Improvement of design of heat networks: parallel-series connection

Abstract

An analytical model of the system has been developed, consisting of a parallel-series connection of thermal devices. On the basis of graph theory, a system of equations of communication between the graphical representation of the system and the table of integers, which are the numbers of nodes and branches, is presented. Mathematical formalization made it possible to create a data representation that describes the distribution of temperatures in the nodes of the system and flows on its branches and reduce the volume of the problem based on expert estimates corresponding to the second law of thermodynamics. The chains and routes of heat distribution in the network of heat exchangers, including input and output elements, distribution and mixing units, are considered. The presented data structure provides an opportunity to programmatically build a system of energy balance equations for the system. The system of equations is supplemented by the hypothesis of the proportionality of the change in the measure of energy in the element to the potential applied to them. As a result, a system of equations is obtained, which forms a complete problem written in matrix form. In the design problem, after determining the requirements for the system, in the mathematical sense, the problem arises of determining the elements of the matrix by the value of the determinant. The requirement of equilibrium in the nodes of mixing flows, together with the conservation law, allows us to formulate a complete system of equations that determines the distribution of flows on the branches of networks. The principle of minimum uncertainty allows us to identify a group of roots that has maximum efficiency when the principle of equilibrium is fulfilled for the elements of the system. An example is presented that implements the exact scheme for solving the design problem for a given ratio of flows at the input to the system and the requirement for its efficiency.