WEBVTT
Kind: captions
Language: en
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ok welcome to the session so will be now we
are looking into the some simplified versions
00:00:23.230 --> 00:00:30.910
of osmotic pressure control filtration and
we have looked into the first simplified case
00:00:30.910 --> 00:00:35.660
there is the no concentration polarization
. and we have looked into the system how the
00:00:35.660 --> 00:00:41.390
system can behave and prediction can be obtained
the second simplified case will be talking
00:00:41.390 --> 00:00:54.330
about is low polarization . and low polarization
can be realized by v w by k exponential
00:00:54.330 --> 00:01:01.590
v w by k is of a in the in a . it gonna
be realized . when . v w by k is much much
00:01:01.590 --> 00:01:07.330
lesser than one that means again in this case
turbulence is quite high so the mass transfer
00:01:07.330 --> 00:01:11.970
co efficient is . so reynolds number will
be quite high very high and therefore the
00:01:11.970 --> 00:01:17.080
mass transfer co efficient will be high compared
to the permeate flux so v w by k will be much
00:01:17.080 --> 00:01:23.400
much less than one and that will allow
us to have an exponential um . expansion of
00:01:23.400 --> 00:01:32.080
e to the power v w by k as one plus v w by
. k plus v w by k square of that one over
00:01:32.080 --> 00:01:38.750
factorial two here plus other higher
. terms we neglect the since v w by k itself
00:01:38.750 --> 00:01:44.740
is less . less much much less than one v w
by k square on the higher terms will be neglected
00:01:44.740 --> 00:01:51.890
and this can be approximated as one plus . v
w by k now what we will be doing next . we
00:01:51.890 --> 00:01:59.860
will be doing will be writing the um
. film theory . the film theory if you remembered
00:01:59.860 --> 00:02:07.750
. this is c a minus c p divided by c naught
minus . c p e to the power v w y k and this
00:02:07.750 --> 00:02:11.680
will be replaced by one plus v w by k
00:02:11.680 --> 00:02:18.280
so these a simplified fraction of the film
theory and then will be writing c p as c m
00:02:18.280 --> 00:02:24.620
into one minus real retention . and then will
be writing the osmotic pressure model as
00:02:24.620 --> 00:02:33.370
v w is equal to l p del p minus del pI and
we have seen earlier how del pI can be replaced
00:02:33.370 --> 00:02:40.340
in terms of real retention and membrane surface
concentration c m now combining these three
00:02:40.340 --> 00:02:52.700
equation will be getting a simplified version
as v w is equal to l p . del p minus b r r
00:02:52.700 --> 00:03:08.430
c naught one plus v w by k divided . by r
r . plus one minus r r into one plus v w by
00:03:08.430 --> 00:03:14.680
k now in this case what will be ultimately
getting so ultimately you will be getting
00:03:14.680 --> 00:03:26.280
a quadratic in v w .
so all the other quantities are known to us
00:03:26.280 --> 00:03:31.830
l p is membrane permeability del p is the
trans membrane pressure drop b is the osmotic
00:03:31.830 --> 00:03:36.819
co efficient known to us r r is real retention
known to us c naught is the fit concentration
00:03:36.819 --> 00:03:41.040
k is the mass transfer co efficient that will
be the we can calculate and we have seen how
00:03:41.040 --> 00:03:42.990
to calculate the mass transfer co efficient
00:03:42.990 --> 00:03:52.370
so we can calculate the v w and from the v
w one . estimate the so one . can
00:03:52.370 --> 00:03:58.730
directly get the expression of c m because
. c m minus c p divided by c naught minus
00:03:58.730 --> 00:04:05.230
c p can be expressed so v w . can be . calculated
independently let us see now how c m will
00:04:05.230 --> 00:04:12.250
be calculate c p will be calculated . so if
you look into the osmotic you know um um um
00:04:12.250 --> 00:04:17.970
in expression film theory . . c m minus c
p divided by c naught minus c p is equal to
00:04:17.970 --> 00:04:25.289
plus v w by k we have already got an explicit
expression in quadratic form of v w so therefore
