Determination of the optimal heat exchanger configuration for wastewater heat recovery

. The work aims to increase the efficiency of the hot water supply system based on local recovery of the heat of wastewater generated in the shower room for preheating cold water. The work uses mathematical modeling of the thermal operation of the heat exchanger under study. Physical modeling of the heat exchange process between media flows in a heat exchanger was carried out (experimental test). Temperatures of media flows were measured. The temperature distribution inside media flows was compared experimentally with data obtained analytically. In conclusion, an analysis and generalization of the results obtained is made. The result of the research was a designed recovery heat exchanger. Data were obtained on the thermal inertia of the device, i.e., about the required time for the device to achieve a stationary thermal regime from the moment it is turned on. Data were obtained on the potential energy effect from introducing a recovery heat exchanger, taking into account its thermal inertia. The number of showers required to pay off the heat exchanger was calculated. Data were obtained on the influence of changes in the geometric and operating parameters of the heat exchanger on the efficiency of wastewater thermal energy utilization. A methodology for designing a heat exchanger for specific operating conditions was developed. The developed method for designing a recovery heat exchanger allows one to determine the optimal configuration of the device under particular operating conditions and mode of use of the heat exchanger, taking into account its thermal inertia. It is planned to continue the work by assessing the energy and economic effect of using local wastewater heat recovery within the heat supply system of a separate settlement/region.


Relevance
Currently, there is a trend in the world towards increasing the efficiency of use of fuel and energy resources.This goal is achieved by improving heat exchange processes [1,2] and apparatus, using thermal insulation to reduce heat losses [3,4], and recycling wastewater.
In the modern world, wastewater generated in showers is almost universally disposed of in sewer networks without any beneficial use.This coolant is relatively pure and has a high heat content [5].
In different countries, this issue has varying degrees of elaboration.Still, it is safe to say that finding ways to use such energy resources is a stage that all countries will go through as part of their individual energy transition [6].
Today, based on existing world experience, the issue of recycling the heat of total wastewater collected from an entire building, micro district, or settlement has been studied in most detail.This recycling method allows you to utilize the heat of a large volume of water.Still, during transportation to the collector, the wastewater cools down, and the thermal potential of the total water drops significantly [7].Also, such large-scale modernization projects or the construction of treatment facilities require serious capital expenditures and organizational work, which is only sometimes consistent with local authorities' current energy-saving policies.
To increase the efficiency of the recycling process, scientists are actively researching the possibility of using thermal energy directly at the water distribution device at the place of its use.Thus, it becomes possible to achieve the maximum available temperature difference between coolants and reduce thermal energy losses during transportation to almost zero [8].
The authors are investigating a method for recycling the heat of wastewater generated in the shower directly next to the tap to preheat cold water planned for use in the shower.A shell-and-tube heat exchanger (HE), part of the shower sewer system, is used for this task.With this method, the water used in the shower is removed from the bathtub (shower tray) first into a heat exchanger, in which heat is removed, and only then into the sewerage system [9].
This method can be used parallel with the centralized recovery of wastewater heat collected from an entire building/district/settlement.But when a private consumer of thermal energy needs to increase the efficiency of its use now, this method will enable him to immediately obtain a significant energy and environmental effect with a relatively short payback period [10].

Degree of scientific development of the topic
When analyzing the work of scientists dealing with the issue of local utilization of the heat of wastewater generated in the shower, it is clear that the case of domestic water is considered comprehensively and in great detail.There are installations put into operation.In their work, scientists Sły´s D. and Kordana S. think about the energy effect of recycling heat from wastewater in an economic sense [11].Novitskaya M.P. in publications he develops the idea of using a recycling device within a separate bathroom [12].The authors Redko A., Redko O., and DiPippo R. consider using low-temperature heat sources in the context of alternative energy sources in utilities [13].A team of authors Wong L.T., Mui K.W., and Guan Y., explores in their works the successful experience of introducing local wastewater heat recovery in residential buildings in Hong Kong [14].The study by Sitzenfrei R., Hillebrand S., and Rauch W. compares centralized and local wastewater heat recovery [15].Feike F., Oltmanns J., Dammel F., and Stephan P. consider the possibility of automating the operation of a heat pump in which the source of low-grade heat is wastewater [16].The team of authors Sadegh Shahmohammadi, Zoran Steinmann, Henry King, Hilde Hendrickx, and Mark A.J. Huijbregts provides a comprehensive study of people's behavior and habits in the context of showering, broken down by country around the world [5].
Analysis of published data on the research topic shows that developing and implementing energy-saving measures to increase the efficiency of hot water supply systems can provide significant energy, economic, and social benefits.

