Harnessing Artificial Neural Networks for Financial Analysis of Investments in a Shower Heat Exchanger

Abstract

This study focused on assessing the financial efficiency of investing in a horizontal shower heat exchanger. The analysis was based on net present value (NPV). The research also examined the possibility of using artificial neural networks and SHapley Additive exPlanation (SHAP) analysis to assess the profitability of the investment and the significance of individual parameters affecting the NPV of the project related to installing the heat exchanger in buildings. Comprehensive research was conducted, considering a wide range of input parameters. As a result, 1,215,000 NPV values were obtained, ranging from EUR −1996.40 to EUR 36,933.83. Based on these values, artificial neural network models were generated, and the one exhibiting the highest accuracy in prediction was selected (R² ≈ 0.999, RMSE ≈ 57). SHAP analysis identified total daily shower length and initial energy price as key factors influencing the profitability of the shower heat exchanger. The least influential parameter was found to be the efficiency of the hot water heater. The research results can contribute to improving systems for assessing the profitability of investments in shower heat exchangers. The application of the developed model can also help in selecting appropriate technical parameters of the system to achieve maximum financial benefits.

1. Introduction

One of the most significant challenges facing contemporary generations is the need to meet the growing demand for energy while simultaneously caring for the environment and minimizing the negative impact of energy systems on the planet [1,2]. Population growth and ongoing urbanization are just some of the factors exacerbating this issue. Additionally, climate change forces communities to alter consumer behaviors [3] and take swift action aimed at reducing greenhouse gas emissions. It is, therefore, evident that the current heating systems should undergo a thorough transformation towards more ecological, sustainable, and efficient solutions [4,5]. However, meeting these challenges will not be possible without developing technologies based on renewable energy sources (RESs). This applies to both well-known RESs, such as solar, wind, and geothermal energy [6], as well as so-called third-generation renewable energy sources, for example, warm wastewater [7]. It is also essential to promote the effective use of energy through continuous public education on the conscious use of resources. In this aspect, the financial efficiency of individual solutions is particularly important [8], as implementing effective energy-saving measures can bring financial benefits to both individual energy consumers and entire communities.

Buildings are of particular importance in terms of the potential for reducing energy consumption and associated costs. As reported by the International Energy Agency (IEA) [9], the use of buildings accounts for up to 30% of global final energy consumption and approximately 26% of energy-related emissions. This should not be surprising, as Ratajczak et al. [10] noted, in some cases, people spend up to 90% of their lives indoors. It should also be noted that over the past few years, the components of the energy usage balance in buildings have been changing. As a result of technological progress and increased environmental awareness in society, buildings are better insulated and equipped with more efficient central heating, air conditioning, and lighting systems. Consequently, the share of domestic hot water (DHW) heating in the overall energy consumption balance in buildings has increased. In Poland, this share averages over 17% [11]. Research from other countries [12] indicates that it can be significantly higher. Much also depends on the type of building because, in passive buildings, the energy demand for preparing domestic hot water may significantly exceed the energy demand for space heating [13]. It is important to note that a significant volume of hot water consumption is used in bathtubs and showers. Some studies suggest that this can be as much as 70% [14]. Therefore, particular attention should be paid to solutions that guarantee a reduction in energy demand for heating water used for these purposes.

Such solutions undoubtedly include shower heat exchangers [15,16]. These devices allow for the recovery of part of the thermal energy present in greywater released from the shower. Warm greywater, flowing through the heat exchanger, transfers its carried thermal energy to the counter-flowing cold water. After preheating, it is then directed to the DHW heater and/or the shower mixing valve. However, the best energy benefits, and thus the highest financial savings, are achieved by preheating all the water used in the shower [17]. It should be noted that the use of greywater heat exchangers can enhance the comfort of using the shower installation by increasing the efficiency of the domestic hot water heater. This can result from both reducing the volumetric flow of heated water and raising the temperature of the water supplied to this device. Considering that ensuring comfortable conditions for using the shower installation with minimal energy consumption is an aspect that cannot be ignored [18], the use of shower heat exchangers seems not only beneficial but also necessary, especially since greywater is available all year round [19], even in the winter season, when energy demand is highest.

The financial efficiency of using a shower heat exchanger in a residential building depends on several factors determining the amount of energy consumption for DHW heating for showering. These primarily include the hot water flow rate and the duration of shower installation use [20]. The input and output water temperatures of the DHW heater, as well as the type and efficiency of this device, are also significant. Besides energy demand for heating water, potential financial savings resulting from the application of a shower heat exchanger are influenced by its effectiveness and unit energy prices. The unit’s effectiveness mainly depends on its type, as well as the functioning conditions of the shower installation and the connection method of preheated water pipes [21]. On the other hand, energy prices depend on the type of fuel used and the applicable tariffs for purchasing and distributing a given energy carrier. Over a longer period of heat recovery system use, geopolitical and economic situations should also be considered. Therefore, the results of the financial analyses conducted by various researchers differ significantly, and they draw contradictory conclusions. For example, Selimli and Eljetlawi [22] noted that the investment in such a device could pay off within two years. Other analyses [20] indicated, however, that depending on the conditions of the installation’s use, this period can be significantly extended. Considering the above, it seems necessary to conduct a thorough analysis identifying the key factors influencing the profitability of investing in a shower heat exchanger. Their relevance should be verified in the context of specific operating conditions of the shower installation and the risk associated with changes in market or operational conditions that may cause deviations in actual financial flows from forecasted values.

