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Ind. Eng. Chem. Res. 2006, 45, 4558-4566
Simulation Model of Polyamide-6,6 Polymerization in a Continuous Two-Phase Flow Coiled Tubular Reactor Renata O. Pimentel† and Reinaldo Giudici*,‡ Rhodia Engineering Plastics, Brazil, and Departamento de Engenharia Quı´mica, UniVersidade de Sa˜ o Paulo, Escola Polite´ cnica, Caixa Postal 61548, CEP 05424-970, Sa˜ o Paulo, Brazil
A mathematical model was developed to simulate an industrial-scale continuous process of polyamide-6,6 polymerization composed by two reactors in series: a coiled tubular reactor operating partially under onephase flow and partially under two-phase flow conditions, followed by a continuous stirred tank reactor. The mass, heat, and momentum balances for the single and two-phase flows were considered in the tubular reactor model. Water and hexamethylenediamine evaporation rates were evaluated according to the thermodynamic vapor-liquid equilibrium equations. Frictional, gravitational, and accelerational terms, as well as fittings for tube expansions, were considered in the pressure drop calculations. The model developed is an improved version of a previous one (Giudici et al., Chem. Eng. Sci. 1999, 54, 3243-3252) and was tested and validated with industrial data under a wider range of conditions, including the effect of catalyst. Introduction Nylon-6,6 or poly(hexamethylene adipamine) is an important polymer produced from the condensation polymerization of hexamethylenediamine and adipic acid, with extensive applications as fibers, film, and engineering plastics. Industrial production of nylon-6,6 is carried out in a variety of different reactors. Mathematical models are usually developed to provide a deeper understanding of the process and are useful for a number of applications such as design, optimization, and control of industrial processes. In more recent years, the development of detailed mathematical models has become essential for the efficient operation of industrial processes. The kinetics and equilibrium of nylon-6,6 polymerization was first studied experimentally by Ogata.2,3 Kumar et al.4 fitted an empirical correlation to Ogata’s data as a function of the initial water content. Steppan et al.5 analyzed Ogata’s data along with some other literature data of nylon-6 polymerization and fitted a thermodynamically consistent empirical model for the equilibrium and kinetics, under broad ranges of temperature and water content. The strongly nonideal behavior of the species causes both the apparent rate constant and the apparent reaction equilibrium constant to vary with composition. Steppan et al.6 also presented a kinetic model for the degradation reactions of nylon-6,6 based on the available experimental observations of nylon degradation studies in the literature. Mallon and Ray7 proposed a new kinetic and equilibrium model for the polymerization of nylon-6 and nylon-6,6 that considers water in the nylon melts to be either free or part of water bridges as hydrated amide complexes. This explains the decrease in the apparent equilibrium constant with the water content. In addition, they also modeled the apparent rate constant as a function of the system dielectric constant that explains the decrease in the rate constant with the increase in water content. Schaffer et al.8 presented a good discussion regarding the previous works on kinetics and equilibrium of nylon polycondensation and obtained new experimental data for melt-phase nylon polycondensation kinetics and equilibrium under high temperatures and low water * To whom correspondence should be addressed. E-mail:
[email protected]. † Rhodia Engineering Plastics. ‡ Universidade de Sa˜o Paulo.
contents. They used nylon-6,12 instead of nylon-6,6 in order to avoid degradation reactions and proposed a simplified kinetic model based on the activity coefficients that is able to correctly represent the observed effect of the water concentration on the apparent equilibrium constant. Their new data was also well represented by the model of Steppan et al..5 More recently, Zheng et al.9 extended the experimental and kinetic modeling results for conditions under different values of end-group balance and moisture level. A comprehensive review on nylon-6,6 degradation was presented by Schaffer et al.10 Coupling a unified general framework for the kinetic modeling of polycondensation reactions presented by Jacobsen and Ray11,12 with a generic reactor model based on a series of tubular reactors with two-phase flow and axial dispersion, Hipp and Ray13 proposed a generalized polycondensation reactor model, that would be able to represent in a generic way a number of different reactor types. Previous works on nylon-6,6 polymerization reactor modeling have been focused on different reactor types: batch autoclaves,14,15 different prepolymerization reactors,16 CSTRs,17 thin or wiped film reactors,18,19 vented twin-screw extruder reactors,20-22 and flasher tubes.1 In the present work, a mathematical model for polyamide6,6 polycondensation in a two-phase flow continuous tubular reactor (flasher tube) followed by a continuous stirred tank reactor under catalyzed and uncatalyzed conditions is presented and validated using industrial data. The model is an improved version over the one previously published by Giudici et al.1 The study was focused in the industrial plant of Rhodia Brazil that produces polyamide-6,6 for applications as engineering plastics. Figure 1 presents a simplified scheme for the industrial process. A nylon salt solution at a concentration of 52 wt % and 52 °C comes through special tank-trucks from another plant located 100 km away from the site. Before being fed to the polymerization process, the nylon salt solution can be preconcentrated in an evaporator, in which nylon salt concentrations up to 66% and temperatures until 143 °C can be reached. No significant losses of hexamethylenediamine (HMD) are observed during this evaporation stage, so that equal, stoichiometric amounts of HMD and adipic acid are present in the feed stream.
