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MODELLING AND SIMULATION OF GREENHOUSE CLIMATE USING DYMOLA
F. Rodríguez
*
, L.J. Yebra
#
, M. Berenguel
*
, S. Dormido
‡
*
Universidad de Almería. Dpto. Lenguajes y Computación. Ctra. Sacramento s/n, La
Cañada, E04120, Almería, Spain. E-mail: frrodrig@ual.es
, beren@ual.es
#
CIEMAT-PSA. Apdo. 22, E-04200 Tabernas, Almería, Spain. E-mail: luis.yebra@psa.es
‡
U.N.E.D. Facultad de Ciencias. Dpto. de Informática y Automática. Avda. Senda del Rey
s/n, E28040, Madrid, Spain. E-mail: sdormido@dia.uned.es
Abstract: This paper deals with the modelling and simulation of greenhouse climate. The
obtaining of greenhouse production models (climate, crop development, etc.) is a subject
of large interest nowadays, as these models can be used for simulation, control and
production optimization purposes. A greenhouse constitutes a complex dynamical system,
which is described in this paper using the object-oriented and equation-based declarative
modelling language Dymola. A systematic procedure has been followed to allow the
development and testing of different submodels, before connecting them to generate the
complete greenhouse model. Comparison results between the real and simulated
greenhouse are provided. Copyright © 2002 IFAC
Keywords: Agriculture, object modelling techniques, dynamic modelling, computer
simulation, physical models.
1. INTRODUCTION
During the last years, a large effort is being devoted
in the obtaining of iclimate models of Mediterranean
greenhouses, both for simulation and control
purposes. This paper deals with the modelling and
simulation of a typical Mediterranean greenhouse
using the object-oriented and equation-based
declarative modelling language Dymola. This model
is of capital interest in the study of the dynamics of
the greenhouse climate under different control
policies, aimed at producing inside climate
conditions which helps to avoid extreme situations
(high temperature or humidity levels, etc.) and to
optimise crop production by achieving adequate
temperature integrals while reducing pollution and
energy consumption.
The dynamic behaviour of the micro-climate is a
combination of physical processes involving energy
transfer (radiation and heat) and mass balance (water
vapour fluxes and CO
2
concentration). These
processes depend on the outside environmental
conditions, structure of the greenhouse, type and
state of the crop and on the effect of the control
actuators (typically ventilation and heating to modify
inside temperature and humidity conditions, shading
and artificial light to change internal radiation, CO
2
injection to influence photosynthesis and
fogging/cooling for humidity enrichment).
Most of the greenhouse models currently used in the
Mediterranean area (made by low cost plastic cover,
taking advantage of favourable outside climatic
conditions) have been developed empirically and are
tailor-made on a hand-craft basis. Some authors of
this paper have developed a nonlinear artificial
neural network (NN) model of a typical
Mediterranean greenhouse (Rodríguez et al., 1999)
and a nonlinear model of both climate conditions and
crop development using physical laws (Rodríguez et
al., 2001). In this paper, the problem of greenhouse
climate modelling is treated by following a modular
modelling approach. The model is composed by six
submodels describing the cover temperature, the soil
surface temperature, the first soil layer temperature,
the inside air temperature and humidity and the PAR
(Photo-synthetically active radiation) radiation. All
of them have been developed independently and have
been validated using measurements from a real plant
sited in Almería (South-East Spain). Once the
submodels have been validated, these are connected
properly to generate the final compound greenhouse
model, which is then simulated and compared again
with the experimental data logged from the real
plant. As has been mentioned, Dymola is used to
describe the structure of the model and its behaviour
in terms of differential-algebraic equations (DAE’s)
with proper language constructs to manage
discontinuities by means of implicit events. In
addition to the symbolic formula manipulation it
Copyright © 2002 IFAC
15th Triennial World Congress, Barcelona, Spain
performs over the DAE system, it gives choice over
several known ODE and DAE solvers. In the
simulations performed in this work, DASSL (Brenan
et. al., 1989), a multistep solver of variable order and
step for stiff systems has been selected.
The paper is organised as follows. In §2, a brief
description of a dynamic model of the greenhouse
climate is formulated. §3 is devoted to introduce a
model of the greenhouse climate using Dymola. In
§4, some simulation results are shown and compared
with real data. Finally, §5 presents some conclusions.
