Pi Theorem Research Articles

An Experimental Approach of Generation of Micro/Nano Scale Liquid Droplets by Electrohydrodynamic Atomization (EHDA) Process

Electrohydrodynamic atomization (EHDA) process is a method of generating liquid droplets through the application of a large electrical potential difference. It has a wide range of industrial applications.This paper investigates the application of EHDA for generation of particles in micro/nano scale. One of the primary objectives of this work is to report the nature and role of electrohydrodynamic during the process. Flow visualization has been conducted to report different modes appearing during the process. The flow is visualized for different operating conditions, both above and below the grounded ring electrode. It is observed that the number of modes of e-spray increases with decreasing electrical conductivity of fluid. Micro particles were generated for different set of fluids and its morphology is analyzed using both stereo microscope and scanning electron microscope (SEM). The sizes of generated particles increases nonlinearly with increasing flow rate for a fluid. It is observed that the charge carried by droplets in DC elecrospray increases/decreases sharply when the mode of e-spray is changes, unlike AC electrospray where generated droplets are neutral in nature. Detailed dimensional analysis has been carried out using Buckingham Pi theorem. The results are reported in terms of dimensionless numbers for distinguishing between various modes.

Sensitivity of driving forces on molten pool behavior and defect formation in high-speed gas tungsten arc welding

The effect of multi-coupled driving forces on molten pool behavior and defect formation in high-current and high-speed gas tungsten arc welding is investigated quantitatively and systematically through combination of sensitivity analysis and experimentally validated numerical modelling. The arc shear stress is the dominant driving force to facilitate free surface deformation of molten pool and backward flow of liquid metal, which masks the effect of Marangoni force. The capillary pressure plays a dominant effect on suppressing free surface deformation. Three dimensionless groups with explicit physical implications are derived to evaluate tendency of high-speed weld bead defect formation using Buckingham π-theorem. The relevant sensitivity analysis shows that arc shear stress and capillary pressure are the most promoting and suppressive factors for defect formation, respectively. High active element concentration induces undercut and humping defects by decreasing capillary pressure.

Equipment performance analysis of a Canadian Kraft mill. Part I: Development of new key performance indicators (KPI)

Most process integration techniques and optimization procedures are based on the implicit assumption that the unit operations and equipment are working efficiently, which is not always the case. Equipment performance analysis by means of key performance indicators is a prerequisite step before undertaking any optimization or improvement measure. However, there are no key performance indicators specifically tailored for the Kraft process equipment, and that take into account both the governing operating conditions and the design parameters. In this paper, new key performance indicators (KPIs) have been developed based on an in depth characterization and dimensional analysis of the Kraft process equipment using the Buckingham Pi (Π) theorem. These KPIs were used to identify process inefficiencies, diagnose causes of inefficiencies and propose remedial actions through a systematic approach. Once adopted, the KPIs proposed could serve to monitor systems performances and allow a tighter process control as they incorporate the design characteristics of the unit operations and their influencing operating conditions.

Mathematical modelling of surface roughness for vapour processing of ABS parts fabricated with fused deposition modelling

Fused deposition modelling (FDM) has number of applications in the field of metal casting but the poor surface finish acts as major obstruction against rapid investment casting of the replicas prepared by this process. The vapour smoothing (VS) has proved to be promising technique as fine details of parts are preserved due to absence of tool-work piece contact and mechanical force. In the present study, the outcome of Taguchi and ANOVA analysis has been used to develop a mathematical model for average surface roughness of vapour treated parts prepared by FDM. The relationship between six input parameters, two of FDM and four of VS has been derived using Buckingham's Pi theorem. The study explored the effect of smoothing time and number of cycles on surface roughness and as confirmed by statistical analysis. The model gives output in form of percentage change in surface roughness as a function of smoothing time and number of cycles where final roughness of vapour exposed FDM parts can be predicted.

