Drop Equation Research Articles

Lubrication-mediated rebounds off fluid baths

We present herein the derivation of a lubrication-mediated (LM) quasi-potential model for droplet rebounds off deep liquid baths, assuming the presence of a persistent dynamic air layer which acts as a lubricating pressure transfer. We then present numerical simulations of the LM model for axisymmetric rebounds of solid spheres and compare quantitatively to current results in the literature, including experimental data in the low-speed impact regime. In this regime the LM model has the advantage of being far more computationally tractable than direct numerical simulation (DNS) and is also able to provide detailed behaviour within the micro-metric thin lubrication region. The LM system has an interesting mathematical structure, with the lubrication layer providing a free-boundary elliptic problem mediating the drop and bath free-boundary evolutionary equations.

Modeling and optimization for the continuous catalytic reforming process based on the hybrid surrogate optimization model

To address the modeling and optimization challenges of the complex reaction system in the continuous catalytic reforming process, a new integrated simulation and optimization framework is presented. First, a detailed mechanism model is established based on a reaction network involving 32 components and 50 reactions, coupled with mass transfer, heat transfer, pressure drop, and catalyst deactivation equations. Then, to solve the differential-algebraic equations in the mechanism model, a multi-objective hybrid optimization method with the adaptive infill strategy is introduced. GAMS and MATLAB are integrated to perform a joint iterative solution. Finally, two cases are conducted with the proposed algorithm. Results show that the mechanism model calculation deviations are below 4 % of reactor temperature, pressure, and composition distribution, and the Pareto front of various production plans is obtained. The accurate simulation and rapid trade-off optimization among the key goals can be achieved to provide scientific decision support for enterprise production.

Pressurised sand bed filtration model: Set up and energy requirements for a filtration cycle

Rapid deep bed filtration is a common process of drip irrigation systems to prevent emitter clogging. The particle retention in the porous media increases the pressure difference between the filter's inlet and outlet. Commercial operational instructions preset a threshold value of this pressure difference to define the end of a filtration cycle. Accurate particle retention models may contribute in the determination of this setpoint to improve the energy efficiency of a filtration cycle. A two-step method was developed to calibrate a phenomenological particle retention model with transient data of the filter pressure drop, and retained mass in different media slices at the end of the filtration cycle. The first step used the vertical profile of accumulated mass in the media to fit the input parameters of the specific deposit rate equation. The second step applied the pressure data to find the input parameters of the pressure drop equation. The use of constant values of the input parameters provided reasonable results for all tests. When the end of the filtration cycle was set by applying a threshold pressure drop, both particle mass accumulation and removal efficiency decreased as the flow rate increased. By using the threshold pressure drop criterion, the energy consumption in a filtration cycle also decreased as the flow rate increased, but the energy consumption per unit of filtered water volume increased. For high setpoint pressure drop values, a minimum of the energy consumption per unit of filtered water volume as a function of the flow rate was found.

New investigation of the analytical behaviors for some nonlinear PDEs in mathematical physics and modern engineering

Numerous complex situations arising from science and engineering are modeled by integral-differential equations (IDEs). Examples of these situations include mathematical modeling of infectious diseases, epidemics, circuit analysis, voltage drop equations, computational neuroscience, intergalactic modeling, and many more. To explain the mathematical structure and the behavior of various phenomena in nature, two equations with simple mathematical structure namely: the (1 + 1)-dimensional integro-differential Ito equation (IDIE) and the (2 + 1)-dimensional integro-differential Sawada-Kotera equation (IDSKE) are considered. In this analytical investigation, we have conducted an in-depth enquiry to trace the distinctive closed form progressive wave solutions of these proposed models using the (G′G′+G+A)-expansion method. The graphical forms of the acquired solutions have been displayed in the bell-shaped soliton, singular periodic, and anti-kink shape soliton solutions after the values for the free parameters were specified. The two non-algebraic solutions, named, exponential and trigonometric function solutions, have emerged by applying our proposed method. Based on the general findings of our investigation for different parametric values, the attained wave solutions revealed can keep a vital role for natural balances and can be engaged in modern physical applications. The parameters have a visible effect on the wave amplitude and the swiftness of the traveling wave. The executed outcomes are innovative and have monumental applications in the present research, especially in the fields of theoretical physics, astrophysics, plasma physics, particle physics, theory of relativity, advance quantum mechanics, string theory, and geotechnical engineering.

