Mass balance equation polymers

By combining thermodynamically-based monomer partitioning relationships for saturation [170] and partial swelling [172] with mass balance equations, Noel et al. [174] proposed a model for saturation and a model for partial swelling that could predict the mole fraction of a specific monomer i in the polymer particles. They showed that the batch emulsion copolymerization behavior predicted by the models presented in this article agreed adequately with experimental results for MA-VAc and MA-Inden (Ind) systems. Karlsson et al. [176] studied the monomer swelling kinetics at 80 °C in Interval III of the seeded emulsion polymerization of isoprene with carboxylated PSt latex particles as the seeds. The authors measured the variation of the isoprene sorption rate into the seed polymer particles with the volume fraction of polymer in the latex particles, and discussed the sorption process of isoprene into the seed polymer particles in Interval III in detail from a thermodynamic point of view. 

The molecular weight distribution can be calculated by solving the mass balance equations for monomer(s), initiator (catalytic sites), and polymeric species with different chain lengths. When quasisteady state assumption is applied to live polymers or propagating active centers, the molecular weight distribution of live polymers is often represented by the Schultz-Flory most probable distribution. However, the calculation of the chain length distribution of dead polymers is in general quite complicated. For some special cases such as... 

Kinetic approaches represent realistic and comprehensive description of the mechanism of network formation. Under this approach, reaction rates are proportional to the concentration of unreacted functional groups involved in a specific reaction times an associated proportionality constant (the kinetic rate constant). This method can be applied to the examination of different reactor types. It is based on population balances derived from a reaction scheme. An infinite set of mass balance equations will result, one for each polymer chain length present in the reaction system. This leads to ordinary differential or algebraic equations, depending on the reactor type under consideration. This set of equations must be solved to obtain the desired information on polymer distribution, and thus instantaneous and accumulated chain polymer properties can be calculated. In the introductory paragraphs of Section... 

In his pioneering study of nonlinear polycondensation [7], Stockmayer has already checked his statistical solution [Eq. (5)] by solving the mass balance equations in a batch reactor for the concentrations of functional groups A and the set of isomeric polymer molecules P with x repeating units X [Eqs. (122)]. 

During the reaction, the mass of monomer in the reactor, m(t), is given by the mass balance equation that relates monomer and polymer mass via ... 

These two types of crosslink density, p iO) and pjfi, ij/), can be derived by a simple mass balance equation with respect to the crosslink point on the primary polymer molecule formed at each instance [22, 23, 45, 46]. 

The MWD functions for the polymers resulting from cationic polymerization can be easily derived based on the following mass balance equations for cationic propagating and dead chains ... 

The additive concentrations at both sides of the interface between the polymer and the solvent are equal, according to a partition coefficient (ratio of additive concentrations in polymer and solvent) of 1. This is justified as the solubility of Irganox 1076 in ethanol is good. In practice, as the solvent volume is large with respect to the polymer volume, the solvent concentration will always be almost equal to zero. Mathematically, the concentration in the polymer at the interface is assumed to be equal to the concentration in the solvent at one time-step earlier. This assumption is justified as long as the step size in time is very small. The concentration in the solvent was calculated using the mass balance (Equation 14.39). 

The behavior of a bead-spring chain immersed in a flowing solvent could be envisioned as the following under the influence of hydrodynamic drag forces (fH), each bead tends to move differently and to distort the equilibrium distance. It is pulled back, however, by the entropic need of the molecule to retain its coiled shape, represented by the restoring forces (fs) and materialized by the spring in the model. The random bombardment of the solvent molecules on the polymer beads is taken into account by time smoothed Brownian forces (fB). Finally inertial forces (f1) are introduced into the forces balance equation by the bead mass (m) times the acceleration ( ) of one bead relative to the others ... 

In the above reactions, I signifies an initiator molecule, Rq the chain-initiating species, M a monomer molecule, R, a radical of chain length n, Pn a polymer molecule of chain length n, and f the initiator efficiency. The usual approximations for long chains and radical quasi-steady state (rate of initiation equals rate of termination) (2-6) are applied. Also applied is the assumption that the initiation step is much faster than initiator decomposition. ,1) With these assumptions, the monomer mass balance for a batch reactor is given by the following differential equation. 

