Pi Theorem Research Articles

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.

Experimental and mathematical analyses of process parameters for roll compaction of steel powder

To reduce these costs and energy consumption, direct powder rolling (DPR) is one of the promising techniques. In this research work, DPR is used at room temperature and the process is evaluated to find out the relation between the process and material parameters. The steel powder was compacted to a green sheet using laboratory-scale powder compaction two high rolling machine equipped with gravity feeder to supply the steel powder. Optical microscopy and image J software were used to determine the porosity (%). A mathematical model using Buckingham pi theorem was formulated to establish the significance of influencing parameters on sheet densification. 3D Surface plot and 2D contour were analyzed to define the maximum sheet densification parameter regime with respect to the employed process variables for powder roll compaction.

A generalized empirical expression for the collapse of worn tubulars with a crescent-shaped wear groove under combined loads

As a large proportion of easily recoverable global hydrocarbon reserves is extracted and fields in operation are maturing, the oil and gas industry requires new methods of ensuring well integrity. Production of the remaining reserves may involve, for instance, extended-reach wells, multilateral wellbores, long horizontal sections and highly deviated trajectories. Moreover, wells in mature fields are being used to increase recovery through side-tracking. Casing wear has a significant influence on the well integrity in such operations and the ability to assess its effects is crucial, both in terms of its impact on health, safety and environment, and in terms of enabling cost-efficiency. This article utilizes the Finite Element Method (FEM) to analyze the stress distribution in worn tubulars with a crescent-shaped wear groove under external pressure. A model for tubular collapse is proposed using generalized empirical expressions for stress concentration factors in worn tubulars. The generalized expressions are based on 640 finite element analyses of different geometries designed to represent typical dimensions used in the oil and gas industry. Comparisons with experimental data and existing models demonstrate improvement in both accuracy and precision using the developed approach. • The influence of wear on elastic and yield collapse is significantly different. • A method for automated finite element analysis is demonstrated. • Results from finite element analysis are generalized using the Buckingham PI theorem. • The new proposed model for collapse of worn tubulars shows excellent performance.

Establishment of mathematical model to predict the dehulling efficiency in miniature pulse splitter machine energized by human powered flywheel motor using dimensional analysis

In India, nearly 58% populations directly or indirectly have their livelihood dependent on farming activity and agricultural food processing. Majority of population in India fulfills their nutritional needs from pulses. India is among the major producer and consumer of pulses contributing nearly 28.34% of pulses globally. The percentage of losses in domestic pulses production nearly equal to its import resulting in decline of pulses per capita. The dehusking and splitting of the grain at the pulse mills are the most important post harvest operation which decides the yield & quality of pulses. The losses at post harvest activities are mainly subjected to the power availability at rural area, availability of technology for use of sustainable energy sources, availability of processing units in reach of farmers and continuous research in post harvesting methods &and techniques & it’s practical viability. The human powered flywheel motor (HPFM) based, pulse producing machine fulfilling the requirements at rural areas is presented here. In the present paper dimensional methodical approach is discussed to determine the effect of the parameters affecting dehulling efficiency of a developed miniature pulse splitter machine energized by HPFM. A dimensional analytical technique using Buckingham Pi-theorem is used to establish an operating relationship between the dependent variable i.e. dehulling efficiency and independent variables which includes diameter of pulse, bulk density, coefficient of friction, feed rate, speed, emery grade, gap between wheel, humidity & air velocity. The present performance of the split pulse machine is improved to its highest value of 80.5% with the feed rate of 45 kg/hr & at the gap of 5 mm between the wheels. The reading obtained from the mathematical data and experimental observation is in good agreement which indicates the minimum error and validation of the mathematical model. The paper presents the method of formulation of mathematical model with the output variable dehulling efficiency which depends on many input variables for the pulse splitting machine. The formulated model can further be used for the other processing units in the agricultural production.

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.

