agronomy
Article
A Parametric Model for Local Air Exchange Rate of Naturally
Ventilated Barns
E. Moustapha Doumbia 1,* , David Janke 1, Qianying Yi 1, Alexander Prinz 1, Thomas Amon 1,2 ,
Martin Kriegel 3and Sabrina Hempel 1
Citation: 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
Academic Editor: Luis
Hernández-Callejo
Received: 17 June 2021
Accepted: 3 August 2021
Published: 10 August 2021
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Attribution (CC BY) license (https://
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4.0/).
1Department of Engineering for Livestock Management, Leibniz Institute for Agricultural Engineering
2Department of Veterinary Medicine, Institute of Animal Hygiene and Environmental Health,
Freie Universität, 14163 Berlin, Germany
3Hermann-Rietschel-Institute, Technische Universität, 10587 Berlin, Germany; [email protected]
*Correspondence: [email protected]
Abstract:
With an increasing number of naturally ventilated dairy barns (NVDBs), the emission
of ammonia and greenhouse gases into the surrounding environment is expected to increase as
well. It is very challenging to accurately determine the amount of gases released from a NVDB
on-farm. Moreover, control options for the micro-climate to increase animal welfare are limited in an
NVDB at present. Both issues are due to the complexity of the NVDB micro-environment, which
is subject to temporal (such as wind direction and temperature) and spatial (such as openings and
animals acting as airflow obstacles) fluctuations. The air exchange rate (AER) is one of the most
valuable evaluation entities, since it is directly related to the gas emission rate and animal welfare.
In this context, our study determined the general and local AERs of NVDBs of different shapes
under diverse airflow conditions. Previous works identified main influencing parameters for the
general AER and mathematically linked them together to predict the AER of the barn as a whole. The
present research study is a continuation and extension of previous studies about the determination
of AER. It provides new insights into the influence of convection flow regimes. In addition, it goes
further in precision by determining the local AERs, depending on the position of the considered
volume inside the barn. After running several computational fluid dynamics (CFD) simulations,
we used the statistical tool of general linear modeling in order to identify quantitative relationships
between the AER and the following five influencing parameters, the length/width ratio of the barn,
the side opening configuration, the airflow temperature, magnitude and incoming direction. The
work succeeded in taking the temperature into account as a further influencing parameter in the
model and, thus, for the first time, in analysing the effect of the different types of flow convection in
this context. The resulting equations predict the barn AER with an
R2
equals 0.98 and the local AER
with a mean
R2
equals around 0.87. The results go a step further in the precise determination of the
AER of NVDB and, therefore, are of fundamental importance for a better and deeper understanding
of the interaction between the driving forces of AER in NVDB.
Keywords:
naturally ventilated dairy barns; air exchange rate; computational fluid dynamics; airflow
convection; general linear model
1. Introduction
Animal husbandry must be animal- and environmentally friendly to be socially
accepted and sustainable [
1
]. The air exchange rate (AER) and the direction of air movement
are key parameters in this context to evaluate animal housing with regard to animal
welfare [
2
] and environmental compatibility [
3
]. The AER is defined as the quotient of
the total volume airflow flow entering a room and the volume of this room. It quantifies
the theoretical air replacement per time unit. Livestock producers have two options
Agronomy 2021,11, 1585. https://doi.org/10.3390/agronomy11081585 https://www.mdpi.com/journal/agronomy
Agronomy 2021,11, 1585 2 of 19
for removing polluting gases, moisture and heat from animal houses—mechanical and
natural ventilation. The economically highly relevant dairy cows sector typically uses
naturally ventilated barns (NVB) across Europe—a housing system which is partly also
used for poultry or pigs [
4
]. The main advantage of these buildings is their energy-saving
property since, in general, natural ventilation does not require electrical energy to operate
fans or a furnace. However, in many cases, fans support natural ventilation under hot
climate conditions. The desire for maximal animal welfare and minimal environmental
impact results in a conflict of goals, since the emission rate (ER) is coupled with the
AER
:ER =C0AER Vt
, with
C0
(in m g m
−3
) as the concentration of a pollutant gas
inside the barn and
Vt
(in m
3
) as the volume of the space. Thus, to only increase the
global AER of a barn to improve the animal welfare results in an increase in the balance-
dependent emissions of an NVB. In contrast, local AER in animal-occupied spaces must be
selectively modified.
