MEIS with variable flows

Development of the flow MEIS with the form reminding the models of nonequilibrium thermodynamics seems to be a very promising direction in equilibrium modeling of physical and chemical systems. Application of these models opens prospects for simpler analysis and solution of many complex problems related to the calculations of processes considered to be irreversible in principle. Certainly the flows in MEIS are interpreted statically as the coordinates of states. Thermodynamic interpretations are naturally extended to the kinetic coefficients that relate these flows with forces. Correctness of such interpretations is confirmed by the application of MP, being the theory of equilibrium states, as the terms for MEIS description. 

MEIS of stationary isothermal flow distribution in a closed (without sources and sinks) multi-loop hydraulic circuit has the form find 

In (13)-(17) Pj)r and Pfov—friction pressure loss and effective pressure (created by a pump or gravitational), respectively, on the i-th branch of the circuit x (x1.xnjl—the vector of volumetric flows in branches A a,j —the (m—1) x n—matrix of incidence of independent nodes and branches a,j = 1, if the flow in the i-th branch in accordance with the direction set in advance nears the y-th node a,j = — 1, if the i-th flow goes from they-th node, and a,j = 0, when node does not belong to the branch i j = functions p, and their limiting values v /,. in this case can be 

The objective function (13) representing the total dissipation of kinetic energy of the flows at isothermal motion of fluid is proportional to the entropy production in the circuit and its transfer to the environment, i.e., proportional to the entropy accumulated by the isolated system (interconnection of the circuit and environment). The matrix equation (14) describes the first Kirchhoff law, which, as applied to hydraulic circuits, expresses the requirement for mass conservation. Equality (15) represents a balance between the energy generated and consumed in the circuit. 

Using model (13)-(17), it is possible to identify the extremality criteria for different cases of interaction between the circuit and environment and reveal the reducibility of the problem of calculating the stationary flow distribution to the CP problem. Let us suppose that for the circuit with lumped parameters the closing relations have the form  

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Development of optimization methods for MEIS with variable flows of a substance participating in chemical reactions and transfer processes of heat, mass, and electric charges. 

Currently the authors are developing three classes of models of extreme intermediate states (MEIS) (1) with variable parameters (2) with variable flows, and (3) those describing spatially inhomogeneous systems. All these classes of the models are formulated and analyzed in terms of MP, which, in the authors opinion, can be defined as a mathematical theory of equilibrium states. It is natural to start the analysis of the created modifications with the MEIS with variable parameters, which is the closest in character to the traditional equilibrium thermodynamics models. 

As opposed to the described MEIS with variable parameters and the mechanisms of physicochemical processes in this case we will try to determine the objective function of applied model for a dissipative system based on the equilibrium principle of conservative systems, i.e. the Lagrange principle of virtual works. Derivation will be given on the example of the closed (not exchanging the fluid flows with the environment) active (with sources of motive pressures) circuit. The simplest scheme of such a circuit is presented in Fig. 3,a. A common character of the chosen example is explained by the easiness of passing to other possible schemes. For example, if at the modeled network nodes there are external... 

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