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Steady-State Multiplicity Behavior Analysis of a High-Impact Polystyrene Continuous Stirred Tank Reactor Using a Bifunctional Initiator Rodrigo Lo´ pez-Negrete de la Fuente and Javier Lopez-Rubio Departamento de Ingenierı´a y Ciencias Quı´micas, UniVersidad Iberoamericana, Prolongacio´ n Paseo de la Reforma 880, Me´ xico D.F., 01210, Me´ xico
Antonio Flores-Tlacuahuac* Department of Chemical Engineering, Carnegie-Mellon UniVersity, 5000 Forbes AVenue, Pittsburgh, PennsylVania 15213
Enrique Saldı´var-Guerra Centro de InVestigacio´ n en Quı´mica Aplicada, BlVd. Enrique Reyna 140, Saltillo, Coahuila, 25100, Me´ xico
In this work, the steady-state multiplicity analysis of a complete and detailed high-impact polymerization system that has been performed in a nonisothermal continuous stirred tank reactor is addressed. The potential effect of manipulated disturbance and design variables on the reactor control is discussed. Through proper combination of certain bifurcation parameters, some ways of increasing monomer conversion without encountering problems related to the gel effect are proposed. It is shown that high-sensitivity dependence problems cannot be removed by making changes in the analyzed design parameters. Some of the best ways of conducting closed-loop reactor temperature control are identified. 1. Introduction Polymer manufacturing involves processes that can exhibit highly nonlinear behavior, including phenomena such as input/ output multiplicities, limit cycles, sustained oscillations, hysteresis, and chaos.1 Nonlinear analysis tools are useful to locate the range of design parameters over which complex operating regimes may occur.2 They also permit, for instance, the setup of control schemes that are able to address issues caused by intrinsic nonlinear behavior.3 Because process plants are now required to be more flexible, a consideration of nonlinearities is necessary to have confidence in model predictions concerning plant operation over a wide range of operating conditions. Commonly, nonlinearities are perceived as an undesired phenomenon, because they may adversely affect product quality and process control, as well as lead to unsafe operations.3 It is widely accepted that processes must be designed and operated not only to optimize static performance, but also to impart the system with accepted levels of controllability and resiliency.4 Thus, nonlinear analysis is potentially helpful in early design stages. Initial efforts to take into account nonlinear behavior in an optimization design framework have been recently reported.5 Even when nonlinearities are commonly considered harmful to process operation, there might be cases when optimal static design calls for operation around highly nonlinear regions. It is a common rationale that, although feasible operation of openloop unstable systems is possible through adequate feedback control, such a situation is usually undesirable, because of operational and safety reasons and should be avoided by proper design. Complicated nonlinear bifurcation behavior has been reported in several types of single polymerization reactors; this phenom* To whom correspondence should be addressed. On leave from Universidad Iberoamericana. Tel./fax: +52(55)59504074. E-mail addresses:
[email protected],
[email protected]. URL: http:// 200.13.98.241/∼antonio.
ena can be predicted from process models, demonstrated in laboratory reactors, and experienced in a wide range of industrial processes.1 Complex kinetic behavior, mass- and heat-transfer limitations, and the gel-effect onset are some sources of complex bifurcation behavior in polymerization reactors.6 Nonlinear behavior of single continuous stirred tank reactors (CSTRs) has been widely studied. Some pioneer work was developed by Van Heerden7 and Aris and Amundson,8 while the complete treatment of nonlinearity and bifurcation of a CSTR with a single reaction was addressed by Uppal et al.9 Advances obtained from these early works motivated studies in more-complex reaction kinetics, such as those found in polymerization reactors. Ray and Villa1 have recently reviewed nonlinearity issues commonly embedded in polymerization reaction systems. The analysis of the nonlinear dynamics of polymerization reactors has been a very important research area. In particular, polymerization processes are of great interest, because they usually present complicated steady-state nonlinear bifurcation behavior. This knowledge is important, because the nonlinear behavior of chemical reactors has an important effect on operation difficulties.2 Furthermore, the knowledge of their behavior might help to optimize the process through the elimination of nonlinearities. The complexity of the nonlinear steady-state bifurcation behavior of polymerization reactors results from very elaborate kinetics, heat- and mass-transfer limitations, etc. For example, in the bulk polymerization of methyl methacrylate (MMA), the emergence of steady-state multiplicity has been associated with the gel effect.6 This occurs in high-conversion regions in which there is a high concentration of polymer and highly viscous polymer mixtures. This causes the molecular weight to increase, because, at such high viscosities, the termination reaction frequency decreases, compared to that of the propagation reactions, because the termination step becomes diffusioncontrolled.