Computational fluid dynamics (CFD) methods are applied with increasing frequency
in agricultural research to model the interior air flow in barns and greenhouses [
5
,
6
]. Due
to the theoretically unlimited spatial and temporal resolution and the defined boundary
conditions, they are an optimal complement to measurements and permit deep insights
into the development of airflow patterns [
7
]. The effect of individual stimuli (e.g., building
design or thermal input) on the airflow patterns can be investigated in detail in order
to develop strategies for the precise control of local AER. The numerical models rely on
discretization schemes and how well the geometric models are divided into tiny parts
called cells (meshing). Model depth and accuracy depends on the research question,
the computational power and the chosen parametrizations of unresolved processes [5,8].
The description of turbulence is of particular relevance. In agriculture, numerical flow
simulations typically rely on the Reynolds Averaged Navier–Stokes (RANS) approach [
9
]. It
provides a good cost–benefit ratio and has been used in scientific and industrial applications
for more than 40 years. State-of-the-art CFD models of air flow through livestock houses
include inflow modeling and the estimation of AER [
10
–
12
]. In a pre-study published
in January 2021 [
13
], we showed that it is possible to model animals as a porous media
that reproduce the pressure drop and heat transfer relatively well (less than 6% error).
The advantage of using porous media is the gained computing time, which is at least 70%
compared to a numerical simulation with 3D modeled animals. The combination of the
RANS with the porous media made the present parametric study timely realizable, in such a
way that 243 simulations have been realized in only three months on 30 CPUs. The number
of simulations comes from the fact that we considered three “values” for three of the
parameter chosen for the study and nine “values” for the other two (coupled) parameters
(3
×
3
×
9 = 243). One of the latest studies on this topic, conducted by Yi et al. [
12
], explored
the influences and possible interactions between the opening ratio, the building length-to-
width ratio, the wind speed and the wind direction. They found that the influence of the
length-to-width ratio was negligible, while the AER was significantly affected by the wind
speed, the wind direction and interaction effects between each two of the three factors.
However, the study was limited to the global AER and the influence of convective flow
regimes was not investigated.
For this reason, the present manuscript, which follows the same direction, wants to
enlarge the scope of the AER prediction by investigating the following hypothesis:
1.
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.
2.
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.
The confirmation of those hypothesizes represents the novelty of the present work.
The present work is subdivided as follow. First, the CFD numerical model will be presented
in detail. Then, the parameters and the motivations behind their choices are explained.
Agronomy 2021,11, 1585 3 of 19
Afterwards, the corresponding scheme and the method used to run and automatize the
simulations are exposed. The next part deals with the evaluation method for global
and local AERs. The last part of the methodology and materials segment introduces the
general linear model as the chosen statistical method for the interpretation of the results.
In the result section, the equations with the highest
R2
describing the AERs are proposed,
following a comparison with the previous findings. Finally, we present conclusion of the
present work and we offer ways to possibly expand on the topic in future.
2. Methodology and Materials
2.1. CFD Model
2.1.1. Domain and Boundary Condition
The present study is based on the same numerical model used in [
13
]. The dimensions
of the numerical domain are summarized in Figure 1. The characteristic lengths, such the
width W and the height H, are based on the ATB barn sizes in Northern Germany [
14
],
W = 34.2 m, H = 11.5 m.
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).
The inlet velocity describes an atmospheric boundary layer (ABL) profile with the
equation [15]:
U(y) = Uref y
yre f !α
(1)
Here,
U(y)
is the inlet velocity profile (in m s
−1
),
Uref
is the chosen velocity (in m s
−1
)
at the height
yre f
(in m) and
α
is a power coefficient (dimensionless). The value of
α= 0.16
was extracted from wind tunnel measurements. The value chosen for the ground roughness
was 0.068 m, which corresponded to a moderately rough terrain and helped to conserve
the form of the ABL profile [
11
]. The distances 4 W for the inlet and 6 W for the outlet
were chosen so that the boundary conditions are far away enough to not influence the flow
inside the barn. The domain boundary conditions were set in relation to the inflow air
angle. Table 1shows the details.
The AOZ of cows, which depict flow obstacles and heat sources, is modelled as a
porous medium. The dimensions of the each AOZ are L
AOZ
= W/2, W
AOZ
= W/4 and
Agronomy 2021,11, 1585 4 of 19
disposed as shown in green in Figure 1. According to the density 2.4 m
2
cow
−1
(taken
from the aforementioned barn in Northern Germany; for a detailed barn description,
see, e.g., Janke et al. [
14
]); the number of cows for the AOZ dimensions is 33. Using the
method descripted in Doumbia et al. [
13
], the AOZ was replaced by a porous medium
with the corresponding pressure drop (in the three main directions) and heat transfer
functions extracted from the 33 cows 3D geometry (randomly distributed with a ratio of
60% lying and 40% standing) numerical simulation. Based on this method, the porous
medium reproduced a pressure drop and heat transfer with fewer than 6% errors, but
can significantly reduce the computation time by up to 70% in comparison with its
corresponding 3D cows model.
