Simulation of Pilot-and Industrial-Scale Vinyl Chloride Batch

Ind. Eng. Chem. Res. 2007, 46, 1179-1196

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PROCESS DESIGN AND CONTROL Simulation of Pilot- and Industrial-Scale Vinyl Chloride Batch Suspension Polymerization Reactors Joris Wieme, Tony De Roo, Guy B. Marin, and Geraldine J. Heynderickx* Laboratorium Voor Petrochemische Techniek, Ghent UniVersity, Krijgslaan 281 (S5), B-9000 Gent, Belgium

A complete model for the simulation of vinyl chloride batch suspension polymerization reactors, capable of calculating pilot-scale and industrial-scale reactors, is developed. The model solves the mass balances and moment equations, on one hand, and the energy balances for the suspension, reactor wall, and cooling jacket fluid on the other hand. The controller loop, which determines the flow rate or temperature of the cooling liquid, is also included. The kinetic model is based on calculated diffusional contributions and experimentally estimated intrinsic rate coefficients. [T. De Roo et al., Macromol. Symp. 2004, 206, 215; T. De Roo et al. Polymer 2005, 46, 8340.] This extended set of equations allows calculation of the monomer conversion, the moments of the molecular mass distribution, the reactor temperature and pressure, the cooling agent flow rate and temperature profiles, and possible thermal runaway and/or reactor vessel explosions. Simulated and experimental results for the pilot-scale reactor are in good agreement. 1. Introduction Poly(vinyl chloride) (PVC) is, by volume, the third-largest thermoplastic that is manufactured in the world, with annual demands close to 30 × 106 tons. Four processes are commercially applied: suspension, bulk, emulsion, and solution polymerization. Approximately 80% of the world’s PVC is produced using the suspension polymerization process, where droplets of vinyl chloride monomer are dispersed in water by a combination of stirring and the use of suspension agents. The polymerization occurs in the suspended droplets. For modeling purposes, each droplet can be treated as a bulk polymerization reactor. Three stages are distinguished during the vinyl chloride suspension polymerization process.3-8 Each stage is characterized by the number of phases present in the polymerization reactor. During the first stage, the polymerization occurs in the monomer phase; this is called the monomer-rich phase. Because the polymer is almost insoluble in its monomer, it almost immediately forms a separate phase in the monomer-rich phase, which is called the polymer-rich phase. This second stage starts at a monomer conversion of ∼0.1%.9-11 During the second stage, polymerization proceeds both in the monomer-rich phase and the polymer-rich phase. The polymer molecules that are formed in the monomer-rich phase are transferred to the polymer-rich phase, resulting in a constant composition of the latter phase of ∼30 wt % monomer and ∼70 wt % polymer. This composition is determined by the solubility of the monomer in the polymer-rich phase. Because of the constant composition of the polymer-rich phase and the increasing conversion of vinyl chloride, the monomer-rich phase decreases in volume, whereas the polymer-rich phase volume increases. At a monomer conversion of ∼65%, which is the so-called critical conversion, the monomer-rich phase disappears and the third stage starts. During this stage, polymerization occurs only in the polymerrich phase, the composition of which now changes because of * To whom correspondence should be addressed. Phone: +32 9 2644532 Fax: +32 9 2644999. E-mail: Geraldine.Heynderickx@ UGent.be.

the further conversion of monomer. As a result, the viscosity of this phase increases notably. According to Gibb’s phase law, there is a reactor pressure drop during the third stage. Because the polymerization of vinyl chloride is a highly exothermic reaction, cooling is needed to retain the required polymerization temperature. The produced thermal power goes through a maximum; therefore, the cooling capacity in industrial suspension polymerization reactors is designed to meet this maximum.12,13 Because of the improvements in heat removal technology, the industrial batch reactor volume has increased from 4 m3 to 150-200 m3, thus reducing the surface-to-volume ratio of the reactor. The objective in an industrial PVC suspension polymerization process is to produce a polymer product that meets the required product specifications in a safe and efficient way. Therefore, there is a need for models that are able to predict the reactor behavior, as well as the development of molecular properties, as a function of polymerization time.14 A correct simulation of a batch suspension polymerization reactor requires the construction and solution of mass balances, moment equations, and energy balances. Furthermore, a reactor control system to alter the flow rate or inlet temperature of the cooling water in the reactor jacket is needed to keep the reactor temperature constantly at the polymerization temperature. Kiparissides et al.5 and Krallis et al.15 performed simulations for smaller-scale reactors. Mejdell et al.16 focused on the cooling circuits in largescale reactors, without implementing diffusion-controlled polymerization kinetics. Bretelle and Macchietto17 and Dimian et al.18 identified adjustable parameters in the kinetic model of Xie et al.3 to obtain agreement with industrially measured data. Pinto and Giudici19 applied the model of Xie et al.3 for the optimization of the polymerization process in a 20 m3 reactor. Nagy and Agachi20 simulated a 20 m3 reactor by applying a model predictive control, including the implementation of heattransfer processes. Lewin21 performed a controllability analysis on the use of different amounts of concentrations of initiators. In this work, a kinetic model that uses intrinsic rate coefficients, which have been determined from a large number of experi-

10.1021/ie0602355 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/17/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 Table 2. Vinyl Chloride Polymerization Reactions in the Monomer-Rich (k ) 1) and the Polymer-Rich Phase (k ) 2), with i, j ) 1, ..., ∞ type of reaction

Figure 1. (a) Pilot-scale and (b) industrial-scale batch vinyl chloride suspension polymerization reactor, showing the reactor geometry and setup configurations (see Table 1 for details). Table 1. Reactor Dimensions of a Pilot-Scale Batch Reactor and Two Industrial-Scale Batch Reactors for Vinyl Chloride Suspension Polymerization Value

reaction fkkd,k

decomposition of the initiator

Ik 98 2R0,k

chain initiation

R0,k + Mk 98 R1,k

propagation

Ri,k + Mk 98 Ri+1,k

chain transfer to monomer

Ri,k + Mk 98 Pi+1,k + Clk

chain initiation by a Cl radical

Clk + Mk 98 R1,k

termination by combination

Ri,k + Rj,k 98 Pi+j,k

termination with a Cl radical

Ri,k + Clk 98 Pi,k

kinI,k

kp,k

parameter

pilot scale

reactor volume, Vr jacket volume, Vj jacket thickness, 1/ (d 2 ew,is - diw,es) wall thickness, 1/ (d 2 iw,es - diw,is) reactor diameter, diw,is Hr/diw,is He/diw,is contact surface area of reactor jacket impeller diameter, dimp

4 m3 0.688 m3 0.042 m

44 m3 4.5 m3 0.078 m

36 m3 4.0 m3 0.078 m

0.014 m

0.027 m

0.027 m

rate coefficientb

1.8 m 1 1/5 9.4 m2

3.1 m 2 1/5 58.2 m2

2.9 m 2 1/5 50.9 m2

0.9 m

1.32 m

1.32 m

for propagation, kp,chem for chain transfer to monomer, ktr,chem for termination ktc,chem ktCl,chem for chain initiation by a Cl radical, kinCl,chem

ktr,k

kinCl,k i,j ktc,k

ktCl,k

industrial scale

al.,2

mental data by De Roo et is combined with a detailed model that describes all the heat-transfer processes occurring in the reactor on one hand and all those occurring in the heating/ cooling jacket of the reactor on the other hand. This model is first applied and validated by comparing simulation results and experimental results for a 4 m3 pilotscale reactor. Next, two industrial-scale reactors, with volumes of 36 and 44 m3, are simulated. 2. Reactor Setup The geometry and control system of the pilot-scale reactor and the industrial-scale reactors are shown in Figure 1. A proportional-integral-derivative (PID) controller is used to keep the reactor temperature constant after the polymerization temperature is attained. For the pilot-scale reactor, this is realized by changing the inlet temperature of the cooling water (Tj,in). For the industrial-scale reactors, the flow rate of the cooling water (Fag) is adapted. The exact dimensions of the reactors are given in Table 1. For the industrial-scale reactors, only the volume was known. The shape of the reactors was considered to be cylindrical, with an elliptical top and bottom.13 The cooling jacket does not cover the top of the reactors, as shown in Figure 1. For the pilot-scale reactor, the cooling jacket is a spiral coil that makes seven wraps around the inox reactor wall. For the industrial-scale reactors, the height-to-diameter ratio is chosen to be 2, as advised for intermediate-sized reactors by Albright and Soni.12 For larger-scale reactors (150-200 m3), a ratio of 3 is advised. For small-scale reactors, such as the pilot reactor, a ratio of 1 is advised. The ratio of the height of the elliptical component to the diameter is chosen to be 1/5.23 The volume of the jacket of the industrial reactors is calculated from the jacket thickness guidelines (0.075-0.080 m) that are given in the literature.13 The cross section is determined from this thickness value: a width and height of 0.078 m are taken. For the reactor wall material, stainless steel is chosen, because it allows good temperature control.22

Table 3. Values of the Reparameterized Pre-exponential Factor and Activation Energy of the Intrinsic Rate Coefficients for Propagation, Chain Transfer to Monomer, Termination, and Chain Initiation by a Cl Radicala A* [m3 mol-1 s-1] E [kJ/mol] 8.7 × 10-1 8.9 × 10-4

24.9 54.3

7.1 × 104 7.9 × 105 6.1 × 10-1

0 0 28.4

a Using the rate coefficient relation k ) A* exp(-E/R(1/T - 1/T h )), where T h ) 323 K. b Values for the decomposition rate coefficients kd,k are given in Tables 11 and 12.

