The experiments were performed in a wind tunnel located in the Air Physics Lab, Aarhus University, using a model building. Figure 1 shows an experimental setup in the wind tunnel. The wind tunnel has double sidewalls and the ceiling surface was removed at the working section. The working section had a dimension of 6.00 m (length) × 1.38 m (width) × 1.55 m (height). 107 wooden blocks with dimensions of 50 mm × 50 mm × 50 mm and 25 mm × 50 mm × 50 mm respectively, were placed in the front of the working section with a distance of 100 mm apart in between to create an amount of roughness on the floor and to make the velocity profile at the working section similar to the natural wind over agricultural fields. The resulting reference wind velocity at 0.5 m high above the floor was equal to 3.2 m/s. and the wind profile can be described in [10]: (1) v = ( 3.46 ± 0.02 ) h 0.17 ± 0.004 ( R 2 = 0.97 )where, v, air velocity, m/s and h, height measured from the floor, m.

A 1:25 scale model building was represented as a sub-section of the stand-alone livestock building. The dimension and layout of the scale model building in the wind tunnel are shown in Figure 1. The roof and wall of the model building were made of a 5 mm thick clear acrylic sheet. The external dimensions of the models were 500 mm in length and 550 mm in width, and ridge height 260 mm. The sidewall under the eave was 130 mm high, and the resulting fully opening size was 100 mm. The roof slope was 25° and volume of the building was 0.11 m³. The internal space of the model building had the width of 270 mm, height of 240 mm and side eave height of 110 mm.

In order to measure the ventilation rate through the ventilated scale model building, the tracer gas constant injection method [11,12] was adopted in this study. The constant method was often used when the ventilation rate remained constant during the measurement period, similar as this study. In the beginning, the injection volume flow of tracer gas was injected to the measured building space at a constant, q(m³/s). After the steady state of the indoor concentration field has been reached, concentration differences between the indoor and outdoor, (C_i − C_o), were monitored simultaneously. The ventilation rate Q (m³/s) was then calculated by Equation (2): (2) Q = q / ( C i − C o ) = q / ( C ⋅ C o )

In this study, C is the dimensionless gas concentration, which is defined as: (3) C = ( C i − C o ) / C owhere, C_i, C_o is the measured indoor and the outdoor concentration, respectively.

Setup of the tracer gas experiment is shown in Figure 1. CO₂ was used as the tracer gas and released from a CO₂ tank (27 L). The air current was guided into a gas regulator and a control valve in order to keep the CO₂ injection rate stable at a constant level of q = 2.5 × 10⁻⁶ m³/s. The pure CO₂ was then mixed with the room air in an air mixing box, where a rotation fan was placed at the bottom to enhance the air mixing. An air pump at the speed of 7.17 × 10⁻⁵ m³/s was used to deliver free air into the box to mix with the pure CO₂. The top of the box was connected to the indoor space through approximately 100 orifices evenly distributed on the floor. The mixed air in the box can be delivered into the building space through these orifices.

The indoor and outdoor CO₂ 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.

Table 1 shows the dimension of openings in different cases of opening state and directions. As shown in Table 1, eight cases of opening states and external wind directions were investigated in the research. Through these cases, different airflow patterns and variable concentration distributions of indoor space can be obtained. During the experiment, the opening states of the building were operated by changing the solid curtain mounted on the sidewall openings. The size of the sidewall opening remained the same in all cases. However, the locations of the openings were different. Two wind directions were investigated, one was the perpendicular and the other was oblique direction. The directions were modified by rotating the scale model building around the center point on the wind tunnel platform as seen in Figure 1(b).

