Physical Model for the Simulation of an Air Handling Unit Employed in an Automotive Production Process: Calibration Procedure and Potential Energy Saving

Abstract

A meticulous thermo-hygrometric control is essential for various industrial production processes, particularly those involving the painting phases of body-in-white, in which the air temperature and relative humidity in production boots must be limited in strict intervals to ensure the high quality of the final product. However, traditional proportional integrative derivative (PID) controllers may result in non-optimal control strategies, leading to energy wastage due to response delays and unnecessary superheatings. In this regard, predictive models designed for control can significantly aid in achieving all the targets set by the European Union. This paper focuses on the development of a predictive model for the energy consumption of an air handling unit (AHU) used in the paint-shop area of an automotive production process. The model, developed in MATLAB 2024b, is based on mass and energy balances within each component, and phenomenological equations for heat exchangers. It enables the evaluation of thermal powers and water mass flow rates required to process an inlet air flow rate to achieve a target condition for the temperature and relative humidity. The model was calibrated and validated using experimental data of a real case study of an automotive production process, obtaining mean errors of 16% and 31% for the hot and cold heat exchangers, respectively, in predicting the water mass flow rate. Additionally, a control logic based on six regulation thermo-hygrometric zones was developed, which depended on the external conditions of temperature and relative humidity. Finally, as the main outcome, several examples are provided to demonstrate both the applicability of the developed model and its potential in optimizing energy consumption, achieving energy savings of up to 46% compared to the actual baseline control strategy, and external boundary conditions, identifying an optimal trade-off between energy saving and operation feasibility.

1. Introduction

Nowadays, energy consumption in the building and industrial sectors accounts for approximately 38% and 30% [1,2] of the total energy consumed worldwide, respectively. Specifically, in buildings, about 40% of energy consumption is attributed to HVAC systems for both heating and cooling purposes [3], whereas, in the industrial sector, a significant amount of energy is needed to maintain optimal temperature levels in various processes and environments. Furthermore, there are certain industrial phases which necessitate a precise temperature control of the surrounding environment. For instance, in automotive production, particularly in body-in-white painting processes, maintaining a specific air temperature and relative humidity levels within fixed intervals is crucial to ensure the quality of the final car produced, by preventing paint defects and avoiding additional costs of re-working [4]. Given that a substantial portion of energy consumption in these sectors still relies on fossil fuels, achieving energy saving is paramount in order to reach the Net Zero Scenario milestone by 2050, limiting the global temperature rise of 1.5 °C [4]. These targets can be achieved by replacing old and energy-intensive systems with new technologies and solutions, by increasing the integration of renewable sources, and by optimizing actual and standard control strategies to achieve higher efficiencies without the replacement of traditional systems. In this regard, having a calibrated and validated digital twin model can be a great advantage in terms of cost-effectiveness and reproducibility compared to an experimental investigation. Moreover, a predictive capability can enable the possibility to employ a digital twin model as a Model Predictive Control (MPC), a modern controller capable of replacing traditional controllers such as simple on/off and proportional integral derivative (PID), offering advantages in terms of energy consumption and operating cost reduction for HVAC systems ranging from 7% to over 50% [5]. The HVAC system controls are actually regulated by the EN ISO 52120-1:2021 for Europe, in terms of evaluation of the control and automation level, and new air flow rate control directives, and by the ASHRAE 62.1 for USA, indicating specific ventilation requirements. Control stability can be an issue also for other application sectors [6,7].

1.1. Literature Review

Unlike a standard controller, the model predictive control (MPC) strategy represents an advanced technique that relies on the solving of an optimization problem that is formulated through models to predict the future behavior of systems and to implement control actions. Numerous literature studies have focused on the development of models and MPC for AHU, and are described in

Table 1. State-of-the-art of works dealing with modeling, MPC, control logics, and monitoring of AHU and HVAC in buildings, industrial, and automotive production sectors.