Table 1. Boundary conditions corresponding to airflow directions.
Inflow Direction North Side South Side West Side East Side
0◦velocity inlet pressure outlet Wall Wall
45◦velocity inlet pressure outlet velocity inlet pressure outlet
90◦Wall wall velocity inlet pressure outlet
2.1.2. Numerical Solver and Mesh
The numerical settings (solvers, turbulence model, meshing) are same as in [
13
].
While the most important information will be summarized here, we suggest that the
reader looks into the pre-cited paper for more details. The simulations were run in steady
state (time independent). The pressure velocity-coupling scheme was set to coupled to
improve convergence. The spatial discretization schemes were kept standard (ANSYS,
2019), i.e., second-order upwind for the momentum and the energy, first-order upwind for
the turbulent kinetic energy and the specific dissipation rate, second-order for the pressure.
The turbulence model chosen for the study is the k-omega model. The mesh is composed
of unstructured (around the barn) and structured (for the domain) (see [
11
]). Based on
the mesh convergence performed in Doumbia et al. [
13
], the chosen cell sizes were 0.4 m
for the refinement body and 0.8 m for the barn, with a growth rate of 1.2. The total cell
number varied from 12 (for the barn
L/W=
2) to 19 million (for the barn
L/W=
4).
The corresponding computing time on 30 CPUs varied from 2 to 4 h for one simulation,
depending on the cell number.
2.2. Parameters
In this section, the parameters chosen for the study, as well as the reasons for their
choice, will be presented in detail. After an extensive look at the literature and based on
the ATB of previous studies [
11
,
12
], the following parameters were retained as potentially
having the most influence on the AER
:
air velocity magnitude (
Vel
), wind direction (
θ
),
temperature gradient between the cows and the ambient air (
Temp
), length-to-width ratio
of the barn (
L/W
) and barn curtain position (
Curt
). Wind obviously plays an important
role in the air exchange of a NVDB. In this context, [
11
] noticed that each incoming air
direction produces a particular pattern inside the barn, with a tendency towards higher
values at the windward side of the building. See Figure 1for the three chosen air incident
angles, 0
◦
, 45
◦
and 90
◦
. We chose to retain the
L/W
ratio for two main reasons, even if
one study, the conclusions of Yi et al. [
12
], suggested that it had negligible influence on the
AER. First, since there is a variety of NVDBs
L/W
ratios [
16
], we want to test if the same
conclusion applied to the local AER. Second, the influence of the interaction of L/Wwith
the temperature (a newly introduced parameter) is still unknown. For the study, the values
L/W= 2 (Figure 2a), 3 (Figure 2b) or 4 (Figure 2c) were retained.
The barn curtain position (or side opening ratio),
Curt
, has been already reported as
an important impact on the AER, Yi et al. [
17
] and Yi et al. [
12
], Gebremedhin et al. [
18
],
Saha et al. [
19
] stated that AER of some configurations can be up to three times the AER
of the standard configuration. The curtain positions retained here are totally open (no
Agronomy 2021,11, 1585 5 of 19
curtain, Figure 2a), open up (curtain from the floor until the middle of the opening height,
Figure 2b
) and open down (curtain from the roof until the middle of the opening height,
Figure 2c).
(a) (b)
(c)
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”.
The velocity magnitude and the temperature gradient, when combined, can create
different convection flow regimes. The Richardson number
Ri
, which is the ratio of the
buoyancy forces and the kinetic forces, was used to distinguish between the three types of
convection [20]
Ri =Gr
Re2(2)
Gr =gβL3
HG
(Ts−Tin)
ν2(3)
Re =u LHG
ν(4)
β=1
Ts+Tin
2
(5)
LHG =Wg +324.52
417.5 (6)
where
Gr
is the Grashof number and
Re
is the Reynolds number (both dimensionless);
LHG
(=2.4 m in our case) the characteristic length is the heart girth used by Wang et al. [
21
]
and based on the cow weight
Wg
(=675 Kg, as the mean cow weight from our ATB barn
in Dummersdorf [
22
]),
Ts
(=311.15 K) is the cow or solid temperature [
23
],
Tin
(in
K
) is the
inlet air temperature,
ν
(=1.652
×
10
−5
m
2
s
−1
) is the kinematic viscosity,
U
(in m s
−1
) is
the inlet velocity and
g
(= 9.81 m s
−2
) is the gravitational acceleration. The
Ri
number
is related to the convection type as the following. When
Ri
> > 1, the buoyancy forces
represented by the Grashof number (
Gr
) prevail over the kinetic forces represented by the
Reynolds number. When
Ri ≈
1, the buoyancy forces are on par with the kinetic forces.