In the beginning of designing the experiment, it is necessary to construct the design space, where the sampling positions are chosen. As seen in Figure 1(a), the cross-section area of the scale model building is enveloped by the roof and sidewall and can be depicted mathematically by a constrained design space: (5) ( 110 mm − 240 mm ) / ( 270 mm − 0 mm ) X + Y − 240 mm ≤ 0 ; ( 110 mm − 240 mm ) / ( − 270 mm − 0 mm ) X + Y − 240 mm ≤ 0 ; − 270 mm ≤ X ≤ 270 mm ; 0 mm ≤ Y

In which, X (mm), Y (mm) is the spatial variables in height and width direction of the building, respectively; the origin point of the X − Y coordinate is the center of the floor. The dimensionless form of the constraint form is shown in Equation (6), where the spatial variables are normalized by the maximum height of the model building (260 mm).

In Equation (4), g(x,y) ≤ 0, is the constraints form of the dimensionless coordinate(x,y). As seen in Figure 1(a), the sampling positions should be chosen from the internal cross-section area of a pitched-roof building. The cross-section area was constructed by the building envelop, which can be expressed mathematically by: (6) − 27 y + 13 x ≤ 24.9231 ; 27 y + 13 x ≤ 24.9231 ; − 1.04 ≤ x ≤ 1.04 ; 0 ≤ y

From the constraints form [Equations (5) and (6)], a design space was constructed, within which the sampling positions for experiment can be chosen. In order to formulate an appropriate model that represents the concentration field of the tracer gas, attention should be paid to the choice of the sampling positions to building up the model. The model accuracy would decrease if there are not sufficient sampling positions within the design space. Moreover, the efficiency of those positions for the model establishment is not verified [15]. Thus, several experimental design methods can be developed to select the sampling positions, such as the factorial design, orthogonal design, central composite design, and optimal design, etc. [17].

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 [16]. The methods with less variance and larger robustness are preferred for the model establishment. However, among those experimental design methods, the optimal design is mostly suitable for this study. That is either because only it can function well when the design space is constrained as similar as this study. Or it can establish the model with low bias and minimum-variance [15].

The optimal design method can find out the sampling positions based on the specified optimality criterion [15]. The basic principle of the optimality criterion is to decrease the model variance and bias to the minimum level. In this study, the IV-optimal design/or IV-optimality was used as the criterion. It sought a way to find the sampling positions with minimum integral prediction variance across the design space. Consequently the model can be built to provide lower prediction variance and less bias [16]. Detailed information of this method and the optimal design can also be found in the literatures [16,18].

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 IV-optimality 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 IV-optimal design to make the sampling positions full filling in the design space [19]. That is because the Distance based design chooses sampling positions in a way that achieves maximum spread throughout the design region.

Thus, as the first step, the IV-optimal 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 [15,20–23].

Followed by design of experiment and conduct of the designed experiment, the corresponding three-order RSM model was established respectively with the general form as depicted in Equation (4) based on the measured data of the response. In order to examine the fitting of the model, such indexes as Adjusted R² and Predicted R² were used in our study [15].

Adjusted R² is the R² adjusted by the number of terms in the model, where, R² is a measurement of the amount of variation around the mean explained by the model: (11) R 2 = 1 − ( SS error / SS Total )

SS_error is the sum of squares of residuals; SS_Total is the total sum of squares.

(12) Adjusted R 2 = 1 − ( SS error / SS Total ) ( df Total / df error ) = 1 − ( SS error / SS Total ) ( ( N − 1 ) / ( N − p ) )where, df_Total, df_error is the degrees of freedom, which are equal to N − 1, N − p respectively. N, p is the number of sampling positions and model terms. The established models are compared by the Adjusted R², not by the R², considering their number of terms, P, was varied.

Predicted R² is also a measure of how good the model predicts a response value. It is computed as: (13) Predicated R 2 = 1 − ( PRESS / ( SS Total ) )where, Predicted Residual Error Sum of Squares (PRESS) is a measure of how the model fits each point in the design. PRESS is calculated by systematically removing each observation from the data set, estimating the regression equation, and determining how well the newly established equation predicts the removed observation. The sum of squares between the observations and predictions are then summed to calculate the value of PRESS.

