Drop Equation Research Articles

Effect of refining and homogenization on nanocellulose fiber development, sheet strength and energy consumption

Nanocellulose has great potential for end use in the pulp and paper industry, but its mechanical production process is often extremely energy intensive. Although various pre-treatment methods have been introduced to significantly lower the energy consumption of nanocellulose production, there still remains a great challenge to match the cost to mechanical strength performance of nanocellulose. This research investigates the relationship between energy consumption with numerous refining and homogenization cycles towards the production of high aspect ratio and low diameter nanocellulose, and whether this is necessary to achieve high strength sheets. Never-dried Bleached Eucalyptus Kraft pulp was mechanically treated with a PFI mill refiner and homogenizer. The actual homogenization energy measured on a lab scale was significantly higher than that predicted only from the simplified pressure drop equation which is inaccurately independent of homogenizer design and size. We measured two aspects of fiber quality believed to be related to sheet mechanical strength: the fiber diameter, which is inversely proportional to the surface area available for bonding, and the fiber aspect ratio, which is believed to control the efficiency of stress transfer within a network. Both refining and homogenization reduced the fiber diameter and increased the aspect ratio, but homogenization was more effective at both. There was surprisingly little correlation between the measures of fiber quality and mechanical sheet strength. All PFI mill refined samples had a tensile index of around 100 Nm/g, while additional treatment with homogenization increased this only by approximately 20%. This was despite the most heavily treated fibers (50,000 PFI revolutions and 3 homogenization passes at 1000 bar) having a median diameter of 12 nm and aspect ratio of 229 compared to the least treated fibers (10,000 PFI revolutions) with a median diameter of 31 nm and aspect ratio of 102. The most heavily treated fibers required 39,000 kWh/t of energy, a tenfold increase over the least heavily treated fibers. The results confirmed that there was small benefit, in terms of mechanical strength, of using very large amounts of energy to produce very low diameter, high aspect ratio nanocellulose fibers.

Electrohydrodynamics of Drops and Vesicles

The 1969 review by J.R. Melcher and G.I. Taylor defined the field of electrohydrodynamics. Fifty years on, the interaction of weakly conducting (leaky dielectric) fluids with electric fields continues to yield intriguing phenomena. The prototypical system of a drop in a uniform electric field has revealed remarkable dynamics in strong electric fields such as symmetry-breaking instabilities (e.g., Quincke rotation) and streaming from the drop equator. This review summarizes recent experimental and theoretical studies in the area of fluid particles (drop and vesicles) in electric fields, with a focus on the transient dynamics and extreme deformations. A theoretical framework to treat the time evolution of nearly spherical shapes is provided. The model has been successful in describing the dynamics of vesicles (closed lipid membranes) in an electric field, highlighting the broader range of applicability of the leaky dielectric approach.

A heuristic approach for nanodrops on a smooth solid surface.

A heuristic approach is developed to obtain a simple equation for the contact angle of a nanodrop on a smooth planar solid surface. First, nanodrops of various fluids in contact with various solid surfaces are considered on the basis of nonlocal density functional theory (DFT). Along with the traditional (apparent) contact angle, θa, which the drop profile makes with the solid surface, another one, θd, formed by the smooth part of the drop profile and the horizontal plane separating that part from the oscillatory part of the profile was examined. For each of the contact angles, a separate simple equation resembling the Young equation for the macroscopic drops but containing, instead of surface tensions, the microscopic parameters of intermolecular interactions, temperature, and average density of the fluid was hypothesized and the parameters of this equation were determined using the results of DFT calculations. It was shown that predictions of these equations coincide with the results provided by DFT.

Accurate Modeling of the Nonlinear Characteristic of a Voltage Source Inverter for Better Performance in Near Zero Currents

In a sensorless vector control method, at low speeds and self-commissioning for AC motors, it is necessary to model dead time, rise/fall time, parasitic capacitors, and semiconductors’ voltage drop accurately. In this paper, voltage drop in each phase due to the nonlinear behavior of the inverter from the control system's viewpoint is precisely modeled by using two lookup tables. These lookup tables make the proposed model robust against the dc-link voltage and carrier frequency variations in the entire current range. In conventional methods at standstill, the measured voltage in the dc current test contains the nonlinearity of two phases. Therefore, to obtain the nonlinear voltage of each inverter phase, it is assumed that the values of inverter nonlinearity voltage at both high current and low current are equal. This simplification leads to distortion in measured voltage in the low currents. However, in this paper, the voltage drop equation of each phase is solved by numerical method that is realizable in conventional processors. The results obtained in actual voltage measurement show the superiority of the proposed method over traditional methods in the low currents. The proposed method is verified by using a voltage source inverter feeding a 2.2-kW three-phase induction motor.

