Simulating nanoparticle concentration using stochastic models to improve indoor air quality in the industry

Research Article Indoor/Outdoor Airflow and Air Quality E-mail: joaquim.ceb[email protected] Simulating nanoparticle concentration using stochastic models to improve indoor air quality in the industry Joaquim Cebolla-Alemany1 (), Marcel Macarulla Martí1, Mar Viana2,3, Santiago Gasso-Domingo1, Verónica Moreno-Martín3, David Bou4, Vicenta San Félix4, Rubén D. López-Carreño1 1. Universitat Politècnica de Catalunya, Department of Project and Construction Engineering, Group of Construction Research and Innovation (GRIC), Terrassa, 08222, Spain 2. Spanish Ministry for Ecological Transition, Pollution Prevention Unit, Madrid, 28071, Spain 3. Institute of Environmental Assessment and Water Research (IDAEA-CSIC), Barcelona, 08034, Spain 4. Institute for Ceramic Technology (ITC-AICE), Castelló de la Plana, 12006, Spain Abstract In industrial scenarios, nanoparticles are incidentally generated in high concentrations during diverse material transformation processes, presenting potential health hazards for exposed workers. Consequently, as an indoor air quality management measure, their concentration is commonly reduced through localized forced ventilation. However, the control of these systems usually relies on traditional rule-based algorithms, which cannot deploy efficient control strategies such as model predictive control. To solve this issue, we propose a novel grey-box reduced order model method, never used before for industrial indoor nanoparticles. This approach can be deployed in model predictive control algorithms in buildings and does not present the data-reliance and transferability issues of black-box modeling. To test this model, a data collection campaign was conducted under real-world operating conditions in an industrial-scale thermal spraying booth, aiming to test the method’s viability for model calibration and validation of indoor total nanoparticle concentration through the maximum likelihood method, statistical validation tests, and physical viability assessment. Results for three different lumped sum models illustrate the effectiveness of grey-box modeling in industrial scenarios with confined processes and forced ventilation systems, handling observations’ noise and background concentration fluctuations, and allowing a performance comparison between models. Further research could be conducted to study the viability of indoor total nanoparticle concentration reduced order models with higher spatial resolution, non-confined sources, and natural airflows. Keywords reduced-order models lumped sum model grey-box modeling industry HVAC Article History Received: 27 September 2024 Revised: 18 December 2024 Accepted: 13 January 2025 © Author(s) 2025 1 Introduction In industrial environments, one of the pollutants which is receiving increasing attention from the European Union concerning indoor air quality (IAQ) is nanoparticles (NP) (Council of the European Union 2024) due to their presence in high concentrations. These particles (also referred to as ultrafine particles (Stone et al. 2017; Kwon et al. 2020)) are suspended in the air in a solid or liquid state and present an aerodynamic diameter below 100 nm. They are incidentally generated (hence named incidental NPs, INP) at high rates by processes such as engine combustion, welding, thermal spraying, laser cutting, grinding, or ceramic tile firing, among others (Bessa et al. 2020; Sonwani et al. 2021). When inhaled in high concentrations, they are a potential source of diverse health hazards (Gwinn and Vallyathan 2006; Kumar et al. 2010; Nemmar et al. 2013). Consequently, industries usually rely on localized ventilation systems to eliminate INPs from the environment (e.g., Jamriska et al. 2003; Hussein et al. 2015). Ventilation systems, however, often rely on traditional reactive rule-based controllers (Gwerder et al. 2010; Borodinecs BUILD SIMUL (2025) 18: 847–862 https://doi.org/10.1007/s12273-025-1245-7 Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 848 et al. 2022) despite the elevated energy consumption of industrial air extraction systems. This approach bolsters the deployment of energy-efficient control strategies (Tashtoush et al. 2005; Prívara et al. 2013; Qian et al. 2024). The emerging tendency to solve this issue in building air extraction and ventilation systems uses model predictive control (MPC) to anticipate and plan the actuation policy based on energy saving, energy flexibility and/or IAQ priorities (see, e.g., Sturzenegger et al. 2016; Afram et al. 2017; Adegbenro et al. 2021; Bird et al. 2022; Wang