Function Of Dimensionless Time Research Articles

The analytic solution of interfacial concentration with observed rejection ratio during dead-end membrane filtration

In this study, we revisit the fundamentals of constant–flux dead–end filtration, develop an analytical solution of the interfacial concentration ϕmτRo as a function of dimensionless time τ and observed rejection Ro, and compare the solution with previous work developed for constant intrinsic rejection, Ri. The excessive concentration, ϕmτRo−1, consists of three nonlinear terms of τ and reaches 4Roτ in an asymptotic limit of τ>1/2. We apply the Robin (mixed) and Dirichlet boundary conditions on the membrane surface and at a far feed–entrance, respectively. The mathematical difficulties for the inverse Laplace transform are resolved using a linear combination of the Laplace transform of error and complementary error functions and applying the convolution theorem. We analytically obtain the unsteady variation of the interfacial concentration after the pressure release using the global mass balance and numerically calculate the required time to reduce the interfacial concentration to a specific limit. More importantly, a relationship between observed and intrinsic rejection ratios is found, such as, Ro≃Ri, and verified using experimental data from the literature.

A non-field analytical method for gas dissolution under forced compression

This paper presents an extension of the non-field analytical method—known as the method of Kulish—to model gas dissolution into a liquid due to forced compression. Solutions are obtained for the time evolution of pressure (and, hence, mass concentration) at the gas–liquid interface. These solutions are in the form of series with respect to fractional differ-integral operators. The asymptotic solutions for the two limiting cases of compression—slow and fast compression—have been established as well. Then several particular examples of the law of gas volume variation are considered. Among them, the law of a linear volume variation is the most interesting for practical purposes, in which case numerical values of the dimensionless pressure as a function of dimensionless time are provided.

Numerical Simulations of a Freely Falling Rigid Sphere in Bounded and Unbounded Water Domains

Abstract Numerical studies were conducted on the hydrodynamics of a freely falling rigid sphere in bounded and unbounded water domains to investigate the drag coefficient, normalized velocity, pressure coefficient, and skin friction coefficient as a function of dimensionless time. The bounded domain was simulated by bringing the cylindrical water container’s wall closer to the impacting rigid sphere and linking it to the blockage ratio (BR), defined as the ratio of the projection area of a freely falling sphere to that of the cross-sectional area of the cylindrical water container. Six cases of bounded domains (BR = 1%, 25%, 45%, 55%, 65%, and 75%) were studied. However, the unbounded domain was considered with a BR of 0.01%. In addition, the k–ω shear stress transport (SST) turbulence model was used, and the computed results of the bounded domain were compared with those of other studies on unbounded domains. In the case of the bounded domain, which has a higher value of BR, a substantial reduction in normalized velocity and an increase in the drag coefficient were found. Moreover, the bounded domain yielded a significant increase in the pressure coefficient when the sphere was half-submerged; however, an insignificant effect was found on the skin friction coefficient. In the case of the unbounded domain, a significant reduction in the normalized velocity occurred with a decrease in the Reynold number (Re), whereas the drag coefficient increases with a decrease in Reynolds number.

On the Diffusion-controlled Transport and Accumulation of Lithium in an Electrode of a Lithium-based Galvanic Cell

The presented formulation in this paper provides the information in the form of lithium concentration profiles and lithium accumulation in the solid-state active material particles of a porous bed cathode of a lithium based galvanic cell as a function of dimensionless time during the cell discharge period under the condition of lithium diffusion in the cathode active material being a predominant process affecting the cell voltage versus time performance. The computed data is deemed to be useful in the predetermination of required cell discharge time corresponding to a desired state-of-charge of a lithium-based galvanic cell under the above-mentioned condition.

