An analysis of electrochemical reactions followed by series or complex chemical reactions

J. Electroanal. Chem., 243 (1988) l-19 Elsevier Sequoia S.A., Lausanne - Printed

in The Netherlands

AN ANALYSIS OF ELECTROCHEMICAL REACTIONS SERIES OR COMPLEX CHEMICAL REACTIONS

KEITH

FOLLOWED

BY

SCOTT

Departmeni of Chemical Engineering, TSI 3BA (Great Britain) (Received

Teesside Polytechnic, Middlesbrough,

5th May 1987; in revised form 9th September

Cleveland

1987)

ABSTRACT

A model for multiple reactions initiated by a charge transfer step at an electrode surface is described. The model incorporates the effect of simultaneous mass transport and chemical reaction in the electrode diffusion film. Two example syntheses, the anodic formation of chlorate and the epoxidation of propylene to propylene oxide, are analysed using the model. For the epoxidation of propylene, application of concentration profiling enables an accurate and straightforward analytical solution to be obtained.

(I) INTRODUCTION

In preparative electrochemistry, a distribution of various compounds is generally obtained even though an important objective is to maximise or optimise the yield of desired product or products by appropriate selection of reaction materials and conditions. The type and relative amounts of products are dictated by competitive and consecutive reaction pathways combining both electron exchanges and chemical changes and the relative rates and orders of reaction. A knowledge of these relationships is of vital importance for electrochemical process engineering. Analysis of reaction mechanisms and rates generally requires a combination of standard electrochemical kinetic techniques and quantitative product distribution data: the relative rates of chemical processes in relation to electron exchange have particular bearing on system behaviour. The location of the chemical reaction, whether mainly in the solution as considered in ref. 1 or mainly within the diffusion layer [2] or located in both regions [3], is important. This knowledge has significant repercussions in electrochemical reaction engineering in that suitable approximations may be adopted which allow simplifications in analysis and simulation. Approximations such as reaction plane or reaction zone concepts are well docu0022-0728/88/$03.50

0 1988 Elsevier Sequoia

S.A.

2

mented in the chemical engineering literature [4,5] for the analogous systems of fluid-fluid reactions and heterogeneous catalysis. If the concept of the film theory as opposed to the penetration theory or surface renewal theory [5] is adopted, steady state analyses are made using the diffusion equation with chemical reaction within the diffusion layer coupled with bulk material balances and appropriate’boundary conditions. Complex chemical reaction sequences consisting of m~tiple series and parallel paths [5-7] have been treated mainly for the area of gas-liquid reactions. Exact solutions can usually be obtained only by numerical techniques; however, by adopting appropriate approximate concentration profiles [5] for one or more of the reactants, analytical solutions can be obtained, often with a high degree of accuracy. In this context the purpose of this paper is to consider the characteristics of an import~t reaction scheme comprised of an initial el~tr~he~~~ reaction followed by consecutive chemical reactions. The analysis is adopted to reconsider a previous interpretation of the mechanism of anodic chlorate or hypochlorite production [8]. In addition, a related reaction system, the electrochemical epoxidation of propylene to propylene oxide via the bromohydrin, is investigated. This is a complex reaction sequence consisting of multiple second order reactions which has recently been considered theoretic~y using elaborate numerical procedures [9]. The mechanism, however, ignored the production of hypobromite and other inorganic halogens. The present work includes this aspect but approaches the problem using approximate solutions. (II) FORMULATION OF THE PROBLEM

The general problem to be considered is that of a series of reactions O=&A%Ekl,R

(1)

the first of which occurs at the electrode surface and de subsequent reactions to E and further to R occur in homogen~us solution. Homogeneous reaction occurs either in the electrolyte bulk or in a stagnant diffusion film of thickness 6 and gives rise to the concentration profiles shown schematically in Fig. 1. The environment in the bulk liquid may be approximately well mixed or in plug flow. The differential equations representing the material balances of the reaction species are based on the film theory: Q, d2C,/dx2 = k,c,

(2)

D, d=cJdx=

= k,c, - k,c,

(3)

D, d2crJdx2

= -k,c,

(4)

The boundary conditions considered here are x=0

dcddx

= 0,

x=6

cE = cEb,

dcR/dx

CR = c&,,

= 0,

dc,/dx

CA= cAb

= -i/nFD_,, (5)

-----

reuctlon

1

in’ayery I

T

Cone

Fig. 1.Schematic representation of concentration profiles.