et al. 2023; Xia et al. 2023; Yue et al. 2023; Klepic et al. 2024). MPC can be deployed through IoT-based solutions employing monitored environmental parameters to adjust ventilation (Sharma et al. 2023) and light models to perform the necessary simulations to find the optimal solution for each situation. This can be conducted through artificial intelligence (AI) algorithms predicting IAQ and optimizing energy consumption (Ren and Cao 2020; Suo et al. 2022), computational fluid dynamics (CFD) models, or multizone models, among others (Feng et al. 2019; Cao et al. 2020). Regarding AI methods, the use of black-box models has also been addressed in the literature through the use of, e.g., artificial neural networks (Xie et al. 2009; Putra et al. 2018; Gaowa et al. 2024) or search algorithms (Zhang et al. 2019; Saini et al. 2022; Al Mindeel et al. 2024). Nonetheless, these algorithms present issues of results transferability to other case studies, not enhancing the comprehension of NPs, and occasionally relying on relatively simple scenarios to show good performance with a fewer amount of data and dynamic requirements, complicating their deployment in industrial buildings (Ren and Cao 2020). Moreover, their data reliance might be critical for INP modeling given the cost and accuracy of current instrumentation. When computational resources and speed are not limiting factors, computational fluid dynamics methods are frequently employed for their high spatial-temporal resolution, enabling highly accurate analyses of ventilation systems (see, e.g., Mousavi 2019; An et al. 2024; Zhang et al. 2024). However, for applications requiring near real-time data and optimization, these models become impractical (Masoumi-Verki et al. 2022; Tariq et al. 2022; Masoumi-Verki et al. 2023). Therefore, reduced-order models emerge as a physicsbased solution to deploy MPC in buildings. Particularly, stochastic (also named grey-box) reduced-order models with lumped sum approaches have been proven competitive for building simulations and management of temperature (Thavlov and Madsen 2015; Brastein et al. 2018), energy (Li et al. 2021), and carbon dioxide concentration (Macarulla et al. 2017, 2018; Wolf et al. 2019a, 2019b). Additionally, they are compatible mass-balance models, highlighted in the previous NP studies due to their transparency and capability to simplify reality without renouncing to include physical processes (Koivisto et al. 2019). However, literature has not yet studied the deployment of grey-box models for other IAQ pollutants, including indoor INP. Previous approaches used for indoor NP concentration estimations (Pirjola 1999; Asmi et al. 2004; Creely et al. 2005; Marquart et al. 2008; Tielemans et al. 2008; Cherrie et al. 2011; Fransman et al. 2011; Alstrup Jensen et al. 2013; Ciuzas et al. 2015; Koivisto et al. 2015; Karl et al. 2022; Macko et al. 2023) rely on the definition of control bands, multiplying factors or other semi-qualitative analyses, including the definition of the user of different system variables. These methods prevent their application on automated control systems or, for example, the definition of necessary airflow rates or extraction time after finishing an INP emitting process to obtain specific concentration levels at working conditions or to clean a polluted space, respectively. These situations are critical in, for example, confined processes with high INP emission rates such as thermal spraying booths, where workers go in and out of the booths while spraying is ongoing to check the pieces or move them. Other explored modeling methods are restrained to highly controlled laboratory conditions, unavailable in industrial settings, to define specific particle processes such as formation (Fonseca et al. 2015), release (Fonseca et al. 2018), deposition (He et al. 2005; Smolík et al. 2005), or coagulation (Seipenbusch et al. 2008). Despite the growing trend in literature to model specific properties of NPs (such as their structure or propagation through different materials for engineered or reinforced nanomaterials (Kim et al. 2008; Zhang 2012; Yan and Xie 2013; Odegard et al. 2017)), research is yet scarce in the modeling of their concentration in complex environments such as indoor industrial scenarios. Regarding environmental NP mass and number concentration monitors, they are based on three broad categories: aerodynamic classification, trajectory observation, and electrostatic classification (Johnson et al. 2020). Within these categories, the used physical principles include optical detection, condensation particle count, scanning mobility particle size, optical