Polymerization process controller

A batch polymerization process controller using inferential sensing to determine the integral reaction heat which in turn is used to indicate the degree of polymerization of the reaction mixture batch. The system uses a reaction temperature compared with a desired temperature, and the result is used as a feedback to monitor and control the process. One version of the process controller also uses a feedforward signal which is an integral reaction heat indication from a process model. In another version of the process controller, the integral reaction heat is compared with a desired integral reaction heat, and result is used as another feedback to monitor and control the process. Heat production and reaction temperature profiles may be used, along with the thermokinetic equations to determine the polymerization process and reactor models which are utilized by the process controller to optimize the polymerization process in terms of efficient use of cooling water and desired polymerization of each mixture batch.

BACKGROUND 
The invention pertains to polymerization reactions, particularly to monitor 
and control the rate and the amount of conversion in such reactions. More 
particularly, the invention pertains to accurately determining the rate 
and amount of conversion at a particular moment in a polymerization 
reaction so as to control the rate of conversion and to optimize cooling 
resources. 
Most polymerization reactions today are run open loop with respect to the 
product quality (end-use) properties. Also operations involved in the 
manufacturing process are scheduled by a simple timer without attention to 
the actual progress of reaction. 
In the last decade, the affordability of powerful computers finally made it 
possible to exploit the advanced control concepts control theorists have 
been developing since the 1960's. As a result, control of continuous 
processes like refinery distillation columns or power generation units has 
seen a rapid evolution from single loop proportional, integral and 
differential (PID) controllers to multivariable predictive controllers 
with built-in constraint optimization whose performance cannot be matched 
by the old PID solutions. 
For a number of reasons, this progress so far has avoided batch processes. 
Control wise, most batches are still run the way they were thirty or more 
years ago. If there was a change, it affected control hardware, but not 
control algorithms. A batch recipe still prescribes time profiles of 
temperatures or pressures to be followed by a batch reactor in order to 
make the product. Feedback controllers, usually PID's, are routinely used 
to make the batch track the recipe in the presence of variations in 
feedstock concentration and purity, catalyst activity, reactor fouling and 
so on. 
Maintaining batch recipe temperatures and pressures is important but it 
should not be the control objective. After all, the process owner does not 
sell batch temperatures or pressures. They are mere process parameters 
and, by themselves, are not even sufficient ones. It is well known and 
exemplified below for the case of polymerization processes, that two 
batches with perfectly identical temperature and pressure profiles can 
still have different rates at which monomer is converted into polymer, and 
thus yield products with inconsistent quality. When comes to the end-use 
parameters of the real product, which are determining its marketable 
quality, most batch processes are still run open loop, with all the 
negative consequences that an open loop recipe execution entails. 
With the present invention, that approach is replaced with a feedback 
controller for polymerization processes that closes the loop using a 
measurement directly tied to the product's marketable quality, and thus 
employs feedback to eliminate quality variations and inconsistencies due 
to the fluctuations of process inputs and operating conditions. 
The invention is a polymerization control that allows the user to specify 
independently the reaction mixture temperature and the degree of monomer 
conversion profiles as a function of time, and execute them under feedback 
control. This both improves the run-to-run consistency of the product and 
reduces the uncertainty of the reaction time and coolant consumption at 
any given instant. Because the coolant availability often is the limiting 
factor of production capacity, the improved predictability of individual 
batch runs offers an opportunity to improve batch planning and scheduling 
and thus increase the plant yield without expensive retrofits. 
SUMMARY OF THE INVENTION 
This invention enables the controller to employ feedback for the control of 
product properties without the need for specialty sensors to measure the 
properties and run the polymerization process on the basis of its inner 
time reflecting its actual progress. As a result, the invention makes it 
possible, first, to manufacture polymers with consistent quality and, 
second, to improve process yield by allowing for better utilization of the 
available cooling capacity without sacrificing process safety. 
The invention includes an inferential sensor, whose concept is based on the 
observation that for polymerization processes, in which heat is released 
by a single reaction, the amount of heat released is proportional, albeit 
in a nonlinear way, to the degree of the monomer conversion. Hence, by 
carefully calculating the reactor's thermal balance on-line one can 
continuously infer the degree of conversion and use it for control. Once 
the actual degree of conversion can be determined and ultimately 
controlled, one can also control the cooling duty of the reactor and thus 
make it conform with the cooling capacity allotted to it by the plant 
scheduler. 
Superficially, an advanced batch control system utilizing the inferential 
sensor looks very much the same as a conventional one. In both cases, 
measurements of temperatures and flows of the reactor coolant as it enters 
and leaves the reactor jacket or cooling coil will constitute the bulk of 
input data. In addition to that data, the inferential sensor may require 
additional data reporting temperatures at some other reactor spots and on 
the amounts and temperatures of feedstocks and catalysts. If some data on 
their composition are available, they can also be used with advantage for 
a more accurate inference. 
The significant difference is in what the controllers do internally with 
the data. In a conventional batch controller, the data are used directly 
to control the reactor mixture temperature by manipulating the incoming 
coolant flow and temperature. In an advanced controller, the data are fed 
into the inferential sensor instead, where they are used to infer the 
current value of the degree of monomer conversion. This quantity is then 
passed to the controller part of the advanced batch controller. 
Even though the inferential sensor could be implemented as a stand-alone 
device and thus resemble physical sensors, this option is unlikely. The 
reason is that the sensor involves a nonlinear dynamic model of both the 
process and the reactor, whose state must be kept in sync or coordinated 
with reality using a state estimation algorithm driven by the measured 
temperatures and flows. Once the model is available, it is shared with the 
advantage of a model-based (nonlinear) controller. 
Polymerization reactions are exothermic (i.e., a chemical change in which 
there is a liberation of heat, such as combustion). The overall amount of 
heat released by a reaction from its start up to a given instant depends 
on how much of the monomer(s) has been converted into polymer. This 
measure of released heat indicating the degree of monomer conversion is a 
more reliable indicator of reaction progress than physical time because 
the same reaction can be running slower or faster depending on the 
initiator (i.e., catalyst) activity, reactant purity and other effects 
that may be difficult to measure directly. Moreover, for many 
polymerization reactions the degree of conversion is linked to the product 
quality and thus can be used for closing the loop for the product quality 
feedback control in place of specialty sensors. 
The degree of conversion is not measured directly, but the invention 
involves inferring its running value by dynamically evaluating the reactor 
heat balance. This invention involves four concepts. First, there is the 
way of inferring the degree of conversion from the dynamic evaluation of 
the reactor heat balance. Second, the use of the degree of conversion 
replaces specialty sensors for feedback control with respect to the 
product quality (end-use) properties. Third, the use of the degree of 
conversion replaces physical time for the timing of process related 
operations like valve opening and closing, controlling the heat 
supply/removal, dosing the reactants, and so forth. Fourth, the sensor 
allows an accurate prediction of the batch evolution and thus makes it 
possible to accurately predict the cooling need profile from the current 
instant out to the batch termination. 
