Homogeneous Continuous-flow Stirred Tank Reactors Research Articles

Solution of the moment closure problem for reversible step-growth polymerization with unequal reactivity of monomer in homogeneous continuous-flow stirred tank reactors

General reversible step-growth polymerization in homogeneous continuous-flow stirred tank reactors (HCSTRs) has been modelled by monomer having different reactivity and the moment closure problem has been solved. From dimensionless mole balance relations for all species, the generation relation has been derived. The resulting equation is a non-linear ordinary differential equation whose analytical solution yields all moments of the molecular-weight distribution to within an arbitrary constant. We have determined this constant through the molecular-weight distribution of the polymer determined in our previous paper. Sometimes high vacuum is applied in order to form polymer of high molecular weights. On doing this, the condensation product as well as the polymer leaves the reaction mass through flashing. We have examined the effect of flashing upon the molecular weight distribution of the polymer formed in HCSTRs.

Analytical solution of molecular-weight distribution in reversible step-growth polymerization in homogeneous continuous-flow stirred tank reactors following unequal reactivity

General unequal reactivity in reversible step-growth polymerization has been modelled by assuming that monomer reacts at different rates. The generation relation for the moment-generating function G has been derived for polymerization in homogeneous continuous-flow stirred tank reactors. It is a non-linear ordinary differential equation and has been solved analytically using the Frobenius method. Analytical solution of the molecular-weight distribution of the polymer is obtained from this in a natural way and is shown to be valid even when there is flashing of condensation product. Subsequently, the molecular-weight distribution at equilibrium has been derived.

Analytical solution of the molecular weight distribution of reversible step growth polymerization under unequal reactivity in homogeneous continuous-flow stirred tank reactors

Relation entre la longueur de chaines et la reactivite inegale, modelee en assumant que le monomere reagit avec d'autres oligomeres (y compris lui-meme) a une vitesse differente

An efficient algorithm of computation of molecular weight distributions and its moments for reversible step growth polymerization in homogeneous continuous flow stirred tank reactors

The reversible step growth polymerization in homogeneous continuous flow stirred tank reactors (HCSTRs), in which the condensation product (W) leaves the reactor through flashing, has been analyzed. The molecular weight distribution (MWD) of the polymer formed is governed by nonlinear coupled algebraic relations to be solved simultaneously. To find the MWD numerically a large number of these are normally solved simultaneously using a suitable iterative procedure. In this paper, these have been decoupled using the technique proposed in our earlier works (1, 2) and the MWD can now be obtained sequentially without any trial and error. This leads to considerable saving in computation time compared to methods currently used. To demonstrate the efficacy of the algorithm, the polycondensation step of the poly(ethylene terephthal-ate) (PET) formed in HCSTRs has been analyzed. The MWD, the average chain length and the polydispersity index of the polymer have been computed and it takes 0.1 CPU seconds on a DEC 1090 as opposed to the earlier method which would take seventy minutes for similar computations. The simple model of the HCSTR for the PET formation gives the effect of reactor temperature and pressure and the quantitative results have been presented in this paper.

Polymerization of melamine and formaldehyde in homogeneous continuous‐flow stirred‐tank reactors using functional group approach: Part A: Conversion of functional groups

AbstractA comprehensive kinetic model using the functional group approach has been proposed for the polymerization of melamine and formaldehyde. The kinetic model is consistent with the basic chemistry of polymerization and involves five rate constants which have been estimated using the experimental data of Tomita. Homogeneous continuous‐flow stirred‐tank reactors (HCSTRs) have been modelled and the mole balance relations for various functional groups have been written. The performance of HCSTRs is governed by algebraic equations and, for any specified residence time, is found by the method of successive substitution using the Brown's algorithm. The computations show that as long as free formaldehyde is present, the reaction mass would consist predominantly of substituted melamine molecules. However, after formaldehyde is completely reacted, larger oligomers are formed in larger concentrations. On comparison of results with batch reactors, it is found that for the same reaction time HCSTRs yield polymer with higher branching.

Polymerization of melamine and formaldehyde in homogeneous continuous‐flow stirred‐tank reactors using functional group approach: Part B: Molecular weight distribution

AbstractPolymerization of melamine and formaldehyde in homogeneous continuous‐flow stirred‐tank reactors (HCSTRs) is reversible and leads to formation of branched polymer due to the hexafunctional nature of melamine. The reversed reaction of branched molecules depends upon the chain structure, and herein a simple model is presented to account for this. The functional group analysis of part A of this series has been extended and the molecular weight distribution (MWD) relations for HCSTRs have been solved. The MWD of the polymer is extremely sharp and the molecules are more branched compared to those for batch reactors and can be explained as follows. A given reacted bond can react with water as well as free formaldehyde through the reversed reaction, and the rate constant for the latter step is much larger. Consequently, the chain growth is limited as long as there is free formaldehyde in the reaction mass. Lower conversion of melamine and formaldehyde in HCSTRs is used to explain the behavior observed.

Effect of intramolecular reactions in multifunctional step growth polymerizations in cascades of continuous‐flow, stirred‐tank reactors

AbstractA simple kinetic model describing intramolecular reactions in multifunctional step growth polymerizations of RAj monomers is presented. The model is far simpler than more detailed kinetic or Monte Carlo models which pose severe numerical difficulties close to gelation. The formulation presented is quite general, but it requires a relationship for the ensemble‐cum‐time average number of rings, rn, per molecule having n RAj units. A very simple yet intuitively correct equation has been chosen for rn, which involves an empirical parameter 𝒜 Results on cascades of homogeneous continuous‐flow stirred‐tank reactors (HCSTRs) show trends which are physically expected. The present formulation could possibly be used for scale‐up of HCSTRs, with 𝒜 obtained experimentally. It is hoped that future studies using rigorous models would lead to more fundamental and improved relationships for rn.

