A THEORETICAL STUDY of the KINETICS of POLYETHYLENE REACTOR DECOMPOSITIONS by GARY MILO GARDNER, B.S

Home , Polymerization, Radical polymerization

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A THEORETICAL STUDY OF THE KINETICS OF POLYETHYLENE REACTOR DECOMPOSITIONS by GARY MILO GARDNER, B.S. in Ch.E.

A THESIS IN CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN CHEMICAL ENGINEERING

Approved

Accepted

August, 1975 AC ~0~ 13 _,. I{; 7~ No,~~ Cnp. )...

ACKNOWLEDGMENT

The author wishes to express his sincere appreciation to Dr. W. J. Huffman for his guidance and expertise in this endeavor. His assistance in research and academic affairs was the most anyone could hope for. Special thanks to Dr. D. C. Bonner for giving a lot of extra time when it was desperately needed. The author acknowledges the Gulf Oil Foundation for their financial support.

i i TABLE OF CONTENTS Page ACKNOWLEDGEMENT...... ii LIST OF TABLES ...... v LIST OF FIGURES...... vi CHAPTER I INTRODUCTION...... 1 Industrial Process ...... 1 Investigations On Reactor Stability ...... 3 Effects of Operating Variables ...... 4 CHAPTER II MATHEMATICAL MODELING TECHNIQUE ...... 8 Reactions Affecting the Overall Polymerization Rate . 8 The Polymerization Rate Expression...... 9 Polymerization Rate Constants ...... 11 Values of Kinetic Parameters ...... 13

Material and Energy Balances. • • ...... 16 Genera 1 Fonn of the Materia 1 and Energy Equations. . 16 Linearization of the Material and Energy Balances. . 17 Analytical Solution and Stability Criteria. . . . 21 Computer Simulation ...... 25

CHAPTER III APPLICATION TO A POLYETHYLENE REACTOR . • • • • • 27

Steady States of a Polymerization Reaction .. • • 29 Transient Responses of the Stability Regions...... 34 Transient Responses in the General Operating Region . 41 Perturbation Analysis ...... 47 Effect of Feed Stream Impurities ...... 53

iii Page Effect of Initiator Choice on Stability ...... 56

Effect of Thermal Decomposition of Ethylene .. . . . 58 Reactor Response to Changes in Viscosity and Pressure 59 Effect of Viscosity and Diffusion-Limited Termination ...... 59

Effect of Pressure ...... 67 CHAPTER IV CONCLUSIONS AND RECOMMENDATIONS ...... 72 Conclusions ...... 72 Recommendations ...... 73 Reactor Operation...... 73 Future Investigations .. 73 LIST OF REFERENCES ...... 74 NOMENCLATURE ...... 78 APPENDIX A ULTIMATE REACTION CONDITIONS ATTAINED DURING A POLYETHYLENE REACTOR DECOMPOSITION ...... 81 APPENDIX B REACTOR OPERATING PARAMETERS AND KINETIC DATA. . . 83 Reactor Operating Conditions and Physical Constants . 84 APPENDIX C FUNCTIONAL RELATIONSHIP OF THE TERMINATION RATE CONSTANT TO VISCOSITY AND PRESSURE EFFECTS ...... 88 Viscosity Effect on Initiator Concentration . . . . . 89 Effect of Pressure on Feed Variations ...... 91 Viscosity Effect to the Transient Responses ...... 92 APPENDIX D STEADY STATES FOR A POLYMERIZATION REACTION. . . . 94

APPENDIX E COMPUTER SIMULATIONS ...... 96

iv LIST OF TABLES

Table Page I PARAMETERS FOR THE STABILITY REGIONS...... • 40 II PARAMETERS FOR TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS CONCENTRATIONS. . • ...... • . . 46 III PARAMETERS FOR TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS TEMPERATURES...... 52 IV EFFECT OF ACETYLENE CONCENTRATION ON THE STEADY STATE PARAMETERS ...... • . . . 55 v EFFECT OF VISCOSITY CHANGES ON THE INITIATOR CONCENTRATION ...... 62 VI EFFECTS OF CHANGES IN OPERATING PRESSURE ON THE STEADY -STATE PARAMETERS ...... 68 A-I ULTIMATE REACTION CONDITIONS ATTAINED DURING A POLYETHYLENE REACTOR DECOMPOSITION ...... 82 B-I REACTION CHEMISTRY AND KINETICS OF POLYETHYLENE DECOMPOSITIONS ...... 85

v LIST OF FIGURES

Figure Page 1 POSSIBLE SEQUENCE FOR INITIATION OF A POLYETHLYENE DECOMPOSITION...... 6 2 SOLUTION OF MATERIAL AND ENERGY BALANCES FOR REVERSIBLE EXOTHERMIC REACTION...... 22

3 UNSTEADY STATE ANALOG SCHEME ...... 26 4 STEADY STATES OF A POLYMERIZATION REACTION . 30 Sa TRANSIENT RESPONSES OF THE STABILITY REGIONS . . . . . 36

Sb TRANSIENT RESPONSES OF THE STABILITY REGIONS . . . . . 37 Sc TRANSIENT RESPONSES OF THE STABILITY REGIONS . 38 Sd TRANSIENT RESPONSES OF THE STABILITY REGIONS . 39 6a TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS CONCENTRATIONS ...... 42 6b TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS CONCENTRATIONS ...... 43 6c TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS CONCENTRATIONS ...... 44 6d TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS CONCENTRATIONS ...... 45 7a TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS TEMPERATURES ...... • • • 48 7b TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS TEMPERATURES ...... • . . . . . • • • 49 7c TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS TEMPERATURES ...... 50 7d TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS TEMPERATURES ...... • • • 51

8 ACETYLENE DECOMPOSITION ...... • • • 54 vi Figure Page 9 EFFECT OF TEMPERATURE SENSITIVE INITIATORS ...... 57 10 VISCOSITY AND PRESSURE EFFECTS ON THE TRANSIENT RESPONSES ...... 65

B-1 COMPARISON OF DATA FOR DECOMPOSITION OF ETHANE •. • • • 87 D-1 EXTENDED PLOT OF THE STEADY STATES FOR A POLYMERIZATION REACT I ON • • • • • • • • • • • • • • • • • • • • • • • • 9 5

vii CHAPTER I

INTRODUCTION

Industrial Process The stability and control of the free-radical polymerization of ethylene at high pressures has become an important area of commercial and theoretical interest. Presently, low density polyethylene (LDPE) is produced commercially in a variety of high pressure industrial re actors. These processes have been described by Raff and Allison (1956), and Smith (1965). They include continuous stirred tank re actors (CSTR), tubular reactors, and several modifications of these basic designs. A high-pressure industrial reactor operates between 15,000 and 45,000 pounds per square inch of pressure. Normal operating tem peratures are between 100°C and 300°C (Miles and Briston, 1965). The process requires a feed stream purity in excess of 99.9 percent ethylene (Hahn, Chaptal, and Sialelli, 1974). The initiation of the polymerization process requires primary free radicals. The ethylene feed is accompanied by an initiator which has the ability to generate these free radicals. Oxygen was employed almost exclu sively as the pioneer initiator (Ehrlich, Pittilo, and Cotman, 1958; Ehrlich and Pittilo, 1960); however, modern industrial initiators are usually organic compounds (Brandrup and Immergut, 1958). Free-radical polymer chains react to form inactive polymer molecules in the termination step of the process. Termination can

1 2 proceed by recombination or disproportionation, although it is widely recognized in the polyethylene industry that recombination dominates (Woodbrey and Ehrlich, 1963). Chain termination as a result of poisoning reactions can usually be ignored because of the purity of the ethylene feed. The frequency with which polymeric chains terminate is described by the termination rate constant. The value of this term is usually assumed to be constant over moderate ranges of tem peratures, densities, and viscosities. The effect of the termination rate constant increases in importance as large deviations of these process parameters are witnessed. Because of the dangers involved, polyethylene decompositions have received significant attention by industrial personnel. The disasterous effects of a decomposition have been monitored, and the magnitude of the explosions is astounding. It is reported that the detonation of an ethylene-air mixture at a pressure of 2.6 kg/cm2 causes explosive damage equivalent to an equal weight of TNT ( 11 Safety in High Pressure Polyethylene Plants .. , 1974). Huffman and Bonner

(1974) have provided_th~oret~cal calculations which indicate that the maximum temperatures and pressures resulting from a decomposition could reach 100,000 pounds per square inch and 1950°C. These values were shown to be. dependent upon initial reactor conditions, but were independent of reactor volumes. On the basis of these calculations, the decomposition gas was predicted to contain large percentages of methane, carbon, and possibly ethane. The numerical results of this study are found in Appendix A. 3

Investigations On Reactor Stability Several studies have been undertaken in an effort to analyze the stability of LOPE systems. Important studies have included Hoftyzer and Zwietering (1961), Warden and Amundson (1962), Goldstein and Amundson (1965), Matsuura and Kato (1966), Goldstein and Hwa (1966), Knorr and O'Driscoll (1970), and Agrawal and Han (1974). Hoftyzer and Zwietering (1961) analyzed stability and control of a high-pressure polymerization reactor which was limited by a half-order reaction rate step. Warden and Amundson (1962) evaluated a CSTR with terminated addition polymerization. Knorr and O'Driscoll (1970) utilized a theoretical study by Matsuura and Kato {1966) to analyze the stability of a concentration-dependent, isothermal re actor. The Matsuura and Kato {1966) analysis considered the con centration stability of liquid-phase autooxidation of isopropyl alcohol. The Goldstein and Amundson (1965) model investigated sta bility in two immiscible phases. The effects on product properties of important reactor variables were studied by Goldstein and Hwa (1966). The Agrawal and Han (1974) model is an application to tubular reactors. Polymer reactor designs are generally based on the steady-state analysis of a reactor system, coupled with the transient response of the system to a perturbation in one of the operating parameters. The steady-state character of a homogeneous reactor is generally obtained by simultaneous solutions of the non-linear material and energy balances. The free-radical polymerization kinetics make 4 algebraic manipulation of these equations cumbersome. It is customary, therefore, to linearize the equations utilizing a Taylor's expansion and then to uie iterative methods of solution. The ultimate goal of a polymer reactor stability study is to obtain the steady-state oper ating points of the system and analyze the effects of disturbances on these steady states. If the reactor responds to changes within a bounded region and then returns to the original operating point, it is considered stable. However, if a perturbation to the system creates an unbounded transient response, the reactor is considered unstable.

Effects of Operating Variables A reactor, at a particular set of operating parameters, is con stantly subject to changes in the operating conditions. These fluc tuations may be caused by changes in the inlet flow rate or by changes of the temperature o~ concentrations in the inlet flow stream. Pressure may also be an important operating parameter. The effects of deviations in operating pressure have not been included in any of the previously cited references. Therefore, it seems pertinent to incorporate the pressure effect in further modeling efforts. It is also possible that many of the impurities in a feed stream can func tion as free-radical generators, and hence, pseudo-initiators. The temperature at which significant initiator decomposition occurs is a basic consideration in reactor design and initiator choice. 5

The previously mentioned parameters are significant in the operation of an industrial polyethylene reactor. Polyethylene re actor control has to be such that the production rates, as well as polymer properties such as density, molecular weight distribution, and average molecular weight, are uniform. The usual values of the magnitudes by which these variables may be manipulated is generally not known because of proprietary considerations; however, it is understood that reactors are extremely sensitive even to small perturbations of the parameters. Control of the parameters is necessary not only for product purposes, but also because of the threat of thermal runaway. Thermal runaway is a form of instability that characterizes autothermic re actors. During a thermal runaway, temperatures and pressures in the reactor increase until an explosion occurs. This catastrophic event is referred to as thermal decomposition, or more commonly, a

11 11 decomp • Huffman and Bonner (1974) have proposed a possible sequence for the initiation of LOPE decompositions (see Figure 1). This scheme suggests several parameters that might initiate a thermal decomposition. These effects include mechanical friction resulting in a reactor hot spot, unstable impurities in the feed, or an ex cess initiator concentration. Each of these deviations from normal operation can create a high rate of polymerization, resulting in increased viscosity within the reactor. In addition, high viscosity in the reactor developed from inadequate mixing or poor feed dis- 6

Unstable Excess Feed Initiator Impurity Concentration Mechanical Inadequate Mixing Friction and Or Poor Feed Hot Spot Distribution

High Cone. of Polymer and High Viscosity

Diffusion-Limited Termination and Generation of Excess Heat

Decomposition

·~ . ~ -.;:.; -·~- ·.,.. .