10.1021/ie051165p CCC: $33.50 © 2006 American Chemical Society Published on Web 05/12/2006
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4559
Figure 1. Simplified scheme of the industrial plant. Table 1. Kinetic Model and Kinetic Parameters5,6 a,b reaction A+C)L+W
kinetics Rp ) Ctkp(xAxC - xLxW/Kap)
C f SE + W L f SE + A
Rd1 ) Ctkd1xC Rd2 ) CtxL(kd2 + kd2cxA)
SE f SB + CO2 SB + 2A f X + 2NH3
Rd3 ) Ctkd3xAxSE0.1 Rd4 ) Ctkd4xAxSB0.3
rate constant Polymerization kp Degradation Reactions kd1 kd2 kd2c kd3 kd4
kj,o (L/h)
Ej (kcal/mol)
To (K)
see below
21.4
473
0.06 0.005 0.32 0.35 10.0
30 30 30 10 50
566 578 578 578 578
A ) amine end-group; C ) carboxyl end-group; L ) amide linkage; W ) water; SE stabilized end-group; SB ) Schiff base; X ) crosslink; xi ) Ci/Ct Ej 1 ∆Hap 1 1 1 Kap ) K0 exp . kp,0 ) exp{2.55 - 0.45 R T T0 R T T0 tanh[25(xw - 0.55)]} + 8.58{tanh [50(xw - 0.10)] - 1}(1 - 30.05xc). ∆Hap ) 7650 tanh[6.5(xw - 0.52)] + 6500 exp(- xw/0.065) - 800. K0 ) exp{[1 0.47 exp(-xw1/2/0.2)](8.45 - 4.2xw)}. a
[ (
where Ct ) CA + CC + CL + CW + CSE + CSB + CX. b kj ) kj,o exp -
The first, main reactor of this industrial process is a 995 m long coiled tubular reactor which is heated externally by a heat transfer medium. The reactor is continuously fed with an aqueous solution of nylon salt, that passes first through a narrower single-phase reactor (fully filled tube). After a given heating length, that depends on the operating conditions, the fluid starts to boil, giving rise to two-phase flow. Along its length, the tubular reactor presents four different diameter sections, varying from 0.04 to 0.15 m, allowing for an increasing amount of the vapor phase to flow along with the increasingly viscous liquid phase due to the water evaporation and polymerization. The coiled tube has upflow and downflow sections with variations in the tube diameter and coil diameter. These kinds of reactors are also called “flashers” or “flash tubes”, and they are similar to those described in the patents of DuPont.23-25 The two-phase flow ends in an atmospheric operated continuous stirred conical tank reactor, called the finisher (or melt pool), that works as a vapor-liquid separator and where additional reaction occurs. The melt polymer that exits the finisher reactor flows to an extruder where it can be mixed with different kinds of mineral fillers, pigments, and additives. This later equipment is not included in the model, though: the mathematical model is focused only on the tubular and the finisher reactors. The model development followed the main ideas given previously in the work of Giudici et al.1 but included some new important improvements. The evaporation of hexamethylenediamine, which may take place simultaneously with the water evaporation, is now considered in a more fundamental basis,
)]
[
(
)]
taking into account the vapor-liquid equilibrium thermodynamics. The viscosity of the liquid phase is now predicted by a more fundamental equation, accounting for the effects of water/ polymer content, polymer molecular weight, and temperature. New empirical correlations were developed for pressure drop corrections, and for mass transfer evaporation coefficients, under a wider range of process conditions. Furthermore, the effect of catalyst concentration in the polymerization kinetics is now considered and included in the model. In addition, the model validation was performed under a wider range of industrial operating conditions. Mathematical Model The mathematical model was developed considering that the tubular and the finisher reactors are operating under steadystate conditions. Some other simplifying assumptions were considered to achieve a model that is at the same time realistic and compatible with the measurements available at the industrial plant. The well-known kinetic scheme given by Steppan et al.5,6 for nylon-6,6 polymerization was adopted to describe the chemical reactions taking place in the liquid phase. Table 1 summarizes the chemical transformations and the corresponding kinetic equations for the polyamide-6,6 polycondensation and degradation reactions, as proposed by Steppan et al..5,6 The species considered are the amine end-group (A), carboxyl endgroup (C), amide linkage (L), water (W), stabilized end-group (SE), Schiff base (SB), and crosslink (X).