All reactions are assumed to be first order with respect to the concentration c, of species j. The boundary conditions at x = 0 consider that components E and R are not electroactive. At x = 8 the boundary conditions are governed by bulk material balances.

(IZ.1) Analysis of the general case Solutions of the general case are presented here and are used later to describe characteristics of an example synthesis. Solutions of eqns. (2), (3) and (5) are obtained by standard procedures as CA

=

CE =

cAb cosh( mtx) i sinh[ m,( 6 - x)] nFm,D, cosh[ m, 61 + cosh( ml&)

%+

\ki exp( -m2x)

+

(6)

W exp( -m,S) - m,nFD,( rnf - rnz)

(m;:;;)

*c‘4

the

1

cosh( m,x)

cosh( m#)

nFDAm,( rn: - m$) - (m: - m;) where rn: = kl/DA, rnz = k,/D,, \k = k,/D,. System performance in terms of the yields of E and P is governed by their respective fluxes from the film into the bulk and the material balances in the bulk, written for example for species A as

DAA,-dc,

klcA + -

V

dx

dc,

-cA,,

x=s +dt=

7

where t is the batch reaction and V is the bulk volume.

time or reactor

(8) residence

time, A, is the electrode

area

4

Under the condition of zero bulk concentration of A the reaction will in most cases be distributed over a reaction zone or layer of thickness /J ( < 8) (see Fig. 1) at which point the concentration of A is zero. Reaction in the film is then divided into two regions 0 < x < p and p < x < 6, and in the latter region the diffusion equation for A applies, i.e. d2c,/dx2

= 0

(9)

Conditions at the “interface” between the reaction layer and the diffusion layer are related by the conservation of flux. This system is now considered further. (II.2) Analysis of the reaction layer In the region 0 x x < p the solutions of eqns. (2) and (3) are i sinh[ m,( p - x)] c A - nFDAm, cosh( m,p) cE= [

\ki exp( - m,p) CE ’ m,nFD,( rnf - m;)

(10)

1

cosh[ m2x] + cosh[m2p]

\ki exp[ -m2x] nFD,m,(mf-rn;)

*CA

- (mf-m:) (11)

The concentration distribution of R in this region is obtained by substituting eqn. (11) into eqn. (4) i.e. C R=

-k,/D,

jj

cE dx dx

(12)

In the region p c x < 6 the concentration

distributions of E and R are obtained

as

03)

(X-S)04) 1 (64

DE(cE,-cE,) tcRb-'R,,) DE cR+-cE=CEb+~+ DR

+

DE% DR

In the above equations the concentrations conservation of flux at x = p, i.e.

D,

cRP and cEP are found

from the

If the concentration of component E is zero at a position in the diffusion film say pi (< S), then a third reaction zone may be included if required.

5

The main objective of the above analysis is to calculate the fluxes of reaction species and determine reactor performance via eqn. (8). Rather than deal with general characteristics, two processes of industrial interest are considered, i.e. sodium chlorate production and propylene oxide production. The analysis of sodium chlorate formation is also applicable to sodium bromate. (III) SODIUM CHLORATE PRODUCTION

The electrochemical production of sodium chlorate place according to the following series of reactions 2X-+X,+2e-

bromate

takes

(16)

X2+HZO%H+ (A)

or sodium

(H)

HOXeXO-+H+ (E) (P) 2 HOX+XO-%X0, (E) (P)

+HOX+X(E)

(17)

(18) (H) +2X-+2H+

(19)

