Microfiltration separates fine solids from a liquid substrate, but requires a process to remove the accumulated solids from the filtering device. In this study, a backwash process was examined, in which a flow of high-pressure air in the direction opposite to the usual fluid flow, is used to dislodge and remove the particles.



The dynamics of the original air penetration of the filtering fibres, the flux balances of the displaced liquid, and the distribution of pressures through a network of modules were all studied. They fall naturally into a classification by increasing spatial scales

Stokes, J.

English

Mathematics in Industry Information Service Eprints Archive

75
MODELLING THE BACKWASH PROCESS IN
MICRO FILTRATION
Microfiltration separates fine solids from a liquid substrate, but requires a
process to remove the accumulated solids from the filtering device. In this
study, a backwash process was examined, in which a flow of high-pressure
air in the direction opposite to the usual fluid flow, is used to dislodge and
remove the particles.
The dynamics of the original air penetration of the filtering fibres, the flux
balances of the displaced liquid, and the distribution of pressures through a
network of modules were all studied. They fall naturally into a classification
by increasing spatial scales.
1. Introduction
The 1986 Study Group considered mathematical aspects of hollow fibre ul-
trafiltration. This is a process in which a liquid is forced from the outside into
the centre of a hollow fibre, typically 350 J.L inner diameter and 650 J.L outer
diameter. The fibres are gathered in bundles (see Figure 1), and the bundles
contained in cartridges of 5cm diameter and 90 cm long. The filtration process
forces the liquid through the pores in the fibre walls, typically 0.1-0.2 J.L in diam-
eter. The 1986 Study Group was concerned, among other things, with various
modes of failure of the tube under this inward pressure.
Fibres
5cm
Figure 1: A filtration cartridge unit
The resulting separation can remove very fine particles, and even achieve
liquid-liquid separation, but to be useful requires that the filtered material can
76 Memtec
be removed. This is done by forcing air flow in the opposite direction at high
pressure (500-700 kPa), called backwash. The lumina are drained, and the air
pressure is applied, with exit of liquid blocked. The opening of a backwash valve
suddenly decreases the liquid pressure, creating a big pressure drop across the
fibre wall. The makers have gone to some trouble to decrease the response time of
this valve, which can be as low as 40 milliseconds, and one of the points of interest
of this investigation is to see whether such effort is justified. This question is
addressed by the development of a suitable microscale model in Section 3. The
answer, according to this model, is that the very fast valves are now faster than
the timescale of the process.
As preparation for the Study Group, a search of the literature was done, and
some useful references are given at the end. The chapter by Porter is a modern
general introduction. Other chapters in the book include a series of three on
membranes, some fundamental papers on membrane transport, an introduction
to ultrafiltration, and a chapter on reverse osmosis. A paper by Doshi, gives some
useful theory on dynamics in the lumen, and the book of which it is part is still a
useful general reference. The Lorne Workshop Proceedings is non-technical, but
is a very interesting exercise in industry-government communication, to which
Memtec has contributed. The paper of Balmann et al. is a very recent study
in volving modelling of fouling, whilst the paper of Klein et al. is one of a number
of articles in a major 1980 symposium.
For a complete study of backwash, three scales of operation need to be con-
sidered:
1. Microscale: The gas-liquid interface is followed as it passes through the
fibre wall. A model for this process is developed in Section 3.
2. Mesoscale: The flow of air and liquid through a bundle of 3000 fibres,
within a tubular envelope (module), is considered. A model for this process
is presented in Section 4.
3. Macroscale: The array consists of a number of modules connected by man-
ifolds into a series and/or parallel grid. The flow dynamics following open-
ing of the backwash valve has been modelled by Memtec, and some sug-
gestions were made during the Study Group to improve the performance