In this invention, the reaction mixture temperature and the integral heat 
rate are treated as two independent process variables. This approach 
provides the user the freedom to specify batch recipes in a way that 
defines the evolutions of either variable during the batch run, and to 
execute them under tight, high performance control. Because the degree of 
monomer conversion is proportional to the integral heat rate for many 
important polymers including PVC, controlling the two variables gives the 
user independent control over two basic determinants of product quality. 
Even more importantly, such control fully defines the heat release at 
every instant of the batch run, thus making it possible to better utilize 
the available cooling capacity through more reliable planning and 
scheduling. To control the temperature and integral heat rate 
independently, the proposed method manipulates the amount of heat added to 
or taken out of the reaction and the amounts of the initiator(s) and 
inhibitor added during the batch run. 
The present invention improves the yield of a PVC or polymerization 
manufacturing plant in two ways. First, this approach provides better 
feedback control of individual reactors, thus reducing the uncertainties 
of the reaction time and coolant consumption at any given instant. Because 
the benefit of plantwide planning and scheduling is dependent on the 
quality of predictions that were used for the plan and schedule 
development, better reactor control is a technological enabler of better 
planning and scheduling. Specifically, more reliable predictions of the 
coolant consumption allow the planner to run the plant with smaller 
cooling capacity margins without sacrificing the plant safety, thus 
increasing the plant yield. The controller of this invention can 
accelerate or decelerate the reaction without changing the reaction 
mixture temperature. Consequently, it can reduce the reaction time without 
sacrificing the product quality by taking advantage of any available 
cooling capacity. Second, this approach improves run-to-run product 
consistency and allows one to tighten the product specifications, which 
also adds to the yield increase, by reducing the off-specification 
production. 
It is well known that some polymers could be produced by reactions running 
at greater speeds without any significant degradation of their quality, if 
only the reactors used could handle the increased heat flow. A good 
example is the manufacturing of PVC by the suspension process. A PVC plant 
in Canada uses water from a river as a coolant for its reactors. In the 
winter, when the water temperature is about 0.6 degrees Celsius (i.e., 33 
degrees Fahrenheit), a batch takes about 5 hours to complete. However, in 
the summer, when the river water temperature raises to 22 degrees C. (72 
degrees F.), the same batch, with comparable product quality, takes 8 to 9 
hours, because the drop in the available cooling capacity forces the plant 
operator to slow down the reaction rate by using smaller amounts of the 
initiator. 
In the above example, the coolant's supply is unlimited and the restriction 
comes from its increased temperature and limited water circulation flow 
through the reactors' jackets. Another example is a PVC plant that uses 
chilled water as the coolant for its jacketed reactors. The plant has a 
centralized utility which supplies water to a dozen or so reactors. 
Because the chilled water is expensive and its supply is limited, water 
exiting the reactors is partly recycled by mixing it with the freshly 
chilled water coming directly from the cooling towers. This creates a 
variable production environment, wherein the availability of the chilled 
water depends on the number of batches currently in progress as some 
reactors are always being charged or discharged, while others are 
temporarily out of service for cleaning and maintenance. Also, the chilled 
water temperature may fluctuate with the weather and the time of the day. 
Before starting a batch, the operator must make a decision on how fast he 
can afford to run it without risking a temperature runaway and choose the 
appropriate amount of the initiator(s) which is then added to the reactor 
charge. To some degree, this decision is guesswork as the operator has to 
consider the effects of gradual deposit buildup on the reactor walls on 
heat removal. Once the reaction gets going, the operator can, in 
principle, speed it up or slow it down manually by adding the initiator or 
inhibitor, respectively, but this is not normally done. Once started, the 
batch is run open loop without further operator interference until its 
completion, which is indicated by the pressure drop in the reactor. 
Given the uncertainty concerning the cooling capacity that will be actually 
available during the upcoming batch run and the impossibility to exactly 
determine the initiator dosing beforehand and to correct it later, the 
operator has to play it safe and make decisions that on the average might 
be overly conservative. This cuts into the reactor yield. Obviously, a 
better control over the actual rates of individual reactors in the plant 
would make it possible to reduce the current technological margins without 
endangering plant safety and thus create an opportunity to employ tighter 
plantwide optimization. 
If one had better control over the reaction progress, then one could even 
think about more unusual ways to increase the plant yield. Currently, for 
each reaction the operator defines its speed before it begins by dumping a 
particular amount of the initiator(s) into the mixture. But there might be 
a window of opportunity when a large amount of chilled water is available, 
say, for two hours, because a couple of other reactors happened to finish 
simultaneously and have to be discharged and recharged. A batch controller 
that would allow the operator to temporarily accelerate the running 
reactions for the two hours to take advantage of the unexpectedly 
available cooling capacity and then bring them back to the original rate 
by applying a suitable amount of an inhibitor, without disturbing the 
reaction mixture temperature, would further improve the plant yield. 
Making the batch follow a given temperature and conversion rate profiles 
not only improves the quality and run-to-run consistency of its product 
but, perhaps even more importantly, enables one to make accurate 
predictions of the heat release during the batch run. As a result, one can 
make reasonable and justifiable provisions for the expected cooling duty 
needed to keep the mixture at the desired temperature all the way up to 
the reaction end, and thus better utilize available plant resources 
through more reliable plantwide planning and scheduling. 
The present advanced batch control, along with the follow-up plantwide 
optimization it enables, may have a major economic impact on plant 
performance. Consider, for example, a plant with fifteen reactors running 
so that the cooling capacity reserve is 10 percent. Since the cooling 
availability is the limiting factor, reducing the reserve to four percent 
would increase its output by six percent, which is almost tantamount to 
adding another reactor to the plant, without the expense of its 
construction and maintenance.

DESCRIPTION OF THE EMBODIMENT 
FIG. 1 shows a conventional batch controller 8. An advanced, high 
performance controller 10 is shown in FIG. 2. Such controller needs more 
information about the process being controlled. One needs to reconcile the 
nonlinear nature of batches that calls for a one-of-a-kind, specialized 
controller for every polymerization process, with business requirements 
preferring a single controller easily customizable for as large a number 
of processes as possible. It is best to bring a garden variety of 
polymerization processes under a common umbrella. 
It is unlikely that there will ever be specialty sensors for all kinds of 
end-use properties various polymers may have. If there is going to be a 
generic controller for batch polymerization processes, then it will have 
to rely on inferential sensing of the properties instead of direct 
measurement of them, and such an inferential sensor will have to be an 
inseparable part of the controller design. The sensor will infer its 
readings from the measurements of generic physical variables such as 
temperatures, pressures, flows, and so forth, that are easily obtainable 
from commodity sensors. Setting up the sensor will constitute a major part 
of the tuning of the controller for a particular application. 
