determining the diffusion coefficient of naphthalene into air. The QCM is placed onto the open top of a closed Stefan tube and its active surface is covered with a thin layer of solid naphthalene. Due to the markedly enhanced QCM resolution of 10−9 g/cm2 over that of the conventional digital electronic balance, the QCM in this study requires much less time than previous studies using digital electronic balances for measuring the diffusion coefficient in a binary gas system. At P=0.1013 MPa and in a temperature range from 278.25 to 315.15 K, the empirical correlations to evaluate both the diffusion coefficient of naphthalene into air at 1 atm pressure and the Schmidt number of naphthalene at various temperatures are presented. The measurement uncertainty of the diffusion coefficient of naphthalene into air with the present system is less than 3%. In addition, measured results verify that the measurement of the diffusion coefficient of naphthalene into air is only slightly affected by the length of the diffusion tube if the QCM is used in the closed Stefan tube for measuring the mass flux in the diffusion tube.