In addition, for
Ri
< < 1, the kinetic forces are dominant. The range and limits 0.2–5 for the
mixed convection are extracted from [
20
] as practical values. The nine values of Richardson
number were considered to cover the three types of convection and the different seasons of
the year, recapitulated in Table 2.
Agronomy 2021,11, 1585 6 of 19
Table 2. Type of convective heat transfer and Ri values.
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
Tair (◦C) 32 30 22 18 15 10 7 0 −2
Ure f (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
2.3. Simulation Scheme and Automatizing Details (with Ansys Fluent and Python)
Figure 3shows the scheme of the parameters composition and their corresponding
values. Due to this composition and the resulting diverse cases, the study required an
enormous number of CFD simulations. The use of the RANS model with the porous model,
instead of the RANS with fully resolved cows, reduces the computational time enormously,
but the number of resulting cases is still very high. It was obvious that manually launching
each simulation run with manual parameter settings was not an option. For this reason,
we used the user interface (UI) provided by ANSYS and developed a fully automated
workflow based on a combination with Python (3.6.4) and BATCH scripts. We coupled a
Python homemade script to modify a so-called ANSYS journal file to provide the parameter
values for the simulation. The journal records ANSYS commands in a text user interface
(TUI) format and can be re-run to automatically execute the recorded commands.
Figure 3. Scheme summarizing all parameters and their “values”.
The ANSYS internal workflow is described in Figure 4. The main script first called
out ANSYS and implemented the mesh (already prepared by the user) in step 1. A loop
was executed for this mesh. In the loop, the journal file was modified according to the
boundary conditions (
Vel
,
θ
and
Temp
), the CFD simulation was executed in ANSYS Fluent
(setup and solution) and, as a second step, the post-processing (analysis of the simulation
results) was performed in CFD-Post. In step 3, the AER of the barn and the local AERs were
calculated internally in ANSYS and then written to a CSV file. In step 4, the journal was
Agronomy 2021,11, 1585 7 of 19
modified by implementing new boundary conditions and then the script returned to
step 2
.
The loop (steps 2-3-4) was run until the simulations for all boundary conditions
Vel
,
θ
and
Temp
were completed. Next, one of the nine cases (3
×L/W×
3
·
curtain positions = 9
possible geometry cases) were run in step 1, and so on. By fully automating the process,
a significant time-saving was achieved and the probability of human typing errors was
drastically reduced.
Figure 4. Description of automatizing process in four steps under Ansys coupled with python.
2.4. AER Evaluation
The AER was evaluated globally, for the barn as a whole, and locally, for each box
of the subdivision. To gain a deeper understanding of the local AER, we decided to
virtually divide the barn into 10 parts (or volumes) relative to the barn sizes (L/W = 2, 3
and 4). The dimensions of the boxes are as follows: L
Box,L/W=i
= L
L/W=i
/5,
WBox = W/2
,
HBox = 2 ×HAOZ = 3.2 m
. These virtual subdivisions can be observed in Figure 5. We
numbered the boxes to identify their position relative to the incoming airflow. The AER is
calculated through the equation:
AER =3600 ˙
V
VAER (7)
Here, AER is the air exchange rate of the volume
VAER
(in h
−1
),
˙
V
is the sum of volume
flow entering
VAER
(in m
3
s
−1
) and
VAER
the considered volume for AER evaluation
(in m3). VAER can be the volume of the whole barn or the volume of a single box.
2.5. Tools and Methods for Statistical Analysis (with R)
The software we employed in this study is the open-source software R for calculations
and statistical analysis. In the following paragraphs, the reasoning for the use of the
principal component analysis and the general linear model are presented, together with
the necessary vocabulary for the study. We would like the readers to keep in mind that
deepening this topic is out of the scope of the present work. We refer the readers to the
wide statistical analysis literature for further details.