(R)

where X = Cl- or Br-. Equation (19) is a slow reaction and takes place mainly in bulk solution, whilst eqn. (18) is a fast equilibrium reaction. Reaction (17) is relatively fast (typical rate constants for chlorine hydrolysis of 0.17 s-* [lo] to 11.0 s-l [ll] and for bromine hydrolysis 110 s-l [ll]) and occurs mainly in the diffusion film. There are a number of undesirable side reactions which can accompany the above series of reactions, such as the electrochemical oxidation of hypohalite (HOX and X0-) 6X0-+3H,O-+2XO;+4X-+6Hi+3/20,+6eIn the case of chlorate (or bromate) production this constitutes a reduction in overall current efficiency, whilst if hypohalite is the desirable species it also causes a loss of selectivity. Previous analyses of these series reactions [8,12] have considered chlorine hydrolysis to occur throughout the diffusion film regardless of its thickness. This is an imprecise picture of the diffusion and reaction mechanism which, although it can give reasonable predictions of chlorate efficiency [8], should be approached by the analysis of Section 11.2. For a typical mass transfer coefficient of 2.5 X lop5 m s-i and with a diffusion coefficient for halogen of 6.7 X lo-” m2 s-‘, halogen hydrolysis occurs solely in the diffusion film for values of the rate constant given approximately by k, z 1.0. In this case the concentration (or activity) of halogen is given by eqn. (11) with the reaction layer thickness p = {m. As reaction (18) is a fast equilibrium process with an equilibrium constant K in the range 3 X 10e8 to 2 X 10m9 mol 1-l at 25OC [ll], the concentration of active

6

species can be obtained from the following differential equations. D, d2c,/dx2

= - k,c,

(20)

D, d2c,/dx2

= 0

(21)

with boundary conditions x = CL,ca = cEpr cP = cP and x = 0, cn = 0, cP = 0. The latter boundary conditions allow for the @electroactivity of both species at limiting current conditions. The solutions are c

-

E-

4

Sir4m,(x-P)] +

2Fm,D,

c

cosh(m,p)

EP-

i,

tanh(m,p) 2Fm,D,

1

x +

F

4

tanCw.4 2m,FD,

(22) (23)

cp = cpsx/G

Combining both equations gives the concentration of “active halide” cs( = [HOX] + [OX-]), which is a convenient way of representing the concentration of reactive species, because reaction (19), which is second order in hypohalite and first order in halite, can be written as an overall third order reaction [13], viz. (rate)R=

k2Kcfi+ i

(K+

cH+)3C

From this reaction rate and the flux of “active chlorine” at the edge of the diffusion film and the reactor material balance (eqn. 8) the performance characteristics can be determined as described by Jaksic [8]. However, the present analysis differs from that of Jaksic [8] due to the effect of the reaction layer region. This will affect, amongst other things, the halide ion oxidation efficiency which will be lowered if reaction (20) occurs. The current density ( i2) for this reaction depends on the flux of active halide at the electrode i.e. _i2 F

CL@

(25)

x=0

From this equation and eqns. (26) and (27) the current efficiency oxidation is obtained as

for chloride ion

1 - DscsbF/i,G

111= 3/2 -

tanNmd.4 2m,6

(b4 - 28 cosh( m,p)

(26)

where i, is the total applied current density and assuming DE = D, = D,. Figure 2 shows the typical effect of hydrodynamics (through the diffusion layer thickness) on the efficiency for chloride ion oxidation. A relatively thin diffusion layer is seen to be detrimental to the efficiency, since this increases the flux of active chlorine to the electrode. The above has considered halogen hydrolysis to be confined to a reaction layer ~1 (K S) which, although this is frequently the case, is not generally applicable to all

loo=‘ E 80. ‘t @ 60. $ 5 40. D 2 20. 01 1o-4

10-3

6/mFig. 2. Effect of diffusion layer thickness csb= 40 mol mm3, k, = 0.17 s-l.

on chloride

ion oxidation

efficiency.