SIMULATION OF DESIGN VARIABLES EFFECT ON PERFORMANCE OF A COMMON BEANS (Phaseolus vulgaris L) PORTABLE THRESHER

In Kenya, threshing of common beans is mainly by traditional method using sticks and animal tramping, which are slow, inefficient and tedious. Consequently, there is a need to develop portable threshers locally available in the market for small and medium-scale farmers. The objective of this study was to simulate design variables effect on the performance of a common beans portable thresher. Sizing of design variables and parameters was key in development of bean thresher. This could be achieved by costly experiments or use of prediction mathematical model equation. The later method was used by developing mathematical models from combination of Buckingham pi theorem and reference to other similar work in literature. The predicting equation for power requirement, grain losses, grain damages, efficiency and throughput capacity were developed and validated using experimental thresher from the same study. The results showed that there was a positive correlation with R2 of 0.9. Based on actual data and 10% absolute residual error interval, the prediction performance of the developed models were above 77%. The results noted that increase in cylinder peripheral speed of the pegs resulted into increase in power requirement, bean grains damages, threshing efficiency and throughput capacity. Also increase in effective cylinder diameter caused increase in threshing efficiency and grain damages.

Mathematical Model for Scaling up Bioprocesses Using Experiment Design Combined with Buckingham Pi Theorem

Scaling up bioprocesses from the experimental to the pilot or industrial scale involves heuristics and scale relationships that are far from the specific phenomena and are usually not connected to the experimental data. In complex systems, the scaling-up methodology must connect the experimental data with the tools of engineering design. In this work, a two-stage gold bioleaching process was used as a case study to develop a mathematical model of bioprocess scaling that combines the design of experiments with dimensional analysis using the Buckingham Pi theorem to formulate a predictive model that allows scaling up bioprocesses. It was found that the C/N, C/K, and T/C ratios are dimensionless factors that can explain the behavior of a system. Using the Pearson Product–Moment bivariate analysis, it was found that the dimensionless factors C/N and C/K were correlated with the leaching potential of the fermented broth at 1060 cm−1. With these results, a non-linear logarithmic model based on dimensionless parameters was proposed to explain the behavior of the system with a correlation coefficient of R2 = 0.9889, showing that the optimal conditions to produce fermented broth comprised a C/N ratio close to 50 and a C/K ratio close to 800, which allows predicting the scaling of the bioprocess.

Development of a mathematical model for prediction of heat transfer coefficient in micro-channel heat sink

The advancement of miniaturization and high-density manufacturing of electronic devices necessitates lightweight and high-performance heat dissipation components. As a result, traditional coolants and cooling technologies are rapidly unable to satisfy the ever conditioning problems and requirements of specific high temperature generated electronic devices. A microchannel heat sink (MCHS) has been extensively researched and commonly used in electronics cooling as a novel system that is highly suitable for applications with high power and small heat dissipation space. This research presents a mathematical model focused on dimension analysis to determine the heat transfer coefficient of a microchannel heat sink in electronics cooling. Buckingham Pi (π)theorem with dimensional analysis was used to determine a functional correlation between the heat transfer coefficient and independent variables. Independent variables comprise; thermal conductivity (copper MCHS), hydraulic diameter (MCHS), length (MCHS), a mass flow rate of nanofluid, the density of nanofluid, inlet temperature of nanofluid, heat flux, % by volume concentration of nanoparticle and thermal conductivity of nanofluid. The established relationship between dependent and independent variables concerning various operating conditions is helpful for analyzing behaviour.

