A Study of Diffusion and Reaction in Unpremixed Step Growth

Ind. Eng. Chem. Res. 1997, 36, 4075-4086

4075

KINETICS, CATALYSIS, AND REACTION ENGINEERING A Study of Diffusion and Reaction in Unpremixed Step Growth Copolymerization in a Microsegregated Continuous Stirred Reactor Gu 1 ray Tosun Central Research & Development, E. I. du Pont de Nemours and Company, Inc., Wilmington, Delaware 19880-0304

The effects of shear rate, residence time, and feed stoichiometry on sequence and chain lengths and polydispersity of product polymer in step growth copolymerization in a microsegregated stirred tank reactor with unpremixed feed streams were investigated. One feed stream contained monomers AA and AX, bifunctional and monofunctional monomers with the reactive group A. The other stream contained monomers BB and DD, short and long types of the bifunctional monomer with the end group B. Mass transport in the reactor was described in terms of diffusion between mobile striations that are stretched by the mean shear rate for time periods that vary according to an exponential RTD. Model results suggest that substantial changes in chain and sequence lengths can be effected when the shear rate is increased at fixed residence time and fixed feed ends-ratio (A to B), particularly when the latter is less than unity. However, the results suggest that when conversion is kept constant by varying the mean shear rate and the mean residence time simultaneously, the influence of shear rate on chain and sequence lengths is minimal. On the other hand, when the feed ends-ratio is varied at constant shear rate and residence time, sequence lengths can vary by appreciable amounts (20-40%). Polydispersity is influenced little by any of the variables that have been studied. Introduction When unpremixed feed streams enter a polymerization reactor in laminar flow, they form striations that retain different compositions that depend on the initial composition and reactor age of the individual striation and the ease and extent of molecular diffusion between striations. These striations may be said to be microsegregated. Step growth copolymerizations between bifunctional monomers such as AA, BB, and DD are common in polyurethane and polyurea chemistry. A particular example is when BB and DD have the same end groups but different chemical structures in between, such as short and long (polymeric) diols reacting with diisocyanates (AA). In this case, due to the different diffusivities of the small and large monomers DD and BB, it can be expected that, in the high-viscosity laminar environment of the reaction, the mean sequence lengths of the DDAA and BBAA sequences will be highly sensitive to the extent of microsegregation in the reactor. Furthermore, product molecular weight distribution (MWD) can also be affected by microsegregation. The effect of microsegregation will be greater, the greater the reaction rates are, since diffusion and chemical kinetics compete for product selectivity in multiple reaction situations under microsegregated conditions. Mean sequence lengths and MWD can have profound effects on product properties in a commercial process. A review of the literature reveals that there have been few attempts to study diffusion and reaction in step growth copolymerizations. Fields and Ottino modeled diffusion and reaction in step growth copolymerizations occurring in laminar striations. In their work, the laminae were stationary with constant thickness (Fields S0888-5885(97)00299-6 CCC: $14.00

and Ottino, 1987a), were stationary with a monodisperse or an arbitrary distribution of striation thickness (Fields and Ottino, 1987b), and stationary with a timevariant striation thickness (Fields and Ottino, 1987c) under presumed shear and elongational flow fields. No attempt was made to account for the movement and age history of the striations in the reactor in terms of a residence time distribution. Atiqullah and Nauman (1990) studied free radical addition copolymerization in a continuous stirred tank with an exponential residence time distribution with striations whose thickness varied by the stretching action of the mean shear field as a function of their age. These striations exchange material by diffusion with an interstitial fluid which has constant concentrations of the diffusing species which represent the time-averaged concentrations that an individual striation would see as it went through the vessel. There appears to be no study of the effects of fluid mechanics on mean sequence lengths and on the molecular weight distribution in random block copolymerizations. The present study examines the effects of shear rate, residence time, and feed composition on number-average sequence lengths and number- and weight-average chain lengths (Xn, Xw) of product polymer in step growth copolymerization in a microsegregated stirred tank reactor with unpremixed feed streams. One feed stream was assumed to contain monomers AA and AX and the other monomers BB and DD, respectively. AA and AX stand for bifunctional and monofunctional monomers with the reactive group A (e.g., isocyanate), while DD and BB stand for the short and the long types of the bifunctional monomer with the end group B (e.g., hydroxyl). © 1997 American Chemical Society

4076 Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997

Chemistry The chemistry involved in the present study is the four-component random block copolymerization thatresults from the reactions of the following monomers:

AA ) bifunctional monomer with reactive ends A AX ) monofunctional monomer with reactive end A DD ) bifunctional short monomer with reactive ends B BB ) bifunctional long monomer with reactive ends B The assumption of equal reactivity of ends was used. That is, a reactive end reacts with the same secondorder rate constant regardless of the length of the chain to which it is already attached. The concentrations of reactive ends and/or product groups that participated in all possible reactions were represented by the following symbols in the model:

A ) A end on an AA monomer B ) B end on a BB monomer D ) B end on a DD monomer C ) A end on a monofunctional AX monomer AE, BE, DE ) free end on an AA, BB, or DD that is bound to a chain of any length ) R-A, R-B, R-D X ) terminal (inactive) link from reaction of a C and a free B end Thus, under the assumption of equal reactivity of ends, any B-type end (i.e., B, BE, D, or DE) could react with any A-type end (i.e., A, AE, or C) to produce a polymeric link as in the following examples: k

