Dimensionless Time Research Articles

Streaming instability with multiple dust species – II. Turbulence and dust–gas dynamics at non-linear saturation

ABSTRACT The streaming instability is a fundamental process that can drive dust–gas dynamics and ultimately planetesimal formation in protoplanetary discs. As a linear instability, it has been shown that its growth with a distribution of dust sizes can be classified into two distinct regimes, fast- and slow-growth, depending on the dust-size distribution and the total dust-to-gas density ratio ϵ. Using numerical simulations of an unstratified disc, we bring three cases in different regimes into non-linear saturation. We find that the saturation states of the two fast-growth cases are similar to its single-species counterparts. The one with maximum dimensionless stopping time τs,max = 0.1 and ϵ = 2 drives turbulent vertical dust–gas vortices, while the other with τs,max = 2 and ϵ = 0.2 leads to radial traffic jams and filamentary structures of dust particles. The dust density distribution for the former is flat in low densities, while the one for the latter has a low-end cut-off. By contrast, the one slow-growth case results in a virtually quiescent state. Moreover, we find that in the fast-growth regime, significant dust segregation by size occurs, with large particles moving towards dense regions while small particles remain in the diffuse regions, and the mean radial drift of each dust species is appreciably altered from the (initial) drag-force equilibrium. The former effect may skew the spectral index derived from multiwavelength observations and change the initial size distribution of a pebble cloud for planetesimal formation. The latter along with turbulent diffusion may influence the radial transport and mixing of solid materials in young protoplanetary discs.

Analytical solutions of advection-dispersion-reaction equation with first decay under constant and time-dependent boundary conditions: Mass transfer shape factor effects

To describe the combined dispersion and convection mechanisms of solute transport through porous media, one may typically refer to the advection-dispersion equation (ADE). In this study, a generalized analytical solution to the ADE combined with first order decay term is derived for the problem of solute transport through porous media in one-dimensional finite region considering transient boundary conditions. First, the addressed convection-dispersion-reaction equation is simplified into a diffusion equation. Subsequently, an analytical solution is generated for a general fractured porous medium for a case where a fracture is viewed as the transient boundary condition. To verify the proposed analytical solution, experimental data acquired from literature as well as numerical techniques of finite difference and element are utilized. Previous studies have utilized fixed values for mass transfer shaper factor which is deployed in dual porosity modeling approach of fractured reservoirs, while in reality this parameter changes over time. In this regard, an attempt is made to attain mass transfer shape factor while considering the impacts of important parameters such as Péclet number, and decay coefficient during both pseudo steady state (PSS) and transient periods. Moreover, for two circumstances of linearly ascending and constant concentration boundary conditions, a correlation is developed to compute PSS mass transfer shape factor as a function of Péclet number, dimensionless time, and decay coefficient. The results illustrated the dependency of PSS shape factor on the decay coefficient, Péclet number, and time while transient shape factor solely depended on time. In addition, the findings showed that the larger the Péclet number, the larger the PSS matrix-fracture mass transfer shape factor. The obtained analytical solution is helpful when validating the numerical simulation results and for performing fast sensitivity analysis. The developed mass transfer shape factor can be applied in the dual porosity modeling part of the commercial reservoir simulation software packages for more accurately modeling the solute transport in the fractured reservoirs.

Numerical study on the influence of liquid viscosity ratio on the hydrodynamics of a single bubble in shear-thinning liquid

The volume of fluid method combined with the adaptive grid method is used to study the influence of shear-thinning liquid viscosity ratio (ε) for various values of rheological index (n) and characteristic time (λ) on bubble shape, rise velocity, and apparent viscosity around a bubble; where 1/λ represents the shear rate at which the change in the viscosity curve occurs from the low shear rate to the power-law region. The results show that, depending on the magnitudes of n and λ, the bubble shape and deformation parameter are strongly related to ε, and bubble shape can exhibit both stable and unstable states. There are two types of deformation phenomena for the unsteady deformation of bubbles. First, the aspect ratio changes with time, whereas the shape remains similar throughout. Second, the shape changes periodically with time. For the given values of n and λ, the bubble rise velocity at large time shows a nonlinear increase as ε increases from 10 to ∞. Under the conditions of n ≤ 0.3, λ* ≥ 68.6, and ε ≥ 100 (λ* is the dimensionless characteristic time), a viscosity blind region (a region of high viscosity, where ε is close to 1) appears behind the bubble. The tendency for the viscosity blind region is to reduce the bubble rise velocity at large time.