of this model (Section 5).
2. Backwash phenomena and general considerations
The events in a backwash sequence are:
Backwash in microfi.ltration 77
Normal Filtering Mode
lOO-400kPa L
.Jl'1ow
Backwash Mode
Shell space - liquid to be filtered Shell space - liquid. later bubbles, 30 kPa
Figure 2: A fibre in cross-section in the two main modes of operation
1. Note the loss of effectiveness of the filter
2. Drain the lumina of the fibres (2 seconds)
3. Shut all shell side valves
4. Pressurise filtrate side with air to 600 kPa
5. Open backwash valve to let liquid escape (0.04-1 second)
6. Air flows for 3 seconds
7. Water is pumped through shell space while air flow continues
8. Pump and air stops
9. Rewet stage
• Rewet with clean fluid in lumen
• Open filtrate valve
• Reapply 600 kPa air to pressurise lumen
78 Memtec
• Open backwash valve 1 second and air valve
10. Resume filtering
When the backwash valve is opened (stage 6), the shell pressure drops to
about 30 kPa, and an air front starts to advance through the fibre walls. This
is relatively slow, although calculations presented later show it still occurs on a
sub-second scale. As the air advances, water is displaced and forced from the
bounding cylinders. The air must do work to impart kinetic energy to this water.
The fibre material is hydrophobic, and therefore presumably surface tension
does not impede the movement of the front. Some discussion occurred in the
group about the experimental observation of "bubble point pressure". This is a
pressure of about 100kPa which must be applied before any air will pass through
the fibre, even after a very long time. Its value is used as a check on the quality
and integrity of the fibres. It was thought to indicate hydrophilic behaviour, but
since it now seems to be established that the fibre material is hydrophobic, the
effect is attributed to the pressure needed to overcome the surface tension in the
very small bubbles formed initially on the outer walls.
3. Backwash on the microscale
The basic equation for movement through the fibre of both phases is Darcy's
law:
V'p= -J.Lvf/k (1)
where p is the pressure, v the fluid velocity, J.L the viscosity of the fluid, f is the
porosity and k the hydraulic permeability. Note that the 'superficial velocity',
which is the volume flux per unit area, is sv. The following complications were
not taken into account:
• The material is deformed by the applied pressure (but only by 2-5%)
• The properties may be changed by the deformation
• The front separating air and liquid may be irregular (fingering)
• The filtrate material may have to be considered as a separate phase
A commonly used figure for permeability is 2 X 10-16 m2 though it varies
during the life of the fibre. In our calculations, the value used for the viscosity
ofwater was J.Lw = 1.0 X 10-3 Pa s and the porosity = 0.7.
Backwash in microfiltration 79
The suffices 0, i, and 1 will denote the inner surface, air/water interface and
outer fibre surface respectively. Since the viscosity of air is much less than that
of water, the pressure at the air-water interface Pi may be taken as the backwash
air pressure, say Po = 600 kPa.
The condition at the outer surface of the fibre is more difficult. At backwash
time, the fibre is surrounded by dirt etc., which creates very great resistance
to radially inward flow. Whether it creates resistance to outward flow is not
so clear; the material would seem unlikely to have much mechanical strength.
The Study Group discussed this question, and decided to neglect the external
resistance to flow; it was partly influenced by the unavailability of measurements.
It would seem quite incorrect to assume that the inward flow resistance applies.
A subsidiary question is whether the material's inertia is significant; again this
is very hard to quantify, because the material does not have to accelerate to the
flow velocity, but merely has to get out of the way.
We ignored the outer resistance, which would have the effect of increasing
the time taken for air to penetrate the fibre wall. We did allow for a slight
pressure at the outer surface of the fibre, say PI = 30 kPa.
With cylindrical symmetry, (1) becomes
(2)
and with incompressibility, assuming k and J.Lconstant:
8(rv)
--=0
r8r
so
rv = rivi
This gives
8p J.LwfriVi
8r = - rk
so
(Po - Pi) = J.Lwrivilog(ri - ro)
and setting Vi = drddt leads to the solution for the time tt for air to reach the
