Ind. Eng. Chem. Res. 1990,29, 1259-1270
y = concentration of any reaction invariant species, mol/L a = concentration of strong acid in wastewater inlet, mol/L
0 = concentration of weak acid in wastewater inlet, mol/L y = concentration of common ion salt wastewater inlet, mol/L 6 = concentration of dissociated HzO, mol/L AG = Gibbs free energy of dissociation of the weak acid, t
cal/mol = concentration of base in the mixed fluids, mol/L
Subscripts f = fictitious
i = inlet streams o = outlet stream sst = steady-state target
Literature Cited Albert, W.; Kurz, H. Adaptive Control of a Waste Water Neutralization Process-Control Concept, Implementation, and Practical Experiences. Conference Proceedings, Intemational Federation of Automatic Control, Adaptive Control of Chemical Processes, Frankfurt am Main, FRG; American Institute of Chemical Engineers: New York, 1985. Balhoff, R. A,; Corripio, A. B. An Adaptive Feedforward Control Algorithm for Computer Control of Waste Water Neutralization. Conference Proceedings, International Federal of Automatic Control, Real Time Digital Control Applications, Guadalajara, Mexico: American Institute of Chemical Engineers: New York, 1983. Economou, C. G.; Morari, M.; Paisson, B. 0. Internal Model Control. 5. Extension to Nonlinear Systems. Ind. Eng. Chem. Process Des. Dev. 1986,25,403-410. Gray, David M. New Solutions to Control Problems. Pollution Engineering; Pudvan Publishing Co.: Northbrook, IL, April, 1984; Reprint C2.1043-RP30-584.
1259
Gustafsson, T. K.; Waller, K. V. Dynamic Modeling and Reaction Invariant Control of pH. Chem. Eng. Sci. 1983,38 (3),389-398. Gustafsson, T. K.: Waller, K. V. Myths About pH and pH Control. AZChE J. 1986,32(2), 335-337. Jeffreson, C. P. Computer Control of Simple Variable Flow Processes. Conference Proceedings, International Federal of Automatic Control Real Time Digital Control Applications, Guadalajara, Mexico; American Institute of Chemical Engineers: New York, 1983. Lee, P. L.; Sullivan, G. R. Generic Model Control-GMC. Comput. Chem. Eng. 1988,12 (6),573. Leeds & Northrup Instruments. 7084 Microprocessor pH Analyzer/Controller, Product Bulletin C2.1213-DS;Leeds & Northrup Instruments: North Wales, PA, 1984. McMillan, G. K. pH Control; Instrument Society of America: Research Triangle Park, NC, 1984. Moore, R. L. Neutralization of Waste Water by p H Control; ISA Monograph Series 1; Instrument Society of America: Research Triangle Park, NC, 1978. Parrish, J. R.; Brosilow, C. B. AIChE J . 1988,34 (4),633. Patwardhan, A. A.; Rawlings, J. F.; Edgar, T. F. Nonlinear Predictive Control Using Simultaneous Solution and Optimization. Presented at the 1988 AIChE National Meeting, Washington, DC, 1988. Rhinehart, R. R.; Choi, J. Y. Process-Model Based Control of Wastewater pH Neutralization. Advances in Instrumentation; Instrument Society of America: Research Triangle Park, NC, 1988;Vol. 43, pp 351-358. Riggs, J. B.; Rhinehart, R. R. Comparison Between Process ModelBased Controllers. Proceedings, American Control Conference, Atlanta, GA; American Institute of Chemical Engineers: New York, 1988. Shinskey, G. pH and PION Control in Process and Waste Streams; John Wiley & Sons: New York, 1973.
Received for reuiew June 21, 1989 Accepted February 6,1990
Thermally Safe Operation of a Semibatch Reactor for Liquid-Liquid Reactions. Slow Reactions Metske Steensma* Akzo Chemicals, Research Centre Deventer, P.O. Box 10, 7400 A A Deventer, The Netherlands
K.Roe1 Westerterp Chemical Reaction Engineering Laboratories, Twente University, P.O. Box 21 7, 7500 AE Enschede, The Netherlands
Thermally safe operation of a semibatch reactor (SBR) implies that conditions leading to large accumulation of unconverted reactants are avoided. The various thermal responses of a cooled SBR, in which an exothermic liquid-liquid reaction takes place, can be precisely defined and represented in a diagram with the kinetics, cooling capacity, and potential temperature rise as the key factors. Reactions of the type “slow, taking place in the dispersed phase,” are found to be more prone to accumulation and dangerous temperature runaway than those of the type “slow, taking place in the continuous phase”. Reaction systems at the brink of dangerous accumulation have a few characteristic features in common such as heat production versus time, conversion at the end of the supply period, amount of accumulated reactants, and adiabatic reaction behavior. Laboratory methods, based on these characteristic features, for the early recognition of such reaction systems can be developed. 1. Introduction
Investigations on the safe operation of a semibatch reactor (SBR) were started by Hugo and Steinbach (1985, 1986) for homogeneous reaction systems. Their work was extended to slow reactions in liquid-liquid systems by Steensma and Westerterp (1988). In this study, we will elaborate on heterogeneous systems and show how dangerous accumulation in a plant-scale reactor can in principle be anticipated by specific laboratory tests. 0888-5885/90f 2629-1259$02.50/ 0
2. Mathematical Model for Liquid-Liquid Reactions in a SBR 2.1. Assumptions and Dimensionless Parameters. In an indirectly cooled SBR, component B is present right from the start, and the second component, A, is added at a constant feed rate until a desired stoichiometric ratio has been reached. Components A and B are either pure or diluted with inert solvents. A and B react under the formation of products C and D. The heat of reaction is
1990 American Chemical Society
1260 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
stopped. Excess or underdelivery of A is obtained by stopping at 6 > 1 or 6 < 1, respectively. The relative volume increase, E, during the supply period is defined as VA CBO e = - &- D ~ D - -Vm VB CAD I
11
.-
..... .
