# Design of a Polystyrene Plant for Differing Single Pass Conversions

**Topics:**Polystyrene, Mass flow rate, Maxwell's equations

**Pages:**9 (2194 words)

**Published:**December 6, 2013

November 25, 2013

Introduction

Polystyrene is one of the most widely used plastics, with applications ranging from food packaging to appliances to manufacturing (Maier). On an industrial scale, polystyrene is derived from its monomer, styrene. This is achieved by free-radical polymerization of a solution of monomer, polymer, and initiator. This reaction is a multistep radical reaction that includes initiation, propagation, and termination (Eq.B.1 through Eq.B.5). Figure A.1 shows a process that produces 1000 kg/hr of polystyrene in an isothermal plug flow reactor at a styrene conversion of lower than 80%. Azobisisobutyronitrile (AIBN) initiator is fed into the reactor at a concentration of 0.022 mol/L. The stream exiting the reactor is fed into an evaporator that perfectly separates unreacted styrene from the mixture. Unreacted styrene is recycled back into the reactor feed at a recycle ratio of 1 (in reference to fresh styrene feed). Costs associated with operating the plant increase with the size of the reactor ($200 per m3), with the rate styrene is fed into the reactor ($70 per kg/hr), and with the power required to pump the viscous mixture through the length of the reactor ($15×10-5 per W). We will determine the optimal single-pass conversion for a polystyrene plant described above.

Design Approach

Qualitatively, to achieve a larger single-pass monomer conversion, a larger reactor volume is need, necessitating more power to pump the reaction mixture through the length of the reactor. Both of these factors will contribute an increase in overall cost. In return, however,, less fresh styrene feed is needed, contributing to a decrease in overall cost. There is a single pass conversion that best balances these cost contributing factors. To determine the single-pass conversion that minimizes plant-operating costs, the following design approach was taken. To summarize the design approach, differential equations relating monomer conversion, initiator conversion, and reactor pressure to reactor volume are solved with a numeric solver. The solutions to these equations result in reactor volume and pressure drop parameters for any given single-pass conversion. Pressure drop is then related to power, while styrene feed is expressed as a function of single-pass conversion. Finally, the three resulting parameters – reactor volume, power, and styrene feed – are related to cost to determine the optimal single-pass monomer conversion. First a differential equation relating monomer conversion to reactor volume is written in terms of known quantities (see appendix D, sec. 1 for full derivation). The process begins with a species mass balance for a differential of a plug flow reactor (Eq. D.1.1). Rate equations for monomer and initiator consumption are applied to their respective species balances (Eq. D.1.7 and Eq. D.1.8). Algebraic manipulation yields a differential equation relating single-pass conversion of monomer to reactor volume (Eq. D.1.10). Next, a differential equation relating pressure to reactor volume must be expressed in terms of conversions and other known quantities (see appendix D, sec 2 for full derivation). A number of substitutions are made using equations from Appendix B. (Eq. D.2.1 through Eq. D.2.10). First, the friction factor equation is rearranged to produce a differential equation relating pressure and volume (Eq. D.2.1). The Blasius relation and the definition of the Reynold’s number are then applied to the equation to remove its dependence friction factor (Eq. D.2.2). Finally, an expression relating the viscosity of the reacting mixture to conversions and known quantities is derived (Eq. D.2.9) and substituted into the differential equation (Eq. D.2.10). We now have a system of three differential equations that describe reactor behavior. This system of differential equations can be solved...

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