Half of the channel height (in case of permeable membrane on both sides of ...... Preferential Sorption Capillary Flow Model (Sourirajan and. Matsuura, 1982).
Mathematical Modelling and Compute~ Simulation of Membrane Processes Involving Synthetic and Natural Membranes
THESIS SUBMITTED
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY IN CHEMICAL ENGINEERING By
Vineet Kumar B.Sc., B.Sc. Engg. (Chern), M.Tech. (Chern.)
Department of Chemical Engineering & Technology Institute of Technology Banaras Hindu University Varanasi 221005 INDIA
1995 Enrol. No.178337
••
Gram : TECH N OLOGY Telex : 0545 /20B
J
Office : 54291 Ext. 324 {Res. : 54291 Ext. 208
Department of Chemical Engineering & Technology INSTITUTE OF TECHNOLOGY BANARAS
HINDU
VARANASI-221
This is to certify that Modelling
the
and Computer Simulation
UNIVERSITY 005 (INDIA)
thesis
entitled
"Mathematical
of Membrane Processes
Involving
a record of bonafide work carried out under our supervision and guidance. The results obtained in this work have not been submitted to any other institution for the award of Degree or Diploma.
Synthetic
fZ·
N
and Natural
Membranes" by Mr. Vineet Kumar is
i{ CMA.~X.e.-l
(Prof. R.N.Pandey) Co-Supervisor
:' .•::(;Y
S ~. N_ '"' L', . [;;(~ifY - ~d OC5 ,N0!.A
Forwarded
a.{ ..
C.\..A.A ... ~~\..~
'"2-9 . >:
cr- S
(Prof. Uma Shankar) Head of the Department
1I"'!fU >:
Hi
•• .,.
it ;';:;;f"equatiotl "·5~5 is to SCpRlRtc oul the effect shear mte (which in conventiQfJaI.1u(xk,ft)s.rcqvites a 1lighstlcss) from the pressure. This call be done by the
"UTP"ot·Crf-, nM:dt04QfOpeiatioo~i.e.,:where the permeate is also puinped, but co-cmrenl 10 the feed, !1lK:h that the pte;.sure llt'(l(' '011 tile '~tentRte and permeate sides arc the same. TIlliS Ihe
.....
~~:~.- .... .........
Oc..;r., ....IS./f'J: U"t;n.•.•.. "'. (Siy""',,rr '!f 11';,,,,,,;,,,·,) l'/aoo.
t~~~ ~~ J
It. t..i "'.... IJ. (c.u..,)
FilII
~'I"I~ .~",.!\.
/h(Jry~ "
.
........
~6.
.......
~
,
.
,MUN·,·/-!.. CII?!"T""'-'lN •••
~
.......
~
•••••••••••••••••••••••• ,
••••
~
~.
~;~~4>""!..t'h
t
-
'\
r
'y'"h '
":If
;'
.,:\ ~~~. I
.
effective tl'allsmembrane pressure i~ slnaU throughout the length oflhe module, but the velocity can be adjusted to whatever is required for the study. witbout having to worry about unequal stresses down the length of the inodule. This mode ofoperation has become popular in many industrial operations now, but it docs require a tubular module that can withstand Ihe back-pressme (e.g., hollow libers or ceramic). As a limit comment •• and this has no bearing o'ntlle quality or acceptability ofthe thesis. It would have been desirable to lise well-known coinmercial nat-sheet membraaes. instead of laboratory-made membranes. We realize the problem and the expense of obtaining stich membranes frolll abroad, but there are Indian-made cellulose acetate membranes available from Uydranautics and Pennionics, both in Gujaral. The application of his models to these membranes would have enhanced the manufacturer's understanding of their membranes alld possibly helped them 10 improve the membrancs. This would lead to widening the scope of al,plicntions of this very valuable technology in Indicl. ('erhal)S this should be the next phase of this study.
: ' t "
~vf. I~ (fS
.
lhale •• ", •••••• 1. .............. .
•
~Oio--
• • • ..
••
~,.
••••• : . . . . . " ••• '* .....
PREFACE ==========================================================================================================================================================================================================================================================================================================================================
Use
of
a
membrane
based
separation
process
as
a
unit
operation is relatively of recent origin but it promises of having a great potential. Due to its lower operating cost and greater selectivity,
in
the
near
future
it
is
likely
to
replace
all
existing separation processes like distillation, fractionation, leaching etc. Its promises are supported by the fact that most of the organs in the living system (kidney, lungs, blood vessels, intestine and even nervous system) rely upon membrane controlled processes. Such an important process is still poorly understood and its potential is only partly exploited. In
the
processes
present
involving
work
a
natural
computer and
simulator
synthetic
for
membranes
membrane has
been
developed using a novel technique. The technique used is slightly different from finite difference and finite element methods where all the parameters are considered at fixed nodal points. In the present approach the entire flow chamber is divided into a large number of elements (similar
to
the
finite
elements)
and
each
element is considered to be an independent unit. All mass flux and velocity components are calculated at the boundaries instead at the center (nodal point) of element (similar to that used in first principle of differentiation). Thus, this method may be termed as the
finite
element
technique
using
first
principle
of
differentiation. It is observed that this method is very fast and computer memory requirement is very low. Even a 10tm long membrane module can be simulated in a small PC (having merely 100 KB of memory) and satisfactory results can be obtained just in 30 hours. Calculations for a 50tcm long module on a mainframe computer are a matter of merely a few seconds. Considering marked difference in the nature and functioning of
natural
and
synthetic
membranes,
i
present
thesis
has
been
ii divided into three sections. First section deals with the general introduction
of
membranes
and
membrane
based
processes
using
synthetic and natural membranes. Second section of the thesis is devoted to the development of the computer simulator, comparison of the results of the simulator with those obtained by analytical solution
for
experimental
different investigation
synthetic of
membrane
filtration
processes
characteristics
and of
cellulose acetate membrane using different polymeric solutions. Finally in the third section, applicability of the simulator in case of natural membranes has been tested. To test the reliability of the simulator, its results have also been compared with the analytical solutions for six different cases. viz. (1) ultrafiltration in a rectangular flow chamber, (2) dialysis unit with constant dialysate concentration, (3) surface reaction on endothelial cell surface in a rectangular duct, (4) surface
reaction a
on
permeation
in
intestinal
epithelial
endothelial
rectangular cell
cell
duct,
surface
surface
(5) in
surface
circular
coupled reaction tube
and
with on (6)
surface reaction on intestinal epithelial cell surface coupled with permeation in circular tube. Analytical solutions obtained
by
previous
workers
(except
those for ultrafiltration) have been reformulated and solved by using
confluent
hypergeometric
function
which
gives
much
more
accurate results. Details of the procedure for the analytical solutions of all of the above mentioned cases are presented at the end of the thesis in Appendices A through D. Appendix E contains short derivations such as formulation of differential equations with boundary conditions, derivations of the velocity profile and pressure distribution in circular tubes, relations between special dimensionless groups, etc.. The computer simulator developed in the present work has been compiled in the form of a demonstration package, a short users’ manual for this package has also been appended in Appendix F. With the help of the simulator developed in this work and experimental data (obtained in present work and those reported by
iii previous
workers),
permeability obtained.
An
a
useful
coefficient analysis
empirical
during of
this
correlation
ultrafiltration correlation
has
for also
reveals
that
the been the
resistance in series model for ultrafiltration may be further modified
to
parallel-series
resistance
model.
Another
useful
correlation developed during the course of the present work is the correlation for osmotic pressure of bovine serum albumin. Main advantage of this correlation over others is that the osmotic pressure
for
a
wide
range
of
concentration
and
calculated more accurately using only one equation.
pH
can
be
ACKNOWLEDGEMENTS ==========================================================================================================================================================================================================================================================================================================================================
I am highly obliged to my supervisors Prof. S. N. Upadhyay (Department
of
Chemical
Engineering
&
Technology,
B.H.U.)
and
Prof. R. N. Pandey (Department of Applied Mathematics, B.H.U.) for their valuable suggestions, guidance and encouragement throughout the course of present work. My sincere thanks are also due to Prof. M. M. Sharma (U.D.C.T., Bombay) for his valuable suggestions and encouragement which lead to complete the present work and to Prof. N. K. Roy (I.I.T., Kharagpur) for taking so much botheration in providing some useful literature well in time. I am thankful to all of the faculty members particularly Prof. Surendra Kumar (Tewari), Dr. G.C.Baral, Dr. A.S.K.Sinha, Prof. Vijay Shankar, Prof. N.S.Garg, Prof. Uma Shankar (Head) and Prof. Y.D.Upadhyaya (Ex-Head) for their whole hearted support and providing all possible facilities in completing the research work. I am thankful to all of my friends (Mr. Anil Kumar Sharma, Mr. Rohit Varma, Dr. B.N.Rai, Dr. Uttam Kumar Ghosh, Mr. Pradeep Kumar Mishra, Mr. Manoj K. Verma and Mr. Pradeep Ahuja) who have directly or indirectly helped in this work. I feel pleasure in acknowledging Mr. Arjun Prasad Gond (Lab. attendant)
for
his
devoted
work
in
fabrication
and
membrane
preparation for the experimental part of the present work.
VINEET KUMAR
iv
NOMENCLATURE ==========================================================================================================================================================================================================================================================================================================================================
A
Constant
A
Constant
B
Constant
C
Concentration (g/ml) *
C
Concentration on the dialysate side (g/ml)
C
Gel concentration (g/ml)
C
Molar concentration (mole/ml)
C
Concentration at the membrane surface (g/ml)
C
Feed concentration (g/ml)
C
Permeate concentration (g/ml)
c
Dimensionless concentration [C/C ] or | o
d
Equivalent pore diameter (cm)
D
2 Diffusivity (cm /s)
G m w o p
#C - C* $ | (-) 3Co- C* 4 ------------------------------
p
Di,j-1 i,j
Diffusivity at the boundary th 2 (i,j-1) elements (cm /s)
i,j-1
DddCy
1
----------
i,j
1
separating
Diffusive flux at the boundary th 2 (i,j-1) elements (g/cm s)
separating
th
(i,j)
and
th
(i.j)
and
F
Function defined in terms of Kummer’s function
G
Function defined in terms of Kummer’s function
h
Half of the channel height (in case of permeable membrane on both sides of the channel) or channel height (in case of permeable membrane on one side of the channel) (cm)
h
Half of the channel dialysis unit (cm)
h
Half of the channel height on the dialysate side of the dialysis unit (cm)
b d
height
v
on
the
blood-side
of
the
vi J
2 Pure solvent flux (ml/cm s)
J
2 Steady state permeate flux (ml/cm s)
J
2 Permeate flux (ml/cm s)
K
Constant of integration
K
Intrinsic membrane conductance [=1/(mR )] (cm/Pa.s) o
K
Mass transfer coefficient (cm/s)
K
Overall membrane conductance (=1/R ) (cm/Pa.s) s
k
First order reaction rate constant based on unit surface area (cm/s)
k
Average mass transfer coefficient across the membrane in dialysis unit (cm/s)
L
Dimensionless axial distance defined as
o s w
o L s
c
volumetric flowrate of filtrate permeating out up to a distance x from the entrance --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
volumetric feed rate
vwx * 2 v x &st (9vw)0i dxi* &2 st (9vw)0i dxi* & & w * = | 7 h8 or 7 R 8 or |7 h |8 or |7 R 8 -------------------------
M
-------------------------
-------------------------------------------------------
-----------------------------------------------------------------
Molecular weight (g/mole)
M(a,b,z) Hypergeometric function [= F (a;b;z)] 1 1 N
Dimensionless number (v /k) (-) w
N
Avogadro’s constant (=6.02252x10
n
Total number of elements along y- or r- axis (-)
DP
Trans-membrane pressure difference (Pa)
DP
Effective trans-membrane pressure difference [DP-p(C)] (Pa)
p
Pressure inside the feed chamber (Pa)
p
Inlet feed pressure (Pa)
R
Radius of the tube (cm)
R , R 1 2
Two components of the fouling layer resistance (Pa.s/cm)
R
Fouling layer resistance (Pa.s/cm)
A
eff
o
c
23
-1
mole
)
vii R
Dynamic intrinsic membrane resistance [m(R +R )] (Pa.s/cm) o m
R
Additional resistance due to solute-membrane interaction 2 -1 (Pa.s /g) or (cm )
d m
R
2 -1 Intrinsic membrane resistance (Pa.s /g) or (cm )
R
Overall membrane resistance [R +R ] (Pa.s/cm) c d
r
Radial distance from the center-line in a tube (cm)
R
Observed rejection, defined by equation (II-2.7) (-)
Rw
Intrinsic rejection, defined by equation (II-2.2) (-)
o s
velocity (rv h/m) w
or
(rh/m)
or
Re
Reynold number based on permeation (rv R/m) (-) w
Re
Reynold number (rR/m) (-)
S
Function defined in terms of Kummer’s function
Sc
Schmidt number [m/(rD)] (-)
Sh
Sherwood number (K h/m) or (k h/m) (-) L c
T
Absolute temperature (K)
T
Function defined in terms of Kummer’s function
o
-
based
on
feed
velocity
U(x)
Average tangential velocity of solution at a distance x from the inlet (cm/s)
u
Longitudinal velocity along the membrane surface (cm/s)
Average feed velocity at the inlet (cm/s)
v
Radial velocity component towards the membrane surface (cm/s)
v*
Dimensionless radial velocity (v/v ) (-) w
vw
Permeation velocity across the membrane (cm/s)
W
Channel width (cm)
x
Distance along x-axis (cm)
y
Distance along y-axis (cm)
z
Parameter used in analytical solution
viii
Greek a
Exponent of velocity resistance (R ) (-) 2
a 2
gradient
term
in
the
dynamic
Dimensionless constant (D/v h) or (D/v R) (-) w w
b
Eigenvalues
d
Boundary-layer thickness (cm)
d
Membrane thickness (cm)
dr
Width of an element along radial direction (cm)
dx
Length of an element along the membrane surface (cm)
dy
Width of the element along radial direction (cm)
e
Porosity of the membrane (-)
m
f
Thiel modulus (k h/D) or (k R/D) (-)
g
Dimensionless radial distance in circular tube (r/R) (-)
h
Dimensionless axial distance defined as ln(1-L)
j
Modified Thiel modulus [(k-v )h/D] or [(k-v )R/D] (-) w w
l
Dimensionless radial distance in rectangular duct (y/h)
m
Viscosity (cp)
n
2 Kinematic viscosity (cm /s)
p
Constant (=3.1415926)
p(C)
Osmotic pressure, at concentration C, (Pa)
q
Parameter representing both f
r
Density (g/ml)
t
2 Shear stress (dynes/ cm )
v
Dimensionless distance representing both x and h (-)
x
Dimensionless axial distance
2
-a
2
and j
D
x
2 h ---------------------------
(-)
(-)
or
D
x
2 R
---------------------------
ix
Subscript i
An integer, used in computer simulator to count number of elements from inlet along x-direction.