The industrial suspension reactors are stirred using a large, flat-blade disc turbine with a diameter of 1.32 m, rotating at a speed of 45 rpm.25 Although the rotation speed is rather low, a sufficiently high heat-transfer coefficient inside the reactor is obtained, stagnant zones in the reactor are avoided, and a uniform distribution of the PVC particles in the suspension is maintained. Note also that the agitation affects the particle size of the dispersed polymer phase and, hence, affects the quality of the product.22 The amount of energy supplied by the agitator is, at most, 10% of that generated by the polymerization22 and is neglected. 3. Model Equations 3.1. Kinetic Model. The reactions taken into account for the vinyl chloride suspension polymerization are summarized in Table 2. Note that termination by Cl radicals is included in the reaction scheme, as discussed by De Roo et al.1 This reaction scheme is valid both in the monomer-rich phase (k ) 1) and the polymer-rich phase (k ) 2). The reactions in the polymerrich phase can become diffusion-controlled, whereas the reactions in the monomer-rich phase are reaction-controlled. Starting from this assumption, the rate coefficients in the monomer-rich phase are equal to the intrinsic rate coefficients (kchem). The rate coefficients in the polymer-rich phase are apparent rate coefficients and are calculated from an intrinsic rate coefficient and a diffusion contribution:26,27

1 1 1 ) + kapp kchem kdiff

(1)

The intrinsic rate coefficients were obtained by regression of the experimental data for monomer conversion and moments of the molecular mass distribution, as reported in an earlier

Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1181 Table 4. Mass Balances for the Different Components in the Suspension Polymerization of Vinyl Chloride in the Monomer-Rich (k ) 1) and the Polymer-Rich Phase (k ) 2)a

a Stage 1: k ) 1; stage 2: k ) 1, 2; stage 3: k ) 2. I ) I , M ) M at t ) 0; R , P ) 0 at t ) 0; R , P ) 0 at start of stage 2 (that is, monomer 1 0 1 0 i,1 i,1 i,2 i,2 conversion X ) 0.1%). I0 and M0 values are taken from Tables 11 and 12.

publication.2 The values of the Arrhenius parameters of the intrinsic rate coefficients for propagation, for chain transfer to monomer, for termination, and for chain initiation by a Cl radical are given in Table 3. For a thorough discussion on the estimation of these intrinsic rate coefficients, the reader is referred to our earlier work.2 The diffusional contribution (kdiff) is consistently modeled with the Smoluchowski28 expression, in which the diffusion coefficients are calculated based on the free-volume theory. We verified the validity of this approach in an earlier work.1 For an extended discussion of the calculation of the diffusional contributions, the reader is referred to our earlier works.1,2 An expression, analogous to eq 1, can be derived for the apparent initiator efficiency:2

1 fapp

)

1 fchem

+

1 kdiff

(2)

The intrinsic initiator efficiency value is dependent on the type of initiator and varies typically between 0.7 and 1. The mass balances for the components appearing in the reactions in Table 2 are given in Table 4. The method of moments is applied to calculate the average properties (Mn, Mm, Mz) of the molecular mass distribution (MMD) of the polymer (Table 5). In these equations, the steadystate approximation is applied to both the Clk and R0,k radicals. The initial conditions are also mentioned in Tables 4 and 5. The apparent rate coefficients for termination are considered to be dependent on the chain length, following the chain-lengthdependent diffusion of a macroradical in the polymer-rich phase. The overall rate coefficient for termination is calculated from the relation ∞

〈ktc,2〉 )

∞

The calculation of the overall termination rate coefficient, as well as the calculation of the termination-related terms in the higher-order moment equations, requires knowledge of the concentration of macroradicals Ri,2. These values are obtained by solving the algebraic equations resulting from the application of the pseudo-steady state approximation to the macroradicals Ri,2.29 To limit the number of equations to be solved simultaneously, a coarse-graining technique is used.2,30,31 3.2. Energy Balances. The energy balances for the reactor, the internal reactor wall, and the cooling jacket are presented in Table 6. The first group of equations (eqs 20-24) are called the open-loop equations. In eq 20, the first term on the righthand side arises from the heat effects of the polymerization reactions, thus coupling the mass and energy balances. The second term on the right-hand side represents the heat transport by conduction over the reactor wall, thus coupling eqs 21 and 24. The temperature of the reactor wall (Tiw) is assumed to be uniform for the entire reactor. Taking into account a separate equation for the reactor wall temperature allows one to correctly calculate the different heat-transfer coefficients in the complete system of a reactor with a cooling jacket. The temperature of the cooling water in the jacket is considered to be dependent on time and space. The corresponding equation (eq 22) is solved using the method of lines, converting the partial differential equation into a set of ordinary differential equations (ODEs), one for each spatial node point in the jacket. Special attention is given to the fact that the heat exchanging surface area of the reactor changes as a function of the polymerization time, because the suspension level in the reactor decreases because of a change in the suspension density:

Aiw,xs ) diw,xsπ(Hs - He) +

∑ ∑ ki,jtc,2Ri,2Rj,2 i)1 j)1 ∞

(

Ri,2)2 ∑ i)1

(9) eecc )

2 πdiw,xs + 4 diw,xs 2 π 1 + eecc ln (18) 5 2eecc 1 - eecc

( ) x ( ) 1-

(

diw,xs/5 2 ) 0.9165 diw,xs/2

)

(19)

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Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007

Table 5. Moment Equations of the Macroradical Distribution (λs,k, s ) 0, ..., 3) and the Polymer Molecule Distribution (µs,k, s ) 0, ..., 3) in the Monomer-Rich Phase (k ) 1) and in the Polymer-Rich Phase (k ) 2)a

a Stage 1: k ) 1; stage 2: k ) 1, 2; stage 3: k ) 2. I ) I , M ) M and λ , µ 1 0 1 0 s,1 s,1 ) 0 at t ) 0 ; λs,2, µs,2 ) 0 at the start of stage 2 (that is, monomer conversion X ) 0.1%). I0 and M0 are taken from Tables 11 and 12.

Table 6. Energy Balances for the Reactor, the Internal Reactor Wall, and the Jacketa

a Energy balances are coupled with the equations given in Table 5. t ) 0: T , T , T ) 298 K; t ) 0, x ) 0: T ) 340 K (heating reactor). T r iw j j set for considered cases is given in Tables 11 and 12, and PID constants are given in Table 7. From polymerization time t, when |Tr - Tset| e 2 K, the closed-loop system equations are solved.

Contrary to systems where the reaction enthalpy is, more or less, constant, the reaction enthalpy changes with the polymerization temperature and, hence, polymerization time. Control of the heat transfer, which is a requirement to control the polymerization process, is of primary interest, because the

polymerization temperature must remain constant. For that goal, a PID controller is used for both the pilot-scale and industrialscale reactor, but in a different way. The PID controller equations for the pilot-scale reactor and the industrial-scale reactor are given in Table 6, under the section labeled “closed-

Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1183 Table 7. Controller (PID) Constants Used in Table 6 To Keep the Reactor Temperature Constant for a Pilot-Scale Reactor and Industrial-Scale Reactors Value PID constant Kp Ti Td

pilot scale

industrial scale

K-1

200 K-1 750 s 0.1Ti s

-100 750 s 0.2Ti s

1 1 1 ) + Ur-iw hr hf,scale

(30)

1 1 1 ) + Uiw-j hj hfo

(31)

( ) (

)

dew,es dew,es 1 1 1 1 dew,es ) + ln + + (32) Uj-a hew,es 2κrw dew,is hj hfo dew,is taking into account the heat-transfer coefficients and scaling factors. In eq 32, hew,es is based on the external wall surface area. In what follows, the calculation of the various heat-transfer coefficients will be discussed. The reactor fouling coefficient (hf,scale) originates from the deposition of polymer on the internal reactor wall,33 which is also called scaling. This causes an additional heat resistance and is calculated from the relation20

κp

(33)

1.410-7t

in which κp is the thermal conductivity of the polymer (see also Table 10, presented later in this work) and t is the polymerization time. Equation 33 is, in principle, an expression that is obtained for 20-m3 reactors, because scaling reduces the heat-transfer capacity in all batch suspension reactors, 12,13,20 it is used for all reactors simulated in this work. It must be noted that, to reduce the polymer deposition on the reactor wall, several techniques have been developed, ranging from cleaning techniques to polishing and treatment of the metal surface of the reactor.22,36 The fouling heat-transfer coefficient in the cooling water jacket (hfo) varies between 1000 and 6000 W m-2 K-1.24,34,35 An average value of 3000 W m-2 K-1 is used during the calculations. The heat-transfer coefficient from the suspension to the reactor wall (hr), and from the reactor wall to the cooling water in the jacket (hj), are calculated from the following correlation:24,34,37,38

