Inverse estimation of the release rate of the heated pollutant source with strong convection and its application to the exhaust system design

Abstract

Accurate release rate of the source is a crucial parameter for the refined design of the exhaust system in the industrial buildings. For the heated pollutant sources emitted by strong convection (SCHP source), it is difficult to accurately measure the source release rate with instruments due to inconsistent emission parameters tested at different locations near the source. In this paper, the three-dimensional CFD (Computational Fluid Dynamics) simulation was used to obtain the hourly concentration of pollutants and to study the applicability of four different regularization methods in the inverse estimation of the release rate of the SCHP source. The influence of the denoising filter and the strong convection of the SCHP source on the accuracy of the inverse estimated source release rate (IESR) was analyzed, and an exhaust flow rate calculation method based on the IESR is proposed. The results show that, compared with the zero-order Tikhonov regularization (ZOTR) and the LSQR methods, the second-order Tikhonov regularization (SOTR) and the truncated SVD (TSVD) methods are more suitable for the inverse estimation of the SCHP source. And it is found that, the introduction of the denoising filter can effectively eliminate the high-frequency or the high-amplitude deviations caused by the regularization method, compared with the SOTR method, the RMSE can be reduced by a maximum of 37.04 %. It is also concluded that the strong convection and the measurement error both have the negative impact on the accuracy of the IESR. Finally, compared to the calculation methods in the existing design manuals, the local exhaust system designed by the IESR method can efficiently capture the pollutants with a 46 % reduction in the exhaust flow rate. This study is useful for the accurately determining the SCHP source release rate and the optimal design of the exhaust system.

Headnote

ABSTRACT

Accurate release rate of the source is a crucial parameter for the refined design of the exhaust system in the industrial buildings. For the heated pollutant sources emitted by strong convection (SCHP source), it is difficult to accurately measure the source release rate with instruments due to inconsistent emission parameters tested at different locations near the source. In this paper, the three-dimensional CFD (Computational Fluid Dynamics) simulation was used to obtain the hourly concentration of pollutants and to study the applicability of four different regularization methods in the inverse estimation of the release rate of the SCHP source. The influence of the denoising filter and the strong convection of the SCHP source on the accuracy of the inverse estimated source release rate (IESR) was analyzed, and an exhaust flow rate calculation method based on the IESR is proposed. The results show that, compared with the zero-order Tikhonov regularization (ZOTR) and the LSQR methods, the second-order Tikhonov regularization (SOTR) and the truncated SVD (TSVD) methods are more suitable for the inverse estimation of the SCHP source. And it is found that, the introduction of the denoising filter can effectively eliminate the high-frequency or the high-amplitude deviations caused by the regularization method, compared with the SOTR method, the RMSE can be reduced by a maximum of 37.04 %. It is also concluded that the strong convection and the measurement error both have the negative impact on the accuracy of the IESR. Finally, compared to the calculation methods in the existing design manuals, the local exhaust system designed by the IESR method can efficiently capture the pollutants with a 46 % reduction in the exhaust flow rate. This study is useful for the accurately determining the SCHP source release rate and the optimal design of the exhaust system.

Keywords: Strong-convection heated pollutant source Inverse estimation of source release rate Tikhonov regularization method Denoising filter Industrial ventilation

(ProQuest: ... denotes formulae omitted.)

1. Introduction

The heated pollutant sources emitted by strong convection (SCHP source) are commonly found in industrial buildings, and the location of these sources is fixed, the local exhaust system near the source is the most high-efficiency control technology, while the crucial parameter for the optimal design of the local exhaust system is the accurate source emission parameters [1]. Compared to the diffusion-dominated source, it is more difficult to measure the release rate of the SCHP source. For example, in the rubber post-treatment screening process, the toxic and hazardous gaseous pollutants are emitted by strong convection. It becomes more difficult to directly test the release rate of such industrial sources, either because of inconsistent emission parameters tested at different locations or because of the large emission area of the source. Meanwhile, the velocity at different locations near the source is different, and if the release rate is calculated based on the emission velocity of the source, the error of the test instrument or the improper test method will lead to a large deviation. To solve such a problem, it is necessary to apply the inverse solution method to accurately calculate the release rate of the SCHP source with a known location.

The inverse solution method has been widely used to estimate the release rate of the pollutant sources in the field of indoor environment control and atmospheric environment management. It has been demonstrated that the inverse solution method is not limited by the location and number of measurement points, and can avoid the miscalculation of the source release rate caused by the error of the instruments and obtain the accurate source release rate [2-7]. Based on the concentration tested in the stable airflow organization, this paper applies the inverse solution method to accurately estimate the release rate of the SCHP source.

According to the different solution ideas of the inverse problem, the inverse solution methods can be divided into two categories, which are iterative optimization method and statistical method. Based on the Gaussian dispersion model, iterative optimization methods, such as genetic algorithms, simulated annealing algorithms and particle swarm optimization algorithms, have been applied to estimate the emission parameters of the source in the atmospheric environment by the global iterative calculation [8-12]. The application of iterative optimization requires sufficient data samples and accurate numerical models of pollutant dispersion. However, for the short-time emitting SCHP source, due to the lack of the accurate pollutant dispersion model under the influence of ventilation systems and the limited concentration data, the application of the iterative optimization method is problematic in terms of high computational cost and large error.

As for the short-time emitting source, the statistical method has been widely used in the inverse identification of the source intensity in confined spaces such as civil buildings and aircraft cabin. The first step of the statistical method is to construct the statistical law of the source release rate and concentration. The concentration response factor and the Markov chain have been used in the inverse solution of pollutant sources in civil buildings [13-17]. Besides, Li et al. applied the transient accessibility of the source to establish the relationship between the source and the concentration of the measurement points under the action of mechanical ventilation, and confirmed that under the steady-state flow field, solving the source intensity-concentration relationship equation with the attachment constraint can effectively obtain the source release rate of the room-temperature source [18,19].

The second step of the statistical method is to calculate the optimal solution by calculating the function between the source and the concentration of the measurement points with the additional constraint method, where the most widely used additional constraint method is the Tikhonov regularization method and the singular value decomposition method (SVD) [20,21]. Akçelik demonstrated the effectiveness of the Tikhonov regularization method for the inverse identification of continuously emitting sources [22]. Liu and Li et al. demonstrated that the SOTR method and LSQR iterative regularization method can be used to accurately estimate the source release rates of gaseous low-momentum pollutant sources under the influence of ventilation systems [23,24]. Zhang et al. applied the concentration response factor and Tikhonov regularization method to the inverse identification of the release rate and the location of a single or multiple sources in an aircraft cabin, and it was also shown that the concentration response factor method is suitable for the inverse identification of low-momentum sources, where the error is less than 10 % [25-29].

