Ind. Eng. Chem. Res. 1994,33, 299-305
299
Snowball Effects in Reactor/Separator Processes with Recycle William L. Luyben Department of Chemical Engineering, Iacocca Hall 111, Lehigh University, Bethlehem, Pennsylvania 18015
In several numerical case studies of some typical recycle processes, Luyben reported the "snowball" phenomenon: a small change in a load variable causes a very large change in the recycle flow rate around the system. It is important to note that snowballing is a steady-state phenomenon and has nothing to do with dynamics. It does, however, depend on the structure of the control system as Luyben demonstrated. This paper presents a mathematical analysis of the problem for several typical kinetic systems. In the simple binary first-order case of A B, an analytical solution can be found for the recycle flow rate as a function of the fresh feed flow rate and fresh feed composition. Two different control structures are explored. It is shown analytically why the control structure proposed by Luyben prevents snowballing and why the conventional structure results in severe snowballing. Two other kinetic systems are studied numerically: consecutive first-order reactions A B C and a second-order reaction A B C.
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1. Introduction
Luyben (1992) presented a numerical case study of a recycle system that exhibited the "snowball" effect. The example process consisted of a continuous stirred-tank reactor followed by two distillation columns. Consecutive first-order reactions A B C occurred in the reactor, with component B as the desired product. The volatilities of the three components were assumed to CYA> CYB> CYC. Unreacted component A was recycled back to the reactor from the top of the first column. Luyben demonstrated that the use of a conventional control structure resulted in a 100% increase in the recycle flow rate for a 10% increase in fresh feed flow rate. Such large changes in the load on the distillation separation section are very undesirable because columns can only tolerate a limited turndown ratio. The conventional control system was the following: 1. Flow control fresh feed flow rate. 2. Control reactor level by manipulating reactor effluent flow rate. 3. Control reactor temperature by manipulating jacket coolant flow rate. 4. Control the impurity of component A in the base of the first column by manipulating heat input. 5. Control reflux drum level in the first column by manipulating distillate flow rate. 6. Flow control reflux in the first column (or use it to control the impurity of component B in the distillate from the first column: the recycle to the reactor). The second column was not involved in the recycle loop and simply separated product (component B) from byproduct (component C). Note that this control structure has both of the flow rates in the recycle loop (reactor effluent F and distillate from the first column D1)set by level controllers. Luyben proposed the generic rule that one flow rate in a liquid recycle loop should be flow controlled. He demonstrated that snowballing could be eliminated by switching the first two loops in the conventional structure. 1. Flow control reactor effluent. 2. Control reactor level by manipulating fresh feed flow rate. This example was presented at the Workshop on the Interaction between Process Design and Process Control that was held on Sept 6-8,1992, at Imperial College. During one of the discussion periods, Morari (1992) suggested
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+
that the snowball effect could be analyzed mathematically. That is precisely the intent of this paper. In section 2, a simple first-order single reaction system is studied. In section 3, a recycle system with consecutive first-order reactions is explored. Finally, in section 4, a second-order reaction is considered. The steady-state changes in recycle flow rates are presented for different disturbances and using different control structures.
2. Process 1: Binary System with Reaction A B
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The simplest possible recycle system is that considered by Papadourakis (1985)which consists of a reactor followed by a distillation column. In the continuous stirred-tank reactor (CSTR) a single irreverisble, first-order reaction A B occurs. Some of the reactant A is consumed. The reactor effluent is a mixture of A and B, and it is fed into a distillation column which takes mostly component B out the bottom and recycles distillate with mostly unreacted component A back to the reactor. The recycle stream is the distillate from the column, D. Fresh feed is introduced into the reactor at a rate FOand composition 20. First-order kinetics and isothermal operation are assumed in the reactor.
