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Modelling reactive absorption processes via film-renewal theory: Numerical schemes and simulation results
Computerschem. Engng Vol. 22, No. 4/5, pp. 515-524, 1998 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain
Pergamon PIh 80098-1354(97)00263-9
0098-1354/98$19.00+0.00
Modelling reactive absorption processes via film-renewal theory: numerical schemes and simulation results Manfred Brinkmann i, Matthias Sch~ffer l, Hans-Joachim Wamecke l* and Jan Prfiss2 J Technische Chemie und Chemische Verfahrenstechnik, Universit~it-GH Paderbom, Paderbom, Germany 2 Fachbereich Mathematik und Informatik, Martin-Luther Universit/it Halle-Wittenberg, Germany
(Received 7 December 1995; revised 7 August 1997) Abstract
Even though film-renewal theory for gas-liquid mass transfer represents the most realistic mass transfer model under turbulent conditions, it is rarely used in practice since this leads to a complex mathematical description and evaluation of reacting systems. However, in contrast with common opinion numerical, evaluation of this class of models is possible nowadays within reasonable time on normal workstations. Therefore, we present a numerical scheme employing this theory which is capable of calculating mass transfer rates, conversion and remaining gas holdup from system and operating parameters. In contrast with the traditional calculation of mass transfer rates based on the enhancement factor, there are no simplifying assumptions concerning the hydrodynamics of operation, and non-linear phase equilibria can be considered as well as more complex reaction schemes. Simulations for reversible and irreversible second order reactions show the parametric sensitivity as well as the influence of different mass transfer models on the predicted mass transfer rates. © 1998 Elsevier Science Ltd. All fights reserved
Keywords: absorption; film-renewal; simulation NOTATION A a
am
Bo c D
Da E H K k
k2 P R r $ t u
interracial area (m 2) specific inteffacial area coefficient (m - 1) Bodenstein number
(m -
V f" x
volume m 3 flowmte (m3s -1) normalized space coordinate
y y z a 8
space coordinate molar fraction variable abbreviation abbreviation film thickness/depth of bulk elements (m)
p • ~"
overall mass flux relative phase ratio mean residence time (s)
7/
phase ratio
i)
concentration (mol m -3) diffusion coefficient ( m 2 S - 1) Damk6hler number enhancement factor distribution coefficient equilibrium constant mass transfer coefficient ( m s - 1) reaction rate constant (2nd order) (I mol - J s - 1) age distribution function (s - 1) mass flux density (mol m - 2 s -i) rate of reaction (mol m -3 s -~) normalized time coordinate time (s) normalized concentration
* Author to whom correspondence should be addressed.
/3
Subscripts: A B c G
g i j L 515
mass transfer component reactant central point gas phase overall phase index component index liquid phase
M. BRINKMA~n~et al.
516 0
inlet conditions
cj.,(t, y) = c~.,(t = 0),
is inconsistent with the previous model assumptions, as the "stagnant" elements are continuously replaced and bulk phase concentrations vary with time. The system is thus reduced to a dynamic CSTR with mass transfer described by the two-film theory. Time-dependent bulk concentrations should be used instead, accounting for 1. Introduction different time scales of diffusive and convective transIn the past, modelling of reactive absorption systems port phenomena. usually has been confined to the calculation of the soIn this work, a numerical treatment of absorption with called enhancement factor E, the ratio of liquid-side reversibe and irreversible second order reactions in a mass transfer coefficients with and without chemical stationary two-phase CSTR is presented. The filmreaction. The overall performance may then be deter- penetration theory has been chosen as an appropriate mined from physical absorption measurements. This mass transfer model as it accounts for turbulent flow common procedure bears a number of disadvantages conditions. Furthermore, the simpler yet less realistic two-fihn and penetration (Higbie, 1935) theories can be though: analyzed as limiting cases of the film-penetration theory. • Analytical solutions for E are available for first order The model consists of the microscopic reaction--diffureactions (Huang and Kuo, 1965), while higher order sion equations and the macroscopic bulk phase balances, reactions can only be treated by means of approximate which are coupled with each other by boundary solutions for a few limiting cases. Alternatively, conditions and the weighted time-average microscopic approximation and/or linearizationmethods have been mass transfer rates. No simplifying assumptions about applied. Despite mathematically complex solutions, the operation regime as in other approaches in literature these methods are of restricted use due to their are required. The numerical scheme developed in this simplifying assumptions such as zero concentration of paper enables the calculation of bulk phase concentraabsorbant in the liquid phase, negligible mass transfer tions as well as mass transfer rates. This procedure is resistance in the gas phase, or equal diffusivities numerically stable and the necessary computing time (Versteeg et al., 1989). remains within reasonable limits. • In case of higher order reactions, the enhancement As a model system, absorption of carbon dioxide into factor is a function of bulk phase concentrations, thus sodium hydroxide solution is investigated with respect to depending on the operation conditions. Since these the importance of various parameters. In case of concentrations are not known in advance but are reversible reactions the influence of diffusivity ratio and dependent system variables, the enhancement factor is equilibrium constant is demonstrated. Also in this more also not known in advance. complicated situation, the Newton technique developed • When based on two-film theory (Lewis and Whitman, for reasons of numerical stability converges quickly into 1924), multiple solutions for E may exist in case of a steady-state solution without further assumptions. heat development (Hoffman et al., 1975) or consecThe striking advantage of our approach is the high utive reactions (Reimus et al., 1990). In such a degree of flexibility. It is very easy to apply any other situation the mass transfer rates resulting from tworeaction scheme, non-linear phase equilibria at the film theory are not uniquely defined, and so this model interface, and energy balances. Also the effect of is inadequate. On the other hand, applying the film viscosity changes due to reaction products has been renewal theory the mass transfer rates are always successfully simulated with the same method; the results uniquely defined. Therefore, in our opinion filmare reported elsewhere (Brinkmann, 1994; Brinkmann et renewal theory should be preferred. al., 1995). The main purpose of this paper is the • Film theory does not account for turbulent flow description of the numerical scheme and the presentation conditions which are typical for technical absorbers. of some simulation results. On the other hand, dynamic mass transfer models such as the film-penetration theory result in even more 2. Mathematical modelling complex equations for E (see e.g. Nagy, 1991).
Superscripts: b bulk phase f feed * final solution
For these reasons numerical approaches appear to be more suitable for the calculation of reactive absorption processes. Versteeg et al. (1989) as well as Romanainen and Salmi (1991) came to the same conclusion. The latter authors attempt to calculate numerically mass transfer rates in a two-phase dynamic CSTR, assuming that mass transfer can be described by filmpenetration theory according to Toor and Marchello (1958). However, their choice of initial conditions (equations 3 and 4) for the diffusion equations,
In this article, absorption rates are calculated by means of a mathematical model which is based on the following assumptions. • Isothermal, isobaric, steady state operation. • Ideally mixed bulk phases. • Absorption is described by the fihn-penetrationtheory with mass transfer resistance on both sides of the interfacial area A. Mass exchanging ("active") bulk elements are being replaced occasionally but simulta-
Modelling reactive absorption processes via film-renewal theory
{ OcG D o2c6
neously on both sides of A. Their age distribution is given by
Oc^ p(t)= 1 e -""
(1)
t>0, - 6~
O2CA
t>0, 0
OCB O2Ca -~ =Da--Oy2 - k2CACB,
T
Active bulk elements are remixed with the bulk phase immediately aRer replacement from the interfacial area. This mechanism of mass transfer is sketched in Fig. 1. • At the interracial area, Henry's law is valid: c~ # ~'= A
517
t>0, 0
with initial conditions
{
cdO, y)=c~ CA(O,y) = C cB(O, y) = e [.