Examination of Averaged Stress Drop Equations Connecting Outer and Inner Fault Parameters for Strong Motion Prediction

Examination of Averaged Stress Drop Equations Connecting Outer and Inner Fault Parameters for Strong Motion Prediction

CO2 Pipeline Design: A Review

There is a need to accurately design pipelines to meet the expected increase in the construction of carbon dioxide (CO2) pipelines after the signing of the Paris Climate Agreement. CO2 pipelines are usually designed with the assumption of a pure CO2 fluid, even though it usually contains impurities, which affect the critical pressure, critical temperature, phase behaviour, and pressure and temperature changes in the pipeline. The design of CO2 pipelines and the calculation of process parameters and fluid properties is not quite accurate with the assumption of pure CO2 fluids. This paper reviews the design of rich CO2 pipelines including pipeline route selection, length and right of way, fluid flow rates and velocities, need for single point-to-point or trunk pipelines, pipeline operating pressures and temperatures, pipeline wall thickness, fluid stream composition, fluid phases, and pipeline diameter and pressure drop calculations. The performance of a hypothetical pipeline was simulated using gPROMS (ver. 4.2.0) and Aspen HYSYS (ver.10.1) and the results of both software were compared to validate equations. Pressure loss due to fluid acceleration was ignored in the development of the diameter/pressure drop equations. Work is ongoing to incorporate fluid acceleration effect and the effects of impurities to improve the current models.

Validation of the CFD-DPM solver DPMFoam in $$\hbox {OpenFOAM}^\circledR $$ OpenFOAM ® through analytical, numerical and experimental comparisons

Multiphase flows are relevant in several industrial processes mainly because they are present in the production of a large diversity of products. Hence, the availability of accurate numerical modeling tools, able to cope with this type of flows, is of major significance to provide detailed information about the system characteristics, in order to guide the design activity. This study presents a detailed assessment of a multiphase flow solver able to couple Eulerian and Lagrangian phases, the last modeled through the discrete particle method. The numerical code is already implemented in the open source computational fluid dynamics software package $$\hbox {OpenFOAM}^\circledR $$ . The solver (DPMFoam) is firstly used to simulate the collision between two particles, for which a good correlation was obtained with the theoretical impulse force value. Subsequently, the solver is employed in the simulation of a pseudo 2D gas-solid flow in a fluidized bed. In this case study, the results obtained for the bubble patterns, time-average flow patterns, bed expansion dynamics and particle phase energy analysis are in agreement with the experimental and numerical results available in the literature. In addition, the numerical pressure drop for the fluidized bed is computed and compared with the analytical Ergun’s pressure drop equation. The accuracy of the numerical results was found to be sensitive to the solid fraction estimation.

An MILP model for the reformation of natural gas pipeline networks with hydrogen injection

A mixed integer linear programming (MILP) model is proposed for the reformation of natural gas pipelines. The model is based on the topology of existing pipelines, the load and pressure at each node and the design factors of the region and minimizes the annual substitution depreciation cost of pipelines, the annual construction depreciation cost of compressor stations and the operating cost of existing compressor stations. Considering the nonlinear pressure drop equations, the model is linearized by a piecewise method and solved by the Gurobi optimizer. Two cases of natural gas pipeline networks with hydrogen injection are presented. Several adjustments are applied to the original natural gas pipeline network to ensure that our design scheme can satisfy the safety and economic requirements of gas transportation. Thus, this work is likely to serve as a decision-support tool for the reformation of pipeline networks with hydrogen injection.

Assessment of two-equation model for simulation of air pocket advancing into a rectangular duct

ABSTRACTThis research aims to evaluate the influence of air on the water surface and pressure profiles in two-phase (air–water) flow. In previous studies, the authors used the continuity and momentum equations for water in a one-dimensional finite volume model. In the present study, the authors additionally incorporate both the mass conservation and momentum equations for incompressible fluids as basic equations for the air portion. Momentum equations were integrated for the free surface and pressurized flow regions after interface observations made at each time step. To correct velocity and pressure in the pressurized portion, the authors used the highly simplified mark-and-cell method. A pressure drop equation is used at the interface of pressurized and free surface flows to deduce the pressure. The Harten total variation diminishing scheme was used to avoid numerical oscillations. The results were compared with experimental data and numerical simulations without considering the air effect.

