Abstract (English):
The monograph presents the results of a study of the digital economy as a new paradigm of economic development, a system of economic relations implemented through the use of digital information computer technologies. It is noted that the main problem in the formation of sustainable economic growth and the successful introduction of digital technologies are the challenges of digitalization of the economy. New digital technologies, innovative business models penetrate into all spheres of the economic life of society, influencing the very essence of the economy, forming qualitative structural changes in it. As a result, a digital economy is being formed as a subsystem of the traditional economy, characterized by the active use of digital technologies and the circulation of specific electronic goods. The monograph is intended for researchers, teachers, graduate students, undergraduates, as well as a wide range of readers interested in topical issues of digitalization of the economy.

Keywords:
digital economy , economic development, digital technologies, engineering applications, dissipative functions

3.1 Analysis of the energy perfection of non-ideal gas compression processes. Compressors are machines designed to increase the pressure of a gas flow. The whole variety of existing compressors can be divided into two broad classes: positive displacement and blade (dynamic) type machines. Volumetric compressors include reciprocating, rotary, membrane compressors. In dynamic compressors, due to the supply of mechanical energy, some kinetic energy is imparted to the gas, which is largely converted into pressure energy. The main varieties of this class are: centrifugal, axial compressors. When studying the real process of gas compression, the following tasks should be solved: 1. Determination of actual energy costs for real, i.e. irreversible process. 2. Determination of the relative efficiency of the process, i.e. calculation of efficiency (efficiency factor). 3. Evaluation of the energy perfection of the compressor unit for energy and resource saving. Equilibrium processes do not take into account the loss of kinetic energy due to friction and correspond to the minimum energy costs. To estimate the actual power of a non-ideal compressor, the values of the indicator efficiency are used, which are the results of bench tests of the compressor. The values of the indicator efficiency depend on the degree of pressure increase and are the passport characteristic of the compressor. The indicator efficiency is the ratio of the reference power corresponding to the ideal process to the value of the actual power consumed in the real process: , (3.1) where ηi – indicator efficiency; – reference power of an ideal, equilibrium process, W; – internal power of the compressor consumed in the real process, W. Let us perform a thermodynamic analysis of the non-equilibrium process of compression of non-ideal gases in a single-stage compressor using specific examples. Task 3.1 Ammonia is compressed in the adiabatic stage of an uncooled compressor. Ammonia parameters at the compressor inlet: Т1 = 306 K, Р1 = 1 bar, gas pressure at the outlet of the compressor stage Р2 = 5 bar. Gas consumption = 1 kg/s, non-equilibrium compression process. The calculation of the characteristics and functions of the state of the gas is carried out according to the Bogolyubov-Mayer virial equation in a truncated form [1]. Define: 1. Internal power consumed by the uncooled compressor stage , kW. 2. Gas temperature at the outlet of the compressor stage T2, K. 3. Exergy efficiency of the compressor unit, . Let us present a schematic diagram of a compressor unit (Fig. 5). Fig. 5. Schematic diagram of the turbocharger: 1 - electric motor, 2 - compressor shaft, 3 - multiplier, 4 - compressor stage. Thermophysical characteristics of ammonia [1], annex (Tables P-3, P-6) М = 17,031 kg/kmol, ТС = 405,6 K; РС = 111,3 atm virial coefficients [b1j], needed to calculate the compressibility factor z of a nonideal gas. m3/kg; m3/kg; m3/kg; m3/kg. Heat capacity series constants 1st stage: Calculation of the power consumed by an ideal compressor stage compressing ammonia. The minimum compression work corresponds to a reversible adiabatic process. The calculation formula has the following form: . (3.2) The results of calculations of the density, as well as isothermal deviations of the thermodynamic functions of ammonia from the ideal gas state according to the parameters of the gas at the inlet to the compressor stage are presented in Table. 3.1. Table 3.1 Thermodynamic functions of the state of ammonia at the compressor inlet Т1, K В1·102, m3/kg , kg/m3 z(0) z ρ1, kg/m3 , kJ/(kg·K) , kJ/kg 306 –1,416 0,6694 0,9905 0,9904 0,676 –13,581 –5,574 The search for the temperature T2S at the outlet of the uncooled ideal compressor stage is carried out based on the process condition X = S = const according to the equation: , (3.3) where , – isothermal deviations of the entropy of ammonia from the ideal gas state in terms of gas parameters at the inlet and outlet of the compressor stage. The zero approximation is set from the condition , further values are found from the calculated relation (2.3) by the method of successive approximation. The calculation results are presented in Table. 3.2 Table 3.2 Calculation of the temperature of ammonia T2S at the outlet of an ideal compressor T2S, K , kg/m3 , kg/m3 , kg/m3 , kJ/(kg∙K) , kJ/(kg∙K) , kJ/(kg∙K) 436,022 2,349 – 5,342 0,987 0,987 2,3799 0,7862 –16,295 – 6,4977 436,798 2,345 – 5,316 0,9875 0,987 2,376 0,7904 – 16,467 – 2,47 Let's choose T2S based on the condition: (table 3.2). K with precision . The calculation of the disposable work of the ideal ammonia compression process is carried out according to the calculated ratio (3.2). The calculation results are presented in Table. 3.3. Table 3.3 Functions of the state of ammonia at the outlet of an ideal compressor T2S, K , kg/m3 , m3/kg , kg/m3 , kJ/kg , kJ/kg 436,798 2,345 – 5,316 0,9875 0,987 2,376 291,434 – 9,789 kJ/kg. The power consumed by an uncooled compressor stage in an ideal process, without taking into account frictional forces, is: kW. 2nd stage: Calculation of the actual power absorbed by a stage of a non-ideal compressor compressing ammonia, . The power absorbed by a stage of a non-ideal compressor , is calculated by the relation (3.1): , where kW – power of an ideal compressor; ηS – adiabatic internal efficiency determined from the diagram, fig. 6. Fig. 6. Dependence of the adiabatic internal efficiency on the degree of pressure increase Р2/Р1. Analyzing the data of the presented diagram, we choose the value of the indicator efficiency, ηS = 0,8 at the degree of pressure increase Р2/Р1 = 5. The power consumed by an uncooled compressor stage in a real process is: kW. 3rd stage: Calculation of the ammonia temperature at the outlet of the uncooled compressor stage in the actual compression process Т2, K. The temperature of ammonia at the outlet of the uncooled compressor stage in a real compression process is determined from the total energy balance equation for a non-equilibrium process (2.2) Analyzing this equation, we obtain the following relation: , (3.4) where , – isothermal deviation of the enthalpy of ammonia from the ideal gas state in terms of gas parameters at the inlet and outlet of the compressor stage; , kW. The temperature T2 is calculated according to the following relation by the method of successive approximation: , (3.5) where – the average heat capacity of ammonia in the temperature range from Т1 to , kJ/(kg∙K); (table 3.1). The calculation of the zero approximation of the final temperature is carried out according to relation (3.5) based on the condition that : , (3.5, а) where ; kJ/kg; kJ/kg (table 3.1). K. Further calculations are carried out according to relation (3.4) with an accuracy: (table 3.4). Table 3.4 Calculation of the temperature of ammonia T2 at the outlet of a non-ideal compressor , K В·103, m3/kg ρ2ид, kg/m3 z(0) z ρ2, kg/m3 , kJ/kg , kJ/(kg∙K) 466,88 – 4,464 2,194 0,9903 0,9901 2,216 – 8,186 2,2557 466,32 – 4,477 2,196 0,9902 0,99 2,218 – 8,208 2,256 Let's choose T2 = 466.32 K with accuracy: . 