Pi Theorem Research Articles

Multi-scale physics-informed machine learning using the Buckingham Pi theorem

Neural networks are a form of machine learning that can be trained to estimate relationships between variables in complex physical processes. They are particularly adept at estimating relationships between variables that lie within the ranges of values for which they have been trained. Their performance often diminishes when tasked with generating estimates of physical processes that lie outside of the region of the input space covered by the training set. In the case of physical systems, the possible relationships between input and output variables are limited. Dimensional variables can be replaced by a smaller number of dimensionless parameters that enforce physical limitations between dimensional input and output variables. This can be accomplished using dimensional analysis and the Buckingham Pi theorem to enforce or test for dynamic similitude between systems operating at different scales. This process can be exploited for two purposes. The first is to reduce the number of variables correlated by a neural network. The second is to allow a dimensionless neural network to be trained to function as an interpolator between dimensionless input and output parameters even though a neural network trained using dimensional data would be required to extrapolate if the dimensional input variables lie outside of the training set. Using dimensionless data to fit an input-output relationship generalizes better, as compared to using dimensional data, when dynamic similitude between systems has been achieved. Examples are presented that demonstrate that the proposed process can enable accurate modeling of the behavior of physically similar systems operating at different scales.

Prediction of the conjugate depth of the hydraulic jump in the trapezoidal channel using Random Forest regression

Prediction of the sequent depths of the hydraulic jump in the trapezoidal channel using the theoretical equation is a challenging task. Therefore, existing studies have attempted to solve the task by conducting experiments or using semi-empirical calculations. The paper proposes a novel method that applies Buckingham's Pi theory and the Random Forest regression to improve the prediction accuracy of the sequent depths of the hydraulic jump in the trapezoidal channel. The study has shown that Machine Learning models can be efficient for the determination of the geometrical features of the jump and have high ability in many real projects.

Understanding Quality of Pinot Noir Wine: Can Modelling and Machine Learning Pave the Way?

Wine research has as its core components the disciplines of sensory analysis, viticulture, and oenology. Wine quality is an important concept for each of these disciplines, as well as for both wine producers and consumers. Any technique that could help producers to understand the nature of wine quality and how consumers perceive it, will help them to design even more effective marketing strategies. However, predicting a wine’s quality presents wine science modelling with a real challenge. We used sample data from Pinot noir wines from different regions of New Zealand to develop a mathematical model that can predict wine quality, and applied dimensional analysis with the Buckingham Pi theorem to determine the mathematical relationship among different chemical and physiochemical compounds. This mathematical model used perceived wine quality indices investigated by wine experts and industry professionals. Afterwards, machine learning algorithms are applied to validate the relevant sensory and chemical concepts. Judgments of wine intrinsic attributes, including overall quality, were made by wine professionals to two sets of 18 Pinot noir wines from New Zealand. This study develops a conceptual and mathematical framework to predict wine quality, and then validated these using a large dataset with machine learning approaches. It is worth noting that the predicted wine quality indices are in good agreement with the wine experts’ perceived quality ratings.

Experimental investigation of the effect of liquid petroleum gas on frictional power loss for performance enhancement of SI engine

Alternative gaseous fuels being clean, economical, and available in large quantity encourage the focus on optimizing various operating conditions for internal combustion engines. Alternative fuel as LPG is being considered to restore the role of alternative fuel to improve fuel economy by reducing fuel consumption and total frictional power loss compared to petrol fuel. The engine is tested using LPG and petrol so that comparison of the engine performance concerning fuel consumption and frictional power loss using Morse test. The conventional 4 Cylinder, 4 Stroke Spark Ignition engine expending a hydraulic dynamometer is used for experimentation. To find the most influencing factors from various input parameters of an engine like engine speed, load on the engine, engine oil viscosity which strongly affects engine frictional power loss. The comparative analysis of petrol and LPG fuels with different multi-grade engine oils shows that the mechanical efficiency obtained from the Morse test are 68.37%, 65.75% and 64.69% at 1750 rpm for petrol fuel with SAE20W30, SAE05W40 and SAE20W50 respectively, while for the same operating condition these values with LPG fuels are 62.99%, 61.85% and 59.85% respectively for SAE20W30, SAE05W40 and SAE20W50 engine oils at1750 rpm. Buckingham Pi-theorem is used to calculate, an experimental data-based mathematical model for which various pi-terms are determined, applying dimensional analysis for selected parameters for predicting outcomes as frictional power loss and fuel consumption rate.