particle count, diffusion size classification, and aerosol spectrometry (Ribalta 2019). Optical detectors use a light source to detect particles through their effect on light scattering (McMurry 2000). Condensation particle counters detect small particles based on liquid condensation on to the particles. Scanning mobility particle sizers (TSI 2022) are composed by a condensation particle counter and a differential mobility analyzer that classifies particles depending on their electrical mobility (see Ribalta (2019)) and references therein). Diffusion size classifier miniatures (Testo Inc 2024) are based on the use of a corona charger for particle positive unipolar charging. Finally, wide range aerosol spectrometers combine electrical mobility principles Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 849 with optical light scattering, using the former for the smaller particle diameters. Nonetheless, there exists a recent concern in the literature to develop new methods to optimize low-cost and compact monitors for ultrafine particles, overcoming the challenges of current devices and enabling deployment of sensor networks to directly measure particle concentration for IAQ (Nishida et al. 2020), consequently enabling MPC deployment. Examples of this trend include tandem electrostatic/inertial classification and electrical detection methods below 100 nm (Kwon et al. 2021), NP-imprinted matrices for air and liquids monitoring up to 200 nm resembling the molecularly imprinted polymer approach (Dery et al. 2022), bipolar charging reaching observation of particles between 44 nm and 1.05 μm (Nishida et al. 2020), the combined use of aerodynamic aerosol classifiers and differential mobility analyzers for the 40–596 nm size range (Johnson et al. 2020). Simultaneously, modeling research tries to better predict NP concentrations in both indoor and outdoor environments. For this, statistical analyses, such as Bayesian linear regression models (Hastie et al. 2017; van Erp et al. 2019) explored to predict spatial variations (Ripley et al. 2022) or artificial neural networks for size-resolved number concentration (Al-Dabbous et al. 2017). Mechanistic approaches have also been studied by coupling chemical and kinetic models to estimate NP growth from volatile organic compounds oxidation in gas-phase (Li et al. 2024) or performing simulations of turbulent transport of NPs (Raja et al. 2016; Vanaki et al. 2016; Shimberg et al. 2022). However, recent literature regarding NP modeling mainly focuses on smallscale chemical and medical perspectives, aiming for disease models and quantification biological responses (see, e.g., Eckmann et al. 2020; Rietberg et al. 2021; Leventi et al. 2023; Minnema et al. 2023; de Roode et al. 2024), or nanomaterials (such as in (Amodeo and Pizzagalli 2021; Choi et al. 2022; Novozhilov et al. 2023; Spelthann et al. 2023)). Consequently, it can be stated that despite the advancements in NP concentration modeling, there remains a significant research gap in applying it to environmental NP concentration in industrial scenarios, specifically in stochastic grey-box models for real-time industrial indoor NP concentration prediction, which would result crucial for energy optimization in industrial settings controlling ventilation systems. Consequently, the novelty of this paper lies in the introduction and validation of a stochastic grey-box modeling approach to accurately predict INP concentrations within industrial environments. This study aims to demonstrate the efficacy and practical applicability of this statistical method, bridging the gap between theoretical modeling and real-world industrial applications involving NP-related indoor air quality. By leveraging the strengths of stochastic grey-box models, we offer a robust solution to model industrial NPs and manage IAQ, enabling more efficient and informed control strategies in environments where traditional NP modeling methods fall short. This paper is organized as follows. Section 2 describes the methodology used to model NP and details the case study under investigation. Section 3 presents the results obtained from the simulations and experimental validation. Section 4 discusses the implications of these results, highlighting their significance in the context of current research and potential applications. Finally, Section 5 provides conclusions and suggests directions for future research. 