Analytical Solutions and Computer Simulations of the Evolution of Flat Temperature Profiles in Spherical FRRPP Systems

The free-radical retrograde-precipitation (FRRPP) process was recently brought into the quantitative areas of work, based on the discovery of possibility of flat temperature profiles in spherical reactive domain systems. With an approximate decoupling analysis of the energy equation from the component-balance equations, these flat temperature profiles were found to be either stable or unstable. Moreover, resulting evolution of the flat profiles has been found to be expressed analytically through the so-called exponential Integral function, which has been shown to be quantitatively inaccurate during the early times of the process. This work tries to resolve this inaccuracy problem, by comparing the exponential integral results with polynomial approximation and numerical results. The result is that for the stable system, the linearized treatment of the evolution of flat temperature profiles is valid at the early 30% 40% in the temperature axis, while the remainder of the evolution curve is well-represented by the application of the exponential integral function. For the unstable system, the only thing that can be generalized is that both linear and cubic polynomial approximations are reasonably accurate at very small times and temperatures close to initial values.


Introduction
A reactive spherical chain-polymerization particulate system with an exothermic heat of reaction is normally expected to exhibit a parabolic temperature profile, as depicted in Figure 1.In these systems, the idea of a flat temperature profile is almost unheard of, and potentially beneficial in terms of operational and product uniformity.Also, the analysis of the model equations can become less complicated for these types of systems, because a field dynamics problem can be reduced to the relative simplicity of a lumped-parameter system.
In recent publications [1,2], it has been conceptually shown that the so-called free-radical retrograde-precipitation polymerization (FRRPP) process can result in flat temperature profiles through a combination of thermodynamic, transport, and reaction-kinetic parameters in the system.This is shown in Figure 2, wherein the profile starts at a uniform Dimensionless T (or θ in Equation ( 6)) of 1.0.The temperature profile ends at a steady-state value of Dimensionless T = 1.83 [3].Note that the dimensionless r in the horizontal axis of Figure 2 is the same as  in Equation ( 7).This means that the reactive fluid will have  ranging from 0 to 1 only.Values of  between 1 and 2 are rescaled so that they correspond to those of the stagnant fluid boundary layer.
The prediction of the possibility of the occurrence of flat temperature profiles from FRRPP systems has been advanced from a pseudo-steady-state analysis of the energy balance of a model spherical reacting system undergoing polymerization-induced phase separation [1].The dimensionless differential energy balance has been shown as and the dimensionless energy source term Φ is expressed as  exp The dimensionless parameters are related to the following dimensional quantities in the energy balance and phase equilibria equations:   The phase behavior for th approximated by the linear rep actions of the monomer and polymer (or just polymer if the monomer conc in Equatio e polymer-rich phase was resentation where X P is the product of the weight fr   the weight fraction of the entration is the same for both polymer-rich and polymer-lean phases at equilibrium, as the case for the PS-S-Ether system [4,5].Since Equation (10) provides a mesoscopic-macroscopic relationship between the temperature and monomer composition, it allows the decoupling of the mesocopic-macroscopic analysis of the time evolution behavior of the thermal aspect of the system from compositional aspects, until the details of the kinetics of microphase separation behavior is incorporated in the field equations.
In order to quantitatively characterize the relatively lower inefficiency of radical maintenance in free-radical polymerization systems, the expression for X P n (10) can be modified to include a polymer radical efficiency factor f P ; thus, Also, a and b in Equation (10) were proposed to be obtained from experimental data poi ria experiments.It should be note ra Φ o and Equation (2) became nts of phase equilibd that the polymer dical efficiency factor, f P , is related to the initiator efficiency, f, in free-radical polymerizations in a relative sense, i.e., for the same monomer, solvent, their concentrations, and operating temperatures, f P is relatively high when an initiator with a relatively high f is used.The reason is that initiation is the starting point in time when inefficiencies of radical production take place.The survival of propagated radicals will later depend on the effectiveness of the FRRPP radical trapping mechanism, which should result in f P < f.For example, in FRRPP of polystyrene-styrene-ether systems using azobis diisobutylonitrile (AIBN) as initiator, f P was found to be in the order of 0.20 [1,4] even though f was known to be around 0.57 [6].
For the expectation of a flat temperature profile, θ = 1 for η = 0; thus, the dimensionless source term was sym-bolized as Then, a combined dimensionless quantity was introduced to quantitatively characterize str ior, wherein the reactive polymer-rich fla ict FRRPP behavdomains attained t temperature profiles.The dimensionless quantity was symbolized by n C (pronounced see-enye), and defined as Values of from computational efforts indicated that for th process, it should be a −1000 for a flat temperature profile [1].

