A comparative analysis of sulfate $\left(S{O}_4^{-2}\right)$ ion concentration via modern fractional derivatives: An industrial application to cooling system of power plant

Abstract

The significance of cooling system of power plant has vividly diverted the scientists, engineers and researchers because of the experimental analyses and numerical approximations on a cooling system of power plant. In fact, the heat exchange processes inside the condenser take worsening place due to uncontrolled increase of the sulfate ion concentration in cooling water which depends upon two main causes (i) an increase in deposition of calcium salts on the surfaces of heat exchangers/cooling towers (ii) the corrosion of power plants in cooling system. In this manuscript, a fractional modeling of sulfate $S{O}_4^{-2}$ ions concentration for circulating water in a closed cooling system of a power plant is based on the contributions of modern differentiations of Atangana–Baleanu and Caputo–Fabrizio types. The governing equation of Sulfate $S{O}_4^{-2}$ ions concentration is converted through the law of conservation of mass for volumetric flow rates using modern fractional differentiations, and then solved analytically by invoking Laplace transform method. An interesting comparative analysis of sulfate $S{O}_4^{-2}$ ions concentration is explored via Atangana–Baleanu and Caputo–Fabrizio fractional operators. Based on both modern differentiation operators our results suggest few similarities and differences for the removal of Sulfate $S{O}_4^{-2}$ ions concentration.

Introduction

There is no refusing fact that 4 million gallons of freshwater is consumed by a 500 MW wet-cooled power plants per day throughout the world. Water is commonly utilized in several industrial processes because it is an excellent coolant in power plants. Commercially, power plants heavily use water for various processes like the cooling installations in boiler auxiliary equipment and the water-steam cycle in power boilers. In order to meet with appropriate chemical and physical properties of industrial processes, one has to require a continuous access of fresh water [1], [2]. On the other hand, the superiority of water has substantial effect on heat production and efficiency of energy as well as failure-free operation of power units. In order to characterize the chemical composition of heat exchangers, water is chief source for heat transfer processes. In simple words, the overall efficiency of a power unit strongly depends upon heat transfer processes. Ryabchikov et al. [3] analyzed commercial heat exchangers in steam turbine units at specific thermal power plants with the individual features of the operation. They tackled new piping systems for commercial heat exchanges with industrial testing. Martín and Martín [4] investigated the impacts of cooling towers location on its size multiperiod optimization formulation for minimum water consumption. They observed the mathematical simulation for the limits in power production as a result of the cooling capabilities in different climates. Condor et al. [5] presented at about 350 samples of cement type 10 and class G and three sulfate concentrations (3000; 6000; and 30,000 ppmW) for periods of 2, 4, 6, 8, and 10 months were prepared and placed into three temperatures conditions (30, 55, and 75 °C). They concluded that the effects of sulfate diffusion on cement deterioration were more pronounced experimentally. Ashane et al. [6] presented a review report on the removal of sulfate ions in contaminated mine water. They also investigated few opportunities and challenges for the removal of sulfate ions in contaminated mine water. Fang et al. [7] observed the study for the factors affecting the removal of sulfate ions and removal of sulfate ions from the sodium alkali FGD wastewater. They appraised the suitability of ettringite precipitation method and suggested that the method is feasible for treating high-concentration sulfate wastewater. Further, the latest study for sulfate ions concentration can be found in recent references as [8], [9].

To model the real world problem in the scientific field of engineering, fractional calculus is utilized as the most powerful mathematical tool. The dynamics of present and past states can be modeled by fractional order derivatives for instance, Caputo, Caputo–Fabrizio, Atangana–Baleanu and many others. Such differential operators allow to model physical processes with long-range interaction and dissipation [10], [11], [12], [13], [14], [15], [16], [17], [18]. Muzaffar et al. [19] utilized Caputo fractional operator on the circular pipe of viscoelastic liquid. Zafar et al. [20] applied the concept of Caputo fractional derivatives on the rate type fluid problem at an infinite plate that applies shear stress to the fluid. Mdallal and Hajji [21] investigated the solution of boundary value problem involving higher order nonlinear differential equation using Caputo fractional derivatives. Nehad and Ilyas [22] observed the effects of heat transfer analysis on second order fluid using Caputo–Fabrizio fractional operator. Nadeem et al. [23] explored the effects of magnetic field on the free convection flow second grade fluid in presence of porosity. They displayed their governing equations in terms of Caputo–Fabrizio fractional operator. Kashif et al. [24] perceived the effects of Caputo–Fabrizio fractional derivative of order [0,1] on the thermal analysis of magnetohydrodynamic Jeffery fluid. Hristov [25] solved the governing equation of steady-state heat conduction by employing Caputo–Fabrizio fractional derivative in which analytical solutions was found by using Jeffery kernel. Kashif et al. [26] suggested the study of Caputo–Fabrizio fractional derivative on the heat and mass transfer problem of Casson fluid with magnetic field and pore structure of plate. Saqib et al. [27] applied the approach of Atangana–Baleanu fractional derivative on molybdenum disulfide nanofluid’s problem of convection. Kashif et al. [28] employed approach of Atangana–Baleanu fractional derivative on convection flow of Maxwell fluid with porosity and magnetism. The same authors solved stokes’ second problem of nanofluid through the approach of Atangana–Baleanu fractional derivative [29]. Progressively, the work on fractional order differentiations is vast but we end here by adherence of few recent studies as [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. The main gap in this analysis is to utilize the modern fractional approaches on the modeling of Sulfate $S{O}_4^{-2}$ ions concentration for circulating water in a closed cooling system of a power plant, because both imposed modern fractional derivatives have the ability of capturing memory effects. The Caputo–Fabrizio fractional derivative is based on the exponential kernel so called non-singular kernel and local kernel. Whilst Atangana–Baleanu fractional derivative has been proposed on the basis of Mittag-Leffler kernel so called the non-singularity and non-locality of the kernel. The main advantage of Atangana–Baleanu fractional derivative is to describe the hidden aspects of accurately on the two main causes (i) an increase in deposition of calcium salts on the surfaces of heat exchangers/cooling towers (ii) the corrosion of power plants in cooling system. In this regard, several numerical and analytical studies can be pertained as [42], [43], [44], [45]. In short, this manuscript presents a fractional modeling of Sulfate $S{O}_4^{-2}$ ions concentration for circulating water in a closed cooling system of a power plant is based on the contributions of modern differentiations of Atangana–Baleanu and Caputo–Fabrizio types. The governing equation of Sulfate $S{O}_4^{-2}$ ions concentration is converted through the law of conservation of mass for volumetric flow rates using modern fractional differentiations, and then solved analytically by invoking Laplace transform method. An interesting comparative analysis is explored via Atangana–Baleanu and Caputo–Fabrizio fractional operators. Based on both modern differentiation operators our results suggest few similarities and differences for the removal of Sulfate $S{O}_4^{-2}$ ions concentration as well.