Considering all possible scenarios of the shower installation’s operation and determining the values of financial efficiency indicators for the full set of input data is, however, time-consuming and laborious. Consequently, financial analyses are often limited to strictly defined values of input parameters corresponding to typical operating conditions of the shower installation. Potential buyers and future users of such devices are deprived of knowledge about the solution’s profitability in case these conditions change. A partial solution to this problem is provided by sensitivity analysis, including scenario analysis, which can give a fuller picture of the investment’s profitability under different conditions [23]. However, even it does not guarantee a comprehensive consideration of all potential conditions and events that may affect the investment’s profitability. It is not always possible to predict future changes, establish precise values of parameters subject to change over time, or consider interactions between scenarios. To gain a fuller picture of the investment’s profitability in a shower heat exchanger and better understand the potential risks associated with it, more advanced techniques and tools are necessary. Such tools undoubtedly include artificial neural networks (ANNs). Thanks to the ANN models’ ability to process large amounts of data, detect patterns, and forecast outcomes, they can help investors make more accurate investment decisions. Their use can also help automate some calculations and eliminate potential human errors. Considering the above, the objectives of this article include the following:

• Evaluation of the financial efficiency of using a horizontal shower heat exchanger;
• Assessment of the possibility of using ANNs to evaluate the profitability of investing in a shower heat exchanger;
• Assessment of the significance of individual parameters influencing the financial analysis results.

This paper continues and expands on the research described by Starzec et al. [21]. The mentioned research [21], however, focused on the use of ANN models to forecast the effectiveness of the heat exchanger, while the research described in this paper relates to the investment profitability expressed by the net present value (NPV).

2. Materials and Methods

2.1. Research Steps

In the first stage of the analyses, a prototype of a horizontal shower heat exchanger installed at the greywater outlet underwent experimental testing. The analysis was carried out for the most energy-justified design variant of the heat recovery system, assuming that preheated water flows to both the DHW heater and the mixing valve. Both the research stand and the technical parameters of the selected shower heat exchanger were presented in detail by the authors in an earlier publication [21] concerning the assessment of the suitability of artificial neural networks for assessing the effectiveness (ε) of a shower heat exchanger. Based on the research results obtained and described by Starzec et al. [21], the temperatures of preheated water were also determined, which formed the basis of the research described in this study (Appendix A, Figure A1). Based on these, research was carried out on the financial efficiency of using a shower heat exchanger for a total of 1,215,000 combinations of input parameters.

Next, ANN models were developed, adopting ten input parameters and one output parameter in the form of the NPV value. Then, using the selected ANN model and the SHapley Additive exPlanation (SHAP) analysis, a hierarchy of significance of the adopted input parameters affecting the financial efficiency of using the horizontal heat exchanger was determined. For this purpose, Python programming language was used. As a result, it was possible to identify the most important parameters that have the greatest impact on financial benefits derived from the usage of this device. Figure 1 illustrates the logical flowchart of the research process.

2.2. Net Present Value

The financial analysis of the investment in a shower heat exchanger was based on the NPV values. Net present value is a key tool in investment analysis and project evaluation, which is why it is widely used in the analysis of projects involving the use of RESs [24,25]. This method allows one to assess the profitability of a project by taking into account future cash flows and discounting them to the current value, which is described by Equation (1).

$$NPV=\sum _{t=1}^n{\displaystyle \frac{{CF}_t}{{\left(1+r\right)}^t}}-{INV}_0,$$

(1)

where NPV is the net present value of a project, EUR; CF_t is the cash flows in year t, EUR; INV₀ is the initial investment outlay, EUR; r is the discount rate; and n is the system lifespan, years.

A positive NPV value indicates that under the given conditions of system operation, the financial benefits from implementing the project will exceed the costs. If the costs are too high compared to the forecast revenues and the NPV turns out to be negative, the investment will be unprofitable.

Regarding investing in a shower heat exchanger, the value of financial flows (CF_t) results from financial savings related to reducing energy consumption in the building. The annual decrease in energy consumption for heating DHW was estimated based on Equation (2).

$$∆EC={\displaystyle \frac{365·l_s·\left[q_{0h}·{\rho }_0·c_{p0}·\left(T_{hw}-T_{cw}\right)-q_{1h}·{\rho }_1·c_{p1}·\left(T_{hw}-T_{pw}\right)\right]}{\eta ·3.6·{10}^9}},$$

(2)

where ΔEC is the annual energy savings used to heat DHW, kWh (MWh); l_s is the total daily shower length, min (s); q_0h is the flow rate of DHW to the shower mixing valve in the variant without a shower heat exchanger, L/min (m³/s); q_1h is the flow rate of DHW to the shower mixing valve in the variant with a shower heat exchanger, L/min (m³/s); T_hw is the temperature of DHW, °C (K); T_cw is the temperature of cold water, °C (K); T_pw is the temperature of preheated water, °C (K); η is the efficiency of a DHW heater; ρ₀ and ρ₁ are the densities of water, kg/m³; and c_p0 and c_p1 are the specific heat capacities of water, J/(kg·K).

The value of the NPV indicator in the case of investing in a shower heat exchanger is influenced by a number of factors related to the conditions of using the shower installation, the costs of supplying the building with energy, and the assumed level of risk expressed in the value of the discount rate (r). The amount of initial investment outlay (INV₀) is also important. In order to create a database of NPV values, based on which artificial neural network models were generated in the next stage of research, 1,215,000 combinations of input parameters were identified and analyzed.

Table 1. Values of the input parameters.

Input Parameter	Unit	Values
Cold water temperature (Tcw)	°C	8, 14, 20
Greywater temperature (Tdw)	°C	30, 35, 40
Initial investment outlay (INV0)	EUR	400, 1200, 2000
Initial energy price (Ce)	EUR/kWh	0.02, 0.10, 0.18, 0.26, 0.34
Annual change in energy price (ie)	%	−2.5, 0, 2.5, 5, 7.5
Domestic hot water heater efficiency (η)	%	80, 90, 100
Linear bottom slope of the shower heat exchanger (i)	%	0, 0.33, 0.66, 1, 2, 2.5, 3.5, 4
Discount rate (r)	%	0, 2.5, 5, 7.5, 10
Mixed water flow rate (q)	L/min	3, 4.5, 6.5, 8.5, 10
Total daily shower length (ls)	min	10, 50, 90
System lifespan (n)	years	15