2. DYNAMIC MODEL OF GREENHOUSE
CLIMATE
The greenhouse climate can be described by a
dynamic model represented by a system of
differential equations which can be represented by:
),,,,,( tCVPUXf
dt
dX
= with
ii
XtX
=
)(
(1)
X=X(t) is a n-dimensional vector of state variables,
U=U(t) is a m-dimensional vector of input variables,
P=P(t) is an o-dimensional vector of disturbances,
V=V(t) is a p-dimensional vector of system variables,
C is a q-dimensional vector of system constants, t is
the time, X
i
is the known initial state at the initial
time t
i
and f=f(t) is a non-linear function based on
mass and heat transfer balances.
The number of equations describing the system and
their characteristics depend on the greenhouse
elements, the installed control actuators and the type
of cultivation method. The model presented in this
paper corresponds with a typical industrial
greenhouse located at the Mediterranean area and has
been developed assuming some general hypothesis:
The greenhouse is divided into four elements:
cover, internal air, soil surface and one soil
layer. The plants are not considered as an
element as no measurements of the leaf
temperature is available at the moment, and thus
they are considered as a source of disturbances.
The state variables of the model are the internal
air temperature (X
ta
) and humidity (X
ha
), cover
temperature (X
tcb
), soil surface temperature (X
tss
)
and soil first layer temperature (X
ts1
). The PAR
radiation onto the canopy (output variable
X
radPAR,a
) is also modelled.
The disturbance inputs of the system are the
outside air temperature (P
text
) and humidity
(P
hext
), wind speed (P
wsext
) and direction (P
wdext
),
sky temperature (P
tsky
), calculated using the
Swinbank formula (Boisson, 1991), deep soil
temperature (P
tds
), outside solar radiation
(P
radsol,ext
), PAR radiation (P
radPAR,ext
),
greenhouse whitening (P
white
) and the
evapotranspiration rate inside the greenhouse via
the leaf area index (P
LAI
).
The control inputs of the system are the natural
ventilation (U
vent
), shade screen (U
shad
) and pipe
heating system (U
heat
).
The heat fluxes are one-dimensional. The model
only considers the vertical dimension.
The following physical processes are included in the
balances: solar and thermal radiation absorption, heat
convection and conduction, crop transpiration,
condensation and evaporation. In what follows, a
brief description of the models is carried out. A full
description can be found in (Rodríguez et al., 2001).
2.1 Model of the PAR radiation
The PAR radiation onto the canopy is modelled using
a static equation, because it is similar to the PAR
radiation outside the greenhouse dimmed by the
different physical elements that absorb the radiation
(cover material, cover whitening and shade screen).
extradPARtrsaradPAR
PVX
,,
⋅= (2)
V
trs
is the greenhouse PAR radiation transmission
coefficient:
=
whiteningshadeCCC
whiteningnoshadeCC
whiteningshadenoCC
whiteningnoshadenoC
V
shadetmwtmcbtm
shadetmcbtm
wtmcbtm
cbtm
PARtrs
,**
,*
,*
,
,.,
.,
.,
,
,
(3)
Where C
tm,cb
is the cover solar transmission
coefficient, C
tm,w
is the whitening solar transmission
coefficient and C
tm,shade
is the shade screen solar
transmission coefficient
2.2 Heat transfer through the cover.
The cover of a greenhouse has two sides with
different temperatures. Due to the fact that the cover
is made using a single material (plastic film) and that
it thickness is of a few microns, the conduction heat
flux is quantitatively not significant compared with
the other fluxes appearing in the balance given in
equation (4) (Garzoli and Blackwell, 1981). So, the
temperatures of the two sides are assumed to be
similar and only one cover temperature has been
modelled (X
tcb
) using the following heat transfer
balance:
cbradcbltecbcvacbcvcbsol
cbt
sarea
cbvol
cbdencbespc
QQQQQ
dt
dX
c
c
cc
,,,,,
,
,
,
,,
+−−−
=
−−
−
(4)
Q
sol,cb
is the solar radiation absorbed by the cover,
Q
cv,cb-e
is the convective flux with the outside air,
Q
cv,cb-a
is the convective flux with the internal air,
Q
lt,cb
is the latent heat produced by condensation only
on the internal side of the cover, Q
rad,cb
is the thermal
radiation absorbed by the cover, c
c-esp,cb
is the specific
heat of the cover material, c
den,cb
is the cover material
density, c
vol,cb
is the cover volume and c
area,s
is the
greenhouse soil surface.
