Identification of source location in a single-sided building with natural ventilation: Case of interunit pollutant dispersion

Abstract

A sudden outbreak of COVID-19 occurred in December 2019 and its rapid spread over the last two years caused a global pandemic. A special airborne transmission via aerosols called interunit dispersion is risky in a high-density urban environment, which needs more attention. In order to identify the source location of pollutants or viruses under the interunit transmission condition with natural ventilation, this study adopted the inverse Computational Fluid Dynamics (CFD) simulation with the adjoint probability method. The detailed process of the inverse modeling was presented. Also, the possible interunit transmission routes of the pollutants or viruses were analyzed. A three-story building model with single-sided openings was built. Six different combinations of fixed sensor locations were tested, and it was determined that setting sensors in the four corner regions of the building was the optimist strategy. A total of 25 cases with five different wind directions ($0°$, $45°$, $90°$, $135°$, and $180°$) were tested to verify the accuracy of the source location with inverse modeling. The results showed that 67%–78% of the rooms in the building can be identified with a limited number of pollutant sensors and all rooms can be identified with one additional sensor in the downstream room of the building under different wind direction. This research revealed that the inverse modeling method could be used to identify the pollutant source in the coupled indoor and outdoor environment. Further, this work can provide guidance for the pollutant monitor positions in the applications.

Greek symbols

$\epsilon$

Turbulent viscous dissipation rate ($m^2/s^3$)

${\eta }_0$

Model constant

$\xi$

Model constant

$\kappa$

Von Karman constant, 0.41

$\rho$

Density ($Kg/m^3$)

$\nu$

Turbulent viscosity ($m^2/s$)

$\phi$

Adjoint probability

$\tau$

Backward time (s)

$\delta (·)$

Dirac delta function

Nomenclature

$A$

Opening area ($m^2$)

$C_1$

Pollutant concentration per unit volume at $\overrightarrow{x}=\overrightarrow{x_1}$ (ppm)

$C_i$

Concentration of the $i^{th}$ measurement (ppm)

$C_{local}$

Measured tracer gas concentration (ppm)

$C_{source}$

Source tracer gas concentration (ppm)

$D$

Molecular diffusion coefficient ($m^2/s$)

$D_W$

Window height, 0.08 m

$D_Z$

Building height, 0.6 m

$h$

Function of the state system

$H_1$

Building height, 0.25 m

$H_2$

Building height, 0.125 m

$k$

Turbulent kinetic energy ($m^2/s^2$)

$K_c$

Non-dimensional concentration

$l_t$

Mixing length, 0.4 m

$M_0$

Total source release mass

$M_{i-j}$

Mass fraction

$p$

Pressure (Pa)

$q_O$

Outflow rate per unit volume

$Q_{source}$

Flow rate of the source emission ($m^3/s$)

$S$

Scale of strain rate

$S_{ij}$

Stain-rate tensor

$t$

Denotes time

$T_i$

turbulence intensity, 4%

$u_i$

Velocity component

$u_{*}$

frictional velocity, 1.068 m/s

$U$

Wind velocity (m/s)

$U_{ref}$

Reference wind speed (m/s)

$Z_0$

Roughness height, 0.0005 m/0.00075 m

1. Introduction

Indoor air quality (IAQ) is closely related to human health because people spend about 80–90% time indoors [1]. Poor IAQ can cause a range of health problems, such as asthma [2], pneumonia [2], and bronchitis [3]. In the last two decades, the outbreaks of the severe acute respiratory syndrome (SARS), bird flu, influenza (H1N1), and COVID-19 [[4], [5], [6]] have seriously affected human health and even cause millions of deaths. Except for the direct-contact transmission and large-droplet contact transmission [7], the airborne transmission is another important route [[8], [9], [10]]. A typical example is the large outbreak of SARS in the Amoy Gardens housing complex, which affected more than 300 residents of this private housing estate in 2003 [11]. Therefore, in order to avoid the spread of pollutants or viruses in buildings, it is crucial to understand the characteristics of airborne transmission. Further, in the situation of a large outbreak, it is essential to find the certain location of the contaminant source or affected points.

Discerning the transmission route of viruses or pollutants in buildings is necessary when finding the locations. Contaminants or bioaerosols are commonly transmitted from the outdoor environment to the nearby indoor environments. Apart from this common route, a special airborne transmission was identified during the outbreak of SARS in Hong Kong, and the cross-transmission between different units in the same building was called interunit dispersion [12,13]. This airborne transmission route is risky, essentially because of its short transportation distance and the possibility of involving infectious aerosols [13,14]. Yu et al. [11] and Li et al. [15] first used CFD and multizone modeling to analyze the spread of the SARS virus in Amoy Gardens, and the results confirmed the high probability of airborne transmission. Using on-site measurement, Niu and Tung [16] concluded that 7% of the exhaust air from the lower floor could re-enter the upper room with buoyancy effects. This indicated that pathogen-laden aerosols could spread to the upper floor through the windows on the same facade from the lower one. Mu et al. [17,18] carried out a set of wind tunnel experiments to investigate the wind induced air pollutant transmission and cross contaminant routes in the multi-story building. Ai et al. [13,19,20] performed a series of numerical simulations to quantify the reentry of exhaust air from each distinct unit to other units in a multistory building and reveal the possible dispersion routes. Recently, Dai et al. [14,21] conducted scale outdoor experiments to investigate the interunit dispersion problem in a 2D street canyon under urban weather conditions. These studies demonstrated a high probability of pollutants spreading between rooms in the same building. Therefore, the dispersion of pollutants or viruses between units cannot be ignored. It is essential to identify the source location in such coupled indoor and outdoor environment.

Currently, there are two main modes to locate pollutant sources. One is the robot active olfaction method, which uses single or multiple mobile robots carrying anemometers and gas sensors to locate pollutant sources, especially odor sources, based on bionic algorithms. Many researchers have continuously improved and innovated the robot active olfaction method to make this method be used for contaminant locating in 2D ventilated rooms [22] and 3D indoor environments [23]. Soon afterward, Yang et al. [24,25] and Feng et al. [26] carried out a large number of experiments, using the robot active olfaction method to locate the contaminant release location in a single room accurately. Recently, Jiang et al. [27] used two multi-robot olfaction methods to identify the time-varying source, which is more difficult to identify than the constant ones. However, due to the limitation of robot movement, it is not suitable to apply the robot to locate the pollution source in the whole building. Therefore, the robot active olfaction method is restrained in the indoor situation.