The present invention is applicable for a large class of polymerization 
processes of practical importance. This approach is based on an 
observation that for many processes, the degree of monomer conversion into 
polymer is proportional to the overall amount of heat released by the 
reaction since its start. Because the speed of conversion, plotted as a 
function of time, has a strong effect on molecular characteristics of the 
resulting polymer chains and, therefore, the polymer end-use properties, 
ensuring repeatable time profiles of the conversion speeds, are one of the 
keys to consistent product quality. One cannot easily measure the degree 
of conversion, but can develop algorithms which permit one to calculate 
estimates of the overall reaction heat, which in turn can be utilized as 
an inferred process variable for feedback control. Unlike the degree of 
conversion, measurement methods needed to monitor the thermal conditions 
of a reaction are independent of a particular polymer being produced, thus 
resulting in a preferred generic approach. 
First, there is a way for inferring the integral reaction heat of a 
polymerization reaction and its use for control. Initially, two uses of 
the inferred integral reaction heat are apparent. First, the integral 
reaction heat is used as another state variable of the process and used to 
improve control of the reaction mixture temperature in the second phase of 
a polymerization process, which is called the temperature tracking mode of 
controller. (The first one is the startup mode, which gives way to the 
temperature tracking mode when the controller switches from heating the 
mixture to cooling it.) FIG. 3 shows a conventional control system 30 
having a reaction mixture temperature feedback link 52. For the sake of 
explanation, the process behavior is characterized by the thermokinetic 
model, which reflects one's understanding of how the process works. The 
sensor's applicability to polymerization process control, as shown in FIG. 
3, is useful for many polymers, but of limited value for others. If the 
reaction mixture temperature T.sub.R is to be held constant throughout the 
batch process, as is the case, for example, in the PVC manufacturing; then 
the temperature control task is rather simple. In other words, for the PVC 
case, after startup there is no demanding temperature profile with steep 
up and down ramps for the control system to track. It is no surprise, 
then, that conventional control systems can maintain the temperature 
within .+-.0.3 degree C. (0.5 degree F.) and PVC manufacturing experts do 
not expect any product quality improvement from a more accurate 
temperature control. 
FIG. 4 shows a system 40 having an improvement. This improvement is a 
feedforward link 53 to a controller 29, which informs the controller about 
anticipated changes of the integral heat rate caused by process 
disturbances before they adversely affect the reaction mixture 
temperature. That information is obtained from a process model 25 which is 
a part of the inferential sensor. Second, the integral reaction heat 
serves as the "inner time" of the reaction, which better reflects its 
actual progress than the ordinary, physical time and is used in this 
capacity to better time operations to be executed during the batch run. 
For example, the timing of stirring or mixing in additives is not derived 
from a clock, but from the integral reaction heat and is thus implicitly 
or impliedly linked to the degree of monomer conversion. 
FIG. 3 shows a feedback 52 execution of a batch recipe in the form of a 
temperature profile which cannot eliminate the impact of process 
disturbances on the integral reaction heat, H(t) and, therefore, product 
quality. In reality, nothing on the process but the reaction mixture 
temperature T.sub.R is accessible for direct measurement. Of particular 
interest are the degree of monomer conversion .delta. and the integral 
reaction heat rate H. 
Block 12 of system 30 is a process model of the heat production in a 
reactor. Input 13 indicates the amount of heat to be removed from the 
reaction in the reactor. Input 14 indicates the amount of heat produced by 
the reaction of the conversion of the monomer to a polymer, or heat rate 
of the reaction (which may be regarded as including disturbances 31). 
Differential amplifier 15 outputs the difference of inputs 13 and 14. This 
output is scaled by a scaling factor (1/c.rho.) 16. c and .rho. may depend 
on H. c and .rho. are heat capacity and specific mass, respectively, of 
the reaction mixture. The scaled output is a rate of reactor temperature 
change dT.sub.R /dt. This rate is processed by integrator 17 resulting in 
an output T.sub.R which is the reaction mixture temperature. T.sub.R is 
input to a heat production rate determiner 18. Block 18 has an integral 
reaction function h which is applied to reactor temperature T.sub.R and 
integral reaction heat H. Also inputs are actual process disturbances 31 
of the reactor which may include the effects of an inhibitor or initiator, 
the condition of the reactor, and other factors. The output of block 18 is 
a rate of heat dH/dt production of the reactor. That output is integrated 
by integrator 19 to provide the integral reaction heat H. That output H is 
fed back into block 18 to be processed as a function of h. Also, H goes to 
a function block 20 that converts or calculates the amount of heat 
determined into the degree of conversion of the monomer into a polymer in 
the reactor, at output 21. 
System 30 controls the reaction in accordance with desired temperature 
T.sub.R desired which is fed along with the actual or calculated integral 
reaction temperature into a summer 22. The output of summer 22 is the 
difference of the actual and desired reactor temperatures which goes to 
controller 23 which determines the amount of heat that should be removed 
which is to effectively control the cooling water or the temperature of 
the reaction. Block 24 covers the feedback temperature control of the 
polymerization process of system 30. 
FIG. 4 reveals system 40 which shows how control of the reaction mixture 
temperature T.sub.R is improved by informing controller 29 about 
anticipated changes of the integral heat rate H. Here feedback temperature 
control is affected by a feedforward signal 53 which is an integral 
reaction heat H indication from integrator 19 of reaction process model 
25. Only the reaction temperature T.sub.R is accessible for measurement. 
In order to estimate H, inferential sensor system 40 involves reaction 
model 25 process which is maintained in sync with the actual 
polymerization process in block 26. Reaction model 25 process is not 
subject to disturbances and nor is the model reaction process temperature 
T.sub.R utilized as a feedback signal for affecting reaction model 25 
process. However, like block 12 of FIG. 3, polymerization process 26 is 
affected by reaction temperature T.sub.R. Feedforward signal 53 indicating 
the amount of reaction heat of model 25 goes to controller 29. Also to 
controller 29 is a difference 48 between the desired reaction temperature 
47 and the actual reaction temperature 52 from summer 22. In the present 
invention of FIG. 5, system 50 has a feedback control of reaction 
temperature T.sub.R 52 and integral reaction heat H 53 (i.e., which 
indicates a degree of conversion). This involves a modification of the 
reaction simulation, that is, polymerization process 26 and reaction 
process model 25, so that they allow for the on-line dosing 32 of 
initiators (and possibly of an inhibitor as well if the user wants or 
needs it) as another manipulated variable in addition to the coolant flow 
to the reactor. This gives the extra degree of freedom needed to control 
simultaneously and independently both the reaction mixture temperature and 
the rate of conversion, which are independent, though coupled state 
variables of the polymerization process. The desired reaction temperature 
T.sub.R desired 47 is input to summer 22 and combined with the reaction 
temperature T.sub.R 52 fed back from the output of polymerization process 
block 26. The output of summer 22 is a difference 48 between the desired 
reaction temperature 47 and actual reaction temperature T.sub.R 52 and it 
goes to controller 35. The desired integral reaction heat production 
H.sub.desired 54 and the actual integral reaction heat production H 53 
from the output of integrator 19 as indicated by the inferential sensing 
of reaction model 25 process, are input to summer 36 which outputs a 
difference 55 of the desired 54 and inferred actual 53 integral reaction 
heat productions H. This difference 55 is input to controller 35. 