Paper	Application Sector	Type of System	Location	Type of Model	Control Strategy Development	Main Goals and Key Findings	Limitations
Huang et al. [8]	Building	VAV	N.A.	First-order time delay model	Yes	Model to improve the robustness of temperature control compared with traditional controllers	No thermodynamic and phenomenological characterization of HEXs. Not related to an industrial process
Alahmer et al. [28]	Automotive Air conditioning	N.A.	N.A.	Thermodynamic	No	Analyze the effect of relative humidity on human thermal comfort in a vehicular air conditioning	Not related to an industrial process, no development of a control strategy
Ogonowski [29]	Painting booth	CAV	N.A.	Least Square polynomial model	Yes	Development of a two-layer control system for a specific commercial spray booth model	No physical insights and no phenomenological component characterization
Siroky et al. [9]	Building	N.A.	Prague, Czech Republic	Thermodynamic	Yes	Analyze the achievable energy saving with an MPC for a building heating system	Not related to an industrial process
Rohdin et al. [22]	Automotive painting booth	VAV	Trollhattan, Sweden	Thermodynamic	No	CFD model to evaluate the potential energy saving and minimum time to reach desired set-point conditions. Case study in Saab, Sweden.	No control strategy, no focus on AHU single components.
Xu et al. [16]	Automotive painting booth	VAV	Detroit, USA	Thermodynamic	Yes	To formulate a scheduling program of the entire automotive painting production process	Simplified approach for the AHU, no phenomenological characterization of HEXs
Alt and Sawodny [12]	Automotive painting booth	CAV	N.A.	Thermodynamic	Yes	Model for the temperature and relative humidity evaluation	No phenomenological characterization of HEXs
Feng and Mears [21]	Automotive painting booth	VAV	N.A.	Thermodynamic	Yes	Model to evaluate the energy saving of the process depending on the set-point tolerance bands	No phenomenological characterization of HEXs
Forbes et al. [25]	Industry	-	-	-	-	Review for MPC in industry processes and applications	-
Afram et al. [5]	Building (residential)	-	-	ANN	-	Review of MPC models for HVAC based on ANN in buildings	Not related to an industrial process
Canova et al. [20]	Automotive painting booth	VAV	Turin, Italy	Thermodynamic	No	Model to provide energy consumption forecast of several industrial processes in a FCA paint shop case study	No phenomenological characterization of HEXs, no development of control strategies
Serale et al. [30]	Building	VAV	N.A.	White, gray, and black box modeling	Yes	Provide an MPC framework for building and HVAC system management	Not related to an industrial process
Ayaz et al. [14]	Automotive painting booth	VAV	N.A.	Thermodynamic	Yes	Comparison between control methods (On/Off, Fuzzy Logic, Adaptive Control) for a HVAC of a painting booth	No comparison with experimental data. No HEX characterization
Nikonczuk and Tuchowski [19]	Automotive painting booth	VAV	N.A.	Thermodynamic	No	Method to evaluate the energy consumption of a heat pump serving the painting process.	Focus only on the heat pump system and not on the AHU
Guan et al. [17]	Automotive painting booth	VAV	Henan, China	Thermodynamic	Yes	To propose a new segmented liquid desiccant air-conditioning system for a painting booth of a bus manufacturing plant	Used fixed values for the HEX efficiencies
Sanz et al. [15]	Automotive painting booth	N.A.	Martorell, Spain	Artificial Intelligence and IoT	Yes	To implement an Industry 4.0 framework in an Automotive PaintShop for control and predictive maintenance	Standard PID controllers without MPC
Yao and Shekhar [23]	Building	-	-	White, gray and black box modeling	-	Review to highlight important design parameters for MPC in buildings	Not related to an industrial process
Giampieri et al. [31]	Automotive painting booth	VAV	Sunderland, United Kingdom	Thermodynamic	Yes	Model to carry out thermo-economical investigations, to reduce energy consumption and costs.	Used a fixed HEX efficiency of 0.5
Velasco-Hernandez et al. [11]	Industrial Painting Booth	N.A.	N.A.	No model	No	Development and implementation of an IoT-based system to monitor temperature and relative humidity	No development either of a predictive model, or of a control strategy
Daniarta et al. [18]	Automotive painting booth	N.A.	N.A.	Thermodynamic	No	Analyze a new concept of ORC combined for power generation using waste heat from a paint shop	Focus on the ORC system and not on the AHU
Taheri et al. [24]	Building	-	-	Thermodynamic, statistic, ANN	-	Comprehensive state-of-the-art review of MPC in HVAC in buildings	Not related to an industrial process
Aruta et al. [10]	Building (Residential)	N.A.	Benevento, Italy	GA, ANN	Yes	Provide optimal values of set-point temperature to minimize heating energy consumption	Not related to an industrial process
Cavalcante et al. [13]	Automotive painting booth	N.A.	N.A.	ANN	Yes	Enhance temperature control in body-in-white parts	Focused on the car body parts and not on the AHU

. Among the reviewed works, most are developed within the building [8,9,10] or other industrial [11] application sectors whereas others focus on automotive body-in-white production processes, simulating temperature and relative humidity [12,13], comparing different control methods [14,15,16], proposing new air conditioning methodologies [17,18], evaluating overall energy consumption [19,20,21,22], and conducting thermo-economical investigations. Moreover, regarding the type of system investigated, most involve Variable Air Volume (VAV) and some involve Constant Air Volume (CAV), although many are not clearly stated. Finally, most of the developed models are thermodynamically based, whereas others rely on artificial intelligence and machine learning techniques [13,15]. Further comprehensive insights can be found instead in dedicated works, both for buildings [5,23,24] and for the industrial sector [25,26,27].

1.2. Research Gap, Novelty, and Paper Structure

The presented case study focuses on an air handling unit (AHU) within a paint shop process in an automotive production plant, which stands as one of the most energy-consuming steps in the production process [32]. The primary objective of this system is to guarantee fixed conditions for the supply air in terms of temperature and relative humidity within predefined tolerance bands. However, these two variables may still be influenced by conditions within the thermal zone itself, due to a heat recovery from the internal air for instance following a sudden and unforeseen variation in the number of car bodies produced or of the paint spray mass flow rate. Therefore, standard control strategies reliant on PID controllers may underperform due to response delays, resulting in increased energy consumption for the analyzed system [33], and making, in this case, the “grey-box” modeling approaches more appropriate. Furthermore, the development of a prediction model could benefit simulating future scenarios, proposing new control logics, simulating company constraints, and evaluating the potential energy saving for each situation.

Hence, the main targets of this paper can be stated as follows:

• To establish a control strategy for an AHU in a paint shop process that remains independent from what is happening in the climatized environment of the building, so from its thermal mass and envelope properties, considering only the air inlet temperature and relative humidity as boundary conditions.
• To demonstrate that, by leveraging well-known physical equations alongside a thorough calibration of field data, it is possible to develop a simple and accurate control strategy, indicating about the requisite procedures and steps.
• To show the potential energy-saving capabilities of such a control strategy, considering an example of the operation optimization of a building chiller for cold water production, in order to ensure the most efficient way to reach a desired air temperature set-point within the controlled thermal zone.

Despite some of the approaches employed in this study resembling those found in other works, as shown in Table 1, the current literature still presents the following research gaps:

• Current literature works have not yet carried out a detailed characterization of all the individual components constituting the paint shop AHU; for instance, heat exchangers.
• Very few of the reviewed papers highlight a detailed procedure to show, from experimental data related to temperature, how relative humidities and mass flow rates could be used to develop a predictive model which considers all the thermodynamic and phenomenological aspects for all the components constituting the system analyzed.
• Regarding the latter point, only the works of Giampieri et al. [31] and Guan et al. [17] introduce detailed physical modeling of an Air Supply Unit for the body-in-white painting booth. However, this work carries out a thermo-economic evaluation for design purposes, without considering any calibration with experimental data and assuming fixed values for the HEX efficiency. Similarly, Ayaz et al. [14] carried out a comparison between different control approaches such as on/off, PID, fuzzy logic, and adaptive control, but without any experimental evidence.

Therefore, the main novelty of our work lies in demonstrating the calibration of a predictive model using unique field data obtained from a low-cost sensor suite and elucidating its utilization for optimization purposes.