Water Distribution Network Optimization Model with Reliability Considerations in Water Flow (Debit)

Water distribution networks (WDNs) are defined as the planning for the development, distribution, and utilization of water resources. The main challenge of WDNs is to preserve limited water resources while providing effective benefits from these resources in accordance with environmental considerations. Water distribution networks use hydraulic components to connect water resources to consumers. The diameter of each pipe, the layout of the pipe network, and the total length of pipes all contribute to the most effective layout for a water distribution system. This study considers the assurance that the flow (discharge) of water is in accordance with what is expected, with such aspects apt to be described as a particular form of reliability. As a result, this study proposes a stochastic optimization model with non-linear probability constraints for overcoming the challenges of water distribution networks while taking water flow reliability into account. The pressure drop equation causes the non-linear shape. The stochastic model of the opportunity constraint is changed to a deterministic multi-objective model using an approach based on integer programming and sample averaging to solve the resulting model. The direct search approach (neighbourhood search) is then applied to tackle the integer part.

Heat transfer correlations and pressure drop dynamics of palm fibre wetting media for evaporative cooling at different air flow rates

The thermal and flow resistance properties of wetting pads play a key role in how effective direct evaporative cooling (DEC) systems Perform. To make an informed decision, it's crucial to study the heat transfer properties and airflow resistance manifested in pressure drop for various types of wetting media. An experimental study was conducted to determine pressure drop and heat transfer correlation constants at three airflow rates for palm fruit mesocarp fibres (PFMF) wetting media. Heat transfer and pressure drop equations were fitted with generated data to obtain correlation constants. The wetting media were effective to lower the ambient temperature at an average depression of 6.6 to 8.2 °C and increase the relative humidity of the inlet air at an average value of 25.54 to 39.32%. PFMF had a low-pressure drop per unit length, ranging from 5.29 to 7.62 Pa/m, which makes it an appealing alternative to other wetting media. New correlation constants generated for pressure drop and heat transfer data fit well with high coefficient of determination (R2) values. The average cooling efficiency ranged from 56.43 to 65.27%, while the average Merkel number was 0.16 to 0.19. The Nusselt number increased with the Reynolds number and showed a laminar flow. Amaranths stored in DEC produced respiratory heat ranging from 290.86 to 336.53 mg/kg/h for all air velocities. Considering the respiratory rate within the range of cooler temperatures, it is recommended to use the DEC only for short periods or for pre-cooling of vegetable storage.

A Coordinated Reconfiguration Strategy for Multi-Stage Resilience Enhancement in Integrated Power Distribution and Heating Networks

The tight coupling of power distribution system (PDS) and district heating system (DHS) makes it imperative to devise a coordinated strategy for the resilience enhancement of integrated electric and heating system (IEHS) in natural disasters. The PDS reconfiguration serves as a critical tool for service restoration by the optimal adjustment of PDS topology with designated tie and sectionalizing switches. Similarly, the DHS reconfiguration provides a viable solution for countering disasters by the optimal adjustment of heating supply topology with targeted tie and sectionalizing valves in DHS. In this paper, a multi-stage resilience enhancement strategy considering the coordinated reconfiguration of PDS and DHS is proposed, which coordinates degradation, fault isolation, and service restoration stages after disasters. The proposed model is formulated as a mixed-integer nonlinear problem, which addresses the coordinated regulation of PDS switches and DHS valves to achieve efficient fault isolation and provide a better service restoration. To make the proposed model tractable, a linearized Distflow model is proposed by introducing current-oriented auxiliary variables and the approximation of power flow and voltage drop equations. Extensive case studies are conducted to validate the efficiency and accuracy of the proposed model and illustrate the performance of coordinated reconfiguration of PDS and DHS on resilience enhancement.