( )

Nu ) CReaPr1/3

η ηsurf

0.14

hr diw,is κs

(35)

FsNimp d 2imp ηs

(36)

ηscp,s κs

(37)

Nur )

loop equations”. The PID constants (Table 7) are determined with the standard Ziegler and Nichols procedure24,32 and tuned by trial and error. 3.3. Heat Transfer. The overall heat-transfer coefficient from the reactor to the internal reactor wall (Ur-iw), from the internal reactor wall to a section of the cooling jacket (Uiw-j), and from the cooling jacket to the environment (Uj-a), are calculated from the relations

hf,scale )

exchanging surface area, ηsurf. The reactor-wall heat-transfer coefficient hr is calculated from eq 34 with empirical coefficients of C ) 0.75 and a ) 2/3 for baffled vessels. The dimensionless numbers are defined as follows:

(34)

In eq 34, C and a are dependent on the configuration of the setup. From the calculated Nu number, the heat-transfer coefficient is calculated. The dependence of the heat-transfer coefficient on the viscosity of the medium is accounted for by taking into account the difference between the viscosity at the bulk temperature, η, and at the temperature of the heat

Rer )

Prr )

The physical constants are calculated at the reactor temperature Tr, except for ηsurf in the correction contribution, which is calculated at the reactor wall temperature Tiw. The value of this convection coefficient changes with the polymerization time. The wall-jacket heat-transfer coefficient hj is also calculated from eq 34, in which C ) 0.023 and a ) 0.8. The dimensionless numbers are calculated as follows:

hj dh κj

(38)

Rej )

FjVj dh ηj

(39)

Prj )

ηjcp,j κj

(40)

Nuj )

The physical constants are calculated at the cooling agent temperature, Tj. The hydraulic mean diameter of the jacket is defined as

dh )

4Vj Aiw,es

(41)

Natural convection causes heat losses to the environment. To calculate the heat transfer from the jacket to the environment, the Grashof number, Gr, is required:

Gr )

Rag∆TL3F2 η2

(42)

in which Ra is the coefficient of volumetric expansion of the environment air equal to 1/T. ∆T is the temperature difference between the heat-transfer surface and the environment. L is a characteristic dimension: L is defined as the height of the cylindrical and bottom portion of the reactor for the heat losses from the external reactor wall, and L is the diameter of the reactor for the heat losses from the top of the reactor. F and η are the density and the viscosity of the environment. The heattransfer coefficients from the cooling jacket and from the top of the reactor to the environment are calculated from38

}

hew-a n 3n-1 htw-a ) C′′(∆T) L

(43)

For the values of the constants reference is made to Table 8. 3.4. Phase Equilibria Calculations. During the polymerization, the monomer-rich phase, the polymer-rich phase, the gas phase, and the water phase in the reactor are assumed to be in equilibrium. This assumption implies that the fugacities of

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Table 8. Values for C′′ and n Used in Eq 43 To Calculate the Heat Losses due to Natural Convection from Vertical and Horizontal Flat Surfacesa GrPr

a

C′′

n

104-109 >109

Vertical Surfaces (Jacket) 1.37 1.24

0.25 0.33

105-2 × 107

Horizontal Surfaces (Top) 1.86

0.25

When no polymer-rich phase is present (stage 1, mm,2 ) 0) or when no monomer-rich phase is present (stage 3, mm,1 ) 0), the amount of monomer in the relevant polymerization phase (mm,1, mm,2) is directly calculated from eq 49, as shown in Table 9. When the two polymerization phases coexist (stage 2), the amount of monomer in the polymer-rich phase (mm,2) is dependent on the amount of polymer in the polymer-rich phase:

mm,2 )

Values taken from ref 38.

the monomer in the four phases are equal:

fm,w ) fm,g ) fm,1 ) fm,2

(44)

The fugacity of the monomer in the gas phase (fm,g) and in the polymer-rich phase (fm,2) is calculated from the following expressions:5,39

{

fm,g ) ymPr exp

}

Pr [B + (1 - ym)2(2Bmw - Bm - Bw)] RT m (45)

fm,2 ) f 0m exp[ln(1 - φp) + φp + χFH φp2]

0 ) ln(am) ) ln

( ) fm,2 fm0

V g ) Vr -

(

(

ωp )

φpFp + (1 - φp)Fm

(48)

where Fm and Fp are the density of the monomer and polymer, respectively. The distribution of the mass of the monomer over the four phases in the reactor (mm,1, mm,2, mm,g, mm,w) can now be determined. The water is distributed over only the water phase and the gas phase (mw,w, mw,g). An overview of the equations to calculate the distributions is given in Table 9. Expressions for the volumes of the four phases (V1, V2, Vg, Vw) are also presented. The total amount of monomer in the reactor at any conversion X is given by

m0(1 - X) ) mm,1 + mm,2 + mm,w + mm,g

(49)

]

)

X23 - mm,g - mm,w ) 0 ωp

mm,1 ) m0 1 -

(51)

(52)

This observation results in

Equation 47 is valid for the polymer volume fraction φp in the polymer-rich phase during the second stage of the polymerization.40 From the volume fraction φp, the mass fraction of polymer in the polymer-rich phase (ωp) is calculated:

φpFp

) [

m0X w0 fw,gVg m0(1 - X) fm,gVg Fw RTFw Fm RTFm Fp

Vr represents the reactor volume. The second term on the righthand side is the liquid water volume in the reactor. The third term on the right-hand side represents the liquid vinyl chloride volume in the reactor at monomer conversion X. The volume of the produced polymer is the last term in eq 51. Rearranging the latter equation yields the expression for Vg that is given in Table 9. The second stage of the polymerization starts at a conversion of X ) 0.1%. The conversion that indicates the end of stage 2 (X23) is attained when there is no monomer left in the monomerrich phase (see stage 2, mm,1 in Table 9):

) ln(1 - φp) + φp + χFHφp2 (47)

(50)

Next, mm,1 in stage 2 is calculated. The mass of monomer in the gas phase (mm,g) is calculated from the fugacity of the monomer fm,g (eq 45) and mm,w is calculated from the solubility constant of the monomer in water.41,42 The water is distributed over the gas phase (mw,g), which is calculated from fw,g (fw,g ) Pr - fm,g), and the water phase. The volume of the monomer-rich phase (V1) and the volume of the polymer-rich phase (V2) are calculated from the masses of monomer and polymer in these phases (see Table 9). The volume of the gas phase (Vg) is derived from the volume contributions in the reactor:

(46)

where Pr is the reactor pressure and ym is the mole fraction of monomer in the gas phase. The parameter fm0 represents the pure monomer fugacity at reactor temperature and saturation pressure, φp is the volume fraction of the polymer in the polymer-rich phase, and χFH is the temperature-dependent and φp-dependent Flory-Huggins interaction parameter. The parameters Bm, Bw, and Bmw are the second virial coefficients of pure monomer, pure water vapor, and monomer-water vapor in the gas-phase mixture. By equating eqs 45 and 46 (see eq 44), the reactor pressure Pr is obtained. The volume fraction of the polymer in the polymer-rich phase (φp) during the second stage of the polymerization is determined from the activity of the monomer, am, which is equal to 1 in the second stage of the polymerization:5,39

( )

1 - ωp m0X ωp

X23 )

ωp (m - mm,g - mm,w) m0 0

(53)

However, the amount of monomer in the gas phase (mm,g) is also a function of the conversion X23. Rearranging eq 53, taking into account the conversion dependency of mm,g (by substituting eq 51 in the equation for mm,g found in Table 9), gives the final expression for X23:

X23 )

ωp m0

m0 - mm,w 1+

fm,g[Vr - (w0/Fw) - (m0/Fm)] RT - [(fm,g/Fm) + (fw,g/Fw)]

ωp fm,g[(1/Fm) - (1/Fp)] RT - [(fm,g/Fm) + (fw,g/Fw)]

(54)

From eq 53, it is concluded that the conversion for the transition of the second stage to the third stage, X23, is determined approximately by the mass fraction of polymer in the polymerrich phase, ωp. X23 will be slightly less than ωp, because the mass of monomer m0 is corrected by the amounts of monomer

Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 1185

present in the gas phase and in the water phase (mm,g and mm,w, respectively). The initiator partition coefficient between the monomer-rich phase and the polymer-rich phase (KI) is defined as3

KI )

I2 I1

(55)

and was determined by the same authors to be equal to 0.77. The latter value is used in all calculations. 3.5. Physical Properties. The most important contribution in the energy balances is, without any doubt, the contribution that is related to the reaction enthalpy. The reaction enthalpy is usually considered to be constant,5,16,43 but it is slightly dependent on the temperature:20

-∆Hr ) (1.025 × 105) + 9.825T

(56)

The correlations used to calculate the physical properties, which is required for the integration of the energy balances, are summarized in Table 10. The suspension density and heat capacity are calculated based on the mass fractions of the monomer, polymer, and water:

Fs-1 )

Ωw Ωm Ωp + + Fw Fm Fp

(57)

cp,s ) Ωwcp,w + Ωmcp,m + Ωpcp,p

(58)

The conductivity of the suspension is calculated from the following empirical correlation:44