DA = 0.67

X

10m9 m s-l,

conditions of hydrodynamics, ionic strength or temperature. Halogen hydrolysis may also occur in the bulk solution and the concentration distribution of halogen is given by eqn. (6). The concentration of active halogen in the diffusion layer is obtained from D, d%s/dx= = - k,c,

(27)

i, sinh( m,6) 2Fm, cosh(m$)

kc*, + rnf cosh(m,b)

_ i sinh[m,(S-x)] _ klcAb cosh( mix) 2 Fm, cosh( ml&) rnf cosh( m,6) From this the flux of active halogen at the electrode can be obtained to give the chloride ion oxidation efficiency as Dscs F _b %A F l_ 1 l-Y-& i,S cosh( ml&) 1 T i 91 = tanh( m,6) 3/2 2m,S

(29)

When cAb= 0 this reduces to the expression derived by Jaksic [8]. (Iv)

PROPYLENE

OXIDE PRODUCTION

The production of propylene oxide from propylene via the bromohydrin route [14] is considered. This system has received considerable attention and the important reaction steps are outlined here. The initial step is the anodic oxidation of bromide (reaction 16). The bromine formed can undergo hydrolysis to the bromohydrin as described in Section III or can react with propylene and OH- to form propylene bromohydrin [15].

Br, + CH,CH=CH,

+ OH-%CH,CH(OH)CH,Br

+ Br-

(30)

8

The bromohyd~n

is then thought to be hydrolysed to propylene oxide

CH,CH(OH)CH,Br

+ OH- %CH&H-CHZ

+ H,O + Br-

(31)

‘0’ A number of side reactions are possible such as the bro~nation propylene dibromide

of propylene to

CH,CH=CHZ + Br,%CH,CHBrHCH,Br

(32)

and the reaction of propylene oxide with water to form propylene glycol. CH,CH-CH,

+ H,O --, CH~CH~HCH~OH

(331

‘0’ With a suitable choice of operating conditions this reaction is insignificant [14] and hence the reaction system can be written as Br, + Propylene~Propylene

bromohyd~n~Propylene

oxide

2 69

P)

’

09

Br, + PropylenepPropylene

dibromide

PO)

(34)

(Q) (PI 4 (A) Br, + H,OpBrOH % OBr- + H+ 3

Reactions (30) and c32) are essentially irreversible and first order in both propylene and bromine concentrations [15]. Reaction (31) is essentially irreversible and first order in both propylene and OH- concentrations [16] and at high pH is approximately pseudo-first order. The differential equations representing the material balances for this simplified picture of propylene oxide production are written as DA d2c,/dx2

= k,c, + k$pCA + k&C,

D, d2c,,‘dx2 = k,c, D,,

- k,c,c,

(35) (36)

= - k,c,

(37)

D, d*c,,‘dx’

= (k, + k,)c,c,

(38)

D, d2cs/dx2

= - k3cA

(39)

d2c,/dx2

Solution of the above set of equations requires numerical procedures, although appropriate appro~mations can be adopted to facilitate solution [S]. Such approximations generally involve the adoption of analytical functions to approximate the concentration profile of one or more reactive species. This technique has been applied successfully in the analysis of the analogous system of two-phase reactors [5]

9 TABLE 1 Steady

state reactor

balances

(CSTR)

(41) -

cEb

-4,

dc,

-=kzC~,-k,c~~c~~+yD~d~ 7

-

-cFQ 7

(42)

x=6

dc

= - k2cEb+ GD-2

(43)

x=6

cQb= cp, - cp b - CEb - cpo b -cs,+A,D 7

V

dc, ‘dx

=?.&_cQ x=6

(44 b

_cEb-cpg

b

(45)