A re-evaluation of design equations for the established flow region of vertical dilute-phase pneumatic conveyors

An analysis of data sets from the literature was completed for particle velocities of granular material in vertical, dilute-phase pneumatic conveyors. The generally accepted equation of motion used to describe such velocity data incorporates the solids friction factor, a variable term based on the additional pressure drop due to solids. The analysis determines that rather being a variable, the term is a constant related to the physical properties of the particle that influence its bouncing behaviour as it progresses along the pipeline. In addition, the analysis reveals that particle drag coefficients determined from the particle velocity data differ significantly from values taken from the standard drag coefficient curve. This observation was confirmed by drag coefficient measurements reported in the literature and made using particle-elutriating equipment. The drag coefficients from the pneumatic conveying experiments and the test equipment were determined under turbulent conditions, which is considered to account for the difference in values from the standard curve determined under quiescent conditions. Relationships are presented that predict the turbulence-influenced drag coefficients and the particle velocity data with good accuracy. These relationships are used to predict pressure drops for two data sets presented with the particle velocity data. The single-term pressure drop equation is used and predicts with good accuracy as well as producing the trough-shaped curve characteristic of the measured pressure drops.

Simplifications for hydronic system models in modelica

Building systems and their heating, ventilation and air conditioning flow networks, are becoming increasingly complex. Some building energy simulation tools simulate these flow networks using pressure drop equations. These flow network models typically generate coupled algebraic nonlinear systems of equations, which become increasingly more difficult to solve as their sizes increase. This leads to longer computation times and can cause the solver to fail. These problems also arise when using the equation-based modelling language Modelica and Annex 60-based libraries. This may limit the applicability of the library to relatively small problems unless problems are restructured. This paper discusses two algebraic loop types and presents an approach that decouples algebraic loops into smaller parts, or removes them completely. The approach is applied to a case study model where an algebraic loop of 86 iteration variables is decoupled into smaller parts with a maximum of five iteration variables.

Modeling and simulation of a pressure–temperature swing adsorption process for dehydration of natural gas

Water vapor in a natural gas stream can result in line plugging due to hydrate formation, reduction of line capacity due to collection of free water in the line, and increased risk of damage to the pipeline due to the corrosive effects of water. Therefore, water vapor must be removed from natural gas to prevent hydrate formation and corrosion from condensed water. Molecular sieves are considered as one of the most important materials that are used as desiccant materials in industrial natural gas dehydration. This paper focuses on modeling of pressure–temperature swing adsorption (PTSA) process in a commercial two-layer adsorption system during sequential steps of cyclic operation to remove water from natural gas. The dynamic model equations were constructed from 4 mol balances; model molecule of water vapor in natural gas, a total mass balance, a pressure drop equation and two energy balances of solid and gas phases in the adiabatic column. Results reveal that about 1 m of the bed was overdesigned by vendor and it doesn’t work in optimum condition. The influential parameters of the process were investigated through a parametric analysis of the process efficiency. For the sake of energy saving some suggestions were proposed for upgrading the design conditions with no significant effect on the purification performance. Results show that optimum conditions will be obtained using adsorption time 22 h and regeneration time 9 h with maximum temperature of 240 °C.

The development of a dynamic single effect, lithium bromide absorption chiller model with enhanced generator fidelity

Single effect, lithium bromide absorption chillers offer the ability to utilize low-pressure steam to produce chilled water for satisfying various comfort cooling needs. Previous attempts have been made to characterize dynamic and steady-state absorption chiller operation. Though these models perform adequately, they are based on hot water driven absorption chillers. Commercially available absorption chillers often can run on both hot water and low-pressure steam. In this paper, the mathematical framework for a dynamic single effect, lithium bromide absorption chiller model capable of using low-pressure steam is presented. The transient thermodynamic FORTRAN model is grounded on mass, energy, and species balances, and builds on prior modeling efforts. Well-known correlations for heat transfer coefficients are used to describe both tube-side and shell-side heat transfer rates in each primary chiller component. To account for the absorption chiller unit receiving steam, a heat transfer model for condensation inside horizontal tubes based on distinct internal condensation flow regimes is incorporated within the generator. This heat transfer model is used with two-phase flow pressure drop equations to establish steam temperature, quality, and pressure along the generator tube bundle. Steam consumption trends are established asa function of fluctuating external conditions. These trends reasonably align with information made available online by the manufacturer, though some deviation does occur at low chiller capacities and cooling water temperatures. Additionally, the transient response of internal and external parameters from a step increase in heat input supplied to the generator mimics results of other dynamic absorption chiller models found throughout literature.