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.% = VRk2
(1)
where .% = rate of consumption of reactant A (mol/h), VR = reactor holdup (lb-mol),k = specific reaction rate (h-l), and z = concentration of reactant A in the reactor (mole fraction A). The concentration of component A in the bottoms product from the column XB is assumed to be held constant by manipulating heat input. The composition of the distillate XD is assumed to be held constant by manipulating reflux flow rate. We consider two control structures for this system. In the first (the conventional structure) the fresh feed flow rate Fo is flow controlled and the reactor effluent flow rate F is manipulated to keep reactor holdup VR constant. In the second, the structure proposed by Luyben (19921, the loops are reversed: reactor effluent is flow controlled and reactor level is held constant by manipulating fresh feed flow rate. In this second structure, throughput changes are accomplished by changing the setpoint of the reactor level controller. Alternatively throughput changes could be achieved by changing the setpoint of the reactor temperature controller if isothermal operation is not
Q888-5885/94/2633-Q299$04.5QlQ 0 1994 American Chemical Society
300 Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994
required by kinetic or other considerations, e.g., the occurrence of side reactions at other temperatures, materials of construction corrosion limitations. 2.1. Conventional S t r u c t u r e (Constant Reactor Holdup). The variables that are costant are VR,k, X D , and XB. The variables that will change when disturbances occur are F , z , and the recycle flow rate D. The steady-state equations that describe the system are given below. overall:
Fo = B
(2)
reactor:
Fo+D=F
(4)
Equations 2 and 3 can be combined to yield eq 6, which shows how reactor composition z must change as fresh feed flow rate FO and fresh feed composition zo change when the conventional control structure is employed (Le., reactor volume is constant).
Equations 4 and 5 can be combined to give recycle flow rate D.
D=
z(Fo + kVR)- F o ~ o XD-2
(7)
Substituting eq 6 into eq 7 gives an analytical expression showing how the recycle flow rate changes with disturbances in fresh feed flow rate and composition.
where @E-
k VR
(9)
%O-'B
It is useful to look at the limiting case in which X D 1 and XB N 0. Under these conditions, eq 8 becomes
This equation clearly shows the strong dependence of the recycle flow rate on the fresh feed flow rate: increasing FOincreases the numerator (as the square) and decreases the denominator. Both effects tend to increase D sharply as Fo increases. Numerical examples will be shown later. 2.2. Reactor Effluent Fixed S t r u c t u r e (Variable Reactor Holdup). The variables that are constant are F , k, XD, and XB. The variables that will change when disturbances occur are VR,z, and the recycle flow rate D. Equations 2-5 still describe the system, but now the value of F is constant while VR is variable. Combining these equations gives
D=F-Fo
Table 1. Parameter Values for Process 1 at normal design conditions fresh feed composition = zo = 0.9 mole fraction component A fresh feed flow rate = FO = 239.5 mol/h reactor holdup = VR = 1250 mol reactor effluent flow rate = F = 500 mol/h recycle flow rate = distillate flow rate = D = 260.5 mol/h parameter values specific reaction rate = k = 0.340 86 h-l bottoms composition = XB = 0.0105 mole fraction component A distillate composition = TD = 0.95 mole fraction component A
(11)
Equation 11shows that the recycle flow rate D will change in direct proportion to the change in fresh feed flow rate and will not change at all when fresh feed composition changes. Equation 12 shows that reactor holdup VR will change as fresh feed flow rate and fresh feed composition change. It is useful to look at the limiting case in which XD N 1 and XB = 0. Under these conditions, eq 12 becomes
This equation shows that reactor holdup changes in direct proportion to fresh feed composition and is less dependent on fresh feed flow rate since the FOterm in the numerator is now only to the first power. Keep in mind that FOis not really a disturbance with this structure since fresh feed is used to control reactor holdup. However, the changes required in the setpoint of the level controller to accomplish a desired change in fresh feed flow rate can be calculated from eq 12. Note that the fresh feed flow rate will change as fresh feed composition changes for a constant reactor holdup. This was demonstrated by Luyben (1992) in his numerical studies. Figures 1 and 2 give numerical resulb for the system considered by Papadourakis (1985). Table 1gives values of design parameters for the base-case conditions. The drastic changes in recycle flow rates when the conventional control structure are used are clearly shown in Figures l b and 2b. Figure l a shows that the reactor compositionz responses differently for the two control structures when fresh feed flow rate changes. Figure 2 shows that neither reactor compositionnor recycle flow rate changes in the constant-F control structure when fresh feed composition changes occur. The only thing that changes is the reactor holdup. 