Figurel. Possible sequence for initiation of a polyethylene decomposition. [after Huffman and Bonner, (1974)] 7 tribution could lead directly to the high concentrations of polymer. At these high viscosities the termination step of the reaction mech anism becomes diffusion-limited and results in the generation of ex cess heat. This sequence of events continues to increase the reaction rate until ultimately, a thermal decomposition occurs. Huffman and Bonner (1974} have reported that the decomposition may be completed within one second. Our theoretical study was undertaken to investigate the stability and control of a high-pressure polyethylene reactor. We propose to extend the work of the earlier CSTR polyethylene reactor models. The extensions were to include (1) the effects of pressure on the reaction kinetics, (2) the possibility. that feed impurities could act as pseudo-initiators, (3) initiator concentrations fluctuations in the feed stream, (4) the utilization of various initiators for increased reactor stability, and (5) reaction dependence on the termination rate constant. It is expected that a model which com bines all of these effects will yield a more realistic estimate of the time scale reported as well as a better assessment of the 11 auto catalytic11 features of the thermal decomposition than previously reported investigations. CHAPTER II

MATHEMATICAL MODELING TECHNIQUE

The mathematical modeling of a free-radical polymerization as performed in a high-pressure polyethylene reactor follows three basic developments. The first of these is the overall polymerization rate and kinetic mechanism with which the polymer forms. The steady states of an autothermic reactor may then be determined by the re quirement that the heat generated by the reaction must be equal to the sensible heat taken away. For a homogenized reactor, this re quirement allows simultaneous solutions of the material and energy balances at steady-state conditions. The final step utilizes the time-dependent material and energy balances to evaluate transient responses to perturbations in the system.

Reactions Affecting the Overall Polymerization Rate The primary free radicals are formed as the initiator undergoes thermal decomposition: k. I 1 2R' (1)

Initiator free radicals (R'), and monomer molecules (M), form monomer free radicals in a rapid chemical reaction. The rate at which initiator molecules {I) decompose is given by the initiation rate constant (k.). Chain initiation can also be induced thermally. 1 It is shown later in this study that pyrolysis is generally neg-

8 9 ligible at the temperatures of a normal reactor. The efficiency by which the free radicals can initiate polymerization has been dis cussed by Flory (1953), and Bamford et !!· (1959). Once a monomer free radical (Rt) has been initiated, the propagation of the poly meric reaction can proceed by several mechanisms which form poly meric free radicals (P').

kp R't + M -~-~· p• (2)

The propagation rate constant k is essentially independent of chain p length after the first few monomer additions (Flory, 1953; Bamford et EJ_., 1958). Free-radical polymer chains react to form inactive polymer molecules (P) in the termination step of the reaction:

2P' (3)

This termination step may proceed by the mechanisms of recombination or disproportionation, although recombination is recognized as the dominant path (Woodbrey and Ehrlich, 1963). The rate at which termination proceeds is given by the termination rate constant (kt). Chain termination due to poisoning is usually negligible because of the purity of the ethylene feed.

The Polymerization Rate Expression Before one employs material and energy balances to evaluate the possible steady states at which a reactor might operate, it is nec essary to have a rate expression describing the reaction. Several 10 authors have derived similar forms of the kinetic rate expression for free-radical polymerizations (Flory, 1953; Bamford et al., 1958; Weale, 1967; Ehrlich and Mortimer, 1970). Derivations of the rate expression utilize the following assumptions: (1) the rates of gener ation and consumption of free radicals are the same; therefore, the steady-state approximation is valid; (2) recombination is the dom inant form of termination; (3) initiation, propagation, and termi nation are determined by the reaction rate constants (that is, there is no poisoning or pyrolysis); (4) the monomer consumption by chain initiation is relatively small. The errors incurred through use of the steady-state approximation have been analyzed by Beisenberger and Capinpin (1972). Their study reveals that relatively small errors are witnessed in the range of perturbations and for the response times (that is, those less than one second) for a study of polyethylene thermal decomposition. The overall polymerization rate then becomes (Weale, 1967)

(4) r pl . = where ki' kp' and kt are the previously defined rate constants. CE and c. represent the concentrations of ethylene and the initiator, 1 respectively. Weale (1967) and Shrier et al. (1965) have both shown that the temperature and pressure dependence of the polymeriza tion rate constants should be expressed as functions of 11 energies and volumes of activation. This effect will be included in this analysis. The initiator efficiency (f.) is usually considered 1 to be unity for modeling purposes (Flory, 1953).

Polymerization Rate Constants Before proceeding with the mathematical derivation, we present a brief discussion of the functional dependence of the polymerization rate constants. Norrish and Smith (1942) first discussed the gel effect in high-conversion, free-radical polymerizations. The gel effect in polymer reactions is due to a decrease in the mobility of molecular free radicals as a result of increasing viscosity. The termination step becomes greatly restricted because it is diffusion limited at the higher viscosities. It may be observed from the polymerization rate expression that a decrease in the termination rate constant, resulting from an increase in viscosity, would lead to accelerated polymerization rates. Unfortunately, the reaction rate expression becomes considerably more complicated as conversion increases (Gladyshev and Rafikov, 1966) because (1) the steady-state approximation may no longer be considered valid, (2) the normal second-order mechanism of the termination rate constant cannot be justified rigorously, and (3) in general, there is believed to be an increasing number of side reactions. It has been observed that in regions of high viscosity there is a significant in crease in the primary free radical recombination effects. This is called the "cage effect" by Gladyshev and Rafikov (1966). Gladyshev 12

and Rafikov (1966) report that one type of recombination is that of the initiator itself, which in effect decreases the magnitude of the initiation rate constant. In terms of efficiency, higher viscosities

contribute to a decrease in the value of initiator efficiency 11 f 11 as the primary reaction becomes limited by hindered free radical move ment. There are studies underway to define further the functional dependency of the propagation rate constant toward inaccessibility to molecular free radicals resulting from diffusion limitations (Nishimura, 1966). At the same time, there is considerable evidence that the major result of high-polymer conversions lies in the diffusion-limited termination step (Biesenberger and Capinpin, 1972). The effect of viscosity on the termination rate constant was utilized quite successfully by Knorr and O'Driscoll (1970) in analyzing high conversions in a polystyrene reactor. For the present polyethylene model, the termination rate constant will be used to describe the effects of viscosity changes in a reactor. It will be assumed that viscosity effects can be interpreted in terms of a diffusion-limited termination step and can be described by the Stokes-Einstein relationship (Einstein, 1906). That is,the diffusivity, and therefore the termination rate constant, is inversely proportional to the viscosity of the solution. This is exemplified in the Stokes Einstein equation (Einstein, 1906) by

(5) 13 where DAB is the diffusivity, k' is the Boltzmann constant, t re presents time, ~ is the viscosity of the solution, and r' is the molecular radius. Since there are no reported experimental data presently avail able on the viscosity LOPE solutions, the viscosity must be estimated indirectly. The viscosity (n) of a polymer is related to the mole cular weight (M) by the relationship

1 2 log n = A + BM 1 (6) where A and B are empirically determined constants. Flory (1953) has suggested that the weight-average molecular weight should be used in evaluating this equation; but more recent work on polyethylene has indicated that the viscosities of mixtures are somewhat lower than those calculated by weight-average molecular wiehgts (Ehrlich and Mortimer, 1970). This equation estimates the values of the solution viscosity, which are then assumed to be maxima. These calculations are very approximate, but they do give considerably better results than using the viscosity of ethylene alone. Average molecular weights will be used in our calculation for solution viscosity.

Values of Kinetic Parameters The difficulty in modeling initiator decomposition is the choice of numerical data to be used. Doehnert et ~· (1959), Raley et ~· (1948), and Ehrlich (1970) have all provided experimental kinetic data for initiator decompositions. Agrawal (1974) has re- 14 ported a very extensive collection of the various rate constants, using kinetic data reported in the literature. It can be assumed that impurities in the feed stream may act as pseudo-initiators and may be included in the overall polymerization reaction. For initial calculations, the ethylene feed stream was assumed to contain quantities of methane, ethane, and acetylene. The ethylene itself was also considered to be a possible source of free radicals. After reviewing the literature, the product distri bution data reported by Towell and Martin (1961), and Pearce and Newsome (1938) were determined to provide a consistent set of kinetic data. The data reported by Pearce and Newsome (1938) were especially important because they appear to represent the only data at pressures approaching normal operating conditions. These data are for the thermal decomposition of hexane at 14,000 psig. The data of Towell and Martin (1961) have significance because they provide a consistent set of data from 700°C to 1200°C for the decomposition of three key reactants: ethylene, acetylene, and ethane. In addition, the rate data are generally consistent with more recent data determined by sophisticated shock-wave techniques such as that reported by Back (1972). A comparison of these two data sets are given in Figure A-1 for the decomposition of ethylene from soooc to 1200°C. The mechanism of the decomposition appears to be consistent in all sets of data. For studies of LOPE reactors, one must distinguish between the main reactant and the initiator. Normally this will create two dif ferent forms of the material balance. It is possible, however, to 15 eliminate one of the material balances by assuming that the main react ant concentration does not vary. This method is especially suited for LOPE systems where the ethylene concentration is found to be nearly constant.

Since we assume the main reactant concentration to be constant, we need only to consider the material balances of the initiators. In many cases the thermal decomposition of the initiator is a first-order re action. The only difference in the material balance for different orders of decomposition lies 1n the exponential term of the rate constant. The overall polymerization kinetics must be written for a main initiator and a set of pseudo-initiators. The reactions become:

(7)

(8)

(9)

-+---- 2CH3 = 2y ( 10)

-~ CH' + H' 2<5 ( 11 ) 3 = For modeling purposes, certain assumptions must be made about these reactions. First, the initiator decompositions and reversible re actions are considered to behave independently. Also, all free radicals generated are assumed to be identical. Making these assumptions, we may write;

R' = 2(R' + a + 8 + y + 8) (12) t 16

So that the propagation reaction becomes: k R' + M P' ( 13) t ~

The termination step will remain unchanged.

Material and Energy Balances It is now possible to derive a mathematical expression utilizing the material and energy balances.

General Form of the Material and Energy Equations A material balance may be written for any of the initiators in the polymerization reaction as:

dC. 1 = -k. C~ exp(-E./RT) +~(C. -C.) ( 14) dt 10 1 1 Vp 10 1

where E. is the activation energy of the initiator, R is the gas 1 constant, T, F, and V represent the temperature, feed rate, and volume of the reactor, respectively. The density is represented

by p. At steady state we may assume that the concentration of the initiator is not a function of time. The material balance at steady state now becomes:

- k . C.n ex p ( - E . I RT ) + VF ( C. - C• s ) (15) 0 = 10 lS 1 S p 10 1

Similarly, we may write the energy balance as:

( 16) 17

where CP is the heat capacity, and ~Hr is the heat of reaction. Which at steady state becomes:

( 17)

Linearization of the Material and Energy Balances The preceding equations are non-linear due to the expon ential and flow rate terms; therefore, analytical solutions cannot be obtained. According to Amundson (1958), a linearized form of these equations is possible. Before proceeding, however, it is desirable to define the material and energy balances in terms of deviations from a steady-state value. At steady state the equations become, Material balance:

F 0 = -k. C.n exp(-E./RT) + _2_ (C. - C.s) (18) 10 1S 1 S p 10 1

Energy balance:

( 19)

For linearization, it will also be beneficial to present the equations in dimensionless form. Using the dimensionless

parameters described in Hoftyzer and Zwietering (1961) one may define the following terms.