Another is the stationary sensor networks method, which uses the contaminant concentration data detected by the pre-installed sensor probe, combined with the CFD model or multi-zone model to estimate the source location by forward or backward calculation. Liu and Zhai [28] summarized the features of existing inverse transport modeling methods and categorized them into the forward method, the backward method, and the probability method. The forward method needs large numbers forward simulation to verify the source location. Accordingly, the forward method is a time-consuming process. In addition, this method is challenging to accomplish when the prior source information is uncertain [28].

The backward method starts from the final contaminant concentration field and uses a negative time step in the simulation process to obtain the location of contaminant sources. Zhang and Chen [29] proposed the quasi-reversible (QR) and pseudo-reversibility (PR) to identify the pollutant sources in indoor environments and successfully found pollutant sources in enclosed cabins [30] and office rooms [31] in their further work. However, the backward method demands important prior source information, such as the source location and the release time, to calculate the magnitude of the contaminant. Eventually, this preliminary source information is challengeable to obtain, which obstructs the application of the backward method.

The probability method obtains the pollutant source information by maximizing or minimizing probability objective functions. Firstly, Neupauer and Wilson [32] proposed an adjoint method to identify contaminant sources in one-dimensional groundwater system using back-in-time location and travel time probabilities. Liu and Zhai [33] introduced the fundamentals of the CFD-based adjoint probability method and used it to track the location of instantaneous sources in an enclosed environment. Then, Zhai et al. [34] formulated a standard adjoint location probability (SALP-D) method to locate a dynamic source in the indoor environment. Wang et al. [35] proposed an improved adjoint probability method to find the source location in the time-dependent airflow field. Zeng et al. [36] employed the probability-based inverse model to characterize the instantaneous pollutant source location and releasing time within a ventilation system and conducted a concentration-measured experiment to prove the feasibility of its applicability. Xue and Zhai [37] exploited an inverse modeling algorithm to track multiple outdoor pollutant sources with a mobile sensor. Recently, Hu et al. [38] applied the joint simulation of CFD and the multizone model to quickly and accurately locate pollutant source in multiple rooms. Liu et al. [39] proposed a synthetic inverse model to quantify the strength of the dynamic release source and identify the source location. Recently, Zeng et al. [40,41] established Marko-chain-based inverse modeling to identify the releasing rate of sources and estimate the source location in buildings with mechanical ventilation systems. The probability method has two main advantages, accepting large random errors in the velocity result [42] and unnecessary to specify a potential source field [42]. Similarly, the probability method requires concentration and airflow field in the building environment as the primary input information. Concentration data can be detected by the sensors installed in the building, and the airflow field can be obtained by CFD simulations before pollutants dispersion. The probability method is widely used when little information is known about the pollutant source, especially without a potential source location.

Therefore, in order to deal with the unexpected contaminant leakage or virus transmission in the building, it is important to identify the source location in a short period. The adjoint probability method can be effective in such circumstances.

Most of the above studies mainly investigated the pollutant source identification in a pure indoor or outdoor environment, and few considered the interunit condition with the coupled indoor and outdoor environment. Natural ventilation, as a passive energy-saving technology, can improve both indoor air quality and indoor thermal comfort [[43], [44], [45], [46]]. In order to identify the source location of pollutants or viruses transmitted between units in buildings by natural ventilation, this study applied inverse modeling in a multistory residential building with the adjoint probability method. A three-storey building model with a single-sided opening was built. In this study, carbon dioxide (${\mathrm{C}\mathrm{O}}_2$) is used as tracer gas to simulate the viruses which is generally effective for inter-unit dispersion of small bioaerosols (order of $5\mu m$ or lower) [[47], [48], [49], [50]]. Based on CFD simulations, the characteristics of airflow and concentration fields in and around the building are investigated. Six different combinations of sensor installation locations were tested based on the interunit dispersion paths of pollutants in the building environment. The optimist strategy, setting sensors in the four corner rooms of the building, was selected. According to the concentration information obtained by the fixed sensor, the pollutant sources in most regions of the building under different wind directions can be identified [51]. The results intend to help prevent the spread of pollutants or viruses in the building environment. Section 2 introduces the CFD method, and the principles of probability-based inverse modeling method. Section 3 demonstrates the validation process of CFD method. Section 4 presents the configuration descriptions of the building model. Section 5 shows the specific analysis of the simulation results. Section 6 offers the discussions of the results, and section 7 concludes the study.

2. Methodology

In order to provide airflow and contaminant transport inside and around the buildings, the CFD method has seen extensive application in recent years. The RNG $k-\epsilon$ model, which is an enhancement over the standard $k-\epsilon$ model [52], has been selected as the turbulence model in this study. This study chooses the adjoint probability method for the inverse modeling, which needs to be combined with CFD forward simulation results.

2.1. CFD methods

The airflow field in the computational domain is modeled as a steady state, three-dimensional Navier-Stokes equations for a confined, incompressible viscous flow of a Newtonian fluid using Eq. (1) and Eq. (2) as following

$$\frac{\partial u_i}{\partial x_i}=0$$ (1)

$$\frac{\partial u_i}{\partial t}+\frac{\partial }{\partial x_j}\left(u_i{u}_j\right)=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left(2v{s}_{ij}\right)$$ (2)

where $u_i$ is the velocity vector in $x_i$ direction, $p$ is the pressure, $\rho$ is the density, $v$ is the molecular viscosity, $s_{ij}$ is the stain-rate tensor as defined in Eq. (3).

$$s_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)$$ (3)

The concentration field in the computational domain is modeled in a transient process, which can be written as following

$$\frac{\partial c}{\partial t}+\frac{\partial }{\partial x_j}\left(c{u}_j\right)=\frac{\partial }{\partial x_j}\left(D\frac{\partial c}{\partial x_j}\right)$$ (4)

where $t$ denotes time, $c$ is the transient concentration, $D$ is the molecular diffusion coefficient. The Renormalization Group (RNG) $k-\epsilon$ model is derived from a rigorous statistical technique, which was proposed by V. Yakhot et al. [53]. It is formally similar to the standard $k-\epsilon$ model, but added the strain-dependent term $R_{\epsilon }$ to the $\epsilon$ equation which improves the accuracy for swirling flow. The added term $R_{\epsilon }$ is shown by the equation as:

$$R_{\epsilon }=\frac{C_{\mu }\rho {\eta }^3\left(1-\eta /{\eta }_0\right)}{1+\xi {\eta }^3}·\frac{{\epsilon }^2}{k}$$ (5)

where $C_{\mu }$, ${\eta }_0$, and $\xi$ are model constants, and $\eta \equiv Sk/\epsilon$ where $S$ is the scale of strain rate.