10.1021/ie0487383 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/31/2006
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Table 1. Design and Rate Constant Information parameter
value or equivalent expression
Tf (K) Ff (kg/L) f ktd (L mol-1 s-1) kfsd (L mol-1 s-1) kt (L mol-1 s-1) kd2 (s-1) ki2 (L mol-1 s-1) kp (L mol-1 s-1) kfs (L mol-1 s-1) kfb (L mol-1 s-1) ki0s (L2 mol-2 s-1) ki0d (L2 mol-2 s-1) ki3s (L mol-1 s-1) ki3d (L mol-1 s-1) kpds (L mol-1 s-1) kpsd (L mol-1 s-1) kpod (L mol-1 s-1) kpdd (L mol-1 s-1) kd1 (s-1) kd3 (s-1) kpss (L mol-1 s-1) kpos (L mol-1 s-1) kts (L mol-1 s-1) kto (L mol-1 s-1) ktss (L mol-1 s-1) ktsd (L mol-1 s-1) ktso (L mol-1 s-1) ktdd (L mol-1 s-1) ktdo (L mol-1 s-1) kfss (L mol-1 s-1) kfds (L mol-1 s-1) kfos (L mol-1 s-1) kfd (L mol-1 s-1) kfdd (L mol-1 s-1) kfod (L mol-1 s-1) kfsb (L mol-1 s-1) kfdb (L mol-1 s-1) kfob (L mol-1 s-1) kfa (L mol-1 s-1) kfsa (L mol-1 s-1) kfoa (L mol-1 s-1) kfda (L mol-1 s-1) ki1s (L mol-1 s-1) ki1d (L mol-1 s-1) c1 c2 c3
333 0.915 0.57 9.8 × 107 exp(-353/T) 2.324 × 108 exp(-9218/T) 1.7 × 109 exp{-843/[T-2(c1x + c2x2 + c3x3)]} 3.47 × 1015 exp(-17127/T) 0.63(9 × 105 exp(-2650/T)) 0.98(9 × 105 exp(-2650/T)) L mol-1 s-1 4.2 × 106 exp(-6477/T) 3.0 × 106 exp(-8950/T) 2.2 × 105 exp(-13809/T) 2.2 × 105 exp(-13809/T) 9 × 105 exp(-2650/T) 9 × 105 exp(-2650/T) 4.9 × 105 exp(-2190/T) 4.9 × 105 exp(-2190/T) 4.9 × 105 exp(-2190/T) 4.9 × 105 exp(-2190/T) 0 kd2 kp kp kt kt kt ktd kt ktd ktd kfs kfsd kfs kfsd kfsd kfsd kfb kfb kfb 26kp 26kp 26kp 26kp kp kp 2.57 - 0.00505T 9.56 - 0.0176T -3.03 + 0.00785T
The aim of this work is to address the operability difficulties faced when operating CSTR high-impact polystyrene (HIPS) reactors. The steady-state operability problem is addressed through the use of nonlinear bifurcation techniques. Even when the industrial production process involves a series of connected reactors, only one CSTR was chosen to perform the analysis, to avoid mathematical modeling complexities. We extend some previous results published by our research group 10 by including a more-sophisticated model description of the HIPS reactor that includes the use of a bifunctional initiator with branching and crosslinking reactions. Also, this model has not been published elsewhere. HIPS is an important commodity material that combines the ease of processing of polystyrene with increased mechanical resistance. It is used in a variety of applications, including packaging, household appliances, etc. Although there are still a few industrial facilities operating the batch masssuspension process for the production of HIPS, most of this material is presently produced via the more-profitable bulk continuous process. There are several variations of this process,11,12 but most of them use a CSTR in series with some type of tubular reactor(s). The residence time distribution of the reactors, as well as the reaction conditions (mixing effectiveness, temperature, viscosity), have an important role in determining the size and morphology of the rubber particles and the final properties of the product. The polymerization can
be performed by thermal or chemical initiation with monofunctional or bifunctional initiators.13 The outline of this paper is as follows. In section 2, the complete set of kinetic reaction steps are fully described. The dynamic mathematical model describing the reactor performance is also described there. Section 3 contains the discussion of the effect of potential manipulated disturbance and design variables on the reactor operability. Finally, in section 4, the main conclusions and contributions of this work are highlighted. 2. Detailed Mathematical Model In the open literature, there are only a few works on the modeling of the HIPS process.14-16 The model used here has been developed by the authors as part of a project aimed at the detailed modeling of the HIPS and the GPPS (general purpose polystyrene) processes. The general model, which will be published elsewhere,17 describes the populations of linear polystyrene (which forms the bulk of the continuous phase), grafted polybutadiene, and crosslinked polybutadiene, and that description has been maintained here, for the sake of completeness, although some of the populations (grafted and crosslinked polybutadiene) are not relevant for the dynamics of the bulk and, therefore, are not discussed in this paper. At this point, it is important to clarify that this paper will focus on the dynamics of the continuous phase for two reasons: (i) the variables of the continuous phase (conversion, molecular weight) are the most important ones, from the point of view of the dynamics of the process, and (ii) the variables of the continuous phase are those more