# Cavity pressure control during the cooling stage in thermoplastic injection molding.

INTRODUCTIONIn injection molding, cavity pressure and polymer melt temperature determine the evolution of the conditions of the polymer inside the mold cavity. The cavity pressure profile and its repeatability have very strong influence on the quality of the molded part, particularly, its dimensions, dimensional stability, mechanical behavior, and surface quality.

The injection molding cycle can be divided into a sequence of three consecutive stages: filling, packing, and cooling, as characterized by the idealized sketch shown in Fig. 1. During the filling stage, cavity pressure rises gradually with the continuation of filling, until the cavity is completely filled. Subsequently, the pressure rises rapidly to the packing pressure level, and it is maintained during the pressure holding phase at a specified level so that additional polymeric material can be "packed" into the cavity to compensate for the material shrinkage due to cooling. This process continues until the gate is frozen. The cooling phase starts immediately after the packing phase and continues to the end of cycle, when the injection mold is opened. In this stage, the polymer melt gradually solidifies, as the heat is removed by the coolant circulating through the cooling channels embedded in the mold. The rate of pressure decrease is determined by the cooling and solidification rates. For semicrystalline materials, solidification is accompanied by crystallization, and crystallization kinetics play an important role.

Controlling cavity pressure to follow desired trajectories (set-point profiles) has been the subject of study of many researchers (1-6, 8-12). Significant progress has been made in controlling cavity pressure during the filling and packing phases. Gao, Patterson, and Kamal (1-4) demonstrated the nonlinear and time-varying characteristics of cavity pressure during filling in relation to the opening of the hydraulic servo-valve. Fixed parameter proportional-integral-derivative (PID) controllers failed to work. Therefore, they developed and implemented a self-tuning adaptive controller for the filling phase (3). Excellent control was obtained with both set-point tracking and disturbance rejection. Later, Gao, Patterson, and Kamal (2-3, 5) extended self-tuning control into the packing stage. Again, very good control was demonstrated. An integral control result for filling and packing using the Gao, Patterson, and Kamal (GPIC) self-tuning controller is shown in Fig. 2. The control of cavity pressure in this case was achieved through manipulation of the hydraulic servo-valve. During the cooling phase, since the gate is already frozen, the cavity pressure can no longer be affected by the servo-valve opening. The only possible manipulated variables are coolant temperature and coolant flow rate. Gao, Patterson, and Kamal (7, 8), have shown that the coolant temperature has a stronger influence on the mold temperature. Thus, it would be a more suitable manipulated variable.

To control cavity pressure through the manipulation of coolant temperature, a coolant temperature control system has to be developed. Heat transfer in injection molding cavities is likely to be a slow process. Therefore, the control of cavity pressure during cooling has to be implemented on a cycle-to-cycle basis. In summary, the objectives of this research are:

(i) To develop a fast coolant temperature control system.

(ii) To develop a cavity pressure control system during the cooling phase, so that the control of cavity pressure can be achieved throughout the injection molding cycle.

SYSTEM IDENTIFICATION AND CONTROL THEORY BACKGROUND

System Identification

An effective control system is largely dependent on the understanding of process dynamics. There are two ways to determine process dynamics: mathematical physically-based modeling and system identification. The first approach attempts to derive dynamic models from analysis of the fundamental mass, energy, and momentum balances. However, the effort needed to develop such a model is usually enormous. Furthermore, many required parameters are unknown and have to be obtained from experimental measurements. On the other hand, system identification produces the dynamic models from collected input and output data of the process. The model is generally valid only for the conditions where the system identification takes place, unless the process is linear and time-invariant (LTI). The system identification approach is used in this investigation.

System identification methods can be classified into classical non-parametric and modern parametric types. The principles of modeling and the practical issues involved have been reported by many researchers (13-15). A commonly used parametric model is the ARX (auto-regressive, or the least squares) model that corresponds to:

A(q)y(t) = B(q)u(t - nk) + e(t) (1)

where A(q) and B(q) are the polynomials in the delay operator [q.sup.-1], A(q) = 1 + [a.sub.1][q.sup.-1] + . . . + [a.sub.na][q.sup.-na], B(q) = [b.sub.1][q.sup.-1] + . . . + [b.sub.nb][q.sup.-nb], na and nb are the order of polynomials A(q) and B(q), and nk is the order of the system delay.