Multivariable predictive batch controller 35 controls the two variables to 
their respective set point profiles in a decoupled fashion using the 
inferential sensor 56 like that of system 40 in FIG. 4, to infer the rate 
of heat production dH/dt 14. The two outputs T.sub.c 58 and f.sub.c 61 of 
controller 35 manipulate the amount of heat, q.sub.tot, via line 13, taken 
out of the reaction by the reactor's cooling/heating system, and the 
amount of the initiator/inhibitor, D, via line 32, added into the reactor. 
Controller 35 accounts for different activities of various initiators or 
inhibitors that may be used. Reaction model 25 process is a part of 
inferential sensor 56 of system 50. 
One should note that the above is a simplified description of controller 
35. Its actual implementation involves more but less significant 
variables. For example, q.sub.tot comprises all heat removal-related 
variables of the process that allow for manipulation, including water flow 
through the cooling jacket 59 and coils, and cooling water temperature at 
inlet 38 and outlet 40, and so forth of the reactor 60 (FIG. 6). 
Minimum reactor 60 instrumentation is needed. One turns the thermokinetic 
equations of a polymerization reaction process 26 into a form of model 25 
that is suitable for real time control. Both the form and reactor model 49 
involve parameters that need to be estimated on the actual process 26, 
some of them off-line, some on-line. In order to do so, one collects data 
on certain process variables. The minimum set of the measured variables is 
schematically shown in FIG. 6. They are: T.sub.R --reaction mixture 
temperature from temperature sensor 27 in reactor 60; p--reaction mixture 
pressure from pressure sensor 28 in reactor 60; T.sub.CI --input cooling 
water temperature from temperature sensor 37 in water input pipe 38 to 
reactor 60; T.sub.CO --output cooling water temperature 58 from 
temperature sensor 39 in water output pipe 40 from reactor 60; f.sub.C 
--cooling water flow rate 61 from flow sensor 41 in pipe 38; P.sub.a 
--agitation power required to rotate agitator or mixing blade 42 in 
mixture 43 in reactor 60 at a rate of n revolutions per minute, from 
sensor 44. Valve 45 provides for the entry of cooling water and the exit 
of the water from jacket 59 of reactor 60. Pump 46 ensures the flow of the 
cooling water to and from reactor 60 jacket 59. These measured variables 
are input to reactor model 49. Induction period length T.sub.I is input to 
reactor model 49. T.sub.I is the period of time or delay before the 
polymerization reaction starts after the initiator is added to the batch 
mixture. An indication of the amount of heat removed from the reaction, 
q.sub.tot, is input to block 25 which processes the thermokinetic 
equations. Also input to block 25 is T.sub.R0 which is the initial 
reaction temperature. H, the indicated integral reaction heat 53, is 
output from block 25. The degree of monomer conversion .delta. 21 of the 
batch mixture is inferred from integral reaction heat H 53. 
For the reactor model 49 development, it is recommended that one have the 
reactor blueprint (or at least a sketch of its physical arrangement and 
dimensions) as well as the placement and quality of the sensors. 
The heat that polymerization reactions produce must be removed from a batch 
reactor by its cooling system. If a reaction is not to end up in a 
temperature runaway, its heat production rate must be matched by the rate 
at which the cooling system removes the heat from a reactor. Because the 
heat production rate is proportional to the rate at which monomer is 
converted into polymer, it is the cooling system capacity that ultimately 
limits the yield of the reactor. The optimal utilization of the available 
cooling capacity of a plant thus becomes a determinant of its product 
yield. 
The direct benefit of using the controller over existing solutions is to 
get a batch under feedback control both with respect to its reaction speed 
and mixture temperature and thus tighten the manufacturing specifications 
which are being constantly threatened by unpredictable variations in the 
initiator activity and feed impurities. The recipes to be used with the 
proposed controller are more accurate, because in addition to the 
temperature profile they also specify the desired degree of monomer 
conversion profile over the batch run. Since for a class of polymers 
specified in the approach in FIG. 4 the degree of monomer conversion 
.delta. is proportional to the integral reaction heat H, the batch recipes 
can be defined directly in terms of H 53, and not of .delta., as a 
function of time, in view of the approach in FIG. 5. This simplifies the 
controller setup because there is no need to develop the nonlinear 
transformation block f(H) 20 shown in FIG. 3. 
Below are stated the thermokinetic equations (4) and (5), modeled by block 
25, of a polymerization reaction. Polymerization reactions are exothermic. 
The amount of heat being released at any given time, however, is not 
constant throughout the reaction but varies in time as a function of the 
reaction mixture temperature and the degrees of conversion of individual 
monomers involved in the reaction. Let one denote the amount of heat 
produced by one cubic meter of the reaction mixture per second as 
EQU h(T.sub.R (t), .delta..sub.1 (t) . . . , .delta..sub.N (t))(1) 
where .delta..sub.1 (t), . . . , .delta..sub.N (t) are degrees of 
conversion of various monomers at the time t, and T.sub.R (t) is the 
mixture temperature, and call it the heat production rate. Its dimension 
is [J/m.sup.3.s]. 
The overall amount of heat, H(t), produced since the reaction was started 
at t=0 is obtained by integrating the differential equation 
##EQU1## 
and is called the integral reaction heat. Its dimension is [J/m.sup.3 ]. 
There is a large class of practically important polymerization reactions, 
which involve either only one monomer or multiple monomers reacting at the 
same kinetic rates. Or, to define the class in the most general terms, one 
can say that its members are distinguished by having a single heat 
producing reaction, which may be chaining up one or more monomers (as is 
the case, for example, in the production of nylon). For such reactions, 
their single degree of conversion, .delta.(t), is in a one-to-one 
relationship, 
EQU .delta.(t)=f(H(t)) (3) 
with the integral reaction heat, H(t), and the equation (2) can be written 
as follows: 
##EQU2## 
The function h(T.sub.R (t),H(t)) is called the heat rate and is specific 
to a particular process. The released heat, H(t), along with the heat 
removed from the reaction mixture 43, q.sub.tot (t), defines the mixture 
temperature, T.sub.R (t): 
##EQU3## 
Here c(t) and .rho.(t) are the heat capacity and specific mass of the 
mixture, respectively. Both of them generally change with time as the 
reaction progresses. Their dimensions are [J/kg..degree.K] and [kg/m.sup.3 
], respectively. 
q.sub.tot (t) is the removed heat flow, that is, the total amount of heat 
removed from 1 m.sup.3 of mixture 43 per second from external sources by 
all means, i.e., through cooling, heat losses, and so forth. If heat is 
added, for example through agitation by blade 42, q.sub.tot (t) might 
become negative. Its dimension is [J/m.sup.3.s]. 