Thresholds for Particle Clumping by the Streaming Instability

Abstract The streaming instability (SI) is a mechanism to aerodynamically concentrate solids in protoplanetary disks and trigger the formation of planetesimals. The SI produces strong particle clumping if the ratio of solid to gas surface density—an effective metallicity—exceeds a critical value. This critical value depends on particle sizes and disk conditions such as radial drift-inducing pressure gradients and levels of turbulence. To quantify these thresholds, we perform a suite of vertically stratified SI simulations over a range of dust sizes and metallicities. We find a critical metallicity as low as 0.4% for the optimum particle sizes and standard radial pressure gradients (normalized value of Π = 0.05). This subsolar metallicity is lower than previous results, due to improved numerical methods and computational effort. We discover a sharp increase in the critical metallicity for small solids, when the dimensionless stopping time (Stokes number) is ≤0.01. We provide simple fits to the size-dependent SI clumping threshold, including generalizations to different disk models and levels of turbulence. We also find that linear, unstratified SI growth rates are a surprisingly poor predictor of particle clumping in nonlinear, stratified simulations, especially when the finite resolution of simulations is considered. Our results widen the parameter space for the SI to trigger planetesimal formation.

Phase change behaviour study of PCM tanks partially filled with graphite foam

• The phase change behaviour of graphite foam partially inserted in PCM was studied. • For constant foam material, a full filling strategy with higher porosity is preferred. • Pertinent scale analysis was performed for PCM melting with fully-filled foams. The aim of this paper is to shorten the melting/solidification time of a high-temperature phase change material (PCM) using graphite foam inserts. Specifically, the phase change behaviour of a two-dimensional rectangular thermal storage tank, containing PCM with graphite foam insert, fully or partially filling the tank, was studied. The composite enclosure was designed assuming it was heated or cooled from the left side wall for charging or discharging, respectively, while the other three walls were perfectly insulated. First, the effect of foam porosity (0.8, 0.85, 0.9, and 0.95), under fully-filled scenarios, was numerically investigated. Then the phase change behaviour of four partially-filled scenarios, with averaged 0.9 porosity, was carried out. The 0.9 porosity foam case showed an excellent cycle performance. With this case, it only takes 68.2 and 65.1 min for entire melting and solidification, respectively. For a tank with no insert, it will take 164/856 min, respectively, to entirely melt/solidify the same mass of PCM as that of the 0.9-porosity-case. As expected, lower porosity values lead to higher heat transfer through conduction. However, our results show that with a fixed mass of foam, it is preferred to increase the foam porosity to fully fill the tank as opposed to a design with a lower porosity foam that only partly fills the tank. Finally, given the high graphite to PCM thermal conductivity ratio, the heat transfer through the foam is mainly due to conduction. Based on this assumption, a theoretical model is presented in parallel to numerical results. Our analysis for a foam-saturated PCM storage tank shows that the dimensionless time taken for completely melting the PCM, expressed as Fourier number, scales with Fo ρ PCM , s ρ PCM , l k eff k PCM , l ε Ste .

Bi-Fo time scaling method in the numerical simulation of transient conjugate heat transfer

Reliable transient thermal analysis plays a very important role in the engine safety analysis. Transient conjugate heat transfer simulation is an important way of temperature analysis. But there exists a great disparity in the time scales between solid conduction and fluid convection. The calculation cost of transient conjugate heat transfer analysis is very huge because of the tiny time step of computational fluid dynamics. The Bi-Fo time scaling method is proposed to improve the computational efficiency of transient conjugate heat transfer. On the one hand, this method carries out a similar transformation on solid heat conduction, scaling the calculation time with the product of density and specific heat capacity to maintain the consistency of Fourier number. On the other hand, it takes very short time for the fluid domain to recover stability after a boundary disturbance. Based on the above characteristic, the flow time is directly compressed to the same as that of the solid domain. It is verified by Mark Ⅱ vane that increasing the solid thermal diffusivity can reduce the time scale of heat conduction. In the situation of rapidly stable flow field, scaling flow time does not affect the solid thermal boundary under corresponding dimensionless time. Within the application scope, the Bi-Fo time scaling method can greatly reduce the time cost of transient conjugate heat transfer simulation while maintaining the accuracy of transient temperature analysis.