Note that in the component mass balance the kinetic rate laws relating reaction rate to species concentrations become important and must be specified. As with the total mass balance, the specific form of each term will vary from one mass transfer problem to the next. A complete description of the behavior of a system with n components includes a total mass balance and n - 1 component mass balances, since the total mass balance is the sum of the individual component mass balances. The solution of this set of equations provides relationships between the dependent variables (usually masses or concentrations) and the independent variables (usually time and/or spatial position) in the particular problem. Further manipulation of the results may also be necessary, since the natural dependent variable in the problem is not always of the greatest interest. For example, in describing drug diffusion in polymer membranes, the concentration of the drug within the membrane is the natural dependent variable, while the cumulative mass transported across the membrane is often of greater interest and can be derived from the concentration. 

Material Balances. The material (mass) balances for the ingredients of an emulsion recipe are of the general form (Accumulation) = (Input) - (Output) + (Production) -(Loss), and their development is quite straightforward. Appendix I contains these equations together with the oligomeric radical concentration balance, which is required in deriving an expression for the net polymer particle generation (nucleation) rate, f(t). 

Within either of the polymer-rich phases, borate esters and diesters of the functional groups are assumed to form with the same association constants as observed for the independent functional groups in aqueous solution. The resulting equations simply describe the borate ester association constants as well as mass balances on boron and polymer-bound functional groups. Wise and Weber used the model to estimate association constants for the borate esters formed with the diols in PVA and to predict the gelation of PVA-borate solutions. As we have independently measured the association constants for the borate esters formed in this work, we have used the model to estimate the radius of gyration of the GP3 dendrimer and to predict both the boron speciation in borate/GP3 solutions and the efficacy of PAUF using these functional dendrimers. 

Through the analysis of the particular selected examples it was shown that it is possible to get a good description of temperature and conversion profiles generated during the cure of a thermosetting polymer. Thermal and mass balances, with adequate initial and boundary conditions, may always be stated for a particular process. These balances, together with constitutive equations for the cure kinetics and reliable values of the necessary parameters, can be solved numerically to simulate the cure process. 

This chapter gave an overview of how to simplify complex processes sufficiently to allow the use of analytical models for their analysis and optimization. These models are based on mass, momentum, energy and kinetic balance equations, with simplified constitutive models. At one point, as the complexity and the depth of these models increases by introducing more realistic geometries and conditions, the problems will no longer have an analytical solution, and in many cases become non-linear. This requires the use of numerical techniques which will be covered in the third part of this book, and for the student of polymer processing, perhaps in a more advanced course. 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

When deriving a material balance equation, the rate of transformation of each component in a reactor is normally governed by the mass action law. However, unlike for the reactions in which only low molecular weight substances are involved, the number of such components in a polymer system and, consequently, the number of the corresponding kinetic equations describing their evolution are enormous. The same can be said about the number of the rate constants of the reactions between individual components. The calculation of such a system becomes feasible because certain general principle can be invoked under the description of the kinetics of the majority of macromolecular reactions. Let us discuss this principle in detail. 

An important quantity which can be calculated at equilibrium conditions is the amount of substance migrated into the food or food simulant at equilibrium, mF,e. Provided that the migration potential in the polymer, i.e. the initial amount of migrant dissolved in the polymer, mP0, is known then from mass balance calculations the following equation can be derived ... 

The simplest estimation of migration is to use the mass balance calculation shown in Eq. (14-1) below. This equation assumes that all of the styrene found in the polymer will migrate into the food instantly. This is of course not realistic but the estimation gives an upper limit to the possible migration that could occur at the end of the product s shelf life. 

Another interesting situation arises when there is proton insertion within the solid film so that protonation of immobile redox centers accompanies electron transfer of the type A -I- H+ -i- e -> HA, described by Wu et al. (1992) for redox polymers. Considering mass balance of protons over an infinitesimal film thickness in the boundary region of the film in contact with the electrolyte solution gives the diffusion equation ... 

For diffusion-mediated release, the amount of drug released from the polymer is proportional to the concentration gradient of the drug in the polymer. By performing a mass balance for drug within a differential volume element in the polymer, the concentration of drug within the polymer as a function of position and time can be described as above (see Equation 10-4) ... 

A mass balance on a differential volume element in the tissue [21] gives a general equation describing drug transport in the region near the polymer [17] ... 

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