The above studies show that the SVD regularization method Tikhonov regularization method and LSQR iterative regularization method have been widely used for the release rate identification of the room-temperature gaseous pollutant sources where the location is known and the diffusion-convection effect in the mass transfer process is equivalent. However, for the SCHP sources in industrial buildings, more research is needed to determine whether the Tikhonov regularization method can be used to accurately inverse estimate the release rate of such sources and to identify the best regularization method, which is also the purpose of this investigation.

In this paper, firstly, by comparing the accuracy, stability and CPU time of the results, the applicability of four different regularization methods in the inverse estimation of the high momentum pollution sources was analyzed. Secondly, for the SCHP sources, the deviation of the inverse result is inevitably too high due to the rapid concentration changes and the high velocity variations at the measurement points, two denoising filters are introduced into the regularization method to improve the stability and accuracy of the inverse results. In addition, the influence of the strong convection on the accuracy of the inverse results is studied, and the best inverse solution method for the SCHP sources with different release rate is determined. Finally, a design method based on the inverse estimated source release rate (IESR) method is proposed to determine the optimal flow rate of the local exhaust system.

2. Methodology

2.1. The relationship between the release rate and the concentration

When the source location is fixed, assuming that the airflow field of passive (neutral buoyant) buoyant jet is a stable forced convection flow field, and the concentration of pollutants is not directly affected by temperature, but only related to the velocity and the emission rate of source, the concentration of pollutants changes linearly with the release rate, following the mass conservation equation of component transfer [16].

... (1)

where C(t) is the time-by-time concentration at any measurement point, ppm; q(f) is the source release rate, m3/h; the concentration response matrix A is a linear matrix that describes the cause-effect relation between the release rate and the concentration. In the linear system, the concentration of any source release rate can also be expressed as the convolution integral between the release rate and the response factor, as shown in Eq. (2).

... (2)

where ... denotes the concentration response factor of a unit impulse release at time t = k [17], in this study, the three-dimensional GED analyses are carried out to calculate the concentration response factor. The concentration response matrix with concentration response factor can be expressed as:

... (3)

2.2. Regularization method for determining the source release rate

2.2.1. Singular value decomposition (SVD) regularization method

As shown in Eq. (3), when the concentration response factor is 0, the matrix A is a rank-loss matrix that cannot be directly inverted. Therefore, it is necessary to solve the inverse matrix by SVD method [21], the expression is as follows.

... (4)

where the matrix U and V are column orthogonal matrices (...) ; the matrix Ľ is a decreasing diagonal matrix of singular values, σ1·≥σ2≥...≥σn≥0.

For the inverse estimation of the SCHP sources, under the influence of the exhaust airflow and ambient airflow, the concentration response factors and the singular values in the concentration response matrix are too small to obtain the accurate inverse results. The zero-order Tikhonov regularization (ZOTR) method and the truncated SVD (TSVD) method can be used to reduce the effect of small singular values on the inverse estimated results [30]. In contrast to the common SVD method, the ZOTR method introduces filter coefficients that take into account the regularization parameter X, which can efficiently screen out the small singular values and improve the accuracy of the results., and the release rate of the SCHP source can be calculated as shown in Eq. (5).

... (5)

where fi is the filter coefficients;ui and vi is the column orthogonal matrix obtained by the singular value decomposition; σi is the singular value; cm is the concentration at any measurement point, ppm; À is the regularization parameter, and the L-curve method is used in this paper to determine the value of the regularization parameter.

The TSVD method preserve the first k terms nonzero singular values, and replace the last n-k terms with exact zeros to obtain a new rank-k matrix Ak, the unique solution is given as shown in Eq. (7). The key factor in the TSVD regularization method is the truncation parameter k. This article uses the L-curve method to determine the truncation parameters [31-32].

... (6)

... (7)

2.2.2. Tikhonov regularization method

Besides, the second-order Tikhonov regularization (SOTR) method with second-order time derivative matrix L can also solve the problem of inverting rank-loss matrix and avoid the overfitting, the release rate can be calculated as shown in Eq. (8) [21].

... (8)

... (9)

2.2.3. LSQR iterative regularization method

The LSQR algorithm is a common iterative method for solving the least squares problem, and it is based on the Lanczos bidiagonalization procedure [33-34]. The iteration number k plays the role of the regularization parameter. At iteration k, LSQR computes the solution ... as Eq. (10) [35].

... (10)

Where k is the number of iterations, and the initial iteration value as shown in Eq. (11) [35]. ...(11)

2.3. Denoising filter to eliminate the result errors

For the SCHP sources in the industrial building, the strong convection of the source and the exhaust airflow will affect the concentration at the measurement point, and cause the large fluctuations in the IESR. Such short-time and high-amplitude data fluctuations will reduce the accuracy of the inverse results, while the application of the denoising filter (i.e., filtering the inverse solution results) can significantly eliminate the error in the results caused by the Tikhonov regularization method [36]. Therefore, in this paper, two filters are introduced after the regularized inverse result, which are:

* Butterworth filter: In this study, an order of filter m = 4 and a normalized cutoff frequency wc =0.1 were chosen.

* Finite Impulse Response (FIR) filter: A least-squares linear-phase FIR filter of order 20 with a transition band between 0 and 0.95 is applied.

The Butterworth filter has been widely used to reduce the adverse effects of data noise, but the phase delay introduced by this filter must be taken into account [37]. The magnitude-squared function of the Butterworth filter is given by Eq. (12). Linearity and stability are the two outstanding advantages of the FIR filter, so it is widely used in system control signal processing [38]. The FIR filter is calculated by Eq. (13) [3941].

... (12)

where wc is the cutoff frequency above which high-frequency error data will be filtered out; m is the order of the Butterworth filter.

... (13)

where bi is the coefficient of the FIR filter and N is the filter length.

3. Method validation

In order to ensure the validity of the two Tikhonov regularization methods used in this paper, based on the concentration obtained by numerical simulation after experimental validation, the IESRs are calculated by MATLAB and compared with the real release rate to determine the validity of the inverse solution model.