2.2. Principles of adjoint probability method

The location probability is a fundamental concept in the adjoint probability method. Releasing a pollutant mass of $M_0$ into the domain and spread to the entire region, if the pollutant concentration per unit volume of $C_1$ is monitored at position $\overrightarrow{x}=\overrightarrow{x_1}$ at $t=T$, then the forward location probability density function is expressed as:

$$f_x\left(\overrightarrow{x_1};t=T,\overrightarrow{x_0}\right)=\frac{C_1}{M_0}=\frac{C\left(\overrightarrow{x},T\right)}{M_0}$$ (6)

where $f_x\left(\overrightarrow{x};t=T,\overrightarrow{x_0}\right)$ is the forward location probability density at $t=T$, $C_1$ is the pollutant concentration per unit volume at $\overrightarrow{x}=\overrightarrow{x_1}$, $M_0$ is the total source release mass, $C\left(\overrightarrow{x},T\right)$ is the resident concentration distribution of the pollutant at $t=T$ after the release of the point source.

Before establishing the backward location probability, a time parameter $\tau =T-t$ is defined as the backward time, so the location probability density can be expressed as

$$f_x\left(\overrightarrow{x};t=T,\overrightarrow{x_0}\right)=f_x\left(\overrightarrow{x};\tau =0,\overrightarrow{x_0}\right)={\phi }_x^{*}\left(\overrightarrow{x_0};\tau =T,\overrightarrow{x}\right)$$ (7)

where ${\phi }_x^{*}\left(\overrightarrow{x_0};\tau =T,\overrightarrow{x}\right)$ is defined as backward location probability.

2.3. Backward adjoint probability method based on CFD method

Based on the principles of the adjoint probability method, in order to obtain the source location probability, it is necessary to combine the backward contaminant transport equation for CFD method. Liu and Zhai [28] proposed the detailed mathematical derivations and provided the adjoint equation as follows:

$$\frac{\partial {\phi }^{*}}{\partial \tau }-\frac{\partial V_j{\phi }^{*}}{\partial x_j}=\frac{\partial }{\partial x_j}\left[v_{c,j}\frac{\partial {\phi }^{*}}{\partial x_j}\right]+\left(-q_0{·\phi }^{*}\right)+\frac{\partial h}{\partial C}$$

$${\phi }^{*}\left(\overrightarrow{x},0\right)=0$$ (8)

$$\frac{\partial h}{\partial C}=\delta \left(\overrightarrow{x}-\overrightarrow{x_w}\right)·\delta \left(\tau \right)$$

where ${\phi }^{*}$ is adjoint probability, $\tau$ is the backward time, $V_j$ is air velocity at $x_j$ direction, $v_{c,j}$ is effective turbulent diffusion coefficient for $C$ at $x_j$ direction, $\overrightarrow{x_w}$ is the place of the measurement. $q_0$ is the outflow rate per unit volume, $h$ is a function of the state system, $\frac{\partial h}{\partial C}$ is the load term specified by Neupauer and Wilson [54]. $\delta \left(\tau \right)$ is the Dirac delta function.

By solving Eq. (8), the calculation results of multiple measuring points can be integrated, and the only possible pollutant source location can be obtained. Lin [55] derived the conditioning equations, the adjoint location probability with $N$ measurements is expressed as follows:

$$f_x\left(X丨C_1,\dots ,C_N;{\tau }_0,X_1,\dots ,X_N,{\tau }_1,\dots ,{\tau }_N\right)=\frac{{\int }_{M_0}{\prod }_{i=1}^{N}P\left(C_i丨M_0,X;{\tau }_0,X_i,{\tau }_i\right)f_x\left(X;{\tau }_0,X_i,{\tau }_i\right)d{M}_0}{{\int }_{M_0}{\int }_x{\prod }_{i=1}^{N}P\left(C_i丨M_0,X;{\tau }_0,X_i,{\tau }_i\right)f_x\left(X;{\tau }_0,X_i,{\tau }_i\right)d{xdM}_0}$$ (9)

where $f_x\left(X;{\tau }_0,X_i,{\tau }_i\right)$ is the SALP of the $i^{th}$ measurement; $C_i$ is the concentration of the $i^{th}$ measurement; $P\left(C_i丨M_0,X;{\tau }_0,X_i,{\tau }_i\right)$ is the probability of measuring concentration with source mass $M_0$ and source location X as conditions, which is called conditioned adjoint location probability and symbolized as CALP.

3. Validation of CFD method

3.1. Validation of the airflow field

The wind tunnel experiment by Jiang et al. [56] is chosen to validate the coupled indoor and outdoor airflow field. The geometric description is shown in Fig. 1 (a), where the building size was length × width × height = 250mm × 250mm × 250 mm, and the thickness of the walls is 6 mm. One opening with size of 84mm × 125 mm (length × height) was at the windward wall of the building. The model is placed in a computational domain with dimensions 21 ${\mathrm{H}}_1$ × 11 ${\mathrm{H}}_1$ × 6 ${\mathrm{H}}_1$ (length × width × height), where $H_1$ is building height (0.25 m), as shown in Fig. 1(b).

Fig. 1.

(a) Schematic view of the model; (b) Computational domain; (c) Sensitivity test for three mesh systems at line Y = 0 m, Z = 0.2 m, X = 0–0.1 m; (d)–(f) The mean wind velocity of the simulated and experimental results.

Mean wind speed adopted logarithmic in the x-direction which fits with the wind experimental by Jiang et al. [56].

$$U\left(z\right)=\frac{u_{*}}{\kappa }\ln \left(\frac{z}{z_0}\right)$$ (3.1)

where $u_{*}$ is the frictional velocity ($u_{*}=1.068m/s$), $z_0$ is the roughness height ($z_0=0.005m$), and $\kappa$ is Von Karman's constant ($\kappa =0.41$).

The profiles of turbulent kinetic energy ($k$) and turbulent kinetic energy dissipation rate ($\epsilon$) are defined by G. Evola et al. [57].

$$k\left(z\right)=\frac{3}{2}{\left(U\left(z\right)·T_i\right)}^2$$ (3.2)

$$\epsilon \left(z\right)=C_{\mu }^{3/4}\frac{{k\left(z\right)}^{3/2}}{l_t}$$ (3.3)

where $T_i$ is turbulence intensity ($T_i=4\%$) and $l_t$ is mixing length ($l_t$ = 0.4 m) proposed by G. Evola et al. [57], $C_{\mu }$ is a constant ($C_{\mu }=0.09$).

