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The rate of hydrogen generation in the electrodeposition of metal powder at gas-evolving electrodes
Surface Technology, 17 (1982) 301 - 307
301
THE RATE OF HYDROGEN GENERATION IN THE ELECTRODEPOSITION OF METAL POWDER AT GAS-EVOLVING ELECTRODES H. VOGT Fachbereich Verfahrens- und Umwelttechnik, Technisehe Fachhochsehule Berlin, D-I O00 Berlin 65 (F.R.G.)
(Received July 8, 1982)
Summary A general equation is presented to predict the current efficiency of hydrogen generation that occurs simultaneously in the electrodeposition of metals at an elevated current density. The equation, which is based on a theoretical mass transfer equation, is compared with numerous experimental data from various workers.
1. Introduction Electrodeposition of metals at sufficiently high current densities is accompanied by hydrogen evolution. There are thus two concomitant cathode reactions. Contrary to the hydrogen-generating reaction, the rate of metal deposition is mass transfer controlled [ 1 ]. With respect to energy consumption, simultaneous hydrogen evolution is of course an unwanted side reaction. It is, however, a well-known fact that formation of gas bubbles at an electrode induces an enormous enhancement of mass transfer. The consequence is an increased rate of metal deposition so that the onset of gas evolution, although formerly considered to signal an upper limit of the current density, may be overstepped to a certain extent. That means from the commercial viewpoint that in electrodeposition of metal, hydrogen generation, although linked with a higher total energy requirement, may be desirable because, owing to the increased production rate, it simultaneously contributes to lower capital and operator cost. A quantitative analysis with commercial consequences for design and operation of the electrochemical reactor is impossible without generally applicable data on the current required for hydrogen generation. In a valuable experimental investigation, Sedahmed e t al. [2] obtained numerous data which confirm that the current efficiency of hydrogen generation is strongly dependent on the total current density and on the bulk concentration of dissolved metal. The scatter of the experimental data 0376-4583/82/0000-0000/$02.75
© Elsevier Sequoia/Printed in The Netherlands
302
clearly shows the difficulties in finding reproducible values of the current efficiency for gas generation. The scatter simultaneously points out the demand for a general relationship sustained by a theoretical basis and confirmed by comparison with experimental results. It is the object of the present paper to give a relationship for the current efficiency of the hydrogen-generating reaction and thus for the rate of h y d r o g e n generation competing with the electrodeposition of metal. 2. T h e o r y The following derivation will be related to metal deposition with simultaneous h y d r o g en evolution. The result, however, will be applicable to every mass-transfer-controlled electrode reaction with c o n c o m i t a n t generation of a substance to be (at least partly) evolved as gaseous phase at the electrode. The total current density of the reactor comprises two terms associated with the metal deposition {subscript 1) and with the hydrogen generation {subscript 2):
J =il +]2
(1)
The current efficiency of the hydrogen-generating reaction may be defined as
e2 -
12
J
Jl = 1 -- - -
J
(2)
As the metal deposition takes place under limiting current condition the pertinent partial cur r ent density can be expressed by
j, = kCA,o nl-F
(3)
PB
The mass transfer coefficient k in eqn. (3) depends on the hydrogen evolution, i.e. on the rate of the com pe t i ng reaction. Out of the available equations for mass transfer at gas-evolving electrodes we use [ 3 ] Sh =
