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Journal articles on the topic 'Empirical equation'

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Published: 4 June 2021

Last updated: 9 June 2025

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Zimdahl, Robert L., Brian K. Cranmer, and Walter W. Stroup. "Use of Empirical Equations to Describe Dissipation of Metribuzin and Pendimethalin." Weed Science 42, no. 2 (June 1994): 241–48. http://dx.doi.org/10.1017/s0043174500080346.

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Four equations were evaluated as predictors of the rate of herbicide dissipation in soil. A biexponential equation was superior to the first-order equation for metribuzin and pendimethalin dissipation under five moisture levels and three temperatures in laboratory and field studies. The Hoerl function, adapted in the course of this work, is also a good descriptor. The first-order equation predicts slower initial and more rapid later dissipation than actually occurs and these deficiencies are not shared by the biexponential or Hoerl equations. The first-order equation ignores small residues remaining late in the dissipation process. These residues are important from an environmental point of view and the Hoerl and biexponential equations are more capable of dealing with them.

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Samdarshi, S. K., and S. C. Mullick. "Analytical Equation for the Top Heat Loss Factor of a Flat-Plate Collector With Double Glazing." Journal of Solar Energy Engineering 113, no. 2 (May 1, 1991): 117–22. http://dx.doi.org/10.1115/1.2929955.

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An analytical equation for the top heat loss factor of a flat-plate collector with double glazing has been developed. The maximum computational errors resulting from the use of this equation are plus or minus three percent compared to numerical solution of the heat balance equations. The equation is considerably more accurate than the currently used semi-empirical equations over the entire range of variables covered. It is found that the computational errors resulting from simplification of the proposed equation by approximation of the individual heat-transfer coefficients are much lower than the errors resulting from the use of semi-empirical equations.

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Sabeti, Ramtin, and Mohammad Heidarzadeh. "A new empirical equation for predicting the maximum initial amplitude of submarine landslide-generated waves." Landslides 19, no. 2 (November 2, 2021): 491–503. http://dx.doi.org/10.1007/s10346-021-01747-w.

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AbstractThe accurate prediction of landslide tsunami amplitudes has been a challenging task given large uncertainties associated with landslide parameters and often the lack of enough information of geological and rheological characteristics. In this context, physical modelling and empirical equations have been instrumental in developing landslide tsunami science and engineering. This study is focused on developing a new empirical equation for estimating the maximum initial landslide tsunami amplitude for solid-block submarine mass movements. We are motivated by the fact that the predictions made by existing equations were divided by a few orders of magnitude (10−1–104 m). Here, we restrict ourselves to three main landslide parameters while deriving the new predictive equation: initial submergence depth, landslide volume and slope angle. Both laboratory and field data are used to derive the new empirical equation. As existing laboratory data was not comprehensive, we conduct laboratory experiments to produce new data. By applying the genetic algorithm approach and considering non-dimensional parameters, we develop and examine 14 empirical equations for the non-dimensional form of the maximum initial tsunami amplitude. The normalized root mean square error (NRMSE) index between observations and calculations is used to choose the best equation. Our proposed empirical equation successfully reproduces both laboratory and field data. This equation can be used to provide a preliminary and rapid estimate of the potential hazards associated with submarine landslides using limited landslide parameters.

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UENO, Masakatsu, and Kametaro ITOH. "Empirical Verification of GROSSMANN'S Equation." Tetsu-to-Hagane 74, no. 5 (1988): 918–24. http://dx.doi.org/10.2355/tetsutohagane1955.74.5_918.

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Wiśniewski, Jerzy Witold. "Empirical Econometric Model of an Enterprise." Folia Oeconomica Stetinensia 16, no. 1 (December 1, 2016): 232–47. http://dx.doi.org/10.1515/foli-2016-0015.

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Abstract This work will present an empirical econometric model describing an enterprise within the category of medium-sized companies (according to European Union classification). The company, code-named ENERGY, carries out a manufacturing, commercial, and service business activity. The statistical data used was in the form of quarterly time series, containing 24 statistical observations from the years 2008–2013. A hypothetical model of the enterprise is a system of interdependent equations. The econometric model is composed of seven stochastic equations. The empirical model is missing the equation describing investments in the enterprise. It results from the fact, that during the years 2008–2013 the company suffered meagre investments. Investment output equation, therefore, does not provide any relevant systemic information for the management, since most statistical information in the time series assumes zero values. An empirical model of the company ENERGY is a system of interdependent equations, with statistically significant feedback between labour efficiency (EFEMP) and the average pay per 1 employee (APAY). Additionally, there is recurrence of the relationships between the fixed assets (FIXAS), employment volume (EMP), and the size of the net sales income (SNET). The empirical equations of the model are characterized by a description accuracy of individual endogenous variables. The model also has good decision-making and forecasting qualities.

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Leary, Rolfe A. "Near-normal, empirical, and identity yield tables for estimating stand growth." Canadian Journal of Forest Research 21, no. 3 (March 1, 1991): 353–62. http://dx.doi.org/10.1139/x91-043.

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Historically, forest growth was estimated using a normal or near-normal yield table as a density standard, and a relative density change equation to estimate approach to the standard. Although normal yield tables have come under intense criticism, critics have generally ignored the relative density change equation. If a yield table captures the "true" relations between volume, age, and site for a species, the relative density change equation can be a simple function of initial relative density and age. If a yield table does not capture the true relations between volume, site, and age, the inadequacy can be overcome by developing more complex relative density change equations, i.e., by transferring representation burden to the change equation. Introduced in the present paper is the concept of an identity yield table (all entries are one), wherein the entire burden of representing the relations between volume, site, and age is transferred from a density standard to a relative density change equation. Modern whole stand (net) growth models are equivalent to historical relative density change equations based on identity yield tables. The conjecture of a continuum of methods to estimate growth from near-normal to empirical to identity yield tables, each with an appropriate relative density change equation, and each equally accurate, is tested on Wisconsin jack pine (Pinusbanksiana Lamb.). The empirical yield table and its relative density change equation were more biased than near-normal and identity-based projection systems.

