2011 Wall effect on pressure drop in packed beds
ABSTRACT The wall effect on the pressure drop in packed beds could be considered by modifying the Ergun equation based on the concept of hydraulic radius. However, the prediction of the two constants involved in the modified Ergun equation, if using the correlations available in the literature, could differ significantly from one another, and all correlations are not applicable for very low bedtoparticle diameter ratios. In this study, a capillarytype model is proposed to be composed of a bundle of capillary tubes subject to a series of local energy losses, the latter being simulated in terms of sphere drag. The formulas derived provide a good description of variations in the two constants for bedtoparticle diameter ratios ranging from 1.1 to 50.5.

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Page 1
Wall effect on pressure drop in packed beds
NianSheng Cheng⁎
School of Civil Environmental Engineering, Nanyang Technological University, Singapore 639798, Singapore
a b s t r a c ta r t i c l ei n f o
Article history:
Received 3 September 2010
Received in revised form 19 March 2011
Accepted 25 March 2011
Available online xxxx
Keywords:
Packed bed
Pressure drop
Ergun equation
Wall effect
Hydraulic radius
The wall effect on the pressure drop in packed beds could be considered by modifying the Ergun equation
based on the concept of hydraulic radius. However, the prediction of the two constants involved in the
modified Ergun equation, if using the correlations available in the literature, could differ significantly from one
another, and all correlations are not applicable for very low bedtoparticle diameter ratios. In this study, a
capillarytype model is proposed to be composed of a bundle of capillary tubes subject to a series of local
energy losses, the latter being simulated in terms of sphere drag. The formulas derived provide a good
description of variations in the two constants for bedtoparticle diameter ratios ranging from 1.1 to 50.5.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
The pressure drop for flow through a packed bed of spheres can be
evaluated using the Ergun equation [1], i.e.
ΔP
L
= AE
1?ε
ε3
ðÞ2
ρνU
d2
+ BE
1?ε
ε3
ρU2
d
ð1Þ
where ΔP is the pressure drop, L is the bed length, U is the superficial
flow velocity (i.e. volumetric flow rate divided by the crosssectional
area of the bed), ρ is the fluid density, ν is the kinematic viscosity of
fluid, d is the particle diameter, ε is the average porosity, AE=150 and
BE=1.75. In terms of the porebased friction factor and Reynolds
number, Eq. (1) can be rewritten to be [2]
fE=
AE
ReE
+ BE
ð2Þ
where fEis the friction factor,
fE=
ε3dΔP
1 ? ε
ðÞρU2L
=
εd
1 ? ε
ðÞ
ΔP
ρ U=ε
ðÞ2L
ð3Þ
and ReEis the Reynolds number,
ReE=
Ud
ð
ν 1?ε
Þ=
εd
1?ε
ðÞ
U= ε
ν
:
ð4Þ
By noting that the hydraulic radius for packed spheres is R=εd/
[6(1−ε)],whichmeasurestheaveragelengthscaleofthepore,andthe
averagevelocityofflowthroughporesisUp(=U/ε),fEandReEserveas
the distinctive parameters for describing flows through the fictitious
pipe that is characterised with the section dimension εd/(1−ε) and
average velocity Up.
The Ergun equation applies only for the packed bed with negligible
wall effects. If the bed diameter D is not large in comparison to the
sphere diameter, say, D/db40, the pressure drop predicted by the
Ergun equation would differ significantly from measurements [3].
Such wall effects on the pressure drop in packed beds have received
limited attention, in spite of the fact that a great deal of studies has
been conducted for flows through porous media.
Thewalleffect is twofold.Inthecreepingflowregime,thepressure
drop may be increased due to the additional wall friction, of which the
effect is dominant in comparison to the increased porosity. On the
other hand, in turbulent flows, the pressure drop might be reduced
due to the increase in the nearwall porosity [4,5]. In other words, the
wall effect is Reynolds number dependent. An increased pressure
drop due to wall effect is usually associated with creeping flows, while
the reduced pressure drop occurs at high Reynolds numbers.
Experimental observations in this respect have been reviewed by
Eisfeld and Schnitzlein [4].