4th stage: Calculation of kinetic energy losses due to friction in a non-equilibrium process, i.e. dissipation estimate , kW. The dissipation is estimated using the following equation, obtained from the balance equation of kinetic and potential energy for a nonequilibrium process (2.4): , (3.6) where kW – internal power expended in the actual process of ammonia compression, taking into account friction forces; – the power of the polytropic equilibrium process, calculated by the formula (3.6 a): (3.6 а) where: – the average value of the polytropic index for the initial and final parameters characterizing the state of the gas at the inlet and outlet of the compressor stage. In the introduction, we emphasized that the real process cannot be represented on a thermodynamic diagram; a real process can be quantified only if the working fluid at the beginning and at the end of the process is in certain equilibrium states. The polytropic process acts as such a process, as the closest to the real one, since the final and initial states of the working fluid. In our case, ammonia, of both processes completely coincide. The calculation of the power of the polytropic process is carried out according to the relation (3.6a). The average value of the polytropic index is calculated from the initial and final parameters characterizing the state of ammonia at the inlet and outlet of the compressor stage. ; кВт. The dissipation value is determined by relation (2.6) kW The dissipation kW is numerically equal to that part of the kinetic energy flow that is converted into internal energy due to overcoming the friction forces that impede the process. The value of the polytropic efficiency of the process is determined by the relation (3.1): The polytropic efficiency of the process ηпол= 0,82 characterizes the degree of energy perfection of the ammonia compression process. Let us represent the polytropic equilibrium process of ammonia compression in the coordinates (PV), (TS) (Fig. 7). Fig. 7. Polytropic equilibrium compression process of ammonia in the coordinates PV(a), TS(b) 5th stage: Analysis of the exergy perfection of the compression process. The exergy analysis of the process is carried out on the basis of the exergy balance equation: , (3.7) where – resulting exergy flow due to the visible, convective movement of the medium, W; (Ex) ̇^(т/о)– exergy flux due to heat transfer, W; – flow of mechanical work, W; – exergy losses due to the irreversibility of the process, are estimated by the Gouy-Stodola formula: . (3.8) The exergy efficiency of the uncooled compressor stage is estimated by the following relation: , (3.9) where – exergy flow at the outlet of the compressor stage; – exergy flow at the inlet to the compressor stage; – transit flows of exergy, i.e. those that make up that part of the exergy of the incoming flows, which passes invariably through the entire apparatus. In this case: We obtain the following calculation formula for the exergy efficiency of the compressor stage: , (3.10) where – exergy losses due to the irreversibility of the process. The results of calculations of the exergy efficiency of the compressor stage are presented in table. 3.5. Table 3.5 Calculation results of the exergy efficiency ( ) of the compressor stage T1, K T2, K , kJ/kg∙K , kJ/kg∙K , kJ/kg∙K , kW 306 466,32 0,9459 –13,581 – 12,812 47,999 0,866 The final loss of exergy, which can only be compensated by external energy carriers (steam, fuel, electricity) kW. The dissipation of the ammonia compression process was kW. The difference between these values is the part of the energy that can be used in the future. In this case, this value has the following value: kW. The possibility of utilizing this energy is limited by climatic and temporal conditions and depends on the average temperature of the process. Since dissipation is quantitatively equal to the heat that is supplied in a polytropic equilibrium process without friction, the average temperature of a polytropic process can be determined from the following relation: , (3.11) where – average temperature of the process, K. Based on the data obtained (Table 2.5), we determine the value of the average process temperature , K: ; K. The difference between the dissipation of kinetic energy for friction and the loss of exergy due to the irreversibility of the process is the part of the energy that can be used in the future. An example of energy recovery is its use in heating systems. Another example of the utilization of this energy can be the transfer to subsequent links in the technological chain, i.e. its use as a secondary energy resource. This is especially effective in large-scale and mass production [9]. 6th stage: Evaluation of the energy perfection of the compressor unit. The following formula is used to determine the capacity of the compressor unit: , (3.12) where – power consumed by the ideal compressor stage; – adiabatic internal efficiency of the compressor stage; – mechanical efficiency of the shaft, which takes into account friction losses between the moving parts of the compressor; – transfer efficiency, which takes into account the presence of a multiplier as an intermediate device between the shaft and the electric motor; – The efficiency of the electric motor. In practice, the values of these efficiency factors are as follows [7, 11]: ; ; . The power of the compressor plant according to formula (3.12) has the following value: кВт. The exergy efficiency of the compressor unit is as follows: . Task 3.2 Ammonia is compressed in a cooled reciprocating compressor, the gas flow rate is = 1 kg/s. The compression process is non-equilibrium, it is known that the ratio of the heat removed and the compression work expended is  = 0,6. Gas parameters at the compressor inlet: , Р1 = 1 bar, gas parameters at the compressor outlet: Р2 = 5 bar. The functions of the state of ammonia are calculated according to the truncated virial Bogolyubov-Mayer equation. The initial data for ammonia (Tc, K; Pc atm; kJ/(kg ∙ K)) are given in the condition of problem 3.1 and are presented in Annex (P-1, P-2). The internal efficiency is Т = 0,7. To define: 1. Internal power consumed by the compressor, , kW. 2. The final temperature of ammonia at the outlet of the compressor, Т2, K. 3. Dissipation of kinetic energy into friction, , kW. 4. Exergy efficiency of the compressor unit in the nominal mode of operation . Let's imagine a schematic diagram of a reciprocating compressor (Fig. 8). Fig. 4. Schematic diagram of a reciprocating compressor: 1 - suction valve; 2 - discharge valve; 3 - cylinder body; 4 - piston; 5 - shaft 1st stage: We determine the internal power of the compressor spent in the real process of gas compression , kW. The internal power of an isothermal compressor, taking into account the indicated efficiency, is calculated according to the following relationship: , (3.12) where - work of an ideal compressor compressing 1 kg of gas in isothermal mode [1]; = 0,7 – internal isothermal efficiency of the reciprocating compressor; kg/s – gas consumption. The results of calculations of the internal power of the real ammonia compression process are presented in Table. 