Finite element modeling of spherical indentation in a poro‐elasto‐plastic medium via step displacement loading

AbstractA hydromechanically coupled finite element method (FEM) algorithm following a mixed continuous Galerkin formulation for displacement and pore pressure is adopted for modeling of spherical indentation in a poro‐elasto‐plastic medium. The fully saturated porous medium is assumed to be isotropic and elasto‐perfectly plastic, obeying a Drucker‐Prager yield criterion with an associative or non‐associative flow rule. The Newton‐Raphson method with the tangent stiffness scheme is employed to deal with plasticity in the solid skeleton. A stabilization scheme, which permits equal‐order interpolation for the displacement and pore pressure fields and suppresses pore pressure oscillation in the incompressible or nearly incompressible limit, is incorporated in this FEM algorithm. The FEM algorithm is extensively benchmarked with poroelastic analytical solutions to the problems of Terzaghi, Mandel, Cryer, and De Leeuw and an analytic solution for one‐dimensional poro‐elasto‐plastic consolidation. Numerical simulations of poroelastic spherical indentation via step displacement loading are conducted to show that the normalized indentation force as a function of dimensionless time has only relatively weak dependence on material properties through a single derived parameter . Such universality is shown for three types of surface drainage conditions. For indentation in a poro‐elasto‐plastic medium, it is shown that even though plasticity occurs immediately at the undrained limit, if cohesion is within a certain range, there is no plastic strain accumulation during the transient period and the normalized force relaxation behavior could be approximated as poroelastic.

On the effect of salinity and nano-particles on polymer flooding in a heterogeneous porous media: Experimental and modeling approaches

In this study, polymer flooding process was simulated using COMSOL Multiphysics® modeling software. Navier–Stokes and phase field equations were solved simultaneously using finite element approach. Effects of salinity, viscosity, and nano-particles were studied on displacement efficiency of oil sample in the heterogeneous porous media. Modeling results for oil recovery factor and breakthrough time were obtained and compared to experimental outputs of another study. Results indicated that there exists a proper accommodation between the results. Errors ranged from 3% to 7% for oil recovery factor and 0.5%–0.9% for the breakthrough time in various injection scenarios. Using predefined structure of the under test micromodel, star-like meshes were developed in this software for the first time. This unstructured mesh is more likely to the pore structures of the sandstone reservoirs. To increase the validity of the results, a boundary layer was designated in the fluid flow path and mesh generation was performed manually in a complicated pore space instead of automatic mesh generation. Experimental and modeling results for the residual oil saturation as a function of dimensionless time were compared. Mesh dependency studies were performed to obtain the best number of mesh elements in which the most accurate results were obtained in the minimum required time. Finally, sensitivity analysis studies were performed as a function of injection rate, mobility ratio, and viscosity and the operating conditions were analyzed in terms of capillary number. All the polymer injection scenarios led to reasonable range for this parameter which verifies the accuracy of the selected operating parameters and the modeling approach.

Compaction in a confined deforming reservoir

A reduction of fluid pressure may lead to altered permeability and reduced thickness. This may lead to reduced well performance and well integrity. In some cases, subsidence and even earthquakes may occur. Although rare, such incidences have occurred many places around the world. The objective of this study is to present results obtained by a simplified analytical model for compacting reservoirs. Based on the constant modulus approach, we present approximate analytical solutions for stress-sensitive deforming reservoirs, and plot the corresponding type curves. We derive the equation for the dynamic behavior of the thickness and show plots of normalized thickness as a function of dimensionless time.

Transient stage comparison of Couette flow under step shear stress and step velocity boundary conditions

Couette flow has been widely used in many industrial and research processes, such as viscosity measurement. For the study on thixotropic viscosity, step-loading, which includes (1) step shear stress and (2) step velocity conditions, is widely used. Transient stages of Couette flow under both step wall shear stress and step wall velocity conditions were investigated. The relative coefficient of viscosity was proposed to reflect the transient process. Relative coefficients of viscosity, dimensionless velocities and dimensionless development times were derived and calculated numerically. This article quantifies the relative coefficients of viscosity as functions of dimensionless time and step ratios when the boundary is subjected to step changes. As expected, in the absence of step changes, the expressions reduce to being functions of dimensionless time. When step wall shear stresses are imposed, the relative coefficients of viscosity change from the values of the step ratios to their steady-state value of 1. But with step-increasing wall velocities, the relative coefficients of viscosity decrease from positive infinity to 1. The relative coefficients of viscosity increase from negative infinity to 1 under the step-decreasing wall velocity condition. During the very initial stage, the relative coefficients of viscosity under step wall velocity conditions are further from 1 than the one under step wall shear stress conditions but the former reaches 1 faster. Dimensionless development times grow with the step ratio under the step-rising conditions and approach the constant value of 1.785 under the step wall shear stress condition, and 0.537 under the step wall velocity condition respectively. The development times under the imposed step wall shear stress conditions are always larger than the same under the imposed step wall velocity conditions.