Powder Technology xxx (2011) xxx–xxx
⁎ Fax: +65 67910676.
Email address: cnscheng@ntu.edu.sg.
PTEC08250; No of Pages 6
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doi:10.1016/j.powtec.2011.03.026
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Please cite this article as: N.S. Cheng, Wall effect on pressure drop in packed beds, Powder Technol. (2011), doi:10.1016/j.
powtec.2011.03.026
Page 2
A simple wall correction approach was proposed early by Mehta
and Hawley [6] by modifying the hydraulic radius as
R =1
6
ε
1?ε
d
M
ð5Þ
where M is the modification factor,
M = 1 +2
3
1
1?ε
d
D:
ð6Þ
Using the modified hydraulic radius, Eq. (2) is revised to be
fw=
Aw
Rew
+ Bw
ð7Þ
or
fE=AwM2
ReE
+ BwM
ð8Þ
where fw=fE/M, Rew=ReE/M, and Aw and Bw are constants.
Comparison of Eqs. (2) and (8) yields that AE=AwM2and BE=BwM.
Mehta and Hawley [6] showed that Eq. (7) with Aw=150 and
Bw=1.75 fits well to experimental data for D/d=7.7–91. However,
Foumeny et al. [7] noted that Mehta and Hawley's approach
overestimated the pressure drop by over 100% at D/d=3.5. Reichelt
[8] argued that Awremained constant but Bwreduced with decreasing
D/d, which for spheres is empirically expressed as
Bw=
1:5
D=d
ðÞ2+ 0:88
"#−2
:
ð9Þ
Eisfeld and Schnitzlein [4] showed that Eq. (9) with slightly
adjusted constants performs better than several other pressure drop
correlations for D/d=1.624–250. Fand and Thinakaran [9] reported
that both Awand Bwincrease with increasing D/d in similar forms for
D/d=1.40–41.28. However, such variations reverse for very small D/
d. Fand et al. [10] observed that AwandBwdecrease with increasing D/
d for D/d=1.08–1.40. Similar experiments for very low ratios of D/d
have also been conducted by Calis et al. [11] and Cheng et al. [12] for
square packed beds. Montillet and Comiti [13] commented that both
porosity and bedtoparticle diameter ratio should be used to take
wall effects into account in relating friction factor to Reynolds
number.
Table 1 summarises various correlations available in the literature
for evaluatingAwand Bw. The formulas by Foumeny et al. [7], Raichura
[14] and Montillet and Comiti [13] are obtained by reformulating their
original correlations for very small or large Reynolds numbers in
terms of fwand Rew. Plotted in Figs. 1 and 2 are the Aw and Bw
correlations, respectively, in comparison with the experimental
results presented by Fand and Thinakaran [9] and Fand et al. [10].
Fand and his colleagues provided two sets of Awand Bw, which were
evaluated for Forchheimer and turbulent flow, respectively. Both data
are plotted in Figs. 1 and 2, showing fluctuations inherent in the two
coefficients. Eisfeld and Schnitzlein's [4] correlation is very close to
that by Reichelt [8] and thus ignored in the figures.
Fig. 1 shows that the data for D/dN1.4 could be approximated as a
constant, as suggested by Mehta and Hawley [6] and Reichelt [8],
while the increasing trend of Awwith increasing D/d is predicted by
Foumeny et al. [7] and Montillet and Comiti [13]. However, Raichura's
[14] Awprediction is much greater than the others and also the data.
Table 1
Empirical formulas proposed for evaluating Awand Bw.
InvestigatorAw
Bw
D/d
Mehta and Hawley [6]
Reichelt [8]
150
150
1.75
h
7–91
1.73–91
1:5
D=d
ðÞ2 + 0:88
D = d
i−2
1
M
Foumeny et al. [7]
130
M2
2:28 + 0:335 D = d
ðÞ
3.23–23.80
Raichura [14]
103
M2
ε
1−ε
??2
6 1−ε
ðÞ +
80
D = d
hi
2:8
M
ε
1−ε1−1:82
D=d
??2
i−2
5–50
Eisfeld and Schnitzlein [4]154
1:15
D=d
ðÞ2 + 0:87
h
1.624–250
Montillet and Comiti [13]
1000a
M2
D
d
? ?0:2
1
1−ε
12a
M
D
d
? ?0:2
3.8–14.5
where a=0.061 for dense packing and 0.050 for loose packing.
