Pi Theorem Research Articles

Efficiency of Dimensional Analysis in Predicting Compression Load–Settlement Relationship of Soft Clay Under a Rigid Foundation

Dimensional analysis is an effective tool for designing laboratory tests, and it is also a successful method to reasonably predict the results of large-scale or field tests, which are difficult and costly to perform, by considering the outcomes obtained from small-scale tests. This paper therefore focuses on predicting the compression load–settlement relationship of soft clay soil under a rigid foundation by dimensional analysis. For that purpose, a series of dimensionless pi (π) terms were first produced for the relevant problem by employing Buckingham’s pi (π) theorem via the MATLAB program. Secondly, the results of a small-scale physical test reported in the literature that investigated the load–settlement relationship of soft clay soil under a rigid foundation subjected to compression load were verified with the finite element method. Thereafter, the small-scale problem was scaled up several times using dimensionless terms, and the compression load–settlement relationship for the large-scale cases was investigated with both dimensional analysis and the validated finite element method. The findings indicate that dimensional analysis produces reasonably successful outcomes considering the computational ease. In addition, the MATLAB program presented in this paper is quite useful for those desiring to determine the dimensionless terms belonging to any sort of problem.

Development of a Mathematical Model for Palm Fruit Digester Design: Integrating Dimensional Analysis and Process Optimization

This investigation delineates a thorough methodology for the formulation of a parametric model about a palm fruit digester apparatus, employing the tenets of dimensional analysis. The palm fruit digester, an essential component within the palm oil manufacturing processes, encompasses intricate interactions among various parameters that influence its operational efficiency and throughput. A palm fruit digester was conceptualised and fabricated locally to attain a more profound comprehension of these interrelationships. The performance parameter was scrutinized as data were amassed during the apparatus's testing, and the Buckingham Pi theorem was utilized to derive dimensionless Pi terms from pertinent parameters. The results of our analysis indicated that R2 values of 0.9956 for both throughput and machine efficiency were achieved. Consequently, the model can forecast the efficiency and throughput of the palm fruit digester. This helps researchers understand that the palm fruit digester's operational efficiency and throughput are essential to the industry's sustainability, financial stability, and quality standards. In addition to helping to optimise these procedures, the dimensional analysis modelling approach advances strategic ideas that have the potential to reshape operational capabilities in the processing of palm oil.

Rapid deposition analysis of inhaled aerosols in human airways

A rapid data-driven method for determining regional deposition of inhaled medication aerosols in human airways is presented, which is patient specific. Inhalation patterns, device characteristics, and aerodynamic particle size distribution of medications are considered. The method is developed using dimensional analysis and Buckingham Pi theorem, and provides total, regional, and lobar distributions of aerosol deposition. 34 dimensionless quantities are selected, of which 22 encode features of the airway trees and segmented lobes, 14 pertain to the device and the drug formulation, and 13 the inhalation profile of the subject. The dimensionless correlations are obtained using a large database of computational fluid dynamics results on patient specific airways. The intraclass correlation coefficient between the current method and its training dataset is 0.92. The difference between the predicted average lobar deposition in the six asthma patients and the in-vivo data is 1.3%. The model has the potential to offer insights into the effectiveness of personalized drug delivery in clinical settings and can aid in drug development cycles.

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.

Similarity law of global and local responses of reinforced concrete beams subjected to impact loading

In the experiment of scaled models to dealing with the impact of reinforced concrete large structures, the strain rate hardening effect can lead to significant differences between the dynamic responses of prototype and model. In the present study, this thorny problem is solved by adjusting the impact velocity. Based on the mechanisms of different dynamic responses, the similarity laws for the global and local responses of reinforced concrete beams subjected to impact loading are proposed by considering the strain-rate effect of materials through Buckingham Π-theorem, where constitutive models of different materials are employed for inferring dynamic stress. In dimensional analysis, multiple control factors need to be considered simultaneously, such as the strain-rate effects of steel and concrete, since reinforced concrete components are typical composite structures. The similarity law cannot be unified when considering multiple control factors. In view of this, it is recommended to estimate the dynamic response of reinforced concrete components subjected to impact loading through the upper and lower bounds of similarity law. The upper bound of similarity law is based on the strain-rate effect of concrete, while the lower bound of similarity law is based on the strain-rate effect of steel bar. The numerical models of reinforced concrete beams are established based on the experimental background. The numerical simulations with different scaling factors indicate that the model modified by the lower bound of similarity law can better predict the global response of prototype, while the model modified by the upper bound of similarity law can more accurately predict the local response of prototype. In addition, the influence of dynamic elastic modulus of reinforced concrete on the similarity law is discussed through comparative analysis.