(2)
(6)
The assumption of continuous concentrations and mass fluxes, combined with the validity of Henry's law at the interfacial area, yields the boundary conditions
• The gas phase is subject to depletion by strong absorption. Assuming a turbulent coalescing system with spherical bubbles, the specific interfacial area can be described as (Brinkmann, 1994)
co(t,- ~)=c~,
CA(t,6t)=c~,
(7)
CB(t, ~ ) = c g , a=am8G.
(5)
(3)
t>O,
and
• In the liquid phase, absorbed gas molecules react according to the scheme
c~(t,O)=HcA(t,0),
A+B~P,
OCB)(t,O)=O,t>O.
the rate of reaction being r=k2CACB.The components B and P are confined to the liquid phase. • The ratio of exiting flow rates equals the ratio of the according phase volumina, i.e.
Oy
Mass transfer within a pair of active bulk elements depends on their age, therefore the mass flux has to be weighted with the age distribution p(t):
(4)
n= ~,{- VL"
(8)
[ co) (
RG=! The local microscopic mass flux inside the active bulk elements is mathematically described by a system of coupled non-linear partial differential equations
--DG ~
(t,O)].p(t)dt.
Similarly, the total amount of A or B reacting in the A
i:i:i:il -9"
¢
x
/ /
x
/
x
¢
G-Bulk
~H
DA,q
Fig. I. Mass transfer according to film-penetrationtheory.
(9)
.rl
DA,~
L-Bulk
518
M. BRINKMANNet al.
liquid-side bulk element during the contact time is given by
l
RB=
(10)
f k~c~(t, y)ca(t, y) dy .p(t) dt.
Boo= ~ ,
BoA=~ ,
BoB=,,rDB ,
od ~o= v~,
The mass flux R~ of component A into the bulk liquid thus becomes
~.,
~,= ~--~,
~:~:'o.
(11)
R A = R 6 - R B.
For stationary operation of a two-phase CSTR as depicted in Fig. 2, the macroscopic mass balances of the bulk phases are given by the system
For the sake of simplicity, the matrix notation F(z)=0 for the bulk equations is favourable, where F(z) takes the form F(z) = g - B(z) - R(z) + L(z)K(T(z)).
(15)
"f ' 0 = Voc of - Vo c 6b - ARG
"f
f
b
"f
f
b
0 = V L ( C A -- C ^) + A R A
- ( VL -- A
b b
¢~L)k2c AC B b
(I2)
b
0 = V L ( C B -- C B) -- A R e - ( Vt - A ~ ) k 2 c Ac B
0 = f/fcg - ~'6cs - A R 6.
The absorption process is thus described by a system of non-linear partial differential equations and a system of non-linear algebraic equations. Both systems are coupled by the initial and boundary conditions of (6)-(8) and a system of integral equations. According to Desch and PrOss (1994) this type of problem is wellposed, and there exists an unique solution, at least for constant interracial area. For computational purposes, the system is normalized using the dimensionless variables
b b f tUG_C~j]CG,
s=t/7,
b b f UAmHCA/CG,
y=x~
resp.
b
b
Here, z contains the final solution of the problem (i.e. the bulk concentrations and the resulting phase ratio). The vector g denotes the entrance conditions (concentrations, feed flow ratio), B(z) stands for the mass flux exiting the reactor, R(z) characterizes chemical reaction in the bulk liquid, and L(z)K(T(z)) denotes the overall mass fluxes.
z=
(14)
y = - x 8 C,
D a a = r k 2 -C~G,
B(z)= 8o=~,
b b CA~ CB
i i i i i i i
L RA
i
_~L is i i i I
~G k2
4 Fig. 2. Mass flow scheme in a two-phase CSTR.
(16)
(13)
f
UB----C~]CB,
f
Da A = ~'k2cf,
u~
uba
(18)
Modelling reactive absorption processes via film-renewal theory
compared to this model for second order kinetics and
0
DaAubu~ R(Z)=(6`L -- flL~O)7"O DaBUbAUb a
"I'>~ 8L/DA, L.
(19)
0 - a~
L(z) = 6`c
aiH 0 r - aly
0
\
- °t2DaA~ -oDaa]
u=T(z), p=K(u).
(20)
z~+l = z . - F ' ( z . ) - i F ( z . ) ,
Instead of the Newton iteration sketched above, a direct iteration procedure has been implemented previously for constant A, i.e. r/=r/0. In systems with dispersed gas phase and good solubility this restriction implies a small concentration of the absorbed species in the gas feed flow, i.e. yf-<0.01. In this case, the bulk mass balance equations simplify to 0 = 1 - u~ - ~tp13ecff~
(21)
The coupled system of integral, algebraic and partial differential equations is solved by Newton Iteration, i.e.
