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Concept# Conservative vector field

Summary

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field under the line integral being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.
Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual path taken.
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This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. (C) 2011 Elsevier B.V. All rights reserved.

2011Laurent Gabriel Félix Alexandre Stéphane Bungener, François Maréchal, Greta Martha Van Eetvelde

Steam is a key energy vector for industrial sites, used for process heating, direct injection and stripping, tracing and cogeneration of mechanical power. Steam networks transport steam from producers to consumers and across different pressure levels. The steam production equipments (boilers, cogeneration units and heat exchangers) should be dimensioned to always supply key consumers as well as to deal with extreme demand caused by exceptional events such as unit startups or extreme weather. An important issue to be dealt with is that of unexpected boiler shutdowns, which can take significant amounts of time to bring back online. In cases where demand surpasses the available production of steam, load shedding is necessary in order to keep the network operable. A penalty cost can be associated to load shedding. A well dimensioned steam network is one which is resilient to such events, being able to overcome extreme demand and unexpected boiler shutdowns at minimum cost. This paper proposes a methodology for evaluating the operability of a steam network when facing unexpected boiler shutdowns. A Monte-Carlo simulation is carried out on a multi-period steam network problem, randomly shutting down boilers according to their failure properties (probability of failure and duration of failure). The aim of this method is to evaluate how resilient a steam network is to boiler shutdowns. The Monte-Carlo simulation is applied to a steam network model built using a Mixed Integer Linear Programming (MILP) formulation, whose objective function is to minimise the operational costs of the steam network and therefore also to minimise the penalty costs associated to load shedding. A case study based on anonymised industrial data is used to demonstrate the method. Two investment propositions are evaluated and compared using the proposed method.

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Steam is a key energy vector for industrial sites, most commonly used for process heating and cooling, cogeneration of heat and mechanical power as a motive fluid or for stripping. Steam networks are used to carry steam from producers to consumers and between pressure levels through letdowns and steam turbines. The steam producers (boilers, heat and power cogeneration units, heat exchangers, chemical reactors) should be sized to supply the consumers at nominal operating conditions as well as peak demand. First, this paper proposes an Mixed Integer Linear Programing formulation to optimize the operations of steam networks in normal operating conditions and exceptional demand (when operating reserves fall to zero), through the introduction of load shedding. Optimization of investments based on operational and investment costs are included in the formulation. Though rare, boiler failures can have a heavy impact on steam network operations and costs, leading to undercapacity and unit shutdowns. A method is therefore proposed to simulate steam network operations when facing boiler failures. Key performance indicators are introduced to quantify the network’s resilience. The proposed methods are applied and demonstrated in an industrial case study using industrial data. The results indicate the importance of oversizing key steam producing equipments and the value of industrial symbiosis to increase industrial site resilience.

2016