3.1. Validation of numerical simulation

In this paper, a physical model with the size of 2m x Im x 1.5m is established according to the prototype of the vibrating screen process in chemical production, as shown in Fig. 1. The size of the source is 0.95m x 0.3m, the angle between the emission speed of the pollution source and the horizontal direction is 40°, and the distance between the emitting surface and the ground is 0.35m, and the height of the side baffles is 0.04mon the two sides of the source. The size of the canopy hood is Im x 0.5m and the vertical height between the canopy hood and the source is 0.36m.

Table 1 shows the detailed release parameters of each case, the source release rates of other cases are determined by increasing and decreasing the release rate of case 3 in the same proportion. The polluted airflow of the source is composed of hexane and air, the temperature and the density of the polluted airflow are 55 °C and 1.288kg/m3. The ratio of buoyancy forces is calculated as N = -0.183 [42] and the dimensionless number ... [43], which represents the ratio of buoyancy and initial momentum of the polluted airflow for case 4, the two ratios indicate that the concentration buoyancy of the polluted airflow is slightly greater than the temperature buoyancy, the source has both negative buoyancy and upward momentum, thus it can be assumed that the polluted airflow in this study is a passive buoyant jet.

The Peclet number, which reflects the contrast between the strength of the convection-diffusion effect of the source, and the calculation formula is shown in Eq. (14). The larger the Pe number, the stronger the convection effect.

...(14)

where lc is the characteristic diameter, m; us is the release velocity of source, m/s; DAB is the mass diffusivity of a mixture of air and hexane, the main gaseous pollutant in the synthetic rubber reprocessing process, m2 Is.

In the synthetic rubber reprocessing process, the pollutant source is mostly released continuously in the non-overhaul period. In order to reduce the amount of concentration data used to calculate the IESR, this paper takes 90s as the source release period and the concentration sampling interval is 0.1s. The source release process of case 3 as an example is shown in Fig. 2.

In this paper, the unsteady Reynolds average method is applied to solve the contaminated airflow flow field distribution of the vibrating screen process in chemical production. The pressure-based PISO algorithm and the body force-weighted pressure term are adopted [44,45]. For the turbulence calculation, the vertical centerline velocity decay predicted by the realizable k-e model, RNG k-¿ model, and standard k-e model were validated with the dispersion equation obtained by Lin [46]. As shown in Fig. 3, the realizable k-e model shows acceptable accuracy. Moreover, the temperature difference between the polluted airflow and the ambient air is greater than 15 °C, the influence of the buoyancy temperature gradient on the velocity field should be considered, so the energy equation is considered. In addition, through calculation, it was found that the Richardson number of each condition in this study is greater than 0.1, which means that natural convection and the gravity term must be considered.

The direction of gravity is the negative direction of the Z-axis, and the absolute value of gravity acceleration is 9.81 m/s2. According to the formula of turbulence intensity I = 0.16(Re)-1/8, the turbulence intensity of the velocity boundary and the pressure boundary are set to 5 % [45]. The side walls are set up as pressure inlets to balance the flow in the enclosed space, and the turbulent viscosity ratio is taken as a default value of 10. The boundary conditions in the simulation were summarized in Table 2.

The distribution of concentration measurement points is shown in Fig. 4. The ratio of the mass fraction of each point to that of the source is calculated to obtain the dimensionless concentration, and the pollu- tant dispersion zone where the dimensionless concentration is greater than 0.2 is defined as the pollutant high-concentration dispersion zone (HCD zone, colored area in Fig. 4(b)), and the rest of the zone is the pollutant low-concentration dispersion area (LCD zone, white area in Fig. 4(b)). The measurement points starting with the letter H represent the points located in the HCD zone, while the points starting with the letter L represent the points located in the LCD zone, and the specific location of each measurement point is listed in Table 3.

In this paper, mesh independence was analyzed using the Grid Convergence Index (GCI) method, which was calculated using the concentration of the ambient measurement point LI 8 [47]. In this study, the unstructured polyhedral grid was used to discretize the computational domain. The size function is used for local mesh refinement and the local mesh refinement is from the source and the opening of the canopy hood to the domain. The Body of Influence (BOI) region was set as a cuboid with length, width and height of 1.2m, 0.6m,0.8m, respectively. Mesh details is shown in Fig. 5. Table 4 presents the GCI results, the comparison of the five different meshes is shown in Fig. 6. ... represents the ratio of the error of the middle and coarse grids to the error of the refine and middle precision grids. If the value is greater than 1, it means that the error of the middle and coarse grids is slightly greater than that of the refine and middle precision grids. The closer the ratio is to 1, the smaller the error. Therefore, the grid number of 4.33 million was chosen for the numerical simulation in this paper.

In this paper, in order to verify the validity of the simulation, the dimensionless diffusion height of the pollutant is used to compare the differences between the simulated results and the experimental test data [48]. The simulation results show a more stable variation of the dimensionless diffusion height compared to the experimental data, as shown in Fig. 7. The simulation results are all within ± 10 % deviation from the experiment except for the simulation result at 9 s. The overall trend of the dimensionless diffusion height is the same, which first increases and then stabilizes at a certain dimensionless diffusion height. Nonlinear fitting of the experimental data gave a stable mean dimensionless diffusion height of 18.72. And the simulated dimensionless diffusion height was 19.06 with a deviation of only 1.8 %.

3.2. Validation of the inverse estimation algorithm

Under the assumption that concentration is not directly affected by temperature, but only depends on wind speed and pollutant emission rate, the concentration response factor method can be used to calculate concentration distribution [16]. Using the example of Case 4, where the polluted airflow is a high-temperature passive buoyant jet, the influence of the source temperature on the concentration distribution was first analyzed. Fig. 8 shows the concentration deviation (...) results along the z-axis height variation. It can be found that within the X<0 area, the concentration deviation at different heights is less than 8 %, while in the X>0 area, the concentration deviation is less than 2 %, the concentration deviations in the X>0 area were significantly smaller than that in the X<0 area. With the influence of the forced plume and the local exhaust system, the X>0 area is in the forced convection flow field. It can be concluded that under the in the forced convection flow field, the concentration of this study is mainly influenced by the flow velocity and the emission rate of the source, the effect of temperature on the concentration distribution is not significant. In addition, the deviation between the simulated concentration Co and the calculated concentration Cm, which is the product of the component transport matrix A and the source strength q, of points Hl6 and LI7 was compared. As shown in Fig. 9, for the same measurement point, the simulated concentration is consistent with the calculated concentration based on the linear equation Co = A * q. Except for the initial 3 s, the relative concentration deviation is less than 5 % at all other times. The deviation is because the polluted airflow is a high-temperature passive buoyant jet rather than isothermal. Therefore, it can be considered that Eq. (1), which has a linear relationship between source intensity and concentration, is valid for the inverse calculation problem of this study.