Implementation of Buckingham's Pi theorem using Python

Buckingham's Pi theorem plays an important role in engineering, applied mathematics, and physics for dimensional analysis. From the given variables, it will be utilised to evaluate the set of dimensionless parameters. It indicates that the validity of the physical law is independent of the specific unit system, and it can be expressed as an identity incorporating only dimensionless variables associated with the law. A python-based function has been developed and reported in this manuscript, which can be utilised to evaluate Pi terms (commonly called π terms) for any fluid flow problem. Smaller moderation in the written function can make it capable of solving any fundamental dimensions. Different fluid mechanics problems are utilised to test and validate the reported function, five of which are presented in this manuscript along with code. Obtained results are in good agreement with the theoretically obtained results.

Dimensionless Characterization to Estimate Horizontal Groundwater Velocity from Temperature–Depth Profiles in Aquifers

The outcome of a dimensionless characterization study in a two-dimensional porous media domain in which groundwater flows at a constant horizontal velocity is presented in this report. Using spatial discrimination, the dimensionless groups that govern the solution patterns are determined from dimensionless governing equations. As a boundary condition on the surface, the case of constant temperature is studied. From the mathematical deduction of the groups, a characteristic horizontal length emerges. This length determines the region in which temperature–depth profiles are affected by flow. Existing analytical solutions have been shown to be invalid due to the severe assumption that the horizontal thermal gradient has a constant value. Therefore, universal solutions based on pi theorem have been obtained for the characteristic horizontal length, temperature field, temperature–depth profiles and horizontal temperature profiles. Dependencies between dimensionless groups have been depicted by universal curves, abacuses and surfaces. These graphical solutions are used in an easy way to estimate groundwater velocity from experimental temperature measurements in the form of an inverse problem. In addition, an easy and fast protocol for estimating fluid flow velocity and groundwater inlet temperature from temperature profile measurements is proposed. This protocol is applied in a scenario of groundwater discharge from a quaternary aquifer to a salty lagoon located in the southeast of Spain.

The effect of key parameter changes on the critical heat flux of spray evaporatively-cooled vibrating surfaces using a single misting nozzle

• • New CHF model shows parameter change influence for cooling of vibrating surfaces. • • Model is constructed using the Generalized Buckingham Π-Theorem. • • Measured data obtained from an electrically-heated test-piece excited by a shaker. • • Surface-to-nozzle distance in presence of vibration significantly influences CHF. A new correlation model is examined for capturing the combined influences of surface-to-nozzle distance and coolant flow rate on critical heat flux associated with spray evaporative cooling of vibrating surfaces. The correlation model is constructed using dimensional analysis by applying the Generalized Buckingham Π-Theorem. The model is calibrated using experimentally-measured spray evaporative cooling data, taken from an electrically-heated horizontal flat circular test-piece excited by a shaker through a range of low and high frequencies of vibration, from small to large amplitude. To understand the combined effect of frequency, amplitude, and surface-to-nozzle distance, at critical heat flux, Vibrational Reynolds Number, Acceleration Number, and Dimensionless Surface-to-Nozzle Distance are used. The results show that surface-to-nozzle distance, in the presence of dynamic effects, significantly influences the critical heat flux, whereas vibration amplitudes and frequencies have differing effects in response to variations in both surface-to-nozzle distance and flow rate. Surface-to-nozzle distance can either increase or decrease the heat transfer, depending on the vibration range. The calibrated correlation model is capable of predicting the effect of surface-to-nozzle distance on the critical heat flux with errors in the range −4.8% and + 10.5%.

Fault identification in a nonlinear rotating system using Dimensional Analysis (DA) and central composite rotatable design (CCRD)

This paper investigates the effect of bearing clearance and external defects on the vibrational behavior of rolling element bearing. An integrated nonlinear model is developed using the Buckingham pi theorem to detect imminent bearing faults. Vibration responses acquainted experimentally by varying speed, clearance, and external defect on the rotor-bearing system are processed using fast Fourier techniques (FFT). Result analysis reveals the dominance of clearance and external defects on the dynamic stability of the rotor-bearing system. The maximum vibration amplitude among the all trials performed was 2.201 mm/sec at 1300 rpm shaft speed, 75 gm unbalance mass, and 0.025 mm bearing clearance. A close agreement of experimentation with model prediction assures accuracy and reliability. Central composite rotatable design (CCRD) shows its effectiveness in performing the design of experiments with understanding procedures. The current approach highlight’s fault identification and interactions with less than 9% error that satisfy the demands of current condition monitoring or diagnosis of industrial rotor-bearing systems.

