# A Parametric Model for Local Air Exchange Rate of Naturally Ventilated Barns

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## Abstract

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## 1. Introduction

- Under the same boundary conditions, local AERs can show considerably different behavior, first, depending on the position inside the barn and second, in comparison with the global AER.
- The interaction between the temperature and velocity (which is directly related to the flow convection type) has a strong influence on the global and local AERs, as well as the interactions between other pre-cited parameters confirmed in previous studies.

## 2. Methodology and Materials

#### 2.1. CFD Model

#### 2.1.1. Domain and Boundary Condition

#### 2.1.2. Numerical Solver and Mesh

#### 2.2. Parameters

#### 2.3. Simulation Scheme and Automatizing Details (with Ansys Fluent and Python)

#### 2.4. AER Evaluation

#### 2.5. Tools and Methods for Statistical Analysis (with R)

#### 2.5.1. Principal Component Analysis

#### 2.5.2. General Linear Model (GLM)

## 3. Results and Discussions

#### 3.1. Clustering with PCA

#### 3.2. AER Formulas

#### 3.2.1. For the Whole Barn

- ${\beta}_{Vel}$ multiplied with $Vel$ produced high equation values. This means that the velocity magnitude has a strong impact on the increase in the AER.
- ${\beta}_{Temp}$ is negative, showing that with increasing temperature the AER tends to decrease. This effect is slightly dampened for the reduced opening configurations (${\beta}_{Curt.Temp}$).
- Another interesting observation is that the reduction in the opening decreases the AER. ${\beta}_{Curt}$ and ${\beta}_{Curt.Vel}$ 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$\xb0$ 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 ${\beta}_{OD.Vel}<{\beta}_{OU.Vel}$ 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 ${\beta}_{OD}$ > ${\beta}_{OU}$, 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 (${\beta}_{Vel.\theta}$ is negative). This effect is even more accentuated for 90$\xb0$ inlet angle, since ${\beta}_{Vel.\theta 90\xb0}$ is around three times ${\beta}_{Vel.\theta 45\xb0}$. 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 ${\beta}_{Curt.Vel.\theta}$. The attenuation is not complete, since $\mid {\beta}_{Vel.\theta}\mid >\mid {\beta}_{Curt.Vel.\theta}\mid $.

#### 3.2.2. For the 10 Boxes Subdivision

- As in the general AER, the velocity magnitude (${\beta}_{Vel}$ at the beginning of the second row) has a strong effect on the increase in the $AE{R}_{box.i}$. However, for the inlet angle $\theta $ is 45$\xb0$(${\beta}_{Vel.\theta 45}$), the effect is attenuated, especially at the upper half of the barn (boxes 5–10). For $\theta $ = 90$\xb0$ (${\beta}_{Vel.\theta 90}$) the AER${}_{box.i}$ 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 ${\beta}_{\theta 45}$ is approximately complementary to ${\beta}_{Vel.\theta 45}$ and also ${\beta}_{\theta 90}$ to ${\beta}_{Vel.\theta 90}$: such a contrast slightly dampens the effect of ${\beta}_{Vel.\theta -}$. 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$\xb0$ and 70$\xb0$ incident wind angle, that the speed and direction of the incident wind significantly influence the air velocity in the AOZ.
- Here, the temperature (${\beta}_{Temp}$) decreases the $AE{R}_{box.i}$ (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, ${\beta}_{Temp.OD}$ and ${\beta}_{Temp.OU}$.
- The similarity with the general AER continues. The coefficients for the curtain levels Open Down and Open Up (${\beta}_{OD}$, ${\beta}_{OU}$, 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.
- ${\beta}_{OD.\theta 45}$, ${\beta}_{OU.\theta 45}$, ${\beta}_{OD.\theta 90}$ and ${\beta}_{OU.\theta 90}$ (third and fourth columns from the left) allowed us to analyze the influence of inlet angle $\theta $ on the reduced opening configurations. We note that with increasing inlet angle, the coefficients also increase. For 45$\xb0$, 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$\xb0$, they all turn positive, with high values complementary to the original coefficient ${\beta}_{OD}$ and ${\beta}_{OU}$. (Where there are high values of ${\beta}_{OD.\theta 90}$, we have low values of ${\beta}_{OD}$. The same applies to ${\beta}_{OU.\theta 90}$ and ${\beta}_{OU}$.)
- Patterns for ${\beta}_{LW3}$ and ${\beta}_{LW4}$ 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 $L/W$ parameter has no significant impact on the general AER, since the local AERs compensate for each other.
- When $L/W$ interacts with the inlet angle $\theta $ (second and third columns from the right), we note that the patterns of ${\beta}_{\theta -.LW3}$ are complementary to ${\beta}_{LW3}$, and the ones of ${\beta}_{\theta -.LW4}$ to ${\beta}_{LW4}$. However, the magnitude of the ${\beta}_{\theta -.LW-}$ is higher than the magnitude of the ${\beta}_{LW-}$, especially for ${\beta}_{\theta 90.LW-}$. This means that for a higher $L/W$, increasing the inlet angle tends to create an opposite effect to that for smaller $L/W$.

