Silo

ro

a

RS, Br

Computer Science Department, University of

a r t i c l e i n f o

Article history:

Received 21 August 2007

Received in revised form

Aeration is widely used in grain stores to cool the grain mass,

ous impurity in the mass, on the configuration and size of the

interstitial space in the mass, on the size and amount of bro-

ken grains, and on the depth of the grain.

The research carried out by Shedd (1953), Brooker (1961,

1969), Brooker et al. (1982), Bunn and Hukill (1963), Pierce and

et al. (2000), Navarro and Noyes (2001), Khatchatourian and

can no longer be assumed homogeneous. Non-homogeneity

can significantly alter the physical parameters involved in

the aeration process, such as air velocity and static pressure

drop. However, there is no research relating compaction of

the grain and the airflow pattern under these conditions.

* Corresponding author. Department of Physics, Statistics and Mathematics, Regional University of the Northwest, Rio Grande do Sul,R. Sao Francisco, 501, 98700-000 IJUI, RS, Brazil.

, [email protected] (M.O. Binelo).

Avai lab le a t www.sc iencedi rec t .com

vi

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8E-mail addresses: [email protected] (O.A. Khatchatourian)to avoid humidity migration, to temporarily conserve the hu-

midity of grains, to remove scents from the grainmass, and to

apply fumigation.

The resistance to the airflow in an aeration systemdepends

on the airflow parameters, on the characteristics of the prod-

uct surface (i.e. rugosity), on the form and size of any extrane-

Savicki (2004), and Khatchatourian and de Oliveira (2006) has

examined the influence of some of these parameters on air-

flow pattern in seeds storage. A recent review of the reported

mathematical models of airflow through grain mass was pre-

sented by Gayathri and Jayas (2007).

With increasing depth of grain storage, the mass of graingood performance. It was considered that themethod could be applied to optimise the perfor-

mance of existing grain stores and lower the engineering costs of new grain stores.

2008 IAgrE. Published by Elsevier Ltd. All rights reserved.

1. Introduction Thompson (1975), Haque et al. (1981), Ribeiro et al. (1983), Jayaset al. (1987), Maier et al. (1992), Weber (1995), Khatchatourian29 May 2008

Accepted 4 June 2008

Published online 21 August 20081537-5110/$ see front matter 2008 IAgrEdoi:10.1016/j.biosystemseng.2008.06.001Cruz Alta, R. Andrade Neves, 308, 98025-810 Cruz Alta, RS, Brazil

Amathematical model and software were developed for the three-dimensional simulation of

airflow throughhigh capacity grain storage bins by considering the non-uniformity of the seed

mass. To validate the proposedmodel, empirical relationships between air velocity and static

pressure drop were obtained for compacted layers of several storage depths for soya bean,

maize, rice and wheat mass. The software was written in ANSI C which is transferable toa variety of platforms. For the construction of 3D geometry and the generation of meshes

free-of-charge software was used. The solver software generated a system of linear algebraic

equations using the finite -elementmethod. Three iterative processeswere carried out: (1) cal-

culation of a local permeability coefficient, using the pressure distribution in the immediately

previous iteration step, (2) search for the systemdesignpoint, located in theperformancecurve

of the aerator fan, and (3) adaptation to refine the mesh. A local criterion to estimate the effi-

ciency of complex aeration system in storage bins was proposed. The simulations showedR. Sao Francisco, 501, 98700-000 IJUI,bazilaDepartment of Physics, Statistics and Mathematics, Regional University of the Northwest, Rio Grande do Sul,Research Paper: SEdStructures and Envi

Simulation of three-dimensionstorage bins

O.A. Khatchatouriana,b,*, M.O. Binelob

journa l homepage : www.e lse. Published by Elsevier Ltdnment

l airflow in grain

er .com/ loca te / i ssn /15375110. All rights reserved.

a product-dependent constant

b product-dependent constant

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8226To simulate the aeration of grain, with any type of air dis-

tribution systems, it is necessary to develop software to pre-

dict the distribution of the parameters, because obtaining

empirical data is very difficult and costly. Most research

on airflow simulation in grain stores is related to one-

dimensional, two-dimensional or axisymmetric cases; al-

C compaction function, dimensionless

c product-dependent constant

G product-dependent constant

H bed depth, m

i order number of corresponding inlet

k permeability coefficient, m3 kg1 sL bed depth, m

LX full length of a trajectory, m

M total number of experimental points

m grain mass, kg

n product-dependent constant; inlet number

n unit vector normal

P pressure, Pa

Pe air entrance or exit pressure in Pa;

Q global airflow rate, m3 s1 kg1

q local specific airflow rate, m3 s1 kg1

R product-dependent constant

S empirical coefficientNomenclature

A surface area, m2though the flow is usually three-dimensional. Even when

the grain mass distribution is two-dimensional or axisym-

metric, the airflow inlets do not satisfy these conditions.

Also, the aeration of large grain stores is frequently carried

out separately in different segments.

The principal objectives of the presentworkwere as follows:

(a) to create a mathematical model, algorithm, and software,

to calculate the static pressure, streamlines, and airflow

velocity distribution in three-dimensions under non-

homogeneous conditions;

(b) to determine the variation in compaction factor for several

depths of grain;

(c) to study the relationship between the air velocity and the

pressure gradient as a function of the compaction factor;

(d) to develop and incorporate into the software a criterion

for system performance based on estimating three-

dimensional air distribution in grain storage bins; and

(e) to carry out numerical simulations of real and hypothetical

grain storeswith aeration to detect areas of operational risk.