Shaking table model tests of concrete containment vessel (CCV) for CPR1000 nuclear power plant

Containment vessel of nuclear power plant is the last barrier to prevent the leakage of radioactive materials, so the seismic performance is of high importance. In order to investigate the seismic performance of the CPR1000 prestressed concrete containment vessel (PCCV) structure under design and beyond-design basis earthquake, a 1/20 scale concrete containment vessel (CCV) model was designed and constructed on the shaking table based on the similitude requirements from the Buckingham's π-Theorem. The horizontal and vertical earthquake waves generated from RG1.60 spectrum were selected as the input earthquake waves in the shaking table model test, and the earthquake waves with peak ground acceleration (PGA) 0.2g, 0.3g, 0.4g, 0.5g, 0.7g, 0.9g and 1.2g respectively were loaded step by step until the macroscopic failure occurred in the model containment vessel. The dynamic responses of the model CCV were obtained from the acceleration sensors and resistance strain gages. The model test results showed that the maximum acceleration magnification factor of the model CCV was 1.974 with PGA of 0.2g, which indicated that the CCV had a good seismic-resistant behavior under 0.2g earthquake. Until the second last test case with the horizontal peak earthquake motion of 0.9g (corresponding to 0.9g for the prototype structure), the fundamental frequency of the model CCV only decreased slightly and the maximum tensile strains at all monitoring points were much smaller than the ultimate tensile strain. It meant that the model CCV as a whole was within the elastic range. In the last case (1.2g), the model fundamental frequency decreased steeply. At the same time, the maximum tensile strains at the heel of the cylinder wall far exceeded the ultimate tensile strain and a circle of cracks occurred at the cylinder wall heel. It meant rupture failure had occurred and the cylinder wall heel of the CCV was the weak region. It agreed well with the numerical analysis results and the visual inspection and camera record at the test site. Although a circle of concrete cracks appeared around the cylinder wall foot, the model CCV still remained integral as a whole and neither cracking parts dislocation nor collapse occurred. Because the test model was a plain concrete structure and the model test results should be conservative estimation, it can be concluded that the prototype PCCV for CPR1000 nuclear power plant had sufficient seismic safety margin.

A Numerical Investigation to Identify Dimensionless Parameters for Dual-Rotor Horizontal Axis Wind Turbines

A numerical study was carried out to identify non-dimensional parameters for dual-rotor horizontal axis wind turbines (DRWTs). Based on some important DRWT parameters such as the rotor speeds, rotor diameters and the distance between the rotors, three dimensionless parameters were derived from the Buckingham Pi theorem. Hypothetical DRWT models were created using geometrically-scaled National Renewable Energy Laboratory (NREL) Phase VI rotor geometry and operating conditions in order to confirm the validity of these parameters. The performance of each turbine was simulated using DR_HAWT, an inhouse prediction tool for single and dual-rotor wind turbines created by the current authors. The variation in normalized output power as a function of the dimensionless parameters suggests that an improved performance of DRWTs can be obtained at lower diameter and gap ratios. The NREL Phase VI rotor equipped with a 5 m geometrically-scaled upwind rotor can generate about 88% of the combined power output of two equivalent single-rotors. In addition, the effect of having an auxiliary upwind rotor reduces the angle of attack along the inboard section of the downwind blade.

An assessment of operational loss data and its implications for risk capital modeling

A mathematical method based on a special dimensional transformation is employed to assess operational loss data from a new perspective. The procedure, which is formally known as the Buckingham (Pi) Theorem, is used broadly in the field of experimental engineering to extrapolate the results of tests conducted on models to prototypes. When applied to the operational loss data considered in this paper, the approach leads to a seemingly universal trend underlying the resulting distributions regardless of how the data set is divided (e.g., by event type, business line, revenue band). This dominating trend, which appears to also acquire a tail parameter of 1, could have profound implications for how operational risk capital is computed.