Goals and objectives, scientific novelty, theoretical and practical significance of the work
The main goal of the work is to increase the efficiency of the hot water supply system based on local wastewater heat recovery.
The object of study is a hot water supply system with an integrated heat exchanger that utilizes wastewater heat to preheat cold water.
The subject of the study is the dependence of the amount of usefully utilized thermal energy on the heat exchanger's geometric, thermophysical, and operational parameters.
To achieve the goal, we solved the following tasks: • analysis of problems leading to a decrease in the energy efficiency of heat supply systems; • development of a schematic diagram of a shell-and-tube type heat exchanger that utilizes the heat of sewer water generated when using a shower; • carrying out structural, thermal, and verification calculations of the heat exchanger; • development of a mathematical model of the thermal operation of a heat exchanger under stationary and non-stationary operating modes; • creation of a working sample of the HE (experimental installation); • verification of the mathematical model of the thermal operation of the recovery heat exchanger; • processing, analysis, and comparison of empirically obtained data on the thermal operation of the installation with data obtained based on a mathematical model; • study of the patterns of changes in the geometric and operating parameters of the heat exchanger on the efficiency of wastewater thermal energy utilization; • development, based on the obtained research results, of an engineering methodology for designing a heat exchange device for recycling heat from wastewater; • effectiveness assessment and technical and economic assessment of the use of recycling HE.
The theoretical significance of the work lies in the development and substantiation of a computational and experimental methodology based on the use of the developed mathematical model of the thermal work of a recovery heat exchanger, assessing the potential of the heat supply system available for the beneficial utilization of thermal energy.
The practical significance of the work lies in: • development of software for calculating the thermal work of a recovery heat exchanger; • development of a universal mobile bench installation, applicable to determine the power and thermal inertia of a heat exchanger that utilizes the heat of wastewater, depending on the operating mode of the water-dispensing device, heat exchange surface area, heat exchanger configuration and operating conditions; • development of recommendations for improving heat supply systems at various levels by integrating recycling heaters into hot water supply networks; • use the obtained research results for further work on heat recovery of a liquid medium using a heat exchanger without an intermediate coolant.

Device Description
The basis for designing the device is the idea of a shell-and-tube heat exchanger consisting of a housing and a bundle of metal corrugated pipes inside the housing.The hot water removed from the shower enters the heat exchanger body through an opening in the side.Water is removed through a hole located on the opposite side of the housing.Both spaces (for water supply and removal) are at the same height level.The movement of the heating medium inside the heat exchanger occurs without pressure.Water in the heat exchanger body is always at the height of the holes for its supply/removal [9].
Cold (heated) water flow is divided into several streams by a collector and moves through a bundle of pipes in a heat exchanger.The lines in the bundle are parallel in the heating water at the same height level.Heat exchange between the heating and heated medium occurs by heat transfer through the pipe wall.
The appearance of the heat exchanger under study is shown in Figure 1.The schematic diagram of the heat exchanger device is presented in Figure 2.  The device under study will be part of a separate shower installation.This method allows you to utilize the heat of wastewater during centralized or individual hot water supply.In any case, when using the heat exchanger under study, the result is heated cold water during a shower.With a centralized hot water supply, the effect of using a recovery heat exchanger will be observed in reducing the share of hot water required to prepare a unit of water at a comfortable temperature since the temperature of cold water in the water distribution device will be higher than without heat recovery [17].With individual hot water supply, the effect of energy-saving measures is expressed in a decrease in the required power of the water heater E3S Web of Conferences 458, 01024 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345801024(the required level of energy consumption by the water heater).A schematic diagram of a shower with a flow-through electric water heater after integrating this heat exchanger into the hot water supply system is shown in Figure 3. Cold water (heated medium) is heated in a heat exchanger and brought to the required temperature in the water heater.After use in the shower, the water enters a heat exchanger, transferring heat to cold water.After this, the heating medium is removed into the sewer network [18].