2. Mathematical model

The problem of incompressible viscous isothermal flow is de-

scribed by the system of equations of continuity [Eq. (1)] and of

NavierStokes [Eq. (2)]:div V 0; (1)

rDVDt

grad P mV2V; (2)

t time, s

U intermediate argument

V velocity vector, m s1

V velocity, m s1

Xi product-dependent constant (i 1, 2, 3)x coordinate located in floor plan, m

y coordinate along airflow axis, m

z coordinate located in floor plan, m

3 porosity factor, dimensionless

r density, kgm3

DP pressure drop, Pa

m dynamic viscosity, Pa s

Subscripts

a air

b bulk

e entrance, exit

g grain

i order number of corresponding inlet

k kernel

L local

X in point X(x, y, z)where V is the velocity vector in m s1; r is the density inkgm3; t is the time in s; P is the pressure in Pa; m is the dy-namic viscosity in Pa s.

The solutions of this system (usually reduced to the non-

dimensional form) depend on the effective Reynolds number

(calculated on apparent velocity taking into account the po-

rosity of the grain mass) and relate to the pressure and veloc-

ity distributions in each point of the integration domain for

each moment in the form of a vector-function V f (grad P),where the components u, v andw of velocity V and P are prim-

itive variables of the initial system.

However, experimental data show that the relationships

between the velocity and pressure gradient are different for

each type of grain, even for the same Reynolds number. This

is probably caused by the factors that cause airflow resistance

to vary, e.g. the geometrical form of the particles since grains

are not spherical they are distinct for different products; zones

within the grain mass exist where there is limited porosity

and there are differences in the rugosity of particle surface.

There are also other factors, e.g. grain layer compaction, var-

iation of humidity content, and the presence of impurities

that create differences between the measured values and

those calculated by solutions of the system described by Eqs.

(1) and (2). This implies that attempts to simulate the airflow

through the grain mass using the equations of continuity

and NavierStokes whilst contributing to our theoretical un-

derstanding of the problem are far from practical. The local

air velocity in an aerated grain storage can vary over a wide

can produce significant errors. For large grain storage bins,

especially with aeration in sections, there are regions with

the increased air velocity and regions where air velocity can

be practically zero. In these cases for calculations of the distri-

bution of pressure and velocity the variation in flow condi-

tions has a significant influence.

It is difficult to describe precisely airflow by means of these

relationships (which depend only on two constants) for all flow

regimes (laminar, transition and turbulent flows). If coefficients

a and b are chosen to accurately describe the transition regime,

the influence of velocity in limiting situations (laminar or

turbulent regime) will be too strong. If limiting regimes are

well described, then, the relationship for the transition regime

is insufficiently exact. Moreover, these relationships when

applied to two-dimensional and three-dimensional cases are

difficult to analyse.

Finding the derivative dln V=dlnjdP=Lj from both Eqs. (7)and (8) respectively:

dln V 1 bVln1 bV ; (9)

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8 227range depending on the cross-sectional area and on the design

of the aerator. Grain stores can have regions of laminar, turbu-

lent and transition flows. This complicates the creation of

mathematical models based on the use of the NavierStokes

equation.

For small velocities corresponding to laminar flow, a propor-

tional relationship exists between the air pressure drop and the

air velocity (i.e. the HagenPoiseuille or BlakeKozeny equation):

dP=dyfV0V kdP=dy; (3)where k is coefficient of proportionality; V jVj is the absolutevalue of velocity, ms1.

Applying logarithms and taking the derivative produces

dln VdlnjgradPj 1: (4)

For the turbulent regime that corresponds to the larger

values of air velocity, the pressure drop is proportional to

the velocity squared (i.e. the BurkePlummer equation) where

dP=dyfV20V kjdP=dyj1=2: (5)Thus, for turbulent flow

dln VdlnjdP=dyj 0:5: (6)

For the transition flows the relationship between the air

pressure drop and air velocity lies between linear and square

law dependency.

There are a large number of nonlinear motion equations in

the literature to describe airflow in porous media (Scheideg-

ger, 1960; Bear, 1988). In most of these equations the gradient

of pressure is expressed as function of velocity by second-order

parabola without a free term, i.e. as the sum of dependences

for the laminar and turbulent regimes.

The fullest recommendations for estimating static pres-

sure requirements are given by Navarro and Noyes (2001)

and the basic results of works for pressure-drop modelling

in stored grain masses are complied in ASAE (2000). In both

these studies the equation of Hukill and Ives (1955) has been

adapted to calculate the pressure gradient:

DPL aV

2

ln1 bV; (7)

where a and b are constants used to describe a particular

grain. However, in addition to Eq. (7), Navarro and Noyes

(2001) also recommend the use of the following equation:

DPL RV SV2; (8)

where R and S are product-dependent constants.

For example, this equation was used for simulation of air-

flow through packed bed of grain in works of Haque et al.

(1981), Haque et al. (1982), and Hunter (1983).

