Dimensionless Products Research Articles

Automated Feedback on Student Attempts to Produce a Set of Dimensionless Power Products from a Set of Physical Quantities that Describe a Physical Problem

Formative feedback is important in learning. Automating the provision of specific, objective, constructive feedback to large cohorts requires complex algorithms that most teachers do not have time to develop, suggesting that a community effort is needed to create a library of specialised algorithms. We present an exemplar algorithm for a class of tasks in ‘dimensional analysis’ relating to the Buckingham Pi theorem. The challenge arises because there are infinitely many valid and invalid answers, but any valid answer is sufficient to complete a task. We present an algorithm that, given one valid reference answer, can evaluate any response to a task. The algorithm uses a vector-space formulation of the Pi theorem. Deployment across seven tasks for 380 students provided feedback 3,090 times, including stating why a response is invalid. The most common reason was that a student-proposed set was not dimensionless, but other educationally relevant reasons were also identified.

Dimensional analysis: Not a recipe!

Dimensional analysis is commonly used to reduce the number of parameters and variables that shall be taken into account in the analysis of a physical problem, by means of the construction of a few non-dimensional products. It is derived from the simple principle that the very concept of a physical law implies that it shall be expressed by mathematical relationships between measures of the involved physical quantities, which must be invariant with respect to any change in the units chosen for measuring these quantities. This principle is expressed mathematically through Vaschy-Buckingham’s theorem, also known as the pi theorem. Inspired from Saint-Guilhem’s papers (Saint-Guilhem R. 1962. Les principes de l’analyse dimensionnelle, invariance des relations vectorielles dans certains groupes d’affinités. Mémorial des sciences mathématiques. Paris: Gauthier-Villars, Vol. 152 Saint-Guilhem R. 1971. Les principes généraux de la similitude physique. Gauthier-Villars: Eyrolles; Saint-Guilhem R. 1985. Sur les fondements de la similitude physique : le théorème de Federman. J Mec Th Appl 4 (3): 337–356; Debongnie JF. 2016. Sur le théorème de Vaschy-Buckingham. [http:// hdl.handle.net/2268/197814] and Barenblatt GI. 1987. Dimensional analysis. New York: Gordon & Breach Sc. Publ) the paper aims at proposing a fairly didactic presentation, where, as strongly advised by Barenblatt, basic concepts such as physical quantities, dimensions, consistent systems of units are first explained. Then, the constitution of dimensionless products is thoroughly developed before specifying the correct number of independent dimensionless products that can be obtained from a given number of physical quantities. Finally, the pi theorem is stated, with a proof in the spirit of Vaschy’s original one. Historical comments evoke Galileo’s analysis, refer to the many contributions by celebrated scholars in the 18th and 19th centuries and conclude with more recent mathematical approaches.

Dimensional analysis of partial discharge initiated by a metallic particle adhering to the spacer surface in a gas-insulated system

Partial discharges (PDs) constitute important phenomena in a Gas-Insulated System (GIS) that warrant recognition (and, subsequently, mitigation) as they are obvious symptoms of system degradation. This paper proposes the application of dimensional analysis, based on Buckingham pi theorem, for characterizing PDs provoked by the presence of metallic particles adhering to the spacer surface in a GIS employing SF6 (Sulphur hexafluoride). The ultimate goal of the analysis is to formulate the relationships that express three PD indicator quantities, namely current, charge, and energy, in terms of six independent quantities that collectively influence these indicators. These six quantities (henceforth referred to as the influencing, determining or affecting variables) include the level of applied voltage, the SF6 pressure, the length and position of the particle on the spacer, the duration of voltage application, and the gap between electrodes. To compute the pertinent dimensionless products, we implement three computational methods based on matrix operations. These three methods produce exactly the same dimensionless products, which are subsequently used for constructing the models depicting the relationships between each of the three PD dependent quantities and the common six determining variables. The models derived provide partial quantitative information and facilitate qualitative reasoning about the considered phenomenon.

