Power Law Equation Research Articles

Improving design and scale-up of columns with structured packings by means of CFD

Structured packings in distillation columns present an opportunity for process optimization. Therefore, current design processes for columns with structured packings are evaluated with respect to their industrial applicability, and a power law equation is proposed for transferring HETP values of a reference system to a test system. A design process that includes CFD simulations is presented, and the CFD simulation setup is introduced. The validity of the simulation setup is examined with regard to the following mixtures: nitrogen/oxygen, chlorobenzene/ethylbenzene, and cyclohexane/n-heptane. Diffusion coefficients are varied in the simulation, and their influence on the HETP value is evaluated. The simulated HETP values are then compared with established mass transfer models and demonstrate a high level of agreement.

High heat producing granites and prolonged extraction of tungsten and tin from melts

The granites associated with world–class W and Sn deposits generally have high concentrations of U, Th, and K and can be classified as high heat producing (HHP) granites (>5 μW m−3) because of high radiogenic heat production, but how long radiogenic heat can prolong the suprasolidus lifetime of magmas and whether radiogenic heat exerts an impact on W and Sn mineralization is poorly understood. To answer these questions, finite element numerical modeling was established to link magma cooling by heat conduction with diffusion of W, Sn, and H2O from silicate melts to coexisting aqueous fluids before the solidus is reached. The magma in the models is emplaced to a depth of 5 km and its size (vertical thickness of 4–5 km and horizontal length of 20 km) is constrained by the granites associated with world–class W and Sn deposits. Fluid convection was not considered in the models. The sensitivity analysis and comparison results suggest that the suprasolidus lifetimes of HHP magmas are positively correlated with the heat production and magma thickness and negatively correlated with the solidus temperatures and the thermal conductivity of host rocks. Average radiogenic heat of 5–10 μW m−3, together with decreased solidus temperatures due to magmatic differentiation and a moderate thermal conductivity of host rocks, can prolong the suprasolidus lifetime of HHP magmas by 49–427 ka (1 Ma = 1000 ka), corresponding to 11–85% of the suprasolidus lifetime of equally sized magmas with normal radiogenic heat (2 μW m−3). From the available gravity modeling data, over half of the granites associated with world–class W and Sn deposits are at least 1 km thicker than the values calculated from the empirical power–law equation, and the modeling results indicate that an increase of 500–1000 m in magma thickness prolongs the suprasolidus lifetimes of HHP magmas (5 μW m−3) by 50–74%. During magma cooling before solidus is reached, the diffusion of W and Sn is at least several orders of magnitude slower than that of H2O in hydrous melts; thus, the equilibrium partitioning of these two metals between silicate melts and coexisting aqueous fluids is not reached, and the metal diffusion from melts is a rate-limiting step. The prolongation of the suprasolidus lifetimes by radiogenic heat and decreased solidus temperatures can allow extraction of 17–95% more W and Sn from silicate melts before the solidus is reached and increase the possibility of producing W and Sn mineralization at later stages. Therefore, the coexistence of HHP granites and world–class W and Sn deposits is not accidental.

Room temperature magnetocaloric effect and electrical transport properties of Co50Fe7V25Ga18Heusler alloys undergoing martensitic transformation

The metamagnetic martensitic transformation from the ferromagnetic austenite phase to the paramagnetic martensite phase was excited in the Co50Fe7V25Ga18 Heusler alloy near room temperature. Accompanying the transformation, the alloy exhibits a good magnetocaloric effect near room temperature with a cooling capacity of 17.02 J/kg under 3 T magnetic field at 302 K and the cusp-type anomalous of the resistivity with the magnetoresistance effect of ∼ 2.27% under a 3 T magnetic field. The power law equations for the temperature dependence of resistivity indicates the actuality of half-metallic properties. Our experimental results indicate that Co50Fe7V25Ga18 Heusler alloy is likely to become a potential candidate for room-temperature magnetic refrigeration and magnetic sensors applications.

