Boško P. Cvetković
PhD Student
University of Belgrade
Faculty of Mechanical Engineering
Dragoslav S. Kuzmanović
Full Professor
University of Belgrade
Faculty of Traffic and Transport Engineering
Predrag A. Cvetković
Full Professor
University of Belgrade
Faculty of Traffic and Transport Engineering
An Application of Continuum Theory
on the Case of One-Dimensional
Sedimentation
The fundamental balance laws of the fluid phase and the disperse phase of
suspension are presented. Constitutive equations are developed and the
method of Lagrangian multipliers is used. The objective of this paper is to
use the continuum theory of two-phase flow to model the one-dimensional
sedimentation of particle through a fluid in the direction of vertical axis.
After using the basic equations of motion and corresponding restrictions,
the diffusion equation of quasi-steady sedimentation is derived. The fact
that the diffusion equation is parabolic, when a diffusivity term is included,
suggests the possibility that sedimentation problem could be solved
numerically and then graphically presented.
Keywords: continuum theory, suspension, constitutive equations, volume
distribution function, free energy, drag coefficient.
1. INTRODUCTION
The sedimentation of particles in liquid is of interest in
many chemical engineering processes. The variables
associated with the liquid and the particles can be
introduced as continuous field from the beginning, and
balance equations and constitutive relations can be
postulated [1-5]. In paper [6] the equations for twophase flow are used to analyze the one-dimensional
sedimentation of solid particles in a stationary container
of liquid. A derivation of the equations of motion is
presented upon Hamilton’s externed variation principle.
The same result was obtained using the continuum
theory for suspension flow presented in [7].
In paper [7-10] the continuum theory for suspension
flow is developed. Fluid, the basic phase of suspension,
and the disperse phase are constituents of the mixture.
Using the basic balance laws of continuum mechanics and
the method of Lagrangian multipliers, a set of constitutive
equations for saturated suspension is deduced.
In this paper, the basic equations of motion of
different phases are derived. The model of onedimensional sedimentation of disperse phase is
considered. Using the basic equations of motion and
corresponding restrictions, the diffusion equation of
quasi-steady sedimentation is derived. This result is the
same as in paper [6].
2. BASIC LAWS OF MOTION
The basic phase (fluid) and the disperse phase
(constituents) of suspension must satisfy the basic laws
of motion of continuum mechanics [7]. In the absence
of chemical reaction between the phases and various
constituents, the following field equations must hold:
Conservation of mass
Received: March 2011, Accepted: May 2011
Correspondence to: Boško Cvetković
Faculty of Mechanical Engineering,
Kraljice Marije 16, 11120 Belgrade 35, Serbia
E-mail: [email protected]
© Faculty of Mechanical Engineering, Belgrade. All rights reserved
(
)
∂ρ r
+ ρ rυir = 0 ,
,i
∂t
(1)
where r = f or α for fluid and α-th constituent of
disperse phase of suspension.
Balance of momentum
ρr
dυir
= tijr , j + ρ r fir + Pir ,
dt
(2)
where tijr , fir and Pir are stress tensor, body force per
unit mass and internal body force, respectively.
Balance of moment of momentum
t rjk = 0 .
[ ]
(3)
Balance of equilibrated force
ρr
(
)
d r r
k n = hir,i + ρ r l r + gˆ r ,
dt
(4)
where k r , n r , hir , l r and gˆ r are equilibrated force,
the volume distribution function (concentration),
equilibrated stress vector, equilibrated force per unit
mass and internal equilibrated force, respectively.
Balance of energy
ρ r er = tklr υlr,k + hkr n,rk + qkr ,k + ρ r r r − gˆ r n r − Pkr ukr , (5)
where qkr and r r are heat flux vector and internal heat
source per unit mass.
