The proper allocation of water resources is a very important practical problem in the field of water network planning. Optimization models that are expeditious and easy to use for all stakeholders of the sector play an important role for water resource management. The present work resumes and reviews a least-cost optimization model proposed by our group (Maiolo and Pantusa in Water Sci Tech-W Sup. https://doi.org/10.2166/ws.2015.114
 , 2016), able to design a water distribution network with multiple supply sources and multiple users. This approach requires of solving an optimization problem based on a nonlinear objective function which is proportional to the cost of the water distribution network. The cost of pre-existing pipelines is considered null. A more realistic scenario, able to consider the maximum flow rate allowed for existing sources-users connections, is considered here. In order to illustrate the usefulness and flexibility of the proposed approach, an application of the model to the real case of the province of Croton, Southern Italy, is presented.

Modelling and optimization of least-cost water distribution networks with multiple supply sources and users

In a recent article (Ann Phys 377:414–445, 2017. https://doi.org/10.1016/j.aop.2016.12.012
 ), Pennisi and Ruggeri proposed a relativistic extended thermodynamics theory of polyatomic gas. It was achieved by adopting the closure procedure for the generalized moments of a distribution function that, as in the classical case, depends on an additional continuous variable representing the energy of the internal modes of a molecule; this permits the theory to take into account the energy exchange between the translational and the internal modes of a molecule in binary collisions. In this paper the attention will be focused on some integrals appearing in the field equations of the relativistic theory of polyatomic gas and their integrability will be proven. In the last part of the paper we consider the case of a monatomic gas and we evaluate the ultra-relativistic limit of the differential system of balance laws.

Integrability properties for relativistic extended thermodynamics of polyatomic gas

This article belongs to the research trend on dense and polyatomic gases which began with a well known article by Profs. Arima, Taniguchi, Ruggeri, and Sugiyama following the idea of considering two blocks of balance equations. Since a model with more moments better fits the experimental data, this case has been considered by Profs. Arima, Mentrelli and Ruggeri in the framework of the kinetic approach. The present article pursues a similar end in the framework of the macroscopic approach which is more general than the kinetic one. The general and explicit closure for the 24 moments model is found up to whatever order with respect to equilibrium. It gets to first base on which to build also the model with an arbitrary number of moments; this more general model heavily needs the results of the present article.

Extended thermodynamics of dense gases with at least 24 moments

To understand the non-equilibrium phenomena beyond the assumption of local equilibrium in rarefied polyatomic gases, a new version of extended thermodynamics has been recently introduced. It proposes two hierarchies of balance equations; the first of these is the usual one, while the second is an “energy block”. In this framework, we investigate here the model with an arbitrary but fixed number of moments and obtain its closure up to every order with respect to equilibrium. This is done in the context of the macroscopic approach. This is not a trivial aspect; in fact, if we impose the conditions up to a given order with respect to equilibrium, there is always the risk to obtain restrictions on the same expressions by imposing the equations at an higher order! An article has been published to prove that this eventuality does not occur. Similarly, if we impose the conditions for a model with a given number of moments, there is always the risk to obtain restrictions on the same expressions by imposing the equations for the model with an higher number of moments! Here this risk is confirmed, but controlled.

Extended Thermodynamics of dense gases with many moments – The macroscopic approach

A new model for Polyatomic Gases with an arbitrary but fixed number of moments has been recently proposed and investigated in the framework of Extended Thermodynamics; the arbitrariness of the number of moments is linked to a number and the resulting model is called an -Model. This model has been elaborated in order to take into account the entropy principle, the Galilean relativity principle, and some symmetry conditions. It has been proved that the solution for all these conditions exists, but it has not been written explicitly because hard notation is necessary; it has only been shown how the theory is self-generating in the sense that if we know the closure of the -Model, then we will be able to find that of -Model. Up to now only a single particular solution has been found in this regard. Instead of this, we find here a numberable set of exact solutions which hold for every fixed number .

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J. M. Sellier

Papers

Extended thermodynamics of dense gases with at least 24 moments

Extended Thermodynamics of dense gases with many moments – The macroscopic approach