Analytical Solution of Unsteady-state Forchheimer Flow Problem in an Infinite Reservoir: The Boltzmann Transform Approach

For several decades, attempts had been made by several authors to develop models suitable for predicting the effects of Forchheimer flow on pressure transient in porous media. However, due to the complexity of the problem, they employed numerical and/or semi-analytical approach, which greatly affected the accuracy and range of applicability of their results. Therefore, in order to increase accuracy and range of applicability, a purely analytical approach to solving this problem is introduced and applied. Therefore, the objective of this paper is to develop a mathematical model suitable for quantifying the effects of turbulence on pressure transient in porous media by employing a purely analytical approach. The partial differential equation (PDE) that governs the unsteady-state flow in porous media under turbulent condition is obtained by combining the Forchheimer equation with the continuity equation and equations of state. The obtained partial differential equation (PDE) is then presented in dimensionless form (by defining appropriate dimensionless variables) in order to enhance more generalization in application and the method of Boltzmann Transform is employed to obtain an exact analytical solution of the dimensionless equation. Finally, the logarithms approximation (for larger times) of the analytical solution is derived. Moreover, after a rigorous mathematical modeling and analysis, a novel mathematical relationship between dimensionless time, dimensionless pressure, and dimensionless radius was obtained for an infinite reservoir dominated by turbulent flow. It was observed that this mathematical relationship bears some similarities with that of unsteady-state flow under laminar conditions. Their logarithm approximations also share some similarities. In addition, the results obtained show the efficiency and accuracy of the Boltzmann Transform approach in solving this kind of complex problem. Doi: 10.28991/HEF-2021-02-03-04 Full Text: PDF

Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities

In this work, we derived a mathematical model for spontaneous imbibition in a Y-shaped branching network model. The classic Lucas–Washburn equation was used for modeling the imbibition process occurring in the Y-shape model. Then, a mathematical model for the Newtonian fluid’s imbibition was derived to reveal the relationship between dimensionless imbibition time and length ratio, radius ratio, and wetting strength. The dimensionless imbibition time in the model was adopted to compare with that of the capillary bundle model. Different length and radius ratios were considered in the adjacent two-stage channels, and different wettabilities were considered in the different branches. The optimal radius ratio, length ratio, and wetting strength were calculated under the condition of the shortest imbibition time. In addition, the shortest dimensionless imbibition time of the three-stage Y-shaped branching network model was calculated when the wettability changes randomly. The results indicate that the imbibition time changed mostly when the wettability of the second branch changed, and the second branch was the most sensitive to wettability in the model.

Development and assessment of generalized drying kinetics in multi-tray solar cabinet dryer

This work evaluates the feasibility and determines a generalized drying characteristic curve in an indirect mode multi-tray solar cabinet dryer. The study considers a combination of tray reordering pattern and dryer performance index (DPI) method to map the influence of four primary factors, which influence the drying kinetics into a generalized curve. Three levels of each of the four factors, such as the mass flow rate (0.03–0.05 kg/s), heat input (500–1000 W/m2), loading density (1.0–2.5 kg/m2), and flake thickness (2–8 mm), represent twelve different drying kinetic curves. Therefore, the DPI value corresponding to the generalized curve would directly indicate the dryer performance and reduce the complexity in assessing the agro-product samples' drying behaviour in multiple trays. This study was initially validated, and it highlighted the merits of a tray reordering pattern in achieving drying uniformity per batch. Dimensionless drying process time (τ), a function of drying constant (k) and drying time (t), resulted in a maximum percentage deviation of 27.93% among the twelve drying kinetic curves. In contrast, expressing the moisture ratio as a function of DPI and drying process time (τ) reduced the maximum percentage deviation to 2.33%, with the value of correlation coefficient (r) above 0.972 for all the twelve drying kinetic curves. These consolidated findings pointed out that the adopted tray reordering pattern greatly influences the DPI method’s success in establishing a generalized curve in a multi-tray dryer.