j
An integer, used in computer simulator to count number of elements from membrane surface along reverse y-direction.
m
indicating parameter based on m
n
th n element (counted from membrane surface), lying near the axis of the channel
n
indicating parameter based on n
th
th
eigenvalue
eigenvalue
CONTENTS ==========================================================================================================================================================================================================================================================================================================================================
PREFACE
i
ACKNOWLEDGEMENT
iv
NOMENCLATURE
v
CONTENTS
x
SECTION-I MEMBRANE SEPARATION PROCESSES I-1
1-10
WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW
General Introduction I-1.1 Synthetic Membrane Reverse Osmosis Ultrafiltration Dialysis I-1.2 Biological Membrane Plasma Membrane Tissues as Membrane Capillary I-1.3 Objectives
SECTION-II SYNTHETIC MEMBRANES
WWW II-1 INTRODUCTION WWW II-2 LITERATURE REVIEW WWW II-2.1 Mathematical Models WWW Models for Reverse Osmosis WWW I.T. model WWW PCS model WWW Concentration polarization WWW Models for Ultrafiltration WWW Pore model WWW Film model WWW Osmotic pressure model WWW Resistance in series model WWW
x
WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW
1 2 2 4 5 5 6 8 9 9
11-110
WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW
11 14 16 16 16 16 17 20 20 21 21 24
xi
WWW WWW WWW WWW WWW WWW WWW WWW
25
WWW II-3.1 Definition of the Problem WWW WWW Velocity Profile in Rectangular Channel WWW Permeable wall on both sides of the channelWWW Permeable wall on one side of the channel WWW Pressure Distribution in Rectangular Channel WWW Velocity Profile in Circular Tube WWW
32
II-2.2 Factors Affecting Limiting Flux Effect of Concentration Polarization Effect of Feed Concentration
WWW
Effect of Membrane Module Length Effect of Temperature Effect of Flow Rate
WWW WWW
Effect of Trans-membrane Pressure Difference II-2.3 Limitations of Published Work WWW II-3 FORMULATION AND SOLUTION OF THE PROBLEM
Pressure Distribution in Circular Tube with Permeable Wall WWW II-3.2 Computer Simulation of Membrane Separation Processes WWW Ultrafiltration Rectangular channel Circular tube Dialysis Unit Blood side Dialysate side
WWW WWW WWW WWW WWW WWW
II-3.3 Analytical Solution of Membrane Separation Processes WWW Ultrafiltration Rectangular channel Circular tube Dialysis Unit II-4 Experimental II-4.1 Materials Chemicals The Membrane II-4.2 Experimental Methods
WWW WWW WWW WWW WWW WWW WWW WWW WWW
Permeability of Distilled Water Permeability of Aqueous Solutions
27 28 29 29 30 30 30
32 33 35 38 38 39
WWW
39
WWW WWW WWW WWW WWW WWW WWW
40
WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW
52
43 43 47 49 49 50
52 52 52 54 55 55 55 55 56 58 58
xii II-5 Results and Discussion
WWW
WWW WWW WWW WWW WWW
II-5.1 Accuracy of the Computer Simulator Comparison With Analytical Solution Rectangular channel Dialysis
WWW WWW
II-5.2 Empirical Correlation for Permeability Coefficient II-5.3 Osmotic Pressure II-5.4 Experimental Results
WWW WWW
WWW WWW WWW WWW WWW
II-5.5 Estimation of Gel-layer Concentration II-6 Conclusions REFERENCES
SECTION-III NATURAL MEMBRANES III-1 Introduction
WWW WWW WWW WWW
III-1.2 Permeation Through Capillary Wall
WWW
III-2.1 Endothelial Cell Response to Fluid Flow III-2.2 Permeation Through Endothelium III-3 Formulation and Solution of the Problem III-3.1 Surface Reaction Without Permeation Rectangular Channel Circular Tube
WWW WWW
III-3.2 Surface Reaction Coupled With Permeation Rectangular Channel Circular Tube III-4 Results and Discussion III-5 Conclusions REFERENCES
WWW WWW WWW WWW WWW
SECTION-IV SUGGESTIONS FOR FURTHER WORK APPENDIX-A APPENDIX-B APPENDIX-C APPENDIX-D APPENDIX-E APPENDIX-F FLOPPY DISK
WWW WWW WWW WWW WWW WWW WWW
59 68 68 74 74 86 87 99 101 102
111-151
III-1.1 Endothelial Cell Response to Fluid Flow III-2 Literature Review
59
WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW WWW
111
WWW
159
112 113 115 116 118 122 123 123 125 127 127 129 132 150 152
A:1-A:13 B:1-B:11 C:1-C:14 D:1-D:11 E:1-E:18 F:1-F:4 Inside back cover
LIST OF FIGURES Fig. No.
TITLE
Page
I-1.1
Particle size range of various processes (Source: Cheryan, 1986; Porter, 1988)
I-1.2
Structure of phospholipid molecule (Source: Vander et al., 1981)
6
I-1.3
Structure of a typical plasma membrane
7
separation
3
II-2.1
Diagram showing different controlling permeate flux
II-2.2
Steps to reduce concentration polarization (Source: Cheryan, 1986)
27
II-3.1
Schematic diagram showing coordinate axes, directions of fluid velocity components and location of semipermeable membrane for various channel geometries
34
II-3.2
Comparison of radial velocity component predicted t th st nd by 0 , 1 and 2 order perturbation solutions in case of permeable wall on both sides of a rectangular channel (Re=100)
36
II-3.3
th Comparison of velocity gradient predicted by 0 , t st nd 1 and 2 order perturbation solutions in case of permeable wall on both sides of a rectangular channel (Re=100)
II-3.4
Diagram showing shape of an element in rectangular channel and circular tube
41
II-3.5
Cross-sectional view of elements in x-r) plane with labelling scheme
(or
42
II-3.6
Mass flux channel
rectangular
43
II-3.7
Effect of variable diffusivity on predicted wall concentration of bovine serum albumin in rectangular channel (C =0.065 g/ml; pH=4.7; h=0.38 o 5 -9 cm; DP=2*10 Pa; =10 cm/s; K =4*10 cm/Pa.s) o Mass flux through a ring element of radius r. in hollow fiber membrane
45
Mass flux through an element in the blood side dialyser
49
II-3.8 II-3.9
through
an
regions
element
xiii
in
and
the
factors
x-y
of
26
37
48
xiv
II-3.10 Mass flux through an element in dialysate side the dialyzer
of
51
II-4.1
Experimental set-up
57
II-5.1
Wall concentration of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (pH=4.7; t5 C =0.05 g/ml; h=0.40 cm; =100 cm/s; DP=2*10 o -9 Pa; K =4*10 cm/Pa.s) o
60
II-5.2
Permeation velocity of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (pH=4.7; t5 C =0.05 g/ml; h=0.40 cm; =100 cm/s; DP=2*10 o -9 Pa; K =4*10 cm/Pa.s) o
61
II-5.3
Wall concentration of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (pH=4.7; t5 C =0.07 g/ml; h=0.40 cm; =100 cm/s; DP=2*10 o -9 Pa; K =4*10 cm/Pa.s) o
62
II-5.4
Permeation velocity of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (pH=4.7; t5 C =0.07 g/ml; h=0.40 cm; =100 cm/s; DP=2*10 o -9 Pa; K =4*10 cm/Pa.s) o
63
II-5.5
Wall concentration of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (pH=4.7; t5 C =0.05 g/ml; h=0.40 cm; =100 cm/s; DP=3*10 o -9 Pa; K =4*10 cm/Pa.s) o
64
II-5.6
Permeation velocity of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (pH=4.7; t5 C =0.05 g/ml; h=0.40 cm; =100 cm/s; DP=3*10 o -9 Pa; K =4*10 cm/Pa.s) o
65
xv
II-5.7
Wall concentration of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (C =0.05 o t5 g/ml; pH=4.7; h=0.40 cm; =50 cm/s; DP=2*10 Pa; t -9 K =4*10 cm/Pa.s) o
66
II-5.8
Permeation velocity of bovine serum albumin predicted by computer simulator using different number of elements in radial direction (C =0.05 o t5 g/ml; pH=4.7; h=0.40 cm; =50 cm/s; DP=2*10 Pa; t -9 K =4*10 cm/Pa.s) o
67
II-5.9
Variation in the predicted value of average permeation velocity with number of elements in y-direction in a rectangular channel of length 50 cm at different values of the ratio dx/dy (C =0.03 o t5 g/ml; pH=4.7; h=0.40 cm; =50 cm/s; DP=3*10 Pa; t -9 K =4*10 cm/Pa.s) o
69
II-5.10 Variation in predicted value of average permeation
70
II-5.11 Change in predicted value
71
II-5.12 Comparison
73
II-5.13 Comparison
75
velocity with number rectangular channel of
of iterations in a length 50 cm. (C =0.03 o t5 g/ml; pH=4.7; h=0.40 cm; =50 cm/s; DP=3*10 Pa; t -9 K =4*10 cm/Pa.s) o of average permeation velocity with the value of the ratio dx/dy in a rectangular channel of length 50 cm. (C =0.03 o t5 g/ml; pH=4.7; h=0.40 cm; =50 cm/s; DP=3*10 Pa; t -9 K =4*10 cm/Pa.s) o of dimensionless wall concentration (proposed by Sherwood et al., 1965) obtained analytically and from computer simulator and the effect of osmotic pressure and fouling-layer resistances. of wall concentration obtained analytically [equation (II-3.46)] and from computer simulation of dialysis unit in case of constant dialysate concentration and the effect of change in dialysate concentration on blood-side concentration in a parallel plate co-current dialysis unit.