2.3 Heat transfer fluxes in the soil layers
The soil (greenhouse thermal mass) plays an
important role on the greenhouse climate control.
During the diurnal time, the soil absorbs the solar
radiation on its surface, heating the deep soil layers.
During the night, the soil transfers heat to the
greenhouse environment from these layers. So the
conductive fluxes are very significant because this
process is the source of the heat fluxes between
them. A simple model of the soil has been
considered, divided in three layers: surface, first
layer and a deep layer with a constant temperature
(calculated as the average of the external air
temperature during one year). Based on these
hypotheses, the temperature of the soil surface
(thickness of 5 cm.) is represented by equation (5).
ssradssltssscdasscvsssol
sst
ssespssdenssespc
QQQQQ
dt
dX
ccc
,,1,,,
,
,,,
+−−−=
=
−−
−
(5)
where Q
sol,ss
is the solar radiation absorbed by the
soil surface based on the crop growth status
(modelling or measuring the Leaf Area Index-LAI),
Q
cv,ss-a
is the convective flux with the internal air,
Q
cd,s-s1
is the conductive flux between the soil surface
and the first soil layer located at 30 cm., Q
lt,ss
is the
latent heat produced by evaporation on the soil
surface, Q
rad,ss
is the thermal radiation absorbed by
the soil surface, c
c-esp,ss
is the specific heat of the soil
surface material, c
den,ss
is the soil surface material
density and c
esp,ss
is the thickness of the soil surface.
In the first soil layer, only the conductive fluxes are
considered and so, the heat balance in this element is
represented by equation (6).
scscdssscd
st
sespsdensespc
QQ
dt
dX
ccc
−−−
−=
1,1,
1,
1,1,1,
(6)
where Q
cd,ss-s1
is the conductive flux between the soil
surface and the first layer of the soil, Q
cd,s1-sc
is the
conductive flux between the first soil layer and the
deep layer at constant temperature, c
c-esp,s1
is the
specific heat of the first soil layer material, c
den,s1
is
the first soil layer material density and c
esp,s1
is the
thickness of this layer.
2.4 Heat transfer fluxes with the internal air
The crop affects the greenhouse air temperature. As
no measurements of the leaf area are available, it is
not possible to use a convective factor in the heat
balance equation using it as a boundary variable. The
effect of the crop on the air temperature is based on
the latent heat due to transpiration of the plants
(Q
trp
), using the work of Stanghellini (1987). The
cultivation method of the tomato crop used in the
installations is NFT (Nutrient Films Technique). The
greenhouse contains non-isolated pools in order to
recycle the fertilized water to maintain the
continuous water flux. The evaporation of the water
of the pools affects the greenhouse climate. In the
same way the transpiration of the crop has been
included in the balances, a new factor has been added
to the latent heat term based on the net radiation and
the water vapour pressure deficit. Based on all these
processes, the greenhouse air temperature can be
modelled using the following heat balance equation:
lossesventrplttrpltacalcv
asscvacbcv
at
sarea
gvol
adenaespc
QQQQQ
QQ
dt
dX
c
c
cc
−−−−+
++=
−
−−−
,,,
,,
,
,
,
,,
(7)
where Q
cv,cb-a
is the convective flux with the cover,
Q
cv,ss-a
is the convective flux with the soil surface,
Q
cv,cal-a
is the convective flux with the heating pipes,
Q
lt,trp
is the latent heat effect of the crop transpiration,
Q
lt,evp
is the latent heat effect of evaporation in the
pools, Q
vent
is the heat lost by natural ventilation,
Q
losses
is the heat lost by infiltration losses and c
c-sp,a
c
den,a
(c
vol,g
/ c
area,s
) is the product of specific heat of
air, air density and effective height of the greenhouse
(greenhouse volume/soil surface area).