This study 185 + 17
ε
1−ε
D
D?d
??2
"#
1
M2
1:3
1−ε
ε
??1=3
+ 0:03
D
D?d
??2
"#
1
M
Fig. 1. Variation of Awwith D/d. The circles denote the data for Forchheimer flow, and
the triangles denote those for turbulent flow.
2
N.S. Cheng / Powder Technology xxx (2011) xxx–xxx
Please cite this article as: N.S. Cheng, Wall effect on pressure drop in packed beds, Powder Technol. (2011), doi:10.1016/j.
powtec.2011.03.026
Page 3
From Fig. 2, it follows that similar to the data for D/dN1.4, Reichelt [8],
Foumeny et al. [7], Raichura [14] and Montillet and Comiti [13] all
predict the increasing trend of Bwwith increasing D/d, in spite of the
different deviations from the data.
Furthermore, it should be mentioned that both Figs. 1 and 2 clearly
show that all correlations are not applicable for very small values of D/
d, say, D/db1.4. As d gradually approaches D, both Awand Bwincrease
significantly, in comparison to the opposite variation observed for D/
dN1.4.
Though the wall effect could be taken into account by modifying
the hydraulic radius, no acceptable theory or experimental correlation
has been established for evaluating Awand Bw. On the other hand, to
understand fluid mechanics implied by the Ergun equation, various
theoretical efforts, such as integrated onedimensional model (which
is concerned here) and differential threedimensional analysis, have
been developed in the literature.
For example, Niven [15] employed a simplified model of pore
conduit, which is subject to a series of expansions and contractions,
and expressed the energy loss as a sum of frictional (laminar or
turbulent) losses along the straight conduit and local losses due to the
expansions and contractions. Blick [16] modelled the constrictions as
a series of orifice plates and thus evaluated the packed bed friction as
pipe friction modified by the orificeinduced drag. This socalled
capillary–orifice model comprised of two parts, a bundle of capillary
tubes and a series of orifice plates spaced along the distance interval
equivalent to the mean pore diameter. By applying this model, the
problem of predicting pressure drops in porous media reduces to one
of determining skin friction coefficients related to the capillary tubes
and drag coefficients associated with the orifice plates. Blick's work
was further developed recently by Choi et al. [17], who evaluated the
orifice diameter and thus drag coefficient by considering two types of
simplified pore structures. Choi et al. demonstrated that the effect of
increased porosity near the wall, which is significant at high Reynolds
numbers, could be taken into account by introducing a dragbased
correction coefficient in the inertial term of the Ergun equation. Both
Blick [16] and Choi et al. [17] show that the capillary–orifice model is
very sensitive to the evaluation of the orifice diameter and its
discharge coefficient, which is generally difficult depending on how to
properly simplify the bed configuration.
This note aims to developan approachfor evaluatingAwandBwfor
a wide range of D/d. By followingthe ideas presented by Blick [16] and
Niven [15], we also assume that the pressure drop comprises two
components, the first being simulated by the fictitious pipe friction
and the other being represented by the local drag or loss. However,
the drag induced by the orifice plates or the local loss by sudden
expansions and contractions is replaced here with that by spheres.
2. Present consideration
Consider a bed of packed spheres, of which the length is L and the
crosssectional area is A. Throughthe bed, the pressuredrop is ΔP, and
its nondimensional form is expressed as the energy slope or
hydraulic gradient, S [=ΔP/(ρgL)]. Similar to ΔP, the energy slope
is assumed to have two parts. The first part Sfis related to the fictitious
pipe friction. Using the porebased velocity or average interstitial
velocity Upand hydraulic radius R, Sfis expressed as
Sf=
fU2
8gR=3f
p
4
U2
gd
p
1−ε
εd
ð10Þ
where f is the friction factor for the fictitious pipe. The second part SL
represents the local loss, which is approximated to be proportional to
sphere drag. In the control volume A×L, the number of packed
spheres is N=AL(1−ε)/(πd3/6). The total drag induced by spheres
is NFD, where FDis the drag force of a sphere. By approximating FDas
CD(πd2/4)(ρUp
associated with the local loss, SLcan be then expressed as the drag
related local energy loss per unit weight of fluid, i.e.
