Paper to the Polymer Processing Society 14th Annual Meeting, Yokohama, Japan, 8th-12th June 1998, paper 2-6.

S F Bush with F G Torres

Introduction

A model proposed by Bush1 has been extended and numerically implemented to produce molecular chain conformations in polymer flows. The effect of shear and extensional flows is studied. The chain shape is defined as a primary transport variable and viscosity is predicted thereof.

Two main approaches have been used in the modelling of polymer flows: the continuum mechanics approach and the molecular approach. The present work is based on the second one. Chain shape is a term used to express the actual conformations of a single entangled polymer chain at a specific time and place in the flow domain. The chain shape and the number of entanglements determine the resistance to flow of a polymer melt. So, viscosity is obtained as a function of the actual chain shape. To allow for the simulation of chain conformations, equivalent chain segments have been defined as primary transport variables, which are convected from one point to another in the flow field. Due to its formulation, the present model allows for polymer viscoelasticity to be represented in a natural way, with the shear and normal stresses being calculated as a function of the chain shape and of the velocity gradients. The treatment followed in this work can be extended in a straightforward manner for the modelling of polymer crystallisation and the prediction of shrinkage in plastic parts.

References

[1] BUSH S F, “Representation of Polymer Chain Shape in Injection Moulding simulation” Poly Proc Soc (PPS European Regional Mtg) Sept 1988, Brunel Univ, UK.

[2] TORRES F G, MPhil Thesis, UMIST, Manchester, UK, 1997.

[3] BUSH S F and DYER P, “The Experimental and Computational Determination of Complex Chemical Kinetics Mechanisms”, Proc Royal Soc, London, A 351, pp 33-53, 1976

See also the section on Mathematics & Computation.

Unpublished paper.

S F Bush with E P B Jongen.

Summary

The paper describes the analysis and computation carried out to model a manufacturing and marketing complex spread over several countries. Without loss of generality the model has been specifically applied to a polymeric materials business encompassing chemical and prepolymer precursors.

The model has been designed to have a variable structure so that changes to the nature and shape of the complex can be read in as data in the same way that parameters such as costs and efficiencies usually are. Because of this facility, the model provides a ready tool not only for optimisation, but also for assessing the strategic effects of possible future conversion technologies and for setting technological targets in the light of likely movements in resources and labour costs, costs of capital, changes in exchange rates and end-use demand.

The model has been designed to support both the planning and budgeting functions. While advantageous in itself, this dual objective was also necessitated by the practical circumstances in which the model was constructed, namely the large quantitities of data which have to be assembled and reconciled if the results from the model are to carry conviction. Accordingly the model output can provide reports on material flows, grade costs, and annual budgets for all or any part of the complex.

An example of the results obtained from optimisation of the full complex (with product details suppressed) is given and compared with the results from the standard sub-optimisation procedures in force. A number of the problems inherent in such procedures – transfer and coproduct pricing, overhead and capital allocation – are highlighted to show where conventional procedures can lead significantly away from global optimisation.

See also the section on Mathematics & Computation.

Paper published in the Proceedings of the Royal Society (A. 351, 33-53) 15th January 1976.

S F Bush with P Dyer

Thanks are due to the late Mr C A J Young, FRS, who communicated the paper.

Summary

Methods for the experimental and computational analysis of complex kinetics problems are described. Two examples which have been applied to industrial-scale design and operation are taken: high temperature chlorocarbon rearrangement and hydrocarbon cracking. Surface mechanisms are included within the treatment.

The experiments were based mainly on the continuous-flow uniform reaction cell which allowed precise control over physical conditions up to the temperature limit of interest, 1000 ^oC. The computational treatment is based on the development of a mathematical model system which permits a model structure to be varied at will, enabling radically different mechanisms to be rapidly examined. Using the methods, many thousands of computations have been carried out on a variety of systems of widely differing structures for the purposes of both research and design.

In Appendix A the model structure was used. In Appendix B the minimisation of the sum of squares by Gauss’s method was used.

See also the other items in this section Mathematics & Computation.