RT x. = dimensionless temperature (20) 1 E.-E 1 18

(-~H ) k.R y. [ r 1 ] 21 n 1 = npC (E.-E) Ci dimensionless concentra- ( 21) p 1 tion of the initiator

z = kp t dimensionless time (22)

In these equations the composite activation energy (E) is de fined by the relationship:

E E + Et/2 + E./2 (23) = p 1 where Ep' Et' and Ei represent the activation energies of the propagation, termination and initiation reactions, respectively. It is also useful to define seven dimensionless parameters:

u. = E (24) 1 E. - E 1

E. - E 1 /\• (25) ' 1 =

(26)

2 (-~H .)k. R-- 2 b. = [ r1 1o ]n (27) 1 npC p (E.-E) 1

d F (28) = pkPov

CEAi g. = (29) 1 kPo

2 nikPo h . = (30) 1 Rk~/2kl/2 10 to 19

Deviation variables are also defined as

X. = X. - X. 1 1 1S temperature deviation (31)

concentration deviation (32)

In the manipulation of the energy equation the simplifying substitution can be made that since all initiators are at the same temperature

x. 1 (33) A·1

Substituting the dimensionless quantities into the material and energy balances, and then simplifying the equations through the dimensionless parameters, one obtains the following forms. Material balance:

n - ( 1+u.) 1 0 = -a.b.y. exp[ x. ] + d u. - d y. (34) 1 1 lS S S 10 S 1S 1

Energy balance:

(E :.E ) ( 1+u.) A. n/2 1 0 = E [g.h. exp[-( P t + A • ) ]y i • 1 1 1 2 Jx?- s 1 s 1 1S (35) d. + ~ (x. - x. ) A l 10 1S

It is now possible to form linear differential equations by taking the first terms of a Taylor series expansion for the 20 dimensionless forms of the material and energy balances. For stability studies this linearized approximation is valid for relatively small deviations from steady state. The Taylor series expansion, without higher order terms is given by

f(ys+Y,xs+X) = f(ys,xs) + Yf~(ys,xs) + Xf~(ys,xs) (36) where f~ and f~ are the partial derivatives of the function. Following the procedure outlined in Hoftyzer and Zwietering (1961) one may derive the following pair of linearized dimension less equations for the unsteady-state material and energy balances. Material balance:

(37)

Energy balance:

dX = E (38) dz 1 where

n.-1 - ( 1+u.) 1 -a.b.n.y. exp ( x. 1 ) - d (39) ali = 1 1 1 1 s 1S

n.-1 - ( 1+u.) 1 -a.b.n.y. exp ( ) 1 1 1 1S ~ xis (l2i = (40) x. 2 1S n./2 1 u.g.h.n.y. exp(-u;/A;X1s) s,; = 1 1 1 1 1S (41) x. 2 lS 21

n 2 2 - 1 g.h .n . Y·s exp(-u./A.X. ) 1 1 1 = 1 1 1 1S (42) 2

Analytical Solution and Stability Criteria The general solutions to the previous set of simultaneous linear differential equations have the forms, Material balance:

(43)

Energy balance:

(44) where B's are the constant's of integration and K's are roots of the characteristic equation. The initial consideration toward the stability of these equations is based on physical interpretations to the solutions of the steady state material and energy balances. The curve describing the energy balance equation does not generally depart significantly from a linear relationship; whereas, the material balance is usually sigmoid in shape (see Figure 2, curves MandL). The intersections of these curves become the steady-state points. Steady-state point b in Figure 2 is clearly unstable, since, for slightly higher temperatures the heat generated is greater than the heat dissipated. As a result, the temperature would continue to climb until steady-state point c was reached. Similarly, a slight decrease in temperature would transpose M

c:: 0 •r- Vl S QJ > c:: 0 u

0 Feed Temperature Temperature

Figure 2. Solution of material and energy balances for reversible exothermic reaction. (Levinspiel, 1972). N N 23 the system to steady-state point a. Therefore, stability requires that the gradient of the energy balance curve be greater than that of the material balance curve at the point of intersection. For most reactions, three steady states occur. It has been reported, however, for the polyethylene system. that as many as ftve steady states are possible (Hoftyzer and Zwietering, 1961). A further description of the stability of the system is obtained from the transient solutions of the equations for a given steady state. A particular steady state is considered stable if deviations in tem perature and concentration return to that steady-state point for in creasing time. This implies that v1 and x1 must approach zero for increasing time. The two coefficients K. are the roots of the characteristic 1 equation and may be determined by 2 ) l/2 (ali + s,i> ± [(ali + 6li) - 4(ali6li - a2i 62i ] K. = (45) 1 2 It is now desirable to introduce the stability criteria as pre sented by Aris and Amundson (1958). Following their nomenclature, one defines

(46) and (47)

Equation 46 and 47 may be substituted into equation 45 to yield 24

= -dfi + (df~- 4dai)l/2 K. 1 (48) 2

One may now introduce the initial conditions that at time z = o,

X; = XiO' Yi = Yio· Defining K1 as the larger of the two roots, one obtains the values of the coefficients as

- K . ) Y . + a . X. (ali 21 10 21 10 8li = (49) Kli K2i

8 = Y. 8 (50) 2i 10 - li

K .) X. - s .v. (Bli - 21 10 21 10 83i = (51) Kli - K2i

8 . = X. - 8 . (52) 41 10 31

For the characteristic equation to represent correctly a stable region (that is, deviations approach zero for increasing time), the real parts of both roots must be negative. This restriction implies that there are two stability criteria for a given steady state:

(53) and (54) d ai > 0

The latter inequality always is a more severe condition than the former. This implies that the curve describing da1 . will always envelop the curve characterizing dfi" 25

Computer Simulation This section presents the two basic methods of computer simulation applied in this study. The first of these utilizes iterative methods to obtain the steady-state points as described by the linearized material and energy balances. The second uses an analog scheme for solutions of the simultaneous differential equations when subjected to deviations in temperature or concentration. The steady-state model (see Appendix E) consists of two basic sections. The first section of the program characterizes the para meters found in the ethylene feed stream. Next, a reactor steady-state temperature is assumed. Using the steady-state material balance and this temperature, a reactor initiator concentration is calculated. The steady-state concentration is then entered in the energy balance from which a temperature may be derived. Comparing the latest tem perature to the assumed temperature, a check is made of the validity of the assumption. The procedure is then repeated for a new assumed temperature. The unsteady-state model (see Appendix E) was solved using an analog technique as outlined in diagram form in Figure 3. The Con tinuous System Modeling Program (CSMP) was utilized for solving the simultaneous differential equations on a digital computer ("Continuous System Modeling Program," 1972). The fourth-order Runge-Kutta routine was used for all integrations. For variable integration interval routines Simpson's Rule was used for error estimation. Step sizes of 0.001 seconds to 0.01 seconds were used when a fixed step method was employed. 26

y (0)

V·1

X(O)

Figure 3. Unsteady state analog scheme. CHAPTER III

APPLICATION TO A POLYETHYLENE REACTOR

Before applying the mathematical model presented in Chapter II, the assumptions employed in the model and their relationship to actual conditions in an industrial reactor will be reviewed. The assumptions were made to simplify the mathematics, but they seem to be realistically justified when the results are compared to the actual physical situation. The study incorporated more realistic aspects of industrial reactors than reported in earlier reactor models by including four, previously unstudied, effects. The effects included: l) the possibility that minor feed components and impurities may react as initiators; 2) pressure dependence of the polymerization rate constant; 3) variation of the termination rate constant with viscosity; and 4) simultaneous changes in flow rate and pressure. The additional effects introduced into the model appear to have increased the applicability of the model beyond that reported in earlier studies (Hoftyzer and Zwietering, 1961). The reactant mixture was considered to be completely homogeneous. This was considered to be a good approximation since the conversion to polymer in the reactor is seldom over twenty mass percent. This assumption is also considered a good approximation since industrial reactors operate in a single-phase regime with polyethylene dis solved in supercritical ethylene with a constant density. The re-

27 28 actor was also considered to be completely back-mixed and adiabatic. The relatively low conversion, constant density, and back-mixed re actor assumptions then lead to the assumption that the fluid pro perties are approximately the same as those of the pure ethylene (Hoftyzer and Zwietering, 1961). The initiation, propagation, and termination pre-exponential factors were assumed to be invariant over a normal range of operating parameters. The termination rate constant was considered to be a function of viscosity following the Stokes-Einstein relationship (Einstein, 1906). This relationship between the termination rate constant and viscosity was used in an analysis of high-conversion polystyrene reactors by Knorr and O'Driscoll (1970), and the validity of the assumption is discussed by Biesenberger and Capinpin (1972). All initiators and impurities in the feed stream were assumed to react independently or follow pure-component, thermal decomposition data. Furthermore, all free radicals generated by the various initiators were considered to have the same reactivity toward poly merization. We assumed that linearization of the material and energy balances by a Taylor series expansion is valid for small deviations from steady state, based on the calculations of Biesenberger and Capinpin (1972). The effect of pressure on the apparent energy of activation was included in our model. An additional contribution of pressure was also incorporated into the transient solutions by adding its effect 29 to the feed terms of the material and energy balances. This effect is discussed in Appendix D. Our study was undertaken to increase the physical and theoretical understanding of a polyethylene reactor behavior and decom position. The engineering goal of our study was to obtain a more realistic idea of how decompositions can be prevented and/or controlled.

Steady States of a Polymerization Reaction According to the modeling technique derived in Chapter II, and the assumptions presented in the previous section, calculations were made to obtain a family of operating points for a polyethylene re actor. The results of these calculations are plotted in Figure 4. This figure defines an operating map of a reactor using a dimension less, steady-state temperature as a function of a dimensionless inlet concentration of the main or dominant initiator. The family of curves represents steady-state operation for specific choices of a dimensionless inlet temperature. At each set of parameters pre sented in Figure 4, there is also a dimensionless steady-state free radical concentration. The values of the steady-state free-radical concentrations were not plotted to simplify the diagram; they are presented in tabular form following the figure. The operating parameters for this diagram are found in Appendix B. The dashed curves seen in Figure 4 describe the stability criteria presented in Chapter II. Curves "b" and "c" represent the points formed by the equations describing the limits for stability 8 Inlet Temperature Curves 1) 0.047 2) 0.052 e 3) 0.060 4) 0.065 c 5) 0.069 6) 0.074 Q) - - ~ 7) 0.080 ~ ~ 8) 0.086 10 ~ Q) c.. E Q) ...... ~

Q) ~ 10 ~ -- Cll I -- -- >, -- "0 1'0 -- Q) ~ Cll --- -- "'Cll X I 4

8 9 1010 1011 107 10 10 y , inlet catalysts concentration 0 w Figure 4. Steady states of a polymerization reaction. 0

, 31

(see Eq. 46 and 47). For further discussion, it is beneficial to use the nomenclature for these regions suggested by Hoftyzer and