Model construction of the suction force of high-sphericity particles based on the PI theorem

Spheroid particles with high-sphericity are more common in particle absorption's applications than spherical particles, and their suction force (Fs) is a crucial mechanical factor in the particle absorption process, which is significantly affected by the spatial position of the particle concerning to the suction hole. Therefore, in this work, a combination of the PI theorem and the idealized model method was employed to establish an Fs model of spheroid particles with high-sphericity. More specifically, the spatial coordinates of the particle relative to the suction hole were introduced, rendering the model more applicable in practical engineering scenarios. Through the validation process, the proposed model exhibited excellent applicability for predicting Fs on spheroid particles of sphericity exceeding 90.24% within conditions of diameter ratio 0.571–1, diameter-distance ratio 0.5–0.786, particle center-tilt ratio 0–0.207, RPp−1 (reciprocal of particle Reynolds number under pressure condition) 3.29–4.25, and the prediction error can be controlled in 10%. Moreover, valuable insights were derived regarding the effective suction domain and constraints associated with the PI theorem in constructing unknown models during the subsequent analysis of the experimental data.

A new paradigm in critical flow analysis: Combining Buckingham Pi theorem with neural network for improved predictions in microchannels

Critical flow phenomena are prevalent in many chemical engineering processes, such as microchannel heat exchangers, nuclear safety analysis, jet pump modeling, and industrial systems utilizing subcooled or two-phase pressurized fluids. Understanding the impact of various physical parameters on critical flow and accurately predicting critical flow mass flux is essential in these processes. This article collects extensive experimental data on subcooled water two-phase critical flow and supplements it with additional experimental data from actual micro-cracked channels, constructing a comprehensive benchmark database. The database covers an inlet pressure range of 1.82–16.29 MPa, an inlet subcooling range of 0.3–119.35 K, and a microchannel hydraulic diameter range of 0.09–2.2 mm. By integrating the classical Buckingham Pi theorem with a data-driven active subspace method, the dominant dimensionless numbers in subcooled water two-phase critical flow were identified, addressing some limitations of the Buckingham Pi theorem by ranking the influences of these dimensionless numbers. Based on these dominant dimensionless numbers, the impact of physical quantities on two-phase critical flow was accurately identified, leading to a concise predictive correlation for critical flow mass flux with a mean absolute percentage error of 20.3 % across 1071 experimental data points, outperforming commonly used critical flow models. Furthermore, leveraging the dominant dimensionless numbers improved existing model results, enhancing the Henry-Fauske model’s overall prediction accuracy from a mean absolute percentage error of 47.6 % to 17.8 %. This study outlines a promising path for discovering physical characteristics and mechanisms in data-driven critical flow, facilitating a deeper understanding of critical flow phenomena.

Modeling powder spreadability in powder-based processes using the discrete element method

Powder-bed fusion (PBF) processes refer to a subset of Additive Manufacturing (AM) techniques where powder is spread on the build-plate before melting (by a laser or electron beam). While PBF processes are attractive due to their ability for realizing complex structures that are either difficult or impossible to create through conventional means, the parts fabricated with these techniques can exhibit defects such as pores, inclusions, and excessive surface roughness. To minimize these defects, much research has been dedicated towards process maturation by optimizing laser or electron beam parameters. However, these developmental efforts typically do not address the recoating process where achieving dense and uniform layers of powder is a necessity for ensuring process repeatability and part quality. While the recoating process can be studied through experimentation, the dynamics of particle movement are difficult to analyze experimentally. Therefore, in this study, powder spreading in PBF was simulated through the Discrete Element Method (DEM) to elucidate the mechanisms that control powder-bed quality. Utilizing the Buckingham Pi theorem, a dimensionless metric referred to as the spreading index is developed that combines powder-bed density, roughness, and particle size to assess the quality of powder layers. The formulated spreading index is then related to several dimensionless quantities that provide insight into the mechanisms dominating powder spreading in PBF. The DEM simulations conducted in this work focused on the scenario where powder is spread onto an existing powder bed and revealed that a reduction in the recoating velocity causes an increase in the spreading index while little to no impact on the spreading index was observed when varying layer thickness from 30 µm to 75 µm. Particle size effects on the powder-bed quality were also investigated.