.. I
'
.
droplets c o n t a n n q
The conversion, fB, of component B-present the start-is
--3cImi
V
Total
,L
une
1'
Homoqeneous Dhase c w tl- B a n d + ner+ c vo
UmP
vc
C
Figure 1. Operation of a heterogeneous liquid-liquid semibatch reactor.
removed by a flow of coolant through a coil or a jacket. The general setup of such a cooled semibatch reactor is sketched in Figure 1. The temperature in the reactor and the concentrations of components A and B as functions of time can be found by solving the heat and mass balances over the reactor. Our assumptions for the mass and heat balances are as follows: (1)The volumes are additive. (2) No phase inversion occurs during the reaction. (3) The reaction rate is first order in the concentration of reactants A and B, and reactions take place in only one phase. (4) The reaction equation is written as vAA + uBB V& + u D D (1) Components B and D remain in the continuous phase, c, the phase already present at the start of the batch. Components A and C remain in the dispersed phase, d, which is usually formed as soon as the supply has been started. C is the desired product. (5) There is only reaction enthalpy, no mixing enthalpy. (6) The product of the heat-transfer area and overall heat-transfer coefficient, UA, is proportional to the liquid volume in the reactor. (7) The starting temperature in the reactor is equal to the mean coolant temperature, T,. The latter temperature remains constant. The initial volume, V,, of phase c is present at the start of the batch: Vm ~ B O / C B O (2) where nBOis the number of kilomoles of B in the reactor at t = 0 and cBois the molar concentration of B in kmol/m3 at t = 0. Phase d containing A is supplied at a constant feed rate &D (m3/s), and it contains cm (kmol/m3) of component A. If phase c contains initially cBSVd (kmol of B), then after some time ( t D ) a stoichiometric amount of A has been supplied if -+
(3)
A dimensionless time, 8, is now introduced by dividing time t by t D : (4)
The stoichiometric ratio is reached if 6 = 1. In this study, we will assume that at 6 = 1 the supply of A is indeed
right from
BO - ~ B ) / ~ B O
(6) The following statements are valid if 8 2 1: for €6, read e; and for 6 - [B or (6 - fB)/6, read 1 - f ~ . The total volume Vat an arbitrary time t (or 0) can now be found: V = V, + $,Dt = Vm(l + €6) (7) For certain reactions during which a considerable change occurs in the mean molecular weight of the phases, e.g., during the hydrolysis of an ester, the simple approach for the individual volumes of the phases c and d, namely, V, = V,,, and vd = dv,, will probably not be accurate enough. For such reactions, we propose the following correlation: V, = Vro(l + ~ B A c J (8) where At, is the relative volume increase of the continuous phase due to reaction. At, can be positive or negative but not larger than e because of the volume taken up by the dispersed phase. The volume of the dispersed phase, v d , is simply V [B
d
(5)
=
VC.
2.2. Heat Balance of the Reactor. The ratio, RH, of the specific heats of the respective phases is introduced: (pcp)D
RH=-"-
(PC,)O
b p ) d
(9)
b,),
The heat of reaction can be expressed as the adiabatic temperature rise. It must be based on the initial mass in the reactor as RH is also based on the initial state of the reactor:
It can also be based on the volume supplied at 6 = 1, ilTad,D,
or be based on the total volume at 6 = 1, AT,,,
The additivity principle (pcpV) = (pc )cVc + (pc )dVd is assumed valid for the heat capacity. T i e thermal %dance for the reactor can now be written, following the scheme used by Steensma and Westerterp (1988), as
[tRH + U * D a ( l
+ d ) ] ( T - T,,*ff)
where the effective coolant temperature is U * D a ( l + d ) T , + tR*TD Tc,eff = u * D a ( i + te) + tRH and Tc,eff
=
Tc,eff/TR
1
(13)
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1261 Table I. Expression for the Conversion Rate, dfB,/de,for a Number of Kinetic Regimes and Atc = 0. Other Limiting Conditions Described in the Texta kinetic slow reaction in the continuous phase; A goes through the interface
expression for r, kmol/(m3.s) mACAdkcC&(l - €d)
expression for dcB/d8 (1 - CB)(~- CB) VA -KmADa
€e
VC
slow reaction in the dispersed phase; B goes through the interface
mBCAdkdC&q
necessary check on validity of the expression Ha = (k,C&A)“*/kl, Ha < 0.3, C A , / C A ~ ~> 0.95
VA
-KmBDa(l -
{B)(e
- {B)
1 Ha = -(kdCADB)’”, kl
VC
Ha < 0.3,CBd/CBid
> 0.95
homogeneous reaction
Da in all expressions equals @RCB,,.
Ha is the Hatta number, defined in Westerterp et al. (1984).
with TD = T for 0 3 1 as there is no longer supply of material to the reactor. Further, T denotes a dimensionless temperature T/TR, U* the cooling capacity (UA),/ kR(cBpcpVr)O, and Da = kRCBOtD the Damkohler number representing the supply time. kR is the reaction rate constant for the reaction under observation at TR. 2.3. Mass Balance. Equation 13 is valid for any reaction mechanism. Under the limiting condition Ae, N 0, explicit expressions, i.e., not containing the conversion fB, for the dimensionless conversion rate dfB/d6 can be directly substituted as the denominator of eq 13 becomes simply 1+ &RH. Explicit expressions for the dimensionless conversion rate can be derived for several fundamental cases. We will start with the following case: low solubility of the reactants in the other phase, no change in the volume of the continuous phase, and slow reaction throughout the temperature range during a reaction. The partial mass balances in the case of low mutual solubility read VB -dnB - - --rV
dt
VC
VA - - - --rV + &&AD dnA
dt VC where r is the conversion rate in the reactor in kmol/(s.m3 liquid volume). The expression for r must reflect the phase where the reaction takes place. The bulk concentrations of A and B in phases d and c, respectively are
n~ (1- C A ) ~ B & ( ~ A /-~ (0 B )- SB)~BO(VA/VB) CAd N - = (16) vd
vd
vd
We can take cACN 0 and cBd cy 0, at least for the “macroscopic” behavior of the reactor. Furthermore, d(1 - ~ B / ~ B o = ) tD -dnB t~ -dfB--= -(VB / vC)r v d0 tD dt nB0 dt nBO (18) The dimensionless activation energy is given by = E/RTR (19)
-
The reaction rate constant is made dimensionless by dividing it by kR, the reaction rate constant at the reference temperature (TR): K = k/kR = exp(-E/RT + E/RTR) = exp(y(1 - l / T ) ) (20)
The distribution coefficients are defined such that a low value means a low solubility in the other phase: mA = CAic/CAid mB = CBid/CBic (21)
Reaction i n t h e dispersed phase
Reaction i n t h e c o n t i n u o u s phase
I
dispersed
continuous
dispersed
continuous
Figure 2. Concentration profiles at the phase boundary, for a slow reaction and low mutual solubility.
