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Finding out the optimal sampling positions for measurement of ventilation rates in a naturally ventilated building using tracer gas is a challenge. Affected by the wind and the opening status, the representative positions inside the building may change dynamically at any time. An optimization procedure using the Response Surface Methodology (RSM) was conducted. In this method, the concentration field inside the building was estimated by a threeorder RSM polynomial model. The experimental sampling positions to develop the model were chosen from the crosssection area of a pitchedroof building. The Optimal Design method which can decrease the bias of the model was adopted to select these sampling positions. Experiments with a scale model building were conducted in a wind tunnel to achieve observed values of those positions. Finally, the models in different cases of opening states and wind conditions were established and the optimum sampling position was obtained with a desirability level up to 92% inside the model building. The optimization was further confirmed by another round of experiments.
The ventilation rate is a crucial parameter for controlling the indoor climate of buildings. It can affect the temperature, humidity and gas concentration around occupants, which is very important for their living or working conditions [
However, determining the actual ventilation rate in a naturally ventilated space with large openings is always a challenge. Tracer gas experiments are commonly conducted to calculate the ventilation rate from the difference between indoor and outdoor concentrations. The tracer gas method prefers the good mixing of tracer gas inside the building. For those buildings, imperfect mixing would lead to large uncertainty of the resulting ventilation rate measurements. It is very difficult to achieve in naturally ventilated buildings [
Thus, research is needed for decreasing the sampling positions and finding out the optimal regions in order to estimate the ventilation rate. Buggenhout
Another difficulty is that the optimal sampling positions may change location when the opening states are modified or the wind direction changes. This is a challenge for us when intending to measure the indoor concentration by using fewer sampling positions. Therefore, it is necessary to determine the sampling positions independent of the opening states and wind directions.
The objective of this study is to find the optimal sampling positions for ventilation rate measurement in a naturally ventilated building. In this research, we focused on the tracer gas measurement method. The investigation includes eight different cases, with four opening states and two wind directions in perpendicular or oblique wind directions. The concentration fields of the tracer gas were modeled with spatial variables and the optimal positions were determined.
The experiments were performed in a wind tunnel located in the Air Physics Lab, Aarhus University, using a model building.
A 1:25 scale model building was represented as a subsection of the standalone livestock building. The dimension and layout of the scale model building in the wind tunnel are shown in
In order to measure the ventilation rate through the ventilated scale model building, the tracer gas constant injection method [
In this study,
Setup of the tracer gas experiment is shown in
The indoor and outdoor CO_{2} concentration was monitored and recorded by an INNOVA 1309 multiplier and analyzer. The sampling duration of each recording was 5 s and the flush time of the instrument chamber between two recordings was 20 s. Three such continuous recordings and flashings were taken for each sampling position.
The Response Surface methodology aimed to formulate an approximated model to predict the response by the related variables and adopts for the optimization process [
Once we have successfully built up an approximate model to predict the concentration field, we can apply it to find out the optimal position in the field for measuring the ventilation rate. In this study, the concentration field can be predicted by the approximate mathematical model that is established by the RSM method. The underlying model has the form:
Normally,
The overall procedure of the RSM method involves following steps [
Designing the experiment for adequate and reliable measurement of the response of interest. It involves: (1) construct the design space, (2) determine the number of design points, (3) locate the design points by experimental design methods.
Developing a polynomial model of the third order response surface with the best fittings.
Finding the optimal set of experimental parameters that produce optimum value of response.
In the beginning of designing the experiment, it is necessary to construct the design space, where the sampling positions are chosen. As seen in
In which,
In
From the constraints form [
The prediction by the established model may involve certain variance compared with the observed response. The variance is found to be related with the chosen experimental design methods [
The optimal design method can find out the sampling positions based on the specified optimality criterion [
Despite the reduction of model bias, the sampling positions should be well distributed in the design space. Thus the “space filling design”, is crucial and should be followed to show extensive information of the overall design space. However, the IVoptimality may not be efficient to achieve that because it searches the sampling that can only decrease the model variance but not consider the spread within the design space. In this study, the Distance based design is often coupled with IVoptimal design to make the sampling positions full filling in the design space [
Thus, as the first step, the IVoptimal design method was adopted to search the design space to find out efficient sampling positions to build up the model. And then the Distance based design fill the gap between those existing positions and make the design space “space filling”. This method had reported to use in some researches in term of its comprehensive and redundant characteristics [
In order to determine the number of sampling positions to develop the model in
The total number of sampling positions that necessarily required constructing a regression model was determined by the experimental design tool contained in the Design Expert ver. 8 software. The design tool is named as the Fraction of Design Space (FDS), which aims to evaluate the efficiency of the chosen number of sampling positions, as well as the quality of design method. In order to do that, this tool will observe what percentage of the design space is under the actual experimental error based on the expected variance of the response at certain statistically significance level. And thus, the expected variance of the response and the actual experimental error was necessary to be known [
The expected change of the response detected by the established model (
According to
The measurement uncertainty of
Thus according to
Based on the minimum expected change of the response and the estimated experimental error, the Design Expert ver. 8 software can predict the distribution of standard error in the design space. The distribution of standard error was modeled by the sampling positions based on the estimated thirdorder model as shown in
It can also be found that large standard error near the boundary of the design space from
Followed by design of experiment and conduct of the designed experiment, the corresponding threeorder RSM model was established respectively with the general form as depicted in
Of all cases, the established model that presented the low
Transformation of the response is an important method when the model is formulated with poor fitting to the observed response [
The desirability function [
The objective function reflects the level of desirability. When the observation is close to the target, the desirability level is high. The range of the desirability was from zero to one (minimum to maximum desirability, respectively).
In this study, for each case as listed in
Our goal was to find the optimal sampling positions for the tracer gas measurement; the goal can be determined as the target value, maximum, minimum or a range within the design space [
Layout of sampling positions for the indoor CO_{2} concentration that obtained from the RSM method is given in
The sampling position of the background concentration was located 0.6 m upstream of the scale model building, 0.1 m above the wind tunnel floor. The gas concentration inside the air mixing box was measured close to the top surface of the air mixing box as seen in
Sampling pipes with a diameter of 3 mm were used to take air samples in all positions. These sampling pipes were mounted on the building endwall and horizontally placed along the building length. Eight sampling orifices with dimensions of 1 mm were evenly distributed along each pipe with an interval of 50 mm. The indoor air at a sampling position was sucked from two ends of the pipe and delivered to the multiplier and the gas monitor through the external pump (230 V, 6.3 × 10^{−5} m^{3}/s).
As seen from
When the
Contour plots of the established RSM models predicting the concentration ratio are shown in
As we see from
Thus, the objective function shown in
For validation purposes, the optimal sampling position obtained for all cases of opening states was tested by an additional new confirmation experiment.
In each case, the estimated concentration ratio of the optimal position is very close to the surface averaged concentration ratio and thus the estimated desirability is very close to 1. After the confirmation experiment, the measured concentration ratio of most cases is also quite close to the surface averaged and the actual desirability is close to 1, except for Case 4. Without considering Case 4, the actual integrated desirability of all other cases equals 0.86 and the estimated one is 0.92. Both are quite close to 1 which indicates the position is optimal for the indoor concentration measurement.
As seen from
However, this method also shows some weakness and uncertainty in this study. In
In order to avoid the failure of such cases, two methods are suggested. One is to increase the amount of tracer gas released into the indoor space to increase the indoor concentration, and thus get a higher difference between the indoor and outdoor concentration. Another way is to decrease the experimental error of indoor concentration measurement.
A challenge of this method is to do a prior experiment before the designed experiment is done. The reason of doing this is because we have to know the actual experimental error, with which we can know how many sampling positions or design points we should define in order to overcome the expected model variance. This indicates that if more accurate prediction by the RSM method is expected, more sampling positions should be added into the design space to construct the model [
The advantage of this method is that it can be able to minimize the number of sampling positions and consequently decrease the workload on measurement. Buggenhout
Challenges also exist in finding the proper experimental design method to seek the sampling positions or design points in the design space. When the design space is regular and not constrained such as rectangular space, a lot of potential methods can be used, such as central composite design and factorial design [
This study used Response Surface Methodology (RSM) to determine the optimal sampling positions for the indoor concentration on the purpose to calculate the ventilation rate accurately.
The RSM modeling technique can be used to optimize the selection of sampling positions for environmental measurement. The target of the optimization was set to be the indoor space average. The optimal sampling positions for the indoor concentration measurement can be seen in the desirability graphs as shown in
Optimal design method was used to search for the design points in a constrained two dimensional design space. The design space mathematically represented a crosssection plane of indoor space in a naturally ventilated building. Thus, this method can be introduced for a naturally ventilated building to searching for the optimal position to measure or monitor several environmental parameters, not only for gas concentration, but also for odor, temperature and humidity,
The required number of design points by RSM to formulate the model is direct proportional to the expected error. In this study, we expected 15% uncertainty of the observed value and managed to formulate a three degree polynomial function. Eighteen design points were selected and the estimated error of 94% fraction in design space was under the actual experimental error.
Statistical tools such as backwards regression and BoxCox transformation were utilized for model establishment in this research. The
The authors would like to thank Jan Ove Johnsen for the technical support and Georgios Ntinas for help in carrying out the experiment. The comments and suggestions for the manuscript preparation from Jan S. Strøm, Senior Scientist emeritus, the Department of Engineering of Aarhus University are greatly appreciated. Thanks also extend to the financial support from the China Scholarship Council (CSC).
(
Distribution of standard deviation error as a thirdorder response surface from the selected sampling positions designed by IV optimal design. The minimum detectable change in the response is 0.15, mean experimental error is 0.087, and significant level is 0.05. Within the design space, 94% section of region area is lower than the surfaceaveraged standard error, which is equal to 0.75.
Desirability graph for the optimization of the concentration ratio within the design space, in which the optimal position = 0.93 shows the optimal position for the concentration measurement of all cases (Cases 1–8). The coordinates of this position are
Experimental setup of different cases (cases 1–8).
mm  mm  mm  mm  °  

