Y. E$ectrochem. Soc.: ELECTROCHEMICAL SCIENCE AND TECHNOLOGY
1714
2. G. M. Rao, D. Elwell, R. S. Feigelson, This Journal,
127, 1940 (1980).
3. R. C. DeMattei and R. S. Feigelson, J. Cryst. Growth,
44, 115 (1978).
4. R. C. DeMat(ei and D. Elwell, Unpublished.
5. J. A. Poris and R. A. Huggins, Private communica%ion.
August I981
6. K. E. Johnson, "High Temperature Technology," pp.
493-98, Butterworths, London (1967).
7. "Phase Diagram for Ceramists," E. M. Levin, C. R.
Robbins, and H. F. McMurdie, Editors, Amer.
Ceram. Soc., Columbus, OH (1964).
8. U. Cohen, J. Electron. Mater., 6, 607 (1977).
9. M. Newberger and S. J. Welles, Silicon, Rept. AD
698342, (October 1969).
Mass Transport Characterization of Donnan Dialysis:
The Nickel Sulfate System
Patrick K. Ng* and Dexter D. Snyder*
General Motors Research Laboratories, Electrochemistr~l Department, Warren, Michigan 48090
ABSTRACT
There is a need for reliable models of the mass transfer characteristics of
hollow tube Donnan dialyzers, to guide application to industrially significant
problems such as recovery from electroplating waste water. This work is
focused on determining mass transport correlations in a shell-and-tube dialyzer, fabricated from ion-selective membranes, used to extract nickel from
dilute nickel sulfate solution with sulfuric acid as the stripping agent. Correlations between mass transport coefficient and Reynolds number a r e r e p o r t e d
for both laminar and turbulent flow regimes.
Donrmn dialysis concentration of ions, using selective
membranes and a chemical potential gradient, has only
recently been considered seriously as an industrial separation and purification technique. Wallace, an early
advocate, used the approach to concentrate uranyl,
strontium, lanthanum, silver, and copper salts (1, 2).
Until very recently, Donnan dialysis has been judged
too slow to be practical. Since separation is diffusion
controlled, one typically needs extremely thin membranes which are effective ion-selective barriers. Membranes available in the late 1960's and early 1970%
were relatively thick and lacked chemical durability.
Better membranes are available today, but there is a
lack of data and models by which to design practical
systems.
Elements of Donnan dialysis concentration of nickel
ions are illustrated in Fig. I. The cation-selective
membrane ideally excludes the anions, but permits
cations to exchange as the system approaches the equilibrium dictated by the Donnan relation, in this case
(aNi I )
~
(amn)
.
frame dialyzer. Their approach involved determining
the membrane resistance at high agitation, and determining the boundary-layer mass transport coefficients
using a computer iteration scheme. For Reynolds numbers up to 500, they observed a dependence of ion
transport rate on the 0.65 power of the Reynolds number.
Eisenmann and Smith (5) carried out an extensive
evaluation of Donnan dialysis softening as a possible
adjunct to distillation desalting of brackish water.
With limited success, they introduced the affinity driving force and built an analytical model of the process
which they used for economic projections. They found
a dependence of the sodium-calcium exchange rate on
a power of the Reynolds number in the range 0.33-0.67
for a tortuous path spacer. Eisenmann (6) recently
reported on the use of Donnan dialysis to recover
nickel sulfate concentrate from dilute plating r i n s e
(1)
i
ii
(II)
(aHI) 2
=
-
(aHn)s
[I]
J
~s
i n which a is the activity of the ions on either side of
the membrane. If the gradient of the stripping chemical, sulfuric acid in this case, is kept high, the nickel
ions can actually be transported against their activity
gradient. In the present case where 0.5M sulfuric acid
is used to strip nickel from 0.0017M nickel sulfate (100
ppm Ni), the nickel ions can be transferred almost
quantitatively. The rate at which this ion exchange
occurs depends both on the membrane properties a n d
on the boundary-layer film at the membrane-solution
interfaces.
Wallace and other workers (1, 3) dealt with relatively slow flows of both feed and stripping streams.
However, the hydrodynamic enviro.nment of the membrane often controls the performance. Lake and Melsheimer (4) approached this issue by examining the
exchange of sodium, potassium, and hydrogen ions
a c r o s s a cation-selective membrane, using a p l a t e - a n d , Electrochemieal Society Active Member.