Comparative evaluation of R1234yf, R1234ze(E) and R450A as alternatives to R134a in a variable speed reciprocating compressor

A comparative energetic evaluation of R1234yf, R1234ze(E) and R450A as alternatives to R134a in a variable speed compressor is carried out. A compressor model based on dimensionless numbers was obtained using the Buckingham π-theorem, which was validated with experimental data; showing that the prediction error of the model is lower than ±10% and ±2 K for temperature. The experimental data were obtained by testing R134a, R1234yf, R1234ze(E) and R450A for a wide range of operating conditions. Results obtained with the validated model, show that the dimensionless approach provides a similar estimation of energy parameters compared with the experimental results, such as power consumption, refrigerant mass flow rate, cooling capacity, COP, discharge temperature and compressor efficiencies for each refrigerant tested using the dimensionless approach proposed. The comparative evaluation of the compressor predictions shows a reduction in the cooling capacity obtained with R1234yf, R450A and R1234ze(E), in comparison with R134a. Also, COP values for R1234yf, R450A, and R1234ze(E) are lower than those obtained from R134a. Finally, results shows that the dimensionless correlation compressor model can be used to predict the performance of other reciprocating compressors, at similar operating conditions for a wide range of compressor rotation speed, with a reasonable accuracy.

Nusselt correlation to predict heat transfer from an oscillated vertical annular fluid column through a porous domain

Experimental evaluation of the heat transfer in oscillating flow under the constant heat flux and constant amplitude fluid displacement conditions is presented for a vertical annular flow through a stainless steel wool porous media. The analysis is carried out for two different heat fluxes and for five different frequencies. The data is acquired from the measurements both in the initial transient period and in the pseudo-steady (cyclic) period by the system. The physical and mathematical behavior of the resulting Nusselt numbers are analyzed, according to data acquired from the experiments and in accordance with the results of the Buckingham Pi theorem. A cycle and space averaged Nusselt number correlation is suggested as a function of kinetic Reynolds number for oscillating flows. The suggested correlation is useful in predicting heat transfer from oscillating flows through highly porous and permeable solid media at low actuation frequencies and at low heat fluxes applied in the wall. The validity of the Nusselt numbers acquired by correlation is discussed using experimental Nusselt numbers for the selected kinetic Reynolds number interval. The present investigation has possible applications in moderate sized wicked heat pipes, solid matrix compact heat exchangers compromising of metallic foams, filtration equipment, and steam generators.

A Fractal Model of Mechanical Seal Surfaces Based on Accelerating Experiment

ABSTRACTThe change in surface topography caused by friction and wear influences the wear properties and service life of mechanical seal. To investigate the wear properties of mechanical seals and obtain the variation in surface topography, it is practical to carry out accelerating experiments by increasing the load, speed, or temperature of the medium than safe life tests in the field device. Based on the assumption that the same fractal roughness corresponds to the same wear properties, the pi theorem was introduced to derive the relationship of wear under experimental and actual conditions. By providing detailed values of constants in the pi theorem through accelerating experiments, a fractal model was established to predict the life expectancy of mechanical seals. Under the state of mixed friction, measured values of surface topography for experimental stationary rings and data on a mechanical seal ring's service life in the field device were in accordance with those of the model. The establishment of the fractal model provides conditions for the performance study of long-period operating mechanical seals.

Scaling analysis and experimental investigation of pulsating loop heat pipes

This paper focuses on performance analysis of pulsating heat pipe (PHP) with fluid having high boiling point like KOOLGARD (an engine coolant). Fluids like Ethanol and Methanol (E&M) mixture also have been used for checking of efficiency enhancement of PHP. Non-dimensional analysis has been carried out for PHP using Buckingham-Pi theorem. Using the obtained non-dimensional numbers, the range of heat flux for some common working fluid has been estimated. Approximate results are obtained before carrying out experimentation. KOOLGARD was found to sustain more than 200W/m2 heat flux, although the thermal resistances offered are high. Flow visualisation has been carried out as well. Different instabilities are visualised in Ethanol and Methanol (E&M) mixture case while no disturbance is visible with KOOLGARD case.

