The paper presents an approach to the design of technical systems, the elements of which are interconnected and carry out an internal exchange of energy. The above analysis showed that for heat-exchange equipment when combining devices into systems, only iterative methods are currently used, a representative of which is Pinch analysis. A limitation of the iterative approach is the impossibility of obtaining an exact solution to such problems, which can only be achieved by analytical methods, which also make it possible to reveal some effects in systems that are practically unavailable for numerical solution. This indicates the absence of a rigorous proof of the existence of a solution and a problem in the construction of approximate solutions, due to the need to involve complementary hypotheses. The topological representation of the system modules allows us to consider the architecture as a network, which contributes to the analysis of the connections between the constituent elements and the identification of their mutual influence. Highlighted the typical connections of network elements such as serial, parallel, contour, which allows to unify the principles of building connections in the system. As an optimality criterion, the NTU parameter was chosen, which includes the heat exchange surface and is usually used when searching for a solution for heat exchangers of moving objects. An analytical solution to the problem of flow distribution and energy exchange efficiency in a system of two series-connected heat exchangers is obtained. His analysis showed that the formulation of the design problem based on the definition of matrix elements in relation to determinants allows not only to meet the requirements for the system, but also to determine the design parameters of its elements that satisfy their extreme characteristics

The paper presents the results of the development of a model of a heat exchange system with a bypass connection of flows. An analytical model of a bypass for two heat exchange elements in the form of a relationship between the temperature ratio differences of flows and efficiency has been developed. The resulting expression for the efficiency of energy exchange in the system depends on the distribution of elements and flows at the entrance to the system and does not depend on the distribution in the mixing unit. It is shown that the key factor determining both the operation of the designed system and its elements is the correspondence of the direction of the processes in the real system with their direction, hypothetically chosen by the designer when specifying its topology. The distribution of the energy potential dictates the conditions for the operability of the system and its elements through the uncertainty of the values of the average energy measures. The statement of the problem of determining the matrix elements that satisfy the requirement of the minimum uncertainty of the average energy measures leads to the determination of the distribution of the efficiency of the system elements in its topological representation in accordance with the requirements of the second law of thermodynamics. The formulated requirements for the minimality of the uncertainty of the average energy measures and the construction, based on the Shannon principle, make it possible to obtain a solution to the formulated problem as a finite subset of the values of the efficiency of the inter-network and intra-network energy exchange. In addition, the extremeness of solutions (minimum uncertainty of average energy measures) ensures the maximum efficiency of energy transfer from the “hot” network to the “cold” network in its elements and the minimum energy dissipation in the mixing nodes. The urgency of the topic is due to the fundamental need to reduce energy costs of systems. The applied aspect is to minimize the mass, dimensions and energy component in enterprises where thermal transformations are significant.

e work is devoted to the development and analysis of a topological model of the interaction of two energy media in a metricspace. The Hausdorff metric space is chosen as the initial set,which obeys the axioms of identity, symmetry and triangle. The real elementof the system in the space of energy measures,designed to transfer energy from one medium to another,is represented in correspondence with its cellular image,defined as a virtual Grassmannian. When energy is transferred from a medium with a higher energy to a medium with a lower energy, energy measures determine the organization of processes in the designed heat exchangeequipment. Informative components of the Grassmannian are also its area and perimeter. An analysis of the structural model, using the modified Heron formula and the Cayley-Merger determinant, showed that,assuming equilibrium at the Grassmannian nodes, its area in the space of energy measures should be equal to zero. At equilibrium, the semi perimeterin terms of energy measures is the energy potential applied to the element under conditions of its minimum. Relationships between the area of the Grassmannian and the potential applied to the element, the temperature efficiency, and the mixing efficiency of two flows are obtained. The study of this relationship shows that the Grassmannian perimeter has an extremum at an area equal to zero, at which the perimeter is equal to theapplied potential. From a design point of view, this means that when specifying flows and inlet temperatures,the requirements for the apparatus are formulated in terms of energy or final temperatures. This essentially determines the required potential and thecorresponding energy exchange efficiency.In this case, the potential takes the minimum required value corresponding to the requirement of the project, and the amount of transferred energy corresponds to the required one at fixed values of flows and energy exchange efficiency.

The mathematical model of the system is considered consisting of a series connection of three heating devices. A system of equations based on the energy conservation law is constructed, which turns out to be incomplete. It is shown that, given the known requirements for the system, expressed only in the efficiency of the system, the formalization of design often becomes insoluble. The system of equations is supplemented with expressions in accordance with the hypothesis of the proportionality of the amount of energy in an element and is presented in matrix form. The design task is reduced to determining the elements of the matrix by the value of the determinants. Analysis of the mathematical model made it possible to obtain an expression for the efficiency of the system as a function of energy exchange in its elements. This made it possible to obtain solutions for flows and their relationships in the elements of the system. In addition, the efficiency of inter-network and intra-network energy exchange has been determined, which satisfy the principles of equilibrium and minimum uncertainty in the values of the average parameters of the system. As an application, one of the main parameters, NTU, is considered, which determines the area of heat exchange with the external environment and the mass and dimensional characteristics of the heat exchange system. Models of direct and opposite switching on of flows with variations of flows and the value of the surface of devices when meeting the requirements for the efficiency of the system are considered. The results of comparing the design process with the iterative calculation method are presented and the advantages of the proposed approach are shown

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 equationsthat 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 itsefficiency.