Although most chemical processes are physically of high order, they can often be modeled satisfactorily as a first order plus delay process (i.e., ARX[11nk]). The model can be expressed as:

y(k + 1) = [a.sub.1]y(k) + [b.sub.1]u(k - nk) + e(k) (2)

The Laplace transforms are commonly used to describe continuous process models. A first order plus delay continuous process model is expressed as:

y(s) = [k.sub.p] / [Tau]s + 1 [multiplied by] [e.sup.-[t.sub.d]s] [multiplied by] u(s) (3)

where [k.sub.p] is the process gain, [Tau] is the process time constant, [t.sub.d] is the process time delay, y(s) is the output in Laplace transform, and u(s) is the input in Laplace transform.

The continuous first order plus delay and discrete first order plus delay models can be easily converted to each other by the following equations:

[a.sub.1] = [Tau] - T / [Tau] (4)

[b.sub.1] = T [multiplied by] [k.sub.p] / [Tau] (5)

nk = [t.sub.d] / T (6)

where T = the sampling period.

Dahlin Control Design

The Dahlin control algorithm specifies that the closed-loop performance of the system should behave similarly to a continuous first-order process plus time delay with respect to the set-point.

[G.sub.CL(s)] = [e.sup.-hs] / [Mu]s + 1 (7)

where [Mu] is the desired closed-loop time constant, h is the time delay of the closed-loop transfer function, and [G.sub.CL] is the closed-loop transfer function. The closed-loop time delay h is usually selected as h = [t.sub.d] = NT (T is the sampling period, N is an integer, [t.sub.d] is the process delay). The time constant [Mu] determines the speed of response of the closed-loop system.

For a first order plus time-delay process, the discrete-time Dahlin controller is (2):

[G.sub.DC] = (1 - A) 1 - [a.sub.1][z.sup.-1] / 1 - [Az.sup.-1] - (1 - A)[z.sup.-N-1] K(1 - [a.sub.1]) (8)

where A = [e.sup.-[Tau]/[Mu]], [a.sub.1] = [e.sup.-[Tau]/[Tau]], K is the process gain, and [Tau] is the process time constant.

COOLANT TEMPERATURE CONTROL SYSTEM

Cooling System Design

A mixing system was designed, as shown in Fig. 3, to control the mold coolant temperature and flowrate. There are three thermocouples (type E) installed in the cooling system to measure the cold water, hot water, and mixed water temperatures. They are represented by the symbol TT in Fig. 3. A paddle wheel flowmeter (Signet 3-8500) (16, 17) was installed to measure the total flowrate into the injection mold. Manual valves 1 and 2, installed in the hot and cold water sides, adjust the maximum flow rates. Manual valve 3 in the side branch is usually shut. It can be opened to provide cooling water to the injection mold, when the cooling system requires maintenance. There is a control valve installed on each of the cold and the hot water supply lines. These control valves {Fisher Control Valve Type 1/2-B-EQ.PCT) (18) have equal percentage trim and are pneumatically operated. The control signals from the computer are first converted to analog signals of 0 to 5 Vdc. The voltage signal is then converted to 4 to 20 mA current by a home-built voltage-to-current converter (V/I). A final conversion gives pneumatic pressures through the I/P converter supplied with the control valves. The millivolt signals from the thermocouples are signal conditioned (i.e. cold junction compensated, linearized, and amplified} before they are connected to the data acquisition board. The Signet flowmeter gives a current signal proportional to the flowrate, which is converted into a 0 to 5 Volts dc signal through a home-built current-to-voltage (I/V) converter.

The computer used for this research is an IBM compatible ALR 486 (33Mhz) computer. All the software was developed in "C" language by the authors running under the QNX real-time operating system (19).

Based on results reported for turbulent flow (7, 8), the coolant temperature has a much larger effect than coolant flow rate on the heat removal rate in injection molding. Therefore, coolant temperature was chosen as the manipulated variable to control cavity pressure during cooling.

The flow rates for both hot and cold water were set equal by adjusting the manual valves 1 and 2, and they were kept constant for all the experiments. The relationship between the two control valves is programmed to the following relation:

C[V.sub.c] = 100 - C[V.sub.h] (9)

where C[V.sub.c] is the control valve opening (in percentage) in the cold water side, and C[V.sub.h] is the control valve opening (in percentage) in the hot water side.