kd
- 0.93Re°'SSc °'4s7
(4a)
DA which is easy to handle and which is based on one of the most elaborate theoretical models available at present. The serviceability of eqn. (4a) was tested in comparison with num er ous experimental data obtained by various workers [ 3 ] . The dimensionless groups are Re -
(5) AVL 12L
Sc -
(6) DA
303
Insertion of eqn. (4a) into eqns. (3) and (2) yields
KI( V d A )°'s e2=1
(7)
J
where K1
=
0.93
~ -0 "013CA ~C
0 '
nl F
(8)
PB
In addition together with the current density of the hydrogen-generating reaction i2 -
T~G P
n2
A RTfG VD
F
(9)
the volume flux density of gas evolved at the electrode can be referred to the total current density from eqn. (1):
]=K 1
+ K2- ~
where
P(n2/VD)F RTfG or, explicitly in the flux density,
(11)
K 2 -
(_~)o.s _ K~
+
1
(12)
Insertion of eqn. (12) into eqn. (7) finally yields the current efficiency of the hydrogen generation expressed as a function of the total current density: (2X + 1) °'s -- 1 e2 = 1 -(13) X where the parameter
X-
2jK2 K12
= 2]
pd(n2/VD)SC °'°26
(14a) R Tfo FDA( 0.93 CA,On1/VB)2 is used for brevity. Equation (13) is plotted in Fig. 1 as the full line. N o t all the hydrogen generated at the electrode is necessarily evolved as gas; fo denotes the fraction of the total amount of hydrogen which is transformed into the gaseous phase (fG ~< 1). At present it is problematic to give reasonable values of fG since, mainly at low values of the partial current
304 I
~
I
0.8
o
0.6 qJ 0.4
c~ 0 2
0 0.01
0.1
1
Fig. 1. Equation (13) (-I, ®) and horizontal ( o A, Sedahmed et al. [2] (o, o, M CuSO4) and Venczel [9]
10 Parameter X
1 100
; 10~
10 ~
) in comparison with experimental data from vertical ($, • ×, +, ~]) electrodes (supporting electrolyte, H2SO4): results from 0.1 M CuSO4;', A, 0.2 M CuSOa; x, +, 0.3 M CuSO4; i, ~, 0.4 (®, 0.033 M Fe3÷).
density J2, an appreciable f r a c t i o n o f the h y d r o g e n is t r a n s p o r t e d f r o m the e l e c t r o d e surface in dissolved f o r m [ 4 ] . It m a y be evolved inside the cell at surfaces d i f f e r e n t f r o m the generating e l e c t r o d e or it m a y be w i t h d r a w n f r o m the i n t e r e l e c t r o d e gap in dissolved f o r m . T h e value o f fG d e p e n d s n o t o n l y on the o p e r a t i o n c o n d i t i o n s b u t also on t h e kind of e l e c t r o c h e m i c a l r e a c t o r [ 5 ] . Nevertheless, various e x p e r i m e n t a l findings c o n f i r m t h a t setting fG = 1 is a reasonable a s s u m p t i o n , e x c e p t for e x t r e m e l y low rates o f the h y d r o g e n generation. E q u a t i o n (14a) can be f u r t h e r simplified by using the a p p r o x i m a t i o n Sh = 0 . 8 5 ( R e S c ) °.s
(4b)
instead o f eqn. (4a). Differences are m o d e r a t e in m e a n values o f the S c h m i d t n u m b e r and remain within the a c c u r a c y n e e d e d for the p r e s e n t application. T h e p a r a m e t e r X t h e n takes the simpler f o r m X =
2.77jpdn2/vD
(14b)
f o R T F D A ( CA,o n 1/UB) 2
It is seen f r o m eqns. (14a) and ( 1 4 b ) that, in a s y s t e m o f c o n s t a n t pressure and t e m p e r a t u r e , the c u r r e n t e f f i c i e n c y of the h y d r o g e n g e n e r a t i o n is a f u n c t i o n o f the total c u r r e n t d e n s i t y , bulk c o n c e n t r a t i o n o f dissolved metal and b u b b l e d e p a r t u r e d i a m e t e r : X = X ( j , CA,O, d)
(15)
(for n o t t o o small values o f the gas g e n e r a t i o n rate V ~ / A because o f the e f f e c t on fo)- Since the b u b b l e d e p a r t u r e d i a m e t e r d can a p p r o x i m a t e l y be c o n s i d e r e d a c o n s t a n t , as will be o u t l i n e d below, eqn. (15) c o n f i r m s the i m p a c t o f quantities as e x p e r i m e n t a l l y f o u n d b y S e d a h m e d et al. [ 2 ] .