Eqs. (7) and (8), i.e. two-parametermodels, present good re-

sults to simulate static pressure drop in silos when the veloc-

ity is similar in all points of a silo and if this velocity pertains

values where constants a and b (or R and S ) have been accu-

rately determined for a particular grain. When calculating

the distribution of air in large grain storage bins with signifi-cant variations in cross-sectional area, when regions of lami-

nar, turbulent and transition flow exist, these relationshipsdlnjdP=Lj 21 bVln1 bV bV

dln VdlnjdP=Lj

1 SRV1 2SR V

: (10)

Both these expressions satisfy to limiting conditions for

laminar and turbulent flow conditions, i.e.

limV/0

dln V

dlnjdP=Lj 1 and lim

V/N

dln V

dlnjdP=Lj 0:5: (11)

This means that Eqs. (7) and (8) are capable of describing the

airflow through grainmass for all flow regimes (laminar, tran-

sition and turbulent flows). Also Fig. 1, where data from Shedd

(1953) and this study are presented, indicates that the data

predicted by Eq. (9) for recommended a and b values show

a significant divergence from experimental data and excessive

dominance by the transient regime. Estimation of Reynolds

number shows that deviation from Darcys law occurs after

Re 10, and the transient regime occurs where 10< Re< 60.

-2 0 2 4 6 8 100.5

0.6

0.7

0.8

0.9

1.0

Turbulentcondition

Transientregime

Laminar-flowcondition

d(ln

V)/d

(ln

|g

radP

|)

ln|gradP|, Pa m-1

Fig. 1 Observed and predicted variation of derivative

dln V=dlnjgradPj[fgradP for airflow through soya beanmass (blue points and curves) and wheat mass (black pointsand curves):,,6, Shedds (1953) data ;-,:, authors data;

predicted by Eq. (12); - - -, predicted by Eq. (9).

and (20), describe the steady-state pressure and velocity distri-

butions in a cross-section of an aerated grain storage.

equations and tool for results three-dimensional presentation

and analysis. Since commercial tools can be costly, free-of-

charge software was used when possible.

i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8Where Re> 60 turbulent flow occurs. These results are in

agreement with the data of Wright (1968) and Bear (1988).

Therefore the use of Eq. (7) is limited, since it is very diffi-

cult to simultaneously achieve good results for both Eqs. (7)

and (9), using only two constants. The same problem concerns

Eqs. (8) and (10). Also, Eqs. (9) and (10) depend on only one con-

stant b and S/R, respectively. To improve accuracy different

values of factors a and b are usually adopted for different in-

tervals and this can be very inconvenient.

Khatchatourian and Savicki (2004) proposed the formula to

describe the variation of the derivative d(ln V)/d(ln(jdP/dyj)) forall the three flow conditions corresponding to the laminar,

turbulent and transition flows:

dln Vdlnjgrad Pj

34 arctanU

2p; (12)

where U(P) a ln(jgrad Pj) b is an intermediate argument;a> 0 and b are constants.

Evidently, when jgrad Pj/ 0, U/N, limu/N3=4

arctanU=2p 1, which corresponds to the laminar flow; andwhen jgrad Pj/N, U/N, lim

u/N3=4 arctanU=2p 0:5,

which corresponds to the turbulent flow, i.e. Eq. (12) satisfies to

limiting conditions Eq. (11).

Fig. 1 shows reasonable agreement between the curve cal-

culated by the Eq. (12) and the experimental data.

Integrating Eq. (12) in relation to the logarithm of the pres-

sure gradient gives the expression for the velocity:

ln V ln1 U2 2U arctanUp 3U4a c; (13)where c is a constant of integration.

As will be shown, this equation, depending on three con-

stants (a, b and c), describes well the experimental data in all

regions. In addition, unlike Eq. (7), explicit dependence of ve-

locity on a pressure gradient in Eq. (13) essentially simplifies

its use together with the equation of continuity for problems

formulated in two-dimensional and three-dimensional.

Finally, the mathematical model of the airflow in the par-

ticular media for the three-dimensional case consists of a sys-

tem of two equations:

div V 0; (14)

V grad Pjgrad Pj exp

ln1 U2 2U arctanUp

3U4a c: (15)The scalar equation (14) is the continuity equation for incom-

pressible fluid. The vector equation (15), which has replaced

the NavierStokes equation, shows that the velocity vector

and pressure gradient are collinear in all points of the airflow

domain and that the ratio of the absolute values of these vec-

tors is a function of the pressure gradient. Expressing the co-

efficient of proportionality k by

kexpln1U22UarctanUp3U4acjgradPj;(16)

and using Eq. (15), the velocity components u, v and w for the

three-dimensional case can be expressed in the form

b i o s y s t em s e n g i n e e r228ukvPvx

; vkvPvy

; wkvPvz; (17)3.1. Geometry construction

The geometry of the system can be constructed in any system

CAD, CAE, or any three-dimensionalmodelling software pack-

age that can export data to a standard format. In this work

Blender3D was used (http://www.blender.org). This software

is available at no-cost under General Public License (GPL). It

is three-dimensional modelling software aimed at artistic

works, but it proved to be very efficient for constructing the

three-dimensional geometry of the storage bins. A user-inter-

face was developed in Lazarus (http://sourceforge.net/pro-

jects/lazarus/) to create geometry choosing basic storage bin

dimensions.