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Tubular Reactor (Flasher) Modeling. A one-dimensional plug-flow model was considered for both the single-phase and two-phase flow regions.1 The balance equations for the onephase flow region are
dFliq A
πdt2
πdt2 ) L 2N˙ evap RA,jRj HMD dz 4 j)1 4 m
∑
(12)
and for the components in the vapor phase, they are
dFliq i
πdt2
)
dz
m
Ri,jRj ∑ j)1
4
dFvap k ) 0 (k ) HMD, W) dz
WLCpL
dT
)
πdt2
dz
4
πdt2 evap dFvap W )+ N˙ dz 4 W
(13)
(2)
πdt2 evap dFvap HMD )+ N˙ dz 4 HMD
(14)
(3)
Water and amine balances in the liquid phase are affected by the evaporation of water and HMD, respectively, and the factor 2 in eq 12 accounts for the fact that one HMD molecule contains two amine end-groups. The energy balance for the two-phase flow is given by
(i ) A, C, L, W, SE, SB, X) (1)
m
(-∆Hj)Rj - Umonofπdt(T - Text) ∑ j)1
dP G2 dH ) -2f - F Lg dz F Ld t dz
(4) (WLCpL + WGCpG)
Pressure drop calculations account for both the frictional term (due to friction at the pipe walls) and the gravitational term (hydrostatic pressure, due to the change in elevations). Tube diameter (dt) and vertical position (or elevation, H) vary as a function of the axial position z. Water and hexamethylenediamine (HMD) are considered the only volatile species contained at the vapor-phase flow due to their evaporation along the tube. The onset of the two-phase flow occurs at the point of the tubular reactor at which the vapor pressure of the liquid phase is equal to the total pressure. The vapor pressure of the solution increases with the length of the tube as the liquid temperature increases, while the total pressure decreases due to the pressure drop. According to the equations given by Russel et al.,14 the vapor pressure of the liquid phase (Pvap) can be evaluated by
Pvap ) max(P1,vap,P2,vap)
( )
(6)
sat P2,vap ) xWPsat W + xHMDPHMD
(7)
where the vapor pressure of pure water and pure HMD can be calculated by
760 1668.21 P ))+ 7.96681 (14.696 T - 45.15 sat w
log10
(8)
-1 Psat HMD ) exp[ -1.164316 + 6237.44(0.002807 - T )] (9)
and nw is the moles of water per mole of fundamental unit (L), i.e., nw ) CW/CL. For the two-phase region, the component mass balances for each component in liquid are
dFliq i dz
) L dFliq W
πdt2 4
( )∑ πdt2 4
4
j)1
(-∆Hj)Rj -
∆Hvap ˙ evap - Ubifπdt(T - Text) (15) k N k
k)w,hmd
evap sat N˙ evap W ) KW av(PW γWΦW - yWP)
(16)
evap sat N˙ evap HMD ) KHMDav(PHMDγHMDΦHMD - yHMDP)
(17)
The inclusion of HMD evaporation is one of the improvements on the mathematical model previously developed for the same industrial plant.1 The terms in the driving force for the evaporation of water and HMD were calculated according to Russell et al.14 and Samant and Ng.16 Activity coefficients for water and HMD in the liquid phase are evaluated by the FloryHuggins theory as
[(
γw ) γhmd ) exp 1 -
)
(i ) C, L, SE, SB, X)
(10)
πdt2 RW,jRj ) L N˙ evap W dz 4 j)1 4
(11)
]
1 Φ + 0.5(Φpoly)2 DPn poly
(18)
where 0.5 is the assumed value for the polymer-solvent interaction parameter,16 DPn is the number-average degree of polymerization
DPn )
1+r 1 + r - 2r
(19)
the number average molecular weight of the polymer can be calculated by
(20)
and the molecular weight of the monomeric unit is
m
∑
m
Mn ) DPnMo
j)1
πdt2
( )∑ πdt2
Equation 15 assumes equal temperature in both vapor and liquid phases at each axial position and also accounts for the heats of vaporization of water and HMD. The evaporation rates of water and of HMD were individually calculated as a function of the individual mass-transfer coefficient and a driving force based on the departure from the vapor-liquid equilibrium relationship:
m
∑ Ri,jRj
dz
) L
(5)
nW 3050 - 10.09 ) P1,vap T
log10
dT
Mo )
vap liq 114((Fliq A )0 - 2Fhmd) + 112(FC )0 liq vap (Fliq A )0 + (FC )0 - 2Fhmd
(21)
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4561