xvi
II-5.14 Schematic diagram of the
resistance
in
78
II-5.15 Effect of shear rate on
components
of
80
the overall membrane resistance with effective pressure difference across the membrane
81
modified series model for ultrafiltration different the overall membrane resistance
II-5.16 Variation of different components of
II-5.17 Variation of different components of
the overall weight at
82
II-5.18 Variation of different components of
the overall weight at
83
II-5.19 Variation of different components of
the
overall
84
II-5.20 Comparison of the correlation [equation (II-5.10)]
88
II-5.21 Comparison of the experimentally observed
89
II-5.22 Comparison of the experimentally observed
90
II-5.23 Comparison of the experimentally observed
91
membrane resistance with constant value of (C N /M) w A
molecular
membrane resistance with molecular constant solute concentration (g/ml) membrane resistance with concentration
with the experimental data (Vilker et al., 1984) of osmotic pressure of bovine serum albumin average permeation velocity of aqueous PEG 9000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.003 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o average permeation velocity of aqueous PEG 9000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.005 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o average permeation velocity of aqueous PEG 9000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.007 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o
xvii
II-5.24 Comparison of the experimentally observed
92
II-5.25 Comparison of the experimentally observed
93
II-5.26 Comparison of the experimentally observed
94
II-5.27 Comparison of the experimentally observed
95
II-5.28 Comparison of the experimentally observed
96
II-5.29 Comparison of the experimentally observed
97
average permeation velocity of aqueous PEG 15000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.003 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o average permeation velocity of aqueous PEG 15000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.005 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o average permeation velocity of aqueous PEG 15000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.007 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o average permeation velocity of aqueous PVA 125000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.003 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o average permeation velocity of aqueous PVA 125000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.005 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o average permeation velocity of aqueous PVA 125000 solution in spiral channel filtration unit with those predicted by the computer simulator for rectangular channel of similar equivalent hydraulic diameter (h) (C = 0.007 g/ml; h= 0.3171 o -8 cm; K = 1.7*10 cm/Pa.s) o
xviii
II-5.30 Comparison
of the experimentally observed permeation velocity from various sources with those predicted by computer simulator
II-5.31 Predicted gel-layer
III-4.1
98
concentration of BSA using computer simulator for various length of rectangular channel (pH=7.4; h=0.32 cm; t -8 K =1.89*10 cm/Pa.s) o
100
Comparison of the dimensionless concentration (C/C ) as a function of radial distance (l) in a o rectangular channel (without permeation) obtained by power-series solution (point values), solution based on Kummer’s function ( ) and computer
133
t
--------------------
2
simulation (-t-t-) at f = 1.0 and different values of dimensionless axial distances (x)
III-4.2
Variation of dimensionless wall concentration (C /C ) with dimensionless axial distance (x) in a w o rectangular channel calculated by four different
t
2
approaches at f
135
= 1 and 4
III-4.3
Comparison of dimensionless wall concentration predicted analytically [equation (III-3.14)] and obtained experimentally (Fig. III-4.4) in a rectangular channel at a distance 2 cm from the inlet as a function of wall shear stress (h=0.0218 t 2 -6 2 cm; D=2.6*10 cm /s; f =1.0)
137
III-4.4
The fluorescence ratio for two individual endothelial cells in a rectangular channel (without permeation) as a function of time. At the time indicated by arrows, the flow was started without ATP and with ATP in the media. (Source: Nollert and McIntire, 1992)
138
III-4.5
Dimensionless concentration (c) as a function of dimensionless radial distance (l) predicted by equation (III-3.38) and the computer simulator at four negative values of j (h=0.1)
139
III-4.6
Dimensionless concentration (c) as a function of dimensionless radial distance (l) predicted by equation (III-3.38) and the computer simulator at four negative values of j (h=0.5)
140
III-4.7
Dimensionless wall concentration (c) as a function of dimensionless axial distance (h) predicted by equation (III-3.38) and the computer simulator at four negative values of j.
142
xix
III-4.8
Dimensionless wall concentration as a function of dimensionless axial distance (x) in a rectangular
143
t
2
channel at f =4 and different values of permeation velocities predicted by equations (III-3.14) and (III-3.38) and computer simulator.
III-4.9
Dimensionless concentration as a function of dimensionless radial distance (g) in a circular
145
t
2
tube at f =1 and x=0.05, 0.10 and 0.50 assuming first order surface reaction (present work) and diffusion controlled surface reaction (Mansour, 1988)
III-4.10 Comparison
of dimensionless wall concentration predicted by equation (III-3.27) and computer simulator as a function of dimensionless axial
t
2
distance (x) in a circular tube at f
146
=4
III-4.11 Dimensionless
wall concentration predicted by equation (III-3.49) and computer simulator (neglecting radial velocity in the bulk) as a function of dimensionless axial distance (L) in circular tube with j=-0.1 and different values ofta
III-4.12 Dimensionless
148
wall concentration predicted by equation (III-3.49) ( ) and computer simulator (neglecting radial velocity in the bulk) (point values) as a function of dimensionless axial distance (h) in circular tube with j=-0.1 and different values of a
149
Screen layout of the demonstration package showing different stages, highlighted items and function box
F:2
--------------------
F-1
xx
LIST OF TABLES Table No.
TITLE
Page
II-1.1 Polymers for MF/UF/RO membranes
12
II-2.1 Mathematical models for reverse osmosis
18
(Source: Nichols and Cheryan, 1986)
II-2.2 Correlations for laminar and turbulent transfer coefficient (Sources: Hwang and Cheryan, 1986)
II-3.1 Correlations for solutes in water
Kammermeyer,
diffusivity
II-3.2 Correlations for different solutes
mass
osmotic
of
22-23
1975;
different
pressure
of
2 (b ) of the characteristic n equation (A-21) for different values of R w and a
A-1
Eigenvalues
A-2
Coefficients (A ) of the equation (A-22) for n different values of R and a w
44 46 A:8-A:10
A:11-A:13
B-1
2 Eigenvalues (b ) at different values of Sh n
B:10
B-2
Coefficients (A ) of equation (B-34) n
B:11
C-1
Eigenvalues
C-2
Coefficients (A ) of equation (C-36) n
D-1
Eigenvalues
D-2
Coefficients (A ) of equation (D-29) n
2 (b ) n equation (C-31)
2 (b ) n equation (D-25)
of
of
the
the
characteristic
characteristic
C:11-C:12
C:13-C:14 D:8-D:9
D:10-D:11
SECTION – I MEMBRANE SEPARATION PROCESSES
I-1
GENERAL INTRODUCTION
==========================================================================================================================================================================================================================================================================================================================================
The most complicated and perhaps the perfect machine known to us is the human body. Surprisingly, almost all activities inside it rely upon a single unit operation
----------
MEMBRANE SEPARATION. Food
taken through mouth enters gastrointestinal tract and nutrients are transported to the blood through epithelium (a semipermeable membrane
made
of
living
cells),
exchange
of
nutrients
and
metabolic end-products of each individual cell of the body with blood takes place through capillary wall (endothelium). Kidney and lungs also are essentially membrane separation units. All hormone secreting glands and even reproductive organs of human body carry out their functions through membrane separation processes. It is the
membrane
facilitate
that
neural
develops activity
electric concerned
potential with
the
across special
it
to
senses
(sight, hearing, taste, etc.), the coordination of muscle activity and the relation between brain activity and consciousness, memory, emotion, etc. Such an important process, which is at the root of life cycle and at the very existence of biosphere, is still poorly understood and only partly exploited. As
an
unit
operation,
membrane
processes
are
being
used
merely as processes for separating different constituents of a mixture.
This
is
clearly
reflected
in
two
commonly
used
definitions of membranes (i) "a region of discontinuity interposed between two phases" (Hwang and Kammermeyer, 1975) and (ii) "phase that acts as a barrier that prevents mass movement but allows restricted and/or regulated passage of one or more species through it" (Lakshminarayanaiah, 1984). Due to marked difference in the role of membranes in the engineered unit operations and those in living systems, membranes
1
2 can be broadly classified into two categories (i) synthetic and (ii) biological.
I-1.1 SYNTHETIC MEMBRANE As far as the laboratory and industrial applications of a membrane are concerned, the primary role of the membrane is to act as
a
selective
barrier
which
permits
passage
of
certain
constituents and retain certain others of a mixture. Passage and retention membrane
of
different
components
characteristics
(pore
of
size,
a
mixture
pore
size
depend
upon
distribution)
molecular weight (shape and size of molecule) and membrane-solute and/or membrane-solvent interactions. Fig.I-1.1
shows
a
classification
of
various
separation
processes based on particle or molecular size, and the primary factors affecting the process of separation. It is seen that the membrane
separation
processes
(i.e.,
microfiltration,
ultrafiltration, reverse osmosis, dialysis and electrodialysis) cover a wide range of particle sizes. It is also clear that membranes are capable of removing both ions and macromolecules from solutions (usually aqueous). Their versatility is matched only
by
centrifugal
processes.
The
later
processes,
however,
require an absolute density difference between the two phases to be separated or require them to be immiscible. Further, large scale separation of very small particles is difficult and costly by centrifugation due to high force requirement (e.g. 90,000 to 100,000 g), whereas ultrafiltration is relatively easy on large scale (Porter, 1988). Stress
requirements
on
the
bowl
of
the
centrifuge prevent the construction of a large ultracentrifuge, whereas it is relatively easy to construct a larger membrane based unit. Presently, reverse osmosis, ultrafiltration and dialysis are the widely used membrane separation processes.
REVERSE OSMOSIS Reverse
osmosis
separation process
(or
used
for
hyperfiltration) separating
very
is
the
small
membrane and
ionic
3
Ul
tlll
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Q.l
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·roi
+I
~fJ...
l'
>."
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en
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(jJ
+I
111
(jJ
(J
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Ul
~
~
i.L~
g 111
~
."::i CT '..-i ..J
.-t." ."CI'll IJ..
L-
111 L..0
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m ..:ti
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C 0
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toC
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4 solutions (particle size between 0.0001 to 0.001 mm or molecular weight
less
than
300).
In
this
process
the
pressure
on
the
solution side of the membrane is raised above its osmotic pressure which increases the chemical potential of the solution more than that
of
the
pure
solvent
and
hence
solvent
molecules
start
permeating through the pores of the membrane. A relation between osmotic pressure, chemical potential and other easily measurable parameters can be developed easily from the basic laws of chemical thermodynamics (Sundstrom and Klie, 1979; Cheryan, 1986; Porter, 1988). It is generally observed that when the solute molecules are of size comparable to that of solvent molecules, separation is difficult. However, in case of desalination process, where brine contains
ionic
species
of
size
comparable
to
that
of
water
molecule, as high as 97% salt rejection has been observed. This has been attributed to the fact that ions are repelled away due to preferential adsorption of water molecule at the membrane surface. Applied
pressure
forces
the
adsorbed
water
molecules
to
pass
through the pores whereas ions are retained in the solution. To attain
such
a
high
salt
rejection
the
pore
diameter
of
the
membrane should be less than four times the diameter of water molecule.
ULTRAFILTRATION Separation
of
solution
containing
particles
of
the
size
between 0.001 to 0.02tmm (MW 300 to 300,000) using membrane is called ultrafiltration. Here the size of solute molecules are normally greater than the pore size of the membrane. Due to larger molecules, the osmotic pressure
is
very
small
and
is
usually
neglected. Considering ultrafiltration relationship
negligible have
between
been solvent
osmotic developed flux
and
pressure, which
models
give
trans-membrane
a
for linear
pressure
difference. However, this is not always the case. Initially the solvent flux increases with increase in pressure difference but
5 often it reaches a maximum value and remains constant with further increase in feed side pressure. This has been attributed to the fact that during ultrafiltration, solvent permeation is high and molecules
having
low
diffusivity
are
used,
the
solute
concentration near the membrane surface often reaches a level where a viscous boundary layer, known as gel layer, is formed. The gel layer has relatively constant composition and do not change with process parameters. Formation of gel layer near the membrane surface offers a very high resistance to the solvent permeation. To eliminate the gel layer or at least to reduce its thickness, mechanical
agitation
or
shear
stress
(by
tangential
flow
of
solution) are used (Shen and Probstein, 1979; Porter, 1979, 1986; Cheryan, 1986).
