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Gas absorption with autocatalytic reaction
Chicd
Engineefing Science.1975,Vol.
30, pp. 1215-1218.
PergamonPress. Printed in Great Britain
GAS ABSORPTION WITH AUTOCATALYTIC REACTION MONG-TECK SIM and R. MANN Department of Chemical Engineering, University of Manchester Institute of Science and Technology, Manchester, England (Received 23 December 1974;accepted 26March 1975)
Abstract-The film theory has been. employed to investigate gas absorption accompanied by an autocatalytic chemical reaction of the form A t R-R + R. This reaction has been taken to be the simplest analogue of the autocatalytic features Of chain reactions encountered in the industrially important gas-liquid oxidation and chlorination reactions. Numerical solutions have been obtained using standard library subroutines for integrating systems of first order equations, having transformed the two describing second order non-linear differential equations into suitable first order initial value form. Predictions of the Enhancement Factor show that it can be very considerably larger than that for the ‘equivalent’ tist order reaction. This is due to the accumulation of radicals within the mass transfer fdm. An asymptotically exact empirical approximation for the Enhancement Factor is also presented. The ‘degree of conversion’ of the absorbed gas across the mass transfer film is shown to be more sensitive to the kinetic and diffusional parameters than is the case for a first order reaction.
INTRODUCTION
Hydrocarbon oxidation and chlorination reactions are basic to the chemical industry, since their products represent important reactive intermediates which are precursors to a wide range of utility chemicals. The reactions in which these intermediates are produced are often two-phase, the oxygen (air) and chlorine being supplied in the gas phase to the liquid phase hydrocarbon. The proper design of such reactors depends upon a detailed knowledge of the individual rate processes that take place and how these should be combined with descriptions of the flow nature of the two phases. The kinetic mechanisms of oxidation and chlorination reactions are believed to be based on chain reactions which have autocatalytic features. The transport process in the liquid phase consists therefore of dissolution followed by diffusion with chain-type reactions. In the present study the autocatalytic reaction
Material balances on species A and 8 over an increment dx of the mass transfer film gives the differential equations 2
$!-$
= kAi
These equations are to be solved to give A(x) and d(x) subject to the boundary conditions: at the interface [
A=A*
solubility relationship non-volatility condition
at the film boundary is chosen as a simple representation for such kinetics in order to determine the effect upon the rate of absorption. Thus species A dissolves at the interface and then reacts autocatalytically with radicals k which are present in the liquid phase. A(g+A(l)
t k(l)---&(l)
x = XL
A =Ab li =i&
the bulk conditions.
The diffusion-reaction eqns (1) and (2) can be cast into a dimensionless form using the reduced variables:
+ ii(l).
SYSTEMOF EQUATIONS Before presentation of the diffusion-reaction equations,
the following assumptions should be noted: (i) gas phase resistance is negligible. (ii) liquid phase radicals are non-volatile. (iii) the concentration of species A at the gas-liquid interface corresponds to equilibrium with the partial pressure of A in the gas phase.
to obtain 2
$j=Mai
1215
(3)
MONO-TECKSIM and R. MANN
1216
It is now only necessary to represent each second order equation, as two simultaneous first-order equations in order to complete the transformation into an initial value problem. In this way the problem is made amenable to solution by a Computer Library Subroutine designed to handle systems of first order equations. In this case, the four differential equations are:
with the boundary conditions changed to
di-o
z=o .
acl.0
z=l
a = qA
z-
f = 1.0.
dyl
x=
Some experimental observations on MEK oxidationll] have suggested that under certain conditions the radicals may be scavenged resulting in a zero order dependence with respect to the absorbing oxygen. If this is the case, then eqns (3) and (4) are uncoupled and they can be solved analytically. However, for the general case no analytical solutions are available and either a co-locational type approximation method [2] or an exact numerical solution has to be employed.