The following entropy inequality is postulated
⎛qf ⎞
ρfrf
k ⎟
−
+
f
⎜θ f ⎟
θ
⎝
⎠, k
ρ f η f − ⎜
N ⎡
⎛ qα
+ ∑ ⎢ ρ αηα − ⎜ k
⎜ θα
⎢
α =1 ⎣
⎝
⎤
⎞
ρ rrr ⎥
≥0,
⎟ −
α ⎥
⎟
⎠, k θ ⎦
FME Transactions (2011) 39, 61-65
(6)
61
where η r is entropy density per unit mass, θ f and θ α
is temperature for fluid and α-th constituent of disperse
phase of suspension.
Using the free energies of phases
ψ r = e r − η rθ r ,
(6a)
the constrains and the method of Lagrangian multipliers
λ for the case of fully saturated suspension flows, the
entropy inequality is used in the form
⎡
1 ⎢ f
− ρ ψ f + η f θ
f ⎢
θ
⎣
(
f
)
+
qkf θ,kf
θ
f
+λ f n f υkf,k − gˆ f − λθ
f
(
1
1
1
+ tklf dlkf + hkf n,kf +
where, as well known quantities, Kα are the drag
coefficients of α-th particulate constituent, αr are positive
material constants, πr are the thermodynamic pressures
and Gr are equilibrated drag coefficients and λ = λθ .
4. GENERAL EQUATIONS OF MOTION
Direct substitution of the constitutive equations for the
stress vectors, equilibrated stress vectors, the internal
forces and the internal equilibrated forces of the basic
phase and the disperse phase into the equations of
balance (1) to (4), yield the general equations of motion:
Momentum of the basic phase
)
− λ f n f −
ρf
⎤
− Pkf + λθ f n,kf ukf ⎥ +
⎥⎦
(
)
(
α
ρα
)
+ hkα n,αk + λ α nα υkα,k − gˆ α − λθ α − λ α nα −
1
1
1
⎤
− Pkα + λθ α n,αk uαk ⎥ ≥ 0 .
⎥⎦
(
)
(7)
{ } and { Aα } be the set of independent
constitutive variables of the basic and disperse phases of
suspension. The following independent variables are
postulated
{ A f } ≡ {ρ f ; n f ; n,if ; n f ; n,if ; dijf ;θ f ;θ,if }
{ Aα } ≡ {ρα ; nα ; n,αi ; nα ; n,αi ; dijα ;θ α ;θ,αi } .
)
( )
= − π r + λ r n r δ kl − 2α r n r n,rk n,rl ,
Pkf =
= 2α
r
(n )
r
n,rk ,
∑ K α (υkα − υkf ) − λ n,kf ,
)
Pkα = K α υkf − υkα − λ n,αk ,
gˆ r = −a0r n r − n r
d ⎛α
⎜
dn r ⎜⎝ n r
(10)
(11)
62 ▪ VOL. 39, No 2, 2011
(14)
(
(
)
)
(15)
Equilibrated force of the basic phase and the
disperse phase
(
) (
) (
)
d r r
k n = 2 α r n,rk
+ ξυr n,rk + ρ r l r −
,k
,k
dt
r ⎞
⎛
d α
−a0r n r − n r ⎜
⎟ n,rk n,rk + λ + λ r − G r n r . (16)
dt ⎜⎝ n r ⎟⎠
Independent variation of variables n r , n,rk and ρ r
(12)
(16a)
ξυr ≡ 0 , G r ≡ 0 ,
(16b)
then (16) may be solved for Lagrangian multipliers, i.e.
⎛ dα r α r ⎞ r r
+
⎟ n n − λ + a0r n r . (17)
⎜ dn r n r ⎟ ,k ,k
⎝
⎠
λ r = −2α r n,rkk − ⎜
For Lagrangian multipliers λr we find λr = πr/nr.
If the thermodynamic pressures πf and πα are
replaced by λfnf and λαnα in (14) and (15), then, by
using (17), these equations become
)
dυi f
dt
f
+ n,kf n,kl
+nf
(13)
kr ≡ 0 , lr ≡ 0 ,
and also
ρ0f n f
⎞ r r
⎟ n,k n,k +
⎟
⎠
+πr + λ + λ r − G r n r ,
+
dυiα
= −πα,l − 2 α α n,αk n,αl
+
,k
dt
(9)
α =1
r
)
(8)
N
(
,k
appear linearly in (16) and then their coefficients can be
neglected. Then it follows that
Using the principle of phase separation by Drew and
Segel [2], after lengthy calculation, we find the
following set of constitutive equations for the stress
tensors, equilibrated stress vectors, the internal forces
and the internal equilibrated forces of the basic phase
and disperse phase of suspension, i.e.
r
E hk
α
ρr
Let the sets A f
(
(
)
+ ∑ K α υkf − υkα − λ n,αk + ρ α fiα .