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Lozano Povis, Arlitt Amy, and Elías Adrián Sanabria Perez. "Experimental Evaluation of an Empirical Equation in a Gaseous Flow." Chemistry & Chemical Technology 18, no. 1 (March 25, 2024): 57–65. http://dx.doi.org/10.23939/chcht18.01.057.

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In this paper, the estimation error of Dr. Pole's empirical equation was evaluated using copper pipes of different diameters (0.00953, 0.0127, 0.01588 m), under different flow pressure conditions (0, 300, 500, 1000, 1500, 2000, 2500, 3000 L/h). To carry out the experiments, the following instruments were used: an air compressor, 2 flow valves, a needle valve, a gas rotameter, copper piping, pressure gauges and transmitters, a Norus data logger with 4 to 20 mA output signals, thermocouples, and thermoresistors. They allow us to establish that the air pressure drops when the flowing through the pipes is higher (380 Pa) for small diameter pipes (0.00953 m), compared to larger diameters (0.01270 m and 0.01588 m) with a maximum of 54 and 28 Pa, respectively; and in relation to the flow rates, the pressure drop increases with a quadratic trend with respect to the flow rate. Finally, the residual errors that the empirical equation has in the pressure drop calculations, in general terms, are not of great magnitude.

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Saglam, Ugur, Kemal Ulutas, Yagmur Parim, Sahin Yakut, and Deniz Deger. "A theoretical approach to conductivity." International Journal of Geometric Methods in Modern Physics 17, no. 01 (December 18, 2019): 2050004. http://dx.doi.org/10.1142/s0219887820500048.

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In amorphous semiconductors and insulators, the using conductivity formulas are semi-empirical and have no satisfying physical explanations. A conductivity equation has been derived by Debye for the response of ideal materials which is rarely observed in practice, but a general conductivity equation which includes the previous empirical equations via a correct choice of arbitrary parameters and moreover totally theoretical derivation had to be generated. Hence, to determine the motion of electrons in the amorphous environment, we defined the equation of motion including viscous forces as a function of coordinates, their derivatives and time variables. We developed a fractional form of this equation over these three variables and finally obtained the most generalized equation of motion, which counts the overall interactions by a fractional form as a variation of two variable. The improved formula, called the stretched Havriliak–Negami equation, has the same form and behavior as the semi-empirical equation and reducible to the Cole–Cole and Cole–Davidson-type of conductivity.

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Awal, Ripendra, Atikur Rahman, Ali Fares, and Hamideh Habibi. "Calibration and Evaluation of Empirical Methods to Estimate Reference Crop Evapotranspiration in West Texas." Water 14, no. 19 (September 27, 2022): 3032. http://dx.doi.org/10.3390/w14193032.

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Evapotranspiration is an essential component of the hydrologic cycle, and its accurate quantification is crucial for managing crop water requirements and the operation of irrigation systems. Evapotranspiration data is key to hydrological and water management research investigations, including studying the impact of various climatic factors on crop water requirements. It has been estimated as the product of the reference crop evapotranspiration and crop coefficient. Daily reference crop evapotranspiration (ETo) can be determined by several methods and equations. The Food and Agriculture Organization Penman-Monteith equation requires complete weather data, whereas empirical equations such as Hargreaves and Samani, Valiantzas, Priestley-Taylor, Makkink, and Stephens-Stewart require limited weather data. This work evaluated different empirical equations for West Texas using the standard FAO Penman-Monteith method and calibrated their parameters to improve ETo estimation. Detailed meteorological data from West Texas Mesonet and high resolution (800 m) Parameter-elevation Regressions on Independent Slopes Model (PRISM) datasets from 2007 to 2016 were used. Daily ETo calculated using the standard FAO Penman-Monteith equation was compared to ETo estimated based on different empirical methods. The results show that all original empirical equations underestimated ETo. Calibration improved the performance of tested equations; however, there seems to be underestimation of ETo in the 8–16 mm range. Overall, the monthly Hargreaves and Samani equation with either original or calibrated values of its parameters outperformed all tested models. This equation seems to be a reasonable estimator, especially under limited weather data conditions.

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Li, Jianqiao, Weidong Song, Jianguo Ning, and Huiping Tang. "Characteristics of impact-generated plasma with different electron temperature and gas temperature." Modern Physics Letters B 28, no. 18 (July 11, 2014): 1450152. http://dx.doi.org/10.1142/s0217984914501528.

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The characteristics of the plasma with difference between the electron temperature and gas temperature were investigated and the relationship between the plasma ionization degree and the internal energy of a system was obtained. A group of equations included the chemical reaction equilibrium equation, the chemical reaction rate equation and the energy conservation equation were adopted to calculate the electron density, the electron temperature and the atom temperature with a given internal energy. These equations combined with Navier–Stokes (N–S) equations is solved by a smooth particle hydrodynamic (SPH) code. The charges generated in hypervelocity impacts with five different velocities are calculated and verified with the empirical formulas. The influence of a critical velocity for plasma generation is considered in the empirical formula and the parameters are fitted by the numerical results. By comparing with the results in reference, the fitted new empirical formula is verified to be reasonable and useful for a wide range of impact velocity.

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Bujak, Damjan, Suzana Ilic, Hanna Miličević, and Dalibor Carević. "Wave Runup Prediction and Alongshore Variability on a Pocket Gravel Beach under Fetch-Limited Wave Conditions." Journal of Marine Science and Engineering 11, no. 3 (March 14, 2023): 614. http://dx.doi.org/10.3390/jmse11030614.