The storage bin geometry data were exported to smash file

format, which is a format that can be read by Tetgen (http://

tetgen.berlios.de/). A Perl script (http://www.perl.org/), used

for exporting data, wasmodified to include exporting facema-

terials. Different face materials were used in order to recog-3. Software description and development

The nonlinear partial differential equation for pressure Eq.

(18) was solved by the finite-element method (Segerlind,

1976) using an iterative process to calculate the permeability

coefficient k using Eq. (16) in each point of the integration do-

main and using the pressure distribution from the immedi-

ately previous iteration step.

The software, developed in ANSI C, consisted of toolsfor geometry construction, mesh generation, generation of

system matrix, solver of obtained system of linear algebraicwhere the y coordinate in m corresponds to vertical direction,

the x and z coordinates are located in the perforated floor plan.

Substituting Eq. (17) in Eq. (14), the nonlinear partial differ-

ential equation is obtained:

v

vx

k vP

vx

vvy

k vP

vy

vvz

k vP

vz

0: (18)

The boundary conditions for the problem considered have

the form:

P Pe Dirichlet condition for air entrance and exit; (19)

n grad P 0 Neumann condition on the walls and floor

of the silo; 20

where Pe is air entrance or exit pressure in Pa; and n is unit

vector normal to the wall or floor surface.

Eqs. (16)(18) along with the boundary conditions, Eqs. (19)nise the surfaces with different bounding conditions, such

as inlets and outlets.

4. Validation of the mathematical model fornon-homogeneous conditions in a grain mass

To validate the proposed mathematical model, the empirical

relationships between air velocity and static pressure drop

were obtained for compacted layers with several grain storage

depths. The coefficients a, b and c presented in themathemat-

ical model were obtained experimentally for soya bean,

maize, rice and wheat grains. In large storage bins, due to

compaction, grain mass is a non-homogeneous medium and

the permeability coefficient varies as a function of the grain

layer depth as well as pressure gradient. Therefore the influ-

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8 2293.2. Mesh generation

For mesh generation Tetgen, available under a GPL license

was used. It generates quality tetrahedral meshes using

Delaunay algorithms. Firstly, a coarse mesh was generated.

To refine the obtainedmesh, a qualitymesh filewas generated

by the solver, then, Tetgen was used to refine the mesh

according to the parameters indicated in this file. The best re-

sults were obtained by dynamic adaptive refinement of the

mesh, based on a tetrahedron size selection in inverse propor-

tion to the tetrahedron pressure gradient. Each tetrahedron

not satisfying the user specified ratio was recursively decom-

posed into eight new tetrahedral elements according to the

method shown in Liu and Joe (1996).

3.3. Problem solving and representation

The developed code is cross-platform and can be compiled in

any ANSI C compatible compiler. The input files to thesolver software are the output files from Tetgen which de-

scribes nodes, faces and tetrahedral elements, and generates

a file describing the boundary conditions and precision re-

quirements. Firstly, the solver software generates the local

matrix for each tetrahedron applying the finite-element

method. Using the local matrix information, the global system

matrix was generated. Since the system order was large and

thematrix was very sparse, a special class was created to han-

dle the matrix, optimising memory and also optimising the

time to access the elements. Instead of using standard sparse

matrix classes, the in-house programming of the special class

enables full advantage to be taken of the system peculiarities,

optimising both memory usage and processor time. The

successive over-relaxation (SOR) method (Hageman and

Young, 1981) was used for resolving the system of linear alge-

braic equations. The developed solver was shown to have

good performance.

The software executes three iterative processes: (1) it cal-

culates the permeability coefficient in each point of the inte-

gration domain, using the pressure distribution in the

immediately previous iteration step, (2) it searches the system

design point, located in the performance curve of the aerator

fan, and (3) it adaptively refines themesh according to the tet-

rahedron size per pressure gradient ratio.

After the system is solved, an output file is generated in

VTK (Visualization toolkit, http://www.vtk.org/) format. This

file includes the nodes and tetrahedral elements. For each

node the value of pressure and for each tetrahedron the velocity

vector is exported. Paraview software (http://www.paraview.

org/), which is open source and available free-of-charge, was

used for to visualise the results.

The velocity vector for the used scheme of a finite-

element method is constant inside the simplex element

(tetrahedron). Using theory of consistent conjugate approxi-

mation (Oden and Reddy, 1973) velocities in all vertices of

the tetrahedrons were calculated, i.e. a continuous vector

field was obtained. Further, for each vertex, the full airflow

trajectory length (from inlet up to outlet) was calculated.

The received values were then used to calculate the local cri-terion introduced in this work to estimate the ventilation

system performance.ence of the grain mass compaction factor on the permeability

coefficient was investigated.

4.1. Experimental equipment

To simulate the aerated grain storage characteristics, the

equipment, described by Khatchatourian and Savicki (2004),

was used to experimentally determine the grain mass com-

paction factor caused by the weight of layers above. The grain

mass porosity varied as a function of the layer depth. The in-

fluence of compaction on the relationship between the airflow

velocity and the static pressure drop was analysed.

Fig. 2 shows the experimental equipment which consisted

of a centrifugal fan, an orifice-plate and small silo composed

of a polyvinyl chloride tube (inside diameter of 0.2 m and

height of 1 m) or a steel tube (inside diameter of 0.11 m and

height of 1 m). Tomodel the conditions at the bottomof a grain

store, a compacting device was developed with a lever, which

made it possible to apply enough force to simulate the depth

up to 50 m. In the tests, soya beans, maize, and wheat had

a moisture content of 1213% and rice had a moisture content

of 10%. Impurities were less than 2%, as determined by the

Laboratory of Seeds Analysis, Department of Agrarian Studies,

Regional University of the Northwest, Rio Grande do Sul UNI-

JUI, Brazil.