Methodology and research methods
The following scientific research methods are used in work: mathematical modeling of the thermal work of the studied heat exchanger in non-stationary and stationary operating modes, local physical modeling of the heat exchange process between media flows in a heat exchanger (experimental test), measurement of the temperature of the cold (heated) water flow at different spatial points in time, abstraction (simplification of the representation of the actual process of heat exchange between two media through the wall of the tube), comparison of the temperature distribution inside the cold water flow, obtained empirically, with the temperature distribution resulting from mathematical modeling, analysis and generalization of the results obtained.
The following materials are used in work: an experimental installation "Recovery heat exchanger" (Fig. 4), an electronic contact thermometer with a probe that has a valid verification certificate, a portable verification installation for checking water meters at the site of their operation or in other stationary conditions (during the experiment it was used for measurements of mass flow of media), having a valid verification certificate, a bathroom in a residential apartment building, utility networks for hot/cold water supply and sewerage networks in an apartment building.

Mathematical model
Based on mathematical modeling, it is necessary to obtain data on the temperature distribution inside the heating and heated medium flows over time under the given operating conditions of the heat exchanger.Thus, the amount of time required to achieve a stationary operating mode of the heat exchanger and the energy effect obtained at each moment of operation of the device will be determined.Of particular interest in the study is the issue of determining the thermal inertia of the device.The device will utilize heat as efficiently as possible only when it reaches a stationary mode.Until this moment, from the beginning of the device's operation, heat exchange will be observed in a non-stationary thermal mode with an ever-increasing value of the heat exchanger power.Since the average person's shower is limited in time and occurs within a specific time interval, it is essential to select the optimal configuration of the device for particular operating conditions.For this purpose, a highquality mathematical model is needed to determine the device's thermal inertia accurately.
At this stage, let us simplify the mathematical model and accept that the temperature of the media changes only along the spatial axis x and in time τ, and the thermal inertia of the tube wall can be neglected.The movement of the flow of the heated medium coincides with the direction of the spatial axis x.The parameters of this stream are indicated by the index "2".The flow of the heating medium moves against the direction of the x-axis, and its parameters are indicated by the index "1".
We will neglect the thermal inertia of the pipe wall and assume that at any moment in time, the amount of heat transferred from the heating medium is equal to the amount of heat received by the heated medium.
The entire volume of the heating medium (in the annulus) is considered a set of flows with cross-section S1, each containing a flow of heated water S2 (Fig. 2).
The form of the general equation of energy and continuity is known, valid for channels of any cross-section, constant along the length.We can write this equation for the case under consideration: where S -cross-sectional area of the medium flow corresponding to the design perimeter u, m 2 ; u -calculated perimeter of the channel, determined as u1 = 2π•d1 /2 for a channel with a heating liquid and u2 = 2π•d2/2 for a channel with a heated liquid, m; qc = k•(t1x − t2x) -power of external heat sources, W/m 2 ; k = 1/(1/α1+δ/λ+1/α2) -heat transfer coefficient, W/m 2 K; α1 and α2 -heat transfer coefficients of the heating and heated medium, respectively, W/m 2 K; δ -pipe wall thickness, m; t1x -temperature of the one-dimensional flow of the heating medium in the x coordinate, C; t2x -temperature of the one-dimensional flow of the heated medium in the x coordinate, C; i -average enthalpy of the liquid over the flow cross section in the x coordinate, J/kg; qv -power of internal heat sources, W/m 3 ;  -thermal conductivity coefficient of pipe material, W/mK;  -liquid density, kg/m 3 ; wx -projection of the medium flow velocity onto the x coordinate axis, m/s.
The third and fourth terms on the right side of equation ( 1) are equal to zero since the diffusion transfer of heat can be neglected in this case, and there are no internal sources of heat release.
Let us write the system of equation ( 2) based on equation (1), replacing the average velocities of the media with their mass flow rates: where S1 = h•a-(π•d1 2 /4) -cross-sectional area of the heating water flow, m 2 ; S2 = (π•d2 2 /4) -cross-sectional area of the heated water flow, m 2 ; d1 and d2 -outer and inner diameter of the pipe, respectively, m; a -distance between the axes of two adjacent pipes, m; h -highaltitude level of heating water inside the heat exchanger housing, m; c -isobaric heat capacity, J/kgK; w1x and w2x -average velocities of the media over the cross section of the flows, m/s; t1 and t2 -average temperatures over the flow cross section, C.
Let us write the final form of the equations for determining the temperature distribution inside the flows: () . x The result of the joint solution of the system of equations ( 3) is the distribution of temperatures inside the media flows over time.To solve this problem, we used the finite difference method (FDM).
As mentioned above, it is necessary to obtain, using mathematical modeling, graphs of temperature distribution inside media flows over time under different operating conditions.To do this, we write down the boundary and initial conditions for solving the system of equations (3) for various heat exchanger operating conditions.
As mentioned above, it is necessary to obtain, using mathematical modeling, graphs of temperature distribution inside media flows over time under different operating conditions.To do this, we write down the boundary and initial conditions for solving the system of equations (3) for various heat exchanger operating conditions.
After a long period of non-use of the shower, the temperature of the media inside the body and heat exchanger pipes will be equal to the air temperature in the room where the device is located [19].Since it is proposed to place the heat exchanger near the water distribution device, the temperature of the media at the initial time is assumed to be equal to the air temperature in the bathroom t0  20 C [20].Let us write the initial condition in the form: Boundary conditions in the context of this problem have a physical meaning of the temperature of the heated and heating water entering the heat exchanger.In this case, the temperature values of the heated and heating medium will depend on the time of year and the temperature of the water removed from the shower.
Let us write the boundary conditions in the form: where t -temperature of the heating medium entering the heat exchanger ( 40 C); ' 2 t -temperature of the heated medium entering the heat exchanger, C; l -length of steel tube with heated medium, m.