Numerical solution of nonlinear dual‐phase‐lag model for analyzing heat transfer in tissue during thermal therapy

This article deals with mathematical modeling and simulation of heat transfer in tissue under periodic boundary condition using nonlinear dual-phase-lag-bioheat-transfer (DPLBHT). We have taken the temperature dependent blood perfusion and metabolic heat source as exponent variation in nonlinear DPLBHT model, both are experimentally validated function of temperature. In this article we applied finite difference method and Runge–Kutta (4,5) scheme to solve nonlinear problem. In particular case the exact solution is obtained and compared with numerical scheme and both are in good agreement. Effect of different parameters are discussed in detail such as blood perfusion rate, dimensionless heat source parameters, relaxation, and thermalization time on dimensionless temperature. The whole article is analyzed in dimensionless form.

Energy transference in time‐dependent Cattaneo–Christov double diffusion of second‐grade fluid with variable thermal conductivity

AbstractThe current investigation communicates the characteristics of upper‐convected second grade nanofluid thin film flow over a time‐dependent stretching sheet with variable thermal conductivity and Cattaneo–Christov double diffusion theory. The mathematical model generates a system of nonlinear partial differential equations (PDEs) for heat, momentum, and mass transfer phenomena. Similarity variables converted the PDEs into nonlinear ordinary differential equations (ODEs). Furthermore, received ODEs are numerically rescued with the built‐in program Bvp4c in MATLAB. The impact of physical parameters like second‐grade fluid , unsteadiness parameter , Prandtl number , Schmidt number , variable thermal conductivity , variable diffusivity , the square of dimensionless film thickness , relaxation time , and retardation time are scrutinized. Tables revealed numerical exposure and graphs depict the geometrical aspect of the current study. The velocity field is improved by heightening the λ, magnification in temperature is found for greater conductivity parameter Greater in the temperature profile goes smaller. The rate of mass transfer diminishes by uplifting the estimations in relaxation time parameter

Numerical analysis of thermofluids inside a porous enclosure with partially heated wall

• Darcy number increases the heat transfer rate and flow strength. • Higher Grashof number at the same Darcy number enhance the fluid temperature. • Porosity increases the flow strength and thermal activity. • Permeability of the medium influenced the flow field and thermal conductivity. In this study, heat transfer in a tall, rectangular permeable cavity with active thermal walls is investigated. Inside the enclosure, the two side walls' central portions are partially cooled at a fixed temperature. The central portion of the footwall is heated. Additionally, the top wall, the remaining portion of the footwall, and the side walls are insulated. The controlling equations are obtained from the Brinkman–Forchheimer-extended Darcy prototypical model using Boussinesq calculations. The leading equations are explained numerically by the finite element Galerkin method of weighted residuals. The computations are executed for some governing and physical parameters. The isotherms, streamlines and average heat transfer rate along with the partially active hot wall are shown for different groupings of governing parameters with respect to dimensionless time ( τ ). The outcomes indicated that the stream and thermal fields are strongly dependent on the considered parameters. It is also established that the average heat transfer rate is a function of these governing parameters.

