208
Ind. Eng. Chem Fundam
1984,2 3 , 208-212
Stability of Uniform Flow Distributions in Multitubular Polymerization Reactors Hendrlk W. Hoogstraten‘ Department of Mathematics, University of Groningen, Groningen, The Netherlands
Geed E. H. Joosten Chemical Engineering Department, University of Groningen, Groningen, The Netherlands
Andr6 P. Rljnaard Department of Mathematics, University of Groningen, Groningen, The Netherlands
Uniform flow distributions in multiiubular polymerization reactors may be unstable as a result of the strong variation in viscosity of the reactants during the reaction. This instability leads to a very nonuniform distribution (maldistribution) of the fluid over the various reactor tubes which is an undesirable situation in practice. I n this paper an earlier stability analysis presented by the authors which is valid only for very small perturbations is extended to arbitrary perturbations. I t turns out that stability against small perturbations does not generally imply stability against any large perturbation. Furthermore, it is found that perturbations of uniform flow distributions in which some of the reactor tubes are forced to operate initially in the “unstable” velocity range need not necessarily result in a maldistribution. I n the Appendix it is shown that the theory developed in this paper can also be applied to the cooling of fluids with temperaturedependent viscosity in multitubular heat exchangers.
Introduction The pressure drop needed to maintain a steady flow of an incompressible fluid of constant viscosity through a tube is an increasing function of the flow velocity. However, if a polymerizing reactant flows through a single reactor tube the dependence of the pressure drop Ap as a function of the flow velocity u may have the form shown in Figure 1.
The striking feature of this case is the presence of a range (V,, V2)of flow velocities for which the pressure drop is a decreasing function of u. In an earlier paper (Joosten et al., 1981) it has been explained that this phenomenon is due to the strong increase in viscosity of the reaction mixture with increasing conversion. The extent of conversion depends on the residence time of the reactants in the reactor tube and, therefore, depends on the flow velocity. A larger velocity corresponds to a shorter residence time, which leads to a smaller extent of polymerization. This implies a reduction of the average viscosity of the fluid in the reactor tube. If this reduction is strong enough a situation may arise in which an increase of the flow velocity leads to a decrease in the pressure drop. On the basis of this observation it is intuitively clear that for a multitubular polymerization reactor the situation in which all of the tubes operate at the same velocity between V , and V 2will be highly unstable. Any perturbation of this uniform flow distribution in the reactor will lead to a maldistribution in which a number of tubes will be filled with nearly stagnant material continuing to polymerize leading to increasing viscosity, the remaining tubes having a large flow rate with, consequently, a small extent of polymerization. This certainly would be an awkward situation in practice. In Joosten et al. (1981) the stability of uniform flow distributions in multitubular polymerization reactors has
been studied only for very small initial perturbations by use of a linearized form of the mathematical model describing the time evolution of the flow in the reactor. It turned out that a uniform flow distribution with velocity u is stable in the above sense if the slope dApldu > 0 for that value of u, the case dAp/du C 0 corresponding to instability. In the latter case we have VI C u < V2in Figure 1. In order to arrive at unconditional stability it was suggested in Joosten et al. (1981) to add an extra resistance to the flow in each tube in such a way that the negative slope in the Ap vs. u relationship vanishes. In practical situations it is important to know whether stability against small perturbations also implies stability against large ones. Therefore, we study in this paper the time behavior of the flow in a multitubular polymerization reactor for arbitrary initial perturbations of uniform flow distributions making use of the fully nonlinear differential equations governing the time evolution of the flow. We find that stability against small perturbations does not imply stability against any large perturbation in general. Another interesting conclusion is that perturbations in which a number of tubes is forced to operate initially at a velocity in the unstable range between V , and V2need not be fatal in the sense that the flow distribution tends to an undersirable (stable) maldistribution.