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Most empirical equations used for wave runup predictions have been developed from measurements at straight sandy beaches in unlimited fetch environments. While there are empirical equations to predict wave runup on gravel beaches, they have not been tested for prediction of wave runup on pocket gravel beaches, in limited-fetch environment, which can be found around Mediterranean. This paper addresses this lack of measurements on this type of beaches and examines the alongshore variability of wave runup. Wave runup measurements were made using video observations along 3 cross-sectional profiles on the pocket beach of Ploče, Croatia. The measurements have shown that the wave runup can vary for about 71% even around the centerline of the pocket beach. This variability is due to beach orientation and alignment of beach profiles to the prevailing wave direction, as well as difference in beach slope. Comparison of wave runup predictions from five well-known empirical equations and field measurements showed significant underprediction (up to NBIAS = −0.33) for energetic wave events, and overall high scatter (up to NRMSE = 0.38). The best performing wave runup equation was used for further refinement outside the original parameter space by including the Goda wave peakedness parameter (Qp). The newly developed empirical equation for wave runup reduced the NBIAS to 0 and the NRMSE by 31% compared to the original equation (developed equation metrics: R = 0.91, NBIAS = 0, NRMSE = 0.2, HH = 0.2 on the study site). This empirical equation can potentially be used for design of coastal structures and artificial beaches in similar environments, but further measurements are needed to test its applicability to a range of forcing and environmental conditions.

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Samdarshi, S. K., and S. C. Mullick. "Generalized Analytical Equation for the Top Heat Loss Factor of a Flat-Plate Solar Collector With N Glass Covers." Journal of Solar Energy Engineering 116, no. 1 (February 1, 1994): 43–46. http://dx.doi.org/10.1115/1.2930064.

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A generalized analytical equation for the top heat loss factor of a flat-plate collector with one or more glass covers has been developed. The maximum computational errors resulting from the use of the analytical equation with several simplifications are ± 5 percent compared to numerical solution of the set of heat balance equations. The analytical equation is considerably more accurate than the available semi-empirical equations over the entire range of variables covered. An additional advantage of the proposed technique over the semi-empirical equations is that results can be obtained for different values of sky temperature, using any given correlation for convective heat transfer in the air gap spacings, and for any given values of fluid (air in the present case) properties.

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Al-Saadi, Balsam J. M., and Hussein A. M. Al-Zubaidi. "Reaeration Coefficient Empirical Equation Selection for Water Quality Modeling in Surface Waterbodies: An Integrated Numerical-Modeling-Based Technique with Field Case Study." Limnological Review 25, no. 2 (April 25, 2025): 15. https://doi.org/10.3390/limnolrev25020015.

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Empirical equations were developed by many investigators to determine the reaeration coefficients (Ka) required for predicting dissolved oxygen concentrations (DO) in surface waters, especially rivers, lakes, and reservoirs. However, these equations yield a wide range of Ka values. In this paper, an integrated numerical-modeling-based technique was developed to check the validity of the equations before using them in water quality modeling for rivers, lakes, and reservoirs. Depending on direct field measurements at the Hilla River headwater (Saddat Al-Hindiyah Reservoir, Iraq), the temporal oxygen mass transport at the water surface was estimated numerically by solving the one-dimensional advection diffusion equation and then using each Ka empirical equation separately in the numerical model obtained the best specific-waterbody equation. The DO modeling results showed that using a reservoir reaeration coefficient of 0.1 day−1 at 20 °C predicts the best DO simulation with low MAEs of 0.4987 and 0.7880 mg/L during the study years 2021 and 2022, respectively, compared to the field data. However, using the Ka empirical equations simulates the DO with wide-ranging statistical errors even though the temporal Ka values have a similar trend during the year. It was noticed that the empirical equations produced maximum Ka values of (0.0080–0.0967 day−1) and minimum Ka values of (0.00052–0.0267 day−1) in 2021 and maximum Ka values of (0.0079 to 0.0951 day−1) and minimum Ka values of (0.00012 and 0.0231 day−1) in 2022. The present equation selection technique revealed that Broecker et al.’s equation followed by Smith’s equation, developed in 1978, are the best selection for water quality modeling at the Hilla River headwater (MAEs: 0.1347 and 0.1686 mg/L in 2021, respectively; and MAEs: 0.1400 and 0.1744 mg/L in 2022, respectively). Hence, it is necessary to find good agreement for the equation-based prediction of DO, DO source–sink, and Ka values compared to the validated model before making selection.

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Lorenzoni, M., D. Giannetto, G. Maio, E. Pizzul, L. Pompei, P. Turin, S. Vincenzi, and A. Crivelli. "Empirical standard mass equation forSalmo marmoratus." Journal of Fish Biology 81, no. 6 (October 19, 2012): 2086–91. http://dx.doi.org/10.1111/j.1095-8649.2012.03459.x.

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Wu, N. S., and W. Wei. "Empirical equation for relative non-overlap." Chromatographia 35, no. 7-8 (April 1993): 471. http://dx.doi.org/10.1007/bf02278606.

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Kolin, Branko, Gradimir Danon, and Tatjana Stevanovic Janezic. "Empirical Equation for Limit of Hygroscopicity." Drying Technology 13, no. 8-9 (January 1995): 2133–39. http://dx.doi.org/10.1080/07373939508917069.

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Wu, N. S., and W. Wei. "Empirical equation for relative non-overlap." Chromatographia 34, no. 9-10 (November 1992): 450–52. http://dx.doi.org/10.1007/bf02290234.

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Cheng, Wen Ting, Shuo Feng, Xiao Qin Cui, and Fang Qin Cheng. "Solubility of Benzoic Acid in Ethanol, Benzene, Acetic Acid and Ethyl Acetate from 291.69 to 356.27 K." Advanced Materials Research 518-523 (May 2012): 3975–79. http://dx.doi.org/10.4028/www.scientific.net/amr.518-523.3975.

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Using a synthetic method designed and installed with laser monitor on line, the solubility values of benzoic acid in ethanol, benzene, acetic acid and ethyl acetate were determined over the temperature range of 291.69-356.27 K. The solubility of benzoic acid in all cases investigated was found to increase with temperature. The two-parameters empirical and λh equations were used successfully to correlate experimental data of benzoic acid solubilities in organic solvents. The mean absolute error σ of 65 data points correlating by two-parameters empirical equation and λh equation was less than 1%. Finally, molar dissolution enthalpy ΔsolH of benzoic acid in organic solvents was determined with the newly obtained empirical equation parameters.

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Watanabe, Yoshimasa, Sumio Masuda, Kiyoshi Nishidome, and Chalermraj Wantawin. "Mathematical Model of Simultaneous Organic Oxidation, Nitrification, and Denitrification in Rotating Biological Contactors." Water Science and Technology 17, no. 2-3 (February 1, 1985): 385–97. http://dx.doi.org/10.2166/wst.1985.0145.