lists the values of the input parameters that were considered when creating the above combinations. These values have been selected to take into account different scenarios and conditions of using the shower installation and to reflect different levels of risk. Considering that among the parameters determining the effectiveness (ε) of the shower heat exchanger, which ranged from 19.20% to 35.44%, the most important is the mixed water flow rate (q) and the linear bottom slope of the shower heat exchanger (i) [21], the analysis included five and eight values of these parameters, respectively. The adoption of a larger number of bottom slope values (i) results from the fact that the impact of changing the value of this parameter on the effectiveness (ε) of the horizontal shower heat exchanger is less predictable and more diverse over the entire range of its values. On the other hand, due to the limited importance of cold water and greywater temperatures (T_cw and T_dw), it was decided to reduce the number of considered values of these parameters to three. As a result, the analysis of the financial efficiency of the investment in the shower heat exchanger was based on 360 preheated water temperatures (T_pw) corresponding to different operating conditions of the greywater heat recovery system (Appendix A/Figure A1). The selected values of the above parameters corresponded to the combinations that, in the previous stage of research, were characterized by the greatest fit to the ANN models [21]. The calculations assumed a constant DHW temperature (T_hw) of 55 °C. This value corresponds to the minimum DHW temperature at the outlet from the sanitary facilities in Polish conditions, which was specified in the Regulation of the Minister of Infrastructure and Development [26].

Additionally, the analysis took into account five different values of the initial energy price (C_e), the annual percentage change in the energy price (i_e), and the discount rate (r). In the case of parameters such as DHW heater efficiency (η), total daily shower length (l_s), and initial investment outlay (INV₀), it was decided to limit the amount of input variable values to three. In the case of the first of the mentioned parameters (η), this is due to the small range of its values. In the remaining cases (l_s, INV₀), the fact that these parameters change the value of the NPV indicator linearly was taken into account. Representative values of the above parameters were selected to consider the different number of system users and their various preferences as to the length of the shower, the varied scope of required installation works, different types of instantaneous DHW heaters (gas, electric), as well as different market trends.

2.3. Artificial Neural Networks

Artificial neural networks can be a useful tool in assessing the effectiveness of greywater heat recovery units. Thanks to the ability to process large amounts of data and identify complex patterns, ANNs can be used to analyze and model thermal processes occurring in these systems. Artificial neural networks can also be used to assess the financial efficiency of using shower heat exchangers. ANN models can predict energy savings and costs associated with installing and operating these systems. Using data on water consumption, energy costs, shower heat exchanger specifications, and operating conditions, ANNs can accurately assess the potential financial benefits of heat recovery in various scenarios. Such analyses can help both individual users and companies make informed investment decisions while ensuring optimization of expenses and maximizing the return on investment in greywater heat recovery technologies.

Artificial neural networks are inspired by the biological structures of the brain, which consist of neurons connected by synapses. ANNs are complex mathematical and computer structures that consist of many layers of artificial neurons capable of learning and processing information. They are widely used in various fields, such as image recognition [27], financial forecasting [28], and in the field of technical sciences [29]. The process of creating an artificial neural network begins with the definition of the problem, i.e., determining the purpose and scope of application of the neural network, for example, whether the network is to recognize images, predict the values of a given parameter, classify data, etc. The next step is to collect data (e.g., laboratory tests) that will be used to learn the ANN models. Next comes data preparation, where data cleanliness and normalization are crucial. The data must be in the right format, and the values must be normalized for the network to learn effectively. The next step is to choose the network architecture, which includes deciding on the number of layers, the number of neurons in each layer, and the type of activation function. Once the architecture is defined, the network training process begins, which involves adjusting neuron weights based on training data using optimization algorithms such as backpropagation. After training, there is a validation and testing stage, where the model is tested on a set of data that was not used during training to assess its performance and generalizability. Finally, the model is put into real use, where it can process new data and provide predictions or classifications according to its intended purpose [30].

In this study, the input data set comprised a total of ten parameters determining the effectiveness of the horizontal shower heat exchanger and the profitability of its application in a residential building (Table 1), assuming a 15-year system lifespan (n). Based on the flow rate (q), the slope of the unit (i), and the temperatures of cold and greywater (T_cw and T_dw), the values of the preheated water temperature (T_pw) were determined through experimental studies. The experimental results of the heat exchanger conducted at the Laboratory of Measurement Techniques and Water and Wastewater Transport Control at Rzeszow University of Technology can be found in the publication by Starzec et al. [21]. The measured T_pw values were then used to calculate the NPV indicator, which was the output variable of the model. The calculated NPV values are presented in Section 3.1. The normalized input data formed the basis for training MultiLayer Perceptron (MLP) neural networks. To ensure the accuracy and reliability of the model, the data set was divided into three parts: training set, validation set, and test set. The training set accounted for 70% of the total data, and the validation and testing sets accounted for 15% each. This division allows for appropriate tuning of the model and its subsequent evaluation of previously unknown data, which allows for a more objective assessment of its effectiveness. The data were divided randomly, which enabled effective verification of prediction results [31]. This approach allows for an easy assessment of how well the model generalizes to unseen data. If it performs well on the training, validation, and test sets, it indicates that it can make accurate predictions based on new data. Therefore, the model that provided the best fit of computational data with the model’s predictions, as well as the most favorable values of model evaluation metrics across all three data sets, was chosen.

The developed artificial neural network models were evaluated using the root mean square error (RMSE) and coefficient of determination (R²). The RMSE measures the average size of the model’s prediction errors. The lower the RMSE value, the better the model fit. The value of the RMSE index is determined according to Equation (3). The R² index measures how well observed outcomes are predicted by the model. Its value ranges from 0 to 1, where 1 means a perfect fit of the model to the data. Equation (4) defines the coefficient of determination (R²) [32].

$$RMSE=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle {\sum }_{i=1}^m}{\left(y_i-{\widehat{y}}_i\right)}^2},$$

(3)

$$R^2=1-{\displaystyle \frac{\sum {\left(y_i-{\widehat{y}}_i\right)}^2}{\sum {\left(y_i-\overline{y}\right)}^2}},$$

(4)

where m is the number of data sets, y_i is the measured value, ${\widehat{y}}_i$ is the predicted value, and $\overline{y}$, is the mean value of data set.