Finally, the IESR and the real release rate of case 0 were compared to verify the validation of the Tikhonov regularization algorithm, and the formula for calculating the average values and the relative error (RE) of the IESR are shown in Eq. (15) and Eq. (16). Besides, the convergence of the inverse estimation results was determined by calculating the root mean square error (RMSE) between the IESR and the actual release rate, and the RMSE calculation formula is shown in Eq. (17). Based on the concentration of the point Hl 5 in the HCD zone, the IESRs of the four where n is the number of concentration data at the measurement point; qreg,i is the inverse estimated time-by-time source intensity, m3/h; Qs is the real source release rate, m3/h.

In Table 5, it was noticed that the RE of the four methods are -1.39 %, -0.21 %, -1.39 % and 0.26 % respectively. The REs are both less than 2 %, thus it can be concluded that the four regularization algorithms are accurate and can be used to inverse estimate the release rate of the SCHP source. The four different regularization methods have similar CPU times and can all be used for the fast inverse solution of the source release rate. Furthermore, it was found that the REs of the SORT method and LSQR method are both less than 1.39 %, compared to the SVD methods, the Tikhonov regularization and LSQR iterative regularization method are more suitable for the diffusion-dominated source. Then, for the SCHP source, the REs and the RMSEs of the four different regularization methods will be discussed in Section 4.1. As shown in Fig. 10, the relative error (RECA) was shown that after 450 iterations, the decrease in the loudness error of the four types of methods is no longer obvious, and it can be concluded that all the four types of regularization methods are able to converge and approximate the real solution of the original problem very well.

4. Results and discussion

4.1. Application evaluation of four different regularization methods

In this subsection, the influence of the regularization methods and the location of the measurement points on the accuracy of the inverse results were compared and analyzed. Four types of regularization methods were separately the ZOTR method, the SOTR method, the TSVD method and the LSQR method. Three typical measurement points were the Hl5 point in the exhaust duct, the Hl6 point in the HCD zone, and the LI 7 point in the LCD zone. Based on the concentrations of case 3, the source release rate was solved using MATLAB software, the inverse estimated results are shown in Fig. 11-13 and Table 6.

The values and the effect factors of the pollutant concentrations are various at different measurement points, which results in the significantly different singular values and the applicable regularization methods. Firstly, as shown in Fig. 11, the point Hl5 located in the exhaust duct, the concentration is less affected by the ambient airflow, there are obvious large-amplitude fluctuations in the results of the ZOTR method, the TSVD method and the LSQR method. In terms of the stability of the lESRs, the SORT method is the best regularization method for point Hl 5. Secondly, point Hl6 is located in the HCD zone, where the concentration is significantly influenced by the SCHP source, which causes the highfrequency fluctuations of the lESRs. Compared with the TSVD method, the maximum amplitudes of the ZOTR, the SORT and the LSQR methods are almost 3-4 times than those of the TSVD method, but the periods of the fluctuations are shorter, as shown in Fig. 12. As far as the stability of the lESRs is concerned, the TSVD method is the best regularization method for the point Hl 6. Finally, as shown in Fig. 13, for the environmental point LI 7, which is in the LCD zone and less affected by the SCHP source, the result shows a short-time large fluctuation at the at the initial moment of the emission. For the SVD methods (namely, the ZORT and the TSVD methods), there are also high-frequency fluctuations at the end of the emission, so the best regularization method for the environmental point LI 7 is the SORT method. Moreover, for each regularization method, there is a phase delay of about 0.5s in the IESR with respect to the real release rate; since the emission of an industrial source is a long period process (much greater than 2s), a small-time delay is acceptable in the current application. The average and the deviation of the lESRs are shown in Table 6.

The RMSE represents the stability of the lESRs, the smaller the RMSE, the better the stability of the results. As shown in Table 6, the numerical results of RMSE are consistent with the analysis results in Fig. 11-13. For the point Hl 5, the RE of the four methods are almost same, taking into account the stability of the lESRs and the CPU time, it can be concluded that the best regularization method is the SORT method. For the point Hl6, the Res of each method are similar, separately are -4.06 %, -3.47 %, -3.16 %, -3.37 %. But there are significant differences in the RMSEs among different methods, the RMSE of the ZOTR, the SORT and the LSQR methods are 5.55, 4.77, 3.92 times than that of the TSVD method. From the comprehensive consideration of the accuracy, the stability of the results and the CPU time, the best regularization method is the TSVD method. Then, with regard to the point LI 7, the RE of the TSVD method is the smallest, while the RMSE of the SORT method is the smallest. In summary, it can be concluded that the SOTR and the TSVD methods are more suitable for the inverse estimation of the SCHP source, and the accuracy and the stability of the IESR decreases as the measurement points move away from the SCHP source.

For the inverse solution, the larger singular value is the key to ensure the accuracy of the results, and the smaller singular value determines the accuracy [49]. In order to improve the accuracy of the lESRs and eliminate the large amplitude fluctuation of the results caused by the Tikhonov regularization method, two denoising filters were introduced into the Tikhonov regularization method to correct the smaller singular values and retain the larger singular values.

4.2. Effects of denoising filter on the inverse results

In this subsection, the Butterworth filter and the FIR filter are introduced into the SOTR and the TSVD methods to form four inverse solution methods, namely the TSVD method with the Butterworth filter (TSVD-BTW method), the TSVD method with the FIR filter (TSVD-FIR method), the SOTR method with the Butterworth filter (SOTR-BTW method), and the SOTR method with the FIR filter (SOTR-FIR method). Fig. 14 and Fig. 15 show the REs of the lESRs based on the concentration of case 3 at point Hl6 and point LI7, respectively. Table 7 shows the RMSEs and REs of the lESRs by applying the six different inverse solution methods.

Comparing the REs of the lESRs at the near-source point Hl6 with that at the environmental point LI 7, it can be found that the inverse solution method including the denoising filter can effectively reduce the deviations and significantly improve the accuracy of the results. Regardless of the high-frequency fluctuation deviations or high-amplitude deviations, both can be effectively reduced with these two filters, as shown in Fig. 14 and Fig. 15. The FIR filter is better than the Butterworth filter in eliminating high amplitude deviations.