The equations (4), (5) are referred to as the thermokinetic equations of a 
given polymerization reaction. FIG. 7, in the box named "reaction model" 
25 shows a schematic of the application of these equations and their 
dynamics. It shows their internal structure and mutual couplings of the 
process variables. Also shown in FIG. 7 are the equations characterizing 
the heat removal from reactor 60, which are explained below. 
For the thermokinetic equations to be valid in the above form, reaction 
mixture 43 must be perfectly mixed so that its temperature, T.sub.R (t), 
is uniform throughout the batch reactor volume. For some processes this 
assumption may be difficult to uphold, particularly toward the end of 
reaction, when the mixture might gel or even glassify. In such cases, the 
lumped parameter model of equations (4), (5) must be replaced by a 
distributed one. For PVC made using the suspension process, a huge water 
volume present in reactor 60 keeps the mixture viscosity low and makes 
good mixing possible, thus justifying the use of the lumped model. It also 
largely suppresses the effects of progressing polymerization on the 
specific heat and weight (i.e., specific mass) of mixture 43. Therefore, 
one assumes c(t) and .rho.(t) to be constant throughout the batch. 
Furthermore, to simplify the notation, one can drop the time t from 
certain symbols in the equations and figures. 
Similar assumptions are applicable for many other polymerization processes 
as well. If some of them are not, then it is possible to modify the 
mathematical form of the exothermic equations to satisfy the 
technicalities, while retaining the idea of using them. 
In order for the thermokinetic equations to be applicable for inferential 
sensing and control, one has to determine the heat rate h(T.sub.R 
(t),H(t)) and the total amount of heat q.sub.tot removed from the process. 
Let one postpone the former until later and outline now how to obtain the 
latter. 
Heat is usually removed in several ways simultaneously, the major one being 
through the cooling system. Others are heat losses to the ambient due to 
lack of reactor insulation, heat escaping in gases intentionally released 
to maintain reactor pressure, heat absorbed by added reactants to bring 
them to the mixture temperature, etc. There might also be heat flow 
entering the reaction, however, most notably through the energy needed to 
agitate mixture 43. Let one consider the total heat to be a sum of three 
major components, 
EQU q.sub.tot =q.sub.C +q.sub.A +q.sub.M, 
where 
q.sub.C is heat 62 removed through the reactor's cooling system, 
q.sub.A is heat 63 escaping into the ambient through the reactor insulation 
(if there is any at all), and 
q.sub.M is a "catch-all" term introduced to account for miscellaneous other 
heat escape routes, termed as miscellaneous heat 64. Depending on 
particular process, some of them may be significant enough to deserve to 
be explicitly modeled. The sum of q.sub.C, q.sub.A and q.sub.M is achieved 
by summing amplifier 68 as an ouput q.sub.tot signal 13 to amplifier 15 of 
reaction model 25. Heat flux 62 crossing the reactor wall from the mixture 
into the cooling medium is 
##EQU4## 
where R.sub.C (H,f.sub.c) is the heat transfer coefficient 65 from mixture 
43 into the coolant, which generally depends on the mixture viscosity 
which, in turn, is related to the degree of conversion and thus to the 
integral heat H as well, and also on the coolant flow f.sub.c 61, 
T.sub.R is the reaction mixture temperature 52, 
T.sub.C is the coolant temperature 58, 
a.sub.C is the area of the wall across which the transfer happens, and 
V is the mixture 43 volume. 
Although the above expression is correct, it cannot be used in this simple 
way to actually calculate q.sub.C because the coolant temperature T.sub.C 
in the jacket is not uniform. Nevertheless, in conjunction with the 
thermokinetic equations it conceptually explains the controller 35 affects 
the polymerization by adjusting the coolant temperature 58. Because the 
actual coolant temperature raises as the coolant flows through the jacket, 
the average temperature can also be decreased by increasing the coolant 
flow f.sub.c 61 and vice versa. 
A similar formula holds for the heat flux 63 escaping into the ambient: 
##EQU5## 
R.sub.A (H,f.sub.c) is the heat transfer function coefficient 66 from 
mixture 43 into the ambient. Of course, there is no way one can manipulate 
the ambient temperature T.sub.A, but its measurement is used as a 
feedforward signal 67 to better estimate this heat loss 63. For highly 
viscous mixtures 43, power 44 needed to agitate them may be large enough 
to demand its explicit inclusion as a heat contributor. 
Reaction model 25 process is synchronized with actual polymerization 
process 26 by model state coordinator 69. Synchronization or coordination 
is cued from a comparison signal 70 from differential summer 72 having as 
inputs reaction temperatures T.sub.R 52 from polymerization process 26 and 
reaction model 25, by coordinator 69 which outputs a synchronizing or 
coordination signal 71 to amplifier 15 of reaction model 25 to maintain 
the state of model 25 to be the same as that of process 26. 
Coolant temperature T.sub.C signal 58 and reaction temperature T.sub.R 
signal 52 are summed by amplifier 73. The output of amplifier 73 is 
multiplied by coefficient a/V 74 where a is the surface area of reactor 60 
where heat exchange is taking place. The output from coeficient block 74, 
along with f.sub.c, signal 61 and H signal 53 goes to heat transfer 
coefficient block 65 which outputs a q.sub.c signal 62. 
Ambient temperature T.sub.A signal 67 and reaction temperature T.sub.R 
signal 53 are summed by amplifier 75. The output of amplifier 75 is 
multiplied by coefficient a/V 76 where a is the surface area of reactor 60 
where heat exchange is taking place. The output from coeficient block 76, 
along with f.sub.c signal 61 and H signal 53, goes to heat transfer 
coefficient block 66 which outputs a q.sub.A signal 63. Amplifier 68 
receives q.sub.c signal 62, q.sub.A signal 63 and miscellaneous heat 
q.sub.M signal 64 and sums them into an ouput q.sub.tot signal 13 fed to 
amplifier 15. 
