The benefit of controlled and modified atmospheres for extending the storage  life of fruits is world wide accepted. However, there are secondary effects such as the  incidence of anaerobic respiration or  the off-favour occurrence which are not  sufficiently known and thus controlled.  This study approaches the knowledge of  those secondary effects by developing a spatio temporal model gathering fluid flow  phenomenon and physic and physiological processes. The lattice Boltzmann model  used as framework for mimicking the fluid flow shows to be a very flexible tool  which reproduces complex macroscopic behaviours on a down up strategy better  than Lattice Gas Cellular Automata (LGCA

Alonso, R.

Barreiro Elorza, Pilar

Bielza Lozoya, Maria Concepcion

Calles, M.

Casasús Latorre, Pedro Luis

Correa Hernando, Eva Cristina

Fabero, J.C.

Ruiz-Altisent, Margarita

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Lattice gas and Lattice Boltzman for spatio-temporal simulation of gases 
in fruit storage chambers  
 
Barreiro, P1; Fabero J.C2; Cassasus P.L3; Calles M4; Bielza C4; Correa, E.C1; Alonso R.1; 
Ruiz-Altisent M.1 
 
1. Physical Properties Laboratory. ETSIA. UPM. pbarreiro@iru.etsia.upm.es 
2. Hardware Architecture and Automation Dept. UCM. 
3. Applied Mathematics Dept. ETSII.UPM 
4. Artificial Intelligence Dept. FI. UPM 
 
 
KEYWORDS: gas flow, physiological processes, modelling, controlled atmosphere, 
simulation 
 
ABSTRACT 
The benefit of controlled and modified atmospheres for extending the storage 
life of fruits is world wide accepted. However, there are secondary effects such as the 
incidence of anaerobic respiration or the off-favour occurrence which are not 
sufficiently known and thus controlled. This study approaches the knowledge of 
those secondary effects by developing a spatio temporal model gathering fluid flow 
phenomenon and physic and physiological processes. The lattice Boltzmann model 
used as framework for mimicking the fluid flow shows to be a very flexible tool 
which reproduces complex macroscopic behaviours on a down up strategy better 
than Lattice Gas Cellular Automata (LGCA). 
 
 
INTRODUCTION 
Fluid Dynamics can be modelled through the Navier-Stokes equations, and there 
are commercial software packages available to do so (Computational Fluid Dynamics, 
CFD). The Navier Stokes description corresponds to a macroscopic definition of the fluid 
motion phenomena, however it is not the unique method to be used. 
During the past 20 years new simulation procedures are emerging from Statistical 
Physics and Computer Science domain. One of them is the Cellular Automata (LGCA) 
method. This procedure corresponds to a microscopic definition of the world. The 
concept here is to simulate complex macroscopic (global) behaviours by means of the 
definition of a limited number of rules operating at a microscopic (molecular) level. For 
LGCA both space and time are defined as discrete variables. Space is defined in a lattice 
were each cell is connected to its nearest neighbours (4 or 6 depending on the lattice 
topology). Time is considered in two steps: movement of particles inside and between cell 
respectively. When appropriate, the  rules defined for both steps enable to reproduce gas 
diffusion in a realistic way. All cells are updated simultaneously for all the lattice (Wolf-
Gladrow, 2000). 
Another simulation procedure for mimicking the fluid motion phenomena is the so 
called Lattice Boltzman method (Rothman and Zaleski, 1997). This corresponds to a 
meso-scale description of the world (Van Der Sman, 2001). In this case, particle 
distributions rather than isolated particles are considered. At each time step a collision 
factor is computed (Ω) for each cell which is used to update the particle distributions at 
every cell of the lattice simultaneously. Simulation of laminar and turbulent motions of 
fluids specially when considering several gas species is still an ongoing research (Shan 
and Chen, 1993). 
Micro and meso-scale based methods correspond to a down-up strategy while 
macroscopic simulation through CFD is considered an up-down procedure (Wolf-
Gladrow, 2000). The main advantage of the former compared to the latter relays on 
facilitating the incorporation of complementary chemical and physiological phenomena to 
the simulation. This fact is a major issue when considering the storage of agricultural 
products such as fruits and vegetables. 
Nowadays, the use of controlled (CA), Low Oxygen (LO) and Ultra Low Oxygen 
(ULO) atmospheres has been recognised as a reliable method to extend the storage life of 
fruits and vegetables. However, small spatial gradients in gas concentration during 
storage may generate internal disorders in the commodities. The accumulation of non 
traceable factors such as intra-batch variations in the maturity stages of the fruit and/or 
local lack of uniformity in the air motion, lead to the erratic occurrence of internal 
disorders.   
The development of tools to facilitate the study of such erratic behaviours will be 
faced in this paper by simulating transpiration, respiration, diffusion and ventilation inside 
fruit storage chambers within micro and meso-scale strategies. 
 