DIALYSIS Dialysis
is
perhaps
the
most
successful
and
widely
used
medical application of membrane separation process. Dialysis and ultrafiltration is used to supplement kidney and liver functions. Hemodialysis is used to treat acute or chronic renal failure and for drug detoxification. In
artificial
circulated
on
one
kidney side
of
using the
(dialysate) circulates on the
dialysis, membrane
other.
The
patient’s and
blood
cleansing
blood
taken
is
fluid
from
an
artery is circulated between the membrane sheet and returns back to the patient through a vain, using pressure available from the patients cardiovascular system. The excess of ionic species and the metabolic wastes are removed from the blood stream to the dialyzing
fluid
through
diffusion
across
the
membrane
due
to
concentration difference. In hemofiltration, on the other hand, blood is filtered at a mild pressure. Plasma water and dissolved micro solutes are removed and proteins and other large solutes are retained (Babb et al., 1968; Bixler et al., 1968; Cheryan, 1986).
I-1.2 BIOLOGICAL MEMBRANES In living systems, there are a large variety of membranes and tissues having different functions besides the separation process
6 associated with them. The content of a cell are separated from the surrounding extracellular medium by a thin layer of lipids and proteins known as the plasma membrane. In most cells there are numerous other membranes which are part of the cell organelle mitochondria,
endoplasmic
reticulum,
lysosomes,
the
Golgi
apparatus, and the nucleus - which divide the intracellular fluid into separate compartments.
PLASMA MEMBRANE In spite of various functions associated with the different membranes, their general structure is similar. In these membranes protein molecules are imbedded in a bimolecular lipid matrix (6 to 10 mm thick). Normally these lipids are phospholipids (Fig. I-1.2) having a charged region at one end and two electrically neutral long chain fatty acids at the other. The individual phospholipids have considerable freedom to move in lateral direction making the lipid matrix more like a fluid than a rigid solid crystal like structure. Thus, unlike synthetic membranes, biological membranes are quite flexible and can easily be bent and folded, however, stretching may rupture their loose phospholipid associations. The bimolecular layer of phospholipids forms a barrier for the movement of ions and polar molecules across it, while some of the
proteins,
organized
in
clusters,
form
pores
(Fig.
I-1.3)
q====================================================================================================e
2
2
H
----------
2
H
O
p
~
C
----------
0
----------
2
2
C
CH
2
----------2
----------
1
CH
2
----------
CH
2
----------
CH
2
----------
CH
2
----------
(CH ) CH 2 n 3 ----------
2 2
1
2
1
2
H
----------
C
O
2
~
----------
0
----------
C
CH
2
----------
CH
2
O
1
1
2
----------
----------
CH
2
----------
CH
2
----------
(CH ) CH 2 n 3
z=========================================================================================================e
2
2
2
2
2
2
CH
2 ----------
1
H
----------
C
----------
0
----------
P p
H
O
----------
1
CH
2
-----
----------
CH
CH
+
~
2
----------
N
3
/ / ----------
CH
2
3
\ \
2
CH
3
2
2
z==================================================================================================================================================================================================================c
Polar end Fig. I-1.2
Structure of phospholipid molecule (Source: Vander et al., 1981)
----------
7
•
9"4
I
I-l
•
t» •••• ta..
8 through which small water-soluble molecules and ions can cross the lipid barrier. Lipid soluble (non-polar) molecules, however, can pass across these membranes easily through phospholipid layer.
TISSUES AS MEMBRANE In
addition
molecules
to
between
providing
the
a
barrier
intracellular
and
to
the
movement
extracellular
of
media,
plasma membranes are involved in interactions between cells to form organized tissues (by joining neighbouring cells through cell membranes)
which
in
turn
may
form
composite
membranes
like
epithelium, endothelium, intestinal wall, blood vessel, etc.. In tissues,
cells
are
packed
tightly
and
are
not
free
to
move,
however, they are not so close to each other that the adjacent cell surfaces are in direct contact. Usually, there exists a space of at least 20 nm between the opposing membranes of adjacent cells, this space is filled with extracellular fluid and provides the pathway for the extracellular diffusion of substances within the tissue (Vander et al., 1981). For a tissue like epithelium separating two compartments, such a large space would make a very leaky barrier that would allow even large protein molecules to diffuse between the two compartments. In fact, however, the intestinal epithelium in an adult is practically impermeable to protein, and the reason is the presence of "tight junctions" joining the epithelial cells near their luminal border. Tight junctions are formed by almost fusion of two adjacent plasma membranes so that there is virtually no space between adjacent cells in the region of the tight junction. This type of fusion extends around the circumference of the cell and greatly reduces (but does not eliminate) the extracellular route
for
the
passage
of
large
protein
molecules
across
the
epithelium. Therefore, in order to cross the epithelium, a lipid soluble molecule must first
cross
the
plasma
membrane
of
the
epithelial cell, pass through the cytoplasm, and exit through the membrane on the opposite side of the cell. Small water or plasma soluble molecules can, however, pass through the extracellular route. Passage of large macro-molecules across such a tissue takes
9 place through large pores created during mitosis or sloughing due to cell death (Weinbaum et al., 1985; Stemerman et al., 1986; Tsay et al., 1989; Yuan et al. 1991).
CAPILLARY Capillaries are responsible for feeding each and every cell of the body. Although, the structure of a capillary varies in different parts of the body, however, their general structure is similar,
i.e.,
a
thin-walled
one
layer
thick
fine
tube
of
endothelial cell, without smooth muscle and elastic or connective tissue.
The
endothelial
cell
layer
acts
like
a
permeability
barrier and modulates the exchange of nutrients and metabolic end-products between blood and neighbouring cells through adjacent vessel wall (Pappenheimer, 1953; Landis and Pappenheimer, 1963; Stemerman et al., 1986; Vander et al., 1981; Yuan et al. 1991). Apart from this the endothelial cell layer is responsible for regulating
several
crucial
physiological
processes
like
maintenance of vascular tone and modulation of platelet adhesion, production endothelium
of
vasoactive
derived
compounds
relaxing
such
factor
as
(EDRF)
prostacycline (Furchgott
and and
Vanhoutte, 1989) in response to specific agonist such as histamine (Hong
et
al.,
1985),
thrombin
(Hallam
et
al.,
1988)
and
adenosine-triphosphate (Pearson and Gordon, 1985 and Pearson et al., 1980).
I-1.3 OBJECTIVES From above discussion of membrane and membrane separation processes it is clear that membranes play an important role both in process industry and living systems. In spite of such a high potential,
the
membrane
separation
process
is
still
poorly
understood. The main hurdle in understanding the process seems to be due to its microscopic nature and influence of several other parameters. The present work is aimed at developing a tool to understand the process in a better way. A computer simulator for the membrane separation process has been developed and results are compared
10 with
those
solutions
for
obtained
experimentally
ultrafiltration
and
and
through
dialysis
in
the
analytical first
part
(Section-II) of the thesis. Applicability of the simulator in case of reacting biological membranes (epithelium) has been tested in the second part (Section-III). The advantage of such a simulator is that the process is simulated
within
the
different
parameters
velocity
profile)
computer
using
determined or
appropriate
either
empirically
equations
theoretically
(osmotic
for
(e.g.,
pressure
and
permeability coefficient). Reliability of the result depends upon selection of these equations. A proper selection gives results that agree closely with experimental values. Finally, analysis of these correlations not only gives better idea about the processes occurring near the membrane surface but also provides ready made tool to a design parameters.
engineer
for
the
quantitative
estimation
of
SECTION – II SYNTHETIC MEMBRANE
II-1
INTRODUCTION
==========================================================================================================================================================================================================================================================================================================================================
The osmosis, a membrane based process, has been known for more than two centuries. In 1748, Abbe Nollet observed that when alcohol and water were separated by an animal bladder membrane, the water passed through the membrane into the alcohol, causing an increase in pressure, but alcohol was not able to pass out into the water. Later during 1827-1832, R. Dutrochet invented the term "endosmosis"
and
"exosmosis".
Subsequently
the
prefixes
were
dropped and the word osmosis (Greek:push) and reverse osmosis were used to describe the flow of water across the membrane (Glasstone, 1946). The reverse osmosis was found to be useful for separation purposes, however, due to unavailability of membranes in plenty and
due
to
applications.
very
low
permeability,
Development
of
thin
it
had
skinned
only
limited
cellulose
acetate
membrane by Loeb and Sourirajan in 1959 marked the modern era of synthetic
membranes
(Sourirajan,
1970;
1977;
Dutka,
1981;
Lonsdale, 1982; Cheryan, 1986). Since then membrane separation technology has made rapid advances particularly in the areas of water purification biomedical applications, etc.. The
basic
requirement
of
a
membrane
for
its
industrial
applications are high flux, high permselectivity (i.e., highly selective to certain component of a mixture to allow permeation) high mechanical strength and ease of fabrication and use. With this in view, various kinds of polymers have been used for the preparation of synthetic membranes. Table II-1.1 shows different polymers used in membrane preparation and their applications in different types of membrane separation process. Although, considerable
the
membrane
attention
separation
and
process
application,
11
the
has
received filtration
12 characteristics problem
in
are
its
still wider
poorly
understood.
application
is
One the
of
the
main
concentration
polarization. A model describing concentration polarization was originally proposed by Brian (1966) for reverse osmosis and later it was extended to ultrafiltration processes by Michaels (1968). According
to
these
models,
during
reverse
osmosis
or
ultrafiltration, solute molecules are wholly or partly retained on the membrane which leads to a concentration gradient near the membrane surface. Due to existence of this concentration gradient a backward diffusion starts and finally a steady state is attained when the solute transport towards the membrane is balanced by solute passage through the membrane and back transport into the bulk of the solution. Due to the formation of concentration boundary layer near the membrane surface a considerable decay in the permeation rate is
Table II-1.1 : Polymers for MF/UF/RO Membranes Application UF RO
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Polymer
MF
---------------------------------------------------------------------------------------------------------
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Cellulose acetate(CA) r r Cellulose triacetate(CTA) r r CA + CTA blend Cellulose esters r Cellulose nitrate r Cellulose (regenerated) r r Gelatin r Polyacrylonitrile (PAN) r Polyvinylchloride (PVC) r PVC copolymer r r Polyamide (aromatic) r r Polysulfone r r Polybenzimidazole (PBI) Polybenzimidiazolone (PBIL) Polycarbonate (track-etched) r Polyester (track-etched) r Polyimide r Polypropylene r Polyelectrolyte complex r Polytetrafluoroethylene (PTFE) r Polyvinylidenefluoride (PVF) r r Polyacrylic acid+Zr oxide (skin) r Polyethyleneimine (PEI) + toluene diisocyanate (TDI) MF- Microfiltration, UF- Ultrafiltration, RO- Reverse Osmosis 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
r r r
r r r r
r r
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
13 observed. The fall in permeability has been attributed to the fact that the concentration buildup forms a more viscous layer at the membrane surface which offers an additional resistance to the passage of solvent and also it increases the osmotic pressure considerably
which
reduces
the
effective
pressure
difference
across the membrane. Solute rejection is also affected due to higher concentration near the membrane surface. Another problem arising from the concentration polarization is the irreversible membrane fouling. The general consensus is that membrane fouling is due to the deposition and accumulation of submicron particles on the membrane and/or the crystallization and precipitation of smaller
solutes
on
the
surface
and
within
the
pores
of
the
membrane itself. The fouling is caused by almost all components of the feed. Its nature and extent, however, depends upon the nature and
surface
chemistry
of
the
membrane
and
solute-solute
and
membrane-solute interactions. To reduce concentration polarization, several methods have been proposed. e.g., vigorous mechanical stirring or tangential flow
of
solution
near
the
membrane
surface.