outer surface
2 2
( ) J.Lwf[ 2 (/) rO-rtPo-PI tt = - 2k rolog rt ro + 2 ] (3)
with
(4)
80 Memtec
Evaluation gives
t _ (0.35/2)2 X 10-6 X 0.7 X 10-3 X (2 log(0.65/0.35) + 1 - (§¥S )2)
1 - - 4 X 2 X 10-16 X 570 X 103
or
t1 ~ 0.057 secs
The initial velocity is
570 X 103 X 2 X 10-16
V = = 1.5 X 1O-3m/s
10-3 X 1.75 X 10-4 X log(6.5/3.5) X 0.7
This may be used to check the assumption that the air pressure drop along the
lumen of the fibre is negligible. Initially, the total flux F at the entry to the
lumen is
F = 0.9 X 211"X 1.75 X 10-4 X 1.5 X 10-3 ~ 1.4 X 10-6m3/s
Use of the formula for Poiseuille flow in a pipe gives
dPo 8J.La X F (5)
dx 1I"r4
8 X 2 X 10-5 X 1.5 X 10-6
(6)= 11"X (1.75 X 10-4)4
~ 100 kPa/m (7)
A check on the validity of this model is that most of the work done by the air
goes in overcoming viscous forces in pushing the water through the fibre. Some
also goes into kinetic energy in the water movement in the cylinder. The power
expended by the air on a module with 1 m2 internal lumen area total is
pressure X area X air velocity = 6 X 105 X 1 X 1 X 10-3 = 600 watts
A very rough comparison with the external liquid kinetic energy is given by
noting that about 0.5 I of water is displaced in 0.3 seconds. If this has to pass
through an orifice of 5 cm2, the average speed would be 3 m/s, and the kinetic
energy about 4 Joules, or an average power of about 13 watts. This figure is less
than estimated above, but the energy increases rapidly as the size of the orifice
diminishes.
The conclusion of this part of the analysis is that it takes about 0.06 seconds
for the air to first reach the outside of the fibre, once the backwash valve is
opened. This is somewhat longer than the opening time of the faster valves.
Backwash in microfiltration 81
4. Mesoscale considerations
Here motion along the length of the fibres, in the lumen, and in the shell
space is analysed. Q is the influx of air into each lumen, in m3s-1, and W is
the return flux of water, in m s-l. Generally, the suffix a indicates reference to
the air phase in the lumen, whilst w refers to the water in the shell space. The
equation for Poiseuille flow in the lumen is
Q = 1rr~ PRdp
8Jla L dz
(8)
Here,
L is half the fibre length
PR is the entry air pressure in Pa
z is the axial coordinate, non limensionalized against L
p is the air pressure, non-dimensionalised against PR
Jla is the viscosity of air
The equation relating movement of water through the fibre wall is, as in the
previous section,
Jlw log(r1/ri) dQ
p- Pw = k 21rLPR dz
and the equation representing volume balance for the return flow in the shell
space is
(9)
W= nQ1-4> (10)
where 4>is the volume fraction occupied by fibres and n is the number of fibres
per m2. The Navier-Stokes equation for the return flow is
W dW + PR dpw + 8JlwW = 0
L dz PwL dz pr~
(11)
The first term expresses the convection of momentum, and the second the force
provided by the pressure gradient. The third term allows for the viscous drag
of the fibres on the liquid by treating it as a body force. The force is obtained
using the expression for Poiseuille pipe flow; the "effective radius" is taken to be
the outer radius of the fibres. This is related to their packing density rather to
an actual role played by that radius.
Gravity has been ignored, as the body force it produces is on this scale
equivalent to a pressure differential of only a few kilopascals. More seriously, the
pressure drop in the exit ducts has also been ignored; some information on this
82 Memtec
became available after the Study Group. Dr Ian Howells, in a separate report,
has made a quantitative analysis taking some of these things into account!
Solving the equations for p gives
d2p dp 2
A dz2 - B(dz) - Cp = H = constant (12)
where
2
C = 1+ (l?JLW ~ ~ 3.57
TO JLa 1- 4>
Here fa is a factor for describing fibre permeability, fa = 1.7 X 1O-15Ik; fa is
about 1 for new fibres and 8.5 for old ones.
For Ti constant, this can be integrated
ds
dp = 2({3s + ,p + H)
where s = (*)2, {3= BIA and, = CIA. This gives
(dp)2 = s = De2{3p _ ,p _ H _ ~
dz (3 /3 2/32 (13)
and on further integration
f1 dp
1- z = Jpa J De2{3p - J -1- ib
By applying * = 0 at z = 0 and using (9) and (14) at z = 1, the extra constants
introduced can be determined.
(14)
The conclusion of this section is that flow in the cartridge can be treated
mathematically, provided the assumption of an "effective resistance" of a fibre