11 11 Zwietering (1961). In the region to the left of curve b , both inequalities described by equations 46 and 47 hold and the reactor is inherently stable. This will be referred to as the SS region since it is completely stable. The points which are contained by curve "c" are in an area where neither inequality described by equations 46 and 47 is satisfied and therefore global instability is prevalent (Aris and Amundson, 1958). This area is signified as the UU region because it is extremely unstable. The zone between curves "b" and "c" satisfies only the inequality described by equation 46 and is considered metastable. Since it lies between the regions designated as SS and UU, it will be referred to as the SU region. There are several important zones in Figure 4 that illustrate the operation of a polyethylene reactor. The region indicated by the dashed lines about point "a" is the zone that incorporates the normal operating conditions of most industrial reactors. The area "a" is here defined by a temperature range of 225 oc to 275°C and by a ten percent deviation in inlet concentration from the assumed value cited in Appendix B. Region "a", of course, must only be considered an approximation of actual operating con ditions as restricted by the simplifying assumptions made in de veloping the model. Nevertheless, it is worthwhile to note that the area falls extremely close to the UU region. This fact has been 32

verified by earlier studies (Hoftyzer and Zwietering, 1961) and suggests that a reactor operating in the lower, right fringes of this region would be expected to be susceptible to unstable decom positions. In the zone of point "d", the reactor is at steady-state tem peratures which are essentially equivalent to the inlet temperature. At the conditions of point "d" there has been little thermal decom positions of the initiator and therefore the polymerization rate is

negligible. Near point 11 e" the entering catalyst is almost com pletely decomposed to free radicals. At such large free-radical concentrations, however, the reactor is completely uncontrollable. At point "f", the reactor and feed temperatures are both high and are practically equal; therefore, most of the initiator is decom posed before entering the reactor. In zone "f'' the inlet initiator concentration is so low within the reactor that very little poly merization takes place. Above the region indicated by point "g", the lines for constant inlet temperature turn off to the right. To show these lines an extension of Figure 4 into regions of higher steady-state temperatures can be seen in Figure D-1. In there gion of high steady-state temperatures, the initiator is almost completely decomposed, and one could suspect high polymerization rates. At these higher steady-state temperatures, ethylene probably decomposes to components such as carbon, methane, and possibly ethane. Thus, the actual polymerization is probably small. In addition, the higher temperatures are above the critical polymeric temperature (Flory, 1953). 33

The diagram of the steady-state values illustrates some other operating conditions. Consider, for instance, a reactor that is operating at a steady-state point where only a relatively small per centage of the initiator is decomposed. If the inlet initiator con centration is decreased at constant inlet temperature, the reactor would seek a new, higher, steady-state temperature. At the higher temperature it is possible for the initiator to decompose further, thus increasing free-radical concentrations. It is probable that in certain regions of the steady-state diagram an increase in poly merization would result from a decrease in the inlet initiator concentration. There is also a corollary to this statement. It would be possible to terminate a polymerization reaction by increasing the inlet catalyst concentration for some operating points of an industrial reactor. These latter statements have not been tested and must be considered as ideal situations which might not be achieved in an actual reactor. The ideas do, however, reveal the importance of the choice of the temperature-dependent initiator. A reactor that is normally.operating in the region of point "a" in Figure 4, may easily enter the UU region by arbitrarily forcing a constant inlet concentration (or operating temperature, or inlet feed temperature), and by changing the remaining parameters. This fact reveals the extreme importance of the interactions of all of the parameters when control of an industrial polyethylene reactor is desired. The effects of these changes will be discussed in detail later in our study. 34

Still another facet of Figure 4 is the number of steady states that exist for given values of inlet initiator concentration. These steady states may be detected by observing the intersections of a constant-inlet concentration line and an inlet temperature curve. Recalling the first stability criterion, one can see from Figure 4 that numerous inlet temperature curves have three possible inter sections. The lowest of these intersections is found in the SS region. If Figure 4 were extended to higher operating temperatures, it would be seen that the last intersection would also lie in the SS region (see Figure D-1). It is interesting to note that a set of five steady states can be found along an inlet concentration line 8 of about 10 (see Figure D-1). This has been reported in previous models as a phenomenon of consecutive or parallel reactions with different activation energies (Hoftyzer and Zwietering, 1961). Other diagrams similar to Figure 4 may be calculated using different numerical parameters and different initiators, but all results show the same general character. In general, then, we conclude that all of the previously discussed stability criteria can be illustrated by a thorough interpretation of this diagram.

Transient Responses of the Stability Regions Each of the stability regions discussed 1n Chapter II has a unique transient response to a perturbation of the reactor steady state temperature or concentration. The entire problem of reactor stability may, however, be determined by the ability of a given 3S

steady state to return to its original position following a per turbation (Aris and Amundson, 19S8). Our analysis was developed by imposing an impulse disturbance on the system at time equal to zero. The unsteady-state equations were then used to characterize the system•s response. Figures Sa and Sb show the transient responses of the reactor to an impulse in temperature. Figures 5c and 5d show similar responses to a concentration perturbation. In each of these illustrations, the other deviation variable was assumed to be zero. The direct responses of the ordinate have been normalized to a 0.0- 1.0 scale by dividing by X(O) and Y(O). This normalization illustrates the reactor stability of a region as deviations swing from 0.0 to 1.0 but then return to zero for longer times. The group Yx /X(O)y is utilized to display 0 0 responses in concentration to changes in temperature. Similarly, the group Xy /x Y(O) is used to relate responses in temperature to changes 0 0 in concentration. These normalizations make the magnitudes of the response instabilities independent of the size of the perturbations. The size of the perturbations has been tested and no changes in the results were observed for different sizes of perturbations. Figure Sa shows that for a temperature perturbation, the SS region swings from 0.0 to 1.0, but then returns to the original steady-state point for increasing time. The temperatures for the SU and UU regions, follow unbounded increases. This implies that these regions are considered inherently unstable. It is also noted that the UU region tends to respond considerably faster to the up- 36

Lt') 0 ~

"'c .,..0 ~ C'l C1J 5- >, .,..+o) .,..~ .J:J tO M ~ 0 ~ "'C1J

-'=+o)

~ "'C"'c 0 0 u C1J C1J "' c "'0 0 "'.. a. ~ C1J E C1J .,.. "'5- 1-- +o) c .,..C1J c "'tO 5- 0 1-- ~

5- tO 0 Lt') ~ 00 C1J C1J 5- X 0::"'-- ~

0 Lt') 0 • 0 0 (O)X X 0.0 I ~ I ...... : I I I I :;;:po= I

1. 0

Response for Y(O) = 0 2.0 0 X(O) > 0 >< >- 0 ~>,

3.0

4.0 uu

2 3 1 • 0 10 10 10 105 Time, seconds

w Figure 5b. Transient responses of the stability regions...... 1 1 • 0 I I

0.5

>-~ Response for Y(O) > 0 X(O) = 0 0.0 ss

2 0. 1 1. 0 10 10 Time, seconds w Figure 5c. Transient responses of the stability regions. CX> uu 0.4

Response for Y(O) > 0 0.3 X(O) = 0

><>,05 >- 0 X 0.2

0. 1

____,.,- I I j I I 3 ::=;....,. 0. 0 I 102 10 4 0.1 1 .0 10 10 Time, seconds w Figure 5d. Transient responses of the stability regions. \0 40

TABLE I PARAMETERS FOR THE STABILITY REGIONS

Region ss su uu

8 8 Yo 1.13xlo 7.00 X 10 1. 09 X 10 l O 1 6 6 1. 08 X 10 O Yss 1.69 X 10 6.53 X 10 2 2 2 xo 7.40 X 10- 6.00 X 10- 6.90 X 10- 2 2 2 7.40 X 10- xss 8.90 X 10- 9.01 X 10-

16 6 -2.28 X 10-l8 al -8.37 X 10- -8.35 X 10-l 31 32 31 -4.72 X l0- a.2 -5.83 X 10- -2.27 X l0- 8 6 s, 1.98xlo-18 5.43 X l0-l 3.01 X 10- -2 -2 3.64 X l0-l6 B2 3.74 X 10 1. 89 X 10

34 34 34 5.22 X 10- -2.34 X 10- -8.28 X 10- 16 16 16 8.35 X 10- 8.31 X 10- -3.6.3 X l0- 41

set than the SU region. This fact reveals the importance of iden tifying the steady-state operating condition of a given commercial reactor. Hoftyzer and Zwietering (1961) indicate that upsets in the SU region can be controlled by adding a proportional plus reset controllers. Although the responses in the SU and UU regions appear to require several minutes to reach an unbounded state, it will be shown later that response times less than one second can be obtained. These results agreed closer with the response times that are generally observed in commercial reactors.

Transient Responses in the General Operating Region From the analysis of the transient response to perturbations in the various stability regions given above, it becomes obvious that the location of a steady-state operating point is the most significant aspect in describing the behavior of a reactor. The general operating region, described by point .. a .. in Figure 4, borders the UU region. Therefore, the response of the system in various areas of this region will be examined more closely. Figures 6a, 6b, 6c, and 6d describe the effect of increasing the inlet catalyst concentration along an isothermal operating line. Ordinates are utilized as previously defined. It is most significant that near y equal to 1010 , and x equal to 0.0866, the steady-state 0 s parameters cross the UU boundary. At this point, the response to a temperature perturbation becomes unbounded in less than one second. The concentration responses are somewhat slower, taking several l. l 0

Response for Y(O) = 0 X(O) > 0 xs = 0.0866 ><~

l. 05

0.3 0.6 l . 0 3.0 6.0 10.0 \ Time, seconds +::. Figure 6a. Transient responses in the operating region at various concentrations. N 0.0 r -~ '

0. 1

>-xo~~0 >,

a Response for 0.2 I- Y(O) = 0 X(O) > 0 xs = 0.0866

0.3 0.6 1. 0 3.0 6.0 10.0 Time, seconds Figure 6b. Transient responses in the operating region at various concentrations. +=w 1. 0

Response for Y(O) > 0 0.5 X(O) = 0 X 0.0866 s = >-~-

0.0 e

3.0 6.0 10.0 30.0 60.0 100.0 Time, seconds

~ Figure 6c. Transient responses in the operating region at various concentrations. ~ 0.3

Response for Y(O) > 0 X(O) = 0 xs.= 0.0866 0.2

><>,0~~ 0 X

0. 1

o. o...... ::;=------;~:::::.._--ia~~~-==:::::::::::~;---~:::=::~-_j3.0 6.0 10.0 30.0 60.0 100.0 Time, seconds

~ Figure 6d. Transient responses in the operating region at various concentrations. <.n TABLE II PARAMETERS FOR TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS CONCENTRATIONS

Curve 9 9 9 8 8 Yo 1 . 09 X 10 l O 2.72 X 10 1.75x10 1 . 09 X 10 2.72 X 10 1. 75 X 10 8 8 7 7 7 6 Yss 5.12 X 10 1. 28 X 10 8.25 X 10 5.12x10 1. 28 X 10 8.25 X 10 -2 -2 -2 -2 -2 -2 xo 3.00 X 10 4.00 X 10 4.70 X 10 5.20 X 10 6.70 X 10 6.90 X 10 -2 -2 -2 -2 -2 -2 xss 8.66 X 10 8.66 X 10 8.66 X 10 8.66 X 10 8.66 X 10 8.66 X 10 6 16 6 16 16 16 al -2.50 X 10-l -2.50 X 10- -2.50 X 10-l -2.50 X 10- -2.50 X 10- -2.50 X 10- 28 28 29 29 29 30 a2 -5.35 X 10- -1.33 X 10- -8.55 X l0- -5.35 X 10- -1.33 X 10- -8.55 X 10- 6 16 X 6 8.79 X 10-l 7 X 17 3.42 X 10-l 7 81 2.81 X 10-l 1. 39 X 10- 1.11 10-l 4.30 10- 5 -4 -4 4 4 4 82 9.72 X 10- 1. 95 X 10 2.43 X 10 3.07 X 10- 6.16 X 10- 7.69 X 10- 32 33 33 33 33 33 Da -1 . 84 X. 10- -8.98 X 10- -7.12 X 10- -5.55 X 10- -2.57 X 10- -1.98 X 10- 17 16 16 16 16 16 of -3.11 X 10- 1.11 X 10- 1.39 X 10- 1. 62 x 1o- 2.07 X 10- 2.16x1o- : \ \

~ m 47 seconds, but they are considerably faster than any of the previously reported response times. It is realized that in such a sensitive region, reactor control or precautionary vent action would be nearly impossible because most controllers and vent valves have time constants on the order of one second. A similar set of curves is presented for movement along a con stant inlet initiator concentration, y equal to 8.0 x 108, for de- o creasing steady-state temperatures (Figures 7a, 7b, 7c, and 7d). The range of this analysis is considerably more restricted than the former analysis, because the operating points lie extremely close to the UU region. Once the reaction enters the area defined as the UU region the low temperatures severely limit the thermal decomposition of the initiator. This thermal limitation creates only small changes in the free-radical concentration, and yields less drastic response changes. These curves, however, show a response time of about six seconds. Fast responses would indicate that a very sensitive control system would have to be developed for an industrial reactor.