Verification of a mathematical model by experimental testing
To verify the mathematical model of the thermal work of the HE, an experimental setup was developed and created in physical form, the image and diagram of which are presented in Figures 4 and 5.  7 -probe of a contact digital thermometer one meter long; 8 -points for measuring the temperature of the cold (heated) water flow; 9 -HE body; 10 -housing of the electronic unit of a contact digital thermometer with a display and control buttons; w1 and w2 -the speeds of the heating and heated medium, respectively.
The experimental setup consists of a trench (9) (simulating a heat exchanger body) with a rectangular cross-section, inside which there are three stainless steel tubes of equal length (1).The heating medium enters the heat exchanger body and is removed from it through openings ( 6) and (3), respectively.The heated medium moves inside the middle tube.The tubes along the edges are filled with water but are motionless.These tubes in the experimental setup are necessary to accurately recreate the flow characteristics of the heating medium in the inter-pipe space.Modeling of media flows is carried out using a residential building's hot and cold water supply networks.
The temperature inside the cold water stream is measured using a contact electronic thermometer probe.The length of the probe is 1 meter; at the end of the probe there is a resistance thermometer.Inside a flow 1 meter long, 11 spatial points were selected to measure temperature, having coordinates x = 0; 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 1 m.By placing the end of the probe at a selected point (in the center of the cross-section of a round pipe), the water temperature was measured every 60 seconds during the first 10 minutes from the start of operation of the heat exchanger.
To measure the temperature at a spatial point over time, we carried out several experimental tests for each point.The gutter and steel pipes were initially filled with water in each experimental test.The temperature distribution inside both media flows at the initial moment was the same T1(x,) = T2(x,) = 18,5 °C, where T1(x,) -function of the temperature of the flow with the heating medium at a spatial point with coordinate x at the moment of time τ, and T2(x,) -temperature function of a flow with a heated medium at a spatial point with coordinate x at time τ.The mass flow rate of both media at the initial moment of time is zero.
The operating conditions and parameters of the heat exchanger under which its experimental testing and mathematical modeling were carried out are presented in Table 1.The largest relative error between the temperature value inside the flow, determined empirically and analytically, was δT(x,τ) = 6 % (Figure 6, x = 0.3m).The average value of the relative error at the studied time points τ was 3.5%.There is a uniform distribution of the relative error in time along the entire flow length, without clearly visible extrema or areas with relatively high/low function values.
Based on the obtained data (empirical and calculated) on the temperature distribution inside the flow over time, the power of the heat exchanger at moments of time τ was determined according to formula (7): where G -mass flow rate of the heated medium in the heat exchanger, kg/s; cp -heat capacity of water J/(kg•K); T2(1,τ) -temperature of heated water at the outlet of the heat exchanger at time τ, determined at point x = 1 m, C; T2(0,τ) -temperature of heated water at the inlet to the heat exchanger at time τ, determined at point x = 0 m, C.
The relative error of the values obtained using formula (7) (convergence of graphs) was determined using formula (8): where Qe() and Qa() -experimental and analytical value of thermal power at time τ, W.
Figure 10 shows graphs of the dependence of the power of the heat exchanger on time (experimental and calculated value) and the convergence of these graphs, expressed in relative error.The power of the heat exchanger, determined based on mathematical modeling, is less than the power value determined empirically throughout the entire operating time of the device under study.This fact in the context of the practical application of the mathematical model (when developing a methodology for designing a recovery heat exchanger for specific operating conditions) can lead to an artificial increase in the actual payback period of an energy-saving measure by the amount of error.
Research carried out on the experimental installation of a "Recovery heat exchanger" made it possible to verify the non-stationary mathematical model of the thermal operation of the recovery heat exchanger and confirm the correctness of the data reproduced by it.The verification criterion was the temperature distribution inside the flow of the heated medium, determined empirically and analytically.Under the given conditions, the maximum relative difference in temperature values δT(x,τ) = 6% was found at a spatial point with coordinate x = 0.3 m at time τ = 60 seconds (Figure 6).
This fact can be explained by the assumptions made when developing the mathematical model and by the increased complexity of the analytical calculation of the actual heat transfer coefficient of heating water in the annulus [21].When a conditionally stationary thermal regime occurs from time τ > 300 seconds, the discrepancy in temperature values does not exceed 5%, which can be accepted as an acceptable error for correct engineering calculations.
A verified mathematical model is the basis for developing software for calculating the thermal operation of a recovery heat exchanger and selecting the optimal device configuration for specific operating conditions.Since the operating conditions of the shower will be different for each consumer (duration of shower, mass flow of water, water temperature, etc.), in the matter of choosing the characteristics of the heat exchanger (heat exchange surface area, length of pipes in the bundle), we can strive to select only the optimal device configuration in each specific case, but not ideal [22].It is to study the influence of geometric, thermal, and design parameters on the efficiency of the heat exchanger under different operating conditions that a verified mathematical model of the thermal operation of the heat exchanger is needed.