2.5.1. Principal Component Analysis
The initial idea of principal component analysis (PCA) is dimensionality reduction
(i.e., explaining as much of the data variance as possible, with the lowest possible number
of variables by conducting an orthogonal projection of the data). At present, this statistical
procedure is often used for data clustering. In our case, it will help to observe if and how
the AERs form clouds. The objective is to extract valuable patterns by displaying the data
in the function of new unrelated parameters, the principal components (PCs). This can
Agronomy 2021,11, 1585 8 of 19
be done by transposing the data into a new orthogonal coordinate system in which the
first principal component is orthogonal to the second principal component, which itself
is orthogonal to the third principal component (if any) and so on (if any). The principal
components are combinations of the parameters/variables influencing the data.
Figure 5.
Local AER volume subdivisions for the different
L/W
ratios, 10 boxes subdivision in red, with their corresponding
number in violet.
2.5.2. General Linear Model (GLM)
The GLM is a regression method that determines only one variable (called the
dependent variable, which, here, is the AER) based on other variables (called independent
variables, which, for us, are the five parameters), which can be multiplied with each
other within the terms of a polynomial-type equation; see the equation below. For the
independent variables, it is important to distinguish between the continuous variables
and the factor variable. While it is easy to understand the meaning of the continuous
variables, the factor variable needs more explanation. Factor variables refer to different
categories and are used for qualitative description instead of a quantitative ranking. They
have different levels (for example, in our case, the levels of the curtain positions are open,
open up and open down) with one of them set as the standard; see Tabel 3.
Table 3. Boundary conditions corresponding to flow directions.
Parameter
Description
Velocity
Magnitude
Ambient
Temperature Inlet Angle Width/Length
Barn Ratio
Curtain
Position
Variable Type Continuous Continuous Factor Factor Factor
Standard - - 0◦LW2 Total open
Agronomy 2021,11, 1585 9 of 19
If we call Ythe dependent variable, the basic equation of the GLM follows a linear
model and looks like this:
Y=α+β1X1+β2X2+..... +βKXK(8)
α
, the intercept is the predicted value of the dependent variable when all the independent
variables are 0 (for the continues variables) and standard (for the factor variables).
Xi
can
be a single independent variable or a combination of those (for example, in our case,
Temp2
,
Temp ×Vel
or
L/W×Vel ×θ
).
βi
, the weights, are quantitative coefficients characterizing
the term of the equation associated with
Xi
. In our study, we are looking for the equations
AER
∼
f(
Vel
,
θ
,
Temp
,
L/W
,
Curt
) through the GLM method that will best describe the
simulation results. Regarding the criterion for the assessment of the accuracy of the model,
we present some that will help locate the best formulas.
The variance is an extremely important notion, as it plays a central role in modern
statistics. The variance of a variable, mathematically speaking, is the average of the squared
deviation of values of this variable from its mean.
The well-known R2, or coefficient of determination, is the proportion of the variance
of the dependent variable that is predictable from the independent variables. The
R2
is
between 0 and 1 and the higher the value, the better the model predicts the data.
R2
can be seen in terms of the percentage of the variance of the dependent variable that the
model predicts.
The residual standard error (RSE) explains how far away, on average, the residuals
are from 0. Residuals are the differences between the actual (original data) and calculated
(from the model) values of the dependent variable. In other words, the RSE can be seen as a
measure that shows the variation in the actual values around the regression line predicted
by the model [
24
]. Like any other “error”, the smaller the RSE, the better the model fits the
data, with zero being a perfect fit.
There is a function in R that selects the best model equation considering the Bayesian
information criterion (BIC). The BIC is another statistical tool that evaluates the quality of
each model, relative to each of the other models based on a likelihood function. The BIC
balances the goodness of fit and complexity of the model.
3. Results and Discussions
3.1. Clustering with PCA
Figure 6summarizes the principal component analysis for the general AER.
Figure 6a
indicates that two principal components (PC1 and PC2) can explain almost all of the
variance in the data; the third one (PC3) is insignificant. The line describes how much of
the data variance is covered by all the PCs. Here, PC1 and PC2 cover almost 100% of the
variance. Figure 6b,c show the projection of the data in the PCA orthogonal coordinate
system, with PC1 as the x-axis and PC2 as the y-axis. As pointed out in Figure 6c by the two
yellow ellipses, the points can be gathered into two main groups. However, the parameter(s)
characterizing those groups were unknown. We tested out all the available parameters.
In Figure 6b, the three colors symbolize the three levels of the parameter air inlet angle
(
θ
), arbitrarily chosen to gain further understanding. It can be observed that the three
levels are all mixed up in the point clouds, showing that the inlet angle is not a suitable
parameter for distinguishing the point clouds. Figure 6c shows that this is not the case for
the parameter Richardson number
Ri
. When considering this parameter, which delimits
the convection type, it can be seen that the points under the natural and mixed convections
(green and blue points) are two groups, forming one group, and the points corresponding
to the forced convection (red points) form another group.