Three mesh systems were adopted with minimum cell widths of 0.001 m, 0.0005 m, and 0.00025 m, respectively. Moreover, the total numbers of hexahedra grids were established around 2.1 million, 3.5 million, and 5.1 million, respectively, to verify the independence of numerical solutions of the different mesh systems. The 2.1 million grid system clearly underestimates the wind velocity, while the 3.5 million grid system and 5.1 million grid system predict the wind velocity in general agreement, as shown in Fig. 3 (c). Therefore, the medium grid system was chosen in the following comparison with the experimental values. The wind tunnel experimental was set up ten vertical measurement lines, three of which (two in the indoor environment X = H/2,3/4H; one in the outdoor environment X = -H/2) were selected as verifications.

Fig. 3.

(a) Schematic view of the model; (b) Wind directions; (c) Computational domain; (d) Sensitivity test for three mesh systems at line Y = 0 m, X = 0 m, Z = 0.4–0.6 m; (e) Sensitivity test for three mesh systems at line Y = 0 m, X = 0.5 m, Z = 0–1.5 m.

The $\mathrm{R}\mathrm{N}\mathrm{G}k-\epsilon$ model was employed as the turbulence model. Comparing with the $\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}k-\epsilon$ model, the $\mathrm{R}\mathrm{N}\mathrm{G}k-\epsilon$ model with standard wall functions predicts the wind velocity around the building more accurately [19,58]. The SIMPLE-consistent (SIMPLEC) is chosen to solve the coupled pressure and velocity fields. The spatial discretization of all parameters chose second-order upwind scheme. The residual to achieve computational convergence is set as ${10}^{-6}$, indicating that the simulation results of each scalar no longer change with the number of iterations.

Fig. 1(d)–(f) present the comparison of experimental and numerical simulations of the non-dimensional velocity on the selected curves. It can be seen that CFD simulations are sufficient to predict the airflow fields inside and around the building.

3.2. Validation of the concentration dispersion

The accuracy of CFD simulation on concentrations is estimated by comparing results with BLASIUS wind tunnel experiment at the University of Hamburg [59]. The result of concentrations in the vicinity of a rectangular building (CEDVAL A1-5) of this wind tunnel experiment is used for validation. The building was modeled at a scale of 1:200. The geometric description is shown in Fig. 2 (a), where the building size was length × width × height = 30 m × 20 m × 25 m, and four source elements with area $A=4.62{cm}^2$ were located on the bottom of the leeward wall. Tracer gas (${CO}_2$) was released from four facing pollutant sources at a velocity of $0.025m/s$. Four measurement lines (horizontal line: Y = 0.15 m, Z = 0.05 m; vertical lines: X = 0.05 m, 0.06 m, 0.075 m, Y = 0 m) were selected. The concentration field is defined in a nondimensional form as:

$$k_c=\frac{C_{local}}{C_{source}}·\frac{U_{ref}{H_2}^2}{Q_{source}}$$ (3.4)

where $C_{local}$ is the measured tracer gas concentration(ppm) with environment background concentration subtracted, $C_{source}$ is the tracer gas concentration (ppm) at the source, $U_{ref}$ is the reference wind speed (m/s) measured at the height of 0.66 m, $H_2$ is the building height ($H_2=0.125m$), and $Q_{source}$ is the flow rate of the source emission (m³/s).

Fig. 2.

(a) Schematic view of the model; (b) Computational domain; (c) Sensitivity test for three mesh systems at line Y = 0.15 m, Z = 0.05 m, X = 0.05–0.2 m; (d)–(f) The concentration distribution of the simulated and experimental results.

In order to simulate the airflow field and concentration field of the building model, a computational domain with a downstream length of 15 $H_2$, an upstream of 5 $H_2$, a lateral length of 5 $H_2$, and a height of 5 $H_2$ was adopted, as shown in Fig. 2(b). Three mesh systems were adopted with minimum cell widths of 0.001 m, 0.0005 m, and 0.00025 m, respectively. Moreover, the total numbers of hexahedra grids were established around 3.2 million, 5.1 million, and 7.3 million, respectively, to verify the independence of numerical solutions of the different mesh systems. Compared with the other two grids, the 3.2 million grid system significantly underestimates the wind velocity, while the 5.1 million and 7.3 million mesh system, show almost identical results of the wind velocity, as shown in Fig. 2(c). Therefore, the medium grid system was selected for the following comparison with the experimental values.

The inlet conditions were obtained by fitting equations wind velocity $U$, turbulent kinetic energy $k$, and turbulent dissipation rate $\epsilon$ with the experimental data [60]. The profiles obtained for $U$, $k$, and $\epsilon$ at the inlet and the coefficients are summarized in Table 1 . The turbulence model and solution methods are similar to those in section 3.1. The time step size of the transient concentration simulation is 0.01s. For transient results, the mean field of parameters was calculated from the mean value within a determined sampling period [61,62].

Table 1.

Boundary conditions.

Domain inlet	$U=\frac{u^{*}}{k}ln\left(\frac{Z+Z_0}{Z_0}\right)$$k=\sqrt{C_1·ln\left(Z+Z_0\right)+C_2}$ $\epsilon =\frac{u^{*}\sqrt{C_{\mu }}}{k\left(Z+Z_0\right)}\sqrt{C_1·ln\left(Z+Z_0\right)+C_2}$
Domain outlet	$\frac{\partial }{\partial x}\left(u,v,w,k,\epsilon \right)=0$
Domain ceiling	$w=0,\frac{\partial }{\partial x}\left(u,v,k,\epsilon \right)=0$
Domain lateral sides	$v=0,\frac{\partial }{\partial x}\left(u,w,k,\epsilon \right)=0$
Domain ground	Standard wall functions
Building surfaces	Non-slip for wall shear stress
Von Karman constant	$k=0.41$
friction velocity	$u^{*}=0.374$
Roughness height	$Z_0=0.00075$
Turbulence model coefficients	$C_1=0.025$ $C_2=0.41$ $C_{\mu }=0.069$

Fig. 2(d)–(f) presents the comparison of experimental and numerical simulations of the concentration dispersion on the selected curves. The simulation of concentration field with the CFD model is consistent with the experimental data, which indicates the CFD simulation can accurately predict the concentration field of the buildings. In the next section, we use CFD simulation to predict the airflow field and concentration field in and around the target building, providing input information for the inverse calculation of locating pollutant sources.