Reduction
in
concentration polarization by tangential flow is further enhanced by using spiral channel membrane filters. In this type of filters shear stress near the membrane surface is further increased due to rotational motion of the fluid. Though, these are widely used techniques to reduce concentration polarization in order to have improved
filtration,
concentration
the
polarization
nature are
and still
extant unknown
of
the
effect
despite
a
of
large
number of experimental as well as theoretical investigations. Thus it is desirable to come out with a general technique for predicting the performance of synthetic membranes. In view of the above, in this part of the thesis, an attempt has been made to develop a computer simulator and compare its results with those obtained
by
analytical
ultrafiltration
and
characteristics
experimental of
cellulose
using aqueous solutions of various solutes.
investigations acetate
of
membranes
II-2.
LITERATURE REVIEW
==========================================================================================================================================================================================================================================================================================================================================
Separation of solid immiscible particles from liquid streams has been an important consideration in both industry and academia. Due to stricter rulings of the government and vigilant voluntary bodies,
manufacturers
minimum
damage
to
are
forced
the
to
environment
make
their
while
products
with
maintaining
their
competitiveness in the market. This needs either improvement in the efficiency of existing separation devices or search for new materials existing
and
methods
separation
for
units
effluent
treatment.
is
cost
less
Improvement
effective
while
in new
technologies like reverse osmosis (for separation of salt and low molecular weight substance), ultrafiltration (for macro molecules) and
microfiltration
promising
(for
particles
and
and
Albrecht,
1989).
(Rautenbach
reverse osmosis is mainly
used
(Wangnick,
microfiltration
1991)
and
the
for
colloids) Among
desalination and
are
more
these,
the
of
to
sea
some
water extent
ultrafiltration are still struggling to become widely accepted as viable
processes
due
to
poor
understanding
of
the
associated
fouling problems (Milisic and Ben, 1986; Finnigan et al., 1987). Ultrafiltration, however, is now being relatively more widely used in industry for separation (Tarleton and Wakeman, 1993) and has great promises for the manufacture of bio-chemicals using membrane bio-reactors
(Hoffmann
et
al.,
1985;
Hatch
et
al.,
1990;
and
ultra-
Jayaraman, 1993). Although, filtration
is
the
fouling
still
problem
poorly
during
understood,
micro-
however,
several
researchers have tried to reduce it to improve filtration rate. A large number of experiments (Madsen, 1977; Shen and Probstein, 1977; 1979; Leung and Probstein, 1979; Baker et al., 1985; Fane, 1986; Fischer and Raasch, 1986; Aimar et al., 1988; Cummings et 14
15 al., 1991; Ilias and Govind, 1993) have been carried out with flat sheet (rectangular channel or rotating disc) and hollow fiber (circular
tube)
membranes
to
study
the
shear
induced
ultrafiltration of several solutions. Experiments have also been carried out for gas-liquid systems (Matson et al., 1983; Cooney and Jackson, 1989; Kreulen et al., 1993) and ion-permselectivity (Tanaka and Seno, 1981; Sata and Izuo, 1989; Saracco et al., 1993; Saracco and Zenetti, 1994). Some researchers have tried to improve the filtration rate by imposed force (Wakeman and Tarleton, 1991) and rotational fluid flow (Holdich and Zang, 1992). However, the published information are so conflicting that even the effect of basic
parameters
such
as
cross-flow
velocity
and
filtration
pressure remains largely unresolved (Tarleton and Wakeman, 1993). The
impact
of
contradicting
results
can
be
easily
seen
on
theoretical work also (Merten, 1963; Sherwood et al., 1965; Gill et al., 1965; Matsuura and Sourirajan, 1981; Zydney and Colton, 1986; Davis and Leighton, 1987; Mansour, 1988; Rautenbach and Schock, 1988; Lebrun et al., 1989; Drew, 1990; Devis and Sherwood, 1990; Jaffrin et al., 1990; Romero and Davis, 1991; Kimura, 1992). Even
after
having
such
a
voluminous
experimental
and
theoretical information, no amicable solution has been arrived at. Smith
et
al.
(1991)
have
shown
that
the
best
existing
model
(Hoogland et al., 1990) also can only reliably predict the flux performance within two order of magnitude of the true value. Typically, during ultrafiltration, permeate flux decays with time,
however,
the
limiting
flux
(i.e.,
the
steady
state
permeation rate) is attained within minutes (Bhattacharyya et al., 1979; Aimar et al., 1988). Since the flux decay is so rapid, the steady state flux is more important than the transient flux and hence in most investigations only the steady state values are reported
(Akay
and
Wakeman,
1993).
The
rapid
fouling
of
the
membrane has been attributed to the concentration polarization or formation of more viscous gel layer near the membrane surface. At the steady state (pseudo- or dynamic- equilibrium), the rate of approach
of
molecules
or
particles
is
equal
to
the
rate
of
16 molecules being washed away by cross-flow. Romero and Davis (1991) termed this as "shear induced hydrodynamic self diffusion". In this model it is hypothesized that a concentrated layer is formed at the membrane surface and a part of it is stationary and a part mobile
depending
on
the
flow
condition.
Due
to
this
the
concentrated boundary layer is often termed as a dynamic membrane. Formation of such a dynamic membrane has been observed by Wakeman and Smith (1992) using a high speed, high magnification video system.
II-2.1 MATHEMATICAL MODELS MODELS FOR REVERSE OSMOSIS Two very different approaches have been used in developing reverse osmosis transport models. The first approach uses the principles of irreversible thermodynamics (IT) to describe the process and the second approach uses physical-chemical-structural (PCS) description of the membrane-solute system. I.T. Model The models based on irreversible thermodynamic approach are: Kedem-Katchalsky
Model
(1958),
Spiegler-Kedem
non-Linear
Model
(1966), Pusch’s Linear Model (1977), Bilayer Model (Kedem and Katchalsky, 1963), and Extended IT Model (Lui,1978). Among these Pusch’s linear model is the simplest and easiest to evaluate. Kedem’s two models (Kedem and Katchalsky, 1958 and 1963)
have
three
transport
parameters
each,
however,
Spiegler-Kedem (1966) non-linear model presents a more realistic picture of the reverse osmosis process. Extended IT model has six parameters and unless the partition coefficient is known it is difficult to use it. PCS Model The models based on this approach are- Solution Diffusion Model (Lonsdale, 1972), Pore Flow Transport Model (Faxen, 1922; Ferry, 1936; Merten, 1966), Finely Porous Model (Spiegler, 1958;
17 Merten, 1966), Combined Viscous Flow-Frictional Model (Jonsson and Boesen, 1978), Solution-Diffusion-Imperfection Model(Sherwood et al., 1967), Diffusion Flow Model (Yasuda and Lamaze, 1971), and Preferential
Sorption
Capillary
Flow
Model
(Sourirajan
and
Matsuura, 1982). Among these the solution-diffusion model, being the simplest, is the most cited and widely used transport model. It is linear and has only two parameters. It has been tested with a large number of solutions and is applicable to very dense membranes in which flux is relatively low and rejection is high. Some of these models are listed in Table II-2.1. Concentration Polarization The solute concentration near membrane becomes dependent of the fluid dynamics and mass transfer properties of boundary layer adjacent to it. Assuming a film theory model, under steady-state the solute transport within the concentration boundary layer can be written as (Sundstrom and Klie, 1979; Cheryan, 1986; Porter, 1988). -
D
dC ---------dy
+ v
w
C
= v
C
= v
(1-R ) C w w
w w
p (II-2.1)
is velocity of permeate through the membrane, D is the w diffusivity of solute, C , C , C and C are solute concentration o p w
where v
t
in feed stream, permeate concentration, bulk concentration and concentration at the membrane wall on the feed side, respectively and
Rw
is the intrinsic rejection defined as u o C 1 p 1 Rw = 1 1 - -------------------1 C m w .
(II-2.2)
Integration of equation (II-2.1) across the concentration boundary layer gives
19
&v / * exp w / K 7 / L8 = --------------------------------------------------------------------------------------------------------------------------------R w + (1-R w) exp7&vw///KL8*
C w -------------------C o
(II-2.3)
is mass transfer coefficient (=D/d) and d is the L concentration boundary layer thickness. For a tubular geometry where
K
under turbulent flow K
L
7
u
----------------------------------------------------------------1/4 2/3 Re Sc
(II-2.4)
where u is longitudinal velocity in the membrane channel. From equations (II-2.3) and (II-2.4)
(
and
)
(vw/ ) Re1/4 Sc2/3| C exp| A w 9 9 / u0 0 --------------- = -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------C ( o ) Re1/4 Sc2/3)| R w + (1-R w) exp|9 A 9(vw/ / u0 0 u 1-R o u v o u 1-R o w 1/4 2/3 w ln 1 ------------------------ 1 = A 1 ---------- 1 Re Sc + ln 1 ------------------------ 1 u R m R . m . m w .
(II-2.5)
(II-2.6)
where A is a constant of proportionality in equation (II-2.4) and
R
is observed rejection factor given by
R
u C o p = 1 1 - ------------------------- 1 C m o .
u 1 - R o By plotting ln1 ----------------------------- 1 m R .
(II-2.7)
(vw/ )0.75 a straight line 9 / u0 t (v )0.75 = 0 (infinite is obtained which when extrapolated to 9 /w/u0 feed velocity) gives
Rw
against
and C . w
Since there is a direct relationship between solvent flux and permeate velocity through the membrane equation (II-2.6) can be solved for the velocity and rearranged at a constant pressure drop to yield J
w
= K
u C - C o w p ln1 ----------------------------------- 1 L C - C m o p .
For a membrane with 100% rejection C
(II-2.8)
p
= 0 and
20
u C o w ln1--------------------1 L C m o .
J
w
= K
(II-2.9)
Equations (II-2.8) and (II-2.9) both indicate that film mass transfer coefficient K
is the controlling factor. Its magnitude L will depend upon flow situation (laminar or turbulent), operating parameters and system geometry. Several workers have attempted to present mathematical analysis of the problem for different flow geometries
(Dresner,
1964;
Brian,
1965;
Gill
et
al.,
1966;
Mastromonica, 1968). Gill (1971) and Hwang and Kammermeyer (1975) presented
a
summary
of
the
various
analyses
for
laminar
turbulent flow situations. It is generally observed that K
L
increased by increasing flow rate.
and
can be
MODELS FOR ULTRAFILTRATION: Pore Model: For membrane with uniformly distributed pores, no fouling, negligible
concentration
polarization
etc.,
Hagen-Poiseuille
equation for flow through channels may be assumed, i.e.,
(II-2.11)
= DP
/(R
J
(II-2.10)
= p e d
4 DP / (32 d m) p eff M
J
w
or w
eff
m)
o
The net pressure DP (DPt-tp(C)),
eff however,
concentration
osmotic
pressure
is
only
DP.
for an ideal membrane process should be in
case
pressure
of is
R
ultrafiltration
almost
in equation o resistance of the membrane such that
negligible
(II-2.11)
4 = (32 d ) / (p e d ) M p
R
o
The model does not et
al.
consider
(1974),
low
and
net
the
static
(II-2.12) the
concentration polarization, hence Blumberg
is
at
secondary it
however,
has used
resistances
limited these
due
to
applications. equations
for
calculating effective pore diameter of the membrane (Blumberg, 1986).