As shown in Table 7, the Butterworth filter has a negative effect on the accuracy of the results. For point Hl 6, the use of Butterworth FIR filters increases the REs of the TSVD-BTW and SOTR-BTW methods by 32.28 % and 8.36 % respectively; similarly, for point LI 7, the REs of the TSVD-BTW and SOTR-BTW methods are increased by 16.56 % and 1.27 % respectively. In addition, the effect of the FIR filter in improving the accuracy of the results was different for the different measurement points. For point Hl6, the REs of the TSVD-FIR and SOTR-FIR methods are increased by 2.44 % and 2.17 % respectively; on the contrary, for point LI 7, the REs of the TSVD-FIR and SOTR-FIR methods are reduced by 80.89 % and 86.78 % respectively. Then, the effect of the two denoising filters on the stability of the results was analyzed. For the environmental measurement point was less affected by the SCHP source, the denoising filter can improve the stability of the results. Compared with the TSVD method, the RMSEs of the TSVD-BTW and the TSVD-FIR methods are reduced by 23.73 % and by 30.93 % separately. Similarly, for the SOTR method, the RMSEs of the SOTR-BTW and the SOTR-FIR methods are reduced by 17.50 % and by 37.04 % respectively. However, for the near-source point, the RMSE of the TSVD method is the smallest among the six inverse solution methods, thus the TSVD method was the best inverse solution method.

Furthermore, it was also found that the inverse estimation results were overestimated by introducing the FIR filter in the regularization methods. When designing the local exhaust system by using the flow ratio method, the magnification factor was usually applied to correct the estimated source release rate in order to ensure the effective capture performance of the exhaust system. Therefore, the FIR filter is more suitable for the inverse estimation of the SCHP source in the industrial building. From the comprehensive consideration of the RMSE and the RE, it can be concluded that, for the inverse estimation of the SCHP source, the TSVD-FIR method is the best inverse solution method for the near-source measurement points; and the SORT-FIR method has the better stability and smaller deviation for the environmental measurement points.

4.3. The effect of measurement error and rounding error on the inverse estimated results

The TSVD-FIR and SORT-FIR inverse solution methods were tested with three levels of measurement error. The perturbance scheme used to generate random errors was defined as Eq. (18) [50,51]. For the high perturbance error, σ = 25%; whereas for the medium and low perturbance errors, the parameters £ was 5.0 % and 1.0 % respectively. The detailed IRSEs of the point LI 7 at different measurement errors were shown in Fig. 16 and Table 8.

... (18)

where, the Cinput is the input concentration, Cm is the tested concentration, σ is the fraction to adjust the level of measurement error, STD is the standard normal deviation.

As presented in Fig. 16 and Table 8, it was found that the accuracy of the inverse results gradually reduced with the increase of the perturbance error. Compared with the absolute value of RE at the measurement concentration without error, for the TSVD-FIR method, the absolute value of RE of the 1.0 %, 5.0 % and 25 % perturbance error separately increased by 3.2 %, 8.5 % and 784.1 %; while for the SORTFIR method, the absolute value of RE of the 1.0 %, 5.0 % and 25 % perturbance error increased by 32.6 %, 65.2 % and 818.5 %, respectively. Besides, for the TSVD-FIR method, the stability of the results became worse with the increase of the perturbance error, but for the SORT-FIR method, the change of the RMSE was too small to be ignored when the perturbance error was less than 25 %. It was found that when the measurement error larger than 25 %, the Mreg (the difference between the average and the actual value of the release rate)were greater than 45 m3/h, the case with this input concentration cannot correctly estimate the release rate of the SCHP source. Therefore, the critical point of the perturbance error for the TSVD-FIR and SORT-FIR inverse solution methods are 25 %.

This paper analyses the effect of numerical rounding error on the accuracy of the inverse estimated results by converting the floating-point numbers into the symbolic variables. The differences in the inverse estimated results based on the measurement point LI 7 are compared when the effective digits are 5, 10 and 15, respectively, to clarify the effect of rounding error on the inverse calculation results and CPU calculation time, the specific results are shown in Table 9.

Table 9 shows that the inverse estimation results are similar under different data storage types, with deviations from the true source strength of about 4.17 %. However, the CPU time required for the iterative calculation of symbolic variables is much longer than for the double precision floating-point method. Therefore, this article uses the double precision floating-point data type for iterative computation and data storage.

4.4. Influence of the strong convection on the accuracy of the inverse results

The significant difference between the pollutant transport of the SCHP sources and other sources in civil buildings is the strength of convection and diffusion effects of the sources. In contrast to the civil sources, the release rate of the industrial sources is much higher and the pollutants transport is dominated by the strong convection. Table 8 and Fig. 17 present the RMSEs and the REs of the IESRs under different Pe numbers, which illustrate the influence of the strong convection on the stability and accuracy of the inverse calculation results, respectively. At the same exhaust flow rate, due to the small source release rate in Case 1 and Case 2, the pollutants have been effectively captured by the canopy hood in the near-field environment, and the concentration at the point LI8 is too small to obtain the available inverse results. Therefore, Table 10 and Fig. 17 only show the RE of the IESRs at point Hl5 and point Hl6 in Case 1 and Case 2.

It can be concluded that the increase of the convection effect of the source has the negative effect on the accuracy of the IESRs, as shown in Fig. 17, the RE of the inverse results increases with the increase of the Pe number at the same point. Firstly, analyzing the IESRs at the point Hl5 located in the exhaust duct, for the case where Pe = 7.705 x 103, almost all of the pollutants have been captured by the canopy hood, the result calculated by the SOTR-FIR method will be larger than the actual release rate; while for the cases where Pe > 2.003 x 104, there are still some pollutants that have escaped from the local exhaust hood, the IS-REs based on the concentration tested in the exhaust duct were smaller than the actual release rate. Secondly, discussing the variation of the lERSs at the near-source point Hl6 which was significantly affected by the strong-convection source, the REs of the SOTR-FIR method were the non-negative, increasing value; while for the TSVD-FIR method, there is an inflection point with RE = 0, which means that except of regularization methods, the optimal solution that infinitely close to the true value can be found by iterative optimization algorithms. And, compared with the REs of the SOTR-FIR method in case 1, the REs in case 2, case 3, and case 4 increased by 6.94 %, 70.20 %, and 127.55 %, respectively.