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Simultaneous organic oxidation and nitrification rates in the RBC are given using a mathematical equation. The equation was derived from a hypothesis stating that intrinsic oxygen uptake rate of the biofilm has a constant value at a fixed temperature , independent of the composition of the aerobic bacteria. Based on this hypothesis, empirical equations are proposed to describe the profile of intrinsic organic oxidation and nitrification rates. Computer simulation of the simultaneous organic oxidation and nitrification was carried out to confirm the empirical equations. The mathematical model of the simultaneous nitrification and denitrification is discussed and was tested by computer simulation.

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Galushkin, Nikolai Efimovich, Natal'ya Nikolaevna Yazvinskaya, and Dmitrii Nikolaevich Galushkin. "Capacitance from Discharge Current Depen-dence Modeling in Nickel-Cadmium Batteries." Electrochemical Energetics 12, no. 3 (2012): 147–54. http://dx.doi.org/10.18500/1608-4039-2012-12-3-147-154.

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It was shown by experiments for SAFT batteries of stationary application and medium rate of discharge, that empirical equations Korovin–Skundin’s, Peukert’s generalized equation, probability integral, porous electrode equation allow to describe the dependence capacitance C(i), given by nickel-cadmium batteries, from discharge current with the accuracy sufficient for practical needs (5–6 %) on all the range of discharge currents. Meanwhile the empirical equations of Peukert and Aguf can be used to calculate the capacitance given away only within the discharge currents, starting from the point of inflexion of experimental curve C(i) infinitely. Libenov equation, which can be used only within the range of discharge currents from the point of inflexion of experimental curve C(i) and up to the discharge currents, wherein C(i)≈ 0.05Cn, is of even bigger specialty.

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Zhao, Yuyang, and Hongbo Zhao. "Symbolic Regression for the Determination of Joint Roughness Coefficient." Mathematical and Computational Applications 30, no. 1 (February 9, 2025): 17. https://doi.org/10.3390/mca30010017.

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In this study, a novel symbolic regression-based empirical equation has been developed to compute the joint roughness coefficient (JRC) value based on the statistical parameters of rock joints. The symbolic regression was adopted to map the nonlinear function, which represents the relation between the JRC and statistical parameters of the rock joint, based on the collected rock joint dataset. It is not necessary to presume the mathematical function form of the empirical equation, which is used to fit the rock joint data while using symbolic regression. The collected rock joint samples from the literature were used to investigate and illustrate the developed symbolic regression-based empirical equation. The performance of the developed empirical equation was compared to the traditional empirical equation. The results show that the generalization performance of the developed empirical equation is better than the traditional empirical equation. They proved that the symbolic regression-based empirical equation characterized the roughness property of rock joints well and that symbolic regression could be used to capture the complex and nonlinear relationship between JRC and the statistical parameters of rock joints. The developed symbolic regression-based empirical equation provides a scientific and excellent tool to estimate the JRC value of rock joints.

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Hitoshi Shoji, Takao Kameda, Kunio Kawada, Okitsugu Watanabe, and Henrik B. Clausen. "An empirical relation between overburden pressure and firn density." Annals of Glaciology 20 (1994): 87–94. http://dx.doi.org/10.3189/1994aog20-1-87-94.

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Two empirical equations for firn densification have been obtained,considering firn porosity as a function of overburden pressure. In the first equation, thereduction ratio of porosity in firn is assumed to be proportional to the increasing ratioof overburden pressure and the mth power of the porosity. The porosity exponent m should be close to -2, so as to have a best-fit with 14 depth-density profiles fromGreenland and Antarctica. In the second equation, the reduction ratio of porosity wasassumed to increase proportionally to the increment of overburden pressure and thenth power of the porosity. The most satisfactory values of the exponent range from -1 to 1. It has been suggested that firn density, determined primarily by overburdenpressure and firn temperature, contribute to a lesser degree.

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Two empirical equations for firn densification have been obtained,considering firn porosity as a function of overburden pressure. In the first equation, thereduction ratio of porosity in firn is assumed to be proportional to the increasing ratioof overburden pressure and the mth power of the porosity. The porosity exponentmshould be close to -2, so as to have a best-fit with 14 depth-density profiles fromGreenland and Antarctica. In the second equation, the reduction ratio of porosity wasassumed to increase proportionally to the increment of overburden pressure and thenth power of the porosity. The most satisfactory values of the exponent range from -1 to 1. It has been suggested that firn density, determined primarily by overburdenpressure and firn temperature, contribute to a lesser degree.

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DEHWAH, AHMAD H., IDRIS A. AJIA, and JOHN S. MARSLAND. "EMPIRICAL FORMULAE FOR EXCESS NOISE FACTOR WITH DEAD SPACE FOR SINGLE CARRIER MULTIPLICATION." Fluctuation and Noise Letters 10, no. 03 (September 2011): 315–21. http://dx.doi.org/10.1142/s0219477511000600.

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In this letter, two empirical equations are presented for the calculation of the excess noise factor of an avalanche photodiode for single carrier multiplication including the dead space effect. The first is an equation for calculating the excess noise factor when the multiplication approaches infinity as a function of parameters that describe the degree of the dead space effect. The second equation can be used to find the minimum value of the excess noise factor for any multiplication when the dead space effect is completely dominant, the so called "deterministic" limit. This agrees with the theoretically known equation for multiplications less than or equal to two.

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Dam, Thang Trong, and Bang Xuan Vu. "Research to establish the equation of the principle of claystone breaking under the seabed." Journal of Mining and Earth Sciences 64, no. 5 (October 31, 2023): 109–19. http://dx.doi.org/10.46326/jmes.2023.64(5).10.