Dimensional analysis and performance laws for organic vapor flow turbomachinery

Non-dimensional groups of variables are obtained for describing turbomachinery performance working with non-perfect gases. The dimensional analysis follows the Rayleigh method and Buckingham-Π theorem. It employs the bulk modulus, the specific gas constant, and the derivative of the compressibility factor regarding pressure at constant entropy. Following a classical analysis from Traupel, this new similarity number is a valuable measure of the non-perfectness of a gas. As an outcome of the present study, it is found that the compressibility factor is not a primary similarity number for non-perfect gas flows. Still, its derivative is part of such a dimensionless number. At least two new similarity numbers are required for describing non-perfect compressible gas flows in addition to the conventional perfect gas similarity numbers. As a limit case of the dimensional analysis, the well-known similarity numbers for a perfect gas result with the isentropic exponent similarity. The theoretical analysis is applied to flow applications utilizing the closed-loop organic vapor wind tunnel test facility CLOWT at Muenster. These applications cover the experimental investigation of the flow of an organic vapor past a circular cylinder and a centrifugal organic vapor compressor's performance. The results of the test cases support the main findings of the dimensional analysis.

Proximate Model of Gear Drive Units Based on Dimensional Analysis for Wear Process Evaluation

Excessive wear of gears will not only cause noise and vibration in the transmission system, but also reduce transmission efficiency and accuracy in severe cases, causing irreversible losses to the transmission system. It is desirable to develop a micro-gear unit model for evaluating the wear process and predicting the failure time of large gear units (such as wind turbine gear units), reducing losses due to sudden failures. Based on the Buckingham pi-theorem of dimensional analysis and Hertz formula, the similarity ratio of each parameter of the gear wear process was proposed. The maximum equivalent stress is calculated by establishing the FEM model and comparing it with the theoretical contact stress calculated by the Hertz formula, and the results were relatively consistent. Two pairs of gear friction and wear experiments with similar parameters were carried out to compare the wear evolution performance of two similar gears. The friction performance process of the test gears was observed by particle counter and analytical ferrograph. The results show that the friction and wear processes of the two groups of gears with similar parameters have a certain correlation, which was consistent with the proposed similarity model. The similarity model combined with the observation results of abrasive particles has a certain application value for the evaluation of the wear state of the transmission system.

Experimental Study on Heating Performance and a Novel Calculation Method of Water Outlet Temperature Based on Air Source Transcritical CO2 Heat Pump System

As a natural refrigerant, CO2 can greatly improve the environmental protection of the heat pump system. Since there is no perfectly suitable heat transfer correlation for supercritical CO2 at present, the water outlet temperature of the transcritical CO2 heat pump system has not been predicted. To study the applications of transcritical CO2 heat pump systems in heating performance and hot water supply, a series of experiments are carried out by an air source transcritical CO2 heat pump test rig. The experimental results show that the main factors affecting outlet water temperature are the system COP, the discharge pressure, the compressor frequency, and the ambient temperature. Based on the experimental results, a dimensionless correlation equation on outlet water temperature is proposed by the Buckingham PI theorem. This equation can be used to calculate the outlet water temperature of the air source transcritical CO2 heat pump systems with different sizes, and the calculation accuracy can be maintained at 13% with experimental results. Finally, the influence factors of the gas cooler water outlet temperature are analyzed based on the novel calculation method. Therefore, this study provides a reference for the prediction of the water outlet temperature of the transcritical CO2 heat pump system.

Data-driven and validated dimensional analysis for rational scale-up of a dual-chamber microbial fuel cell system for water-energy nexus exploitation

Mathematical modelling of microbial fuel cells (MFC) facilitates their scale-up by maintaining dimensionless parameters across reactor volumes for consistent performance. This study developed data-driven correlations to predict areal power density for a batch-fed dual-chamber MFC using hybridised first-principle mechanistic model and Buckingham’s Pi theorem. The established correlations were validated using experimentally-derived data for pre-enriched electroactive biofilm from mixed cultures. The biochemical model parameters are infilled with stoichiometric and thermodynamics estimations. Results showed that the correlations using logistic kinetics (Nash-Sutcliffe Efficiency, NSE = 0.59) outperformed Monod kinetics (NSE = 0.52) as the latter was not suitable for representing the first-order biochemical kinetics under limited substrate conditions. Sensitivity analysis on varying pH and bicarbonate concentration improved model predictions by ± 50%, though relative absolute error was ± 20% due to inherent error of estimated biochemical parameters. The application of hybridised approach for modelling MFC provides renewed perspectives for their rational design and scale-up applications.