A model of humidity (water vapour content of the
greenhouse air, Kg
H2O
/Kg
air
) is based on a water mass
balance equation. The main sources of vapour in a
greenhouse are the crop transpiration, the
evaporation of the soil surface and pools and the
water influx by fogging or cooling. The vapour
outflow takes place through the condensation on the
internal side of the cover, the ventilation and the
vapour lost by infiltration losses. The evaporation
from the soil surface is neglected as it is mulched
(Stanghellini, 1995). As artificial water influxes
(cooling, fogging, etc.) were not installed, the mean
water vapour content of the greenhouse air, X
ha
,
(absolute humidity) is modelled using the water mass
balance equation given by equation (8).
lossesOHvenOHcndOH
evpOHtrpOH
ah
aden
sarea
gvol
MMM
MM
dt
Xd
c
c
c
,,,
,,
,
,
,
,
222
22
−−−
−+=
(8)
where M
H2O,trp
is the crop transpiration flux, M
H2O,evp
is the evaporation flux from the pools, M
H2O,cnd
is the
condensation flux from the cover, M
H2O,ven
is the
outflow by natural ventilation and M
H2O,losses
is the
vapour lost by infiltration losses.
The values of the coefficients in previous equations
have been obtained by using a large set of
input/output data covering different operating
conditions obtained at the real greenhouse and by
iterative search in the range of values given by
different authors using genetic algorithms. The
previous equations are the basis for the formulation
of a nonlinear simulation model using Dymola, as it
is commented in the following section.
3. MODELLING AND SIMULATION OF THE
GREENHOUSE USING DYMOLA
3.1 Modelling of dynamical systems using Dymola
Object-oriented modelling is a method for structuring
models by applying the following ideas: use of
declarative models, modularity, abstraction,
encapsulation, information hiding, use of classes and
inheritance. This methodology makes possible to
describe models of many different domains like
electrical circuits, mechanics, thermo-dynamics, etc.
in an uniform way. A description of this
methodology can be found in (Andersson, 1994).
In this work, the Dynamic Modelling Language
Dymola (Elmqvist, 1978) has been used to model the
greenhouse climate, taking advantage of the features
of the object-oriented modelling approach. In
addition to the modelling environment, Dymola
implements automated formulae manipulation
techniques such as: solving the causality assignment
problem, generation of the equations that result from
the couplings between different objects, automatic
reduction of higher index systems and treatment of
algebraic loops that often result from subsystem
couplings and that also appear as a consequence of
the reduction of higher index models. The motivation
for using Dymola in this work is not due to the
necessity of solving higher index problems (which is
not a problem in this application), but that of using
and object-oriented modelling language that allows
the obtaining of noncausal descriptions of the
subsystems components that form the final system to
be simulated. The characteristics of object-oriented
languages allow the development of models with
reusing possibilities. In addition to the symbolic
manipulations performed by Dymola over the whole
system, it offers a rich repertory of ODE and DAE
solvers that brings enough guaranties that the
simulation has been performed correctly.
3.2 Modelling of greenhouse climate
The first stage of the modelling approach followed in
this work has been to identify the model components
and to define their interfaces (set of variables that
connect the model with its environment). The
behaviour of each component is described by DAE’s,
interconnected by their interfaces. Those models may
be decomposed in submodels components allowing
the use of hierarchical modelling techniques.
As has been mentioned in
§
2, the greenhouse climate
dynamics may be decomposed in six submodels
corresponding with the main state and output
variables, interconnected via their interfaces. This
methodology has many advantages, as helps to study
each submodel independently and to debug it before
connecting all of them to create the final compound
model of the whole system. The submodels describe
the PAR radiation, the inside air humidity and the
temperature of the cover, the soil surface, the soil
layer and the internal air. As will be mentioned in the
following section, another advantage of the
decomposition is that each submodel can be
substituted by real measurements when these are
available (e.g., measurements of the soil surface
temperature).
The next subsections describe the classical model
decomposition in interface and behaviour in object-
oriented modelling of physical systems, in the case
of the greenhouse climate model. For the sake of
space, only the model of the internal air and the
complete compound model are briefly described.
The inside air temperature model
Interface: formed by X
tss
, U
heat
, P
text
, P
wsext
,
P
radsol,ext
, U
vent
, U
shad
, X
h_inv
, X
t,cb
, P
white
and P
LAI
.
Behaviour: described by ODE in equation (8).
The compound model: composed of the six
mentioned submodels interconnected via their
interface variables (Fig. 1).