Enhanced Mathematical Model for Producing Highly Dense Metallic Components through Selective Laser Melting.

Featured Application This article provides proper fitting parameters, for a wide variety of metallic alloys, to apply the derived mathematical expression (which allows to determine the scanning speed value needed, with respect to laser power) to produce highly dense components through selective laser melting. In this work, a previously developed mathematical model to predict bulk density of SLMed (produced via Selective Laser Melting) component is enhanced by taking laser power, scanning speed, hatch spacing, powder’s thermal conductivity and specific heat capacity as independent variables. Experimental data and manufacturing conditions for the selective laser melting (SLM) of metallic materials (which include aluminum, steel, titanium, copper, tungsten and nickel alloys) are adapted from the literature and used to evaluate the validity of the proposed enhanced model. A strong relation between dependent and independent dimensionless products is observed throughout the studied materials. The proposed enhanced mathematical model shows to be highly accurate since the computed root-mean-square-error values (RMSE) does not exceed 5 × 10−7. Furthermore, an analytical expression for the prediction of bulk density of SLMed components was developed. From this, an expression for determining the needed scanning speed, with respect to laser power, to achieve highly dense components produced via SLM, is derived.

A Mathematical Dimensional Model for Predicting Bulk Density of Inconel 718 Parts Produced by Selective Laser Melting.

Featured Application Mathematical tool for tuning Selective Laser Melting process parameters for achieving highly dense components. In this work, dimensional analysis is used to develop a general mathematical model to predict bulk density of SLMed components taking volumetric energy density, scanning speed, powder’s thermal conductivity, specific heat capacity, and average grain diameter as independent variables. Strong relation between dependent and independent dimensionless products is observed. Inconel 718 samples were additively manufactured and a particular expression, in the form of a power-law polynomial, for its bulk density, in the working domain of the independent dimensionless product, was obtained. It is found that with longer laser exposure time, and lower scanning speed, better densification is attained. Likewise, volumetric energy density has a positive influence on bulk density. The negative effect of laser power in bulk density is attributed to improper process conditions leading to powder particle sublimation and ejection. A maximum error percentage between experimental and predicted bulk density of 3.7119% is achieved, which corroborates the accuracy of our proposed model. A general expression for determining the scanning speed, with respect to laser power, needed to achieve highly dense components, was derived. The model’s applicability was further validated considering SLMed samples produced by AlSi10Mg and Ti6Al4V alloys. This article elucidates how to tune relevant manufacturing parameters to produce highly dense SLM parts using mathematical expressions derived from Buckingham’s π- theorem.

Modeling Coronavirus Spread Rate Utilizing Dimensional Analysis via an Irredundant Set of Fundamental Quantities

The phenomenon of spread of a (pathogenic) virus involves many physical variables, and is not amenable to satisfactory analysis via conventional methods. Dimensional Analysis (DA) is singled out as a simple and accessible way that can determine (at least qualitatively) how virus spread is related to seven physical quantities that are thought to influence it. However, classical DA deduces four dimensionless products only, none of which incorporates temperature and humidity, despite the obvious relevance of these two meteorological factors. This paper proposes an alternative version of dimensional analysis using a novel irredundant set of three fundamental quantities only. This new DA version produces five dimensionless products, four of which are essentially a replication of the old ones, while the fifth is a novel product that relates both humidity and temperature to other influencing factors. Our novel DA solution is a significant contribution, since it provides a more realistic model for virus spread rate, and it does not ignore any of the essential influencing factors. Such a model might lead to a better understanding of the determinants of spread for the novel coronavirus SARS-CoV-2 that causes the ongoing COVID-19 fatal pandemic.

