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Introduction of Explicit Equations for the Estimation of Surface Tension, Specific Weight, and Kinematic Viscosity of Water As a Function of Temperature

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Introduction of Explicit Equations for the Estimation of Surface Tension, Specific Weight, and Kinematic Viscosity of Water As a Function of Temperature Fluid Mechanics research International Journal Research Article Open Access Introduction of explicit equations for the estimation of surface tension, specific weight, and kinematic viscosity of water as a function of temperature Abstract Volume 4 Issue 1 - 2020 Three simple empirical models were proposed to predict surface tension, specific weight, and kinematic viscosity of water as a function of temperature. The formulations were Kaan Yetilmezsoy derived within the framework of the nonlinear regression analysis based on the Richardson’s Department of Environmental Engineering, Yildiz Technical extrapolation method and the Levenberg–Marquardt algorithm. The estimations were University, Turkey proven to be satisfactory with very high determination coefficients above 0.999. The Kaan Yetilmezsoy, Professor, Department proposed formulations were developed using a total of 155 data points and compared Correspondence: of Environmental Engineering, Faculty of Civil Engineering, Yildiz against different equations from the literature. Moreover, illustrative examples and ® Technical University, Davutpasa Campus, 34220, Esenler, Istanbul, relevant MATLAB scripts were presented to demonstrate the applicability of the present Turkey, Tel +90 212 383 53 76; equations. The statistical results clearly corroborated that the proposed equations were Email accurate enough to be used in estimation of the present fluid mechanics-related parameters. The computational analysis yielded simple mathematical structures to be easily used for Received: December 19, 2019 | Published: February 05, 2020 educational and practical purposes. Keywords: fluid mechanics, surface tension, specific weight, kinematic viscosity, water, temperature, nonlinear regression, statistical analysis Introduction liquid viscosities are much less derived than for gases, and their implementation is restricted to a qualitative definition. The behaviour Surface tension (σs in N/m or kgf/m in meter-kilogram-second of liquids is noticeably different from gases, and the viscosity decreases 1 (MKS) unit system) is resulted from the attraction effect between with increasing temperature for the liquids. This relationship can be 2,3 the molecules of the liquid due to various intermolecular forces. mostly expressed in the form of an Arrhenius-type formulation. The Because of this effect, the surface layer of the liquid behaves like a surface tension, specific weight, and kinematic viscosity of water are stretched elastic membrane that allows the insects (e.g. water strider, important parameters for many biological or industrial processes.4–7 some spider species) to walk on the liquid and causes capillary action. The use of computational techniques for various aims (e.g. modelling, Moreover, a change in temperature (T in°C or K) causes a change in simulation, and solving problems) has recently received considerable surface tension of a liquid. When temperature increases, kinetic energy attention, particularly among researchers in science and engineering. of liquid molecules increases, resulting a decrease in intermolecular Modelling not only helps develop a comprehensive understanding of 2 forces. Thus, surface tension decreases with increase in temperature. a process, but also has the potential to predict and solve problems The weight per unit volume of a substance is called the specific in specific processes.8 Mathematical modelling and computer 3 weight γ (in kgf/m ) and is determined from γ=ρg, where g is the simulation are also valuable and powerful tools for describing and 2 2 4 4,5 acceleration of gravity (9.807m/s ) and ρ (in kgf.s /m ) is density. In evaluating their performance under both dynamic and steady-state MKS unit system, the specific weight of water at 4°C is approximately conditions. On the other hand, there are few systematic articles on 3 1000kgf/m , and the density (mass density or specific mass) is about the implementation of forecasting models that can be used directly 2 4 101.97kgf.s /m . When the temperature changes from either greater for both design and educational purposes.9,10 However, to the best of or less than 4°C, the density will become less than this value. Water the author’s knowledge, there are no systematic papers specifically has the maximum density only when it is pure water. Other factors devoted to the development of deterministic equations for prediction influence water’s density such as whether it is tap or fresh water or of the present fluid mechanics-related parameters (e.g. surface 6 