4.2. Experimental results

The experimental results, presented in Fig. 3, show the rela-

tionship between airflow velocity and static pressure drop in

the soya beans, shelled maize, rice and wheat mass. Table 1

shows the values of empirical model coefficients a, b and c,Fig. 2 Sketch of the experimental equipment.

-2

-1

0

(dP

/d

y), P

a m

-1

0.2

0.3

0.4

0.5

Velo

city, m

s

-1

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8230obtained by minimising the residual error between experi-

mental and simulated data. The simulations based on these

coefficients satisfactorily described the experimental data

(Fig. 3).

Experimental data in Fig. 4 show the significant influence

of the storage layer depth on the aerodynamic resistance of

the grain mass over the studied depths (from 1 m up to 50 m).

Fig. 5 presents a reduction of themeasured porosity factor 3

with storage depths for soya bean, maize and rice, where 3 is

the ratio of the void volume to the total bed volume. Experi-

mental porosity valuesweremeasured using a specially devel-

oped and adjusted pycnometer. The relationship between the

reduction in porosity and layer depth H can be presented as

3 4 5 6 7 8 9-4

-3

ln

lnV, m s-1

Fig. 3 Relationship between air velocity (V) in m sL1 and

air pressure drop (dP/dy) in PamL1; ,, soya bean,

coefficient of correlation R2[ 0.9954; 6, shelled maize

R2[ 0.9982; B, rice, R2[ 0.9934;>, wheat, R2[ 0.9972; d,

predicted by Eq. (13).3

30 eSH50

n

; (21)

where 30 is the porosity factor for H 1 m dimensionless. Theempirical coefficients S and n, which were obtained by least-

squares method, are presented in Table 2.

The analysis of the measurements of the porosity factor 3

for various bed depths indicated that the effective velocity in-

crease due to reductions in the porosity factor in the deepest

layers was not sufficient to explain and calculate the pressure

losses under these conditions. The values in Fig. 4, calculated

by using a porosity reduction for H 50 m, are significantlydifferent from the corresponding experimental points.

Table 1 The empirical coefficients a, b and c with 95% confidecoefficient of determination (R2) and root mean squared error (

a b

Soya bean 0.82 0.12 3.57 0.66 Maize 0.61 0.07 2.92 0.39 Rice 0.51 0.13 3.08 0.82 Wheat 0.86 0.15 5.49 0.98 However, the greater porosity did not guarantee smaller re-

sistance to airflow in the grain mass. For example, the rice in

the husk (or paddy) had a resistance greater than themaize al-

though the rice porosity was greater. It is possible that free

volumes of air between husk and the grain increased porosity

but did not increase cross-sectional area for airflow.

It must be concluded that, besides the global non-unifor-

mity defined by the alteration of the mean porosity factor

with the depth variation, there is local non-uniformity caused

by the seed form and that this does not significantly alter the

porosity factor value of themedium. Probably, the compaction

of non-spherical seeds creates local dense regions through

0 500 1000 1500 2000 2500 30000.0

0.1

dP/dy, Pa m-1

Fig. 4 Influence of bed depth (H ) on the air velocity (V) in

m sL1 as function of air pressure drop (dP/dy) in PamL1

(one-dimensional storage), shelled maize: -, H[ 1 m; ,,

H[ 10 m; C, H[ 20 m; B, H[ 30 m; :, H[ 40 m; 6,

H[ 50 m;d predicted; $, predicted by porosity reduction

for H[ 50 m; - - -, predicted by Eq. (25).which airflow is hindered.

The experimental data presented in Fig. 6 show that for the

studied velocity and depth variation intervals, the relative

pressure gradient increment C (jgrad PHj jgrad P0j)/jgrad P0jcan be considered as being independent of air velocity and de-

pends only on the storage layer depth H, where H is a distance

between the upper seed surface (free surface) and the layer

under consideration. This hypothesis was confirmed by

multi-factorial analysis of variance and by a nonparametric

association test of the Spearman rank order correlation

(Table 3).

The function C C(H ) relates to the initial pressure gradi-ent jgrad P0j, where P0 corresponds to grain depth H 1 m,and the pressure gradient jgrad PHj for considered depth H at

nce bounds for different seeds, sum squared error (SSE),standard error) for Eq. (13)

c SSE R2 RSME

2.77 0.12 0.5013 0.9954 0.04802.75 0.08 0.1304 0.9982 0.02892.23 0.13 0.3526 0.9934 0.05252.18 0.06 0.1296 0.9972 0.0348

effect of bed depth on resistance to airflow of grain for all

range of airflow with permanent values X1, X2 and X3 and

0.95

0.96

0.97

0.98

0.99

1.00

0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

C=

(g

radP

- g

radP

0)/g

radP

0

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8 231same velocity. This function, designated as the compaction

function in this work, was presented in the form

CH G1 eaH; (22)where G and a are empirical product-dependent constants

presented in Table 4 for soya bean, maize and rice.