The subscript i in eq 21 denotes the concentration at the interface. The concentration drop of the component just transferred through the interface, being (CBid - cm) or ( c h - cAC),is important, and we must check under which conditions this drop is more than e.g. 5% of the concentration CBd or cAC.If this is the case, the simplification cA, N chC or CBd cB~! becomes less accurate. Figure 2 gwes typical concentration profiles at the phase boundary, for a slow reaction with a low solubility of the reactants in the other phase. For the time being, we assume that the above-mentioned concentration drop in the reaction phase is indeed relatively small. For the time being, we will also assume that the transport of a reactant through the reaction film is not chemically enhanced. Table I gives a survey of the expressions for slow reactions under the limiting conditions just described, for a second-order reaction. The homogeneous reaction system is included for comparison purposes. For the evaluation of the homogeneous reaction, the bulk concentrations defined by eq 16 and 17 must be adapted by taking V instead of V, or V,+ Equation 13 and the relevant expressions for the dimensionless conversion rate, in the form of two simultaneous differential equations, are solved by using a fourth-order Runge-Kutta integration method (Simmons, 1974) with a variable integration step in order to increase the calculation speed and accuracy. The computer results are data sets [r,fB,6]. For the interpretation of the computer results, two new parameters, namely, the relative number of kilomoles of M of A and B and the dimensionless concentrations (I’) of A and B in their own phase, are introduced
1262 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 Ad nA - (B rAd ===CAD
(22)
VrOt6CAD
cB
These parameters show specific limiting values: (1) 0 if there is no reaction at all, and thus, I'h and MB remain 1, MA increases linearly with 6, until 6 becomes 1, and r A d remains 1. (2) CB = 6 for 6 < 1 if the reaction is infinitely fast, and thus, r A d and MA remain 0 and r B c and MB decrease linearly with 6, until they become zero at 6 = 1. The parameter MA seems to be very suitable to quantify the accumulation of substance A. When useful, the reference lines MA = 6, MB = 1 - 6, r A or r B = 1- 6, and (B = 6, for 6 < 1 are given in the figures for concentration, conversion, and accumulation versus dimensionless time. The Figures 3-5 give an insight into what can happen in the reactor. Three as yet undefined reaction conditions are represented: (i) marginal ignition resulting in poor conversion but no runaway reaction; (ii) slight overcooling with a temperature runaway near 6 = 1; (iii) good start of the reaction, long period of practically constant reaction temperature, and fair conversion. The kinetic regime is a slow reaction in the dispersed phase, and only the cooling temperature is varied in the three cases. Condition i of Figure 3 looks reasonably safe but is undesirable from a practical point of view as the conversion remains low. If one looks closely at Figure 4, it turns out that condition i is in fact very close to condition ii: obviously a runaway reaction is a matter of a few degrees kelvin difference in starting temperature. In a runaway situation, there is a temperature jump simultaneous with a conversion jump after a severe accumulation of unreacted A. In the example of Figure 4, about 60% of the maximum possible amount of A has accumulated at the point where the runaway reaction shows up. Condition iii of Figure 5 leads to a good start of the reaction-though the mixture is not immediately at the reaction temperature due to our choice To = T,-an extended period of practically constant temperature, and a fair conversion at the end of the dosing period, in this case about 90%. Such a situation will be referred to as a QFS condition (Quick onset, Fair conversion, Smooth temperature profile). 3. The Phenomenon of Accumulation 3.1. Derivation of the Accumulation Criterion. Some accumulation of unreacted A will inevitably occur, only infinitely fast reactions will not show accumulation. There is no easy guide as to what is an acceptable accumulation of reactant A. A situation such as in Figure 5, which is intuitively judged "reasonably safe", shows no temperature jump, no sudden conversion of reactant A, no large accumulation of A, and a smooth course of the reaction temperature after the warming-up period. Basically the choice of a quantifiable criterion to describe such a reasonably safe situation is somewhat arbitrary. We feel that the obtaining of a smooth, stable, and realistic target temperature is more important than a sharp limitation of the accumulated mass, MA, or a limitation of the maximum dimensionless conversion rate, dCB/d8. Analysis of a great many QFS situations where a realistic target temperature is not exceeded reveals at the same time that the accumulation MA is limited to 0.2 f 0.1 and the maximum dimensionless conversion rate to 2.5 f 1, whereas in the case of no accumulation at all, dlB/d0would be 1.
a-
t
t
I
0
'
-
1
:
dimension l e s s time
-
, 1
1
B
0
!
I
dimensionless time
-
i
Figure 3. Reaction behavior in the case of marginal ignition, for a slow reaction in the dispersed phase. Parameters: e = 0.4, DamB = 1.8,U*Da/t = 10,ATad = 0.6, = 38,TR= 300 K,T,= 283.5 K,RH = 1.
Figure 6 can be used in the definition of the target temperature for a reaction carried out in a SBR. It shows that the temperature courses in the case of a high initial reaction rate, here obviously for T, = 310 K and T, = 317 K, do not basically differ. At T,= 302 K, there is a modest temperature excursion. The curves at still lower temperatures give a strong runaway and an overcooling, respectively. Figure 7 represents the effect of starting temperatures above the coolant temperature, which here is T, = 302 K, and shows clearly that the reaction system goes to a steady-state temperature, which is the same for all starting temperatures. After 6 N 0.7, the effect of depletion of the reactants becomes visible. At 6 = 1,the conversion of B is found to be the same for all starting temperatures. Obviously the target temperature, T,, is the steady-state temperature for a well-ignited reaction, started at Tho. Steinbach (1985) stated that the target temperature can be derived from the mass and heat balances by assuming simultaneously dT/dO = 0
dS',/d8 = 1
The latter condition means that component A is converted as soon as it is supplied. The first condition dT/dO = 0 is not rigidly valid in our m e as the effective cooling area increases during the supply period, leading to a slightly decreasing target temperature until 6 = 1. For engineering purposes, the target temperature is approximated very well by
Expression 23 replaces the numerical solution of eq 13 for
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1263 350
400
,
I
i
T (K)
1“
T t arget
300
’
0
.5
-+ +
Relative ,
I
............................................................................
npi
0 0
i
’
dimensionless time
number o f moles M
-
’ 1 1 .
Relative
concentrat ions
Re1 a t i v e con cent r a t i
.5
,
...................:, 1
;.m
r
II
.......................................................
....
I.
0
’
0
0
dimensionless time
MB.”
Re I a t ; v e
number o f moles M
’
0
dimensionless t i m e -
0
1
’
1
dimensionless t i n e
dimensionless time-
Y
1
dimensionless t i m e 0
-+
I
.’
’
dimensionless time-
Figure 4. Reaction behavior in the case of slight overcooling followed by a runaway, for a slow reaction in the dispersed phase. Parameters as in Figure 3, but T,= 286 K. Table 11. Definitions Related to Accumulative Behavior relation with the target temp type of behavior no ignition reactor temp does not approach the target temp, even for 0 >> 1 marginal ignition reactor temp curve approaches the target temp line; approach can take place just before, at, or after 0 = 1 target temp line is undesirably exceeded; the temp runaway term undesirably can be quantified by eq 26 reactor temp curve approaches the target QFS reaction temp line rather rapidly; QFS stands for quick onset, fair conversion, and smooth temp profile
dfB/dO = 1,for the interval 0 < 8 < 1and for any realistic combination of t, RH,ATad,O,y, and U*Da. We selected eq 23 by plotting a number of reactor temperature curves versus time, starting from various temperatures in the neighborhood of the expected target temperature, together with the candidate expression for the target temperature. For 0 > 1in eq 23, we assume that the target temperature will not decrease any longer, as no more cooling area becomes available. The definition of the target temperature, T,(B), enables an exact description of the various types of thermal behavior with respect to accumulation as shown in Figure 6. See Table I1 where the described phenomena occur a t ever increasing coolant temperatures. 3.2. Influenceof the Cooling Strategy. The simulation of the cooling system of a SBR by taking To= T,,,n = TDand full cooling without any control loop represents a worst case which is nevertheless sometimes found in industrial practice. Therefore, it must be checked how the characteristic phenomena of Table I1 are influenced by the starting conditions.