AE + BE 98 link and k

BE + C 98 X Mass Transport and Reaction in a Microsegregated Continuous Stirred Reactor (MSCSR) When unpremixed fluid streams enter a polymerization reactor in laminar flow, they form striations that are microsegregated. The extent of this microsegregation depends on the diffusivities of the molecules that make up the striations and on the fluid mechanics of the reactor vessel. Thus, a general rule of thumb for increasing diffusive mass transport between unpremixed fluid streams entering a polymerization reactor in laminar flow is the generation of striations that are as thin as possible. This is usually accomplished by means of applying shear (simple laminar, extensional, or elongational) over a sufficiently long period of time. Hence, shear rate and the time over which the shear rate has been applied are the two most important factors in laminar mixing. The basic challenges to any mathematical description of diffusive transport in laminar flow therefore are (1) the difficulty of describing the shear field as it varies with time and space, (2) the difficulty of accounting for striation thickness distribu-

tion, and (3), perhaps the most important of all, the difficulty of describing the frequency and the length of contacts between different striations that all have different reactor ages and thicknesses. When chemical reaction accompanies diffusion in laminar flow, the problem becomes further complicated due to the fact that physical properties change with local composition. Any mathematical treatment of diffusion and reaction in laminar flow therefore has to be based on some simplifying assumptions. The present study consisted of a mathematical description of diffusive transport and step growth copolymerization between two unpremixed streams in a continuous flow reactor with agitation that can generate a broad range of mean shear rates (10010 000 s-1). The basic assumptions of the model are given below. Basic Assumptions. The following basic assumptions were made: When two fluids containing inert scalars A and B are fed separately to a stirred vessel, a lamellar structure of alternate A- and B-rich striations form on a microscale. These striations never lose their identities although they become stretched by the shear field and exchange molecules of solute by diffusion. Thus, the reaction vessel is said to be microsegregated. The reaction vessel is completely backmixed; i.e., the striations are instantly dispersed throughout the vessel and have the exponential residence time distribution characteristic of complete backmixing (Levenspiel, 1972). The initial striation thickness of each feed stream is equal to the feed tube diameter. Only small monomer molecules, i.e., AA, AX, and DD, can diffuse. Long monomer BB or any reacted molecules do not diffuse. This assumption had to be made in order to keep the computation times from becoming excessively long (see discussion later). It is possible to relax this assumption and allow the long monomer BB also to diffuse. Doing this would increase the number of unknown boundary values and closure equations (see discussion below) by one. However, the polymeric species that are of comparable molecular size to BB would then have to be allowed to diffuse as well for the sake of consistency. This, in turn, would necessitate mathematically tracking those species individually by means of partial differential equations assigned to each one, as opposed to the method used in the present approach which tracks end-group concentrations for all nonmonomer ends. The (mean) shear rate γ˘ is uniform throughout the vessel. Each striation is stretched by the mean shear as it goes through the reactor. Thus, it becomes thinner in time and can break up into sections of high L/D. It is assumed that the striation also becomes longer in order to conserve matter. As the striations have a high length to width ratio, only diffusion in the transverse direction is considered. Each striation exchanges diffusing species with an assumed interstitial space of negligibly small volume. This interstitial space has constant concentrations AS, CS, and DS of the three diffusing species. These concentrations may be visualized as time-averaged concentrations that each striation “sees” at its walls as it goes through the reactor. These concentrations, which are the unknown boundary conditions of the convective diffusion equations, are to be determined by means of the mass conservation requirements to be discussed. This key assumption, which was adopted

Ind. Eng. Chem. Res., Vol. 36, No. 10, 1997 4077

from Atiqullah and Nauman (1990), will henceforth be referred to as the Atiqullah-Nauman boundary condition. Equal reactivity of ends is assumed; i.e., the rate constant for the reaction of the two ends is independent of the type of molecule whether it is a monomer or a chain of arbitrary length. The reaction is assumed to take place in solvent at dilute enough concentrations of reactants such that diffusion of solvent between striations can be neglected. This assumption can be relaxed at the expense of adding one more diffusing component to the model. Development of the Convective Diffusion Equation. Under the foregoing assumptions, as the striation gets stretched by the shear field, the local time-dependent shrinkage velocity vx within a striation is a linear function of position in the x-direction and is given by (Ranz, 1979)

vx )

x dδ δ dt

( )

(1)

where δ is half of the striation thickness and t is the time. The convective diffusion equation for any species j in a striation can then be written as (Ou and Ranz, 1983)

( ) (

)

∂Cj ∂Cj ∂Cj ∂ + vx ) Dj + Rj ∂t ∂x ∂x ∂x

[

]

( )

( )

Djk ∂C ∂2 C ) [1 + 0.5(γ˘ t)2] 2 + R 2 ∂t δ0 ∂ξ

(7)

where C ) {Cjk}, R ) {Rjk}, Djk ) diffusivity for species j in striation type k. Equation 7 is applied to all reactive species in striation type 1 and striation type 2. The first term on the right-hand side is used only for the diffusing species. The equations for all species in each striation are solved simultaneously subject to the initial and boundary conditions to be given in the next section. Initial and Boundary Conditions for Primary Reactive Species. It is assumed that there are two feed streams. Stream 1 contains monomers BB and DD. Stream 2 contains monomers AA and AX. The seven different reactive ends that are found on the monomers or chains, i.e., A, B, C, D, AE, BE, and DE, will be referred as the “primary species” in the following discussion. All seven primary species exist in striation type 1. But due to the assumption of nondiffusivity for reacted species, there can be no B or BE ends in striation type 2. Thus, the initial and boundary conditions for the concentrations of the primary species are given below:

striation type 1 (2)

where Dj ) diffusivity for species j, Cj ) concentration of species j, Rj ) chemical source term for species j, and x ) direction perpendicular to longitudinal axis. The average value of δ after time t can be related to the initial half-thickness δ0 by (Bourne and GarciaRosas, 1985) -1/2 δ 1 ) 1 + (γ˘ t)2 δ0 2

In matrix notation, it can be written as

(3)