0=u f - UbA-- (EL0=
1 -
ug -
(eL --
flL(e13))DaA.rOUbAU~+OtlHp13S13
g(eB))OaB¢oU~U[ -- a 2 D a ~ 6 ` ~
r/o 6"13-----1+~/o
(22)
where
1
6`L= l+,r/° •
F'(z) = - B'(z) - R'(z) + L'(z)K(T(z))
(24)
+L(z)K'(T(z))T'(z) is the Fr6chet-derivative of E Especially, T'(z) denotes the solution of the system of (5)-(8) differentiated with respect to the macroscopic variables c~, 77. Using an arbitrary vector z, (e.g. uy and 7/for zero mass transfer), the PDE system for u and u' is simultaneously solved by means of the FORTRANNAG-Routine D03PCF. This routine is based on spatial discretization and the method of lines, solving the resulting set of ODEs using Gear's method (NAG, 1991). With the solutions u and u',
u=
519
UA(S,X) a n d u ' =
Ou---~j , i , j = G , L , B ,
uB(s, x) the overall diffusional mass fluxes K(T(z)) and K(T'(z)) are calculated by the routine D01GAF, using a method described by Gill and Miller (1972). The new vector z,+ is obtained by means of the routine F04ATF, a solver for systems of linear algebraic equations which is based on LU-factorization with partial pivoting and iterative refinement. This vector z,,+~ is either used as initial/ boundary condition for the next iteration step, or it contains the solution z* in case the required accuracy is met. The accuracy of this method has been checked for two different limiting cases. According to Reineke (1993), the liquid-side mass transfer coefficient for pseudo-first order kinetics (i.e. VLCB>> ' f f Vocc) - f r is given by
/DA'L .~/~(l+Da), kL= "~/ 7" (l +Da) coth ~/ /']A'LT
(23)
with Da=kr and k=kECfB. In this case it is possible to calculate z analytically. Bulk phase concentrations obtained from analytical and numerical methods differ by at most 3% from each other, depending on k: and r. In another investigation, the results obtained from a similar model based on two-film theory (Kube, 1992; Kube et aL, 1994) differ by no more than 2% when
Starting from initial values for the bulk concentrations, the PDEs are solved via the NAG routine, and from their solutions the normalized mass fluxes pj are determined. The new values for the bulk concentrations are then obtained from solving the algebraic equation (24). This way, only three instead of 12 PDEs have to be solved in one iteration step, and the cpu time for this is reduced from about 30 s to only 3 s.* However, the advantage of the Newton iteration procedure is shown in Fig. 3. For the parameters listed in Table 1, the refined algorithm requires at most three iteration steps with an overall calculation time of 90 s whereas the simpler method reveals oscillatory behaviour and does not reach the same accuracy after 25 steps and a total of 75 epu s. Furthermore, with increasing interfaeial area numerical instability occurs as shown in Fig. 3b. Starting off from the same steady state condition at am=5 cm - t, an increase of am to 7.5 cm -J can only be handled by the Newton method, while the simple procedure predicts increasingly unfeasible conditions until the systems physical boundaries are exceeded. For large mole fractions, e.g. yf=0.5, the strong interaction between mass transfer rates, gas holdup and interfacial area impairs the application of the "direct" iteration procedure, which has been dropped for these reasons.
3. R e s u l t s
The refined iteration procedure presented above enables study of the influence of chemical reaction on mass transfer in dependence of the rate of reaction. Furthermore, it is possible to distinguish different mass transfer models by varying the ratio e J r with TD=82L/DA,Lmaintaining the same mass transfer coefficient for physical absorption. * Convex C3400, multiuser operation, no compiler optimization.
520
M. BRINKMANNet al.
3.1. A: irreversible reaction
There are only minor differences depending on k2, especially in the range of several hundreds 1 mol - ms as well as for very fast reactions (kz>8000 1 mol -i s -t).
The dependence of the normalized bulk phase concentrations u ~ and u ~ on the rate of reaction, k2, are shown in Fig. 4. Mass transfer coefficients and other relevant parameters are listed in Table 2. They are characteristic for the CO2 absorption process as described by Deckwer (1985), However,/co has to be adjusted such that at given diffusion coefficient 8o does not exceed the bubble radius. The parameter a m has been determined experimentally in a jet loop reactor (Brinkmann et al., 1994). Besides the reaction rate constant, the mean residence time of active bulk elements has been varied according to Table 3 in order to show the negligible influence of the mass transfer theory on absorption rates for this reaction mechanism. For given values of kL and ~, the depth of the active bulk elements is calculated from Eq. (25). 11
I
i
= ~/-DA.~z a r c t h (ki~-~-OA, i),
Similar results have been reported by Huang and Kuo (1965), Danckwerts and Kennedy (1954) and Uchida et al. (1981) for first and second order irreversible reactions. This situation changes drastically when the liquid viscosity changes due to reaction products as shown by Bdnkmarm et al. (1995). The influence of other parameters (diffusivity DE and inteffacial area) on mass transfer rates has been studied by means of a sensitivity analysis as shown in Fig. 5. The parameters listed in Table 4 are chosen such that
i
I
I
I
[
Newton
0.9
(25)
~
I
m e t h o d
direct method ~ - -
0.8 0.7
0.6 ::' ::
:... O O
0.5 f 0.4 0.3
0
I
I
I
10
20
30
1
I
40 50 cpu-time (s)
1.2 1
.'~
I
I
70
80
90
~, ,,~, , :~ Newtonmethod :~ directmethod ~ ' -
,
0.8
I
60
..
-.
~
~
•
,
.
0.6 ( 0.4
..
.
.
.
.
.
0.2 0
I
0
10
20
30 cpu-time (s)
40
50
60
Fig. 3. Comparison of "direcf' solution and Newton iteration: (a) oscillatory behaviour; (b) increase of interracial area impairs direct solution. Table 1. Simulation parameters in Fig. 3 c~ (mol 1") 0.01825 TL (5)
l0
cg (mol 1- i) 0.073
H 2.34
am (cm -i) 5
kL(cm s -i) 0.023
k~ (era s -i) 7
D G (cmz $ - ' )
D A(cm 2 s =') 1.6X 10 -s
D a (cm 2 s =') 2.56× 10 -5
¢/o 0.25
Y~ 0.01
0.14
Modelling reactive absorption processes via film-renewal theory
521
1.00 0.95 0.90 0.85 0.80
0.75 0.70
0.65 0.60 I" 0.55
t~It
0.50 0.45 0
2000
4000 6000 k2 [l/(mol s)]
8000
10000
Fig. 4. Normalized bulk phase concentrations for different mass transfer theories. absorption of a pure gas and equal molar feed rates of A and B is simulated for ¢=¢D. A phase ratio of 0.25 indicates zero absorption rates, while a phase ratio of 0 is equivalent to infinitely fast mass transfer. Starting from the central point, each parameter is varied by multiples of two, covering approximately two orders of magnitude while the others are kept constant. For the given set of parameters, it is the interfacial area which has the strongest effect on mass transfer. Increasing k2 has a small effect as compared with am. Further simulations have shown that the situation is quite similar in the range of slow reactions (k2 = 10 1 mo1-1 s -i). It should be pointed out that for irreversible reaction, the numerical method is capable to readily calculate bulk phase concentrations as well as remaining phase ratios for the whole range of operating conditions and system parameters, without previous knowledge of the range of operation. Choice of initial conditions is inessential, as long as they are kept within their physical bounds.
3.2. B: reversible reaction In order to show the general applicability of the numerical procedure, a reversible reaction of the type A+B-¢~2 P has been implemented, the elementary reactions being A +B-*2P,
r=k2c^ca
2P---,A+ B,
r=k_2c~.