4. Case design

In order to analyze the pollutant transmission in an isolated building and then locate the pollutant source by inverse modeling, a three-storey building model with three rooms on each floor was constructed, as shown in Fig. 3 (a). The building model was at a reduced scale of 1:200. The dimension of each room was length × width × height = 0.2 m × 0.2 m × 0.2 m. An opening of 0.08 m × 0.08 m was fixed on the windward wall of each room. Each room has a point contaminant source (${\mathrm{C}\mathrm{O}}_2$) near the opening, and the pollutant concentration in each room will be monitored. In this paper, Reynolds number based on the building height (${Re}_H=\left(U_{ref}{D}_z/v\right)$) was 125000, and Reynolds number based on the window height (${Re}_W=\left(U_{ref}{D}_w/v\right)$) was 16667. The values of ${Re}_H$ and ${Re}_W$ exceed the threshold values of $4.8\times {10}^4$ and $1.4\times {10}^4$ proposed by Dai et al. [61], respectively, which meet the requirement of the Reynolds number independence. Noted that, in this study, we used a scaled model to represent the real building, the nine rooms of the model can be also regarded as nine zones of the real building. Identifying the sources in the nine zones can help narrow the target area of the source localization for the whole building.

The computational domain size was length × width × height = 21H × 11H × 6H, as shown in Fig. 3(c). Three different mesh systems were constructed with near-wall minimum grid widths of 0.001 m, 0.0005 m, and 0.00025 m, respectively. The grid numbers were around 3.8 million, 5.5 million, and 7.6 million. Fig. 3(d) and (e) show the comparison of wind velocity for these three grids on the indoor measurement line 1 (X = 0 m, Y = 0 m, Z = 0.4–0.6 m) and the outdoor measurement line 2 (X = 0.5 m, Y = 0 m, Z = 0–1.5 m), respectively. The prediction of wind velocity by these three grid systems is generally consistent. The Root Mean Square Error (RMSE) was 1.14% and 3.62% between the 3.8 and 5.5 million grid system, and 2.57% and 4.88% between the 3.8 and 7.6 million grid system for measurement lines 1 and 2, respectively. RMSE less than 5% would lead to a reasonable prediction [63]. Therefore, in order to save computing time and obtain the airflow field at a relatively fast speed, a coarse grid with minimum cell widths of 0.001 m and a total grid number of 3.8 million was selected [64]. The boundary condition followed the above method, as shown in Table 1. The $\mathrm{R}\mathrm{N}\mathrm{G}k-\epsilon$ model is also used as turbulence model to provide an airflow and concentration field. In this study, five outdoor incident wind directions ($0°$, $45°$, $90°$, $135°$, and $180°$) are considered, as shown in Fig. 3(b).

In this paper, FLUENT is chosen as the CFD simulation tool to provide airflow and concentration fields in and around the building. The forward simulation process obtains the steady-state flow field and transient concentration field to provide input conditions for the subsequent inverse calculation process. The time step size of the transient concentration simulation is 0.1s. Then, using the built-in UDF function, the program is written to inverse the airflow field. A unit mass of tracer gas is released in the monitoring room under the condition of the inverse velocity field, and the distribution of SALP is obtained by solving Eq. (8). Finally, combining the concentration data in the three monitoring rooms and the SALP of the corresponding rooms, the distribution of CALP is obtained by solving Eq. (9). Fig. 4 shows a brief procedure of CFD-based method for locating pollutant source.

Fig. 4.

Flow chart of CFD-based method for locating pollutant source.

5. Results of the forward and inverse simulations

In this section, the results of the forward simulation and the inverse simulation will be presented. The forward simulation obtains the steady-state airflow field and transient concentration field, which will provide the initial conditions for the inverse simulations. In addition, it reveals the pollutant transmission routes in the building environment and provides guidance for selecting the location of the pollutant sensors. Sections 5.1, 5.2, 5.3 mainly explore the airflow field and concentration field in and around building under normal outdoor wind direction ($0°$), and reveal the dispersion of pollutants in various building zones. Section 5.4 tested six different combinations of pollutant sensor locations and selected the optimist strategy to analyze the results of source identification under normal wind direction ($0°$). Further, the source identification results of four different incident wind directions ($45°$, $90°$, $135°$, and $180°$) were acquired by using a limited number of pollutant sensors.

5.1. Airflow field ($\mathbf{0}°$ wind direction)

The airflow pattern is an essential factor affecting the pollutant dispersion between units by wind effect. The openings of the rooms result in the exchange of indoor and outdoor airflow, which leads to more complex airflow in and around the building. Fig. 5 presents the airflow pattern in and around a single-sided ventilation building. The wind-flow pattern around an isolated building is similar to the previous studies [65,66]. Vortices formed at the top, bottom or both sides of the building when outdoor wind pushing against the building. The highest wind pressure on the windward side occurs at two-thirds of the height of the building, resulting in the stagnation zone as shown in Fig. 5(a). The airflow in the upper part of the stagnation zone vertically upwards along the wall surface and moves to the leeward surface through the top of the building. Conversely, the airflow in the lower part of the stagnation zone moves in the opposite direction and eventually forms a vortex at the down in front of the building. From Fig. 5(b), it can be observed that the outdoor wind is separated in the center of the building and flows along the building wall towards both sides. The airflow pattern is important to the pollutant dispersion between units. In section 5.3, a detailed analysis of pollutants transmission routes will be conducted in conjunction with the results of this section.

Fig. 5.

Velocity magnitude and streamlines of vertical and horizontal planes of the building.

5.2. Ventilation ($\mathbf{0}°$ wind direction)

The air exchange rate (ACH) is an important parameter to evaluate the room ventilation performance. The integral method is used to calculate the ACH calculation for each room: $ACH=3600\left(0.5{\int }_0^A\left|V_X\right|dA\right)/{Vol}_R$ [56], where the $V_X$ is the velocity normal to the openings ($m/s$), $A$ is the area of the opening ($m^2$), and ${Vol}_R$ is the room volume of each unit ($m^3$). This study only focuses on the air exchange rate of the windward room. The ACH values of each room are shown in Table 2 . It can be seen that the lowest ACH values are found on the first-floor rooms. This is due to the outdoor wind below the stagnation zone being vertical down the wall, resulting in a small horizontal component into the room through the opening. At the same time, the ACH values of the rooms on both sides of the same floor are higher than those in the middle room. Therefore, room B1 in the center of the lowest floor is the worst ventilated room, and the best ventilated room is room C3 on the third floor.

Table 2.

ACH values of each unit in the building on the windward side.

