Helmholtz Free Energy

The effectiveness of Helmholtz resonators (HRs) inserted in various configurations along a pipeline system is investigated under conditions of acoustic resonance. It is shown that the acoustic damping achieved by a single, large volume HR can be achieved using multiple, smaller HRs. The attenuation is found to be dependent upon the location of the HR along the standing wave, with maximum attenuation achieved at the acoustic pressure antinode and minimal attenuation at the node. Mean flow introduced into the pipeline is shown to slightly reduce the effectiveness of the HRs. Additionally, the use of multiple HRs placed at strategic intervals along the pipeline is shown to achieve significant damping, eliminating the need for determining the standing wave formation in order to determine damping device placement. The results show the potential for using HRs in industrial systems, where space may be limited and characterizing the acoustics inside the pipelines is impractical.

The statistical thermodynamic perturbation equation of state. originally developed for fluids with Lennard-Jones intermolecular potential energy is improved. and then it is modified for the prediction of real fluid properties. The improved equation of state, when applied to Lennard-Jones fluid. predicts the properties of this fluid consistently better than the original perturbation equation of state. In order to modify the perturbation equation of state for the prediction of real fluid properties both pair-, and triplet-interaction potential functions are considered in the formulation of this equation of state. For the pair-interaction potential function the Kihara spherical-core function is used, and for the triplet-interaction potential function the Axilord-Teller function is used. Comparisons are made between the calculated and the experimental thermodynamic properties of argon. methane and neo-pentane in liquid and in vapor states. These comparisons indicate that the perturbation equation of state is very promising in predicting the properties of real fluids in condensed states.

A variational technique which is based on two different inequalities for the Helmholtz free energies is used to calculate the equilibrium thermodynamic properties of simple fluids. A system with hard-sphere potential function is used as the reference system. Helmholtz free energy of the original system is calculated by variation around the Helmholtz free energy of the reference system, and the other thermodynamic properties are calculated from free energy. By choosing a hard-sphere reference system, it is possible to calculate the equilibrium thermodynamic properties of fluids from very low densities to densities close to solid, and from high temperatures in the gas phase to low temperatures in the liquid phase, in the ranges where experimental and machine-calculated data are available. It is shown that the present variational technique is a better approach to the prediction of the equilibrium thermodynamic properties of liquids and vapor–liquid phase transition than any other approach so far developed. While the variational calculation based on a hard-sphere reference system does not predict the liquid–solid phase transition, it is argued that this might be due to the neglect of the orientation, or ordering in the formulation of the working inequality for fluids.

In this paper, we present the point symmetry group of three-dimensional homogeneous Helmholtz equation, when we consider the cylindrical coor-dinate system. In continuation, we present a complete set of functionally independent invariants of the equation along with ...