IMRICH KLEIN DONALD 1. MARSHALL convenient ways to describe a physical 0neprocess :of the most or system is by an algebraic or differential equation representing a model of the process. The general form of the model is usually derived by analytic means, and to assure the model's usefulness it is important to determine its validity. A statistical tool devised for this purpose many.years ago by K. F. Gauss is known to most of us as the least squares procedure or regression analysis. The technique enables one to determine the values of the coefficients in the model by fitting the model to experimental data. The practicality of using this technique has grown in recent years due to the decreasing cost of digital computer time which allows elimination of the manual computation work. Regression analysis is also used to get a quantitative picture of the degree of fit or misfit of the model for its possible improvements. This procedure is therefore an important tool in building a valid model for the process., The only difficulty is that it can handle only line& models. I n certain cases a nonlinear model can be linearized by transforming the variables; however, this is only possible if the model is linear with respect to the coefficients. I n other cases linearization is still possible by other mean-for example, taking the logarithm of both sides of the equation. But often the model is nonlinear and cannot be linearized. Usually the process or system to be modeled is so complex that it can only be described by a set of simultaneous algebraic and/or differential equations. These equations are often implicit functions where the dependent variable or response cannot be easily expressed in terms of the other variables. Further complication is still possible when several dependent variables or responses are also interdependent and sometimes also in the form of implicit functions. These six types of modeling problems are shown in Table I. A technique to handle complex cases such as these is the nonlinear estimation method (2). This technique; instead of calculating directly, which is not possible in nonlinear cases, searches for the combination of levels of all the coefficients that cause the model to best fit the experimental data. 36
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
TABLE 1. SIX BASIC TYPES OF MODELING PROBLEMS LINEAR MODEL Y
= a,X,
+ ox,+ orXs + etc.
NONLINEAR MODEL CAN BE LINEARIZED BY TRANSFORMATION OF VARIABLES
NONLINEAR MODEL Y= f nicdz COMPLICATED MODELS USING SIMULTANEOUS EQUATIONS
IMPLICIT SIMULTANEOUS EQUATIONS
IMPLICIT SIMULTANEOUS EQUATIONS WITH SEVERAL RESPONSES
I n our work, while trying to build a mathematical model for plastic melt extrusion, we found that the complexity of the analytical model required the use of
I
MATHEMATICAL MODELS OF EXTRUSION
A proved technique for simplification of mathematical modeling of complex systems is presented, with an illustrative example in the plastics extrusion field which verifies the practicality of the technique
L
1
I
Figure 1. Physical conjguration of the plm&s axmrdn
nonlinear estimation. This paper presents a procedure used in building and improving such a mathematical model for a complex system; the example used is the model for a plastic melt extruder. The Extrusion Model
The extruder usually consists of a screw rotating in a heated b a r d and is schematically shown in Figure 1. Solid plastic pellets fed through a hopper into the machine slowly melt, partly because of the barrel heat and partly because of heat generated from friction. After all the plastic has melted, it is pumped by the last section of the screw through a die. I t is thii melt pumping section of the extruder that was treated analytically and for which some sort of a model exists (4). Measurement of pressure and temperature profiles (5, 6)through the melt pumping section of the extruder enabled us to upgrade the analytical model based on the experimental data. The basic equation of the melt extruder model is:
where Q = flow rate of plastic, and 01 and B are constants dependent on the geometry of the screw. bP/bL is the pressure rise in the axial direction, N = screw speed; p = true viscosity of plastic, which we expressed as (6) :
log.&
=
+ A11og.+ + AnT +
A@
Amp
log."+ + + AinT log. +
A11
(2)
where T = temperature, y = true shear rate, and A,, = coefiients.
The first term on the right-hand side of Equation 1 is the drag flow and the second, the pressure flow term. While Equation 1 is simplified and derived for parallel plate flow under isothermal conditions in all directions, several corrections to it are available and incorporated in the model. They account for curvature and shape of the channel and for radial temperature profile. In addition to these equations, another one for heat balance and one for heat transfer through the barrel are also included in the model. The correction of the model for temperature gradient in the down-channel or axial direction of the extruder is done by subdividing the axial dimension into short sections. Each such section is handled as isothermal in the axial direction and its average temperature calculated by trial and error from the model until the finaltemperature is reached. Model Buildinp Details
After the model was assembled in the form of a computer program, all the experiments were simulated by it using a viscosity equation, Equation 2, the coefficients of which were determined from independent experiments on a capillary rheometer. However, the simulated temperature and pressure profiles differed widely from the ones found experimentally. I t follows that the coefficients of the flow equation should not be determined independently, but directly from the extruder model utilizing the experimental data. This procedure will upgrade the existing mathematical model for melt extrusion and test its validity over exnerimental range. Because the extruder model is complex and the VOL 58
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computer program long-running, it became necessary to limit the size of the problem by keeping to a minimum the number of coefficients for which a search was made by nonlinear estimation. I n the search for the coefficients or parameters, an initial guess is made for each of them, as well as for the size of the unit step by which they should be changed. T h e search for the best combination of values for the coefficients or parameters is systematically done by adjusting the values according to a rotatable central composite design. This has proved to be a most effective technique, and is illustrated in Figure 2 for a threeparameter case. The coefficients or parameters to be determined by this method are summarized in Table 11, which includes a correction for the heat transfer coefficient calculated by Jepson's (3) model. The table also includes a correction for the drag and pressure flow terms in Equation 1, necessitated by the presence of thermocouples in the extruder channel. Two of the coefficients of Equation 2-the absolute term and log,-are also searched for, as is a correction in the form of the pressure flow term in Equation 1. From this last term the pressure rise across the melt extruder is being computed.