00:04:25.289 --> 00:04:32.919
these c m can minus c p can be replaced
in terms of . c m so this will be nothing
00:04:32.919 --> 00:04:40.680
but c m times r r because one of the definition
of real retention one minus c p by c m so
00:04:40.680 --> 00:04:45.280
c m minus c p is nothing but c m time r r
. and c naught will be nothing c p will be
00:04:45.280 --> 00:04:51.940
the denominator c p can be replaced c m
into one minus r r . is equal to one plus
00:04:51.940 --> 00:04:57.440
v w by k
so just rearrange c m so this become c m r
00:04:57.440 --> 00:05:07.970
r is equal to one plus v w . by k times c
naught minus one minus r r one plus v w by
00:05:07.970 --> 00:05:23.090
k times c m so c m will be . r r plus one
minus r r one plus v w by k and here it will
00:05:23.090 --> 00:05:32.590
be one plus v w by k times c naught or one
can estimate the value of c m as one plus
00:05:32.590 --> 00:05:43.240
v w by k time c naught divided by . the denominator
r r plus one minus r r into one plus v w by
00:05:43.240 --> 00:05:49.690
k so v w you have estimated already as a quadratic
in the . in the only few minutes back and
00:05:49.690 --> 00:05:53.510
k is known to as c naught is known to as r
r is known to as and one can estimate the
00:05:53.510 --> 00:05:58.800
value of c m once the c m is estimated then
permeate concentration can be estimated as
00:05:58.800 --> 00:06:05.080
well through their definition of real . . retention
so we can have an independent system prediction
00:06:05.080 --> 00:06:11.110
in terms of . you know um permeate concentration
or permeate flux for a low polarization case
00:06:11.110 --> 00:06:15.940
where in the polarization is not low whether
one it is significant we have already seen
00:06:15.940 --> 00:06:20.670
that we will be a we will be you have to solve
two equations . and two unknown three equations
00:06:20.670 --> 00:06:26.110
in three unknown system by using newton raphson
. after combining all the equations we will
00:06:26.110 --> 00:06:32.290
be getting will be landing of . with a single
nonlinear algebraic equation which will be
00:06:32.290 --> 00:06:35.760
solved . by using the newton raphson method
00:06:35.760 --> 00:06:41.000
so we will be doing another simplification
so that the things will be physically quite
00:06:41.000 --> 00:06:49.169
significant and apparent to us the third simplified
case will be the low polarization . and completely
00:06:49.169 --> 00:06:59.050
retentive membrane . low polarization that
is p w by k is much much less than one and
00:06:59.050 --> 00:07:14.220
completely retentive membrane . that means
. c p is equal to zero .
00:07:14.220 --> 00:07:19.830
c p is equal to zero . c p is equal to zero
so in this case permeate flux will be l p
00:07:19.830 --> 00:07:33.669
del p minus del pI del pI will be b c naught
b c m minus c p and c p is be it will be equal
00:07:33.669 --> 00:07:44.500
to zero so you will be having l p del p minus
b c m and from the . from the film theory
00:07:44.500 --> 00:07:52.400
what you will be getting . for film theory
you will be getting c m by c naught because
00:07:52.400 --> 00:08:00.099
all c ps are zero it is a completely retentive
membrane exponential v w by k so it will be
00:08:00.099 --> 00:08:11.870
. one plus . v w by k so you will be getting
permeate flux is equal to l p del p minus
00:08:11.870 --> 00:08:24.919
b c naught one plus . v w by k now if you
bring v w to the other side and write everything
00:08:24.919 --> 00:08:34.509
in terms of that so you will be getting l
p del p . minus b c naught divided by one
00:08:34.509 --> 00:08:46.250
plus b c naught divided by into l p divided
by mass transfer coefficient k and now if
00:08:46.250 --> 00:08:53.040
I replace l p the membrane permeability in
terms of membrane hydraulic resistance r m
00:08:53.040 --> 00:09:00.120
then I will be getting a concrete expression
of permeate flux as delta p minus b c naught
00:09:00.120 --> 00:09:11.850
is equal to . . mu r m . plus b c naught divided
by k .