The main objectives of this paper are addressed in the following section: Section 2 and Section 3 are describe the case study and the model equation employed. The detailed procedure indicates how to calibrate the model based on experimental data, and how to use it in simulation mode, as well as the detailed resolution algorithm are presented in Section 4 and Section 5, respectively, addressing objective number 1 (see above) of this paper. Section 6.1 and Section 6.2 show the results in terms of application of the model and of the potential energy saving between a MPC and a standard PID control approach, respectively. Finally, Section 6.3 analyzes the potential use of the predictive model for the optimization of the supply cold water temperature in order to minimize the energy consumption of the production chiller, so addressing the objective number 2 of this work.

2. Case Study Description

This case study analyzed an air handling unit (AHU) serving a booth in which the painting process of car bodies is performed. This use-case is examined within the framework of the EnerMan H2020 European Project [34]. A schematic diagram for the entire system is provided in Figure 1. The objective of the system is to condition the inlet air, characterized by a temperature $T_{a,in}$ and relative humidity ${\varphi }_{a,in}$, to achieve a predetermined target for temperature and relative humidity at the inlet of the painting booth ($T_{target},{\varphi }_{target}$). The AHU comprises three heat exchangers (HEX) and a humidifier, with all processed volumetric flow rates originating from the external environment. However, heat recovery with the internal air may be performed by an air/air heat recovery unit before entering the first HEX of the analyzed system (which was not considered in this case). The first HEX, a hot fin-and-tube type, serves to provide a heat flux (${\dot{Q}}_{hot,pre}$) for a pre-heating operation in the heating mode. The second HEX needs to provide the heat load (${\dot{Q}}_{cool}$) for cooling and de-humidification processes in cooling mode, and it is composed of a collecting tank at the bottom to remove the condensing water. The humidifier functions to spray a continuous water mass flow rate, increasing the level of air humidity if requested. Lastly, the third heat exchanger is utilized for post-heating (${\dot{Q}}_{hot,post}$) during cooling mode operations. Both hot heat exchangers are supplied with water flows (${\dot{m}}_{w,pre},{\dot{m}}_{w,post}$) coming from a hot water line of the plant (indicated in red in Figure 1), presumably produced by means of an industrial boiler, while the cold heat exchanger is supplied with a cold mass flow rate (${\dot{m}}_{w,cool}$) also coming from a dedicated cold water line (indicated in blue in Figure 1), produced by a chiller. Humidification is provided through an ambient temperature industrial water line (in green in Figure 1). Regulation of the water supply mass flow rates is achieved through two-way valves installed on the supply of each component. Furthermore, each component can be deactivated if the process does not need a specific moist air process, in accordance with a fixed control strategy that will be presented in the next sections.

3. Model

In this section, a thermodynamic physics-based approach is described, along with all the equations on which the model relies. We assume a steady-state condition for all the sub-transformation of the entire process, neglecting the inertial effects of the AHU external structure. The heat addition coming from the supply fan was also considered; however, the result was almost always negligible compared to the actual heating load requested by the air conditioning process. Moreover, the specific heat for air ($c_{air}$) and water ($c_w$) were assumed constant and equal to 1.01 kJ/kgK and 4.19 kJ/kgK, respectively, which is reasonable due to the limited temperature variations occurring in this investigation, according to Borgnakke and Sonntag [35]. The model was implemented using the software MATLAB [36].

3.1. Mass and Energy Balances in Each Component

The enthalpies and specific humidities, employed in the following equations of this section, derive from the measures of dry bulb temperature and relative humidity through the classic moist air thermodynamic theory.

For the first pre-heating heat exchanger, energy balances both on the air side and the water side can be expressed as follows:

$${\dot{Q}}_{hot,pre}={\dot{m}}_{air}·(h_{out,pre}-h_{a,in})={\dot{m}}_{w,pre}·c_w·(T_{w,in,pre}-T_{w,out,pre})$$

(1)

where ${\dot{m}}_{air}$ represents the total air mass flow rate circulating in the system. $h_{out,pre}$ and $h_{a,in}$ denote the specific enthalpies of the moist air at the outlet and inlet of the pre-heating HEX, respectively. Similar considerations can also be applied to the post-heating heat exchanger, with the energy balance being expressed as follows:

$${\dot{Q}}_{hot,post}={\dot{m}}_{air}·(h_{target}-h_{out,hum})={\dot{m}}_{w,post}·c_w·(T_{w,in,post}-T_{w,out,post})$$

(2)

In this case, $h_{out,hum}$ and $h_{target}$ represent the specific enthalpies of moist air at the humidifier outlet and at the post-heating HEX outlet, respectively. The specific humidities at the inlet and the outlet of both processes have been supposed as constant, as no water additions or removals occur.

For the cooling heat exchanger, both water mass and energy balances can be expressed as follows:

$${\dot{m}}_{w,cond}={\dot{m}}_{air}·({\omega }_{out,pre}-{\omega }_{out,cool})$$

(3)

$${\dot{Q}}_{cold}={\dot{m}}_{air}·(h_{out,pre}-h_{out,cool})={\dot{m}}_{w,cold}·c_w·(T_{w,out,cool}-T_{w,in,cool})$$

(4)

where ${\omega }_{out,pre}$, ${\omega }_{out,cool}$, $h_{out,pre}$, and $h_{out,cool}$ represent the specific humidities and enthalpies at the inlet and outlet of the cooling HEX, respectively. ${\dot{m}}_{w,cond}$ is the water flow rate condensing in the collecting tank of the HEX, subsequently removed from the system. This fraction has been considered negligible in terms of energetic contribution; thus, the condensation term is disregarded in the energy balance. Moreover, the moist air at the outlet of the cold battery is assumed to be in saturated conditions, with a relative humidity of ${\varphi }_{out,cool}=100\%$ without accounting for any bypass influence.

Finally, for the humidifier, a water mass balance was considered as follows:

$${\dot{m}}_{w,hum}={\dot{m}}_{air}·({\omega }_{out,hum}-{\omega }_{out,cool})$$

(5)

where ${\dot{m}}_{w,hum}$ represents the water mass flow rate sprayed by the humidifier, whereas ${\omega }_{out,hum}$ and ${\omega }_{out,cool}$ are the outlet and inlet specific humidities of the humidifier, respectively. Since humidification occurs with liquid water, the entire process has been supposed to be isenthalpic.