2.3. Impossible Control Structure. Before we leave this process example, it might be useful to point out that there is an impossible control structure for this system. As discussed by Downs (19921,the followingcontrol structure could be proposed: 1. Flow control the fresh feed. 2. Control the reactor level by manipulating the reactor effluent flow rate. 3. Control the reactor temperature by manipulating the jacket coolant flow rate. 4. Control the composition in the reactor by manipulating the recycle flow rate. 5. Control the impurity of component A in the base of the first column by manipulating the heat input. 6. Control reflux drum level in the first column by manipulating reflux flow rate. Although this strategy looks on the surface like a viable candidate structure, it is fatally flawed. The problem is
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 301
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Reeycb, A B; s o l ~ m t mVn; t d.lbsdrOmmt F
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0.82
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Rcsh Feed Flow R.to (molany)
086
0.9
088
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0.94
0.96
0.98
1
Fresh Feed Cornpolition (m.t mrnpnent A) Recycle: A = E solidrmmtant VR,dashcdsconstantF
Fresh Feed Cornpition (nf. component A) Recyck; A = B solid-mmont VR,drshcd=mnstantF
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Figure 1. Process 1 (a, top-c, bottom): changes in reactor composition, recycle flow rate, and reactor holdup with two control structures for changes in fresh feed flow rate.
that it attempts to hold constant all three variables in the reactor (z, k, and VR)and at the same time set the fresh feed flow rate of component A into the system Fo.Fixing these three variables fixes the rate of consumption of component A in the system, and the control system on the base of the column prevents component A from leaving the system. Thus it is impossible to independently set the flow rate of component A into the system. One of the variables in the reactor that affects the total reaction rate must change as the amount of component A fed into the system changes. 3. Process 2: Consecutive Reactions In this section we study a process in which two simultaneous reactions occur involving three compo-
.-
:
1 A,,::,,pq . i.., . ....................j............. ........................................ ; j
...... 1150 -. ............ , CUB> ac. We assume first-order, isothermal kinetics for both reactions.
a, = VRklZ1
(14) (15) where Rj = reaction rate of j t h reaction (lb-mol/h), k, = specific reaction rate (h-l), and zj = concentration of j t h component in the reactor (mole fraction). We will use the nomenclature that component A is number 1,component = V~k2Z2
302 Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994
B is number 2, and component C is number 3. Component A is consumed by the first reaction. Component B is produced by the first reaction and consumed by the second. Component C is produced by the second reaction. Fresh feed enters the reactor at a rate Fo and composition 20,. The normal steady state is FO= 100 lb-mol/h and z01 = 1(pure component A in the fresh feed). Distillate from the top of the first distillation column is recycled back to the reactor at a flow rate D1 and with a composition XDlj. Reactor effluent F is fed into the first distillation column. Bottoms B1 from the first column is fed into the second column in which components B and C are separated into product streams. Since the second column is not involved in the recycle loop, we need to look only at the reactor and the first column. We assume that the control system on the first column will maintain the distillate composition at X D ~ I= 0.95, XD12 = 0.05, X ~ 1 3= 0 (no component C goes overhead in the first column) by manipulating reflux. At the other end, we assume that the impurity of component A in bottoms is held at X B H = 0.01 by manipulating heat input. We will look at two control structures. In the first, the reactor holdup VRis held constant by manipulating reactor effluent flow rate. In the second, the reactor effluent flow rate is held constant, and reactor holdup is controlled by manipulating fresh feed flow rate. 3.1. Constant Reactor Holdup Structure. The variables that are constant are VR, kl, 122, XD11, XD12, and ~ ~ 1 The 1 . variables that will change when disturbances occur are F, z 1 , z 2 , XB12, and the recycle flow rate D1. The steady-state component balances that describe the recycle section of the plant are given below.