Y2
and dy2 d2a
x=z=Mat
(10)
representing eqn (3), plus
dys
&=
(11)
Y4
NUMERICALSOLUTION
A major diSiculty in numerically solving eqns (3) and (4) arises from the fact that one of the boundary conditions is a flux condition. If it could be assumed that P were known, the boundary value problem is transformed into an initial-value problem. This can be seen by combining the diffusion-reaction eqns (3) and (4) to obtain
and dy, _ d2+ - Mai
(12)
dz-z=~
representing eqn (4). This system of first order differential equations has initial conditions: y1= 1-o
This is a linear homogeneous second-order differential equation which can be integrated twice to obtain
y2 =
qA
-
1+ (1 - i*)qayR atz=O.
y3 = i* a + qr& = c, + c*z
(6)
where Cl and CZ are integration constants. Applying the boundary conditions gives c1= 1+ qIt’y,G* c2
=
qA
+
qR,%
-
(1 +
qh,d*)
and substituting back in eqn (6), there results a + qltpi = (9.4 + gayId + (1 + qi&*)(l-
2).
Now, provided aand i are differentiable at z = 0 di da t q&y@-&&= (4.4+ qrw) dt I r-O
-
(1+ 9mi~*)*
(7)
)u=o
1
Equations (9)-(12) reveal that film theory solutions for autocatalytic reaction depend upon the parameters M and qti’ya.For a particular choice of these parameters the set of four first order differential equations can be integrated from the initial conditions at the interface (z = 0) provided that the unknown interface radical concentration r* is assigned a value. The correct value for )* will generate profiles that are compatible with the boundary conditions at z = 1, i.e. at the bulk boundary. Film theory solutions were developed using Runge-Kutta-Mersons method for the integration and the method of bisection to search out the appropriate +*. (The NAG library subroutines D02ABF and COSACF[3] were employed in the computation.) Further details are given in Ref. (4). ARSORPTIONPREDICTIONS
Thus, since di dr I Z_O=O the above equation reduces to da = (4A - 1)+ (1 - i*)q*yd. dr I r-0
Typical concentration profiles for the case q* = 1, q,t = 0 and yti = 1 are presented in Fig. l(a-d). Fig. l(a) shows the profiles developed under relatively slow reaction conditions when f varies only slightly with position in the film and the reaction is approximately pseudo 1st order. However, as the autocatalytic reaction velocity is increased (m increasing) a large buildup of radicals is observed at the interface in Fig. l(d) and a
1217
Gas absorption with autocatalytic reaction
(b)
I.0 t
0
0.0
(b)
3.0
200
2.0
IO.0
(cl
(d) t
I
0
I.0
0.0
I.0
I.0 t i
0
l2LLd_a
0.0
0.2
0.4
0.6
0.0 0.6
’ 0.0
I.0
0.2
I
I
I
0.4
0.6
0.6
ho.0 I.0
Z.X XL
Fig.2.Multiplefunctionsolutionsfor q,t = 1,‘y,t= 1,d/M = 4. Fig. 1. Concentration profiles for increasing reaction velocity
corresponding enhancement of the absorption rate is expected. Some difficulties were experienced due to the existence of multiple roots of 3*, which appear to be present when the initial gradient of A is smaller than -10 for any parameter qkyh, i.e. under relatively fast reaction conditions. This is a property of non-linear equations which has been discussed by Ames[5]. A typical example for q,tyd, = 1 and a= 4 can be used to illustrate this. It has been found for this case that the function roots are 15.7863, 15.8267and 1.7976. The concentration profiles for A and ‘R that correspond to these roots are shown in Fig. 2(a-& It can be seen that the three roots satisfy the four boundary conditions but only the one shown in Fig. (2b) is the proper solution because the remaining two solutions have profiles that oscillate within the defined film and furthermore the dimensionless concentration a( = A/A *) exceeds 1-Owhich is physically impossible. Fig. (2b) shows that the profile of a decreases exponentially towards the film boundary and i(= &I&,) decreases linearly as soon as a has been consumed. This is taken to be the only physically proper solution and due to this complication it was necessary to plot out the concentration profiles in order to check the correct i* value. The Enhancement Factor for autocatalytic reaction is given by
and is simply the initial value of eqn (9) provided the correct value of i* is chosen. Numerical results were obtained up to values of E of about 16. Above this value the concentration profiles were so steep that the computing time required was prohibitive. Computed results are shown as continuous lines in Fig. 3.