Non-dimensional analysis for heat pipe characteristics in the heat pipe heat exchanger as energy recovery device in the HVAC systems

Heat pipe heat exchangers (HPHE) were widely applied as a recovery energy device in HVAC systems and significantly can reduce the energy consumption of HVAC systems. This study performs experimental and non-dimensional analysis using the Buckingham Pi theorem to develop a correlation of heat pipe performance as an energy recovery device in air HVAC systems. The experiment was conducted to investigate heat transfer characteristics in the heat pipe heat exchangers. The fresh-air temperature in the evaporator section (Te,in) was varied at 30–45 °C, fresh air velocity varied at 1.5–2.5 m/s, and the number of rows HPHE varied at 3, 6, and 9 rows. While the return air temperature was entering the condenser's inlet side, HPHE (Tc,in) was kept at 23 °C. The procedures of the Buckingham Pi theorem were adopted to determine non-dimensional parameters and found a correlation between Pi forms. Each non-dimensional parameters value was calculated based on the measured data. The correlation between non-dimensional parameters can be solved simultaneously using the multiple linear regression with the logarithmic transformations method. The new correlation has been found in the form Sp Number. The results show that Sp numbers is affected primarily by the number of rows (n), effectiveness (ε), and Reynold numbers (Re,D). The equation Sp number can be used to predict the thermal resistance of a single heat pipe and is very useful to know the performance of a heat pipe in the design and operating stages.

Corrected similarity laws of stiffened panels considering geometric distortion under low speed impact load

In the experiment of scaled models to dealing with the impact of large structures, the strain rate hardening effect can cause large differences between dynamic responses of prototype structure and scaled models. It is almost impossible to completely scale models due to manufacturing restrictions. In the present study, a corrected method based on VSG (V represented initial impact velocity, S represented dynamic flow stress and G represented impact mass) system is developed through Buckingham-Π theorem. The dimensionless analysis focus on factors related to energy absorption, in which the static moment of the cross-section is regarded as the dominant one. Through introducing the corrected velocity factor βv which considered the strain rate hardening effect and geometric distortion, experimental and numerical tests are established and consistent results show the method is effective to relate dynamic responses of scaled models even with distortion to the prototype structure. Additionally, as the distortion factor affects scaled models design greatly, the sensitivity of geometry distortion is discussed and it is shown that a more notable difference occurred when scaled models under the negative βD. Furthermore, it is confirmed in the paper that although exact similitude is not generally possible through the Cowper-Symonds constitutive model, good accuracy is nevertheless achievable regardless of structure facture.

Numerical simulation and experimental study on hematite production by oxidation of mill scale

Mill scale is a waste product from steel hot rolling processes and containing a high amount of FeO and Fe3O4. It is crucial to recycle and reuse this waste for the recovery into a metallic iron or its single oxide derivative. One of the end products of mill scale oxidation is hematite which has multitude of uses in various application. In this present study, mill scale is converted to hematite by roasting a mixture of mill scale at a specific set of conditions at 900°C for a varied time under ambient air environment by addition of pelletized limestone as a heat storage media. In this work, the Dimensionless Degree of Oxidation Prediction Model (DDOPM) has been constructed to gain hematite purity approach in the resulting powder by using Matlab. The Buckingham Pi Theorem is used to find dimensionless parameters considering the effect of different parameters including the geometric factor of tubular horizontal furnace used, characteristic of mill scale, ambient air factor, and thermal characteristics of limestone. The degree of oxidation from experimental result was obtained from XRF analysis and compared to the result of DDOPM to show the correlation between the experimental and numerical.

Formulation of mathematical model using dimensional analysis approach for plastic shredding process through Human powered flywheel Motor (HPFM)

The practical feasibility and productive viability of a Human-powered flywheel machine (HPFM) system having the capacity to produce 3 to 5 Horse Power (H.P) is preferred over motorized machines for rural-based applications. The present work deals to establish precise dimensionless equations for four parameters i.e. Duration of Energy consumption, Quantity of Material Processed/Time, Quality of shredding parts, Process Horse Power (H.P), and Instantaneous Load Torque using the Buckingham Pi-theorem technique for the waste monomer plastic shredding operation using HPFM. The independent and dependent variables affecting the shredding operation have been identified to form the experimental data-based model equations for the above-mentioned response variables. These equations will further help to understand the phenomenon of shredding plastic using HPFM in the rural sector and they will also help in developing the capacity of the machine according to the various requirements. A distinct experimental plan has been established for calculating the kinetic energy of the system and the different variables identified for the process.