κs 2 + (κdrop/κw) - 2Φdrop[1 - (κdrop/κw)] ) κw 2 + (κdrop/κw) + Φdrop[1 - (κdrop/κw)]

(59)

κdrop ) φmκm + φpκp

(60)

A special problem is posed by the suspension viscosity calculation. The suspension viscosity increases as a function of polymerization time or conversion. The Taylor equation, based on the viscosity of the water and suspension droplets,45 is used:

(

)

(

)

ηw + 5/2ηdrop ηw + 7.54ηdrop ηs )1+ Φdrop + Φdrop2 ηw ηw + ηdrop ηw + ηdrop (61) The quadratic term in this correlation results in a more accurate estimate of the suspension viscosity.46 The viscosity of the water

Table 9. Calculation of the Amount of Monomer in the Monomer-Rich Phase, the Polymer-Rich Phase, the Gas Phase and the Water Phase (mm,1, mm,2, mm,g, mm,w), of the Amount of Water in the Water Phase and in the Gas Phase (mw,g, mw,w), the Volume of the Monomer-rich Phase, the Polymer-rich Phase, the Gas Phase and the Water Phase (V1, V2, Vg, Vw), and of the Volume Fraction and the Mass Fraction of Polymer in the Polymer-rich Phase (φp and ωp) Stage 1, 0 e X < 0.1%

variable

Stage 2, 0.1% e X < X23

(

)

X - mm,g - mm,w ωp

Stage 3, X23 e X

mm,1

m0(1 - X) - mm,g - mm,w

m0 1 -

mm,2

0

( )

mm,g

fm,gVgmmm RT

id. stage 1

id. stage 1

mm,w

Km,w

id. stage 1

id. stage 1

mw,g

fw,gVg RT

id. stage 1

id. stage 1

mw,w

w0 -

id. stage 1

id. stage 1

V1

mm,1 Fm

id. stage 1

V1 ) 0

V2

0

mm,2 m0X + Fm Fp

id. stage 2

Vg

Vr - (m0/Fm) - (w0/Fw) + m0X[(1/Fm) - (1/Fp)]

id. stage 1

id. stage 1

id. stage 1

id. stage 1

Vw

φp

ωp

1 - ωp m0X ωp

( ) Pr

Psat m

mw,w

fw,gVg RT

1 - [1/(RT)][(fm,g/Fm) + (fw,g /Fw)] mw,w mm,w + Fw Fm

0 ) ln(1 - φp) + φp + χFHφp2 φpFp φpFp + (1 - φp)Fm

0

m0(1 - X) - mm,g - mm,w

m0X/Fp (mm,2/Fm) + (m0X/Fp) id. stage 2

1186


Table 10. Physical Properties of Vinyl Chloride, Poly(vinyl chloride), Water, and Wall Material variable

value Vinyl Chloride, VC 94.469/0.27071+(1-T(K)/432)0.2716 -1.651 102 + 5.165T 10-3 exp(9.373 - 648.32/T - 4.294 10-2T + 4.316 10-5T2) 0.172(0.229 + 1.529(1 - T/425)2/3)

Fa [kg/m3] cpa [J kg-1 K-1] ηf [Pa s] κe [W m-1 K-1]

Poly(vinyl chloride), PVC 1394 - 0.203(T - 273.15) - 2.19 10-3(T - 273.15)2 1220(0.64 + 1.2 10-3T) 1.164 10-1 + 1.373 10-4T

Fd [kg/m3] cpc [J kg-1 K-1] κd [W m-1 K-1] Fa [kg/m3] cpa [J kg-1 K-1] ηb [Pa s] κb [W m-1 K-1]

Water 98.349/0.30541+(1-T(K)/647.13)0.081 1.534 104 - 116.0T + 0.4510T2 - 7.835 10-4T3 + 5.201 10-7T4 10-3 exp(-24.71 + 4209/T + 4.52710-2T - 3.37610-5T2) -3.838 10-1 + 5.254 10-3T - 6.36910-6T2

Fa [kg/m3] cpa [J kg-1 K-1] ηa [Pa s] κa [W m-1 K-1]

Air 1.969 - 2.750 10-3T 1007 8.589 10-6 + 2.973 10-8T 1.021 10-2 + 4.83510-5T

F [kg/m3] cp [J kg-1 K-1] κ [W m-1 K-1]

Wall 7930 460.5 16.3

a Data taken from ref 24. b Data taken from ref 47. c Data taken from ref 48. d Data taken from ref 49. e Data taken from the Missenard equation in ref 47. f Data fitted for the equation with 213-413 K data from ref 50.

(ηw) is calculated from the correlation in Table 10, and the viscosity of the suspension droplets (ηdrop) is calculated from

ηdrop 5 ) 1 + φp + 7.54φp2 ηm 2

()

Table 11. Pilot-Scale Batch Reactor (4 m3): Reaction Conditions of Vinyl Chloride Suspension Polymerization with the Initiator Dicetylperoxydicarbonate (CEPC) condition

(62)

which is a reduced form of eq 61, because the monomer in the suspended droplets has a negligible viscosity ηm, compared to the polymer viscosity. The suspension droplets are thus represented by monomer droplets that contain polymer droplets with a volume fraction φp, and, because of the high viscosity of the polymer, eq 62 is only a function of the volume fraction of the polymer in the monomer droplets. 4. Results and Discussion 4.1. Pilot-Scale Reactor: Validation of the Model. The operating conditions for the pilot-scale suspension polymerization batch reactor are given in Table 11. The selected initiator, dicetylperoxydicarbonate (CEPC), has a half-life of ∼2.5 h, under the applied reaction conditions. The reactivity of this initiator is such that the required amount of initiator is not too high, while it is still possible to adequately remove the heat from the reactor. A lower amount would be required if a morereactive initiator is used, but it should be taken into account that the reactivity of the initiator also influences the rate of polymerization and, hence, the rate of heat generation.22 The heating of the pilot-scale reactor to the polymerization temperature is conducted within 3000 s with hot water at a temperature of 340 K. The PID controller is started when the reactor temperature is 2 K below the polymerization temperature. The experimental data and the calculated values for the monomer conversion X, the reactor temperature Tr, and the cooling water inlet temperature (Tj,in) are compared in Figure 2. The calculated and the experimental conversion profiles, as a function of polymerization time, show good agreement, with a final conversion of 91.2% and an experimental value of 89.7%. The simulated reactor temperature Tr, after 3000 s, remains constant, as in the experiment. The calculated cooling water

value/comment Initial

temperatures Tr, Tiw, Tj Tj,in vinyl chloride amount water amount initiator amount VC/H2O jacket agent flow rate, Fag

293 K 340 K 1490 kg 1788 kg 1.49 kg 0.83 kg/kg 2.2 kg/s

Other reactor volume 4 m3 temperatures Tj,in at t where |Tr - Tset| e 2 K from PID Tset 331.65 K water added to reactor, 0.074 kg/s t ) 10800-16200 s kd,ka (CEPC) 3.02 × 1015 exp(-124300/RT) s-1 fchem 1.0 115.7 × 10-6 m3/mol V ˜ /ij a

Data taken from ref 51.

inlet temperature Tj,in follows the same trend, as a function of polymerization time, and is similar to the experimental value. The minimum value is reached near the transition from the second stage to the third stage of the polymerization (∼67% conversion2). During the third stage, some differences are observed between the experimental and calculated cooling water inlet temperature. The calculated conversion profile is flatter than the experimentally observed profile, which results in a steeper increase in the temperature of the cooling water. The release of less reaction enthalpy requires less cooling, and the calculated cooling water inlet temperature quickly increases. The MMD moments and the corresponding K values, which are a measure in industry for the polymer molecular mass,52 as a function of polymerization time, are given in Figure 3a. At low temperatures, high-molecular-mass PVC is produced; therefore, during the heating of the reactor, the moments of the MMD decrease as a function of polymerization time. Shortly after the


Figure 2. Monomer conversion (X), reactor temperature (Tr), and cooling water temperature (Tj,in), as a function of polymerization time, for the pilotscale reactor. Symbols represent experimental data ((9) X, ([) Tr, and (2) Tj,in. Solid lines are calculated by integrating the equations given in Tables 5 and 6 with the reaction conditions given in Table 11. The Arrhenius parameters for the intrinsic rate coefficients are given in Table 3.