These constants were obtained by minimising

mina;G

XMi1

G1 eaHi grad Pi grad P0

grad P0

2; (23)

where M is the total number of experimental points for se-

lected grain type.

Fig. 7 shows the variation of compaction function with

layer depth for soya bean and maize. The compaction func-

tion allowed the influence of the depth H to be included in

0 10 20 30 40 500.92

0.93

0.94

Bed Depth, m

Fig. 5 Porosity reduction with bed depth; -, soya bean;

:, maize; d, predicted; - -, 95% confidence bounds for

prediction.the model through the intermediate argument U, substituting

the pressure gradient without compaction jgrad P0j for the ex-pression jgrad PH /(1 C )j:U a lnjgrad PH =1 Cj b: (24)As a result, Eqs. (13) and (22) and with the intermediate argu-

ment in Eq. (24) relate the air velocity for the storage layer lo-

cated in the depth H, and the necessary pressure gradient.

Fig. 4 shows close agreement between observed and predicted

data.

To take account of the grain bulk density (and in implicit

form the bed depth) the ASAE Standards 2000 recommends

using the equation obtained by Bern and Charity (1975):

Table 2 Porosity factor 30 for H[ 1 m and empiricalcoefficients S and n of Eq. (21) for different seeds

Seed type 30 S n Coefficient ofdetermination (R2)

Soya bean 0.43 0.0680 0.5261 0.9985

Maize 0.44 0.0736 0.4683 0.9997

Rice 0.61 0.0664 0.5134 0.9989DPL X1 X2

rb

rk

2V

1 rb

rk

3 X3rbrk

V2

1 rb

rk

3; (25)

where DP is pressure drop, Pa; L is bed depth, m; rb is product

bulk density, kgm3; rk is product kernel density, kgm3; X1,

X2 and X3 are constants.

Eq. (25), based on the equation from Ergun (1952), repre-

sents a three-parameter model and describes the relationship

between airflow and pressure drop better than Eqs. (7) and (8).

Unfortunately, as Fig. 4 shows, it is impossible to describe the

Velocity, m s-1

Fig. 6 Variation of the compaction function

C[(jgradPHjL jgradP0j)/jgradP0j for shelled maize with beddepth H at various air velocities V in m;,, H[ 10 m; C,

H[ 20 m; B, H[ 30 m; :, H[ 40 m; 6, H[ 50 m;

d predicted.Eq. (25). The empirical coefficients X1, X2 and X3 for this simu-

lation were obtained by minimisation of the residual error be-

tween observed and simulated data, using Eq. (21) and the

relationship

rb=rk 1 3: (26)Navarro and Noyes (2001) recommended calculating the aver-

age value of the grain bulk density during filling a silo by

Table 3 Influence of bed depth H and air velocity V oncompaction factor C

Variable F-value Probability> F R2

Soya bean

Depth, m 76.6 0.002 0.958

Velocity, m s1 0.54 0.855 0.035

Shelled corn

Depth, m 221.6 0.001 0.954

Velocity, m s1 5.05 0.103 0.031

Rice

Depth, m 147.2 0.001 0.958

Velocity, m s1 4.03 0.138 0.045

means of known weight of loaded grain and the calculated

volume which will occupy this grain in a silo. Representing

the depth-static pressure of the grain using a nomograph is

another recommended way of estimating static pressure re-

quirements. The basic values for loose grain were increased

as follows: 30% for wheat, 34% formaize, and 41% for sorghum

and soya beans.

For silo aeration systems this procedure gives admissible

results. For large grain storage bins, for bins with significant

variation in cross-sectional area, and for bins using aeration

in parts, such estimations can be unacceptable.

was made for case 1 (air inlet ducts installed in the base of

the storage system). Although in this case the storage bin

has two axes of symmetry and it is possible to consider only

Table 4 The empirical coefficients G and a of compactionfunction C, Eq. (22), for different seeds

Seed type G a Coefficient ofdetermination (R2)

Standard error(RSME)

Soya bean 0.6865 0.0345 0.9943 0.0111Maize 0.8514 0.0155 0.9858 0.0176Rice 0.7652 0.0171 0.9872 0.0155

Fig. 8 Outline sketch of simulated store bin.

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8232In our view, estimating the influence of bed depth on pres-

sure increase by means of Eq. (22) is preferable to using only

one formula. Also, calculations and experiment have shown

that this dependence is very useful for inverse problem solu-

tion, when for known integral parameters of airflow (e.g. air

consumption and total pressure head) define the characteris-

tics of grain mass which influence aerodynamic resistance.

Coefficient G in Eq. (22) can be single variable parameter since

the sensitivity of coefficient a in Eq. (22) is insignificant in

comparison with sensitivity G.

To simulate the storage bins of complex layouts, the soft-

ware was operated using an iterative process which deter-

mines the equilibrium between the fan output and the

resistance of the aeration system to airflow, i.e. the operating

point of the aerator fan. The software then calculates: (1) the

pressure needed to get the required airflow rate (estimating

static pressure requirements); (2) the airflow rate, knowing

0.2

0.3

0.4

0.5

0.6

C10 20 30 40 500.0

0.1

Layer Depth, m

Fig. 7 Variation of the compaction function

C[ (jgradPHjL jgradP0j)/jgradP0j with bed depth (H ) in mfor soya bean and shelled maize: C, soya bean, observed

data; 6, shelled maize, observed data; , 95% confidence

bounds for prediction; - -, non-simultaneous bounds for

observation; , predicted.14 th of total storage, the simulation was carried out for com-

plete domain, because generally symmetry conditions do not

exist.the initial pressure; and (3) the pressure and airflow rate in

an iterative process for the chosen fan and electric motor (by

estimating system design point).