3. CONSTITUTIVE EQUATIONS
r
E tkl
= −π,fl − 2 α f n,kf n,lf
Momentum of the disperse phase
)
(
dt
+ ∑ K α υkα − υkf − λ n,kf + ρ f fi f .
qα θ α α α
1 ⎡ α α
⎢ − ρ ψ + η α θα + k ,k + tkl
dlk +
α
θα
α =1 θ ⎢⎣
N
+∑
(
dυif
f
= 2α f n f n,kkl
+ 2n f
d 2α f
n,kf n,kf n,lf
2
f
dα f
dn
f
(n
+
− 2a0f n f n,lf + λ,l n f +
( )
+ ∑ K α (υlα − υl f ) + ρ0f n f bl f ,
α
d n
f f
,l n,kk
(18)
FME Transactions
ρ0α nα
dυiα
dα α α α
= 2α α nα n,αkkl + 2nα
n,l n,kk +
dt
dnα
Neglecting derivatives of higher order and using the
total time derivatives for velocities, equations of
momentum in one-dimensional form are
(
)
+ n,αk n,αkl + nα
d 2α α
α α α
− 2a0α nα n,αl + λ,l nα +
n,k n,k n,l
α 2
( )
+ K α (υl f − υlα ) + ρ0α nα blα .
⎛ ∂υ f
ρ0f ⎜
d n
⎜ ∂t
⎝
(19)
∂υif ⎞
∂n f
⎟ = ∇λ − 2a0f
+
∂x ⎟
∂x
⎠
+ υkf
i
+∑
Kα
5. THE ONE-DIMENSIONAL SEDIMENTATION
⎛ ∂υiα
ρ0α ⎜
We apply the general equations of motion (18) and (19)
to suspension flow. For the reason of simplicity, we
suppose that the material constants α r and a0r are
independent of the fluid volume distribution function
and volume distribution function of α-th particulate
constituent. In this case (Fig. 1), the equations of motion
become:
Mass
(
)
∂n r
+ ∇ n rυir = 0 .
∂t
dυif
dt
dt
n0
∂υiα
∂x
(υ
f
l
f
0 bl
,
(24)
⎞
∂nα
+
⎟ = ∇λ − 2a0α
⎟
∂x
⎠
)
− υlα + ρ0α bl .
(25)
The linearized form of the equations of mass
conservations (20) are given by
∂n r
+ n0r ∇υir = 0 .
∂t
(26)
Summing (26) for all species and using the
α
2 f
= ∇λ + 2α ∇∇ n
f
Kα
f
α n0
ρ0
α
f
l
saturation condition given by n f = 1 − n ( n = ∑ nα ),
+∑
α
α dυi
Kα
+
(20)
Momentum
ρ0f
+ υkα
⎜ ∂t
⎝
(υα − υ ) + ρ
l
f
α n0
(υ
α
l
− υl
)
f
− 2a0f ∇n f
+ ρ0f bl
we find
+
,
⎛
∇ ⎜ n0f υi f + ∑ n0α υiα
⎜
α
⎝
(21)
2 α
α
α
= ∇λ + 2α ∇∇ n − 2a0 ∇n +
+
K
n0α
(27)
from where it is
α
α
⎞
⎟⎟ = 0 ,
⎠
(υ
l
f
)
− υlα + ρ0α bl ,
υif = −
(22)
where it is assumed that the body force accelerations are
the same, i.e. bl f = blα = bl .
If we keep a record, as pointed out in the
introduction that the one-dimensional sedimentation in
the x direction will be considered, then the equations of
masses are given by
(
)
∂n r ∂ r r
+
n υi = 0 .