least below
In the unsteady-state analysis of the FRRPP system, the occurrence of the flat temperature profile from FRRPP systems was reinforced, with ad r stability of steady-state behavior [2].For a stable steady-state system, it was found that the quantity α < 0 even for an insulated reactive domain system.For  > 0, the reactive system was found to be under control with the possibility of a flat temperature profile through relatively ineffective heat removal from the fluid.With more aggressive heat removal from the fluid, the temperature profile in the reactive solid becomes more of a parabolic one.
When the monomer concentration drops to low enough values, the reactive system automatically reverts to a stable negative number [1].This happens either locally (due to relatively fast reaction rate compared to diffusion rate, or polymer domain densification [2]) or in the overall reactive system, due to depletion of monomer molecules.
With assumption of a flat temperature profile, the field energy balance equation was simplified to a mixing-cup ordinary differential equation and later analyzed for its ability characteristics [2].Results from the field partial differential equation correspond to the reacting sphere with insulated surfaces.An analytical expression for the modified dimensionless temperature () vs dimensionless time (τ) was also obtained for the flat temperature profile reactive spherical domain with insulated boundary surfaces as in which the dimensionless time was expressed as and the modified dimensionless temperature (   ), and   are obtained as Note that T b is the bulk fluid temperature while T s is the reactive particle surface temperature (Figure 1).
The function, Ei or the Expo [7], is a special transcendental function which has been fo nential Integral Function und to occur in only a few experimental systems; thus, It can be evaluated through the following infinite series expansion for real positive arguments (x > 0): The Euler-Mascheroni constant (also called Euler's constant) has been cited to be equal to 0.577215664 should be noted from Equation (20) that the Ei fun approaches −∞ at its argument (x) approaching zero from th ppr ted for the asometry ameter an dimensional form the lumped-9...It ction e positive side (0 + ).This is reflected in inaccuracies in results from direct evaluation Equation (14) at τ → 0 + .
In this paper, inaccuracies of the analytical evaluation of Equation ( 14) are addressed, by comparing predictions of the evolution of flat temperature profiles in FRRPP systems from numerical solutions and analytical a oaches.Numerical solutions used include the evaluation of the field energy equation and its lumped-parameter version for a flat temperature profile.Analytical methods employed here include the use of the exponential integral function (Equation ( 14)), as well the Taylor-Series-based polynomial approximations.

Generalized Mathematical Model of FRRPP Domains
Since the diffusive term could be neglec sumption of a flat temperature profile, then the ge of the system does not matter and a lumped-par alysis can be made.In parameter energy balance around a reactive particle can be more conveniently expressed using the convective heat transfer coefficient, h, as where A is the heat transfer area.For insulating boundary surfaces, h = 0, and thus Thus, the integral operation for the determination of τ at any value of  is sed as the starting material for polynomial approximation at relatively small values of dimensionless time, τ, wherein the app with the integrand.etailed field simulation results were also obtained.The following conditions have been de-

Computer Simulation Results of IVP-PDE
In order to validate results of polynomial approximations of Equation ( 23), d duced for the occurrence of a flat temperature profile in FRRPP systems: 1) 1000   , and h = 0 Note that only Conditions 2 and 3 are covered in this paper.