305 3. Comparison with experimental data Numerous experimental data for copper powder deposition with simultaneous hydrogen evolution as recently reported by Sedahmed et al. [2] can be referred to to test the suitability of the general equation (eqn. (13)). The data are particularly useful for a check since the metal bulk concentration was varied in rather wide limits (between 0.1 and 0.4 M CuSO4). They are peculiar in that the current efficiency of the hydrogen-generating reaction exhibits a strongly inconsistent behaviour as the total current density was varied (from 400 to 1800 A m-2). The following data were used to calculate the parameter X: p = l 0 s Pa, T = 20 °C, d = 50/~m. Data for the kinematic viscosity and diffusion coefficient were taken from Wilke e t al. [6]. The bubble departure diameter was assumed to be independent of the current density. In fact, the behaviour of the diameter is a question not completely settled so far. The problem is discussed in detail in refs. 5 and 7. The diameter used is, however, a reasonable value with satisfactory accuracy for this field of application and agrees with the careful investigations carried out by Janssen and Hoogland [8]. Results are plotted in Fig. 1 together with the theoretical line. In spite of the scatter, the experimental results suggest that the theoretical line is quite serviceable. The undecided tendency of the experimental data with respect to the total current density as interpreted by Sedahmed et al. [2] is revealed to be experimental scatter and involves no further significance. Another set of current efficiency data presented previously by Venczel [9] was obtained under significantly different conditions. Not only was the kind of solution different {0.033 M Fe2(SO4)3 + 1 M H2SO4) but also its concentration. Furthermore, the total current density was varied in a much wider range (5 - 8000 A m-2). The following data were used to calculate values of X: p = 105 Pa, T = 25 °C, DA = 0.54 X 10 -9 m ; s-I, Cfe3+ CA.0---- 33 mol m -3. The bubble departure diameter used was again d = 50 pm. Larger diameters as reported by Ibl and Venczel [ 10] disagree strongly with observations of various other workers and appear doubtful to the present author. Owing to the smaller concentration and the larger current density, most of the values of X are larger than in the experiments of Sedahmed et al. [2] and are useful to check the upper range of eqn. (13). Results are plotted in Fig. 1 and they confirm the theoretical relationship. It must be pointed out that the above model involves a fundamental departure from reality in that it supposes that mass transfer of the metal ions to the electrode is a l w a y s controlled by gas evolution. In reality, at current densities below the onset of gas evolution, other mechanisms such as natural convection due to concentration gradients must be active [11] and, at extremely low rates of the gas evolution, combined'action of several mechanisms is decisive. Therefore the model can only represent a deformed description of reality in the range of these regions of low current density. The range is, at any rate, inaccurately represented owing to the uncertainty in fG as discussed. However, examination of Fig. 1 discloses that the impact ~-
306 o f t h i s u n c e r t a i n t y is s u b o r d i n a t e , a n d t h e r e s u l t i n g e q u a t i o n s s e e m t o b e serviceable over the total range of current density with satisfactory accuracy.
4. Results (1) The theoretical relationship of eqn. (13) together with eqn. (14b) or (14a) allows the current efficiency of a gas-generating reaction (of any kind o f gas) c o m p e t i n g w i t h a n y m a s s - t r a n s f e r - c o n t r o l l e d r e a c t i o n t o b e p r e d i c t e d . (2) Equation (13) represents a general relationship based on a fully t h e o r e t i c a l m a s s t r a n s f e r m o d e l ( m i c r o c o n v e c t i o n m o d e l ) w h i c h is s u p p o r t e d by various mass transfer data. (3) Equation (13) was shown to be verified by various experimental data. ( 4 ) E q u a t i o n ( 1 3 ) t o g e t h e r w i t h cell v o l t a g e d a t a , e.g. as r e p o r t e d b y S e d a h m e d et al. [ 2 ] , e n a b l e s t h e p r e d i c t i o n o f a n e c o n o m i c a l o p t i m u m o f operation for those electrochemical reactors which balance the energy loss d u e t o s i m u l t a n e o u s gas e v o l u t i o n w i t h t h e b e n e f i t o f a r a i s e d p r o d u c t i o n rate owing to enhanced mass transfer.