saltwater, and these variations of water changes its density. A liquid tension, specific weight, and kinematic viscosity of water) in the same can be considered to consist of molecular layers superimposed on study. From both engineering and educational perspectives, this work one another. Although the liquid flows when a shear force is applied, offers a specific computer-based implementation (including nonlinear however, the frictional forces between the fluid layers (e.g. two parallel regression-based analysis, detailed statistical assessment, and special layers moving in a liquid) provide a resistance to this flow. Therefore, MATLAB®-based simulations) specifically aimed at exploring 2 2 viscosity (μ in kgf.s/m or ν in m /s, respectively, for dynamic and explicit formulations for the estimation of these parameters as a kinematic viscosity) can be described as a magnitude of the resistance function of temperature (i.e. a single input multiple outputs (SIMO)- 2–5 of a fluid to deformation under shear stress. In fluid mechanics, type modelling study).In consideration of the foregoing facts, the the ratio of dynamic viscosity (μ) to density (ρ) appears frequently. overall objectives of this study were: (i) to derive simple and explicit For convenience, this ratio is given the name kinematic viscosity (ν) equations for practicing engineers, researchers, and students, which 4,5 and is expressed as ν=μ/ρ. It is noted that models for estimating makes them possible to precisely forecast surface tension, specific Submit Manuscript | http://medcraveonline.com Fluid Mech Res Int J. 2020;4(1):7‒17. 7 ©2020 Yetilmezsoy. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Introduction of explicit equations for the estimation of surface tension, specific weight, and kinematic Copyright: 8 viscosity of water as a function of temperature ©2020 Yetilmezsoy. weight, and kinematic viscosity of water; (ii) to verify the models’ coefficient of variation (CV), are utilized as helpful mathematical predictions in terms of several statistical performance indicators; tools to evaluate the prediction performance of any prediction model.12 (iii) to evaluate the predictive capabilities of the developed equations Determination coefficient (R2) denotes that how much of the observed by comparing the computer-based outputs with the results from the variability is explained for by the prediction model.12 Besides, a high existed formulations reported in other studies; and (iv) to demonstrate value of R implies a significant correlation between the observed data the applicability of the proposed equations on the specific examples in and the predicted values.12 Additionally, RMSE is one of the most the field of fluid mechanics and hydraulics. common indicators used with artificial intelligence-based models (e.g. artificial neural networks (ANN), adaptive neuro-fuzzy inference Nonlinear regression-based analysis system (ANFIS)) and can be allocated into systematic (RMSES) and unsystematic (RMSE ) components using the least squares fitting. In the present study, a nonlinear regression-based analysis was U Among them, RMSE describes the part of the error due to the model conducted for the derivation of the temperature-dependent equations S (linear bias). Therefore, a low value implies a good model. RMSEU [σs=f(T), γw=f(T), νw=f(T)] for surface tension, specific weight, and kinematic viscosity of water at the same time. For this aim, the relevant describes the part of the error which is due to the random noise and data sets obtained from the open literature (e.g. several articles, online cannot be captured by the model. Moreover, MAE is the simplest of calculators, and web-based tables) were recorded in Microsoft® the numerical goodness measures. It is simply the mean of the absolute Excel® Office 365 (Microsoft Inc., Redmond, WA). Thereafter, for errors taken over the set of the estimate. Furthermore, PSE is another estimator that gives the ratio of squared systematic and unsystematic the modelling purpose, the data sets were imported from this open 13 database connectivity data source into the DataFit® (V8.1.69, Oakdale errors. Thus, a lower value implies a better model. Moreover, IA Engineering, PA, US) multiple regression software package running is regarded as a dimensionless relative measure limited in the range under Windows 10 system on a Casper Excalibur (Intel® CoreTM i7- of 0–1. Therefore, it is ideal for making cross-comparisons between models. It is a measure of the degree to which model predictions are 7700HQ CPU, 2.81 GHz, 16 GB of RAM, 64-bit) PC. In the nonlinear 14 regression-based analysis, the convergence criteria were implemented free of error. FA2 provides the percentage of forecasted cases in for the following values of the solution preferences: (a) regression which the values of the ratio O/P (observed/predicted) in the range of tolerance=1×10–10,
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