Perturbation Analysis It was determined that any movement from the operating region into the UU region will create an uncontrollable transient response. The next step in this study was to consider possible manipulations of other operating parameters that would move the system into the UU region. 1.10

><~- 1. 05

Response for Y(O) = 0 X(O) > 0 8 Y 8 X 10 0 =

0.6 1 . 0 3.0 6.0 10.0 Time, seconds

~ Figure 7a. Transient responses in the operating region at various temperatures. 00 0.0 I I ~ ...... I I I I

0. 1

~0~0 Response for 0.2 Y(O) = 0 X(O) > 0 8 -=-\ y = 8 X 10 0

~en -4 ~'.;f-l c~ ~ c. efJ 0.6 1.0 3.0 6.0 10.0 ~ ~~ Time, seconds ~ ~ Figure 7b. Transient responses in the operating region at various temperatures. \.0 1. 0

g h

0.5 Response for Y(O) > 0 X(O) = 0 8 y = 8 X 10 0 >-~

0.0

30.0 60.0 100.0 300.0 600.0 1000.0 Time, seconds

U'1 Figure 7c. Transient responses in the operating region at various temperatures. 0 0.3

0.2

>,OB>- Response for >< 0 >< Y(O) > 0 X(O) = 0 8 y = 8 X 10 0. 1 0

0.0 30.0 60.0 100.0 300.0 600.0 1000.0 Time, seconds Figure 7d. Transient responses in the operating region at various temperatures. U'1__.. TABLE III PARAMETERS FOR TRANSIENT RESPONSES IN THE OPERATING REGION AT VARIOUS TEMPERATURES

Curve g h ; j k 8 8 8 8 8 y 8.00 X 10 8.00 X 10 8.00 X 10 8.00 X 10 8.00 X 10 0 8 7 7 7 7 Yss 1. 32 X 10 7.20 X 10 3.92 X 10 2. 13 X 10 1 . 17 X 10 2 -2 2 -2 2 xo 6.60 X 10- 6.20 X 10 6.00 X 10- 5.40 X 10 5.20 X 10- -2 -2 -2 -2 2 xss 8.30 X 10 8.50 X 10 8.66 X 10 8.80 X 10 8.99 X 10- 17 6 16 6 5 etl -6.94 X 10- -1.65 X 10-l -3.20 X 10- -5.64 X 10-l -1.18 X 10-l 29 29 29 29 29 a2 -4.02 X 10- -5.02 X 10- -5. 15 X 10- -4.78 X 10- -5.29 X 10- 3 -5 -4 -4 -4 1. 85 X 10- e1 7.98 X 10 1. 95 X 10 4.17xl0 8.30 X 10 7 7 16 X 7 7.98 X 10-l 7 8.95 X 10-l 9.39 X 10-l 1.10 X 10- 82 6.23 10-l 33 33 33 32 32 Da -1.12x1o- -3.33 X 10- -7.22 X 10- -1.33 X 10- -3.29 X 10- 18 17 16 6 5 Df 7.08 X 10- 8.47 X 10- 2.31 X 10- 4.70 X 10-l 1.07 X 10-l

U"1 N 53

Effect of Feed Stream Impurities It is widely understood by industrial personnel that im- . purities in the ethylene feed stream can have a significant effect on the operability of an LOPE reactor system. As the exact details of the effects of the various impurities are not well understood, three possible feed stream impurities were introduced into the system as pseudo-initiators. Methane, ethane, and acetylene were all considered as sources of free radicals. The ethylene itself was also deemed a possible free radical source. The thermal decomposition of methane and ethane were insignificant at the operating temperatures of the model. The ethylene decomposed less than one tenth of one

11 11 percent at the conditions of interest around point a • The amount of acetylene which has decomposed, on the other hand, is considerably higher. At normal operating conditions (temperatures of 250°C and 35,000 psia pressures) only five percent of the acetylene decomposed. However, as much as thirty to forty percent of the acetylene may decompose at slightly higher temperatures (see Figure 8). It is apparent that, of the impurities studied, acetylene is the most likely source of uncontrolled free radicals. Table IV shows the effect of an increase in acetylene concentration in the ethylene feed stream on the steady-state parameters. There were only small changes in the operating temperatures for concentrations of less than 100 ppm. Above these conditions, the acetylene becomes extremely 54

0 0 ('i")

0 0'\ N

0 CX) N

0 ,...... r::: N .,....0 .,....+.l (/) 0 c. 0 E '-0 0 N u u 0 .. Q) Q) "Q) ~ r::: ~ Q) 0 ,....- L.O +.l N n::s ~ ~ Q) Q) c. u E c( Q) 1-- • 0 CX) ~ N Q) ~ ~ C') .•r- LL.. 0 ('i") N

0 N N

0 0'\ CX) . 0 0 0 UO ~ SJtMUOJ - l 55

TABLE IV

EFFECT OF ACETYLENE CONCENTRATION ON THE STEADY-STATE PARAMETERS xo = 0.054 Y = 8.00 x 108 0 Acetylene Steady-state temperature (ppm) xs

1 0.0866 5 0.0866 10 0.0866 25 0.0866 50 0.0865 75 0.0865 100 0.0864 250 0.0863 500 0.0861 750 0.0858 1000 0.0854 2500 0.0848 5000 0.0839 7500 0.0824 56

important, moving the system into the UU region at concentrations in the range of 1,000 ppm. As has been verified industrially, acetylene concentration in the feed stream becomes a very signif icant parameter--especially at high concentrations. For this reason it is recommended that LOPE reactors be protected from random "slugs" of acetylene or other easily decomposed compounds.

Effect of Initiator Choice on Stability The choice of the main initiator on the polymerization reaction is one of the most important considerations in reactor design. Initiators are usually divided into classes according to their elemental structure and dependency of their decomposition on temperature. DTBP is one of the higher temperature initiators, and it follows that lower decompositions of this initiator are realized at lower steady-state temperatures. This results in greater fluctuations in free-radical concentrations due to fluctuations in reactor temperature. A low temperature initiator would be completely decomposed and therefore would not be affected by the temperature fluctuations. Figure 9 gives an idea of how the operating region of a reactor may be transposed from the UU region by the utilization of a low temperature initiator such as azobisisobutyronitrite. Approximately ten times as much of this low temperature initiator has decomposed as that of DTBP at the operating tem peratures of a normal reactor. This drastic change in actual \ \ \X = 0.052 \ x =0.060 0 \ 0.09~ \ s...C1J .....,:::J I ItSs... C1J df 0.. • E ( ~~ -- C1J -- ....., - - -- .....,C1J \ ( .....,ItS (/) "'-. I ' "'-. ~ ' ' ItS """" """"'- .....,C1J ~ (/) -- .. 0.07 ------(/) ------d >< -- a------0.06 r

8 9 lo 10 1010 y , inlet catalysts concentration 0 0'1 Figure 9. ·Effect of temperature sensitive initiators...... 58 free-radical concentration explains the shift to a more stable region. It is not possible to compare di-rectly the dimension less parameters of the two different initiators because Figure 9 may not be viewed in a quantitative sense. This is a result of the different kinetic constants. The figure does disclose that extreme care should be used in the choice of polymerization initiators. Certainly, an initiator must be chosen to maximize conversion and product properties. This analysis suggests that an additional criterion for choosing an initiator is reactor stability. The results indicate that for maximum stability, the ideal initiator is one that completely decomposes at the operating conditions. This improved stability is achieved at such conditions because the reaction is already occurring at its maximum rate. The system is, therefore, consuming a max imum concentration of free radicals, assuming of course that no other initiators are present.

Effect of Thermal Decomposition of Ethylene Sets of LOPE steady states were also calculated to estimate the effect of the pyrolysis of ethylene or thermal polymerization (that is, no other initiators present). These calculations showed that yields of less than one percent conversion to polymer would be expected, even at temperatures far exceeding the ranges of industrial applicability (temperatures above 750°C). Low conversions are mainly due to the fact that ethylene is less 59 than one tenth of one percent decomposed at most conditions of interest. This means that only small concentrations of free radicals are present within the reactor to initiate polymeri zation. At extremely high temperatures (above 750°C), slightly larger conversions (still less than 5%) were calculated by the mathematical model; however, at these high temperatures the ethylene is more likely to decompose into methane, carbon, and possibly ethane.

Reactor Response to Changes in Viscosity and Pressure The calculations reported above do not approach the time scale of the unbounded responses that have been witnessed in dustrially (with the exception of those responses in the UU region). Two approaches could be applied to develop a more realistic model. One approach would be to investigate the effect of the nonlinearity of the system. Secondly, the analysis could be extended to include effects of viscosity and pressure or feed flow changes that might occur in a re actor. The second approach was chosen because mixing effects (hot spots, bearing failures, and two phase perturbations) are reported as major sources of decompositions. Furthermore, these effects have not been previously evaluated.

Effect of Viscosity and Diffusion-Limited Termination The possible effect of the termination rate constant, high conversion or high viscosity has been discussed in 60

Chapter I of this study. Briefly, it was proposed that high conversions or high viscosities within a polyethylene reactor could result in the diffusion-limited termination of a polymerization reaction. The diffusion-limited re action would in turn result in the generation of excess heat through the propagation step to further increase the conversion to polymer. Subsequently, this causes a further increase in viscosity by the interactions. This sequence would continue until a thermal decomposition occurred. To quantify the viscosity effect, it was assumed that the termination rate constant is inversely pro portional to viscosity according to the Stokes-Einstein equation (Einstein, 1906). This relationship between the termination rate constant and viscosity was used in an analysis of high-conversion polystyrene reactors by Knorr and O'Driscoll (1970). The validity of this assumption is discussed by Biesenberger and Capinpin (1972). These previous studies both assume that the termination rate constant is inversely proportional to some function of viscosity. Knorr and O'Driscoll (1970) assumed that the relationship is a simple inverse proportionality. Biesen berger and Capinpin (1972) made calculations using several, more severe effects of viscosity. Their model included inverse exponential and inverse power relationships. For this analysis, the more conservation relationship of 61

Knorr and o•oriscoll (1970) was assumed to adequately de scribe the viscosity effect on the termination rate con stant.

The viscosity effect was incorporated in the termin ation rate constant by using the following equation

(55) where kt is the termination rate constant and ~ is the viscosity. The constant, kto' was estimated using the termination rate constant as the accepted viscosity 9 (kto = 1.0 x 10 at~= 11.0 cp)(Flory, 1953). The dependence of the termination rate constant on viscosity may be interpreted in terms of concentration by the following relationship (for details of this derivation, see Appendix D).

where Ynew is the concentration at the current viscosity of interest and Yoriginal is the concentration before the viscosity change was incorporated. The term ~R represents the ratio of the new viscosity to the original viscosity. Using this equation, the effective change in initiator concentration as a function of viscosity may be calculated. The results of these calculations are shown in Table V. These results reveal dramatic changes in the effective 62

TABLE V

EFFECT OF VISCOSITY CHANGES ON THE INITIATOR CONCENTRATION

Weight Percent Relative Effective Polyethylene Viscosity Concentration (14.1% as basis)

0.0 1 . 0 0.158

3.9 6.6 0.392

7.5 14.0 0.549

14. 1 79.0 1.0

20.0 113.0 1.085

30.0 249.0 1.23 63 concentration of the initiator at extremely high and very low polymer conversions. For example, for a relative viscosity change from 1.0 to 6.6 the effective concentration doubles. The effect is less significant for viscosity changes of less than ten percent. The viscosity effect may also be incorporated into the transient response equations. This was accomplished by including the viscosity effect in the original material and energy balances through the termination rate constant (for details of this derivation, see Appendix D). Again, the termination rate constant is assumed to follow the Stokes-Einstein relationship (Einstein, 1906). This mod ification does not affect the basic form of the unsteady state material balance and the linearization technique leads to an extra term in the unsteady-state energy balance. For example, the unsteady-state energy balance becomes

dX = ~ (8 .Y. + e .X. + e !X.) (57) dz i 11 1 2 1 1 3 1 1 where is a dimensionless steady-state parameter which e3i couples the viscosity to a temperature deviation. The calculations performed with this modification re veals important differences when compared to the pre vious responses. As an example, curve 11 b", in Figure 6a (y = 2.72 x 10 9 , x = 4.00 x 10 -2 , xs = 0.0866 ) has a 0 0 64 response time of six seconds. Figure 4 shows that this steady-state operating point is in the SU region. When the viscosity effect is included, however, Figure 10, curve