Similitude Analysis of Experiment and Modelling of Immiscible Displacement Effects with Scaling and Dimensional Approach

This paper presents the use of scaling and dimensional analysis to assess the viability of conventional modelling of immiscible displacement occurring when water is injected into the oil-saturated, porous rock—a conventional secondary oil-recovery method. A brief description of the laboratory tests of oil displacement with water performed on long core sets taken from wells operating on a Polish oil reservoir was presented. A dimensionless product generator based on dimensional analysis and Buckingham Π theorem was used to generate all possible combinatorial sets of dimensionless products for physical variables describing the phenomenon. The mathematical model of the phenomenon was transformed to its dimensionless form, using a selected set of the products. The results of the laboratory tests were analyzed as functions of the products. Statistically verified quantities describing both dependent and independent experiment variables were subject to a regression analysis to study dependencies of the experimental results upon selected dimensionless products. The degrees of the dependencies were determined and compared with the model coefficients. The conclusions are drawn for the purposes of model application to correctly describe the laboratory and, consequently, field scale processes of immiscible oil displacement by water.

Investigation of the Corona Discharge Problem Based on Different Computational Approaches of Dimensional Analysis

Although corona discharge is notorious for its detrimental effects, it is also used in many beneficial practical applications. Despite the existence of a variety of sophisticated theoretical and experimental methods for investigating corona discharge, we explore yet a much simpler method that relies on the use of Dimensional Analysis (DA). The DA method does not demand profound knowledge of the underlying phenomenon or its governing equations, as it only needs the correct identification of the variables influencing the phenomenon, and the specification of their physical dimensions. The classical and well-known Gauss-Jordan elimination method is compared with other matrix-oriented computational approaches in analyzing the pertinent dimensional system. This method relies upon solution-preserving elementary row operations, i.e., operations that one can use on a matrix without spoiling the solution set for an associated matrix equation. A distinct advantage of this method is that it does not pre-suppose a particular value of the matrix rank but tackles the task of determining this rank while proceeding towards its ultimate result. Moreover, this method can also be used to find the inverse of a regular (invertible) matrix or to determine the solution (if any) of a system of linear equations. As a bonus, novel results of numerical investigations of bases, regimes, and dimensionless products are extensively presented in this paper. A remarkable observation made herein is that the set of variables pertinent for a particular phenomenon cannot be arbitrarily partitioned into basis (input) variables and regime (output or isolated) variables. The paper running example deals with expressing a specific variable, viz. ozone generation rate per unit length of wire (r0) through dimensionless products in terms of a set of determining or influencing variables.

New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary Temperature Model

A recent study has revealed that the Earth’s natural atmospheric greenhouse effect is around 90 K or about 2.7 times stronger than assumed for the past 40 years. A thermal enhancement of such a magnitude cannot be explained with the observed amount of outgoing infrared long-wave radiation absorbed by the atmosphere (i.e. ≈ 158 W m-2), thus requiring a re-examination of the underlying Greenhouse theory. We present here a new investigation into the physical nature of the atmospheric thermal effect using a novel empirical approach toward predicting the Global Mean Annual near-surface equilibrium Temperature (GMAT) of rocky planets with diverse atmospheres. Our method utilizes Dimensional Analysis (DA) applied to a vetted set of observed data from six celestial bodies representing a broad range of physical environments in our Solar System, i.e. Venus, Earth, the Moon, Mars, Titan (a moon of Saturn), and Triton (a moon of Neptune). Twelve relationships (models) suggested by DA are explored via non-linear regression analyses that involve dimensionless products comprised of solar irradiance, greenhouse-gas partial pressure/density and total atmospheric pressure/density as forcing variables, and two temperature ratios as dependent variables. One non-linear regression model is found to statistically outperform the rest by a wide margin. Our analysis revealed that GMATs of rocky planets with tangible atmospheres and a negligible geothermal surface heating can accurately be predicted over a broad range of conditions using only two forcing variables: top-of-the-atmosphere solar irradiance and total surface atmospheric pressure. The hereto discovered interplanetary pressure-temperature relationship is shown to be statistically robust while describing a smooth physical continuum without climatic tipping points. This continuum fully explains the recently discovered 90 K thermal effect of Earth’s atmosphere. The new model displays characteristics of an emergent macro-level thermodynamic relationship heretofore unbeknown to science that has important theoretical implications. A key entailment from the model is that the atmospheric ‘greenhouse effect’ currently viewed as a radiative phenomenon is in fact an adiabatic (pressure-induced) thermal enhancement analogous to compression heating and independent of atmospheric composition. Consequently, the global down-welling long-wave flux presently assumed to drive Earth’s surface warming appears to be a product of the air temperature set by solar heating and atmospheric pressure. In other words, the so-called ‘greenhouse back radiation’ is globally a result of the atmospheric thermal effect rather than a cause for it. Our empirical model has also fundamental implications for the role of oceans, water vapour, and planetary albedo in global climate. Since produced by a rigorous attempt to describe planetary temperatures in the context of a cosmic continuum using an objective analysis of vetted observations from across the Solar System, these findings call for a paradigm shift in our understanding of the atmospheric ‘greenhouse effect’ as a fundamental property of climate.