The Basic Equations Consider a multitubular reactor consisting of N parallel tubes of equal cross-sectional area A and length L. The tubes are fed with a given constant total volumetric feed rate 4”. The average flow velocity in the ith tube at time t is denoted by u i ( t ) . Then the mass balance gives N
CAu,(t)= 4”
1=1
0196-4313/84/1023-0208$01.50/00 1984 American Chemical Society
(1)
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
209
Table I. Values o f Parameters Used in the Calculation a=
8.1
dh = 1 . 5 x k =2x
"1
v2
+
v
0.u
Figure 1. Pressure drop Ap over a reactor tube as a function of flow velocity u when the viscosity of the fluid increases strongly with conversion of the reactants.
and the momentum balance gives the N differential equations
where p denotes the (constant) density of the fluid and p the time-dependent inlet pressure of the tubes, the outlet pressure being taken equal to zero. The unknown pressure p can be expressed in terms of the vis by adding together eq 2 and noting that du1 d ~ 2 dVN dt dt dt as a consequence of eq 1. We find for p
- + - + ... + - = 0
Thus, at any time t , the inlet pressure is equal to the average of the %teadyn pressure drops over the N tubes at that moment. For a steady-state solution we should have p = Ap(U1) = A ~ ( u , = ) ... = A ~ ( u N ) (4) The uniform distribution is a special steady state with equal flow velocities for all tubes
(5) However, for A p ( u ) of the form as indicated in Figure 1, several other steady states are possible in general. For those states the flow velocities are not the same for all tubes; that is, these steady states correspond to maldistributions. The number of possible maldistributions increases with increasing N . We shall see later that a perturbation of an unstable uniform distribution of the type (5) will cause the reactor to evolve toward one of the (stable) maldistributions. The particular expression giving A p as a function of the velocity u depends on the properties of the fluid and the geometry of the tube and its internals, if any. We will consider the case of isothermal flow of a solution of polymerizing monomers. The solution is considered to behave as a Newtonian fluid whose viscosity is a function of the polymer content and molecular weight. In view of the flow velocities and liquid viscosities likely to be encountered, the flow is assumed to be laminar in the case of a tube without internal devices. In this case there would be a pronounced radial velocity profile with lower velocities near the tube wall than near the tube axis. This in its own right gives rise to an extensive spread in residence time of the fluid in the reactor, usually affecting the conversion and molecular-weight distribution of the material. In a polymerizing fluid the velocity profile may have another consequence. The faster moving material near the
lo-* m s-l
L=12m p = 800 kg/m3 qo = 4 x lo-*
Ns/m2
axis of the tube will not have reacted as much in travelling a certain distance down the tube as the material near the wall. As the viscosity of the solution increases sharply with polymer content and molecular weight, the material near the wall will have an increased viscosity as compared with material near the axis. This radial viscosity gradient may eventually lead to either complete plugging of the reactor tube or to channeling of the monomer (solution) through a narrow core formed within high-molecular-weight material being almost stagnant near the wall. This aspect of flow instability has been studied by Lynn and Huff (1971). From the above it is clear that a tube without internal devices does not always seem to be the most attractive reactor for polymerizations. In many cases it is essential to reduce the radial gradients. This may effectively be done by placing static mixers in the tube, e.g., Kenics mixers (Chen and MacDonald, 1973) or Sulzer mixers (Kaluza, 1975; Tauscher, 1976). In tubes with these internals the flow closely approaches plug flow, and radial concentration gradients are virtually absent. For these reasons Sulzer mixers are used on an industrial scale, for instance, in the production of nylon-6 from caprolactam (Sulzer, 1976; Tauscher, 1976). Under plug-flow conditions the pressure drop for steady-state flow in a single reactor tube can be written as an integral along the tube L
AP(u) =
G u s 0 o ( l / u ) dl
(6)
where the local viscosity 77 depends on the local residence time llu. The constant C1 depends on the tube geometry and on any internals in the tube. For instance, for a circular tube with Sulzer mixers C1 = 73/dh2 where dh denotes the hydraulic diameter of the tube (Tauscher and Schutz, 1973). In order to arrive at a specific A p vs. u relationship with negative slope we consider a hypothetical situation in which reactants with viscosity qo are entering the tubes. To simulate a practical polymerization process in a realistic way we assume a strong dependence of the viscosity Q on the fractional conversion x of the reactants 7 = q,,eax
(7)
with a being a moderately large number. The conversion x is taken to be related to the residence time l l u according to the relationship for a first-order irreversible reaction with reaction-rate constant k x = 1 - exp(-kl/u) (8) Thus, by eq 7 and 8, we can write oU/u) = 70 exp[all - exp(-kl/u)ll
(9)
Using (9) in eq 6 we have computed the scaled pressure drop F = A p / ( p L )as a function of u (see Figure 2). The values of the various parameters used in the calculations are given in Table I. It is interesting to note that the phenomenon of a A p vs. u relationship with negative slope similar to the one derived above can also occur in the cooling of fluids with temperature-dependent viscosity in multitubular heat exchangers. (The authors are grateful to a referee for pointing out the possibility of applying the present theory
210
Ind. Eng. Chem. Fundam., Vol. 23, No. 2 , 1984 h/*
2t
f
L
-
1
2
e
C
I
2L
TIpP/s
01
-\.