∂t ∂x
(23)
1
∑ n0αυiα ,
(28)
n0f α
where integration constant C can be neglected.
Eliminating ∇λ between (24) and (25) and using
(28), we find
∂υ α
∂υ α
ρ0α i + ρ0α υkα i +
∂t
−
ρ0f
(n )
f
0
∂x
nα υ α n β
2∑ 0 i ∑ 0
α
β
ρ0f
∑ n0α
n0f α
∂υiβ
∂x
+ 2∑ a0α
α
Kα ⎛ f α
⎜ n υ + ∑ n0β υiβ
α f ⎜ 0 i
α n0 n0 ⎝
β
= −2∑
(
∂υiα
−
∂t
)
∂nα
=
∂x
⎞
⎟+
⎟
⎠
+ ρ0f − ρ0α bi .
(29)
The linearized form of (29) and (23) is
ρ0α
∂υiα ρ0
∂υ α
∂nα
+
=
n0α i + 2∑ a0α
∑
∂t
∂t
∂x
n0f α
α
f
Kα ⎛ f α
⎜ n υ + ∑ n0β υiβ
α f ⎜ 0 i
n
n
α 0 0 ⎝
β
= −2∑
Figure 1. Sedimentation configuration
FME Transactions
(
)
+ ρ0f − ρ0α bi ,
⎞
⎟+
⎟
⎠
(30)
VOL. 39, No 2, 2011 ▪ 63
Particle concentration profiles
∂υ α
∂nα
+ n0α i = 0 .
∂t
∂t
0.18
(31)
0.16
0.14
Eliminating υiα between (30) and (31), yields the
α
ρ0α ∂ 2 nα
n0α ∂t 2
= 2∑
α
−
ρ0f
∑
n0f ∂ 2 nα
n0f α n0α ∂t 2
K
α
( n0 )
+ 2∑ a0α
α
α
α 2
α
∂ 2 nα
∂x 2
=
α
∂ 2 nα
∂x 2
0
K α ∂nα
Kα
∂n β
.
a0
=
+
∑
2 ∂t
∂x 2
n0α n0f β ∂t
n0α
2
α ∂ n
( )
(34)
∂n β
= 0,
β ∂t
(34a)
(35)
the same as in paper [6], where the diffusion
coefficient
2
,
(36)
out of which we can conclude that this equation is
parabolic.
The fact that (34) is parabolic suggests the
possibility that certain classes of sedimentation
problems could be solved.
A brief illustration of the theory for the particle
concentration and the diffusion coefficient, using (34)
and (35), yields.
Figure 2 shows the measured particle concentration
as a function of height in to the tube after 20,000 s of
elapsed time, using values from literature of a0α = 15
kg/m2, K α = 3.1 · 107 kg/m3s, and the diffusion
coefficient β in boundaries from 10–7 (data 1) to 10–10
m2/s (data 4).
Figure 3 shows the diffusion coefficient as a
function of the local particle concentration using values
64 ▪ VOL. 39, No 2, 2011
0.3
0.4
0.5
0.6
Particle concentration
0.7
0.8
0.9
1
-7
5
Diffusion coefficient profiles
x 10
4.5
4
3.5
3
2.5
2
0
-0.2
0
0.2
0.4
0.6
0.8
Local particle concentration
1
1.2
Figure 3. Diffusion particle coefficient
∂nα
∂ 2 nα
,
=β
∂t
∂x 2
of a0α and K α in (35).
0.2
1
we obtain the diffusion equation
Kα
0.1
0.5
∑
0
0
1.5
For the case of simple suspension [3]
nα
α ( 0 )
β =a
data4
Figure 2. Particle concentration profiles
( )
i.e.
data3
0.02
K α ∂nα
Kα
∂n β
, (33)
+∑
∑
2 ∂t
α f
α nα
α n0 n0 β ∂t
0
=∑
data1
data2
(32)
Then, if the acceleration terms in (29) are neglected
(the case of quasi-steady sedimentation), (32) becomes
the generalized diffusion equation
∑ a0α
0.08
0.04
β
∂n
K
∂n
.