Table 2. Parameter Values for Process 2 at normal design conditions fresh feed composition = 201 = 1mole fraction component A fresh feed flow rate = FO = 100 lb-mol/h reactor holdup = VR= 150 lb-mol reactor effluent flow rate = F = 324 lb-mol/h recycle flow rate = distillate flow rate = D1 = 224 lb-mol/h reactor composition = z1 = 0.6600 mole fraction component A z2= 0.2324 mole fraction component B parameter values specific reaction rates = k l = k2 = 1h-' bottoms composition = ~ ~ = 10.011 mole fraction component A distillate composition = X D I ~ =0.95 mole fraction component A ~ ~ = 0.05 1 2 mole fraction component B XD13 = 0 mole fraction component C
overall:
Then 2 3 1 2 can be found from eq 24, and z 2 can be found from eq 25. Figures 3 and 4 show how the recycle flow rate D1, reactor composition z1, and reactor holdup VR vary for the two different control structures as fresh feed flow rate and composition are varied. Table 2 gives the values of parameters used in this numerical case. The large changes in recycle flow rates when the conventional structure is used are clearly shown. When the constant-F control structure is used, feed composition changes produce no changes in either reactor composition or in the recycle flow rate.
Fo = B, FOzOl
(16)
= B I X B l l + 'RklZ1
(17)
first column:
F = D,
+ B,
(19)
z2 = D1XD12 + ' B l P O F Numerical examples are given below. 3.2. Constant Reactor Effluent Flow Rate Structure. The variables that are constant are F, k ~ ~, ~ , X D I I , XD12, and ~ ~ 1 1The . variables that will change when disturbances occur are VR,z 1 , z 2 , ~ ~ 1and 2 , the recycle flow rate D1. Equations 16-21 can be combined to solve sequentially for the variables of interest.
Dl=F-Fo z1 = 'DllF - FO(xD1l - 'Bll) F
(26) (27)
4. Process 3: Second-OrderReaction
Equations 16-21 give us six equations. There are six unknowns: B1, XB12,z1,z2, F, and the recycle flow rate D1. These equations can be solved analytically and sequentially to find the flow rate of the recycle. Equation 16 gives us B1. Equation 17 can be solved for z1. - 'Bll)
z1 = FO(zOl
'lVR
Combining eqs 19 and 20 gives
D,= Bl(zl - X B l l ) 'Dll-
1 '
Then F can be found from eq 19. Combining eqs 18 and 21 yields
In this section we study the process considered by Tyreus and Luyben (1993) with the second-order reaction A + B C. Two fresh-feed makeup streams are introduced into the reactor: FOAand FOB. The concentrations of the two reactants in the reactor are z1 and Z Z . Neither reactant undergoes complete one-pass conversion in the reactor. Therefore both reactants must be recycled. The volatilities of the A, B, and C components dictate how many recycle streams will be required and where they will be produced in the separation system. In this paper we assume that the volatilities are CYA > CYC > CYB;i.e., component C is intermediate between components A and B. Hence, two columns and two recycle streams will be required. We assume that component B, the heaviest, is recycled from the bottom of the first column (in stream B1) back to the reactor. Component A, the lightest, is recycled from the top of the second column (in stream D 2 ) back to the reactor. Second-order kinetics were assumed.
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Note that "moles" are not conserved in this system. The
Ind. Eng. Chem. Res., Vol. 33, No. 2, 1994 303 Recycle, A-B-C, solid-coastant VR, Qahed-conrtant F
Rcsyclc; A-B-C; aolid-mmW vR;-P
065-
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94
96
98
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102
104
106
108
110
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0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Fresh Feed Composition ml (m f component A)
Frcsh Fad Flow Rate (moleflu) l&yclc: A=B=C, solid=cOnStnnt VR;da8h%%onstont P
.,’~
150 rlXCU,n /”
5
f
144
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4
140138
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92 94
%
98
100
102
16
loa
loa
,Io
RwhFecdPlowRate(mkwhr)
Figure 3. Process 2 (a, top-c, bottom): changes in reactor composition, recycle flow rate, and reactor holdup with two control structures for changes in fresh feed flow rate.