00:09:11.850 --> 00:09:17.310
so these expression clearly shows that this
is the effective driving force there is the
00:09:17.310 --> 00:09:21.990
. the trans membrane pressure drop applied
trans membrane pressure drop minus osmotic
00:09:21.990 --> 00:09:27.940
pressure divided by there will be two resistances
in series one is membrane hydraulic resistance
00:09:27.940 --> 00:09:33.810
other is the resistance due to the mass transfer
in the mass transfer boundary layer so two
00:09:33.810 --> 00:09:37.870
resistance in series that will be appearing
in the denominator and in the numerator you
00:09:37.870 --> 00:09:42.570
will be having the effective trans membrane
pressure drop which will be nothing but the
00:09:42.570 --> 00:09:47.310
. actual trans membrane shut drop minus the
osmotic pressure that is developed in to the
00:09:47.310 --> 00:09:53.380
system so this clearly shows that mass transfer
resistance and membrane resistance they will
00:09:53.380 --> 00:10:01.709
be acting in series in an actual system ok
so before going into the . other variance
00:10:01.709 --> 00:10:07.209
of solution diffusion model I would like to
now in . at this point of time I would
00:10:07.209 --> 00:10:13.230
like to include or introduce . velocity
variation technique or . one more technique
00:10:13.230 --> 00:10:19.029
the if you remember in an earlier class we
have said that there are two ways or two experimental
00:10:19.029 --> 00:10:24.149
methods to estimate the real retention of
the system now will be really looking into
00:10:24.149 --> 00:10:30.149
how to . we have seen . one method already
how to conduct a separate set of experiments
00:10:30.149 --> 00:10:35.959
so that the real retention of the system can
be estimated by under low polarization condition
00:10:35.959 --> 00:10:42.860
that is low operating pressure low fit concentration
high turbulence now into in this . will be
00:10:42.860 --> 00:10:48.470
looking into one more method to estimate the
velocity variation technique which will
00:10:48.470 --> 00:10:59.700
be utilizing the film theory model to estimate
the real retention of the membrane . technique
00:10:59.700 --> 00:11:19.100
. .
to estimate . r r of membrane solute . solvent
00:11:19.100 --> 00:11:20.399
system
00:11:20.399 --> 00:11:33.149
so we take request to the film theory model
and it's equation . so c m minus c p divided
00:11:33.149 --> 00:11:44.020
by c naught c p is equal to exponential v
w by k now next what we will be doing will
00:11:44.020 --> 00:11:50.230
be writing the definition of real retention
and observe retention so real retention is
00:11:50.230 --> 00:12:04.089
one minus c p by c m . so c m can be can
be written as . so c p by c m will be one
00:12:04.089 --> 00:12:15.040
minus r r c m by c p will be one over one
minus r r so therefore c m is nothing but
00:12:15.040 --> 00:12:24.410
c p divided by one minus r r . similarly from
the definition of observe retention r naught
00:12:24.410 --> 00:12:32.519
is one minus c p by c naught and you can estimate
the value of c naught in terms of c p so
00:12:32.519 --> 00:12:38.870
c naught will be nothing but c p divided by
one minus r naught so now we are going to
00:12:38.870 --> 00:12:46.070
replace the expression of the the . c m in
favor of c p and r r and c naught in favors
00:12:46.070 --> 00:12:52.230
of c p and r naught in these equation and
see what we get .