3.6. Table 3.6 Calculation of the internal power of a non-ideal compressor Т1, K , kg/m3 , kg/m3 , kg/m3 , kJ/kg , kW 306 0,676 –1,416 3,523 –234,56 –335,086 The power consumed by the compressor in the real process is: kW. 2nd stage: We determine the temperature of ammonia at the outlet of the compressor Т2, K. The temperature T2 is calculated by the method of successive approximation from the relation obtained on the basis of the equation for the balance of the total energy of the nonequilibrium , (3.13) where , kJ/kg; kJ/kg (table 3.1); – isothermal deviation of the enthalpy of ammonia from the ideal gas state according to the parameters of the gas at the outlet of the compressor, kJ/kg; kJ/kg (table 3.6). The value of the zero approximation is determined based on the condition: ; . We get the following value: . Subsequent values are determined based on the value , obtained from equation (3.13). We solve by numerical approximation method with accuracy: . The calculation results are presented in Table. 3.7. Table 3.7 Calculation of the temperature of ammonia at the outlet of a non-ideal compressor kg/m3 kJ/kg kg/m3 kJ/kg kg/m3 , kJ/kg kJ/kg kJ/kg 0,676 –5,574 3,523 –234,56 373,087 2,8099 145,493 –15,423 –1,6087 0,676 –5,574 3,523 –234,56 372,345 2,816 143,8562 –15,525 0,1291 Selecting a value with precision . 3rd stage: Calculation of dissipation of kinetic energy on friction . We estimate dissipation using the balance equation for kinetic and potential energy for a nonequilibrium process (2.4). , where kW – internal power of the nonequilibrium process; – power of the equilibrium polytropic process, kW, – power of the equilibrium polytropic process, kW, - the average value of the polytropic index for the initial and final parameters characterizing the state of the gas at the inlet and outlet of the compressor. The results of calculations of viscous dissipation are presented in Table. 3.8. Table 3.8 Analysis of the polytropic equilibrium process of ammonia compression kg/m3 kg/m3 0,676 372,345 2,815 1,128 –261,27 –335,086 73,816 Let us represent the equilibrium polytropic process of ammonia compression (1–2 ) in coordinates (PV), (TS) (table 9). Рис. 9. Polytropic equilibrium process of ammonia compression with heat removal in the coordinates PV(a), TS(b). We determine the polytropic efficiency, the value of which characterizes the degree of energy perfection of the gas compression process . . 4th stage: Exergy analysis of ammonia compression process with heat removal. The exergy analysis of the ammonia compression process with heat removal is carried out on the basis of the exergy balance equation (3.7). Heat flow exergy is calculated based on equation (3.14): , (3.14) where – heat flow removed from gas, kW, То.с.= 298,15 K – ambient temperature, under standard conditions, – average thermodynamic temperature of the process, K, The exergy balance equation (2.7), taking into account relation (3.14), takes the following form: , (3.15) where – exergy flow due to the convective movement of the gas flow, kW, – heat flow exergy, kW, – internal power of the compressor, kW, – internal exergy losses due to irreversibility of the process, kW. The calculation of internal exergy losses , due to the irreversibility of the process itself is carried out on the basis of the following relationship: , (3.16) where – dissipation of kinetic energy due to friction, kW, – ambient temperature under standard conditions, – average thermodynamic temperature of the process, K. The results of calculations of the ammonia exergy flux due to convection are presented in Table. 3.9. The calculation of isothermal deviations of the enthalpy , entropy of ammonia from the ideal gas state is carried out according to the calculated ratios presented in the author's manual [1]. Table 3.9 Calculation of loss of specific exergy of ammonia due to convection kJ/kg kJ/(kg∙K) kJ/kg kJ/kg kJ/(kg∙K) 306 372,345 144,00794 0,4255 –5,574 –15,522 –13,581 –30,657 –246,538 The calculation of the average thermodynamic temperature of the process is carried out on the basis of the exergy balance equation (3.15). . The calculation of exergy losses due to the irreversibility of the process is carried out according to the calculated ratio (3.16): . The value is the final loss of exergy, which can only be compensated by external energy carriers. The difference between the values of dissipation of kinetic energy and internal losses of exergy, due to the irreversibility of the process, is that part of the exergy that can still be usefully used in the future. . The calculation of the exergy efficiency of the ammonia compression process in a cooled compressor, provided that the removed heat flow is usefully used in the future, is carried out according to the relation (3.10): . In the event that the thermal energy of the cooling agent is not used, then the exergy losses are calculated according to the equation: , (3.17) where – total exergy losses, kW, – internal exergy losses, kW, – external exergy losses, kW. The calculation of the exergy efficiency of the compression process with heat removal in this case is carried out according to the following relation: . (3.18) The results of calculations of the exergy efficiency of the ammonia compression process without utilization of the removed heat are presented in Table. 3.10. Table 3.10 Calculation of the exergy efficiency of of the process without disposal of removed heat flux –201,0516 –335,086 23,278 65,27 337,19 0,736 The exergy analysis of the process indicates that it is advisable to use the thermal energy of the cooling agent as a secondary energy resource. The possibility of practical implementation of heat exergy in each specific case is decided individually. [9, 10]. Task 3.3 To obtain high-pressure gas, multistage compressors are used, between the stages of which heat exchangers are installed to provide cooling of the gas compressed in the previous stage. Let's consider the process of gas compression in a two-stage turbocharger with intermediate cooling in a refrigerator at constant pressure. The schematic diagram of the compressor unit is shown in fig. 10. Fig. 10. Schematic diagram of a two-stage adiabatic compressor with intercooling: 1 - electric motor; 2 - the first stage of the compressor; 3 - heat exchanger cooled by recycled water; 4 - the second stage of the compressor Gas with initial parameters T1, P1 enters the first stage 2 of the compressor unit, where an adiabatic