Conserving heat consumption by modeling and optimizing efficiency of complex heat exchanger systems

. Many needs have to be met in human life. One of the key needs is to provide living comfort, which is clearly associated with heat. Analyzing the amount of produced heat, it should be emphasized that the higher the development of a country, the greater the demand for heat. In the era of the debate on the impact of human activities on the climate, it is impossible not to emphasize the importance of conserving energy and heat, and thus the rational management of these goods. The paper proposes a mathematical description of the complex systems of heat exchangers in the form of linear differential equations. Their analytical solution is presented in the form of temperature change of the heat carrier along the heating surface of a four-thread heat exchanger. Analysis of the possible heat exchange cases for eight possible flow systems is presented. In addition, the most effective minimization of heat losses was found.


Introduction
The implementation of the linear model of the economy in the world for decades has caused a significant reduction in natural resources, as well as a huge demand for energy. The processes of manufacturing goods were based on raw materials newly obtained from the natural environment, and to a negligible extent on raw materials from recycling. The same was true for the energy demand. New energy generation units were being created, while the amount of energy recovered in the whole structure of energy demand accounted for a small percentage. This linear production concept is associated with enormous environmental costs. The transition to the implementation of a closed-circuit economy eliminates these burdens to a significant extent. In general, there are heat exchangers in the processes of energy consumption between the energy generation system and its direct use. They are devices whose efficiency is high, nevertheless, not 100%. A slight improvement in the efficiency value of these devices, taking into account the amount of consumed energy, would result in huge amounts of conserved energy. What is more, it is well known that nothing is as profitable, both economically and ecologically as conserved energy. An increase in the efficiency of heat exchangers is also the easiest way to "obtain" the cheapest fuel.

Heat exchangers in the power industry. Finding conserving in energy consumption
Heat exchangers are widely used in human lifeamong others in construction, refrigeration, heating, air conditioning, cars, computers or central heating systems for residential buildings. They are generally used where there is heat exchange between the generation system and the reception system. They have the form of large-sized devices, but also small computer exchangers. Another exchanger for many applications in industry and heating systems is the shell and tube exchanger. It is built of a shell in which a tube bundle is placed, where the heat transfer medium flows in the coil, whereas the medium that receives the heat is in the shell.
Energy processes occurring during energy generation or processing are characterized by relatively low efficiency. A significant part of the chemical energy of the fuel is expelled in the form of waste energy, and one of the methods of minimizing energy consumption is to optimize heat recovery. Recycling at least a small part of this lost energy and reusing it for the needs of any other process or technology is another simple way of "obtaining" the cheapest fuel [1]. The recovered heat can be used, for example, for heating a building, providing technological heat for ventilation units, heating hot water and all other technological processes.

Mathematical model of a multi-element heat exchanger with any flow structure of medium
Each surface heat exchanger is a four-pole object with two incoming and two outgoing elements ( Figure 1). The heat flux q d  transmitted by the warmer medium to the colder medium can be expressed by means of the Peclet equation [2,3,4,5]   where: q d elementary heat flux, W/m 2 ; kheat transfer coefficient, W/m 2 K; dFelementary surface of heat exchange, m 2 ; t1warmer medium temperature, K; t2cooler medium temperature, K. Whereas the energy balance of mediums, corresponding to the elementary surface of the exchanger, takes the form [2,3,4,5]   cspecific heat, J/(kgK); m mass flux, kg/s. After the equations (2) and (3) are transformed, equations of temperature changes of warmer and colder factors along the surface of the exchanger are obtained where coefficients a1 and a2 are expressed as follows Equations (4) and (5) form a system of differential equations. Their solution is the following relationships [6]   The above system of equations can be written in matrix form [7,8,9,10]  tvalues of temperatures of warmer and cooler medium at the entry to the system of exchangers (or multistage heat exchanger), K. In order to further generalize the described model, taking into account any configuration of the flux sequence of mediums in the multistage exchanger system, the following assumptions were made: -two external mediums with known temperatures are given to the system, -the system also leaves two external mediums. For each exchanger constituting an element of the system, a coefficient matrix can write [7,8,9,10] (24) where A -block matrix defining the structure of the exchanger system.

Calculation results
The matrix of temperatures of working media T and the matrix of coefficients A for the variant of the system from Fig. 2a are presented in the form: where: 1 1 results of which are presented in the form of temperature change dependencies along the heating surface for four heat carriers.
For each of the eight possible cases of media flow, shown in Fig. 1, the systems of linear differential equations were formulated by analogy. The results of the solutions for each of the eight cases are also shown in Fig. 2 in the form of graphs of media temperature changes along the heating surface.

Conclusions
The heat exchange occurring between two media can be assessed by different efficiency criteria. In the case of a warmer medium, the efficiency of heat exchange can be identified by the value of its final temperature. The lower its value, the more heat has passed to the colder medium, and thus the greater efficiency of the exchanger. At the same time, it means conserving the heat consumption and "acquiring" the cheapest fuel.
The conducted computational analysis showed that the most effective media flow system corresponds to code 100, which makes it possible to ensure a minimum temperature value of the hot medium at the outlet of the device with a minimum value of heat exchange surface.