	A1	A2	A3	B1	B2	B3	C1	C2	C3
ACH(${\mathrm{h}}^{-1}$)	11.5	16.8	23.0	10.4	12.9	15.9	11.9	16.9	22.6

5.3. Concentration field ($\mathbf{0}°$ wind direction)

In this section, each room is set individually as a source room to release tracer gas, and the transmission routes of pollutants in each case are analyzed. This paper only focuses on the pollutant dispersion between the windward side rooms. Due to the symmetrical distribution of the airflow pattern and pollutant transmission route of an isolated building, the left side room is used to represent both side rooms.

Fig. 6 presents the concentration contours in the vertical plane when the left and middle rooms are set as source rooms. It can be seen that for the rooms located above the stagnation zone, the pollutants disperse upwards to the top of the building, and few pollutants will re-enter the lower rooms. The lowest rooms are affected by the outdoor airflow, which makes the tracer gas difficult to re-enter the upper room. When the rooms on both sides of the building are as source rooms, most of the pollutants spread to the sides of the building and hardly affect the rooms on the horizontal plane.

Fig. 6.

Non-dimensional concentration fields on the plane of X = -0.1 m (Note that plane X = -0.1 m is shown in Fig. 3(c)).

Concentration data for all building regions in the forward simulation are shown in Table 3 . It can be clearly seen that the strength of each room of the building to receive the intrusive pollutant is different. It is worth noting that the concentration information in Table 3 is used to evaluate the installation location of the pollutant sensors, and only part of the information is used as input information for the inverse calculation.

Table 3.

The tracer gas concentration monitoring in other rooms (ppm), ○ represents the source room and × represents no tracer gas concentration detected above the critical value.

source room	room number
	A1	A2	A3	B1	B2	B3	C1	C2	C3
A1	○	×	×	×	×	×	×	×	×
A2	1.34	○	×	×	×	×	×	×	×
A3	×	×	○	×	×	×	×	×	×
B1	1.92	0.12	×	○	0.16	×	1.84	0.13	×
B2	104.9	19.11	×	330.0	○	×	81.04	15.83	×
B3	×	×	10.25	×	×	○	×	×	8.17
C1	×	×	×	×	×	×	○	×	×
C2	×	×	×	×	×	×	1.46	○	×
C3	×	×	×	×	×	×	×	×	○

To evaluate the tracer gas transportation between the source and other rooms, the quantifying method proposed by Niu and Tung [16] is adopted. The reentry ratio is defined as the fraction of the tracer gas from the source room that reenters another room. It can be calculated by the following equation: $R_k=M_{i-j}\frac{{Vol}_j{\left(ACH\right)}_j}{{Vol}_i{\left(ACH\right)}_i}$, where the $M_{i-j}$ is the mass fraction, which is expressed as the ratio of tracer gas at infected unit ($C_j,kg/m^3$) to concentration at the source unit ($C_i,kg/m^3$).

Fig. 7 shows the re-entry ratios of tracer gas from the source to other units; the red dot represents the location of the source. Since the transmission route of pollutants between units is mainly affected by outdoor airflow, this section combines above analysis of airflow field to obtain the main transmission route of pollutants. When the source is located in the rooms on both sides of the building, affected by the outdoor airflow, its main transmission route is to spread to both sides of the building to the leeward side. This leads to the source room are located in A1, A3, C1, and C3, the tracer gas rarely re-enters the other rooms, with $R_k$ values less than 0.001% as negligible. It can be seen that when source rooms are located in rooms B1 and B3, only a small amount of the tracer gas spreads along the lateral direction. When the source room is located in room B2, the pathways of tracer gas transmission increases, and almost all rooms except the upper room are affected. It is also found that the $R_k$ value of tracer gas along the longitudinal direction to the lower room is one order of magnitude higher than that the transverse direction to the rooms on both sides. This indicates that the ability of pollutants to travel in the vertical direction is more significant than in the horizontal direction. This also indicates that the rooms in the lower part of the stagnant windward region are susceptible to the influence of the rooms above.

Fig. 7.

Re-entry ratios of tracer gas from the source room to other units.

5.4. Result analysis of locating pollutant source

For the inverse modeling, we put sensors in four rooms instead of all nine rooms of the building to monitor the concentrations. This strategy aims to 1) reduce the number of sensors in the building, 2) locate the source more precisely. In this paper, we tested five cases with different outdoor wind directions (case A is $0°$ outdoor incident wind direction, case B is $45°$ outdoor incident wind direction, case C is $90°$ outdoor incident wind direction, case D is $135°$ outdoor incident wind direction, and case E is $180°$ outdoor incident wind direction).

Case A

$\mathbf{0}°$ wind direction

In this case, we selected four building rooms as the fixed monitoring rooms based on the forward-simulated contaminant transmission routes, and the other five were used as source rooms to release tracer gases. Due to the large outdoor wind speed, the tracer gas disperses and dilutes quickly, and the proportion of tracer gas re-entering other rooms is not large. In order to make the result more obvious, the tracer gas ${CO}_2$ was released at a rate of 12.5 $g/s$ in the middle of each unit. According to previous studies [[67], [68], [69], [70]], the threshold value of hazardous gases is mostly in the order of 0.1 (ppm) to 100 (ppm), so we choose the lowest concentration limit of 0.1 (ppm) as the critical value; data exceeding this value will be recorded.

The CALP calculation needs at least two points of the tracer gas concentration to locate the source; therefore, we used concentration data of two or three rooms with higher concentrations in four monitoring rooms when identifying the source location for the other five rooms. The purpose of selecting four monitoring rooms instead of three is to acquire more concentration data and increase the error-tolerant rate of the source localization. Choosing only three monitoring rooms would lead to the shortage of the pollutant concentration information. However, with one more monitoring room (a total of four rooms), the pollutant concentration information could be obtained substantially. The monitoring rooms should be selected as the rooms most susceptible to the influence of other rooms rather than the rooms with more significant impacts on others, so that we can obtain as many pollutant concentration values as possible with a limited number of pollutant sensors. According to the above analysis of pollutant transmission routes in section 5.3, it can be seen that the rooms on both sides of the building are most susceptible to others with more opportunities to test the concentrations. Therefore, we chose four rooms on both sides of the building, a total of six rooms for different combinations, to test the success rate of identifying the source location; the combination of the test and the results are shown in Table 4 .

Table 4.