Key words: membrane, interfaces, connection, ion exchange.
o
l
[[
C~H
CNi
c~
II
CNi
!
I
I
:
,
NISO 4
3
MEMBRANE
H2S0 4
Fig. 1, Donnan dialysis
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Vol. 128, No. 8
DONNAN
1715
DIALYSIS
water. Using perfluorosulfonic acid m e m b r a n e s , in
hollow t u b e form, he m e a s u r e d n i c k e l fluxes in t h e
r a n g e 0.3-3 • 10 - 3 m o l / ( c m 2. sec) for a feed in t h e
r a n g e 25-100 p p m nickel ion. His e x p e r i m e n t s s p a n n e d
t h e Reynolds n u m b e r r a n g e 50-300.
T h e p r e s e n t w o r k was focused on d e t e r m i n i n g mass
t r a n s f e r correlations for the b o u n d a r y layer, and on
estimating the interdiffusion coefficient of nickel in the
m e m b r a n e . T h e t w o - c a t i o n system offers results w h i c h
c a n be i n t e r p r e t e d and c o m p a r e d w i t h previous results,
a n d the nickel waste is also a significant i n d u s t r i a l
p r o b l e m w o r t h y of b e t t e r solutions t h a n n o w exist
commercially. T u b u l a r cation-selective m e m b r a n e s
w e r e selected for t h e i r s u i t a b i l i t y in designing r e c o v e r y a n d recycle modules w i t h high s u r f a c e - t o v o l u m e ratios, giving p o t e n t i a l l y h i g h e r v o l u m e t r i c
nickel s e p a r a t i o n rates. Both sulfuric acid a n d nickel
waste s t r e a m s w e r e c i r c u l a t e d continuously t h r o u g h
the dialyzer.
Experimental
Membrane dimension measurements.--Table I shows
the dimensions of the m e m b r a n e s used in this study.
Dimensions w e r e d e t e r m i n e d b y first soaking a piece
of the t u b i n g in a m i x t u r e of nickel sulfate (100 p p m
in nickel) and 1N sulfuric acid. The tubing was t h e n
cut open a n d its thickness m e a s u r e d w i t h a m i c r o m e ter. Its outside d i a m e t e r was m e a s u r e d using a m i c r o scope a n d the inside d i a m e t e r was s u b s e q u e n t l y calculated.
Apparatus.--The d i a l y z e r is p i c t u r e d in Fig. 2, a n d a
schematic d i a g r a m of the e x p e r i m e n t a l setup is shown
in Fig. 3. M e m b r a n e s used w e r e c o m m e r c i a l perfluorosulfonic acid type. The s h e l l - a n d - t u b e module w a s
17.15 cm long and the inside d i a m e t e r of the shell w a s
0.48 cm. F o r m e m b r a n e 810, o n l y a single m e m b r a n e
t u b e was used. I n the case of the s m a l l e r m e m b r a n e
811, t h r e e m e m b r a n e tubes w e r e used to g u a r a n t e e an
a p p r e c i a b l e change in n i c k e l concentration o v e r the
e x p e r i m e n t period. Nickel sulfate solution (100 p p m
in nickel) was circulated t h r o u g h the t u b e side, while
sulfuric acid (1, 2, or 4N) w a s p u m p e d c o n t i n u o u s l y
t h r o u g h the shell side. E a c h s t r e a m contained i n i t i a l l y
250 ml solution, and its flow r a t e was controlled b y a
t u b i n g p u m p and m o n i t o r e d w i t h a flowmeter.
P r e l i m i n a r y results suggested t h a t t h e b o u n d a r y l a y e r resistance on t h e feed solution side (nickel sulfate) was dominant. I n practice, the process does n o t
d e p e n d on w h i c h side of the m e m b r a n e each solution
contacts. However, w i t h the feed solution being the
p r i n c i p a l t a r g e t of this study, one could achieve h i g h e r
l i n e a r velocities w h e n this feed w a s p a s s e d t h r o u g h
t h e tube, and this e x t e n d e d the r a n g e of the correlation
w h i c h was u l t i m a t e l y established.