A New Method for Research on Unsteady Pressure Dynamics and Productivity of Ultralow-Permeability Reservoirs

In the numerous low-permeability reservoirs, knowing the real productivity of the reservoir became one of the most important steps in its exploitation. However, the value of permeability interpreted by a conventional well-test method is far lower than logging, which further leads to an inaccurate skin factor. This skin factor cannot match the real production situation and will mislead engineer to do an inappropriate development strategy of the oilfield. In order to solve this problem, key parameters affecting the skin factor need to be found. Based on the real core experiment and digital core experiment results, stress sensitivity and threshold pressure gradient are verified to be the most influential factors in the production of low-permeability reservoirs. On that basis, instead of a constant skin factor, a well-test interpretation mathematical model is established by defining and using a time-varying skin factor. The time-varying skin factor changes with the change of stress sensitivity and threshold pressure gradient. In this model, the Laplace transform is used to solve the Laplace space solution, and the Stehfest numerical inversion is used to calculate the real space solution. Then, the double logarithmic chart of dimensionless borehole wall pressure and pressure derivative changing with dimensionless time is drawn. The influences of parameters in expressions including stress sensitivity, threshold pressure, and variable skin factor on pressure and pressure derivative and productivity are analyzed, respectively. At last, the method is applied to the well-test interpretation of low-permeability oil fields in the eastern South China Sea. The interpretation results turn out to be reasonable and can truly reflect the situation of low-permeability reservoirs, which can give guidance to the rational development of low-permeability reservoirs.

Evaluation of the residence time distribution (RTD) for flow in ducts with velocity profile of two independent variables

The correct information on RTD can help in system design and evaluation. The RTD corresponding to the velocity profile is known only for certain cases, where the velocity profile depends on one coordinate only. In this research, a general procedure for derivation of RTD corresponding to a known velocity profile is introduced. The RTD of laminar flows in different ducts as elliptic, equilateral triangular, moon-shaped and rectangular ducts are derived. Also, it is shown that the final RTD for laminar flow in any duct, can be estimated using relation E(θ) = K θmin/θn that is similar to laminar flow in the pipe, with their own dimensionless minimum time, , where is defined as the required time for traveling the duct with the maximum velocity in unit of the space-time. The values of K and n is calculated to meet the condition of . Besides, the values of for different cross-sections are studied. The results show that the RTD of elliptic ducts is precisely similar to the pipe flow. In the case of other shape ducts, the proposed model shows a suitable estimate of the numerical values. The previously published experimental data and precise analytical solutions agree with the proposed model with an acceptable consistency, except for very little time say θmin < θ < 0.7.

Analysis of Double-diffusive natural convection in a solar distiller embedded with PCM and cooled with external water stream

BackgroundThe increase demand of water for irrigation and human activities leads to a global development of alternative methods to produce distillate water especially from sea. In this context, the use of solar distiller instead of the classic distillation process is a good solution to preserve environment and reduce the use of fuel. This study explores the advantages of the classic solar still by studying the effect of the PCM-RT42. Hence, a detailed investigation of the heat and mass transfer of a single slope solar still incorporation PCM-RT42 as heat storage system will be presented. MethodsThe governing equations are solved numerically with employing the finite volume method. The next scopes of parameters are utilized in the computations: dimensionless time (τ = 4.2×10–3, 25.2×10–3, and 50.4×10–3), Rayleigh number (104 ≤ Ra ≤ 106), buoyancy ratio (1 ≤ N ≤ 5). Mixture Prandtl number (Prm), Schmidt number (Sc) and Lewis number (Le) are presumed as fixed to 0.705, 0.613 and 0.869, respectively. Significant FindingsThe results show the good effect of the PCM-RT42 on the transfer of heat and mass compared to the initial time for low Ra. However, for Ra=105 and 106, a reduction in the efficiency of condensation and the performance of the solar distiller is observed.

A Semi‐Analytical Solution to Evaluate the Spatiotemporal Behavior of Diffusive Pressure Plume and Leakage From Geological Storage Sites

AbstractAnalytical and semi‐analytical approaches provide an easy‐to‐use tool to verify the success of long‐term and large‐scale deployment of subsurface CO2 storage. Existing classical solutions lack significant features that are essential in the interpretation of the spatiotemporal evolution of pressure plume. We, obviating the limiting assumptions, report a novel general semi‐analytical solution to a physical problem where a storage layer and a caprock are embedded in an infinite medium. We exploited the features of an integral transform technique and Laplace transform to derive the new solution. An effective numerical treatment is suggested to evaluate the obtained solutions. We identify a distorted Gaussian spatiotemporal pressure plume for the cases when the overburden/underburden permeabilities are less than that of the storage layer. We investigate the nature of temporal behavior of leakage rates and identify a scaling relation in the form of Ld ∝ td1/2 for dimensionless leakage rate and time. We also present a benchmark pressure trend to detect the pressure anomaly signaling potential brine and CO2 leakage. This theory will be invoked as a benchmark analytical study in fields where diffusion is of immense interest.