Figure 3. (a) Calculated (solid lines) moments of the MMD (Mn, Mm, Mz) and (dotted line) K value (from eq 63), as a function of polymerization time, for the pilot-scale reactor; the square symbol (9) denotes the experimental K value, with the indicated standard deviation. Solid lines are calculated by integrating the equations given in Tables 5 and 6 with the reaction conditions given in Table 11. The Arrhenius parameters for the intrinsic rate coefficients are given in Table 3. (b) Calculated reactor pressure Pr (eqs 44-46, and equations given in Table 9) and polymer volume fraction in the polymer-rich phase φ2 (see Table 9), as a function of polymerization time, for the pilot-scale reactor; the dotted line indicates the time at which the experimental Pr starts to decrease.

polymerization temperature is attained (3000 s), their values remain constant, as a function of polymerization time. The K value has been calculated from an experimentally obtained correlation with the mass-averaged moment (Mm) of the MMD, which has been determined in this study:

K ) (3.9808 × 101) + (3.560 × 10-1)Mm (4.792 × 10-4)Mm2 (63) The calculated value of K is 66.4, as compared to the experimental value of 66.0 ( 1.0. If no water is added to the reactor (this is done from 10 800 s to 16 200 s, as seen in Table 11), the suspension level in the reactorsand, thus, the heatexchanging surface areasdecreases. It is calculated that, in case of a decreasing suspension level, the cooling water inlet temperature must be reduced by 3 K from the start of the decrease of the suspension level onward, to keep the reactor temperature constant. The calculated reactor pressure profile and polymer volume fraction in the polymer-rich phase are shown in Figure 3b. The calculated reactor pressure Pr decreases from the start of the third stage at a calculated polymerization time of 15 560 s, while, experimentally, it is observed that the reactor pressure starts to decrease after 15 600 s. 4.2. Industrial-Scale Reactor. 4.2.1. Results. Three industrialscale suspension polymerization batch reactor simulations have been performed. The operating conditions for the three cases

are given in Table 12. The selected initiators, tert-butyl peroxyneodecanoate (TBPD) and bis-(2-ethylhexyl)peroxydicarbonate (EHPC) have half-lifes of 2.2 and 3.8 h, respectively, at 330 K. The reactivity of these initiators is such that the required amount of initiator is rather low, but the heat generation rate does not become too high, as will be discussed below. For the three industrial-scale simulation cases, the monomer conversion profiles, the moments of the MMD, and the K-value, as a function of polymerization time, are presented in Figure 4a, c, and e. The final conversion is ∼90% for all three cases, after ∼7 h of polymerization time, which is a typical value for a batch operation.5,13,53 At the start of the polymerization, the moments of the MMD are high, and then they decrease as a function of polymerization time, because of the heating of the reactor. After the reactor temperature becomes constant, the values remain rather invariant. The apparent rate coefficients, which are determined by the intrinsic rate coefficients and the diffusional contributions,2 are also presented in Figure 4b, d, and f, as a function of monomer conversion. Both for the termination between macroradicals (〈ktc,2〉) and for the termination between macroradicals and Cl radicals (ktCl,2), the gel effect, which accounts for the influence of diffusion on termination reactions between macroradicals in the polymer-rich phase, is observed in the third stage. From the start of the third stage, the cage effect, which accounts for the fact that not all initiator-derived radicals in the polymerrich phase are able to initiate polymerization, becomes more pronounced and causes the initiator efficiency in the polymerrich phase (f2) to decrease. The glass effect, which accounts for the influence of diffusion on the reaction of a macroradical and a monomer molecule in a propagation step, hardly affects the propagation rate, and the apparent rate coefficients for propagation (kp,2) and for chain transfer to monomer (ktr,2) remain constant or decrease only slightly, as a function of monomer conversion.1,54 A detailed discussion of case 1 is presented further. The calculated temperatures are shown in Figure 5a. During the operation, the calculated temperatures at a given position in the cooling jacket wrapped around the industrial-scale reactor (90 m and 650 m are shown in the figure) decrease, mainly because of the increasing flow rate, as shown in Figure 5. At the beginning of the third stage of the polymerization, the cooling water flow (Fag) reaches a maximum (see Figure 5b), resulting in a minimum for the cooling water temperature (see Figure 5a). The change in the flow rate of the cooling water is determined by the closed-loop equations for the PID controller (see Table 6). The polymerization rate, as a function of conversion, is presented in Figure 6, in both phases, separately and for the entire reactor. The maximum in the polymerization rate at the beginning of the third stage follows from the interaction of the gel effect and the cage effect, as is shown in Figure 6. This maximum in the polymerization rate corresponds to a maximum in the amount of heat that must be removed from the reactor. The maximum in the polymerization rate profile is strongly dependent on the amount and type of initiator that is used, which, therefore, must be chosen carefully. Albright and Soni12 suggested that the initiator be chosen based on the value of the ratio of the maximum heat generation rate to the average heat generation rate over the entire polymerization run. Common values for this ratio are in the range of 1-2. The closer the ratio is to a value of 1, the easier the removal of heat that is generated during the polymerization process. In the presented simulation case, this ratio is ∼1.5. The glass effect only affects the polymerization rate from high conversions onward.1,54

1188


Table 12. Industrial-Scale Reactor: Reaction Conditions of the Vinyl Chloride Suspension Polymerization with the Initiators tert-Butyl Peroxineodecanoate (TBPD) and bis-(2-ethylhexyl) Peroxydicarbonate (EHPC) variable

case 1

case 2

case 3

Initial temperatures Tr, Tiw, Tj Tj,in vinyl chloride amount water amount initiator type amount of initiator VC/H2O heating agent flow rate, Fag

298 K 360 K 13000 kg 16400 kg TBPD 7 kg (98%) 0.79 kg/kg 6 kg/s

298 K 360 K 12250 kg 16200 kg TBPD 9.5 kg (98%) 0.76 kg/kg 6 kg/s

298 K 360 K 12500 kg 14300 kg EHPC 17 kg (75%) 0.87 kg/kg 6 kg/s

44 m3

44 m3

36 m3

283 K 331 K

283 K 324 K

283 K 327 K

PID

PID

PID

1.52 × 1014 s-1 115.47 kJ/mol 0.7 158.1 × 10-6 m3/mol no

1.52 × 1014 s-1 115.47 kJ/mol 0.7 158.1 × 10-6 m3/mol no

1.83 × 1015 s-1 122.45 kJ/mol 1.0 149.5 × 10-6 m3/mol no

Other reactor volume temperatures Tj,in at t where |Tr - Tset| e 2 K Tset cooling agent flow rate Fag at t where |Tr - Tset| e 2 K [kg/s] kd,k A E fchem V ˜ ij* water added to reactor

Initially, in the third stage, the apparent overall termination by combination rate coefficient 〈ktc,2〉 decreases faster than the apparent initiator efficiency f2, which causes an initial increase in the polymerization rate. Later, in the third stage, f2 decreases faster and the termination of macroradicals with Cl radicals (and the residual termination) starts to dominate and the polymerization slows. The physical properties of the suspension are presented in Figure 7. The density and the conductivity of the suspension increase as a function of the polymerization time. This corresponds with the literature data.5 The suspension viscosity initially decreases, because of the heating of the reactor (0-3000 s). It then increases by ∼20%. This is rather limited, compared to the literature data. Kiparissides et al.5 calculated an increase in the suspension viscosity by 1 order of magnitude. Dimian et al.18 calculated a factor-of-5 increase. The variation of the physical properties of the suspension with polymerization time causes the reactor-wall heat-transfer coefficient hr to increase, as a function of polymerization time, as can be observed in Figure 7. This seems to be contradictory, because of the increasing suspension viscosity as a function of polymerization time, but can be explained. From the expression for the dimensionless Nusselt number for the reactor (Nur; see eq 34), it follows that

hr ∝ ηs-1/3 hr ∝ Fs2/3 and

hr ∝ κs2/3 Therefore, the increase of Fs and κs in Nur dominates over the increase of ηs. As a result, hr increases as a function of polymerization time. Despite the increase of hr, the overall internal reactor wall heat-transfer coefficient (Ur-iw) decreases, because of the scaling effect, which is the growing thickness of the solid layer of polymer that is formed against the internal reactor wall, resulting in a fast decrease of hf,scale. The overall heat-transfer coefficient from the internal reactor wall to the jacket, at different positions in the jacket, is presented

in Figure 8 (at 90 m and 650 m). The differences in the overall heat-transfer coefficients are entirely attributed to the changes in the physical properties of the cooling water in the jacket. The changes in the flow rate and temperature of the cooling water influence these properties. The effect of fouling in the cooling water jacket cannot be underestimated, as observed from Figure 9. In this figure, the calculated values, as a function of polymerization time for different values of the fouling heat transfer coefficient hfo in the jacket, are presented. As mentioned previously, this coefficient can vary over a range of 10006000 W m-2 K-1. A lower value implies a thicker fouling layer. Above 40 kg/s, Fag reaches an absolute maximum and thermal runaway of the batch suspension reactor is calculated. Heat losses to the environment over the reactor jacket and over the reactor top, as calculated, are presented in Figure 10. The losses over the top are almost constant, as a function of polymerization time. The losses over the jacket are highest when the temperature of the cooling water in the jacket is lowest (this is the highest driving force for heat transfer). A simulation was also performed with a failing PID controller system, which resulted in an adiabatic operation of the exothermic suspension polymerization batch reactor. At the corresponding uncontrolled high reactor temperature, the pressure in the reactor exceeds the design pressure of the vessel and the vessel (reactor) explodes. The adiabatic temperature, pressure, and conversion profiles are shown in Figure 11a. The reactor temperature and pressure runaway start at a monomer conversion of ∼25% (after 8000 s of polymerization time). Explosion of the reactor vessel occurs at 9200 s. An additional calculation is then performed, in which a safety valve is added to the industrial reactor. This safety valve opens when the reactor pressure reaches a value of 18 bar. During the calculations, a linear relationship between the reactor outlet flow rate of vinyl chloride and the safety valve opening is used. As observed in Figure 11b, the reactor pressure increases to 26-27 bar at approximately the same rate as that during the adiabatic operation, because of the assumption that the vinyl chloride in the monomer-rich phase will vaporize and leave the reactor before the vinyl chloride in the polymer-rich phase. Because of the fact that, in this stage of the polymerization, most of the reaction already occurs in the polymer-rich phase, the reactor