Modeling for prediction of design parameters for micro-hydro Archimedean screw turbines

The present research aimed to develop a model for predicting the optimal efficiency of Archimedes Screw Turbines (ASTs) under different conditions using Response Surface Methodology (RSM). In RSM, the effect of several independent parameters (inputs) on a response parameter (output) is investigated. Among different geometric and flow parameters of an AST, screw’s outer diameter, blade number, inclination angle, rotation speed, available head, and flow were used to define the efficiency of ASTs. The well-known Buckingham Pi theorem was used to describe non-dimensional relationships between independent and response parameters. A Face-Centered Central Composite Design response surface was used for the investigation. To this end, the desired ranges of dimensionless effective parameters were considered. Computational Fluid Dynamics (CFD) was employed to determine AST’s efficiency in the considered ranges. Finally, a quadratic prediction model was developed, with R2 of 0.9755 and a maximum P-value of 0.0001. The maximum value of the relative error was 4.75%, and the maximum efficiency was obtained as 83.4% while blade number, inclination angle, dimensionless diameter were 4, 27°, and 0.731, respectively. The accuracy of the developed model, which is only valid for small laboratory-scale Archimedes screws due to the utilized laboratory-scale screw data, was reasonable.

Applying Criteria Equations in Studying the Energy Efficiency of Pump Systems

This paper presents a method for evaluating the energy efficiency of pump systems used to transport fluids .It is mainly scientifically applied and engineering-applied in nature and aims to propose a new approach (method) to researchers in their study of the energy efficiency of such systems. By applying the well-known scientific method of Dimensional Analysis (Buckingham π-theorem), dimensionless complexes (π-criterions and their relevant equations, which are original (innovative) and are offered for the first time in the scientific literature), used in accomplishing an energy assessment and analysis of such systems, are obtained. The criterion ∏1 = ev/ρgD represents specific energy consumption in kWh/m3 for a given pipe system with an exemplary diameter D. The criterion ∏2 = Q/[n(Hp − Hst)D2] represents a generalized parameter which is characterized by the selected method of flow rate (Q) regulation for a pump system with given static head Hst—by changing the speed of rotation (VFD, Variable Frequency Drive), by throttling, leading to an increase of the system hydraulic losses hv = (HpHst) or by diverting a part of the flow, known as “by-pass”, where the pump operates with the required system head Hp, but ensures higher flow rates, i.e., Qp > Qs. The flow rate criterion ∏3 = Q/(νD) characterizes the flow rate for a pipe system with an exemplary diameter D, used to transport a liquid with known viscosity ν. An example for applying these dimensionless complexes in accomplishing a quantitative evaluation of the energy efficiency of a given pump system is presented. A method for determining the main parameters forming these criterions, used to describe the different methods of flow rate regulation, has been developed. To demonstrate the application of this method, newly proposed by the authors, including obtaining the relevant criteria equations of the type ∏1 = f(∏2, ∏3), a certain pump system was used. This original approach for studying pump systems used to transport fluids can be used both to accomplish an energy analysis of such systems as well as to solve for optimization or other engineering problems.

Utilization of thermoelectric technology in converting waste heat into electrical power required by an impressed current cathodic protection system

Supplying an energy source for impressed current cathodic protection is considered as the main challenge for pipelines corrosion protection in remote areas. The purpose of this study is to construct and evaluate the performance of an impressed current cathodic protection system using waste heat to replace the external electrical power supply. The system consists of four thermoelectric modules producing the electrical power from the hot combusted gas stream. Mathematical modeling and Buckingham Pi theorem were applied to obtain five dimensionless parameters and several models to estimate the system performance. According to the experimental results, the generated voltage, electrical current, and temperature difference have been increased rapidly during the experiment up to 201 mV, 44 mA, and 20 K, respectively. Moreover, mathematical modeling and Buckingham Pi theorem were also utilized to obtain three equations, two nonlinear and one dimensionless equation, to calculate the generated electrical voltage with the maximum error of 1.22%, 9.11%, and 10%, respectively. The results indicate that the temperature difference (between inlet gas and environment) and the figure of merit have a direct effect, and heat sink thermal resistance has an inverse effect on the generated electrical voltage.