Mathematical models of temperature in electric discharge rolling of metals

The model of analytical calculation of the average volumetric temperature of plastic deformation in the conditions of applied impulse current was developed on the base of rolling energy balance. Additionally, the formula for determination of the contact temperature between rolls and strip as well as temperature gradient through strip thickness and its variation in time of electroplastic process conduction was concluded on the base of solution of the heat conductivity differential equation in Fourier partial derivatives. This equation is valid for the boundary condition taking into account joint heating with impulse current and plastic deformation. Conducted temperature calculations for dimensionless time in the deformation area during electric discharge rolling of steel 3 and VT-6 titanium alloy displayed that temperature rise up to its maximal value has temporal displacement to strip exit side of deformation area. It is connected with temporal retarding of heat forming comparing with the time of elastic-plastic deformation. As soon as the contact square of rolls with strip has two different areas, where bonding and slipping are realized via rolling friction and slipping friction, immediate temperatures will also differ respectively. The obtained models allow to conduct calculations of the contact temperatures in the areas of roll bonding and slipping relating to the rolling strip. Electric discharge rolling reduces conventional cold rolling technology via elimination of heating with intermediate heat treatment (annealing) and improves thereby thin strip quality via healing of its cracks and defects by electric impulses, as well as increases productivity by 30 %.

Mixing Time and Scale‐up Investigation of a Moving‐Baffle Oscillatory Baffled Column

AbstractMixing time and scale‐up behavior were investigated for a moving‐baffle oscillatory baffled column (MB‐OBC). Computational fluid dynamics software was used to investigate the effects of the oscillation conditions and changes in the column scale on the dimensionless mixing time. The results indicate that small amplitudes combined with high frequencies are detrimental to the mixing efficiency and that the scale‐up of an MB‐OBC is linearly dependent on the column diameter. This work, in combination with previous studies, proves that the predictability of the hydrodynamic behavior during scale‐up is inherent to oscillatory baffled columns. This trait is desirable for adapting these columns for industrial‐scale applications.

Horizontal Motion of a Superhydrophobic Substrate Affects the Drop Bouncing Dynamics.

While the drop impact dynamics on stationary surfaces has been widely studied, the way a drop impacts a moving solid is by far less known. Here, we report the physical mechanisms of water drops impacting on superhydrophobic surfaces with horizontal motions. We find that a viscous force is created due to the entrainment of a thin air layer between the liquid and solid interfaces, which competes with the capillary and inertia forces, leading to an asymmetric elongation of the drop and an unexpected contact time reduction. Our experimental and theoretical results uncover consolidated scaling relations: the maximum spreading diameter is controlled by both the Weber and capillary numbers D_{max}/D_{0}∼We^{1/4}Ca^{1/6}, while the dimensionless contact time depends on the capillary number τ/τ_{0}∼Ca^{-1/6}. These findings strengthen our fundamental understandings of interactions between drops and moving solids and open up new opportunities for controlling the preferred water repellency through largely unexplored active approaches.

On the Imbibition Model for Oil-Water Replacement of Tight Sandstone Oil Reservoirs

A model suitable for evaluating a tight sandstone reservoir is established. The model includes two oil-water replacement modes: capillary force mode and osmotic pressure mode. The relationship between oil-water displacement rate and dimensionless time under different parameters is drawn considering the influence of capillary force, osmotic pressure, production pressure difference, and starting pressure gradient. Results indicate that the higher the relative permeability of the water phase, the lower the relative permeability of the oil phase, the smaller the oil-water viscosity ratio, and the higher the oil-water replacement rate. The relative permeability of the water phase also affects the infiltration stabilization time. Low salinity fracturing fluid infiltration helps to improve the oil-water replacement rate.