Forced oscillations in continuous flow stirred tank reactors with nonlinear step growth polymerization

AbstractThe self‐step growth polymerization of RAf monomers in homogeneous, continuous flow stirred tank reactors (HCSTRs) is simulated under conditions of periodic feed concentration (with frequency ω and amplitude α). By having periodic operation, the polydispersity index of the polymer is found to increase by about 35% over the values at steady state. Periodic operation of HCSTRs is found to lead to gelation only for certain values of the frequency and the dimensionless residence time τ*. Gelling envelopes have been obtained to give conditions under which HCSTRs should be operated. These envelopes can be described in terms of two critical dimensionless residence times, τ and τ such that nongelling operation is always ensured when τ* < τ. For τ* > τ, periodic operation always leads to gelation, and HCSTRs cannot be used. For τ < τ* < τ, the gelling behavior is found to depend on the functionality f, amplitude α, and the dimensionless residence time τ*.

Modelling of condensation polymerization of novolac‐type phenol–formaldehyde in homogeneous, continuous‐flow, stirred‐tank reactors

AbstractA detailed kinetic model for the novolac‐type phenol–formaldehyde polymerization has been proposed in terms of five reactive sites. These sites have been shown to have different reactivities. Mass‐balance equations for homogeneous, continuous‐flow, stirred tank reactors (HCSTR) for each of the species have been written, and these have been solved numerically. A comparison of the results with those for the batch reactor shows that, for a given residence time, smaller but more polydispersed and branched chains are formed.

Computational scheme for the calculation of molecular weight distributions for nylon 6 polymerization in homogeneous, continuous‐flow stirred‐tank reactors with continuous removal of water

AbstractAn efficient computational scheme has been established for obtaining the molecular‐weight distributions (MWDs) for Nylon 6 polymerization in homogeneous, continuous‐flow stirred‐tank reactors (HCSTRs) with water vaporization. A three‐step procedure is used: the moment equations are solved by a sequential application of Brown's algorithm, the moments so calculated are used to give an empirical Flory‐Schultz distribution and, finally, this empirical distribution is used as a starting guess in Brown's algorithm applied to species mass‐balance equations to give the exact MWD. The moment‐closure approximations normally used for hutch reactors have been found to work equally well with HCSTRs for residence times of industrial significance.

Simulation of reversible nylon-66 polymerization in homogeneous continuous-flow stirred tank reactors

Experimental data of Ogata1 has been curve-fitted to obtain the forward and reverse rate constants for nylon-66 polymerization. Its molecular weight distribution (MWD) has been simulated in homogeneous continuous-flow stirred tank reactors (HCSTR) for 11 h of residence time when the reaction mass is very close to equilibrium. The set of algebraic equations have been solved using Brown's algorithm,2 which was found to be more efficient compared to the Gauss-Jordon techniques of solution. The MWD thus obtained is compared with our earlier simulation of the molecular weight distribution from batch reactors3 and was found to differ significantly. In HCSTR, the weight fraction distribution does not undergo a maximum and the polydispersity index ρ of the polymer formed is much higher than that obtained from batch reactors. The number and weight average of the polymer formed in HCSTR is found to be significantly lower.

Molecular weight distribution of polyethylene terephthalate in homogeneous, continuous‐flow‐stirred tank reactors

AbstractPoly(ethylene terephthalate) (PET) formation in homogeneous, continuous‐flow‐stirred tank reactors (HCSTRs) operating at steady state has been simulated. The feed to the reactor is assumed to consist of the monomer bis‐(hydroxyethyl) terephthalate and monofunctional compound (MF1) cetyl alcohol. The overall polymerization is assumed to consist of the polycondensation, reaction with monofunctional compounds, redistribution, and cyclization reactions. At a given time, the reaction mass consists of polyester molecules (Pn), polyester molecules with an ending of molecules of monofunctional compound (MFn), and cyclic polymers (Cn). A mass balance for each of these species in the reactor gives rise to a set of algebraic equations to be solved simultaneously. The MWD calculations show that the redistribution reaction plays a major role and cannot be ignored, This result is in contrast lo the observation for semi‐batch reactors, for which redistribution becomes important when the cyclization reaction is included. For the same residence times of semi‐batch and HCSTRs, the latter gives considerably lower‐number average molecular weight, Nav, and polydispersity index, ρ. However, for the same conversions, the ρ for CSTR is higher. The concentration of the monofurctional compound, [MF1]0, in the feed and the reactor temperature both influence ρ, but the effect is small within the range studied.

Molecular weight distribution in novolac‐type polymerization in homogeneous, continuous‐flow stirred tank reactors

AbstractThe kinetic model for irreversible novolac type phenol formaldehyde polymerization and equations governing the molecular weight distribution (MWD) of the polymer formed in homogeneous continuous‐flow stirred tank reactors (HCSTR) have been derived. The set of algebraic equations involves three reaction parameters R1, R2, and R3 and have been solved using Brown's algorithm which was found to be more efficient than the Gauss‐Jordon technique of solution. A sensitivity analysis of different reaction parameters has been carried out and the reactivity of the para position was found to be an important factor affecting the MWD. In view of the fact that parameters R1 and R2 have a negligible effect on MWD, the phenomenon of molecular shielding described by Drumm et al. (1) can be neglected. The novolac formation can then be equivalently described by assuming R1 = 1 and R2 = R3 and the kinetic description so obtained is completely based on the experimentally observed different reactivities of ortho and para sites. The results for HCSTR have been compared with those for batch reactors and the former is found to give polymer of lower average molecular weights having higher polydispersity index.