where 114 is the mass of an A-A segment within the polymer chain, 112 is the mass of a C-C segment within the polymer molecule, is the extent of reaction in terms of the limiting end-group, r is the feed ratio, or imbalance ratio (r < 1), Fliq i is is the molar flow rate of species i in the liquid phase, and Fvap i the molar flow rate of species i in the vapor phase (the index 0 refers to at the feeding conditions). Russel et al.14 provided the adaptation of Flory’s theory in order to take into account the HDM vaporization as well as the degradation reactions that change the imbalance ratio. When carboxyl is the limiting end-group, the imbalance ratio between carboxyl and amine end-groups (r), the conversion of the limiting end-group (), and the fraction of polymer molecules that are HMD (n1) are given by
r)
(Fliq C )0
(22)
vap (Fliq A )0 - 2FHMD
Vkxk (k ) W, HMD, poly) VHMDxHMD + VWxW + Vpolyxpoly (32)
Correlations for the molar volume of each species in the liquid phase as a function of temperature were taken from Russel et al.14 Pressure drop calculations for the two-phase flow accounts for the frictional term (due to friction at the pipe walls), the gravitational term (hydrostatic pressure, due to the change in elevations), and an additional accelerational term (due to the change of mixture density and gas expansion). The LockhartMartinelli correlation26 was used in the two-phase frictional pressure drop calculations:
() dP dz
liq liq (Fliq C )0 - FC - FSE
)
Φk )
(23)
(Fliq C )0
dP dP dP dP + + ) dz dz fric dz acel dz grav
( )
) -Wtot2
acel
(dPdz )
grav
n1 )
(1 - r)2r-1 1+r
-1
(24)
- 2
If, however, amine is the limiting end-group, the corresponding equations are
r)
vap (Fliq A )0 - 2FHMD
liq vap (Fliq A )0 - FA - 2FHMD
)
(26)
vap (Fliq A )0 - 2FHMD
n1 )
(1 - )
2
(27)
1 + r-1 - 2
Mole fractions of water and HMD in the vapor phase are
yk )
Fvap k vap Fvap HMD + FW
(k ) W, HMD)
xk )
Fliq k
(k ) W, HMD, poly) (29)
liq liq Fliq W + Fpoly + FHMD
where
(Fliq C
Fliq poly ) (1 - n1)
(Fliq C
Fliq HMD ) (n1)
+
Fliq A
+
Fliq SE
+
Fliq SB
-
Fliq X)
2 +
Fliq A
+
Fliq SE 2
+
Fliq SB
-
Fliq X)
(30)
(31)
The volume fraction of water, HMD, and polymer in the liquid phase are
2 x2 d (1 - x) + dz FLL FG(1 - L)
( )(
dH dz
tt
)
]
(34) (35) (36)
where the weight fraction of the vapor phase in the two-phase mixture is
x)
WG WG ) Wtot WG + WL
(37)
and the liquid hold-up (volume fraction of the liquid phase in the two-phase mixture) is calculated by26,27
{
0.24Xtt0.80 0.175Xtt0.32 L ) (1 - G) ) 0.143Xtt0.42 1/[0.97 + (19/Xtt)]
0.01 < Xtt < 0.5 0.5 < Xtt < 5 (38) 5 < Xtt < 50 50 < Xtt < 500
where the Lockhart-Martinelli Xtt is defined by
(28)
and mole fraction of water, HMD, and polymer in liquid phase are
(33)
G2 C 1 ) - 2f 1+ + 2 F d X X fric L t tt
(25)
(Fliq C )0
( )
) -[FG(1 - L) + FLL]g
() dP dz
[
( )
Xtt )
x
(dP/dz)L
(dP/dz)G
(39)
where (dP/dz)L and (dP/dz)G are the pressure gradients for the flow of the liquid alone and the gas alone, respectively, at the same volumetric flow rate of the two-phase system. The parameter CLM of the Lockhart-Martinelli correlation is given by27
{
5 10 CLM ) 12 20
ReL e 1000 ReL > 1000 ReL e 1000 ReL > 1000
ReG e 1000 ReG e 1000 ReG > 1000 ReG > 1000
(40)
The corrections for the non-Newtonian behavior of the liquid phase are accounted for using the equations reported by Chhabra and Richardson.27 Pressure drops in pipeline expansions (changes in tube diameter) are also locally considered. The friction factor and heat transfer coefficient for the singlephase flow were predicted using the equations reported by Raman28
{
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Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
f) 16 Re
{
[
{
{
Re < 2100
( )
/dt 5.02 3.7 Re /dt 5.02 /dt 13 log + log 3.7 Re 3.7 Re Re 3.6 log 0.135[Re(/dt) + 6.5]
- 4 log
Nu )
[
()
µL 0.085Gz 2/3 µ 1 + 0.047Gz w µ 0.14 1/3 L 1.86Gz µw
()
log10(µL) ) 4.7 log10 Mn - 22.35 +
)]} ]}
(
hdt ) kL
3.66 +
The viscosity of the liquid phase is calculated using the equation14,30
0.14
-2
2100 < Re < 4000 4000 < Re < 108 (41)
Re < 2100 and Gz e 100 Re < 2100 and Gz > 100
[ ( ) ]( )
0.023Re0.8Pr1/3
dt L
2/3
() µL µw
µL µw
0.14
2100 < Re < 104
0.14
Re > 104 (42)
