PreprintNumber04{15
APPLICATION OF THE ADVECTION-DISPERSION
EQUATION TO CHARACTERIZE THE HYDRODYNAMIC
REGIME IN A SUBMERGED PACKED BED REACTOR
ANT
ONIOALBUQUERQUE,AD
ERITOARA
UJO ANDERC
ILIA SOUSA
Abstrat: The hydrauli harateristis of a laboratory submerged paked bed,
lled with a volani stone, pozzuolana, have been experimentally investigated
through traer tests. Sets of essays at ow rates from 1 to 2:5 l/h in lean on-
ditions were performed. The results showed a onsiderable amount of dispersion
throughthelterasthehydrauliloadingwashanged, indiatingamultipliityof
hydrodynamistates, approahingits behaviorto plugow.
An analytialsolutionfortheadvetion-dispersionequationmodelhave beende-
veloped for a semi-innitesystem and we have onsideredan appropriate physial
boundaryondition. Anumerialsimulation usingnitediereneshemes isdone
taking into aount this partiular boundary ondition that hanges aording to
the ow rates. Proper formulation of boundary onditions for analysis of olumn
displaementsexperimentsinthelaboratoryis ritiallyimportantto theinterpre-
tationofobserveddata, aswellasforsubsequentextrapolationoftheexperimental
results to transportproblemsinthe eld.
Keywords: advetion-dispersionequation, nitedierenes, hydrauliloading.
1.Introdution
The experiments were arried out on a pilot sale paked bed (Fig. 1)
made of tubular aryli glass with 7 m internal diameter, 41 m total pak-
ing length, submerged with 3 m of water level. The lter was lled with
a homogeneous pozzuolana material with 4 mm of eetive diameter and
porosity of 0.52. Five ports have been used to ollet samples. The ow
rates were measured by a peristalti pump.
Experiments have been performed at ow rates of 1.0, 2.0 and 2.5 l/h, at
dierent arbon onentrations for a 33 m paking length. These experi-
ments will allow the studying of the hydrodynami harateristis along the
lter.
We injeted 10 ml of a traer (Blue Dextran) impulse immediately above
the liquid level being the response evaluated by measuring the absorbane at
610 nm of olleted samples at equal time periods.
Reeived May4,2004.
Figure 1. Shemati representation of the experimental apparatus
In vertial olumns, espeially if the ratio length/diameter is too large
(Bedient et al [1℄), the eets of liquid ow in the horizontal diretion x is
onsidered not important ompared with the ux in the vertial diretion z.
In these onditions, the mehanism of advetion, dispersion and exhange
reation in an isotropi and homogeneous paked bed under steady-state
onditions, are generally desribed by the well-known advetion-dispersion
equation, see for instane, Ogata and Banks [5℄, van Genuhten and Alves
[6℄, van Genuhten and Parker [7℄, Levespiel [3℄, Bedient et al [1℄,
R C
t + V
C
z
= D
2
C
z 2
; (1)
where C is the solute onentration, D is the dispersion oeÆient, V is
the average pore-water veloity, t is the time and z is the distane. The
parameter R aounts from possible interations between the hemial and
the solid phase of the soil. Here,we onsider there is no interationsbetween
the hemial and the solid phase and therefore R = 1.
We onsider a dimensionless parameter, alled Pelet number,
Pe = VL
D
; (2)
The Pelet number desribes the relative inuene of the eets ara-
terisedbyadvetion-dispersionproblemswhihinvolveanon-dissipativeom-
ponent and a dissipative omponent. The Pelet number also determines
the nature of the problem, that is, the Pelet number is low for dispersion-
dominated problems and is large for advetive dominated problems.
2.The model problem
Our interest is in the solution of
C
t +V
C
z
= D
2
C
z 2
(3)
for t > 0;z 0 with an initial ondition
C(z;0)= f(z) (4)
and subjet to the boundary onditions
lim
z!1
C(z;t)= 0 and C(0;t)= g(t); t 0. (5)
Theexatsolutionoftheproblem(3)-(5)anbe foundusingLaplaeTrans-
forms in t and we will get the solution
C(z;t) = 1
p
Z
t
0
g(t ^)G
(z;^)d^ + 1
p
Z
+1
Vt z
2 p
Dt
f(z V t+2 p
Dt )e
2
d
1
p
Z
+1
Vt+z
2 p
Dt
f( z Vt+2 p
Dt )e Vz=D
e
2
d; (6)
where the funtion G
(z;^) is given by
G
(z;^) =
z
2 p
D^ 3=2
e
(z V^) 2
=4D^
.