Development of a methodology for designing a heat exchange device for specific operating conditions
Let's consider the issue of selecting the optimal heat exchanger configuration for specific operating conditions (heat exchanger design methods).Operating conditions are flow rate and temperature of water in the shower, duration of the shower, the temperature of cold water, physical space available to accommodate the heat exchanger and cost of thermal energy for heating water.
All these factors influence the energy effect of introducing energy-saving measures [23].The device under study utilizes heat only when the shower is running.From the moment of switching on, the device operates in a non-stationary thermal mode and only reaches maximum available power.
Let's simulate the temperature distribution inside the heat exchanger's media flows (heating and heating) within 14 minutes after the shower starts operating [5].Based on the resulting distribution, the power of the heat exchanger is determined at different times when the shower is operating, as well as the average power of the heat exchanger for the entire operating time of the device.
At the initial moment, the temperature of the heating and heating medium is the same along the entire length of the flows and is equal to the air temperature in the bathroom (20 ° C).For calculation, we assume that the average person's water temperature in the shower is 40 °C; it follows that wastewater also has a temperature of approximately 40 °C.The characteristics of the heat exchanger under study and its operating conditions are presented in Table 2. Based on the obtained temperature distribution in the flow of the heated medium over time, the power of the heat exchanger is determined at each moment when the shower is operating according to the formula (9).The calculation results are presented in Figure 11.Let's assume that an electric water heater is used to heat water in the shower.Then, the amount of thermal energy required to heat water with a temperature of 18.5 °C in a shower operating for 14 minutes is determined by the formula: where G -mass flow rate of the heated medium in the heat exchanger, kg/s; cp -heat capacity of water J/(kg•K); When using a heat exchanger, the amount of usefully recovered heat into the HE, used to heat cold water, is determined based on the data presented in Figure 11, using the formula: where Q(τ) -power of the heat exchanger at time τ for a given number of tubes (1÷12), W.
Let us determine the relative decrease in the required thermal energy for a single shower using the expression: where E -amount of usefully recovered heat in the heat exchanger during shower operation, J; ΔQ -amount of thermal energy required to heat water in the shower, J. Based on formula (12), the relative reduction in the required amount of thermal energy for a single intake is determined depending on the number of pipes in the heat exchanger.The calculation results are presented in Figure 12.Capital costs in conventional monetary units (c.m.u.) for energy-saving measures are determined by formula: where K1 -cost of materials for creating a heat exchanger, c.m.u.; K2 -cost of installation work for assembling and installing a recovery heat exchanger, c.m.u.; K3 -cost of transport services, c.m.u.; K4 -cost of commissioning works, c.m.u.
Capital costs were calculated based on average prices for countries part of the CIS.The results of calculating capital costs for energy-saving measures depending on the heat exchange area used (number of pipes) are presented in Figure 13.The number of showers required to recoup the capital costs of an energy-saving measure is determined by formula ( 14 where ΔQ -the amount of thermal energy required to heat water during a single shower use, J; E -amount of usefully recovered heat in the heat exchanger during a shower, J; Ccapital costs for energy-saving measures, c.m.u.; m -average cost of 1 J of electrical energy according to current tariffs for thermal energy in the European part of Russia.
The calculation results using formula 14 for different numbers of tubes used in the heat exchanger are presented in Figure 14.