easily measured and, therefore, are more amenable to being controlled variables in a control scheme. The properties of the dispersed phase, such as gel content and swelling index, are also important parameters for the quality of the product; however, their dynamics are mostly driven by the dynamics of the bulk phase and, therefore, their study is not of primary importance in this work. It is worth mentioning that, in industrial practice, given the high operating temperatures used in the second and third reactors, all the dispersed phase is trapped as crosslinked material and it is separated from the continuous (linear polystyrene phase) before analysis in the laboratory. The molecular weight measured in the laboratory corresponds only to the continuous linear phase. This determines that the only population considered for calculations of the average molecular weight in this paper is that corresponding to the continuous linear phase; the model, however, is able to estimate also the properties of the dispersed phase. An important simplification that is used throughout the model is to assume that the system is homogeneous. Although this simplification is clearly not accurate at a local level, it is a reasonable lumped approximation, given the relatively small fraction of polybutadiene added to the reactor; besides, as noted by other researchers,15 given the low solubility of styrene and polystyrene (PS) in the rubber-rich phase, and the low content of rubber in the formulation, most of the chemical reactions occur in the continuous phase consisting of PS and styrene, and, therefore, the homogeneous approximation is not far from reality. Also, under this assumption, the mathematical model for the GPPS process can be conceived as a subset of the more general model for the HIPS process, in which the polybutadiene feed is set to zero. Other features of the model are as follows: (i) autothermal initiation of styrene is included; (ii) initiation via monofunctional and bifunctional initiators is described; (iii) the set of polymerization reactions are conducted in a nonisothermal CSTR assuming perfect mixing; (iv) the quasi-steady state assumption
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Figure 1. (a) Conversion, (b) number-average molecular weight, (c) weight-average molecular weight, and (d) reactor temperature using the feedstream flow rate as the main bifurcation parameter and the cooling water temperature as the secondary parameter: curve 1, TJ0 ) 373.15 K; curve 2, TJ0 ) 294 K; and curve 3, TJ0 ) 273.15 K.
is used for the living species; (v) the long chain hypothesis is valid; (vi) volume variations due to polymer contraction are taken into account; and (vii) the kinetic mechanism involves initiation, propagation, transfer, termination, grafting to polybutadiene, and crosslinking reactions. Polybutadiene is added to guarantee the desired mechanical properties by promoting grafting reactions, and a general model includes network formation reactions, because they may lead to an undesirable excess of crosslinking in the rubber particles. The grafted molecules have a tendency to go to the interphases between rubber particles and the continuous polystyrene phase, where they reduce the interfacial energy and help to keep a stable dispersion. The model does not count the number of grafts of each polybutadiene molecule but, instead, gives an average description. Polybutadiene chains are considered to be part of the crosslinked material when two polybutadiene molecules are linked through a polystyrene “bridge” or when they are directly linked. There is some debate about the reactions that cause
crosslinking. Kesskula18 has proposed, as the main reaction responsible for crosslinking, the reaction that occurs between two just activated polybutadiene radicals. Other authors14,15 have assumed that coupling between two radicals in the active ends of graft chains is the main contributor to crosslinking. In this model, both possibilities are included. Moreover, the branching reactions are caused by hydrogen abstraction, either by a primary radical coming from the initiator or from a polystyryl growing radical. Hydrogen abstraction creates a polymeric radical, which can be located on the main chain or on the tip of a growing branch. When these radicals terminate with other radicals by combination, they produce an “X”-type or “H”-type crosslink. The kinetic mechanism, which includes the reactions just discussed, is summarized in Appendix A. Our industrial experience with the steady-state behavior of the process indicates that the previous approximations are reasonable ones, in the sense that the resulting mathematical model is able to reproduce the observed industrial behavior.