2.4. SHapley Additive exPlanations

SHAP analysis seems to be a valuable tool for assessing the financial efficiency of using shower heat exchangers. Thanks to this method, it is possible to understand in detail how particular features influence the model’s predictions [33], which is crucial when assessing the costs and benefits associated with the implementation of this type of technology. SHAP analysis allows for the identification of key factors influencing energy and financial savings, which enables more accurate investment planning and optimization of installation processes. Thanks to this tool, investors and decision-makers can make more informed decisions, minimizing risks and maximizing economic benefits from the use of shower heat exchangers.

SHAP analysis is a tool for explaining the results of complex machine learning models. SHAP is based on the concept of Shapley values from game theory, which is used to separate the output of a model into the contributions of individual features. In the context of machine learning, these values represent the contribution of each input feature to the model’s prediction. Thanks to this, SHAP analysis allows for the interpretation of the performance of models that are often perceived as “black boxes” due to their complexity.

The basis of SHAP analysis is the calculation of the Shapley value for each feature. These values measure how adding a given feature to all possible combinations of other features affects the model result. In practice, SHAP algorithms calculate these values by simulating all possible combinations of features and their impact on the result. In this way, SHAP values provide global and local explanations of the model’s performance. Global explanations show which features are most important overall, and local explanations help understand how particular features affect specific predictions.

One of the key benefits of SHAP analysis is its universality and accuracy. This method is applicable to a wide range of models, including decision trees, artificial neural networks, and linear models. Moreover, SHAP values are intuitive: the sum of individual feature contributions plus the “base value” equals the model output, which makes the interpretations easy to understand even for people who are not machine learning experts.

SHAP analysis is widely used in various fields where machine learning is used. SHAP analysis is used in power engineering [34], construction [35], environmental engineering [36], socio-economic sciences [37], and many other areas where understanding how predictive models work is crucial to making informed decisions.

3. Results

3.1. Net Present Value

Based on the data presented in Table 1, 1,215,000 NPV values were determined, which ranged from EUR −1996.40 to EUR 36,933.83 (Figure 2). Such a large dispersion of results indicates significant uncertainty as to expected future financial flows. Changes in the forecast values of input parameters, both those related to the costs of supplying energy to the property and those characterizing water consumption for showering, may significantly affect the outcomes of the financial analysis. This confirms the validity of analyzing the profitability of a project in a wide range of input data for effective strategic planning and risk management. The statistical analysis of the results showed that for the entire studied data set, the median net present value was EUR −90.70. Therefore, it is clear that in the case of most of the combinations of input parameters considered, the project would turn out to be unprofitable. This results from both the relatively low effectiveness of the heat exchanger and the wide range of adopted values of input parameters, including extremely unfavorable conditions of use of the heat recovery system. For a more complete picture of the distribution of NPV values, quartile values were also determined. The first quartile was equal to EUR −965.60, which means that lower NPV values were obtained for 25% of the parameter combinations. On the other hand, the third quartile was equal to EUR 1433.33. This shows that in 25% of cases, higher net present values were obtained. The distribution of NPV values is, therefore, right-skewed. This suggests that despite the predominance of cases for which NPV < 0, there are a significant number of scenarios with very high profitability, which significantly increases the predicted average NPV value. This distribution suggests a high level of risk while indicating the possibility of achieving significant financial profits under appropriate conditions of the system’s operation.

To thoroughly comprehend the impact of individual input variables on the profitability of the project and the dispersion of the obtained results, Figure 3 shows the distribution of the NPV values depending on the adopted values of these parameters. The analysis of individual box-and-whisker plots, prepared on the basis of 12.50–33.33% of all results (depending on the number of considered values of a given parameter), confirms that the median NPV value is close to zero in many cases. However, there are exceptions indicating a significant impact of selected parameters on the profitability of the project. Such exceptions undoubtedly include initial energy price (C_e), total daily shower length (l_s), and initial investment outlay (INV₀). The research analyzed initial energy prices ranging from 0.02 to 0.34 EUR/kWh. Adopting such a wide range of these parameters allows for the estimation of the profitability of the project with respect to various energy carriers. Analyzing Figure 3d, it can be seen that in the case of the lowest C_e values, the investment would be unprofitable in the vast majority of the considered scenarios because the third quartile turned out to be negative. For the next value considered (C_e = 0.10 EUR/kWh), the median NPV still remained negative. It is worth noting that this price is slightly higher than the current frozen price of natural gas in Poland. It can, therefore, be concluded that the implementation of a shower heat exchanger in a system where DHW is prepared using a gas water heater is unfavorable in financial terms. However, such low energy prices are unlikely over an extended period, which creates an opportunity to increase the financial efficiency of using this device. An increase in the initial energy price is equivalent to an increase in the value of the NPV. Already at C_e = 0.18 EUR/kWh, the median of the results becomes positive, and at the highest C_e value considered, almost the entire box is above the zero value. In the case of Poland, higher prices refer to electricity prices. It can, therefore, be concluded that shower heat exchangers will be a much more attractive option in the case of installations equipped with an electric DHW heater. In some cases, it will still be necessary to consider the risk of loss, but as the initial energy price increases, this risk will decrease while the potential return on investment simultaneously increases. An increase in the initial energy price additionally results in greater dispersion of results, especially the highest NPV values. This proves that the net present value is significantly sensitive to this parameter. Even small changes in the initial energy price can lead to significant changes in the projected NPV values. On the other hand, it confirms that there is the potential to achieve very high profits if the conditions of use of the installation and market conditions are appropriate.