compression process takes place from the initial pressure P1 to the intermediate pressure P2. Then, the gas with temperature T2, pressure P2 is sent to the intermediate cooler 3, where it is cooled to the initial temperature T1 at a constant pressure with water from the circulating water supply. The resistance of the cooler along the gas path is made small in order to save energy spent on compression, which makes it possible to consider the gas cooling process as isobaric. After the cooler 3, the gas is sent to the second stage 4 of the compressor unit, where the adiabatic compression process takes place from the intermediate pressure P2 to the specified final pressure P3. In multi-stage compression, in order to select the optimal intermediate pressures at which the work would be the least, the distribution of the load on each stage is calculated according to the following relation: , (3.19) where 1, 2, … n – the degree of gas pressure increase in the first stage, second stage, n-th stage of the multistage compressor; n – number of compressor stages; общ = – total degree of increase in gas pressure in the compressor with the number of steps n from the initial pressure Рнач to the specified final pressure Ркон. If this condition (3.19) is met, the pressure ratio in all stages is the same, which is favorable not only for the power consumption, but also for the discharge temperatures in reciprocating compressors, which in this case are lower than with different pressure ratios in the stages. With an increase in the number of stages and intermediate coolers of the compressor unit, the compression process is more and more close to isothermal, i.e. to the most advantageous in terms of energy consumption. This does not exhaust the advantages of the multi-stage compression process. In reciprocating compressors, a decrease in the discharge temperature is achieved, and the risk of ignition of lubricating oils is reduced [10]. In the practice of compressor construction, there are very different relationships between the number of stages and the final pressure. The number of compressor stages of medium and high efficiency should be chosen so that the pressure ratio in each stage of the turbocharger does not exceed the value , equal to four. When compressing polyatomic gases, such as ammonia, it is advantageous to take higher pressure ratios than for compressors compressing diatomic gases, such as nitrogen. In compressors for gases with a low specific gravity, such as hydrogen, it is advantageous to adopt a reduced pressure ratio ( = 1,5 2), since the pressure losses between stages are below average. To increase the efficiency of compressors, they strive for the most complete cooling of the gas in intermediate coolers. The limit of possible cooling is determined by the initial temperature of the cooling water. When using water from a circulating water supply system, this temperature is determined by climatic and weather conditions. In modern designs of multistage compressors, the difference between the final and initial temperatures of the cooling water is 5–10 °C. The choice of the most advantageous number of stages should be carried out, guided not only by the desire for the minimum energy consumption, but also by considerations of a general economic nature. Let us give a specific example of calculating gas compression in a two-stage compressor unit. A two-stage ammonia turbocompressor with an intermediate isobaric cooler 3 serves to compress ammonia to a final pressure Р3 = 7 bar (fig. 10). The compression process is non-equilibrium, the consumption of ammonia is kg/s. The parameters of ammonia at the inlet to the first stage 2 of the compressor unit are as follows: Р1 = 1 bar, Т1 = 306 K. Cooling water from the circulating water supply is used to cool the ammonia compressed in the first stage 2 to an intermediate pressure P2. The cooling of ammonia after the first stage 2 in the isobaric cooler 3 is achieved to the initial temperature Т3 = Т1 = 306 K. Water heating is ТВ = 5 K. The value of the adiabatic efficiency of each stage of the compressor unit is assumed to be the same and equal to the following value: . The functions and parameters of the state of ammonia are calculated according to the equation of state of an ideal gas with a constant heat capacity. To define: 1. The power absorbed by each stage of the compressor unit , , kW. 2. Heat flow removed in the intermediate heat exchanger 3 , kW. 3. Exergy efficiency of the compressor unit . Stage 1: Calculation of the power absorbed by the first stage of the compressor unit, , kW. The determination of the power spent on compressing the gas from the initial pressure P1 to the intermediate pressure P2 in the first stage of the compressor unit is carried out according to the calculated ratio: , (3.20) where: = 4,5RM T2 = T1– – ammonia temperature at the end of the nonequilibrium compression process in the first stage, K; – temperature of ammonia at the end of the adiabatic equilibrium compression process in the first stage, K. The determination of the intermediate pressure of ammonia after the first stage Р2 is carried out according to the calculated relation (3.19). The calculation results , are presented in Table. 3.11. Table 3.11 Calculation of internal power of compressor steps ( ; ) , K , K , K , kW , kW , kW 306 380,82 399,53 2,196 –205,39 –205,39 –205,39 2,646 Since the ammonia compression ratio in both stages of the compressor unit is the same, the gas temperatures at the inlet to the first and second stages are equal to each other, and the adiabatic efficiency of the stages is the same, the internal power in both stages will be the same and equal to the following value : = = –205,39 kW. 2nd stage: Calculation of the heat flow , removed in the intermediate isobaric cooler 3, cooled by circulating water, kW. The calculation is carried out in accordance with the total energy balance equation (2.2) for heat exchanger 3 with respect to ammonia: , (3.21) where – the enthalpy flow of ammonia, kW; – heat flow supplied by the cooling circulating water, kW; = –205,39 кВт – heat flow removed from ammonia in the intermediate cooler 3. To determine the flow rate of cooling water, we use the total energy balance equation (1.25) for heat exchanger 3 with respect to water: , (3.22) where = 205,39 kW – heat flow supplied by ammonia compressed in the first stage; – enthalpy flow of water coming from the circulating water supply to heat exchanger 3, kW; = 4,19 [3, nomogram XI] – the value of the heat capacity of water at an average temperature = 302,5 K. The cooling water flow is as follows: kg/s. Stage 3: Calculation of the exergy efficiency of the compressor unit, . The calculation of the exergy efficiency of the compressor unit is carried out according to relation (3.12): , where – ammonia exergy flow at the outlet of the compressor unit, kW; – transit flow of exergy, kW; – technical power of the first and second stages, kW; = 9,81 кВт – power absorbed by the pump for pumping circulating water; = 298,15 K – ambient temperature. . In the quasi-static approximation, the actual compression process in the compressor can be represented as an equilibrium polytropic process, the initial and final states of which completely coincide with the real process. Let's represent this process in the coordinates PV(a) and TS(b), (Fig. 11). Fig. 11. Polytropic equilibrium process of two-stage ammonia compression with intermediate cooling in coordinates PV(a) and TS(b). Due to the cooling of ammonia in the intermediate cooler 3 at a constant pressure (isobaric process 23) the overall compression process in the compressor unit approaches isothermal, i.e. most advantageous in terms of energy savings. For a non-ideal gas, the problem of economic distribution of compression between stages becomes more complicated. In this case, when the pressure ratios are equal, the work flow in individual stages is different. It is greater in the last stages if the final pressure is high enough. In order to save energy in compressor installations, an automatic process control system is used, which ensures the regulation of those parameters, the deviation from which requires the compressor to be stopped in order to protect against an accident. 