We repeated the PCA for the AERs of the 10 boxes’ subdivision and the results in
Figure 7present the data for all the 10 boxes. The findings for the clustering of the general
AER obtained from PCA were also valid here. However, it is interesting to notice that,
when comparing Figure 6b with Figure 7b (or Figure 6c with Figure 7c), the distribution of
the variability between the two PCs is rather different for the general AER and the local
Agronomy 2021,11, 1585 10 of 19
AER (different slops). This allows us to envisage that the general AER and the local AER
will have different predictable regression models.
(a) (b) (c)
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).
These PCAs also confirm the second hypothesis, that flow convection type has a strong
influence on the general and local AERs.
Based on those observations, we decided to split the data into the three groups—(1)
natural convection (Nat), (2) mixed convection (Mix) and (3) forced convection (For)—for
the subsequent analysis, to derive accurate and representative AER formulas.
(a) (b) (c)
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).
3.2. AER Formulas
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. Tables 4and 5show the intercepts
α
and the coefficients
associated with the term of the sum (the weights
βi
) for each groups. The
β
for the
curtain nonstandard cases were noted as OU for open-up and OD for open-down curtain.
Agronomy 2021,11, 1585 11 of 19
As expected,
αall 6=αnat−mix 6=αforced
and
βi,all 6=βi,nat−mix 6=βi,forced
. The resulting RSE
and
R2
are, consequently, different, and are revealed in Table 4as well. The readers may
have noticed the small values of the coefficients
βTemp
and
βCurt.Temp
. 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.
AER =α+βCurt ×δ(Curt) + βTemp ×Temp +βVel ×Vel +βVelU ×Vel ×δ(θ)
+βCurt.Vel δ(Curt)×Vel +βCurt.Temp δ(Curt)×Temp
+βCurt.Vel.Uδ(Curt)×Vel ×δ(θ)
(9)
with δ(Curt) = (1, Curt ∈ {OU,OD}
0, Curt 6∈ {OU,OD}and δ(θ) = (1, θ∈ {45◦, 90◦}
0, θ6∈ {45◦, 90◦}
Table 4. Residual standard error (RSE) and R2for all cluster groups and corresponding alphas and first part of the betas.
Cluster Group Coefficients RSE R2
α βTemp βVel βCurt βCurt.Vel βCurt.Temp βVel.U
All 40.25 −
1.42
61.98 OU −19.06 −32.81 0.54 45◦−
12.32
4.58 0.98
OD −12.85 −39.09 0.84 90◦−
44.94
Nat-Mix 43.83 −
1.01
50.87 OU −20.25 −29.07 0.33 45◦−
17.17
3.31 0.96
OD −12.85 −28.74 0.58 90◦−
36.47
Forced 36.99 −
0.76
56.92 OD −25.86 −30.00 0.50 45◦−
11.53
4.04 0.99
OU −5.06 −34.96 0.09 90◦−
46.33
Table 5. Continuation of Table 4listing the remaining betas.
Cluster Group Coefficients
βCurt.Vel.U
βOU.U45◦βOU.U90◦βOD.U45◦βOD.U90◦
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
Looking at the extracted Equation (9), it appears that the parameter
L/W
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
L/W
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
Agronomy 2021,11, 1585 12 of 19
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:
•βVel
multiplied with
Vel
produced high equation values. This means that the velocity
magnitude has a strong impact on the increase in the AER.
•βTemp
is negative, showing that with increasing temperature the AER tends to decrease.
This effect is slightly dampened for the reduced opening configurations (βCurt.Temp).
•
Another interesting observation is that the reduction in the opening decreases the AER.
βCurt
and
βCurt.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
◦
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
βOD.Vel <βOU.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
βOD
>
βOU
, 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 (
βVel.θ
is negative).
This effect is even more accentuated for 90
◦
inlet angle, since
βVel.θ90◦
is around
three times
βVel.θ45◦
. 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
βCurt.Vel.θ
. The attenuation is not complete, since
|βVel.θ|>|βCurt.Vel.θ|.