Figure 2. Scaled pressure drop F (N/kg) as a function of flow velocity u (mm/s) used in our computations.
to multitubular heat exchangers.) We refer to the Appendix for further details. The basic eq 2 and 3 can now be written as a set of N nonlinear first-order differential equations for the N flow velocities u, dui dt
-=
1N Nj=l
- C F ( u j ) - F(UJ (i = 1, ..., N)
3
10 23 30 NLMBER OF TUBE
Figure 3. Velocity distributions (mm/s) at different times t (s) for linear perturbation of unstable uniform flow distribution with V = 5 mm/s and N = 30.
(10)
In the sequel we shall use these equations to compute numerically the time-behavior of the flow in the reactor for given initial perturbations of uniform flow distributions. Numerical Results The stability of a given uniform flow distribution u1 = u2 = ... = uN = V against nonsmall perturbations can be studied by solving the N differential eq 10 for various sets of initial values uJ0) = + w ; (i = 1, ..., N)
v
where w i represents the initial disturbance of the flow velocity in the ith tube. Mass conservation requires that the w ishould be prescribed such that
;
0
c
10 20 W M B E R OF T U B E
d
30 5
\ 0
10 20 30 *UMBER OF TUBE
I O - o NUMBER O F TUBE
Figure 4. Velocity distributions (mm/s) at different times t (s) for various linear perturbations of conditionally stable uniform flow distribution with V = 20 mm/s and N = 30. Maximum initial deviations are 10 mm/s (a), 14 mm/s (b), and 15 mm/s (c).
N
cwi = 0 i= 1
We have performed some numerical computations for N = 30 by use of a standard second-order Runge-Kutta technique with time step 0.015 s. The initial disturbance has been chosen “linearly” W , = b(15.5 - i) (i = 1, ..., 30) for various values of the parameter b. Other types of initial perturbations can be investigated in a similar way and yield qualitatively the same behavior. Figure 3 shows a result for V = 5 mm/s which is typical of the behavior in the unstable regime, that is, F’( V) < 0. In reality the graphs are step functions with 30 steps but they have been smoothened to show a clearer picture. The initial linear velocity profile evolves rapidly into a maldistribution with 13 tubes operating at 11.5 mm/s and the remaining 17 tubes operating at such an extremely small velocity that the plot coincides with u = 0. This situation is reached within 5.1 seconds. Similar behaviour is found for other initial perturbations, so we may conclude that uniform flow distributions with velocity V in the range F‘(V) < 0 are always unstable. For uniform distributions with velocity V such that F’( V) > 0 the situation is different. Figures 4a, b, and c show some results for V = 20 mm/s which is in this range. In Figure 4a a linear initial perturbation with maximum deviation 10 mm/s from the undisturbed value V = 20
mm/s is seen to return to the original uniform distribution in 13 s. If the maximum initial deviation from V = 20 mm/s is increased a situation occurs (see Figure 4b) in which some tubes operate initially at an unstable velocity (roughly, below 10 mm/s) and still the reactor returns in 13.5 s to the original uniform distribution. So, if some of the tubes are forced to operate at an unstable velocity at some instant, this need not be fatal in the sense that the flow evolves into a maldistribution. However, if the maximum initial deviation is increased still further, the flow distribution may tend to a maldistribution quite rapidly, as is shown by Figure 4c. In this maldistribution only one tube has a near-zero velocity. Computations have shown that for increasing maximum initial deviations the number of tubes finally operating at near-zero velocity in the limit maldistribution also increases. We conclude that the behavior in the “stable” regime F’( V) > 0 as illustrated by the above results for V = 20 mm/s is in fact only conditionally stable: the uniform distribution is stable against small and moderately large perturbations but unstable for too large ones; that is, it is not stable inthe-large. However, for both extremely small and extremely large values of V (which are not very interesting from a practical point of view) stability in-the-large is found. The computations of this section have been performed for a reactor consisting of 30 tubes. They can easily be repeated for any other number of tubes provided sufficient
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
I G
a fraction of N
1
Figure 5. Typical piecewise constant initial perturbation of uniform flow distribution with flow velocity V.