0.01
04
IO
fi Fig. 3. Enhancement factors for autocatalytic reaction.
An important discovery which enables a more rapid evaluation of the Enhancement Factor when performing a gas-liquid reactor calculation is that the implicit approximate relationship
1218
MONG-TECK SIMand R. MANN
E=
d”E3 tanh JM[d-=-J
(13)
is asymptotically coincident with exact numerical evalua-
tions. (Apart from a sign change eqn (13) closely resembles the approximate expression proposed by Van Krevelyn and Hoftijzer[6] for second order reaction of the type A t B+products.) Moreover, the difference between exact and approximate predictions is only significant over small ranges of a. The maximum difference is about 55% for qti?R = lo-’ and only about 10% for qti-yti= 10-l. For smaller values of q.e’ytithe difference declines to zero as the behaviour becomes closer to pseudo-first order. Results based on eqn (13) are shown as broken lines in Fig. 3. It is noteworthy that in contrast to second order reactions of the type A t B-+products, the production and accumulation of radicals within the mass transfer film can give rise to enhancement factors which are markedly greater than would be expected on the basis of a ‘naive’ pseudo first order analysis. Thus at a value of X&= 1 an analysis based on neglect of radical accumulation would give an enhancement of only O-5,whereas if qayfi = lo-* the rate of absorption would be increased a hundredfold. These theoretically large enhancement factors may go some way towards explaining the often large absorption rates encountered in chlorinators and oxidisers. In substitutive hydrocarbon chlorination for example, it is quite common for the vented hydrogen chloride gas to contain only trace amounts of the feed chlorine and the vented gas is then a potentially useful high purity by-product stream. In hydrocarbon oxidisers the vented gas necessarily should contain only a small trace of oxygen otherwise explosive mixtures may form in the ullage. Finally the results have been treated to determine the fraction of A that is reacted within the mass transfer film, which is given by
Fig. 4. Balance between film and bulk reaction.
NOTATION concentration
of species A
dimensionless concentration of A given by A/A* diffusion coefficient autocatalytic reaction rate constant the grouping x,~(k&)/D., dimensionless ratio of bulk concentrations to interfacial concentration of A concentration of radical saecies dimensionless concentration of & given by i/R, position in liquid film liquid film thickness representations for 1st and 2nd differentials of a and + dimensionless position within liquid film ratio of diffusion coefficients of d and A Superscripts
* -interfacial conditions Subscripts b bulk phase conditions
The behaviour of Y as a function of q/M and qtiy* is shown in Fig. 4. It is interesting to note that for q&y*< 0.1 the changeover from entirely film to entirely bulk reaction takes place within a very narrow range of d/M. In other words, transition from ‘slow’ to ‘fast’ reaction is very sensitive to d/M. This observation might well be important in relation to the selectivity behaviour of a gas-liquid reactor in which an autocatalytic reaction proceeds in parallel with say a fist order reaction.
REFERENCES [I] Hobbs C. C. et al., Ind. Engng Chem., Prod. Res. Deu. 197211 220. [2] Juvekar V. A., Chem. Engng Sci. 197429 1842. [3] Nottingham Algorithms Group, Library Manual, Mark 3,1973. -(41 _ Sim M-T.. M.Sc. Dissertation. University of Manchester, 1973. [5] Ames W. F., Non-Linear OrdinaryDifferential Equations in Transport Processes, Academic Press, New York 1%8. [6] Van Krevelen D. W. and Hoftijzer P. J., Rec. Trau.Chim. 1948 61 563.