Transient pressure analysis of fractured well in bi-zonal gas reservoirs

For hydraulic fractured well, how to evaluate the properties of fracture and formation are always tough jobs and it is very complex to use the conventional method to do that, especially for partially penetrating fractured well. Although the source function is a very powerful tool to analyze the transient pressure for complex structure well, the corresponding reports on gas reservoir are rare. In this paper, the continuous point source functions in anisotropic reservoirs are derived on the basis of source function theory, Laplace transform method and Duhamel principle. Application of construction method, the continuous point source functions in bi-zonal gas reservoir with closed upper and lower boundaries are obtained. Sequentially, the physical models and transient pressure solutions are developed for fully and partially penetrating fractured vertical wells in this reservoir. Type curves of dimensionless pseudo-pressure and its derivative as function of dimensionless time are plotted as well by numerical inversion algorithm, and the flow periods and sensitive factors are also analyzed. The source functions and solutions of fractured well have both theoretical and practical application in well test interpretation for such gas reservoirs, especial for the well with stimulated reservoir volume around the well in unconventional gas reservoir by massive hydraulic fracturing which always can be described with the composite model.

Transient Natural Convection in Vertical Annuli: Numerical Modeling and Heat Transfer Correlation

A finite volume model is developed for analyzing transient natural convection in vertical annuli, which are assumed to be isothermally heated (or cooled) from the inner surface and insulated from horizontal and outer surfaces. Based on the numerical results, the dependence of transient heat transfer on governing dimensionless parameters is investigated. It is found that the dimensionless accumulated heat transfer as a function of dimensionless time can be satisfactorily approximated by a simple correlation, which explicitly includes the effects of the Rayleigh and Prandtl numbers as well as the enclosure aspect ratio and height-inner radius ratio.

Approximation of Transient 1D Conduction in a Finite Domain Using Parametric Fractional Derivatives

A solution to the problem of transient one-dimensional heat conduction in a finite domain is developed through the use of parametric fractional derivatives. The heat diffusion equation is rewritten as anomalous diffusion, and both analytical and numerical solutions for the evolution of the dimensionless temperature profile are obtained. For large slab thicknesses, the results using fractional order derivatives match the semi-infinite domain solution for Fourier numbers, Fo∊[0,1/16]. For thinner slabs, the fractional order solution matches the results obtained using the integral transform method and Green’s function solution for finite domains. A correlation is obtained to display the variation of the fractional order p as a function of dimensionless time (Fo) and slab thickness (ζ) at the boundary ζ=0.

Generation of dynamic biochemical signals with a tube mixer: effect of dispersion in an oscillatory flow

The generation of controlled dynamic biochemical signals has many applications in the life sciences. This paper presents an analysis of the dispersion of an oscillatory biochemical signal in an incompressible viscous oscillatory flow in a mixing tube. By using the method of Gill and Sankarasubramanian, the dispersion coefficients \(K_i(\tau)\;(i=1, 2,\ldots)\) are determined as the functions of dimensionless time τ. With the assumptions of quasi-steady flow and steady flow, the dispersion coefficients \(K_i(\tau)\;(i=1, 2)\) are simplified. The effects of the frequencies of the oscillatory flow and the oscillatory biochemical signal, and the length of the mixing tube on the average solute concentrations, θm, over the tube cross-section at the outlet of the mixing tube are analyzed by numerical simulations with and without the assumptions of the quasi-steady and steady flows. It is concluded that for the dispersion in an oscillatory flow, an excellent accuracy can be achieved by using quasi-steady flow assumption while the steady flow assumption would lead to inaccurate results. However, if the frequency of oscillatory flow is sufficiently high, the steady flow assumption can be used to further simplify the calculation while still maintaining sufficient accuracy. These results are of practical importance in producing dynamic biochemical signals as the stimuli of biological cells by a tube mixer.