The corelation between creep resistance and microstructure in AA6111 alloy strengthened by synergistic Er, Zr addition

In this research work, the creep behavior of AA6111 alloy strengthened by Er, Zr microalloying was investigated at 473–573 K/70–90 MPa. The results showed that the improvement on creep resistance induced by synergistic Er, Zr addition was better than the only Er addition, which can be attributed to the formation of Ll2-ordered Al3(Er, Zr) precipitates with special core-shell structure. The outer shell composed of slowing-diffusing Zr could act as a diffusion barrier, decreasing the coarsening rate and maintaining the thermal stability of Al3(Er, Zr) nanoprecipitates at high temperature. Thus, these nanosized Al3(Er, Zr) precipitates could hinder the movement of dislocations as well as pin grain boundaries effectively during creep process. Besides, coarse intermetallics embedded on grain boundaries were also refined to tiny pieces assisted by Er, Zr modification, which is an important reason for such prominent creep performance. Based on the modified power-law equation, the true stress exponents for all experimental alloys were measured to be 5, indicating the creep deformation was dominated by dislocation climbing mechanism.

Permeability modeling of Upper Cretaceous Bahariya Formation rocks, Abu Sennan field, Western Desert, Egypt

The present work aims to describe in detail the petrophysical properties of the Bahariya Formation (Abu Sennan field) through processing the core sample data and finding out the possible relations between permeability and other reservoir parameters such as porosity, specific internal surface, formation factor, and nuclear magnetic resonance to obtain reliable permeability prediction model of the Upper Cretaceous formation. The core samples have a length range of 2–4 cm with a diameter of 2.54 cm. Many petrophysical reservoir parameters were measured, such as permeability, porosity, density, formation factor, and special measurements such as spectral induced polarization, nuclear magnetic resonance, and specific internal surface. The samples cover a wide range of porosity (11.82–23.68 %) and permeability (0.06–133.65 mD). The porosity-permeability relationship indicates the increase of the permeability with porosity increase for the studied samples. The reservoir quality index ranges from 0.02 to 0.78 μm. The flow zone indicator ranges from 0.09 to 3.39 μm where pore throat radius (R35) ranges from 0.10 to 7.40 μm. The reservoir quality index (RQI) and flow zone indicator (FZI) cross-plot show a better understanding of the formation evaluation. The hydraulic flow units are (0.09 μm < FZI ≤ 3.39 μm) and RQI < (1 μm). NMR longitudinal relaxation time T1 range is 1.21–175 ms, NMR transversal relaxation time T2(100) range is 0.70–91.77 ms, and T2(600) range is 0.74–75.86 ms. Models that incorporate porosity and NMR relaxation times will be more accurate and suitable for permeability prediction. Using multiple regression to determine the exponents of the power-law equation and replacing the porosity with the formation factor can improve the permeability prediction for the samples under investigation. The utilizing of NMR in predicting the permeability through a robust model considers an additional value for the Upper Cretaceous rocks in Egypt.

Unveiling the generalization of the derivative order with a novel application of the fractional order model to green soybean oil extraction

Green solvents have represented suitable options for substituting nonrenewable substances applied to vegetable oil extraction processes. The general power-law equations used for this modeling present poor results. For better modeling, the generalization of the order of the equation is required. Scientific literature and laboratory data were collected in this study. The fractional model was derived from the standard power law first-order model using the Caputo fractional derivative and the Mittag-Leffler function. The fractional model (FOKM) and first-order model (FOM) were compared after data reconciliation using various statistical metrics and tests. The FOKM achieved the best statistical performance with Radj2 over 0.9841 compared to 0.9043 for FOM. The AIC for FOKM was higher in almost all cases, and Fisher's exact test showed that FOM is not statistically similar to FOKM. Both models presented the residues' normality by the Shapiro-Wilk normality test; however, a higher correlation between the parameters was found for the FOKM. It must be emphasized that the fractional model produced statistically significant improvements when fitting the data, showing the relevance of generalized order equations. This novel application of FOKM in green soybean oil extraction may help other researchers understand this phenomenon and provide a better representation of the process for industrial scale-up and optimization.