in (11) is acceptable.
IThis valuable report could not be included here; it has however been communicated to
Memtec. Ed.
Backwash in microfiltration 83
5. Macroscale
For fluid motion through an array of modules it is useful to be able to model
the pressure losses in the manifolds and their connections external to the car-
tridges. The dynamic response of the cartridges is also important. Mathematical
modelling should, in principle, be able to resolve important design questions such
as how many cartridges can be treated by one backwash valve (synchronizing
several may not be easy), and what is their optimal arrangement.
The Study Group did not have time, nor really enough information, to treat
this problem in detail. Nevertheless, there was active discussion. The existing
method treats the system as analogous to a network of electrical resistances. It
was pointed out that this is a way of representing the solution of the ordinary
differential equations for the pressures in the various components by Laplace
transformation, and if this is borne in mind, it is possible to incorporate more
ad vanced dynamics and better solution. The following suggestions were made:
• One should be wary of the effect of highly transient events, for example, the
transition to turbulence of accelerating flows. Even if the model correctly
accounts for them, they will cause numerical difficulties.
• Taking the differential equation point of view, the existing solution method
is simple Euler; a higher order method such as fourth order Runge-Kutta
would allow longer time steps with only a little extra effort.
• In the electrical analogy, it is simple and desirable to allow reactive ele-
ments (capacitance and inductance). These represent stored energy, which
is certainly present in the system. For example, the work done in creating
kinetic energy is presently regarded as having been dissipated but, if it
is regarded as having been done against an inductance (stored energy =
~L X [2, L = inductance, I = current ), this is formally analogous to the
kinetic energy (!mv2).
• Attention was drawn to the usefulness of Milman's Theorem in network
analysis
for the combined effect of a number of voltage sources with internal resis-
tance connected in parallel.
84 Memtec
6. Conclusion
1. On the microscale, the model developed in Section 3 shows that the time
taken for air to penetrate the fibre walls during backwash is at the smaller
end of the range of backwash valve opening times, suggesting that it may
be justified to use the existing fast valves but not to expend resources on
seeking further speed. This conclusion is affected by uncertainty about the
effect of the fouling material.
2. A mesoscale model was developed and used to calculate solutions. These
solutions have qualitatively correct features, although detailed interpreta-
tion requires data which was not available at the Study Group.
3. The macroscale networks of cartridges was discussed in terms of models
developed by Memtec. The Memtec models involved systems of ODEs,
and suggestions were made for improvement of the solution procedure.
Acknowledgements
The group was exceedingly indebted to Ian Howells, especially for his thor-
ough mesoscale analysis. Jay Darzi from Memtec made many timely suggestions
and patiently kept us on the rails. It was a lively and creative group with valu-
able contributions from Steve Spencer, John Mooney, Neville Fowkes, David
Jenkins and Patrick Howden.
References
M.C. Porter, "Microfiltration" in P.M. Bungay et al. (eds), Synthetic Mem-
branes: Science, Engineering and Applications, NATO ASI Series (D Rei-
del, Dordrecht, Netherlands, 1986), 225-248.
M.R. Doshi, "Boundary layer removal in ultrafiltration", in A.R. Cooper (ed.),
Ultrafiltration Membranes and Applications in Polymer Science and Tech-
nology 13 (Plenum Press, NY, 1980),231-247.
A.J. Jones (00) "Membrane Technology - The Lorne Workshop" (DITAC, Can-
berra 1987).
H. de Balmann, P. Aimar and V. Sanchez, "Membrane partition and mass
transfer in ultrafiltration", Separation Sci ('3 Tech 25 (1990), 507-534.
E. Klein, F.F. Holland, K. Eberle and R.P. Wendt, "Ultrafiltration rates and re-
jection of solutes by cellulosic hollow fibres" in A.F. Turbak (ed.) Synthetic
Membranes, Vol 2, A Chem Soc Symposium Series 153 (1981), 75-108.


Modelling the backwash process in micro filtration

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