Nomenclature A CA,O
d
DA WG F
J
k K1 K2
n
p R Re Sc Sh T
~2G X C2 P 1)L
total electrode area (m 2) bulk concentration of metal (mol m -3) bubble departure diameter (m) diffusion coefficient of dissolved metal (m 2 s-1) fraction of the total amount of hydrogen evolved as gas 96 500 A s tool -I , Faraday constant current density (A m -2) mass transfer coefficient (m S 1) parameter in eqn. (8) (A s°'s m -2"s) parameter in eqn. (11) (A s m -3) charge number of the electrode reaction pressure (kg2f m -1 S- 2 ) 8.314 kg m s -2 mo1-1 K -1, gas constant Reynolds number (eqn. (5)) Sehmidt number (eqn. (6)) Sherwood number (eqn. (4a)) temperature (K) volume flow rate of evolved gas (m 3 s -1 ) dimensionless parameter in eqn. (14a) or (14b) current efficiency of the hydrogen generation stoiehiometrie number kinematic viscosity of the electrolyte (m 2 s - l )
Subscripts A B D
dissolved metal deposited metal hydrogen
307
References 1 2 3 4 5 6 7 8 9 10 11
C. Wagner, Adv. Electrochem. Electrochem. Eng., 2 (1962) 1. G. H. Sedahmed, Y. A. E1-Taweel and O. A. Hassan, Surf. Technol., 14 (1981) 109. K. Stephan and H. Vogt, Electrochim. Acta, 24 (1979) 11. H. Vogt, Gas-evolving electrodes. In J. O'M. Bockris (ed.), Comprehensive Treatise o f Electrochemistry, Vol. 6, Plenum, New York, pp. 445 et seq. H. Vogt, Fortschr. Verfahrenstech., 20 (1982) 369. C. R. Wilke, M. Eisenberg and C. W. Tobias, J. Electrochem. Soc., 100 {1953) 513. H. Vogt, Fortschr. Verfahrenstech., 16 (1978) 297. L. J. J. Janssen and J. G. Hoogland, Electrochim. Acta, 18 (1973) 543. J. Venczel, lJber den Stofftransport an gasentwickelnden Elektroden, Dissertation, EidgenSssische Technische Hochschule, Zurich, 1961. N. Ibl and J. Venczel, Metalloberfliiche, 24 (1970) 365. N. Ibl, Adv. Electrochem. Electrochem. Eng., 2 (1962) 49.
301
THE RATE OF HYDROGEN GENERATION IN THE ELECTRODEPOSITION OF METAL POWDER AT GAS-EVOLVING ELECTRODES H. VOGT Fachbereich Verfahrens- und Umwelttechnik, Technisehe Fachhochsehule Berlin, D-I O00 Berlin 65 (F.R.G.)
(Received July 8, 1982)
Summary A general equation is presented to predict the current efficiency of hydrogen generation that occurs simultaneously in the electrodeposition of metals at an elevated current density. The equation, which is based on a theoretical mass transfer equation, is compared with numerous experimental data from various workers.