+ 2∑
α f ∑ ∂t
∂t
α n0 n0 β
0.1
0.06
Diffusion coefficient
−
Height [cm]
single equation for n , i.e.
0.12
6. CONCLUSION
In this paper, the continuum theory has been applied to
the case of two-phase (suspension) flow. The
suspension is defined as a mixture of the basic phase
(fluid) and the disperse phase (rigid particles). The basic
equations of motion of different phases are derived as
the model of one-dimensional sedimentation of rigid
particles through the fluid in the direction of vertical
axis. After using the basic equations of motion, the
diffusion equation of quasi-steady sedimentation is
derived and then graphically presented. The analysis of
the obtained results shows that the suspension
concentration distribution of rigid particles depends
upon the diffusion coefficients. The diffusion equation
is the same as in the paper by Hill and Bedford but
derived using Hamilton’s externed variation principle.
REFERENCES
[1] Bedford, A. and Hill, C.D.: A mixture theory for
particulate sedimentation with diffusivity, AIChE
Journal, Vol. 23, No. 3, pp. 403-404, 1977.
[2] Drew, D. and Segel, L.: Averaged equations for
two-phase flows, Studies in Applied Mathematics,
Vol. 1, No. 2, pp. 205-231, 1971.
[3] Ahmadi, G.: A generalized continuum theory for
multiphase suspension flows, International Journal
FME Transactions
[4]
[5]
[6]
[7]
[8]
[9]
of Engineering Science, Vol. 23, No. 1, pp. 1-25,
1985.
Drew, D.: Mathematical modeling of two-phase
flow, Annual Review of Fluid Mechanics, Vol. 15,
No. 3, pp. 261-291, 1983.
Hill, C.D. and Bedford, A.: Stability of the equations
for particulate sedimentation, The Physics of Fluid,
Vol. 22, No. 2, pp. 1252-1254, 1979.
Hill, C.D., Bedford, A. and Drumheller, D.S.: An
application of mixture theory to particulate
sedimentation, Journal of Applied Mechanics, Vol.
47, No. 2, pp. 261-265, 1980.
Cvetković, P.: The continuum theory applied of
suspension flow, in: Proceedings of the XVII
Yugoslav Meeting of Theoretical and Applied
Mechanics, 02-09.06.1986, Zadar, Croatia, pp. 125128.
Bedford, A. and Drumheller, D.S.: Theories of
immiscible and structured mixtures, International
Journal of Engineering Science, Vol. 21, No. 8, pp.
863-960, 1983.
Jarić, J., Cvetković, P., Golubović, Z. and
Kuzmanović, D.: Advances in Continuum
Machanics, Monographycal booklets in applied &
computer mathematics, MV-25/PAMM, Budapest,
2002.
FME Transactions
[10] Hutter, K. and Schneider, Lukas: Important aspects
in the formulation of solid-fluid debris-flow
models. Part II. Constitutive modelling, Continuum
Mechanics and Thermodynamics, Vol. 22, No. 5,
pp. 391-411, 2010.
О ПРИМЕНИ ТЕОРИЈЕ КОНТИНУУМА НА
СЛУЧАЈ ЈЕДНОДИМЕНЗИЈСКЕ
СЕДИМЕНТАЦИЈЕ
Бошко П. Цветковић, Драгослав С. Кузмановић,
Предраг А. Цветковић
У раду примењена је теорија континуума на кретање
суспензија. За дисперзну фазу узети су састојци
мешавине, који су расподељени у основној фази,
флуиду. Користећи основне законе баланса
механике континуума и Лагранжеве множиоце,
изведене су конститутивне једначине засићене
суспензије. Добијене конститутивне једначине се
користе за извођење опште једначине кретања
основне и α-ог састојка дисперзне фазе у правцу
вертикалне осе. Користећи опште једначине кретања
и одговарајућа ограничења, изводи се једначина
дифузије за случај квазистатичке седиментације која
се и графички представља.
VOL. 39, No 2, 2011 ▪ 65