Mass balances for the product P have to be introduced in the systems (5) and (12), and the terms for chemical reaction have to be changed accordingly. Initial and boundary conditions for the product P are similar to those of the reactant B. In Fig. 6, z is again varied in order to show the effect of the mass transfer model on calculated gas phase concentrations. In contrast with irreversible reactions, the different models give increasingly diverging results at higher reaction rate constants and K = I . Again, predictions by penetration theory are close to those obtained from film-penetration theory. The influence of the equilibrium constant K=k2/k_2 is shown in Fig. 7 for Dp=DA and r=0.05 s. In case of reverse reaction (K= 1), the maximum conversion is much lower than in the case of irreversible reaction (K=I000). Reaction rate constants higher than about 1001mo1-1 s -]) do not significantly improve mass transfer any more, while the irreversible reaction first is limited by interracial area at higher k2 values. The diffusivity ratio Dp/D^ becomes most important for K= 1. Different diffusivities of reactant and product must be accounted for at k2>400 1 mol - ~s - ~), as can be seen from Fig. 8. A small diffusion coefficient of the product leads to accumulation of P (and thus A) at the interface, limiting mass transfer of A. 4. S u m m a r y
A mechanistical model based on the film-penetration theory of Toor and Marchello has been developed in
Table 2. Simulation parameters in Fig. 4 c[3 (mol I"1) 0.009125 7L (s) I0
% (mol I -i) 0.073
H 2.34
am (cm -') l0
kL(cm s -') 0.023
ko (cm s -') 7
D~ (cm 2 s -') 0.14
D^ (cm 2 s = l) 1.6X I0 -s
D e (cm 2 s -') 2.56X I0 -s
7/o 0.25
y~ 0.5
M. BLrNKMANNet al.
522
order to calculate gas-liquid mass transfer rates with chemical reaction. It accounts for mass transfer resistance in both phases and depletion of the gas phase due to strong absorption. A simple yet theoretically reasonable function correlating the interfacial area with the gas holdup is also included. Bulk phase concentrations can directly be calculated from operating conditions and system parameters using a
Table 3. Residence times ¢ and resultingdiffusiontimes
r (s)
8L (cm)
8o (cm)
¢D/¢
0.030246 0.05 0.5
0.00455 0.00093 0.00070
0.0207 0.0204 0.0200
43 1.08 0.06
0.25
--o-~ --Q--k, ×
0.20
a
0.15
0.10
0.05
0.00
,
I
-3
,
-2
I
i
-1
I
,
I
0 2"
,
I
1
2
i
Fig. 5. Influence of De, k2 and a,~ on phase ratio 7/at k2= 1000 1mol - ' s - ~. Table 4. Parameters for sensitivity analysis c[3,o(mol I") 0.01825
TL(S) I0
cg (mol I-') 0.073
H 2.34
am,0 (cm -i) I0
kL (cm s -t) 0.023
ko (cm s -') 7
¢ (s) 0.05
DG(cm2 s-I ) 0.14
D^( cm2 s-I ) 1.6'I0-s
Ds,o(cm2 s-I ) 2.56'I0-s
7]o 0.25
Y~ 1,0
k2,o(lmOl-I s-I ) I000
0.7751 0,770! 0.765
---0---
r
• --+--
¢
= 0.030246s =O.05s
---0--
~
= 0.5s
0.760 _~
0.755
ca
0.750 0.745
0.740 0.735 0.730 0,725
,
I
500
,
I
1000 k2 [1/(tool s)]
Fig. 6. Influence of mass transfer model on gas phase concentrations.
,
I
1500
,
2000
Modelling reactive absorption processes via film-renewal theory
523
0.78 -----O.---K = 1
0.77
" " + ' " K = 1000 0.76
cb ~
0.75 0.74 0.73 ".4=, "'.,
0.72
° -.~..., . . . . .+. . . . . . . . .4.. . . .
0.71 I
0.70
200
,
I
I
i
400
600
J
I
i
800
1000
k2 [1/(mol s)] Fig. 7. Influence of K on mass transfer.
0.775 - - - o - - o , / D , •o.1 ---+.- D , / D , = 1.0 ---V--O~/D~ = 10
0.770 l 0.765 0.760 0.755 0.750 0.745 0.740
0.735
0
200
400
600
800
1000
k2 [1/(mols)l Fig. 8. Influence o f diffusivity ratio at K = 1.
Newton iteration technique. This technique is favourable since it avoids numerical instabilities which occur in direct iteration. In contrast with the traditional calculation of "enhancement factors", there are no restrictions for applicability of this method and the whole range of operating conditions can be covered. Results obtained by film theory or penetration theory as limiting cases can be approximated by varying the ratio of z and ZDfor given kL and k~. Due to the numerical calculation of mass transfer rates, it is possible to extend the procedure to more complex reaction schemes as well as to non-linear phase equilibria. The method has successfully been applied to reactive systems with conversion dependent viscosity. For irreversible reactions, different mass transfer
models predict virtually the same absorption rates. This result agrees with findings of other authors. A sensitivity analysis shows the strong effect of the interfaeial area and the decreasing importance of the reaction rate constant in the regime of fast reaction. For reversible reactions, the mass transfer model becomes important at higher reaction rates. At an equilibrium constant of about unity, different diffusivities of reactants and product have to be accounted for as well. For successful application of film-penetration theory on existing absorption systems, determination of 8L, and ~"is crucial. Comparison of mass transfer rates in physical absorption with (pseudo-)first order reacting systems will be useful to determine these parameters
524
M. BRn~g_~tANNet al.
since analytical expressions for mass transfer rates under these conditions are available. Gas-side mass transfer resistance is often negligible in absorption processes due to the high diffusivity in the gas phase. However, the calculation procedure presented in this work is not limited to gas-liquid systems. Its application to extraction processes is generally feasible and should be considered.
Acknowledgements This work has been supported financially by Deutsche Forschungsgemeinschaft. We also express our appreciation to Dr. Sergej Gwosdew-Karelin for his help on numerical calculations.