-
Figure 5. Reaction behavior in a QFS situation, for a slow reaction in the dispersed phase. Parameters as in Figure 3, but T,= 302 K.
-
0
~
1
Peranlet e r 5
dimensionless t i m e
-+
Figure 6. Reaction in a SBR at stepwise increased coolant temperatures. Kinetic regime: slow reaction in the dispersed phase. 350
388v j .......=‘coo I ant
0
I
dimensionless time
-
Figure 7. Effect of higher starting temperatures in a QFS situation. Kinetic regime: slow reaction in the dispersed phase. Parameters as in Figure 6.
Figure 8 represents a reaction with moderate accumulation tendency and shows that a runaway occurs if the starting temperature, To,is taken below the target temperature, T,,o, but not if the starting temperature is in the neighborhood of T,,,. The conclusions of similar calculations for the other thermal phenomena are that the phenomena become more moderate if To> T,, that dangerous conditions are more difficult to achieve, and that
1264 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1
L
(K>
3fi0
‘ I
348
320
“ *
388
280
narg i n a l
i
’
runauay a t @ < I
t-
B
’
dlmensioniess time
Figure 8. Effect of the starting temperature for a slow reaction in the dispersed phase. Condition leading to a runaway if TeW = T, = 286 K.
Ok-
2
3
Exothermicity
4
runauay a t e = l
5
-t
Figure 10. Characteristic phenomena in the boundary diagram. Slow reaction in the dispersed phase. RH = 1, U*Da/c = 10, and Aec = 10.
Ti a r g e t temp.
f
’
B
Type
t
dimensionless time
b y Type
-
Ry =
I : OFS-reaction
I
temp.
dimensionless time
2:
marginal
ignition at
-
81
tamp.
f dimensionless time
Type 4 : Runway, tmrr a t
to the parameters a and b used earlier (Hugo and Steinbach, 1985,1986),but adapted for a heterogeneous system, will be used for the construction of a diagram:
-
Ex =
(VA/VC)DaKc,eff(mAor mB) tRH + U*Da ( m O~r mB)(VA/vc)Da exP(’Y(1- 1/TC,&)) (24) tRH + U*Da temp rise term ATad,oY TR2 =cooling capacity Tc,effP tRH U*Da
+
(25)
Whether to take mAor mBin the evaluation of Ry depends on the kinetic regime. In the definitions of Ex and Ry, the + U*Da) arises from the mass and cooling capacity (aH heat balances, eq 13. The definitions contain the cooling capacity in the denominator since, with regard to Ex, a large cooling effect leads to a nearly isothermal mode of operation and counteracts the danger of a large value of ATad and, with regard to Ry, a large cooling effect also diminishes the reaction rate. The point (Ex,Ry) = (0,O) of the diagram is hence a fully isothermal condition, whereas large positive values of (Ex,Ry) indicate an increasingly adiabatic condition. The numerator in the definition of R y stems from the expression for the dimensionless conversion rate dfB/d8 as given in Table I. A generalized form for all expressions for the conversion rate is
8>1
Figure 9. Characterisic reactor temperature profiles.
the situation seems safe if To = Tao, as long as full cooling is applied. 3.3. Development of the Boundary Diagram. Figure 9 shows in a nutshell four characteristic types of thermal behavior in a cooled SBR in which an exothermic reaction takes place. The target temperature line, given by eq 23, is the general reference line. A diagram indicating the boundaries between the various types of thermal behavior will be based on the dimensionless parameters in a suitable combination. We first investigate the influence of the parameter At,, the relative volume increase of the continuous phase due to reaction. The influence of At, is found to be small the change in T, to get a specific thermal behavior is around 0.1 K if Atc is taken to be f0.256, instead of zero. See Appendix 4 and Table VI. Therefore, this effect will be neglected in further calculations, and the simplified expressions for the conversion rate as mentioned in Table I will be used. As a first approach, the combinations R y (reactivity number) and Ex (exothermicity number) corresponding
where f(6,f’B) is a function dependent on the kinetic regime, and RE is the reactivity enhancement factor, also dependent on the kinetic regime. The dimensionless activation energy, y, and the effective coolant temperature, Td, can best be placed-found by trial and error-in the definition of the exothermicity in the way shown. Diagram for a Slow Reaction in the Dispersed Phase. In Figure 10 the areas of the various thermal phenomena are represented in the Ry-Ex diagram. The conversion rate for this kinetic regime is given by dtB/dfl = (vA/vc)da(l - S‘B)(e - fB)mB The following range of parameter values is used to determine the boundary lines: RH = l , U*Da/c = 10,0.3 < t < 0.55,0.29 < Arad,o < 0.70, 32 < 7 < 42, VA/.U, = 1, and 0.025 < DamB < 14. Noteworthy in Figure 10 is the small distance, only a few degrees kelvin, between the lines “marginal ignition at 6 > 1”and “runaway at B = 1”.The latter line merges with the line marginal ignition at 8 > 1 at the point marginal ignition at 0 = 1. The use of the parameters R y and Ex according to eq 24 and 25 gives a
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1265
end o f
OF5-I . . ine
,/
,
mar inal ignft'ion
\ -OFS-
-+
Exothermicity
+
Figure 12. Influence of RH on the boundary diagram. Slow reaction in the dispersed phase, at U*Da/c = 20.
smooth diagram but a few minor deviations of ideality remain, such as marginal ignition at 8 = 1 is a small area rather than a point, and the lines QFS and marginal ignition at 8 < 1 slightly protrude into each other. These small deviations are probably caused by putting all characteristic phenomena in a single two-dimensional diagram. Attempts to remove these deviations lead to irregularities elsewhere in the diagram. A nearly adiabatic situation, given by tRH + U*Da e 0, is an interesting case. R y and Ex are large then, and according to the boundary diagram, the thermal behavior is "harmless accumulation". This is logical: if a reaction can normally be carried with little or no cooling, and the reaction would occasionally show accumulation due to wrong starting conditions, this will not harm, as no higher final temperatures than usual can be reached. Figure 11 is the boundary diagram for a slow reaction in the dispersed phase for three values of the parameter U*Da/c. Figures 10 and 11 enable conclusions about how to perform and interpret scaled-down experiments. Generally U* will not remain constant but decrease during scale-up. In order to draw conclusions about the thermal phenomena for the scaled-up situation, one must take a smaller value of Da in the laboratory experiment in order to keep U*Da constant during the scale-up. The reactivity number cannot be kept the same now because Ry contains the parameter Da. Complete thermal similarity can only be obtained if in the scaled-down reactor the amount of heat-exchange area is intentionally reduced. We conclude that the definitions of reactivity and exothermicity as given by eqs 24 and 25 are adequate. Influence of Different Specific Heats of the Phases. The influence of the specific heats of the phases is checked for a slow reaction in the dispersed phase at U*Da/t = 20 and three values of RH, namely, 0.4,1.0, and 2.5; see Figure 12.