The tracer gas concentration monitoring in sensor rooms, ○ represents the sensor room, × represents the predicted source room location deviates from the real source location, and √ represents the predicted source room location is the real source location.

sensor rooms	room number
	A1	A2	A3	B1	B2	B3	C1	C2	C3
A1, A2, A3, C1	○	○	○	✓	✓	×	○	×	×
A1, A2, A3, C2	○	○	○	×	✓	×	×	○	×
A1, A2, A3, C3	○	○	○	×	✓	✓	×	×	○
A1, A3, C1, C3	○	×	○	✓	✓	✓	○	×	○
A1, A3, C1, C2	○	×	○	✓	✓	×	○	○	×
A1, A3, C2, C3	○	×	○	×	✓	✓	×	○	○

The optimist combination of all results is rooms A1, A3, C1, and C3, that is, the four corners of the building. This combination is successful in positioning the three regions in the middle of the building, with an overall success rate of 78%. It can be observed that neither combination can identify the source location when the source room are on the side. This also implies that the impact of the both side regions of the building on other windward regions can be ignored.

The concentration results exceeding the threshold for each room are shown in Table 5 . These concentration values provide essential input information for the inverse tracking process. Fig. 8 shows the results of identifying the other five rooms when rooms A1, A3, C1, and C3 are used as fixed monitoring rooms. The black dot represents the location of the real contaminant source. The red dot represents the highest probability of predicting source location. It can be seen that the source location can be identified when the source rooms are the middle rooms of the building, as shown in Fig. 8 (a), (b), and (e), which is consistent with the above analysis. These three building rooms are the most influential region for the entire building, so more pollutant concentration data (two or three values) can be obtained in the four fixed monitoring rooms; while the source rooms are on the sides (A2 and C2), only one value can be obtained in the four monitoring rooms, which cannot offer enough input data to locate the source.

Table 5.

The tracer gas concentration monitoring in sensor rooms (ppm), × represents no tracer gas concentration detected above the critical value. ○ represents the sensor room.

Sensor room	Source room
	A2	B1	B2	B3	C2
$0°$ wind direction
A1	1.34	1.92	104.90	×	×
A3	×	×	×	10.25	×
C1	×	1.84	81.04	×	1.46
C3	×	×	×	8.17	×
$45°$ wind direction
A1	×	6.74	3.81	×	7.62
A3	×	×	0.50	55.4	0.18
C1	×	×	×	×	×
C3	×	×	×	×	×
A2 (additional)	○	0.47	108.6	0.15	12.93
$90°$ wind direction
A1	0.41	44.9	3.22	×	1.30
A3	×	×	×	8.05	0.45
C1	×	×	0.78	×	0.41
C3	×	×	0.39	×	×
A2 (additional)	○	1.80	23.37	16.22	18.05
$135°$ wind direction
A1	×	×	×	×	×
A3	×	×	×	×	×
C1	×	22.62	×	×	×
C3	3.81	×	24.88	×	×
A2(additional)	○	×	×	×	×
$180°$ wind direction
A1	×	1.45	×	×	×
A3	12.84	2.08	0.33	×	×
C1	×	0.69	×	×	×
C3	×	1.22	0.29	×	22.10
A2(additional)	○	24.88	×	×	×

Fig. 8.

Prediction of source room location probability. (a) Source room of B1. (b) Source room of B2. (c) Source room of A2. (d) Source room of C2. (e) Source room of B3 ($0°$ outdoor incident wind direction).

When the source room is located in room A2, it can be seen that the predicted source location is not entirely in room A2, as shown in Fig. 8 (c). This is due to the contaminants in room A2 only diffusing into room A1 under normal outdoor win direction ($0°$), so only the pollutant concentrations in room A1 over the limit can be detected. However, even if the inverse calculation result of the source room A2 is unsatisfactory, we can also determine that the source room is in A2 because only room A1 has monitored the abnormal value of the pollutant concentration in this situation.

When the source room is C2, the situation is similar to A2, so that the same speculative results can be used for room C2.

In summary, based on the pollutant concentration data detected by the selected four fixed monitoring rooms, the middle room can be accurately identified. At the same time, the location of the source rooms on either side can also be roughly predicted based on the concentration information.

Case B

$\mathbf{45}°$ wind direction

In order to simulate a more realistic outdoor environment, we transformed the outdoor wind direction into an oblique wind direction ($45°$). In this circumstance, we still select rooms A1, A3, C1, and C3 as fixed monitoring rooms and the other five rooms as source rooms.

In the case of the different source rooms, the concentration results exceeding the threshold for each room are shown in Table 5. The CALP distribution indicating the location probability is shown in Fig. 9 . The result of case C is similar to case B. The source rooms B2 and C2 can accurately locate the source location. Similarly, rooms A2, B1, and B3 cannot identify the source location due to the lack of input information. In order to solve this problem, the same method is used as adding one more sensor in the downstream region A2 of the building. Based on their concentration data after adding one more sensor, the predicted results for source rooms B1 and B3 are shown in Fig. 10 . This also implies that source identification of the entire building could be achieved by adding only one pollutant sensor (five sensors over nine rooms) in the downstream region of the building in $45°$ outdoor incident wind direction.

Case C

$\mathbf{90}°$ wind direction

Fig. 9.

Prediction of source room location probability with four monitoring rooms. (a) Source room of B2. (b) Source room of C2 ($45°$ outdoor incident wind direction).

Fig. 10.

Prediction of source room location probability with five monitoring rooms. (a) Source room of B1. (b) Source room of B3 ($45°$ outdoor incident wind direction).

Case C also used four corners of the building to monitor the tracer gas concentration and tested the accuracy of source localization results.

In the case of the different source rooms, the concentration results exceeding the threshold for each room are shown in Table 5. The CALP distribution indicating the location probability is shown in Fig. 11 . The results showed that only two of five source rooms (room B2 and C2) could obtain more than two pollutant concentration information, which means that the source location could not be predicted when room A2, room B1, and room B3 were the source rooms. This is because only one monitoring room can detect the concentration data that exceeds the limit value and result in lacking important input information. For this situation, it is only necessary to add one more pollutant sensor in room A2 downstream of the building; room B1 and room B3 can be accurately identified as the source room, as shown in Fig. 12 . At the same time, room A2 can be easily located based on the concentration value of the sensor when it is the source room. The other two rooms (rooms B2 and C2) can also accurately identify the source location using of a limited number of pollutant sensors. The results implied that 67% of the building could be identified with four fixed pollutant sensors while adding one more sensor in the downstream room of the building could identify 100% of the building region.

Case D

$\mathbf{135}°$ wind direction

Fig. 11.

Prediction of source room location probability with four fixed monitoring rooms. (a) Source room of B2. (b) Source room of C2 ($90°$ outdoor incident wind direction).

Fig. 12.