21
Film Model Film concept is one of the simplest and widely used theories for modelling the limiting flux. Rate of arrival of solute at the membrane surface is given by equation (II-2.1). Neglecting axial concentration gradient, the rate of back diffusion of solute may be given by (Cheryan, 1986)
D
-
dC -------------------- = J C dz w
which gives J
(II-2.13)
u C o G ln1--------------------1 L C m o .
w
= K
(II-2.14)
where C
is bulk concentration and C is the gel concentration. o G According to equation (II-2.14) flux (J ) can be improved only by w improving the value of mass transfer coefficient (K ) which L depends upon the flow situation and channel geometry. Several authors (Merten, 1966; Spiegler and Kedem, 1966; Sherwood et al., 1967;
Yasuda
and
Lamaze,
1971;
Lonsdale,
1972;
Pusch,
1977;
Jonsson and Boesen, 1978) have reported different correlations for evaluating K . Some of these are listed in Table (II-2.2). L Equation
(II-2.14)
suggests
that
the
flux
will
decrease
exponentially with increasing feed concentration. It also suggests that the flux is controlled by the rate at which solutes are transported back from the membrane surface into the bulk fluid. Osmotic Pressure Model: This model assumes that the deviation from pure water flux occurs solely due to the rise in osmotic pressure at the membrane surface (Cheryan, 1986; Porter, 1988), i.e., DP - p(C ) w = -------------------------------------------------w R o
J where
(II-2.15)
p(C ) is the osmotic pressure of the solute at wall w concentration (C ). For most macromolecules, the osmotic pressure w data can be expressed by the virial expression
22
N
N
o
o
o
o
......
VI
lV
"""
N.... ..•.•.. ...•....•
Il')
-
en
t"')
t"')
o ~ o
CO')
..•.•..
-.•.. lV
tIX)
o·
(00
~
~
Pol
"-
I
4)
P4 >
.
-.-f
S::0 ....-10
II .p ....-1014)
o s::~
00...-1....-1 G).p...-l >::S4) ...-II::: 01::: 'H co I'd o ,c 0 I::: I:::O~ 0....-1I'd rn.p...-l ....-I1'd::S ~.otll
0
.-I
CAp/np)
0
N
I
+U8!POj~
I")
I
Al!JOI8/\
-.to I
1'd~S:: PI::S1'd s.p.p O~O t.)4) 0 0 ~ ,0
0
I{)
I
0
,.....
-... U X
X •.....
•.... '0
•.....
x
f'.
II ~
'S;:
.iii :J
4-
:t 0
..•... C 0
.•... (f)
c 0
0 I
,,
~ 0 0
.
0
I
, it!
"Ct"
r--..
.••....•...
II
lH 0
~
s:: 0 ...-f +> it! 1-1 +> s:: Q.'I 0 s:: 0 0
bd
II
0
,S a1
0
0
~
"
A
?
r-i r-i it!
~
:~
it!
:t
(fJ
:J
0 f")
:t: .0
....--..
E
u
.........,.
V
X
.a
~
0
Q)
'C
u c:
0
>
0 0""'"
N.~ £:)
~
'CS IQ Q.'I 0 ~ +> 0 ~ ...-f II 'CS Q.'I ~ 1-1 .,r-i1X) +>Q.'IC"':) ...-fS:: >S::O ...-f it! II a1~~ ::s 0 lH lH 1-1t...-fit! 'CSr-i"l:f'
· ·
::s
II
Q.'I tG:r:: r-is::1J4 ,0 it! it! +> ...-f 0 r-i 1-IQ.'IS it! 1-1' > tG s:: lH...-f1Q CD 0 S::O +>...-f OSO Q.'I ::s II lH,o 0 lHr-it) rz:lit! ••.•.•. oft
0 •.....
·
0
0
I,{)
0
0 ~ 0
.
0
1"'1
0
0
. 0
N
0
•.....
. 0
(IW/6) UO!lOj'lUaOuo8 110M
0 0
0
r-• (Y) I Jool Jool
•
t1I
'"ta.
46
ui-1,1 .Ci-1,1 .dy + vi,2 .Ci, 2 .dx + D i,1 .C .dx/dy i,2 i,2 C = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------i,1 ui,1 .dy + vw.R .dx + D i,1 .d x /dy i,2
(II-3.16)
and n -1 ui-1,n .Ci-1,n .dy + D i, .C. dx/dy i, n i,n-1 C = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------i,n ui,n .dy + vi,n .dx + D i,n-1 dx/dy i,n
(II-3.17)
Although, equations for velocity components discussed earlier were developed for constant permeation velocity throughout the channel
length,
however,
in
the
present
computer
simulator
constancy of v
is being considered for a single strip (i.e. at a w fixed i) only, thus equation (II-3.12) becomes
t
&v *= &K * &DP * (II-3.18) 7 w8i 7 s8i 7 eff8i t & * is the local membrane permeability coefficient near where K 7 s8i th & *t is the net pressure the membrane in (i,1) element and DP 7 eff8it difference across the membrane wall [=DP t-tp(C ) ]. The osmotic i w i pressure
[p(C ) ] at the concentration C (evaluated by w i w extrapolating C , C and C to the wall) is determined by an i,1 i,2 i,3
t
appropriate equation listed in Table II-3.2 for different solutes in water. To calculate the value of
& * 7Ks8ian empirical correlation has
been developed using the computer program based on above approach
Table II-3.2 Correlations for osmotic pressure of different solutes -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Solute Osmotic pressure Source ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#
$
(176 * pH + 1524) exp{(1.04 * pH + 4.1)C }-1 3 w 4 Present work 9 2.49*10 -3 2 PEG ---------------------------------------- C + 7.2*10 C Brandrup and Immergut (1967) M w w 9 2.49*10 -4 2 PVA ---------------------------------------- C + 4.5*10 C Brandrup and Immergut (1967) M w w ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------BSA
47 and the present experimental data for polyethylene glycol (MW 9,000 and 15,000) and polyvinyl alcohol (MW 125,000) and the data obtained by Shen and Probstein (1977; 1979) and Madsen (1977) for bovine serum albumin. A detailed discussion about this correlation will be made in the results and discussion section of this part of the thesis (S II-5.2). In order to calculate concentration in individual element ) one may start from i=1 and j=1 (Fig.tII-3.5) and with
(C
i,j
C
t
=C . o
All
o,j
t
(II-3.15), C
i,j+1
t
variables
(II-3.16)
on
and
the
(II-3.17)
are known. To evaluate C
the first approximation C
right
i,j+1
t
i,j
t
at
hand a
side
of
particular
equations i,
except
iterative method is used. For
is taken to be equal to C
. i-1,j+1
t
It is observed that four iterations are safe enough to proceed further
to
next
i
(along
the
x-direction).
As
in
equations
(II-3.15) through (II-3.17), there is no term containing i+1 as a suffix, iteration over a single i (before evaluating concentration at i+1) gives the same result that were obtained by iterating over the entire channel length. Here, it should be noted that the solution technique and equations
for
concentration
profile
for
ultrafiltration
in
rectangular channel with permeable wall on one side or both sides of the channel are exactly the same. The only difference between them is that correlations for velocity profile [equations (II-3.1) through (II-3.4)] are different. Circular Tube In case of ultrafiltration in a semipermeable circular tube (i.e., hollow fiber) the problem is very similar to that explained in Fig.tII-3.1t(a) for rectangular channel except that the flow chamber is considered to be of circular cross-section and y is replaced by r (Fig.tII-3.1tb) and the fluid velocity profile is different [equations (II-3.8) and (II-3.9)]. Considering a ring element
(i,j)
of
radius
r,
thickness
dx
and
width
dr
(Fig.
II-3.4tb), all six mass fluxes associated with this element are shown in Fig.tII-3.8.
48 Again adopting similar procedure for circular tube one may get following equations for concentration in different elements (see Appendix-E, S E-2.2)
# ui-1,jCi-1 , j2rdr + vi , j+ 1 Ci,j+1(2r - d r)dx $ | | | | #(D i,j C i,j-1 )*2r | + + D C | 39 i,j+1 i,j+ 1 i,j i,j - 1 0 | | | ( i, j -1C i,j )*dr$dx/dr|| | + D - D C 3 9 i, j i,j-1 i,j+1 i , j+10 4 4 C = ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------i,j i,j-1) ui,j 2r d r + vi,j (2 r +d r )dx + 2r(9D ii ,j + D dx/dr ,j+1 i,j 0 (II-3.19)
ui-1,1 .Ci-1,1 .dy + vi,2 .Ci, 2 .dx + D i,1 .C .dx/dy i,2 i,2 C = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------i,1 ui,1 .dy + vw.R .dx + D i,1 .d x /dy i,2
(II-3.20)
and n -1 ui-1,n .Ci-1,n .dy + D i, .C. dx/dy i, n i,n-1 C = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------i,n ui,n .dy + vi,n .dx + D i,n-1 dx/dy i,n
(II-3.21)
dx 1J--------------------------------------------------L1 i,j
C1 Ddd----------r1
I 31 (vC)i,j+12p(r-dr/2)dx .2p(r-dr/2)dx ! i,j+1 !6 1 < u------------------------------------------------------------o p p p 1 ppp p 2 p (uC) 2prdr --------------------Lp C p--------------------L (uC) .2prdr i-1,j p i,j p i,j p p p p p p m------------------------------------------------------------. I 41 i,j-1 ! d C1 D---------2p(r+dr/2)dx!5 1 (vC) 2p(r+dr/2)dx dr1i,j < i,j
----------I 1 1 1dr 1 1 ----------
L[)"..d r-I v r-I V r-I00«S:4 «S «S....-l :4 OO'O~ ....-l 0 +l d I >......-ldO .~ 0 r-I «s+=' d....-l V «S d 01.' +=' ~t:fJ«s '0 d r-I vO')«SAi d....-l.c: ....-l it,) 0 r-I «S >.. v +lr-l r-I ..o«stHr-I 0....-l0«S va '0 :4 d «S o "H +=' Ai ....-l00 +=' V«s «SdtH :40"Hd +=,....-lV....-l d+=' v «S d Or-iVO d~.c:....-l OS+='+=' O....-l «S 0') :4 r-I 'O+l r-I:4dd «S V «S 01.' :l:+=' 0 "H &d§ 0 SOo O....-l d o+='v «S'O 0 0')S:4....-l ....-l o +=' it,) :4:4dl «S"H'3)'O 00 Ai S'Odo OdOr-l t) «S 0 ..0 1"""'1
,--
/ 0
/
0
c
oww
c 0
-+-'
::J 0
(f] 0
0
,--
u :;:;
>. 0
c
and R
1
becomes
1
and R
2
2 #__ _ + C N o e m A 3/2 b M
K
--------------------------------------------------
1
This indicates that when shear stress is low [i.e., when
to be at
(II-5.9) a
du dy
1
1
---------1
is
1
less than the term on the right hand side of inequality relation (II-5.9)], at low pressure the overall resistance becomes equal to the
static
resistance
(R ) which increases with increasing 1 pressure. Finally when the overall resistance becomes very high so that the permeation velocity becomes very low (thereby making concentrated layer loosely packed), the effect of shear stress and hence dynamic resistance comes in picture. On the other hand, when shear stress is high, there will not be any static fouling layer
86 near the membrane surface. Figures II-5.17 and II-5.18 show the effect of change in molecular weight on the different resistances at constant value of (C N /M) (i.e., at constant value of number of molecules per unit w A volume) and at constant mass concentration (g/ml), respectively. In both these cases it is observed that with increase in molecular weight
the
overall
resistance
(R ) varies from the static s resistance (R ) to pore resistance (R ) following the dynamic 1 o resistance (R ). In case of constant mass concentration, the 2 static resistance is independent of molecular weight whereas for constant value of (C N /M), R itself increases linearly with w A 1 increase in molecular weight. Figure II-5.19 shows the variation of all three resistances with mass concentration (g/ml). It is observed that the value of R
is equal to the pore resistance for very small mass s concentration and it is equal to the secondary resistance at higher values of concentration.
II-5.3 OSMOTIC PRESSURE As explained earlier in this section, osmotic pressure has considerable
effect
ultrafiltration.
The
on
the
permeation
correlations
for
velocity
osmotic
during
pressure
of
polyethylene glycol and polyvinyl alcohol has been obtained from standard text. All the correlations used in the present work are listed in Table II-3.2. However, in case of certain proteins (like bovine serum albumin, BSA) even the equations with two virial coefficients do not give satisfactory result. Vilker et al. (1984) observed that concentration and
pH
have
strong
effect
on
the
osmotic pressure of these proteins. During present investigation following empirical correlation for the osmotic pressure of BSA has also been developed in terms of concentration and pH on the basis of data obtained by Vilker et al. (1984). p(C,pH) = (176WpH + 1524) [exp{(1.04WpH + 4.1)Wc} - 1] (II-5.10)
87 where
C
is
the
concentration
(g/ml)
and
p(C,pH)
is
osmotic
pressure (Pa). Comparison of Vilker’s data, present correlation, equation
proposed
by
Leung
and
Probstein
(1979)
and
osmotic
pressure obtained using two virial coefficients are shown in Fig. II-5.20. It is clear from this figure that the present correlation is best among these and it covers wide range of pH.