A1(ξ,0) ) 0, D1(ξ,0) ) D0, C1(ξ,0) ) 0 B1(ξ,0) ) B0, AE1(ξ,0) ) 0, BE1(ξ,0) ) 0 DE1(ξ,0) ) 0

(8)

A1(1,t) ) AS, D1(1,t) ) DS, C1(1,t) ) CS ∂A1 ∂C1 ∂D1 (0,t) ) (0,t) ) (0,t) ) 0 ∂ξ ∂ξ ∂ξ striation type 2

where γ˘ ) mean shear rate. We define the dimensionless distance as

A2(ξ,0) ) A0, D2(ξ,0) ) 0, C2(ξ,0) ) C0 AE2(ξ,0) ) 0, DE2(ξ,0) ) 0

x ξ) δ(t)

(4)

Since Cj ) f(ξ,t), eq 2 can now be transformed by the rules governing partial derivatives and by use of eqs 1 and 4 and then eq 3 into

( )

( ) 2

∂Cj Dj ∂ Cj ) [1 + 0.5(γ˘ t)2] + Rj 2 ∂t δ0 ∂ξ2

(5)

Equation 5 describes the mass balance for a reacting species j in a striation at dimensionless position ξ at time t. In the case of two unpremixed streams in a segregated stirred tank, the above equation has to be applied to each species j in each striation k (k ) 1, 2). Thus, eq 5 can be written as a system of equations as follows:

( )

( )

∂Cjk Djk ∂2Cjk 2 ) [1 + 0.5(γ˘ t) ] + Rj ∂t δ02 ∂ξ2

(6)

A2(1,t) ) AS, D2(1,t) ) DS, C2(1,t) ) CS

(9)

∂A2 ∂C2 ∂D2 (0,t) ) (0,t) ) (0,t) ) 0 ∂ξ ∂ξ ∂ξ Equations 7-9 constitute a coupled system of ordinary and boundary value equations with unknown values AS, CS, and DS for A, C, and D at the boundary ξ ) 1.0. Thus, a “shooting” type of solution is needed. However, one still needs three more equations that define the unknown boundary values. Determination of the Unknown Boundary Values. In order to solve the model equations for the primary reactive species defined above, the unknown boundary values AS, CS, and DS of the system of partial and ordinary differential equations must be solved simultaneously with all the model equations in such a way that satisfies mass conservation constraints that arise from the basic premises of the model. These constraints arise from the requirement that the three diffusing species A, C, and D can only react within the two types of striations originating from the two feed


streams. That is, even though the interstitial space is assumed to contain A, C, and D at fixed concentrations, AS, CS, and DS, its volume is assumed to be negligibly small so that no net loss or gain of A, C, or D can occur on account of reaction in the interstitial space. These mass conservation constraints can be mathematically expressed as follows. Mass conservation constraint for end group A:

1-

( )( )∫ ∫ ( )( )∫ ∫ ( )∫ ∫ rF 1 A0 ht

∞

0

1

0

A1(ξ,t) dξ e-t/th dt -

1 1 A0 ht rF A0

∞

1

0

0

∞

A2(ξ,t) dξ e-t/th dt ) 1

0

0

RA1 dξ e-t/th dt -

( )∫ ∫ 1 A0

∞

0

1

0

RA2 dξ e-t/th dt (10)

Mass conservation constraint for end group C:

1-

( )( )∫ ∫ ( )( )∫ ∫ ( )∫ ∫ rF 1 C0 ht

∞

0

1

0

C1(ξ,t) dξ e-t/th dt -

1 1 C0 ht rF C0

∞

1

0

0

∞

0

C2(ξ,t) dξ e-t/th dt ) 1

0

RC1 dξ e-t/th dt -

( )∫ ∫ 1 C0

∞

0

1

0

RC2 dξ e-t/th dt (11)

Mass conservation constraint for end group D:

1-

( )( )∫ ∫ ( )( )∫ ∫ ( )∫ ∫ ( )∫ ∫ 1 1 D0 ht

∞

0

1

0

D1(ξ,t) dξ e-t/th dt -

1 1 rFD0 ht 1 rFD0

∞

0

1

0

∞

0

D2(ξ,t) dξ e-t/th dt ) 1

0

RD1 dξ e-t/th dt -

1 rFD0

∞

0

1

0

RD2 dξ e-t/th dt (12)

Here rF ) Q1/Q2, Q1, Q2 ) volumetric flow rates, ht ) mean residence time of MSCSR, and RAk, RCk, and RDk ) source terms for A, C, D in striation type k ()1, 2). Solution Procedures. A two-stage solution procedure had to be followed for the MSCSR model. In the first stage, a special type of shooting method was employed whereby the matrix eq 7 for the primary species, subject to the initial and boundary conditions of eqs 8 and 9, was solved simultaneously with eqs 1012 The two matrices of eq 7 were

[] [] A1

A2

RA1

RA2

B1

0 C2

RB1

0 RC2

C1

R C1

C ) D1 D2 AE1 AE2

and R ) RD1 RD2 RAE1 RAE2

BE1 0 DE1 DE2

RBE1 0 RDE1 RDE2

(13)