Interface: formed by U
shad
, P
radsol,ext
, U
vent
, P
text
,
P
wsext
, P
habse
, P
tds
, P
white
and P
LAI
.
Behaviour: described by the equations that
‘connect’ the submodel components and the set
of interface variables not connected between
them to the compound model interface variables.
Dymola code for the compound model: the list
included in Fig. 2 contains a piece of code for
the compound model, where the declaration of
the submodels and their connection by different
equations is shown.
Fig. 1. Block diagram of the compound model
4. RESULTS
4.1 Real greenhouse
The experiments were carried out in an “Araba”
greenhouse located in El Ejido (Almería, South-East
Spain) near the sea (Fig. 3). It is a two symmetric
curved slope roof with five North-South oriented
naves of 7.5 x 40 m (1500 m
2
of soil surface and 5.5
m. high). The covering material is a PE film of 200
microns thick, laid on a structure made of galvanized
steel. The control actuators are vents (lateral and
roof), shade screen and hot water pipes heating.
Several sensors were installed for data acquisition
purposes. The soil temperature measurements were
Plai
P
rad,ex
U
sha
V
whit
P
te
P
wse
U
ve
U
he
P
la
P
he
P
td
Inside PAR
Radiation
Cover
Temperature
Model
Inside Air
Temperature
Model
Inside Air
Humidity
Model
Soil Surface
Temperature
Model
Soil Temp.
Model
P
rad,ex
X
tcb
X
ta
X
ha
X
tss
X
ts1
s
xt
i
at
nt
xt
xt
e
de
t
carried out through semiconductor sensors at
different depths (both side of the mulching,
immediately under the soil surface layer and at a
depth of 50 mm.). The greenhouse air temperature
thermoresistent sensor and the air relative humidity
capacitive sensor were placed at the top of the crop.
Eight semiconductor contact sensors were installed
on both cover sides to calculate the inside and
outside cover temperatures on average. An external
meteorological station was installed at 6 m. height
for acquiring measurements of temperature, relative
humidity, solar and PAR radiation, wind speed and
direction and rain. The system also acquires
measurements of the actuators status
.
Data are
sampled and stored each minute.
4.2 Simulations
To test the compound model with Dymola it is
necessary to define a experiment in which all the
interface variables from an instantiation of the
greenhouse_compound
model are assigned to
experimental data registers, obtained from the data
acquisition equipment installed at the greenhouse, as
has been previously mentioned. Once the formulae
manipulation has been performed by Dymola over
the compound model, it generates a sequence of 115
equations (without algebraic loops), 5 of them
ODE’s and the rest are algebraic ones. This system
of DAE’s is discontinuous and these discontinuities
are handled properly in Dymola by the definition in
the submodels components of implicit state events.
The selected interval of integration with DASSL is
13 days covering data obtained from the
experimental greenhouse.
The first simulation shown in this section
corresponds with the results obtained using the
internal air submodel excited individually with data
obtained from the experimental registers, and it is
compared with the real temperature in order to
validate the submodel. Fig. 4 represents both the
simulated (
Air_Xta
) and real (
Xta
) internal air
temperatures (ºC).
The same procedure has been applied with the other
output variables (cover temperature, soil
temperature, etc.) in such a way that individual
submodels have been simulated and validated. All
these submodels can be integrated within the
compound model, in such a way that the simulation
of the greenhouse climate is a result of the
interaction between the different submodels. Fig. 5
represents the evolution of the different output
variables of the compound model, compared with the
real values. Notice that the dynamic behaviour of the
complete model is obtained by the interaction of
simulated submodels connected properly. The
variables in the plots are:
Variable Real Simulated
Cover temp.
Xtcb Cover_Xtcb
Soil surface temp.
Xts Soil_Xts
Soil layer 1 temp.
Xtsp Soil_Xtsp
Inside air temp.
Xta Air_Xta
Inside air hum.