Total daily shower length (l_s) is also important (Figure 3h). The lowest value of the l_s corresponds to the situation when the shower is used by one person or two people who prefer short showers. In such conditions, water consumption is low, even in the case of shower heads with a high flow rate of mixed water. This causes the investment to have low financial efficiency. In over 75% of the considered scenarios, the project turned out to be unprofitable. It can, therefore, be concluded that in the case of single-person households, investing in a shower heat exchanger will be financially unprofitable. Increasing the time of water consumption from the shower head to 50 min per day resulted in the median of the results reaching a positive value. This means that in the case of several-person households, investing in a horizontal shower heat exchanger may prove profitable, especially when the DHW is heated with an electric water heater. Increasing the total daily shower length to 90 min means that approximately 75% of all scenarios guarantee profits from the implementation of the installation. The implementation of the analyzed shower heat exchanger will, therefore, be particularly beneficial for people who prefer long showers. It is also worth considering the possibility of installing a common heat exchanger on the greywater outflow from two neighboring apartments. Favorable financial effects could also be achieved if the unit was installed in buildings with higher water consumption than residential buildings, for example, sports facilities, health care facilities, campsites, dormitories, prisons, etc.

Clear differences in the determined NPV values and their distributions are also visible in the case of the initial investment outlay (INV₀). In the case of a relatively cheap heat exchanger, the initial investment outlay varies depending on the scope of required installation and renovation work. If the heat exchanger has been considered at the building design stage or is to be located in the immediate vicinity of the shower and the DHW heater, these costs will not be excessive. However, if the installation of the heat exchanger requires significant modifications to the internal water supply and sewage system as well as related construction and installation works, the initial investment outlay may increase significantly, and the costs of installing the unit may even exceed the cost of its purchase several times. In the case of the lowest investment costs considered (INV₀ = EUR 400), most scenarios turned out to be profitable (68.40%). Additionally, even in the event of extremely unfavorable circumstances, the potential financial losses will not be significant. An increase in the amount of initial investment outlay results in a clear reduction in the profitability of the investment. Even with expenditures of EUR 1200, more than half of the considered scenarios turned out to be financially unfavorable (55.44%). In the case of an increase in INV₀ to EUR 2000, approximately 67.45% of the parameter combinations were considered unfavorable. It follows that the implementation of greywater heat exchangers in Polish conditions will require subsidies, tax reliefs, or other forms of financial support for investments in solutions that increase the energy efficiency of buildings.

Upon reviewing the distributions illustrated in Figure 3, it becomes clear that the remaining input parameters have a much lower impact on the results of the investment profitability analysis. It is clearly visible that in the case of a decrease in the bottom linear slope of the heat exchanger (i), the flow rate of water and greywater through the unit (q), the annual change in the energy price (i_e), and the greywater temperature (T_dw), the directions of changes in these parameters are consistent with the direction of changes in the NPV. On the other hand, in the case of parameters such as the discount rate (r), cold water temperature (T_cw), and DHW heater efficiency (η), these directions are opposite. To precisely determine the influence of parameter changes on the analysis outcomes, it is necessary to use more advanced tools. For this reason, artificial neural networks were used in the next stage of research.

3.2. Artificial Neural Networks

To evaluate the feasibility of using machine learning methods to assess the financial efficiency of using a horizontal shower heat exchanger, artificial neural network models were generated. To evaluate these models, the RMSE and R² coefficients were used, which were determined in accordance with Equations (3) and (4). These indicators were computed for the training, validation, and test datasets.

During the research process, several neural network models were developed and tested. Among them, the model that was selected showed optimal values of the RMSE and R² indicators, which proved its highest efficiency. The values of the RMSE and R² indices for the selected model are summarized in

Table 2. Values of R² and RMSE indicators for the selected artificial neural network model.

ANN Model Architecture	Training Set		Validation Set		Testing Set
	R2	RMSE	R2	RMSE	R2	RMSE
10-15-9-11-1	0.999	56.745	0.999	57.205	0.999	56.981

.

The selected artificial neural network model achieves a coefficient of determination (R²) close to 1.0 and a low root mean square error (RMSE) for all three data sets. In practical terms, this implies that the model explains the variability of the data very well, and almost all the data are accurately fitted by the model (Figure 4). When both of these metrics are consistent across all data sets, we can conclude that the model is highly accurate in prediction. Furthermore, the lack of significant differences in the results for the training, validation, and testing sets suggests that the model is not overfitted. Overfitting usually leads to high R² on the training set but low on the testing set. The analysis results prove that the developed ANN model is effective in predicting data and can be used with a high level of confidence.

3.3. SHAP Analysis

In order to explain the impact of individual input parameters on the financial efficiency of the project, a SHAP analysis was conducted. The results are presented in Figure 5. Figure 5a presents global feature importance, and Figure 5b presents a local explanation summary. Global SHAP analysis aims to understand the overall behavior of the model, while local analysis focuses on understanding individual predictions. The Y-axis in Figure 5b shows the individual input parameters whose impact on the NPV value was analyzed, while the X-axis shows the SHAP values. The order of the input parameters along the vertical axis corresponds to the impact on the profitability of investing in a horizontal shower heat exchanger. The legend attached to the chart indicates the value of the input parameter in a given observation in relation to the entire range of its values adopted for the analysis. Each point corresponds to a single analyzed case, with the points on the right side indicating a positive impact on the NPV value, while the points on the left side represent a negative impact on the analysis results.

The SHAP analysis confirmed that in the case of the considered shower heat exchanger, total daily shower length (l_s) and initial energy price (C_e) are of key importance for the profitability of the project. This is evidenced by the average SHAP values, which for both of these parameters exceed EUR 800. Slightly higher values were obtained for the first parameter. It should be noted, however, that the annual change in energy price (i_e) will also have an impact on energy prices in the subsequent years of system operation. Although this parameter ranks only fifth in the hierarchy of importance of input parameters, the sum of average SHAP values for C_e and i_e will exceed EUR 1200. Therefore, it can be concluded that energy prices throughout the entire operational period will be of key importance for the achieved NPV values. Initial energy prices and their annual changes are related primarily to the type of basic energy carrier, as well as production and distribution costs, geopolitical conditions, and implemented government policies. For this reason, the course of changes in energy prices over the entire operational period is difficult to predict. Although attempts are made to create advanced models for forecasting future energy prices, these forecasts are always subject to a certain degree of uncertainty. The average user of a shower heat exchanger is not able to predict the dynamics of these changes on his own. For this reason, analyzing a wide range of potential energy prices is crucial for assessing the profitability of investing in a shower heat exchanger. The analysis should also focus on various total daily shower lengths (l_s), as this input parameter also has a considerable impact on the financial viability of the project. The habits of system users should be analyzed, and potential changes in their numbers in the future should be considered.

