Gas phase polymerization reaction temperature is a critical process parameter that influences both catalyst productivity and resin properties, and reactors often run near the limits as set out by resin melting temperature for economical and product specification reasons.
For consistency and safety reasons, the temperature controllers are typically tuned at steady state conditions, leading to a set of tuning parameters that work well for steady state, but may fail under disturbances such as production rate changes.
For PID based controllers, the typical failure pattern is one where the proportional gain is too small.
Case Study
In the gas phase polymerization process, polyethylene or polypropylene is produced in a fluidized bed reactor operating at near steady state for extended periods.
Reactor temperature directly impacts polymer molecular weight distribution, melt flow rate (MFR), and catalyst activity. With the polymerization reaction being highly exothermic, increasing catalyst activity requires a robust temperature control system to offset the additional heat generated in the reactor.
Control of exothermic reaction vessels is known to be highly nonlinear, as can be attributed to Arrhenius kinetics where the reaction rate constant tends to grow exponentially with temperature.
where k is the reaction rate constant, k_0 is the pre-exponential factor, E_a is the activation energy, R is the universal gas constant, and T_R is the reaction temperature.
As temperature increases, as it might during a production rate increase, reaction rates grow exponentially, causing temperature to grow even quicker. Such systems can very suddenly and unexpectedly experience a reaction rate excursion, raising temperature to unsafe levels.
For the reasons listed above, reaction temperature control must be robust to rein in any possible rate excursions. Often, the manipulated variable is the temperature of the utility stream of the cycle gas heat exchanger.
In this case study, we have identified a set of K_p, T_I, and T_D that does great at setpoint tracking and have stress tested expected temperature moves with great success, as shown below.
The Problem
For the gas phase polymerization reactor, temperature setpoint changes are rare. Disturbances, however, are frequent. Typical disturbances affecting reactor temperature include:
1. Feed composition changes
- Ethylene purity fluctuations
- Hydrogen or comonomer concentration shifts
2. Catalyst feed and productivity variations
- Step changes in catalyst feed rate
- Step changes in catalyst productivity
3. Throughput or production-rate changes
- Increases in production rate directly raise reaction heat load
These are classic load disturbances on the temperature control loop.
Near-miss example
Using the same set of PID tuning parameters from before, the simulated reactor went through a series of throughput increases, as shown in Figure 2.
Catalyst feed was raised to 8 kg/hr at hour 51. An hour later, at around hour 52, the reactor temperature began oscillating in an unstable fashion, which would have eventually led to reactor shutdown/kill to prevent melting the entire bed.
Seeing the temperature oscillation, we decisively raised the controller gain by 20%, which temporarily reduced the oscillation.
We then tried to tune the integral and derivative time constants, which did not see significant results.
Finally, raising the controller proportional gain by another 20% stabilized the control loop.
Technical Analysis of Stability Criteria
Why did the tuning that worked well for temperature setpoint tracking almost cause a temperature runaway event when raising production rate? We will approach this problem using a simplified model for the gas phase polymerization reaction loop.
Simplified Model
The reactor loop can be thought of as three coupled effects:
1. The reactor generates heat from polymerization.
2. The cycle gas cooler changes the recycled gas temperature before it returns to the reactor.
3. The cooled recycled gas removes heat from the reactor.
Reactor Model
For a small deviation around a steady operating point s, write the reactor temperature deviation as
where T_R is the actual reactor temperature and T_{Rs} is the original steady-state reactor temperature.
Similarly, write the reactor inlet, the recycled gas, temperature deviation as
The simplified deviation form reactor energy balance is then
where:
- C is the effective thermal capacitance of the reactor. A larger C means the reactor temperature moves more slowly. This term accounts for the large resin inventory present in the reactor which can "soak up" a lot of heat.
- \sigma T is the extra reaction heat caused by the reactor being hotter than the original operating point. This term accounts for the nonlinear effects from the Arrhenius kinetics.
- q_d = \Delta H_p \Delta \dot m_p is the heat disturbance caused by a production-rate change, assuming temperature effects do not come into play. This term is the linear baseline without the Arrhenius kinetics.
- F is the recycle gas heat capacity flow. That is, F = \dot m c_p, where \dot m is the mass flow rate of the recycle gas, and c_p is the heat capacity of the recycle gas.