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
−1
and 198 h
−1
. While all
R2
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 test1, we find the equation:
AER =αtest1+βCurt,test1×δ(Curt) + βTemp,test1×Temp +βVel,test1×Vel
+βVel.U,test1×Vel ×δ(θ) + βCurt.Vel,test1×δ(Curt)×Vel
+βCurt.Temp,test1×δ(Curt)×Temp +βCurt.Vel.U,test1×δ(Curt)×Vel ×δ(θ)
(10)
with δ(Curt) = (1, Curt ∈ {OU,OD}
0, Curt 6∈ {OU,OD}and δ(θ) = (1, θ∈ {45◦, 90◦}
0, θ6∈ {45◦, 90◦}
We tested how well the equation with those coefficients can predict the remaining
5% of the data by looking at the
R2
and RSE. This operation is repeated ntimes (in our
case n= 20). We consequently obtained a series of
αtest.i
and
β−,test.i
(with
i=
1
. . . n
). 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 8for 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
R2
Agronomy 2021,11, 1585 13 of 19
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.
Figure 8. Coefficients distribution based on 20 test for the “all” group.
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
R2and RSE (for an AER range theoretically between 121 h−1and 1033 h−1):
AERBox.i=αi+βCurt.i×δ(Curt) + βθ.i×δ(θ) + βL/W.i×δ(L/W) + βTemp.i×Temp
+βVel.i×Vel +βVel.θ.i×Vel ×δ(θ) + βCurt.Temp.i×δ(Curt)×Temp
+βCurt.θ.i×δ(Curt)×δ(θ) + βL/W.θ.i×δ(L/W)×δ(θ)
(11)
with
δ(Curt) = (1, Curt ∈ {OU,OD}
0, Curt 6∈ {OU,OD};
δ(θ) = (1, θ∈ {45◦, 90◦}
0, θ6∈ {45◦, 90◦};
Agronomy 2021,11, 1585 14 of 19
δ(L/W) = (1, L/W∈ {3, 4}
0, L/W6∈ {3, 4}
Equation (11) should be seen as a prediction of the AER of one individual box
i
(with
i=
1
. . .
10, 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
α
and
βk
. There are
ten equations with the same independent variables (in red and green) but with location-
specific coefficients
α
and
βk
. 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
L/W
appears twice. This leads to the
conclusion that, for local AERs,
L/W
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
R2
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
R2
and RSE values, as well as the
coefficients
αi
and
βki
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 (
βVel
at the beginning of the second
row) has a strong effect on the increase in the
AERbox.i
. However, for the inlet angle
θ
is 45
◦
(
βVel.θ45
), the effect is attenuated, especially at the upper half of the barn (boxes
5–10). For
θ
= 90
◦
(
βVel.θ90
) 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
βθ45
is approximately complementary to
βVel.θ45
and also
βθ90
to
βVel.θ90
: such a
contrast slightly dampens the effect of
βVel.θ−
. 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 (
βTemp
) decreases the
AERbox.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, βTemp.OD
and βTemp.OU.
•
The similarity with the general AER continues. The coefficients for the curtain levels
Open Down and Open Up (
βOD
,
βOU
, 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.
•βOD.θ45
,
βOU.θ45
,
βOD.θ90
and
βOU.θ90
(third and fourth columns from the left) allowed
us to analyze the influence of inlet angle
θ
on the reduced opening configurations. We
Agronomy 2021,11, 1585 15 of 19
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
βOD
and
βOU
. (Where there are high values
of βOD.θ90, we have low values of βOD. The same applies to βOU.θ90 and βOU.)
•
Patterns for
βLW3
and
βLW4
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
θ
(second and third columns from the
right), we note that the patterns of
βθ−.LW3
are complementary to
βLW3
, and the
ones of
βθ−.LW4
to
βLW4
. However, the magnitude of the
βθ−.LW−
is higher than the
magnitude of the
βLW−
, especially for
βθ90.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.
Figure 9. RSE, R2and coefficients of the AERBox.iof Equation (11) corresponding to the forced convection group.
Equation (12) represents the fitted model for
AERBox.i
of the mixed convection group.
Again, the green-colored terms are the ones that can be found in Equation (9) of the general
Agronomy 2021,11, 1585 16 of 19
AER. The underlined terms correspond to the terms matching with Equation (11). Here,
we note that both equations are more similar to curt ×L/Wthan curt ×temp.