Dynamics of Coolant Temperature

An experiment was carried out to test the designed cooling water mixing system. Some step changes of the opening of the control valve (C[V.sub.h]) in the hot water side were introduced. The cold, hot, and mixed water temperatures, as well as the coolant flowrate, were recorded. The sampling rate for this experiment was 200 ms. The top graph of Fig. 4 shows the step changes of the control valve (C[V.sub.h]) opening. The bottom graph of the same Figure shows the responses of the hot ([T.sub.h]), cold([T.sub.c]), and mixed ([T.sub.m]) water temperatures and total flowrate. As shown in Fig. 4, the change in the flowrate is insignificant. Therefore, it is reasonable to assume that the flowrate is independent of the changes in control valve opening.

The response of the coolant temperature (i.e. the mixed water temperature) to the control valve opening approximates a first order system plus delay, as shown in Fig. 4. In Laplace transform, the model is

[T.sub.m](s) / C[V.sub.h](S) = [K.sub.p] / [Tau]s + 1 [e.sup.(-[t.sub.d]s)] (10)

where [k.sub.p] is the process gain in [degrees] C/% control valve opening, [t.sub.d] is the process time delay in seconds, and [Tau] is the process time constant in seconds. The process parameters for the last modeling experiment were obtained by the least squares fitting, and they are listed in Table 1.

The process gain ([k.sub.p]) obtained in Table 1 depends on the hot and cold water temperatures. These temperatures depend on the supply conditions and are measured directly so that the mixed temperature can be non-dimensionalized as [T.sub.mnd]:

[T.sub.mnd] = [T.sub.m] - [T.sub.c] / [T.sub.h] - [T.sub.c] (11)

Figure 5 gives the responses of [T.sub.mnd] to the step changes shown in the top graph of the control valve opening. It is again apparent that the response can be modeled as a first order system plus delay:

[G.sub.nd] (s) = [T.sub.mnd](s) / C[V.sub.h(s)] = [K.sub.pnd] / [[Tau].sub.nd]s + 1 [e.sup.([-t.sub.dnd]s)] (12)

where [k.sub.pnd] is the process gain of [T.sub.mnd] in relation to C[V.sub.h], [[Tau].sub.nd] is the process time constant, and [t.sub.dnd] is the process time delay.

The parameters for the above model are shown in Table 2. They vary slightly, showing some degree of nonlinearity. This is attributed to the nonlinearity and hysteresis of the valves in the cooling system. The averaged parameters are given in the same Table. They were used in the coolant temperature controller design.

Coolant Temperature Control

The Dahlin controller (Eq 8), with the desired closed-loop time constant ([Mu] = 1) and a sampling interval (T) of 0.2 s, was chosen to control the coolant [TABULAR DATA FOR TABLE 1 OMITTED] temperature. The process gain K in Eq 8 is the actual process gain, K = [k.sub.pnd] [multiplied by] ([T.sub.h] - [T.sub.c]). Therefore, the closed-loop control, in essence, is a gain scheduling system with respect to the varying temperatures [T.sub.c] and [T.sub.h]. An experiment, carried out to verify the controller effectiveness, started with a coolant temperature of about 15 [degrees] C. The set-point of the control system was initially 25 [degrees] C for 8 s, then it jumped to 45 [degrees] C and stayed at that value for 8 s, then stepped down to 20 [degrees] C and remained there. The experimental results are shown in Fig. 6. The top graph shows the set-point changes (solid line) and the coolant temperature responses (dashed line). It clearly shows that the coolant temperature closely follows the set-point profile. The coolant temperature reaches a new set-point within 2 s in the worst case. Figure 6B gives the corresponding control valve opening in percent on the hot water side. It is seen that with the coolant temperature at a fixed value, the valve opening tends to drift. This is due to the hot-cold water temperature difference drifting at the beginning of the experiment, when the water streams have been just turned on.

EXPERIMENTAL CONDITIONS

The work was done on a Danson Metalmec 2-1/3 oz, reciprocating screw injection molding machine, model 60-SR. The melt temperature was determined by setting the four barrel temperature set-points to 235, 215, 180, and 150 [degrees] C from the nozzle to the hopper, respectively. The sampling rate for pressure during the cooling stage was 50 ms. The total injection molding cycle was set to be 32 s. The filling pressure was controlled at a constant ramp of 1.73 MPa/s (250 psi/s). The packing pressure was controlled at 20.7 MPa (3000 psi) for a duration of 4 s.