00:12:52.230 --> 00:12:59.610
so if you really do that you will be getting
c p one minus c p divided one minus r r minus
00:12:59.610 --> 00:13:12.940
c p . is equal to c p divided by one minus
r naught minus c p exponential . v w by k
00:13:12.940 --> 00:13:18.680
so this can be so c p will be canceling out
from from from both the numerator and denominator
00:13:18.680 --> 00:13:23.769
and you will be getting an expression of . observe
retention and real retention and permeate
00:13:23.769 --> 00:13:28.170
flux and mass transfer coefficient if you
simplify this equation I am just writing the
00:13:28.170 --> 00:13:39.100
final form l n r naught divided . by one minus
r naught . is equal to l n r r . by one minus
00:13:39.100 --> 00:13:52.050
r r minus v w by k now in this expression
. if if we have the real retention as constant
00:13:52.050 --> 00:13:58.450
and . let us see how if you if you change
your operating conditions operating conditions
00:13:58.450 --> 00:14:02.779
means here we cannot change the . operating
conditions in terms of pressure because the
00:14:02.779 --> 00:14:07.600
mass transfer coefficient is appearing here
and as you have already seen that mass transfer
00:14:07.600 --> 00:14:13.019
coefficient is a strong function of reynolds
number and reynolds number will be basically
00:14:13.019 --> 00:14:16.649
. will be a strong function of the cross flow
velocity
00:14:16.649 --> 00:14:21.840
so these is this method is known as velocity
variation technique what will be doing will
00:14:21.840 --> 00:14:27.270
be will be applied different cross flow velocity
in the system that will alter the mass transfer
00:14:27.270 --> 00:14:31.899
coefficient once the mass transfer coefficient
will be altered the permeate flux will be
00:14:31.899 --> 00:14:35.950
altered and once the permeate flux will be
altered as well as the it will be affecting
00:14:35.950 --> 00:14:41.459
the permeate quality so observe retention
because permeate concentration also be altered
00:14:41.459 --> 00:14:46.440
so observe retention will be altered so that
is therefore we change the cross flow velocity
00:14:46.440 --> 00:14:52.410
and get different type different values of
permeate . mass transfer coefficient and permeate
00:14:52.410 --> 00:15:01.550
flux and then we plot l n r naught by one
minus r naught versus v w by k . . so now
00:15:01.550 --> 00:15:14.480
if a plot . l n r naught divided by one minus
r not versus . v w by k you will be getting
00:15:14.480 --> 00:15:26.910
a curve something like this . so similarly
actually these will be hitting the y axis
00:15:26.910 --> 00:15:32.329
the curve will be something like this and
similarly it will be for the other other values
00:15:32.329 --> 00:15:40.579
of . mass transfer coefficient will be will
be getting this so ultimately all these points
00:15:40.579 --> 00:15:47.610
will be will be going to the intercept r naught
by . one minus r naught so this is for a particular
00:15:47.610 --> 00:15:49.649
k this is that is for another particular k
00:15:49.649 --> 00:15:55.110
so you will be getting the by from
the intercept what is the intercept the intercept
00:15:55.110 --> 00:16:03.550
. is l n r r by one minus r r so from the
intercept one can estimate the . value of
00:16:03.550 --> 00:16:11.839
real retention so r r can be estimated so
what is the drawback of this system the major
00:16:11.839 --> 00:16:18.339
draw back of this system is that the experiments
should be very very accurate if there is slight
00:16:18.339 --> 00:16:23.880
in accuracy in the . in the experiments then
in the log scale these will be even magnified
00:16:23.880 --> 00:16:29.389
and therefore the it will giving a very wrong
estimation so . there will be number of more
00:16:29.389 --> 00:16:34.850
number of experiment one has to conduct not
only that not only the more number of experiments
00:16:34.850 --> 00:16:40.000
the experiments has to be very very accurate
then only one can get an estimation of real
00:16:40.000 --> 00:16:45.709
retention appropriate estimation of real
retention by this velocity variation method
00:16:45.709 --> 00:16:49.300
ok
so next will be looking into as you have discussed
00:16:49.300 --> 00:16:55.070
that the third equation if the if the lets
look into a once again the modeling equation
00:16:55.070 --> 00:17:06.620
for the osmotic pressure control filtration
one dimensional model . .
00:17:06.620 --> 00:17:14.850
so we have at the film theory v w is equal
to k l n c m minus c naught c p minus c naught
00:17:14.850 --> 00:17:22.049
we have the darcys law v w is equal to del
p minus del pI and we have two equations two
00:17:22.049 --> 00:17:27.579
unknown c m c p v w these are the three unknown
and two equations though other equation is
00:17:27.579 --> 00:17:39.879
the connection between the c m and c p .