2 Materials and methods The process followed in this research to apply grey-box modeling to INP concentration simulations in indoor industrial environments is conducted in three stages: industrial data collection in operative conditions, total INP number concentration modeling, and model validation. During data collection stage, total nanoparticle number concentration is measured and situations of INP generation, forced ventilation, and natural airflow presence are monitored. Then, for INP concentration modeling, an ordinary differential equation (ODE) is proposed to describe the deterministic behavior of the system, subsequently transformed into a stochastic differential equation (SDE) to conduct parameter calibration and model simulation. Finally, models are validated through significance and true optima tests of their parameters, a feasibility assessment of the calibrated values, the absolute residuals analysis, a white noise test, and their representation in a Taylor diagram. 2.1 Experimental design and data collection The location under study is a precision mechanics industrial facility belonging to the company T.M. Comas, located in Blanes, Girona, Spain. This company specializes in the repair of high-precision components for various industries, including the petrochemical, pharmaceutical, and energy sectors. Several activities take place in this facility to repair parts, such as welding, sandblasting, cutting, or plasma spraying. Focusing on the plasma spraying processes, a workshop of approximately 375 m2 is dedicated to this activity. This workshop is separated from the rest of the plant by a wall with two doors: one for workers and another for forklift trucks, usually kept closed. This area contains three thermal spraying booths of different volumes where two different processes are conducted: high-velocity oxy-fuel coating spraying and atmospheric plasma spraying (APS). This space was previously studied by Ribalta et al. (2019b), but it has since undergone modifications to isolate the thermal spraying processes from the worker area (WA). Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 850 The present study focuses on booth #2, where APS is conducted with dichromium trioxide (Cr2O3) at high temperatures (5000–20000 °C) and low spraying velocity (200–500 m/s). The process was carried out intermittently for two consecutive days in July 2022 during the reparation of different pieces series. This booth has a volume of 102 m3 and a ventilation system shared with booth #3 with a total airflow of 15000 m3/h, split or not depending if both are working or not in function of clients’ requests, having the extraction flow activation manually controlled by the worker while spraying and five more minutes to clean the booth after finishing the APS. During the data collection, booth #3 had no activity, so booth #2 used the total ventilation capacity, resulting in roughly 147 air changes per hour. The air for the ventilation is obtained directly from the WA through two different points at both sides of the booth, as seen in Figure 1. These points are connected through an air duct to the rest of the booth by its center. Then, the airflow leaves the booth through the ventilation system on the opposite wall. All booths work independently of the others and follow irregular and intermittent routines organized by the priority for the client companies, the size and number of pieces, the occupancy of the booths, and the preparation of the (a) (b) Fig. 1 (a) Thermal spraying booth #2 photography and (b) schematic diagram of the thermal spraying area and modeled booth at T.M. Comas facilities with dimensions (m) components, among other criteria. In parallel, the forklift truck’s access door opens a few times daily to transport large pieces from or into the WA, affecting its concentration. Therefore, Booleans describing the activation of the spraying pistol, mechanical ventilation, and booth’s door opening were manually monitored during the experiment. The instruments used to collect NP concentration data were two NanoScan SMPS TSI Model 3910 (NS), an electrical mobility spectrometer that measures particle number concentration between 100 and 106 particles/cm3 and particle size distribution from 10 to 420 nm (TSI 2022). One was placed inside the booth and another in the WA in front of the booth’s door. To be able to measure concentrations over the instrument’s limit inside the booth, the 1:100 dilution system DIL 550 (Topas 2019) was used. These instruments captured size-resolved concentration data with 1-minute time resolution, which was thereafter converted through direct sum into total concentration data. The manufacturer’s inversion algorithm was used to convert the monitor’s output to particle size distribution data (TSI 2022). The reliability of this instrument for occupational and indoor air quality monitoring was