Abstract
A new yet simple transformation is proposed to significantly improve the accuracy of computational fluid dynamics (CFD) modeling and simulations of free radical polymerization (FRP) reactions carried out especially in flow microreactors. The new transformation makes the kinetic rate coefficients dimensionless in terms of concentration. To that extent, the chemical data (chemical species concentration and kinetic rate coefficients values) can be fed in original molar form instead of usual mass form to CFD software package while simulating chemical species as passive scalars. The normalization of various variables (passive scalars) helps in reducing the numerical stiffness as well as numerical errors during simulations. Another advantage of this new transformation is that the expression for transformed reaction rate equations remains unchanged thus enabling an easy coding and debugging process. The new transformation was first validated through numerical simulation against theoretical analytical solution of FRP for homogeneous batch reactor. It was then validated through CFD simulation against published experimental data for FRP in coiled tube microreactor under steady-state flow condition. It has also been demonstrated that in CFD simulations of FRP in flow microreactors, significant error arises for the prediction of number-average chain length (and thus \({\text{MW}}_{n}\), number-average molecular weight) from the use of chemical data in mass form instead of original molar form. This new transformation is thus found to be more suitable for CFD simulations in flow reactors compared to previous Zhu’s transformation.
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Abbreviations
- \(A\) :
-
Chain transfer agent concentration at any time \(t\) (mol/l)
- \(A_{\text{H}}\) :
-
Area for heat transfer (m2)
- \(C_{\text{A}}\) :
-
\({=} \frac{{K_{fa} }}{{K_{p} }}\) (dimensionless)
- \(C_{\text{b}}\) :
-
Bulk monomer concentration (mol/l)
- \(C_{\text{M}}\) :
-
\({=} \frac{{K_{{\text{fm}}} }}{{K_{\text{p}} }}\) (dimensionless)
- \(C_{\text{p}}\) :
-
Specific heat capacity of mixture [cal/(g °C)]
- \(C_{\text{S}}\) :
-
\({=} \frac{{K_{{\text{fs}}} }}{{K_{\text{p}} }}\) (dimensionless)
- \(C_{\text{T}}\) :
-
\({=} \frac{{K_{{\text{td}}} }}{{K_{{\text{tc}}} }}\) (dimensionless)
- \({\text{DP}}_{n}\) :
-
Number-average degree of polymerization
- \(I\) :
-
Initiator concentration (mol/l)
- \(K_{d}\) :
-
Dissociation rate coefficient (min−1)
- \(K_{{\text{fa}}}\) :
-
Transfer to CTA rate coefficient [l/(mol min)]
- \(K_{{\text{fm}}}\) :
-
Transfer to monomer rate coefficient [l/(mol min)]
- \(K_{{\text{fs}}}\) :
-
Transfer to solvent rate coefficient [l/(mol min)]
- \(K_{\text{i}}\) :
-
Kinetic rate constant for initiation (s−1)
- \(K_{\text{p}}\) :
-
Propagation rate coefficient [l/(mol min)]
- \(K_{{\text{pr}}}\) :
-
\({=} K_{\text{p}} + K_{{\text{fm}}} = (1 + C_{\text{M}} )K_{\text{p}}\) [l/(mol min)]
- \(K_{\text{t}}\) :
-
\({=} K_{{\text{tc}}} + K_{{\text{td}}}\) [l/(mol min)]
- \(K_{{\text{tc}}}\) :
-
Termination by combination rate coefficient [l/(mol min)]
- \(K_{{\text{td}}}\) :
-
Termination by disproportionation rate coefficient [l/(mol min)]
- \(L\) :
-
Kinetic chain length, \({=} \frac{{K_{{\text{pr}}} M\lambda_{0} }}{{2fK_{\text{d}} I}}\)
- \(\bar{L}\) :
-
\({=} L\left( {\frac{{1 - R_{{\text{MM}}} }}{{1 + R_{\text{P}} L}}} \right) = L\left( {\frac{{1 - R_{\text{M}} }}{{1 + R_{\text{P}} L}}} \right)\)
- \(M\) :
-
Monomer concentration (mol/l)
- \({\text{MW}}\) :
-
Molecular weight (g/mol)
- \({\text{MW}}_{n}\) :
-
Number-average chain length of polymer (g/mol)
- \({\text{MW}}_{\text{w}}\) :
-
Weight averaged chain length of polymer (g/mol)
- \({\text{PDI}}\) :
-
Polydispersity index (dimensionless)
- \(P_{n}\) :
-
Dead polymer chain length of n no. of monomer units
