Introduction of explicit equations for the estimation of surface tension, specific weight, and kinematic viscosity of water as a function of temperature

Surface tension (σs in N/m or kgf/m in meter-kilogram-second (MKS) unit system)1 is resulted from the attraction effect between the molecules of the liquid due to various intermolecular forces.2,3 Because of this effect, the surface layer of the liquid behaves like a stretched elastic membrane that allows the insects (e.g. water strider, some spider species) to walk on the liquid and causes capillary action. Moreover, a change in temperature (T in°C or K) causes a change in surface tension of a liquid. When temperature increases, kinetic energy of liquid molecules increases, resulting a decrease in intermolecular forces. Thus, surface tension decreases with increase in temperature.2 The weight per unit volume of a substance is called the specific weight γ (in kgf/m3) and is determined from γ=ρg, where g is the acceleration of gravity (9.807m/s2) and ρ (in kgf.s2/m4) is density.4,5 In MKS unit system, the specific weight of water at 4°C is approximately 1000kgf/m3, and the density (mass density or specific mass) is about 101.97kgf.s2/m4. When the temperature changes from either greater or less than 4°C, the density will become less than this value. Water has the maximum density only when it is pure water. Other factors influence water’s density such as whether it is tap or fresh water or saltwater, and these variations of water changes its density.6 A liquid can be considered to consist of molecular layers superimposed on one another. Although the liquid flows when a shear force is applied, however, the frictional forces between the fluid layers (e.g. two parallel layers moving in a liquid) provide a resistance to this flow. Therefore, viscosity (μ in kgf.s/m2 or ν in m2/s, respectively, for dynamic and kinematic viscosity) can be described as a magnitude of the resistance of a fluid to deformation under shear stress.2–5 In fluid mechanics, the ratio of dynamic viscosity (μ) to density (ρ) appears frequently. For convenience, this ratio is given the name kinematic viscosity (ν) and is expressed as ν=μ/ρ. 4,5 It is noted that models for estimating liquid viscosities are much less derived than for gases, and their implementation is restricted to a qualitative definition. The behaviour of liquids is noticeably different from gases, and the viscosity decreases with increasing temperature for the liquids. This relationship can be mostly expressed in the form of an Arrhenius-type formulation. The surface tension, specific weight, and kinematic viscosity of water are important parameters for many biological or industrial processes.4–7 The use of computational techniques for various aims (e.g. modelling, simulation, and solving problems) has recently received considerable attention, particularly among researchers in science and engineering. Modelling not only helps develop a comprehensive understanding of a process, but also has the potential to predict and solve problems in specific processes.8 Mathematical modelling and computer simulation are also valuable and powerful tools for describing and evaluating their performance under both dynamic and steady-state conditions. On the other hand, there are few systematic articles on the implementation of forecasting models that can be used directly for both design and educational purposes.9,10 However, to the best of the author’s knowledge, there are no systematic papers specifically devoted to the development of deterministic equations for prediction of the present fluid mechanics-related parameters (e.g. surface tension, specific weight, and kinematic viscosity of water) in the same study. From both engineering and educational perspectives, this work offers a specific computer-based implementation (including nonlinear regression-based analysis, detailed statistical assessment, and special MATLAB®-based simulations) specifically aimed at exploring explicit formulations for the estimation of these parameters as a function of temperature (i.e. a single input multiple outputs (SIMO)type modelling study). In consideration of the foregoing facts, the overall objectives of this study were: (i) to derive simple and explicit equations for practicing engineers, researchers, and students, which makes them possible to precisely forecast surface tension, specific