where the elements of the matrices C and R are, respectively, the concentrations of primary reactive species and the source terms for the primary species in the two types of striations. The source terms were written down for each element of the matrix C as the algebraic sum of the mass action rate expressions of the reactions that consumed or produced that reactive species. The equations for A, C, and D in each striation were partial differential equations; the rest were ODEs. However, all equations had to be solved simultaneously by the method of lines with central differencing on a 20-node grid across the dimensionless space interval ξ ) 0-1.0. The solution procedure is summarized in Figure 1 and described below. 1. In order to find good initial guesses for AS, CS, and DS (see next step) solve the ideal CSTR case. This was done by providing subroutines in the code for the ideal CSTR case which solved seven nonlinear algebraic equations corresponding to the well known CSTR equation for each of the seven primary species involved. With the CSTR solutions for A, C, and D in hand, lower and upper boundaries for AS, CS, and DS were specified as the CSTR value multiplied by a factor (usually ranging from 0.01 to 20). 2. Solve for the unknown variables AS, CS, and DS by a special shooting technique that consisted of solution of eq 7 for the primary species in both striations by method of lines starting with initial guesses of the unknown boundary values (AS, CS, and DS) and utilizing the minimization routine B2LSF from NAG library to solve for the locally optimal values of AS, CS, and DS from eqs 10-12. B2LSF uses a modified MarquartLevenberg method to solve nonlinear least-squares problems subject to simple bounds on the variables to be optimized. The bounds and the initial guesses for the unknown variables are provided by the user. The bounds were described above. The initial guesses for AS, CS, and DS were selected from the range specified by the lower and upper bounds by a random number generator. A random number between 0 and 1 was multiplied by the range to get the initial guess for each one. An objective function FNORM was defined as the square root of the sum of the squares of the three residual functions based on eqs 10-12, defined as

residual (i) ) lhs(i)/rhs(i) - 1.0 where i ) 1-3 for eqs 10, 11, and 12, respectively. The optimal values of the three unknown boundary concentrations that minimized FNORM locally were then determined by B2LSF. The value of FNORM was then compared to a pre specified tolerance, which was set equal to 0.0001. If FNORM was equal to or less than the tolerance, the solution process progressed to the next step. If FNORM was higher than the tolerance, then new initial guesses were provided by means of a new random number. This process could be repeated a specified number of times. However, in some cases, the only local solution would be on the bounds, and the bounds had to be changed by respecifying the factors described in step 1 above. Upper integration limits on the first integral in eqs 10-12 was set equal to 20 × mean residence time based on experience with the model. It was found that the value of the integral did not change noticeably beyond 20 residence times. On average, convergence to the desired tolerance could be


Figure 1. Flow diagram for solution procedure.

achieved within a CPU time of about 60 min on a CRAY C90. 3. Upon convergence, the optimal values of AS, CS, and DS were fixed in the model, and a final solution with the known interstitial boundary values was started. This solution obviously did not require the shooting method described above. It consisted of simultaneous solution of initial and boundary value equations for the dependent variables of interest. In the final solution, in addition to the primary species, some additional variables were introduced as will be discussed below. Additional Species in the Final Solution. In the final solution, reactive species of specific significance were defined and solved in order to calculate the mean

sequence lengths (see discussion below). These, to be referred to as “secondary species” were

AB ) R-BB-AA ) chain-end AA linked to a BB AD ) R-DD-AA ) chain-end AA linked to a DD CB ) R-BB-AX ) chain-end AX linked to a BB CD ) R-DD-AX ) chain-end AX linked to a DD BAD ) R′-BB-AA-DD-R ) internal link joining -BBAA- and -DDAA- sequences


Furthermore, the first three moments of chain length distribution (CLD) of the six types of chains were computed as will be described below. The secondary species defined above and the moments of the chain length distribution in both types of striation are nondiffusing. Therefore, eq 7 applied to the secondary species and the moments would have only the source term on the rhs. The local values of all species (primary and secondary) and the moments at every position of the ξ-grid can then be computed by the simultaneous solution of eq 7 at every grid position provided the interstitial boundary values of the diffusing species and the source terms for all the nondiffusing species and moments are known. Source terms for the secondary species were written under the same assumptions made for the source terms of the primary species. Source terms for the moments required development of population balances and moment generating functions as will be described below. Development of Conservation Equations for Moments Number- and weight-average chain length for the product polymer were calculated by use of the first three moments of the chain length distribution by the so called method of moments. In order to simplify the mathematics, no distinction was made between reacted monomers BB and DD in the polymerized chain. Thus, the number- and weight-average chain lengths were based on repeating BB or DD units in the polymer chain, which were both described as BB in the moments treatment to follow. With this simplification, six types of chains can be distinguished in the product polymer which can be expressed as

AAn ) (AABB)nAA BBn ) (BBAA)n-1BB

Population Balances. One can write population balances for the rates of generation of the above six chain types by using the same assumptions that were used for the reactive ends. Thus, remembering the definitions of the reactive ends A, C, B and D, one can write for striation type 1 (subscript k ) 1 dropped for simplicity)

d(AAn)

) -2k(B + D)AAn + k(A)ABn +

dt

∞

n

2k

∑

AAmABn-m - 2kAAn(2

m)1

∑ BBn +

n)2 ∞

∑

∞

ABn +

n)1

d(ABn)

n

4k

n

∑ BBmAAn-m + km)1 ∑ ABmABn-m m)1

∞

kABn(2

∞

∑ BBn + m)1 ∑ (2ABm + 2AAm + BXm + AXm)) n)2 (15)

d(BBn)

) -2k(A)BBn - 2k(C)BBn +

dt

∞

n

2k

∑

BBmABn-m - 2kBBn(

m)1

∑ (2AAn + ABn + AXn))

n)1

(16) d(AXn) dt

) -k(B + D)AXn + k(C)ABn + 2k

ABn ) (AABB)n

∑

n

AAn-mBXm + k

m)1

∑ AXmABn-m -

m)1 ∞

AXn ) (AABB)nAX

XXn ) XA(BBAA)n-1BBAX

kAXn( d(BXn) dt 2k

∞

∑ (ABn + BXn) + 2n)2 ∑ BBn) n)1

(17)

n

) -k(A)BXn - k(C)BXn + k

∑ ABmBXn-m +

m)1 ∞

n

where n ) 2, 3, 4 for BBn and n ) 1, 2, 3 for all except BBn. Note that n indicates the number of repeating BB units in each species. It is clear from the above indexing convention that none of the six chain types includes the monomers AA, BB, DD, and AX. Thus, the moments to be calculated will belong to polymeric species exclusively. Moments for each of the six types of chains had to be calculated. Derivation of the species balance equations for the moments will be given in this section. Briefly, the procedure consists of writing population balances for the six chain types, taking into account all chemical events that generate or consume each chain type, converting these population balances to species balances for moment generating functions (MGF), and converting these equations for MGF to species conservation equations for the zeroth, first, and second moment of each chain type, as will be described in the next section.