METHODS 
Four different gases have been considered for modelling transpiration respiration 
and ethylene emission: oxygen, carbon dioxide, water vapour and ethylene. Transpiration 
affects the level of water vapour, respiration modifies both oxygen and carbon dioxide 
concentrations, and ethylene emission affects the concentration of this phyto-hormone in 
the air. 
 
Psychometric model 
The ASAE model (ASAE, 1983) is used for establishing the vapour ratio (g/kg dry 
air) based on relative humidity, temperature and atmospheric pressure. 
 
Transpiration model 
The water loss of the fruit towards the surrounding air is considered, due to the 
difference in water potential between each piece of fruit (ψf) and the ambient. The water 
potential of the air (ψ0) is a function of the temperature (T) and the relative humidity 
(RH). The water potential of the fruit (ψf) is considered in a first approximation strictly 
dependent on the osmotic potential of the fruit (ψπf), and thus related to the concentration 
of solutes in the juice (De Smedt, 2000). For such transpiration model, the relative 
humidity term referred to the global relative humidity of the ambient since no spatial 
effect was considered. At current stage, the relative humidity for the boundary layer of the 
skin for each fruit is computed, which changes according to the diffusion and transport 
phenomena modelled with LGCA or Lattice Boltzmann. 
 
  
Eq.1 
  
Eq.2 
 
 
Respiration model 
Respiration refers to hexose consumption (C6H12) by means of oxygen (O2) which 
generates water vapour (H2O) and carbon dioxide (CO2). This process together with 
transpiration has already been faced in previous studies (De Smedt, 2000) miss-regarding 
spatial effects. Respiration decreases the osmotic potential inside the fruit and thus 
facilitates transpiration. Whenever the carbon dioxide in the air reaches a certain 
threshold (species and variety dependant) anaerobic respiration may occur leading to 
abnormal storage features in the fruits. The chemical reaction for respiration allows a 
proper calculation of mass conversion between all those chemical species. 
 
Definition of simulated media 
To proceed with the simulation of gas motion, there is a need of defining the 
properties of the different media: air, recipients (boxes and pallets) and fruits. Thus, in the 
air cells only motion phenomena may occur, recipients are considered non porous, that is 
impermeable cells unless perforated. Concerning the fruits, the physiological activity is 
concentrated towards the skin, while the inner part is considered similar to the boxes. 
Beside the mentioned dominia when trying to simulated forced air flows, two new 
dominia were also defined: suction area and fan. There is a direct mass transfer between 
the suction areas and the fans. Also the impulsion direction is tuneable. 
 
 
Figure 1. Hexagonal topology used for the simulation of gas motion. The figure to the 
right indicates the maximum number of gas particles allowed for the LGCA model. 
  
⎟⎠
⎞⎜⎝
⎛
+=ΨΨ HW
WLn
V
RT
M
fruit π~
⎟⎠
⎞⎜⎝
⎛=Ψ
1000
HRLn
V
RT
M
Gas motion with LGCA  
This procedure allows a proper definition of the diffusion process inside the cold 
storage chamber, while fails to reproduce forced air motion due to particle density 
restrictions inside the cells. Figure 1 shows the configuration of a hexagonal topology of 
the spatial lattice. Inside each cell, seven particles may concur . As stated in the previous 
paragraph, movement occurs in two steps: diffusion inside and between cells. 
 
Gas motion with Lattice Boltzmann 
The model proposed by Shan and Chen in 1993 demonstrates the feasibility of 
reproducing large range of Reynold numbers. Moreover, it enables to consider the 
behaviour of multiple mixable gas species. In this procedure particle distributions (naσ) 
are considered instead of isolated particles as for the LGCA model. At each time step a 
collision factor is computed (Ωaσ) for each cell which is used to update the particle 
distributions at every cell of the lattice simultaneously. An overview of eq. 1 to 10 
indicates that the speed (u) is computed globally for all the gas species considered 
weighed by mass and relaxation time of each chemical. The particle distribution in the 
equilibrium maximises for a direction “a” (naσ(eq)) whenever its direction vector (ea) is 
parallel to the global speed vector (u). It also increases in relation to square module of 
global speed (u2). In spite of it, the particle distribution in the equilibrium at rest (n0σ(eq)) 
increases for increasing compressibility coefficient (d0) and decreases in relation to square 
module of global speed (u2). 
 