As shown in Table 10, the optimal inverse solution varies under different Pe numbers. For the low release rate SCHP source in Case 1, comparing the RMSE and the difference of the results, the optimal inverse solution method is the SOTR-FIR method at two different typical measurement points. For cases 2 and 3, the accuracy of the result is related to the location of the measurement points. For the near-source point Hl 6, the TSVD-FIR method is the best method; while for the measurement points located in the exhaust duct and the environment, the SOTR-FIR method is the best method. In summary, for the heated pollutant sources with strong convection, the SORT-FIR method has better accuracy and stability.

4.5. The exhaust flow rate optimal design method based on the ISER

The flow rate ratio method, which is a linear function of the source release rate, has been commonly used to calculate the exhaust flow rate in the engineering design. As shown in Eq. (19), the critical flow rate ratio KL and the exact release rate of the source Qs can be economically reasonable to ensure that the high efficiency control performance of the local exhaust hood [52]. Among them, the most common method to determine the source release rate is to multiply the measured release velocity by the emitted area. However, the velocity of the SCHP source was difficult to measure directly. The average velocity of the buoyant jet emitted from the source can be calculated from the temperature difference between the source and the ambient air (AVBJ method), or the release velocity can be estimated based on pollutant diffusion phenomena (VDP method) [1,48]. Based on the above estimated source release rate, the exhaust flow calculated by the flow rate ratio method may result in poor control performance or high exhaust flow rate of the local exhaust system. Therefore, this paper proposes a new calculation method based on the IESR and using the flow rate ratio method to accurately calculate the exhaust flow rate of the local exhaust system.

... (19)

where Qe is the exhaust flow rate that calculated by the flow rate ratio method, m3/h; Qs is the release rate of the SCHP source, m3/h; n is the safety factor considering the lateral disturbance airflow, can be calculated according to the Eq. (20); ... is the limit value of the flow rate ratio considering the temperature difference between the pollutant and the ambient air, can be calculated according to the Eq. (21) and Eq. (22).

... (20)

where, v' is the velocity of the lateral airflow, m/s; us is the source release velocity, m/s.

... (21)

... (22)

where, At is the temperature difference between the source and the ambient air, K; H is the distance from the source to the canopy hood, m; E is the width of the source, m; / is the length of the source, m;F is the width of the canopy hood, m.

Taking the rubber post-treatment screening process as an example, the temperature of the pollutant source is 328K, the release velocity of the source is 0.25 m/s according to the average velocity calculation of the upward buoyant jet. And according to the diffusion phenomenon of the pollutant, the minimum and the maximum release velocity of the source could be 0.3m/s and 1.0m/s, respectively [52]. Table 11 shows the exhaust flow rate calculated by the flow rate ratio method, while the release rates were estimated separately by the AVBJ method, the VDP method and the IESR method. Among them, like the VDP-min method, the VDP-max method calculates the source release rate by multiplying the emitted area by the maximum release velocity, which was determined according to the pollutant diffusion phenomenon, by the area.

As shown in Table 11, for the SCHP sources, the source release rate estimated by the AVBJ method was the smallest, and as a direct result, the flow rate of the local exhaust system will be too small to effectively capture the pollutants. Meanwhile, the VDP-min and VDP-max methods rely on the engineer's experience, and the estimated exhaust flow rate takes on a wide range of values. With the same source release rate, Fig. 18 and Fig. 19 show the concentration distribution of the flow field and the capture efficiency of the canopy hood at different exhaust flow rates. The formula for calculating the capture efficiency is shown in Eq. (23). ... (23)

where, Ce is the concentration in the exhaust duct, ppm; Cs is the concentration of the source, ppm; Qe is the exhaust flow rate, m3/h; Qs is the release rate of the SCHP source, m3/h.

Comparing the concentration distribution and the variation of the capture efficiency at different exhaust flow rates, as shown in Fig. 18 and Fig. 19, it can be determined that the exhaust flow rates designed based on the AVBJ method and the VDP-min method are too small to capture the pollutants efficiently, the capture efficiency is only 55.03 % and 59.56 %. Using the VDP-max method, the capture efficiency of the designed canopy hood can reach 97.41 %, only a few pollutants escape from the canopy hood. Moreover, compared with the control performance and exhaust flow rate of the case D where Qe = 4500m3/h, the case C can also effectively control the pollutants within the capture area of the canopy hood, the capture efficiency was more than 85.0 %, and the exhaust flow rate was reduced by 46 %.

In addition, from the analysis of Fig. 19, it can be observed that as the flow rate of the canopy hood increases from 1125m3/h to 2415m3/h, the capture efficiency increases approximately linearly with the exhaust flow rate, and the rate of the increase rate is about 2.49 x 10-4; while the growth rate of the capture efficiency decreases to 5.10 x 10-5 as flow rate varies from 2415m3/h to 4500m3/h. Hence, the IESR method can be considered as an effective design method for calculating the exhaust flow rate, and the local exhaust system designed according to the IESR method can simultaneously meet the dual objectives of efficient pollution control and energy conservation.

5. Conclusion

When designing an industrial ventilation system to efficiently capture the pollutants, the local exhaust hood should be located as close to the source as possible, and the accurate source release rate is the important factor that directly affects the near-source environment and the flow rate of the exhaust system. Unlike the pollutant sources in civil buildings, for the SCHP sources, it is difficult to directly test the source release rate, either because of the inconsistent emission parameters tested at different locations or because of the large area of the source. To solve this problem, this paper analyzes the applicability of four different regularization methods in the inverse estimation problem of the SCHP sources, and studies the influence of the denoising filter and the strong convection effect on the stability and accuracy of the IESRs. At the same time, a new design method based on the IESR is proposed to calculate the flow rate of the canopy hood. The conclusions are as follows:

(1) The accuracy of the inverse results was closely related to with the regularization methods and the location of the measurement points. Compared with the ZOTR, and the LSQR methods, the SOTR and the TSVD methods are more suitable for the inverse estimation of the SCHP source, and the SORT method is the best regularization method for the point less affected by the SCHP source, while the TSVD method is the best method for the point close to the source.

(2) It can be concluded that the inverse solution method including the denoising filter can significantly improve the accuracy of the results, regardless of the high-frequency fluctuation deviations or highamplitude deviations, both can be effectively reduced, and the FIR filter is better than the Butterworth filter in eliminating high amplitude deviations. From the comprehensive consideration of the RMSE and the RE, it can be concluded that, the TSVD-FIR method is the best inverse solution method for the near-source measurement points; and the SORT-FIR method has the better stability and smaller deviation for the environmental measurement points.