and Umonof ) h, i.e., it is assumed that the heat transfer resistances in the tube wall and at the external side are negligible. The heat transfer coefficient for the two-phase flow was calculated using the equations presented by Hewitt et al.29
(43)
0.00122(Twall - Tsat)0.24(Pwall - Psat)0.75CpL0.45FL0.49kL0.79 σ0.5∆Hvap0.24FG0.24µL0.29 (44)
{
2.35(1/Xtt + 0.213)0.736 (1/Xtt) > 0.1 (1/Xtt) < 0.1 1
(45)
1 1 + 2.53 × 10-6(ReLF1.25)1.17
(46)
Available corrections for helical coiled tubes for both the friction factors and the heat transfer coefficients were also included in the model.29
(
dt hcoiled_tube ) hstraight_tube 1 + 3.5 dc
[
fcoiled_tube ) fstraight_tube 1 + 0.14
() dt dc
0.97
)
Re1-0.644(dt/dc)
(47)
]
0.312
(48)
Values of the viscosity index (IV), a routine measurement used to monitor the polymer quality in the industrial plant, can be obtained by the following correlation as a function of the carboxyl and amine end-group concentrations:
IV ) 0.008494
(
2090 wS (50) T
)
where ws is the weight fraction of water soluble components (water, adipic acid, and HMD) and Mn is the number average molecular weight. Additional equations were employed to evaluate the physical and transport properties (heat capacity, density, and thermal conductivity) of the components (water, nylon salt, and polymer) along with adequate mixing rules to calculate the properties of the solution. CSTR (Finisher Reactor) Modeling. The finisher reactor (located at the end of the tubular reactor) is a continuous stirred tank reactor where both the liquid and vapor phase streams are separated and additional polycondensation and degradation reactions may occur. Experimental observations have shown that the amount of evaporation in the finisher is negligible in comparison to the evaporation in the tubular reactor. Therefore, evaporation was not considered to occur in the finisher. Equation balances for components in the liquid phase and an energy balance are given by m
liq liq - Fi,exit + Vfinisseur 0 ) Fi,in
Ri,jRj ∑ j)1
(i ) A, C, L, W, SE, SB, X) (51) 0 ) WLCpL(Tin - T) + Vfinisseur
FZ )
S)
Mn - 18.82 +
m
Ubif ) FUmonof + SFZ
F)
10
-2
0.116(Re2/3 -
125)Pr2/3 1 +
(4.7 log
3190 T
)
2 × 106 CA + C C
(49)
(-∆Hj)Rj ∑ j)1
UfinisseurAfinisseur(T - Text) (52) Parameter Estimation A number of new experiments were designed and performed in the industrial plant in order to obtain pertinent data for model validation. To enlarge the data basis and to cover a wider range of operation variables than that usually practiced in the plant operation, the data set included also the experiments formerly reported in the work of Giudici et al.1 The main variables manipulated in these experiments were the feed flow rate, the temperature, and the concentration of the nylon salt solution. The measured response variables are the carboxyl (CC) and amine (CA) end-group concentrations in the produced polymer, the viscosity index of the produced polymer (IV), and the pressure drop in the tubular reactor (DP). Especially in the tubular reactor, the variables are strongly correlated. For instance, pressure drop and temperature affect the driving force for water evaporation, which in turn creates the driving force for the polymerization. As the polymerization proceeds, strong changes in liquid viscosity arise, which affects the heat transfer as well as the evaporation rate and the pressure drop. As a result, the system of balance equations is highly coupled. Due to the high complexity of the system under consideration (two-phase flow, Non-Newtonian liquid phase, simultaneous flow, heat transfer, mass transfer, and reaction), some model parameters had to be adjusted in order to fit the model to the experimental data. The adjusting of a certain set of parameters can be used to compensate for model uncertainties, which may be expected to occur in such a complex system. The number of
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4563
Figure 2. Model prediction versus experimental plant data. Open symbols refer to uncatalyzed runs; black symbols refer to catalyzed runs.