Data-driven dimensional analysis of critical heat flux in subcooled vertical flow: A two-stage machine learning approach

This study presents a novel two-stage machine learning algorithm that identifies dominant dimensionless numbers for critical heat flux (CHF) prediction, circumventing the limitations of traditional dimensional analysis by integrating the Buckingham Pi theorem with advanced computational methods, including the active subspace method and machine learning. When applied to extensive datasets, including 1438 data points from subcooled departure from nucleate boiling (DNB) conditions and validated against a non-overlapping 24,284-point lookup table dataset, our algorithm effectively discerns and validates the dimensionless numbers critical to the CHF phenomena. Notably, it identifies a significant negative correlation between a newly discovered dominant dimensionless number and the boiling number Bochf = q″w,chf / (G hfg), and introduces a new simplified scaling law for water under earth's gravitational conditions, outperforming existing models with substantial predictive accuracy. This scaling law and the identified dimensionless numbers, derived from an initial dataset of 1438 data points, exhibit exceptional generalization capabilities when validated against a significantly larger dataset of 24,284 points. These findings also emphasize the nonlinear effects of inlet quality on the boiling number Bochf. The contribution of this research lies in its potential to deepen the understanding of CHF based on data-driven fashion, with the potential to enhance thermal management and design optimization across various applications.

Scale up validation tests for Buckingham Pi theorem applied to leaf test experiments for iron ore slurry

Bench tests were conducted on ore slurry filtration using the leaf test method, varying scales, pressures, and water/solids ratios. These results were compared with analytical outcomes derived from applying the dimensionless Buckingham pi method to the general equation of slurry filtration with incompressible cakes under constant pressure. The aim was to validate the tests by corroborating experimental findings with calculated parameters for scale-up purposes. Vacuum pressure was employed to facilitate slurry dewatering, forming a cake comprising solids from the slurry. This cake’s thickness evolved over time, introducing specific resistivity that impeded filtering efficiency. Parameters such as cake resistance and filter media resistance were computed. Filtrate volume was measured over time to determine the filtration rate. These values were then adjusted using factors obtained through dimensional analysis. These factors facilitated comparisons to validate the feasibility of employing the Buckingham pi method for predicting slurry filtration results at larger scales. The results indicated lower errors when analyzing the filtrate rate over time with a solids/water ratio of 20/80%. When evaluating cake resistance, the method yielded lower errors at a ratio of 70/30%, while results for medium resistance were inconclusive. It’s important to consider the ratio and associated errors when predicting the behavior of an industrial filter based on bench test results, aiming to achieve theoretical scaled filtration rates and cake resistances. However, it’s not advisable to rely on the Buckingham pi method for scale-up purposes when evaluating medium resistance.

Data-driven dryout prediction in helical-coiled once-through steam generator: A physics-informed approach leveraging the Buckingham Pi theorem

A dimensionally consistent physics-informed neural network, named DimNet, has been developed to predict dryout quality in helical coils. Central to its design is the automated and optimizable dimensionality reduction technique, leveraging the Buckingham Pi theorem, which transforms 11 dimensional physical quantities into 8 dimensionless groups. Rigorous 5-fold cross-validation and cross-fluid testing affirm its performance, with an mean absolute error of 0.0540 and 0.198, respectively. The strategic incorporation of noise during training elucidates pronounced improvements, accentuating the model's adaptability. In comparison to three other neural network architectures, DimNet consistently displays superior accuracy. Ablation experiments have underscored the efficacy of each module within the model's design. Ultimately, while DimNet presents as a promising tool for enhancing thermal efficiency in helical-coiled steam generators, its design principles also shed light on the broader potential and versatility of dimensionally consistent neural architectures.

Improved estimation of uniaxial compressive strength and elastic modulus for carbonate rocks using back propagation neural network and buckingham pi theorem

Improved estimation of uniaxial compressive strength and elastic modulus for carbonate rocks using back propagation neural network and buckingham pi theorem

Mass flow rate prediction of a direct-expansion ice thermal storage system using R134a based on dimensionless correlation and artificial neural network

Accurately predicting the refrigerant mass flow rate through the electronic expansion valve (EEV) is crucial for improving the system's performance and achieving intelligent control. However, the refrigerant mass flow rate model applicable to the direct-expansion ice thermal storage (DX-ITS) system for a wide range of flow rates is rare in the open literature. In this study, Buckingham-π theorem and artificial neural network (ANN) are adopted to predict the refrigerant mass flow rate through the EEV of the DX-ITS system using R134a. The dimensionless π-groups and optimal number of neurons in hidden layers of ANN model are obtained. The obtained ANN model shows good accuracy, and over 95.7 % of the predicted data are within the 15 % error band. Results indicate that the EEV outlet pressure has a significant impact on the refrigerant mass flow rate compared with the inlet pressure, the average increase in refrigerant mass flow rate is 43.76 kg/h with an outlet pressure increase of 0.05 MPa. Moreover, the refrigerant mass flow rate gently rises with the increase of superheat temperature under fixed EEV inlet and outlet pressure. On average, every 2 °C increase in superheat temperature leads to an approximately 3.98 kg/h increase in refrigerant mass flow rate.