Conventional batch controls cannot guarantee product consistency. Assume 
that a recipe specifies a desired batch temperature profile, 
T.sub.Rdesired (t), 47, and a conventional temperature control system 
ensures that this profile is perfectly followed by actual reactor 
temperature T.sub.R 52, i.e., 
EQU T.sub.R (t)=T.sub.Rdesired (t) 
Now consider two batches in which different amounts of the initiator 
(catalyst), as prescribed by signal 32, were used. Different initiator 
concentration causes the two reactions run at different speeds and, 
consequently, their heat production rates as well as their integral heats 
will differ: 
EQU h.sup.(1) (T.sub.R,H.sup.(1)).noteq.h.sup.(2) (T.sub.R,H.sup.(2)) 
Their temperature profiles T.sub.R (t), however, can still be maintained 
perfectly identical by the temperature control system, which properly 
manipulates the amounts of heat, q.sup.(1).sub.tot and q.sup.(2).sub.tot, 
taken out of the mixture by the reactor cooling so that the equations (5) 
corresponding to each batch hold despite variations in H.sup.(1) (t), 
H.sup.(2) (t): 
##EQU6## 
Obviously, unless the control system keeps track of how much heat was added 
to or removed from the mixture as a result of its control actions, then 
one may not even find out that the two seemingly identical batches 
actually ran at different conversion speeds. 
The present contention is that feedback batch temperature control still 
runs the batch open loop with respect to the integral heat rate 14, H(t), 
and also as to the degree of conversion 21, .delta.(t), as revealed in 
FIG. 3. Process disturbances 31 affecting the heat rate, h(T.sub.R,H), 
which, in turn, also affect the mixture temperature 52, are seen by 
controller 23 as external disturbances 31 and their thermal effects are 
rejected, while they are free to let H(t) 53 and, consequently, .delta. 21 
fluctuate without any correction. Feedback execution of a batch recipe in 
the form of a temperature profile cannot eliminate the impact of process 
disturbances on H(t) and, therefore, product quality. 
A concept of advanced polymerization control has a main idea which is to 
close the loop using integral reaction heat 53, H(t), output of integrator 
19, as a substitute for the desired degree of conversion, .delta.(t), 
which would be extremely difficult to measure directly and would require 
specialty sensors for different polymers. The integral reaction heat H(t) 
is directly related to product quality. Unless process disturbances 31 are 
entering the nonlinear mapping f(H) 20, this simplification permits 
skipping modeling without degrading the control performance. For instance, 
reaction mixture 43 agitation so vigorous that it impacts the distribution 
of polymer chains lengths might be an example of such a disturbance. In 
contrast, FIG. 3 shows temperature T.sub.R feedback 52 rather than the 
heat H feedback 53. 
Usual process disturbances, be they fluctuations of the initiator activity 
or monomer concentration, effects of impurities and others all show up as 
changes of the heat rate h(T.sub.R,H). Because the heat capacity and 
weight of the mixture are either known or can be easily determined, 
h(T.sub.R,H) is the only missing piece of knowledge. This function of two 
variables represents a mathematical encoding of chemical facts about the 
given reaction and therefore will be different for different processes. 
Some of the facts represent the knowledge that can be extracted beforehand 
(i.e., off-line) from books or historical data collected on the process. 
Some other facts, such as initiator activity, depend on so many 
unmeasurable contributions that the only way to find out is to either 
measure or estimate it on-line from data collected while the batch to be 
controlled is already in progress. 
To estimate with a reasonable accuracy a function of two variables with no 
prior information about a process functional form is a formidable task 
whose completion would require large amounts of data, more than can ever 
be collected from a single batch run. On the other hand, because a model 
is needed for controlling the very batch on which data is being collected, 
one has to develop h(T.sub.R,H) very early into the batch, while it still 
is in its initial phase, without the luxury of having data from a complete 
run. This presents a dilemma. On one hand, one knows that h(T.sub.R,H) 
will always contain something that is specific for each batch run and thus 
implying that h(T.sub.R,H) be estimated anew in every run. On the other 
hand, it is clear that data collected at the beginning of a single run 
would never suffice to compute a good functional estimate. 
The answer to resolving this dilemma lies in factoring h(T.sub.R,H) into 
two components. One component will contain only knowledge that is 
permanent, independent of a particular run. This component can be 
established beforehand, using both chemical theories and historical data. 
Because its computation is done off-line, one is free to use as much data 
as one pleases and employ sophisticated, but time-consuming algorithms to 
do the job. The other component will contain only knowledge which is valid 
only in a particular run and thus needs to be extracted on-line. Because 
the amount of available run time data is small, the amount of this 
knowledge must be kept to an absolute minimum. One can safely say that the 
ability to do the h(T.sub.R,H) factorization correctly is the key to 
success of the proposed approach. This is the place where expertise in 
polymer chemistry is necessary in order to come up with factorizations 
that stand a chance both to reflect the chemical reality and to be fitted 
using limited amounts of data. 
The heat production rate h(T.sub.R,H) is factored. For the sake of 
explanation, let one think of the estimation of h(T.sub.R,H) as a two step 
problem. The first step is to estimate the integral reaction heat flow, 
dH(t)/dt, knowing the removed heat flow, q(t), and the mixture 
temperature, T.sub.R (t). The specific heat, c, and weight, .rho., of the 
mixture are assumed to be known. 
This is a trivial problem once one knows q(t). Because heat is an integral 
quality, it cannot be directly observed and must be computed from 
temperature and flow measurements. The computation of q(t) for a given 
reactor is the real challenge in this step. This computation is discussed 
below. 
The second step is to estimate h(T.sub.R,H), knowing the integral reaction 
heat flow, dH(t)/dt, from the first step and the mixture temperature, 
T.sub.R (t). 
The formidable task of function estimation mentioned above, which, in the 
present application, cannot be solved without a suitable factorization. 
Here one proposes to factorize h(T.sub.R,H) as a product of two functions 
of a single variable, s.sub.T (T.sub.R) and h.sub.H (H), and a scaling 
constant s.sub.0, 
EQU h(T.sub.R,H)=s.sub.0 .multidot.s.sub.T (T.sub.R).multidot.h.sub.H (H)(6) 
and proposes s.sub.T (T.sub.R) to be of this particular form 
##EQU7## 
which can be justified by assuming the reaction rate to accelerate with 
reaction temperature according to Arrhenius. Here 
E is the activation energy of the polymerization reaction, 
R.sub.G is the universal gas constant, 
T.sub.R (t) is the actual temperature profile of the reaction mixture, 
T.sub.0 is the constant reference temperature, 
h.sub.H (H) is the nominal heat production rate observed at the reference 
temperature, 
s.sub.T (T.sub.R) is the temperature factor due to temperature differences, 
s.sub.0 is a "catch-all" scaling factor called the reactivity to account 
for other variations. 
The value of equations (6) and (7) is in allowing one to transform data 
collected during batches, whose temperature profiles differ from the 
nominal one, as if they were collected under the nominal conditions. Once 
one knows how to remove the effects of temperature variations, data from 
many batches to estimate the nominal heat production rate, h.sub.H (H), 
can be used. However, before one can remove such effects, knowledge of how 
to eliminate the effects of initiator activity is needed. 