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The rule of breaking rock has practical significance in the process of controlling the blasting for the optimization of mine production or construction of blasting works. The equations describing the particle size distribution after the explosion are established in an experimental form corresponding to the specific conditions of the blasting work. It only has the meaning of a mathematical regression function and does not reflect the characteristics of the rock type, explosive conditions and explosive results. This research has exploited the theory of rock breaking from the point of view of probabilistic energy to establish the equation of rock breaking of claystone in water. This is a form of semi-empirical equation, which contains 4 empirical coefficients reflecting rock characteristics, explosive conditions, explosive energy absorption characteristics of rocks and actual destruction volume, determined for each type of rock. Through conducting experiments, processing experimental data and solving a system of four equations, four coefficients of the equations that characterize the breaking properties of claystone under the seabed have been found. The error of the average particle size between the actual and calculated values from the equations is less than 7% to 10%. Tt is possible to confirm the accuracy of the semi-empirical equation describing the particle size distribution rule when the claystone is exploded underwater.

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Kemp, Jonathan, Benoit Vandeputte, Thomas Eccleshall, Richard Simons, and Peter Troch. "A MODIFIED HYPERBOLIC TANGENT EQUATION TO DETERMINE EQUILIBRIUM SHAPE OF HEADLAND BAY BEACHES." Coastal Engineering Proceedings, no. 36 (December 30, 2018): 106. http://dx.doi.org/10.9753/icce.v36.papers.106.

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When designing any artificial beach, itâ€™s desirable to avoid (or minimise) future maintenance commitments by arranging the initial beach planshape so that it remains in equilibrium given the incident wave climate. Headlands bays, or embayments, where a sandy beach is held between two erosion resistant headlands, tend to evolve to a stable beach planshape with little movement of the beach contours over time. Several empirical bay shape equations have been derived to fit curves to the shoreline of headland bay beaches. One of the most widely adopted empirical equations is the parabolic bay shape equation, as it is the only equation that directly links the shoreline positions to the predominant wave direction and the point of diffraction. However, the main limitation with the application of the parabolic bay shape equation is locating the downcoast control point. As a result of research presented in this paper a new equation, based on the hyperbolic tangent shape equation was developed, which eliminates the requirement of placing the down coast control point and relies on defining a minimum beach width instead. This modified equation was incorporated into a new ArcGIS tool.

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Fernandes, Lázaro Costa, Célia Maria Paiva, and Otto Corrêa Rotunno Filho. "Evaluation of six empirical evapotranspiration equations - case study: Campos dos Goytacazes/RJ." Revista Brasileira de Meteorologia 27, no. 3 (September 2012): 272–80. http://dx.doi.org/10.1590/s0102-77862012000300002.

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The evapotranspiration is a component of the water balance constituting a major challenge in its quantification. The complex physical processes involved in its effective determination on a large scale have spurred scientists to often make use of empirical equations, which have inherent limitations with regard to their applicability as descriptors of the evapotranspiration behavior in different regions across the world. This study was performed for the Campos dos Goytacazes region, in Rio de Janeiro state. It is proposed to investigate and to evaluate the performance of six empirical equations in contrast to FAO56-Penman-Monteith equation. The results indicated that the differences observed between the values obtained using the empirical models applied in this study and the values calculated by the FAO56-Penman-Monteith equation were greater than 10%, which means an error of about 0.5 mm.day-1.

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Folmar, Norman D., and Arthur C. Miller. "Development of an Empirical Lag Time Equation." Journal of Irrigation and Drainage Engineering 134, no. 4 (August 2008): 501–6. http://dx.doi.org/10.1061/(asce)0733-9437(2008)134:4(501).

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Saeki, Susumu. "Empirical equation of state for supercritical fluids." Journal of Supercritical Fluids 8, no. 1 (March 1995): 30–45. http://dx.doi.org/10.1016/0896-8446(95)90048-9.

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Ferrer, Gerard, Stefan Gschwander, Aran Solé, Camila Barreneche, A. Inés Fernández, Peter Schossig, and Luisa F. Cabeza. "Empirical equation to estimate viscosity of paraffin." Journal of Energy Storage 11 (June 2017): 154–61. http://dx.doi.org/10.1016/j.est.2017.03.002.

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Toscani, Siro, and Henri Szwarc. "An empirical equation of state for liquids." Journal of Chemical & Engineering Data 38, no. 4 (October 1993): 591–97. http://dx.doi.org/10.1021/je00012a031.

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Girko, V. L. "Canonical spectral equation for empirical covariance matrices." Ukrainian Mathematical Journal 47, no. 9 (September 1995): 1341–55. http://dx.doi.org/10.1007/bf01057509.

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Zhenyu, Tan, and He Yancai. "An empirical energy loss equation of electrons." Scanning 24, no. 1 (December 6, 2006): 46–51. http://dx.doi.org/10.1002/sca.4950240107.

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Imai, Keisuke, and Eiji Ōsawa. "An empirical extension of the karplus equation." Magnetic Resonance in Chemistry 28, no. 8 (August 1990): 668–74. http://dx.doi.org/10.1002/mrc.1260280803.

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Yuen, Pong Kau, Cheng Man Diana Lau, and Kuok In Gabriel Yuen. "Empirical Buswell’s Equation for Identifying Anaerobic Digestate." International Journal of Chemistry 17, no. 2 (March 17, 2025): 1. https://doi.org/10.5539/ijc.v17n2p1.

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Anaerobic digestion is a promising circular economic technology. Using organic matters as feedstocks, Buswell&rsquo;s equation can represent anaerobic digestion in accordance with the elemental composition of any organic matter. An organic feedstock is biodegradable to biomethane, biogenetic carbon dioxide, and digestate, but the management of anaerobic digestate encounters some environmental and technological challenges. Currently there is a research gap between Buswell&rsquo;s concept and the general organic elemental composition of unknown anaerobic digestate. To bridge the gap, this research developed an empirical Buswell&rsquo;s equation for identifying anaerobic digestate through the integration of theoretical Buswell&rsquo;s equation and experimental biomethane potential. This model can identify the organic elemental composition and characteristics of any anaerobic digestate, as well as reveal the correlation between an organic matter and its anaerobic digestate. It also discovers a higher heat value of anaerobic digestate which is greater than that of its corresponding organic matter. In addition, the empirical Buswell&rsquo;s equation can be used for assessing the validity of the empirical formula of organic feedstock.