An injection molding grade high density polyethylene resin, Sclair 2907 (DuPont Canada), was used in this study.

The cooling system designed in the earlier section was used to provide quick manipulation of coolant temperature while keeping the coolant flowrate constant. The coolant temperature follows the new set-point within 2 s, in the worst cases.

A fan-gated rectangular cavity, as shown in Fig. 7, was used for the experiments. The pressure transducer installed near the cavity gate was used for measurement of cavity pressure throughout the experiments.

The real-time computer control system developed for this research is described elsewhere (19).

DEFINITION OF REPRESENTATIVE VARIABLES DURING COOLING

Once the injection gate is frozen, the cavity pressure can only be influenced by the coolant temperature or coolant flowrate. Owing to the large thermal mass of the injection mold, it is not possible to manipulate the pressure during cooling within a cycle. However, it is possible to change it over successive cycles.

Figure 8 gives two experimental cavity pressure profiles with coolant temperatures of 45 [degrees] C and 15 [degrees] C, respectively. The packing pressure for both experiments was 20.7 MPa (3000 psi). It is seen that the coolant temperature has a strong effect on the pressure profile during cooling. In both cases, the pressure starts to decrease slowly from the packing pressure. The rate of decrease becomes larger until the pressure reaches zero.

Several alternative quantitative methods for describing the cooling process are possible. The merits and disadvantages of these methods are discussed below. Figure 9 gives a graphical representation of these methods.

[TABULAR DATA FOR TABLE 2 OMITTED]

Function Fitting

This describes the cooling pressure profile with a mathematical function (e.g. a polynomial or some form of exponential function). It gives an accurate description of the pressure profile during cooling, but the computation power needed to do this during every injection cycle is large. Therefore, this method was not chosen.

Pressure Cooling Time to Inflection ([PCT.sub.i])

This is defined as the time from the end of packing to the point where the pressure profile passes through the point of inflection. It can be determined from the maximum of the derivative of the pressure profile. The difficulty is that differentiation of a signal increases the noise-to-signal ratio, which makes it difficult to determine the maximum.

Pressure Cooling Time ([PCT.sub.0])

[PCT.sub.0] is defined as the time from the end of packing to the time at which the pressure reaches a value which is (close to) zero. The [PCT.sub.0] is easy to measure, requires little computation time and indicates an "average" pressure loss rate. However, this variable only gives information about both ends of the pressure profile. The nature of the pressure profile between these two values is not taken into consideration.

Multiple Pressure Cooling Times ([PCT.sub.x]) and Controlled Pressure Cooling Time (CPCT)

The pressure cooling time ([PCT.sub.x]) is defined as the time from the end of packing to the time at which the pressure decreases to x percent of the pressure at the end of packing, [P.sub.PE].

To remedy the problem associated with [PCT.sub.0], multiple measurement of [PCT.sub.x] is proposed. Instead of measuring the time from the end of packing to a near zero pressure, this method employs a vector of times from the end of packing to a vector of predetermined pressures. This method has all of the advantages of [PCT.sub.0]. It also gives an approximate characterization of the pressure-time profile. However, multiple pressure cooling times give multiple (a vector of) times instead of a single scalar value. The vector size and the vector of predetermined pressure percentage values need to be determined. One practical simplification is to define a Controlled Pressure Cooling Time ([Delta][PCT.sub.67-33], or CPCT) which is the time required to progress from 2/3 (67%) of the packing pressure to 1/3 (33%) of the packing pressure.

DYNAMIC MODELS RELATING CPCT RESPONSE TO COOLANT TEMPERATURE

An open-loop experiment was conducted to test the dynamic model relating CPCT to the coolant temperature. The coolant temperature varies according to the top graph of Fig. 10. The bottom graph gives the corresponding change in CPCT. The total cycle time for this experiment is 32 s.

The response of CPCT to a change in coolant temperature can be approximately modeled as a first order system. The small variation of CPCT is due to the measurement resolution and, possibly, incomplete gate freezing at the end of packing. The sampling rate for pressure measurement during cooling is 50 ms, which determines the resolution of [PCT.sub.x] and CPCT ([Delta][PCT.sub.67-33]).