based on existing literature reviews (Stabile et al. 2014; Fonseca et al. 2016; Jørgensen 2019). Particle number size distributions describing observations inside and outside the booth during the measurements are represented in Figure 2, calculated following the procedure described in Cebolla-Alemany et al. (2024) with a Pearson coefficient for the fitting to a lognormal distribution exceeding 0.996 in both cases. Additionally, total concentration data was averaged every two minutes to reduce noise before being used for model calibration and parameterization. A summary of the system’s most relevant data is available in Table 1, while monitored data (booth’s concentration, WA concentration, and Booleans for generation, ventilation, and booth door opening) can be seen in Figure 3. Fig. 2 Particle number size distributions inside (Nin) and outside (Nout) the booth during the measurement Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 851 Table 1 Main features of the case study Parameter Value Booth’s volume (m3) 102 Booth’s door height × width (m) 2.4×2.2 Ventilation flow rate (m3/h) 15000 Minimum particle size (nm) 10 Maximum particle size (nm) 420 Process APS Spray material Cr2O3 2.2 Modeling process Grey-box models integrate information from observations with physical knowledge of a system (Bacher and Madsen 2011). The system’s physics is represented through a set of first-order stochastic differential equations consisting of a drift term and a diffusion term. The drift term describes the deterministic processes governing the system. For this study, three models (M1, M2, and M3) describing NP concentration inside the modeled booth have been proposed and evaluated. The first studied model (M1) considers generation, forced ventilation, and diffusion phenomena. It is the simplest of the three and it is based on well-established literature (see, e.g., Ribalta et al. (2019b) and references therein). Its equation is: in source ventilation diffusion dd NJJ J t=+ + (1) where dNin/dt is the particle number concentration variation through time inside the booth, Jsource is the INP generation term inside the booth, Jventilation is the term that represents the change when mechanical ventilation for air extraction is on, and Jdiffusion describes the variation for diffusion processes between the booth and the WA when the door is open. These terms can be further developed, resulting in Equation (2): ()() out in ven out in dif in ddgS v N N Q d N N Q N tV ⋅+⋅-⋅+⋅-⋅ = (2) where Nin is the number concentration inside the studied booth, Nout the number concentration in the WA, S is the source strength or emission rate of the process, V is the volume of the booth, Qven is the flow rate that comes in from the WA and leaves the booth through the ventilation system when it is on, Qdif is the flow rate that occurs by diffusion between the booth and the WA when the booth’s door is open, g is a Boolean variable describing when there is generation, v is another Boolean indicating when mechanical ventilation occurs, and d is another Boolean that activates the door of the booth is open. Equation (2) stems from previously used ideal gasses’ mass-balance reduced-order models (Macarulla et al. 2017, 2018) and similar models have already been used in literature for aerosol and other pollutants concentration, not restricted to NPs (see, for example, Furtaw et al. (1996) and Jensen et al. (2018)). (a) (b) Fig. 3 (a) Boolean inputs for the model training representing the instants of generation (g), ventilation (v), and door opening (d) in the booth; and (b) concentration data collected inside (Nin) and outside (Nout) the booth Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 852 The other two models under study (M2 and M3) include a losses term (Jlosses) representing aggregation and deposition phenomena, extensively described in the literature (see, e.g., Jacobson (2005); Hussein et al. (2009); Yu et al. (2013)), although some studies concluded that these losses are not significant (Ribalta et al. 2019b). For these two models, Equation (1) changes to Equation (3): in source ventilation diffusion losses dd NJJ J J t=+ + + (3) The losses term is defined in two different ways as shown in Equation (4) and Equation (5), resulting in the first model M1 without losses, a second with a linear decrease of particles (M2), and a third with a quadratic loss rate (M3), where Kloss is the particle loss rate that includes both deposition and aggregation over the measured size range, simplifying the aggregation and deposition components used in Jensen et al. (2018), for example, as it can be observed in Equation (6), where