Development of an efficient numerical model for two-phase flows in air-lift pumps and its application to deep-sea mining

Air-lift pump system has been selected as a potential application to convey the Rare-Earth mud in deep-sea mining processes due to the superior reliability and schematic simplicity. It is a major focus of current research to study the behavior and mechanism of the two-phase flow. Based on the drift flux model and modified Akawa pressure drop equation, mathematical model controlling the numerical simulation of two-phase flow within air-lift pump system is proposed and constructed. Comparison with experimental data concerning steady and unsteady two-phase flows in vertical pipes shows the high precision of the model. Combing with the robustness test results, it can be found that the present model can accurately predict the maximum flow rate of the two-phase flow at the outlet of the gas lift pump, as well as the average gas and liquid flow rates of the two-phase flow at the outlet in both steady and unsteady states. On this basis, several operation laws of the deep-sea mining airlift pump were studied. The relationship between input and output in the mining process is established and the output of the gas lift pump system is predicted. It is found that changing the position of the injected gas, the time for the two-phase flow in the vertical pipe to reach dynamic equilibrium varies equiprimordially. And changing the maximum amount of gas injected, the time required to reach dynamic equilibrium shows a symmetric relationship with the volume fraction after dynamic equilibrium. A series of laws closely related to the prediction of rare earth mud at the outlet of gas-lift pumps in deep-sea mining are investigated to be proposed for subsequent studies.

Investigating the Relationship between the Time Constant Ratio and Plug-Flow Behaviour in the Pneumatic Conveyance of Biomass Material

This study introduces a novel methodology to evaluate the behaviour of biomass material by examining the ratio of aeration and deaeration time constants. To this end, a series of tests were conducted on four different materials, namely, cottonseed, wood chips, wood pellets, and wheat straw, in order to investigate their aeration and deaeration behaviours. The study derives the aeration and deaeration pressure drop equations, and discusses the corresponding time constant expression. Subsequently, the four materials were conveyed in 12 m long batch-fed and continuous pneumatic conveying pipelines to examine their behaviour in longer pipelines. The results indicate that the aeration and deaeration time constants increased with an increase in air superficial velocity. However, the ratio of the aeration and deaeration time constants was identified as a unique number, where a value close to 1 indicates a higher likelihood of plug flow. On the basis of the results, cottonseed, with the lowest ratio of time constant, was more likely to form a stable plug flow in both batch-fed and continuous pneumatic conveying. Given the unique properties of biomass and the limited research on the pneumatic conveyance of biomass, this methodology represents a novel approach for predicting modes of flow in materials with complex properties.

A spectral boundary integral method for simulating electrohydrodynamic flows in viscous drops

A weakly conducting liquid droplet immersed in another leaky dielectric liquid can exhibit rich dynamical behaviors under the effect of an applied electric field. Depending on material properties and field strength, the nonlinear coupling of interfacial charge transport and fluid flow can trigger electrohydrodynamic instabilities that lead to shape deformations and complex dynamics. We present a spectral boundary integral method to simulate droplet electrohydrodynamics in a uniform electric field. All physical variables, such as drop shape and interfacial charge density, are represented using spherical harmonic expansions. In addition to its exponential accuracy, the spectral representation affords a nondissipative dealiasing method required for numerical stability. A comprehensive charge transport model, valid under a wide range of electric field strengths, accounts for charge relaxation, Ohmic conduction, and surface charge convection by the flow. A shape reparametrization technique enables the exploration of significant droplet deformation regimes. For low-viscosity drops, the convection by the flow drives steep interfacial charge gradients near the drop equator. This introduces numerical ringing artifacts that we treat via a weighted spherical harmonic expansion, resulting in solution convergence. The method and simulations are validated against experimental data and analytical predictions in the axisymmetric Taylor and Quincke electrorotation regimes.