Also, analyzing individual cases, it can be seen that omitting the l_s and C_e parameters results in the highest SHAP values. For example, the maximum difference between the forecasted net present value considering ten input variables and the forecasted NPV without taking into account the total daily shower length (l_s) amounted to EUR 9226.15. In the case of the initial energy price (C_e), it was EUR 7349.38. In both cases, the directions of changes in the input parameters and the net present value are consistent, which means that an increase in the value of each of them increases the profitability of the project. An analogous pattern was also observed in the case of a mixed water flow rate (q), annual change in energy price (i_e), greywater temperature (T_dw), and linear bottom slope of the heat exchanger (i). In the case of the latter, however, the influence on the investment’s economic viability is negligible. The average SHAP value does not reach EUR 150, and the maximum value for individual cases slightly exceeds EUR 1000. This is surprising because, in the previous stage of the analysis, the course of the impact of changes in this parameter on the effectiveness of the heat exchanger turned out to be the most diverse and least predictable of all the considered input parameters [21]. However, in the case of estimating financial savings, parameters that were not associated with the effectiveness of the shower heat exchanger turned out to be more significant. In addition to the linear bottom slope (i), cold water and greywater temperatures were also relatively low in the hierarchy. The only exception is the mixed water flow rate, for which SHAP values range from EUR −3064.84 to EUR 3556.30. This is probably due to the fact that the flow rate of water and greywater through the heat exchanger determines the amount of heat that can be recovered. A high flow rate (q) can increase the heat transfer rate of the heat exchanger, leading to greater energy savings and, consequently, increased financial savings and NPV value. For this reason, an increase in the value of q results in an increase in the net present value, even though the effectiveness of the heat exchanger is reduced in such a situation.

The parameters whose directions of change are opposite to the direction of changes in the NPV indicator include the initial investment outlay (INV₀). The average SHAP value, in this case, is close to EUR 600, which places this parameter in third place in the hierarchy of influence on the analysis results and confirms the need for co-financing the investment in a shower heat exchanger. Potential buyers should also look for savings in the costs of installing a heat recovery unit, e.g., by selecting a location that will limit the scope of required installation work. The parameters that had a negative impact on the model result also included the discount rate (r) and cold water temperature (T_cw), for which the average strength of the influence turned out to be similar. These parameters ranked sixth and seventh in the hierarchy, respectively. The last position went to the efficiency of the hot water heater (η). Also, in this case, an increase in the value of the input parameter results in a decrease in the value of the NPV. However, its influence on the model prediction result is negligible, which may be due to the fact that the efficiency of hot water heaters is relatively high in the entire adopted range of values of this parameter.

This study additionally analyzed local SHAP values for three characteristic cases (two extreme cases and the central one). The SHAP values for these individual observations are presented in Figure 6. For the parameter configuration shown in Figure 6a, the sum of the SHAP values is EUR 2820.76 with the calculated and predicted NPV values of EUR −1996.40 and EUR −2024.82, respectively. The initial investment outlay (INV₀), which reduces the NPV value in relation to the base value of EUR 795.94 by as much as EUR 906.81, has the greatest impact on the model’s predictions. It is worth noting that INV₀ takes the highest considered value in this configuration. On the contrary, total daily shower length (l_s) and initial energy price (C_e), which also have a major influence on the prediction results, have the lowest values in the analyzed ranges. The remaining seven input parameters are responsible for reducing the base value by a total of EUR 577.81.

Figure 6b refers to the median of the obtained SHAP values. In this case, the forecast NPV is EUR 94.91, and the parameter that has a key impact on this result is total daily shower length (l_s). Adopting the lowest l_s value reduces the base value by almost EUR 1250. The low flow rate (q), high cold water temperature (T_cw), and no changes in the energy price during system operation compared to the initial price also have a negative impact on the NPV prediction result. On the other hand, adopting a relatively low initial investment outlay (INV₀) results in an increase in the base value by over EUR 700. A positive, although less significant, impact on the NPV value is also demonstrated by the lack of changes in the value of money over time, high greywater temperature, high initial energy price, slightly exceeding the current price of electricity for households in Poland, and low efficiency of the DHW heater. The arrangement of the heat exchanger with a slight slope in the direction of greywater flow also has an impact, but it is negligible.

The last figure (Figure 6c) concerns the extremely favorable case where all input parameters contribute to an increase in the predicted net present value with respect to the base value. The hierarchy of significance of individual parameters in terms of the strength of their impact on the subject of analysis differs significantly from that presented for previous observations. Although the very long total daily shower length (l_s) and the high initial energy price (C_e) have the highest impact on the subject of the analysis, the discount rate (r) also ranks relatively high. Adopting a zero value of this parameter results in no reduction in the value of financial flows (savings) in the subsequent years of operation compared to the current year. A high annual change in energy price (i_e) and a high flow rate of media through the unit (q) guarantee an increase in the base value by over EUR 3000. Next in the ranking were cold water and greywater temperatures (T_cw and T_dw), linear bottom slope (i), and DHW heater efficiency (η). It may seem surprising that the initial investment outlay (INV₀) has the smallest impact on the model prediction in this case. This is probably because, due to significant water consumption for showering and high energy prices, the potential financial savings are so high that the low value of the initial investment outlay loses its importance.

Analyzing the selected observations, it was noticed that the strength of the impact of individual input variables on the NPV prediction result may differ significantly from the average values presented in Figure 5a. These differences may be due to the unique conditions prevailing in each case. In a situation where the projected financial savings are very high, the importance of initial investment outlay will not be so significant. However, if the building has low water consumption and energy prices are low, as is the case with the use of gas hot water heaters, even a low investment outlay may turn out to be a key factor determining the financial efficiency of the project. The average SHAP values shown in Figure 5a only provide a general picture. However, if you want to better adapt the recommendations to specific conditions, an individual analysis should be carried out.