aggregation and deposition phenomena are a function of the Fuks coagulation coefficient Kaggregation between particles (Fuks 1964), k is each possible deposition direction, Ak is the available area in each direction k, and vk, is the velocity of particle deposition. loss in losses KN JV ⋅- = (4) 2 loss in losses KN JV ⋅- = (5) aggregation deposition aggregation in in 11 2kk k JJ K vNAN V ⋅⋅--⋅+= å (6) System inputs, that is, concentrations inside and outside the booth (from the concentration monitors) and the Booleans (monitored in the field), serve to train and validate the models as previously described on grey-box modeling literature (see, e.g., Thavlov and Madsen 2015; Macarulla et al. 2017, 2018; Brastein et al. 2018; Tugores et al. 2024). These inputs are represented in Figure 3. These equations assume a known background concentration (since several non-monitored activities act as sources influencing the INP concentration of air coming from the plant). Other assumptions are that the air entering the booth equals the air leaving, that generation and airflows are constant through time, and there is a perfect mixture of NPs within the booth and in the WA between timesteps (Jayjock et al. 2011; Jensen et al. 2018). It is also assumed that diffusion prevails over natural airflows. Consequently, when there is no forced ventilation, particles always move from zones with high concentrations to zones with lower values. This hypothesis is reasonable given the absence of windows and doors connected to the outdoors in the WA causing wind airstreams. Additionally, Equations (4) and (5) assume that losses caused by deposition and aggregation are irreversible, and no resuspension is contemplated (Jensen et al. 2018). To transform an ODE into an SDE, the diffusion term representing measurement noise and the impact of unmodeled or unknown disturbances is incorporated. This term consists of a function describing the disturbances of the system and a Wiener process representing the white noise. Moreover, observations in discrete time include a Gaussian-distributed measurement error. Consequently, a set of first-order SDEs has to be defined. The generalized equations (Macarulla et al. 2017, 2018; Tugores et al. 2024) of this system are: () () d,,,d,d ttt t X f X U θt t Gθt W=⋅+⋅ (7) () ,,, k tttt YhXUθte=+ (8) where Xt is the vector of system states, Ut is the vector that contains experimental measurements, θ is a vector with the parameters that have to be identified, k t Y is the INP concentration measurement from inside the booth in the instant tk, h(Xt, Ut, θ, t) is the relationship between measurements and the state variables, and et is a vector describing the white noise from the measurements. In this system, the drift term is f(Xt, Ut, θ, t)·dt, where f(Xt, Ut, θ, t) equals Equation (1) for M1 and Equation (3) for M2 and M3; the diffusion term is G(θ, t), and dWt is a Wiener process. Rewriting the state-of-space system for M1, the result is: () in source ventilation diffusion dddNJ J J tσw=+ + ⋅+⋅ (9) in, kk ttk YN e=+ (10) where dw is the Wiener process, σ is the incremental variance in the Wiener process, in, k t N is the measured NP number concentration inside the booth, and ek describes the noise of the measurements. Table 2 summarizes the three exposed models and their estimated parameters (excluding the initial concentration for the simulations): emission rate S, airflow rates Qven and Qdif, Kloss, σ, and ek. The Continuous Time Stochastic Modeling library for R (CTSM-R) (Juhl et al. 2016) has been used for the model calibration and simulation, including the calculation of the booth’s initial concentration (N0). This library uses the maximum likelihood method for this purpose, commonly employed in the literature (Thavlov and Madsen 2015; Andersen et al. 2000), solving equation systems in Table 2 through the gathered experimental data. This method requires the definition of an initial assumption for each estimated parameter and the lower and upper bounds for the search Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 853 of the optimal solution for the simulation. Therefore, the initial values and bounds for each calibrated parameter can be seen in Table 3. Moreover, this library also provides the means for conducting the statistical tests described in Section 2.3. Table 3 Initial values and bounds Initial value Lower bound Upper bound N0 (particles/cm3) 60000 40000 500000 ek 5 0 100 σ 0.01 0 10 Kloss (cm6/(particles·min)) 1000 10 10000 Qven (cm3/min) 108 106 109 Qdif (cm3/min) 106 104 108 S (particles/min) 1015 1013 1016 2.3 Model validation The proposed validation process is based on