overall reaction rate depends on the product of the two concentrations z1 and 22. As we will see later in numerical examples, if one composition decreases, the other has to increase to keep the same productivity. This interplay between the two compositions can produce large changes in the required recycle flow rates, as we will demonstrate below. Two control structures will be considered. In both schemes, reactor holdup is held constant. In the first control structure, the flow rates of one or both of the recycle streams are changed in order to change the throughput. In the second control structure, the recycle flow rates are held constant, and throughput changes are achieved by changingthe setpoint of the reactor temperature controller. Since the reactivity term always contains the product of
-
,,/’
,,.‘....
,/ /’ Fixed F
...... ...........
134 Fresh Fced Compositionml (m.f. component A)
Figure 4. Process 2 (a, top-c, bottom): changes in reactor composition, recycle flow rate, and reactor holdup with two control structures for changes in fresh feed composition.
k and VR,we can change the specificreaction rate or reactor holdup in the same proportion. In this example we will illustrate how reactivity can be controlled by changing reactor temperature (Le., changing k). Thus in this example we assume that it is not necessary to maintain the reactor at a constant optimum temperature. 4.1. Variable Recycle Structure. The process is described by three component balances for each unit: reactor, column 1, and column 2. There are a number of ways to solve the nine nonlinear algebraic equations, but the following procedure was found to be straightforward and involved no iteration. 1. Specify the throughput by fixing the flow rate of the product stream B2 leaving the bottom of column 2. The
304
Ind. Eng. Chem. Res., Val. 33, No. 2, 1994 Recycle; A+B-C: aalld:B2=90; daakd-100; dotted-110
base case value is 100 lb-mol/h. The composition of this ) 0.01, and stream was specified to be ~ g 2 ( 1=) 0.01, X B ~ ( Z= ~ ~ 2 (= 3 )0.98. 2. Set the flow rate of the light recycle stream Dz from the top of column 2 (to be varied later). The composition of this recycle stream was specified to be X D Z ( ~ )= 0.99, XDZ(Z) = 0, and ~ ~ z ( =3 0.01. ) 3. Guess a value of the flow rate of the heavy recycle stream B1 from the bottom of column 1. The composition of this recycle stream was specified to be XBI(I) = 0, XBl(2) = 0.99, and XB1(3) = 0.01. 4. Calculate the feed to the second column:
40 -
30 -
-
20
D, = B,
+ D,
1
(30)
15
w20
25
30
35
40
45
50
55
i 60 60
Heavy Recycle Flow Rate B1 (moleihr) Recycle; A+B=C; solid,BZ=% dashed=lOa;dotted410 045,
5. Calculate the feed to the first column:
0251
I
B2.110
(35)
I -. 02-
9
:: 2
6. The rate of production of product C is equal to B z x B ~ (and ~ ) ,this must be equal to the rate of generation of component C in the reactor (assuming the two fresh feed streams contain no component C). Therefore the reactor volume can be calculated:
015nt.
0 0515
20
25
30
35
40
45
50
55
Heavy Recycle Flow Rate E1 (moleihr) 4,
,
Recy,cle; A+B;C; solid:B2=90;dashed=lOO;dotted=llO
,
(36) 0.35 -
7. If the calculated reactor holdup does not equal the
actual reactor holdup (2970.82 lb-mol in the numerical example), go back to step 3 and reguess the value of the heavy recycle stream B1. A simple interval-halving convergence technique was used. 8. Calculate the fresh-feed makeup flows of both component A (FoA)and component B (FOB). FOA
= Fzl + VRkZ1Z2 - DZXD2(l)
(37)
FOB
= Fz2 +
- BlxBl(Z)
(38)
'RkZIZZ
0.3
~
E
v
';J c
0.25 -
3
8
5 i
0.2
~
0.15 -
O1 "."