1  50  50  105  105  90 
2  50  50  55  105  90 
3  50  50  105  55  90 
4  50  50  55  55  90 
5  50  50  55  55  45 
6  50  50  105  55  45 
7  50  50  105  105  45 
8  50  50  55  105  45 
List of sampling positions of tracer gas experiment, by IV optimal design from software Design Expert ver. 8.
8  1  1.04  0.42  270  110 
3  2  1.04  0.00  270  0 
2  3  0.00  0.00  0  0 
5  4  0.54  0.19  140.4  49.1845 
1  5  −1.04  0.00  −270  0 
11  6  0.00  0.65  0  168 
6  7  0.00  0.35  0  92 
9  8  −0.61  0.62  −159.3  162 
4  9  −0.54  0.19  −140.16  49.0287 
10  10  0.62  0.62  161.244  162.364 
12  11  0.00  0.92  0  240 
7  12  −1.04  0.42  −270  110 
14  13  0.44  0.00  113.4  0 
15  14  1.03  0.21  267.3  54 
16  15  −1.04  0.21  −270  55.0898 
13  16  −0.46  0.00  −118.8  0 
18  17  0.52  0.41  135  106.8 
17  18  −0.51  0.41  −132.3  106.506 
Experimental result of all cases (Cases 1–8) listed in
 