2/2), where CD is the equivalent drag coefficient
SL=
NFD
ALερg=3CD
4
1−ε
ε
U2
gd:
p
ð11Þ
Therefore, the total energy slope is
S = Sf+ SL=3 f + CD
ðÞ
4
1−ε
ε
U2
gd
p
ð12Þ
and the corresponding friction factor is
fE=
εdSg
1−ε
ðÞU2
p
=3
4f + CD
ðÞ:
ð13Þ
To evaluate f and CDincluded in Eq. (13), the knowledge of pipe
flow and sphere drag is applied. Although the friction function for
regular pipe flows and the drag coefficient related to isolated sphere
are generally not applicable for the packed bed model here, the
fundamental relationships in terms of dimensionless parameters
would remain for very small or large Reynolds numbers.
2.1. Friction factor for fictitious pipe
Consider two extreme cases. For the creeping flow at very low
Reynolds numbers, the pipe friction factor can be expressed as
fL=C1
Re
ð14Þ
where C1is a constant. By taking Re=4UpR/ν and R=εd/[6(1−ε)],
fL= C2
1−ε
ð
εUpd
Þν
ð15Þ
where C2is a constant.
On the other hand, if the inertial effect is dominant at high
Reynolds numbers, the friction factor is a function of the relative
roughness height, which is in the order of R/d or (1−ε)/ε. A further
approximation can be made based on the observation of flow
resistance in rough open channels. For the latter, the ratio of the
Fig. 2. Variations of Bwwith D/d. The circles denote the data for Forchheimer flow, and
the triangles denote those for turbulent flow.
3
N.S. Cheng / Powder Technology xxx (2011) xxx–xxx
Please cite this article as: N.S. Cheng, Wall effect on pressure drop in packed beds, Powder Technol. (2011), doi:10.1016/j.
powtec.2011.03.026
Page 4
mean velocity V to shear velocity u*can be empirically related to the
relative roughness height [18],
V
u?
= C3
R
ks
? ?1=6
ð16Þ
where C3 is a constant, R is the hydraulic radius and ks is the
roughness height that is in the order of the grain size d. In terms of
friction factor, Eq. (16) can be rewritten as
fH= C4
1−ε
ε
??1=3
ð17Þ
where C4 is a constant, and subscript H denotes high Reynolds
number.
2.2. Drag coefficient for spheres
In a packed bed, a sphere is confined by its identical neighbours
and thus the modified drag differs from that for an isolated sphere.
However, it is still reasonable to assume that the fundamental
relations, except for the numerical coefficients, remain at least for two
extreme regimes, i.e. creeping flow and inertiadominant flow.
In the regime of creeping flow, the drag coefficient is assumed
inversely proportional to the Reynolds number,
CDL= E1
ν
Upd
ð18Þ
where E1is a constant. If the inertial effect at high Reynolds numbers
is dominant, the drag coefficient is independent of the Reynolds
number,
CDH= E2
ð19Þ
where E2is a constant.
Furthermore, it is noted that the drag is increased for a single
particle settling in a confined tube. The previous studies on the
confined settling [18] suggested that the ratio of the reduced settling
velocity to that with no wall restriction is proportional to (D−d)/D.
Therefore, it may be assumed that the increased drag coefficient CDw
in the presence of wall is related to CDas follows,
?
CDw= E3CD
D
D−d
?m
ð20Þ
where E3is a coefficient and m is an exponent.
2.3. Formulas for evaluating Awand Bw
For the case of low Reynolds numbers, using Eqs. (15) and (18)
with CDreplaced by CDwgiven in Eq. (20), Eq. (13) can be rewritten as
fEL=3
4fL+ CDL
ðÞ = α1
ν
Upd
1−ε
ε
+ α2
ν
Upd
D
D−d
??m
ð21Þ
where α1and α2are constants. Comparing Eq. (21) with Eq. (2) for
low Reynolds number yields
AE= α1+ α2
ε
1−ε
D
D−d
??m
ð22Þ
and thus
Aw= α1+ α2
ε
1−ε
D
D−d
??m
??