Of all cases, the established model that presented the low Adjusted R² and Predicted R² were required to be enhanced [28]. Thus in this study, backward regression method [21] was utilized to revise the established model by eliminating the non-significant terms. In the beginning of the regression process, the full model was established with the form as shown in Equation (4). We pre-defined the threshold value p equals to 0.10, and those terms lower than this p value would be remained. The elimination of terms of the model would terminate until most of them can satisfy the p-value. By this method, each term of the full model will give a chance to be included in the model and thus the model can be enhanced.

Transformation of the response is an important method when the model is formulated with poor fitting to the observed response [15]. The quality of fitting can be seen from how the errors (residuals) of the model go with the magnitude of the response (predicted values). Box-Cox Transformation [15] is a commonly applied transformation method, especially when the residuals are not stabilized and the statistical distribution does not present normal distribution. In this study, we evaluated square Root, natural log, inverse square root and inverse etc. of transformation function to enhance the established model. After the backward regression and Box-Cox Transformation, the RSM model with best quality of fit with the observations would be used for the optimization process.

The desirability function [13,29] was established based on the RSM model and used for optimization process. It made use of an objective function, which depend on the target goal defined by the users.

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 Table 1, the object function d_i (1 ≤ i ≤ 8, i ∊ N) came from the according RSM model. The simultaneous objective function, which showed the overall disability distribution within the design space, was equal to a geometric mean of all cases: (14) D = ( d 1 × d 2 × … × d 8 ) 1 / 8and d_i(Ĉ_i) was defined as: (15) C ^ i ( x , y ) < L i , d i ( C ^ i ) = 0 ; L i ≤ C ^ i ( x , y ) ≤ T i , d i ( C ^ i ) = ( C ^ i ( x , y ) − L i T i − L i ) s ; T i ≤ C ^ i ( x , y ) ≤ U i , d i ( C ^ i ) = ( C ^ i ( x , y ) − U i T i − U i ) t ; C ^ i ( x , y ) > U i , d i ( C ^ i ) = 0where, Ĉ_i, Ĉ_i(x,y) was the estimated concentration ratio by the established RSM model. And L_i, T_i and U_i was the low limit, target value and upper limit of Ĉ_i(x,y) of each case, respectively. The exponent s and t represented the degree of importance to hit the target value, respectively. In this study, both values equal 1 because of the same importance as the values getting close to the target value. If any of the responses or factors is not in their desirability range, the overall function becomes zero.

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 [15]. For simultaneous optimization, the response of each case is assigned a low, high limit value and a target. In our study, we set the target zone of each case where the local concentration was close to the surface average: (16) ∬ S f ( x , y ) dxdy ∬ S dxdywhere, S is the constrained region by spatial variables x and y. f(x,y) is the estimated response by the model regressed from the observed experimental data. Finally, the optimal position was confirmed by another new experiment, and the desirability and observed response values were compared with the estimation by the regression model.

Table 3 shows the direct measurement results of all cases (Cases 1–8) as listed in Tables 1 and 2. For all the cases, the relative error of the concentration measurement is lower than 8%, which shows a reasonable experimental error. The relative error of the position in the mixing box is below 0.5%, which indicates a stable source of tracer gas during the experiment. The direct measurement results are then turned into dimensionless results by Equation (3).

Table 4 shows the established models in all cases of opening states and wind directions. The actual responses of the model are the dimensionless concentration, C. The RSM models in Table 4 are different from the full model as shown in Equation (4) because they are reduced from the full regression model by eliminating the non-significant terms within it. Meanwhile, the observed responses are enhanced by using transformation functions such as natural log and square root.