Dimensions and Degree of Containment of Waterflood-Induced Fractures From Pressure-Transient Analysis

Summary It is well established within the industry that injection of (produced) water almost always takes place under fracturing conditions. Particularly when large volumes of very contaminated water are injected—either for voidage replacement or disposal—large fractures may be induced over time. This paper aims to provide a methodology for injection-falloff (IFO) test analysis of fractured (produced) water-injection wells. Some essential elements of IFO for fractured water injectors include the closing fracture, (early) transient elliptical reservoir-fluid flow, finite fracture conductivity, and fracture face skin. An exact semianalytical solution is presented to the fully transient elliptical fluid-flow equation around a closing fracture with finite conductivity, fracture face skin, and multiple mobility zones in the reservoir surrounding the fracture. This solution also captures the case that during closure, the fracture is generally shrinking from adjacent geological layers under higher in-situ stress. Based on this solution, type curves of the dimensionless bottom-hole pressure as a function of dimensionless time are provided, covering both the period during fracture closure/shrinkage and the period after fracture closure. The shape of these type curves is studied as a function of the different relevant parameters, in particular the fracture compliance, the height of in-situ stress contrasts, fracture face skin, fracture closure time, and injection period. It is shown how the fracture length and height and the degree of fracture containment (in combination with the heights of the stress contrasts) can be derived from these types of curves. It is also demonstrated that the analyses based on the storage flow and linear formation flow regimes need to be integrated into one analysis method to obtain consistent results. Finally, the concepts developed in this paper are applied to a number of field examples, in which the dimensions and degree of containment of the induced fractures are derived from the analysis of the IFO data.

Modeling of liquid desiccant drying method for gelcast ceramic parts

Gelcasting is a promising new technology for manufacturing of advanced structural ceramic components. The process involves drying of the “green” gelcast part before densification. Conventional methods in drying green gelcast part, using dry air or gas with a specific condition of humidity and temperature usually are confronted with many difficulties. Non uniform and differential drying in various regions due to the solvent gradient induces structural and residual stresses, that cause defects such as cracking, warping, bending and the other malformations, which make the part useless. Liquid desiccant drying method, as a novel method, has been used for drying green gelcast ceramic parts in the purpose of increasing the drying rate and removing the defects due to the release of residual stresses. In this study, the kinetics of one-dimensional drying of green gelcast ceramic parts through this method is described by a Fickian model, which accounts for the movement of the gel surface. For a constant mutual diffusion coefficient, D m, the fractional approach to equilibrium, F, is only a function of dimensionless time τ 0 ∗ , and the equilibrium volume ratio, Φ. By using simple method experimental values of D m which account for the movement of the gel boundary are obtained. Experimental data for alumina gelcast cylinders having diameters of 7 and 9 mm are well fitted by the model, and the amount of ceramic loading was 5, 10, 20, 30 and 35% by volume, respectively. Various concentrations of aqueous solution of PEG1000 were used as liquid desiccants.

Fumigation of pollutants in and above the entrainment zone into a growing convective boundary layer: a large-eddy simulation

Fumigation of a passive plume located in or above the entrainment zone (EZ) into a growing convective boundary layer (CBL) has been simulated by large-eddy simulation (LES). Three non-dimensional parameters, α(= w e0/ w *0), z 0/ z i0, and σ z0 / z i0, are used to classify different cases, where w *0 is the convective velocity scale, w e0 the initial entrainment velocity, z i0 the initial CBL height, z 0 the initial plume height, and σ z0 is the initial plume half-depth. Forty cases have been run and analysed. The crosswind-integrated concentrations have been compared with existing laboratory data from a saline convection tank. The results show that LES is a promising tool to reproduce fumigation processes. With a relatively coarse grid mesh near the EZ, LES derives reliable results that are in a good agreement with the laboratory data. The first parameter, α, containing the effects due to inversion strength, plays an important role in determining C 0( T), the ground-level concentration (GLC) as a function of dimensionless time, T. For large α (say >0.03, corresponding to fast entrainment), variation of α gives significant change in C 0( T); whereas for a wide range of α between 0.01 and 0.02 (corresponding to slow entrainment), C 0( T) is almost independent of α. The starting time of fumigation does not vary significantly with the second parameter, z 0/ z i0 (relative height of plume), although C 0( T) is, in general, smaller for a higher plume. This confirms laboratory findings that the traditional notion of zero fumigation for a high plume (say above 1.10 z i) is not correct. The effect of the third parameter, σ z0 / z i0, is on the magnitude of C 0( T); thinner initial plumes have higher GLCs.