The air mass flow rate ${\dot{m}}_{air}$, assumed to be constant in each component due to the steady-state condition hypothesis, has been evaluated from the inlet volumetric air flow rate ${\dot{V}}_{air}$, considering the air density corresponding to the actual external conditions at the AHU inlet.

3.2. Heat Exchanger Phenomenological Equations

For the heat exchangers characterization, the ε-NTU method was employed [37], and all the relevant phenomenological equations are presented herein. Specifically, the relationship between the heat exchanger efficiency (ε) and the number of transfer units (NTU), for fin-and-tube heat exchangers, with a number of tube rows exceeding four was used, in accordance with that described by Liang et al. [38]:

$$\epsilon =1-\exp \{NT{U}^{0.22}·[\exp \left(-C^{*}·NT{U}^{0.78}\right)-1]/C^{*}\}$$

(6)

This relation is applicable to all the mass flow rates of air and water, independently from which fluid has the highest thermal capacity. The variable $C^{*}$ represents the ratio between the minimum (${\dot{C}}_{min}$) and maximum (${\dot{C}}_{max}$) thermal capacities of the two fluids, and is evaluated as follows:

$$C^{*}={\displaystyle \frac{{\dot{C}}_{min}}{{\dot{C}}_{max}}}$$

(7)

$${\dot{C}}_w={\dot{m}}_w·c_w$$

(8)

$${\dot{C}}_{air}={\dot{m}}_{air}·c_{air}$$

(9)

From the experimental data, ε values are determined using the actual thermal power (e.g., assessed via an energy balance on the water side) and the maximum potential thermal power, as follows:

$$\epsilon ={\displaystyle \frac{\dot{Q}}{{\dot{Q}}_{max}}}$$

(10)

$$\dot{Q}={\dot{m}}_w{c}_w(T_{in,w}-T_{out,w})$$

(11)

$${\dot{Q}}_{max}={\dot{C}}_{min}(T_{in,w}-T_{in,a})$$

(12)

Moreover, experimental UA values can be obtained from the definition of NTU, as follows:

$$UA=NTU·{\dot{C}}_{min}$$

(13)

4. Model Calibration and Validation

In this section, the calibration of the heat exchanger models, and corresponding validation results are presented. Particularly, the employed calibration database collected from the field is described in Section 4.1, whereas the calibration of an equation to evaluate the heat exchanger global conductance, as well as a comparison between experimental and predicted HEX water mass flow rates are provided in Section 4.2.

4.1. Database Description

In the case of the study’s processes, a sensor suite was installed to collect various data, including temperatures, relative humidities, and mass/volumetric flow rates, both in summer and in winter modes. These data come from continuous monitoring of the case study investigated, along a six month period of operations including both heating and cooling seasons, and a measurement frequency of 15 min was considered. Specifically, resistance temperature detectors (RTDs) and humidity sensors were used to measure temperatures and relative humidities at the inlet ($T_{a,in},{\varphi }_{a,in}$) and outlet ($T_{target},{\varphi }_{target}$) of the AHU, whereas other RTDs were used to measure the inlet and outlet temperatures of the hot pre ($T_{w,in/out,pre}$), cold ($T_{w,in/out,cool}$), and hot post ($T_{w,in/out,post}$) water fluxes. Volumetric flowmeters were used to measure both the air (${\dot{V}}_{air}$) and water (${\dot{V}}_{w,pre},{\dot{V}}_{w,cool},$ and ${\dot{V}}_{w,post}$) flow rates. Air temperatures and relative humidities between two consecutive components, which are not directly measured, were obtained for calibration purposes by isolating the case in which only one component is operating or, in the case of reliable measured conditions, by solving the other components and obtaining the desired values across each component of the analyzed system.

4.2. Calibration of the Heat Exchangers Global Conductance

The global conductance (UA) of all the heat exchangers was calibrated using the experimental database presented above. Due to a lack of individual data for the hot-post heat exchanger, both hot batteries have been assumed equal, resulting in a single calibration for both. Additionally, a separate calibration was conducted for the cold heat exchanger.

Experimental UA values were determined using the following procedure: given the inlet and outlet air and water temperatures between the heat exchanger, along with all the mass flow rates, both $\dot{Q}$ and ${\dot{Q}}_{max}$ were evaluated through Equations (11) and (12), respectively. Subsequently, the experimental efficiency ε was determined using Equation (10), together with $C^{*}$ obtained from Equation (7). These values were then used in Equation (6) (solved iteratively) to evaluate the experimental NTU, and subsequently, the UA was calculated through Equation (13).

The experimental values were then utilized to calibrate an expression for the evaluation of the global conductance, as a function of both water and air mass flow rates, as follows:

$$UA={\displaystyle \frac{a}{\frac{1}{b·{\dot{m}}_w^{0.8}}+\frac{1}{c·{\dot{m}}_{air}^{0.8}}}}$$

(14)

where a and b are the two calibration coefficients determined through a comparison with the experimental data. Assuming negligible thermal resistance due to conduction, the global conductance is proportional to the heat transfer coefficients at the air side and the water side, which are in turn supposed to be proportional to the known mass flow rate with a power law and exponent of 0.8, typical for turbulent flows [39]. This approach allows for the definition of saturation zones, in which, despite increases in air and water mass flow rates, UA no longer increases, making the heat exchanger unable to satisfy the user’s thermal power requirements under certain conditions.

Figure 2a shows the comparison between the fitted equation and the experimental points for the hot pre heat exchanger. Calibration was performed using 1600 experimental points obtained during the heating mode in January, yielding a coefficient of determination (R²) of 0.85. The model validation in the simulation model is shown in Figure 2b, illustrating the experimental and evaluated water mass flow rates, along with the statistical indexes for the mean absolute (MAPE) and relative (MRPE) percentage errors, maximum error ($Er{r}_{max}$), percentage of points within the ±20% error band (${\delta }_{\pm 20\%}$) and Root Mean Square Error (RMSE), also defined in the nomenclature section. This evaluation was based on 2000 experimental points collected during the month of February, yielding a MAPE of approximately 16%. This error may be influenced also by the fact that the production line is not a totally controlled environment, so there can be several unknown external factors that could covertly influence the process. For this reason, along with the uncertainty of instruments commonly employed for such applications, lower errors are difficult to achieve, also by means of a more detailed modeling structure. For confidentiality reasons, all the plots were normalized by dividing the results for the maximum experimental values obtained for the hot pre-HEX, of the global conductance ($U{A}_{max}$), water (${\dot{m}}_{w,max}$), and air (${\dot{m}}_{a,max}$) mass flow rates, respectively.