fitting parameters were kept as low as possible, and the selected adjustable parameters were those that strongly affect the model predictions (sensitivity analysis) and those to which there is some lack of fundamental knowledge or of reliable correlations. The following model parameters were adjusted: (a) the parameter C of the Lockhart-Martinelli correlation; (b) the mass-transfer coefficient for water evaporation, Kevap W av; (c) the evap av. The mass-transfer coefficient for HMD evaporation, KHMD first parameter affects the pressure drop along the two-phase flow in the tubular reactor, as well as the driving force for water evaporation, which in turn ultimately affects the driving force for the polycondensation reaction. The other two parameters are related to the kinetics of water and HMD evaporation, thus also affecting the conversion of monomer into polymer and the evolution of the polymer molecular weight. At first, the three parameters were simultaneously adjusted for each operating condition using the Nelder-Mead optimization method.31 The parameter values thus obtained have presented some dependence on experimental conditions; therefore, empirical expressions based on surface response methodology (i.e., low order multivariable polynomials) were developed to describe such a relationship. The obtained empirical equations for these parameters as a function of the feed flow rate (W0), the feed nylon salt concentration (C0), and the feed temperature (T0) have the following form:
C ) CLM + W0(2.2786 - 0.0037W0 + (1.9416 × 10-6)W02 - 3.3483C02 - 0.0001T02 + 0.0348T0C0) (53) -0.6 (0.3744 + 0.0010W0 - (6.2461 × Kevap W av ) IV
10-7)W02 + 0.4724C02 + (3.5989 × 10-7)T02 0.0012W0C0 + 0.0010C0T0)-2 (54) -0.6 (-2.5260 - 0.0013W0 + 25.7977C02 + Kevap HMDav ) IV
0.0007T02 + 0.0042W0C0 - (1.0023 × 10-5)W0T0 0.2432C0T0) (55) The last two equations include a dependence of the evaporation coefficients on the viscosity index of the polymer, correctly reflecting a hindrance in the evaporation rate as the degree of polymerization (represented by IV) increases. Figure 2 presents the comparison between the model predictions and the industrial plant data, showing that the model is
able to represent the industrial data with quite satisfactory quantitative agreement. Effect of Catalyst. To evaluate the effect of catalyst on the process and to include such an effect into the model, two additional industrial runs were performed with a specified quantity of catalyst (a phosphorus based compound). To keep the polymer produced under the market specifications, these runs with catalyst were carried out using also different quantities of monofunctional monomer (acetic acid). The monofunctional monomers are used to control the polymer growth in polycondensation reactions, as an additional process variable that is independent of the reaction conversion. The presence of monofunctional monomer (acetic acid) is easily taken into account in the model in the calculation of the viscosity index and the average molecular weight of the polymer. The effect of the presence of catalyst was included in the model by modifying the equation for the polymerization rate in the following form:
Rp ) (1 + kcatCcat)Ctkp(xAxC - xLxW/Kap)
(56)
The first factor in this expression takes into account that the catalyst increases the polymerization rate, and in the absence of catalyst, the expression reduces to the original one presented in Table 1. The concentration of catalyst increases along the tubular reactor as the flow rate of the liquid phase decreases due to water evaporation. This is taken into consideration by evaluating Ccat as a function of the axial position in the tubular reactor. The additional parameter kcat was then fitted using the experimental data of these two additional runs carried out with catalyst. This fitting was done using a one-dimensional line search, and the obtained value was Kcat ) 2.23 × 103 (kg liquid/ kg phosphorus). It is interesting to mention that, in such fitting, no additional modification on the other parameters was necessary; all other parameters were kept at their previously fitted values. This would be, of course, expected, as the presence of catalyst should only accelerate the polycondensation reaction. Since all phenomena are strongly coupled in this system, the effect of catalyst on the pressure drop and on the water and HMD evaporation could appear, but only indirectly, through the faster evolution of the polycondensation reaction along the reactor. This is observed indeed, and all these changes were