Modeling and Simulation of Low Density Polyethylene (LDPE) High Pressure Tubular Reactor

Low Density Polyethylene (LDPE) is produced by free radical polymerization. Industrially, LDPE polymerization process is usually carried out in a continuous mode (tubular or autoclave reactors) at high pressures (1300-3400 atm) and temperatures (50-340 0 C). LDPE exhibits a number of desirable properties, including high tensile strength, flexibility, high impact resistance, low crystallinity, low density, high resistance to solvents, chemicals, and oxidating agents. In the present study, existing mathematical model for LDPE process is improved by tuning the model parameters. This model consists of mass, momentum and energy balance equations along with reaction kinetic equations and property correlations totalling to 25 coupled, nonlinear and highly stiff differential equations. The mathematical models comprise of a set of 13 model parameters (including activation energies for 8 reactions, volumetric flow rate of 4 jackets and a constant used in viscosity estimation), which are tuned by minimizing the sum-of-squares of the normalized error, between the model-predicted and the actual experimental values obtained from industry using a population based search algorithm named Differential Evolution (DE). The model is simulated using ODE15s (Gears technique) routine of MATLAB. The model predicted results are compared with the data available in the literature and the industrial data. Such model is useful in analysing the process plant by further carrying out multi-objective optimization study considering different sets of conflicting objectives. INTRODUCTION Since past more than 30 years several mathematical models have been developed and simulated for high pressure low density polyethylene (LDPE) reactor with varying degrees of configuration. Industrially polyethylene (PE) is produced by both high pressure (free radical) and low pressure (ionic addition) ethylene process. Two kinds of reactor configurations are employed in high pressure free radical production process; those are tubular technology and multi impeller stirred autoclave technology. The quality of PE mainly depends on density and degree of branching. Depending upon the density, the PE can be classified into LDPE (0.91-0.925 kg/m 3 ), medium density polyethylene (MDPE) (0.926-0.94 kg/m 3 ) and high density polyethylene (HDPE) (0.941-0.965 kg/m 3 ). The density of PE is also related to the degree of short chain branching (SCB). The higher the degree of SCB, the lower is the density of PE. In LDPE, typically 10-40 SCB and 0.3-03 long chain branching (LCB) exists respectively per thousand backbone carbon atoms. The model used in the present study is based on well known and widely accepted kinetic scheme for free radical polymerization of ethylene to polyethylene. The model includes the reaction schemes such as initiation, propagation, termination by various mechanisms such as by combination, by thermal degradation, by chain transfer to monomer and by chain transfer to the solvent. Second order moment equations are used to define the concentration of growing and dead polymer species. The mathematical models comprises of 25 numbers of coupled nonlinear and highly stiff differential equations and property correlations. Heat balance is simulated by considering constant temperature of the coolant on the shell side. The stiffness in the differential equation is due to wide variation in the rates of various species involved in the overall reaction scheme. Physical properties of the reaction mixture such as density, viscosity and thermal conductivity are assumed to vary along the reactor length. Due to the increased demand and in order to improve the performance of LDPE production process, several studies are reported in the literature on modeling and simulation of polyethylene. Agarwal and Han [1] focused on axial mixing analysis of tubular reactor by incorporating the axial mixing parameter, the Peclet number. They studied effects of various operating parameters such as feed conditions, axial mixing, and chain transfer to dead polymers etc. on the reactor performance. However, Chen et al. [2], in their study, summarized not to include axial mixing due to high value of Peclet number and excessive turbulence. Gupta et al. [3] carried out simulation of tubular LDPE reactor with intermediate feeds. Effects of various operating parameters such as jacket temperature, feed temperature, initiator concentration, wall heat transfer coefficients and reactor diameter was tested on the reactor performance. It was concluded that reactor operation becomes inherently unstable under certain operating conditions. Shirodkar and Taien [4] presented a mathematical model for tubular reactor and compared the performance of model with actual plant data. Brandoline et al. [5] proposed a mathematical model for high pressure polymerization of ethylene in tubular reactor. Kinetic parameters were determined by non linear regression analysis based on measured reaction temperature and molecular weight distribution properties. The model proposed by this group gave model predictions in agreement with the industrial data in the range with certain degree of error. Katz and Siedel [6] presented a comprehensive method of moments for representing size distribution of free radical polymerization. Lee et al. [7] studied decomposition phenomena in an industrial high pressure autoclave polyethylene reactor. Kalyon et al. [8] made a simulation based study of high pressure polymerization of ethylene and its rheological behavior. Data from commercial reactor was used having a reactor length of 720 m. Model predictions and experimental properties for the long and the short chain branching for three different polyethylene resin was obtained in their study. Rheological behavior of three different resins was studied with respect to apparent viscosity, shear stress, modulus of elasticity etc. More recently Kim and Iedema [9] carried out modeling of the branching density and the branching distribution in the low density polyethylene polymerization. They concluded that the concentration of long chain branching (LCB) are close to those of first branching moments in both the CSTR and the tubular reactor systems. SIMULATION AND OPTIMIZATION Brandoline et al. [10] presented a comprehensive model which proved to be a well known model which included several mechanisms of termination reactions. They also included the effect of the pulse valve in their study. Agarwal et al. [11] made use of model proposed by Brandoline et al. [10]. They carried out the tuning of important rate parameters involved in the model. When we tried to use the model predicted parameters reported by Agarwal et al.[11], we observed a slight deviation in the exit concentrations of various species. However, the trend followed by each species remains same, as reported in the literature [10, 11]. ODE15s routine of Matlab library is used in present study. ODE15s in MATLAB 7.0 library is a variable order solver based on the numerical differentiation formulas (NDFs). Optionally it uses backward differentiation formula (BDFs, also known as Gears method) that is usually less efficient. For sufficiently small step size, NDF can achieve the same accuracy as BDF with step size about 26% bigger. [12]. Agarwal et al. [11] used the Gears Routine (D02EJF of NAG library) in their simulation runs. The slight deviation in the in the exit concentration of various species encouraged us to fine tune the model in order to find the rate law parameters associated with the several reactions. A complete set of 13 model parameters used to fine tune used for model equation is available in literature [11]. The Differential Evolution (DE) algorithm [13, 14, 15] is used to minimize the sum of square of the normalized error, I, between the model-predicted values and the industrial values (Eq. 1),