3.2 Analysis of the energy perfection of the processes of expansion and cooling of non-ideal gases. 3.2.1 Features of the use of low-temperature thermal resources. In most cases, the analysis of the effectiveness of actual processes in a particular device is carried out using the method of classical thermodynamics, which is able to determine predictions that are important for practice at the initial design stage. In this case, it is possible not only to predetermine the consumption of energy and material resources in a real unit, but also to get an idea of a number of engineering factors, such as the weight of the apparatus, the dimensions of individual components and the cost of their manufacture. The desire to reduce the cost of primary and traditional energy sources (consumption of fuel, electricity) without reducing or even increasing the return of energy to the end consumer due to its more rational transformation is the main trend of modern technology. The questions raised are reflected both when considering the features of the energy of low-temperature processes in chemical technology, and in a wide variety of industries. The need to use artificial cold arises in all cases when the task is to remove heat from a technological object at temperatures below ambient Тср. The variety of chemical industries, in which almost all known physical and chemical processes are carried out using substances with a wide variety of properties, gives rise to a variety of specific technological problems that can be solved with the help of cold. However, some typical applications of cold in chemical technology can be identified: 1. The need for cooling in exothermic reactions of chemical interaction, and we are talking not only about preliminary cooling of the starting materials to a given temperature and ensuring the removal of process heat, but also about direct control of the rate and direction of the reaction. The highest yield of dichloroethane in the reaction of ethylene chlorination is fixed in the range from 243 to 253 K. Even the production of polymers of high molecular weight is expedient in the low-temperature regime of the polymerization reaction, in particular, butyl rubber with desired properties is formed during the catalytic copolymerization of isobutylene and isoprene at 173 K. 2. Liquefaction of low-temperature gases and gas mixtures and associated with these processes low-temperature distillation and fractional condensation. 3. In the processes of polythermal crystallization, by changing the temperature, one can control the speed of the process, adjust the size and shape of the crystals. Fractionated crystallization at low temperatures is used in the production of aromatic compounds in the separation of para- and meta-xylenes (200 K), in the production of mineral fertilizers in the freezing of calcium nitrate (263 K), in dewaxing processes in the production of petroleum oils (240 K). 4. Large consumers of cold in the chemical industry are drying processes, including freeze-drying, industrial air conditioning systems. In all cases, the introduction of cold allows you to create new technological processes, intensify production, increase product yield and quality. In addition, it becomes possible to reduce the level of toxicity of industrial emissions and create more comfortable working conditions. It should be noted that obtaining artificial cold is an expensive and energy-intensive process, in connection with which the questions of the economic justification of technological processes using cold arise very sharply. Modern chemical industries are the largest consumers of cold. At the same time, chemical enterprises have a huge amount of secondary energy resources (SER) in the form of flue and waste gases, gas flares, waste steam of low parameters. The use of these types of energy to produce cold in absorption refrigeration machines can drastically reduce electricity consumption, which is an important way to create economical (energy-saving) chemical-technological systems. More than half a century of history of the development of refrigeration has led to its differentiation and specialization. Currently RU are classified: – according to the temperature range of work; – by type of energy used; – according to the state of aggregation of the working fluid; – by methods of obtaining a cooling effect. There are areas distinguished: Moderate cold 150 К <Тх<Тос : ТсублимацииСО2 = 195 K; ТнвNH3 = 240 K; ТнвSO2 = 263 K; Deep cold 70 K<Тх< 150 K: Тнв О2 = 90 K; ТнвAr = 87 K; ТнвN2 = 72 K; Cryogenic cold 3–5 K<Тх< 70 K: Тнв Н2 = 20 K; Тнв Не = 27 K. In refrigeration engineering, to obtain low temperatures, phase transformations, adiabatic expansion of gases and vapors with the return of external work, throttling, vortex effect, thermoelectric and thermomagnetic effect, and desorption are used. Consider the energy consumption of the most used methods for obtaining the effect of cooling gases and vapors in the chemical industry. 3.2.2 Analysis of the adiabatic expansion of gases in the expander. The processes of expansion of gases and vapors are widely practiced in equipment of chemical industries, for example, in turbines (steam, gas, hydraulic), expanders (piston, turbo expanders), nozzles, throttle devices. Adiabatic expansion of gases and vapors with the return of external work in expanders is the most effective method of internal cooling. Expanders are low-temperature expansion machines that serve to produce cold by expanding the working fluid with a decrease in temperature and the return of external work (energy). The term "expander" comes from the French word "de'tendre", which means to reduce pressure. In practice, there are basically two classes of expanders: 1. Expansion machines of volumetric action, such as piston, screw and rotary expanders (Fig. 12). Fig. 12. Scheme of a piston expander: 1 - piston; 2 - cylinder; 3 - inlet valve; 4 - exhaust valve; 5 - crank mechanism 2. Expansion machines of dynamic type (kinetic action), turbo-expanders (Fig. 13). Fig. 13. Scheme of a centripetal jet turboexpander: 1 - spiral gas supply; 2 - directing nozzle apparatus; 3 - rotor; 4 - outlet diffuser In volumetric expanders, gas expansion occurs due to a direct change in the volume of the working fluid by moving a piston or some