Forourpartiularproblem theinitial onditionisgivenby f(z) = 0andwe
need to determine the boundary ondition, g(t), whih represents the solute
onentration on the inow boundary.
We have the following physial parameters: V
inj
denotes the volume of
injeted traer; V
sl
is the volume of the liquid on the top of the paked bed;
M
0
is the mass injeted; C
sl
is the onentrationof the liquid level where the
traer is absorbed before going into the paked bed through the media top
We have that
C
sl
=
M
s
V
inj +V
sl
(7)
and the physial boundary ondition is given by the following exponential
deay
g(t) = C
sl e
Qt=V
sl
: (8)
This onditionis obtainedonsidering that the inow onentrationis gov-
erned by the dierential equation,
dg
dt
= Q
V
sl
g with g(0) = C
sl
(9)
whih desribes the inow deay by a rate of Q=V
sl .
Note that for our spei ase where the initial ondition is given by
C(z;0) = 0 and the inow is governed by (8) we have the analytial so-
lution
C(z;t) = 1
p
Z
t
0
g(t ^)G
(z;^)d^: (10)
3.Numerial solution using a nite dierene sheme
To derive a nite dierene sheme we suppose there are approximations
U n
:= fU n
j
g to the values C(x
j
;t
n
) at the mesh points
x
j
= jx; j = 0;1;2;::::
If we hoose a uniform spae step x and time step t, there are two di-
mensionlessquantitiesveryimportantinthepropertiesofanumerialsheme
=
Dt
(x) 2
; =
Vt
x :
The quantity is usually alled the Courant (or CFL) number.
We use the usual entral, bakward and seond dierene operators,
0 U
j :=
1
2 (U
j+1 U
j 1
); U
j := U
j U
j 1
; and Æ 2
U
j := U
j+1 2U
j +U
j 1
to desribe the nite dierene sheme.
Consider the approximation formula
U n+1
j
= [1
0 +(
1
2
+)Æ 2
+(
1
2
)Æ 2
℄U
n
j
: (11)
This sheme was rst proposed by Leonard [2℄ using ontrol volume argu-
ments. However, it an also be obtained using a ubi expansion by interpo-
lating U n
j 2
as well as U n
j 1 , U
n
j
and U n
j+1
, as we an see in Morton and Sobey
[4℄.
The model problem we are interested in is dened on the half-line with an
inow boundary ondition
C(0;t)= g(t); (12)
where g(t) is dened by (8). Consequently we onsider
U n
0
= g(nt): (13)
The sheme (11)is ahigher order sheme and it uses twopoints upstream.
Therefore it an not be applied on the rst interior point of the mesh. At
this partiular point we need to apply a numerial boundary ondition. To
determine the numerial boundary ondition we use for interpolation the
pointsU n
0 , U
n
1 , U
n
2
and U n
3
and we bring in a forward third dierene instead
of a bakward third order dierene to yield
U n+1
1
= [1
0 +(
1
2
2
+)Æ 2
+( 1
6
2
6
)Æ 2
+
℄U n
1
; (14)
where
+
is the forward operator dened by
+ U
j
:= U
j+1 U
j
. For more
informationonthis andother numerialboundaryonditions seefor instane
Sousa and Sobey [8℄.
Theuseofthisdownwindthirddierene doesnotaetauraysinestill
based on a ubi loal approximation. However,it does have some penalties
interms of stability. Some more interesting disussions ould be done on the
right hoie of the numerialboundaryonditionwhih is independent of the
physial boundary ondition (13).
4.Numerial results versus experimental results
In this setion we present the numerial results that adjust the essays for
three dierent ow rates.