correctly predicted using the same set of other parameters previously fitted. Then, these observations can be seen as an additional check of the predictive capacity of the model. Figure 2 also includes the comparison between the model prediction and the industrial plant data for the experiments with catalyst (black dots). Fairly good agreement between model predictions and experimental data can be seen for the catalyzed runs as well as for the uncatalyzed runs. Additional Validation with Independent Data. Figures 3 and 4 present a comparison of model predictions and experimental results of additional runs not previously included in the data set used in the parameter estimation. It is interesting the note that the predictions are fairly close to the experimental data in these new industrial runs. In these new runs, an additional pressure sensor was installed in an intermediate point inside the tubular reactor. Such a measurement was not available at the time the model was developed, and all model development steps and reported parameter estimation were done without this information. It is noteworthy that the model correctly predicts the trend of the pressure profile, and the agreement with the measurement is quite satisfactory. Such intermediate measure-
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Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
Figure 3. Comparison between model predictions (continuous line) and experimental data (points) of pressure along the flasher tube. Experimental data was not used in the parameter estimation.
Figure 4. Comparison between model predictions and experimental data for experiments not used in the parameter estimation.
ment, although limited to one single point, provides an additional fair validation to the model developed. Further model validation with independent data was performed using experimental data taken in industrial experiments in which the level of melt polymer in the finisher was varied. Three different degrees of filling in the finisher were tested: the standard level, the minimum level, and an additional experiment with an empty finisher (zero level). The measured values of the viscosity index and the corresponding values predicted by the model are compared in Table 2, showing reasonable agreement. This is an additional indication of the model validity and, in particular, of the hypothesis of negligible water evaporation in the finisher. Typical Simulation Results Typical model predictions for a representative set of operating conditions are shown in Figure 5. In the example shown, the two-phase flow starts at about 250 m, where the saturation
Figure 5. Typical profiles predicted by the model along the tubular reactor. Table 2. Comparison between Model Predictions and Experimental Data of Runs Varying the Degree of Filling of the Finisher
degree of filling of the finisher standard minimum zero
viscosity index (mL/g) exp model 136.7 130.0 124.0
135.9 132.6 121.6
amine end-group conc (mol/Mg) exp model
carboxyl end-group conc (mol/Mg) exp model
43.3 46.0 47.8
79.2 86.6 89.2
41.1 42.5 48.1
83.9 85.6 91.6
pressure surpasses the total pressure. From this point on, the temperature presents a little decrease, as the HMD and water
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evaporation take place. Note that there are some humps and discontinuities at the expansions of the tube diameter. The model shows that the velocity of the vapor phase increases along the tube, except at the positions where the tube diameter changes. The velocity of the liquid phase shows a maximum at a section in the middle of the reactor, as a result of the effects of decreasing the liquid-phase flow rate and decreasing the liquid holdup along the tube length. Carboxyl and amine end-group concentrations start to differ significantly only at the final part of the tubular reactor as a result of the degradation reactions. Conclusions A representative model for the industrial process of polyamide-6,6 polymerization in a two-phase flow coiled tubular (flasher) reactor was developed and validated with industrial data. This model, which is an improvement over the previously presented model1 in terms of liquid-vapor equilibrium equations, increased amount of different operational conditions, and addition of catalyst experiments, was validated with an enlarged data basis. The model was based on well-known phenomenological equations and some empirical expressions specially developed to correct some model parameters. The necessity of including these empirical equations to