Engineefing Science.1975,Vol.
30, pp. 1215-1218.
PergamonPress. Printed in Great Britain
GAS ABSORPTION WITH AUTOCATALYTIC REACTION MONG-TECK SIM and R. MANN Department of Chemical Engineering, University of Manchester Institute of Science and Technology, Manchester, England (Received 23 December 1974;accepted 26March 1975)
Abstract-The film theory has been. employed to investigate gas absorption accompanied by an autocatalytic chemical reaction of the form A t R-R + R. This reaction has been taken to be the simplest analogue of the autocatalytic features Of chain reactions encountered in the industrially important gas-liquid oxidation and chlorination reactions. Numerical solutions have been obtained using standard library subroutines for integrating systems of first order equations, having transformed the two describing second order non-linear differential equations into suitable first order initial value form. Predictions of the Enhancement Factor show that it can be very considerably larger than that for the ‘equivalent’ tist order reaction. This is due to the accumulation of radicals within the mass transfer fdm. An asymptotically exact empirical approximation for the Enhancement Factor is also presented. The ‘degree of conversion’ of the absorbed gas across the mass transfer film is shown to be more sensitive to the kinetic and diffusional parameters than is the case for a first order reaction.
INTRODUCTION
Hydrocarbon oxidation and chlorination reactions are basic to the chemical industry, since their products represent important reactive intermediates which are precursors to a wide range of utility chemicals. The reactions in which these intermediates are produced are often two-phase, the oxygen (air) and chlorine being supplied in the gas phase to the liquid phase hydrocarbon. The proper design of such reactors depends upon a detailed knowledge of the individual rate processes that take place and how these should be combined with descriptions of the flow nature of the two phases. The kinetic mechanisms of oxidation and chlorination reactions are believed to be based on chain reactions which have autocatalytic features. The transport process in the liquid phase consists therefore of dissolution followed by diffusion with chain-type reactions. In the present study the autocatalytic reaction
Material balances on species A and 8 over an increment dx of the mass transfer film gives the differential equations 2
$!-$
= kAi
These equations are to be solved to give A(x) and d(x) subject to the boundary conditions: at the interface [
A=A*
solubility relationship non-volatility condition
at the film boundary is chosen as a simple representation for such kinetics in order to determine the effect upon the rate of absorption. Thus species A dissolves at the interface and then reacts autocatalytically with radicals k which are present in the liquid phase. A(g+A(l)
t k(l)---&(l)
x = XL
A =Ab li =i&
the bulk conditions.
The diffusion-reaction eqns (1) and (2) can be cast into a dimensionless form using the reduced variables:
+ ii(l).
SYSTEMOF EQUATIONS Before presentation of the diffusion-reaction equations,
the following assumptions should be noted: (i) gas phase resistance is negligible. (ii) liquid phase radicals are non-volatile. (iii) the concentration of species A at the gas-liquid interface corresponds to equilibrium with the partial pressure of A in the gas phase.
to obtain 2
$j=Mai
1215
(3)
MONO-TECKSIM and R. MANN
1216
It is now only necessary to represent each second order equation, as two simultaneous first-order equations in order to complete the transformation into an initial value problem. In this way the problem is made amenable to solution by a Computer Library Subroutine designed to handle systems of first order equations. In this case, the four differential equations are:
with the boundary conditions changed to
di-o
z=o .
acl.0
z=l
a = qA
z-
f = 1.0.
dyl
x=
Some experimental observations on MEK oxidationll] have suggested that under certain conditions the radicals may be scavenged resulting in a zero order dependence with respect to the absorbing oxygen. If this is the case, then eqns (3) and (4) are uncoupled and they can be solved analytically. However, for the general case no analytical solutions are available and either a co-locational type approximation method [2] or an exact numerical solution has to be employed.