A review of frictional pressure drop characteristics of single phase microchannels having different shapes of cross sections

Abstract This paper theoretically studied pressure drop variation in microchannels having different cross sections (circular, rectangular, square, trapezoidal, triangular, elliptical, parallel plate, co-centric circles, hexagonal, wavy, smoothed or rounded corners cross sections, and rhombus) for single phase Newtonian fluid (gas and liquid) flow. Based on 41 years (approximately) prior literature (1981–till now), 249 articles were studied and number of correlations of pressure drop calculation in microchannels with or without friction factor equation for four cross sections i.e., rectangular, square, circular, trapezoidal, wavy and triangular is collected and also mentioned their limitations at one place. Other than these four cross sections, there is very few experimental/numerical works was present in the literature. A comparable study was performed for laminar as well as turbulent friction factor to calculate the pressure drop with the help of classical theory for gas and liquid flow in microchannels with circular and rectangular cross sections. Results show wonderful outcomes i.e., correlations of laminar pressure drop study can be extendable for transition and turbulent regime in both types (circular and rectangular) of cross sections of microchannels. In different types of flow regime, it is suggested that for each type of cross section (circular and rectangular) we can go for single correlation for gas/liquid system. It is also investigated that the macro channels pressure drop equations can be used for microchannels up to the certain values of Reynolds number. Basically, this paper provides all possible equations of friction factor related to the microchannels that helps to calculate the pressure drop, is collected at one platform also compared their deviation with conventional channels.

Dynamic modelling of trickle bed reactor: Case study of arabinose oxidation

Trickle beds are among the most used reactors in different sectors of chemical industries. In this work the aim was to develop a general heterogeneous multiscale model for continuous trickle bed reactors, which enables the calculation of instantaneous concentration and temperature changes as well as the stationary behaviour of the reactor. The model development was based on solving simultaneously both the energy and mass balances for the continuous gas and liquid phases and for the stagnant catalyst particles. Mass transfer coefficients and pressure drop equations from well-established correlations were used. The developed model is aimed to be as general as possible in order to be used as a framework for other kind of bed reactors and arbitrary reaction schemes.The model can be simplified taking into consideration only one or two phases, and in this case it was applied to catalytic sugar oxidation to sugar acid and the effect of catalyst shape and size was investigated. Simulation results revealed that complete sugar conversion was achieved only for small catalyst particles, also the length of the reactor affects the conversion of arabinose more than the residence time or the bed radius: in fact for much bigger particles the conversion can also be achieved for longer reactor beds.

Development and validation of a general two-fluid thermal-hydraulic analysis code for reactors with plate-type fuel assemblies

A general two-fluid thermal-hydraulic analysis code, TTHAC-PFA (Two-fluid Thermal-Hydraulic Analysis Code for reactors using Plate-type Fuel Assemblies), has been developed. The code was developed on basis of the fundamental governing equations. Reasonable mathematic and physical models for parallel rectangular channels and fuel plates have been built up to meet the command of the safety analysis for plate-type fuel reactors. The coolant distribution is conducted on basis of the pressure drop equation theory. Normal subchannel approach is adopted in the code. Fortran language is adopted. The RA-6 experiment is used to evaluate the heat transfer models. The max deviation of the simulations from the measured value is about 5.7 °C. Then, experiments on two-phase flow pressure drop are adopted to validate the code. Approximately 97% of the data points are within ±20% of the deviation range. In addition, CARR (China Advanced Research Reactor) is adopted as the benchmark for normal condition validation. Flow distribution validation and temperature profile validation are conducted. The flow validation presents negligible mean discrepancy and the temperature differences are within the scope of 6 °C. All validations are in good agreement, which indicates the developed code meet the requirements of analyzing thermal-hydraulic characteristics for parallel rectangular channels in reactors. Furthermore, TTHAC-PFA is applied to the investigate channel blockage accident occurred in the CARR. The mass flow is redistributed among channels due to the obstruction. In the obstructed channel and two nearby channels, the coolant temperatures increase a lot. The coolant temperature in the obstructed channel rises by 60 °C. In the rest of the channels, the coolant temperatures decrease with the increase of coolant flow. Besides, the two fuel plates, which form the blocked channel, are asymmetrically cooled due to flow maldistribution. The temperature difference between the cladding on both sides of the fuel plate can reach 40 °C. As a result of partial blockage, the cladding temperature on the side of the blocked channel rises by 63 °C. During the blockage accident, the plate temperature is under the melting temperature. The results indicate the significance of studying the effects of the adjacent channels near the obstructed channel for reactors with plate-type fuel elements.