- \(R\) :
-
Universal gas constant (1.986 cal/mol/K)
- \(R_{0}\) :
-
Zero order radical obtained from initiator dissociation
- \(R_{\text{A}}\) :
-
\({=} \frac{{C_{\text{A}} }}{{1 + C_{\text{M}} }} = \frac{{K_{\text{fa}} }}{{K_{\text{pr}} }}\)
- \(R_{\text{AM}}\) :
-
\({=} \frac{{C_{\text{A}} }}{{1 + C_{\text{M}} }} \frac{A}{M} \approx \frac{{C_{\text{A}} }}{{1 + C_{\text{M}} }} \frac{{A_{0} }}{{M_{0} }}\)
- \(R_{\text{M}}\) :
-
\({=} \frac{{K_{\text{fm}} }}{{K_{\text{p}} + K_{\text{fm}} }} = \frac{{K_{\text{fm}} }}{{K_{\text{pr}} }} = \frac{{C_{\text{M}} }}{{1 + C_{\text{M}} }}\)
- \(R_{\text{MM}}\) :
-
\({=} R_{\text{M}}\)
- \(R_{n}\) :
-
Live polymer chain length of n no. of monomer units
- \(R_{\text{P}}\) :
-
\({=} R_{\text{MM}} + R_{\text{SM}} + R_{\text{AM}} = R_{\text{MM}} + R_{\text{SA}}\)
- \(R_{\text{S}}\) :
-
\({=} \frac{{C_{\text{S}} }}{{1 + C_{\text{M}} }} = \frac{{K_{\text{fs}} }}{{K_{\text{pr}} }}\)
- \(R_{\text{SA}}\) :
-
\({=} R_{\text{SM}} + R_{\text{AM}}\)
- \(R_{\text{SM}}\) :
-
\({=} \frac{{C_{\text{S}} }}{{1 + C_{\text{M}} }} \frac{S}{M} \approx \frac{{C_{\text{S}} }}{{1 + C_{\text{M}} }} \frac{{S_{0} }}{{M_{0} }}\)
- \(R_{\text{T}}\) :
-
\({=} \frac{{K_{\text{tc}} }}{{K_{\text{tc}} + K_{\text{td}} }} = \frac{{K_{\text{tc}} }}{{K_{\text{t}} }} = \frac{1}{{1 + C_{\text{T}} }}\) (dimensionless)
- \(S\) :
-
Solvent concentration any time \(t\) (mol/l)
- \(T\) :
-
Temperature (K)
- \(T_{\text{bath}}\) :
-
Temperature of heat sink (K)
- \(U\) :
-
Overall heat transfer coefficient (W/m2/K)
- \(V_{\text{R}}\) :
-
Volume of solution at any time t (l)
- \(V_{{{\text{R}}_{0} }}\) :
-
Initial volume of solution at t 0 (l)
- \(f\) :
-
Initiator efficiency (dimensionless)
- \(f_{\text{s}}\) :
-
Solvent volume fraction (dimensionless)
- \(t\) :
-
Time (min)
- u :
-
Velocity (m/s)
- \(x_{\text{M}}\) :
-
Monomer conversion (dimensionless)
- \(y\) :
-
\({=} {\text{e}}^{{\frac{{ - K_{\text{d}} \cdot t}}{2}}}\), variable evaluated in the analytical solution
- \(\Delta H_{\text{P}}\) :
-
Heat of reaction (cal/mol)
- \(\beta\) :
-
Ratio of solvent volume to non-solvent volume (dimensionless)
- \(\varepsilon\) :
-
Volume contraction factor corrected for solvent volume fraction (dimensionless)
- \(\varepsilon_{0}\) :
-
Volume contraction factor without solvent volume fraction (dimensionless)
- \(\lambda_{0}\) :
-
Zeroth order moment for live polymer chain concentration (mol/l)
- \(\lambda_{1}\) :
-
First order moment for live polymer chain concentration (mol/l)
- \(\lambda_{2}\) :
-
Second order moment for live polymer chain concentration (mol/l)
- \(\mu_{0}\) :
-
Zeroth order moment for dead polymer chain concentration (mol/l)
- \(\mu_{1}\) :
-
First order moment for dead polymer chain concentration (mol/l)
- \(\mu_{2}\) :
-
Second order moment for dead polymer chain concentration (mol/l)
- \(\rho\) :
-
Mixture density (g/cm3)
- \(\varPhi\) :
-
Volume fraction (dimensionless)
- \(\eta\) :
-
Dynamic viscosity (cP)
- \(\left[ \eta \right]\) :
-
Intrinsic viscosity of the polymer (dl/g)
- M :
-
Monomer
- P :
-
Polymer
- S :
-
Solvent
- I :
-
Initiator
- 0:
-
At time t = 0
References
Achilias DS, Kiparissides C (1992a) Development of a general mathematical framework for modeling diffusion-controlled free-radical polymerization reactions. Macromolecules 25(14):3739–3750
Achilias DS, Kiparissides C (1992b) Toward the development of a general framework for modeling molecular weight and compositional changes in free-radical copolymerization reactions. J Macromol Sci Part C Polym Rev 32(2):183–234
Baillagou PE, Soong DS (1985a) Major factors contributing to the nonlinear kinetics of free-radical polymerization. Chem Eng Sci 40(1):75–86
Baillagou PE, Soong DS (1985b) Free-radical polymerization of methyl-methacrylate in tubular reactors. Polym Eng Sci 25(4):212–231