Y2
and dy2 d2a
x=z=Mat
(10)
representing eqn (3), plus
dys
&=
(11)
Y4
NUMERICALSOLUTION
A major diSiculty in numerically solving eqns (3) and (4) arises from the fact that one of the boundary conditions is a flux condition. If it could be assumed that P were known, the boundary value problem is transformed into an initial-value problem. This can be seen by combining the diffusion-reaction eqns (3) and (4) to obtain
and dy, _ d2+ - Mai
(12)
dz-z=~
representing eqn (4). This system of first order differential equations has initial conditions: y1= 1-o
This is a linear homogeneous second-order differential equation which can be integrated twice to obtain
y2 =
qA
-
1+ (1 - i*)qayR atz=O.
y3 = i* a + qr& = c, + c*z
(6)
where Cl and CZ are integration constants. Applying the boundary conditions gives c1= 1+ qIt’y,G* c2
=
qA
+
qR,%
-
(1 +
qh,d*)
and substituting back in eqn (6), there results a + qltpi = (9.4 + gayId + (1 + qi&*)(l-
2).
Now, provided aand i are differentiable at z = 0 di da t q&y@-&&= (4.4+ qrw) dt I r-O
-
(1+ 9mi~*)*
(7)
)u=o
1
Equations (9)-(12) reveal that film theory solutions for autocatalytic reaction depend upon the parameters M and qti’ya.For a particular choice of these parameters the set of four first order differential equations can be integrated from the initial conditions at the interface (z = 0) provided that the unknown interface radical concentration r* is assigned a value. The correct value for )* will generate profiles that are compatible with the boundary conditions at z = 1, i.e. at the bulk boundary. Film theory solutions were developed using Runge-Kutta-Mersons method for the integration and the method of bisection to search out the appropriate +*. (The NAG library subroutines D02ABF and COSACF[3] were employed in the computation.) Further details are given in Ref. (4). ARSORPTIONPREDICTIONS
Thus, since di dr I Z_O=O the above equation reduces to da = (4A - 1)+ (1 - i*)q*yd. dr I r-0
Typical concentration profiles for the case q* = 1, q,t = 0 and yti = 1 are presented in Fig. l(a-d). Fig. l(a) shows the profiles developed under relatively slow reaction conditions when f varies only slightly with position in the film and the reaction is approximately pseudo 1st order. However, as the autocatalytic reaction velocity is increased (m increasing) a large buildup of radicals is observed at the interface in Fig. l(d) and a
1217
Gas absorption with autocatalytic reaction
(b)
I.0 t
0
0.0
(b)
3.0
200
2.0
IO.0
(cl
(d) t
I
0
I.0
0.0
I.0
I.0 t i
0
l2LLd_a
0.0
0.2
0.4
0.6
0.0 0.6
’ 0.0
I.0
0.2
I
I
I
0.4
0.6
0.6
ho.0 I.0
Z.X XL
Fig.2.Multiplefunctionsolutionsfor q,t = 1,‘y,t= 1,d/M = 4. Fig. 1. Concentration profiles for increasing reaction velocity
corresponding enhancement of the absorption rate is expected. Some difficulties were experienced due to the existence of multiple roots of 3*, which appear to be present when the initial gradient of A is smaller than -10 for any parameter qkyh, i.e. under relatively fast reaction conditions. This is a property of non-linear equations which has been discussed by Ames[5]. A typical example for q,tyd, = 1 and a= 4 can be used to illustrate this. It has been found for this case that the function roots are 15.7863, 15.8267and 1.7976. The concentration profiles for A and ‘R that correspond to these roots are shown in Fig. 2(a-& It can be seen that the three roots satisfy the four boundary conditions but only the one shown in Fig. (2b) is the proper solution because the remaining two solutions have profiles that oscillate within the defined film and furthermore the dimensionless concentration a( = A/A *) exceeds 1-Owhich is physically impossible. Fig. (2b) shows that the profile of a decreases exponentially towards the film boundary and i(= &I&,) decreases linearly as soon as a has been consumed. This is taken to be the only physically proper solution and due to this complication it was necessary to plot out the concentration profiles in order to check the correct i* value. The Enhancement Factor for autocatalytic reaction is given by
and is simply the initial value of eqn (9) provided the correct value of i* is chosen. Numerical results were obtained up to values of E of about 16. Above this value the concentration profiles were so steep that the computing time required was prohibitive. Computed results are shown as continuous lines in Fig. 3.