computer capacity is available. Nevertheless, for large N the computations become quite time-consuming and it is interesting to know that statements about stability can be made independent of the number of tubes and without actually solving the set of differential equations 10 when we restrict ourselves to a special type of initial perturbations, viz., piecewise constant perturbations. This will be worked out in the next section. Piecewise Constant Perturbations There exists a certain type of initial perturbations for which the response of the reactor can be easily computed independent of N , the total number of tubes. For these “piecewise constant” perturbations a fraction a of the total number of tubes operates initially at equal velocity P, the remaining (1- a)N tubes operating initially at the velocity
V - ap 1-CY
(45
v)
as dictated by the mass balance. A typical perturbation of this class is shown in Figure 5. It turns out that at any later time t > 0 the flow velocities in all tubes belonging to the first group are equal, and the same holds for the velocities in the tubes of the second group. This means that the velocity distribution is similar to that shown in Figure 5 at any time. One could say in this case that the reactor behaves like a reactor consisting of only two tubes with respective cross-sectional areas aNA and (1- a)NA. This behavior is possible because eq 10 possess solutions of the form ~ i ( t=) u(t) (i = 1, 2, ..., CYN) (W
-
v - cy&) 1-ff
a.u
1
I
0
i
211
(i = aN
+ 1, aN + 2, ...)N)
(llb)
where the members of the two groups of tubes have been identified by the labels 1, 2, ..., aN and aN 1,aN + 2, ...,N , respectively. Indeed, substituting (11)into eq 10 and adding together either the first OrN differential equations or the remaining (1- a ) N equations, we obtain in both cases the following single differential equation for the flow velocity u ( t )
+
du dt with initial condition u(0) = p. Note that eq 12 and its initial condition do not depend on N . The right-hand side G of eq 12 vanishes for u = V; this corresponds to the uniform distribution the stability of which is to be investigated. A specific choice of the parameters cy and p determines a particular piecewise constant initial perturbation. The subsequent behavior of the flow in the reactor is determined by the other zeroes of G, that is, possible steady states corresponding to maldistributions to which u may tend as t tends to infinity. A
Figure 6. Graph of G typical of a conditionally stable uniform flow distribution with flow velocity V (represented by zero no. 2). Zeroes 1 and 3 correspond to conditionally stable maldistributions, and zeroes 4 and 5 to unstable maldistributions.
1
mm/s
100
80
P
------ _ - _ _ _ _ _ _
0.L I 0.6 I -0.8 r-_ 1 -a Figure 7. a,p diagram for the conditionally stable uniform flow distribution with V = 20 mm/s.
00
0.2 I
graph of G which is typical of the conditionally stable case to be dealt with below is shown in Figure 6. The sign of the derivative G’at the zeroes of G determines the stability of the steady states: G’ < 0 implies conditional stability, G’ > 0 instability. It is illustrative to show an a,p diagram for the uniform distribution with V = 20 mm/s, which lies in the conditionally stable regime (see Figure 7). All possible initial perturbations are represented by points in the a,@plane in the region bounded by the lines a = 0, a = 1,p = 0 and the hyperbola ab = V. For each a between 0 and 1 the function G has 5 zeroes, one of which is at u = 20 (conditionally stable). Two zeroes correspond to unstable maldistributions and have been graphically depicted as functions of a by interrupted lines. The final two zeroes correspond to conditionally stable maldistributions. They are lying so close to the curves /3 = 0 and cup = 20 that they have not been shown. The region in the a,@diagram bounded by the two interrupted lines is the set of “stable” a,/3 values defining initial perturbations after which the flow in the reactor returns to the original uniform distribution V = 20. Complete blocking of one or more tubes (0 = 0) is seen to be fatal. However, a number of tubes may have reduced velocity initially even down into the “unstable” regime below 10 mm/s and still the uniform distribution will be restored. The diagram thus illustrates again the conditional stability of the uniform distribution V = 20. Furthermore, it offers for any fixed choice of a between 0 and 1a lower and an upper bound for the velocities in the tubes between which the reactor can operate safely. Appendix The Cooling of a Fluid with Temperature-Dependent Viscosity in a Multitubular Heat Exchanger. Consider a steady plug flow (with velocity u ) of a hot fluid
212
Ind. Eng. Chem. Fundam., Vol.