When the Adjusted R² is closer to 1, the scattered sampling positions in the actual-predicted plot would accumulate in a line, indicates the predicted values are well in accordance with the actual values. It is noted that high Adjusted R² and Predicted R² of an established model do not occur simultaneously. As the degree of polynomial increases, the Adjusted R² will increase; however, it is not always the same for the Predicted R². Even a high order polynomial function can obtain a high Adjusted R², but may suffer from the low Predicted R². Thus, in order to establish an accurate model, it is necessary not just to focus on the Adjusted R² solely, but also consider the Predicted R² comprehensively. That is because the latter is also crucial to determine the quality of an established model. It is reported that the difference between Adjusted R² and Predicted R² should be within 0.2 [28]. From Table 4, six of the eight cases are in accordance with this requirement. Even though Case 5 and Case 6 do not fulfill it, however, both established models show relative high and positive Predicted R² and Adjusted R². This indicts that the models in Case 5 and 6 would be somehow efficient among all cases and can be adapted.

Contour plots of the established RSM models predicting the concentration ratio are shown in Table 5.

As we see from Table 5, the concentration field varies between cases because of different opening states and wind direction (Table 2). For Cases 1–4, the wind direction is perpendicular to the building sidewall. As seen in Table 5, the concentration ratio is low behind the windward opening, particularly with the opening close to the floor. High concentrations are found in the region close to the floor and near the leeward sidewall. With the windward opening in its high position the high concentrations are found near the floor under the opening. This phenomenon also exists in Cases 5–8, in which the incoming wind direction is 45° to the sidewall. It may indicate that location of the windward sidewall opening effects the distribution of response. Besides, Case 3 and Case 6 which have different wind direction and the same location of opening, show that the wind direction effects the distribution.

Table 6 shows the distribution of desirability for concentration measurement of each case. The desirable sampling positions should be representative and equal the volume average. Those red zones with the desirability close to 1 indicate that they are good and the blue zones close to 0 are bad for sampling positions. In separated cases, there are a big amount of candidate positions suitable for the measurement within the indoor space.

Thus, the objective function shown in Equation 15 may be utilized to calculate the geometric mean of desirability of all cases. Through this, we can obtain this adaptable function to predict the optimal positions for all cases.

Figure 3 shows the desirability graph of all cases for the optimization of sampling position selection. The integrated desirability distributes as a convex surface with the spatial variables. The red zone lies in the leeward part of the indoor space and the maximum desirability equals 0.93 lies in the position of x = −0.2, y = 0.27.

For validation purposes, the optimal sampling position obtained for all cases of opening states was tested by an additional new confirmation experiment. Table 7 presents the result of the confirmation test at the sampling position that shows the maximum desirability (x = −0.2, y = 0.27).

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 Table 4, the RSM model has shown desirable potential in predicting the concentration field. In Table 7, we can see that the optimal positions predicted by the model present high desirability for ventilation rate measurement. All of these can highlight the RSM modeling method in producing promising optimization and accurate modeling.

However, this method also shows some weakness and uncertainty in this study. In Table 7, we can find that Case 4 is distinct from other cases may because of the low estimated and actual concentration ratio in the optimal position. As seen in Equation (3), the low concentration ratio, C, may come from a low indoor and outdoor concentration difference, C_i − C_o. Besides, from Equation (10), if we kept the measurement error of the indoor concentration a constant, the relative error of C would be relative larger. These would make the actual response value difficult to detect in the experiment and thus lead to large uncertainty for Case 4.

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 [14–16].

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 et al. divided the measured zone equally into 36 partitions and measured the concentration in each partition [5]. A linear interpolation performed between the measurements to visualize the spatial distribution. However, compared to the method, the sampling method for RSM modeling can be more efficient and statistically supported [15,17,23]. Moreover, this research investigated the concentration field with 3D spatial variables and was different from 2D in our study. Further research is recommended for 3D cases based on the RSM method.

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 [16]. However, in terms of constrained design space as this study, many classified design methods such as central composite design, full factorial design cannot be applied. In this context, optimal design method can be used to build up the model. This research use IV Optimality combined with Distance Based Design for sampling. Further study can be focused on other optimal design method in varied optimality criterion, such as D-optimality, A-optimality and so on.