Incorporating Darcy's law for pure solvent flow through porous tubes: Asymptotic solution and numerical simulations

A generalized solution for pressure‐driven, incompressible, Newtonian flow in a porous tubular membrane is challenging due to the coupling between the transmembrane pressure and velocity. To date, all analytical solutions require simplifications such as neglecting the coupling between the transmembrane pressure and velocity, assuming the form of the velocity fields, or expanding in powers of parameters involving the tube length. Moreover, previous solutions have not been validated with comparison to direct numerical simulation (DNS). We comprehensively revisit the problem to present a robust analytical solution incorporating Darcy's law on the membrane. We make no assumptions about the tube length or form of the velocity fields. The analytic solution is validated with detailed comparison to DNSs, including cases of axial flow exhaustion and cross flow reversal. We explore the validity of typical assumptions used in modeling porous tube flow and present a solution for porous channels in Supporting Information. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

Spontaneous imbibition of liquids in glass‐fiber wicks. Part I: Usefulness of a sharp‐front approach

Spontaneous imbibition of a liquid into glass‐fiber wicks is modeled using the single‐phase Darcy's law after assuming a sharp flow‐front marked by full saturation behind the front occurring in a transversely isotropic porous medium. An analytical expression for the height of the wicking flow‐front as a function of time is tested through comprehensive experiments involving using eight different wicks and one oil as the wicking liquid. A good fit with experimental data is obtained without using any fitting parameter. The contact‐angle is observed to be important for the success of the model—lower contact angle cases marked by higher capillary pressures were predicted the best. The proposed model provides a nice upper bound for all the wicks, thereby establishing its potential as a good tool to predict liquid absorption in glass‐fiber wicks. However, the sharp‐front model is unable to explain region of partial saturation, thereby necessitating the development of part II of this article series (Zarandi and Pillai, Spontaneous Imbibition of Liquid in Glass fiber wicks. Part II: Validation of a Diffuse‐Front Model. AIChE J, 64: 306–315, 2018) using Richard's equation. © 2017 American Institute of Chemical Engineers AIChE J, 63: 294–305, 2018     

Darcy's law‐based model for wicking in paper‐like swelling porous media

The wicking of liquid into a paper‐like swelling porous medium made from cellulose and superabsorbent fibers was modeled using Darcy's law. The work is built on a previous study in which the Washburn equation, modified to account for swelling, was used to predict wicking in a composite of cellulose and superabsorbent fibers. In a new wicking model proposed here, Darcy's law for flow in porous media is coupled with the mass conservation equation containing an added sink or source term to account for matrix swelling and liquid absorption. The wicking‐rate predicted by the new model compares well with the previous experimental data, as well as the modified Washburn equation predictions. The effectiveness of various permeability models used with the new wicking model is also investigated. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

On mixing rules for the hydraulic permeability of fiber mixtures

We perform numerical simulations of Stokes flow through ordered arrays of fibers of square cross-section. The results for the permeability of single-size-fiber arrays are in very good agreement with the well-known correlation of Jackson and James, provided that an effective radius of equivalent cylindrical fibers is used with the same cross-sectional area. Then, we consider arrays of mixtures of different-sized fibers and test several mixing rules, proposed for estimating the composite permeability. The two most well-known, namely, the unweighted- and the volume-averaged resistivity rules give good predictions for not too large size-ratios, yet they systematically overestimate and underestimate the numerical results, respectively. Then, it appears almost self-evident that a combination of the above two mixing rules would provide a better prediction. Indeed, the geometric mean gives excellent agreement with the numerical results over an extended range of size ratios and relative volume fractions.     