23, No. 2, 1984
through a tube of length L surrounded by coolant fluid of constant temperature To. A simple heat balance leads to the following expression for the fluid temperature T as a function of the axial coordinate 1 T(1) = To + (Ti, - To)exp(-cl/u) (AS) where Ti,(>To)is the inlet temperature and c is a constant depending on the geometry of the tube, the fluid properties, and the heat-transfer coefficient. The fluid viscosity q is assumed to vary with the (absolute) temperature T according to Andrade's law q ( T ) = q m exp(B/T)
tA.2)
Then the pressure-drop formula becomes the same as eq 6, that is L
M u ) = C 1 u l q ( l / u ) dl 0
(A.3)
where q is now given by
On the basis of these relations the dimensionless pressure drop has been computed as a function of the dimensionless velocity u/(Lc) for the cooling of glycerol with inlet temperature Ti, = 410 K and coolant temperature To = 273 K. Note that the group Lc/v represents the number of heat transfer units in the heat exchanger. In eq A.2 we have taken the values qm = 9 X Ns/m2 and B = 6145 K. These values have been obtained by fitting relation A.2 with the viscosities 0.5 and 0.02 Ns/m2 for glycerol a t temperatures 305 and 363 K, respectively. The behavior of the pressure drop was found to be completely analogous to that found in the polymerization case: a steep increase from zero at zero flow velocity to a maximum value at u/(Lc) = 0.2 (corresponding to about 5 heat transfer units), a subsequent decline to a minimum value of about 5% of the maximum value at u / ( L c ) = 2.8 (correspondingto about 3.5 heat transfer units), and finally a slow monotone increase for larger values of the dimensionless velocity. This means that the theory developed in this paper for the flow stability in multitubular polymerization reactors may apply as well to the cooling of fluids with temperature-dependent viscosity in multitubular heat exchangers.
Nomenclature A = cross-sectional area of reactor tube, m2 a = dimensionless constant defined by eq 7 b = parameter occurring in the values of w l , mm/s B = constant defined by eq A.2, K c = constant defined by eq A.1, s-l C1 = constant, m-2 dh = hydraulic diameter of reactor tube, m F = scaled pressure drop, N/kg G = right-hand side of eq 12, mm/sz ij = integers, ranging from 1 to N k = first-order reaction-rate constant, s-l L = length of reactor tube, m 1 = axial coordinate, m N = number of reactor tubes p = inlet pressure, N/m2 t = time, s T = fluid temperature, K Ti, = inlet fluid temperature, K To = coolant fluid temperature, K u = flow velocity, mm/s V = flow velocity for uniform flow distribution, mm/s V,, Vz = flow velocities defined in Figure 1, mm/s u = flow velocity, mm/s u, = flow velocity in ith reactor tube, mm/s ul = initial disturbance of flow velocity in ith tube, mm/s x = fractional conversion Greek Symbols a = fraction of total number of tubes p = initial flow velocity, mm/s Ap = pressure drop, N/m2 q = fluid viscosity, Ns/m2 qo = fluid viscosity at zero conversion, Ns/m2 q m = viscosity defined by eq A.2, Ns/m2 p = fluid density, kg/m3 4" = volumetric flow rate, m3/s Literature Cited Chen, S.J.; MacDonald, A. P. Chem. Eng. Mar 1973, 80, 105. Joosten, G. E. H.;Hoogstraten, H. W.; Ouwerkerk, C. Ind. Eng. Chem. Process Des. Dev. 1981, 2 0 , 177. Kaluza, H. J. Engineering Foundation Conference on Mixing Research, Rindge, New Hampshire, Aug 17-22, 1975. Lynn, S.;Huff, J. E. AIChE J. 1971, 17, 475. Suizer Technical Review, VT 10018le, June 1976. Tauscher, W. Verfahrenstechnik 1076, 10, 258. Tauscher, W.; Schiitz, G. Sulzer Technical Review 211973, 1973.
Receiued for review January 27, 1983 Revised manuscript receiued December 20, 1983 Accepted January 12, 1984