5. Numerical simulations

Fig. 8 shows the structural layout of real V-form floor storage

bin, used in the state of Rio Grande do Sul, Brazil. The storage

bin has a maximum width of 30 m and length of 95 m. Three

air inlet systems were analysed: (1) a central inlet system; (2)

a systemwith central and upper lateral inlets; and (3) a system

with central, lower lateral and upper lateral inlets. The aera-

tion simulations in storage bins, for different layouts, were

generated using the global airflow rate of Q 9 m3 h1 t1(2.5 106 m3 s1 kg1), which is the most commonly recom-mended value for aerated grain storage.

Firstly, airflow simulation in the V-form floor storage binFig. 9 Surface wireframe of the tetrahedral mesh.

Fig. 9 shows part of the computational mesh used. The grid

had a higher density in regions where the pressure gradient

was greater. For the layout under consideration the number

of tetrahedrons was approximately 500,000.

Isobaric surfaces for storage bin section with central, lower

lateral and upper lateral inlet systems are shown in Fig. 10. It

can be seen that airflow in lower storage section was three-

dimensional in character. In the upper section of the storage,

the character of the airflow approached the two-dimensional

case.

The simulation results of three aeration systems under

consideration are shown in Fig. 11. Analysis of the pressure

distribution (left column) showed that the installation of

Fig. 10 Isobaric surfaces in storage bin section with

central, lower lateral and upper lateral inlet systems.Fig. 12 Schematic model for determination of the local

specific airflow rate.

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8 233Fig. 11 Comparison of three simulated aeration systems: distr

(right column).ibution of pressure (left column) and risk regions

between the total airflow rate and the total productmass. This

Fig. 13 Visualisation of regions with inadequate ventilation (qL< 4.5) for three air inlet systems: (1) central inlet system; (2)central and upper lateral inlet systems; and (3) central, lower lateral and upper lateral inlet systems; Q[ 9 m3 tL1 hL1.

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8234criterion is suitable for simple silo designs with constant

cross-sectional area, when the air velocity is uniform through-

out the storage. If variations in the cross-sectional area are

significant, or the aeration distribution system is complex

(e.g. case 3), this criterion is not suitable.

To evaluate aeration efficiency for storage bins with vari-

able cross-sectional area and with complex air distribution

system, a local specific airflow rate is proposed. For simplelateral ducts essentially equalised the airflowwhen compared

with the same storage bin without lateral ducts and reduced

the initial pressure head. To analyse the distribution of pa-

rameters, the software was used to show the storage sections,

which satisfied certain conditions. For example, the frame-

works in Fig. 11 (right column) show only cells where velocity

is 18) for three air inlet systems: (1) centralcentral, lower lateral and upper lateral inlet systems;

Fig. 15 Distribution of local specific airflow rate, Q[ 9 m3 hL1 tL1 (2.53 10L6 m3 sL1 kgL1): (a) central inlet and (b) lower

lateral inlet.

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8 235A local criterion multiplied by aeration time has additive

properties. This allows the quality of aeration to be calcu-

lated for all parts of the storage bin even if the ventilation

is carried out separately at each of the inlets over different

periods of time.

5.2. Numerical simulation results

To visualise risk domains in the grain storage bin, the distribu-

tion of local specific airflow rates was studied (Figs. 1319).Fig. 13 shows the visualisation of domains with inadequate

ventilation (qL< 4.5) for three air inlet systems: (1) a central

Fig. 16 Distribution of local specific airflow rate, Q[ 9 m3 hL1

central, lower lateral and upper lateral inlets with identical initinlet; (2) central and upper lateral inlets; and (3) central, lower

lateral and upper lateral inlets. As simulations show, the sys-

temwith central, lower lateral and upper lateral inlets consid-

erably improved the conditions of storage in regions close to

walls when compared with other inlet systems. For all cases

considered there was an area of risk in the uppermost part

of grain mass.

The regionswith the raised intensity of ventilation (qL> 18)

are shown in Fig. 14. The results obtained show that the sec-

ond system (central and upper lateral inlets) has a smaller vol-

ume with excessive intensity of ventilation in comparisonwith others, i.e. has improved efficiency.

tL1 (2.53 10L6 m3 sL1 kgL1): (a) upper lateral inlet and (b)

ial pressures.

Fig. 17 Distribution of local specific airflow rate, Q[ 9 m3hL1 tL1 (2.53 10L6 m3 sL1 kgL1): (a) central and lower lateral

inlets with different initial pressures and (b) central, lower lateral and upper lateral inlets with different initial pressures.