"b1.. , shows that the response is decreased to 0.63 seconds. For these deviations, all perturbations were made through the temperature deviation variable. This calculation implies that high viscosity coupled with diffusion-limited termination of the reaction may be a major factor in the in stability of the polyethylene system, i.e., the response time was decreased by a factor of 9.0. The effect of viscosity was incorporated into the transient response equations for other sets of steady- state parameters. As an example, for curve "a" in Figure 9 -2 6a (y = 1.09 x 10, x = 3.00 x 10 , xs = 0.0866), the 0 0 response has been previously calculated as about two seconds. Figure 4 shows that this steady-state operating point is in the UU region. By incorporating the viscosity effect into the transient equations, this response time i.s reduced to 0.28 seconds. In general, then, the effect of incot·porating the viscosity into the unsteady-state equations is to decrease the response times by an order of magnitude less than the same calculations made while omitting the viscosity effect. This implies that responses for steady-state points within the SU region will be re duced by an order of magnitude, but does not mean that all operating points become totally unstable. There- 1. 10 b2 bl

Response for Y(O) = 0 X(O) > 0 xs = 0.0866

><~- 1. 05

0.3 0.6 1 . 0 3.0 6.0 Time, seconds

0'\ Figure 10. Viscosity and pressure effects on the transient responses. ()1 66 fore, control is considerably more sensitive than pre viously discussed. Once the steady-state point moves into the UU region, however, control becomes impossible. The diffusion-limited termination of the polymerization reaction has been reported / as a major concern of LOPE decompositions by Huffman and Bonner (1974}. Their proposed scheme is for deviations from normal operating conditions to create a high rate of polymerization, resulting in in creased viscosity within the reactor. At high viscosities the termination step of the reaction mechanism becomes diffusion-limited and results in the generation of excess heat. The generation of heat continues to increase the reaction rate until a decomposition occurs. Huffman and Bonner (1974) have reported that the decomposition may be completed within one second. Our study reveals that the diffusion-limited termin ation of the reaction is the most significant effect ob served. It is realized that these results could be limited by the parametric sensitivity of the analysis. However, the magnitude and time frame of these responses, using realistic viscosity and kinetic data (Flory, 1953) indicate the relative importance of the effect. It is important to note that many factors could ultimately lead to high viscosities within the polyethylene reactor, such as mechanical friction resulting in a reactor hot spot, 67 failures in mixing, phase changes, unstable impurities in the ethylene feed stream, or excess initiator. This in vestigation suggests that none of these effects could cause uncontrolled responses by themselves at their normal op erating values. However, when the effects are coupled to the diffusion-limitations of the termination rate constant, the analysis implies that all can become percursors of a decomposition.

Effect of Pressure The effects of discrete levels of operating pressure on the steady-state operation of a reactor were also cal culated. The numerical results of an operating pressure change in the system are given in Table VI. It was ob served that the steady-state operating point moved into the UU region for relative pressure increases of approximately 10% (that is, a pressure increase of 3,000 psia for a 35,000 psia original operating pressure). A 10% deviation should be an observable change within the accuracy control systems; hence the effect is probably only a secondary cause of decompositions. It would be important, however, in establishing the original conditions for a steady-state operation. The pressure effect may also be incorporated into the transient response equations. This was accomplished 68

TABLE VI

EFFECTS OF CHANGES IN OPERATING PRESSURE ON THE STEADY-STATE PARAMETERS

X0 = 0.054 8 Y0 = 8.00 x 10 Operating Relative Steady-state temperature Pressure Pressure xs (psi a)

35,000 1. 00 0.0880 36,000 1. 02 0.0871 37,000 1. 05 0.0862 38,000 1. 08 0.0854 39,000 1 . 11 0.0846 40,000 1 . 14 0.0837 41,000 1 . 17 0.0834 42,000 1. 20 0.0826 43,000 1. 23 0.0817 44,000 1. 26 0.0814 45,000 1.29 0.0806 46,000 1 . 31 0.0803 47,000 1. 34 0.0797 48,000 1. 37 0.0792 49,000 1. 40 0.0789 50,000 1. 43 0.0782 69 by including the pressure effect into the flow rate terms of the material and energy balances to simulate a loss in compressor efficiency and sonic flow through a discharge valve. For this analysis, the deviations in density may be assumed small. This was made because the density versus pressure curve is relatively flat in the range of pressures under consideration {pressures from 1.0 atm to 3,000 atm) (Breedreld and Prausnitz, 1973). The deviation in pressure can then be coupled to the deviation variable for temperature through an equation of state (for derivation, see Appendix D). The linearization technique simply leads to another term in the unsteady-state energy balance. The modified form of the unsteady-state energy balance becomes

dx = ~ (e .v. + e .x. + s ~x. + s ~x.) (sa) dz 1 111 2 11 3 11 411 where is a dimensionless steady-state parameter which e4i couples the pressure or feed stream effect to the temperature deviation. A similar modification may be made to the material balance. Figure 10 curve "b ", shows the added effect of pres 2 sure on the transient response time when the system witnesses a temperature perturbation, and when the vis cosity effect also included in it. The response time for this transient solution is 0.42 seconds versus 0.63 seconds for the viscosity effect alone. Therefore, the pressure 70 has an important effect on the response times when it is incorporated into the transient equations. The calculation also reveals that 0.42 seconds after the perturbation in terupts the system, the reactor pressure is in excess of 66,000 psia. For solutions of the transient equations which have faster response times, the magnitudes of the reactor pressure increase even more. For example, the pressure is above 80,000 psia for the response noted in 9 2 Figure 6a (y - 1.0 X 10 , X = 3.00 X 10- , X = 0.0866) 0 0 s at 0.42 seconds after the perturbation. The general form of the material and energy balances also shows that the decrease of flow by the ethylene feed would be almost instantaneous for a decomposition at the previously calculated conditions. For the response shown in Figure 10,

11 11 curve b2 , the ethylene feed is almost zero within 0.1 seconds after the perturbation. The dischange flow is not effected by the perturbation, but is limited by the velocity at which the pressure wave propagates through the reactor. Therefore, our calculations substantiate the instantaneous pressure rises that are witnessed in commercial reactors during a thermal decomposition. The viscosity and pressure effects appear to be the most important aspects of a thermal decomposition. If the effects are included with the possible operating fluctuations calculated in the early sections of this 71 analysis (that is, temperatures, initiator concentration, and feed stream impurities) they greatly increase the ap plicability of the model. Several sets of industrial data have been obtained from industrial sources. From these data a comparison was made of the results of this study. The magnitude and response times for the math ematical model were in good agreement. The dashed lines in Figure 10 show the range of the response times obtained from these data for an industrial polyethylene decomposition. Therefore, an overall conclusion is that this study has substantiated the sequence of events leading to a thermal polyethylene decomposition proposed by Huffman and Bonner (1974). CHAPTER IV

CONCLUSIONS AND RECOMMENDATIONS

The following conclusions may be drawn within the limitations of the mathematical model used in our analysis.

Conclusions 1. A polyethylene polymerization reaction appears to be in a metastable state at normal or commercial operating con ditions. 2. The addition of a diffusion-limited termination reaction coupled with viscosity changes was the most significant effect in approximating the extreme sensitivity generally observed during a decomposition. 3. A polyethylene polymerization reaction can be shifted from the metastable state to a completely unstable one by per turbations which exceed 500 ppm, and step changes in reactor pressure which exceed 10%. The thermal decomposition of ethylene itself was not found to be a significant variable in initiating either a decomposition or the polymerization reaction. A low temperature initiator shifted the reaction away from unstable operating zones.

72 73

Recommendations Reactor Operation 1. A reactor should be protected from random slugs of acetylene or other impurities which decompose to free radicals. 2. Low temperature initiators should be used wherever consistent with product quality to minimize the pro bability that a decomposition will occur. 3. In order to reduce further the probability of de compositions, those factors which affect viscosity or conversion, and hence the termination reaction, should be closely monitored.

Future Investigations 1. The non-linear model of the reactor system should be investigated to define further the precusors of polyethylene reaction decompositions. 2. An experimental study should be initiated to obtain a more accurate time frame and magnitude of temperatures and pressures during a polyethylene decomposition. LIST OF REFERENCES

Agrawal, S. C., uuse of Function Space Methods for the Analysis of Tubular Reactors with Axial Mixing, .. Ph.D. Thesis (Chern. Eng.), Polytechnic Institute of New York, Brooklyn, N.Y., (1974).

Agrawal, S. C., and C. D. Han, 11 Analysis of the High-Pressure Poly ethylene Tubular Reactor with Axial Mixing, .. Ph.D. Thesis (Chern. Eng.), Polytechnic Institute of New York, Brooklyn, N.Y., (1974). Aris, R., and N. R. Amundson, "Analysis of Chemical Reactor Stability and Contra 1," Chern. Eng. Sci., ]_, 121 ( 1958). Back, M. H., "Pyrolysis of Hydrocarbons, .. NBS Report {Special) 357, June, (1972).

Bamford, C. H., W. G. Barb, A. D. Jenkins, and P. F. Onyon, 11 The Kinetics of Vinyl Polymerization by Radical Mechanisms, .. Academic Press, New York, (1958).

Benson, S. W., and G. R. Haugen, 11 Mechanism for Some High-Temperature Gas Phase Reactions of Ethylene, Acetylene, and Butadiene," J. Phys. Chern., l_l, 1735 (1967). Biesenberger, J. A., and R. Capinpin, "The Quasi Stationary-State Approximation in Polymerization Kinetics," J. App. Polym. Sci., .l§_, 695 (1972). Brandrup, J., and E. H. Irrmergut, "Polymer Handbook, .. Interscience Publishers, New York, N.Y., (1958).

11 Continuous System Modeling Program, .. 5th ed., IBM Corporation, Technical Publications Dept., White Plains, N.Y., (1972). Cvetanovic, R. J., and R. S. Irwin, 11 Kinetic Reactions of the Methyl Free-Radical, .. J. Chern. Phys., 46, 2694 (1967).

Dingle, J. R., and D. J. LeRoy, 11 Kinetics of the Reaction of Atomic Hydrogen with Acetylene, 11 J. Chern. Phys. , ]_§_, 1632 ( 1950) ; p. 242 in "Chemical Kinetics of Gas Reactions, 11 V. N. Kondratiev (ed.), Pergammon Press, New York, (1964). Doehnert, D., 0. Mageli, and S. Stengle, "Correlation of Peroxide Half-Life with Polymerization," Ind. Plast. Mod.,.§_, 50 (1959). Ehrlich, P., and G. A. Mortimer, .. Fundamentals of the Free-Radical Polymerization of Ethylene," Adv. Polym. Sci., ?_, 386 (1970).

74 75

Ehrlich, P., and R. N. Pittilo, "A Kinetic Study of the Oxygen-Initiated Polymerization of Ethylene," J. Polym. Sci., 43, 389 (1960).

Ehrlich, P.,.R. ~· Pittilo, and J.D. Cotman, "The Oxygen-Initiated Polymer1zat1on of Ethylene and the Oxidation of Ethylene at High Pressure, " J. Polym. Sci., 32, 509 ( 1958). Einstein, A., Ann. Physik, 19, 371 (1906); p. 127 in "Mass Transport Phenomena,'' C. J. Geankoplis, Holt, Rinehart, and Winston, Inc., (1971}.