0.01
04
IO
fi Fig. 3. Enhancement factors for autocatalytic reaction.
An important discovery which enables a more rapid evaluation of the Enhancement Factor when performing a gas-liquid reactor calculation is that the implicit approximate relationship
1218
MONG-TECK SIMand R. MANN
E=
d”E3 tanh JM[d-=-J
(13)
is asymptotically coincident with exact numerical evalua-
tions. (Apart from a sign change eqn (13) closely resembles the approximate expression proposed by Van Krevelyn and Hoftijzer[6] for second order reaction of the type A t B+products.) Moreover, the difference between exact and approximate predictions is only significant over small ranges of a. The maximum difference is about 55% for qti?R = lo-’ and only about 10% for qti-yti= 10-l. For smaller values of q.e’ytithe difference declines to zero as the behaviour becomes closer to pseudo-first order. Results based on eqn (13) are shown as broken lines in Fig. 3. It is noteworthy that in contrast to second order reactions of the type A t B-+products, the production and accumulation of radicals within the mass transfer film can give rise to enhancement factors which are markedly greater than would be expected on the basis of a ‘naive’ pseudo first order analysis. Thus at a value of X&= 1 an analysis based on neglect of radical accumulation would give an enhancement of only O-5,whereas if qayfi = lo-* the rate of absorption would be increased a hundredfold. These theoretically large enhancement factors may go some way towards explaining the often large absorption rates encountered in chlorinators and oxidisers. In substitutive hydrocarbon chlorination for example, it is quite common for the vented hydrogen chloride gas to contain only trace amounts of the feed chlorine and the vented gas is then a potentially useful high purity by-product stream. In hydrocarbon oxidisers the vented gas necessarily should contain only a small trace of oxygen otherwise explosive mixtures may form in the ullage. Finally the results have been treated to determine the fraction of A that is reacted within the mass transfer film, which is given by
Fig. 4. Balance between film and bulk reaction.
NOTATION concentration
of species A
dimensionless concentration of A given by A/A* diffusion coefficient autocatalytic reaction rate constant the grouping x,~(k&)/D., dimensionless ratio of bulk concentrations to interfacial concentration of A concentration of radical saecies dimensionless concentration of & given by i/R, position in liquid film liquid film thickness representations for 1st and 2nd differentials of a and + dimensionless position within liquid film ratio of diffusion coefficients of d and A Superscripts
* -interfacial conditions Subscripts b bulk phase conditions
The behaviour of Y as a function of q/M and qtiy* is shown in Fig. 4. It is interesting to note that for q&y*< 0.1 the changeover from entirely film to entirely bulk reaction takes place within a very narrow range of d/M. In other words, transition from ‘slow’ to ‘fast’ reaction is very sensitive to d/M. This observation might well be important in relation to the selectivity behaviour of a gas-liquid reactor in which an autocatalytic reaction proceeds in parallel with say a fist order reaction.
REFERENCES [I] Hobbs C. C. et al., Ind. Engng Chem., Prod. Res. Deu. 197211 220. [2] Juvekar V. A., Chem. Engng Sci. 197429 1842. [3] Nottingham Algorithms Group, Library Manual, Mark 3,1973. -(41 _ Sim M-T.. M.Sc. Dissertation. University of Manchester, 1973. [5] Ames W. F., Non-Linear OrdinaryDifferential Equations in Transport Processes, Academic Press, New York 1%8. [6] Van Krevelen D. W. and Hoftijzer P. J., Rec. Trau.Chim. 1948 61 563.