Procedures.--The dialyzer was first flushed w i t h
distilled w a t e r to r e m o v e a n y nickel sulfate a n d sulfuric acid l e f t in the system. This w a t e r was t h e n
d r a i n e d and both streams of electrolyte w e r e p u m p e d
t h r o u g h t h e i r respective loops to s t a r t the e x p e r i m e n t .
S a m p l e s w e r e t a k e n f r o m each r e s e r v o i r at t i m e i n t e r vals, a n d t h e n i c k e l concentration of each was d e t e r -~
m i n e d b y atomic absorption analysis. To minimize the
e r r o r of changing volume, the total a m o u n t t a k e n for
s a m p l i n g o v e r the e x p e r i m e n t was l i m i t e d to less t h a n
10% of the initial r e s e r v o i r v o l u m e (25'0 m l ) . A t t h e
e n d of the e x p e r i m e n t (3-4 h r ) , the v o l u m e of each
s t r e a m was m e a s u r e d b y d r a i n i n g the solution. In a d dition, both t e m p e r a t u r e and p H of the solutions w e r e
measured. Results indicate t h a t pI-I of t h e t u b e - s i d e
Table I. Membrane dimensions
Membrane
810
811
Thickness
Inside diam
Outside diam
0.0368
0.0127
0.2383
0.0637
0.3119
(cm)
(cm)
(cm}
0,0891
Fig. 2. Dialyzer
1
C•
rF+
,0
(NiS04 )(Ci,VR)]
i ii!i
Fig. 3. Schematic diagram of the experiment
reservoir changed typically from about 5.5 to 2.2. The
shell-side p H remained relatively constant, with a
m a x i m u m increase of about 0.I p H unit. Due to heat
generated by the tubing pumps, temperature increases
f r o m 2~ (low flow rates) to 7~ (high flow r a t e s )
w e r e found. A l l e x p e r i m e n t s i n d i c a t e d a v o l u m e d e crease for the t u b e - s i d e r e s e r v o i r of 5-6 ml. This can
be e x p l a i n e d b y a combination of evaporation, osmotic
transport, a n d the m i g r a t i o n of h y d r a t i o n w a t e r w i t h
the ions.
Results and Discussion
This w o r k was focused on s e p a r a t i n g the m e m b r a n e
a n d b o u n d a r y - l a y e r effects in the D o n n a n dialysis
concentration of n i c k e l sulfate f r o m an e x t r e m e l y
dilute s t a r t i n g solution. This has b e e n a p p r o a c h e d b y
s t u d y i n g the n i c k e l t r a n s p o r t r a t e as a function of the
h y d r o d y n a m i c condition of the m e m b r a n e dialyzer,
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Specifically, t h e nickel c o n c e n t r a t i o n - t i m e r e l a t i o n is
d e t e r m i n e d for a m e m b r a n e tube dialyzer o p e r a t e d
u n d e r p r e s e t flow conditions.
Rate constant of the tube-side reservoir.--Figure 4
shows typical results for the 810 m e m b r a n e . The mass
of n i c k e l in t h e r e s e r v o i r is here p l o t t e d instead of
concentration to e l i m i n a t e a n y discrepancy i n t r o d u c e d
b y change in v o l u m e due to sampling. Qualitatively,
h i g h e r nickel feed rates f a v o r faster r e m o v a l of nickel
f r o m the reservoir. The nickel mass increase in the
sulfuric acid r e s e r v o i r was found consistently to be a
m i r r o r of t h e decrease in the feed reservoir. Thus, the
m e m b r a n e a p p e a r s to be n e i t h e r source nor sink for
nickel in this process. These nickel m a s s - t i m e results
(Fig. 5) suggest t h a t the r e s e r v o i r can be r e g a r d e d as
a reactor w i t h a f i r s t - o r d e r r a t e constant (m) e q u a l ing the slope of the best fit line.
E x p e r i m e n t s were also conducted to. s t u d y the effects
of s h e l l - s i d e flow r a t e a n d sulfuric acid concentration
on the t r a n s f e r rate, and the results are shown in Fig.
6a and b. F r o m e x a m i n a t i o n of Fig. 6a, it is clear that
both acid-side flow r a t e and acid concentration (1-4N)
a r e u n i m p o r t a n t in affecting the nickel t r a n s p o r t rate.