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Bagi, Richárd, Filip Alamaa, and Felix Ryde. "Empirical relations in thermal Comptonization." Monthly Notices of the Royal Astronomical Society 536, no. 1 (November 29, 2024): 603–8. https://doi.org/10.1093/mnras/stae2664.

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ABSTRACT In many astrophysical systems, photons interact with matter through thermal Comptonization. In these cases, under certain simplifying assumptions, the evolution of the photon spectrum is described by an energy diffusion equation such as the Kompaneets equation, having dependencies on the seed photon temperature, $\theta _i$, the electron temperature, $\theta _e$, and the Compton y-parameter. The resulting steady-state spectrum is characterized by the average photon energy and the Compton temperature, which both lack analytical dependencies on the initial parameters. Here, we present empirical relations of these two quantities as functions of $\theta _i$, $\theta _e$, and y, obtained by evaluating the steady-state solution of the Kompaneets equation accounting for energy diffusion and electron recoil. The relations have average fractional errors ${\sim} 1~{{\ \rm per\ cent}}$ across a wide range of the initial parameters, which make them useful in numerical applications.

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Oliveira, Vanessa Vaz de, Marcos Vinícius Mateus, Julio Cesar De Souza Inácio Gonçalves, Alex Garcez Utsumi, and Marcius Fantozzi Giorgetti. "Prediction of the longitudinal dispersion coefficient for small watercourses." Acta Scientiarum. Technology 39, no. 3 (July 6, 2017): 291. http://dx.doi.org/10.4025/actascitechnol.v39i3.29397.

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Longitudinal dispersion coefficient (DL) is considered an essential physical parameter to water quality modeling in rivers. Therefore, the estimation of this parameter with high accuracy guarantees the reliability of the results of a water quality model. In this study, the observed values of longitudinal dispersion coefficient are determined for natural streams (with discharge less than 2.84 m3s-1), based on sets of measured data from stimulus-response tests using sodium chloride as a tracer. Additionally, a semi-empirical equation for prediction of DL is derived using dimensional analysis and multiple linear regression technique. The performance of the produced equation was compared to five empirical prediction equations of DL selected from literature. It presented correlation coefficient r2 = 0.87, suggesting that this equation is suitable for the estimation of DL in streams. It also presented better results for predicting the DL than the five equations from literature, showing an accuracy of 71%.

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Zerihun, Yebegaeshet T. "A Study of the Sedimentation and Storage Capacity Depletion of a Reservoir." Slovak Journal of Civil Engineering 31, no. 2 (June 1, 2023): 37–47. http://dx.doi.org/10.2478/sjce-2023-0011.

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Abstract Theoretical and empirical equations have previously been proposed to compute the sediment trap efficiency of a reservoir. The empirical equations were developed by analyzing a limited amount of field data for reservoirs in selected areas; thus, their suitability is restricted to reservoirs in regions with similar hydrological characteristics. In this study, a sediment trap-efficiency equation, which accounts for the dynamic behavior of sedimentation and the hydraulic conditions of a reservoir, was deduced from the principle of mass conservation. Using this equation, the approximate analytical expression for the storage capacity of the reservoir was obtained. Both equations were applied to investigate real-life problems involving the impacts of reservoir sedimentation, and their predicted solutions were compared with the field data and the results of the theoretical and empirical equations. The results of this investigation reveal that the proposed equations are capable of simulating the sediment trapping and storage capacity depletion characteristics of large reservoirs, which are a measure of the adverse effect of sedimentation on the performance of hydraulic structures. These equations can potentially be incorporated into a large-scale watershed model for estimating a sediment budget.

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Cano, Nicolas D., Antonio P. de Camargo, Gustavo L. Muniz, Jonesmar de Oliveira, José G. Dalfré Filho, and José A. Frizzone. "Performance of models to determine flow rate using orifice plates." Revista Brasileira de Engenharia Agrícola e Ambiental 25, no. 1 (January 2021): 10–16. http://dx.doi.org/10.1590/1807-1929/agriambi.v25n1p10-16.

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ABSTRACT This study aimed to evaluate three methodologies for orifice-plate water-flow estimation by quantifying errors in the flow determinations to propose an appropriate measurement range for each evaluated condition. Two orifice-plate models (nominal diameters of 100 and 150 mm) with 50% restriction in the flow section were evaluated. In the theoretical equations, the discharge coefficient was obtained using the Reader-Harris/Gallagher equation (Method 1) and approximated from experimental data using the angular coefficient of a zero-intercept straight line (Method 2). The recommended measurement ranges for errors that were lower than 5% for the 100 and 150 mm plates were 30 to 65 m3 h-1 and 70 to 130 m3 h-1 using the theoretical equation and 20 to 65 m3 h-1 and 40 to 130 m3 h-1 using the empirical equation, respectively. The Reader-Harris/Gallagher equation (Method 1) adequately estimated the discharge coefficient of the orifice plates; however, the use of empirical equations (Method 3) demonstrated smaller measurement errors and greater rangeability of the evaluated flow meters.

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Tang, Han, and Wenfei Li. "Empirical study for uncertain finance." Journal of Intelligent & Fuzzy Systems 40, no. 5 (April 22, 2021): 9485–92. http://dx.doi.org/10.3233/jifs-201955.

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Interest rate, stock and option are all important parts of finance. This paper applies uncertain differential equation to the study of the evolution of interest rate and stock price separately. Based on actual observations, we estimate the parameters in uncertain differential equation with the method of moments. Using the introduced interest rate and stock models, we price European options and compare the results with actual observations. Finally, a paradox of the stochastic financial model is stated.

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Gordeev, D. G., L. F. Gudarenko, M. V. Zhernokletov, V. G. Kudel’kin, and M. A. Mochalov. "Semi-empirical equation of state of metals. Equation of state of aluminum." Combustion, Explosion, and Shock Waves 44, no. 2 (March 2008): 177–89. http://dx.doi.org/10.1007/s10573-008-0024-y.

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Jayadev, M. "Predictive Power of Financial Risk Factors: An Empirical Analysis of Default Companies." Vikalpa: The Journal for Decision Makers 31, no. 3 (July 2006): 45–56. http://dx.doi.org/10.1177/0256090920060304.