Figure 4. (a, c, e) Monomer conversion, moments of the MMD (Mn, Mm, Mz), and K value (represented by the dotted line), as a function of polymerization time, for industrial-scale reactors (cases 1-3): (a) case 1, (c) case 2, and (e) case 3. (b, d, f) Apparent rate coefficients for propagation (kp,2), overall termination by combination between macroradicals (〈ktc,2〉), termination between macroradicals and Cl radicals (ktCl,2), chain transfer to monomer (ktr,2) and apparent initiator efficiency f2 in the polymer-rich phase, as a function of monomer conversion, for industrial-scale reactors (cases 1-3): (b) case 1, (d) case 2, and (f) case 3. Calculations for cases 1-3 are done by integrating the equations given in Tables 5 and 6 for the conditions listed in Table 12, with intrinsic rate coefficients from Table 3. The apparent rate coefficients and initiator efficiency are calculated from eq 1.

pressure initially experiences a further increase. Finally, the polymerization rate and the reactor pressure decrease as the vinyl chloride in the polymer-rich phase is vaporized. Nevertheless, an explosion is avoided as the pressure remains within the design limits. 4.2.2. Increase of Cooling Capacity. The polymerization of vinyl chloride is highly exothermic; therefore, cooling of the reactor is an important issue. Increasing the cooling capacity can be realized by increasing the heat-transfer area between the suspension and the cooling jacket, by modifying the reactor jacket geometry. Other possibilities to increase the cooling capacity imply the combination of heat removal through a reflux condenser and/or an additional internal heat-transfer surface.55

To investigate the effect of these changes in the setup on the behavior of an industrial-scale reactor, additional calculations are performed. The reaction conditions taken into account in these simulations are identical to the reaction conditions used in case 1, as mentioned in Table 12. Because of the fact that, in these calculations, the reactor temperature will be kept constant (at the same value of case 1), no changes in the final conversion and moments of the MMD occur. 4.2.2.1. Modified Cooling Jacket Geometry. Simulations including the application of an improved cooling jacket geometry are performed. An arrangement of a half-pipe coil inside the reactor is chosen, as shown in Figure 12.56 This modification results in an improvement of the heat transfer as the heat-

1190


Figure 5. (a) Calculated temperature of the reactor (Tr), the internal reactor wall (Tiw), and the agent in the jacket at different positions (Tj), as well as reactor pressure (Pr), for the industrial-scale reactor (case 1). (b) Reactor internal heat-transfer surface area Aiw,is (eq 18) and jacket cooling water flow rate Fag, as a function of polymerization time, for the industrial-scale reactor (case 1); solid line represents calculated data accounting for a decrease in liquid level due to an increase in suspension density, whereas the dotted line represents calculated data with a constant liquid level in the reactor. Temperatures and jacket agent flow rates are calculated by integrating the equations given in Tables 5 and 6. Reactor pressure (Pr) is calculated from eqs 44-46. Reaction conditions are given in Table 12.

Figure 8. Overal heat-transfer coefficients from the internal reactor wall to the jacket (eq 31) for the industrial-scale reactor, case 1: (s) Uiw-j at the inlet of the jacket (90 m), and (- - -) Uiw-j further in the jacket (650 m). Calculated by integrating the equations given in Tables 5 and 6.

Figure 9. Flow rate of cooling water (Fag), as a function of polymerization time, for the industrial-scale reactor, case 1, calculated for three fouling heat-transfer coefficients in the jacket (hfo): (1) 1000, (2) 3000, and (3) 6000 W m-2 K-1. Data calculated by integrating the equations given in Tables 5 and 6. Figure 6. Polymerization rate in the monomer-rich phase (Rpol,1) and in the polymer-rich phase (Rpol,2), as well as the total polymerization rate (Rpol,tot) (denoted by solid lines), for the industrial-scale reactor, case 1. Volumetric polymerization rates (RVpol,1 and RVpol,2), as a function of monomer conversion (denoted by dotted lines), for the industrial-scale reactor, case 1. Data calculated using eq 4 during integration of the equations given in Tables 5 and 6.

Figure 10. Calculated heat losses to the environment for the industrialscale reactor, case 1: (s) over the reactor jacket (Qj) and (- - -) over the reactor top (Qtop). Heat losses given in units of watts. Data calculated from the equations given in Tables 5 and 6. Figure 7. (a) Calculated suspension viscosity (ηs), density (Fs), and conductivity (κs), and (b) the reactor-wall heat transfer coefficient (hr), the reactor scaling coefficient (hf,scale), and the overall reactor heat transfer coefficient (Ur-iw), as a function of polymerization time, for the industrialscale reactor, case 1. Data calculated using eqs 30 and 57-59 during integration of the equations given in Tables 5 and 6.

exchanging surface area between the reactor suspension and the cooling jacket is drastically increased. The diameter of the pipe is chosen to be equal to the thickness of the external jacket used in the case 1 simulation, as mentioned in Table 1. The coil has a total volume of 1.79 m3, makes 45 wraps inside the reactor, and has a total length of 750 m. Figure 13a presents the flow rate of cooling agent in the halfpipe coil required to control the reactor temperature. From this

figure, it can be concluded that replacing the external cooling jacket by an internal arrangement, using a half-pipe coil, results in a significant decrease in the flow rate of cooling agent. The maximum required flow rate of cooling agent decreases from 13.5 kg/s to 7.9 kg/s. This decrease is mainly due to the fact that the alternative cooling geometry provides more heat-transfer surface area. Furthermore, the desired polymerization temperature is attained at a lower polymerization time when the alternative cooling geometry is applied (see Figure 13a). This can be explained by the fact that, for both simulations, an identical heating agent flow rate of 6 kg/s was used (see Table 12). As the heat-transfer surface area is increased, the polymerization temperature is reached in less time when the same heating flow rate is used.


of the fact that the initiator must dissolve in the monomer droplets before the polymerization is initiated once more, an initiator concentration that is too low decreases the polymerization rate. A simulation is performed to investigate the effect of implementing a reflux condenser in the reactor configuration of an industrial-scale reactor, as shown in Figure 14a. Besides eq 20, which must be modified to account for the removal of heat by the reflux condenser, all the model equations remain unchanged (see section 3). The reflux condenser starts to operate when |Tr - Tset| e 2 K, resulting in eqs 64 and 65. Figure 11. (a) Reactor temperature (Tr), reactor presssure (Pr), and monomer conversion (X), each as a function of polymerization time, for the industrial-scale reactor, case 1, calculated for adiabatic operation of the reactor. (b) Reactor pressure for adiabatic operation of the reactor, including a safety valve opening at 18 bar. X and Tr data are obtained from integration of the equations given in Tables 5 and 6; Pr data are calculated from eqs 44-46 and the equations given in Table 9.

dTr (VFcp)s

dt

2

) (-∆Hr)

V (-Rpol,k )Vk ∑ k)1

(when |Tr - Tset| g 2 K) (64) 2 dTr V )Vk - Qcond ) (-∆Hr) (Rpol,k (VFcp)s dt k)1 (when |Tr - Tset| e 2 K) (65)

∑

In eq 65, Qcond is the heat removal via the reflux condenser. Typically, in a reflux condenser, ∼30% of the total heat produced is removed by condensing the vinyl chloride monomer. Thus, Qcond is calculated from Figure 12. Schematic of an industrial-scale batch vinyl chloride suspension polymerization reactor, including a half-pipe coil cooling jacket.

Figure 13b shows the calculated temperature of the cooling agent in the alternative cooling jacket. Analogous to the simulation discussed in section 4.2.1, the calculated temperature of the cooling agent at a given position in the cooling jacket decreases mainly because of the increasing flow rate (see Figure 13a). At the beginning of the third stage of the polymerization, the flow rate of cooling agent reaches a maximum, and, hence, the cooling agent temperature reaches a minimum. 4.2.2.2. Reflux Condenser. To increase the cooling capacity of intermediate- to large-sized reactors, reflux condensers are applied. Because of the condensation of the vinyl chloride monomer, a significant amount of heat is removed. The latent heat of evaporation of the vinyl chloride monomer varies with temperature. Values of 336 and 279 kJ/kg are obtained at 293 and 353 K, respectively.8 The fraction of cooling by the reflux condenser is limited to ∼30%, because of the low concentration of initiator in the condensed vinyl chloride that is refluxed into the reactor.8 Because

2

Qcond ) 0.3(-∆Hr)

V V k) ∑ (Rpol,k k)1

(66)

Figure 15a shows the flow rate of cooling agent required for an industrial-scale reactor that includes a reflux condenser. This figure shows that the additional removal of heat by means of a reflux condenser results in significantly lower values for the cooling agent flow rates that are required to keep the reactor temperature constant. By removing 30% of the produced reaction heat through a reflux condenser, the maximum required cooling agent flow rate decreases from 13.5 kg/s to 6.5 kg/s. Lower flow rates result in higher residence times of the cooling agent in the jacket, and, hence, the jacket cooling agent temperature is significantly higher when a reflux condenser is applied, as can be seen from Figure 15b. As the reflux condenser starts to operate when |Tr - Tset| e 2 K, the desired polymerization temperature is attained at the same polymerization time as that when no reflux condenser is applied (see Figure 15a).