(14)

) -k(A + B + D)ABn - k(C)ABn +

dt

n

BXn ) (BBAA)n-1BBAX

∑ BXn)

n)1

∑

BBmAXn-m - kBXn(

m)1

∑ (ABn + 2AAn + AXn))

n)1

(18) d(XXn) dt

n

) -k(C)BXn + k

∑ AXmBXn-m

(19)

m)1

where k is the rate constant, assumed to be the same for all reactions. Population balance equations for striation type 2 are similar to those given above for striation type 1 with the exception that (B + D) is replaced by D in each occurrence since striation type 2 contains no B under the assumptions of the model. Also, in the equations for striation 2, B in ABn, BBn, and BXn represents only reacted DD for the same reason. Derivation of Source Terms for the Moments. In general, a moment generating function (MGF) G(s) for


moments λ0, λ1, and λ2 of the polymeric species Pn (Ray, 1972)

a polymeric species Pn is defined as ∞

G(s) )

snPn ∑ n)1

(20)

where s is an arbitrary variable. Then the moment generating function for each of the above chain types can be defined as

∑ s AAn n

(21)

n)1 ∞

BB(s) )

snBBn ∑ n)1

(22)

∞

AB(s) )

snABn ∑ n)1

(23)

∞

AX(s) )

snAXn ∑ n)1

BX(s) )

∑ snBXn n)1

(24)

∞

(25)

∞

XX(s) )

snXXn ∑ n)1

(26)

From the definition of the kth moment of the chain length distribution, the kth moment for each of the six chain types is given by ∞

∑ n AAn k

RRk )

(27)

n)1 ∞

ββk )

[ ] dG(s) ds

∑ nkBBn n)1

(28)

s)1

(33)

2

ds

+

s)1

(34)

) λ1

[ ] [ ] d2G(s)

∞

AA(s) )

G(s ) 1) ) λ0

dG(s) ds

s)1

) λ2

(35)

With these relationships on hand, the population balance eqs 14-19 above can be transformed to moment balance equations as follows: Multiply both sides of eqs 14-19 by sn and sum both sides from n ) 1 to ∞. Utilizing the definitions of the moment generating functions above (eqs 21-26) and some mathematical manipulation, transform eqs 14-19 to ODEs in terms of the six moment generating functions and the concentrations of the monomers A, C, B, and D. Finally utilize eqs 33-35 to transform these equations to equations for the first three moments of each of the six chain types. Detailed description of the method of moments was given by Fields and Ottino (1987a). As it happens, the ODEs derived for the moments for the present problem have particularly lengthy right hand sides. Therefore, only the equations for one type of chain (AAn) are given below. Derivation of the others are straightforward. Also, subscripts for the striation type are dropped for simplicity. The equations below are exact for striation type 1. For striation type 2, the same equations apply but the rhs would have only D terms instead of the (B + D) terms.

source terms for moments of AAn in striation type 1 d(RR0) ) -2k(B + D)RR0 + k(A)Rβ0 + 2kRR0Rβ0 dt 2kRR0(2ββ0 + Rβ0 + βχ0) (36) d(RR1) ) -2k(B + D)RR1 + k(A)Rβ1 + 2k(RR0Rβ1 + dt Rβ0RR1) - 2kRR1(2ββ0 + Rβ0 + βχ0) (37)

∞

Rβk )

nkABn ∑ n)1

(29)

∞

Rχk )

nkAXn ∑ n)1

βχk )

nkBXn ∑ n)1

(30)

∞

(31)

∞

χχk )

nkXXn ∑ n)1

(32)

where k ) 0, 1, 2 would denote the zeroth, first, and second moments, respectively. It can be shown that the following relationships exist between the moment generating function G(s) and its derivatives (evaluated at s ) 1) and the first three

d(RR2) ) -2k(B + D)RR2 + k(A)Rβ2 + 2k(RR0Rβ2 + dt Rβ0RR2 + 2RR1Rβ1) - 2kRR2(2ββ0 + Rβ0 + βχ0) (38) The 36 ODEs (k ) 0, 1, 2, for six chain types in two striations) that resulted from the above procedure were in terms of the four monomers A, C, B, and D and the 36 moments defined above. These ODEs were the source terms for the moments for eq 7. The initial conditions for the moments were similar to those for the nondiffusing species shown in eqs 8 and 9; i.e., all moments were equal to zero at time zero at any value of ξ. Calculation of Product Properties Once concentration of each reacting species or moment was known at every node of the ξ-grid for all values of time, then the exit concentrations of the 12 reacting species and the exit values (concentrations) for


the 18 moments were computed from the general relationship

Cje )

( )( )∫ ∫ ( )( )∫ ∫ rF 1 rF + 1 ht

∞

0

1

0

Cj1(ξ,t) dξ e-t/th dt +

1 1 rF + 1 ht

∞

0

1

0

Cj2(ξ,t) dξ e-t/th dt (39)

where Cje represents the exit concentration of the reactive group or the moment. Cj1 and Cj2 represent the concentration of the particular group or moment in striation type 1 or type 2. With the exit concentrations known, conversion XB based on B end groups or XA based on A end groups, mean sequence length for BBAA sequences (〈BBSL〉), and mean sequence length for DDAA sequences (〈DDSL〉), could be calculated by the relationships