- F(T - T_{ret}) is the heat removed by the recycle gas.
The most important term is \sigma, which determines how much additional reaction heat is generated at the same operating condition if reactor temperature increases by 1 K. For example, if \sigma = 2\ \text{MW/K}, then a 1 K temperature rise creates an additional 2 MW of reaction heat due to additional production rate.
\sigma is derived by taking the partial derivative of the heat released from the reaction with respect to temperature:
where:
- Q_{rxn,s} is the steady reaction heat release.
- T_s is the absolute reactor temperature at the linearization point/steady state.
As Q_{rxn,s} increases with production rate, \sigma also increases. That is, the reactor becomes more thermally sensitive at high rates.
Recycle Gas Cooler Model
The cooler/recycle side is approximated as a first-order lag, still in deviation form:
where:
- \tau_c is the effective cooler response time.
- \varepsilon is the cooler effectiveness.
- \varepsilon = 0 physically means the cooler provides no cooling, as can be seen by the steady state solution of T_{ret,s} = T, where u has no effect.
- \varepsilon = 1 represents a perfect heat exchanger, as can be seen by the steady state solution of T_{ret,s} = -u, where the cooler process side inlet temperature T has no effect.
- u is the cooling controller output, which can be thought of as the negative of the utility side inlet temperature.
The controller is written in the following PI form:
with
where T_{sp} is the reactor temperature setpoint deviation.
Setpoint Tracking Is Ok
For a temperature setpoint increase, the controller is being asked to move the reactor from one temperature to another.
The heat-generation increase caused by a 1 K temperature rise is
For the simulated case, \sigma \approx 2\ \text{MW/K}, so a 1 K setpoint increase adds about 2 MW of reaction heat. Not trivial, but still a relatively small change compared with the total heat load of a high-rate polymerization reactor, as you will see in a few paragraphs.
Production Rate Changes May Not Be Ok
Production rate increase directly increases the heat generated by reaction.
where:
- \Delta H_p is the heat released per kg of polymer produced, for which we will use a representative value of \Delta H_p \approx 3\ \text{MJ/kg}.
- \Delta \dot m_p is the change in polymer production rate in kg per hour.
At 40 tonne/hour, a 20% production increase is 8 tonne/hour, or an additional heat load of approximately 7 MW, significantly more than the expected heat generation from temperature setpoint increase.
Stability Versus Performance
The simplified linear model can be used to derive a Routh stability criterion for the PI-controlled loop.
Skipping over some tedious math, the Routh Stability Criteria says that for the PI controller from before, the requirements on the PI tuning parameters are:
and
The first condition says the controller gain must increase as the reactor heat sensitivity \sigma approaches or exceeds the effective cooling authority F\varepsilon. In plain terms: higher production rate makes the reactor more thermally self-amplifying, so the same gain eventually stops being enough.
Still, stability for the linearized system alone is a weak requirement. A stable loop can still allow an unacceptable temperature excursion, during which the Arrhenius kinetics might lead to rate runs.
Using the simplified model, we ran various scenarios and compared three gain requirements:
1. Minimum K_p for mathematical stability according to the Routh Stability Criteria.
2. Minimum K_p for acceptable setpoint tracking, where acceptable is defined as "no more than 0.2 K offset from temperature setpoint during a slow temperature setpoint ramp."
3. Minimum K_p for acceptable production disturbance rejection, where acceptable is defined as "no more than 1.5 K offset from temperature setpoint during a production rate load disturbance."
As can be seen in Figure 3, the minimum controller gain K_p changes as production rate increases. Furthermore, the controller tuning that is good enough for setpoint tracking may not necessarily be strong enough to reject a production rate disturbance.
Figure 4 shows a specific example of disturbance rejection when production rate increased steadily by 20% over approximately an hour, where the "good" setpoint tracking gain led to an additional 0.34 kelvin overshoot in peak temperature.
Keep in mind that the linearized models start losing authority the further the states deviate from their original steady state value. In the example above, the temperature increase may very well have caused the production rate to increase beyond what the cooler can handle, leading to rate runs.
Learnings
Temperature setpoint tracking is not a sufficient test of reactor temperature controller robustness, and a set of PID tunings that work well for setpoint tracking may not be strong enough to recover when subjected to production rate disturbances in a typical exothermic fluidized bed reactor.