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Figure 2. (a) Conversion, (b) number-average molecular weight, (c) weight-average molecular weight, and (d) reactor temperature using the feedstream temperature as the main bifurcation parameter and the feedstream flow rate as the secondary parameter: curve 1, Q ) 0.5000 L/s; curve 2, Q ) 1.6566 L/s; and curve 3, Q ) 2.0000 L/s.
Because of industrial secrecy, most details of the model validation with industrial data cannot be offered; however, it is possible to mention that the behavior of molecular weight and conversion was fitted by the model, using the set of kinetic parameters included in this work. Also, a previous simplified version of this model has been used for a steady-state and bifurcation analysis of HIPS reactors. In that study,10 some limited comparison of the model with steady-state industrial data is offered. An important difference between the work reported in ref 10 and the present work is the use of a bifunctional initiator. A bifunctional initiator allows one to operate the process at higher temperatures and, therefore, higher production rates (with respect to the process with a monofunctional initiator), while preserving relatively high molecular weights in the product. A model that incorporates both the monofunctional initiator and the bifunctional initiator descriptions is of great utility to determine the conditions under which the same
grades of polymer produced with a monofunctional initiator can be produced at higher production rates (higher temperatures) with a bifunctional initiator. The kinetic mechanism is given in Appendix A, and the dynamic mathematical model is given in Appendix B; all the kinetic rate constants are shown in Table 1. Most of the kinetic rate constant information has been taken from ref 10. The additional rate constants were fitted to reflect plant reactor operation. Because of model size and complexities, the available continuation packages (Matcont, XPP/Auto,etc) were unable to generate the bifurcation diagrams that we intended to compute. As a consequence, all continuation diagrams in this paper were generated using a Matlab code that was written especially for this purpose. The NLEQ119 solver was used to solve the set of algebraic nonlinear equations obtained from eliminating the time derivatives for each value of the main continuation parameter.
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Figure 3. (a) Temperature and conversion and (b) manipulated variable (feedstream temperature) for the first transition (A to B). (c) Temperature and conversion and (d) manipulated variable (feedstream temperature) for the second transition (A to C). The solid line represents temperature data, the dashed line represents conversion data, and the dashed-dotted line represents set-point data.
3. Results and Discussion Feedstream Flow Rate. As displayed in Figure 1, under nominal operating conditions, the steady-state operation point, which is denoted by the symbol “o”, turns out to be open-loop unstable. This is consistent with previous results obtained using a shorter model version.10 The implications are that the HIPS reactor requires a closed-loop control system to be operated under such unstable conditions. However, care should be taken, because there have been reports20 that closed-loop control (i.e., proportional integral (PI) control) might introduce additional nonlinearities that previously were not present. Figure 1 clearly shows that, under nominal operating conditions, the HIPS reactor exhibits a high-sensitivity parametric region. According to calculations, a maximum of three output steady states are present. The low-conversion steady state is open-loop stable, but, because of small monomer conversion, the reactor would be hardly operated there. On the other hand, the high-monomer-conversion steady state turns out to be openloop stable as well. However, because of the gel-effect onset, operation of the reactor is not recommended near this steady
state. Thereby, operation of the reactors around stable steady states is discarded, based on the previously discussed grounds. HIPS manufacturing is normally conducted in a series of connected reactors. In this work, only the first reactor of the train reaction is analyzed. When working with the first reactor, only modest monomer conversions are normally reached. Therefore, the intermediate conversion steady state represents the operation region around which the HIPS reactor should be operated. From Figure 1a, the implications of feedstream flow rate disturbances can be readily appreciated. If a 35% reduction in the feedstream flow rate occurs, the monomer conversion would decrease ∼15%, with no apparent operability problems (except the fact that the disturbance should be rejected). However, the same change in the opposite direction would result in a 60% monomer conversion, resulting in potential operability problems, because of the gel-effect onset. This behavior is a direct consequence of the high-sensitivity parametric region around the nominal operating region. Also, Figure 1a shows that the effect of variations in the cooling water feedstream temperature
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Figure 4. (a) Conversion, (b) number-average molecular weight, (c) weight-average molecular weight, and (d) reactor temperature using the feedstream temperature as the main bifurcation parameter and the initiator feedstream flow rate as the secondary parameter: curve 1, Qi2 ) 4.1581 × 10-4 L/s; curve 2, Qi2 ) 0.4500 L/s; and curve 3, Qi2 ) 0.6000 L/s.