(B + BE + D + DE) XB ) 1 β(B0 + D0) XA ) 1 -

(A + AE + C + CB + CD)

〈BBSL〉 )

〈DDSL〉 )

(1 - β)(A0 + C0) βB0 - B AB + BE + CB + BAD βD0 - D AD + DE + CD + BAD

(40)

(41)

(42)

(43)

where

β)

rF 1 + rF

With the exit values for the moments known, the moments for the mixture of the six chain types, i.e., the product polymer, can be calculated simply from the additivity property of the moments as

λ0e ) RR0e + ββ0e + Rβ0e + Rχ0e + βχ0e + χχ0e

(44)


(45)


(46)

where the subscript e stands for the exit value from eq 39. Once the first three moments of the polymer chain population are known, number and weight-average chain lengths can be calculated from the well-known relationships

Xn ) λ1/λ0

(47)

Xw ) λ2/λ1

(48)

Discussion of Results Table 1 summarizes the parameters used for the base case calculation. These are realistic parameters for a typical industrial case of block copolymerization occurring in a high-shear agitated vessel such as a rotorstator device. Numerical Experiments. Inspection of eq 7 reveals that at constant temperature, time and mean shear rate

Table 1. Base Case Parameters for the Computations stream 1 flow rate (m3/s) stream 2 flow rate (m3/s) BB in stream 1 (kg mol/m3) DD in stream 1 (kg mol/m3) AA in stream 2 (kg mol/m3) AX in stream 2 (kg mol/m3) k (m3/kg mol s) Ea (J/(kg mol)) mean shear rate (1/s) diffusivity of AA, AX in striation type 1 (m2/s) diffusivity of DD in striation type 1 (m2/s) diffusivity of AA, AX in striation type 2 (m2/s) diffusivity of DD in striation type 2 (m2/s) initial half thickness of striation type 1 (m) initial half thickness of striation type 2 (m) reactor volume (m3) mean residence time (s) temperature (°C)

1.6 × 10-5 4.0 × 10-6 0.15625 0.09375 1.00000 0.05000 7.161 × 1016 8.4 × 107 100 1.0 × 10-11 5.0 × 10-12 1.0 × 10-9 5.0 × 10-10 1.0 × 10-2 5.0 × 10-3 2.0 × 10-3 100 80

are the two independent variables that strongly affect the rate of diffusive transport. Therefore, in the case of a fast chemical reaction, it stands to reason that time and mean shear rate might have significant influence on mean chain lengths and monomer sequence lengths in unpremixed step-growth copolymerization in a stirred vessel reactor. The dependent variables of eq 7 in this work were the concentrations of the reactive species defined earlier and the moments of the chain length distribution. It can also be argued that feed composition, which defines the initial values for the concentrations, could have significant influences on the product properties of interest. It is relevant to point out that there are no experimental data or other modeling studies available in the open literature for relating the effects of diffusive transport, as defined by shear rate and residence time, or the effects of feed stochiometry on product chain length distribution and sequence length distribution in step-growth copolymerization with mono- and bifunctional monomers in a continuous reactor with unpremixed feeds. Experiments to investigate such relationships are complicated and costly. Although mathematical modeling and analysis are no substitute for experimentation, so called “numerical experiments” with a plausible model can be performed with relative ease and they can provide valuable insights into the system which may not be available otherwise and which can be used to design some key experiments for validation of the model. In the following discussion, the term “numerical experiments” is used, for lack of a better term, to indicate sets of computations which eventually yielded the converged results that are reported here. Numerical experiments were carried out with the model in order to investigate the effects of mean residence time, mean shear rate, and molar ratio (Φ) of A (in AA and AX) to B (in BB and DD) in the two feed streams on the following product properties: numberaverage chain length Xn, ratio of weight-average to number-average chain length (polydispersity Zp), and number-average sequence lengths of the short monomer DD and the long monomer BB. Equation 7 is a Lagrangian convective diffusion relationship for a given striation where the variable “time” is the age of the striation. Since the mean residence time is the most probable exit age for the fluid elements, it was chosen as the integral time variable for plotting purposes. For convenience only, the mean residence time in parts a and b of Figure 2 is shown as a dimensionless multiple of the base case residence time that corresponded to the flows in Table 1.


Figure 2. Model predictions for (a) conversion, number-average chain length (Xn), and sequence lengths and (b) Xn and polydispersity (Zp) vs residence time at constant shear rate (γ ) 100 s-1) and feed ends-ratio (Φ ) 1.05).

Figure 3. Model predictions for (a) conversion, number-average chain length (Xn), and sequence lengths and (b) Xn and polydispersity (Zp) vs shear rate at constant residence time (θ ) 1.0) and feed ends-ratio (Φ ) 1.05).

In the interest of avoiding repetitive use of certain somewhat lengthy groups of terms in the following discussion, the following verbal shortcuts will be employed: The term “residence time” is used to indicate the dimensionless mean residence time referenced to the base case as discussed above. Similarly, the term “mean” will be dropped in referring to the mean shear rate; and the term “number-average” will be dropped in referring to the sequence lengths since only the number-average value was computed. Also, since the discussion of the number- and weight-average chain lengths Xn and Xw is handled in terms of Xn and Zp (polydispersity or the ratio of Xw to Xn), the numberaverage chain length will be henceforth referred to simply as the chain length. Six sets of numerical experiments were performed. In the first series of experiments, the residence time was varied from the base value by varying the base case volume. All other parameters were kept at their base case values (Figure 2). In the second through fourth sets of experiments, the shear rate was varied at a fixed feed ends-ratio (Figures 3-5). Three ends-ratios Φ were used for the second through fourth sets: 1.05, 0.95, and 0.90. As Table 1 shows for the base case, feed stream 1 consisted of long (BB) and short (DD) monomer in 5:3 ratio while the feed stream 2 consisted of bifunctional monomer AA and monofunctional monomer AX in a 20:1 ratio. These ratios were kept unchanged in all experiments. However, the ratio of total A groups (in AA and