Process intensification through the application of centrifugal field: An experimental study of absorption processes using the falling film microreactor

In this work, liquid and gas side mass transfer characteristics were investigated within a centrifugal field using the falling film microreactor (FFMR). Enhancement in mass transfer rates were assessed by relating the absorption efficiency of the processes conducted within the centrifugal field to those in which liquid flows are purely-driven by gravity. The liquid side rotational mass transfer coefficients (kL) determined were considerably higher than those without rotation. Similarly, relative increase in rotational gas side mass transfer coefficient (kG) was also significantly high (up to 42%).The liquid side experimental data were well predicted by the modified correlation (a power law equation modified to consider the effect of the applied external force, geometry of the system, hydrodynamic characteristics and physical properties of the fluids employed) at lower rotational speed but deviated considerably at higher rotational speeds while, the gas side experimental data were well predicted by the correlation.

Room-Temperature Strengthening, Portevin-Le Chatelier Effect, High-Temperature Tensile Deformation Behavior, and Constitutive Modeling in a Lightweight Mg-Gd-Al-Zn Alloy

To explore room-temperature strengthening and high-temperature ductility, a lightweight novel Mg-1.85Gd-0.64Al-0.62Zn alloy was fabricated by innovative multidirectional forging and a hot-rolling technique. Microstructures and mechanical properties were studied at room and elevated temperatures with an optical microscope, an X-ray diffractometer, and a tensile tester. An ultimate tensile strength of 260 MPa, yield strength of 171 MPa, and elongation of 20.4% were demonstrated at room temperature. The room-temperature strengthening mechanisms were identified by strengthening the model estimation. A type C Portevin-Le Chatelier effect was discovered and elucidated in this alloy. X-ray diffraction analysis revealed that the phase composition is α-Mg solid solution and (Mg, Al)3Gd, Al7Zn3, and Al2Gd intermetallic compounds. Examination of the microstructure at elevated temperatures revealed that dynamic recrystallization and dynamic grain growth occur. In particular, it was discovered that bimodal microstructures or incomplete dynamic recrystallization microstructures exist in high-temperature deformation. A maximum quasi-superplasticity of 228.4% was demonstrated in this alloy at 673 K and 5.0 × 10-4 s-1. Flow stress curves showed that the present alloy exhibits Sotoudeh-Bate curves or a long intermediate strain-hardening stage followed by a strain-softening stage. A modified Zerilli-Armstrong constitutive equation incorporating the number of dislocations was established. The power-law constitutive equation was established to identify the deformation mechanism. Both constitutive models had good predictability. At 673 K and 5.0 × 10-4 s-1, the stress exponent was 4, and the average deformation activation energy was 104.42 kJ/mol. The number of dislocations inside a grain was 146. This characteristic evidence confirmed that dislocation motion controlled by pipe diffusion dominates the rate-controlling process under this condition.

Creep strengthening mechanisms of a superlattice γ′-strengthened Co–Al–W–Ta–Ti single crystal superalloy at steady-state conditions under compressive stresses

The strengthening mechanism of steady-state creep is a critical factor in developing advanced superalloys. In this study, the steady-state creep mechanisms of a superlattice γ′-strengthened Co–Al–W–Ta–Ti single crystal superalloy are investigated with the creep conditions of 750 °C/800 MPa, 900 °C/420 MPa and 1000 °C/137 MPa. Results suggest that the microstructures of steady-state creep samples are significantly different under the above three experimental conditions. A detailed TEM study shows that, Lomer-Cottrell locks and stacking faults interactions are responsible for the low creep rate at low temperature and high stress creep regime, dislocations cross-slip and the dislocation tangles contribute the main creep resistance at intermediate temperatures and moderate stress creep regime, interactions of stacking faults and dislocation networks are the strengthening mechanism for the high temperature and low stress creep regime. The steady-state creep rate of investigated alloy follows the classical power law equation, and the creep activation energy calculated by the power law equation is 378.46 kJ/mol. The results of this study provide insights into the effects of temperature and stress on steady-state creep mechanisms, and high activation energy makes the γ′-strengthened Co–Al–W–Ta–Ti single crystal superalloy an attractive candidate for further study as a high temperature structural material.