are almost negligible. They hardly change the steady-state multiplicity map, except at large excursions of such temperature, as displayed in branch 3 of this figure. However, branch 3 is only included for demonstration purposes. Because of the fact that cooling water comes from reservoir tanks or from the plant pipeline system, such large temperature variations are not expected to occur. Figure 1d shows that the feedstream flow rate may be a good candidate for reactor temperature control. However, process gains between these two variables have different magnitudes, depending whether the set point is higher or lower than the nominal operating point. Figures 1b and 1c display the open-loop behavior of the Mn and Mw molecular weights, respectively. It may be concluded that molecular weights are roughly insensitive to wide variations in the main and secondary bifurcation variables. Although such molecular weights are not normally directly on-line controlled, they define important HIPS properties. Care should be taken when designing a control system to attain the desired molecular weights values.
Feedstream Temperature. Figures 2, 4, and 5 display the steady-state multiplicity maps that use the feedstream temperature as the main bifurcation parameter and the feedstream, initiator and cooling water flow rates, respectively. Figure 2a displays an interesting fact. It shows that monomer conversion could be increased by reducing the feedstream temperature. For instance, by reducing the feedstream temperature to 300 K, the monomer conversion increases ∼40%. Normally, operation problems associated with the gel-effect onset should not matter near this conversion region. This increase in monomer conversion is not obtained by modifying the process output rate, because the monomer feedstream flow rate is kept constant, as can be seen from branch 2 of this figure. Figure 2a also shows that the process output rate could be increased by operating along branch 3 of this figure. In fact, without changing additional operating conditions, except for the monomer feedstream flow rate, the process will turn out to be more profitable, in terms of increased monomer conversion and polymer output rate. More-
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Figure 5. (a) Conversion, (b) number-average molecular weight, (c) weight-average molecular weight, and (d) reactor temperature using the feedstream temperature as the main bifurcation parameter and the cooling water flow rate as the secondary parameter: curve 1, Qw ) 0.500 L/s; curve 2, Qw ) 1.000 L/s; and curve 3, Qw ) 4.000 L/s.
over, an additional reduction in the feedstream temperature, so the final reactor temperature is ∼300 K, will lead to a monomer conversion of 60% where, hopefully, operational problems that are due to the gel-effect onset are not yet important. On the other hand, branch 1 of Figure 2a shows that reducing the monomer feedstream flow rate will lead to a reduction in both monomer conversion and polymer output rate. This discussion highlights the fact that the feedstream temperature may be used as a manipulated variable for monomer conversion control. However, industrially, on-line monomer conversion is not normally available for process control purposes. But, as Figure 2d displays, monomer conversion closed-loop control could be forced by controlling the reactor temperature. However, the control of reactor temperature by manipulating the feedstream temperature should be tight; otherwise, the process may be operating near the high-conversion region, where gel effects are important. For instance, suppose that, to increase monomer conversion and reactor output rate, the feedstream flow rate is increased, so the reactor ends up operating along branch 3 of
Figure 2d. By reducing the feedstream temperature along this branch, monomer conversion is increased. However, a decrease of ∼25 °C in the feedstream temperature would result in a situation where, if the reactor temperature control loop is not tightly tuned, the reactor would ultimately be operating around the high-monomer-conversion region. From Figures 2b and 2c, it is clearly observed that molecular weights exhibit a smooth dependence on both bifurcation parameters. To corroborate this, a simple PI feedback controller was used to control the reactor temperature. The feedstream temperature was selected as a manipulated variable, and the feedstream flow rate was chosen as a possible perturbation. Initial values for the controller parameters were selected according to the classic control theory, and these values were improved by trial and error, because of the complex behavior of the system. In Figure 3a, the reactor closed-loop dynamic response to a change in the set point is shown. It can be seen that the controller efficiently makes the transition within ∼4 h, featuring 40% monomer conversion, as shown in Figure 2. Moreover, in Figure
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Figure 6. (a) Conversion, (b) number-average molecular weight, (c) weight-average molecular weight, and (d) reactor temperature using the initiator feedstream flow rate as the main bifurcation parameter and the cooling water flow rate as the secondary parameter: curve 1, Qw ) 0.500 L/s; curve 2, Qw ) 1.000 L/s; and curve 3, Qw ) 4.000 L/s.