1. Introduction Electrodeposition of metals at sufficiently high current densities is accompanied by hydrogen evolution. There are thus two concomitant cathode reactions. Contrary to the hydrogen-generating reaction, the rate of metal deposition is mass transfer controlled [ 1 ]. With respect to energy consumption, simultaneous hydrogen evolution is of course an unwanted side reaction. It is, however, a well-known fact that formation of gas bubbles at an electrode induces an enormous enhancement of mass transfer. The consequence is an increased rate of metal deposition so that the onset of gas evolution, although formerly considered to signal an upper limit of the current density, may be overstepped to a certain extent. That means from the commercial viewpoint that in electrodeposition of metal, hydrogen generation, although linked with a higher total energy requirement, may be desirable because, owing to the increased production rate, it simultaneously contributes to lower capital and operator cost. A quantitative analysis with commercial consequences for design and operation of the electrochemical reactor is impossible without generally applicable data on the current required for hydrogen generation. In a valuable experimental investigation, Sedahmed e t al. [2] obtained numerous data which confirm that the current efficiency of hydrogen generation is strongly dependent on the total current density and on the bulk concentration of dissolved metal. The scatter of the experimental data 0376-4583/82/0000-0000/$02.75
© Elsevier Sequoia/Printed in The Netherlands
302
clearly shows the difficulties in finding reproducible values of the current efficiency for gas generation. The scatter simultaneously points out the demand for a general relationship sustained by a theoretical basis and confirmed by comparison with experimental results. It is the object of the present paper to give a relationship for the current efficiency of the hydrogen-generating reaction and thus for the rate of h y d r o g e n generation competing with the electrodeposition of metal. 2. T h e o r y The following derivation will be related to metal deposition with simultaneous h y d r o g en evolution. The result, however, will be applicable to every mass-transfer-controlled electrode reaction with c o n c o m i t a n t generation of a substance to be (at least partly) evolved as gaseous phase at the electrode. The total current density of the reactor comprises two terms associated with the metal deposition {subscript 1) and with the hydrogen generation {subscript 2):
J =il +]2
(1)
The current efficiency of the hydrogen-generating reaction may be defined as
e2 -
12
J
Jl = 1 -- - -
J
(2)
As the metal deposition takes place under limiting current condition the pertinent partial cur r ent density can be expressed by
j, = kCA,o nl-F
(3)
PB
The mass transfer coefficient k in eqn. (3) depends on the hydrogen evolution, i.e. on the rate of the com pe t i ng reaction. Out of the available equations for mass transfer at gas-evolving electrodes we use [ 3 ] Sh =
kd
- 0.93Re°'SSc °'4s7
(4a)
DA which is easy to handle and which is based on one of the most elaborate theoretical models available at present. The serviceability of eqn. (4a) was tested in comparison with num er ous experimental data obtained by various workers [ 3 ] . The dimensionless groups are Re -
(5) AVL 12L
Sc -
(6) DA
303
Insertion of eqn. (4a) into eqns. (3) and (2) yields
KI( V d A )°'s e2=1
(7)
J
where K1
=
0.93
~ -0 "013CA ~C
0 '
nl F
(8)
PB
In addition together with the current density of the hydrogen-generating reaction i2 -
T~G P
n2
A RTfG VD
F
(9)
the volume flux density of gas evolved at the electrode can be referred to the total current density from eqn. (1):
]=K 1
+ K2- ~
where
P(n2/VD)F RTfG or, explicitly in the flux density,
(11)
K 2 -
(_~)o.s _ K~
+
1
(12)
Insertion of eqn. (12) into eqn. (7) finally yields the current efficiency of the hydrogen generation expressed as a function of the total current density: (2X + 1) °'s -- 1 e2 = 1 -(13) X where the parameter
X-
2jK2 K12
= 2]
pd(n2/VD)SC °'°26
(14a) R Tfo FDA( 0.93 CA,On1/VB)2 is used for brevity. Equation (13) is plotted in Fig. 1 as the full line. N o t all the hydrogen generated at the electrode is necessarily evolved as gas; fo denotes the fraction of the total amount of hydrogen which is transformed into the gaseous phase (fG ~< 1). At present it is problematic to give reasonable values of fG since, mainly at low values of the partial current
304 I
~
I
0.8
o
0.6 qJ 0.4
c~ 0 2
0 0.01
0.1
1
Fig. 1. Equation (13) (-I, ®) and horizontal ( o A, Sedahmed et al. [2] (o, o, M CuSO4) and Venczel [9]
10 Parameter X
1 100
; 10~
10 ~
) in comparison with experimental data from vertical ($, • ×, +, ~]) electrodes (supporting electrolyte, H2SO4): results from 0.1 M CuSO4;', A, 0.2 M CuSOa; x, +, 0.3 M CuSO4; i, ~, 0.4 (®, 0.033 M Fe3÷).