4. Discussion

4.1. Investment Profitability

Considering the current geopolitical situation, potential energy crises, as well as ongoing climate change, saving energy in buildings should be a priority for both local government officials and ordinary citizens. Meanwhile, as noted by Su et al. [38], people prefer to place greater emphasis on issues related to building construction, while rising energy costs and the instability of energy supplies require urgent actions to reduce dependence on fossil fuels. The introduction of technologies that reduce energy consumption for water heating, such as shower heat exchangers, can significantly reduce energy consumption in households, contributing to improved energy security. Considering that financial and environmental issues are interconnected and mutually reinforcing [39], it can be assumed that investments in greywater heat recovery technologies can support both household budgets and sustainable development. The assessment of the financial efficiency of investments is particularly important, as noted by Ober et al. [40], because some user groups place less emphasis on pro-environmental actions. In such cases, the profitability of the investment project becomes a key aspect of its evaluation.

For the considered horizontal heat exchanger, the net present value ranges in a relatively wide range from EUR −1996.40 to EUR 36,933.83, indicating significant uncertainty regarding future cash flows. The analysis showed that more than half of the calculated NPV values were negative, which is an unfavorable result compared to other types of shower heat exchangers [17]. However, even for this heat exchanger, there is a significant number of scenarios with very high profitability. With the highest initial investment outlay (INV₀ = EUR 2000), the discounted payback period for the most profitable cases does not exceed two years. With lower INV₀ values, it may be shorter than a year. Therefore, it is crucial to have a thorough understanding of the parameters influencing the NPV value and to manage them in a way that maximizes potential benefits and minimizes risks when assessing the financial efficiency of such an initiative. A comprehensive evaluation of the influence of these parameters on the profitability of the project can help balance uncertainties regarding the outcomes. It can also ensure achieving stable and profitable results.

The results of the analyses evaluating the efficiency of investments in horizontal shower heat exchangers showed that not all parameters considered in calculating the NPV have a notable impact on the project’s financial viability. The SHAP analysis proved that statistically, the total daily shower length and the initial energy price have the greatest impact on the NPV value. It is also worth noting that these parameters were significant in both global and local analysis, and their impact on the prediction result was consistently positive. This means that as their values increase, the profitability of the investment also increases. Increasing the total daily shower length leads to higher water consumption, which increases the energy demand for its heating, thereby potentially enhancing energy savings through the application of a shower heat exchanger. Similarly, a higher energy price enhances the profitability of investments in energy-efficient technologies, resulting in increased savings and higher NPV values. Additionally, although the annual change in energy price ranks fifth in the hierarchy of input parameter significance, the sum of average SHAP values for C_e and i_e exceeds EUR 1200. This indicates the crucial importance of energy prices and, therefore, the type of DHW heater used throughout the entire life cycle of the shower heat exchanger.

The average user of a shower heat exchanger does not influence energy prices and is unable to independently predict the dynamics of their changes. However, the implementation of technologies based on renewable energy sources can reduce dependence on conventional energy sources and minimize the risk associated with fluctuations in their prices. On the other hand, it is different for parameters characterizing the use of the shower installation, including l_s. As mentioned earlier, the longer the total daily shower length, the higher the water consumption in the installation, and consequently, the greater the financial savings. An increase in this parameter can result, for example, from an increase in the number of system users [13]. In contrast, in the opposite situation, the NPV value would decrease. A similar issue can arise when the flow rate of mixed water from the showerhead is reduced. While the average impact of q on predicting NPV values is significantly lower than l_s, in some cases, it can still be very significant, as confirmed by the analysis of individual observations. Reducing energy consumption for heating water due to a lower flow rate from the showerhead can result from installing flow limiters or replacing showerheads with water-saving models. In extreme cases, it can also result from a reduction in the available pressure in the water supply network. Research conducted in Austria [41] indicated that actual energy consumption for water heating can be significantly lower than the design values. Therefore, during the planning stage of investing in a shower heat exchanger, it is crucial to take into account the investment risk related to reducing water consumption in the shower installation, especially since minimizing the use of potable water in buildings is a component of the circular economy model [42].

The SHAP analysis also highlighted the significant importance of initial investment outlay, which is one of the main weaknesses of shower heat exchangers [43]. The average SHAP values classify this parameter as third in the ranking. However, the analysis of individual observations revealed that depending on the magnitude of the projected financial savings, INV₀ can be either the most or the least significant parameter in the analysis. This proves that although global analysis is useful, only local analysis can provide the detailed information necessary to make optimal investment decisions. If the initial investment outlay is significant, it may be necessary to conduct a benefit–cost analysis [44].

It is worth noting that other parameters also influence the analysis, but their impact is generally significantly smaller. However, in some cases, their significance can prove to be more substantial. This confirms the need to develop tools that will enable future users of greywater heat recovery systems to precisely determine the NPV value in relation to the given specific conditions. Thanks to the developed artificial neural network model, it will be possible to create an application that allows potential users of shower heat exchangers to compare NPV values under different operating conditions. As a result, the financial forecasts will be tailored to the individual needs of the users. The application, developed as part of future research, should also provide the opportunity to assess how changes in input conditions will impact the financial efficiency of the investment. The development of such a tool will significantly increase the efficiency of investment decisions, contributing to better resource management and cost optimization.

It should also be taken into account that each study may encounter certain limitations that need to be taken into account when analyzing the results. In this analysis, this limitation is undoubtedly the analysis of only the horizontal heat exchanger with relatively low effectiveness. Therefore, the next step in the research should include an investigation of a vertical heat exchanger, which is characterized by higher heat recovery effectiveness [45]. Such a comprehensive assessment will provide a more complete picture of the influence of the considered parameters on the financial efficiency of investments in shower heat exchangers.