3D numerical and experimental modelling of multiphase flow through an annular geometry applied for cuttings transport

Accurate estimation of volume fraction and pressure gradient is considered indicating parameters of efficient cuttings transportation. It is vital, in this regard, to consider all parameters that can affect cuttings volume fraction and pressure drop during enrollment of the drilling process. The analysis was conducted based on the turbulent flow of a power-law fluid through an annular domain by employing the Finite Volume Method. In addition, dimensional relationships were developed with the Buckingham-π theorem. Before carrying out simulations, the numerical schemes were validated using actual measurements made with the flow loop system available in Texas A&M University at Qatar. The simulation results demonstrated the followings: (i) using a power-law type drilling fluid with a shear-thinning character would reduce energy consumption for an inclination greater than 45°; (ii) inclination angles from 45° to 60° would be least desirable for an effective cuttings transportation with a turbulent Ostwald-de Waele fluid.

Development and validation of a model for soil wetting geometry under Moistube Irrigation

We developed an empirical soil wetting geometry model for silty clay loam and coarse sand soils under a semi-permeable porous wall line source Moistube Irrigation (MTI) lateral irrigation. The model was developed to simulate vertical and lateral soil water movement using the Buckingham pi (π) theorem. This study was premised on a hypothesis that soil hydraulic properties influence soil water movement under MTI. Two independent, but similar experiments, were conducted to calibrate and validate the model using MTI lateral placed at a depth of 0.2 m below the soil surface in a soil bin with a continuous water supply (150 kPa). Soil water content was measured every 5 min for 100 h using MPS-2 sensors. Model calibration showed that soil texture influenced water movement (p < 0.05) and showed a good fit for wetted widths and depths for both soils (nRMSE = 0.5–10%; NSE ge 0.50; and d-index ge 0.50. The percentage bias left( {PBIAS} right) statistic revealed that the models’ under-estimated wetted depth after 24 h by 21.9% and 3.9% for silty clay loam and sandy soil, respectively. Sensitivity analysis revealed agreeable models’ performance values. This implies the model's applicability for estimating wetted distances for an MTI lateral placed at 0.2 m and MTI operating pressure of 150 kPa. We concluded that the models are prescriptive and should be used to estimate wetting geometries for conditions under which they were developed. Further experimentation under varying scenarios for which MTI would be used, including field conditions, is needed to further validate the model and establish robustness. MTI wetting geometry informs placement depth for optimal irrigation water usage.

Dimensional Analysis Model Predicting the Number of Food Microorganisms.

Predicting the number of microorganisms has excellent application in the food industry. It helps in predicting and managing the storage time and food safety. This study aimed to establish a new, simple, and effective model for predicting the number of microorganisms. The dimensional analysis model (DAM) was established based on dimensionless analysis and the Pi theorem. It was then applied to predict the number of Pseudomonas in Niuganba (NGB), a traditional Chinese fermented dry-cured beef, which was prepared and stored at 278 K, 283 K, and 288 K. Finally, the internal and external validation of the DAM was performed using six parameters including R2, R2adj, root mean square error (RMSE), standard error of prediction (%SEP), Af, and Bf. High R2 and R2adj and low RMSE and %SEP values indicated that the DAM had high accuracy in predicting the number of microorganisms and the storage time of NGB samples. Both Af and Bf values were close to 1. The correlation between the observed and predicted numbers of Pseudomonas was high. The study showed that the DAM was a simple, unified and effective model to predict the number of microorganisms and storage time.

Utilizing Buckingham Pi theorem and multiple regression analysis in scaling up direct contact membrane distillation processes

Predicting the performance of a full-scale direct contact membrane distillation (DCMD) module based on experimental lab-scale results is rather difficult, since the DCMD performance is dependent on many different process parameters. Hence, there is a need for a methodology to perform DCMD system up-scaling based on lab-scale experimental results. In this study, we devise an approach to scale up the performance of DCMD systems by using the Buckingham's Pi theorem to group the DCMD process parameters into eight relevant dimensionless groups. Experimental data obtained from literature at various module dimensions were used to evaluate the developed dimensionless groups. An experimentally validated computational fluid dynamics (CFD) model was also developed and used to extend the coverage of operational parameters beyond the available experimental data. Then, two empirical dimensionless correlations were created, using multiple nonlinear regression analysis, and then validated, to enable the prediction of flux and pressure drop in DCMD systems at any scale.