1
M2:
ð23Þ
Similarly, for the case of high Reynolds numbers, using Eqs. (17)
and (19) with CDreplaced by CDwgiven in Eq. (20), Eq. (13) is
rewritten as
fEH=3
4fH+ CDH
ðÞ = β1
1−ε
ε
??1=3
+ β2
D
D−d
??m
ð24Þ
where β1and β2are constants. By comparing Eq. (24) with Eq. (2) for
high Reynolds numbers, one gets
BE= β1
1−ε
ε
??1=3
+ β2
D
D−d
??m
ð25Þ
and thus
Bw= β1
1−ε
ε
??1=3
+ β2
D
D−d
??m
??1
M:
ð26Þ
In the foregoing section, both Aw and Bw are formulated by
considering pipe friction and sphere drag for the extreme conditions.
However, it should be noted that no particular values of the relevant
frictionfactorand dragcoefficient are used in the analysis.Instead,the
five unknown parameters, α1, α2, β1, β2and m, included in Eqs. (23)
and (26) are to be calibrated using experimental data collected for
confined packed beds. This is detailed in the following section.
Fig. 3. Variation of porosity with D/d, in comparison with Eq. (27).
Fig. 4. Comparison of Eq. (23) (with α1=185, α2=17 and m=2) with data. The circles
denote the data for Forchheimer flow, and the triangles denote those for turbulent flow.
4
N.S. Cheng / Powder Technology xxx (2011) xxx–xxx
Please cite this article as: N.S. Cheng, Wall effect on pressure drop in packed beds, Powder Technol. (2011), doi:10.1016/j.
powtec.2011.03.026
Page 5
3. Comparisons
From Eqs. (23) and (26), it follows that both Awand Bwdepend on
both ε and D/d. In general, the average porosity increases with
reducing D/d though different packing configuration may lead to
porosity variation for given D/d. Fig. 3 plots various measurements of
the porosity, together with the relevant formulas presented previ
ously [7,10,17]. It is interesting to note that the measurements
reported by Calis et al. [11] and Cheng et al. [12] for square packed
beds are also consistent with the data measured for circular packed
beds.
With the data presented in Fig. 3, the following formula is
proposed to empirically describe the relationship between ε and D/d,
ε = ε−3
1
+ ε−3
2
??−1=3
ð27Þ
where ε1is an asymptote for small D/d,
ε1= 0:8
D−d
d
??0:27
ð28Þ
and ε2approximates the D/ddependence for large D/d,
ε2= 0:38 1 +
d
D−d
??1:9
??
:
ð29Þ
It should be noted that the porosity in the nearwall region, say,
D/db2 may strongly depend on channel shape and the location of the
peak porosity may also vary.
With the porosity calculated using Eq. (27), Eqs. (23) and (26) are
compared with the experimental data provided largely by Fand and
Thinakaran [9] and Fand et al. [10], as shown in Figs. 4 and 5,
respectively. Additional data points plotted in the figures are due to
Reichelt [8], Calis et al. [11] and Cheng et al. [12]. Both Calis et al. [11]
andChengetal.[12]investigatedspherespackedinsquaretubes.Calis
et al. [11] performed experiments for flows at medium Reynolds
numbers, and then extrapolated experimental results with CFD
techniques to the flows at low and high Reynolds numbers. In
comparison,Chengetal.'s[12]experimentswereconductedforawide
range of Reynolds number (=Ud/ν), which varied from 2 to 5550
covering both linear and nonlinear flow regimes. From their results,
thetwoconstants,AwandBw,arecalculated,assummarisedinTable2.
Here, the pipe diameters are calculated by considering equivalent
circular pipes with the same crosssectional areas of the square tubes,
and the correction factor M is taken to be 1+2 d/[3(1−ε)s], where s
is the side of the square cross section.
By trial and error, we get α1=185, α2=17, β1=1.3, β2=0.03
andm=2. As shownin Figs.4 and5, Eqs.(23) and(26)representwell
the experimental data in the wide range of D/d. However, further
improvement could be made if more data are available, in particular,
for small values of D/d.