II-5.4 EXPERIMENTAL RESULTS During
present
investigation,
permeation
velocities
of
aqueous solutions of different polymers (polyethylene glycol, MW 9000
and
15000
and
polyvinyl
alcohol,
MW
125000)
have
been
obtained. A few experimentally determined permeation velocities (point
values)
equation
and
(II-5.5)
calculated and
computer
values
(continuous
simulator,
are
lines)
shown
in
using Figs.
II-5.21 to II-5.29 for different values of tangential velocity and inlet concentration. All the experimental runs were conducted in a spiral channel membrane separation unit. Although, the computer simulator has been developed for a hollow fiber or rectangular channels only, but the predicted values are found to be reasonably good even in case of spiral channel. To calculate permeation velocity, the channel height is assumed to be equal to the height of a channel having
the
same
equivalent
hydraulic
diameter
as
the
spiral
channel used. The length of the channel is taken to be equal to the total length of the spiral path. Due to secondary flow in the spiral channel, the permeation velocity has been reported to be higher, however, in the present work no attempt has been made to predict this secondary flow pattern. The effect of secondary flow might have been adjusted in the exponent (a) of the velocity gradient term in the equation for dynamic resistance (equation II-5.4). Comparison of experimental observations and calculated values is shown in Fig. II-5.30. Most of the data points obtained during present work and those obtained by other investigators lie within
+t20% error band.
~ , ",,",
0.0
o Ii
o. ,
I
1
t--r:': Ii
-, I
0.3
Concentration (g/ml)
0.2
I I
----
•
I
...
7?7C• ~ -
pH=7.4 pH=5.4 'il pH=4.5 - - - - pH=4 .7 (Leung and Probstein. 1979) Present Work
•
o
f
• •
o
0.4
I ill
I
2 \li('\ol Coe/I.
•
'"
I
0.5
I I
Fig. 11-5.20 Comparison of the correlation [equation (11-5.10)] with the experimental data (Vilker at al .• 1984) of osmotic preS8ure of bovine serum albumin
0
E
.,
0
~
0
1l...2
\...
\...
'
100000 150000 Applied Pressure (Po)
---
"
" " "
D
0
200000
1.72 em/s 2.44 cmjs 3.15 cmi's
Tangential Velocity
o
~
c __ _ "'D-
D
"
250000
- -------
Computer Simulator
o
"
o
_...- _ _ _ _ D
-8
(Co = 0.005 K/ml; h= 0.3171 om; Ko = 1.1X10
-
om/Pa.s)
Fig. 11-5.22 Comparison of the experimentally observed av.ra~e permeation velocity of aqueous fEG BOOO solution in sp1ral channel filtrat10n un1t with those predioted by the oomputer si~lator far reotangular ohannel of similar e~u1valent hydraulic d1ameter (h)
(L
w
E '-
w
:g
o
c
o w >
'()
~
"--'
u 0.020
"'E"
E
C
r--...
0.030
'"o
,
Fig.
Il.
(IJ
E L
Q>
~ o
o
c
>
Q>
o o
:':::
>,
() '../
o
,
0
./"
",,/ ./
D
___
.--
-6
o .,-
~-
0
--
0
C
200000
e m/s em!s
1.72 cm/s 2.44 3. 15 4
0
o
-
0
'-
langent;ol Velocity
0
-11
0
Expe ri mental Data Points
100000 1500 00 App lied Pressure ( Pa)
0
/~ _________....-----o-..-
6/
m/ , / '/'
50000
J
,
~~
~
---
0
a
0
25000 0
Simulator
Computer
0
,
-0-- - -
(C o = 0.007 SimI; h= 0 . 3171 em; K0
-
= 1.7x lO -8
cm/Pa.s)
observed average permeation velocity of aqueous PEG 9000 solut ion in spiral channel filtration unit with those predicted by the computer sim~lator for rectangular ohannel of similar e quivalent hydraulic diameter (h)
0:
t
i
1... 1
II-5.2:3 Comparison of the experimentally
0.000
0010 .
E 0. 0 20
'-,
E
c
.......
0.030
-'"
0.000
(
/
I
I
I
/
I
I
!l
/0
;"
I
I
o
ri i
1/
I
I
'/
I
Q
:J
~
f1
'-
(l)
....,
/
i
,
, J
/
/
0
6
0
/
/
/
tlOOOO
I 11 -1 I
/
/
/
,
---
I i i
100000
I I I
o
"
I
average
-
a
--:.
0
I
I
at
with
of
those 5imilar
ve l ocity
250000
i -I
-- --
permeation
-
Simulator
Computer
a
4.......
aqueous PEG 15000 solution in spiral channel filtration unit predloted by the comp~ter 5im~lator tor rectangular ohannel equivalent hyd~aulio diameter (h) -8 (C = 0.003 g/ml; h= 0,3171 om; Ko : 1,7x lO om/Pa.5) o
obse rved
I I I
200000
I
3.15 cm/s 0
I I i i I I
1. 72 cm/s
2.44 cm/s
_
~
150000
I I I I i I I
~
o
- ---- -
Tangent ial Velocity
o
_ _6
o
0
Experimental Data Points
e
"-
o
o
--- ---
App lied Pressure ( Po)
j
Fig. II-5.24 Comparison of the experimentally
[L
(I)
L
E 0.005
Q)
:.=; o
o
c
>
Q)
00.0 10
U
+-'
>,
~
o
E
~
E 0.0 15
c::
.-..
0.020
N
'"
0.000
o
o
;:,
L.
0)
?o
I
/
I
A /
I /
/
/
/
Q
"
/
/
,
50 000
/
/
~
0/
/
/
o
A
r
/0
~
-
o
-
A
--"
_0
0
A
200000
-
250000
----- -
-
-
-
-
1.72 cm/s 2.44 em/s 3. 15 cm/s
."..
D
0
A
0- - -
Com pute r Sim ulator
-
Tangen tial Velocity
o
.
0
Expe rimenta l Data Point s
o
'"
is'
100000 150000 Applied Pressu re (Po)
--
~
o
- t'J - -
observed average permeation velocity of aqueou 5 PEG 15000 solution in spiral channel filtration unit with those predicted by the computer simulator f o r rectangular channel of similar equivalent hydraulic diameter (h) (Co: 0 . 005 g/ml; h: 0.3171 em ; Ko : l . Tx 10- 8 cm/ Pa.s)
/
o /
'/O I ,/
Q
I/.
I
~
I!)
, .>
Fig. II -6.26 Comparison of the experimentally
0..
(l)
E 0.005 L
Q)
o
:;::;
o
c
>
Q)
2 0. 01 0
u
>,
....
'-.,.;
u
E
"-...
E 0.0 15
r--. C
0 .020
'"w
/
/
/
/
/11
"
/ 0 ,/
/
50000
/
/
/
" ~
~
11,
c
~
--
"
~
-- -[>
0- -
o
/l-
0
a
A
0
Experimental Doto Po ints
n
--6. - -
o
100000 150000 Appli ed Pressu re (Pa)
/
~
~-
o
1.72 cm/s
200000
3 .1 5 em /s
2.44 em/s
o
-/1
D
-
-
250000
-
--- - - -
Comput er Simulator
o
o
- - ---- --
Tan gential Velocity
'"
11
o
Fig. 11-5.26 Comparison of the experimentally observed average permeation velOCity of aqueous FEG 15000 s o lution in spiral channel filtrati on uni t with those predicted by the computer sim~lator for rectangular c hannel of similar equ ivalent hydrauli c diameter (h ) -8 (C = 0 .00 7 s/ml; h= 0, 31 71 cm; Ko = 1 . 7x iO e m/ Fa, s) o
o
I/.
/ /0 1/
1 11 /
/
0...
I
o / I
0.0 00
:J
L
OJ
0 0 ,010
()
.....>-.
'-/
0
E
"-.
E 0.0 15
c:
r---
0 ,020
'"""
0 .0 03
0 .0 00
0.00 1
o
D
I
r
I /
I
I
:::J
1:' a..
~
,I
I
I
I
L Q) ~
/
I
/
o
/'
0
/
/'
0
I>
--
50000
/'
Cl
0
-0
[J
0
I>
D
1500 0 0
[J
200000
1.72 em /s 2.4 4 em!s 3. 15 em / s 0
t.
0
I>
Tan'j entiol Velocity
0
I>
a
0
-
0
I>
u
-
-
25 0000
-
-- - - - -
Comput er Simulator
0
I>
0
-- - -- - - - - -
Exp eri ment al Dot o Pcints
0
I>
App li ed Press ure ( Po)
100000
0
"
0
0
-- --
observed average permea tion ve l ocity of aqueous PVA 1 25000 s olution in spira l channe l filtration unit with those predicted by the c omputer simulator for r ectangular channel of s i milar e quival e nt hYd raulic d iameter (h ) -8 ( Co~ 0. 00 3 g/ ml ; h = 0 . 31 7 1 e m; Ko = 1 . 7x 10 c m/Pa . s )
I
/
/
/
/
/
Fig. II-5.27 Comparison of the experiment a l ly
J:
E L
(j)
0
:,::;
0
c 0 .002
>
(j)
0
'0
...,
>.
'-'
()
"E
E
'20.004
0,005
'" Ul
0.000
o
a...
:::J
~
o 3:
\...
V ....,
/
/
~
r
/
b
II
50000
---
"
0
II
100000
0
-ll
0
0
II
o
0
II
°
250000
Com puter Simulator
0
II
0
observed average permeation velocity of aqueous PYA 125000 solution in spiral channel filtration unit with those predicted by the computer sim~lator for rectangular channel of similar equivalent hydraulic diameter (h) -6 ( Co = 0.005 g/ml ; h= 0.3171 cm; Ko 1.7x lO cm/ Pa.s)
200000
3.15 c m/ s
1.72 c m/s
2.44 c m/s
o
Tangen tial Velocity
0
II
a
11
150000
=
0
II
°
Experi menta l Data Points
0
II
E-
Appli ed Pressure (Po)
-------
", 6-
g -- -
_ D_
Fi g. II -6.28 Comparison of the experimentally
0....
Q)
E 0.00 1 '-
Q)
o
-+-'
o
c
>
Q)
o 0.002
U
-+-'
>,
'-.../
u
E
""""'-.
"""" c E 0.003
0 .004
'"'"
o
I '
!I'
(,
1&
/
I /
j
Iff!
"/
~
50000
/
/
/
/
---
/
.-r.
[J ~
0
I
II
I I
150000
I
0
A
0
o
0
App li ed PresslJ re (Pa)
1COOOO
0
0 .34 e m/g 0.4B cm/s 0 .62 em/s
Tangentia l Ve locity
0
A
0
200000
I I I I 1' 1, -1
-- ~-
Expe rimenta l Data Points
a
0 A
- A
I I III 1 I I 1 I I I I
a
IV
[J
/
0
[J
--
" o
A
-
I
-
II
of with those of similar veloci~Y
250000
j
-
-- - -- -
Si mul at or
Com puter
o
A
[J
- --
exper1men~allY observed average permeation aqueous PVA 125000 solution in spiral channel filtration unit predicted by the computer simulator for rectangular channel equ1valent hYdrau11c diameter (hl -6 (C = 0.007 g/ ml; h= 0.3171 cm; K = 1.7xl0 om/Pa.s)
j
/
/
,,"
/
/
fig. 1l-5.E:9 Compar1son of the
0...
1/
0
1/
I
."
/
I
"-
•..
x
e
•..0
8:
~ b b 0
"'
.."~ +'
,., 0
...•
"•..
"
+'
"§" •• ;; 0
01
c
.".,
Ul
Cl> U
L..