and includes the use of constant concentration [17], linear [5], exponential and hyperbolic functions [18]. The adoption of a suitable approximate procedure depends on the relative rates of reaction and mass transport in the diffusion layer as well as the complexity of the reaction sequence. In this respect systems are categorised in terms of reaction rates being slow, intermediate, fast or instantaneous. These categories and the solution methods in relation to propylene oxide formation are discussed briefly below. This discussion and the later analysis is based on the reactor operating as a continuous stirred tank reactor (CSTR). (IV. 1) Slow reaction In this case the reaction rates are so slow that the mass transport of bromine (A) from the electrode is so fast that no reaction takes place in the diffusion layer. Thus in the material balances (Table 1) the flux terms at the edge of the diffusion layer are zero for all components other than bromine. The solution of the set of simultaneous equations is easily achieved. (IT/.2) Intermediate reaction This is the general situation where reaction takes place in both the diffusion layer and bulk solution. A “limiting” case of this is when there is little or no reaction in bulk solution and the reaction is considered “fast” occurring in a limited region of the diffusion layer. Numerical solution of eqns. (35)-(39) is required in this region although approximate solutions based on “concentration profiling” frequently pro-

10

vide the desired degree of accuracy. Two such approaches are based on constant concentration and hyperbolic profiles. (ZV.3) Constant concentration profile The constant concentration profile effectively converts a second order reaction to a pseudo-first order form. A strict application of the constant concentration profile approximation first proposed by Van Krevelin and Hoftijzer [17] and outlined in ref. 5, to propylene in the diffusion layer will give the solution previously obtained for the set of reactions (1) and (16)-(18). In this case reactions (30) and (32) become pseudo-first order with rate constants of k,‘= k,Z, and k;= k4Cp, where Cr. is the effective constant concentration propylene in the diffusion layer. Thus eqn. (35) is written as DA d3c,/dx2

= k *cA

where k*=k +k;+k’ The m&no&r conce:tration

(46) C, is obtained from material balances [5] as

(47) where rn: = k*/D,.

When m,6 > 2, eqn. (47) simplifies to

The essence of this method is that an approximate concentration profile is assigned to one reactant species, i.e. bromine, and subsequent solutions of eqns. (35)-(39) are developed. The assigned profile satisfies the limiting case of a -first order reaction and the appropriate boundary conditions. (ZV.4) Hyperbolic profile solutions for propylene

oxide production

The use of a hyperbolic concentration profile for the approximate solution of the problem of simultaneous diffusion and reaction of the two components was proposed by Juvekar [18]. The hyperbolic approximation has been applied to the simultaneous absorption of two gases accompanied by a complex reaction [6] and the degree of accuracy obtained is typically within 5% of the numerical solution. The reaction system [6] is similar to the present case involving simultaneous diffusion of propylene and bromine and in view of this the above “profiled solution procedures” may be applied to this problem. The form of the profile is based on the analogy with the diffusion of a single solute accompanied by a fast reaction in that it satisfies the limiting case of a pseudo-first order reaction (the concentration of the other reacting species remains constant). A proposed profile for bromine is:

sinh[m(s-x)] cosh( ma)

i cA= FD,m

where m is a constant. This satisfies the following boundary conditions for the set of eqns. (35)-(39): x=0

dc, -=-_ dx

x=6orp

CA= 0

-i 2FDA’

dcp _ o de, _ ~CPO dx - dx dx ’

“=O

With the assumed profile for A as in eqn. (49), the set of differential equations is solved using standard procedures. For example, eqns. (35) and (38) are subtracted and solved in the form DAcA + D,c, =

1.i

c,dxdx+a,+b

(50)

thus enabling cp to be expressed as a function of x and hence allowing analytical solutions of eqns. (36), (37) and (39). The concentration distribution of active bromine is obtained as in Section III and is given in Table 1. Propylene dibromide production is obtained from an overall material balance. The concentration profiles obtained require a value for m which is obtained in the following way. Equation (43) is differentiated and eqn. (35) integrated to give two expressions for the bromine flux. These two expressions are matched at the boundaries x = 0 and x = S or x = p to give a final parametric equation in m. The equation requires an iterative procedure to evaluate m, which for fixed conditions of mass transport and temperature is a function of the bulk concentration of propylene and hence reactant conversion. (IVS) Instantaneous reaction An instantaneous reaction is one where as soon as two reacting components meet in the diffusion layer they are immediately consumed to produce desired product(s). Reaction therefore only takes place at a plane a distance p from the electrode (Fig. 3). For a single second order reaction the fluxes of components to that plane are given by D~c~,/k‘