Toward predicting blast-induced flyrock: a hybrid dimensional analysis fuzzy inference system

A hybrid dimensional analysis fuzzy inference system approach was introduced to predict blast-induced flyrock in surface mining, by integrating a dimensional analysis procedure and Mamdani’s fuzzy inference system. In the dimensional analysis, the blast-induced flyrock was considered as a function of the most effective parameters. Hence, a number of dimensionless products resulted and were used as input and output parameters of Mamdani’s fuzzy inference system. The capability of the hybrid approach was determined by comparing its results with the real measurement of flyrock in the case of a copper mine, based on a number of 320 in situ blasting datasets. Predictions by the system were close to the real measurements. Sensitivity analysis of the hybrid dimensional analysis fuzzy inference system showed that the most effective dimensionless products on flyrock distance were spacing, the multiplication of rock mass rating and hole length, and the subtraction of burden and hole length multiplication and stemming length.

Determination of Material Removal Rate and Radial Overcut in Electro Discharge Machining of AISI 304 Using Dimensional Analysis

Electro discharge machining (EDM) is a most commonly used machining process among all the non-conventional machining process which removes materials via electrical and thermal energy. The primary goal of EDM is to get more material removal rate (MRR) with lower radial overcut (ROC). Normally, the responses are predicted using empirical models which are limited to only machining parameters and they do not consider the effects of work material properties on the process performance. Therefore in this study, a model has been developed including machining parameter as well as thermo-physical property of work material. In this investigation, a semi-empirical model has been established for the material removal rate (MRR) and radial overcut (ROC) by adopting the dimensional analysis technique. Dimensional analysis is a technique of dimensions and a mathematical technique that deals with the physical quantities concerned with the experiments to formulate a model for the response in terms of response control parameters as well as some physical properties of the materials. Buckingham’s л theorem is a main theorem in dimensional analysis and it is a signification of Rayleigh’s method of dimensional analysis. The theory is applied to gather each and every variable presenting the problem in a number of the dimensionless products. For this study, the thermo-physical properties viz. density, thermal conductivity and coefficient of thermal expansion and machining parameters like peak current, pulse on time, gap voltage and duty cycle are considered as input factor. AISI 304 stainless steel used as work material and Tungsten carbide is used as tool material for this investigation.

Empirical Correlation of the Primary Stability Variable of Liquid Jet and Liquid Sheet Under Acoustic Field

The investigation focuses on optimizing the length of wind-pipe that transmits acoustic energy from the compression driver to the cavity of twin-fluid atomizers. To accomplish this objective, the primary variable of stability, that is, the breakup length of liquid jet and sheet under acoustic perturbations has been experimentally characterized for a range of wind-pipe length and liquid velocity. The analysis considers liquid phase Weber number in the range of 0.7–8, and the results are compared with primary breakup data without acoustic perturbations. The range of Weber number tested belongs to Rayleigh breakup zone, so that inertia force is negligible compared to surface tension force. It shows the existence of unique stability functions based on dimensionless products up to an optimum wind-pipe length, which extends greater for liquid sheet configuration. The present results may find relevance in atomizer design that utilizes acoustic source to enhance liquid column breakup processes.