Similarly, Figure 3a shows the comparison between the fitted surface and the experimental points for the cold heat exchanger, while Figure 3b depicts the validation in terms of predicted and experimental cold water mass flow rate. However, due to limited available data (the calibration has been carried out only on 139 experimental points), the model accuracy is lower compared to the one achieved for the hot pre-HEX, yielding a calibration R² of 0.65 and a MAPE of approximately 30%. As in the previous case, the data have been normalized for the maximum experimental values of the cold HEX of the global conductance ($U{A}_{max}$), water (${\dot{m}}_{w,max}$), and air (${\dot{m}}_{a,max}$) mass flow rates, respectively.

An overview of the calibration coefficients a, b, and c for the two hot and cold heat exchangers is reported in

Table 2. Calibration coefficients and statistics parameters indicating the goodness of prevision, for the hot pre/post and cold heat exchangers.

Heat Exchanger	a	b	c	${\mathit{R}}^2$ Calibration	MAPE [%]	MRPE [%]	${\mathit{\delta }}_{\pm 20\mathit{\%}}$ [%]	RMSE * [-]
Hot Pre/post	21.85	0.554	81.99	0.851	15.58	9.22	78.11	$0.091·{\dot{m}}_{w,pre,max}$
Cold	26.38	0.164	91.96	0.649	30.82	13.86	43.80	$0.30·{\dot{m}}_{w,cold,max}$

* Normalized for the maximum values of the hot pre/cold water mass flow rates.

, as well as all the calibration and validation goodness statistics indexes.

5. Simulation Mode, Resolution Algorithm and Control Logics

5.1. Inputs, Outputs and Resolution Algorithm

The development of the MPC investigated in this paper includes the direct implementation of all the equations shown in the methodology section (Equations (1)–(13)), applying them with a specific logic that we explain. A list of inputs and outputs of the developed model for use in simulation mode is provided in Figure 4 (above). The primary objective of the model is to predict the heat fluxes (${\dot{Q}}_{hot,pre/post},{\dot{Q}}_{cool}$) and water mass flow rates (${\dot{m}}_{w,pre/post/cool}$) of all the heat exchangers, as well as the humidification and condensing mass flow rates (${\dot{m}}_{hum/cond}$) necessary to process an air volumetric flow rate (${\dot{V}}_{air}$) to achieve a target set-point in terms of temperature and relative humidity ($T_{target},{\varphi }_{target}$) within specified tolerances ($to{l}_T,to{l}_{\varphi }$), as a function of the air inlet temperature and relative humidity ($T_{a,in},{\varphi }_{a,in}$) and from other boundary conditions in terms of inlet water temperatures ($T_{w,in,pre/post/cold}$) of heat exchangers. The main aim of the MPC is to evaluate the heating and cooling needs in specified boundary conditions, and to directly intervene on HEX water regulation valves in a way to guarantee the desired water mass flow rates. In Figure 4 (below), an indication of the resolution algorithm employed by the model is provided. Initially, based on the inputs of the model and the external conditions, a thermo-hygrometric zone is determined, which is described in the following section. Subsequently, depending on the thermo-hygrometric zone identified, the thermal powers and humidification/de-humidification mass flow rates are evaluated by means of the energy balances across each system component (Equations (1)–(5)). Finally, heat exchangers are iteratively solved, by determining the required water mass flow rates to achieve the desired value for the global conductance and heat rates, utilizing Equations (6)–(13) presented in Section 3.2.

5.2. Operating Thermo-Hygrometric Zones Evaluation Criteria

The determination of necessary processes that moist air must undergo to reach the set-point zone starting from the external conditions of temperature and relative humidity is crucial to deciding which mass and energy balance should be applied and which component should operate. With this aim, depending on the external ambient conditions entering in the AHU ($T_{amb},{\varphi }_{amb}$), six different operating thermo-hygrometric zones were delineated, as illustrated on the psychrometric chart in Figure 5.

• Zone 1 refers to the external conditions where the moist air necessitates both pre-heating and humidification processes.
• Zone 2 refers to conditions which require only humidification.
• Zone 3 necessitates solely cooling, with potential humidification if the air inlet specific humidity is lower than that of point a (which is an uncommon occurrence).
• Zone 4 requires a pre-cooling and dehumidification process, along with a post-heating operation.
• Zone 5 necessitates solely heating to reach the setpoint zone.
• Zone 6 refers to external conditions already within the tolerances of the target set-point, obviating the need for any moist air transformations.

An overview of the system’s components utilized for each thermo-hygrometric zone is presented in

Table 3. Control logic in terms of components of the AHU turned ON or OFF based on the actual thermo-hygrometric zone.

Zone	Hot Pre-HEX	Cold HEX	Humidifier	Hot Post HEX
1	ON	OFF	ON	OFF
2	OFF	OFF	ON	OFF
3	OFF	ON	OFF if ${\omega }_{amb}>{\omega }_{point,a}$, otherwise ON	OFF
4	OFF	ON	OFF	ON
5	ON	OFF	OFF	OFF
6	OFF	OFF	OFF	OFF

. Specifically, when a component is marked as “ON”, both thermal balances and phenomenological equations associated with that component are applied. Conversely, if the component is marked “OFF” for a given thermo-hygrometric zone, the air temperature and relative humidity at the inlet and outlet of that component are supposed to remain constant, and all the thermal powers and water mass flow rates are set to zero. The components employed for each zone are as follows: the hot pre-HEX and humidifier for zone 1, the humidifier for zone 2, the cold HEX (and optionally the humidifier) for zone 3, the cold and hot post-HEXs for zone 4, and the hot pre-HEX for zone 5. Zone 6, on the other hand, does not require any component to be activated. It should be noted that the effective target point considered in the mass and energy balances has been assumed to be the first point inside zone 6, which is the closest to each external investigated condition. This approach minimizes the overall path on the psychrometric chart, consequently reducing the total energy and water consumption.