4. Conclusions
In applying the Ergun equation to the evaluation of pressure drop
in packed beds of limited size, the two constants involved can be
modified based on the concept of hydraulic radius. However, the two
constants cannot be predicted correctly using the correlations
available in the literature, in particular for very low bedtoparticle
diameter ratios. In this study, a capillarytype model is assumed to
comprise a bundle of capillary tubes subject to a series of local energy
losses, which is represented by sphere drag. Using the model, the
dependence of the two constants on porosity and bedtoparticle
diameter ratio is formulated. Being compared with experimental data
of packed spheres, the formulas proposed describe the wall affected
constants well for bedtoparticle diameter ratios ranging from 1.1 to
50.5.
References
[1] S. Ergun, Fluid flow through packed columns, Chemical Engineering Progress 48
(1952) 9–94.
[2] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd ed. J.Wiley, New
York, 2002.
[3] M. Winterberg, E. Tsotsas, Impact of tubetoparticlediameter ratio on pressure
drop in packed beds, AICHE Journal 46 (5) (2000 May) 1084–1088.
[4] B. Eisfeld, K. Schnitzlein, The influence of confining walls on the pressure drop in
packed beds, Chemical Engineering Science 56 (14) (2001 Jul) 4321–4329.
[5] R. Di Felice, L.G. Gibilaro, Wall effects for the pressure drop in fixed beds, Chemical
Engineering Science 59 (14) (2004 Jul) 3037–3040.
[6] D. Mehta, M.C. Hawley, Wall effect in packed columns, Industrial & Engineering
Chemistry Process Design and Development 8 (2) (1969) 280–282.
[7] E.A. Foumeny, F. Benyahia, J.A.A. Castro, H.A. Moallemi, S. Roshani, Correlations of
pressuredrop in packedbeds taking into account the effect of confining wall,
International Journal of Heat and Mass Transfer 36 (2) (1993 Jan) 536–540.
[8] W. Reichelt, Calculation of pressuredrop in spherical and cylindrical packings for
singlephase flow, Chemie Ingenieur Technik 44 (18) (1972) 1068–1071.
[9] R.M. Fand, R. Thinakaran, The influence of the wall on flow through pipes packed
with spheres, Journal of Fluids EngineeringTransactions of the ASME 112 (1)
(1990 Mar) 84–88.
[10] R.M. Fand, M. Sundaram, M. Varahasamy, Incompressible fluidflow through pipes
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[11] H.P.A. Calis, J. Nijenhuis, B.C. Paikert, F.M. Dautzenberg, C.M. van den Bleek, CFD
modelling and experimental validation of pressure drop and flow profile in a
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(2001 Feb) 1713–1720.
[12] N.S. Cheng, Z.Y. Hao, S.K. Tan, Comparison of quadratic and power law for
nonlinear flow through porous media, Experimental Thermal and Fluid Science 32
(8) (2008 Sep) 1538–1547.
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Fig. 5. Comparison of Eq. (26) (with β1=1.3, β2=0.03 and m=2) with data. The
circles denote the data for Forchheimer flow, and the triangles denote those for
turbulent flow.
Table 2
Awand Bwfor spheres packed in square tube.
InvestigatorRatio of square
side to d
Equivalent
D/d
ε
Aw
Bw
AE
BE
Calis et al. [11]1.00
1.15
1.47
2
2
1.025
1.13
1.30
1.66
2.26
2.26
1.16
0.48
0.60
0.68
0.48
0.54
0.50
166.5
128.3
81.9
126.0
240.8
127.3
0.377
0.269
0.275
0.488
0.780
0.410
867.1
769.4
478.5
339.3
716.1
674.0
0.861
0.659
0.665
0.801
1.345
0.95Cheng et al. [12]
5
N.S. Cheng / Powder Technology xxx (2011) xxx–xxx
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powtec.2011.03.026
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Please cite this article as: N.S. Cheng, Wall effect on pressure drop in packed beds, Powder Technol. (2011), doi:10.1016/j.
powtec.2011.03.026