previous studies (Thavlov and Madsen 2015; Brastein et al. 2018; Wolf et al. 2019b; Li et al. 2021). It includes statistical tests to assess parameter significance, and if calibrated values are true optima. Then, the physical feasibility of parameter estimations is analyzed. Afterwards, residuals are studied to verify the white noise assumption in the measurements, and absolute residuals are compared with the instruments’ tolerance. Finally, the Taylor diagram representation completes the validation through a visual comparison of the models’ performance. First, a set of statistical tests are used to validate the grey-box models through the CTSM-R library (Andersen et al. 2000; Kristensen et al. 2004; Bacher and Madsen 2011). The p-value of each parameter in the model, obtained from the t-test scores, has to be less than 0.05 for the parameter to be significant, assuming a significance level of 95%. Following, the calculation of the derivative of the objective function with respect to the initial conditions (dF/dPar) assesses whether the solution obtained is a true optimum. The result is close to zero when the value is the true optimum. Then, the penalty function’s derivative with respect to the initial state (dPen/dPar) is used to know if the parameter or its initial state is too close to one of its limits. Second, absolute residuals of the simulations are calculated to test if the model’s accuracy falls within the instrument tolerance range. Moreover, the one-step-ahead residuals analysis shows if the white noise residuals assumption is fulfilled through the autocorrelation function (ACF) residuals and the cumulated periodogram (Andersen et al. 2000; Bacher and Madsen 2011). Then, the physical feasibility of the estimations is studied given the known system’s information coming from design and equipment specifications. Finally, the Taylor diagram (Taylor 2001) for all the models is also calculated to compare their performance visually. This graph simultaneously represents the rootmean-square error, the normalized standard deviation, and the correlation coefficient between modeled data and observations. Each model is shown as a point on the diagram and thus allows for an intuitive and comprehensive assessment of multiple performance metrics in a single plot. 3 Results In this section, presented models are validated and intercompared. Feeding the three models in Table 2 with observations in Figure 3, and given the initial values and boundary conditions for the calculations shown in Table 3, the estimated parameters, their standard deviation (SD), and the values from the validation process are shown in this section. Table 4 summarizes data involving M1. Observing the p-values, all of them are below 0.05 and so are significant. Taking a look at the derivative of the objective function with respect to the initial conditions and the penalty function’s derivative with respect to the initial state, all parameters are near 0, concluding that they are true optima and are not Table 2 Studied grey-box models Model State space representation Estimated parameters M1 ()() out in ven out in dif in ddd gSvN NQ dN NQ Ntσw V ⋅+⋅ - ⋅ +⋅ - ⋅ =⋅+⋅ in k tk YNe=+ S, Qven, Qdif, σ, ek M2 ( ) ( ) out in ven out in dif loss in in ddd gSvN NQ dN NQ KN Ntσw V ⋅+⋅-⋅+⋅-⋅- =⋅+⋅ ⋅ in k tk YNe=+ S, Qven, Qdif, Kloss, σ, ek M3 ()() 2 out in ven out in dif loss in in ddd gSvN NQ dN NQ KN Ntσw V ⋅+⋅-⋅+⋅-⋅-⋅ =⋅+⋅ in k tk YNe=+ S, Qven, Qdif, Kloss, σ, ek Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 854 Table 4 Estimation results for M1 (no losses) Estimate Std. error p-value dF/dPar dPen/dPar N0 (particles/cm3) 60000 15 0.0000 −1.2362×10−3 −0.0006 ek 26.6 0.1 0.0000 1.8807×10−2 0.0000 σ 0.01 0.00 0.0000 7.0650×10−4 0.0000 Qven (cm3/min) 2.45×108 0.45×108 0.0000 2.9107×10−3 0.0000 Qdif (cm3/min) 1.00×106 7900 0.0000 −9.7668×10−2 0.0000 S (particles/min) 1.38×1015 0.25×1015 0.0000 8.0814×10−4 0.0000 limited by their distance to the boundary values of the calculations, respectively. Regarding its estimations, the initial concentration N0 of 60000 particles/cm3 is coherent with the measurements, which start with values between 60000 and 65000 particles/cm3; the ventilation flow rate Qven of 2.45 × 108 cm3/min with an SD of 0.45 × 108 cm3/min shows a reasonable value within the system design value of 2.5 × 108 cm3/min, resulting in an air change per hour rate (ACH) of roughly 144 h−1 considering the booth volume of 102 m3; the diffusive flow rate Qdif through the booth’s door shows an airflow speed below 0.1 m/s (having a speed of 0.007 m/s if considering the most usual situation of just half door open), confirming