These findings a g r e e w i t h the w o r k of L a k e and M e l s h e i m e r (4), w h e r e t h e y find that the b o u n d a r y - l a y e r
resistance on the acid side is small. T h e i n d e p e n d e n c e
from sulfuric acid concentration is r e a s o n a b l e because
the
acid g r a d i e n t r e m a i n s high in this 1-4N range.
T e m p e r a t u r e effects can be considered to be s m a l l
because the m a x i m u m change at high flow r a t e was
only 7 ~C.
Boundary layer and membrane resistances.--Resul~s
in Fig. 6 indicate t h a t the t u b e - s i d e b o u n d a r y - l a y e r
resistance and the m e m b r a n e resistance a r e the controlling factors for the nickel transport. The r e l a t i v e
values of these two resistances can be e s t i m a t e d w i t h
a series model used b y Helfferich (7), G r e g o r (8), and
others. F o r the case of nickel and hydrogen, we will
assume complete colon exclusion and define diffusion
coefficients b y the following flux equations
DNi-H (C-UNi
JNi = V
DNi-H (C'H
JH -- ~
C-~,Ni)
_
-
-
C--;'H)
Re
9
= :tO
~ 0
Dtt
= T
[3]
(CHI -- C'H)
C'm
~m
(C'H)"
- (~'a~)
-
U"m
(CHII) $
=
-
-
(~"a) ~
9ztCl -- X = 0
Membrane 810
[5]
on feed side and
on acid side.
E l e c t r o n e u t r a l i t y w i t h i n the m e m b r a n e is
by
4701
2351
C6]
expressed
[7]
w h e r e X" is the concentration of fixed negative charges
in the w e t t e d m e m b r a n e . W i t h zero n e t current, the
fluxes a r e r e l a t e d b y
zzdi = 0
[8]
h
_~8
~
The interracial concentrations can be w r i t t e n as follows b y r e a r r a n g e m e n t of Eq. [2] a n d [3], a n d b y u s e
of Eq. [8]
u)
C'Nl :
Z
o
[2]
A t each interface, the ions are assumed to be dist r i b u t e d according to the Donnan equilibrium, i g n o r ing activity effects
-
(tube-side)
DNi
= T
(CNiI -- C'Nl)
DNi-H, a n d DNi and DH a r e the diffusion coefficients
in the m e m b r a n e and the b o u n d a r y layer, respectively.
F i g u r e 1 gives m e a n i n g to the o t h e r symbGls. A n effective mass t r a n s f e r coefficient (Kin) can also be defined
for the m e m b r a n e as
DNi-H
Km -[4]
l
CNiII
•4"
August 1981
J. Electrochem. Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
1716
8JNi
- Dm
28Jm
o
C'H = - -
Time (rain)
DH
Fig. 4 Transient variation of nickel mass in the reservoir
Membrane810
Re (shell-side): 1064
~
Z ~
Re (tube-side)
9 294
9 4701
9 2351
o
~.a
o.o
Time ( m i n )
Fig. 5. Transient varioti0n of log nickel moss in the reservoir
+ CNiI
+ Ca x
[9]
[10]
Equations [2], [5]-[7], [9], and [10] can now b e
solved for fluxes and i n t e r f a c i a l concentrations as
functions of b u l k concentrations, diffusivities, and
m e m b r a n e a n d b o u n d a r y - l a y e r thicknesses. F i g u r e s 7a
and b show the result of such a calculation using
p a r a m e t e r values close to those expected in the e x periments. F o r b u l k n i c k e l concentration less t h a n
10 -5 m o l / m l , the b o u n d a r y l a y e r is the m a j o r controlling factor for the nickel transfer. The m e m b r a n e r e sistance ~s negligible until the b u l k nickel c o n c e n t r a tion reaches 10 -4 m o l / m l . Helfferich (7) gives a d e tailed description of these resistances.
F i g u r e 7 suggests that t h e t u b e - s i d e b o u n d a r y - l a y e r
resistance is t h e d o m i n a n t l i m i t a t i o n for nickel t r a n s fer in the concentration r a n g e studied in this work.
