3.2.1. For the Whole Barn
Equation (
9) was found by using the BIC for all the data, and then fitting the individual natural-mixed and forced groups.
Table 4 and
Table 5 show the intercepts
and the coefficients associated with the term of the sum (the weights
) for each groups. The
for the curtain nonstandard cases were noted as OU for open-up and OD for open-down curtain. As expected,
and
. The resulting RSE and
are, consequently, different, and are revealed in
Table 4 as well. The readers may have noticed the small values of the coefficients
and
. However, remember that those coefficients have to be multiplied by the temperature, with values lying between (−1 and 32). This means that their corresponding term reaches the same range as the other terms of the equation.
with
and
Looking at the extracted Equation (
9), it appears that the parameter
ratio is missing. This indicates that the ratio has no significant effect on the general AER. This is consistent with the findings of [
12]. This means that the convection type added to the study does not significantly interact with the
ratio to influence the general AER. Further comparison with [
12] results means we can observe that here, too, there is notable impact from the wind speed, the wind direction, opening configuration and the interaction effects between them. This is also consistent with earlier experimental and numerical studies, which particularly highlight the importance of opening height and wind incident angle. For example, ref. [
25] found, in wind tunnel experiments, a reduction in the air exchange rate of around 70% when closing half of the sidewall openings. Wind tunnel experiments reported by De Paepe et al. [
26,
27] indicated that the AER via the outlet opening was reduced by up to 85% when the upper 12% of the sidewall opening was closed, and it was reduced by about 40% when moving from 0
to 90
wind incident angle. Numerical studies by [
10] indicated a reduction in the AER of around 20% when closing the lower third, and by around 35% when closing the upper third of the sidewalls. [
28] even reported a reduction in the air exchange by almost 90% when only bottom part of the sidewalls was open, and a reduction of 60% when moving from a 0
to 90
wind incident angle, based on numerical studies. However, all of those studies considered only isothermal conditions and focused on forced convection regimes. The present work, as a novelty, investigated the role of temperature and showed that it has also a non-negligible impact.
A closer look at the coefficients lets us notice the following:
multiplied with produced high equation values. This means that the velocity magnitude has a strong impact on the increase in the AER.
is negative, showing that with increasing temperature the AER tends to decrease. This effect is slightly dampened for the reduced opening configurations ().
Another interesting observation is that the reduction in the opening decreases the AER.
and
are both negative and have the same magnitude as the intercept. This goes together with the observation of [
10]. In his paper, he investigated the influence of several opening configurations on the flow pattern and the airflow rate of an NVDB for high-velocity magnitude at the 0
airflow direction. He also observed that the AER of the curtain Open Down is slightly less than the Open Up configuration. It is also consistent with the wind tunnel experiments of [
26], who also reported a significant reduction in the AER when closing parts of the sidewall opening, where a lower air exchange was observed in the open-down case than in the open-up case. The same can be noted in our study when looking at how
for the forced convection group (high velocity), which means that the higher the velocity, the more the AER is reduced for the Open Down curtain compared to the Open Up curtain. On the other hand, as
>
, for lower velocities (in this setting around 1 m/s), the air exchange in the open-up and open-down cases will be nearly equal, but still considerably reduced compared with the fully open case. The latter fits well with the observations in the wind tunnel experiments of [
25].
Changing in the incoming air inlet angle also reduced the AER (
is negative). This effect is even more accentuated for 90
inlet angle, since
is around three times
. This is also consistent with previous reports on the effect of wind incident angle [
27,
28]. However, when the opening is reduced, we observed an attenuation by the positivity of
. The attenuation is not complete, since
.
Additionally, the RSE of the groups natural-mixed and forced is smaller than the RSE for the “all” group (original data group without clustering) with a corresponding AER range between 18 h and 198 h. While all are very high (extremely close to the upper limit 1), the natural-mixed group has the smallest one, suggesting that the equation fits the data relatively well.
To minimize the risk of over-fitting, we further tested the robustness of Equation (
9) by randomly taking 95% of the available data and fitting the corresponding coefficients. For example, if we call one test test
, we find the equation:
with
and
We tested how well the equation with those coefficients can predict the remaining 5% of the data by looking at the
and RSE. This operation is repeated
n times (in our case
n = 20). We consequently obtained a series of
and
(with
). To provide an idea of how much the fitted model depends on the selection of training data, we chose to plot a distribution for each coefficient as a boxplot in
Figure 8 for the group “all”. The reference for each coefficient boxplot was the coefficients from the 100% data (the coefficients of
Table 4, first horizontal line, group “all”). Looking at
Figure 8, we observed that the individual coefficients diverge very little from their reference values. The mean
over the 20 tests was equal to 0.984 (0.985 and 0.983) for the 95% training data and 0.975 (between 0.911 and 0.993) for the 5% test data.
3.2.2. For the 10 Boxes Subdivision
Finding a good fitting model for all three convection groups was not possible. Here, the clustering played an importing role. As it was first formed by looking separately at each group, better results were obtained. Using the BIC as the model selection criterion, we found the following equation to best fit the data of the forced convection group in terms of
and RSE (for an AER range theoretically between 121 h
and 1033 h
):
with
Equation (
11) should be seen as a prediction of the AER of one individual box
i (with
, see
Section 2.4 for the subdivision on the barn into ten equal volumes, called boxes). Each box i possesses its own values for the coefficients
and
. There are ten equations with the same independent variables (in red and green) but with location-specific coefficients
and
. There are some similarities when comparing Equation (
11) to Equation (
9). Five of the nine terms are identical (the ones in green), demonstrating that there is a relationship between the general AER and the local AERs. However, these are not completely the same, since the variable
appears twice. This leads to the conclusion that, for local AERs,
cannot be ignored, thus validating our choice to use this parameter in the analysis even if previous studies concluded that it was irrelevant.