Model for the outer cavity of a dual‐cavity die with parameters determined by two‐dimensional finite‐element analysis

A coating die forms liquid layers of uniform thickness for application to a substrate. In a dual‐cavity coating die an outer cavity and slot improves flow distribution from an inner cavity and slot. A model for axial flow in the outer cavity must consider the ever‐present cross flow. A 1‐D equation for the pressure gradient for a power‐law liquid is obtained as a small departure from a uniform flow distribution and no axial flow. The equation contains a shape factor dependent on cavity shape, Reynolds number, and power‐law index. The shape factor for five triangular cavity shapes is obtained by finite‐element analysis and correlated for application to die design up to the onset of flow recirculation which arises at the junction of the cavity and outer slot. The performance of the combined cavity and slot is considered and the most effective design determined. © 2017 American Institute of Chemical Engineers AIChE J, 64: 708–716, 2018     

Filtered two‐fluid models of fluidized gas‐particle flows: New constitutive relations

New constitutive relations for filtered two‐fluid models (TFM) of gas‐particle flows are obtained by systematically filtering results generated through highly resolved simulations of a kinetic theory‐based TFM. It was found in our earlier studies that the residual correlations appearing in the filtered TFM equations depended principally on the filter size and filtered particle volume fraction. Closer inspection of a large amount of computational data gathered in this study reveals an additional, systematic dependence of the correction to the drag coefficient on the filtered slip velocity, which serves as a marker for the extent of subfilter‐scale inhomogeneity. Furthermore, the residual correlations for the momentum fluxes in the gas and particle phases arising from the subfilter‐scale fluctuations are found to be modeled nicely using constitutive relations of the form used in large‐eddy simulations of single‐phase turbulent flows. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3265–3275, 2013     

Dynamic effects on capillary pressure–Saturation relationships for two‐phase porous flow: Implications of temperature

Work carried out in the last decade or so suggests that the simulators for multiphase flow in porous media should include an additional term, namely a dynamic coefficient, as a measure of the dynamic effect associated with capillary pressure. In this work, we examine the dependence of the dynamic coefficient on temperature by carrying out quasi‐static and dynamic flow simulations for an immiscible perchloroethylene–water system. Simulations have been carried out using a two‐phase porous media flow simulator for a range of temperatures between 20 and 80^°C. Simulation domains represent 3‐D cylindrical setups used by the authors for laboratory‐scale investigations of dynamic effects in two‐phase flow. Results are presented for two different porous domains, namely the coarse and fine sands, which are then interpreted by examining the correlations between dynamic coefficient(s) and temperature, time period(s) required for attaining irreducible water saturation, and the dynamic aqueous/nonaqueous phase saturation and capillary pressure plots. The simulations presented here maintain continuity from our previous work and address the uncertainties associated with the dependency of dynamic coefficient(s) on temperature, thereby complementing the existing database for the characterization of dynamic coefficients and subsequently enabling the users to carry out computationally economical and reliable modeling studies. © 2012 The Authors. AIChE Journal, published by Wiley on behalf of the AIChE. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. AIChE J, 58: 1951–1965, 2012     

Wicking and evaporation of liquids in porous wicks: A simple analytical approach to optimization of wick design

Wicking and evaporation of volatile liquids in porous, cylindrical wicks is investigated where the goal is to model, using simple analytical expressions, the effects of variation in geometrical parameters of a wick, such as porosity, height and bead‐size, on the wicking and evaporation processes, and find optimum design conditions. An analytical sharp‐front flow model involving the single‐phase Darcy's law is combined with analytical expressions for the capillary suction pressure and wick permeability to yield a novel analytical approach for optimizing wick parameters. First, the optimum bead‐radius and porosity maximizing the wicking flow‐rate are estimated. Later, after combining the wicking model with evaporation from the wick‐top, the allowable ranges of bead‐radius, height and porosity for ensuring full saturation of the wick are calculated. The analytical results are demonstrated using some highly volatile alkanes in a polycarbonate sintered wick. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1930–1940, 2014     

Radial diffusion model for fragrance materials: Prediction and validation

A predictive model based on Fick's second law for radial diffusion is proposed and validated for modeling the diffusion of fragrance materials. A pure component, two binary systems, and a ternary system were used for validation. The model combines the prediction model to represent the liquid phase nonidealities, using the UNIFAC group contribution method, with the Fickian radial diffusion approach. The experimental headspace concentrations were measured in a diffusion chamber using the solid-phase microextraction technique and quantified using gas chromatography with a flame ionization detector. The numerical solutions were obtained along with an analytical model considering constant surface concentration. The odor intensities of the studied systems were calculated using Stevens' power law and the strongest component model, respectively. The numerical simulation presented good adherence to the experimental gas concentration data. The proposed methodology is an efficient and validated tool to assess the radial diffusion of fragrance and volatile systems.     