4.2. Impact on the Environment

An important aspect of analyzing the feasibility of installing a shower heat exchanger is undoubtedly the financial efficiency of such a project. Nevertheless, in times of growing ecological awareness of society, environmental issues are becoming more and more important, especially since, as noted by Szalay [46], the operation of buildings is responsible for over a quarter of energy-related CO₂ emissions. Even when the use of a shower heat exchanger turns out to be financially unprofitable, its use may be justified for environmental reasons. This is due to the fact that the direct effect of reducing energy consumption for heating domestic hot water is reducing emissions resulting from the combustion of fossil fuels. The amount of this reduction depends primarily on the type of primary energy source used in the DHW preparation installation. Under the established operating conditions of the shower installation, the expected reduction in CO₂ emission is almost three and a half times higher when the DHW is heated with an electric water heater than in the case of a gas water heater. Figure A2 and Figure A3 (Appendix B) show the CO₂ emission reduction for selected shower use conditions in the building. The values of annual CO₂ emission reduction (E_r) presented in the plane diagrams were determined based on emission factors for the considered energy carriers, i.e., gas fuel [47] and electricity for end users [48]. It is clearly visible that the increase in water consumption related to the increase in water flow rate (q) and shower length (l_s) results in an increase in the annual reduction of CO₂ emission. With water consumption for showering at the level of 30 L per day and a minimum temperature difference between the media flowing through the heat exchanger, the annual reduction in CO₂ emission will not exceed 7 kg and 22 kg, respectively, for the gas and electric DHW heater. However, if more people use the shower and the shower head has a higher flow rate (q), the E_r value will be significantly higher. For example, for l_s = 50 min, the annual CO₂ emission reduction is from 32 to 398 kg for a gas water heater and from 106 to 1312 kg for an electric water heater. These differences result primarily from different water flow rates (q) but also from the wide range of water and greywater temperatures at the entrance to the shower heat exchanger. The greater the difference between these temperatures, the greater the reduction in CO₂ emission, which results from the greater energy demand for water heating and the higher effectiveness of the heat exchanger. The E_r values determined for T_cw = 8 °C and T_dw = 40 °C are more than three times higher than those determined for the lowest considered temperature difference between cold and greywater (T_cw = 20 °C and T_dw = 30 °C). It follows that the ecological benefits of using shower heat exchangers will be particularly visible in regions with a temperate or cold climate, where the temperature of cold tap water will be relatively low. If l_s = 90 min and the DHW is heated by an electric device, under the most favorable conditions, an annual reduction in CO₂ emission of almost 2400 kg can be expected. Analyzing Figure A2 and Figure A3, one can also notice the impact of the unit bottom slope on the E_r value, which results from increasing effectiveness (ε) with increasing slope (i). This impact is not as pronounced as in the case of water flow rate (q) and shower length (l_s), but increasing the slope (i) from 0% to 4% can contribute to increasing the annual emission reduction by more than 50%.

Apart from the stage that covers the period of operation of the shower heat exchanger, the environmental analysis of the investment related to its installation should also consider its production and end of operation. In the production phase, emissions related to the extraction of raw materials, transport, material production, and assembly are analyzed. In the case of emissions related to the production of the analyzed shower heat exchanger, an important element of the environmental analysis is to take into account emissions related to the extraction and preparation of copper for its production, which is used due to its heat conduction properties and durability. The environmental impact analysis should also take into account emissions resulting from the shower exchanger production process, which involves forming the exchanger elements from copper, which are brazed using high-temperature techniques. In the end-of-life phase of the shower heat exchanger, emissions related to its dismantling and those related to the disposal, recycling, or storage processes should be considered. In the case of end-of-life emissions, the environmental impact of transporting the exchanger to its end-of-life location must also be taken into account. The collected data on the consumption of energy and raw materials necessary for production and end-of-life are converted into CO₂ equivalent emissions using appropriate factors. Analysis of the investment’s impact on the environment allows for the identification and minimization of the largest emission sources already at the design stage. As Pei et al. [49] emphasize, considering the analysis of the life cycle impact of a given solution on the environment in the building sector is intended to guide designers to more sustainable solutions. It also contributes to the achievement of global goals of reducing greenhouse gas emissions [50].

5. Conclusions

This paper evaluated the financial efficiency of investing in the horizontal shower heat exchanger and examined the possibility of using artificial neural networks to assess the profitability of such a solution. Additionally, the magnitude of influence exerted by individual input parameters on predicting the net present value of this project was examined. Based on the analyses, the following conclusions were formulated:

• ANNs are effective for assessing the profitability of investing in a shower heat exchanger, providing accurate forecasts by considering key parameters like energy prices and initial investment, as well as variables with less significant impact;
• The profitability analysis showed that most of the analyzed cases were unprofitable. This was due to the wide range of input data, mainly energy prices and total daily shower length. Nevertheless, there was a certain group of scenarios that were highly profitable;
• Machine learning methods identify complex patterns, extract scenarios, and provide detailed analyses of parameter influences on NPV predictions. This advanced approach is necessary due to NPV’s sensitivity to key input changes, confirming the importance of detailed case analysis and the consideration of a wide range of input values for effective planning and risk management.

The manuscript also identifies directions for future research, which, according to the authors, should focus on the following:

• Assessment of the impact of input parameters on the profitability of investments for various models of shower heat exchangers with different characteristics affecting financial efficiency;
• Analyzing the feasibility of using shower heat exchangers in high water consumption buildings, such as sports facilities, campsites, or dormitories, and assessing the impact of input parameters on their effectiveness in these facilities;
• Conducting profitability analysis for devices with varying effectiveness and price, as well as buildings with specific shower usage conditions, to determine the most beneficial investment scenarios and adapt the technology to users’ needs;
• Developing an application to compare NPV values under various conditions and for different devices, which is crucial for potential users;
• Conducting a detailed Life Cycle Assessment (LCA) analysis of the shower heat exchanger, considering the production, use, and disposal phases, and comparing this solution with other domestic hot water heating techniques.