(3) The effect of the measurement errors on the accuracy of the ISREs was tested, it was found that the accuracy of the inverse results gradually reduced with the increase of the perturbance error. When the measurement error larger than 25 %, the A^regwas greater than 45 m3/h, the case with this input concentration cannot correctly estimate the release rate of the SCHP source.

(4) It can be concluded that the increase of the convection effect of the source has the negative effect on the accuracy of the IESRs, the RE of the inverse results increases with the increase of the Pe number at the same point. Compared with the REs of the SOTR-FIR method in case 1, the REs in case 2, case 3, and case 4 increased by 6.94 %, 70.20 %, and 127.55 %, respectively.

(5) The optimal inverse solution method is different for the pollutant sources with different convection strength. For the low release rate SCHP source, the optimal inverse solution method is the SOTR-FIR method. For cases with , the accuracy of the result is related to the location of the measurement points. For the near-source point, the TSVD-FIR method is the best method; while for the measurement points located in the exhaust duct and the environment, the SOTR-FIR method is the best. In summary, for the heated pollutant sources with strong convection, the SORT-FIR method has a better accuracy and stability.

(6) By comprehensively considering of the capture performance and the exhaust flow rate of the canopy hood, it can be concluded that the IESR method can be considered as an effective and economical design method for calculating the exhaust flow rate. Compared with the local exhaust system designed according to the VDP-max method, the local exhaust system designed according to the IESR method can also effectively control the pollutants within the capture area of the canopy hood, the capture efficiency was 86.80 %, but the exhaust flow rate was reduced by 46 %.

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.

CRediT authorship contribution statement

Lei Cao: Writing - review & editing, Writing - original draft, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. Yi Wang: Supervision, Resources, Funding acquisition, Conceptualization. Yanqiu Huang: Writing - review & editing, Supervision, Resources, Methodology, Funding acquisition. Shengnan Guo: Writing - review & editing, Investigation. Junwei Guo: Writing - review & editing, Validation, Investigation. Yingke Zheng: Investigation.

Acknowledgement

This research was supported by the National Natural Science Foundation of China (Grant No.52178089), and the Science and Technology Plan Project of Yulin (CXY-2021-139).

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ARTICLE INFO

Received 1 May 2023; Received in revised form 6 February 2024; Accepted 12 February 2024

Available online 14 February 2024

References

References

[1] H.D. Goodfellow, Y. Wang, Industrial ventilation design guidebook, Engineering Design and Applications 2, Academic press, 2021.

[2] N.K. Ala, P.A. Domenico, Inverse analytical techniques applied to coincident contaminant distributions at Otis Air Force Base, Massachusetts, Ground Water 30 (2) (1992) 212-218.

[3] H. Jia, H. Kikumoto, Source term estimation in complex urban environments based on Bayesian inference and unsteady adjoint equations simulated via large eddy simulation, Build. Environ. 193 (2021) 107669.

[4] P. Kathirgamanathan, R. Mckibbin, R. Mclachlan, Source Term Estimation of Pollution from an Instantaneous Point Source, Modsim 3 (2002) 59-67.

[5] X. Liu. University of Colorado at Boulder, Identification of Indoor Airborne Contaminant Sources With Probability-Based Inverse Modeling Methods, (2008).

[6] J. Wang, B. Wang, J. Liu, W. Cheng, J. Zhang, An inverse method to estimate the source term of atmospheric pollutant releases, Atmos. Environ. 260 (2021) 118554.

[7] G.C. Efthimiou, I.V. Kovalets, C.D. Argyropoulos, A. Venetsanos, S. Andronopoulos, K.E. Kakosimos, Evaluation of an inverse modelling methodology for the prediction of a stationary point pollutant source in complex urban environments, Build. Environ. 143 (2018) 107-119.

[8] L.C. Thomson, B. Hirst, G. Gibson, S. Gillespie, P. Jonathan, K.D. Skeldon, M.J. Padgett, An improved algorithm for locating a gas source using inverse methods, Atmos. Environ. 41 (6) (2007) 1128-1134.

[9] H. Kouichi, P. Ngae, P. Kumar, A.A. Feiz, N. Bekka, Optimization of an urban monitoring network for retrieving an unknown point source emission, Geosci. Model Develop. Discuss. (2018) 1-24.

[10] D. Ma, J. Gao, Z. Zhang, Q. Wang, An improved firefly algorithm for gas emission source parameter estimation in atmosphere, IEEe Access. 7 (2019) 111923-111930.

[11] D. Ma, Z. Zhang, Contaminant dispersion prediction and source estimation with integrated Gaussian-machine learning network model for point source emission in atmosphere, J. Hazard. Mater. 311 (2016) 237-245.

[12] R. Wang, B. Chen, S. Qiu, L. Ma, Z. Zhu, Y. Wang, X. Qiu, Hazardous source estimation using an artificial neural network, particle swarm optimization and a simulated annealing algorithm, Atmosphere (Basel) 9 (4) (2018) 119.

[13] L. Zeng, J. Gao, L. Lv, B. Du, Y. Zhang, R. Zhang, W. Ye, X. Zhang, Localization and characterization of intermittent pollutant source in buildings with ventilation systems: development and validation of an inverse model, Build. Simul. 14 (3) (2020) 841-855.

[14] L. Zeng, J. Gao, L. Lv, R. Zhang, Y. Chen, X. Zhang, Z. Huang, Z. Zhang, Markov-chain-based inverse modeling to fast localize hazardous gaseous pollutant sources in buildings with ventilation systems, Build. Environ. 169 (2020).

[15] K. Hiyama, Y.I. DrEng, S.K. DrEng, Coupling 3D transient pollutant transport in a room into a flow network model with concentration response factor method, ASHRAe Trans. 114 (2) (2008) 119-129.

[16] Y. Ishida, S. Kato, H. Huang, X. Zhang, Calculation method for transient concentration response on the steady state flow field : part 1-calculation method and examples of transient concentration response in an air-conditioned room with return air duct system, Neuroimmunomodulation. 1 (1) (1994) 14-22.

[17] Y. Ishida, S. Kato, Method for Coupling 3D Transient Pollutant Transport within Room into ID Transport Simulation with Flow Net Work Based on Concentration Response Factor, Ashrae Winter Meeting, 2008.

[18] H. Cai, X. Li, L. Kong, X. Ma, X. Shao, Rapid identification of single constant contaminant source by considering characteristics of real sensors, J. Cent. South. Univ. 19 (3) (2012) 593-599.