6. Results

The model serves two different purposes. Firstly, it can simulate the future behavior of the analyzed system by evaluating the thermal powers and mass flow rates required in each thermal regulation zone, given fixed boundary conditions of the inlet water temperature for each heat exchanger. The second purpose is to provide feedback to the line operator, enabling them to select optimal values for the inlet hot/cold water temperature. This aims to reduce the energy consumption of the energy conversion system (boiler and heat pump) used to produce hot and cold water for the industrial lines, while ensuring the proper functioning of the system itself. The results of the first purpose are presented in Section 6.1, whereas the second part is reported in Section 6.3.

6.1. Application of the Model for Each Thermo-Hygrometric Zone

This section outlines the results obtained from applying the model to each thermo-hygrometric zone. Various input data were selected, including air inlet and target temperatures, relative humidities, air volumetric flow rate, and inlet water temperatures. Specifically, a target temperature and relative humidity of 24 ± 4 °C and 50 ± 5%, respectively, were set, and an air volumetric flow rate of 80,000 m³/h was assumed for all the investigated cases. Target setpoint values, with corresponding tolerances, were chosen empirically by the manufacturer in a way to avoid as far as possible the body-in-white painting defects and were in accordance with typical values used also in other references [17,31]. The input values chosen for each example are provided in

Table 4. Input data considered for each example and thermo-hygrometric zone investigated in the result section. These data are different from the ones employed by the company, which cannot be declared for confidentiality issues.

Input Data
Example N°	Thermo-Hygrometric Zone	${\mathit{T}}_{\mathit{a},\mathit{i}\mathit{n}}$ [°C]	${\mathit{\varphi }}_{\mathit{a},\mathit{i}\mathit{n}}$ [%]	${\mathit{T}}_{\mathit{t}\mathit{a}\mathit{r}\mathit{g}\mathit{e}\mathit{t}} \pm \mathit{t}\mathit{o}{\mathit{l}}_{\mathit{T}}$ [°C]	${\mathit{\varphi }}_{\mathit{t}\mathit{a}\mathit{r}\mathit{g}\mathit{e}\mathit{t}} \pm \mathit{t}\mathit{o}{\mathit{l}}_{\mathit{\varphi }}$ [%]	${\dot{\mathit{V}}}_{\mathit{a}\mathit{i}\mathit{r}}$ $[{\mathbf{m}}^3/\mathbf{h}]$	${\mathit{T}}_{\begin{array}{c}\mathit{w},\mathit{i}\mathit{n},\\ \mathit{p}\mathit{r}\mathit{e}\end{array}}$ [°C] *	${\mathit{T}}_{\begin{array}{c}\mathit{w},\mathit{i}\mathit{n},\\ \mathit{c}\mathit{o}\mathit{o}\mathit{l}\end{array}}$ [°C] *	${\mathit{T}}_{\begin{array}{c}\mathit{w},\mathit{i}\mathit{n},\\ \mathit{p}\mathit{o}\mathit{s}\mathit{t}\end{array}}$ [°C] *
1	1	10	50	$24\pm 4$	$50\pm 5$	80,000	80	-	-
2	2	35	20	$24\pm 4$	$50\pm 5$	80,000	-	-	-
3	3	40	25	$24\pm 4$	$50\pm 5$	80,000	-	10	-
4	4	28	60	$24\pm 4$	$50\pm 5$	80,000	-	3	80
5	5	20	75	$24\pm 4$	$50\pm 5$	80,000	80	-	-

* A numeric value is reported whenever the component is turned ON, according to Table 3, otherwise ‘-’.

. It should be clarified that these input values are tailored for the purpose of the current research and do not reflect the real conditions of the company case study plant, which cannot be disclosed due to confidentiality issues.

For thermo-hygrometric zone 1, an air inlet temperature and relative humidity of 10 °C and 50%, respectively, was considered typical of a winter season. Additionally, an inlet water temperature of 80 °C was assumed. Figure 6a depicts the transformations on the psychrometric chart (red line), alongside the air inlet and target points (in green and red, respectively). The results in terms of required thermal powers are reported in

Table 5. Output data from the mass and energy balances application, for all the investigated examples and thermo-hygrometric zones in this paper.

Output Data (Mass and Energy Balances)
Example N°	Thermo-Hygrometric Zone	${\dot{\mathit{Q}}}_{\mathit{h}\mathit{o}\mathit{t},\mathit{p}\mathit{r}\mathit{e}}$ [kW]	${\dot{\mathit{Q}}}_{\mathit{h}\mathit{o}\mathit{t},\mathit{p}\mathit{o}\mathit{s}\mathit{t}}$ [kW]	${\dot{\mathit{Q}}}_{\mathit{c}\mathit{o}\mathit{l}\mathit{d}}$ [kW]	${\dot{\mathit{m}}}_{\mathit{h}\mathit{u}\mathit{m}}$ [kg/s]	${\dot{\mathit{m}}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{d}}$ [kg/s]
1	1	475	0	0	0.076	0
2	2	0	0	0	0.080	0
3	3	0	0	308.9	0	0
4	4	0	265.2	344.12	0	0.031
5	5	140.3	0	0	0	0

, whereas Figure 6b illustrates the balance between the available heat rate of the HEX, depending on the water mass flow rate (red line), and the exact value requested by the user (black line). It is noteworthy that a water mass flow rate of approximately 2 kg/s is necessary to satisfy the required heat rate, which amounts to approximately 500 kW. The results pertaining to the hot water mass flow rate, global conductance, outlet water temperature, and HEX efficiency are reported in

Table 6. Output data from the heat exchanger phenomenological relations application, for all the investigated examples and thermo-hygrometric zones in this paper.