c m and c p and we have already seen that
00:17:39.879 --> 00:17:45.999
one way of connecting c m and c p is the
real retention that is nothing but a partition
00:17:45.999 --> 00:17:51.080
co efficient between across the membrane
. of the solute partition co efficient across
00:17:51.080 --> 00:17:55.799
the membrane now we have already seen the
all the formulations and the method for solving
00:17:55.799 --> 00:18:02.259
. these third equation to get a system . prediction
of the system . performance now we are what
00:18:02.259 --> 00:18:08.179
will be doing will be doing will be replacing
the . if the expression of real retention
00:18:08.179 --> 00:18:13.020
that will be valid more valid for the alter
filtration system and various cut off alter
00:18:13.020 --> 00:18:18.940
filtration system by the solution diffusion
model . for the for the reverse osmosis
00:18:18.940 --> 00:18:20.179
nano filtration system
00:18:20.179 --> 00:18:24.289
so real retention is a concept which will
be more valid for the alter filtration system
00:18:24.289 --> 00:18:30.720
but for the reverse osmosis and nano filtration
will not . able to use the expression of real
00:18:30.720 --> 00:18:36.359
retention as as partition co efficient one
minus c p by c m you have to use the other
00:18:36.359 --> 00:18:40.700
variance of the connection between the
c p and c m that is the solution diffusion
00:18:40.700 --> 00:18:49.210
an it's various variations . or modifications
so first will be talking about the . the
00:18:49.210 --> 00:19:03.659
solution diffusion model for reverse osmosis
and nano filtration system . .
00:19:03.659 --> 00:19:11.580
for r o n f system so now let us look in to
the one of the various comforting equations
00:19:11.580 --> 00:19:27.100
so one equation will be the . film theory
. is v w is equal to k l n c m minus c
00:19:27.100 --> 00:19:35.590
p so is not c p c naught minus c p c m minus
c p divided by c naught minus c p the other
00:19:35.590 --> 00:19:43.600
one is the darcys law . which is nothing
but the solvent flux through the membrane
00:19:43.600 --> 00:19:51.320
v w is equal to l p del p minus del pI and
the third one is the solute flux through
00:19:51.320 --> 00:20:02.830
the membrane which is the solution diffusion
model . . so these will be giving to the solvent
00:20:02.830 --> 00:20:07.529
flux through the . mass transfer boundary
layer solvent flux to the porous membrane
00:20:07.529 --> 00:20:17.090
and the solute flux to the porous membrane
v w c p is equal to b c m minus c p so we
00:20:17.090 --> 00:20:23.809
will be having three equations and three unknowns
. v w c m and c p and can get the system prediction
00:20:23.809 --> 00:20:30.690
so I will just give a . small algebraic
manipulation in order to solve this set of
00:20:30.690 --> 00:20:31.739
equations
00:20:31.739 --> 00:20:40.249
so v w can be the . solvent flux to the
membrane can be written as v w is equal
00:20:40.249 --> 00:20:50.840
to l p del p minus del pI so this will be
if u take l p out del p out so it will be
00:20:50.840 --> 00:21:04.049
giving one minus del pI and del pI is lets
say pI is equal to a c . so um del pI will
00:21:04.049 --> 00:21:17.470
be basically a by delta p c m minus c p
. so I write v w is equal to v w naught one
00:21:17.470 --> 00:21:28.330
minus alpha . c m minus c p so this is my
equation number one and what is v w naught
00:21:28.330 --> 00:21:39.129
v w naught is basically l p times del p it
is a pure water flux . pure water flux and
00:21:39.129 --> 00:21:43.799
what is the parameter alpha alpha so these
will be v w naught will be will be known us
00:21:43.799 --> 00:21:48.720
because l p is known to us and delta p is
the trans membrane pressure drop so what is
00:21:48.720 --> 00:21:54.820
alpha alpha is the parameter a by delta p
so it is non dimensional so there will be
00:21:54.820 --> 00:21:59.960
equation number one then will be equating
these with the film theory equation so v w
00:21:59.960 --> 00:22:12.559
. naught one minus alpha c m minus c p . will
be equal to k l n c m minus c p divided by
00:22:12.559 --> 00:22:22.940
. c naught minus c p so these will be the
combination of the film theory and the
00:22:22.940 --> 00:22:27.970
and the osmotic pressure model darcys
law so I write it a this is a equation number
00:22:27.970 --> 00:22:34.070