Cuttings transport behavior in directional drilling using computational fluid dynamics (CFD)

As one of the most crucial concerns in hole cleaning and especially in directional drilling, cutting transport optimization eventuates in the reduction of drilling costs and increase in oil recovery. Cutting removal is, however, affected by different parameters including fluid flow rate, drill pipe rotation, cutting size and inclination which have to be considered simultaneously. In the current report, Liquid-Solid model is implemented to solve the governing equations in the Eulerian CFD framework. Results are compared with experimental data and successful validation with experimental data is achieved. On the basis of design of experiment algorithm, further simulations have been carried out to investigate the effect of the as mentioned key parameters on cutting transport behavior. Results show that hole inclinations between 45 and 60° are the most difficult angles in hole cleaning process. Moreover, increase in values of flow rate and drill pipe rotation will effectively improve the drag effects leading to superior cutting removal. Besides, it has been proven that cutting size affects cutting transport less than the other parameters. For industrial applications, an empirical correlation is developed for estimating cutting concentration. Buckingham-π theorem along with multi-variable regression technique is applied to establish the correlation.

On the Prediction of Transient Wear

During the running-in process, a friction pair experiences drastic evolution in many of its tribological parameters, such as surface roughness, wear rate, and coefficient of friction until steady-state is attained. In this paper, we present a model for predicting the behavior of the running-in process. Specifically, we determine a general relationship between the wear loss and surface roughness during the running-in stage and test the validity of its prediction of wear rate by comparing to available experimental results. We show, by using a dimensional analysis and applying the Buckingham Pi theorem, that there exists a linear relationship between the transient dimensionless wear, the dimensionless initial surface roughness, and dimensionless running-in time.

A comparison of the modern Lie scaling method to classical scaling techniques

Abstract. In the past 2 decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics, including but not limited to overland flow, groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non-dimensionalization, and the Vaschy–Buckingham-Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is made with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables, which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

Critical Concentration Ratio for Solar Thermoelectric Generators

A correlation for determining the critical concentration ratio (CCR) of solar concentrated thermoelectric generators (SCTEGs) has been established, and the significance of the contributing parameters is discussed in detail. For any SCTEG, higher concentration ratio leads to higher temperatures at the hot side of modules. However, the maximum value of this temperature for safe operation is limited by the material properties of the modules and should be considered as an important design constraint. Taking into account this limitation, the CCR can be defined as the maximum concentration ratio usable for a particular SCTEG. The established correlation is based on factors associated with the material and geometric properties of modules, thermal characteristics of the receiver, installation site attributes, and thermal and electrical operating conditions. To reduce the number of terms in the correlation, these factors are combined to form dimensionless groups by applying the Buckingham Pi theorem. A correlation model containing these groups is proposed and fit to a dataset obtained by simulating a thermodynamic (physical) model over sampled values acquired by applying the Latin hypercube sampling (LHS) technique over a realistic distribution of factors. The coefficient of determination and relative error are found to be 97% and ±20%, respectively. The correlation is validated by comparing the predicted results with literature values. In addition, the significance and effects of the Pi groups on the CCR are evaluated and thoroughly discussed. This study will lead to a wide range of opportunities regarding design and optimization of SCTEGs.

A Mathematical Model for Predicting the Thermal Efficiency of a Gas Turbine Power Plant Using Dimensional Analysis Based on the Buckingham’s Pi Theorem

A Mathematical Model for Predicting the Thermal Efficiency of a Gas Turbine Power Plant Using Dimensional Analysis Based on the Buckingham’s Pi Theorem

Dimensional analysis and experimental study of gas penetration depth model for submerged side-blown equipment

A dimensionless penetration depth equation related to modified Froude number, Reynolds number and density ratio of gas–liquid was established by dimensional analysis and Buckingham pi theorem. Based on the experimental data of side-blown penetration, extracted by MATLAB digital image processing algorithm in this scaled water model experiment, a dimensionless expression which serves as a convenient way to determine the penetration depth under different operation conditions was developed by the Least Square Estimation (LES) method. Meanwhile, detailed comparison of different empirical equations from this study and other related researches has also been sufficiently investigated. At last, the influence of each dimensionless number on the penetration depth was discussed by partial correlation analysis.