- Again, as for the general AER, the velocity magnitude (${\beta}_{Vel}$ at the beginning of the second row) has a significant effect on the increase in the $AE{R}_{box.i}$. However, for the inlet angle $\theta $ 45$\xb0$ (${\beta}_{Vel.\theta 45}$) the effect is attenuate. For $\theta $ = 90$\xb0$ (${\beta}_{Vel.\theta 90}$), $AE{R}_{box.i}$ 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 $AE{R}_{box.i}$ 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 (${\beta}_{OD}$, ${\beta}_{OU}$, 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 ${\beta}_{OD.\theta 45}$, ${\beta}_{OU.\theta 45}$, ${\beta}_{OD.\theta 90}$ and ${\beta}_{OU.\theta 90}$ (second and third columns from the left), we note that with increasing inlet angle, the coefficients also increase. For 45$\xb0$, the boxes at the rear half of the barn (and for box 1), the coefficients are still negative, but for 90$\xb0$, 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 ${\beta}_{LW3}$ and ${\beta}_{LW4}$ vary between high and low from one half of the barn to the other. When the openings are reduced (${\beta}_{Curt.LW-}$), 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 ${\beta}_{LW-}$.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NVDB | Naturally Ventilated Dairy Barn |

AER | Air Exchange Rate |

CFD | Computational Fluid Dynamics |

RANS | Reynolds Averaged Navier-Stokes |

CPU | Control Processing Unit |

ER | Emission Rate |

ABL | Atmospheric Boundary Layer |

AOZ | Animal Occupied Zone |

TUI | Text User Interface |

PCA | Principal Component Analysis |

RSE | Residual Standard Error |

AIC | Akaike information criterion |

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**Figure 1.**Numerical domain dimensions and boundary conditions, example for the barn $L/W=4$, total open (see Section 2.2 for the chosen parameters).

**Figure 2.**3 of the 9 possible barn configurations for the study, the AOZs in green and the curtain in red. (

**a**) barn $L/W$ = 2 without curtain, (

**b**) barn $L/W$ = 3 with curtain “open up”, (

**c**) barn $L/W$ = 4 with curtain “open down”.

**Figure 5.**Local AER volume subdivisions for the different $L/W$ ratios, 10 boxes subdivision in red, with their corresponding number in violet.

**Figure 6.**PCA and clustering for the AER of the barn, (

**a**) is the percentage of variance explained by the PCs, (

**b**) is the datapoints displayed in the first two PCs’ orthogonal system and colored according to the three levels of the inlet angle, (

**c**), similar to (

**b**), but this time the datapoints are colored with respect to the three different convection types: forced (For), mixed (Mix) and natural (Nat).

**Figure 7.**PC for the 10 boxes’ subdivision. (

**a**) is the percentage of variance explained by the PCs, (

**b**) is the datapoints displayed in the first two PCs’ orthogonal system and colored according to the three levels of the inlet angle, (

**c**), similar to (

**b**), but this time the datapoints are colored with respect to the three different convection types: forced (For), mixed (Mix) and natural (Nat).

**Figure 9.**RSE, ${R}^{2}$ and coefficients of the AER${}_{Box.i}$ of Equation (11) corresponding to the forced convection group.

**Figure 10.**RSE, ${R}^{2}$ and coefficients of the AER${}_{Box.i}$ of Equation (12) corresponding to the forced convection group.

Inflow Direction | North Side | South Side | West Side | East Side |
---|---|---|---|---|

0$\xb0$ | velocity inlet | pressure outlet | Wall | Wall |

45$\xb0$ | velocity inlet | pressure outlet | velocity inlet | pressure outlet |

90$\xb0$ | Wall | wall | velocity inlet | pressure outlet |

Ri Cases | |||||||||
---|---|---|---|---|---|---|---|---|---|

Type of Convection | Forced | Mixed | Natural | ||||||

Ri domain | Ri < 0.2 | 0.2 < Ri < 5 | Ri > 5 | ||||||

Ri values | Ri = 0.048 | Ri = 0.078 | Ri = 0.198 | Ri = 0.8 | Ri = 1.9 | Ri = 3 | Ri = 12 | Ri = 28 | Ri = 51.75 |