References Brinkmann, M. (1994) Wechselwirkung von Hydrodynamik, Stofftransport und Chemischer Reaktion im G-L-Strahldiisenschlaufenreaktor. PhD thesis, VDI-Fortschrittberichte, Reihe 3, Nr. 371, VDIVerlag, Diisseldorf, Germany (1994). Brinkmann, M., Meier, D., PrOss, J. and Wameeke, H.-J. (1994) Influence of liquid viscosity on hydrodynamics and mass transfer in a G-L jet loop reactor, 3rd German--Japanese Symposium "Bubble Columns", June, Schwerte, Germany. Brinkmarm, M., Wamecke, H.-J. and Press, J. (1996) Modellierung reaktiver Stoffaustauschprozesse. Chemie, Ingenieur, Technik, 68, 239-253. Danckwerts, P. V. and Kennedy, A. M. (1954) Kinetics of liquid-film process in gas absorption. Part 1: models of the absorption process. Trans. Instn. Chem. Engrs. 32, 49-53. Deckwer, W.-D. (1985) Reaktionstechnik in Blasensi~ulen. Otto Salle Verlag, Frankfurt. Desch, W. and Priiss, J. (1994) Station~es nnd dynamisches Verhalten yon StofftibergangsmodeUen. Differential and lntegral Equations 7, 767-793. Gill, P. E. and Miller, G. F. (1972) An algorithm for the integration of unequally spaced data. Comput. J. 15, 80--83. Higbie, R. (1935) The rate of absorption of a pure gas
into a still liquid during short periods of exposure. Trans. Amer. Inst Chem. Engng 31, 365-389. Hoffman, L. A., Sharma, S. S. and Luss, D. (1975) Steady state multiplicity of adiabatic reactors: I. The single reaction case. AIChE J. 21, 318-326. Huang, C.-J. and Kuo, C.-H. (1965) Mathematical models for mass transfer accompanied by reversible chemical reaction. AIChE J. 11, 901--910. Kube, C. (1992) Mathematische Modellierung eines oxidativen Absorptionsprozesses. PhD Thesis, Universitat Paderbom. Kube, C., Blume, T., Warnecke, H.-J. and Priiss, J. (1994) Chemical absorption of mercaptan in an aerosol operated loop reactor. Canad. J. Chem. Engng 72, 1000-1006. Lewis, W. K. and Whitman, W. G. (1924) Principles of gas absorption. Ind. Engng Chem. 19, 1215-1220. Nagy, E. (1991) Gas absorption with chemical reaction using the film-penetration theory. Hung. J. Ind. Chem. Veszpr. 19, 237-244. NAG (The Numerical Algorithms Group Ltd.) (1991) The NAG Fortran Library Introductory Guide, Mark 15, 1st edn. Oxford Univ. Press. Reimus, A. M., Carta, G. and Hudson, J. L. (1990) Effects of interracial diffusive transport on the dynamics of oscillating reactions. Chem. Engng Sci. 45, 665-677. Reineke, E (1993) Gasphasendispersion im aerosolbctriebenen Strahld0sen-Sehlaufenreaktor. PhD Thesis, Universit~t Paderbom, Germany. Romanainen, J. J. and Salmi, T. (1991) Numerical strategies in solving gas-liquid reactor models-- 2. Transient films and dynamic tank reactors. Computers Chem. Engngng 15, 783--795. Toot, H. L. and Marchello, J. M. (1958) Film-penetration model for mass and heat transfer. AIChE J. 4, 97-101. Uchida, S., Miyachi, M. and Ariga, O. (1981) Penetration model of gas absorption into slurry accompanied by an instantaneous irreversible reaction. Canad. J. Chem. Engng 59, 560-562. Versteeg, G. E, Kuipers, J. A. M., van Beckum, F. P. H. and van Swaaij, W. E M. (1989) Mass transfer with complex reversible chemical reactions - - Part I. Single reversible chemical reaction. Chem. Engng Sci. 44, 2295-2310.
Pergamon PIh 80098-1354(97)00263-9
0098-1354/98$19.00+0.00
Modelling reactive absorption processes via film-renewal theory: numerical schemes and simulation results Manfred Brinkmann i, Matthias Sch~ffer l, Hans-Joachim Wamecke l* and Jan Prfiss2 J Technische Chemie und Chemische Verfahrenstechnik, Universit~it-GH Paderbom, Paderbom, Germany 2 Fachbereich Mathematik und Informatik, Martin-Luther Universit/it Halle-Wittenberg, Germany
(Received 7 December 1995; revised 7 August 1997) Abstract
Even though film-renewal theory for gas-liquid mass transfer represents the most realistic mass transfer model under turbulent conditions, it is rarely used in practice since this leads to a complex mathematical description and evaluation of reacting systems. However, in contrast with common opinion numerical, evaluation of this class of models is possible nowadays within reasonable time on normal workstations. Therefore, we present a numerical scheme employing this theory which is capable of calculating mass transfer rates, conversion and remaining gas holdup from system and operating parameters. In contrast with the traditional calculation of mass transfer rates based on the enhancement factor, there are no simplifying assumptions concerning the hydrodynamics of operation, and non-linear phase equilibria can be considered as well as more complex reaction schemes. Simulations for reversible and irreversible second order reactions show the parametric sensitivity as well as the influence of different mass transfer models on the predicted mass transfer rates. © 1998 Elsevier Science Ltd. All fights reserved
Keywords: absorption; film-renewal; simulation NOTATION A a
am
Bo c D
Da E H K k
k2 P R r $ t u
interracial area (m 2) specific inteffacial area coefficient (m - 1) Bodenstein number
(m -
V f" x
volume m 3 flowmte (m3s -1) normalized space coordinate
y y z a 8
space coordinate molar fraction variable abbreviation abbreviation film thickness/depth of bulk elements (m)
p • ~"
overall mass flux relative phase ratio mean residence time (s)
7/
phase ratio
i)
concentration (mol m -3) diffusion coefficient ( m 2 S - 1) Damk6hler number enhancement factor distribution coefficient equilibrium constant mass transfer coefficient ( m s - 1) reaction rate constant (2nd order) (I mol - J s - 1) age distribution function (s - 1) mass flux density (mol m - 2 s -i) rate of reaction (mol m -3 s -~) normalized time coordinate time (s) normalized concentration
* Author to whom correspondence should be addressed.
/3
Subscripts: A B c G
g i j L 515
mass transfer component reactant central point gas phase overall phase index component index liquid phase
M. BRINKMA~n~et al.
516 0
inlet conditions
cj.,(t, y) = c~.,(t = 0),
is inconsistent with the previous model assumptions, as the "stagnant" elements are continuously replaced and bulk phase concentrations vary with time. The system is thus reduced to a dynamic CSTR with mass transfer described by the two-film theory. Time-dependent bulk concentrations should be used instead, accounting for 1. Introduction different time scales of diffusive and convective transIn the past, modelling of reactive absorption systems port phenomena. usually has been confined to the calculation of the soIn this work, a numerical treatment of absorption with called enhancement factor E, the ratio of liquid-side reversibe and irreversible second order reactions in a mass transfer coefficients with and without chemical stationary two-phase CSTR is presented. The filmreaction. The overall performance may then be deter- penetration theory has been chosen as an appropriate mined from physical absorption measurements. This mass transfer model as it accounts for turbulent flow common procedure bears a number of disadvantages conditions. Furthermore, the simpler yet less realistic two-fihn and penetration (Higbie, 1935) theories can be though: analyzed as limiting cases of the film-penetration theory. • Analytical solutions for E are available for first order The model consists of the microscopic reaction--diffureactions (Huang and Kuo, 1965), while higher order sion equations and the macroscopic bulk phase balances, reactions can only be treated by means of approximate which are coupled with each other by boundary solutions for a few limiting cases. Alternatively, conditions and the weighted time-average microscopic approximation and/or linearizationmethods have been mass transfer rates. No simplifying assumptions about applied. Despite mathematically complex solutions, the operation regime as in other approaches in literature these methods are of restricted use due to their are required. The numerical scheme developed in this simplifying assumptions such as zero concentration of paper enables the calculation of bulk phase concentraabsorbant in the liquid phase, negligible mass transfer tions as well as mass transfer rates. This procedure is resistance in the gas phase, or equal diffusivities numerically stable and the necessary computing time (Versteeg et al., 1989). remains within reasonable limits. • In case of higher order reactions, the enhancement As a model system, absorption of carbon dioxide into factor is a function of bulk phase concentrations, thus sodium hydroxide solution is investigated with respect to depending on the operation conditions. Since these the importance of various parameters. In case of concentrations are not known in advance but are reversible reactions the influence of diffusivity ratio and dependent system variables, the enhancement factor is equilibrium constant is demonstrated. Also in this more also not known in advance. complicated situation, the Newton technique developed • When based on two-film theory (Lewis and Whitman, for reasons of numerical stability converges quickly into 1924), multiple solutions for E may exist in case of a steady-state solution without further assumptions. heat development (Hoffman et al., 1975) or consecThe striking advantage of our approach is the high utive reactions (Reimus et al., 1990). In such a degree of flexibility. It is very easy to apply any other situation the mass transfer rates resulting from tworeaction scheme, non-linear phase equilibria at the film theory are not uniquely defined, and so this model interface, and energy balances. Also the effect of is inadequate. On the other hand, applying the film viscosity changes due to reaction products has been renewal theory the mass transfer rates are always successfully simulated with the same method; the results uniquely defined. Therefore, in our opinion filmare reported elsewhere (Brinkmann, 1994; Brinkmann et renewal theory should be preferred. al., 1995). The main purpose of this paper is the • Film theory does not account for turbulent flow description of the numerical scheme and the presentation conditions which are typical for technical absorbers. of some simulation results. On the other hand, dynamic mass transfer models such as the film-penetration theory result in even more 2. Mathematical modelling complex equations for E (see e.g. Nagy, 1991).
