Non-dimensional Groups Research Articles

Unsteady bipolar diffusion charging in aerosol neutralisers: A non-dimensional approach to predict charge distribution equilibrium behaviour

High total particle concentration and small particle size are common features of aerosols encountered in the field of aerosol-based nanotechnology that can potentially lead to non-equilibrium issues in the neutraliser upon SMPS characterisation, resulting in large errors in size distribution measurements. Experiments show that the commonly assumed n⋅t product rule fails to predict equilibrium behaviour in aerosol neutralisers under these conditions, as it does not capture the influence of total particle concentration and particle size. The aim of this work is to provide an equilibrium indicator that identifies situations where equilibrium is not reached in the neutraliser as a function of residence time, ion generation rate, total particle concentration, and particle size. Bipolar diffusion charging equations are solved numerically in a one-dimensional model first, and a non-dimensional analysis of the results is carried out in order to map equilibrium behaviour as a function of two non-dimensional groups, the non-dimensional ion concentration, and the non-dimensional neutraliser residence time. Solving the three-dimensional form of the charging equations in the geometry of the neutraliser then enables one to find good agreement in terms of equilibrium behaviour between experiments and predictions from the non-dimensional model. The three-dimensional model captures the complexity of the physics of unsteady particle charging inside a neutraliser. This work then discusses this as a new approach to non-equilibrium behaviour prediction in neutralisers, providing a tool supplementing the n⋅t product rule that can be used in practice.

КОМПЬЮТЕРНЕ МОДЕЛЮВАННЯ ХОЛОДИЛЬНОЇ МАШИНИ СТІРЛІНГА

In present numerical research, the mathematical model for precise performance simulation and detailed behavior of Stirling refrigerating machine is considered. The mathematical model for alpha Stirling refrigerating machine with helium as the working fluid will be useful in optimization of these machines mechanical design. Complete non-linear mathematical model of the machine, including thermodynamics of helium, and heat transfer from the walls, as well as heat transfer and gas resistance in the regenerator is developed. Non-dimensional groups are derived, and the mathematical model is numerically solved. Important design parameters are varied and their effect on Stirling refrigerating machine performance determined. The simulation results of Stirling refrigerating machine which include heat transfer and coefficient of performance are presented.

Natural dispersion revisited

This paper presents a new semi-empirical model for oil droplet size distributions generated by single breaking wave events. Empirical data was obtained from laboratory experiments with different crude oils at different stages of weathering. The paper starts with a review of the most commonly used model for natural dispersion, which is followed by a presentation of the laboratory study on oil droplet size distributions formed by breaking waves conducted by SINTEF on behalf of the NOAA/UNH Coastal Response Research Center. The next section presents the theoretical and empirical foundation for the new model. The model is based on dimensional analysis and contains two non-dimensional groups; the Weber and Reynolds number. The model was validated with data from a full scale experimental oil spill conducted in the Haltenbanken area offshore Norway in July 1982, as described in the last section of the paper.

On the rupture of a round liquid jet: the primary atomization process

The present study analyzes the process of fragmentation of a liquid volume into a cloud of droplets, a well-known phenomenon usually denoted as primary atomization. The experimental work was performed in a single-orifice pressure-swirl atomizer. For the first time, a transition was observed in the droplet size distribution function (DSDF) that transforms from a Nukiyama–Tanasawa type to a power law dependence described by the function d −x . A non-dimensional group including atomizing pressure was identified as the controlling parameter in the transition, as in other similar fracture processes, for example in solid materials. The value obtained for the critical exponent of a power law function that best fitted the DSDF for the maximum atomizing pressure is a quantitative evidence of the three-dimensional characteristics of this process.

Analysis and improvement of Brinkman lattice Boltzmann schemes: bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media.

