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Chemical Equilibria: Systematic Approach to Equilibrium Calculations01:21

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Equilibrium calculations for systems involving multiple equilibria are often complex. For example, to calculate the solubility of a sparingly soluble salt in an aqueous solution in the presence of a common ion, one must consider all the equilibria in this solution. Calculations for these systems can be complicated and tedious, so a systematic approach with a series of steps is often helpful. The process is detailed below.
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AES is a powerful analytical technique, especially effective when used with plasma sources, producing abundant spectra in characteristic emission lines. The Inductively Coupled Plasma (ICP), in particular, yields superior quantitative analytical data due to its high stability, low noise, low background, and minimal interferences under optimal experimental conditions. However, newer air-operated microwave sources are emerging as promising alternatives that could be more cost-effective than...
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Algorithms for Solving the Equilibrium Composition Model of Arc Plasma.

Zhongyuan Chi1, Yuzhang Ji1, Ningning Liu2

  • 1School of Metallurgy, Northeastern University, Shenyang 110819, China.

Entropy (Basel, Switzerland)
|January 24, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces two algorithms, HLMA and PV-LMA, to solve complex plasma equilibrium composition models. PV-LMA demonstrates superior computational efficiency and accuracy for high-temperature plasma analysis.

Keywords:
HLMAPV–LMAarc plasmaequilibrium composition model

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Area of Science:

  • Plasma Physics
  • Computational Chemistry
  • Thermodynamics

Background:

  • Plasma equilibrium composition models are crucial for understanding high-temperature systems.
  • These models often result in nonlinear systems of equations with weakly singular Jacobian matrices.
  • Solving these systems requires robust numerical methods capable of handling coefficient discrepancies.

Purpose of the Study:

  • To develop and analyze novel algorithms for solving high-temperature plasma equilibrium composition models.
  • To address the challenges posed by weakly singular Jacobian matrices in these models.
  • To compare the efficiency and accuracy of the proposed algorithms.

Main Methods:

  • Formulation of the equilibrium composition model using Saha and Guldberg-Waage equations, Dalton's law, stoichiometric equilibrium, and charge conservation.
  • Development and application of the Homotopy Levenberg-Marquardt Algorithm (HLMA) and Parameter Variation Levenberg-Marquardt Algorithm (PV-LMA).
  • Analysis of the algorithms' principles, processes, and suitability for nonlinear systems with weak singularities.

Main Results:

  • Both HLMA and PV-LMA successfully solve the plasma equilibrium composition model.
  • PV-LMA exhibits superior computational efficiency compared to HLMA.
  • Both algorithms achieve high computational accuracy, with equation system variance below 1 × 10⁻¹⁵.

Conclusions:

  • HLMA and PV-LMA are accurate and feasible methods for solving high-temperature plasma equilibrium composition models.
  • The homotopy calculation principle provides reliable initial values for iterative solutions.
  • PV-LMA is recommended for its enhanced computational performance in this context.