AVG 
STDEV 
Rel. 
AVG 
STDEV 
Rel. 
AVG 
STDEV 
Rel. 
AVG 
STDEV 
Rel.  
1,072.2  2.5  0.2  1,177.8  11.9  1.0  1,084.0  3.9  0.4  1,136.0  6.9  0.6  
2,722.5  113.9  4.2  1,064.3  20.9  2.0  2,815.7  156.6  5.6  1,066.5  18.2  1.7  
1,844.1  106.3  5.8  1,519.5  37.0  2.4  1,805.6  80.6  4.5  1,396.6  25.6  1.8  
2,114.4  129.5  6.1  1,064.7  6.2  0.6  2,147.9  141.3  6.6  1,045.0  4.2  0.4  
1,383.9  12.8  0.9  1,697.8  32.2  1.9  1,398.6  21.6  1.5  1,906.9  38.4  2.0  
1,343.9  13.6  1.0  1,104.1  7.7  0.7  1,359.4  23.1  1.7  1,070.2  7.2  0.7  
1,526.8  27.2  1.8  1,170.2  12.2  1.0  1,550.4  43.4  2.8  1,131.1  7.5  0.7  
1,293.6  11.1  0.9  1,173.5  15.5  1.3  1,292.7  14.3  1.1  1,119.4  8.2  0.7  
1,424.5  23.1  1.6  1,416.5  29.5  2.1  1,439.0  19.6  1.4  1,306.8  24.5  1.9  
1,173.3  16.4  1.4  1,157.9  7.8  0.7  1,186.4  14.7  1.2  1,146.1  9.7  0.9  
1,261.9  19.9  1.6  1,096.8  15.3  1.4  1,257.8  15.0  1.2  1,086.4  6.7  0.6  
1,307.5  15.2  1.2  1,264.1  21.0  1.7  1,304.1  19.8  1.5  1,150.9  10.6  0.9  
3,201.4  303.9  9.5  1,101.5  23.2  2.1  3,113.7  202.9  6.5  1,072.9  14.4  1.3  
2,556.9  87.4  3.4  1,073.5  7.6  0.7  2,573.0  116.9  4.5  1,063.2  6.1  0.6  
1,391.3  17.0  1.2  1,632.2  34.3  2.1  1,337.9  16.4  1.2  1,227.9  12.9  1.1  
1,573.8  77.1  4.9  2,436.2  108.9  4.5  1,602.2  92.2  5.8  2,238.0  84.3  3.8  
1,516.7  27.0  1.8  1,091.4  8.1  0.7  1,520.7  33.1  2.2  1,086.8  4.9  0.5  
1,374.8  19.2  1.4  1,198.9  16.7  1.4  1,379.4  17.2  1.2  1,151.9  11.3  1.0  
1,006.2  3.5  0.3  1,013.9  8.2  0.8  1,018.8  3.3  0.3  1,021.3  2.9  0.3  
47,770.8  246.3  0.5  49,327.1  139.7  0.3  48,670.9  128.8  0.3  47,981.8  73.3  0.2  
 