=Dpcpb/@

- CL)

(51)

The concentration profiles of A and P are assumed to be linear. In cases where the reactant A undergoes a second (parallel) reaction which is reasonably fast then this will modify the concentration profile as shown in Fig. 3. The flux balance at the reaction plane then becomes

‘h -DAdx

=b x=p

DPCP

(52) WI.4

12 re

P

ctlon

p’?ne

TCone

A+P-Q

reactlon

Fig. 3. Concentration profiles for instantaneous reaction.

where the flux of A is obtained Section II.

from concentration

profiles

similar to that derived

in

(V) EXAMPLE CALCULATIONS

(V.1) Rate data and selection of operating regime Values of kinetic data and physical properties used are summarised in Table 2. The reactions of propylene to propylene dibromide and to propylene bromohydrin are irreversible with second order rate constants given by k, =fii

and

k, = (1 -f)k;

(53)

TABLE 2 Kinetic and physical data for propylene oxide production Parameter

Value

k; k; k,

3.4 X103 4.56 x~O-~ 110 1.3 x10-9 0.987~10-~ 1.18 x~O-~ 1.44 x10-9 0.67 x~O-~

DA DE

DPo DP

Ds

Ref. m3 s-’ mol-’ m3 s-l mol-’ s-1 m2 s-’ m2 s-l m2 s-l m2 s-l m2 s-l

15 15 11 19 9 9 20 8

13

where f is a function

of bromide

f= 0.967 - 0.583c,,-

- 0.368(~,,-)~

ion concentration

car- (mol 1-l) given by [15] (54)

Clearly the sum of the two rates appearing in eqns. (35) and (38) is independent of concentration. Although the bromide ion concentration will vary in the diffusion layer, it would be reasonable to assume a constant value considering the relatively low concentrations of bromide used (of the order of 0.2 mol 1-i) to minimise dibromide and bromate formation. At these levels of bromide the function f is not too sensitive to variations in car- and an appropriate value should be based on a mean of the bulk and electrode surface concentrations. The rate constant k2 for propylene oxide formation is assumed pseudo-first order and written as k, = k; cOH-. A suitable concentration of hydroxide ion used in this constant will be a mean value based on a Van Krevelin and Hoftijzer approximation [17]. It is worth noting that the critical reaction step in propylene oxide production is propylene bromohydrin formation in competition with propylene dibromide and hypobromite. Hence the assumption of a constant OH- concentration should not affect model predictions adversely regarding propylene oxide formation in view of the possible further reaction of bromohydrin outside the cell environment. The selection of the most appropriate operating regime and the most suitable solution depends on the values of a parameter $J given by

A saturation concentration of propylene of 7.56 mol m-3 [9] and with a typical diffusion layer thickness of 10e5 m gives a value of $B= 44.5 for the reaction between bromine and propylene. A value of $ greater than 2 is the condition for a fast reaction and hence with the estimated value, reaction is virtually instantaneous. Propylene oxide production will therefore be analysed on this basis in the following section. The hyperbolic profile solution could also be used but leads to more complex iterative procedures. The linear profile is likely to lead to considerable inaccuracies due to the low concentration of propylene in the bulk solution. In addition, its range of applicability from eqn. (48) is restricted to a value of 6 = 2.7 x 10e6 m, equivalent to a mass transfer coefficient of approximately 5 x lop4 m s-l or greater. In the above analyses a number of approximations are applied which clearly could be improved upon by introducing additional material balances. This, although giving greater scientific credibility, adds significantly to the complexity of the problem and is outside the scope of this paper, the main object of which is to describe approximate solution methods to the general problem of diffusion and simultaneous complex reactions in electrochemical reactors. Using the above discussed kinetic data, the validity of the approximate model and solution procedure will now be assessed by comparing predicted behaviour with the results of Alkire and Lisius [9], obtained by exact numerical solution of a more rigorous reactor model based on a continuous stirred tank reactor.

a 3-

/

l-

!