  
Eq.3 
  
Eq.4 
  
Eq.5 
  
Eq.6 
  
Eq.7 
),(),(),( txtxnttexn aaaa
σσσ Ω+=Δ++
( ) ttxntxn eqaaa Δ×−×−=Ω ),(),(1 )(σσσσ τ
⎥⎦
⎤⎢⎣
⎡ +⋅++⋅+−×= 22420)( 22
)2(1)()( u
bc
Duuee
bc
DDue
bc
D
b
dxnxn aaa
eq
a
σσ
⎥⎦
⎤⎢⎣
⎡ −×= 220)( 1)(),( ucdxntxn
eq
o
σσ
)2,1(  ),2,1(   ;   ==××=⋅ jiuuQuuee jiaijaa
  
 
Eq.8 
  
 
 
 
Eq.9 
  
 
 
Eq.10 
 
Figure 2 shows an example of forced air flow, where a suction area and a fan have 
been defined. The speed vector field is visualised after being computed with Lattice 
Boltzmann. 
 
 
 
 
 
wall 
air 
suction 
fan 
 
 
 
 
Figure 2. Example of forced air flow where a suction area and a fan have been defined. The 
image to the right represents the speed field computed for each cell with the Lattice 
Boltzmann model. 
 
∑
∑∑
=
σ σ
σ
σ
σ σ
σ
σ
σ
τ
τ
nm
enm
u
a
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
×=
)3/5sen()3/5cos(
)3/4sen()3/4cos(
)3/3sen()3/3cos(
)3/2sen()3/2cos(
)3/sen()3/cos(
)0sen()0cos(
ππ
ππ
ππ
ππ
ππ
cea
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
−⋅⋅
⋅−⋅
=
D
eeee
ee
D
ee
Q
aaaa
aaaa
a 1
1
2212
2111
Global model 
The fluid flow phenomena has been combined with relevant processes in fruit 
storage such as the mentioned transpiration, respiration and ethylene emission. The 
psychometric model defined by ASAE together with several geometric models for the 
different type of fruits are used. The definition of a scale factor (L, m) for the simulation 
window is basic in order to derive realistic data from draw fruits. 
 
  
Eq.11 
  
Eq.12 
  
Eq.13 
 
RESULTS OF SIMULATION 
A simulation software has been implemented which combines a drawing tool with 
user friendly definition of the parameters of global model. Figure 3 shows the evolution 
of the relative humidity in the boundary layer of the fruit for two geometric models 
(grapes and kiwi), the fact that the relative humidity in the skin of the grapes increases at 
a lower rate leads to higher weight losses of the fruit. In the time evolution shown in this 
figure, both transpiration though the skin and diffusion to neighbour cells can be 
identified.  
 
Table 1. Geometric models used for the different types of fruit in order to derive fruit 
mass and external surface form the drawing tool. 
GEOMETRIC 
MODEL 
TYPE OF 
FRUIT 
PARMETERS VOLUME 
(V, voxels) 
EXTERNAL 
SURFACE 
(S, pixels2) 
Sphere Apple, peach, 
citrus 
radius, r 3
3
4 rV ×= π  24 rS ×= π  
Ellipsoid Kiwi, pear semi axes a, b 
baV 2
3
4 ×= π  abS ×= π4  
 
2
2
512
)(  __ ⎟⎠
⎞⎜⎝
⎛×= LSmSurfaceExternalFruit
3
9
512
 10)/( ⎟⎠
⎞⎜⎝
⎛= LcelllVoxel μ
3
3
10
512
)(  ⎟⎠
⎞⎜⎝
⎛×= LVkgMassFruit
2/3 Toroid  Banana radios: internal 
external r1 y r2 
1
2
2 23
2 rrV ππ ××=
 
12 223
2 rrS ππ ×=
 
Cone Strawberry radius of base, R 
apothem, a 
hRV 2
3
1 ×= π
22 Rah −=
)( RaRS +×= π
 
Set of spheres Grapes number of 
spheres, N and 
radius, r  
3
3
4 rNV ××= π 24 rNS ××= π
 
Experimental data of individual fruits can be used for adjusting relevant 
parameters such as the mass transfer coefficient (h, mol/m2Pas). The fitting procedure 
consists of modifying the mass transfer coefficient whenever the difference between 
simulated and experimental data mismatch beyond a 3% of total variance. The mass 
transfer coefficient is increased whenever the simulated relative humidity is lower than 
the experimental, and decreased on the contrary situation. 
 