Impact of the numerical domain on turbulent flow statistics: scalings and considerations for canopy flows

Large eddy simulations (LES) are widely used to study the effects of surface morphology on turbulence statistics, exchange processes and turbulence topology in urban canopies. However, as LES are only approximations of reality, special attention is needed for the computational model set-up to ensure an accurate representation of the physical processes of interest. This paper shows that the choice of the numerical domain can significantly affect the accuracy of turbulent flow statistics, potentially causing a mismatch between numerical studies and experimental data. The study examines the influence of cross-stream aspect ratio (YAR), streamwise aspect ratio (XAR) and scale separation (SS) on first- and second-order flow statistics and turbulence topology. It is found that domains with a low YAR underestimate the velocity variance, while those with a low XAR overestimate the variance value. The study proposes a new approach based on the Buckingham Pi theorem to evaluate the effect of SS, as the existing method has major limitations for canopy flows. The results suggest that domains with small SS underpredict the variance value. To minimise the artificial impact of the numerical domain on turbulent flow statistics, the study recommends guidelines for future research, including a YAR of 3 or more, an XAR of 6 or more and an SS of 12 or more. Error tables are presented to allow researchers to select smaller domains than recommended, depending on their research interests in specific parts of the flow.

Dose Distribution Scaling and Validation of Ultraviolet Photoreactors Using Dimensional Analysis.

Ultraviolet (UV) disinfection is commonly applied in the treatment of drinking water and wastewater. The performance of UV disinfection systems is governed by the UV dose distribution delivered to the fluid, which is an intrinsic characteristic of the reactor under a given operating condition. Current design and validation approaches are based on empirical methods that are expensive to apply and provide limited information about the UV photoreactor behavior. To address this issue, a dose distribution scaling method was developed based on dimensional analysis (i.e., application of the Buckingham-π theorem). Three dimensionless groups representing UV dose, reactor geometry, and UV absorption behavior were defined. Using these groups, the approach was applied for the analysis of 15 operating conditions, defined by process variables of volumetric flow rate, UV transmittance, and lamp power. The approach was demonstrated to allow scaling of the dose distribution with these critical, dimensionless variables and yielded close agreement between predictions of disinfection efficacy against MS2 and E. coli based on the scaling approach with conventional CFD-E' modeling results. The approach thus provides a low-cost, rapid method for predicting the performance of UV disinfection systems across a wide range of operating conditions and against essentially any microbial challenge agent.

Data Analytics and Development of a Wellbore Cleaning Coefficient Model

In this study, wellbore cleaning coefficient (WCC) correlations were developed for three conventional coiled tubing sizes (2.375”, 2.625”, and 2.875”). These sizes correspond to roughness to internal (ε/D) ratios of 0.000460828, 0.000510637, and 0.000572517, respectively. Dimensional analysis, applying the Buckingham-π theorem and a database from 150 wells in the Spraberry formation in West Texas, was used. Key performance indicators (KPIs) that influence flow in a cased pipe around an object (coil tubing) were identified and employed in model development. These KPIs are (1) slick water density ( f ñ ), (2) slick water viscosity ( µ f ), (3) hydraulic diameter ( c t d - d ) between casing inner diameter (dc ) and coil tubing outer diameter ( t d ), (4) average annular velocity ( v ) and (5) cleaning pressure gradient(∆P) . The cleaning pressure gradient is the ratio of the circulating differential pressure (pu -pd ) to measured depth (MD). A global model that relates WCC to the Euler number and the inverse of the Reynolds number was attempted at first. A low coefficient of multiple determination R2 of 0.626 was obtained. To better explain the physics of the cleaning process and improve the model fit, data segregation was performed by separating data into three data sets, a set for each ε/D ratio. R2 of 0.974, 0.945, and 0.877 were obtained. It was decided to separate the database further and create models that would be used to identify “clean” and “not clean” wellbores. These equations addressed operational conditions since in data partition threshold values of annular velocity, Euler and Reynolds numbers were applied to describe laminar and turbulent flow conditions. The predictive equations showed excellent degrees of fit with R2 of 0.979, 0.822, and 0.897, for clean wells for the three ε/D ratios, respectively. This study’s findings were also validated using cumulative debris versus elapsed time data from 12 Woodford wells.