Parametric models of first order with no delay, and with one molding cycle delay, are given in Table 3. Details of the methods used to estimate these parameters are given in references (13-15). The final prediction error (FPE) is an indicator of the model accuracy. The smaller it is, the better is the model. The results indicate that the first order with no delay model gives a better representation of the system.

Based on FPE analysis, the parametric model ARX[110] was therefore selected. The model parameters were then converted to the Laplace domain based on Eqs 4, 5, and 6.

[TABULAR DATA FOR TABLE 3 OMITTED]

EXPERIMENTAL CONTROL OF CPCT

An experimental closed-loop control to track a changing set-point profile was carried out. The controller tuning parameter [Mu] = 1 was used (i.e. one injection cycle). In this experiment, the CPCT set-point started at 2.5 s. At the 20th cycle, the set-point decreased at a rate of 50 ms a cycle until the 34th cycle. Then, it stayed at 1.8 s until cycle 54. Subsequently, it stepped back to 2.5 s and stayed at this value until cycle 74. It then stepped again to 1.8 s and stayed at this value until the 94th cycle, after which it increased at a rate of 50 ms per cycle until it reached 2.5 s, and remained at that value. The set-point is graphically represented as the solid line in the top graph of Fig. 11. This set-point profile involves both step and ramp changes in both the up and down directions. The dashed line is the experimental response of the closed-loop control to the set-point changes. It is clear that CPCT follows the set-point closely. The bottom graph of Fig. 11 shows the corresponding coolant temperature changes.

One more experiment was carried out to test the effectiveness of the control system in handling melt temperature changes. The melt temperature for this experiment was changed to 205 and 235 [degrees] C in the previous experiment. The set-point for this experiment was started at 2.5 s. It changed to 1.8 s at cycle 20. The solid and dashed lines in Fig. 12 are the set-point and CPCT response, respectively. The corresponding coolant temperature changes are shown in the bottom graph of the same Figure. CPCT is seen to follow the set-point closely, indicating that the control system can handle the melt temperature changes successfully.

The pressure cooling profiles of the last experiment of CPCT control are given in Fig. 13. It shows the profiles of the cycles 15, 17, 19, 21, 23, and 25, just before, during, and after a step change of the CPCT set-point was introduced at cycle 20. The pressure cooling profiles of cycles 15, 17, and 19 superimpose, before the CPCT set-point change was introduced. The pressure cooling profiles of cycles 23 and 25 are also very similar after the set-point change of CPCT occurred. The pressure cooling profile of cycle 21 coincides with the transition of the controlled pressure cooling time (CPCT). It is clear, from Figs. 12 and 13, that the control of CPCT is effective in controlling the pressure cooling profiles.

Figure 14 shows the cavity pressure control results over two complete cycles. The filling and packing pressure was controlled using the self-tuning adaptive control techniques described in references (2, 3), and the cooling cavity pressure control was achieved by controlling CPCT as discussed in this paper. The cavity pressures for two different cycles follow each other very closely. With the cavity pressure control throughout the injection molding cycle, the part weight variations for over 100 cycles was found to be within 0.05% (2).

SUMMARY

A coolant temperature control system has been successfully designed and implemented. Several alternative variables for pressure control during cooling have been proposed. The dynamics of one of these, CPCT, were investigated and modeled. A successful control system for CPCT was Implemented and tested using coolant temperature as the manipulated variable. Cavity pressure during the cooling phase can be controlled effectively by using CPCT control.

REFERENCES

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2. F. Gao, PhD thesis, McGill University, Montreal (1993).

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14. T. Soderstrom and P. Stoica, System Identification, Prentice-Hall, New York (1989).

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16. Instruction Manual, Compack[TM] 8500 Flow Transmitter, Signet Scientific, El Monte, Calif.

17. Instruction Manual, MK 588, Digital Flowmeter, Signet Scientific, El Monte, Calif.

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19. F. Gao, H. Fusser, M. R. Kamal, and W. I. Patterson, SPE ANTEC Tech. Papers, 36, 1887 (1992).

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Author: | Gao, F.; Patterson, W.I.; Kamal, M.R. |
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Publication: | Polymer Engineering and Science |

Date: | Oct 15, 1996 |

Words: | 4133 |

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