two then we will be having the solution diffusion
model . for the solute flux through membrane
00:22:34.070 --> 00:22:50.320
. that will be v w c p is equal to b c m minus
c p . so now we can combining these two
00:22:50.320 --> 00:22:56.779
equations and finally will be getting into
these so we can we can we can . we will be
00:22:56.779 --> 00:23:04.179
combining the equation number three solution
diffusion model . and . darcys law that
00:23:04.179 --> 00:23:12.549
is equation one you can combine these two
equation . and see what we will get if you
00:23:12.549 --> 00:23:23.279
combine this two equation you will be getting
v w naught . one minus alpha c m minus
00:23:23.279 --> 00:23:33.330
c p is equal to b c m minus c p divided by
c p so these equation can be written in this
00:23:33.330 --> 00:23:43.599
form one minus alpha times c m plus alpha
times c p is equal to beta c m minus c p divided
00:23:43.599 --> 00:23:54.330
by c p where . alpha we already defined and
beta is b by v w naught ok so from these equation
00:23:54.330 --> 00:24:01.080
the membrane surface concentration is obtained
in terms of permeate concentration . so from
00:24:01.080 --> 00:24:08.710
these we can get c m membrane surface concentration
as function of permeate concentration . c
00:24:08.710 --> 00:24:17.090
p into one plus one plus beta plus alpha c
p .
00:24:17.090 --> 00:24:24.759
now these expression of c m can be put into
the uh the equation that we have already
00:24:24.759 --> 00:24:29.639
talked about equation number two we are going
to put the value of c m that we are obtained
00:24:29.639 --> 00:24:34.759
in the in these equation equation number two
if you really do that . put into equation
00:24:34.759 --> 00:24:41.919
number two you will be getting one nonlinear
algebraic equation in the form of I be giving
00:24:41.919 --> 00:24:53.460
the final expression beta v w naught alpha
c p plus beta minus k l n c p . divided by
00:24:53.460 --> 00:25:04.860
alpha c p plus beta into c naught minus c
p . is equal to zero again these expression
00:25:04.860 --> 00:25:11.969
is nothing but a nonlinear algebraic equation
in c p and again one has to take care go go
00:25:11.969 --> 00:25:20.859
go by a newton raphson method iteratively
so one has to have an iterative solution iterative
00:25:20.859 --> 00:25:32.299
solution of c p and the method can be . used
as newton raphson method . once you get the
00:25:32.299 --> 00:25:41.659
value of c p you will be getting the value
of c m .
00:25:41.659 --> 00:25:47.479
c p is equal . will be getting the because
from this from this equation you will be getting
00:25:47.479 --> 00:25:52.399
the value of c m because c p you have already
estimated once you know the value of c p and
00:25:52.399 --> 00:25:57.840
c m then one can go through the film theory
equation and can get the or expression of
00:25:57.840 --> 00:25:58.879
permeate flux
00:25:58.879 --> 00:26:03.749
so again in this method also . . we can land
up with three equations only difference is
00:26:03.749 --> 00:26:08.809
the definition of real retention which basically
acts as a partition co efficient between the
00:26:08.809 --> 00:26:13.580
membrane surface concentration solute concentration
in the membrane surface and in the permeate
00:26:13.580 --> 00:26:17.919
will be replaced by the solution diffusion
equation which is basically nothing but the
00:26:17.919 --> 00:26:24.369
solute . transport equation for the solute
flux through the membrane and then again these
00:26:24.369 --> 00:26:28.950
three equations can be solved and can be recast
in the form of single equation which will
00:26:28.950 --> 00:26:33.700
be algebraic nonlinear equation and that can
be solved by using an iterative method and
00:26:33.700 --> 00:26:38.999
by that one can get the permeate concentration
membrane surface concentration and permeate
00:26:38.999 --> 00:26:44.659
flux ultimately so I will stopping in this
class in the next class next session will
00:26:44.659 --> 00:26:50.409
be looking into the other variance of solution
diffusion model and how to use these solution
00:26:50.409 --> 00:26:56.169
diffusion model the . what there will be the
there will be a modifications of we need
00:26:56.169 --> 00:26:59.369
in order to get the system prediction perfectly
00:26:59.369 --> 00:26:59.720
thank you very much . .