This resistance depends on the h y d r o d y n a m i c c o n d i tions and is best characterized b y a mass t r a n s f e r coefficient, KBL. KBL c a n be c a l c u l a t e d from the rate
constant of the r e s e r v o i r based on a model w i t h the
following assumptions: (i) Nickel concentration at t h e
m e m b r a n e surface is negligible c o m p a r e d to the b u l k
concentration; (ii) the t u b e - s i d e solution is w e l l m i x e d
and its concentration is thus u n i f o r m a n d equal to t h e
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Voi 128, No. 8
DONNAN
1717
DIALYSIS
~&~,~Tube Reynolds Number=250
Membrane 810
Tube Reynolds Number=1000
Tube Reynolds Number=9404
Shell Reynolds Number=1064
Membrane 811
E?-
:I
Tube Reynolds Number=4000
o
~
non
~
~
~
Shell Side Reynolds Number
~o
O~
~
in
~
Sulfuric Acid Normality
u
Fig. 6. Dependence of mass transfer on secondary factors. (a) Shell s|de (1N H2S04) Reynolds number; (b) sulfutlc acid concent~.iion
~ Boundary Laye
r~////
~.
~"
~9 Z
Membrane
2
~
jt._Boundery Layer....jL
~
Control
r
/
Membrane --.---411
Control - -
~
~
1o-s
. lx10 -3 real/am3
/
~
.1o-6oo,o
/ _~
lo-i
i
H
Z 10 -9
/
' 2"6x10-5 cm2/s
, 3x10 -3 cm
itial
H~
/
, 3x10 -2 cm
//.
10 -5
Nickel Concentration
10 -3
I0-7
in Feed (mol/mL)
10 -6 cm2/s
DNi
9 8.7x10 -6 cm2/s
/]
DII
-2.6X10 -5
/
6
. 3x10 -3 cm
~
, 3x10 -2 cm
C~
, ixl0
.
~oncentration
10 -7
9l~10 -3 mol/o~3
I
O
i0 -5
cm2/s
mo i/mL
10 -3
Nickel Concentration
in Feed (raol/mL)
Fig. 7. (a, left) Dependence of flux on bulk nickel concentration; (b, right) dependence of interfacial nickel concentration on bulk
nickel concentration.
outlet concentration; (iii) at each time, the dialyzer
operates at steady state.
T h e first assumption implies that a n y nickel t r a n s ferred to the m e m b r a n e surface will be transported
i m m e d i a t e l y t h r o u g h the m e m b r a n e , and its v a l i d i t y
is illustrated in Fig. 7b. Assumptions (ii) and (iii) are
reasonable if the flow rates are r e l a t i v e l y high. Ref e r r i n g to the schematic d i a g r a m i n Fig. 3, one can
o b t a i n a mass balance equation for nickeI i n the tube
side of the dialyzer, as follows
Q (Ci -- Co) : K B L A C o
[11]
where A is the m e m b r a n e area based on the inside
diameter a n d Q is the volumetric flow rate. Ci and Co
represent the b u l k concentration of Ni i n the tube side
a n d are understood to be the same a s C N i I in Fig. 1.
Similarly, a mass balance of the reservoir gives
dCi
VR--~-- = Q (Co - Ci)
[12]
where VR is the reservoir volume. The results in Fig.
5 also i m p l y that
Ci : Ci~ exp (mr)
[13]
w h e r e Cin is the initial nickel concentration i n the
reservoir, a n d m is the rate constant. Combining Eq.
[11]-[13], one obtains the relationship b e t w e e n KBL
and m, t h a t is
-- fnVR
KBL =
[14]
mV~
A(I+-w- J
The flux J can also be calculated b y
Q
- mVR
Cl
[15]
With %he KBL values calculated b y Eq. [14], mass
transfer correlations were s u b s e q u e n t l y obtained and
are shown in Fig. 8. The e x p o n e n t on the Schmidt
n u m b e r (Sc) was t a k e n as 1/3 according to convective
mass transfer in the pipe. Kinematic viscosity of w a t e r
a n d diffusion coefficient of NiSO4 at infinite dilution
(DNisO4 --~ 8.674 • 10-6-cmF/sec) were used to estimate
Sc. The t r a n s i t i o n f r o m l a m i n a r flow to t u r b u l e n t flow
occurs at Reynolds n u m b e r s in the r a n g e 1000-2000.
