A mathematical analysis of bubble growth in an expanding foam is presented. The analysis is based on a cell model whereby the foam is divided into spherical microscopic unit cells of equal and constant mass, each consisting of a liquid envelope (or shell) and a concentric spherical gas bubble. Expansion occurs by diffusion of a dissolved gas from the supersaturated envelope into the bubble. This cell model is capable of describing important qualitative features of a real system of numerous bubbles growing in close proximity to one another, and is intended as the building block of a global analysis of macroscopic foam expansion. The coupled algebraic and differential equations governing the growth of a cell are derived and solved numerically. Five dimensionless parameters are identified for the case of constant temperature and pressure outside the cell, and their effects are demonstrated through computer simulations of the system. Of these parameters, surface tension and initial radius prove to be of relatively little importance in the practical cases considered. The other parameters are the thermodynamic driving force, the cell mass (inversely proportional to the number density of bubbles), and the ratio of characteristic times for mass and momentum transport.

D e p a r t m e n t o f Chemical Engineering University of D e l a w a r e N e w a r k , D e l a w a r eA cell model describing the diffusion-induced growth of closely spaced single bubbles developed in an earlier publication is now used as the microscopic building block of an analysis of low-pressure structural foam molding. Heat transfer, solidification, and bulk flow of the foam in the mold cavity are included in this macroscopic analysis. An experimental method is described whereby the bulk density of an expanding thermoplastic polymeric foam is measured as a function of time when the expanding foam is prepared by the short shot injection molding process. Instantaneous and final bulk foam density data obtained for various values of the controlling process parameters are compared with the predictions of the model. These predictions are found to be in qualitative agreement with the experimental measurements reported here. However, quantitative differences exist between the two. These differences are explained mainly by the omission of melt elasticity and bubble coalescence in the model.

The fluid motion at a free surface advancing into a mold or duct is appreciably different from its steady state behavior in well-developed flow; this affects the residence time distribution and structure of macromolecular fluids as they are frozen in injection molding processes. In this work such motion is treated numerically and measured precisely for Newtonian fluids. While the three-phase contact line represents a special problem conceptually and analytically, a very simple algorithm is seen to represent the fluid motion in this region accurately.Good agreement is found over wide ranges of the governing dimensionless groups (the Reynolds, Jeffrey, and capillary numbers). Since viscous forces are dominant under the circumstances studied, this finding is not surprising but it confirms the applicability of the numerical methods developed herein to the modeling of these flows under conditions of actual interest. As a result, simulations may be made with confidence to predict flow patterns encountered in practice but difficult to reproduce in laboratory experiments.

The isothermal flow of a Newtonian liquid in a co‐rotating twin screw extruder having screw elements with three tips has been analyzed when the effect of the intermeshing zone on flow can be neglected. It was found that values for four dimensionless parameters must be specified in order to obtain a unique relationship between the dimensionless axial pressure gradient and the dimensionless volumetric flow rate. These parameters included the number of screw tips, the helix angle, the ratio of the clearance to the screw radius, and the ratio of the distance between screw centers‐to the screw radius Values for the dimensionless throughput and pressure gradient were computed for a range of helix angles at fixed values for all other dimensionless parameters. Shape factors were also computed and it was found that the shape‐factor for pressure How is substantially less than that for drag flow. Asymptotic values for both these factors at large values of the channel width (large helix angles) were found to be less than unity.

A bubble inflation technique for establishing equal biaxial extensional flow in viscoelastic materials is presented. This technique was used to measure biaxial extensional viscosity and elastic properties of a polyisobutylene at room temperature (23°C). A theoretical experimental procedure, based on certain idealizations, was developed for establishing bubble growth under constant stress. Modifications of the experimental design were introduced to correct for the nonidealities encountered in practice. The accuracy and reliability of the measurements are tested, and sources of error, and possibilities for future work are discussed. Several constitutive equations are examined with respect to biaxial extensional flow. Predicted behavior patterns are compared with experimental results.

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The various types of extensional flows and extensional viscosities are defined and methods of measurement discussed. The role that each of these extensional viscosities plays in various polymer fabrication processes is discussed with examples. Finally, it is shown how engineering analyses of these complex flow fields are conducted using simplified phenomenological equations for the rheological behavior. This approach is recommended for use until such time as tensorially correct, mathematically tractable constitutive equations that are based on molecular theory are available.

Theoretical and experimental studies have been conducted on the mass transfer rates which occur during the continuous devolatilization of bubble-free polymeric solutions in a corotating twin-screw extruder. Experimental measurements of the mass transfer rates, when reported in terms of the product of the individual liquid phase mass transfer coefficient and the surface area for mass transfer per unit volume of empty extruder, were found to correlate well with screw speed in accordance with the predictions of a model based on penetration theory. It is shown, however, that significant differences exist between the measured and predicted values for the rates of mass transfer, which leads to the conclusion that the surface area for mass transfer is substantially smaller than the value computed on the basis that continuous films are formed on the surfaces of the screws and the barrel wall. G. P. COLLINS SCOPEThe removal of volatile components from polymeric solutions, commonly known as devolatilization, is a unit operation that finds importance in both the synthesis of polymers and the shaping of polymeric materials to form finished goods. Condensation polymerizations, for example, require that (volatile) by-products be continuously removed during polymerization so that reverse reactions will not prevent the formation of high molecular weight products (Odian, 1970). And, in food packaging applications which involve the use of polymeric materials, residual monomer must be reduced to levels of the order of parts per million since some of these monomer are thought to be carcinogenic (Todd, 1974;Hess and Eise, 1977).I n commercial practice when polymeric solutions of high viscosity are to be stripped of volatile components, the devolatilization operation is almost always conducted in geometries which generate a wiped film, and here single-screw extruders have played a major role. In recent years, increasing attention has been given to conducting these operations in twin-screw extruders, especially in circumstances when a chemical reaction is involved, and this has led to significant advances in the technology related to the use of these machines for stripping operations. As judged by the published literature, however, a scientific understanding of stripping operations in twin-screw extruders is far from complete, thereby bringing into question whether the engineering analysis and design of these processes can be performed in a rational manner and with confidence.The research reported here was undertaken as a first step toward ultimately developing a more complete understanding of polymer devolatilization processes in twin-screw extruders. Mass transfer rates were determined experimentally at various liquid flow rates and screw speeds using a solvent-free inert gas at atmospheric pressure to strip a volatile component from a polymeric solution for which the diffusion coefficient and Henry's constant were known. These results are reported in terms of kLa, the product of the individual liquid phase mass transfer coefficient...