Table1showsthevaluesofdierent parametersneessarytothe evaluation
ofthe inowboundaryonditiondenedby(8). Weanobservethatwe have
Q (l/h) V
inj
(ml) V
sl
(ml) M
s
(mg) C
sl
(mg/l)
1 10 112 32.5 267.26
2 10 112 32.5 267.26
2.5 10 112 32.5 267.26
Table 1. Parameters related to the determination of the physial inow
boundary ondition.
0 20 40 60 80 100 120
0 10 20 30 40 50
t
C
Q=1 Q=2.5 Q=2
0 20 40 60 80 100 120
0 10 20 30 40 50
t
C
Q=1 Q=2.5 Q=2
(a) (b)
Figure 2. (a) Experimental results for ow rates Q = 1;2;2:5;
(b) Numerial simulation for ow rates Q = 1;2;2:5.
0 20 40 60 80 100 120
0 10 20 30 40 50
t
C
0 20 40 60 80 100 120
0 10 20 30 40 50
t
C
0 20 40 60 80 100 120
0 10 20 30 40 50
t
C
(a) (b) ()
Figure 3. The same as Fig. 2 but with the numerial results
and experimental results in the same gure:
(a) Q = 1: V = 0:00828, D=VL = 0:065, Pe = 15:3
(b) Q = 2: V = 0:01440, D=VL = 0:056, Pe = 17:8
We show, in Fig. 2 and Fig. 3, the experimental results and the numer-
ial simulations for dierent ow rates. The numerial results allow us to
determine the Pelet number, that is helpful in the haraterization of the
hydrauli onditions.
5.Conlusion
The results lead us to onlude that, aording to the range of hydrauli
loading applied, a large amount of diusion ours in the lter bed. This
ourrene is assoiated to the likely ombination of fators suh as dead
zones, immobile zones, short-iruiting and diusion (both mehanial dis-
persion and moleular diusion).
The analytial solution represented by (6) for the semi-innitive system
an aurately predit the experimental urves and may be applied to re-
sults from nite experiments as the one here mentioned. To the numerial
simulation we use a numerial sheme quite appropriated sine when we
have signiant values of diusion we need a larger stability region, that is,
we need the method to onverge to the analyti solution in a region where
we an have great auray and at the same time we are allowed to have
signiant diusion.
More experiments are in progress onsidering dierent organi loadings at
dierent hydrauli loadings.
Referenes
[1℄ P.Bedient,H.Rifai,C.Newell(1999)Groundwaterontamination{transportandremediation.
2nd edition,Prentie HallPTR,New Jersey, USA.
[2℄ B.P.Leonard(1979), Astableandaurateonvetivemodelingproedure,Computer Methods
in Applied Mehanisand Engineering19 59-98.
[3℄ O. Levenspiel (1986) The Chemial Reator Omnibook, O.S.U., Book Store In, New York,
USA.
[4℄ K.W. Morton and I.J. Sobey (1993), Disretisation of a onvetion-diusion equation, IMA
Journal of Numerial Analysis 13141-160.
[5℄ A. Ogata, R.Banks(1961), A solutionof theDierential equation ofLongitudinalDispersion
inPorous Media,U.S. Geol.Survey, Paper411-A, 7pp.
[6℄ M.vanGenuhten,J.Parker(1984),Boundaryonditionsfordisplaementexperimentsthrough
shortlaboratorysoilolumns. J.Soil Si.So. Ame., 48 , 4,703-708.
[7℄ M. van Genuhten, W. Alves (1982), Analytial solutions of the one-dimensional onvetive-
dispersive solute transport equation.TehnialBulletin N 1661.Agriultural Researh Servie.
USDARiverside,California. USA,149pp.
[8℄ E. Sousa and I.J. Sobey (2002), On the inuene of numerial boundary onditions, Applied
Numerial Mathematis41 325-344.
Ant
onioAlbuquerque
DepartamentodeEngenhariaCivileArquitetura,UniversidadedaBeiraInterior,Por-
tugal
Ad
erito Ara
ujo
Departamento de
Matem
atia, Universidadede Coimbra,Portugal
Er
lia Sousa
Departamento deMatem
atia, Universidadede Coimbra,Portugal