This work focuses on the numerical solution of the Stokes-Brinkman equation for a voxel-type porous-media grid, resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude. It is first analytically demonstrated that the lattice Boltzmann method (LBM) and the linear-finite-element method (FEM) both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity. This numerical artefact may lead to an apparent negative viscosity in low-permeable blocks, inducing spurious velocity oscillations. The two-relaxation-times (TRT) LBM may control this effect thanks to free-tunable two-rates combination Λ. Moreover, the Brinkman-force-based BF-TRT schemes may maintain the nondimensional Darcy group and produce viscosity-independent permeability provided that the spatial distribution of Λ is fixed independently of the kinematic viscosity. Such a property is lost not only in the BF-BGK scheme but also by "partial bounce-back" TRT gray models, as shown in this work. Further, we propose a consistent and improved IBF-TRT model which vanishes viscosity correction via simple specific adjusting of the viscous-mode relaxation rate to local permeability value. This prevents the model from velocity fluctuations and, in parallel, improves for effective permeability measurements, from porous channel to multidimensions. The framework of our exact analysis employs a symbolic approach developed for both LBM and FEM in single and stratified, unconfined, and bounded channels. It shows that even with similar bulk discretization, BF, IBF, and FEM may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no-slip conditions. While FEM enforces them on the grid vertexes, the LBM prescribes them implicitly. We derive effective LBM continuity conditions and show that the heterogeneous viscosity correction impacts them, a property also shared by FEM for shear stress. But, in contrast with FEM, effective velocity conditions in LBM give rise to slip velocity jumps which depend on (i) neighbor permeability values, (ii) resolution, and (iii) control parameter Λ, ranging its reliable values from Poiseuille bounce-back solution in open flow to zero in Darcy's limit. We suggest an "upscaling" algorithm for Λ, from multilayers to multidimensions in random extremely dispersive samples. Finally, on the positive side for LBM besides its overall versatility, the implicit boundary layers allow for smooth accommodation of the flat discontinuous Darcy profiles, quite deficient in FEM.

Experimental convective heat transfer in a geometrically large two-dimensional impinging synthetic jet

A jet can be synthesized by ejecting and injecting fluid from and to an orifice or channel. When actuated, a synthetic jet delivers positive momentum but no net mass flow per cycle. Small, compact synthetic jet actuators can be fabricated to operate in the subaudible acoustic range and can be packaged in orientations that allow them to deliver cooling air flow to electronic devices. For cooling, the most promising orientation is one that delivers the jet flow in a direction normal to the heated surface such that it impinges on the surface as a periodic jet. In previous studies, numerical simulations were performed by the authors, utilizing an idealized canonical geometry, with the goal of eliminating actuator artifacts from the fundamental physics that drive the problem. The present paper reports on laboratory experiments that were undertaken in order to nearly replicate the idealized synthetic jet geometry and thus allow validation of the previous numerical investigations. The experiment was designed at a geometrically large scale, with jet slot widths of 4.2 and 6 mm. The amplitude and frequency at which the jet was actuated determined the Reynolds (Re) and Womersley (Ω) numbers which are the dominant non-dimensional groups. By maintaining Re and Ω in the laboratory experiments to match those of the small scale actuators, the laboratory experiments were geometrically scaled up to allow highly resolved measurements of the unsteady velocity field and the local time-dependent Nusselt number on the target heated surface. Experiments were performed at variable jet Re, Ω, and height from the target surface. The dependence of the surface averaged Nu to jet parameters agrees with the computational results when the convective heat transfer is significantly larger than the experimental heat losses, and when the fluid remains coherent with respect to the forcing frequency. Discrepancies found between numerical and empirical local data at large spacings, high Ω and low Re, suggest flow transition to turbulence and possible emergence of three dimensional effects, which were not accounted for in the computational analyses.

Computational Intelligence on Hydrodynamic Performance Characteristics of Emerged Perforated Quarter Circle Breakwater

Protecting the lagoon area from the wave attack is one of the primary challenges in coastal engineering. Due to the scarcity of rubble and to achieve economy, new types of breakwaters are being used in place of conventional rubble mound breakwaters. Emerged Perforated Quarter Circle Breakwater (EPQCB) is an artificial concrete breakwater consisting of a curved perforated face fronting the waves, a vertical wall on back and a base slab resting on a low rubble mound base. The perforated curved front face is having advantages like energy dissipation and good stability with less material as it is hollow inside. Computational Intelligence (CI) can be adopted for the evaluation of performance characteristics like reflection, dissipation, run-up and rundown which are complex, time consuming and expensive to perform in laboratory. The paper presents the work carried out to predict the reflection coefficient (Kr) for input parameters, wave period (T) beyond the data range used for training and of wave height (H) along with the data on input parameters of water depth (d), spacing-perforation ratio (S/D) and radius (R) of the EPQCB. The data on various parameters are taken in two categories for training and testing of ANN as mentioned below in order to understand the effect of using non-dimensional data in place of parametric values: 1) Input in the form of parametric data (H, T, d, R, S, D), and 2) Input in the form of non-dimensional values (H/gT2, d/gT2, S/D, R/H). Better correlation was found when individual dimensional parametric data was used instead of non-dimensional group values in both the methods of prediction. Similarly, the correlation between the beyond the data range prediction and actual values was found to be good in both methods of prediction.

