N.Daribayeva,
A.Serikbayeva,
Kazakh National
Research Technical University named after K. Satbayev, Kazakhstan, Almaty
Hydrodynamic
modeling of a rotating filtration process
Abstract.Rotative filtration is an
innovative tangential filtration process that allows
concentration or
clarification of suspensions at lower cost. A rotative filter is made of two
coaxial cylinders, an outer fixed cylinder and an inner porous rotating
cylinder. A suspension is injected at the top of the device and flows out at
the bottom. During the descent, part of the liquid is evacuated radially
through the inner porous wall and the
particle concentration increases. This system is already use for milk skim separation, in fermentation
processes and separation of oil, biological suspension, blood plasma, and even
for waste water treatment in spatial stations.[1]
The
main advantage of this technology is that the presence of Taylor vortices that
appear in the liquid flow if the inner cylinder rotates at sufficiently high
velocity. This secondary flow, which results from a centrifugal instability may
delay significantly the fouling of the filter, since it keeps the particles in
motion, away from the filtering wall. As a consequence, a smaller filtering
surface is necessary to attain the same performances as a static filter. From a
hydrodynamic point of view, the flow in a rotating filter can be seen as the
superposition of a circular Couette flow V(r), an annular Poiseuille flow and W(r) a radial flow U(r).
The objectives of this work is to study the
stability of CouettePoiseuilleradial flow and in particular the appearance of
Taylor vortices. To achieve these
goals, a triple approach, theoretical, numerical and experimental, will be
performed. An overview of research work was made in France, Nancy in Lorraine
University (ENSIC) in laboratory of Gemico under the direction of professors Cherif Nouar and CecileLemaitre.
Keywords: NonNewtonian
fluid, circular Couette flow, shearthinning fluid,
stability analysis.
A TaylorCouette flow, the flow between a
rotating inner cylinder and stationary outer cylinder becomes unstable when the
rotation speed of the inner cylinder exceeds a given critical value, and this
because of the centrifugal force. However, the stability is found to be
impaired if one of these two cylinders is porous and a radial flow is imposed.
The study of the linear stability for the axisymmetric flows shows that
a flow stream can be stabilized by: a
radial flow from the outside inside and a strong radial flow, but from the
inner to the outside.
On the
contrary, a small radial flow of the inner cylinder to the outer cylinder can
slightly destabilize the system. All physical phenomena inherent to the effects
of stabilization or disturbance of the radial flow are not yet fully defined.
It is therefore the superposition of a radial flow a TaylorCouette flow
between two rollers which modifies the linear system stability.
Figure
1. (a) Scheme of a rotative filter [2]. (b) In this filter, liquid flows in the
axial, azimuthal and radial directions [2]
We consider the flow of
a purely viscous nonNewtonian fluid between two infinitely long concentric
cylinders. The inner cylinder of radius _{}is rotating with a constant angular velocity _{}> 0. The outer
cylinder of radius _{}is rotating with a constant angular velocity_{}, with _{} > 0
for corotating cylinders and _{}< 0 for counter
rotating cylinders. The conservation of mass for
an incompressible fluid reads:
_{}
(1.1)
And the
conservation of momentum reads:
_{}
(1.2)
Here _{}s the velocity, _{} is the pressure
and _{}is the deviactoric extrastress tensor.
The
stress tensor is expressed as a function of the _{}
_{}
The
shear rate tensor is defined with the gradient of velocity.
_{}
The
associated boundary conditions reads:
_{}
_{}
_{} is a positive constant
To be able to easily compare our results with
those produced already in the literature, we
nondimensionalize
the equations by using the following reference scales:

_{} a
reference viscosity

_{} the
reference length scale

_{}
velocity scale

_{} time
scale

_{}the quantity for stresses and pressure scale
So
that_{}; _{} ; dimensionless _{}; _{}; _{};
The
quantities denoted with a hat (ˆ.) are dimensional while quantities
without a hat are nondimensional.
The
conservation equations thus become
_{}
_{}
(1.3)
The
non dimensional corresponding boundary conditions read:
_{}
Where
the following nondimensional parameters have appeared
_{} the
radius ration; _{}the Reynolds number Re;
_{} ; _{}
In this work, we consider only the case of
shearthinning fluids, i.e., fluids for which the effective viscosity _{} decreases as
the shear rate increases. For the numerical computation, we consider
rheological model of Carreau.
The
Carreau model is given by :
_{} (1.4)
Where _{} the viscosity
at low shearrat and _{} is the
viscosity at high shear rate.
_{}is a time constant of the fluid. The location of the transition from the
Newtonian plateau to the shearthinning regime is determined by 1/_{}. The infiniteshear viscosity _{} is generally associated with a breakdown of
the fluid, and is frequently significantly smaller (_{}to_{} times smaller) than _{}. Thus _{} will be
neglected in this study.
_{} is the
second invariant of the shear rate.
_{}
The evolution of _{} as a function _{} is shown in
Figures 7 and 8.
It is non dimensional
form, the Carreau law leads:
_{}, where
_{}
Figure 2. Rheogram _{}for _{}(dimensionless quantities) and varying index n
Figure 3. . Rheogram _{}for _{}(dimensionless quantities) and varying lambda
The
flow is considered to be a stationary. Due to the cylindrical symmetry, we have
assume that the velocity field is independent of the angle _{}. In order to have a velocity field independent of
z , which is requirement to perform a
stability analysis hereafter, we ensure a radial flow both at the inner and the
outer cylinder. This hypothesis reads:
_{}
Where_{} is the radial
flow rate per unit height.
We
therefore assume the following form of the velocity
_{}.
Mass
conservation gives:
_{}
_{};
Which
leads to:
_{} considering the boundary conditions
Conservation
of momentum leads to:
_{} (1 .5)
The
components of the stress tensor (_{}) read :
_{}
Boundary
conditions for the base state:
_{}
Formulation for collocation points
To solve these equations
to nonlinear, we turn to a resolution digital. The cylindrical geometry of the
problem and the behavior of the fluid to the walls lead us to use a
distribution point GaussLobatto.
_{}
_{} is the
number of points selected for the discretization of the space.
Since the _{} equation has
already been resolved, only two last equations of system are considered and
becomes , in matrix form:
_{}
Note that this system
is in the form:
_{}
At each iteration, the
matrices _{}are completely determined. The resolution is therefore
up to the inversion of a matrix. These equations are solved in ascending order.
Integration
of the boundary conditions
For the resolution of
this system , it is necessary to include the boundary conditions.
The translation of
matrix is:
_{}
Thus are modified the first
and last lines of Kv, Lv, and Lw, Kw of the way in follows:
_{} _{}
A program
implementing this numerical collocation method was thus developed with the
software Matlab.
We solve problem for
eigenvalues _{} in the
particular case of a Newtonian fluid undergoing TaylorCouette flow. We
take the following dimensional parameters:
After non dimensionalization, we obtain the
following parameters
_{}
The real part of these
values is represented as a function of the imaginary part in figure. On the
same figure 17 are plotted the
eigenvalues obtained with the program Newtonian taylor Couette of Cherif Nouar
and Cecile Lemaitre.
Figure4.
Eigenspectrum of a
Newtonian fluid in TaylorCouette flow.
Real part of eigenvalues vs. imaginary part.
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