Spontaneous imbibition of liquid in glass fiber wicks,Part II: Validation of a diffuse‐front model

In Part I (Zarandi MAF, Pillai KM. Spontaneous Imbibition of Liquids in Glass‐Fiber Wicks. Part I: Usefulness of a Sharp‐Front Approach. AIChE J, 64: 294–305, 2018), a model based on sharp liquid‐front was proposed where a good match with the experimental data was achieved. However, the model failed to account for partial saturations in the wicks. Here, Richard's equation to predict liquid saturation is tried where the equation is solved numerically in 3D using COMSOL and analytically in 1D using Mathematica for glass‐fiber wicks after treating them as transversely‐isotropic porous media. As a novel contribution, relative permeability and capillary pressure are determined directly from pore‐scale simulations in wick microstructure using the state‐of‐the‐art software GeoDict. The saturation along the wick length is determined experimentally through a new liquid‐N₂ based freezing technique. After including the gravity effect, good agreements between the numerical/analytical predictions and experimental results are achieved in saturation distributions. We also validated the Richard's equation based model while predicting absorbed liquid‐mass into the wick as function of time. © 2017 American Institute of Chemical Engineers AIChE J, 63: 306–315, 2018     

Simulation,model‐reduction,and state estimation of a two‐component coagulation process

The issue of state estimation of an aggregation process through (1) using model reduction to obtain a tractable approximation of the governing dynamics and (2) designing a fast moving‐horizon estimator for the reduced‐order model is addressed. The method of moments is first used to reduce the governing integro‐differential equation down to a nonlinear ordinary differential equation. This reduced‐order model is then simulated for both batch and continuous processes and the results are shown to agree with constant Number Monte Carlo simulation results of the original model. Next, the states of the reduced order model are estimated in a moving horizon estimation approach. For this purpose, Carleman linearization is first employed and the nonlinear system is represented in a bilinear form. This representation lessens the computation burden of the estimation problem by allowing for analytical solution of the state variables as well as sensitivities with respect to decision variables. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1557–1567, 2016     

Investigating liquid fronts during spontaneous imbibition of liquids in industrial wicks. Part I: Experimental studies

A large set of industrial wicks made from sintered polymer beads and fibrous materials were studied. Three types of liquid fronts were observed: sharp, semi-sharp, and diffuse. The bead-like microstructure with lower porosity is found to support sharp fronts, while the aligned-fiber microstructure with higher porosities leads to a smudging/diffusing of the fronts. The previously developed sharp-front model is found to predict the front height as a function of time quite well in the sintered-bead wicks and the two “semi-sharp” fibrous wicks. The liquid-front height calculated from the liquid mass-gain data was validated through visual observations for all but the two fibrous wicks generating the diffuse fronts. These latter cases, observed to start off as a sharp fronts, faded with time.     

Natural convection in a porous medium at high rayleigh numbers — application of prandtl's analogy

The problem of natural convection heat transfer from an isothermal vertical cylinder to a saturated porous medium is solved by the integral method making use of one-seventh profiles for velocity and temperature variation in the boundary layer. Prandtl's analogy is used to obtain the wall heat transfer coefficients. The results obtained for Rayleigh numbers greater than 10⁹ and Prandtl numbers 4 and 0.12 are compared with experimental data from literature show satisfactory agreement. Also an approximate closed form solution is presented for the case of a flat wall to predict local Nusselt numbers.     

Absorption of radiation by substances at “high” concentrations: A new equation and process monitoring applications

Beer-Lambert's law has a wide range of applications, spanning the fields of pure sciences and engineering, and is commonly used for the determination of the concentration of an optically sensitive compound, at “low” concentrations. In this report it is shown that a log-log expression, log(T/T₀) = K log(C/C₀), accurately models experimental data (±3%) for four different absorbing species, at “high” concentrations. The dimensionless constant K is shown to be only a characteristic of the absorbing species, and independent of the spectrophotometer/cuvette system used. The new equation is used for on-line monitoring of dynamic biomass concentration in a continuous bioreactor (CSTBR) for a step change in dilution rate.     