AX) to total B groups (in BB and DD) were varied as 1.05, 0.95, and 0.90. (The base case described in Table 1 corresponded to the ratio 1.05.) The actual numerical values of the concentrations can be seen in Table 1. In the fifth set, residence time and shear rate were varied (at constant Φ) simultaneously so as to maintain a constant conversion (Figure 6). In the sixth set, Φ (feed ends-ratio) was varied at constant residence time and shear rate (Figure 7). Due to the long computation times, only limited numbers of numerical experiments could be performed for each set of experiments. Figure 2a shows model predictions for conversion (XB), chain length (Xn), sequence length for short monomer DD (〈DDSL〉), and sequence length for long monomer BB (〈BBSL〉) vs residence time at a constant shear rate of 100 s-1 and ends-ratio (Φ) of 1.05. As the residence time increases from 0.25 to 20, conversion increases asymptotically as would be expected. Chain length Xn and sequence lengths 〈DDSL〉 and 〈BBSL〉 also follow the asymptotic trend which cannot be predicted a priori. These results indicate that mean residence time is an important factor in polymerization under microsegregated conditions in that given enough time, chain and sequence lengths build up along with conversion regardless of the shear rate that is provided by agitation. Figure 2b shows Xn and Zp, polydispersity, vs the residence time again at a constant shear rate of 100 s-1 and ends-ratio of 1.05. The polydispersity drops from




2.56 to 2.36 as θ varies from 0.25 to 1.0, but it remains nearly stable up to a θ of 20 with only a slight drop to 2.26. These results imply that under microsegregated conditions polydispersity is only slightly affected by residence time except perhaps at very short residence times where it shows a tendency to rise. Figure 3a shows model predictions for conversion (XB), chain length (Xn), sequence length for short monomer DD (〈DDSL〉), and sequence length for long monomer BB (〈BBSL〉) vs the shear rate at a constant dimensionless residence time of 1.0 an ends-ratio of 1.05. As the shear rate increased from 50 to 10 000 s-1, conversion increased in an asymptotic fashion from 0.86 to 0.99. Chain length Xn also followed the asymptotic trend, which again cannot be predicted a priori. Sequence lengths 〈DDSL〉 and 〈BBSL〉, on the other hand, exhibit an asymptotic rise to about 4000 s-1 followed by an imperceptibly slow decline. This trend which certainly cannot be predicted a priori becomes much more pronounced at different feed stochiometries as will be seen below. It is interesting to note that Figures 2a and 3a also show that constant values of the product γ˘ θ, average total strain (Edwards, 1985), do not produce identical values of product properties, e.g., chain length. For instance a γ˘ θ value of 1200 would correspond to an Xn of 14.5 in Figure 2a and of 13.4 in Figure 3a. Similar differences in conversion and sequence lengths that can

be shown would warn against using total average strain as a basis for design and scaleup in similar problems. Figure 3b shows Xn and Zp vs the shear rate at a constant residence time of 1.0 and feed ends-ratio of 1.05. Again, Zp exhibits a very slight downward drift as shear rate increased from 50 to 10 000 s-1, i.e., from 2.39 to 2.28. Figures 2b and 3b suggest that polydispersity has only a very slight dependency on residence time or shear rate. Figure 4a shows model predictions for conversion (XA), chain length (Xn), sequence length 〈DDSL〉, and sequence length 〈BBSL〉 vs the shear rate at a constant dimensionless residence time of 1.0 and ends-ratio of 0.95. Due to the feed ends-ratio being less than unity, XA instead of XB is shown in the plot since A and not B is the limiting reagent. As the shear rate increases from 100 s-1, chain length Xn, and sequence lengths 〈DDSL〉 and 〈BBSL〉 all exhibit maxima while conversion increases steadily. The chain length reaches a maximum in the 2000-4000 s-1 range while the sequence lengths 〈BBSL〉 and 〈DDSL〉 exhibit very pronounced maxima around 6000 s-1 that are about 2-fold and 5-fold the initial values, respectively. These trends in product properties are greatly different from those seen in Figure 3a where the only difference was that Φ, the feed ends-ratio, was 1.05 vs 0.95 in Figure 4a. Figure 4b shows Xn and Zp, polydispersity, vs the shear rate at a constant residence time of 1.0 and ends-


Figure 6. Model predictions for (a) sequence lengths and (b) number-average chain length (Xn) and polydispersity (Zp) vs shear rate at constant conversion (XB ) 0.955) and feed ends-ratio (Φ ) 1.05).

ratio of 0.95. Results for Zp are qualitatively very similar to those in Figures 2b and 3b with a slight initial drop (2.29 to 2.14) followed by a slight downward drift (2.14 to 2.02) over the entire range of shear rates. Figure 5a shows model predictions for conversion (XB), chain length (Xn), sequence length 〈DDSL〉, and sequence length 〈BBSL〉 vs the shear rate at a constant dimensionless residence time of 1.0 and ends-ratio of 0.90. Again due to the feed ratio being less than unity, XA instead of XB is shown in the plot. As in the case of Φ ) 0.95 (Figure 4a), as the shear rate increases from 100 s-1, Xn, 〈DDSL〉, and 〈BBSL〉 all exhibit very pronounced maxima while conversion increases steadily. Again 〈BBSL〉 and 〈DDSL〉 exhibit very pronounced maxima that occur after the maximum in Xn (around 1300 vs 800 s-1) that are about 2-fold and 4-fold the initial values, respectively. These trends in product properties are greatly different from those seen in Figure 3a where the only difference was that Φ, the ends-ratio, was 1.05 vs 0.90 in Figure 5a and qualitatively very similar to those in Figure 4a where the endsratio was 0.95. The results in Figures 4a and 5a seem to suggest that when the ends-ratio drops below unity the product chain and sequence lengths develop great sensitivity to the shear rate. Figure 5b shows Xn and Zp vs the shear rate at a constant residence time of 1.0 and ends-ratio Φ of 0.90. Zp undergoes a slight initial drop from 2.16 to 2.11 as

