Abstract
The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F α -fractal calculus, in this paper, we derive exact F α -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility.
Highlights
The complicated shapes of known phenomena are described with a parameter called “fractal dimension”
In addition to fractal geometry, the fractional calculus and fractal calculus can be helpful in the description of phenomena, with the difference that the operators in the latter calculi are respectively non-local and local [6,7,8,9,10,11,12,13,14,15,16,17,18]
In spite of the valuable efforts made to apply measure theory and harmonic analysis in fractals [19,20], the main step in the foundation of fractal calculus was taken by Parvate and Gangal
Summary
“Euclidean geometry” is able to model a limited number of known phenomena precisely. Introducing the concept of “fractal” provided a framework of modeling called “fractal geometry”. Computer graphics has provided the possibility of making complicated and beautiful fractal shapes by applying the mathematical language in the system of the iterative function for such shapes [4]. This prepares the ground for simulation and numerical solving of various problems with complicated geometries [3,5]. In spite of the valuable efforts made to apply measure theory and harmonic analysis in fractals [19,20], the main step in the foundation of fractal calculus was taken by Parvate and Gangal They suggested the algorithmic and Riemann-like method calculus on the fractal that can be mathematical models for many phenomena in nature [16,21]. By applying this geometry for ideal gases [23], the relations associated with exact differential equations for the physical quantities of temperature, pressure, and volume are presented
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