Superscripts: b bulk phase f feed * final solution
For these reasons numerical approaches appear to be more suitable for the calculation of reactive absorption processes. Versteeg et al. (1989) as well as Romanainen and Salmi (1991) came to the same conclusion. The latter authors attempt to calculate numerically mass transfer rates in a two-phase dynamic CSTR, assuming that mass transfer can be described by filmpenetration theory according to Toor and Marchello (1958). However, their choice of initial conditions (equations 3 and 4) for the diffusion equations,
In this article, absorption rates are calculated by means of a mathematical model which is based on the following assumptions. • Isothermal, isobaric, steady state operation. • Ideally mixed bulk phases. • Absorption is described by the fihn-penetrationtheory with mass transfer resistance on both sides of the interfacial area A. Mass exchanging ("active") bulk elements are being replaced occasionally but simulta-
Modelling reactive absorption processes via film-renewal theory
{ OcG D o2c6
neously on both sides of A. Their age distribution is given by
Oc^ p(t)= 1 e -""
(1)
t>0, - 6~
O2CA
t>0, 0
OCB O2Ca -~ =Da--Oy2 - k2CACB,
T
Active bulk elements are remixed with the bulk phase immediately aRer replacement from the interfacial area. This mechanism of mass transfer is sketched in Fig. 1. • At the interracial area, Henry's law is valid: c~ # ~'= A
517
t>0, 0
with initial conditions
{
cdO, y)=c~ CA(O,y) = C cB(O, y) = e [.
(2)
(6)
The assumption of continuous concentrations and mass fluxes, combined with the validity of Henry's law at the interfacial area, yields the boundary conditions
• The gas phase is subject to depletion by strong absorption. Assuming a turbulent coalescing system with spherical bubbles, the specific interfacial area can be described as (Brinkmann, 1994)
co(t,- ~)=c~,
CA(t,6t)=c~,
(7)
CB(t, ~ ) = c g , a=am8G.
(5)
(3)
t>O,
and
• In the liquid phase, absorbed gas molecules react according to the scheme
c~(t,O)=HcA(t,0),
A+B~P,
OCB)(t,O)=O,t>O.
the rate of reaction being r=k2CACB.The components B and P are confined to the liquid phase. • The ratio of exiting flow rates equals the ratio of the according phase volumina, i.e.
Oy
Mass transfer within a pair of active bulk elements depends on their age, therefore the mass flux has to be weighted with the age distribution p(t):
(4)
n= ~,{- VL"
(8)
[ co) (
RG=! The local microscopic mass flux inside the active bulk elements is mathematically described by a system of coupled non-linear partial differential equations
--DG ~
(t,O)].p(t)dt.
Similarly, the total amount of A or B reacting in the A
i:i:i:il -9"
¢
x
/ /
x
/
x
¢
G-Bulk
~H
DA,q
Fig. I. Mass transfer according to film-penetrationtheory.
(9)
.rl
DA,~
L-Bulk
518
M. BRINKMANNet al.
liquid-side bulk element during the contact time is given by
l
RB=
(10)
f k~c~(t, y)ca(t, y) dy .p(t) dt.
Boo= ~ ,
BoA=~ ,
BoB=,,rDB ,
od ~o= v~,
The mass flux R~ of component A into the bulk liquid thus becomes
~.,
~,= ~--~,
~:~:'o.
(11)
R A = R 6 - R B.
For stationary operation of a two-phase CSTR as depicted in Fig. 2, the macroscopic mass balances of the bulk phases are given by the system
For the sake of simplicity, the matrix notation F(z)=0 for the bulk equations is favourable, where F(z) takes the form F(z) = g - B(z) - R(z) + L(z)K(T(z)).
(15)
"f ' 0 = Voc of - Vo c 6b - ARG
"f
f
b
"f
f
b
0 = V L ( C A -- C ^) + A R A
- ( VL -- A
b b
¢~L)k2c AC B b
(I2)
b
0 = V L ( C B -- C B) -- A R e - ( Vt - A ~ ) k 2 c Ac B
0 = f/fcg - ~'6cs - A R 6.
The absorption process is thus described by a system of non-linear partial differential equations and a system of non-linear algebraic equations. Both systems are coupled by the initial and boundary conditions of (6)-(8) and a system of integral equations. According to Desch and PrOss (1994) this type of problem is wellposed, and there exists an unique solution, at least for constant interracial area. For computational purposes, the system is normalized using the dimensionless variables
b b f tUG_C~j]CG,
s=t/7,
b b f UAmHCA/CG,
y=x~
resp.
b
b
Here, z contains the final solution of the problem (i.e. the bulk concentrations and the resulting phase ratio). The vector g denotes the entrance conditions (concentrations, feed flow ratio), B(z) stands for the mass flux exiting the reactor, R(z) characterizes chemical reaction in the bulk liquid, and L(z)K(T(z)) denotes the overall mass fluxes.
z=
(14)
y = - x 8 C,
D a a = r k 2 -C~G,
B(z)= 8o=~,
b b CA~ CB
i i i i i i i
L RA
i
_~L is i i i I
~G k2
4 Fig. 2. Mass flow scheme in a two-phase CSTR.
(16)
(13)
f
UB----C~]CB,
f
Da A = ~'k2cf,
u~
uba
(18)
Modelling reactive absorption processes via film-renewal theory
compared to this model for second order kinetics and
0
DaAubu~ R(Z)=(6`L -- flL~O)7"O DaBUbAUb a
"I'>~ 8L/DA, L.
(19)
0 - a~
L(z) = 6`c
aiH 0 r - aly
0
\
- °t2DaA~ -oDaa]
u=T(z), p=K(u).