Gas‐phase flow resistance of metal foams: Experiments and modeling

Based on experimental results of gas flow resistance through two metal foams, NC 2733 and Ni 2733, a modeling is performed to adjudicate governing flow mechanism. Two essential models are considered: developing laminar flow within short capillary channel (i.e., foam pore) and flow around solid body (foam strut modeled as cylinder or sphere), each of them of some variants. Foam geometry was studied using computer microtomography. The model of flow around a sphere (diameter equal to strut thickness) gives the best conformity with experiments. © 2017 American Institute of Chemical Engineers AIChE J, 63: 1799–1803, 2017     

Effects of protein properties on adsorption and transport in polymer‐grafted ion exchangers: A multiscale modeling study

We use multiscale modeling to study how the molecular properties of a protein affect its adsorption and transport in ion exchange chromatography matrices with either open pores or charged polymers grafted into the pore structure. Coarse‐grained molecular dynamics (MD) simulations of lysozyme, bovine serum albumin, and immunoglobulin show that higher protein net charge leads to greater partitioning into the polymer‐grafted pore space but slower diffusion there due to favorable electrostatic interactions, while larger size decreases both pore space partitioning and diffusion due to steric effects of the polymers. Mass transfer simulations based on the MD results show that the polymer‐grafted systems can enhance the adsorption kinetics if pore space partitioning and diffusion are both sufficiently high. The simulations illustrate that to achieve fast adsorption kinetics, there is a tradeoff between favorable binding and rapid diffusion which largely depends on the charge and size of the protein. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4564–4575, 2017     

Dynamic effects in capillary pressure relationships for two‐phase flow in porous media: Experiments and numerical analyses

Well defined experiments and numerical analyses are conducted to determine the importance of dynamic effect in capillary pressure relationships for two‐phase flow in porous media. Dynamic and quasi‐static capillary pressure‐saturation (P^c‐S_w) and, ?S_w/?t‐t curves are determined. These are then used to determine the dynamic effects, indicated by a dynamic coefficient (τ) in the porous domains which establishes the speed at which flow equilibrium (?S_w/?t = 0) is reached. τ is found to be a nonlinear function of saturation which also depends on the medium permeability. Locally determined τ seems to increase as the distance of the measurement point from the fluid inlet into the domain increases. However, the functional dependence τ‐S_w follows similar trends at different locations within the domain. We argue that saturation weighted average of local τ‐S_w curves can be defined as an effective τ‐S_w curve for the whole domain which follows an exponential trend too. © 2012 The Authors. AIChE Journal, published by Wiley on behalf of the AIChE. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. AIChE J, 58: 3891–3903, 2012     

Synthesis and characterization of novel poly(acrylamide‐co‐acrylic acid‐co‐2‐hydroxy ethyl actylate) hydrogels: Mathematical approach in investigation of swelling kinetics and diffusion models

In this study, a series of poly(acrylamide‐co‐acrylic acid‐co‐2‐hydroxy ethyl actylate) [AM‐co‐AA‐co‐HEA] hydrogels have been synthesized by varying the acrylic acid (AA)content over eightfold in feed in the range of 33.34–93.76% by keeping other monomer constant. These hydrogels were characterized by FTIR, SEM analysis, elemental analysis, residual acrylic acid analysis, network parameters, and dynamic swelling behavior. The swelling study showed that equilibrium swelling ratio was nonlinearly increased with increasing AA content. Interestingly, the equilibrium swelling ratio decreased from 53.42 to 48.52 for 75–80% AA content hydrogel. The swelling data were found to satisfactorily fit Fick's second law, demonstrating that diffusion rate of water uptake was primarily Fickian. From model fitting, it was observed that early model was applicable for first 30% water absorption, and late model was applicable for latter 70% water absorption for increasing AA content from 33.34–90.90%. For 93.76% AA, early‐time model was extended up to first 50% of water absorption and late model was contracted for latter 50% water absorption, indicating that excessive AA content affects the applicability range of early‐time and late‐time diffusion models for water absorption. Etters model was best applicable to all type of hydrogels and followed over all swelling range. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012