(; 1·25
o
1·50
with 1)J.m ATP
4 dynes Isq em
1.75r,-----------------------------------------------.
~
w 00
1.20
O.BO
0.00
,
~--
-1 _"*, _
0.20
~-
*- -
.. .>t--)
-
--
..- ...... .....- -
==;::-:::::-:: ...... - - --- ---+- - - ~ - - -
~_
0.40 0.60 Dimensionless Dista nce
i u __ =*~-
...-..--
---
I
.... .,..-
/'
....:t -
,"-
-
/'
-
"
I -'-''''--'-'-'"
0.80
r,
~ --*_ ~ _
--*---
_>to '"
,....-* ... /
./
,Y-
/'
y/
/
1.00
....
-~
!IV
concentration (cl as a function of dimensionless radial distance (~l predicted by equation (111-3.38) and the oomputer simulator at four negative values of ~ (~=O.ll
-
IjI
-0 .10 -0.50 - 075 - 1.00 (1/ = 0.1)
Fig. 111-4.5 Dimensionless
0
E 1.00
Q)
c
(f)
0
C
CD
(f) (f)
U
0
c
Q) ()
c
~
C; 1.40 ....
0
c
0' 1.60 u '--.. u '-'
1.80
~
W 0.0
-'1>-' -
_. -
---*~--
" .--,r
•
1
0 .00
I
I
I
I j
--- ~
0.20
I
~~--~
I 0 .40
I i
I
I I
I
I
I I
--'~
I
I 0 .60
I
I
I
I
I I
I 0 .80
I
I I
I
1.00
I
-~ - --~--~ - --~---~
Dim ensionless Distance (>J
I
~---~-
.......
__ _ ~:r--------1f-
-- -'
•
*
concentration (c) as a function of dimensionless radial distance (~) predicted by equation (III-3.38) and the computer simulator at four negative values of ~ (~=O.5)
j
--
-~ ~~: ~ ~ ~~ :~ ~~~ :~ ~ :~:~~=~-..-" -,-' .--- --.-------,,-
(1/
lfI
- 0 . 10 -0 .50 - 0 .75 -1.00 0.5)
Fig. 1II-4.6 Dimensionless
0 .50
Ci 1.00
E
Q)
c
(j)
o
c 1.50
(j) if) Q)
o u
g 2.00
C
+-'
L.
o
+-'
o
c 2 .50
'-'
u
'-.,.
u
o
.---.. 3.00
3.50
""
~
o
141 axial positions (h=0.1 and 1.0, respectively) as a function of dimensionless radial distance (l) at different values of j when permeation
velocity
is
larger
than
the
surface
reaction
rate
(i.e., when j is negative). Results obtained by the analytical solution are shown by lines and those of the computer simulator are shown as point values. In these figures it is seen that the concentration near the non-reacting surface is almost equal to the inlet concentration (Fig. III-4.5) and it increases near the cell surface. Further down the length of the chamber (Fig. III-4.6), the
concentration
of
ATP
is
higher
and
consequently
the
concentration gradient near the wall is decreased. This indicates that less amount of ATP is diffusing back in the bulk and hence higher wall concentration is observed. Figure III-4.7 shows the dimensionless wall concentration as a function of dimensionless axial distance (h) at different values of j. It is seen that the wall concentration is almost equal to the inlet concentration up to h=0.01 beyond that it increases rapidly. Also, it is clear from this figure that when j decreases (i.e., when permeation velocity increases) the wall concentration also increases at all values of h. Figure III-4.8 shows the dimensionless concentration at the cell surface as a function of dimensionless distance (x) in case of complete rejection (R =1). It is clear from this figure that at w N=1 i.e., when permeation velocity equals the surface reaction rate term, concentration in the flow chamber predicted by both analytical solution [equation (III-3.38)] and computer simulation (without
neglecting
radial
velocity
term
in
the
bulk,
see
Appendix-E, S E-1.3) remains equal to inlet concentration all along the length [for a relation between L and x see Appendix-E, equation (E-7.5)]. Also, the wall concentration predicted by both methods are in excellent agreement in case of reaction without permeation [equation (III-3.14) and computer simulator with v =0]. w However, for the coupled reaction-permeation problem, the wall concentration deviates considerably at higher values of x. When permeation
rate
is
smaller
than
the
reaction
rate
(N=0.75),
analytical solution under predicts the wall concentration whereas
142
o
'I'
oQ) ~ 0
c o
-+-' (j)
o (j) (j)
Q)
c ,., 0 I ._
o
(j)
~
Q)
c
E o
OOLDO
'1
~ '""":Ll.!r:q
o
000...-
I I I I
o o
o o
o .q-
to (OJ/MJ)
o o
o
o C'J
UO!fOJlU88UO:)
110M
SS8IUO!SU8W!O
o
o o
143
~ ...-i
'HOC O~ ~
~
ro ..••.. 4)
ro
r-l'"
r-l
)
~ ~ ~
(J)
,:::
Q)
0
E
4)
,:::
0 0 r-l r-l
ro ro
4) r-l
,:::
0
...-i ro
U1
...-i A
4)
:::J
S
00
I
OJ 0
r:I$
,:::
0
0 0
'"
::s
'H
't:
0 .q-
,:::.-1 •
~ I 0 ...-i 'O;fl •••• II •••• ~ 0 N •••• ~ ~ •.....•
~ ~
c'-
0
to
Q)
-
00
0 0 0 CillOQ)cQ)O u u u 0 ............ 0 ............ 0 ............
E ~-::s 0
Q) :J
n
Q)
E
:;:;
(f)
c
..•...• C :::;".-..,. :J 0
E c'-
(f)
a
~3=
C 0 ..c.~ ~ :::;~:;:; ~ :::; 3: 3::::J 3:
0
'~ r:I$
ro 0
...-i
+>
r-lr:l$
::s
...-i
o II
;= C.......
~>;':::J
Q)
0
:;:;0
:.;::;Dc E .- ........... C:::Jl.Lc..co...co..co
~ 0
..•...• 0 l....
+-'
..•....
c 0... C 0 Q)~ :;::; ~ 0
0
I
0
'0
..-(J)
C 0 ............
C 0
c
..•.... +-' en:::;
'H~ ~r:I$
.c:>..
0
••••
4) ~CJI ,::: 4)
'"
r:I$
0
•
~ I ~ ~ ~
•
C)
••••
t&.
~
0..0
'0
~ IV ~+>
r-l0 ::$...-i ~'O ~
4)
r:I$ ~
+>At 0
4) ~4)
ro
...-i
+>~
r:I$...-iO
~g~
...-ir-lr-l 4)::S ..••.. >S
...-i
loV
••.•..•~ ro 0
IV...-i~
0+>4) ':::r:I$+> r:l$4)::S +>SAt ro~S ...-i4)0 ocAto
144 in case of large permeation rate (N=1.1) this method over predicts the
concentration.
This
is
due
to
the
analytical solution, radial velocity liquid
has
been
neglected
while
fact
term
axial
that
in
case
of
in
the
bulk
of
(v)
velocity
(u)
has
been
assumed to be decreasing along the x-axis [equation (III-3.28)]. This leads to a net accumulation of mass (and solute) in the bulk. Also, absence of radial velocity term neglects the solute movement towards the cell surface by convection. Thus when v
is large, the w error due to accumulation of mass is dominant which reduces back diffusion and hence a higher wall concentration is predicted. On the
other
convective
hand,
when
term
is
concentration
vw is small, error due to absence of
pronounced
prediction.
This
which
again
leads
to
shows
a
that
lower
wall
analytical
solution has serious limitations whereas computer simulator gives better result for a wider range of problems.
III-4.3 CIRCULAR TUBE: SURFACE REACTION WITH OR WITHOUT PERMEATION Figure III-4.9 shows the dimensionless concentration in the bulk of liquid as a function of dimensionless radial distance
gt(=r/R) in a circular tube at three different axial positions (x=0.05, 0.1 and 0.5) assuming first order surface reaction. Dotted lines are obtained by the solution proposed by Mansour (1988)
assuming
diffusion
controlled
surface
reaction
with
permeation. Solid lines are obtained by the present solution given by equation (III-3.27) and results obtained from the simulator are shown as point values. It is observed that the concentration is always lower when diffusion controlled surface reaction is assumed and results of the simulator are in excellent agreement with the analytical solution. A comparison of the dimensionless wall concentration obtained by the computer simulator and analytical solution in circular tube at
f2=4
is
presented
in
Fig.
III-4.10
as
a
function
of
dimensionless axial position (x). Results obtained by two methods -4 are in good agreement, however, for very small x ( c
X
«
Q) Q1 (l)
r--.
I')
I
..- c
0).-
0
-o
...I
L
0) O-+-'
.-
::J
0
-+-'0...
;E
000
(OJ/MJ)
N 0
UO!+DX).U8~UOJ IIDM SS8IUO!SU8W!O
~~
01 01
....o:s
S
V..-f
((
N~
~
o:s
v
,0 J.t
s::
....o:s
..-f4)
0 J.t
o ~
::s
01~
vp.
0
..-f0
o:s
"00
"0
r+-fS::
o o:s s::or-
..-fs::
'•....•. V
..-f~C") . 0s:: o:s V
01c\]
••• o o
'o:t'
::s ::s ~
•••• 01
0
o
~
....•
P.H SH OH 0-
0
~
o:s
•..• J.t «S 0
«SI
c 0 «0
ro
0 0 01
S S
E
:2(J)
o:s
s::
(l)
c
...•....E
s::
4) 0
::s
(f)
-0 0 0-
..c::l
J.t
~r+-f
s:: ::s ..-f
0
:;:;
o
(f) (f) (l)
0-
c
'-../0
.•... ~
(f)
0
::l
Lf)
c
~o «S s::
•
"d'I
1-4 1-4 1-4
•
0
'1"1
e-
~
01
..-f
"0
147 The
dimensionless
wall
concentration
reaction coupled with permeation (with
Rw=1)
in
case
of
surface
and negative value of
j (=-0.1) in a circular tube has been plotted in Figs. III-4.11 and III-4.12. These figures show that when the concentration is plotted
against
dimensionless
distance
L
(Fig.
III-4.11),
different lines for different values of a are obtained, whereas when the same concentration is plotted as a function of h (Fig. III-4.12) we get a single line (for relation between h, x and L see Appendix-E, S E-7.3). In Figs. III-4.11 and III-4.12, it is observed that the wall concentration predicted through analytical solution deviates considerably in the entrance region (x or h < 0.05).
On
the
other
hand,
results
predicted
through
computer
simulator are in good agreement with the analytical solution for x or h > 0.05 and are realistic (i.e., C/C
o
~ 1) near the entrance.
148
-----0---
_
---
- -0-
-
-
o
-
,g
t I t~~~~/~~~~~~~~~~~~~~~~~~~~
I
I
0.00000000' 0.00000000 12.64252879 17.19955115 50.17199082 67.35739105 111.81813651 149.64652540 197.52824913 264.00549454 307.27941679 410.40880549 441.05955377 588.84322096 598.86141211 799.30087835 780.67748957 1041.7754766 986.44382943 1315.4899771
I
~
Table A-I Eigenvalues (~~) of the characteristic equation (A-21) for different values of ~ and a
1 2 3 4 5 6 7 8 9 10 11 12 13
f n
a :: 0.05
1.49998879 4.87779854 8.99280546 14.11984320 20.46620994 28.17109326 37.32725450 47.99492827 60.21122670 73.99799113 89.36774086 106.32567736 125.09355386
« :: 0.05
1.49997462 4.87654026 8.97665678 14.03953301 20.24054641 27.73138778 36.65067330 47.09756051 59.12518758 72.75587013 87.99681140 104.85230470 123.39961744
I
II
9 10 11 12 13
B
1 2 3 4 5 6 7
n
Ie
I
a :: 0.50
1.38022545 6.99094161 19.76050948 40.47650922 69.18165981 105.88132266 150.57725585 203.27038021 263.96115302 332.66319489
« :: 0.25 1.20213026 10.98506889 36.36058579 77.72529480 135.08463174 208.44038340 297.79354607 403.14474566 524.49440192 661.75332044
« :: 0.50
W
fJ