(20)
z~+l = z . - F ' ( z . ) - i F ( z . ) ,
Instead of the Newton iteration sketched above, a direct iteration procedure has been implemented previously for constant A, i.e. r/=r/0. In systems with dispersed gas phase and good solubility this restriction implies a small concentration of the absorbed species in the gas feed flow, i.e. yf-<0.01. In this case, the bulk mass balance equations simplify to 0 = 1 - u~ - ~tp13ecff~
(21)
The coupled system of integral, algebraic and partial differential equations is solved by Newton Iteration, i.e.
0=u f - UbA-- (EL0=
1 -
ug -
(eL --
flL(e13))DaA.rOUbAU~+OtlHp13S13
g(eB))OaB¢oU~U[ -- a 2 D a ~ 6 ` ~
r/o 6"13-----1+~/o
(22)
where
1
6`L= l+,r/° •
F'(z) = - B'(z) - R'(z) + L'(z)K(T(z))
(24)
+L(z)K'(T(z))T'(z) is the Fr6chet-derivative of E Especially, T'(z) denotes the solution of the system of (5)-(8) differentiated with respect to the macroscopic variables c~, 77. Using an arbitrary vector z, (e.g. uy and 7/for zero mass transfer), the PDE system for u and u' is simultaneously solved by means of the FORTRANNAG-Routine D03PCF. This routine is based on spatial discretization and the method of lines, solving the resulting set of ODEs using Gear's method (NAG, 1991). With the solutions u and u',
u=
519
UA(S,X) a n d u ' =
Ou---~j , i , j = G , L , B ,
uB(s, x) the overall diffusional mass fluxes K(T(z)) and K(T'(z)) are calculated by the routine D01GAF, using a method described by Gill and Miller (1972). The new vector z,+ is obtained by means of the routine F04ATF, a solver for systems of linear algebraic equations which is based on LU-factorization with partial pivoting and iterative refinement. This vector z,,+~ is either used as initial/ boundary condition for the next iteration step, or it contains the solution z* in case the required accuracy is met. The accuracy of this method has been checked for two different limiting cases. According to Reineke (1993), the liquid-side mass transfer coefficient for pseudo-first order kinetics (i.e. VLCB>> ' f f Vocc) - f r is given by
/DA'L .~/~(l+Da), kL= "~/ 7" (l +Da) coth ~/ /']A'LT
(23)
with Da=kr and k=kECfB. In this case it is possible to calculate z analytically. Bulk phase concentrations obtained from analytical and numerical methods differ by at most 3% from each other, depending on k: and r. In another investigation, the results obtained from a similar model based on two-film theory (Kube, 1992; Kube et aL, 1994) differ by no more than 2% when
Starting from initial values for the bulk concentrations, the PDEs are solved via the NAG routine, and from their solutions the normalized mass fluxes pj are determined. The new values for the bulk concentrations are then obtained from solving the algebraic equation (24). This way, only three instead of 12 PDEs have to be solved in one iteration step, and the cpu time for this is reduced from about 30 s to only 3 s.* However, the advantage of the Newton iteration procedure is shown in Fig. 3. For the parameters listed in Table 1, the refined algorithm requires at most three iteration steps with an overall calculation time of 90 s whereas the simpler method reveals oscillatory behaviour and does not reach the same accuracy after 25 steps and a total of 75 epu s. Furthermore, with increasing interfaeial area numerical instability occurs as shown in Fig. 3b. Starting off from the same steady state condition at am=5 cm - t, an increase of am to 7.5 cm -J can only be handled by the Newton method, while the simple procedure predicts increasingly unfeasible conditions until the systems physical boundaries are exceeded. For large mole fractions, e.g. yf=0.5, the strong interaction between mass transfer rates, gas holdup and interfacial area impairs the application of the "direct" iteration procedure, which has been dropped for these reasons.
3. R e s u l t s
The refined iteration procedure presented above enables study of the influence of chemical reaction on mass transfer in dependence of the rate of reaction. Furthermore, it is possible to distinguish different mass transfer models by varying the ratio e J r with TD=82L/DA,Lmaintaining the same mass transfer coefficient for physical absorption. * Convex C3400, multiuser operation, no compiler optimization.
520
M. BRINKMANNet al.
3.1. A: irreversible reaction
There are only minor differences depending on k2, especially in the range of several hundreds 1 mol - ms as well as for very fast reactions (kz>8000 1 mol -i s -t).
The dependence of the normalized bulk phase concentrations u ~ and u ~ on the rate of reaction, k2, are shown in Fig. 4. Mass transfer coefficients and other relevant parameters are listed in Table 2. They are characteristic for the CO2 absorption process as described by Deckwer (1985), However,/co has to be adjusted such that at given diffusion coefficient 8o does not exceed the bubble radius. The parameter a m has been determined experimentally in a jet loop reactor (Brinkmann et al., 1994). Besides the reaction rate constant, the mean residence time of active bulk elements has been varied according to Table 3 in order to show the negligible influence of the mass transfer theory on absorption rates for this reaction mechanism. For given values of kL and ~, the depth of the active bulk elements is calculated from Eq. (25). 11
I
i
= ~/-DA.~z a r c t h (ki~-~-OA, i),
Similar results have been reported by Huang and Kuo (1965), Danckwerts and Kennedy (1954) and Uchida et al. (1981) for first and second order irreversible reactions. This situation changes drastically when the liquid viscosity changes due to reaction products as shown by Bdnkmarm et al. (1995). The influence of other parameters (diffusivity DE and inteffacial area) on mass transfer rates has been studied by means of a sensitivity analysis as shown in Fig. 5. The parameters listed in Table 4 are chosen such that
i
I
I
I
[
Newton
0.9
(25)
~
I
m e t h o d
direct method ~ - -
0.8 0.7
0.6 ::' ::
:... O O
0.5 f 0.4 0.3
0
I
I
I
10
20
30
1
I
40 50 cpu-time (s)
1.2 1
.'~
I
I
70
80
90
~, ,,~, , :~ Newtonmethod :~ directmethod ~ ' -
,
0.8
I
60
..
-.
~
~
•
,
.
0.6 ( 0.4
..
.
.
.
.
.
0.2 0
I
0
10
20
30 cpu-time (s)
40
50
60
Fig. 3. Comparison of "direcf' solution and Newton iteration: (a) oscillatory behaviour; (b) increase of interracial area impairs direct solution. Table 1. Simulation parameters in Fig. 3 c~ (mol 1") 0.01825 TL (5)
l0
cg (mol 1- i) 0.073
H 2.34
am (cm -i) 5
kL(cm s -i) 0.023
k~ (era s -i) 7
D G (cmz $ - ' )
D A(cm 2 s =') 1.6X 10 -s
D a (cm 2 s =') 2.56× 10 -5
¢/o 0.25
Y~ 0.01
0.14
Modelling reactive absorption processes via film-renewal theory
521
1.00 0.95 0.90 0.85 0.80
0.75 0.70
0.65 0.60 I" 0.55
t~It
0.50 0.45 0
2000
4000 6000 k2 [l/(mol s)]
8000
10000
Fig. 4. Normalized bulk phase concentrations for different mass transfer theories. absorption of a pure gas and equal molar feed rates of A and B is simulated for ¢=¢D. A phase ratio of 0.25 indicates zero absorption rates, while a phase ratio of 0 is equivalent to infinitely fast mass transfer. Starting from the central point, each parameter is varied by multiples of two, covering approximately two orders of magnitude while the others are kept constant. For the given set of parameters, it is the interfacial area which has the strongest effect on mass transfer. Increasing k2 has a small effect as compared with am. Further simulations have shown that the situation is quite similar in the range of slow reactions (k2 = 10 1 mo1-1 s -i). It should be pointed out that for irreversible reaction, the numerical method is capable to readily calculate bulk phase concentrations as well as remaining phase ratios for the whole range of operating conditions and system parameters, without previous knowledge of the range of operation. Choice of initial conditions is inessential, as long as they are kept within their physical bounds.