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8236Additional more detailed comparative analyses of the effi-

ciency of different aeration systems were made for the same

grain storage bin with the same global specific airflow rate

Q 9 m3 h1 t1 (2.5 106 m3 s1 kg1). The number of inputs(from one up to three), their position (upper lateral, lower lat-

eral and central inlets), and ratio of pressure between various

inputs were varied. Using the additive property of the local

specific airflow rate, estimations of ventilation system efficiency

were carried out separately using each of inlets during the

different periods of time. Relationships between the durationFig. 18 Distribution of resultant local specific airflow rate with

(2.53 10L6 m3 sL1 kgL1): (a) upper lateral, lower lateral and cen

lateral, lower lateral and central inlets with different applicatioof ventilation time through each inlet and the airflow rates

were chosen so that the global specific airflow rate Q was equal

to 9m3h1 t1 (2.5 106 m3 s1 kg1). The simulation resultsfor the distribution of local specific airflow rates in a plane of

symmetry in the grain storage bin are presented in Figs. 1518.

As the simulation results presented in Figs. 15 and 16(a)

show, if only one airflow inlet is used, there is always a large

area with superfluous ventilation. Since the airflow tends to

leave the grainmass through the line of least resistance, mov-

ing the airflow inlet from the lower position to the upperseparated functioning inlets, Q[ 9 m3hL1 tL1

tral inlets with equal application times (1:1:1) and (b) upper

n times (1:2:2).

of

f si

:2)

b i o s y s t em s e n g i n e e r i n g 1 0 1 ( 2 0 0 8 ) 2 2 5 2 3 8 237position provokes a pressure reduction and a deterioration in

ventilation uniformity. Therefore, there is area with excessive

ventilation close to the upper lateral inlet if all of three inlets

operate together with identical pressure (Fig. 16(b)).

By selecting the appropriate pressures ratio for the inlets it

is possible to considerably improve the system of air distribu-

tion in the storage bin. This is demonstrated in Fig. 17(a) for

two inlets and in Fig. 17(b) for three inlets.

In high capacity storage, the grain ventilation is usually

carried out stage by stage, serially using air inlets located in

different storage sections. Under these conditions the advan-

tage of using of local specific airflow rates for ventilation effi-

ciency estimation is especially great. In these cases the

resultant local specific airflow rate qL in each point of storage

bin can be calculate by the expression

qL Pn

i1 tiqiPni1 ti

; (29)

where qi is the local specific airflow rate corresponding to ven-

tilation with only one inlet (order number i); ti is ventilation

Fig. 19 Visualisation of domains with the lowered intensity

ventilation (qL> 18; left); qL was obtained by superposition olateral and central inlets with different application times (1:2timewith only one inlet (i); n is total number of inlets; i is order

number of corresponding inlet.

For example, Fig. 18 shows the distribution of resultant lo-

cal specific airflow rates with separated operation of the upper

lateral, lower lateral or central inlets. In case (a) the applica-

tion time is the same for each inlet, and the resultant local

specific airflow rate at each point of the storage bin can be cal-

culate by

qL 13q1 13q2 13 q3; (30)

where q1, q2 and q3 are local specific airflow rates correspond-

ing to upper lateral, lower lateral or central inlets.

The simulations presented in Figs. 16(b) and 18(a) indicate

the significant advantage of ventilation carried out in turn by

each of inlets in comparison with the simultaneous use of all

inlets at equal pressures. This improvement is caused because

the capacity for air to penetrate to all zones under the domi-

nant influence of each inlet results in amore uniformdistribu-

tion of qL. By varying the duration of aeration for each inlet, itis possible to find a optimumdistribution of qL for a given stor-

age bin design.

For example, Fig. 18(b) shows the distribution of qL in the

grain storage bin with alternate use of upper lateral, lower lat-

eral or central inlets for durations of aeration varying as

t1:t2:t3 1:2:2. As results showed, this distribution had thefewest regions with insufficient or excessive aeration. These

regions are shown in Fig. 19 for whole grain storage bin.

Unfortunately, where only one inlet was used there was an

inevitable increase in head pressure.

6. Conclusions

Amathematical model of three-dimensional airflow in an aer-

ated grain storage system was developed for non-uniform

conditions of the seed mass. Experiments were conducted to

obtain the relationship between air velocity and pressure gra-

dient and the values of the porosity factors for different seed

types and different storage layer depths. A local criterion

was proposed to estimate the efficiency of complex aeration

ventilation (qL< 4.5; right) and with excessive intensity ofmulations for separated functioning of upper lateral, lower

; Q[ 9 m3hL1 tL1 (2.53 10L6 m3 sL1 kgL1).system in grain storage bins.

Software was developed to determine the velocity, pres-

sure and local specific airflow rates distributions, the global

airflow rate or initial pressure head in the grain mass store

for three-dimensional cases. The aeration system efficiency

of several stored seeds was analysed to provide the airflow

distribution uniformity and the static pressure head values

that generate the appropriate airflow rate for safe storage.

It was shown that the aeration system of grain storage bin

can be essentially improved by the use of inlets system with

different initial pressures selected for each inlet. Also, it was

shown that it is possible to optimise air distribution in a grain

storage bin by operating each inlet in turn and by selecting

a suitable aeration period for each inlet.

Acknowledgements

The authors would like to thank CNPq for the financial sup-

port for this work (process No. 464380/00-6).

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Simulation of three-dimensional airflow in grain storage binsIntroductionMathematical modelSoftware description and developmentGeometry constructionMesh generationProblem solving and representation

Validation of the mathematical model for non-homogeneous conditions in a grain massExperimental equipmentExperimental results

Numerical simulationsCriterion for describing the efficiency of aeration systemNumerical simulation results

ConclusionsAcknowledgementsReferences