other device. In kinetic action expanders (turbo-expanders), gas expansion occurs due to the force interaction of the expanding gas with the blades of the impeller when the gas flow moves in a specially profiled channel in which a rotating lattice of the blade apparatus (impeller) is installed. With the help of the rotating blades of the impeller, the internal and kinetic energy of the gas flow is converted into mechanical energy of the rotating lattice of the blade apparatus. This mechanical energy is converted into electrical or thermal energy, and then transferred to the rotation of the blower or compressor impeller. Expanders of both volumetric and kinetic action, depending on the pressure of the working fluid used at the inlet, are divided into high, medium and low pressure expanders. High pressure expanders have an inlet pressure of more than 10 MPa; low-pressure apparatus, not more than 1.5 MPa. In accordance with the working gas used, the devices are divided into air, nitrogen, hydrogen, helium. Structurally, both volumetric expanders and expansion machines of kinetic action are very diverse. Main energy characteristics of expanders Expanders are characterized by an adiabatic (internal) coefficient of performance (COP), which is the ratio of real power, i.e. non-equilibrium process in the nominal mode of operation (excluding mechanical losses), to the reference power of the ideal expansion process: , (3.23) where – the adiabatic (internal) efficiency of the expander; – the power of the expander in the actual expansion process, kW; – the mass flow rate of the working fluid, kg/s; – the external work of the expander in the nonequilibrium expansion process, kJ/kg; – external, disposable work of the expander in an ideal, equilibrium expansion process, kJ/kg; – the power of the expander in the ideal expansion process, kW. For piston expanders, the values of the indicator (internal) efficiency are , for turbo-expanders the higher values are – . Expanders are widely used in refrigeration cycles, i.e. reverse circular processes designed to transfer heat from less heated bodies to more heated ones. Expander refrigeration cycles are characterized by the following energy characteristics: refrigeration capacity, power absorbed by refrigeration unit, and coefficient of performance (Fig. 14). Fig. 14. Schematic diagram of the expander refrigeration cycle: 1 - electric motor; 2 – adiabatic compressor; 3 – heat exchanger for cooling compressed gas; 4 - pump for water supply from the circulating system; 5 – turbo expander; 6 - electric motor; 7 - heat exchanger, where heat is removed from the cooling object The principle of operation of the expander refrigeration cycle is based on the adiabatic expansion in the turboexpander 5 of pre-compressed gas (for example, nitrogen) in the adiabatic compressor 2 and then cooled in the isobaric heat exchanger 3. As a heat carrier that removes heat from the working fluid of the refrigeration cycle (for example, nitrogen) in the heat exchanger 3, the water supplied by the pump 4 from the circulating water supply circuit is used. There is a transfer of heat taken from nitrogen to the external environment. The technology provides for control of the water temperature at the inlet to the cooled heat exchanger. The prepared nitrogen enters the turbo expander 5, where the adiabatic expansion of nitrogen takes place with the return of external work . The expansion of the gas is accompanied by a decrease in its temperature, external work is removed in the form of electricity in the electric motor 6, sitting on the same shaft with the expander 5. Next, nitrogen is sent to the isobaric heat exchanger 7, where it is used as a coolant that removes heat from the cooling object . Hydrocarbons (ethane, propane), air can serve as coolants. Let us consider in detail the real process of adiabatic expansion of nitrogen in an expander. When studying the expansion process in expanders, the following tasks should be solved: 1. Determination of the integral cooling effect, i.e. degree of cooling of the working fluid; 2. Determination of the technical capacity of the expander in the non-equilibrium expansion process; 3. Determination of the exergy efficiency of the process. The following is a specific example of calculating and analyzing the adiabatic expansion of gas in a turboexpander: Task 3.4. Nitrogen expands in an adiabatic turboexpander. The expansion process is non-equilibrium, the nitrogen consumption is 1 kg/s. Parameters of nitrogen at the expander inlet: , Р1 = 5 bar, nitrogen pressure at the expander outlet Р2 = 1,5 bar, nitrogen state functions are calculated according to the truncated Bogolyubov-Mayer virial equation (Fig. 15). To define: 1. Integral temperature effect of the expansion process , K. 2. Technical capacity of the expander, , kW. 3. Exergy efficiency of the nitrogen expansion process in the expander, . Fig. 15. Schematic diagram of the turboexpander stage: 1 – expander stage body; 2 - rotating elements (blades) of the impeller; 3 - shaft; 4 - electric motor The thermophysical characteristics and reference data of nitrogen are as follows (Annex, Table P.1): М = 28,013 kg/kmol; Тс = 126,2 К; Рс = 33,5 atm; ω = 0,04; virial coefficients of nitrogen: m3/kg; m3/kg; m3/kg; m3/kg. The heat capacity series constants have the following values (Annex, Table P.3): d_0=1,113 kJ/(kg·К); d_1=–4,846·10^(-4) kJ/(kg·К^2 ) ; d_2=9,573·10^(-7) kJ/(kg·К^3 ); d_3=–4,173·10^(-10) kJ/(kg·К^4 ). 1st stage: calculation of the integral effect of nitrogen cooling in an ideal equilibrium expansion process, , K. The integral effect of nitrogen cooling of an ideal expansion process is determined by the equation: , (3.24) where – nitrogen temperature at the expander inlet, К; T2S – nitrogen temperature at the end of the adiabatic equilibrium expansion process, К. The temperature T2S is calculated based on the process condition X = S = const, according to the equation: (3.25) The zero approximation is given from the condition that the isothermal entropy deviation from the ideal gas state of nitrogen . The calculation of nitrogen density, isothermal deviations of enthalpy and entropy is carried out according to the Bogolyubov-Mayer equation in a truncated form. The calculation results are presented in Table 3.12. Table 3.12 Т1, K В1∙104, m3/kg , kg/m3 Z(0) Z , kg/m3 300 –1,778 5,616 0,999 0,999 5,622 –3,493 –1,137 is calculated by the formula (3.25), provided . Further approximations are found from the calculated relation (3.25) by the method of successive approximations. The value is chosen , based on the condition: The calculation results are presented in Table. 3.13. Table 3.13 , K , kg/m3 В∙103, m3/kg Z(0) Z , kg/m3 211,97 2,384 –1,024 0,997 0,997 2,391 – 0,3631 –3,348 –5,635 212,65 2,377 –1,0925 0,997 0,997 2,384 – 0,3597 –2,478 –1,355 = 212,65 K is chosen with accuracy The integral temperature effect of cooling of of an ideal equilibrium expansion process is calculated by formula (3.24). . 