Results and discussion
Heat extraction efficiency from wastewater directly depends on the heat exchange surface area.In the case under study, when only the number of tubes with the heated medium changes, as the number of tubes increases, the speed of the heated liquid in a single tube decreases.For this reason, the thermal inertia of the heat exchanger increases, that is, the time required for the device to achieve a conditionally stationary thermal regime.Also, this is due to the natural increase in the volume of water in the inter-tube space inside the heat exchanger housing, which takes more time to replace with hot wastewater.
An increase in the thermal inertia of the heat exchanger is observed in the figure: when using one tube with a heated medium, the device reaches a conditionally stationary thermal operating mode in approximately 2 minutes, and with a subsequent increase in the number of tubes, the duration of reaching the stationary mode increases.
It is essential to select the heat exchange area so that the heat exchanger operates in a stationary mode for a relatively large part of the total operating time of the shower.Also, the requirement should be specified that the energy-saving measure provides a significant effect within the framework of a separate one-time shower (after a long period of non-use of the shower) [24].This is because some objects consume hot water in the heat, where the shower works almost constantly or with short breaks (sports complexes, domestic premises, and industrial facilities).These objects should be considered separately, considering the specifics in each specific case, since if the operating mode of the object deviates from the design one, the heat exchangers will likely lose their potential for thermal energy utilization [25].
Based on formulas ( 9) -( 12), the relative decrease in the required thermal energy for taking a shower was determined.Figure 12 shows that the relative savings in thermal energy uniformly increase with increased heat exchange surface area.
The data shown in Fig. 12 and 13 make it possible to determine the minimum number of showers with given operating conditions for the payback of the energy-saving measure (Fig. 14).It is worth noting that when calculating the payback period, the cost of servicing the device after installation was not taken into account.It is more rational to replace the time required to pay for installing a heat exchanger with the number of showers.This will make calculating the economic effect for each specific case more accurate.
Fig. 14 highlights the area of the curve in which a further increase in the heat exchange area does not reduce the payback period (the area when using 9-12 tubes).
When analyzing the data in Fig. 14, it is difficult to determine the optimal number of tubes in the heat exchanger for a given case.This decision is complex and depends on many factors: • available funds for the implementation of energy-saving measures; • required energy, economic, and social effect; • available physical space to accommodate the recovery heat exchanger; • features of the policy implemented at the heat supply facility regarding modernizing existing systems.