Flory, P. J., "Principle of Polymer Chemistry," Cornell University Press, Ithaca, N.Y., (1953). Frost, A. A., and R. G. Pearson, "Kinetics and Mechanism," 2nd ed., John Wiley, New York, (1962). Gladyshev, G. P., and S. R. Rafikov, "Bulk Polymerization of Vinyl Monomers at High Extents of Reaction," Russian Chern. Rev., 35, 405 (1966). Goldstein, R. P., and N. R. Amundson, "An Analysis of Chemical Reactor Stability and Control-Xa," Chern. Eng. Sci., 20, 195 (1965). Goldstein, R., and C. S. Hwa, "Modeling and Analysis of Complex Re actors by Computer Simulation," paper presented at the American Chemical Society Symposium on Analysis of Complex Reaction Systems, Pittsburgh, PA., March 23-26, (1966). Hahn, A., A. Chaptal, J. Sialelli, "Why Olefin Specs Have Changed," Hydrocarbon Proc., 54, 89 (1975). Holftyzer, P. J., and T. N. Zwietering, "The Characteristics of a Homogenized Reactor for the Polymerization of Ethylene," Chern. Eng. Sci., ji, 241 (1961). Huffman, W. J., and D. C. Bonner, "Decompositions in Polyethylene Reactors: A Theoretical Study," paper presented at 67th Annual Meeting of the American Institute of Chemical Engineers (Paper No. 63d), December 3, (1974). Kerr, J. A., "Bond Dissociation Energies by Kinetic Methods," Chern. Rev., 66, 465 (1966). Knorr, R. S., and K. F. O'Driscoll, "Multiple Steady States, Viscosity, and High Conversion in Continuous Free-Radical Polymerization," J. Appl. Polym. Sci., }i, 2683 (1970). 76

Kungi, T., T. Sakai, K. Soma, Y. Sasai, "Kinetics and Mechanism of the Thermal Reaction of Ethylene," Ind. Eng. Chern., 8, 1969 (1969). Laidler, K. J., and L. F. Loucks, "Chemical Kinetics," C. H. Bamford a(nd C. F. Tipper (eds.), Vol. 5, Elsevier Pub. Co., New York, 19 72) . Levenspiel, 0., "Chemical Reaction Engineering," John Wiley and Sons, I nc . , New York , N. Y. , ( 19 72 ) . Matsuura, T., and M. Kato, "Concentration Stability of the Isothermal Reactor, .. Chern. Eng. Sci., 22, 171 (1967). Miles, D. C., and J. H. Briston, "Polymer Technology," Chemical Pub. Co., New York, (1965). Nishimura, N., "Kinetics of Diffusion-Controlled Free-Radical Poly merization, .. J. Macromol. Chern.,!, 257 {1966). Norrish, R. G. W., and R. R. Smith, "Mobility of Long-Chain Free Radical Polymers," Nature, 150, 336 {1942). Pearce, J. N., and J. W. Newsome, "Thermal Decomposition of Hexane at High Pressures," I.E.C., 30{5), 588 {1938). Raff, A. V., and J. B. Allison, "Polyethylene," Interscience Pub., New York, (1956). Raley, J. H., F. F. Rust, W. E. Vaughan, "Decompositions of Di-t-Alkyl Peroxides," J. Amer. Chern. Soc., 70, 88 {1948). ~- "Safety in High Pressure Polyethylene Plants," paper presented at 64th National Meeting of the American Institute of Chemical Engineers, Tulsa, Oklahoma, March 12-13, {1974). Shrier, A. L., B. F. Dodge, and R. H. Bretton, "Free-Radical Polymer ization of Ethylene at High Pressures, 11 paper presented at the American Institute of Chemical Engineers and British Institute of Chemical Engineers- Joint Meeting, June, (1965).

Smith, M. V., 11 Manufacture of Plastics," Reinhold Pub. Co., New York, (1964). Towell, G. D., and J. J. Martin, "Kinetic Data From Non-Isothermal Experiments: Thermal Decomposition of Ethane, Ethylene, and Acetylene," AIChE J., .lli}_, 693 (1961). Trotman-Dickenson, A. F., "Entropy Changes in Free-Radical Reactions," J. Chern. Phys. , n_, 211 ( 1953). 77

Trotman-Dickenson, A. F., and E. W. R. Stacie, 11 The Reactions of Methyl Radicals," J. Chern. Phys., ~, 169 (1951).

Van Heerden, C., 11 Steady States in Continuous Exothennic Processes, 11 Chern. Eng. Sci., 8, 133 (1958).

Voevodsky, V. V., and V. N. Kondratiev, "Progress in Reaction Kinetics, 11 G. Porter (ed.), Vol. 1, Pergammon Press, London, (1961). Warden, R. B. and N. R. Amundson, "Stability and Control of Addition Polymerization Reactions," Chern. Eng. Sci., .!.Z., 725 (1962). Weale, K. E., "Chemical Reactions at High Pressures," E. and F. N. Span, Ltd., London, England, (1967). Woodbrey, J. C. and P. Ehrlich, "The Free-Radical, High-Pressure Poly merization of Ethylene. II. The Evidence for Side Reactions from Polymer Structure and Number Average Molecular Weights," J. Amer. Chern. Soc., 85, 1580 (1963). NOMENCLATURE

A Flory polymer solution constant a dimensionless reaction rate constant B Flory polymer solution constant b dimensionless variable of the energy equation c concentration D diffusivity d dimensionless feed parameter E activation energy F molar feed rate g dimensionles~ energy balance variable h dimensionless energy balance variable

~Hr heat of reaction I initiator molecule

K root of a characteristic equation k reaction rate constant k1 Boltzman constant M monomer molecule M average molecular weight n order of the initiator decomposition P inactive polymer molecule P reactor pressure p• polymeric free radicals R gas constant 78 79

R' initiator free radicals Rt monomer free radicals r rate of polymerization r' molecular radius T absolute temperature t time u dimensionless material balance variable V reactor volume X temperature deviation variable x dimensionless temperature Y initiator concentration deviation variable y dimensionless initiator concentration z dimensionless time

Greek Letters a dimensionless variable of the linearized material balance S dimensionless variable of the linearized energy balance n relative viscosity A dimensionless energy of activation parameter

~ viscosity p density

Subscripts AB molecule A with respect to molecule B a of the UU region boundary E ethylene 80 f of the SS region boundary i of initiation

1 initiator number or identity p of polymerization r new with respect to original s at steady state t of termination

...... - APPENDIX A ULTIMATE REACTION CONDITIONS ATTAINED DURING A POLYETHYLENE REACTOR DECOMPOSITION

81 TABLE A-I ULTIMATE REACTION CONDITIONS ATTAINED DURING A POLYETHYLENE REACTOR DECOMPOSITION

Initial Ultimate Conditions Density* Reaction [1] Reaction [2]

Po ~T ~P/po ~T ~P/po g/cm 3 OK atm•cm 3/g OK atm•cm 3/g

0.45 1700 11338 540 3182

0.50 1310 8299 370 702

0.55 810 6184 350 -1197

* An initial temperature of 250°C was assumed for these calculations. This value approximates severe operating conditions where decompositions may be expected. Source: Huffman and Bonner (1974)

82 APPENDIX B

REACTOR OPERATING PARAMETERS AND KINETIC DATA

83 Reactor Operating Conditions and Physical Constants

Feed Stream Characteristics (at normal conditions): Component Volume Percent Ethylene 99.8 Methane 0.1 Ethane 0.1 Acetylene 5.0 (ppm)

Feed Rate - 250 mole/min. Reactor Volume - 575 cm3 Reactor Pressure - 35,000 psia Feed Temperature - 325°K 3 Density (ethylene) - 17.17 gm/cm Reactor Temperature - 250°C Gas Constant- 1.987 cal/mole oK Heat Capacity- 14.1 cal/mole °K Heat of Reaction - 22,350 cal/mole

84 TABLE B-1 REACTION CHEMISTRY AND KINETICS OF POLYETHYLENE DECOMPOSITIONS

Heat of Source Reactions Arrenhius Data Reaction 11V* Activation Preexponential At 25°C cm 3/mo1e I. Initiation Energy kea1/mo1e Factor kea1/mo1e 1 Initiator -+ 2R"" 30-40 1018 - 1019 + 5-10 + 9 5 2 2C2H4 -+ C2H3 + C2Hs 65 1014.7 + 69 -+ 8 2 3 2C2H2 -+ C2H"" + C2H3(=s) 60 1014.7 60.6 -+ 8 10 1014.9 4 CH'+ -+ CHi + H"" 103 +104 + 5 9 1017.9 5 C2H6 -+ 2CHi 80 88.2 + 5 1,10,11

II. ProQogation a= 7.5 a=107.2a a= -20 6 Pn_ R"" + C2H'+ abpfi ± 22 5 1 b=29.5 b=1014.6 b= + 4 a= 7.5 a=107.2a a= -20 P 8... + C2 H 4 a b Prfl ± 22 5 7 m-1 b=29.5 b=1014.6 b= + 4 1012 8 C2Hs + C2H4 -+ C2H6 + C2Hi 11 9.3 -10 8 1011 9 C2Hi + C2H4 -+ C4H6 + H"" 7 - 2.1 -10 8 1013.3 10 H"" + C2H4 -+ H2 + C2Hi 6.6 3.4 -10 13 1011 11 CH3 + C2H4 -+ CH4 + C2Hi 10 3.6 -10 12 1014.9 12 C2Hi -+ C2H2 + H"" 31.5 - 2. 1 + 5 2 (X) . 1014·4 01 13 C2Hs -+ C2H4 + H"" 41 38.6 + 5 9 TABLE B-1--Continued.

14 CHi + C2H6 -+ CH4 + C2Hs 11 . 8 1012.2 - 5.7 -10 3,7 15 H""' + C4H6 -+ C2H4 + C2Hi 1 . 0 109.8 2.1 -10 2 16 C2Hi + H2 -+ C2H4 + H""' 7.4 109.9 - 3.4 -10 2 17 H""' + C2H2 -+ H2 + C2H""' 5 1013 7.7 -10 18 C2H ... -+ 2C + H""' 38 1014.5 - 61.9 -15 10 19 H""' + CH4 -+ H2 + CH3 14.0 1014.2 - 0.2 -10 8 III. Termination 9 20 2Pfi -+ P2n 1. 0 10 -- +11 5 9 21 2Pm -+ P2m 1. 0 10 -- +11 5 22 2H ... -+ H2 0 10163 -104.2 + 5 6 23 2C2Hi -+ C4H6 2 101 1 -109.7 + 5 6 24 2CH3 -+ C2H6 0 101 33 - 88.2 + 5 1 25 CH3+ H' -+ CH4 0 1013 -104.0 + 5 Sources: 1. Back (1972) 11. Trotman-Dickenson (1953) 2. Benson and Haugen (1967) 12. Trotman-Dickenson and Stacie (1951) 3. Cvetanovic and Irwin (1967) 13. Voevodsky and Kondratiev (1961) 4. Dingle and LeRoy (1950) 5. Ehrlich and Mortimer (1970) 6. Frost and Pearson (1962) 7. Kerr (1966) 8. Kungi et al. (1969) 9. Laidler and Loucks (1972)

10. Towell and Martin (1961) ()) "' 87

104 r-~------r------.------~-----

\ X \ \ \ X \ \ 'X \ \ \~ \ \

'x\ X Towell & Martin, 1961 ~ \ A Back, 1972 ,, c H ~ 2CHJ \ 2 6 \

6 8 10 12 4 1/T, °K X 10 I I 1394 977 727 560 T, oc

Figure B-1. Comparison of data for decomposition of ethane. APPENDIX C FUNCTIONAL RELATIONSHIP OF THE TERMINATION RATE CONSTANT TO VISCOSITY AND PRESSURE EFFECTS

88 Viscosity Effect on Initiator Concentration The relationship for the rate of the polymerization reaction has been described in Chapter II as

(C-1)

This equation may be differentiated with respect to kt to obtain

= (C-2)

The defining expression for the Stokes-Einstein equation is

(C-3)

Assuming kt as proportional to DAB' one obtains

(C-4) where kto is the termination rate constant at a reference viscosity. Differentiation of this latest expression with respect to viscosity yields

-kto = =2 (C-5) ll where ll is the average viscosity over the range of ~ll· Combining the two differential equations by eliminating kt gives

(C-6) 90 where kt is the average termination rate constant. One may now define

rnew = roriginal + ~r (C-7)

and incorporate Eq. C-5 to obtain

kt 0 ~ll r = r (1 + ) (C-8) new 2l.12

or in terms of initiator concentration, and by defining llR = llnew/

lloriginal'

(C-9) Effect of Pressure on Feed Variations

For feed and output flow variations, the material and energy balances must be modified in the following manner. Material Balance:

dCi F C. FC. = -k. c~ exp{-E./RT) + 0 10 --1 (C-10) dt 10 1 1 p p

Energy Balance:

(C-11)

The variations of flow in these equations may be described as, Input flow:

(C-12) where ~p = p - p s (C-13) 5 e: = 1 X 10-

Output flow:

(C-14)

The equation of state utilized in the analysis is:

P = zpRT (C-15)

where P is the pressure, z is the compressibility factor, P is the density, R is the gas constant, and T is the temperature.