The generally accepted R e y n o l d s n u m b e r transition
value for smooth tubes is about 2100, b u t v i b r a t i o n
of this m e m b r a n e t u b i n g could well introduce t u r b u lence at lower Reynolds n u m b e r . Least square fits to
the two regions give the following correlations
Sh ----0.00272 Re 1.024 Sc 1/3 103 < Re < 104
[16]
S h ----0.1663 R e 0-47~Sc 1/8
Re < 103
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3. EZectrochem. Sac.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
1718
Augus$ 1981
Table II. Membrane interdiffuslon coefficient
Sh = 0.00272 Re I~024 SC I/3
O/0
Sh = 0.1663 Re 0"475 Sc I/3
l0
I~
I0 0
o
9
,
-
0
U
Q
,
- , , ,
9
10 2
9
/
,
-
Membrane 810
Membrane 811
,-, 9
103
104
Feed-Side Reynolds Number
Fig. 8. Mass transfer correlations
A n i m p o r t a n t p r o p e r t y of the m e m b r a n e is the i n t e r diffusion coefficient of nickel and h y d r o g e n ions
(Dsi-H). A l t h o u g h e x p e r i m e n t s w e r e p e r f o r m e d w h e r e
b o u n d a r y - l a y e r resistance is controlling, DNi-H can be
e s t i m a t e d b y e x t r a p o l a t i n g results at h i g h flow rates.
As t h e flow r a t e increases, the m e m b r a n e becomes
m o r e d o m i n a n t and the i n t e r f a c i a l concentrations can
b e a p p r o x i m a t e d b y b u l k concentrations. Equations
[2], [5], [6], and [7] can then be used to calculate Km
at a n y flow r a t e and b u l k nickel concentration. By
e x t r a p o l a t i n g the d a t a in a plot of 1/Re vs. 1/Km, one
can o b t a i n the v a l u e of Km at infinite flow r a t e w h e r e
m e m b r a n e is the only resistance. F i g u r e 9 shews the
result of this exercise for m e m b r a n e s 810 a n d 811 at
b u l k nickel concentrations of 100 and 55 ppm. I n t e r diffusion coefficients (DIn-H) thus calculated a r e s u m m a r i z e d in Table II a n d t h e i r m a g n i t u d e agrees w i t h
t h e r a n g e r e p o r t e d in literature. The d e p e n d e n c e of
D---Ni-Hon b u l k concentration needs f u r t h e r investigation.
Conclusions
D o n n a n dialysis, even w i t h its low t r a n s f e r rate, is
n o w a p r o m i s i n g s e p a r a t i o n technique w i t h the a v a i l a b i l i t y of hollow fiber m e m b r a n e s from which modules
can be built to give high surface a r e a - t o - v o l u m e ratio.
T h e t r a n s f e r r a t e depends, in general, on both m e m b r a n e p r o p e r t i e s a n d h y d r o d y n a m i c conditions outside
Membrane
Bulk Ni
concentration
mol/ml ( • 1 0 6 )
D~I-H
cm~/sec ( x 10e)
810
810
811
1,7
0.935
1.7
7.063
3,244
1,75
t h e m e m b r a n e . F o r r e m o v a l of nickel f r o m dilute sulfate solution, this w o r k c h a r a c t e r i z e d these two d e pendencies, p a r t i c u l a r l y t h a t for h y d r o d y n a m i c conditions a n d g e n e r a t e d correlations for design purposes.
Within the 0-0.0017M range of n i c k e l concentration
studied, the results indicate t h a t the b o u n d a r y l a y e r
is the m a j o r resistance to nickel transport. M e m b r a n e
resistance is i m p o r t a n t only at high flow rates and high
concentrations. A series m o d e l was e m p l o y e d to estim a t e the r e l a t i v e i m p o r t a n c e of these two resistances
u n d e r different e x p e r i m e n t a l conditions.
T h e b o u n d a r y - l a y e r effect is c h a r a c t e r i z e d b y m a s s
t r a n s f e r correlations based on a mass balance for the
ctial'yzer. In dimensionless quantities, these correlations
a r e given b y
Sh -- 0.00272 Re 1.~ Sc 1/s
Sh ----0.1663 Re ~
S c 1/3
108 < R e < 104
R e < 108
and the exponents a r e in reasonable a g r e e m e n t w i t h
values in the literature.