Figure 7. Model predictions for (a) sequence lengths and (b) number-average chain length (Xn) and polydispersity (Zp) vs feed ends-ratio at constant residence time (θ ) 1.0) and shear rate (γ ) 100 s-1).

shear rate increases from 100 to 400 s-1 after which it is stable within reproducibility over the entire range of shear rates. These results are qualitatively similar to the previous results at Φ of 0.95 and 1.05 in Figures 3b and 4b. Figures 3-5 show model predictions for product properties against shear rate at a fixed residence time of 1.0. Therefore, conversion is different for every single numerical experiment shown in Figures 3-5. An interesting case is that where the residence time is also varied as the shear rate varies such that the conversion is kept constant. However, such numerical data are very costly to obtain since one has to perform numerical experiments at a number of different combinations of γ˘ and θ until the target conversion is achieved. Therefore, only three such data sets could be generated. Parts a and b of Figure 6 show the results where the conversion is constant at 0.955 as shear rate (and residence time) were varied at constant Φ of 1.05. Figure 6a shows that at constant conversion, shear rate (mixing intensity) has a very slight effect on the sequence lengths as the shear rate changed from 100 to 800 s-1, even though in the previous figures (Figures 3a-5a) with varying conversion great changes could be observed in sequence lengths as shear (or mixing intensity) was increased. Similar to the trends in Figure 6a, Figure 6b shows that


Xn exhibits a very modest variation while Zp hardly varies at all as the shear rate changed from 100 to 800 s-1. Parts a and b of Figure 7 show model predictions for sequence lengths, Xn, and Zp against ends-ratio at constant shear rate of 100 s-1 and residence time of 1.0. Note that as Φ varies at constant shear and residence time, the conversion is again not constant. In Figure 7a, both sequence lengths display a steady decline as the feed ends-ratio increases from 0.90 to 1.05 This decline corresponds to a variation coefficient (range/ mean) of about 0.4 for the DD sequences and about 0.20 for the BB sequences indicating that flow control problems can bring about appreciable changes especially in the DD sequence lentos. In Figure 7b, Xn shows a very modest decrease and Zp an even more modest increase as the feed ratio increases. Conclusions A plausible model, developed for unpremixed stepgrowth copolymerization in a microsegregated continuous stirred tank reactor, has been used in performing numerical experiments under a limited range of experimental variables due to the high demands on supercomputer times. Model predictions have not been validated against experimental data due to the absence of such data. It is not possible to draw generalized conclusions without validation. Only a summary of the more significant results will be given. The reader is reminded that other systems with different chemistry, reactor residence time distribution, feed tube geometry, and feed composition may very well produce significantly different results. Model predictions suggest that reactor residence time, shear rate and feed ends-ratio influence the product properties number-average chain length and monomer sequence lengths in ways that cannot be predicted a priori. The results suggest that substantial changes in chain and sequence lengths can be effected when shear rate is increased at constant residence time and constant feed ends-ratio particularly when the latter is less than unity. On the other hand, if the conversion is kept constant by varying shear rate and residence time together, the influence of shear rate on chain and

sequence lengths is minimal. Model predictions also suggest that when the feed ends-ratio is varied at constant shear and residence time, sequence lengths can vary by appreciable amounts (20-40%). According to the model predictions, polydispersity is influenced little by any of the variables that have been studied. Acknowledgment Mr. Anatoly Genin collaborated in the development of the computer code for the model described here. His contribution is gratefully acknowledged. Literature Cited Atiqullah, M.; Nauman, E. B. A Model and Measurement Technique For Micromixing In Copolymerization Reactors. Chem. Eng. Sci. 1990, 45, 1267. Bourne, J. R.; Garcia-Rosas J. Laminar Shear Mixing In Reaction Injection Molding Polym. Eng. Sci. 1985, 25, 1. Edwards, M. F. Laminar Flow and Distributive Mixing. In Mixing in the Process Industries; Harnby, N., Edwards, M. F., Nienow, A. W., Eds.; Butterworths: London, 1985. Fields, S. D.; Ottino, J. M. Effect of Segregation on the Course of Unpremixed Polymerizations. AIChE J. 1987a, 33, 959. Fields, S. D.; Ottino, J. M. Effect of Striation Thickness Distribution on the Course of an Unpremixed Polymerization. Chem. Eng. Sci. 1987b, 42, 459. Fields, S. D.; Ottino, J. M. Effect of Stretching Path on The Course of Polymerizations: Applications to Idealized Unpremixed Reactors. Chem. Eng. Sci. 1987c, 42, 467. Levenspiel, O. Chemical Reaction Engineering; John Wiley & Sons: New York, 1972; Chapter 9. Ou, J.-J.; Ranz, W. E. Mixing and Chemical Reactions. Chem. Eng. Sci. 1983, 38, 1005. Ranz, W. E. Applications of a Stretch Model to Mixing, Diffusion, and Reaction in Laminar and Turbulent Flows. AIChE J. 1979, 25, 41. Ray, W. H. On the Mathematical Modeling of Polymerization Reactors J. Macromol. Sci.sRev. Macromol. Chem. 1972, C8, 1.

Received for review April 25, 1997 Revised manuscript received July 21, 1997 Accepted July 22, 1997X IE9702993

X Abstract published in Advance ACS Abstracts, September 1, 1997.

A Study of Diffusion and Reaction in Unpremixed Step Growth

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