Conclusions
In the study, a schematic diagram of a heat exchanger was developed for local utilization of the thermal energy of wastewater generated in the shower.The heat exchanger preheats cold water for use in the shower by removing heat from the wastewater in the shower.
A mathematical model of the thermal operation of a heat exchanger in unsteady and stationary modes has been developed and verified by experimental testing.The mathematical model makes it possible to obtain the temperature distribution inside the flows of the heated and heating medium over time under different device configurations and operating conditions.
Based on mathematical modeling, data were obtained on the dependence of the thermal inertia of the device on the used heat exchange surface area.The data obtained are necessary when calculating the energy, economic, and potential social effects of implementing energysaving measures.
An example of a methodology for determining the characteristics of a heat exchanger depending on the operating mode of the shower environmental and investment policies about a specific energy consumption object is presented.It is worth noting the fact that the purpose of this technique can only be to determine the optimal characteristics, and not the ideal ones, since in each specific case, the operating conditions of the shower may be different, and the selection of the device configuration must be carried out based on the individual parameters of the heat-consuming object.
Data were obtained on the potential energy effect from the implementation of energysaving measures on the number of showers required to recoup capital costs.
The data obtained makes it possible to conclude that when implementing this method of wastewater heat recovery near a water distribution device, it is possible to predict the achievement of a significant energy and economic effect.It is planned to continue the work by calculating the energy and economic effect of introducing waste heat exchangers in the hot water supply network within the heat supply system of a separate settlement/region [26].Also of particular interest is the issue of assessing the potential social effect of using this method of wastewater heat recovery in the form of intensification of settlement in areas with high prices for energy resources.

Fig. 1 .
Fig. 1.Heat exchange device model: 1 -tube with heated water; 2 -hole for supplying/removing heating medium from the HE body.

Fig. 2 .
Fig. 2. Diagram of the HE device with parameters for mathematical modeling of heat exchange between two media flows: 1 -steel tube with heated water; 2 -holes for supplying/removing the heating medium from the HE body; 3 -high-altitude level of heating water inside the HE body; d1 and d2 are the outer and inner diameters of the tube, respectively; S1 and S2 are the cross-sectional area of the flow of the heating and heated medium, respectively.

Fig. 3 .
Fig. 3. Shower diagram after integration of the recovery HE device: 1 -recovery heat exchanger; 2instantaneous electric water heater; 3 -shower installation (water dispensing device); 2 t  -temperature of heated water at the entrance to the HE; 2 t  -temperature of heated water at the outlet from the HE;

Fig. 5 .
Fig.5.Scheme of the experimental setup: 1 -corrugated stainless steel tube with diameter d1 with cold (heated) water; 2 -high-altitude level of the heating medium inside the HE housing; 3 -hole for removing heating water from the heat exchanger body; 4 -drainage networks of an apartment building; 5 -water supply networks of an apartment building, with the help of which the flows of the heating and heated medium are modeled; 6 -hole for supplying hot (heating water) to the HE inter-tube space;

Figures 6 -where 2 t
show graphs of temperature distribution inside the heated flow at different times, obtained empirically and based on mathematical modeling.Also, the figures show the value of the relative error δT(x,τ) between these temperature distributions, determined by formula (6):     

Fig. 10 .
Fig. 10.Dependence of heat exchanger power on time; 1 -experimental; 2 -calculation; 3 -error, %.Let's look at Figure10.The mathematical model predicts the achievement of a stationary thermal operating mode of the heat exchanger in 180-240 seconds.An experimental test achieves a stationary thermal mode in approximately 300 seconds.The largest discrepancy between the empirical and calculated value of the heat exchanger power was δQ(τ) = 5%.The average discrepancy within the first 10 minutes after switching on was 3.5%.

E3SFig. 11 .
Fig. 11.The power of the heat exchanger depending on time with a different number of tubes with a heated medium used; 1 -12 -number of tubes used.

' 1 t 2 t
= 40 -temperature of the water used in the shower (wastewater temperature), C; ' = 18,5 -cold water temperature at the water heater inlet, C; Δτ -shower duration, s.

Fig. 12 .
Fig. 12. Dependence of the relative decrease in the required heat for a single shower on the number of tubes.

Fig. 13 .
Fig. 13.Dependence of capital costs on a heat exchanger on the number of pipes.

Fig. 14 .
Fig. 14.The number of showers required to recoup the capital costs of implementing an energy-saving measure.

Table 1 .
Conditions and parameters of heat exchanger operation.

Table 2 .
Conditions and parameters of heat exchanger operation.