3.2. B: reversible reaction In order to show the general applicability of the numerical procedure, a reversible reaction of the type A+B-¢~2 P has been implemented, the elementary reactions being A +B-*2P,
r=k2c^ca
2P---,A+ B,
r=k_2c~.
Mass balances for the product P have to be introduced in the systems (5) and (12), and the terms for chemical reaction have to be changed accordingly. Initial and boundary conditions for the product P are similar to those of the reactant B. In Fig. 6, z is again varied in order to show the effect of the mass transfer model on calculated gas phase concentrations. In contrast with irreversible reactions, the different models give increasingly diverging results at higher reaction rate constants and K = I . Again, predictions by penetration theory are close to those obtained from film-penetration theory. The influence of the equilibrium constant K=k2/k_2 is shown in Fig. 7 for Dp=DA and r=0.05 s. In case of reverse reaction (K= 1), the maximum conversion is much lower than in the case of irreversible reaction (K=I000). Reaction rate constants higher than about 1001mo1-1 s -]) do not significantly improve mass transfer any more, while the irreversible reaction first is limited by interracial area at higher k2 values. The diffusivity ratio Dp/D^ becomes most important for K= 1. Different diffusivities of reactant and product must be accounted for at k2>400 1 mol - ~s - ~), as can be seen from Fig. 8. A small diffusion coefficient of the product leads to accumulation of P (and thus A) at the interface, limiting mass transfer of A. 4. S u m m a r y
A mechanistical model based on the film-penetration theory of Toor and Marchello has been developed in
Table 2. Simulation parameters in Fig. 4 c[3 (mol I"1) 0.009125 7L (s) I0
% (mol I -i) 0.073
H 2.34
am (cm -') l0
kL(cm s -') 0.023
ko (cm s -') 7
D~ (cm 2 s -') 0.14
D^ (cm 2 s = l) 1.6X I0 -s
D e (cm 2 s -') 2.56X I0 -s
7/o 0.25
y~ 0.5
M. BLrNKMANNet al.
522
order to calculate gas-liquid mass transfer rates with chemical reaction. It accounts for mass transfer resistance in both phases and depletion of the gas phase due to strong absorption. A simple yet theoretically reasonable function correlating the interfacial area with the gas holdup is also included. Bulk phase concentrations can directly be calculated from operating conditions and system parameters using a
Table 3. Residence times ¢ and resultingdiffusiontimes
r (s)
8L (cm)
8o (cm)
¢D/¢
0.030246 0.05 0.5
0.00455 0.00093 0.00070
0.0207 0.0204 0.0200
43 1.08 0.06
0.25
--o-~ --Q--k, ×
0.20
a
0.15
0.10
0.05
0.00
,
I
-3
,
-2
I
i
-1
I
,
I
0 2"
,
I
1
2
i
Fig. 5. Influence of De, k2 and a,~ on phase ratio 7/at k2= 1000 1mol - ' s - ~. Table 4. Parameters for sensitivity analysis c[3,o(mol I") 0.01825
TL(S) I0
cg (mol I-') 0.073
H 2.34
am,0 (cm -i) I0
kL (cm s -t) 0.023
ko (cm s -') 7
¢ (s) 0.05
DG(cm2 s-I ) 0.14
D^( cm2 s-I ) 1.6'I0-s
Ds,o(cm2 s-I ) 2.56'I0-s
7]o 0.25
Y~ 1,0
k2,o(lmOl-I s-I ) I000
0.7751 0,770! 0.765
---0---
r
• --+--
¢
= 0.030246s =O.05s
---0--
~
= 0.5s
0.760 _~
0.755
ca
0.750 0.745
0.740 0.735 0.730 0,725
,
I
500
,
I
1000 k2 [1/(tool s)]
Fig. 6. Influence of mass transfer model on gas phase concentrations.
,
I
1500
,
2000
Modelling reactive absorption processes via film-renewal theory
523
0.78 -----O.---K = 1
0.77
" " + ' " K = 1000 0.76
cb ~
0.75 0.74 0.73 ".4=, "'.,
0.72
° -.~..., . . . . .+. . . . . . . . .4.. . . .
0.71 I
0.70
200
,
I
I
i
400
600
J
I
i
800
1000
k2 [1/(mol s)] Fig. 7. Influence of K on mass transfer.
0.775 - - - o - - o , / D , •o.1 ---+.- D , / D , = 1.0 ---V--O~/D~ = 10
0.770 l 0.765 0.760 0.755 0.750 0.745 0.740
0.735
0
200
400
600
800
1000
k2 [1/(mols)l Fig. 8. Influence o f diffusivity ratio at K = 1.
Newton iteration technique. This technique is favourable since it avoids numerical instabilities which occur in direct iteration. In contrast with the traditional calculation of "enhancement factors", there are no restrictions for applicability of this method and the whole range of operating conditions can be covered. Results obtained by film theory or penetration theory as limiting cases can be approximated by varying the ratio of z and ZDfor given kL and k~. Due to the numerical calculation of mass transfer rates, it is possible to extend the procedure to more complex reaction schemes as well as to non-linear phase equilibria. The method has successfully been applied to reactive systems with conversion dependent viscosity. For irreversible reactions, different mass transfer
models predict virtually the same absorption rates. This result agrees with findings of other authors. A sensitivity analysis shows the strong effect of the interfaeial area and the decreasing importance of the reaction rate constant in the regime of fast reaction. For reversible reactions, the mass transfer model becomes important at higher reaction rates. At an equilibrium constant of about unity, different diffusivities of reactants and product have to be accounted for as well. For successful application of film-penetration theory on existing absorption systems, determination of 8L, and ~"is crucial. Comparison of mass transfer rates in physical absorption with (pseudo-)first order reacting systems will be useful to determine these parameters
524
M. BRn~g_~tANNet al.
since analytical expressions for mass transfer rates under these conditions are available. Gas-side mass transfer resistance is often negligible in absorption processes due to the high diffusivity in the gas phase. However, the calculation procedure presented in this work is not limited to gas-liquid systems. Its application to extraction processes is generally feasible and should be considered.
Acknowledgements This work has been supported financially by Deutsche Forschungsgemeinschaft. We also express our appreciation to Dr. Sergej Gwosdew-Karelin for his help on numerical calculations.
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