Theoretical and experimental research on the grout fronts diffusion considering filtration effect

Based on the seepage continuity equation, Darcy law, mass conservation equation, linear filtering law and pressure drop equation, the theoretical model of grout fronts diffusion was established. And through the model, the filtration mechanism of cement particles and the diffusion law of slurry fronts were analyzed. By applying the model test system developed independent, the dynamic variation law of the pressure of grout fronts is studied and compared with the theoretical value. The results show that, the permeability coefficient of the precipitated cement increases along the grout diffusion distance under the filtration effect. The change rate of frontal velocity is negatively correlated with the permeability coefficient of precipitated cement. At the specific diffusion distance, the frontal pressure drop p increases first and then decreases, while the frontal pressure p and the average unit pressure drop Δ P show the trend of decreasing first and then increasing. The deviation rate of frontal pressure value between theory and test is only from –10% to 26%, which reveal a high degree of coincidence. The conclusions can be used to guide the grouting design, which has certain application value to the practical engineering.

Dripping to jetting transition for cross-flowing liquids

We experimentally study drops formed from a nozzle into an immiscible, cross-flowing phase. Depending on the operating conditions, drops are generated either in dripping or jetting mode. We investigate the impact of the continuous and dispersed phase velocities, dispersed phase viscosity, and interfacial tension on the drop generation mode and size. We find that a dripping to jetting transition (DJT) takes place at a critical inner Weber number, a function of the outer capillary and Ohnesorge numbers. Two jetting regimes occur depending on the phase velocity ratio. When the continuous phase velocity is significantly greater (respectively lower) than the dispersed phase velocity, jet narrowing (respectively widening) occurs. In jet widening, the critical inner Weber number depends little on the outer capillary number whereas in jet narrowing, it sharply decreases as the outer capillary number increases. We propose a comprehensive model to describe the DJT based on the attached drop equation of motion. The model satisfactorily predicts the DJT and the effect of the outer capillary number on the critical inner Weber number. It also well accounts for the drop diameter in jet narrowing.

Analytical and Numerical Calculation of the Orifice Minimum Temperature Due to Joule - Thomson Effect

High pressure drop generated by a restriction orifice may result in a very low temperature, which can affect the piping material and may cause catastrophic piping failure if the operating temperature becomes lower than the minimum design temperature. This minimum design temperature is stated by piping ASME B31.3 code as -48°C. In such piping research branch, there has been relatively little investigation of very low temperature effect on pipelines. As well as, sizing the orifice with implementing temperature control to match piping material has a few analytical explanations, particularly in investigating the influence of Joule - Thomson effect on piping damage. Most commercial orifice sizing software ignore Joule - Thomson effect even though in choked flow condition. The objective of the present research is to compare a derived analytical equation with 3-D computational calculations by using ANSYS 16.0 for Joule - Thomson temperature drop through the orifice. As well as correlate the analytical equation to be safely considered as a good prediction tool for the lowest temperature at orifice throat instead of misleading ISO 5761 fully developed Joule - Thomson temperature drop. The analytical equation correlation has been carried out based on non-linear regression by grouping flow conditions, fluid properties, and orifice geometry, for minimum temperature prediction at orifice Vena-contracta. The numerical temperature differences in the fully developed flow regime after the office have been compared with EN ISO 5761-Part 3 Joule - Thomson temperature drop equation. Three orifices with β ratios, 0.3, 0.4, and 0.5 have been chosen for such study and numerical simulations have be carried out using k-e and k-ω turbulence models. As a corollary of this study, it was concluded that the k-e and k-ω models predict well both the flow and the fully developed temperature drop as compared with ISO 5761 equations. The errors are generally accepted at all conditions and both values give good agreement. The derived equation successfully predicts the lowest minimum temperature at Vena-contracta and can supersede ISO 5761-Part 3 Joule - Thomson temperature drop at fully devolved region.