Accurate Prediction of the Proppant Distribution in a Hydraulically Fractured Stage.

One of the greatest challenges in stimulating a single hydraulic fracturing stage with multiple clusters is ensuring that the proppant is equally distributed across all clusters. In this paper, the Buckingham pi theorem was applied to implement a dimensional analysis to establish an empirical correlation. The experimental correlation was developed by acquiring and integrating the data of various independent variables, such as different proppant characteristics (i.e., size, density, and concentration), using a wide range of internal diameters of the horizontal wellbore and multiple perforation configurations, and utilizing many carrier fluids with different viscosities. This work presents a newly enhanced experimental correlation for the distribution of proppants by incorporating the effect of gravity on proppant particles. The correlation proved its reliability in forecasting the proppant distribution with an average percentage error of less than 10%. The developed correlation has the potential to serve as a tool for forecasting proppant distributions among multiple clusters in multistage hydraulic fracturing treatments in the field.

Microfluidic Droplet-Generation Device with Flexible Walls.

Controlling droplet sizes is one of the most important aspects of droplet generators used in biomedical research, drug discovery, high-throughput screening, and emulsion manufacturing applications. This is usually achieved by using multiple devices that are restricted in their range of generated droplet sizes. In this paper, a co-flow microfluidic droplet-generation device with flexible walls was developed such that the width of the continuous (C)-phase channel around the dispersed (D)-phase droplet-generating needle can be adjusted on demand. This actuation mechanism allowed for the adjustment of the C-phase flow velocity, hence providing modulated viscous forces to manipulate droplet sizes in a single device. Two distinct droplet-generation regimes were observed at low D-phase Weber numbers, i.e., a dripping regime at high- and medium-channel widths and a plug regime at low-channel widths. The effect of channel width on droplet size was investigated in the dripping regime under three modes of constant C-phase flow rate, velocity, and Capillary number. Reducing the channel width at a constant C-phase flow rate had the most pronounced effect on producing smaller droplets. This effect can be attributed to the combined influences of the wall effect and increased C-phase velocity, leading to a greater impact on droplet size due to the intensified viscous force. Droplet sizes in the range of 175-913 µm were generated; this range was ~2.5 times wider than the state of the art, notably using a single microfluidic device. Lastly, an empirical model based on Buckingham's Pi theorem was developed to predict the size of droplets based on channel width and height as well as the C-phase Capillary and Reynolds numbers.

Modeling the Ultrasonic Micro-Injection Molding Process Using the Buckingham Pi Theorem.

Dimensional analysis through the Buckingham Pi theorem was confirmed as an efficient mathematical tool to model the otherwise non-linear high order ultrasonic micro-injection molding process (UMIM). Several combinations of processing conditions were evaluated to obtain experimental measurements and validate the derived equations. UMIM processing parameters, output variable energy consumption, and final specimen's Young modulus were arranged in dimensionless groups and formulated as functional relationships, which lead to dimensionless equations that predict output variables as a function of the user-specified processing parameters and known material properties.

Optimisation of the damping properties of steel structures using adhesively bonded joints

AbstractAdhesives offer both good damping properties and very high strengths. The possibilities for using the positive damping properties in particular to optimise dynamically stressed steel structures have rarely been explored to date. In this paper, comprehensive numerical and experimental investigations are presented. In numerical investigations on dynamically loaded, representative steel structures, the potential for damping of structures by adhesively bonded joints is analysed. Numerical investigations show that the vibration amplitudes of adhesively bonded structures are more than 95 % lower than those of welded reference structures in the relevant resonance case. Based on this, the damping properties of adhesively bonded tubular steel joints in a scale relevant for construction industry were experimentally investigated. Different geometry and test boundary conditions are investigated in a parameter study. The loss factor of tubular joints is a maximum of 0.38, which is relevant for steel construction. Based on the results of experimental investiga‐tions, a methodology is presented for the analytic determination of the damp‐ing properties of adhesively bonded tubular joints for various specimen ge‐ometries and stress boundary conditions. For this purpose, dimensional anal‐ysis is carried out using the Buckingham Pi Theorem.