It is worth noting that the average of the
over the ten boxes is equal to 0.93 (between 0.86 and 0.98), which is lower than the 0.98 for the Equation (
9) on the general AER, but still translates to a good fit with the data. All
and RSE values, as well as the coefficients
and
for the boxes are summarized in
Figure 9. The chosen color code is dark blue—white—dark red to easily distinguish the variation between the boxes. When, for a single coefficient, all box-values are positive, the minimum value is associated with the white color and the maximum value with dark-blue. For the opposite case, when all box-values are negative, the maximum value is associated with white and the minimum with the dark-red. Finally, for the case when the box-values are both positive and negative, dark-blue was assigned to the maximum, white to zero and dark-red to the minimum. The following points summarize the conversations on the influence of each parameter on the individual boxes:
As in the general AER, the velocity magnitude (
at the beginning of the second row) has a strong effect on the increase in the
. However, for the inlet angle
is 45
(
), the effect is attenuated, especially at the upper half of the barn (boxes 5–10). For
= 90
(
) the AER
are decreased even more, but this time for the boxes of the rear half of the barn (boxes 2, 4, 6, 8, 10). We noted that the pattern of
is approximately complementary to
and also
to
: such a contrast slightly dampens the effect of
. This is consistent with contemporary knowledge of the indoor air flow pattern of naturally ventilated cattle buildings. For example, [
29] concluded, from on-farm measurements in the AOZ with 25
and 70
incident wind angle, that the speed and direction of the incident wind significantly influence the air velocity in the AOZ.
Here, the temperature () decreases the (except at boxes 1 and 3), but the effect is more pronounced in the rear and upper half of the barn (boxes 2, 4, 6, 7, 8, 9, 10). This effect is the same for the reduced openings; see the last column, and .
The similarity with the general AER continues. The coefficients for the curtain levels Open Down and Open Up (
,
, second row) are both negative, meaning that the reduced opening size reduces the local AER (OU more than OD, with stronger effects at the rear half of the barn). This goes together with the indoor air flow pattern observed by [
17] wind tunnel experiments with different opening configurations. The study reported a slightly lower wind speed in the AOZ in the Open Down case compared to the open case and a considerably lower wind speed in the AOZ in the Open Up case compared to the Open case. While, in our study, the reduction effect on the local AER was generally stronger in the rear half of the barn, the changes in the air flow pattern reported by [
17] were more pronounced in the front part of the barn. However, this difference might be explained by the fact that [
17] considered only one cross-section through the building and a model with a L/W ratio of nearly 1:1.
, , and (third and fourth columns from the left) allowed us to analyze the influence of inlet angle on the reduced opening configurations. We note that with increasing inlet angle, the coefficients also increase. For 45, the boxes 2, 4, 6 of the rear half and 7, 9 of front half are still negative, while their counterparts on the other side are positive. For 90, they all turn positive, with high values complementary to the original coefficient and . (Where there are high values of , we have low values of . The same applies to and .)
Patterns for and can hardly be recognized. However, we noticed that the values switch from positive to negative (or high to low) from a box of one half-barn side (front or rear) to the other half-barn side, except for box 1 and 2, which have almost same values. An average over all the boxes gave a value near to zero. This might explain why the parameter has no significant impact on the general AER, since the local AERs compensate for each other.
When interacts with the inlet angle (second and third columns from the right), we note that the patterns of are complementary to , and the ones of to . However, the magnitude of the is higher than the magnitude of the , especially for . This means that for a higher , increasing the inlet angle tends to create an opposite effect to that for smaller .
Equation (
12) represents the fitted model for
of the mixed convection group. Again, the green-colored terms are the ones that can be found in Equation (
9) of the general AER. The underlined terms correspond to the terms matching with Equation (
11). Here, we note that both equations are more similar to
than
.
with
Analyzing the coefficients of the Equation (
12) summarized in
Figure 10, we can extract the following information:
Again, as for the general AER, the velocity magnitude ( at the beginning of the second row) has a significant effect on the increase in the . However, for the inlet angle 45 () the effect is attenuate. For = 90 (), are decreased, especially for the boxes of the rear half of the barn (boxes 2, 4, 6, 8, 10).
Here, too, the temperature decreases the but the effect is more pronounced in the front half of the barn (boxes 1, 3, 5, 7, 9).
The coefficients for curtain levels Open Down and Open Up (, , second row) are both negative, meaning that the reducing opening size reduces local AER (OD reduces more in the front half of the barn, except for box 1, with a pattern that is complementary to alpha).
For , , and (second and third columns from the left), we note that with increasing inlet angle, the coefficients also increase. For 45, the boxes at the rear half of the barn (and for box 1), the coefficients are still negative, but for 90, they all turn positive. However, the lowest values are retained for the boxes in the rear half of the boxes and box 1.
As in the forced convection case, the values of and vary between high and low from one half of the barn to the other. When the openings are reduced (), there are small changes in most of the boxes (light colors which means close to zero). The three boxes with high values (5, 7, 9) are complementary to their respective original .
Here, the mean over the 10 boxes is about 0.875 (between 0.81 and 0.92), which is lower than that for the forced convection group but still acceptable. The mean RSE is about 26 (minimum of 17 and maximum of 34), while the range is between 151 h and 530 h.
Unfortunately, we could not find a fitting equation for the natural convection with an acceptable and RSE. We tried a logarithmic transformation, but it was still not satisfactory. This means that the local AER prediction for natural convection does not follow a linear “rule” with respect to the chosen independent parameter. Further investigations are needed to explore other possibilities.