T${}_{air}$ ($\xb0$C) | 32 | 30 | 22 | 18 | 15 | 10 | 7 | 0 | −2 |

U${}_{ref}$ (m s${}^{-1}$) | 3.2 | 2.8 | 2.5 | 1.4 | 0.97 | 0.85 | 0.45 | 0.33 | 0.25 |

Season | Summer | Fall-Spring | Winter |

Parameter Description | Velocity Magnitude | Ambient Temperature | Inlet Angle | Width/Length Barn Ratio | Curtain Position |
---|---|---|---|---|---|

Variable Type | Continuous | Continuous | Factor | Factor | Factor |

Standard | - | - | 0$\xb0$ | LW2 | Total open |

**Table 4.**Residual standard error (RSE) and ${R}^{2}$ for all cluster groups and corresponding alphas and first part of the betas.

Cluster Group | Coefficients | RSE | ${\mathit{R}}^{2}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | ${\mathbf{\beta}}_{\mathit{T}\mathit{e}\mathit{m}\mathit{p}}$ | ${\mathbf{\beta}}_{\mathit{V}\mathit{e}\mathit{l}}$ | ${\mathbf{\beta}}_{\mathit{C}\mathit{u}\mathit{r}\mathit{t}}$ | ${\mathbf{\beta}}_{\mathit{C}\mathit{u}\mathit{r}\mathit{t}.\mathit{V}\mathit{e}\mathit{l}}$ | ${\mathbf{\beta}}_{\mathit{C}\mathit{u}\mathit{r}\mathit{t}.\mathit{T}\mathit{e}\mathit{m}\mathit{p}}$ | ${\mathbf{\beta}}_{\mathit{V}\mathit{e}\mathit{l}.\mathit{U}}$ | |||||

All | 40.25 | −1.42 | 61.98 | OU | −19.06 | −32.81 | 0.54 | 45$\xb0$ | −12.32 | 4.58 | 0.98 |

OD | −12.85 | −39.09 | 0.84 | 90$\xb0$ | −44.94 | ||||||

Nat-Mix | 43.83 | −1.01 | 50.87 | OU | −20.25 | −29.07 | 0.33 | 45$\xb0$ | −17.17 | 3.31 | 0.96 |

OD | −12.85 | −28.74 | 0.58 | 90$\xb0$ | −36.47 | ||||||

Forced | 36.99 | −0.76 | 56.92 | OD | −25.86 | −30.00 | 0.50 | 45$\xb0$ | −11.53 | 4.04 | 0.99 |

OU | −5.06 | −34.96 | 0.09 | 90$\xb0$ | −46.33 |

**Table 5.**Continuation of Table 4 listing the remaining betas.

Cluster Group | Coefficients | |||
---|---|---|---|---|

${\mathbf{\beta}}_{\mathit{C}\mathit{u}\mathit{r}\mathit{t}.\mathit{V}\mathit{e}\mathit{l}.\mathit{U}}$ | ||||

${\mathbf{\beta}}_{\mathit{O}\mathit{U}.\mathit{U}\mathbf{45}\xb0}$ | ${\mathbf{\beta}}_{\mathit{O}\mathit{U}.\mathit{U}\mathbf{90}\xb0}$ | ${\mathbf{\beta}}_{\mathit{O}\mathit{D}.\mathit{U}\mathbf{45}\xb0}$ | ${\mathbf{\beta}}_{\mathit{O}\mathit{D}.\mathit{U}\mathbf{90}\xb0}$ | |

All | 5.10 | 26.65 | 10.16 | 30.50 |

Nat-Mix | 10.35 | 22.85 | 11.00 | 20.22 |

Forced | 4.23 | 27.27 | 10.02 | 32.19 |

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**MDPI and ACS Style**

Doumbia, E.M.; Janke, D.; Yi, Q.; Prinz, A.; Amon, T.; Kriegel, M.; Hempel, S. A Parametric Model for Local Air Exchange Rate of Naturally Ventilated Barns. *Agronomy* **2021**, *11*, 1585.
https://doi.org/10.3390/agronomy11081585

**AMA Style**

Doumbia EM, Janke D, Yi Q, Prinz A, Amon T, Kriegel M, Hempel S. A Parametric Model for Local Air Exchange Rate of Naturally Ventilated Barns. *Agronomy*. 2021; 11(8):1585.
https://doi.org/10.3390/agronomy11081585

**Chicago/Turabian Style**

Doumbia, E. Moustapha, David Janke, Qianying Yi, Alexander Prinz, Thomas Amon, Martin Kriegel, and Sabrina Hempel. 2021. "A Parametric Model for Local Air Exchange Rate of Naturally Ventilated Barns" *Agronomy* 11, no. 8: 1585.
https://doi.org/10.3390/agronomy11081585