New Propped-Fracture-Conductivity Models for Tight Gas Sands

Summary In this work, we developed two models of fracture conductivity in the presence of proppant-pack damage. The models are derived from the analysis of experiments relating dynamic fracture conductivity to flowback rate, reservoir temperature, polymer loading, presence of breaker, closure stress, and proppant concentration. The first model is purely empirical, and it is modeled after fracture-conductivity studies conducted in previous work. The experimental design and planning table were derived from fractional-factorial designs. This meant that we could uniquely quantify the effect of all the dependent variables and also ensure that the factors are not correlated. We tested the empirical model for adequacy—that is, ensuring the model residuals are normally distributed with no obvious trend and approximately constant variance. Because empirical models are generally constrained by the data on which they are built, we developed a semiempirical model dependent on dimensional analysis and nonlinear regression. Dimensionless products from the dimensional analysis procedure form the backbone of this semiempirical model. We used subset-regression analysis to determine the optimal regression model and found that model was adequate. We also found two dimensionless groups and used them to develop the semiempirical model. We thereafter corrected the semiempirical model for the effect of temperature. Both the purely empirical model and the semiempirical model match experimental data with reasonable accuracy. These models can be used as a first approximation for the short-term fracture conductivity of field hydraulic fractures. We also present methodology by which engineers can develop empirical models, assess uncertainty, and test for regression-model adequacy given noisy data.

Effects of Grease Characteristics on Sound and Vibration of a Linear-Guideway Type Recirculating Ball Bearing

This paper deals with the effects of grease characteristics on sound and vibration of a linear-guideway type ball bearing. First, sound, vibration, and temperature of a linear-guideway type ball bearing were measured by changing 16 types of greases. Next, in order to explain the effects of grease characteristics (absolute viscosity η0 of the base oil in the grease at atmospheric pressure, pressure–viscosity coefficient ξ of the base oil, and penetration P of grease) and linear velocity V on the sound pressure p and vibratory acceleration a of the linear bearing, dimensional analyses were carried out. The dimensional analyses derived dimensionless products Pp/(η0V) and η0ξV/P for the sound, and dimensionless products Pa/V2 and η0ξV/P for the vibration. The plot of the measured data using dimensionless products showed that both sound pressure p and vibratory acceleration a (under a certain linear velocity) decreased as the absolute viscosity η0 of the base oil in the grease at atmospheric pressure or the pressure–viscosity coefficient ξ increased. Whereas the sound pressure p decreased as the penetration P decreased, the vibratory acceleration a decreased as P increased.

Metric for the measurement of the quality of complex beams: a theoretical study.

We present a theoretical study of various definitions of laser beam width in a given cross section. Quality of the beam is characterized by dimensionless beam propagation products (BPPs) Δx·Δθ(x)/λ, which are different for the 21 definitions presented, but are close to 1. Six particular beams are studied in detail. In the process, we had to review the properties for the Fourier transform of various modifications and the relationships between them: physical Fourier transform (PFT), mathematical Fourier transform (MFT), and discrete Fourier transform (DFT). We found an axially symmetric self-MFT function, which may be useful for descriptions of diffraction-quality beams. In the appendices, we illustrate the thesis "the Fourier transform lives on the singularities of the original."

Using dimensional-regression analysis to predict the mean particle size of fragmentation by blasting at the Sungun copper mine

A methodology was founded on the basis of a dimensional analysis procedure, together with multivariate nonlinear regression analysis which is used to predict mean particle size of fragmentation at the Sungun surface mine. A practical database was made through a number of blasting operations in various levels of the mine with geomechanical investigations and experimental tests. The mean particle size is first considered to be a function of various controllable and uncontrollable variables. Then by setting up a nonlinear correlation among the independent dimensionless products obtained from the dimensional analysis, a fundamental equation has been deduced. The equation can be practically used by mining engineers in all situations where the mean particle size of fragmentation by bench blasting should be predicted. Capability of the proposed method is determined by comparing its predictions with the real measurement (observation) of sieve analysis, a standardized image-processing technique, and the predictions by the Kuz–Ram model as the most applied model, together with the modified version of the Kuz–Ram model. The results obtained from the proposed method are closer to the real results of the sieve analysis than those of the other methods. Finally, the methodology was found to be strong and better in prediction than in image processing and also in both versions of the Kuz–Ram model.