AERBox.i=αi+βCurt.i×δ(Curt) + βθ.i×δ(θ) + βL/W.i×δ(L/W) + βTemp.i×Temp
+βVel.i×Vel +βVel.θ.i×Vel ×δ(θ) + βCurt.θ.i×δ(Curt)×δ(θ)
+βL/W.θ.i×δ(L/W)×δ(θ) + βCurt.L/W.i×δ(Curt)×δ(L/W)
(12)
with
δ(Curt) = (1, Curt ∈ {OU,OD}
0, Curt 6∈ {OU,OD};
δ(θ) = (1, θ∈ {45◦, 90◦}
0, θ6∈ {45◦, 90◦};
δ(L/W) = (1, L/W∈ {3, 4}
0, L/W6∈ {3, 4}
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 (
βVel
at the beginning of the
second row) has a significant effect on the increase in the
AERbox.i
. However, for the
inlet angle
θ
45
◦
(
βVel.θ45
) the effect is attenuate. For
θ
= 90
◦
(
βVel.θ90
),
AERbox.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
AERbox.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 (βOD,βOU, 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
βOD.θ45
,
βOU.θ45
,
βOD.θ90
and
βOU.θ90
(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
βLW3
and
βLW4
vary between high
and low from one half of the barn to the other. When the openings are reduced
(
βCurt.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 βLW−.
Here, the mean
R2
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
AERBox
range is between 151 h
−1
and 530 h−1.
Unfortunately, we could not find a fitting equation for the natural convection with
an acceptable
R2
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.
Agronomy 2021,11, 1585 17 of 19
Figure 10. RSE, R2and coefficients of the AERBox.iof Equation (12) corresponding to the forced convection group.
4. Conclusions
In the present work, the prediction of the general and local AER of NVDBs based
on five independent parameters was investigated using the statistical method general
linear model. In comparison to previous works, here, the temperature was added as an
influencing parameter, leading to consideration of the three different flow convection
types. We first found that the AER dataset for both the general and local cases could be
clustered into three groups, which perfectly correspond to the convection types. This
confirms the second hypothesis stated in the introduction, regarding the importance of the
temperature–velocity interaction for the AER.
Furthermore, an equation modelling the general barn AER with high accuracy
(R2= 0.98)
was found, which was in agreement with previous studies. Our study confirmed
that the barn’s length-to-width ratio (
L/W
) did not have a significant influence on the
general AER and that reducing the opening size reduces the AER. In addition, we observed
that, with increasing inlet angle (
θ
), the latter effect was dampened. This scenario could
also be seen for the local AER. The velocity magnitude (
Vel
) had a significant impact on
the general and local AER ,while the temperature tended to decrease the AER. Interactions
between the considered parameters were also noted to influence the AER.
This present work also investigated the AER locally, in the form of ten equal volume
subdivions (called box). It was noted that the only possibility of finding a reliable modelling
was by considering each convection group separately. The equations for the forced and
mixed convection groups indicated a similar behavior to the general group. Nevertheless,
at the same time, this behavior is more complex. This time,
L/W
has a non-negligent impact
on the AER
Box
, in such a way that the positive and negative box-coefficients associated
with this parameter compensate for themselves. This could explain the absent
L/W
for
the general AER model. We also noticed some patterns where, depending on the equation
term, the front half of the barn was more strongly affected than the rear half of the barn,
or vice-versa. These findings allow us to confirm the first hypothesis, that the local AERs
have a different behavior depending on their position and relative to the general AER.
Agronomy 2021,11, 1585 18 of 19
While it was possible to extract modelling equations for the forced and mixed groups, it
was hard to find a representative model for the natural convection group.
This leaves exciting expectations for future works. For example, a non-linear model of
the natural convection group could be explored. In addition, a more precise subdivision
could be investigated: for example, 40 boxes instead of 10. Another possible direction could
be the implementation of position coordinates in the modelling of a local AER equation.
Author Contributions:
Conceptualization, E.M.D., D.J., Q.Y. and S.H.; methodology, E.M.D., D.J.,
Q.Y. and S.H.; software, E.M.D., A.P.; validation, E.M.D.; formal analysis, E.M.D., D.J., Q.Y. and S.H.;
investigation, E.M.D.; writing—original draft preparation, E.M.D.; writing—review and editing, D.J.,
Q.Y., A.P., T.A., M.K. and S.H.; visualization, E.M.D.; supervision, D.J., T.A., M.K. and S.H.; project
administration, S.H.; funding acquisition, D.J. and S.H. All authors have read and agreed to the
published version of the manuscript.
Funding:
The research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation), Grant Number 397548689. We further thank Sai Krishna Danda for his support in
typesetting the paper.
Data Availability Statement:
The data presented in this study are available from the authors on
request. Storage in the open respository Zenodo is in preparation.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or
in the decision to publish the results.
Abbreviations
The following abbreviations are used in this manuscript:
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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