Output Data (Heat Exchangers)
Example N°	Regulation Thermal Zone	Hot PreHeat Exchanger				CoolingHeat Exchanger				Hot PostHeat Exchanger
		${\dot{\mathit{m}}}_{\mathit{w}}$ [kg/s]	$\mathit{U}\mathit{A}$ [kW/K]	${\mathit{T}}_{\mathit{w},\mathit{o}\mathit{u}\mathit{t}}$ [°C]	$\mathit{\epsilon }$ [-]	${\dot{\mathit{m}}}_{\mathit{w}}$ [kg/s]	$\mathit{U}\mathit{A}$ [kW/K]	${\mathit{T}}_{\mathit{w},\mathit{o}\mathit{u}\mathit{t}}$ [°C]	$\mathit{\epsilon }$ [-]	${\dot{\mathit{m}}}_{\mathit{w}}$ [kg/s]	$\mathit{U}\mathit{A}$ [kW/K]	${\mathit{T}}_{\mathit{w},\mathit{o}\mathit{u}\mathit{t}}$ [°C]	$\mathit{\epsilon }$ [-]
1	1	1.90	20.22	20.3	0.85	/	/	/	/	/	/	/	/
2	2	/	/	/	/	/	/	/	/	/	/	/	/
3	3	/	/	/	/	6.33	18.93	21.7	0.41	/	/	/	/
4	4	/	/	/	/	9.71	26.64	11.47	0.53	1.13	13.34	24.3	0.90
5	5	0.59	7.96	23.41	0.94	/	/	/	/	/	/	/	/

“/”: Component not in use for the current regulation zone.

.

For zone 2, an air inlet temperature and relative humidity of 35 °C and 20%, respectively, have been assumed, typical of a hot dry day in the summer season. In this scenario, only humidification is required to attain the target zone, as depicted by the air transformation from the external conditions shown in Figure 7. Given that the process is assumed to be isenthalpic, only a mass balance across the humidifier has been applied, with no additional phenomenological equations considered for the heat exchangers. Consequently, a humidification mass flow rate of approximately 0.080 kg/s has been determined, as shown in Table 5.

For thermo-hygrometric zone 3, an air inlet temperature of 40 °C and a relative humidity of 25% was considered. These conditions are typical of an extremely hot and dry summer period, in which a cooling process is needed without any de-humidification. While humidification could potentially be required in such scenarios, it is relatively uncommon (occurring primarily in extremely hot and dry climates). Hence, a single thermo-hygrometric zone was considered for simplicity. In this case, the cold water flow was set at 10 °C. The cooling transformation is shown in Figure 8a, while Figure 8b illustrates the available heat rate of the cold HEX depending on the water mass flow rate, along with the requested thermal power of approximately 300 kW, as detailed in Table 5. To meet the user’s needs, a cold water mass flow rate of approximately 6 kg/s is necessary, as also indicated in Table 6.

For thermo-hygrometric zone 4, an air inlet temperature and relative humidity of 28 °C and 60%, respectively, have been considered, representing typical conditions of a warm and humid summer period. For the cooling and de-humidification processes, the cold water flow was set at 3 °C, whereas for the post-heating, a hot water flow was set at 80 °C. The air transformation on the psychrometric chart is depicted in Figure 9a, while Figure 9b,c illustrates the available heat rate for the cold and post-heating HEXs depending on the corresponding water mass flow rates. It is noteworthy that the post-heating HEX can meet the user’s requirements in terms of the heating load of approximately 250 kW, with a water mass flow rate of nearly 1 kg/s, while the cold HEX is capable of providing the requested cooling load of approximately 500 kW, with a water mass flow rate of approximately 9.7 kg/s. Similarly to previous instances, all the results regarding heat rates and de-humidification mass flow rates are presented in Table 5, whereas Table 6 reports the characteristic variables obtained for the investigated operating condition for the HEXs.

Finally, for zone 5, an air inlet temperature and relative humidity of 20 °C and 75%, respectively, were assumed, representing typical conditions of a mild and humid climate. In this scenario, simple heating was required, with a requested heating load of approximately 140 kW, as indicated in Table 5. The corresponding moist air transformations are depicted in Figure 10a. Figure 10b instead illustrates the heat rate provided by the heat exchanger, and it is worth noting that, with an 80 °C water source, a water mass flow rate of about 0.6 kg/s is sufficient to meet the user’s needs, as presented in Table 6.

6.2. Example of Energy Saving Between MPC and Standard Control Approaches

In this section, the potential energy savings achievable through the implementation of an MPC compared to a standard control strategy are investigated. Particularly, an experimental condition characterized by an air volumetric flow rate of 80,000 m³/h, an air inlet temperature of 10.3 °C, and a relative humidity of 45.6% was considered. The target conditions were set to 20 ± 4 °C and 30 ± 5%, respectively, as shown in Figure 11. The currently employed standard control strategy comprises PID controllers able to act on the hot/cold water mass flow rates by automatically adjusting the water valve section openings (represented in Figure 1), based on feedback from the air temperature at the outlet of each heat exchanger. Conversely, air dampers were considered fixed since the AHU is CAV type. This control strategy results in an air outlet temperature and relative humidity of 19.5 °C and 29.2%, respectively, as shown in Figure 11a. The experiment conducted shows that a pre-heating thermal power of 296 kW and a humidification water mass flow rate of 0.016 kg/s were required. Conversely, the proposed control strategy, making use of the developed predictive model, is able to already know the required water flow rate. Therefore, this can be controlled by adjusting the water valve section or by means of a calibration law, or with a controller which acts directly on the mass flow rate and not anymore on a temperature, being more responsive and limiting useless superheating or subcooling.

When simulating a control strategy based on a predictive model for the same air inlet and target conditions, an outlet condition of 16 °C and 31.4%, respectively, for temperature and relative humidity have been obtained. In this scenario, a pre-heating thermal power of 160 kW is necessary to achieve the desired target conditions, which is approximately 46% lower compared to the same value obtained with the experiment, in which the standard PID control strategy was employed. The transformation of moist air performed under the two control strategies is also illustrated in Figure 11b. It should be clarified that, despite it being demonstrated how the model-based control strategy can yield significant energy savings by limiting unnecessary superheating and water consumption compared to standard PID control strategies, the example presented here is merely illustrative and based on a single experimental point. A comprehensive assessment and comparison of the two approaches can only be conducted once the MPC strategy is implemented on the production line, a task we plan to undertake in future research works.