2nd stage: the power of the expander is determined under the conditions of the adiabatic equilibrium process of nitrogen expansion. The calculation of the power of the equilibrium adiabatic process of nitrogen expansion in the expander is carried out according to the relation: , (3.26) where – power of equilibrium adiabatic (isoentropic) expansion of nitrogen in the expander, kW; – difference between the enthalpy values of nitrogen in terms of parameters at the inlet and outlet of the expander in an ideal gas state, ; – difference between the isothermal deviations of the nitrogen enthalpy from the ideal gas state in terms of gas parameters at the inlet and outlet of the expander working zone (stage), . The calculation results are presented in Table. 3.14. Table 3.14 , K , K , kg/m3 , kg/m3 300 212,65 5,622 2,384 –91,287 –1,137 – 0,666 90,816 90,816 3rd stage: calculation of the integral effect of nitrogen cooling in real, i.e. non-equilibrium expansion process in the expander, , K. The determination of the integral effect of cooling is carried out according to the calculated ratio: , (3.27) where Т1 – the nitrogen temperature at the expander inlet, K; Т2 – the nitrogen temperature at the expander outlet during the nonequilibrium expansion process, K; – the integral cooling effect, K. The determination of the temperature value T2 is carried out by the method of successive approximation based on the relationship: , (3.28) where – the adiabatic efficiency of the expander ( = 0,83); – mass flow rate of nitrogen, kg/s; , , – isothermal deviation of the nitrogen enthalpy value from the ideal gas state in terms of the parameters at the inlet and outlet of the expander, ; The determination of the zero approximation of the temperature value is carried out according to the relation below, provided that =0. , (3.29) where Срид = 3,5 RM. Further calculations are carried out with accuracy: according to the ratio: , (3.30) where – the average heat capacity within the temperature range from to , . The results of calculating the temperature T2 at the outlet of the expander are presented in table 3.15. Table 3.15 , , , 300 226,34 2,237 1,0444 –1,137 –593,37 90,816 300 227,33 2,227 1,0444 –1,137 –588,98 90,816 The value Т2 = 227,33 K is selected with an accuracy of: . The integral effect of cooling is the following value according to the formula (3.27): . The actual power of the turboexpander in the real expansion process in the nominal operating mode is determined by the following ratio: (3.31) The calculation results are presented in Table 3.16. Table 3.16 , , , , , kW 300 227,33 –1,137 –588,98 –75,899 75,351 75,351 4th stage: calculation of kinetic energy losses due to friction in a non-equilibrium expansion process, i.e. evaluation of the viscous dissipation of the process , kW. The calculation of kinetic energy losses due to friction in a non-equilibrium process is carried out according to the equation obtained on the basis of the integral equation for the balance of kinetic and potential energy of a non-equilibrium process: , (3.32) where =75,351 kW – technical power removed in the actual process of nitrogen expansion, i.e. taking into account friction forces; – power of the polytropic equilibrium process, calculated by the formula: , (3.33) where n ̅= (ln P_2/P_1 )/(ln ρ_2/ρ_1 ) – average value of the polytropic index for the initial and final parameters characterizing the state of the gas at the inlet and outlet of the expander stage. The calculation results are presented in Table 3.17. Table 3.17 , bar , bar , kg/m3 , kg/m3 , kW , kW , kW 300 227,33 5 1,5 5,622 2,227 1,3 93,57 75,351 18,219 The viscous dissipation of the nonequilibrium expansion process , according to the calculated relation (3.32), is the following value: kW The polytropic equilibrium process of nitrogen expansion, carried out through the initial and final states, which completely coincide with the real non-equilibrium process, can be represented in the form of diagrams (PV), (TS) (fig. 16). а b Fig. 16. Polytropic equilibrium process of nitrogen expansion in PV (a), TS (b) diagrams. 5th stage: exergy analysis of the nitrogen expansion process in the expander. The calculation of the exergy efficiency of the nitrogen expansion process is based on the calculated ratio, excluding transit exergy flows: η_ex= ((Ex) ̇_вых- (Ex) ̇_транзит)/((Ex) ̇_вх- (Ex) ̇_транзит ), (3.34) where – exergy flux at the output of the expander stage; – exergy flux at the inlet to the expander stage; – transit flows of exergy, i.e. constituting that part of the exergy of incoming flows, which passes invariably through the entire apparatus. The nitrogen exergy flux at the expander inlet is calculated by the relation: , (3.35) where Тос = 298,15 K – temperature value in the reference state; – change in the enthalpy of nitrogen in the ideal gas state, ; – change in the entropy of an ideal gas depending on the change in temperature, ; – isothermal deviation of the nitrogen enthalpy from the ideal gas state according to the parameters at the expander inlet, ; – isothermal deviation of the entropy of nitrogen from the ideal gas state in terms of the parameters T1, P1, . The calculation results are presented in Table 3.18. Table 3.18 Т1, K Р1, Bar , , , , еx1, , W 300 5 1,928 6,448 –1,137 –3,439 141,16 141,16 The exergy flow at the outlet is the sum of the exergy flows due to the movement of nitrogen (〖Eх〗_2 ) ̇ and the flow of mechanical work . According to the exergy balance equation, the calculated ratio for the exergy of nitrogen at the outlet of the expander is obtained: , (3.36) where = 141,16 кВт – nitrogen exergy flow at the expander inlet, kW; – exergy loss due to irreversibility of the process, kW. The calculation formula for the exergy efficiency taking into account relations (3.34), takes the form: , (3.37) Exergy losses due to the irreversibility of the process are calculated using the Hui-Stodola formula: The results of calculations of the exergy efficiency of the process are presented in Table 3.19. Table 3.19 Т1, K Т2, K , , , , kW 300 227,33 –289,7 –3,493 –2,021 20,602 141,16 0,85 . To increase the exergy efficiency of the nitrogen expansion process, it is advisable to increase the efficiency of the expander, i.e. reduce exergy losses due to the irreversibility of the process itself. This is achieved by improving the gas dynamics of motion, namely by reducing friction losses in the gas path of the turboexpander. 3.2.3 Analysis of throttling processes. The problem of internal cooling can be solved using the throttling process. Throttling is an irreversible adiabatic process of reducing the pressure of a gas (steam) flow, when passing through a narrowed hole (throttle), and the working fluid does not perform external work. For such a process, the total energy of the flow remains unchanged. Let us consider this process during the outflow of a gas flow using the example of a diaphragm as a throttle device (Fig. 17). Fig. 17. Scheme of throttling the working flow along the channel profile In the figure, d1, d2 – the cross section of the channel before and after the diaphragm; d0 – diaphragm (throttle) section of the channel. Since throttling is carried out without external work W ̇ and without supply (removal) of heat flow Q ̇ when gas (steam) flows through local hydraulic resistance, then for sections (1–1) and (2–2) that are sufficiently remote from the throttle, where the values of the parameters can be assumed to be steady, is obtained from the total energy balance equation (2.2): the following equation: m ̇(h1 + (〖 