density J2, an appreciable f r a c t i o n o f the h y d r o g e n is t r a n s p o r t e d f r o m the e l e c t r o d e surface in dissolved f o r m [ 4 ] . It m a y be evolved inside the cell at surfaces d i f f e r e n t f r o m the generating e l e c t r o d e or it m a y be w i t h d r a w n f r o m the i n t e r e l e c t r o d e gap in dissolved f o r m . T h e value o f fG d e p e n d s n o t o n l y on the o p e r a t i o n c o n d i t i o n s b u t also on t h e kind of e l e c t r o c h e m i c a l r e a c t o r [ 5 ] . Nevertheless, various e x p e r i m e n t a l findings c o n f i r m t h a t setting fG = 1 is a reasonable a s s u m p t i o n , e x c e p t for e x t r e m e l y low rates o f the h y d r o g e n generation. E q u a t i o n (14a) can be f u r t h e r simplified by using the a p p r o x i m a t i o n Sh = 0 . 8 5 ( R e S c ) °.s
(4b)
instead o f eqn. (4a). Differences are m o d e r a t e in m e a n values o f the S c h m i d t n u m b e r and remain within the a c c u r a c y n e e d e d for the p r e s e n t application. T h e p a r a m e t e r X t h e n takes the simpler f o r m X =
2.77jpdn2/vD
(14b)
f o R T F D A ( CA,o n 1/UB) 2
It is seen f r o m eqns. (14a) and ( 1 4 b ) that, in a s y s t e m o f c o n s t a n t pressure and t e m p e r a t u r e , the c u r r e n t e f f i c i e n c y of the h y d r o g e n g e n e r a t i o n is a f u n c t i o n o f the total c u r r e n t d e n s i t y , bulk c o n c e n t r a t i o n o f dissolved metal and b u b b l e d e p a r t u r e d i a m e t e r : X = X ( j , CA,O, d)
(15)
(for n o t t o o small values o f the gas g e n e r a t i o n rate V ~ / A because o f the e f f e c t on fo)- Since the b u b b l e d e p a r t u r e d i a m e t e r d can a p p r o x i m a t e l y be c o n s i d e r e d a c o n s t a n t , as will be o u t l i n e d below, eqn. (15) c o n f i r m s the i m p a c t o f quantities as e x p e r i m e n t a l l y f o u n d b y S e d a h m e d et al. [ 2 ] .
305 3. Comparison with experimental data Numerous experimental data for copper powder deposition with simultaneous hydrogen evolution as recently reported by Sedahmed et al. [2] can be referred to to test the suitability of the general equation (eqn. (13)). The data are particularly useful for a check since the metal bulk concentration was varied in rather wide limits (between 0.1 and 0.4 M CuSO4). They are peculiar in that the current efficiency of the hydrogen-generating reaction exhibits a strongly inconsistent behaviour as the total current density was varied (from 400 to 1800 A m-2). The following data were used to calculate the parameter X: p = l 0 s Pa, T = 20 °C, d = 50/~m. Data for the kinematic viscosity and diffusion coefficient were taken from Wilke e t al. [6]. The bubble departure diameter was assumed to be independent of the current density. In fact, the behaviour of the diameter is a question not completely settled so far. The problem is discussed in detail in refs. 5 and 7. The diameter used is, however, a reasonable value with satisfactory accuracy for this field of application and agrees with the careful investigations carried out by Janssen and Hoogland [8]. Results are plotted in Fig. 1 together with the theoretical line. In spite of the scatter, the experimental results suggest that the theoretical line is quite serviceable. The undecided tendency of the experimental data with respect to the total current density as interpreted by Sedahmed et al. [2] is revealed to be experimental scatter and involves no further significance. Another set of current efficiency data presented previously by Venczel [9] was obtained under significantly different conditions. Not only was the kind of solution different {0.033 M Fe2(SO4)3 + 1 M H2SO4) but