First, accounting of the effects of initiator activity is needed. Many 
polymerization reactions will not run at all or only at very low 
conversion speeds unless an initiator (catalyst) is used. Because the 
initiator activity defines the reaction speed, its effects must be somehow 
encoded in h.sub.H (H). But the initiator activity is a very volatile 
aspect which may change from run to run and cause large fluctuations in 
the batch behavior. Unless one explicitly removes the initiator activity 
effects out of the historical data and thus, in effect, renormalize the 
data to a "standard initiator activity profile", the resulting estimate of 
h.sub.H (H) might easily be a flat average which is not particularly good 
for any actual batch. Also, if the estimate of h.sub.H (H) is to be of any 
use for control, one must know how to customize it for every future batch 
using its actual initiator activity profile. 
Before incorporating the initiator effects into h.sub.H (H), a few facts 
about what happens to the initiator in the course of a polymerization 
reaction need to be known. Although polymerization reactions do not 
consume the initiator, its concentration nevertheless keeps falling in the 
course of a reaction mainly due to two effects. As to the first effect, 
immediately after adding the initiator to the mixture, the polymerization 
reaction does not take off immediately. It appears that the initiator's 
activity is temporarily hindered by impurities present in the input 
stocks. Among chemical engineers, this delay (denote it T.sub.I) is known 
as the induction period. Because the amount of impurities in a batch is 
virtually impossible to determine beforehand, both the induction period 
and the actual initiator concentration at the true beginning of the batch 
reaction are unknowns that must be estimated on-line. 
As to the second effect, many initiators are high energy compounds and as 
such tend to be chemically unstable. During a batch run, the initiator 
typically undergoes a spontaneous disintegration which exponentially (or 
even faster) lowers its concentration in the mixture. Also, long storage 
of the initiator compound tends to degrade it and unless the initiator 
activity is measured before its use, the degradation will introduce an 
unmeasured disturbance into the batch process. 
To account for those phenomena, one assumes the following three points. 
First, the nominal heat production rate is proportional to the initiator 
concentration, [I]: 
EQU h.sub.H (H)=[I].multidot.h.sub.I (H) (8) 
where h.sub.I (H) is also called the nominal heat production rate but this 
time normalized with respect to both the reaction temperature and the 
initiator concentration. 
Second, the polymerization reaction starts only after the induction period. 
From the modeling point of view, the period is a pure delay. 
The initiator's spontaneous disintegration proceeds as the first order 
reaction 
##EQU8## 
If the initiator is added in the course of a reaction, the equation needs 
be amended by an appropriate driving force term. 
As factorization (8) implies, in order to develop the estimate h.sub.H (H), 
one has to know both the solution, [I] (t), of equation (9) and the 
estimate h.sub.I (H). Whereas the latter is always estimated off-line and 
thus one's choice of a suitable computational method is constrained 
neither by data availability nor real-time concerns; this is not true for 
[I] (t). As discussed above, the initiator concentration can vary from 
batch to batch and thus must be estimated on-line from scarce real time 
data collected early into the batch run. This severely limits the present 
options as to how to proceed. 
Equation (9) is linear but with a time-varying coefficient, because the 
kinetic rate, k.sub.I (T.sub.R (t)), depends on the reaction temperature 
T.sub.R (t). Its solution 
##EQU9## 
shows that disintegration is exponential in time and generally accelerates 
with growing temperature. To fully determine the solution, one has to 
estimate the initial concentration [I](T.sub.I) at the end of the 
induction period (i.e., at the beginning of the polymerization reaction), 
the induction period length T.sub.I and the variable kinetic rate k.sub.I 
(T.sub.R). The first two are scalar numerical values, which are easy to 
estimate, the third one is a function of a single variable T.sub.R. 
If nothing is known about a function being estimated, then one has to 
construct its estimate by a nonparametric estimation method. However, to 
produce reliable results such methods generally require so much data that 
one can rule them out in these circumstances. The only sensible approach 
seems to be to replace the nonparametric function estimation by a 
parametric one by inserting a piece of chemistry knowledge regarding the 
likely form of the function k.sub.I (T.sub.R). Because any estimation 
based on a limited amount of data is always a tricky business, every piece 
of knowledge one can get about the resulting function up front without 
having to estimate it goes a long way in stretching the value of the 
available data because it greatly reduces the amount of knowledge that 
needs to be correctly extracted. Below is a possibility for which one can 
put forth some chemical justification: 
Assume the disintegration kinetics to be of the Arrhenius type. Then, one 
can express k.sub.I as 
##EQU10## 
where k.sub.I0 is the kinetic rate constant at the nominal temperature 
T.sub.0, and 
E.sub.I is the activation energy. 
Since E.sub.I can be assumed to be known, the original problem of 
estimating the function k.sub.I (T.sub.R) boils down to the estimation of 
the single numerical constant k.sub.I0. 
The initiator concentration drops over time since the moment the initiator 
was added into the reaction mixture at the time 0. Although the 
concentration [I] (0) at this moment can perhaps be calculated from the 
input stock amounts and then projected into the future value [I](T.sub.I) 
using the equation (10), the actual amount of the initiator at the time 
T.sub.I can be a way off from such projection due to other unmodeled 
effects taking place during the induction period. These effects can be 
brought into consideration by expressing the actual value [I](T.sub.I) as 
a function of the value [I] (0) as follows: 
##EQU11## 
The scaling constant s represents an uncertainty due to the unmodeled 
effects and as another multiplicative constant might be eventually 
included in the reactivity s.sub.0 introduced earlier in (6). 
The nominal specific reaction heat h.sub.I (H) can be estimated. Finally, 
one is in a position to devise a complete approach to estimate the nominal 
heat production rate h.sub.H (H). Recall that one began by explicitly 
introducing the effects of the initiator concentration [I] in the form 
EQU h.sub.H (H)=[I].multidot.h.sub.I (H) 
followed by assumptions concerning the initiator disintegration and the 
delayed onset of the polymerization reaction, 
##EQU12## 
in which one made yet another assumption to convert the estimation from 
functional to parametric to make the problem solvable using small data 
sets: 
##EQU13## 
The proposed factorization of the heat production rate outlined above 
should be thought of as a working hypothesis that may need amendments to 
account for specifics of any particular polymerization process. Hence its 
validation is the first step to take when applying the approach to a new 
process. 
Now one will consider two estimation problems whose objective is to obtain 
the factors from experimental data. In both, the function h.sub.H (H) is 
known from the first step (see the discussion above on factoring the heat 
production rate), at least for H(t) for t ranging from t=0 up to the 
current time t or, in the case of archived data, till the end of the batch 
run, T.sub.B. Also, one assumes that the temperature profiles T.sub.R (t) 
and T.sub.0 (t) are available, and the activation constant E.sub.I is 
known. (If not, then it can be either established independently or 
estimated like any other parameter.) 