T h e m e m b r a n e interdiffusion coefficient (DNi-~) c a n
be e s t i m a t e d b y extrapolation, using d a t a at high flow
rates. Its m a g n i t u d e (10 -6 cm2/sec) agrees w i t h t h e
r a n g e r e p o r t e d in l i t e r a t u r e for other m e t a l ions. As
L a k e and M e l s h e i m e r (4) have i n d i c a t e d in t h e i r
work, the e x t r a p o l a t i o n technique gives m o r e r e l i a b l e
results at high concentrations. However, at h i g h concentrations, the m e m b r a n e a n d the a c i d - s i d e b o u n d a r y
l a y e r m a y contribute significantly to the nickel t r a n s fer rate, a n d s t u d y of this more complex system is b e y o n d the scope of this work.
M a n u s c r i p t s u b m i t t e d Nov. 18, 1980; r e v i s e d m a n u script received ca. March 9, 1981. This was P a p e r 619
p r e s e n t e d at the Hollywood, Florida, Meeting of t h e
Society, Oct. 5-!0, 1980.
A n y discussion of this p a p e r will a p p e a r in a Discussion Section to be p u b l i s h e d in the J u n e 1982 JOURNAL.
A l l discussions for the J u n e 1982 Discussion Section
Should be s u b m i t t e d b y Feb. 1, 1982.
Publication costs of this article were assisted by
General Motors Research Laboratories.
L I S T O F SYMBOLS
a c t i v i t y of ions, m o l / m l
m e m b r a n e surface area, cm ~
n i c k e l concentration in reservoir, m o l / m l
t u b e - s i d e nickel concentration, m o l / m l
initial nickel concentration, m o l / m l
concentration in the m e m b r a n e , m o l / m l
C ~ C n b u l k concentration, m o l / m l
surface concentration on side I, m o l / m l
C"
surface concentration on side II, m o l / m l
d
m e m b r a n e diameter, c m
D
diffusion coefficient in b u l k solution, cm2/sec
I)
diffusion coefficient in m e m b r a n e , cm2/sec
J
flux, m o l / ( c m ~ 9 sec)
KBL b o u n d a r y - l a y e r mass t r a n s f e r coefficient, c m / s e c
Km m e m b r a n e t r a n s f e r coefficient, c m / s e c
1
m e m b r a n e thickness, c m
m
r e s e r v o i r r a t e constant, sec -1
Q
v o l u m e t r i c flow rate, m l / s e c
Re
Reynolds n u m b e r ----vd/v, dimensionless
Sc
S c h m i d t n u m b e r -~ v/D, dimensionless
Sh
S h e r w o o d n u m b e r = KBLd/D, dimensionless
t
time, sec
v
solution velocity, c m / s e c
VR r e s e r v o i r volume, cm 8
fixed charge concentration in m e m b r a n e , e q / m l
a
A
Ci
Co
Cio
~.
oo~:~;~o ~. . . . . . . .
o.o
2:0
4:0
6:o
s:o
0
RcmlprocalReynoldsNumber
Fig. 9. Extrapolation of membrane resistance
x 10 - 4
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Vol. I28, No. 8
zi
charge of ions
Greek Characters
v
8
1719
DONNAN DIALYSIS
kinematic viscosity, cm~/sec
boundary layer thickness, cm
REFERENCES
1. R.M. Wallace, IEC Proc. Des. Dev., 6, 423 (1967).
2. R. M. Wallace, U.S. Pat. 3,454,490 (1969).
3. W. E. Rush, M.S. Thesis, University of South Carolina (1973).
4. M. A. Lake and S. S. Melsheimer, AIChE J., 24, 130
(1978).
5. J. L. Eisenmann and J. D. Smith, "Donnan Softening
as a Pretreatment to Desalination Processes,"
OSW R&D Progress Report No. 506 (1970).
6. J. L. Eisenmann, Second Annual EPA-AES Conference on Advanced Pollution Control, Kissimmee,
Florida (1979).
7. F. Helfferich, "Ion Exchange," Chap. 8, McGrawHill, New York (1962).
8. M. A. Peterson and H. P. Gregor, This Journal, 106,
1051 (1959).