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This paper provides empirical evidence on the significance of financial risk factors in predicting default companies. Traditionally, credit decision process is built on accounting ratios derived from financial statements of the borrower. Combining various ratios through application of multivariate statistical techniques and testing their predictive power has been popular in credit risk quantification. Altman's Z-score model is the most acceptable model in this category. In this paper, three forms of Z-score models are applied: The first equation is developed by surveying the internal credit rating models of the Indian banks and the ratios selected are: current ratio, debt-equity ratio, and operating margin. The second equation is similar to that of Altman's (1968) original equation with a slight modification: instead of debt-to-market value of equity, debt-to-book value of equity is considered. The other three ratios of the second equation are working capital to total assets, retained earnings to total assets, and earnings before interest and taxes to total assets. The third equation is called as Altman, Hartzell and Peck's ‘Emerging Market Score Model.’ Except the asset turnover ratio, all the ratios of the second equation are considered. In all the three equations, the coefficients are estimated by using the development sample of 112 companies. The dominant variables discriminating the default companies from non-default ones are: current ratio, debt-equity ratio, operating margin, working capital to total assets, earnings before interest and tax to total assets, net worth to debt, and asset-turnover ratio. The classification accuracy of the second and the third equations is 82 per cent while that of the first equation is only 57 per cent. It implies that the most widely used two ratios — current ratio and debt-equity ratio — are relatively poor in predicting the default companies. Similarly, the ROC accuracy ratio is the highest for Altman's equation whereas the variables considered in internal credit rating models of banks is having a relatively low accuracy ratio. To test the ability of the model in identifying the defaulting companies correctly, an unbiased diagnostic test of the model is conducted on two separate sets of defaulted firms. The results reveal the following : The Altman's model is capable of predicting default in most of the sample companies. The hold-out sample accuracy results show that the selected variables are capable of predicting default. The analysis shows that the financial risk factors being considered by banks in their internal rating models are not very effective in comparison to other two models in discriminating the firms into default and non-default categories. Banks can map the internal ratings with the Z-scores and scale this up to assign various credit ratings. By arriving at the coefficients on the basis of their own database, banks can develop Z-score calculators for various segments of borrowers.

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Zhao, Ming-Xian, Gui-Ming Le та Jianyong Lu. "Can We Estimate the Intensities of Great Geomagnetic Storms (ΔSYM-H ≤ −200 nT) with the Burton Equation or the O’Brien and McPherron Equation?" Astrophysical Journal 928, № 1 (1 березня 2022): 18. http://dx.doi.org/10.3847/1538-4357/ac50a8.

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Abstract We input the solar wind parameters responsible for the main phases of 15 great geomagnetic storms (GGSs; ΔSYM-H ≤ −200 nT) into the empirical formulae created by Burton et al. (hereafter the Burton equation) and by OBrien & McPherron (hereafter the OM equation) to evaluate whether these two equations can correctly estimate the intensities of GGSs. The results show that the intensities of most GGSs estimated by the OM equation are much smaller than the observed intensities. The rms error between the intensities estimated by the OM equation and the observed intensities is 203 nT, implying that the estimated storm intensity deviates significantly from the observed one. The rms error between the intensities estimated by the Burton equation and the observed intensities is 130.8 nT. The relative error caused by the Burton equation for storms with intensities ΔSYM-H<−400 nT is larger than 27%, implying that the absolute error will be large for storms with ΔSYM-H < −400 nT. The results indicate that the two equations cannot predict the intensities of GGSs correctly. On the contrary, the intensity of a GGS estimated by the empirical formula created by Wang et al. can approximate observations better if we select the right weight for the solar wind dynamic pressure, proving that solar wind dynamic pressure is an important factor of GGS intensity. This pressure is overlooked by the ring current injection terms of the Burton and OM equations. This is the reason why the two equations do not effectively estimate GGSs.

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Hurdle, V. F., and Dominique Lord. "Analysis of Two Left-Turn Equations from the Highway Capacity Manual." Transportation Research Record: Journal of the Transportation Research Board 1646, no. 1 (January 1998): 71–78. http://dx.doi.org/10.3141/1646-09.

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The left-turn procedures in the Highway Capacity Manual are complex, and some of the equations are presented with little explanation of how they were obtained and the assumptions they embody. This paper is an analytic exploration of two of the four equations used to estimate gq, the amount of green time needed to discharge the opposing queue, and gf, the amount of green time available for through vehicles before the first left-turning vehicle enters the intersection. The investigation reveals surprising hidden assumptions underlying Equation 9-17 for gq, which lead to errors when the opposing flow includes left turns. In the case of Equation 9-20 for gf, the theoretical results are consistently about 5 s larger but otherwise in reasonable agreement with the empirical equation and provide insight into how it could be improved. An analytic approximation is offered either as a replacement or as a framework for an improved empirical model.

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Meng, Q., Y. Li, and J. Xu. "New empirical stiffness equations for corner-filleted flexure hinges." Mechanical Sciences 4, no. 2 (October 15, 2013): 345–56. http://dx.doi.org/10.5194/ms-4-345-2013.

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Abstract. This paper investigates the existing stiffness equations for corner-filleted flexure hinges. Three empirical stiffness equations for corner-filleted flexure hinges (each fillet radius, r, equals to 0.1 l; l, the length of a corner-filleted flexure hinge) are formulated based on finite element analysis results for the purpose of overcoming these investigated limitations. Three comparisons made with the existing compliance/stiffness equations and finite element analysis (FEA) results indicate that the proposed empirical stiffness equations enlarge the range of rate of thickness (t, the minimum thickness of a corner-filleted flexure hinge) to length (l), t/l (0.02 &amp;leq; t/l &amp;leq; 1) and ensure the accuracy for each empirical stiffness equation under large deformation. The errors are within 6% when compared to FEA results.

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Hosseini, S. M. "Densities and derived properties of nanofluids from new empirical model based on Pack-Cho scheme." High Temperatures-High Pressures 53, no. 4 (2024): 323–38. http://dx.doi.org/10.32908/hthp.v53.1611.