A Look on Fang Number

In this study, a look on Fang number (Fa) is presented. The Fa was introduced recently in 2013 by Professor Xiande Fang to provide great facilitation in describing flow boiling heat transfer coefficients. It is defined as the product of two terms. The first term is the ratio of buoyancy force to gravitational force, which has effects on bubble departure. The second term is the ratio of surface tension force to inertial force, which affects bubble formation. As a result, Fa is associated with the formation and departure of bubbles. The Fa will be expressed by using a combination of the Eötvös number (Eo), Froude number (Fr), and Weber number (We). Based on this study, it is clear that existing dimensionless numbers in literature, i.e., Eötvös number, Froude number, Weber number, and their combinations can be used to describe flow boiling heat transfer coefficients. This combination of existing non-dimensional groups (Eo, Fr, and We) leads to good correlation with flow boiling data of different working fluids such as CO2, R134a, and R22.

The coupled effects of working fluid and solid wall on thermal performance of micro heat pipes

By incorporating the solid wall conduction, together with the continuity, momentum, and energy equations of the liquid and vapor phases, a mathematical model is developed based on the conservation laws and is solved to yield the heat and fluid flow characteristics of micro heat pipes. This work provides a comprehensive and insightful analysis on the effects of working fluid and solid wall on the thermal performance of micro heat pipes. The characteristics and performance of different types of working fluid and solid wall are elucidated. A well-defined exposition of the circulation effectiveness of the working fluid is proposed and the operation regime map for different types of working fluid is conceived for the identification of the optimal operating conditions. The coupled role of working fluid and solid wall is quantified by deriving a new non-dimensional group, which can be used to characterize the contribution of working fluid and the solid wall to the heat transport rate. The present study serves as a useful analytical tool in the micro heat pipe design and performance analysis, associated with the selection of both working fluid and solid wall material for specific operating conditions.

The Role of Solid Mechanics in Electrochemical Energy Systems such as Lithium-ion Batteries

The effect of stress on storage particles within a lithium-ion battery, while acknowledged, is not fully understood. In this study we identify the importance of solid mechanics in the performance and reliability of the system. We identify three non-dimensional parameters that govern the stress response within a spherical storage particle, and we describe the results of numerical simulations that characterize particle stresses. The non-dimensional groups contain system parameters such as the diffusion coefficient, the particle radius, the lithium partial molar volume and the host material's Young's modulus. Stress maps are presented for various values of these parameters for fixed rates of insertion and extraction, with boundary conditions applied to particles similar to those found in a battery. Stress and lithium concentration profiles for various values of these parameters show that the coupling between stress and concentration is magnified depending on the values of the parameters.

Impact Failure of Planetary Materials:

The dynamic fragmentation of a fine grained granitoid material has been examined. Target thicknesses ranged from 7 to 40 mm and impact energies from 12 to 2,500 J. Combined particle image velocimetry and image enhancement techniques are introduced and have been used to measure the size and velocity of material ejected laterally from the rear of the target during impact testing. Non-dimensional groups were formed and fitted with coefficients to predict median values of the distribution of mass and kinetic energy among radial distance, R, from the impact centre, ejecta velocities, v, and ejecta lengths, L. The statistics are well correlated with increasing non-dimensional impact energy (positive correlation for radial distance and velocity, and negative correlation for ejecta length). Median values were used to collapse cumulative distributions and non-centred Gaussian fits were used to describe these curves. Approximately 85 % of the total mass and kinetic energy is captured between R/R50 % = 0.3 to 2, v/v50 % = 0.3 to 2, and L/L50 % = 0.2 and 3. This data facilitates a better comparison among a wide range of test conditions, especially when attempting to extrapolate principal features of impacts into brittle materials at higher velocities. The ejecta tracking techniques and methodologies can be used to improve current impact testing experiments and computer modelling validation.

Mathematical model of an air-filled alpha stirling refrigerator

This work develops a mathematical model for an alpha Stirling refrigerator with air as the working fluid and will be useful in optimizing the mechanical design of these machines. Two pistons cyclically compress and expand air while moving sinusoidally in separate chambers connected by a regenerator, thus creating a temperature difference across the system. A complete non-linear mathematical model of the machine, including air thermodynamics, and heat transfer from the walls, as well as heat transfer and fluid resistance in the regenerator, is developed. Non-dimensional groups are derived, and the mathematical model is numerically solved. The heat transfer and work are found for both chambers, and the coefficient of performance of each chamber is calculated. Important design parameters are varied and their effect on refrigerator performance determined. This sensitivity analysis, which shows what the significant parameters are, is a useful tool for the design of practical Stirling refrigeration systems.