Numerical Simulation of the Dynamics of Entropy Production in a One-Dimensional Unsteady, Thermally Inhomogeneous System

where d is the plate thickness. The integral in Eq. (3) was calculated by the values of temperature at the coor- dinate net points for different moments of time t . The table gives the values of the ratio of entropy production given by Eq. (3) to its value in the steady state as functions of dimensionless time for different values of δ defining the behavior of the temperature dependence of thermal conductivity given by Eq. (2). In the case of δ = -2, in full accord with the Prigogine the- orem, a monotonic decrease in the entropy production is observed down to the transition of the system to the

Mass Transfer Model for Supercritical Fluid Extraction Tests

A mass transfer model using a linear equilibrium isotherm is presented for the static and dynamic stages of supercritical fluid extraction tests. In the static stage the fraction extracted is a function of a dimensionless time and a dimensionless equilibrium parameter. In the dynamic stage the fraction recovered is a function of a different dimensionless time, the same equilibrium parameter, and a dimensionless mass transfer parameter. Fitting the model to experimental extraction curves showed that increasing the pressure or density increases the equilibrium constant and decreases the overall mass transfer coefficient.

Sorption of diffusants by polymeric cylinders under isothermal, finite bath conditions: Advances in mathematical modeling

Diffusion-controlled, isothermal sorption (or desorption) of diffusants by cylindrical polymeric materials can be modeled for finite bath conditions by the use of the conventional diffusion equation solutions of Wilson or Crank when certain limiting assumptions apply. Iterative determination of diffusion coefficients by the use of these advanced equations, however, requires accurate “seed roots” for efficient solution. An empirical seed root generating function has been found that is so very accurate that it can be used in place of the conventional diffusion equation solutions for all but the most extreme levels of definitude. The parameters of the empirical equation as functions of equilibrium bath exhaustion are given, and values of fractional equilibrium uptake, M1/Mx, as a function of dimensionless time, Dt/r2, found by the use of both the diffusion equation solutions and the empirical equation are compared. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 63: 1237–1242, 1997

Experimental and Numerical Investigation of Soil Vapor Extraction

In recent years soil vapor extraction (SVE) has been used extensively to remove volatile organic compounds (VOCs) from the vadose zone. In order to investigate processes limiting the removal of VOCs during the later stages of SVE operations, multicomponent soil‐venting experiments were performed at different water contents in a sand tank (80 × 66 × 5 cm) in the absence of a liquid organic phase. Four chlorinated VOCs were used as the model compounds. A homogeneous packing of quartz sand was used as the model soil. Gas phase concentrations were measured at several locations with different water saturations during each experiment. The compounds did not adsorb onto the sand. Volatilization was the key process for VOC removal. Gas concentrations decreased more slowly at locations with high water saturation and for compounds having a small Henry's law constant. Gas concentrations observed for experiments conducted at low water content were found to be a function of dimensionless time. Tailing in gas concentration distributions at large dimensionless times was attributed solely to diffusion in interparticle water and suggested that local nonequilibrium conditions prevailed. This hypothesis was supported by numerical simulations based on the local equilibrium assumption (LEA) and a first‐order kinetics approach. For the experiments conducted at low water saturations the LEA was valid only for small dimensionless times. Tailing in the gas concentration distributions could then be described quite well by means of a first‐order kinetic approach using calibrated mass transfer coefficients.

Calculation of Temperature Distributions during the Solidification Stage in Ceramic Injection Molding

A finite difference procedure for calculating unsteady‐state temperatures in a molded ceramic–polymer suspension during solidification is described. The method incorporates temperature‐dependent thermal diffusivity and enthalpy of fusion of the polymer. An experimental method is used to assess the surface heat transfer coefficient h at the mold wall. It is found that at low injection pressures h varies with time as the molded body shrinks from the wall, but at high injection pressures h can be treated as constant throughout the solidification stage. Using an analytical method, graphical charts are produced for dimensionless temperature as a function of dimensionless time for values of Biot's modulus in the region relevant to ceramic injection molding.