Figure 12 teaches us that though RH is included in the definitions of reactivity and exothermicity it has an additional influence on the position of the boundary lines not covered by R y and Ex themselves.
3
high
~
e(i
Exothermicity
accumu I a t ion)
r e a c t ion
m o d e r a t e ',,
ISOTHERNAL OPERATION
Figure 11. Influence of U*Da/c on the boundary lines, for a slow reaction in the dispersed phase, with RH = 1, Ae, = 0, and e < 0.6.
( Harm I ess
+\
'
RUNMAY
Ex, in
Exothermicity
-+
Figure 13. Thermal behavior in a SBR.
Exotherm!city
Figure 14. Overheating $- for a slow reaction in the dispersed phase, at U*Da/c = 10 and a temperature overshoot taking place at e = 1.
General Interpretation of the Diagram. Figure 13 gives a survey of our findings. The temperature excursion above the target temperature is relatively small if Ex < Exmin. The Exminvalue corresponds to the situation marginal ignition at 8 = 1. The dotted line is the route through the diagram for a given combination c, Da, ..., y if only T,is varied. A marginal ignition at 19 = 1 situation is a rare phenomenon and most combinations of c, Da, ..., y cannot be forced through this point by taking a suitable value of T,. Figure 14 gives the magnitude of temperature overshoots ($ma) taking place in the runaway area of the boundary diagram as a function of the distance to Exmin,where $,is given by $ma= = ( 7'" - Tc)/ (Tt,- Tc) (26) For a slow reaction in the dispersed phase and U*Da/c = 10, the value of Exminvaries between 3.15 and 3.30. Diagram for a Slow Reaction in the Continuous Phase. The conversion rate for this regime, under the assumption of very low mutual solubility, is given by
In this regime, component A is transferred through the interface, and hence mA is the relevant distribution coefficient. The limiting case, 0 = 0, for eq 27 can be solved by using L'H6pital's theorem; this yields, after a few arrangements,
Figure 15 gives the boundary diagram for the slow reaction in the continuous phase, for about the same range of parameter values as in the previous diagrams. For the purpose of comparison, the slow reaction in the dispersed phase and the homogeneous reaction are also given.
1266 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 Table IV. Correction on the Effective Cooling Temperature To Get a Specific Thermal Behavior, as a Function of the Distribution Coefficient of A” AT, (K) in ATc (K) in the case of mA
reaction /” in the d i s p e r s e d phase
0.01L L 0
\--. t h e continuous s l o w reaction i n
I I
1
2
3
4
Exothermicitg
5
Figure 15. Boundary diagrams for homogeneous and heterogeneous reactions in a SBR, with RH = 1 and U*Da/t = 10. Table 111. Approximate Hatta Numbers for a QFS Reaction. Kinetic Regime: Slow Reaction in the Dispersed Phase
mB
1
0.1 0.01 0.001
potential for a small temp overshoot (Ex,Ry) = (3.2,0.8)
potential for a large temp overshoot (Er,Ry) = (8,0.3)
H%ki 0.011 0.035
Haski 0.007 0.022 0.07 0.22
0.11 0.35
Hatarget
0.04 0.13 0.41
1.3
Hatarget
0.13 0.4
1.3 4
Comparing the two types of slow reactions, we observe that the area enclosed by the boundary lines, where overheating will occur, is the smallest for a reaction in the continuous phase. As a matter of fact, this is caused by the large value of the ratio reaction volume to nonreaction volume for this kinetic regime, notably for small 0 values, which gives the system a better reaction rate at the start than in the case of a slow reaction in the dispersed phase. 4. Check on the Validity of the Model for Slow Reactions 4.1. Check on the Concentration Drop at the Interface. Use of the interfacial concentration instead of the bulk concentration is only allowed, for the component being transferred through the interface, if the concentration drop from the interface to the bulk of the reaction phase is relatively small. For the component already present in the reaction phase, there is no such restriction, as its concentration drop is relatively very small (see Figure 2). The maximum dimensionless conversion rate at a QFS condition is typically around 2.5. An evaluation in Appendix 1yields that, for a QFS situation in a well-stirred reaction mixture, the neglect of the concentration drop in the reaction phase does not lead to great inaccuracy, unless the distribution coefficients are very low, typically less than 0.005. If the distribution coefficient is very low, the mass transfer through the interface, required to maintain a low accumulation of A or B, will lead to a concentration drop in the film that is too large. 4.2. Check on the Validity of the “Slow Reaction” Regime. This check is done using the data of the boundary diagram for a slow reaction in the dispersed phase, Figure 10. For a system with U*Da/c = 10, a high temperature excess, J/ > 3, is possible by taking the combination (Ex,@) = (8,0.3) and only a small overshoot, J/ = 1,by taking the combination (Ex,Ry) = (3.2,0.8). Use of these typical (Ex,Ry)combinations, in Appendix 2, leads to the approximate Hatta numbers given in Table 111. Table I11 shows that the boundary diagram is valid as long as the distribution coefficients are larger than about 0.01-0.001. If the distribution coefficients are smaller, the
0 0.001 0.01 0.03
QFS situation 0
0.1-0.2 0.7-1.5 2.0-3.5
marginal ignition a t 0 > 1 0 0.05-0.1 0.3-0.4 0.7-0.9
a Indication of the required temperature increase of the cooling medium. AT,, to achieve a specific thermal behavior, all other parameters remaining the same. Reference situation: m A = 0.