〗_1^2)/2 + φ1) = m ̇(h2 + (〖 〗_2^2)/2 + φ2) (3.38) This equation is a special case of the integral total energy balance equation. Let's analyze the presented ratio. Since the mass flow rate of the working fluid in each section remains constant and the area of the flow section of the channel (1–1) does not change before and after local constriction (2–2), the flow velocities remain practically unchanged 〖 〗_(1 )≈ 〖 〗_2. Assuming the specific gravitational potential over the flow cross section to be constant φ_1 = φ_2, since the area of the control section is insignificant, the following integral throttling condition is obtained: h1=h2 (3.39) In essence, this is a non-equilibrium process of poorly organized outflow in non-profiled channels, when the kinetic energy of the expanded gas is not converted into external work, as happens in expanders, but is transformed due to turbulent eddies into dissipation energy ψ, which, in turn, is used to restore enthalpy. For analysis, it is advisable to replace such an outflow process with a quasi-static irreversible process that has the same integral result of the constancy of enthalpy (3.39). In this case, however, it becomes possible to apply the analytical apparatus of classical equilibrium thermodynamics. Task 3.5. Determine the integral effect of throttling ∆T12h, K of nitrogen during the outflow of the flow through the diaphragmatic constriction of the pipeline (Fig. 17). The process is irreversible, adiabatic. Nitrogen parameters in the section (1–1), before the throttle device: P1=5 bar, T1=300 К, nitrogen parameters in the cross section and (2–2), after the throttle device: P2=3 bar. The functions of the state of nitrogen in the section (1–1) should be determined by the truncated virial equation of Bogolyubov – Mayer. In the section (2–2) it is permissible to use the ideal gas model. Based on equation (3.39), taking into account the equation of state of the gas, the following expression is obtained: сp(T1 – T2h) + ∆h_1^∂- ∆h_2^∂ = 0 (3.40) The integral throttling effect is determined by the relation: ∆T12h = T1 – T2h , (3.41) where T1, К – temperature of nitrogen in the section (1–1), before entering the diaphragmatic constriction; T2h, К – the temperature of nitrogen in the section (2–2), after the diaphragmatic constriction, when the pipeline section becomes full again. The integral effect ∆T12h, К can be found on the basis of the thermal equation of state for a nonideal gas. For an ideal gas, the throttling effect is zero. The calculation of T2h should be carried out using equation (3.40) together with the truncated Bogolyubov-Mayer equation of state in virial form. From the above it follows: T2h = T1+ (∆h_1^∂- ∆h_2^∂)/C_p , (3.42) where ∆h_1^∂, ∆h_2^∂ – the isothermal deviation of the nitrogen enthalpy from the ideal gas state in terms of the parameters in the section (1–1) and (2–2), kJ/kg. The change in gas temperature during throttling, due to the deviation from the ideal gas state, is called the Joule-Thompson effect. To reduce the load on the numerical apparatus and gain qualitative experience in the analysis of the problem described, it is proposed to calculate nitrogen in the cross section (2–2) according to the ideal gas equation. Therefore, the temperature value T2h is determined by the following relation: T2h = T1 + (∆h_1^∂)/C_p (3.43) The results of the calculation are given in table 3.20. Table 3.20 T1, K В1∙104,m3/kg , kg/m3 Z(0) Z , kg/m3 C_p(T1), 300 –1,778 5,616 0,999 0,999 5,622 1,0425 –1,137 The value of the gas temperature T2h is found: T2h = 300 + (–1,137)/1,0425 = 298,9 К. The intensity of temperature change, called the differential Joule-Thompson effect, is characterized as 𝛼h = 〖(∂T/∂P)〗_h. The value of this characteristic is determined from the energy balance of the closed system dh = TdS – VdP together with the Maxwell equation 〖(∂S/∂P)〗_T= - 〖(∂V/∂T)〗_P: 𝛼h = (T(∂V/∂T)_p- V)/C_p = 𝛼S – V/C_p , (3.44) where 𝛼S – differential cooling effect for an ideal reversible gas (steam) expansion. Relationships for determining the magnitude of the differential and integral effects of gas cooling in the adiabatic equilibrium process of gas expansion from pressure P1 to the final value P2: 𝛼s = (k-1)/k·T_1/P_1 , (3.45а) ∆T_12S=T_1·[1-(P_2/P_1 )^((k-1)/k)] (3.45b) Attention should be paid to the dependence of the value of the adiabatic exponent k in relation (3.45) on the thermal equation of state of the gas. The calculation of this quantity for a non-ideal gas, considered in the example of the model of the truncated virial Bogolyubov – Mayer equation, is carried out according to the formula: k=(1+2Bρ)/(1+Bρ)·C_p/C_V , (3.4b) where В – the second virial coefficient. The calculation results are given in Table 3.21. Table 3.21 T1, K В1∙104, m3/kg , kg/m3 C_p(T1), k 𝛼s·104, К/Pa 𝛼h·106, К/Pa 300 –1,778 5,622 1,0425 1,3966 1,717 1,078 Since α_S>0 and V/C_p >0, then α_S>α_h, which is a consequence of the irreversibility of the process. Obviously, for an ideal gas, α_h = 0. If the gas is not ideal and the value α_SV/C_p , then the throttling effect will be positive α_h>0, i.e., the expansion of the gas will be accompanied by cooling. The state of matter in which the differential effect changes sign is called the point of inversion. The set of such points forms an inversion line, the equation of which is as follows: 〖T(∂V/∂T)〗_P=V (3.47) An analysis of this relationship shows that in the T–P coordinates the inversion line has a maximum and limits the range of T and P, where α_h>0 and gas cooling occurs. For any pressure P0 are exchanged through the surface dF by the elementary heat flux δQ. Considering the set of two interacting bodies as an isolated TDS, the entropy of which must increase during irreversible heat transfer, the growth of entropy within the process or the produced entropy is determined by the entropy balance equation (2.5): m ̇(S_1-S_2 )+inS ̇_12=0 Using thermodynamic laws and heat transfer equations, we obtain a relation for calculating the entropy produced (3.49): inS ̇_12=(K∆T_12^2 F)/(T_1 T_2 ) (3.49) where: K – heat transfer coefficient, Вт/(м^2 К); F – the area of the heat exchange surface. An analysis of relation (3.49) shows that the losses from irreversible heat transfer depend on the squared temperature difference and on the temperature zone of heat exchange. The smaller the product T_1 T_2, the more significant the effect of the temperature difference in low-temperature aggregates. Usually in refrigeration plants ∆T does not exceed 7-10 ℃, and even lower in the cryogenic area. Let us estimate the amount of losses arising due to the imperfection of thermal insulation. In this case, from the environment to the cold elements of the installation with a temperature T, a heat flow is supplied that does not contain exergy δQиз. Considering this process as an irreversible heat transfer at ∆T=(T_ср-T) and T_1=T_ср , we find 〖δQ〗_из=T_ср·δS_из^н=〖δQ〗_из (T_ср/T-1) (3.50) The lower the temperature of the machines and apparatus of the refrigeration system, the greater the loss of exergy, the more perfect the thermal insulation should be. Cryogenic systems use high-vacuum and vacuum-powder insulation with the lowest values of effective thermal conductivity.

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