The first problem is estimating h.sub.I (H) from archived data. This is an 
off-line procedure. From a set of N batch records, one first extracts the 
nominal heat production rates h.sub.H (H).sup.(n), n=1, . . . , N. 
Although the rates all are outcomes of what should be the same 
polymerization reaction, they will generally differ as a result of 
different initiator concentration drop rates and other disturbances. 
Whereas the same desired heat h.sub.I (H) must be universally valid for 
all batch runs, the constants T.sub.I.sup.(n), [I.sub.0 ].sup.(n), 
k.sub.I0.sup.(n) are not. One highlights the fact by marking them with the 
superscript denoting their batch number. Even though the constants are of 
no use, one has to estimate them for each batch in order to get h.sub.I 
(H) right and then one simply tosses them. 
A possible formulation of the problem is as follows. Let the h.sub.I (H) 
estimate be expressed in the finite series form 
##EQU14## 
where .PHI..sub.m (H), m=1, . . . , M is a suitable set of basis functions 
and w.sub.0, . . . , w.sub.M are unknown weights. 
Because the shape of h.sub.I (H) is basically given by the order of the 
reaction kinetics, the basis set needs to be defined so that its members 
reflect the reaction order as well as effects not covered by kinetics 
(e.g., gelation). 
Find 
EQU W.sub.0, . . . , W.sub.M, T.sub.I.sup.(1),[I.sub.0 
].sup.(1),k.sub.I0.sup.(1), . . . , T.sub.I.sup.(N), [I.sub.0 
].sup.(N),k.sub.I0.sup.(N) 
such that they minimize the fitness criterion 
##EQU15## 
Aside from its apparent complexity, this is conceptually a plain nonlinear 
problem. 
The second problem is estimating T.sub.I,[I.sub.0 ],k.sub.I0 from real time 
data. This is an on-line procedure used during real time batch control. 
Its objective is to find the likely values of T.sub.I,[I.sub.0 ],k.sub.I0, 
which characterize the initiator activity in the current batch. Now the 
heat h.sub.I (H) is already known. A possible mathematical formulation of 
the problem is to find T.sub.I,[I.sub.0 ],k.sub.I0 such that they minimize 
the fitness criterion is 
##EQU16## 
Notice that now the integral runs only up to the current time t. Ideally, 
the minimum will be recomputed over and over as the batch progresses thus 
allowing for continuous updates of the initiator activity model. Another 
approach can incorporate the nonlinear extended Kalman filter theory. 
One may develop algorithms to solve the above problems. The removed heat 
flow q is calculated in the first step on factoring the heat production 
rate. Once one knows it, the estimation of the heat flow dH/dt is simple. 
Now one can focus attention on how to calculate qt. 
There are several items to note. First, heat is an integral property and 
thus has to be calculated from instantaneous measurements like those of 
temperatures, pressures, flows and so forth. Second, heat generated by the 
polymerization reaction can be stored in the reactor in a number of ways. 
In other words, the heat that one is measuring as exiting the system 
through the cooling medium or thermal losses may be only a fraction of the 
heat that is actually being generated at any given time. To produce a 
realistic picture of all heat flows requires a meticulous analysis of the 
production equipment, especially for continuous processes. Third, when it 
comes to heat transfer, reactors are not uniform. Some parts of reactor 
walls or coolers do a better job than others and, as a result, one might 
be forced to describe the reactors by finite element models, even though 
the simpler, lumped parameter models are sufficient for a number of 
processes. The models involve heat transfer coefficients that must be 
experimentally determined to make sure that they agree with the physical 
reality. 
The reactor provides an environment in which the polymerization reaction is 
executed. It interfaces with the reaction through heat transfer, thus 
changing the reaction mixture temperature T.sub.R and, consequently, the 
speed of monomer conversion. However, the interaction works in the other 
direction as well. Because the heat transfer coefficient at the boundary 
layer between the mixture and reactor walls, R.sub.RW (H), depends on the 
mixture viscosity, the amount of heat removed from the reaction will 
depend on the current degree of conversion and thus on the integral 
reaction heat H, all other things being equal. On the outer side of the 
wall, the heat is removed by the coolant flowing through the reactor 
jacket or cooling coils. Because the heat transfer coefficient between the 
wall and the coolant, R.sub.WC (f.sub.C), changes with the coolant flow, 
f.sub.C, which generally is not constant, one must also know the 
relationship R.sub.WC (f.sub.C) for a range of flow values. 
The heat transfer coefficient R.sub.WC (f.sub.C) (or, rather, heat transfer 
coefficient function) is not related to a particular polymerization 
process. Similarly, if the mixture-to-wall coefficient is expressed as a 
function of viscosity, R.sub.RW (viscosity), instead of the integral 
reaction heat, R.sub.RW (H), then it would be largely independent of the 
process, too. It may happen that both coefficients have already been 
established in process-independent forms by the reactor manufacturer and 
are provided in the reactor documentation, or they have been measured 
earlier as a part of some other project. If this is not the case, then the 
coefficients must be determined experimentally. Of course, if R.sub.RW 
(viscosity) is available, then one still has to determine experimentally 
the relation between the integral reaction heat and viscosity for the 
given process: 
EQU viscosity=v(H) (17) 
Sometimes the reactor is cleaned up periodically but not after every batch 
and a gradual buildup of deposits on the reactor walls may occur. The 
deposit layer impacts the coefficient R.sub.RW (H) which then requires 
on-line corrections. 
When the reaction is in progress, an algorithm involving the concepts 
described above, but extended to handle both the thermokinetic and reactor 
model equations as well as the state update in addition to the mere 
parameter estimation, will keep the models in agreement with the real 
process. Because of their continuous update, the models are far more 
accurate than any off-line model established once the system is 
commissioned. Such models not only improve the performance of model-based 
process controllers, but have applications beyond the classic control 
theory field. In particular, the controller can share models with the 
production planning and scheduling packages. This vertical integration is 
perhaps the most economically attractive feature of this invention. 
In the PVC manufacturing (and possibly other kinds of manufacturing), a 
comparable product quality can be achieved at different conversion rates 
as long as the mixture temperature is kept constant. This makes it 
possible to accelerate or decelerate the reaction, with the only 
restriction being the ability to remove the released heat. The cooling 
capacity, though, is always limited. In principle, the scheduler either 
knows or can decide how much cooling capacity can be allocated to each 
reactor at any particular time a few hours ahead. An optimizer then can 
custom-design the integral heat (or degree of conversion) recipe for the 
reactor so that while tracking it, the controller will make its cooling 
duty output match the allocated cooling capacity and dose the 
initiator/inhibitor so as to maintain the mixture temperature constant. 
This will guarantee the fastest reaction for the desired temperature, and 
thus will maximize the reactor throughput. At the same time, better models 
will make planning and optimization more reliable, thus reducing the need 
for large safety margins without endangering the plant safety.