Implications of Buckingham’s Pi Theorem to the Study of Similitude in Discrete Structures: Introduction of the RFN, μN, and SN Dimensionless Numbers and the Concept of Structural Speed

AbstractMotivated by the need to evaluate the seismic response of large-capacity gravity energy storage systems (potential energy batteries) such as the proposed frictional Multiblock Tower Structures (MTS) recently discussed by Andrade et al. (2021, “Seismic Performance Assessment of Multiblock Tower Structures As Gravity Energy Storage Systems,” ASME J. Appl. Mech., Submitted), we apply Buckingham’s Pi theorem (Buckingham, E., 1914, “On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” Phys. Rev., 4, pp. 345–376) to identify the most general forms of dimensionless numbers and dynamic similitude laws appropriate for scaling discontinuous multiblock structural systems involving general restoring forces resisting inertial loading. We begin by introducing the dimensionless “mu-number” (μN) appropriate for both gravitational and frictional restoring forces and then generalize by introducing the “arbitrary restoring force number” (RFN). RFN is subsequently employed to study similitude in various types of discontinuous or discrete systems featuring frictional, gravitational, cohesive, elastic, and mixed restoring forces acting at the block interfaces. In the process, we explore the additional consequences of inter and intra-block elasticity on scaling. We also formulate a model describing the mechanism of structural signal transmission for the case of rigid MTS featuring inter-block restoring forces composed of elastic springs and interfacial friction, introducing the concept of “structural speed.” Finally, we validate our results by demonstrating that dynamic time-histories of field quantities and structural speeds between MTS models at various scales are governed by our proposed similitude laws, thus demonstrating the consistency of our approach.

Dimensional analysis applied to jacketed shell and tube heat exchangers modeling

Dimensional analysis was utilized on this research to establish a shortcut model for predicting hydrogen sulphide gas discharge temperature in jacketed shell and tube heat exchangers. Since the equipment belongs to an online industrial facility, the passive experimental method was applied. Selection of the heat transfer process parameters was followed by application of the Buckingham Pi-theorem and the repeating-variables technique. After formulation of the dimensionless groups, approximation of the explicit model equation was carried out through a least-squares multivariate linear regression. The model predictive ability performance was appraised by comparing predictions versus measured discharge temperatures, hence attaining a Pearson correlation of 97.5 %, a mean absolute error of 2.1 K, and 1.7 % maximum deviations. The explicit equation that was obtained is pertinent to studied heat exchangers, when 0.55 ≤ ṁ1 ≤ 0.60, 1.06 ≤ ṁ2 ≤ 1.09, and 0.22 ≤ ṁ3 ≤ 0.24 (fluids flowrate, kg/s). It can be used as an alternative calculation method for quick anticipation of the equipment performance, which overcomes computation of the overall heat transfer coefficients.

Mechanistic modelling of Chinese hamster ovary cell clarification using acoustic wave separator

Acoustic wave separation (AWS) is gradually emerging as an effective method for continuous clarification of Chinese hamster ovary (CHO) cells. In this paper, we present a mechanistic model for AWS based on population balance model (PBM) along with acoustic agglomeration kernel to predict cell size distribution as a function of time. Results show that more than 90% CSE was achieved for flowrates <5 mL min−1 and acoustic power >12.5 W. Experimental results confirm that the mechanistic model offers an accurate prediction of the CHO agglomeration process. Furthermore, a generalized correlation has been developed using the Buckingham Pi theorem between the dimensionless numbers composed of relevant process and geometrical variables to predict the agglomeration efficiency. To our knowledge, this is the first such study to model the agglomeration dynamics of CHO cells in acoustic standing waves.