A Numerical Model for the Identification of the Structural Damages in Rolling Contact Bearings Using Matrix Method of Dimensional Analysis

The damages to the structural elements, viz., inner race, outer race, rollers, cage, etc., of rolling contact bearings if not detected in time can cause tragic failures of the machineries supported by these bearings. The operating parameters like variations in the machinery speed, unbalance, operating load, etc., can cause a bearing to vibrate at higher energy levels and consequently will accelerate its wear. An attempt is made in this study, and a generalized model is developed using matrix method of dimensional analysis (MMDA) that predicted the response and correlated the dependent parameter, i.e., response with the significant independent parameters. Combined use of response surface methodology (RSM) is made to explore the dependence of various factors such as size of the defect, unbalance, speed, and their interactions on the vibration characteristics of the bearings. It is observed from the study that the model developed based on the MMDA has provided an efficient approach in recognizing the damaged bearing state, which can be easily implemented in the condition-based preventive maintenance strategies. Also, the effectiveness of MMDA as compared to the conventional Buckingham's pi theorem in the dimensional analysis (DA) practice, especially in the problems involving multiple variables, is shown in this study.

Image Processing for Measuring Damage and Delamination in Glass Reinforced Epoxy

This work proposed an image processing method for the measurement of damage and delamination in layered materials. Measurement of the damage and delamination area in composite materials has a significant importance as it provides an estimate of factors for implementation of fracture mechanics formulation. It is not always easy to measure delamination as it usually exists inter laminar. In this method, an image of the damaged composite material was processed by commercially available software and the delamination area was estimated within each layer. A strong back light was used for capturing images with prominent damaged areas and clear demarcation of such areas. This method proved itself simple, efficient, and cost effective. It has many advantages over simpler techniques like post experiment visual inspection. At the same time, it is cost effective for composites where standard non-destructive techniques may cost many folds more than the cost of the specimen. The results obtained for damage and delamination in cylindrical glass reinforced epoxy specimen were presented. Buckingham pi theorem was employed to develop a scale model consisting of four scales. Damage and delamination growth rate collapsed onto the same trend when scaled with the factors of n3 and n4, respectively. Accuracy and confidence in the measurement of delamination area reduced as the number of layers increased; however, overall damage area can be estimated precisely. This technique was used for in situ measurements while specimens were subjected to load, which provided damage and delamination growth rate, too.

Non-dimensional physics of pulsatile cardiovascular networks and energy efficiency.

In Nature, there exist a variety of cardiovascular circulation networks in which the energetic ventricular load has both steady and pulsatile components. Steady load is related to the mean cardiac output (CO) and the haemodynamic resistance of the peripheral vascular system. On the other hand, the pulsatile load is determined by the simultaneous pressure and flow waveforms at the ventricular outlet, which in turn are governed through arterial wave dynamics (transmission) and pulse decay characteristics (windkessel effect). Both the steady and pulsatile contributions of the haemodynamic power load are critical for characterizing/comparing disease states and for predicting the performance of cardiovascular devices. However, haemodynamic performance parameters vary significantly from subject to subject because of body size, heart rate and subject-specific CO. Therefore, a 'normalized' energy dissipation index, as a function of the 'non-dimensional' physical parameters that govern the circulation networks, is needed for comparative/integrative biological studies and clinical decision-making. In this paper, a complete network-independent non-dimensional formulation that incorporates pulsatile flow regimes is developed. Mechanical design variables of cardiovascular flow systems are identified and the Buckingham Pi theorem is formally applied to obtain the corresponding non-dimensional scaling parameter sets. Two scaling approaches are considered to address both the lumped parameter networks and the distributed circulation components. The validity of these non-dimensional number sets is tested extensively through the existing empirical allometric scaling laws of circulation systems. Additional validation studies are performed using a parametric numerical arterial model that represents the transmission and windkessel characteristics, which are adjusted to represent different body sizes and non-dimensional haemodynamic states. Simulations demonstrate that the proposed non-dimensional indices are independent of body size for healthy conditions, but are sensitive to deviations caused by off-design disease states that alter the energetic load. Sensitivity simulations are used to identify the relationship between pulsatile power loss and non-dimensional characteristics, and optimal operational states are computed.