Creep lifetime of Al 6061-T6 pressurized rotating friction stir welded tube subjected to internal pressure and rotational velocity: Welding and creep lifetime optimization

Aluminum 6061-T6 alloys have numerous applications in the industry due to their unique properties such as higher corrosion resistance and strength-to-weight ratio. Friction stir welding is also a technique that can be used in aluminum joints because of its numerous advantages. In this paper, optimum friction stir welding parameters that lead to a higher failure load are obtained using tensile test data and analysis of variance technique. According to the analysis of variance results, it is found that the tool rotational speed has a major effect among the welding parameters on the failure load. Then, creep tests are carried out on non-welded and welded specimens to predict their creep behavior. The Norton power law equation is used as a creep constitutive equation to model the creep behavior of the specimens. The model is used to predict the creep lifetime of the welded and non-welded rotating pressurized vessels. The results show that an optimally welded vessel can only sustain up to 0.75 of pressure for the non-welded vessel with the same rotating speed. Since this comparison is not straightforward because of the number of affecting parameters. Hence, the reference stress method is used to produce a set of diagrams that provide design conditions and optimum creep lifetime for the friction stir welded Al 6061-T6 rotating pressurized vessels.

Novel green flexible rice straw nanofibers/zinc oxide nanoparticles films with electrical properties

In the current work, rice straw nanofibers (RSNF) with the width of elementary fibrils (~ 4–5 nm) were isolated from rice straw. The isolated nanofibers were used with zinc oxide nanoparticles (ZnONPs) to prepare flexible nanopaper films. Tensile strength and electrical properties of the prepared RSNF/ZnONPs nanopaper were investigated. The addition of ZnONPs to RSNF nanopaper did not deteriorate its mechanical properties and showed a slight improvement in tensile strength and Young's modulus of about 14% and 10%, respectively, upon the addition of 5% of ZnONPs. Microscopy investigation using scanning electron microscopy (SEM) showed the inclusion of the ZnONPs within the RSNF. Electrical conductivity and dielectric properties as a function of frequency at different temperatures were studied. The ac‐electrical conductivity increased with frequency and fitted with the power law equation. The dc‐ electrical conductivity of the samples verified the Arrhenius equation and the activation energies varied in the range from 0.9 to 0.42 eV. The dielectric constant decreased with increasing frequency and increased with increasing temperature, probably due to the free movement of dipole molecular chains within the RSNF nanopaper. The high values of the dielectric constant and conductivity of the prepared nanopaper films support their use in electronic components.

The study of composition-properties relationships for composite hydrogels based on poly(methacrylic acid) and high concentrations of MFI zeolite

High concentrations of hydrophobic MFI zeolite (26.0–51.3 wt%) were successfully embedded in poly(methacrylic acid) (PMAA) network in simple, one-pot synthesis of PMAA-MFI composite hydrogels. The effects of zeolite concentration regarding pure PMAA xerogel, for all of the investigated properties (primary structural parameters, morphological properties, water swelling ability, thermal degradation mechanism and stability, and mechanical properties) were determined and discussed. The PMAA-MFI composite hydrogels had remarkably higher density than PMAA xerogel (1660/1420 kg m−3), higher crosslinking density (29/1.0 × 10−4), improved mechanics in the dry and swollen state (up to 4.9 times higher storage modulus) and lower swelling degree (42/420 g g−1), average molar mass between the network crosslinks (3.6/110 × 104 g mol−1), and distance between the macromolecular chains (64/710 nm). Despite high zeolite concentration, the composites kept porous hydrogel structure with uniformly dispersed zeolite particles. FTIR spectra disclosed the existence of functional groups of PMAA network and MFI zeolite that interacted mutually and led to increased crosslinking density. The effect of zeolite concentration on the thermal degradation mechanism was established. The composition-properties correlations were determined and described by the adequate linear or power law equations revealing the additive nature of the composites and the fractal structure of the networks. These correlations enable the achievement of desired properties by simple adjustment of the initial composition, i.e. zeolite concentration. Therefore, this work presents the base for further development of tailor-made PMAA-MFI composite hydrogels aimed for specific application.