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This work aimed to develop a new empirical model for density and derived properties of nanofluids using a new empirical equation based on Pack-Cho scheme. The previously developed Tammann-Tait based model by the author has suffered a mathematical drawback to properly represent isothermal compressibility and isobaric expansivity confident values with respect to the nanoparticle fraction. Then, that drawback has been fixed through the above-mentioned approach. The base fluid contributions were obtained from Tammann-Tait equation of earlier work, whereas the nanoparticle contribution was expressed in terms of both fraction and density of nanoparticle as well as some binary parameter. High-pressure densities of 8 nanofluids based on ethylene glycol, poly ethylene glycol, water, poly ethylene glycol + water and ethylene glycol + water were correlated using the new empirical equation mentioned above in 273–363 K range and pressures up to 45 MPa. From 2115 data points examined, the AARD of 0.049% was obtained. The derived thermodynamic coefficients of studied nanofluids were also estimated by the proposed approach. Finally, results of the present empirical model have also been compared with the earlier work and Tammann-Tait type equations of literature, as well.

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Khalid, Muhammad Ashraf, Abdul Razzaq Ghumman, and Ghufran Ahmed Pasha. "A Comparative Analysis of Sediment Concentration Using Artificial Intelligence and Empirical Equations." Hydrology 11, no. 5 (April 27, 2024): 63. http://dx.doi.org/10.3390/hydrology11050063.

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Morphological changes in canals are greatly influenced by sediment load dynamics, whose estimation is a challenging task because of the non-linear behavior of the sediment concentration variables. This study aims to compare different techniques including Artificial Intelligence Models (AIM) and empirical equations for estimating sediment load in Upper Chenab Canal based on 10 years of sediment data from 2012 to 2022. The methodology involves utilization of a newly developed empirical equation, the Ackers and White formula and AIM including 20 neural networks with 10 training functions for both Double and Triple Layers, two Artificial Neuro-Fuzzy Inference System (ANFIS), Particle Swarm Optimization, and Ensemble Learning Random Forest models. Sensitivity analysis of sediment concentration variables has also been performed using various scenarios of input combinations in AIM. A state-of-the-art optimization technique has been used to identify the parameters of the empirical equation, and its performance is tested against AIM and the Ackers and White equation. To compare the performance of various models, four types of errors—correlation coefficient (R), T-Test, Analysis of Variance (ANOVA), and Taylor’s Diagram—have been used. The results of the study show successful application of Artificial Intelligence (AI) and empirical equations to capture the non-linear behavior of sediment concentration variables and indicate that, among all models, the ANFIS outperformed in simulating the total sediment load with a high R-value of 0.958. The performance of various models in simulating sediment concentration was assessed, with notable accuracy achieved by models AIM11 and AIM21. Moreover, the newly developed equation performed better (R = 0.92) compared to the Ackers and White formula (R = 0.88). In conclusion, the study provides valuable insights into sediment concentration dynamics in canals, highlighting the effectiveness of AI models and optimization techniques. It is suggested to incorporate other AI techniques and use multiple canals data in modeling for the future.

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Soriano, C. M., and P. E. Quinay. "Development of Fragility Curves for the Underground PVC Water Distribution Pipes in Quezon City and Damage Estimation under a Magnitude 7.2 Scenario Earthquake." IOP Conference Series: Earth and Environmental Science 1091, no. 1 (November 1, 2022): 012005. http://dx.doi.org/10.1088/1755-1315/1091/1/012005.

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Quezon City is traversed by the West Valley Fault System (WVFS) that has the capability of generating a magnitude 7.2 earthquake known as “The Big One”. It has an extensive water distribution network that is very susceptible to damages that will be caused by the ground shaking component of the M 7.2 earthquake. This study focuses on determining the behavior and estimating the damage of underground polyvinyl chloride (PVC) water pipes due to the M 7.2 earthquake by using appropriate empirical repair rates (RR) and developing fragility curves. The appropriate empirical RR equation was determined by comparing the results of selected PVC RR equations and the simulation using line-element modeling. The PGV ranges from 23.10 cm/s to 64.49 cm/s as determined using the Boore and Atkinson (2008) ground motion prediction equation. Using the results from the empirical and simulations methods, the equation by the American Lifelines Alliance (2001) was determined to be the appropriate empirical RR equation for the study area. The expected average repair rate of PVC is 0.05 repairs/km length of pipe or an estimated 84 total PVC pipe repairs in the city. Three fragility curves were generated showing the relationship of PGV and RR which is an important tool in estimating the underground pipe damages with respect to the M 7.2 earthquake and other future earthquakes of similar properties.

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Le, Giang Song, and Hung Manh Le. "PRELIMINARY RESULT OF DEVELOPMENT OF 3D NUMERICAL MODEL FOR MORPHOLOGICAL PROCESS." Science and Technology Development Journal 12, no. 18 (December 15, 2009): 5–11. http://dx.doi.org/10.32508/stdj.v12i18.2376.

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The paper presented a 3D numerical model for calculation of flow and sediment transport in channels. The flow is calculated by solving the Reynolds equation with hydrostatic assumption. Suspended-load is simulated by transport equation while bed-load is calculated using empirical formula. Bed transformation is obtained by solving an overall mass-balance equation. All equations are solved using finite volume method. The model is applied for study of bed deformation of VamNao river in the Mekong delta.

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Gubashi, Karim R. "Using Hydraulic Parameters to Estimate Longitudinal Dispersion Coefficient in Open Channel." Tikrit Journal of Engineering Sciences 16, no. 1 (March 31, 2009): 42–62. http://dx.doi.org/10.25130/tjes.16.1.04.

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A comparative analysis of previous theoretical and empirical equations is applied to evaluate their behavior in predicting dispersion coefficient in open channel. A new simplified method for predicting dispersion coefficients using hydraulic parameters is developed. A nonlinear multiple regression method is prepared to derive a new equation of dispersion coefficient. This equation is proven to be superior in explaining dispersion coefficient of open channel more precisely, as compared to existing equations. Keywords: Longitudinal dispersion, hydraulic parameters, dispersion coefficients.

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