Dynamic fragmentation of planetary materials: Ejecta length quantification and semi-analytical modelling

The dynamic fragmentation of planetary materials during impact into finite targets has been examined. A particle tracking algorithm was implemented to estimate the size and velocity of fragments ejected from the rear of the target. A total of 76 experiments were performed for four materials, target thicknesses of 7 mm–55 mm, and impact energies of 10 J–6810 J. Semi-empirical models were developed from non-dimensional groups to predict key experimental results. This includes the transformation of incoming projectile kinetic energy to the ejecta kinetic energy. The amount of impact energy converted to kinetic energy of ejecta was found to increase from 2% to 18% over the range of test conditions. Energy dissipated into expanding the field laterally was found to be small in comparison to the streamwise direction (∑KEy/∑KEx = 4%).Percentiles of the distribution of mass, momentum and kinetic energy with respect to ejecta lengths were also examined. Percentile ejecta lengths decrease for increasing normalized impact energy. Fits of the non-dimensional ejecta lengths provide reasonable collapse for the percentile values. Lastly, the cumulative distributions of mass, momentum and kinetic energy among normalized 50% length values were quantified. Exponential function forms were found to fit all of the data over the range over normalized length scales of 0.3–4. When integrated, this predicts the probability density distribution of mass, momentum, and kinetic energy among ejecta lengths for the range of experimental conditions in this study. This data is important in the development and validation of numerical models.

Convective Heat Transfer in an Impinging Synthetic Jet: A Numerical Investigation of a Canonical Geometry

Synthetic jets are generated by an equivalent inflow and outflow of fluid into a system. Even though such a jet creates no net mass flux, net positive momentum can be produced because the outflow momentum during the first half of the cycle is contained primarily in a vigorous vortex pair created at the orifice edges; whereas in the backstroke, the backflow momentum is weaker, despite the fact that mass is conserved. As a consequence of this, the approach can be potentially utilized for the impingement of a cooling fluid onto a heated surface. In previous studies, little attention has been given to the influence of the jet's origins; hence it has been difficult to find reproducible results that are independent of the jet apparatus or actuators utilized to create the jet. Furthermore, because of restrictions of the resonators used in typical actuators, previous investigations have not been able to independently isolate effects of jet frequency, amplitude, and Reynolds number. In the present study, a canonical geometry is presented, in order to study the flow and heat transfer of a purely oscillatory jet that is not influenced by the manner in which it is produced. The unsteady Navier–Stokes equations and the convection–diffusion equation were solved using a fully unsteady, two-dimensional finite volume approach in order to capture the complex time dependent flow field. A detailed analysis was performed on the correlation between the complex velocity field and the observed wall heat transfer. Scaling analysis of the governing equations was utilized to identify nondimensional groups and propose a correlation for the space-averaged and time-averaged Nusselt number. A fundamental frequency, in addition to the jet forcing frequency, was found, and was attributed to the coalescence of consecutive vortex pairs. In terms of time-averaged data, the merging of vortices led to lower heat transfer. Point to point correlations showed that the instantaneous local Nusselt number strongly correlates with the vertical velocity v although the spatial-temporal dependencies are not yet fully understood.

Prediction of the maximum air velocities created by metro trains using an artificial neural network approach

The maximum air velocity created by a moving train inside a tunnel is obtained using an artificial neural network approach. A neural network model is developed to represent a single train travelling in a single tunnel. A set of non-dimensional groups, which are known to influence the induced flow characteristics, is used for the training of the neural network. Various test runs are compared with the results of the authoritative software, Subway Environmental Simulation. The presence of ventilation shafts within a tunnel is included in the model by defining an aerodynamically equivalent single tunnel using major head loss characteristics of different parts of the system. This approach eliminated the requirement to train the neural network for a large number of possible tunnel/shaft configurations.

Modelling the Doctor Blade-Roller Tribosystem for Improving the Cleaning Performance During Paper Production

Doctor blades are commonly used in paper machines to keep the surface of rollers clean. Due to higher demanding conditions, the requirements for doctor blades have steadily increased. The wear rates must remain low, while simultaneously their cleaning function has to be ensured. For this reason, the paper industry has developed a high degree of empirical knowledge concerning the cleaning of roller surfaces. However, up to now, no systematic approach has been successfully applied to optimize the cleaning performance of the doctor blade-roller tribosystem. This study presents an attempt to model the system based on the force equilibrium conditions at the blade tip between hydrodynamic and contact forces. The change of the blade geometry due to wear is also taken into account. By these means, a non-dimensional group involving the key parameters is obtained. This allows for a systematic improvement of the cleaning efficiency, by targeted changes of the process parameters.