91 Viscosity Effect to the Transient Responses

The general energy balance is presented in Chapter II by Eq. 16 as

( C-1 6)

The rate of polymerization is given 1n Chapter II by Eq. 4 as

k. f. 1/2 r . = k ( 1 1) CE(c1.)n ( C-1 7) p1 p kt

The general energy balance may be manipulated to include the viscosity effect in the following manner. We can utilize the Stokes-Einstein relationship for the functional dependency of the termination rate constant on viscosity (Einstein, 1906). This procedure was discussed in Chapter II. We may then perform the Taylor's series expansion with respect to temperature, initiator concentration, and viscosity. Also, by incorporating the dimensionless parameters presented in Chapter II we obtain

dX = L (s .v. +.s .x. + s .M.) (C-18) dz is 11 1 21 1 3 1 1 where M l..1 - (C-19) = 0 l.ls s,i 8 = (C-20) 3i 2 l/2 lls

In this equation, Mdefines a viscosity deviation variable, and s3; 92 93 represents a constant that is a function of the steady-state parameters. We seek a constant that will mathematically relate the viscosity de viation to the temperature deviation. By assuming the concentration of ethylene to be constant with respect to time, and also assuming that the system is adiabatic, one obtains (Levenspiel, 1972) c !J.X = Mf- !J. T ( C- 21) r where !J.X is the deviation of conversion from a steady-state value. We also let the deviation in viscosity be a function of changes in conversion by

{C- 22)

Performing the necessary algebra, one obtains

M = ( C- 23)

The final form of the energy balance then becomes

dX E {C-24) = • I {slivi + s2ixi + s3ixi) dz 1 s

where l/2 ~s cpsli = {C- 25) S3; !J.H r APPENDIX D STEADY STATES FOR A POLYMERIZATION REACTION

94 xo = 0.074

Q) s.. :;::, I I I +-> co s.. r I Q) ~ ,.,1 I E ' Q) +-> ' Q) +-> co +-> (/) I >, ""C co Q) +-> (/).. (/) X

7 8 9 10 10 10 y , inlet catalyst concentration 0

~ U1 Figure D-1. Extended plot of the steady states for a polymerization reaction. APPENDIX E COMPUTER SIMULATIONS

96 ITERATIVE DI~ITAL SOLUTION FOR MATERIAL AND E~ERGY BALANCE

****************************************************** ******************************************************

A PROGRAM DESIGNED TO EVALUATE THE STEADY-~TATE OPERATING POINTS lJF A POLY!:THYLENE REACTO~ ****************************************************** ******************************************************

REAL KO,LAMO,K,NEWXSS DIMENSION N(5),E(7l,K0(7l,OELV(7),C0(7l,U(7),X0(7l, l G( 7), A( 7) ,8( 7) ,EP( 7) ,EPT( 7), XSS( 7), YSS( 7), 2 Z(7),X(7l,ALP1(7),/\LP2(7),BET1(7l,~ET2(7), 3 AK1(7) ,AK2(7) ,CAPX0(7) ,CAPY0(7),1\Rl (7), 4 YO( 7) ,LAMO( 7) ,YSSOYO( 200) ,OA( 7) ,OF( 7) ,AB2 (7) 5 ,AB3(7),A84(7),HH(7),CAPX(7),CAPY(7)

INPUT DATA N = REACTION ORO!:~ E = ENERGY OF ACTIVATION OELV = ACTIVATION V~LU~~ co = INITIATOR CONCE~TRATION F = FEED RATE v = REACTOR VOLUME KO = INITIATIO'J RATE CONSTANT -- CATA N,E,OELV,CO,F,V,K0/1,2,~.1,1,35000.,65000., * 62CCQ.,80000.,l03000.,7500.,1.,9.,-8.,-8.,5.,5.,-20. * ,11.,3.1639E-08,C.Ol717,.17E-09,.17E-09,.17E-09,0.0, * 0.0,25Q.0,575.179,2.8El4,5.011868Ell,5.011868Ell, * 7.943275El7,7.q43275El4,1.90546E7,l.OE09/ 1 FORMAT(4F1Q.O) 100 FORMAT('l',7El6.7,//) 101 FORMAT(5X,lF7.2,//) 102 FORMAT(5X,5El6.7,////) 103 FORMAT(5X,6El6.7,///)

1/0 DATA p = REACTOR PRESSURE TO = FEED TE MP Er{ AT U R C.: CE = ETHYLENE CONCENT~ATION RHO = DENSITY GASC = GAS CONSTI\~T HT C.~ P = HEAT CAPACITY D!:LHT = HEAT CF REACTION

READ( 5,1 l P, TO, CE, RHO 97 READ (5,11 GASC, HTCAP, DELHT 98 CONV = 0.005

DEFINE CCM~CSITE ACTIVATION ~NERGY

co 10 J= 1' 7 EP(Jl=E{J)+P*OELV(J) 10 CONTII\J~E

THIS IS THE INLET SECTIO~ 0~ THE PROG~~M THIS SECTIO"J SETS THE VALUES OF THE PARAMETERS I'\1 THE INLET FLOW STREAM OF THE REACTOR. C=V*Rt--C*K0(6) D=FIC DO 20 J= l, 5

A(J)=KO{J)IKC(6) EPT(Jl=EP(6)-EP(7)12 + EP(Jll2 Z( ll = l.OElC Z ( 2 ) =E:: X P ( 0 3. 2 3- 3 2. 6 7 IT \I ) Zf3);EXP(l3.43-37.101T0) Z(4);EXP(23.2l-40.281TG) l(5)=[XP( 14.0l-372.61fCJ) K=KO ( 6) * ( KO ( J) I ( ( KO ( J) I Z ( J)) +KO ( 7))) **:). 5 A(J)=(DELHT*K*GASCli(N(J)*RHO*HTCAP*fEP(J)- * EPT(J))) U( Jl=EPT(J )I(EP{J )-[PT(J)) XO(J) = ~ASC * TOifEP(Jl-EPT(J)) YO(J) = (R(J)**(211\J(J)l)*CO(J) t-- H ( J ) ; ( ~.J ( J ) *K 0 ( 6 ) * * 2 ) I ( R * ( K 0 ( J ) * t< 0 ( 7 ) ) * * 0 • 5 ) LA~O(J) = fEP(l)- EPT(J)) I (EP(J)- EPT(l)) G(J) ; CE * LA~D(J) I K0(6) GtJ) = G(J) * HH(J) 20 CONTINUE T = 425. THIS SECTION OF THE PROGRAM SOLVES THE MATERIAL AND E~ERGY BALANCE THIS IS AN ITERATIVE SOLUTION SECTIO~ A TEMPERATURE IS CHOOS~N TO SOLVE THE MAT=RIAL BALANCE , THE RESULTII\JG CONCEf\JTRATION IS THEN USED IN THF ENERGY BALANCE AS ~ TE~PERATURE CHECK

00 40 ITRATE = 1, 20 TOTAL=O T = T + 5.0 co 2 5 J = 1 '5 XSS ( J l= ( G.\SC*T) I ( EP( J )-EPT( J)) Z(l) = l.OE10 Z(2l=EXP(03.23-32.671T) Z t 3 ) =EX P ( 1 3. 4 3- 3 7. 10 IT ) Zf4)=EXP(23.21-40.281T) Z(5l=EXP(l4.01-372.61Tl co 30 J = 1 '5 99 K = K 0 ( 6 ) * (K 0 ( J ) I ( ( K 0 ( J ) I l { J ) ) + K 0 ( 7 ) ) ) * *J • '> 8(J)=(DELHT*K*GASC)I(~(J)*RHO*HTCAP*(EP(J)- * EPT(J))) L=N(J) IF(CO(J)-0.0)33,33,34 34 CO~TINUE CONA=-A( J )*( FXP (-( l+U( J)) I( LAMO(J l*XS * s ( 1 ) ) ) ) IF{~(Jl.EQ.2l:ON~=-A(J)*(EXP(-(l+U(J) * )I(XSS(1))))1B(J) CONB=-0 CO·\JC=D*YO(J) GO TO (31,32),L 31 Y S S ( J ) =-C0 N C I ( C 0 '·J A+ C0 N A ) GO TC 36 32 YSS(J)=f-CONB-SQOT(0**2-4*CONA*CJNC) * )1(2*CONA) IF ( ( -CONB-S12l-l )) *EXP *(-U(J)I(LAMO{J)*XSS(1)))*0.5 0A(J)=OETl(J)*ALPl(J)-BET2(Jl*ALP2(J) 0F(J)=-AETl(J)-ALP1(J) CAPXO(J)=ABS(XO(J)-XSS(J)) CAPYO(J)=ABS(YO(J)-YSS(J)) 37 CONTINUE wRITE(6, 101 lTP W~ITE(6,100) XSS(l),(Y<)S(IJl,IJ=l,5),~EWXSS W~ IT E ( 6, 1 0?) ( Y S SO YO ( I J) , I J = l , 5 ) wRITEC6,102) (XSS( IJ ), IJ=l, 5) wRITE(6,102)(XQ(IJ),IJ=1,5) WR I T E ( 6 , 1 () 2 ) ( Y0 ( I J ) , I J = 1 , 5 ) WRITE(6,102)(0A( IJ),IJ=1,5) wRITE(6,102) (DF( IJ ), IJ=l,5) 40 CONTINLE ANALOG SOLUTION FOR UNSTEADY-STATE CALCULATION ******************************************************

A PROGRhM TO MODEL THE U~STEAOY STATE RESPONSE OF A POLYETHYLE~E REACTCR UTILIZI~S THE CSMP PO~PJTER ANALOG SCHEME ******************************************************

TITLE POLYETHYLENE REACTO~ LABEL UNSTEADY STATE RESPJNSf

INTEGRATION TECH~IQUE - RUNGE-KUTTA FOURTH O~DER METHOD RKSFX

INPUT DATA OERIVEC FROM THE STEADY STATE ~no=L INITIAL

I~CCN CAPXO = 0.00173 I~CCN CAPYO = 0.0 CONSTANT AKlJ = 2.8E+l4 CCNSTANT ALPl = -2.50E-16 CCNSTAt\T ALP2 = -1.33E-?8 CONSTANT AETl = 1.39E-16 CONSTANT BET 2 = 1. g5E-04 CCr'\STAt\T BET3 = 3.237E-16 CONSTANT BET4 = 6.51E-17

SI~ULTANEOUS CIFFERENTIAL EQUATIONS OYNA~IC

ZT = TIME/AKO CYOZ = ALPl*YDEV+ALP2*XDEV OXOZ = BETl*XDEV+BET2*YDEV+RET3*XDEV+BET4*XDEV YOEV = INTGRL(CAPYO,OYOZ) XOEV = INTGRL(CAPXO,OXOZ)

TERMINAL

TIMER CELT=l.OE+l3, FI~TIM=l.OE+l5, OUTDEL=2.0E+l3 PRTPLT XOEV(ZT) PRTPLT YDEV(ZT) END STOP

100