1,213.1  15.8  1.3  1,098.9  7.5  0.7  1,086.4  17.8  1.6  1,253.1  14.9  1.2  
1,196.6  26.7  2.2  2,215.1  118.4  5.3  2,264.5  151.4  6.7  1,211.4  33.7  2.8  
2,437.4  182.9  7.5  2,650.9  188.7  7.1  2,869.2  197.4  6.9  2,549.0  186.8  7.3  
1,165.2  16.5  1.4  1,867.3  76.0  4.1  1,905.9  76.3  4.0  1,183.6  22.5  1.9  
1,736.0  40.1  2.3  1,534.2  36.8  2.4  1,420.5  26.9  1.9  1,630.2  47.3  2.9  
1,214.4  30.4  2.5  1,621.3  31.3  1.9  1,686.8  28.4  1.7  1,320.4  51.4  3.9  
1,371.9  25.9  1.9  1,812.8  39.5  2.2  1,879.7  44.3  2.4  1,514.0  68.4  4.5  
1,200.4  15.5  1.3  1,473.3  17.3  1.2  1,501.3  18.6  1.2  1,240.7  18.3  1.5  
1,776.5  63.2  3.6  1,646.2  47.3  2.9  1,738.1  54.5  3.1  1,980.9  61.9  3.1  
1,192.3  14.9  1.2  1,489.8  26.9  1.8  1,525.5  27.9  1.8  1,248.5  26.1  2.1  
1,127.3  20.8  1.8  1,459.1  20.2  1.4  1,502.0  23.4  1.6  1,154.7  18.9  1.6  
1,265.5  37.0  2.9  1,410.5  13.1  0.9  1,481.1  12.1  0.8  1,382.0  27.1  2.0  
1,602.3  84.6  5.3  2,767.6  148.3  5.4  2,793.0  111.7  4.0  1,641.1  93.1  5.7  
1,038.2  14.0  1.3  1,886.3  116.0  6.1  2,015.3  109.3  5.4  1,038.7  10.1  1.0  
1,374.0  14.5  1.1  1,452.8  14.8  1.0  1,483.0  19.9  1.3  1,725.7  39.2  2.3  
3,080.9  165.2  5.4  3,779.8  273.1  7.2  4,092.6  236.9  5.8  3,614.4  222.3  6.2  
1,177.0  24.6  2.1  1,599.2  36.3  2.3  1,655.9  34.0  2.1  1,250.2  33.0  2.6  
1,380.9  42.2  3.1  1,613.7  27.1  1.7  1,672.6  46.8  2.8  1,534.0  39.0  2.5  
1,023.7  10.4  1.0  1,020.1  7.0  0.7  1,006.3  5.0  0.5  1,017.8  7.0  0.7  
45,217.5  102.2  0.2  47,192.6  124.3  0.3  47,539.6  91.7  0.2  46,146.3  88.4  0.2 
Established RSM models and their quality of fit of varied cases (Case 1–8). Adj. R^{2} and Pred. R^{2} represent the
Case 1  0.76  0.63 
 
Case 2  0.93  0.82 
 
Case 3  0.73  0.58 
 
Case 4  0.67  0.54 
 
Case 5  0.75  0.52 
 
Case 6  0.69  0.47 
 
Case 7  0.59  0.40 
 
Case 8  0.90  0.78 

Actual values indicate the observations of the response.
Contours of the response suface of the established enhanced regression model for different experiemental cases (Cases 1–8). The black bars in the plots represent the obstacles in the sidewall openings. In the plot, the incoming wind blows from the right side to the left.
Case 1 

Case 7 

Case 2 

Case 8 

Case 3 

Case 6 

Case 4 

Case 5 

Contours of the response suface of the desirability function for different experiemntal cases (Cases 1–8). The black bars in the plots represent the obstacles in the sidewall openings.In the plot, the incoming wind blows from the right side to the left.
Case 1 

Case 7 

Case 2 

Case 8 

Case 3 

Case 6 

Case 4 

Case 5 

Confirmation of the optimal sampling position (
0.53  0.56  0.95  0.43  0.74  
0.25  0.21  0.97  0.41  0.84  
0.53  0.53  1.00  0.42  0.76  
0.49  0.36  0.92  0.37  0.99  
0.94  0.76  0.91  0.70  0.91  
1.24  0.91  0.85  0.76  0.82  
0.79  0.50  0.86  0.62  0.94  
All  0.93  0.80 