1

1

I

1

10

100

0

1000

‘0 t-l-/ ma1me3

04

02 ‘6;

/ml

c-6

dm-3

Fig. 4. Effect of electrolyte hydroxide ion concentration on propylene oxide production and selectivity. -3, i =lOOO A m-*. (a) c,/c,; (b) c,/co; (c) cpg. (0) r=lOs, A,/V=lOOm-‘, c,,-=5OOmolm Alkire and Lisius [9]; (0) instantaneous reaction. Fig. 5. Effect of bromide ion concentration. 7 =lO s, AJV cog; (b) CP; (C) CQ; (4 CE; (e) +O/CBrOH,

=lOO m-‘, pH = 14, i = 1000 A m-*. (a)

IOOr dlffuslm;

L Layer

06-.__

,

’ _.._.____ - - - PO

Ob--e-cG 'Bi /mol

Fig. 6. Concentration profiles during propylene oxide production. ( -) instantaneous reaction. Fig. 7. Current efficiency of propylene oxide formation. experimental data; (x) instantaneous reaction model.

(0)

dmb3

Exact solution; (------)

Experimental

data [22]; (0) scaled

15

(V. 2) Predicted reactor performance

To illustrate the reactor performance characteristics of propylene oxide production, the cell is modelled as a continuous stirred tank reactor (CSTR) which is a reasonable approximation to cell designs based on the sieve plate absorber [21] or sparged tank. The influence of reaction in a cathode diffusion layer is ignored. The steady state reactor material balances for the major products from the bromohydrin route are given in Table 1. The theoretical development for the instantaneous reaction model is given in the Appendix. An important consideration with the proposed reactor model is to compare its predictive ability with the exact numerical solution. Alkire [9] has obtained such solutions for a rigorous reactor model which incorporates reactions which are second order and reversible. In most cases the reverse reaction is not significant and has not been considered in the present model. In propylene oxide production via the bromohydrin there are two parameters which in particular affect product distribution, i.e. bromide ion and hydroxide ion concentration. Low bromide concentration and high hydroxide concentration both tend to favour the formation of propylene oxide. The effect of hydroxide ion concentration on selectivities of propylene oxide and its bulk concentration are shown in Fig. 4. The agreement between predictions of the instantaneous reaction model and those of the exact solution is good. Similarly, predictions of the effect of bromide ion concentration on reactor exit concentration and selectivity (Fig. 5) are also in reasonable agreement. The instantaneous reaction model applied here assumes that both bromine and propylene concentration distributions in the diffusion layer are linear. A comparison between the predicted concentrations from the instantaneous reaction model and exact solution is given in Fig. 6. Considering the simplicity of the approximate model, agreement is good. The exact model exhibits a “reaction plane” at x/6 = 0.82. The instantaneous reaction model would come more in line with this if the consumption of bromine to hypobromite in the region 0 < x < p were taken into account using the modified flux balance of eqn. (46). (V.3) Comparison with experimental

data

Figure 7 shows a comparison of the current efficiencies obtained with the instantaneous reaction model and the experimental data of Bejerano et al. [22] for propylene oxide production in a bipolar rod cell. A general agreement between model and experiment is seen with regard to the effect of bromide concentration. The comparatively low experimental values can be explained by the effect of current bypass in the rod cell which increases significantly as the bromide ion concentration increases. An approximate scaling of the experimental results, based on the current efficiencies of identified products, which allows for current bypass, gives a closer agreement between experiment and theory.