[19] X. Shao, Research On Demand-Oriented Mechanism For Building Non-Uniform Indoor Environment, Tsinghua University, 2013 Beijing.

[20] A.N. Tikhonov, A. Goncharsky, V.V.e. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Springer, Netherlands, 1995.

[21] R.C. Aster, B. Borchers, C.H. Thurber, Parameter Estimation and Inverse Problems, Elsevier, 2018.

[22] X. Liu, Z. Zhai, Inverse modeling methods for indoor airborne pollutant tracking: literature review and fundamentals, Indoor. Air. 17 (6) (2007) 419-438.

[23] X. Liu, F. Li, H. Cai, K. Zhang, J. Liu, J. Xu, X. Li, Dynamical source term estimation in a multi-compartment building under time-varying airflow, Build. Environ. 160 (2019).

[24] F. Li, X. Liu, J. Liu, H. Cai, H. Wang, K. Zhang, C. Dai, Solutions to mitigate the impact of measurement noise on the air pollution source strength estimation in a multi-zone building, Build. Simul. 13 (6) (2020) 1329-1337.

[25] S. Yin, Quantitatively Identify Unsteady Gas Pollutant Releases in Indoor Environment By Inverse CFD Modeling, Dalian University of Technology, Dalian, 2011.

[26] T. Zhang, S. Yin, S. Wang, An inverse method based on CFD to quantify the temporal release rate of a continuously released pollutant source, Atmos. Environ. 77 (2013) 62-77.

[27] Q. Chen, Z. Zhai, X. You, T. Zhang, Inverse Design Methods for the Built Environment, Routledge, 2017.

[28] T. Zhang, H. Zhou, S. Wang, Inverse identification of the release location, temporal rates, and sensor alarming time of an airborne pollutant source, Indoor. Air. 25 (4) (2015) 415-427.

[29] Y. Wei, H. Zhou, T. Zhang, S. Wang, Inverse identification of multiple temporal sources releasing the same tracer gaseous pollutant, Build. Environ. 118 (2017) 184-195.

[30] P.C. Hansen, Rank-deficient and Discrete Ill-Posed problems: Numerical Aspects of Linear Inversion, SIAM, 1998.

[31] J.L Kunpu, Y.Z. Shen, in: Unbiased Estimation of Unit Weight Variance by TSVD Regularization, 45, Geomatics and Information Science of Wuhan University, 2020, pp. 626-632.

[32] P.C. Hansen, The Truncated SVD as a Method for Regularization, BIT Numerical Mathe. 27 (4) (1987) 534-553.

[33] C.C. Paige, M.A. Saunders, LSQR, An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software 8 (1) (1982) 43-71.

[34] Y. Huang, Iterative Regularization Methods For Large-Scale Discrete Ill-Posed Problems Dissertation Submitte, Tsing unversity press, 2015 Beijing.

[35] T. Min, X. Han, Hybrid regularization LSQR algorithm for image denoising, J. Math. 40 (3) (2020) 332-340.

[36] M. Li, F. Li, Y. Jing, K. Zhang, H. Cai, L. Chen, X. Zhang, L. Feng, Estimation of pollutant sources in multi-zone buildings through different deconvolution algorithms, Build. Simul. 15 (5) (2022) 817-830.

[37] W. Liang, R. Quinte, X. Jia, J.Q. Sun, MPC control for improving energy efficiency of a building air handler for multi-zone VAVs, Build. Environ. 92 (2015) 256-268.

[38] X. Yu, K.S. Skeie, M.D. Knudsen, Z. Ren, L. Imsland, L.J.B. Georges, Environment, Influence of data pre-processing and sensor dynamics on grey-box models for space-heating, Anal, using field measure. 212 (2022) 108832.

[39] L. Tan, J. Jiang, Digital Signal processing: Fundamentals and Applications, Academic Press, 2018.

[40] Y. Yu, V. Loftness, D. Yu, Multi-structural fast nonlinear model-based predictive control of a hydronic heating system, Build. Environ. 69 (2013) 131-148.

[41] P. Cheng, Digital Signal Processing Course, Tsing unversity press, 2001 Beijing.

[42] S. Chen, J. Toelke, M. Krafczyk, Numerical investigation of double-diffusive (natural) convection in vertical annuluses with opposing temperature and concentration gradients, Int. J. Heat Fluid Flow 31 (2) (2010) 217-226.

[43] B.R. Morton, Forced plumes, J. Fluid Meeh. 5 (1959) 151-163.

[44] Y.Q. Huang, K. LU, J.W. Guo, Y. Wang, C.X. Zhao, Y.N. Zhai, Study on ventilation performance of lateral exhaust hood under the influence of two high-temperature buoyant jets, Build. Environ. 177 (2020) 106849.

[45] Y.Q. Huang, K. LU, Y. Wang, J. Chuang, L. Cao, Y. Liu, Study on limit flow ratio method for a lateral exhaust hood above high-temperature buoyant jets, Sustain. Cities. Soc. 54 (2020) 101969.

[46] W.X. Lin, S.W. Armfield, Direct simulation of weak axisymmetric fountains in a homogeneous fluid, J. Fluid. Meeh. 403 (2000) 67-88.

[47] P. J. Roache, Perspective: a method for uniform reporting of grid refinement studies, J. Fluid Eng. 116 (3) (1994) 405-413.

[48] Y.Q. Mao, A.G. Li, J.G. Che, T.Q. Wang, C.Q. Yang, L.F. Che, J. Liu, Investigation of heavy gas dispersion characteristics in a static environment: spatial distribution and volume flux prediction, Build. Environ. 242 (2023) 110501.

[49] D. Lin, J. Zhu, Y. Song, Y. He, Construction method of regularization by singular value decomposition of design matrix, Acta Geodaetica Et Cartographica Sinica 45 (8) (2016) 20150134.

[50] H. Cai, X.T. Li, Z.L. Chen, M.Y. Wang, Rapid identification of multiple constantly-released contaminant sources in indoor environments with unknown release time, Build. Environ. 81 (2014) 7-19.

[51] Z.Q. Zhai, Q. Jin, Identifying decaying contaminant source location in building HVAC system using the adjoint probability method, Build. Simul. 11 (5) (2018) 1029-1038.

[52] T. Hayashi, R.H. Howell, Industrial Ventilation and Air Conditioning, Crc Press Inc, Boca Raton, 1985 Fl.

Footnote

* Corresponding author.

E-mail address: [email protected] (Y. Huang).