A study on the mathematical model for predicting the peel removal efficiency of a cassava peeler

A mathematical model for predicting the peeling efficiency of a cassava peeler was developed using a dimensional analysis based on Buckingham' s pi theorem. The model was validated using data from experimental studies which revealed a maximum coefficient of determination of R2 = 0.8366 between the measured and predicted values. The developed model proved appropriate in estimating the peel removal efficiency for a cassava peeler by up to 83.66%. There was no significance difference between the experimental and predicted values at a 0.05 significance level.

Experimental investigations on nucleate pool boiling over micro-finned cylindrical surfaces

The nucleate pool boiling heat transfer over micro-finned cylindrical surfaces has application in the heat exchangers used in thermal power plants and chemical industries. The estimation of boiling heat transfer coefficient is an important parameter in the design of two-phase heat exchangers using micro-finned cylindrical surfaces. In the present work, related experimental investigations on four micro-finned cylindrical surfaces with different surface geometry using refrigerant R-141b at atmospheric pressure are conducted to determine the boiling heat transfer coefficient over micro-finned cylindrical surfaces. A correlation is developed by dimensional analysis wherein the effects of geometrical parameters, operating pressure and thermo-physical properties of fluids are taken into consideration and dimensional analysis conducted using Buckingham π-theorem. The correlation developed utilizes experimental data obtained over the present study as well as from previous studies by various researchers including experimental data for water over different micro-finned cylindrical surfaces at 1 bar by Mehta and Kandlikar, experimental data for R123 at 0.97 bar by Saidi et al. and experimental data for R134a over micro-finned cylindrical surface at 6.1 bar, 8.1 bar, 10.1 bar and 12.2 bar by Rocha et al. The heat flux ranging from 5 to 1100 kW/m2 are considered for the analysis. The data points have been compared with the proposed correlation and the absolute average deviation of the whole data set was obtained as 13.43% with root mean square deviation of 0.0273. All the predicted values were within ±15% of the experimental values of the boiling heat transfer coefficient.

Effect of roller diameter on the production rate in hot rolling process: Mathematical modeling and experimentation approach

Metal forming techniques are critical in the industrial sector. Metal forming procedures are used to create components with cross-sections such as conical, elliptical, cylindrical, oval, etc. As mentioned earlier, one of the procedures for forming the shapes is plate bending. The functioning of a plate rolling mill is related to various loading restrictions, while metals are bit with the lateral rollers of a reversed roller stand. Hot rolling is a vital phase in the manufacture and fabrication of steel sheets, resulting in a product with a high degree of flexibility. The paper justifies the experimental model, which verifies the experimental result using a mathematical model for the hot rolling process production rate. The mathematical model was developed using the Buckingham Pi (π) methodology and the dimensional analysis method. In addition, the influence of roller diameter on production rate was calculated at a diameter of 225 mm, 285 mm and 345 mm. The maximum production rate of 46.2 tons, 49.5 tons and 53.0 tons was observed at a roller diameter of 225 mm, 285 mm and 345 mm respectively. The percentage error between experimental and mathematical readings varies from 4.1 to 10.0%, 3.7 to 8.6% and 2.3 to 8.2% at roller diameter of 225 mm, 285 mm and 345 mm respectively. The experimental results are closely related to mathematical a modeling value, which further verifies the developed mathematical model using the Buckingham Pi (π) approach.

A study on dimensional analysis modeling of crater size during wire electrical discharge turning process by using Buckingham Pi theorem

The investigation on material removal by thermal erosion of discrete spark and vaporization in wire electrical discharge turning process was made to understand the crater size variation on turned components. In this study the modelling is done to establish the relationship between dependent and independent variables through Buckingham’s Pi-theorem, to predict the variations of crater diameter depending on physical and thermal properties, subsequently the dimensional model was validated by conducting experiments on wire-electrical discharge turning process for two distinct density variant materials such as Aluminium 6061 and INCONEL 718 super alloys. The density, enthalpy of vaporization, radius of spark, specific heat and other quantities effect on crater diameter have been discussed in this research paper.