The effect of polypropylene fiber on the curing time of class G oil well cement and its mechanical, petrophysical, and elastic properties

The cement paste is subjected to various loads throughout a well’s life, which may compromise some of its essential characteristics and impair its performance. When the cement paste is first being formed and the cement’s characteristics have not yet fully matured, these loadings take on greater importance. In this study, the early properties of cement used in oil wells that contains polypropylene fiber are assessed. Five different curing times were used to prepare ten cement samples (6, 12, 24, 48, and 72 h). Five samples contained polypropylene fiber, while the remaining five samples were without polypropylene fibers. After the samples were prepared, the examination of several early cement properties took place. Nuclear magnetic resonance (NMR) was used to describe each sample in order to determine how the curing times affected the cement’s porosity. The findings demonstrated that both cement systems’ compressive and tensile strengths increased with curing time, and that adding polypropylene fiber enhanced the cement’s strength. The porosity and permeability of the cement specimens were significantly reduced with the incorporation of polypropylene fibers, as well as with time during the curing process for both cement samples. The reduction of Young’s modulus and the increase in Poisson’s ratio show that the addition of polypropylene fibers also makes the cement more elastic. To express variations in porosity as well as compressive and tensile strengths, logarithmic relationships were constructed. While the Poisson’s ratio, Young’s modulus, density variations, and permeability were precisely modeled by power-law equations.

The Effect of Visible Light on the Postharvest Life of Tomatoes (Solanum lycopersicum L.)

Tomatoes (Solanum lycopersicum L.) are widely cultivated and consumed, but ripening should be carried out in controlled storage conditions to extend their shelf life and avoid economic losses. The aim of this study was to investigate the effects of visible artificial light on the ripening and quality of fresh market tomatoes stored at a low temperature and high humidity. The postharvest performance with respect to the ripening of organically grown tomatoes in the Toscano cultivar, with a long storage life, was studied in the presence and the absence of visible LED light. The maturation kinetics of the tomatoes was modeled using the Power Law equation. Results showed that tomatoes stored in the presence of light exhibited an increased respiration rate and a faster preclimacteric phase. Lycopene content, total soluble solids, and maturity index increased in the presence of light. Hence, light increased the postharvest ripening of tomatoes, affecting their shelf life.

Dynamic aspects of suspended-sediment-concentration recession curves

Understanding and modeling suspended sediment concentration (C) dynamics in river channels are essential for many theoretical as well as practical purposes. The sediment rating curve approach that relates C with river discharge (Q) is among the most followed modeling approaches, mainly because of its simplicity and effectiveness. Although studies in the past have suggested the possibility of C-Q relationship for a basin not being unique, how the relationship evolves with time is not yet fully understood. In this study, we limit our analysis to recession periods when both C and Q continuously decrease with time and show that the coefficient of C-Q power-law equation changes across recession events. Furthermore, we propose to express -dC/dt as a function of C for recession periods: -dC/dt=fCe. The exponent e is assumed to be related to static catchment characteristics and the coefficient f to dynamic factors, namely catchment-water storage and sediment availability condition. Assuming the dynamic factors controlling f to evolve slowly with time, we propose that it is possible to predict f for a recession event considering only past C data. Once f is estimated, we can predict C for anytime during the recession event. The advantage of the proposed model is that it can predict future suspended sediment concentration at a river cross section during a recession event. We applied the proposed model to 35 US Geological Survey basins and found the 25th, 50th, and 75th values of NSE, respectively, to be 0.55, 0.64, and 0.78, and the corresponding values for the rating curve model - 0.05, 0.01 and 0.20. The results here essentially highlight the importance of recognizing the dynamic nature of suspended sediment concentration-discharge relationship.

Numerical implementation and application of an internal state variable model to analyze the time-dependent behavior of mining excavations in rock salt

The geomechanical behavior of rock salt has been investigated extensively over the years. Experimental studies have identified distinctive features associated with nonlinear inelastic response that shows strong time and loading history dependencies. In this paper, a relatively simple constitutive model is presented and implemented into the numerical code FLAC, and then applied to analyze the time-dependent behavior of excavations in salt mines. The unified creep-plasticity model includes an internal state variable (ISV) with an evolution law that induces progressive strain hardening (SH) until a steady state is reached. Numerical analyses are performed with the proposed ISV–SH model to evaluate material parameters, based on creep test results on natural rock salt, and to simulate the response of a circular opening and of rectangular mining excavations created in sequence. The results are analysed and compared with those obtained with the well-known Norton power law equation, commonly used in salt mine engineering. The time-dependent calculation results illustrate key aspects of rock salt behavior and highlight the major influence of transient inelastic behavior and stress redistribution on the response of underground openings. The proposed modelling approach constitutes an advantageous alternative to analyses based on stationary creep laws, often used in rock salt mining operations.