reaction rate constant, k,at the starting temperature must be very large to still obtain a QFS condition. As a result, the regime of chemically enhanced transfer is entered and the reaction in the film becomes predominant. Then, also, the interfacial area, for slow reactions only relevant if the concentration drops too much, becomes an important factor for the total mass transfer. 4.3. Effect of Neglect of “Physical Loss” to the Other Phase. Until now we assumed in the model that practically all of substance A is found in phase d and practically all of substance B in phase c. If the amounts of A in phase c and/or B in phase d are no longer negligible, the effective concentrations of A and B will somewhat decrease, and the condition for a QFS reaction is more difficult to obtain, i.e., by a somewhat higher effective cooling temperature, which in turn influences the boundary diagram. A study has been made in Appendix 3 to assess this effect, with the following conclusions: The effect of varying m B between 0 and 0.03 for both types of the slow reaction is very small: at most the effective cooling temperature must be taken as 0.05 K higher to get a QFS or marginal ignition condition if all other parameters remain the same. This is logical: in the important initial period where the temperature must rise, not much of material B can be lost to the dispersed phase, as it has a small volume. The influence of mAis that systematically higher temperatures are needed to get a desired behavior. This is also logical, as in the important initial period the loss of A to the other phase due to dissolution can be relatively large, as there is not yet a “reservoir” of A. The required temperature increases are indicated in Table IV. 5. Early Recognition of Dangerous States A full set of kinetic and physicochemical data is not always available when in actual practice a reaction system is suspected of having a tendency to accumulation under unfavorable conditions. Four indicative laboratory experiments to find the possible occurrence of such dangerous behavior are evaluated. 5.1. Adiabatic Calorimetry. The early stage of a reaction is crucial for the later thermal behavior: if the initial reaction rate is too low, then accumulation may occur followed by a temperature overshoot. Adiabatic situations can simply be simulated in our model by taking U *Da= 0. The reaction temperature versus 0 in such an adiabatic case for a QFS reaction is given in Figure 16. The reference curve in Figure 16 is the temperature course for an infinitely fast reaction, starting at Tc,eff, carried out adiabatically, given by
The temperature during the QFS reaction has reached
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1267
t
I
0
’
dimensionless time
-
Figure 16. Indication of sufficient initial reaction rate by means of adiabatic calorimetry. Slow reaction in the dispersed phase.
s
a
I
I
I
I
5
I0
15
20
cooling
capacity
U’Da/,
-9
Figure 18. Usual maximum values of MA,the accumulation of compound A during a QFS reaction. Kinetic regime: slow reaction.
4
m o + J O W L O
c
.-
n L
L O
Ql
--.:: + .- ..0 010
IQ C O
0 W
L
0 Y 0
1
dimensionless time
-
8
0.7
0
5
10
15 U’Db/,
20
Figure 17. Conversion rate or normalized heat production in a practically isothermal reactor with U*Da/c = 400. Use of the same reaction parameters, but U*Da/c = 10, e.g., at a larger scale production, yields a QFS situation. Reaction at three representative temperatures.
Figure 19. Approximate conversion at the end of the supply period. QFS condition for a slow reaction in the dispersed phase.
between 0 = 0.25 and 0 = 0.50 the temperature line given by eq 29. These 0 values are observed for QFS reactions with U*Da/c being 20 and 10, respectively. Such a test only requires a temperature registration and an agitated Dewar vessel and can be executed in every laboratory. 5.2. Isothermal Heat Measurement in a Bench Scale Reactor. Reaction calorimetry is nowadays a well-established technique for the study of (dangerous) reactions (Bjornberg et al., 1984; Hub, 1975),and the usual mode of operation is the practical isothermal one. To simulate the isothermal reaction calorimeter, we have plotted in Figure 17 the computed conversion rate versus the dimensionless time for a practical isothermal situation by taking U*Da/e = 400. Figure 17 refers to a QFS situation, if U *Dale is 10 instead of 400 and all other parameters are kept the same. The heat production curve is similar to the conversion rate curve as long as there is no mixing enthalpy. The reaction temperatures taken in Figure 17 are the target temperature, the effective cooling + T,)/2. temperature, and in between the temperature (T, 5.3. Allowable Accumulation of Supplied Component A. The maximum value for the accumulation MA in the case of a QFS reaction is found to be dependent on the value of U *Dale; see Figure 18. The maximum accumulation in a QFS reaction is particularly sensitive to a small decrease of the cooling temperature, as then the region of accumulation followed by runaway is entered. Therefore, it is good practice to aim at somewhat lower values of MA than indicated in Figure 18. Figure 18 shows an increase of the maximum accumulation for lower values of U*Da. 5.4. Conversion of B at the End of the Supply Period. Imminent danger of accumulation can also be expected if the conversion of B, at the end of the supply period is still low. Indicative values of at 0 = 1 for a QFS reaction are given in Figure 19. At higher values of AT+,, the conversion at 6 = 1 is higher because regions of higher temperatures are passed.
6. Discussion and Conclusions The boundary diagrams contain the key parameters dosing rate, cooling capacity, potential temperature rise, and starting temperature. The scenario in which the starting temperature is set equal to the cooling temperature is a “worst case”, which can be used as a reference situation to assess the effectiveness of various possible safety measures for a semibatch reaction with accumulation tendencies. These might be, e.g., interruption of the supply when exceeding a certain temperature, increase or decrease the cooling rate at a certain temperature, or a certain dosing rate profile as a function of time. The calculations showed the predominant influence of the droplet size. We assumed a liquid-liquid mixture with technical-gradechemicals mixed with a powerful disperser, resulting in droplet sizes in the range from 50 to 100 pm. The small droplet size leads to a large interfacial area. Large droplet sizes can only limit the overall reaction rate if at the same time the distribution coefficient is very low. The mass transfer required to keep pace with the dosing rate will in such a case cause a large concentration drop of the reactant transferred through the interface to the reaction phase. A further essential factor is the distribution coefficient defined such that a low value means a low solubility of the reactant in the other phase. The distribution coefficient greatly determines the possible concentrations in the reaction phase and, consequently, the reaction rate. We assumed, as we deal with liquid-liquid reactions, rather low distribution coefficients, starting from down to lower values. A consequence of this choice for the material balance is that very little material is physically present in the other phase, though it may react there. Accumulation refers to substance A in the dispersed phase. The reactivity according to eq 24 is proportional to the relevant distribution coefficient. The parameter reaction rate constant Itl,l,d or kl,l,c, the most variable factor in the Hatta number, determines
cB,
cB
cooling capacity
------.
1268 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
Table V. Complicating Factors in the Use of the Diagrams, a a a Function of the Distribution Coefficient ~
IO-*
indicative value of mB or mA increasing physical loss to the other phase: corrections in Table IV 1 inaccuracy due to more than 5-10% concn drop in the reaction phase, for slow reactions reaction becomes of the type “fast”, when approaching the target temp, for systems with a potential for a large temp overshoot reaction becomes of the type “fast”, when approaching the target temp, for systems with a potential for a small temp overshoot reaction becomes of the type fast, even at the low starting
7
temp
Table VI temp (T,)for stated effect combin no. 1
2
3 4
5 6 7
8
AfC=
thermal behavior QFS reaction marginal ignition at 8 > 1 runaway at 8 = 1 QFS reaction marginal ignition at 8 = 1 QFS reaction runaway at 8 = 1 marginal ignition at B < 1
Ac, = 0 305.3 295.4 290.6 317.3 301.7 316.7 287.4 295.0
0.25~ 305.3 295.4 290.4 317.4 301.7 316.7 287.4 294.9
AcC = -0.25~ 305.3 295.4 290.3 317.3 301.7 316.7 281.4 294.9
Table VI1 no.
t
1 2 3 4 5 6 7 8
0.30 0.40 0.55 0.30 0.30 0.40 0.35 0.60
DamB 0.30 1.20 1.7 0.30 0.68 1.0 2.0 2.0
y
36 36 34.5 36 36.7 36 42 36
U*Dafc 5 5 5 10 10 20 20 10
ATad,o
0.50 0.30 0.45 0.45 0.30 0.50 0.65 0.45
whether the reaction regime is fast or slow. Having conditions leading to a QFS reaction, or a reaction with even less accumulation, is beyond doubt the most important objective for a chemical engineer. Consultation of the relevant boundary diagram learns which combination of reaction rate, distribution coefficient, installed cooling capacity, and dosing rate is correct with respect to this goal. Table V summarizes the various complications and limitations in our composition of the boundary diagram. Certain results for the fast reaction regime are already incorporated (Steensma and Westerterp, 1990).