The significance of drop non-sphericity in sprays

This paper presents a new framework to model drop dynamics in Lagrangian sprays. The framework builds on the Taylor Analogy Breakup (TAB) model. Real-fluid (gas-liquid) thermodynamics applicable to multicomponent systems are combined with Gradient Theory to facilitate detailed calculations of drop surface tension forces, oscillations, and breakup processes. This is combined with a more detailed treatment of deforming drop dynamics to construct more accurate representations of the local interfacial exchanges of mass, momentum, and energy. The framework is derived using an energy balance equation that explicitly enforces drop momentum conservation during the breakup process. This facilitates development of a refined set of drop equations that address current shortcomings in the prediction of drop properties over a wide range of relevant breakup conditions. The resulting drag forces, evaporation, and heating rates deviate significantly from the predictions given by contemporary drop models used in modern simulations. These deviations are quantified using Large Eddy Simulation (LES) with a Lagrangian-Eulerian modeling approach. The analysis demonstrates how the model improvements in the new framework provides a more detailed representation of physical complexities that are largely neglected in modern studies.

Application of a Pore Fraction Hot Tearing Model to Directionally Solidified and Direct Chill Cast Aluminum Alloys

Hot tearing susceptibility is commonly assessed using a pressure drop equation in the mushy zone that includes the effects of both tensile deformation perpendicular to the thermal gradient as well as shrinkage feeding. In this study, a Pore Fraction hot tearing model, recently developed by Monroe and Beckermann (JOM 66:1439–1445, 2014), is extended to additionally include the effect of strain rate parallel to the thermal gradient. The deformation and shrinkage pore fractions are obtained on the basis of the dimensionless Niyama criterion and a scaling variable method. First, the model is applied to the binary Al-Cu system under conditions of directional solidification. It is shown that for the same Niyama criterion, a decrease in the cooling rate increases both the deformation and shrinkage pore fractions because of an increase in the time spent in the brittle temperature region. Second, the model is applied to the industrial aluminum alloy AA5182 as part of a finite element simulation of the Direct Chill (DC) casting process. It is shown that an increase in the casting speed during DC casting increases the deformation and shrinkage pore fractions, causing the maximum point of pore fraction to move towards the base of the casting. These results demonstrate that including the strain rate parallel to the thermal gradient significantly improves the predictive quality of hot tearing criteria based on the pressure drop equation.

Numerical–analytical assessment on Manzanares prototype

In this study an appropriate expression to estimate the output power of solar chimney power plant systems (SCPPS) was considered. Recently several mathematical models of SCPPS were derived, studied for a variety of boundary conditions, and compared against CFD calculations. An important concern for modeling SCPPS is the accuracy of the derived pressure drop and output power equation. To elucidate the matter, axisymmetric CFD analysis was performed to model the solar chimney power plant and calculate the output power for different available solar radiation. Both analytical and numerical results were compared against the available experimental data from the historical Manzanares power plant. We also evaluated the fidelity of the assumptions underlying the derivation and present the output power characteristics of Manzanares prototype under a range of solar irradiation, mass flow rate and collector efficiency. This research provides an approach to estimate the output power with respect to available radiation to the collector.

Modeling of a forward osmosis and a pressure-retarded osmosis spiral wound module using the Spiegler-Kedem model and experimental validation

Abstract A mathematical model is developed for a spiral wound module for both forward osmosis (FO) and pressure-retarded osmosis (PRO) applications. This model is obtained by combining the membrane transport, external concentration polarization, and internal concentration polarization models with the pressure drop and mass balance equations for both the feed as well as the draw solutions. The Spiegler-Kedem (SK) model is used for describing the local mass transport in the active layer of the membrane. In addition, a new expression is established for determining the internal concentration polarization in the support layer of the membrane. A nonlinear constrained optimization technique is then applied, along with the experimental data for FO and PRO, to predict the unknown parameters of the mathematical model for the spiral wound module. To ensure the validity of the proposed model and predicted parameters, the model is again used to predict and compare the performance of the module at a variety of operating conditions. Apart from this, the product recovery and the energy extraction efficiency of the module at a variety of operating conditions are also analyzed. Lastly, we have tried to answer the question, whether the Solution Diffusion (SD) based model or the SK based model is better for design and analysis, and for obtaining the optimal operating conditions for both FO as well as PRO applications.