The Thermodynamics Cycles with a Reversible Chemical Reaction
Issue:
Volume 12, Issue 2, March 2023
Pages:
14-20
Received:
31 March 2023
Accepted:
17 April 2023
Published:
9 June 2023
Abstract: The relevance: In the modern world, there is an urgent need for the efficient use of all possible heat sources for the subsequent production of mechanical work or electrical energy. The gradual depletion of fossil fuels on the planet is bringing humanity closer to a large-scale energy crisis. Since the conversion of heat into freely convertible work or electrical energy is possible with the help of heat engines, it is necessary to look for new ways to improve them. One of these ways can be the use of thermodynamic cycles with reversible chemical reactions. The main aim of the investigation of thermodynamic cycles with reversible chemical reactions, comparison and analysis of the results obtained and formulation of conclusions. Object: thermodynamic cycles of Carnot, Brighton and Stirling; mixtures of gases capable of changing their composition as a result of a reversible chemical reaction; chemical work. Methods: solving the problem of finding the efficiency coefficient using analytical methods solving the problem of finding the efficiency coefficient using analytical methods. Results: A Brighton, Carnot and Stirling thermodynamic cycles is considered in which the working substance is a chemically reacting gas with molar weight and heat capacity changing as a result of a reversible chemical reaction. By way of example, the reactions N2 + 3H2 →2NH3 and CO + 2H2 ↔ CH3OH is considered. For a constant heat supply, the cycles is characterized by the lower (Tlow) and upper (Ttop) temperature boundaries of existence; between these boundaries, the efficiency η can change from 0 to 1. Such peculiar properties are manifested because of two factors: reversibility of the chemical reaction and the special role of the chemical work in the conversion of heat into mechanical work, which minimizes the heat loss to the surrounding space in a closed thermodynamic cycles. The possibility of achieving a thermal efficiency in them equal to 1 in a limited temperature range does not depend on the type of cycle. The value of efficiency in Carnot and Stirling machines depends on the method of heat supply and removal at isotherms. Each of the Carnot, Stirling and Brighton cycles is characterized, respectively, by its parameter αC, αSt and αBr, which determines the condition for achieving efficiency η = 1. Heat engines operating on thermodynamic cycles with reversible chemical reactions are most efficient when using low potential heat sources.
Abstract: The relevance: In the modern world, there is an urgent need for the efficient use of all possible heat sources for the subsequent production of mechanical work or electrical energy. The gradual depletion of fossil fuels on the planet is bringing humanity closer to a large-scale energy crisis. Since the conversion of heat into freely convertible wor...
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Approximation of the Sum of a Power Series by Its First Four Terms
Issue:
Volume 12, Issue 2, March 2023
Pages:
21-29
Received:
16 May 2023
Accepted:
5 June 2023
Published:
20 June 2023
Abstract: The paper develops a three-parameter method for approximating the sum of the McLaurin series by its first four expansion terms, which allows obtaining analytical approximations for functions that are expanded into a power series. The expressions for the approximation parameters (a, b, c) of the exact sum ∑(S) of the geometric power series-base are obtained in general form and are determined by the coefficients at the second (A), third (B), and fourth (C) terms of the McLaurin series. For series that converge rapidly {their coefficients satisfy the inequality (аn)2≥(an‒1×an+1)}, the new method gives the real values of the sum ∑(S), and for series that converge slowly {for them (аn)2<(an‒1×an+1)}, the method gives the complex-conjugate roots of the parameters of their sum ∑(S). The paper presents examples of approximate determination of series sums by both three-parameter and two-parameter methods based on the analysis of series coefficients. The accuracy of the two- and three-parameter methods of approximation of ∑(S) is evaluated on the basis of determining the approximate sums of known numerical series (for the number , the number e, etc.). The new three-parameter method was used to approximate the sum of a series whose first terms were obtained by Lord Rayleigh when refining the method of determining the capillary complex of a liquid by the capillary rise method.
Abstract: The paper develops a three-parameter method for approximating the sum of the McLaurin series by its first four expansion terms, which allows obtaining analytical approximations for functions that are expanded into a power series. The expressions for the approximation parameters (a, b, c) of the exact sum ∑(S) of the geometric power series-base are ...
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