Nomenclature Normal Symbols a = interfacial area, m2/m3 A = area of cooling jacket or coil, m2 c = molar concentration, kmol/m3 = heat capacity, J/(kg.K) = diffusivity, m2/s E = activation energy, J/kmol AH, = reaction enthalpy, J/kmol J = molar transport rate per unit area, kmol/(s.m2) k = reaction rate constant, m3/(kmol.s) k l = mass-transfer coefficient, m/s m = distribution coefficient, eq 28 M = relative molar amount of a compound, eq 22 n = number of moles of a compound, kmol R = gas constant = 8314 J/(kmol.K) r = conversion rate in the reactor, kmol/(m3.s) T = temperature, K t = time, s
8
AT,d = adiabatic temperature rise, K U = overall heat-transfer coefficient, W / (m2.K) V = liquid volume in the reactor, m3 Subscripts A, B, C = refers to component A, B, or C, respectively ad = adiabatic b = in db, droplet diameter c = continuous (e.g., in V,) or coolant (in T,) crit = critical d = dispersed phase D = in tD,dosing time to get the stoichiometric ratio D = component D eff = effective H = in heat capacity ratio RH i = (at the) interface max = maximum value of a parameter min = minimum value of a parameter r = (at the) reaction R = (at the) reference temperature ta = target v = volume in 0 = (at the) start of the supply period Greek Symbols and Dimensionless Numbers r = relative concentration, eq 22 y = E/(RTR),the dimensionless activation energy Da = Damkohler number, Da = kRCBotD AcC = relative volume increase of the continuous phase t = relative volume increase at the end of dosing t d = volume fraction of the dispersed phase Ex = exothermicity, or potential temperature rise, eq 25 Ha = ( k c C B $ A ) 1 / 2 / k 1 or ( k d C A D B ) ’ / 2 / k l , Hatta number { = conversion 0 = t/tD, dimensionless time K = k / k R , dimensionless reaction rate constant p = U*Da/t,dimensionless cooling intensity u = stoichiometric coefficient p = density, kg/m3 R H = ( ~ C ~ ) ~ / ( P=C (pc,)d/(pcp),, ~ ) ~ ratio of specific heats RE = reactivity enhancement factor R y = initial reactivity parameter, eq 24 T = T/TR, dimensionless temperature U* = dimensionless cooling capacity, U * = ( U A ) , / (PC, V R C B ) O ~ R $ = temperature overshoot parameter, eq 26 4” = flow rate, m3/s
Appendix 1: Check on a Sufficiently Small Concentration Drop at the Interface for a QFS Reaction. Calculation Example The derivation will be given for a slow reaction in the dispersed phase. At the point where the maximum conversion rate occurs, the following parameter values are typical: d{B/d6 2.5, {B 0.3 at 6 0.5, and hence r A d N 0.4 and r B c N 0.7 (the relative concentrations of A and B in their own phase). Further it is assumed that an organic compound with a supply concentration CAd = 8.0 kmol/m3 reacts with an aqueous compound with initial concentration cB0 = 2.0 kmol/m3. The relative volume increase at the end of the supply period is given by t = 0.5. The starting volume ( VRo)is 1 m3, and the supply period is 3600 s. Because of our assumption of a reaction in the dispersed phase, the compound B is transferred through the interface, and the molar flow (JB) through the total interfacial area ( a V ) in the reactor is
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1269 Insertion of the above data gives J B = 9.3 X 104db (kmol/(s.m2)). The predominant factor in eq Al-1 is the magnitude of the droplet diameter in the liquid-liquid dispersion. The transport JBresults in the largest concentration drop in the reaction phase for the component being transferred through the interface. Assuming k , = m / s , we find for th;ls concentration drop (ACB)d
Or ( A c A ) ~= J ~ / k 1= 9.34,
(kmol/m3)
increases. Equation A2-4 can be combined with the definition of the reactivity according to eq 24 and A2-2 to yield an expression for the reaction rate constant (k)at T,, which is needed for the calculation of the Hatta number at Tmm:
(Al-2) L
The concentration at the interface is given by (CBi)d
CBmB
The concentration drop is less than 5% if (ACB)d/(CBi)d < 0.05
1
This yields the following condition for a sufficiently small concentration drop at the interface: d b < 0.0054 (CBmB) During a QFS reaction, the maximum conversion rate is limited to approximately dcB/d6 = 2.5. The bulk concentrations at that moment are calculated according to cB = rBCB@ Assuming the droplet diameter is 60 X lo* m (van Heuven and Beek, 1970), we find mB > 0.008
A similar derivation for a slow reaction in the continuous phase yields mA > 0.003 This result indicates that in a well-stirred reaction mixture, with a reaction rate sufficiently high to prevent dangerous accumulation, the concentration drop in the reaction phase is relatively small unless the distribution coefficient is very low, typically less than 0.005.
Appendix 2: Check on the Validity of the Assumption of Slow Reaction Regime The maximum value of Ha, the Hatta number, must be derived for a QFS reaction, as this situation yields the highest reaction rate, much more than the marginal ignition situation. The Hatta number must be correlated to the distribution coefficient(s1 and be calculated for the maximum relevant reaction temperature, T,, being the target temperature at approximately 0 = 0.5:
In order to eliminate A'Tad,ofrom eq A2-1, this equation is combined with the definition of the exothermicity of the system according to eq 25:
T,,
=
RH + U*DU RT: Tc + 1.05 -EX R"__ + U*Da(l + 0.56) E
+
(ERH+ U*Da)ExT-RTc/E 1 1.05 cRH + U*Da(l + 0.5~)
Further
)] \ m
(A24
and the Hatta number is hence
By combining eqs A2-5 and A2-6 and taking c = 0.4, RH = 1, tD = 3600 S, E/RTc = 36, CBO = 2.0 kmol/m3, vA/vc = 1, and (cADB)1/2/kl= 0.5, we obtain the following: (i) for a system with a small temperature overshoot, given by the combination (Ex,Ry) = (3.2,0.8), Ha,, = 0.041mB-1/2 attained at the target temperature and Hastart= 0.011mB-1/2attained at the starting temperature, Tc; (ii) for a system with a large temperature overshoot, given by the combination (Ex,Ry) = (8,0.3), Ha,, = 0.131mB-1/2 attained at the target temperature and Hastart= 0.0068mB-1/2attained at the starting temperature, Tc. These figures have been used to obtain Table 111.