Photoelectrochemical Properties of n-Type
J. F. McCann 1
Chemistry Discipline, School oS Physical Sciences, Flinders University, Bedford Park 5042, South Australia
and J. O'M. Bockris
Chemistry Department, Texas A & M University, College Station, Texas 77843
ABSTRACT
The electrochemical and photoelectrochemical properties of single crystal
n-type In203 were examined in 1M NaOI-I and 1N H2SO4. The photocurrentwavelength response indicated the absorption edge is an indirect transition of
about 2.3-2.5 eV. The flatband potentials were determined from Mott-Schottky
plots to be --0.72 and 0.22V (vs. SCE) in 1M NaOH and 1N H2SO4, respectively.
The photocurrent from the n-ln~Os electrode was steady during 13 hr while
illuminating the sample in 1M NaOH at 0.3V (vs. SCE).
A wide range of semiconductors have been investigated to assess their performance as photoelectrodes
in solar photoelectrolysis cells (1). In this work we
have applied standard electrochemical and photoelectrochemical techniques to examine single crystal
n-type In203 electrodes. To the authors' knowledge
the photoelectrochemicaI properties of n-type In2Os
have not been previously reported, although a study
of the dark current-voltage characteristics has been
published (2).
Experimental
Single crystals of In2Os w h i c h had been grown by
two different techniques were donated by Dr. J. H. W.
De Wit of the Inorganic Chemistry Department, Unid
versity of Utrecht, Utrecht, The Netherlands. The first
sample was grown in a PbO-B~O3 flux as described
elsewhere (3). Two electrical contacts were made by
soldering indium onto the back of the crystal. The
i-V relation was linear indicating that the In contacts
were ohmic. The resistivity of this sample was determined as 0.004 ~.cm. The color of pure In2Os powder
is light yellow, whereas this crystal was black. This
observation suggests that the crystal had occluded
impurities from the flux medium during its growth.
In subsequent photoelectrochemical studies of this
sample using medium intensity xenon irradiation
(,~ 26 mW cm-2), the photocurrent was negligible as
a function of applied voltage.
The second sample was an In2Os single crystal
grown from the vapor phase as described in Ref. (3).
The crystal was green and was transparent. An indium
contact was soldered onto the back of the crystal. The
sample was then fabricated into an electrode by
soldering a tin-coated copper wire to the indium contact and fixing the sample in a Teflon holder with
* Electrochemical Society Active Member.
XP r e s e n t address: School of Physics, University of New South
Wales, Kensington, N.S.W. 2033, Australia.
Key words: indium oxide, photoelectrolysis, oxide electrochemistry, e n e r g y c o n v e r s i o n .
epoxy. The resistivity of the sample was measured
from a high frequency impedance plot and calculated
as N 1200 ~ cm which was much higher than that of
the flux grown In2Os crystal. However, since the
photoelectrochemical response of the electrode fabricated ~rom the vapor grown crystal was substantial,
the results reported in this study were derived from
this sample.
A Wenking potential stepping motor control Model
SMP 69 and a PAR Model 173 potentiostat fitted with
a PAR Model 176 current-to-voltage converter w e r e
used to control the electrode potential and also to
perform current-voltage scans. A 900W xenon lamp
was used as the light source. An infrared absorbing
filter (Oriel G-776-7100) was placed between the
lamp and the cell to reduce heating effects. The illumination intensity was measured with a HewlettPackard Radiant Flux Detector Model 8334A. Monochromatic light was obtained by passing xenon light
through a high intensity Bausch and Lomb scanning
monochromator. Monochromatic light intensities were
measured with a Carl Zeiss calibrated thermopile
Model VT Q3/A. Capacitance measurements were
made with the use of a General Radio impedance
bridge Model 1608-A fitted with a General Radio
tuned amplifier Model 1232A. The reference electrode
was saturated calomel (SCE) and the counterelectrode was Pt. Rectified alternating photocurrent voltammograms were performed using the technique and
apparatus to be reported elsewhere (4).
Results and Discussion
A flux grown n-In2Os single crystal electrode examined in this study exhibited negligible photoelectrochemical activity. Other flux grown semiconductor
electrodes including Fe2TiOs, CoTiO3, and PbTiO3
which were examined by the authors, but have not
been reported here, have also lacked photoelectrochemical activity. It is probable that the absence of
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