Figure 13. (a) Cooling agent flow rate (Fag) and reactor temperature (Tr), as a function of polymerization time. Solid lines represent data obtained with a cooling jacket, as shown in Figure 1b; dotted lines represent data obtained with a cooling jacket, as shown in Figure 12. (b) Temperature of the agent in the cooling jacket as shown in Figure 12.

1192


Figure 14. Schematic of the industrial-scale batch vinyl chloride suspension polymerization reactor including (a) a reflux condenser and (b) a helical coil as the internal heat-transfer surface.

4.2.2.3. Internal Heat-Transfer Surface. The insertion of an internal heat-transfer surface in the reactor configuration allows additional removal of heat. The simplest and leastexpensive heat-transfer equipment inside a vessel is a helical coil, as shown in Figure 14b.34 A simulation is performed to investigate the effect of implementing an internal heat-transfer surface in the configuration of an industrial-scale reactor. The pitch and diameter of the helical coil are chosen to suit the application, taking into account prescriptions as described by Sinnot.34 The diameter of the coil is 2 m, the diameter of

the pipe is diw,is/30, and the coil pitch is twice the pipe diameter. This results in an internal heat-transfer surface area of ∼16 m2, compared to the reactor-jacket contact area of ∼50 m2. The pipe is composed of stainless steel and has a thickness of 0.01 m (the thickness is computed as 1/2(dc,es - dc,is)). The internal heat-transfer surface is used for cooling purposes only, that is, when |Tr - Tset| e 2 K. The cooling agent flow rate in the coil (Fag,c) is chosen to be 1.5 kg/s. The temperature of the cooling water in the coil (Tc) is considered to be dependent on time and space. An additional balance to calculate the temperature profile of the cooling water in the coil is added to the set of energy balances in Table 6:

(VFcp)c

∂Tc ∂Tc ) -LcFag,ccp,ag + Ur-cAr-c(Tr - Tc) ∂t ∂x

(67)

Equation 67 is solved using the method of lines. At the start of the cooling process (that is, the first time |Tr - Tset| e 2 K), the temperature of the cooling agent is considered to be uniform over the coil length, having a value of 283 K. The temperature at the inlet of the coil is 283 K over the entire polymerization time. The heat-transfer coefficient from the reactor suspension to the internal coil (Ur-c) is calculated from

Figure 15. (a) Flow rate of jacket cooling water (Fag), and reactor temperature (Tr), as a function of polymerization time, for an industrial-scale batch vinyl chloride suspension polymerization reactor, including a reflux condenser. (b) Calculated temperature of the agent in the jacket at different positions (Tj) for an industrial-scale batch vinyl chloride suspension polymerization reactor, including a reflux condenser. Solid lines represent data that do not account for the reflux condenser, whereas dotted lines represent data that do account for the reflux condenser. Temperatures and jacket agent flow rates are calculated by integrating the equations given in Table 6. Reaction conditions are given in Table 12.

Figure 16. (a) Flow rate of jacket cooling water (Fag), as a function of polymerization time, for an industrial-scale batch vinyl chloride suspension polymerization reactor, including an internal heat-transfer surface. (b) Calculated temperature of the agent in the jacket at different positions (Tj) for an industrial-scale batch vinyl chloride suspension polymerization reactor, including an internal heat-transfer surface. Solid lines represent data that do not account for the internal heat-transfer surface area, whereas dotted lines represent data that do account for the internal heat-transfer surface area. Temperatures and jacket agent flow rates are calculated by integrating the equations given in Table 6. Reaction conditions are given in Table 12.


( ) (

)

dc,es dc,es 1 1 1 dc,es 1 1 ) + + ln + + Ur-c hr-c hf,scale 2κc dc,is hfo hc dc,is

(68)

In eq 68, hr-c represents the heat-transfer coefficient from the reactor suspension to the external coil surface, hf,scale is the external coil surface scaling coefficient (calculated from eq 33), hfo is the internal coil surface fouling coefficient, and hc represents the heat-transfer coefficient inside the coil. The parameter hr-c is calculated from eq 34, with values of C ) 1.10 and a ) 0.62.34 The heat-transfer coefficient inside the coil (hc) is also calculated from eq 34, using values of C ) 0.023 and a ) 0.8. The heat balance that accounts for the reactor temperature calculation (eq 20) is corrected for the additional removal of heat via this internal helical coil, resulting in eqs 69 and 70.

dTr V )Vk ) (-∆Hr) (Rpol,k (VFcp)s dt k)1 (when |Tr - Tset| g 2 K) (69) 2

∑

2 dTr V )Vk - Qc ) (-∆Hr) (Rpol,k (VFcp)s dt k)1 (when |Tr - Tset| e 2 K) (70)

∑

In eq 70, Qc is the heat removal from the reactor via the internal heat-transfer surface, and it is calculated from the relation

Qc )

∫0L Ur-cAr-c(Tr - Tc) dx c

(71)

Figure 16a shows the flow rate of cooling agent required for the simulation of an industrial-scale reactor that includes an internal heat-transfer surface. This figure shows that an additional heat removal by means of an internal heat-transfer surface results in lower values for the flow rate of cooling agent that is needed to keep the reactor temperature constant. By applying a cooling agent flow rate of 1.5 kg/s in the coil, the maximum required flow rate decrease from 13.5 kg/s to 10.2 kg/s. As these lower flow rates result in higher residence times of the cooling agent in the jacket, the temperature of the cooling agent inside the jacket is higher, as can be observed from Figure 16b. From this figure, it is concluded that, near the end of the polymerization process, all heat is removed through the internal heat-transfer surface, as a constant temperature of the jacket cooling agent is calculated, as a function of polymerization time. 5. Conclusions Simulations of a pilot-scale reactor and three industrial-scale reactors for vinyl chloride batch suspension polymerization have been performed. The energy balances for the suspension, the reactor wall, and the cooling agent in the reactor jacket are solved simultaneously with the mass balances for the initiator and the monomer and the moment equations in both the monomer-rich and polymer-rich phases. Furthermore, the proportional-integral-derivative (PID) controller equations are considered. The applied kinetic model is based on calculated diffusional contributions and estimated intrinsic rate coefficients. This complete model is validated, based on an excellent agreement between experimental data and calculated values for the pilot-scale reactor. For the industrial-scale reactors, conversion and moments of MMD, as industrially experienced, are calculated within the required polymerization times. Contrary to that commonly found

in literature, the suspension viscosity increases only slightly as a function of polymerization time, resulting in an increase of the convection coefficient in the suspension, as a function of polymerization time. Fouling in the cooling jacket of the reactor can result in thermal runaway of the reactor. A control failure of the reactor that leads to an adiabatic operation may result in an explosion of the reactor vessel if no safety valve is present in the reactor. Increasing the cooling capacity of the reactor by changing the cooling jacket geometry, adding a reflux condenser, or adding an internal heat-transfer surface area results in noticeable changes in the flow rates of the cooling agent. Acknowledgment J.W. works as Research Assistant of the Fund for Scientific Research, Flanders, Belgium (F.W.O.). T.D.R. acknowledges the contribution of A. Palacios to this work. List of Symbols Nomenclature am ) activity of the monomer cp,j ) heat capacity of agent in the reactor jacket [J kg-1 K-1] cp,m ) heat capacity of the monomer [J kg-1 K-1] cp,p ) heat capacity of the polymer [J kg-1 K-1] cp,s ) heat capacity of the suspension [J kg-1 K-1] cp,w ) heat capacity of water [J kg-1 K-1] dc,is ) internal diameter of the coil pipe [m] dc,es ) external diameter of the coil pipe [m] dew,is ) internal diameter of the external reactor wall [m] dew,es ) external diameter of the external reactor wall [m] dh ) hydraulic mean diameter of the jacket [m] dimp ) diameter of impeller [m] diw,is ) internal diameter of the internal reactor wall [m] diw,es ) external diameter of the internal reactor wall [m] e(t) ) PID error, expressed as the difference between the reactor temperature and required polymerization temperature; e(t) ) Tr - Tset [K] eecc ) eccentricity of an Ellips fapp ) apparent initiator efficiency fchem ) intrinsic initiator efficiency fk ) apparent initiator efficiency in phase k fm,k ) fugacity of the monomer in phase k [N/m2] f0m ) fugacity of the monomer under standard conditions of reactor temperature and vapor pressure [N/m2] g ) gravitational constant [m/s2] hc ) heat-transfer coefficient inside the helical coil [W m-2 K-1] hfo ) fouling heat-transfer coefficient in the reactor jacket [W m-2 K-1] hew,es ) heat-transfer coefficient from the external reactor wall to the environment [W m-2 K-1] hj ) heat-transfer coefficient from the internal reactor wall to the jacket [W m-2 K-1] hr ) internal reactor wall film heat-transfer coefficient [W m-2 K-1] hr-c ) heat-transfer coefficient from the reactor suspension to the external coil surface [W m-2 K-1] hf,scale ) scaling heat transfer coefficient [W m-2 K-1] hew-a ) heat-transfer coefficient from the cooling jacket to the environment [W m-2 K-1]