6.3. Inlet Water Temperature Optimization

Here, an example of the employment of the model to carry out optimization of values for boundary conditions is presented. Particularly, the model was used to aid operators in selecting the optimal value for the supply water temperature to the cold heat exchanger, produced using a centrifugal chiller, in order to ensure a proper operation of the entire system avoiding extra energy costs. A similar approach could also be applied for both the hot heat exchanger, by considering the hot water produced by an industrial gas boiler or a high-temperature heat pump [40].

The model was queried for different values of supply cold water temperature (from 1 and 15 °C) and mass flow rate (from 0.1 to 5 kg/s), by means of the simulation mode. The procedure is clarified in Section 5.1. Therefore, the several trends obtained by means of a brute-force research, of available cooling thermal power (${\dot{Q}}_{cold}$) suppliable by the cold HEX are reported in Figure 12a, as a function of the two aforementioned variables. In the case of this example, the air inlet temperature and relative humidity were set to 28 °C and 60%, respectively (corresponding to the thermal regulation zone 4), with all the other boundary conditions fixed as specified in Table 4. It is noteworthy that if the inlet water temperature is too high (close to 15 °C) the heat exchanger would be unable to provide the requested thermal load, or the water mass flow rate would be excessively high compared to the design values of the heat exchanger. Conversely, if the inlet water temperature is too low (close to 1 °C), there would be extra energy consumption from the chiller to produce the cold water flow (the energy consumption of circulation pumps has been neglected in this paper). The energy consumption needed to produce the cold water was estimated by simulating the behavior of an R1234yf centrifugal chiller, following the same approach of Pelella et al. [41,42,43]. Zero outlet evaporator superheating and condenser subcooling were considered, along with fixed minimum temperature differences at the condenser and the evaporator of 5 °C and 10 °C, respectively, and a fixed compressor efficiency of 0.7. Figure 12b reports the trends for the compressor electric power (red line) and the chiller COP (blue line) in providing the thermal load requested by the user. A maximum value for the inlet cold water temperature can be determined ($T_{w,max}$), beyond which the cold heat exchanger may not meet the user’s needs (and for that reason the zone has been filled in the background in Figure 12). Conversely, for water temperatures below the maximum value, the system does not operate at its maximum COP conditions. Therefore, an optimal value depending on the external boundary conditions can be identified, ensuring the minimal energy consumption of the chiller while maintaining the correct functioning of the entire system. In this specific example, the highest chiller performance is achieved with an inlet water temperature of approximately 8 °C, having a water mass flow rate of 15 kg/s.

Finally, by querying the model for various boundary conditions in terms of external temperature and relative humidity, performance maps can be generated, as shown in Figure 13. These maps offer the operator the optimal values of the inlet water cold temperature to impose, ensuring the proper operation of heat exchangers without incurring unnecessary energy consumption. Figure 13a shows the optimal inlet water temperature for thermo-hygrometric zone 4, which significantly decreases with the increase in the air inlet temperature for the same relative humidity, due to the increase in the thermal load. Conversely, it decreases with the rise in relative humidity, attributed to a higher condensation effect required to achieve the desired target conditions. On the contrary, for thermo-hygrometic zone 3, as shown in Figure 13b, the optimal cold water temperature increases with both the air inlet temperature and the relative humidity, up to a saturation curve in which the air inlet condition assumes a higher specific humidity than the one shown in Figure 5. Therefore, under such conditions, the cooling load becomes independent from the relative humidity and relies solely on the air inlet temperature.

7. Conclusions

This paper examines the development of a mechanistic physics-based model capable of predicting the future behavior and energy consumption of an AHU utilized in an automotive production facility. The primary objective is to establish a thermodynamic model that, through the solution of thermodynamic and heat exchanger phenomenological equations, serves as a model predictive control for the analyzed system, replacing traditional control strategies such as manual or PID controllers. The key findings are reported below:

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Expressions for evaluating the global conductance of hot and cold heat exchangers have been calibrated using experimental data from a real case study, depending on the air and water mass flow rates. The fitting processes yielded a coefficient of determination R² of 0.85 for the hot heat exchanger, with a MAPE of about 15% in predicting the effective water mass flow rate. However, lower accuracy was achieved for the cold heat exchanger, with a fitting R² of 0.65 and a MAPE of about 30% in predicting the mass flow rate, due to limited data availability in cooling operating mode.

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The model inputs and outputs for the simulation mode have been determined. Particularly, the model employs as inputs the air inlet temperature and relative humidity, air mass flow rate, target temperature and relative humidity set-points (with relative tolerances), and the boundary conditions of inlet water temperatures of the heat exchangers. Model outputs comprise heat exchanger thermal powers, water mass flow rates, and humidification and condensing water flows.

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A logic based on six thermo-hygrometric zones was established, depending on the external conditions of temperature and relative humidity. Particularly, for each thermo-hygrometric zone, the processes of the moist air to be performed are defined, as well as the employment and status of each component of the system analyzed.

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The model was utilized for two distinct purposes. Firstly, it was employed to simulate the future behavior of the analyzed system in five different examples for each of the thermo-hygrometric zones investigated, evaluating the required heat and mass flow rates of the heat exchangers to meet the user needs under fixed boundary conditions. In this case, comparing the proposed control approach with the standard PID employed by the company in a numerical example, a potential energy saving of approximately 46% has been obtained.

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Subsequently, the model was utilized to optimize the cold inlet water temperature of the heat exchanger, depending on the external operating conditions, in order to ensure the proper functioning of the heat exchanger avoiding too off-design water mass flow rates and working in the maximum performance conditions for the cold water production chiller. A similar approach could also be applied to the hot water production system.

It is worth clarifying that the presented physics-based method always ensures a fair predictability compared to other MPC approaches such as the ones based on machine learning tools. However, the limitation of the model in this case could always be related to the numerosity of data at the disposal of a real case study. In fact, to limit extrapolation and to increase the prediction accuracy, it would be necessary to collect numerous data coming from several different heating and cooling operating conditions, for different external temperatures and relative humidities. In future works, we plan to execute a further re-calibration of the model with a higher amount of data at the disposal both for the heating and for the cooling season. Moreover, a direct implementation of the model predictive control approach on the production line will be considered, in order to assess and demonstrate the effectiveness of the model in terms of energy saving compared to the baseline control strategy.