Cabral PA, Melo PA, Biscaia EC, Lima EL, Pinto JC (2003) Free-radical solution polymerization of styrene in a tubular reactor—effects of recycling. Polym Eng Sci 43(6):1163–1179
Chen CC (1994) A continuous bulk-polymerization process for crystal polystyrene. Polym Plast Technol 33(1):55–81
Chen CC (2000) Continuous production of solid polystyrene in back-mixed and linear-flow reactors. Polym Eng Sci 40(2):441–464
Costa EF, Lage PLC, Biscaia EC (2003) On the numerical solution and optimization of styrene polymerization in tubular reactors. Comput Chem Eng 27(11):1591–1604
Dreher S, Engler M, Kockmann N, Woias P (2010) Theoretical and experimental investigations of convective micromixers and microreactors for chemical reactions. In: Bockhorn H, Mewes D, Peukert W, Peukert H-J (eds) Micro and macro mixing, heat and mass transfer. Springer, Berlin, pp 325–346
Garg DK, Serra CA, Hoarau Y, Parida D, Bouquey M, Muller R (2014a) Analytical solution of free radical polymerization: derivation and validation. Macromolecules 47(14):4567–4586
Garg DK, Serra CA, Hoarau Y, Parida D, Bouquey M, Muller R (2014b) Analytical solution of free radical polymerization: applications-implementing gel effect using CCS model. Macromolecules. doi:10.1021/ma501251j
Garg DK, Serra CA, Hoarau Y, Parida D, Bouquey M, Muller R (2014c) Analytical solution of free radical polymerization: applications-implementing gel effect using AK model. Macromolecules 47(21):7370–7377
Hui AW, Hamielec AE (1972) Thermal polymerization of styrene at high conversions and temperatures. An experimental study. J Appl Polym Sci 16:749–769
Keramopoulos A, Kiparissides C (2002) Development of a comprehensive model for diffusion-controlled free-radical copolymerization reactions. Macromolecules 35(10):4155–4166
Konstadinidis K, Achilias DS, Kiparissides C (1992) Development of a unified mathematical framework for modelling molecular and structural changes in free-radical homopolymerization reactions. Polymer 33(23):5019–5031
Liu RH, Yang J, Grodzinski P, Pindera MJ, Athavale M (2002) Bubble-induced acoustic micromixing. Lab Chip 2(3):151–157
Mandal MM, Serra C, Hoarau Y, Nigam KDP (2011) Numerical modeling of polystyrene synthesis in coiled flow inverter. Microfluid Nanofluidics 10(2):415–423
Senn SM, Poulikakos D (2004) Laminar mixing, heat transfer and pressure drop in tree-like microchannel nets and their application for thermal management in polymer electrolyte fuel cells. J Power Sources 130(1–2):178–191
Serra C, Sary N, Schlatter G (2005a) A multiphysics approach of the styrene free radical polymerization modeling performed in different microreactors. In: Proceedings of conference FemLab, Paris (France), 15 Nov
Serra C, Sary N, Schlatter G, Hadziioannou G, Hessel V (2005b) Numerical simulation of polymerization in interdigital multilamination micromixers. Lab Chip 5(9):966–973
Serra C, Schlatter G, Sary N, Schonfeld F, Hadziioannou G (2007) Free radical polymerization in multilaminated microreactors: 2D and 3D multiphysics CFD modeling. Microfluid Nanofluidics 3(4):451–461
Stutz MJ, Poulikakos D (2005) Effects of microreactor wall heat conduction on the reforming process of methane. Chem Eng Sci 60(24):6983–6997
Zhu S (1999) Modeling of molecular weight development in atom transfer radical polymerization. Macromol Theor Simul 8(1):29–37
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The financial support by ANR Grant No. 09-CP2D-DIP2 is greatly appreciated.
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Garg, D.K., Serra, C.A., Hoarau, Y. et al. New transformation proposed for improving CFD simulation of free radical polymerization reactions in microreactors. Microfluid Nanofluid 18, 1287–1297 (2015). https://doi.org/10.1007/s10404-014-1527-3
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DOI: https://doi.org/10.1007/s10404-014-1527-3