Testing and Evaluation of Surge Flow Gate For Discharge Measurement Under Different Operating Conditions

With surge irrigation, applying surges to an area is a cheap labor task. In industrialized nations, a variety of electronic valves for discontinuous water application (surges) are available; but their pricing prevents their usage in emerging regions. Additionally, these valves are too advanced for use by growers in underdeveloped nations. As a result, a basic Automatic Surge Gate was developed and tested in the labs to assess its effectiveness in terms of producing on-off surges. The reservoir was built to hold or collect low inflows for barrier functioning, and the gate was positioned on the suction side of the reservoir. Inflow rates, pinions, and poundage placements on the gate lever were the working parameters defined for the gate functioning. In the current investigation, three inflow rates 134, 169, and 187 l/s were employed. By leveraging the low inflows to the reservoir, the automatic surge gate demonstrated the possibility for automating the on-off action. The surge gate produced sizeable outflows even at the lowest inflow rate of 134 l/s into the reservoir. Power-law equations were discovered to be a good representation of the behavior in the statistical models that were also created using quasi statistical method.

Convergence from power-law to logarithm-law in nonlinear fractional Schrödinger equations

In this paper, we show a connection between fractional Schrödinger equations with power-law nonlinearity and fractional Schrödinger equations with logarithm-law nonlinearity. We prove that ground state solutions of power-law fractional equations, as p → 2+, converge to a ground state solution of logarithm-law fractional equations. In particular, we provide a new proof to the existence of a ground state of logarithm-law fractional Schrödinger equations.

A (Simplified) Biogeochemical Numerical Model to Predict Saturation, Porosity and Permeability During Microbially Induced Desaturation and Precipitation

AbstractMicrobially Induced Desaturation and Precipitation (MIDP) through denitrification is an emerging ground improvement method in which indigenous nitrate reducing bacteria are stimulated to introduce biogas, biominerals and biomass in the soil matrix. In this study, a numerical model is developed to evaluate the effect of biogas, biominerals and biomass on the hydraulic properties of soils treated with MIDP. The proposed model couples the biochemical conversions to changes of porosity and water saturation and predicts changes in permeability through two separate power law equations. Experimental studies from the literature are used to calibrate the model. Comparing the results with other studies on bioclogging or biomineralization in porous media reveals that the combined production of biogas, biomass, and biominerals results in efficient clogging, in the sense that only a small amount of products leads to a substantial permeability reduction. Based on this comparison, the authors postulate that biogenic gas bubbles preferably form within the larger pore bodies. The presence of biogenic gas in the larger pore bodies forces calcium carbonate minerals and biomass to be formed mainly at the pore throats. The interaction between the different phases results in more efficient clogging than observed in other studies which focus on a single product only.

Evaluation of Numerous Kinetic Energy-Rainfall Intensity Equations Using Disdrometer Data

Calculating rainfall erosivity, which is the capacity of rainfall to dislodge soil particles and cause erosion, requires the measurement of the rainfall kinetic energy (KE). Direct measurement of KE has its own challenges, owing to the high cost and complexity of the measuring instruments involved. Consequently, the KE is often approximated using empirical equations derived from rainfall intensity (Ir) inputs in the absence of such instruments. However, the KE–Ir equations strongly depend on local climate patterns and measurement methods. Therefore, this study aims to compare and evaluate the efficacy of 27 KE–Ir equations with observed data. Based on a re-analysis, we also propose an exponential KE–Ir equation for the entire Korean site, and the spatial distribution of its parameter in the equation is also discussed. In this investigation, we used an optical disdrometer (OTT Parsivel2) to gather data in Sangju City (Korea) between June 2020 and December 2021. The outputs of this study are shown as follows: (1) The statistically most accurate estimates of KE expenditure and KE content in Sangju City are obtained using power-law equations given by Sanchez-Moreno et al. and exponential equations published by Lee and Won, respectively. (2) The suggested KE–Ir equation applied to the entire Korean site exhibits a comparable general correlation with the observed data. The parameter maps indicate a high variance in geography.