Calculation Model of Bottom Hole Flowing Pressure of Double-Layer Combined Production in Coalbed Methane Wells

Bottom hole flowing pressure (BHFP) is the key factor to determine a reasonable working system and achieve long-term stable production of coalbed methane (CBM) wells. However, there is no special BHFP model for double-layer combined production (DLCP). Generally, the constant mass model (CMM) for single-layer production is applied to treat the double reservoirs as a whole, ignoring the changes of fluid mass in each section and the acceleration pressure drop in the reservoir section. The calculation results have great errors, and the BHFP of the lower reservoir is used to adjust the production system of the two reservoirs, which does not meet the requirements of the upper reservoir. In this paper, the expression of acceleration pressure drop assumed to be zero in CMM is decomposed and derived, the relationship between acceleration pressure drop and unit length radial flow is established, and then the pressure drop formula of reservoir section with radial inflow is obtained. The reservoir is divided into several sublayers, and the pressure drop equation for each sublayer is established. According to the water flow and gas flow in the reservoir and nonreservoir sections, the corresponding velocity equations of water phase and gas phase are derived. The above equations are combined to establish the variable mass model (VMM) for DLCP with three stages. The field data are substituted into the VMM and the CMM, and the accuracy of the new model is verified. The results show that in the stages of double-layer water production and double-layer gas production, the errors of the two models are less than 5%, while in the stage of gas-water coproduction, the error of the VMM is 2.75%-6.58%, and the error of the CMM is 7.15%-15.18%. The VMM is more accurate. In addition, in the stages of water production and gas-water coproduction during DLCP, the BHFP of the two reservoirs differs greatly, with a maximum difference of 49.1%. Therefore, the two reservoirs need to adjust the production rule according to their respective BHFP. To sum up, the VMM can accurately give the BHFP of each reservoir, which is more realistic. It also solves the problem that one BHFP cannot accurately adjust the production rule of the two reservoirs, so as to provide technical support for the formulation of optimal production rule and the realization of high production.

Study on the Cavitation Suppression Mechanism of Axial Piston Pump

Gaseous and vaporous cavitation is extremely harmful to axial piston pumps, such as reducing flow rate, increasing flow pulsation, increasing vibration, increasing noise, and shortening life. To suppress the cavitation and improve the performance of axial piston pumps, a mathematical model for suppressing cavitation in the plunger chamber with a constant theoretical flow rate is innovatively established by combining the flow equation of the plunger pump and the pressure drop equation of the plunger chamber. Based on the model, two methods to suppress cavitation in the plunger chamber under the condition of a constant theoretical flow rate are proposed. The first method is to increase the distribution circle radius and correspondingly reduce the rotation speed, and the second method is to increase the plunger chamber radius and correspondingly reduce the rotation speed. To verify the effectiveness of these two methods, the CFD model of the axial piston pump is established, and the correctness of the model is verified by experiments. The results show that the two methods can effectively suppress cavitation in the plunger chamber, improve the actual flow rate, and reduce the flow pulsation under the condition of a constant theoretical flow rate. The research results can provide an important reference for the design and optimization of the plunger pump.

Stepwise mathematical derivation of the Herschel–Bulkley laminar fluid flow equations—in pipes

Stepwise derivation of flow equations of the Herschel–Bulkley (HB) model is not available in the literature. These equations are crucial for mechanical, chemical and petroleum engineering academia and industries where fundamental works on non-Newtonian fluids may be done to reach future models and estimation methods. Therefore, this work focuses on derivation of laminar flow equations and estimation methods of HB fluids through pipes. In this work, first, stepwise derivation of the HB fluid flow parameters consisting of fluid velocity, flow rate, average velocity and relative velocity equations is presented, followed by a straightforward mathematical model for use in numerical solution. Next, stepwise mathematical derivation of the laminar pressure drop equations by Merlo et al. (An innovative model for drilling fluid hydraulics. Paper presented at the SPE Asia Pacific oil and gas conference, Kuala Lumpur, Malaysia, 1995) and Gjerstad and Time (SPE J 20:1–18, 2014) is presented, and finally practical and user-friendly calculation procedures for different estimation methods are presented. The step-by-step derivation procedures presented in this work contribute to effective learning for engineering students and practitioners in addition to providing a clear example derivation guideline for future researchers to reach other more accurate non-Newtonian hydraulics models and estimation methods.