Appro tio Ea s Behav rly-Time ior
Let the fun Taylor series expansion at the initial dimensionless time of zero and dimensionless temperature  = 1 leads to The software Mathematica ® is used to deriv ous derivatives in Equation (25), and the following results were obtained up to the cubic derivative.e the vari- In order to understand the F() function, it is plotted in Figure 3 at typical parameter values for both stable and unstable systems, wherein the flat temperature profile is proposed to occur.
From this figure, the area under the curve can be obta resulting plots of the Dimensi r.This exercise is very important for va ined to be equal to the dimensionless time, .Note that for the stable system, the value of  approaches infinity at the steady-state value of 5.With the two sets of parameters in Figure 3, the onless temperature, , vs Dimensionless time, , is shown in Figure 4.
Since the use of the Exponential Integral function, Ei, as defined in Equation (19), is not well understood in FRRPP systems, comparisons with straight numerical solution and Taylor Series approximations are made and reported in this pape lidation of a new conceptual phenomenon that is expressible with a rarely used transcendental function.These plots indicate the validity of the use of the numerical approaches, compared to the Ei approximation method using Equation ( 14).If one uses the polynomial approximation, for  < 0, the linear approximation would be quite accurate in the initial 60% -70 e cubic approximation would be accurate in the initial 20% -range.For  > 0, linear and cubic approximations seem to be accurate in the small-times range, or when   1.A point of caveat here is that the polynomial approximation does not seem to be asymptotic, until the cubic terms are reached.The final obvious point to be made here is that for  < 0, the analytical-based Ei function is reasonably accurate at the above 20% -range.

Conclusio
linear Taylor series approximation up to 30% -40% of n he quantitative analysis of the occurrence of the flat in spherical FRRPP systems indicates hes to predict its evolution would the -range, and continuing to steady-state condition with the use of the Exponential Integral, Ei, function.For T temperature profile that numerical approac be reasonably accurate.If an analytical approximation is to be done, for  < 0, the approach would be to use the  > 0, the only conclusion that can be made is that both linear and cubic polynomial approximations are reasonably accurate at very small times and low -values, i.e., at τ → 0 and   1.

Figure 1 .Figure 2 .
Figure 1.The model used for the analysis of the effect of exotherm when chain polymerization occur within a spherical par-s ticulate system that is immersed in a fluid bath.The uniform fluid bath temperature is T b and the particulate surface temperature is T s .

1 Figure 3 . 1 Figure 4 .
Figure 3. Plots of F() vs  from Equation (24) for typical sets of parameters representing stable and unstable systems.The legend indicates the triplet of parameters     , ,    , parameters wherein the stable system represents the set of with negative .

Figures 5 ( 4 .
Figures 5(a) and 6(a) show polynomial approximation results of  vs  for flat temperature profile histories involving typical stable and unstable parameter system which could be compared to numerical results in Fig 4.This stable system curve in Figure 4, in turn, has b found to compare very well with Figure 5(b), which is the numerical result of the solution of the full differential equation in Equation (22).For a representative unstable system, the dashed-curve result in Figure 4 is compared with polynomial approximation results in Figure 6(a).This representative unstable system curve in Figure 4, in turn, has been found to compare well with Figure 6(b), which is the numerical result of the solution of the full differential equation in Equation (22).In addition, exponential integral results are shown for given parameter sets in Figures 5(a) and 6(a).Figures 7(a) and 7(b) show expanded abscissa scales for Figures 5(a) and 5(b), for additional comparisons.On the other hand, Figures 8(a) and 8(b) show shorter time behavior for conditions associated with Figures 6(a) and 6(b).These plots indicate the validity of the use of the numerical approaches, compared to the Ei approximation method using Equation (14).If one uses the polynomial

Figure 5 .Figure 6 .
Figure 5. (a) Approximation results of  vs  for flat temperatu istories involving a typical stable parameter s stem, represented by re profile h y