correctly predict the behavior of some parameters reflects the lack of reliable equations, either fundamental or empirical, about the complex phenomena taking place in this tubular two-phase flow reactor. Nevertheless, the model provided a better understanding of this process and proved useful for the plant engineers to carry on studies of optimization, debottlenecking, and prediction of new operation conditions of the industrial process. For instance, this model was indeed used to find a new operational condition that guarantees at the same time increase of the industrial production, either with or without addition of catalyst, maintenance of the polymer quality (i.e., the same value for the viscosity index), and decrease of the energy spent per kilogram of polyamide produced in this plant. Acknowledgment The authors are thankful to Rhodia Engineering Plastics, FAPESP (Fundac¸ a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo), and CNPq (Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico) for supporting this work. Nomenclature av ) area of liquid-vapor interface per unit volume of reactor, m2/m3 C ) modified parameter of Lockhart-Martinelli correlation C0 ) initial mass fraction of nylon salt Ccat ) concentration of catalyst, kg phosphorus/kg liquid phase Ci ) concentration of species i, kmol/m3 CLM ) original parameter of Lockhart-Martinelli correlation CpG, CpL ) heat capacity of the gas phase and liquid phase, respectively, J/(kg K) DP ) pressure drop, Pa DPn ) number-average degree of polymerization dc ) coil diameter dt ) tube diameter, m f ) Fanning’s friction factor liq Fi ) mole flow rate of species i in the liquid phase, kmol/s vap Fi ) mole flow rate of species i in the gas phase, kmol/s g ) gravitacional acceleration, m/s2 G ) mass velocity, kg/(m2 s)
Gz ) Graetz number ) udt2CpLFL/(kLL) GTA ) amine end group concentration, mol/Mg GTC ) carboxyl end group concentration, mol/Mg H ) height, m IV ) viscosity index, mL/g Kap ) apparent equilibrium constant for polycondensation reaction kcat ) parameter correction for catalyzed reaction rate, g liquid phase/g phosphorus kj ) rate constant of reaction j, kmol/(m3 s) Kevap W ) mass transfer coefficient for water evaporation, mol/(h m2 bar) evap KHMD ) mass transfer coefficient for HMD evaporation, mol/ (h m2 bar) kL ) thermal conductivity of liquid phase Mn ) number-average molecular weight, kg/kmol Mo ) molecular weight of the monomeric unit, kg/kmol n1 ) fraction of polymer molecules that are HMD ) evaporation rate of species i, mol/(h m3 bar) Nevap i nw ) mole ratio, mol of water/mol of amide linkage Nu ) Nusselt number ) hdt/kL P ) pressure, bar sat ) vapor pressure of HMD, bar PHMD Pvap ) vapor pressure, bar Psat w ) vapor pressure of water, bar Pr ) Prandtl number ) CpLµL/kL r ) feed ratio or imbalance ratio between amine and carboxyl end-groups ReG, ReL ) Reynolds number for gas phase and for liquid phase Rj ) rate of reaction j, mol/(m3 h) T ) temperature, K T0 ) temperature of salt nylon solution fed to the reactor, K Text ) temperature of external fluid, K Twall ) wall temperature (assumed equal to Text) Ubif ) two-phase flow heat transfer coefficient, J/(m2 K s) Umonof ) single-phase flow heat transfer coefficient, J/(m2 K s) Vi ) molar volume of species i, m3/kmol W0 ) feed flow rate of nylon salt solution, L/h WG, WL ) mass flow rate of the gas phase and the liquid phase, respectively, kg/s Wtot ) mass flow rate of the liquid and gas phases, kg/s xi ) mole fraction of i in the liquid phase Xtt ) Martinelli parameter yk ) mole fraction of k in the vapor phase z ) axial position, m Rij ) stoichiometric coefficient of component i in thereaction j G, L ) volume fraction of gas and liquid phases ) extent of the reaction in terms of the limiting end-group (/dt) ) relative rugosity of the tube FL, FG ) density of liquid phase and gas phase, respectively, kg/m3 γi ) activity coefficient of species i φi ) volume fraction of species i ) heat of vaporization of species i (i ) water, HMD), ∆Hvap i J/kmol ∆Hj ) heat of reaction j, J/kmol σ ) surface tension of liquid phase µL ) viscosity of the liquid phase, kg/(m s) Literature Cited (1) Giudici, R.; Nascimento, C. A. O.; Tresmondi, A.; Domingues, A.; Pellicciotta, R. Mathematical modeling of an industrial process of nylon6,6 polymerization in a two-phase flow tubular reactor. Chem. Eng. Sci. 1999, 54, 3243-3249.
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ReceiVed for reView October 18, 2005 ReVised manuscript receiVed April 12, 2006 Accepted April 13, 2006 IE051165P