TABLE I I . PARAMETERS T O BE SELECTED FOR METERING SCREW OPTIMUM MODEL 1. Correction for heat transfer coefficient Drag and pressure flow correction factors for presence of thermocouples 3. Coefficients of flow equation, Ao, AI, A3 4. Correction for pressure rise
2.
The surface described by this index as a function of the parameters was then represented by a second-order regression function which was further analyzed by the technique of response surface analysis ( 7 ) using the index as the response. The levels of the various parameters used for the simulation were then changed until a minimum combined response was reached. T h e values for the several parameters associated with the minimum response represent the new improved model. A zero value for a parameter at the minimum response meant the absence of the term from the model, whereas the presence of a term absent from the original model was indicative of the modification introduced into the analytical model. Table I11 summarizes the model building computer program written for this purpose.
TABLE 1 1 1 . SUMMARY O F MODEL BUILDING PROGRAM 1 . Read data for simulation of experiments 2. Select a combination of parameters 3. Simulate all experiments 4. Calculate sum of squares of deviation for temperature and pressure profiles 5. Change levels of parameters and repeat 3 and 4 for all levels of the parameters 6. Combine temperature and pressure error responses 7 . Fit second-order equation to combined response ts. all parameters 8. Perform response surface analysis and minimize the combined response 9. Select new range for parameters around the minimum response 10. Repeat above until absolute minimum response is reached -
T h e objective of model building is to find a model which yields the minimum sum of squares of deviation (SS) between the measured and calculated response. I n the case of the plastic melt extruder two responses, plastic temperature and pressure, are measured. The sums of squares of deviation for each can be represented as a second-order function with respect to the parameters. A response surface analysis of these two functions can then yield the particular levels of the parameters involved for which simulated temperature and pressure profits best represent the experimental values. A less complicated method for combining the two sums of squares of deviation with equal weight was devised and the model optimized with respect to this index of misfit. This simplified technique proved to be quite satisfactory for the majority of problems. T h e index of misfit is actually the combination of the two sums of squares of deviations, each normalized with respect to its extreme values as summarized in Equation 3.
Y = 50
SSAT - SSAT,,, - SSAT,i,
(SSAT,,,
SSAP - SSAP,i, + SSAP,,, - SSAP,,i, )
(3) For each set of parameters tried for the simulation, a value for the index of misfit was, therefore, obtained. 38
INDUSTRIAL A N D ENGINEERING CHEMISTRY
-
I n the building of the model for the plastic melt extruder, it appeared that the minimum response associated with the optimum model obtained was higher than the one considered acceptable for a final model. T h e standard error for predicted temperature profiles was 8.5" F. and that for pressure 1075 p.s.i. Analysis of the experimental data showed that by separating the experiments according to the three extruder screws on which they were run, each differing from the other only by the depth of its channcl, H,that the three minimum misfits were considerably less than the one obtained when all the experiments were treated together. However, the three minimum responses for each screw denoted as A , B, and C do not occur at the same combination of levels of the Parameters. T h e procedure for establishing this difference in degree of fit for different screws can be illustrated by selecting only two coefficients and plotting the main features of error responses for each screw. For each screw, the response surface showed a stationary trough where a line of minimum error could be found. Figure 3 illustrates a simplified but typical situation of three separate stationary troughs. One of the parameters was selected at the average level of the three for A , B , and C denoted as X I , and the corresponding values for the other parameter was found as the
presence of thermocouples in the basic channel equation (Equation 1) and showed that no such correction is needed in the pressure flow term of the analytical model. The flow equation (Equation 2) needs some modification because AI is a function of channel depth instead of being a constant; also, the negative value of As, coefficient of T X 0 (0 = residence time) term, is indicative that some sort of thermal degradation affects the plastic while it passes through the extruder.
TABLE IV.
FINAL MODEL COEFFICIENTS
For drag flows For pressure flow* Ai
Initial Guess None None -0.7717
An As
2.055 0.00
Variable
d Figure 2. A three-pmmncter central rotatab& cmnposite &sign
a H-S
AP Heat transfer coefficient
i
a
(r.p.m.)
Final Value 0.842 None -0.7666+ 0.2030 H / D 2.06 -0,055 a H-1.0"-16.9 slo a (r.p.m.)O.N
I
Figwe 3. Method of deriving equatiom missing from thc model
point where the particular stationary trough intersected the dotted line of constant XI,. The three values for parameter X2 were plotted as a function of its corresponding channel depth. This function was interpreted as an equation missing from the original model.
Measured temperature range Standard error of model Measured pressure range Standard error of model
174.9" F. 4.4'F. 4743 p.s.i. 290 u.s.i.
BIBLIOGRAPHY Model Speciflcatlons
The technique of nonlinear estimation is used to derive an improved model for the plastic melt extruder. I t was modified for a complex model with several differential and algebraic equations and two measured mutually interdependent responses-namely, plastic melt temperature and pressure profile across the screw channel. As can be seen from Table IV, the final model, derived by the procedure of nonlinear estimation, yielded a value for the drag flow correction owing to the
(1) Box, G. E. P., Bionuhk 10,
16 (1951).
(2) Box, 0.E. P. Bvllefio dc I'Inrtitutc Infanational de Statistiquc 36, Part 3, Stockholm (1956. (3) Jepron, C.H., IND. ENO.CHIY.45 (5), 992 (1953). (1) McKclvcv. J. M.. "Polmu Roc-." W i l m New York. 1962.
Imrich Klein is Senior Engines and Donald I. Marshall is Research &ads for th Plastics Processing Research Department at Western Electric Co.'s Engikeering Pesearch Center.
AUTHORS
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