Phabsi Refi::hume.XH_absa
Fig.2.Example of Dymola code for compound model
Fig. 3. Araba greenhouse
Table 2 provides some significant figures of the
results obtained with both models, in terms of
maximum (
Max_abs
), mean (
Mean_abs
) and standard
deviation (
Std
) absolute errors when compared with
model class greenhouse_compound
{Submodels}
submodel (greenhouse_humidity) hum
submodel (greenhouse_thermal_cover) cover
submodel (greenhouse_thermal_soil) soil
submodel (greenhouse_thermal_air) air
{Interface}
input Ushad, Prade, Uvent, Pte, Pve, Phabse, Ptds, Pwhite,
Plai
{Equations describing the connections between submodels}
{Connection of the greenhouse_humidity model}
hum.Xta = air.Xta-273
hum.Xtcb = cover.Xtcb-273
hum.Xlai = Plai
hum.PHabse = PHabse
hum.Pve = Pve
hum.Pte = Pte
hum.Prade = Prade
hum.Uvent = Uvent
hum.Ushad = Ushad
hum.Pwhite=Pwhite
{Connection of the greenhouse_thermal_cover model}
cover.Prade = Prade
cover.Pve = Pve
cover.Pte = Pte+273 {Kelvin}
cover.Xta = air.Xta {Kelvin}
cover.Ushad = Ushad
cover.Pwhite=Pwhite
cover.XH_absa = hum.XH_absa
cover.Xts = soil.Xts
cover.Xtcal = Uheat
{Connection of the greenhouse_thermal_soil model}
soil.Ushad=Ushad
soil.Pwhite=Pwhite
soil.Xlai=Plai
soil.Prade=Prade
soil.Pte=Pte+273 {Kelvin}
soil.Pve=Pve
soil.Xta=air.Xta {Kelvin}
soil.Uvent=Uvent
soil.Xtcb=cover.Xtcb {Kelvin}
soil.Xtcal=Uheat
soil.Ptds=Ptds+273 {Kelvin}
{Connection of the greenhouse_thermal_air model}
air.Xts=soil.Xts {Kelvin}
air.Xtcal=Uheat
air.Pte=Pte+273 {Kelvin}
air.Pve=Pve
air.Prade=Prade
air.Uvent=Uvent
air.Ushad=Ushad
air.Pwhite=Pwhite
air.Xlai=Plai
air.XH_absa=hum.XH_absa
air.Xtcb=cover.Xtcb {Kelvin}
end
real data. Moreover, as can be seen from mean
values, maximum errors tend to be spurious values
when compared with general behaviour of the model.
The model behaves quite well and constitutes an
important tool in the analysis of the greenhouse
dynamics and in the development of control schemes
for optimising crop production. The methodology
and way in which the model has been constructed
allows it extension to other types of greenhouses.
5. CONCLUSIONS
The obtaining of adequate models for greenhouse
climate modelling from mass and energy balances is
a hard and time consuming task. Once the equations
are adequately formulated, the use of modelling
environments as Dymola and systematic procedures
for decomposing the complete model in submodels,
which can be independently validated, has shown to
facilitate the implementation of the compound model
(as an integration of the single submodels) and its
extension to other types of greenhouses.
ACKNOWLEDGEMENTS
Authors would like to acknowledge CICYT for
partially funding this work under grants QUI99-
0663-C02-02 and DPI2001-2380-C02-02.
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crops
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Stanghellini, C., de Jong, T. (1995). A model of humidity
and its applications in a greenhouse. Agricultural
and Forest Meteorology, 76, pp. 120-148.
Table 2. Comparative results
Air temp. submodel
Max_abs:2.8002 Mean_abs: 0.5006 Std: 0.5154
Compound model
Xta Xha Xtcb Xtss Xts1
Max_abs
6.1037 0.0079 5.1827 4.2122 1.3648
Mean_abs
1.4418 0.001 0.8929 0.7003 0.3432
Std
1.0913 0.0009625 0.7973 0.6323 0.2702
0 5000 1E4 1.5E4
20
25
30
35
40
45
50
55
Xt a
air_Xta
Fig. 4. Simulation with the air temperature submodel
0 5000 1E4 1.5E4
15
20
25
30
35
40
45
50
55
cover_Xtcb
Xt c b
0 5000 1E4 1.5E4
24
28
32
36
40
44
soil_Xts
Xt s
0 5000 1E4 1.5E4
28
28.5
29
29.5
30
30.5
31
31.5
32
soil_Xtsp
Xt s p
0 5000 1E4 1.5E4
20
25
30
35
40
45
50
55
air_Xta
Xt a
0 5000 1E4 1.5E4
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
refi::hume.XH_absa
PHabsi
Fig. 5. Simulation results with the compound model