16 (VI) CONCLUSIONS

A general analysis for chemical processes occurring in electrochemical reactors which are initiated by an electron exchange is developed. The analysis is applicable to fast and slow reactions and can be adapted to consider more complex reactions involving reversible, instantaneous and second order processes. Application of concentration profiling techniques enables solutions to be worked out without the need of elaborate numerical integration of the boundary value equations. Such approximations have been demonstrated in the production of propylene oxide via the bromohydrin route and give an acceptable degree of accuracy. APPENDIX:

INSTANTANEOUS

REACTION

MODEL

Reaction scheme (34) is modelled as occurring in a CSTR. The reaction between propylene and bromine occurs at the reaction plane x = p and produces two products, propylene bromohydrin (E) and propylene dibromide, in the ratio of concentrations

CQ,_ _- k, k

I-

%

1-f f

(Al)

The reaction of bromine to hypobromide is assumed negligible due to the relatively low rate of reaction. The flux balance between A and P (eqn. 51) is solved to give the reaction plane position as

where cA is the concentration of bromine at the electrode. The material balance (diffusion) equations for components model are D, d=c,/dx= D,,

E and PO in this

= k,c, = - k,c,

d=c,o/dx=

(A3) 644)

These equations are solved in the regions 0 < x < p and p < x < S to give: (i) 0 < x < p c

=c E

cosh(m24

W)

5 =w

$2

where I#J= = m2p.

1

17

(ii) p < x < 6 CE = A, exp(m,x)

646)

+ B, exp( -m2x)

where cE, exd@2)

-cE,

exd+l)

A,=

(A7) exd249

> - ew@h)

and B, =

cE,

exP(+,)

-4

(A8)

exP(%)

with cpl = m,S. (iii) 0 < x < p cpg = cpo, +

4

(A9)

YCE

m2

where rn: = k2/D,,.

C PO =

3[cE]

+A,x

(‘w

+B,

where ‘PO, +

(All)

A,= a-11 and Dice

6412)

L-A,8 D

B~=cPo,+

PO

To evaluate the concentration distributions of products in the diffusion layer and the bulk concentrations, eqns. (A5)-(AlO) above are combined with the material balances of Table 1 with the condition cA, = 0. The bulk concentration of E is thus obtained as

CEb -= c%

2ex~h+O2) 2 ev(G2)

where P =

- 0

+P)(exdW2)

V(1 + kg) rA,DEm2

*

(A13) -exp(%))

18

The bulk concentration

of PO is obtained as

r~,~,m;

CPO,=WEb+

m

2cE, expb + +22) - c&v

WI + exp2+2)

v

2

e4W2)

The bulk concentration

-

exd%)

of propylene is given by eqns. (48) and (A2) as

1

CP, -= CP,

(Al%

43’~~

( l+

I/(&-/k)

1

In eqns. (A5)-(A14) there are two unknown concentrations cn, and cpg, and hence two further equations are required. These are obtained as follows. (a) At x = S flux of P in = flux of all products out. This yields the expression DPCP,

= &o(CPOr

-

CPO,)

+ D&E&,

-

CE,)

+ DQ(CQ,

-

CQ,)

(b) At x = p the flux of PO is continuous. That is dc,/dx (A9) and (AlO) are matched to give cE,

[

2 exp(2e2)

tanh(+2)+1-

exp(2cp2)

-

exp(244

(AW

obtained from eqns.

%,exr+i% +G2) m2 1 =

-A3q

-

exp(2+,)

- exp(2+,)

(Al?

The bulk concentration balance cP, = cP, + cPO,

+ cE,

of propylene dibromide (co,)

+ CQb

NOMENCLATURE

4 CJ 0, F i

4 i2 h

k k LA ml m2 n

electrode area/m2 concentration of species j/m01 mm3 diffusion coefficient of species j/m2 s-l Faraday number/C mol - ’ current density/ A m - 2 current density for halogen production/A mm2 current density for hypohalite oxidation/A rnp2 total current density/A me2 rate constant of step i/s-’ mass transfer coefficient of A/m s-l Jm/m-‘/2 ’ {m/m-1/2 number of electrons

is obtained from a material

(AN

19

f V

x T s V CL \k +

reaction time or residence time/s electrolyte volume/ m3 distance from the electrode surface/m yield of species j diffusion layer thickness/m current efficiency reaction layer thickness/m (W&)/m-’ qg7x

Subscripts

b S

P S

bulk condition concentration at the electrode surface reaction layer condition active chlorine

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