also its concentration. Furthermore, the total current density was varied in a much wider range (5 - 8000 A m-2). The following data were used to calculate values of X: p = 105 Pa, T = 25 °C, DA = 0.54 X 10 -9 m ; s-I, Cfe3+ CA.0---- 33 mol m -3. The bubble departure diameter used was again d = 50 pm. Larger diameters as reported by Ibl and Venczel [ 10] disagree strongly with observations of various other workers and appear doubtful to the present author. Owing to the smaller concentration and the larger current density, most of the values of X are larger than in the experiments of Sedahmed et al. [2] and are useful to check the upper range of eqn. (13). Results are plotted in Fig. 1 and they confirm the theoretical relationship. It must be pointed out that the above model involves a fundamental departure from reality in that it supposes that mass transfer of the metal ions to the electrode is a l w a y s controlled by gas evolution. In reality, at current densities below the onset of gas evolution, other mechanisms such as natural convection due to concentration gradients must be active [11] and, at extremely low rates of the gas evolution, combined'action of several mechanisms is decisive. Therefore the model can only represent a deformed description of reality in the range of these regions of low current density. The range is, at any rate, inaccurately represented owing to the uncertainty in fG as discussed. However, examination of Fig. 1 discloses that the impact ~-
306 o f t h i s u n c e r t a i n t y is s u b o r d i n a t e , a n d t h e r e s u l t i n g e q u a t i o n s s e e m t o b e serviceable over the total range of current density with satisfactory accuracy.
4. Results (1) The theoretical relationship of eqn. (13) together with eqn. (14b) or (14a) allows the current efficiency of a gas-generating reaction (of any kind o f gas) c o m p e t i n g w i t h a n y m a s s - t r a n s f e r - c o n t r o l l e d r e a c t i o n t o b e p r e d i c t e d . (2) Equation (13) represents a general relationship based on a fully t h e o r e t i c a l m a s s t r a n s f e r m o d e l ( m i c r o c o n v e c t i o n m o d e l ) w h i c h is s u p p o r t e d by various mass transfer data. (3) Equation (13) was shown to be verified by various experimental data. ( 4 ) E q u a t i o n ( 1 3 ) t o g e t h e r w i t h cell v o l t a g e d a t a , e.g. as r e p o r t e d b y S e d a h m e d et al. [ 2 ] , e n a b l e s t h e p r e d i c t i o n o f a n e c o n o m i c a l o p t i m u m o f operation for those electrochemical reactors which balance the energy loss d u e t o s i m u l t a n e o u s gas e v o l u t i o n w i t h t h e b e n e f i t o f a r a i s e d p r o d u c t i o n rate owing to enhanced mass transfer.
Nomenclature A CA,O
d
DA WG F
J
k K1 K2
n
p R Re Sc Sh T
~2G X C2 P 1)L
total electrode area (m 2) bulk concentration of metal (mol m -3) bubble departure diameter (m) diffusion coefficient of dissolved metal (m 2 s-1) fraction of the total amount of hydrogen evolved as gas 96 500 A s tool -I , Faraday constant current density (A m -2) mass transfer coefficient (m S 1) parameter in eqn. (8) (A s°'s m -2"s) parameter in eqn. (11) (A s m -3) charge number of the electrode reaction pressure (kg2f m -1 S- 2 ) 8.314 kg m s -2 mo1-1 K -1, gas constant Reynolds number (eqn. (5)) Sehmidt number (eqn. (6)) Sherwood number (eqn. (4a)) temperature (K) volume flow rate of evolved gas (m 3 s -1 ) dimensionless parameter in eqn. (14a) or (14b) current efficiency of the hydrogen generation stoiehiometrie number kinematic viscosity of the electrolyte (m 2 s - l )
Subscripts A B D
dissolved metal deposited metal hydrogen
307
References 1 2 3 4 5 6 7 8 9 10 11
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