that the WA, separated to the rest of the plant by a wall and two usually closed doors, does not contain major air streams when ventilation is off inside the booths; the source’s emission rate S of 1.38 × 1015 particles/min is two orders of magnitudes higher than previous analysis in literature (Ribalta et al. 2019b). However, these results are incomparable given a substantial change in the system’s layout: the enclosure was different in the previous setting, as was the ventilation system. Additionally, used data in Ribalta et al. (2019b) to calculate the emission rate was from the WA, not from the booth. Consequently, they defined their particle emission rates as transported emission rates to the WA from the booths, so they did not quantify the process’ emission, but the transported INP from the booth to the WA. Therefore, since the rest of the calculated parameters in M1 are consistent with reality and design specifications, the estimation is considered reliable. Observing M2 information in Table 5, p-values also show that all parameters are significant for the system, and the penalty function’s derivative with respect to the initial state confirms again that the estimations and their initial values are not too close to their limits. Initial concentration, both flow rates, and the emission rate are the same as the calculated in M1, additionally showing similar SDs. The loss rate Kloss remains at the starting value of 1000 cm6/(particles·min), although the derivative of the objective function with respect to the initial conditions test shows that the solution is a true optimum, like all the other parameters. In M3 (see Table 6), the p-values show that Kloss is not significant, and all the estimations are true optima far Table 5 Estimation results for M2 (linear losses) Estimate Std. error p-value dF/dPar dPen/dPar N0 (particles/cm3) 60000 13 0.0000 1.1439×10−3 0.0006 ek 26.6 0.07 0.0000 −4.2010×10−3 0.0000 σ 0.01 0.00 0.0000 −8.6994×10−5 0.0000 Kloss (cm6/(particles·min)) 1000 0.6 0.0000 −1.8664×10−4 0.0000 Qven (cm3/min) 2.45×1080.21×108 0.0000 −9.2949×10−3 0.0000 Qdif (cm3/min) 1.00×106800 0.0000 −1.0066×10−1 0.0000 S (particles/min) 1.38×1015 0.12×1015 0.0000 1.1087×10−3 0.0000 Table 6 Estimation results for M3 (quadratic losses) Estimate Std. error p-value dF/dPar dPen/dPar N0 (particles/cm3) 60000 643 0.0000 4.0134×10−3 −0.0006 ek 26.6 0.06 0.0000 −3.4132×10−4 0.0000 σ 0.01 0.00 0.0000 −3.9153×10−7 0.0000 Kloss (cm6/(particles·min)) 11.2 7.4 0.1289 2.3045×10−3 −0.0079 Qven (cm3/min) 9.37×1082.82×108 0.0010 8.9533×10−3 0.0240 Qdif (cm3/min) 1.00×1060.50×106 0.0000 −2.8549×10−2 0.0000 S (particles/min) 5.63×1015 1.35×1015 0.0000 2.3030×10−4 0.0003 from their calculation limits. In this model, the initial concentration and the diffusion flow rate are similar to the estimations for the other two models, but their standard errors increase. The source’s emission rate and the ventilation flow rate roughly quadruple in value and present an SD of their order of magnitude, resulting in an extraction beyond the physical capabilities of the system and increasing the uncertainty of the model. The loss rate Kloss decreases with respect to M2 to 11.2 cm6/(particles·min) and also gets a high SD. Observing the models’ simulations and their absolute residuals in Figure 4, all models exhibit similar behaviors. Concentration rapidly ascends and descends when generation is turned on and off, and it slowly decreases parabolically when the door opens (and, therefore, diffusion occurs). Steady states with generation and mechanical extraction activated present a concentration of 5.7 × 106 particles/cm3. When the door is closed and no generation or ventilation occurs, M1 and M2 present almost identical behaviors. For M1, concentration remains invariant in these situations. For M2, concentration subtly decreases linearly. For M3, concentration diminishes quadratically when the door is closed and, once it opens, the drop accelerates, as can be clearly observed around 19:00 on July 19. Since all these differences occur at lower concentrations, absolute residual plots show minor differences, having high values at the beginning and the end of the simulations due to the high order of magnitude in the particle concentration affected by Cebolla-Alemany et al. / Building Simulation / Vol. 18, No. 4 855 measurement noise amplified due to the dilution system’s effect on the instrument. Analyzing ACF residuals and cumulated periodograms (Figure 5), they present high similarity among them. Some lags of the ACF residuals are outside the confidence interval of 95%. 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