Optimal and off-design operations of acoustic dampers using perforated plates backed by a cavity

The frequency of thermo-acoustic instabilities may vary during operation of combustors, a feature that must be included in the design of robust acoustic dampers tuned to damp specific unstable modes. This problem is tackled here by using perforated plates traversed by a bias flow and backed by a resonant cavity in the absence of grazing flow. A methodology is proposed to find the optimal dimensionless products maximizing absorption in two limit regimes. In these two asymptotic regimes, the optimal bias flow velocity is only controlled by the plate porosity while the size of the back cavity fixes the peak absorption frequency. The present analysis also includes effects of the plate thickness. A Helmholtz resonance and a narrow absorption peak in the frequency space characterize the first absorption regime reached at high Strouhal numbers. Conversely, the second absorption regime reached at low Strouhal numbers operates with a quarter-wave resonance type. This regime requires larger cavity depths but offers a wider absorption bandwidth around the peak absorption frequency. In both cases, total absorption is achieved at the nominal conditions identified. The way the absorption coefficient is reduced when variations of the bias flow velocity around the optimal value are considered is then investigated. Theoretical predictions are validated by measurements in the two nominal regimes identified and at off-design operation. The expressions derived in this study can be used to improve the design of robust acoustic dampers.

Sizing the slenderness of free-standing rocking columns to withstand earthquake shaking

This paper investigates the problem of sizing the width of tall free-standing columns with a given height which are intended to rock, yet shall remain stable during the maximum expected earthquake shaking. The motivation for this study is the emerging seismic design concept of allowing tall rigid structures to uplift and rock in order to limit base moments and shears. The paper first discusses the mathematical characterization of pulse-like ground motions and the dimensionless products that govern the dynamics of the rocking response of a free-standing block and subsequently, using basic principles of dynamics, derives a closed-form expression that offers the minimum design slenderness that is sufficient for a free-standing column with a given size to survive a pulse-like motion with known acceleration amplitude and duration.

Artificial intelligence applied to plane wave reflection at submerged breakwaters

The reflection process at submerged breakwaters is investigated by means of an artificial neural network (ANN) model. This model estimates the reflection coefficient based on three dimensionless products which are functions of the wave parameters and the characteristics of the submerged breakwater. The data used for training the model were obtained from an extensive experimental campaign in which seven models of submerged breakwaters were tested under irregular waves combinations. To choose the neural network architecture best suited for this problem, the performances of 400 ANN models involving 10 different architectures are assessed. Having selected the most appropiate architecture, the model was succesfully trained and validated. Excellent agreement was achieved between the model's results and the experimental data. The new artificial intelligence model can be used as a virtual laboratory to predict the reflection coefficient without a need for a physical model test.

Simplified Physically Based Model for Estimating Effective Floc Density

This paper applies dimensional analysis to propose an alternative model for estimating the effective density of flocs (Δρf). The model takes into account the effective density of the primary particles, in addition to the sizes of the floc and primary particles, and does not consider the concept of self-similarity. The model contains three dimensionless products and two empirical parameters (αf and βf), which were calibrated by using data available in the literature. Values of αf=0.7 and βf=0.8 were obtained. The average value of the primary particle size (Dp) for the data used in the analysis, inferred from the new model, was found to vary from 0.05 μm to 100 μm with a mean value of 2.5 μm. Good comparisons were obtained in comparing the estimated floc-settling velocity on the basis of the proposed model for effective floc density with the measured value.