Energy dissipation in unsteady turbulent pipe flows caused by water hammer

Energy dissipation and turbulent kinetic energy production and its dissipation in unsteady turbulent pipe flows due to water hammer phenomena are numerically studied. For this purpose, the two-dimensional governing equations of water hammer are solved using the method of characteristics. A k–ω turbulence model which is accurate for two-dimensional boundary layers under adverse and favorable pressure gradients is applied. The numerical results are in good agreement with the experimental data. Through an order of magnitude analysis, two dimensionless parameters have been identified which can be used for the evaluation of viscous and turbulent shear stress terms. The influence of these non-dimensional parameters on pressure oscillations, wall-shear-stress, dissipation rate as well as profiles of velocity, turbulent production and dissipation are investigated. The non-dimensional parameter P, which represents time scale ratio of turbulence diffusion in the radial direction to the pressure wave speed, is used to study the structure and strength of turbulence. It is found that for the case of P≈1, for which the values of the non-dimensional groups are larger, the peaks of turbulence energy production and dissipation move rapidly away from the wall and turbulence structure is significantly changed. For the case of P≫1, for which the values of non-dimensional parameters are smaller, these variations are found to be small.

On the scaling of steady structured waves in heterogeneous materials

Large amplitude steady waves in materials have been observed to display certain scaling relationships between the strain rate and the stress amplitude. In many homogeneous materials, strain rate scales with stress to the fourth power. However, scaling of strain rate with stress to the first, second, and fourth power has been found for different classes of heterogeneous materials. We examine wave structures for three classes of heterogeneous materials through mesoscale simulations that resolve the scale of heterogeneity explicitly. We utilize these simulations to gain insight into the scaling phenomena observed and to identify the critical non-dimensional parameters for the phenomena. These parameters are then applied to the available experimental data for the three classes. The same set of non-dimensional groups is found to be appropriate for layered and particulate composite materials, while somewhat different groups are found for granular materials. Two different types of simulations lead to different conclusions on the need for the inclusion of strength in the non-dimensionalization for granular materials. The groups formed are found to collapse the experimental data quite well when the strength parameter is not included. Finally, a simple model for granular materials demonstrates that the crucial aspect of their behavior that controls the scaling of waves is the need for mass transfer to close voids in the material.

On the role of inserts in forced convection heat transfer augmentation

The role of inserts in internal forced convection has been widely acknowledged as a passive device in the heat transfer enhancement. The present study is aimed to empirically investigate the heat transfer enhancement in a tube fitted with a square-cut circular ring insert in the transitional and the fully turbulent flow regimes. By performing an in-depth analysis on the experimental data, the role of insert has been quantified by deriving a new non-dimensional group. This new non-dimensional group is proposed to characterize the effect of inserts on the heat transfer enhancement. While the findings show the incorporation of insert in the flow passage enhances the heat transfer rate, the characteristics of the flow in the transitional and the fully turbulent flow regimes induced by the effect of insert are distinct. The new non-dimensional group provides interesting insights into the role played by the insert. The physical significance of the non-dimensional number which provides a measure of the change of enthalpy relative to the change of flow energy in the flow direction can be used to explain the decrease of heat transfer augmentation in the turbulent flow regime relative to the transitional flow regime. Based on the analysis of the non-dimensional group, it can be deduced that the contribution of the axial pressure drop in the heat transfer augmentation is marginal albeit not negligible compared to the temperature rise in the characterization of the heat transfer augmentation with the incorporation of insert. The evaluation of heat transfer augmentation efficiency based on the rate of change of internal energy shows that the performance efficiency of an insert would be identical in different flow regimes, contradictory to the widely held axiom that the effect induced by the insert on the heat transfer augmentation diminishes in the turbulent flow regime.

Experimental heat transfer to supercritical carbon dioxide flowing upward vertical tube with highly conducting surroundings

This paper presents heat transfer characteristics of supercritical carbon dioxide flow inside vertical circular pipe surrounded by highly conducting material, and develops an adequate tool to test the performance of available heat transfer correlations with. The possible situations are illustrated for the nuclear power plant to which the above-mentioned geometric configuration might be applicable. An experimental loop with vertical circular geometry is designed and constructed to test the upward flow in supercritical state when the axial heat transfer is enhanced by the surrounding metals, resulting in a wall boundary condition between the constant heat flux and temperature. The set of correlations and important findings are critically reviewed from extensive literature survey. Incorporating nondimensional groups resorting to past insights from the available literature, a convective heat transfer correlation is proposed. The optimization procedure is described which utilizes a random walk method along with the in-house finite element heat transfer code to determine the coefficients of the proposed heat transfer correlation. The proposed methodology can be applied to evaluation of heat transfer when the heat transfer coefficient data cannot directly be determined from the experiment.