 
CONCLUSIONS 
The flow of miscible fluids can be modelled through Lattice Boltzmann. For each 
cell of the lattice a overall speed is computed at each time step, which is the motor of 
redistribution of the particles of all the gas species considered. 
The Lattice Boltzmann allows to simulate forced fluid flows. The numerical 
model has several parameters which are susceptible to be adjusted such as the 
compressibility factor (d0) or the relaxation time for each species (τσ). 
Different dominia may be defined easily with very different flow properties 
without numerical stability problems. 
The fact that discrete space and time are considered enables to gather other 
physical (psychometric and fruit geometry models), and physiological models 
(transpiration, respiration, ethylene emission) 
This type of simulation is a useful tool to design most relevant experiments in 
relation to fruit storage. 
Optimisation of the computation time and increasing the lattice definition beyond 
256.000 voxels remain at a research stage. 
 
40
45
50
55
60
65
70
75
80
85
90
0 50 100 150
time (s)
R
el
at
iv
e 
H
um
id
ity
 (%
)
grapes kiwi
transpiration
difussion
 
40
50
60
70
80
90
100
110
0 10 20 30 40 50
time (minutes)
R
el
at
iv
e 
H
um
id
ity
 (%
)
h=1.087E-14
h=3.34e-15
 
Figure 3. Time evolution of the relative 
humidity in the boundary layer of 
several fruits corresponding to 
different geometric models. 
Figure 4. Adjustment of the mass transfer 
coefficient (h) from experimental 
data, 24 iterations were needed with 
3.34e-15 mol/m2Pas as final value. 
ACKNOWLEDGEMENTS  
To the Polytechnic University of Madrid for founding of the Project: Simulation 
of gases in fruit cold storage chambers. 
 
 
REFERENCES 
ASAE. 1983. Agricultural Engineers Yearbook of STANDARDS. ASAE Data: D271,2. 
De Smedt V. (2000). Measurement and modelling of mealiness in apples. PhD Thesis. 
Katholieke University Leuven . 
Shan, X; Chen, H. (1993). Lattice Boltzman model for simulating flows with multiple 
phases and components. Physical Review Letters Vol. 47(3) pp: 1815-1819 
Rothman D.H; Zaleski S. 1997. Lattice-Gas Cellular Automata. Simple models for 
complex hydrodynamics. Ed. Cambridge. ISBN 0 521 55201 X 
Van Der Sman R. 2001. Mesoscale modelling of structured foods. 1st International 
Workshop on Mathematical an Computing Techniques for Agro-Food Technologies, 
CIMNE. Barcelona 
Wolf-Gladrow D.A. 2000. Lattice Gas Cellular automata and Lattice Boltzman Models. 
An Introduction. Ed Springer. ISBN 35406697 
 
 
NOTATION FOR THE LATTICE BOLTZMANN MODEL 
a Spatial direction 0..6 
b Maximum number of directions (6) 
σ Chemical species (1..4) 
ea Array of direction vectors. 6 × 2, six directions times 2 components  
| ea | Module of direction vectors 
NOTATION FOR THE LATTICE BOLTZMANN MODEL 
Δt Time step (s) 
c Distance between cells divided by the time step considered (m/s) 
nσ sum of particle densities inside a cell for each chemical species (moles) 
naσ(eq)(x) Value of particle densities in the equilibrium (moles) 
naσ(x,t) Particle density for each chemical species, cell and time (moles) 
Ωaσ(x,t) Collision factor (moles/s) 
τσ Relaxation time (s) 
D Spatial dimension (2) 
d0 Compressibility coefficient (moles/moles) 
u(x,t) Overall fluid speed in a cell (m/s), two components  
m Fluid mass (molar weight, kg) for each chemical species 
 


Lattice gas and lattice Boltzman for spatio-temporal simulation of gases in fruit storage chambers

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Lattice gas and lattice Boltzman for spatio-temporal simulation of gases in fruit storage chambers

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