3b, it can be seen that the control action is mirrored on the manipulated variable; the temperature reaches 300 K, when it stabilizes. As was previously mentioned, this temperature is expected at a monomer conversion of 40%. Also, with this controller, it was possible to make the transition from point A up to point C, as shown in Figure 2d. This was done by applying a change in the perturbation variable from 1.6566 L/s to 2 L/s, and making a change in the reactor temperature set point, from its nominal value to ∼420 K (see Figures 3c and 3d). In this case, the controller performed as expected, as exhibited by the behavior of the manipulated variable in Figure 3d. The controller action clearly was to change the feedstream temperature to achieve the transition; however, finally, at the new set point, it stabilizes at ∼333 K, which is the nominal feedstream temperature. This new steady state is the one just above the nominal operating point on branch 3 of Figures 2a and 2d. The intention of this short discussion is to show that open-loop continuation behavior can be a useful tool to address closed-loop control problems.
Figure 4 is especially important, because it displays the steady-state multiplicity maps, using two variables that are commonly used for closed-loop control purposessfeedstream temperature and initiator flow ratesas bifurcation parameters. As shown in Figure 4a, under nominal operating conditions, an increase in the initiator flow rate (as represented by branches 2 and 3) would result in monomer conversions of ∼45% and ∼57%, respectively. This situation can be easily explained by recalling that, when the initiator flow rate is increased, there will be more free radicals to increase the monomer conversion. Note that monomer conversion could be increased by a combination of increasing the initiator flow rate and reducing the feedstream temperature, without affecting the polymer output rate. In fact, Figure 4a shows that such a process combination turns out to be feasible. Of course, this combination of process variables should avoid operation around the gel-effect onset region. It is also interesting to note, from Figures 4b and 4c, that both molecular weight values remain slightly sensitive to variations in both bifurcation parameters. As a matter of fact,
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Figure 7. (a) Conversion, (b) number-average molecular weight, (c) weight-average molecular weight, and (d) reactor temperature using the residence time as the main bifurcation parameter and the feedstream flow rate as the secondary parameter: curve 1, Q ) 0.5000 L/s; curve 2, Q ) 1.6566 L/s; and curve 3, Q ) 2.0000 L/s.
if the feedstream temperature is reduced to ∼300 K, monomer conversion will increase up to 40%. However, the same change in feedstream temperature would result in small variations in both molecular weights. In a sense, molecular weight values are more “robust”, from the point of view that they almost remain constant for wide variations of the bifurcation parameters. Again, Figure 4d shows that the reactor temperature control loop, using the feedstream temperature as a manipulated variable, should be tight enough. Otherwise, the reactor will ultimately be operating around the high-conversion region. Figure 5a clearly shows that monomer conversion could be increased by simply reducing the feedstream temperature. In fact, under nominal operating conditions, if the feedstream temperature is reduced to 300 K, monomer conversion would increase ∼40%. Moreover, if starting from this point, the flow of cooling water is increased up to 4 L/s (branch 3), then monomer conversion would be higher, ∼47%. Even without decreasing the feedstream temperature, it is clear that by simply
increasing the cooling water flow rate, monomer conversion could be increased to ∼33%. This behavior is especially relevant, because cooling water is usually an inexpensive plant service. From Figure 4d, it is interesting to note that using the cooling water flow rate as a manipulated variable for reactor temperature control would allow larger-sized feedstream disturbances. In fact, even when the flow rate of cooling water is increased to 4 L/s, the feedstream temperature should decrease to