Modelling the minimally fluidized state under reduced pressure

Several aspects of numerically modelling a minimally fluidized gas–solid system have been investigated in this work. The numerical results show that voidage and the resulting pressure drop are not a function of the fluidizing cycle. More interestingly, the pressure drop was not impacted by introducing the lateral axis of gas and solid flow in the 3D models. Under a reduced pressure environment, none of the well-known drag models could capture the effect of the slip flow. A relatively new but not well-known slip flow drag model showed the ability to capture the impact of the slip flow regime. However, improvements in its overall accuracy are desirable. To this extent, the Ergun pressure drop equation was modified to introduce the effect of the slip flow regime. The losses in the slip flow regime were captured by deriving a new correlation using experimental work that predicted a linear relationship between the laminar coefficient and the Knudsen number. The modified Ergun equation showed notable improvement in its pressure drop accuracy. Furthermore, the modified Ergun equation was implemented as a modified Gidaspow drag model. It showed better accuracy in predicting pressure drop and minimum fluidization velocity at reduced pressure for various alumina particle sizes.

Hydrodynamics and Scaling Laws of Gas–Liquid Taylor Flow in Viscous Liquids in a Microchannel

The hydrodynamics of gas–liquid Taylor flow in microchannels is vital to the microreactor’s application, but its characteristics in viscous liquids are rarely reported. Accordingly, this study concentrates on the characteristics of bubble behavior in viscous liquids (45.6–240.5 mPa·s). The results show that the viscosity effect could dominate the bubble morphology. Importantly, considering the liquid film distribution, a novel parameter as the equivalent liquid film thickness is proposed for the square microchannel and it is strongly related to the capillary number and gas–liquid flow rate ratio. Besides, the bubble velocity first decreases to a minimum value and then becomes faster with the flow rate of the liquid, and the ideality of the multiphase system decreases with the increase of liquid viscosity and pressure drop. Finally, the viscosity term is introduced to the pressure drop equation based on the Lockhart–Martinelli type. This work enriches the understanding of the viscosity effect on gas–liquid microflows.

Generalized optimization-based synthesis of membrane systems for multicomponent gas mixture separation

Synthesizing a membrane system to separate multicomponent gas mixture is challenging due to the combinatorial number of feasible configurations and the difficulties in describing the multicomponent permeators. We present a mixed-integer nonlinear programming (MINLP) model for synthesizing membrane systems for multicomponent gas mixture separation. The approach employs a richly connected superstructure to represent numerous potential system configurations, and different physics-based surrogate permeator models, such as countercurrent flow or crossflow, to be used in each stage. Moreover, to describe realistic systems, pressure drop equations can be included. We also present solution methods to accelerate the solution process. Through a case study of natural gas sweetening, we demonstrate that the proposed approach is able to obtain good solutions using an off-the-shelf global optimization solver. Finally, we expand the conventional membrane system synthesis problem by introducing feed variability in our model through a case study of an integrated reactor-separation system.

Pressure drop and optimization meta-models for arbitrary low-height pleated filter shapes and flowrates

This study delivers equations useful for low-height pleated fibrous filter design: two pressure drop equations and one set of optimum design equations applicable to arbitrary pleated filter shapes and flowrates. The pressure drop equations were derived to predict the pressure loss of the pleated filter. They were made through regression analysis with a total of 1024 CFD data. The set of optimum design equations was developed to find the optimum filter shape minimizing pressure drop. All equations were validated through the 8-fold cross-validation method and were accurate enough to replace the CFD simulations. Additionally, novel contour plots were made to describe how optimum filter geometry changes due to flowrate, height, and media permeability. The delivered equations were applied to an actual filter design problem and verified with additional CFD simulations. This study allows filter designers to predict the pressure loss and to design the optimum filter shape without any simulations.