Prediction of source room location probability with five monitoring rooms. (a) Source room of B1. (b) Source room of B3 ($90°$ outdoor incident wind direction).

When source rooms are located on the leeward side of the building, the contaminant transport is completely different from the windward situations. Due to the influence of airflow from the leeward side of the building, pollutants in the source room rarely re-enter other rooms and diffuse directly to the downstream area. When the outdoor wind direction is $135°$, none of the source rooms was detected two or more concentration values that exceeded the limit. It is worth noting that all other rooms, including non-sensor rooms, did not detect two or more concentration information. This also implies that at $135°$ outdoor incident wind direction, the impact of re-entered pollutants from any room is negligible.

Case E

$\mathbf{180}°$ wind direction

When the wind is coming from the back side of the building perpendicularly, the pollutant dispersion is similar to case D. However, in contrast to case D, source rooms B1 and B2 can be monitored for two or more concentration values above the limits of the sensor rooms. The concentration results exceeding the threshold for source rooms B1 and B2 are shown in Table 5. The CALP distribution indicating the location probability is shown in Fig. 13 . The results show that source rooms B1 and B2 that affect other rooms can be accurately identified.

Fig. 13.

Prediction of source room location probability with five monitoring rooms. (a) Source room of B1. (b) Source room of B2 ($180°$ outdoor incident wind direction).

6. Discussions and limitations

In this paper, we conducted both forward and inverse modeling to locate the pollutant source in the coupled indoor and outdoor conditions. In five different outdoor wind directions, a total of 25 different cases were analyzed to investigate the accuracy of adjoint probability method under this scenario. Discussions can be made as follow.

In this paper, a $3\times 3$ building model with a scale ratio of 1:200 was adopted to conduct the inverse modeling, which is generic and representative of carrying out the pollutant tracking process. The $3\times 3$ building model can represent the whole building divided into nine regions; each room corresponds to a region. In the real environment, pollutant sensors are rarely set in each point of the buildings. Under this condition, we tested a combination of six different regions and found the optimist strategy. We set up pollutant sensors in four selected building regions to obtain pollutant concentration data and use them as input information for the inverse calculations. For the pollutant source in different building regions, we can predict the approximate location; the pollutant source location can be narrowed to a part of the building area.

For case A, we compared six sensor room combinations under normal wind direction ($0°$). Finally, we used the combination with the highest success rate of rooms A1, A3, C1, and C3 as fixed monitoring rooms. In this paper, we focus on whether we can identify the location of the pollutant source room, the precise point of the source in the room is not necessary in this study; that is, the case of predicted location at the source room is considered as a successful identification. Therefore, the success rate of the inverse modeling method under normal outdoor wind direction ($0°$) is around 78%. Since the pollutant concentration monitoring points are only set up in the rooms on the windward side, we focused on the conditions that pollutants can spread to other units. This leads to that when room A2 and room C2 are the source rooms, the concentrations in the other rooms, except those directly below, cannot be detected. Because the pollutant disperses quickly to the rear of the building, the inverse calculation process cannot be used to locate the pollutant source due to the lack of input information. However, this also shows that these two regions have a negligible impact on the other rooms.

For both case B and case C, also using rooms A1, A3, C1, and C3 as fixed monitoring rooms, the success rate of the inverse modeling method was 67%. The results of these two cases are similar. With the addition of a new pollutant sensor located in a room downstream of the building, it is possible to identify the location of the source of contamination in any area of the building.

For case D and case E, when the source room is on the leeward side of the building, only rooms B1 and B2 have an effect on other rooms with the outdoor incident wind direction $180°$. The two source rooms are accurately identified using the adjoint probability method. It also shows that the impact of re-entered pollutants on other rooms is negligible when the source room is located on the leeward side of the building.

In practice, when a contaminant leak occurs at influential regions in an envelope building, it is adequate to identify the source location with the pollutant sensors in the four corners of the building. in order to achieve a more detailed identification of the entire building, one more pollutant sensor in regions A2 or C2 needs to be added.

This paper still contains some limitations. This paper only considers the windward side rooms of a single-sided building, however, in the real urban environment, the leeward side rooms also have the possibility to get involved, which will be studied in the future.

Further, this paper only considers pollutant dispersion by wind effect. With the influence of strong wind, the dilution and dissipation of pollutants or viruses is fast and the cross transmission is not obvious in the actual environment. However, buoyancy effect is an important factor with the low-wind conditions, the characteristic of the pollutant dispersion would be complex and different from the pure wind conditions. The inverse modeling with the combined buoyancy and wind effect in the building environment will be investigated in the future.

7. Conclusions

This paper applied the inverse CFD simulation with the adjoint probability method to find the pollutant source location in a single-sided building by natural ventilation. A total 25 different cases with five different wind directions ($0°$, $45°$, $90°$, $135°$, and $180°$) were set to test the accuracy of the source identification results. Through the analysis of the pollutant transmission route and the results of source location, the following conclusions can be drawn:

• 1.By analyzing the airflow field and the re-entry ratio ($R_k$) of pollutants in an isolated building under $0°$ outdoor wind direction, it can be concluded that the longitudinal dispersion of pollutants is stronger than the lateral dispersion. Below the stagnation zone, the pollutants from the upper room will re-enter the lower room. When the source room is located in the middle of the building, the pollutants will disperse to the surrounding room by the wind effect. The pollutant concentration from the rooms on the lateral sides of the building, especially those at the corners of the building, to other rooms can be ignored.
• 2.According to the pollutant transmission routes, we tested the combination of six different pollutant sensor installation locations and finally determined the optimist combination as A1, A3, C1, and C4 rooms in the building. Based on the concentration information from these four fixed sensors, it is possible to identify the location of the pollutant source in approximately 67%–78% of the building in different outdoor wind directions.
• 3.When the outdoor wind direction is $45°$ and $90°$, the source location in the whole building can be identified when one more pollutant sensor in the downstream region (A2 or C2) of the building is added. When the outdoor wind direction is $135°$, the source room has a negligible effect on other rooms. When the outdoor wind direction is $180°$, the adjoint probability method could also accurately identify rooms B1 and B2 that impact other rooms. This shows that the location of source room can be successfully identified with limited sensors. This also indicates that the adjoint probability method is capable of source identification in coupled indoor and outdoor environments.

Author statement

Yuwei Dai: Conceptualization, Methodology, Investigation, Writing- Reviewing and Editing, Funding acquisition. Fuyao Zhang: Visualization, Investigation, Data curation, Software, Validation, Writing- Original Draft. Haidong Wang: Conceptualization, Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.