Appendix 3: Effect of Solubility in the Other Phase The effect of more than negligible solubility in the other phase is that the molar amounts of A and B are no longer considered to be only present in phases d and c, respectively, but are divided between the two phases, according to (A3-1) nA = CAdVd + cAc vc and (A3-2) nB = CBcVc + CBdVd By invoking the definitions of the distribution coefficients and neglecting the small concentration drop at both sides of the interface, which is allowed under the conditions derived in Appendix 1, we find (A3-3) nA = cAd(vd + mAVc) and nB = cBd(Vc/mB
(A2-2)
\
+ vd)
(A3-4)
Effect for a Slow Reaction in the Dispersed Phase. The relevant expression for the reaction rate in this case is
and DU = tDCBOkR
(A2-4)
The relevant distribution coefficient for a slow reaction in the dispersed phase is mB. A QFS condition for a slow reaction in the continuous phase is obtained at a much lower value of the reactivity, and hence, a slow reaction in the dispersed phase is more critical with respect to entering the fast reaction regime when the temperature
or
Substitution of eq A3-6 into the general expression eq 18 for the conversion rate, under the assumption of v d = tev,
Ind. Eng. Chem. Res. 1990, 29, 1270-1278
1270
and V , = V,, yields after a few rearrangements €6 d {B _ dfl - ( ’ A / u c ) K m B D a ( f l - {B)(l - l B ) ( e o + mA)(tomB + 1) (A3-7)
The latter part of eq A3-7 is obviously the correction factor on the original expression as found in Table I. Its value is 1 if mA = mB = 0. Effect for a Slow Reaction in the Continuous Phase. Similarly for the kinetic regime of slow reaction in the continuous phase, with reaction rate r according to =
I z l , l , ~ c A ~ c B ~-( ltd)
the following expression for the conversion rate is obtained: B - {=
dfl
(A3-8)
The effect of mBis small, as in the initial period not much of material B can be lost to the other phase: mgte is always > m A .
Appendix 4: Effect of Changing Volume of the Continuous Phase, Ac, For a number of parameter combinations, we calculate the change in the coolant temperature needed to recover the specific thermal behavior at At, = 0 (Table VI). The value of At, is limited to f0.25~. For a slow reaction in the dispersed phase, eq 18 is rearranged to V A (fl - CB)(1 - {B) d_ TB - mBI3a-K do uc 1 + l~&,
The parameter combinations given in Table VI are shown in Table VII. The conclusion of these sample calculations is that the effect of variable volume of the continuous phase due to reaction is negligible.
Literature Cited Bjornberg, H.; Nilsson, H.; Silvegren, C. A Calorimetric Study of Heterogeneous Styrene Polymerization. Thermochim. Acta 1984, 72, 239-244. Hub, L. Entwicklung von Prufmethoden zur Erhohung der Sicherheit bei der Durchfuhrung chemischer Prozesse. Ph.D. Dissertation, ETH Zurich, 1975. Hugo, P.; Steinbach, J. Practically oriented representation of thermal safe limita for an indirectly cooled semi-batch reactor. Chem. Ing. Tech. 1985, 57 (9), 780-782. Hugo, P.; Steinbach, J. A comparison of the limits of safe operation of a SBR and a CSTR. Chem. Eng. Sci. 1986, 41, 1081-1087. Simmons, G . F. Differential Equations: Tata McGraw-Hill: New Delhi, 1974. Steensma, M.; Westerterp, K. P. Thermally Safe operation of a cooled semi-batch reactor. Slow liquid-liquid reactions. Chem. Eng. Sci. 1988, 43 (8), 2125-2132. Steensma, M.; Westerterp, K. R. Thermally Safe operation of a cooled semi-batch reactor. Fast liquid-liquid reactions. In preparation, 1990. Steinbach, J. Untersuchung zur thermischen Sicherheit des indirekt gekuhlten semibatch-Reaktors (Investigation into the thermal safety of an indirectly cooled semi-batch reactor). Ph.D. Dissertation, D83, Technical University, Berlin, 1985. van Heuven, J. W.; Beek, W. J. Vergrotingsregels voor turbulente vloeistof-vloeistof dispersies in geroerde vaten (Scale-uprules for turbulent liquid-liquid dispersions in stirred vessels). 1ng.Chem-Tech. 1970, 44 (6), 41-60. Westerterp, K. R.; Swaay, W. P. M. van; Beenackers, A. A. C. M. Chemical Reactor Design and Operation, 2nd ed.; John Wiley & Sons: Chichester, U.K., 1984.
Received for review May 17, 1989 Revised manuscript receiued December 18, 1989 Accepted January 2, 1990
Radiation Field inside a Tubular Multilamp Reactor for Water and Wastewater Treatment Orlando M. Alfano, Manuel Vicente,+Santiago Esplugas,+and Albert0 E. Cassano* INTEC,t Casilla de Correo No. 92, 3000 Santa Fe, Argentina
This paper studies the radiant energy distribution inside a multilamp tubular photoreactor used for water and wastewater treatment. The system is a continuous tubular reactor irradiated from outside by four germicidal lamps symmetrically located about the reactor. The equations governing the energy transfer were formulated and solved numerically for two limiting cases: (i) a diactinic medium and (ii) a medium for which the attenation coefficient is assumed independent of the extent of the reaction. T o do so, the Extense Source model with voluminal emission was applied. The radiation field inside the reactor was then analyzed as a function of position and of the radiation absorbing characteristics of the medium. Through the analysis of the results obtained in this work, it is possible to draw useful criteria to improve the design and operation of this type of photoreactor. Photochemical processes are taking an active role in processing polluted streams of different origin. Moreover, in some cases they seem to be one of the few, if not the unique, possibilities to degradate undesired pollutants. UV Present address: Departamento de Ingenieria Quimica, Facultad de Quimica, Universidad de Barcelona, Barcelona, Espaiia. f Instituto de Desarrollo Tecnolijgico para la Industria Quimica (INTEC). Universidad Nacional del Litoral (UNL) and Consejo Nacional de Investigaciones Cientificas y Tgcnicas (CONICET), Santa Fe, Argentina. 0888-5885/90/2629-1270$02.50/0
radiation alone or combined with an oxidizing agent in photooxidizing processes has been frequently used (Peyton et al., 1982a,b; Coburn et al., 1984; Legan, 1982). One of the best photoreactor configurations to ensure water potability is the multilamp tubular photoreactor enclosed inside a reflecting device. A typical commercialized design for medium-scale water treatment has a cylindrical quartz tube irradiated from outside by four cylindrical germicidal lamps symmetrically located about the reactor. One cylindrical reflector of circular shape surrounds each lamp, giving the overall system a four-peak 0 1990 American Chemical Society