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htw-a ) heat-transfer coefficient from the top of the reactor to the environment [W m-2 K-1] i ) chain length kapp ) apparent rate coefficient [m3 mol-1 s-1] kchem ) intrinsic rate coefficient [m3 mol-1 s-1] kd,k ) initiator decomposition rate coefficient [s-1] kdiff ) diffusion contribution [m3 mol-1 s-1] kinI,k ) apparent chain-initiation rate coefficient for the primary radicals R0,k [m3 mol-1 s-1] kinCl,k ) apparent chain-initiation rate coefficient for the Cl radicals [m3 mol-1 s-1] kp,k ) apparent propagation rate coefficient [m3 mol-1 s-1] i,j ktc,k ) apparent rate coefficient for termination by combination between two radicals Ri and Rj with chain length i and j [m3 mol-1 s-1] 〈ktc,2〉 ) overall apparent termination rate coefficient for termination by combination [m3 mol-1 s-1] ktr,k ) apparent chain transfer to monomer rate coefficient [m3 mol-1 s-1] ktCl,k ) apparent termination rate coefficient for termination between a Cl radical and radicals Ri with chain length i [m3 mol-1 s-1] m0 ) initial amount of monomer in the reactor [kg] mmm ) molecular mass of the monomer [kg/mol] mx,k ) mass of component x (x ) m,w) in phase k (k ) 1, 2, g, w) [kg] rm,k ) reaction rate of the monomer in phase k [mol m-3 s-1] t ) polymerization time [s] w0 ) initial amount of water in the reactor [kg] ym ) mole fraction of monomer in the gas phase yw ) mole fraction of water in the gas phase A* ) reparameterized pre-exponential factor of intrinsic rate coefficient [m3 mol-1 s-1] Aew,is ) internal surface of the external reactor wall [m2] Aew,es ) external surface of the external reactor wall [m2] Atop,es ) external surface of the top of the reactor [m2] Aiw,is ) internal surface of the internal reactor wall [m2] Aiw,es ) external surface of the internal reactor wall [m2] Ar-c ) external surface of the internal heat transfer surface [m2] Bx ) second virial coefficient of component x [m3] Bmw ) second virial coefficient of the mixture of monomer and water vapor [m3] Clk ) concentration of Cl radicals in phase k [mol/m3] E ) activation energy [J/mol] Fag ) cooling agent flow in the reactor jacket [kg/s] Gr ) Grashof number Hr ) height of the reactor [m] Hs ) suspension level in the reactor [m] He ) height of elliptical top or bottom part of the reactor [m] I0 ) initial amount of initiator [mol] Ik ) initiator concentration [mol/m3] K ) K value, defined as a property of the polymer, as a measure for molecular mass Km,w ) solubility of the monomer in the water phase Kp ) gain of PID controller [K-1] KI ) initiator partition coefficient over monomer-rich phase and polymer-rich phase L ) characteristic dimension in the Grashof number [m] Lj ) characteristic dimension of the cooling jacket [m] Lc ) characteristic dimension of the internal heat-transfer surface [m] M0 ) initial amount of the monomer in the reactor [mol] Mi ) molecular mass of polymer molecules with chain length i [kg/mol]

Mk ) concentration of the monomer in phase k [mol/m3] M h m ) mass-averaged molecular mass of polymer molecules [kg/ mol] M h m,k ) mass-averaged molecular mass of polymer molecules in phase k [kg/mol] M h n ) number-averaged molecular mass of polymer molecules [kg/mol] M h n,k ) number-averaged molecular mass of polymer molecules in phase k [kg/mol] M h z ) z-averaged molecular mass of polymer molecules [kg/ mol] M h z,k ) z-averaged molecular mass of polymer molecules in phase k [kg/mol] NA ) Avogadro number [mol-1] Nimp ) impeller speed [s-1] Nu ) Nusselt number Pi,k ) concentration of polymer molecules with chain length i [mol/m3] Pm ) partial pressure of the monomer [N/m2] Pr ) reactor pressure [N/m2] Pw ) partial pressure of the water [N/m2] 2 Psat x ) vapor pressure of component x (x ) m,w) [N/m ] Pr ) Prandtl number Qc ) heat removed via internal heat transfer surface [W] Qcond ) heat removed via reflux condenser [W] Qj ) heat losses from the jacket of the reactor [W] Qtop ) heat losses from the top of the reactor [W] R ) universal gas constant [J mol-1 K-1] R0,k ) initiator derived radical (or primary radical) concentration [mol/m3] Re ) Reynolds number Ri,k ) concentration of macroradicals with chain length i [mol/ m3] V Rpol,k ) volumetric polymerization rate in phase k [mol m-3 s-1] Rpol,k ) polymerization rate in phase k [mol/s] Rpol,tot ) total polymerization rate; Rpol,tot ) Rpol,1 + Rpol,2 [mol/ s] T ) temperature [K] Ta ) temperature of the environment [K] Tc ) temperature of the cooling agent inside the helical coil [K] Td ) PID controller characteristic, representing the derivative time [s] Ti ) PID controller characteristic, representing the integral time [s] Tiw ) internal reactor wall temperature [K] Tj ) temperature of the agent in jacket [K] Tj,in ) temperature of the agent in jacket at the inlet [K] Tr ) reactor temperature [K] Tset ) required reactor temperature [K] Uj-a ) overall average jacket-environment heat-transfer coefficient [W m-2 K-1] Uiw-j ) overall internal reactor wall-jacket heat-transfer coefficient [W m-2 K-1] Ur-j ) overall average reactor-jacket heat-transfer coefficient [W m-2 K-1] Ur-c ) overall average reactor-coil heat-transfer coefficient [W m-2 K-1] Ur-iw ) overall reactor-internal reactor wall heat-transfer coefficient [W m-2 K-1] Ut-a ) overall top-environment heat-transfer coefficient [W m-2 K-1] Vj ) volume of reactor jacket [m3]


Vj,i ) volume of reactor jacket section i [m3] Vk ) volume of phase k [m3] Vr ) reactor volume [m3] Vs ) suspension volume [m3] V ˜ ij* ) hole-free volume required for 1 jump per mole of jumping units of the initiator-derived radicals [m3/mol] X ) monomer conversion Greek Symbols Ra ) coefficient of volumetric expansion of the environment [K-1] φx ) volume fraction of component x (x ) m, p) in the suspension droplets φx,k ) fugacity coefficient of component x in phase k φp ) volume fraction of component x (x ) m, p) in the polymerrich phase η ) liquid or polymer solution viscosity [N s m-2] ηdrop ) viscosity of the suspension droplet [N s m-2] ηj ) viscosity of the agent in the reactor jacket [N s m-2] ηm ) monomer viscosity [N s m-2] ηs ) suspension viscosity [N s m-2] ηsurf ) viscosity of a liquid calculated at the wall surface temperature [N s m-2] ηw ) viscosity of water [N s m-2] κdrop ) thermal conductivity of the suspension droplets [W m-1 K-1] κc ) thermal conductivity of the coil material [W m-1 K-1] κw ) thermal conductivity of water [W m-1 K-1] κj ) thermal conductivity of the agent in the jacket of the reactor [W m-1 K-1] κm ) thermal conductivity of the monomer [W m-1 K-1] κp ) thermal conductivity of the polymer [W m-1 K-1] κs ) thermal conductivity of the suspension [W m-1 K-1] κrw ) thermal conductivity of the reactor wall [W m-1 K-1] λs,k ) moments of the macroradical distribution (s ) 0, ..., 3) [mol/m3] µs,k ) moments of the polymer molecule distribution (s ) 0, ..., 3) [mol/m3] Fj ) density of the agent in the reactor jacket [kg/m3] Fs ) density of the suspension [kg/m3] Fiw ) density of reactor wall [kg/m3] Fw ) density of water [kg/m3] σm ) reaction distance [m] -∆Hr ) reaction enthalpy [J/mol] Φdrop ) volume fraction of the droplets in the reactor suspension χFH ) Flory-Huggins thermodynamic interaction parameter ωm ) mass fraction of monomer in the polymer-rich phase ωp ) mass fraction of polymer in the polymer-rich phase Ωw ) mass fraction of water in the reactor suspension Ωm ) mass fraction of monomer in the reactor suspension Ωp ) mass fraction of polymer in the reactor suspension Subscripts ag ) (heating/cooling) agent in jacket app ) apparent c ) coil cond ) reflux condenser chem ) intrinsic diff ) diffusion es ) external surface ew ) external wall i ) chain length; initiator-derived radical is ) internal surface iw ) internal wall j ) reactor jacket

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ReceiVed for reView February 24, 2006 ReVised manuscript receiVed November 10, 2006 Accepted November 23, 2006 IE0602355

Simulation of Pilot-and Industrial-Scale Vinyl Chloride Batch

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