Study of Riser Reactor Design Using Simultaneous Ordinary Differential Equations (ODE)

Ø

Catalyst activity

ϵ_g

Hydrocarbon gases void fraction

ρ_g

Density of gas phase

A

Notation for vacuum gas oil

B

Notation for gasoline

C

Notation for coke

D

Notation for gas

A_C

Crossectional area of the riser reactor

ΔV

Volume of the reactor shell

Δz

Thickness of the reactor shell

r_j

The rate of consumption of reactant j per unit catalyst volume (j = A, B, C, D)

k_ij

Reaction rate constants between species i and j, where i ≠ j (K_AB, K_AC, K_AD, K_BC and K_BD)

F_j

Hydrocarbon gases mass flow rate of j specie in the riser (j = A, B, C or D)

y_i

Weight percent of hydrocarbons in the riser (i = A,B,C, or D)

The mathematical modelling of chemical reactions contributes to the design of more efficient chemical reactors. Reactor designs for elementary chemical reactions were successfully implemented by carrying out material balance and reaction kinetic expressions. Although the chemical reactors represented the most dangerous operational units in the chemical industry, the performance of these reactors can be managed during the production phase by using mathematical models (Gupta and Subba, 2003a, 2001b; Lee et al., 1989; Reza, 2000). The fluid catalytic cracking unit, FCC, helps to increase the gasoline production of the refining process by increasing the conversion of the heavy petroleum fractions into lower molecular-weight products. Many improvements have been made in FCC operation; these include feed preparation, catalyst development, equipment design, and operating strategies (Ali et al., 1997; Vogt and Weckhuysen, 2015). The chemical reaction initiated and completed in a short-contact-time inside a vertical reactor that is called a riser (Mahfud, 2016; Reza, 2000). Inside this riser, the catalyst is pneumatically conveyed from the bottom to the top by steam and hydrocarbon vapours, this leads to efficient mixing of catalyst and hydrocarbons and as a result of that the catalytic cracking reactions will be enhanced. However, the hot catalyst vaporizes the feed and catalyses the cracking reactions to produce components including liquefied petroleum gases, gasoline, coke, and light gas oil (Ali et al., 1997; Gupta et al, 2010). Since the cracking reaction is endothermic, burning coke can generate the required energy. The regenerator in the FCC unit used to burn the coke deposited on the catalyst to reactivate the catalyst and to supply the heat required for cracking reaction in the riser (Gupta et al., 2010). The accurate prediction of the amount of coke deposited on the catalyst helps to estimate the endothermic heat needed in the riser.

For modelling of cracking kinetics, 3-lump model proposed and lumped reactant and all products into three major groups; feed, coke plus C1-C4 hydrocarbons, and gasoline (Gupta and Subba, 2003). A more complicated and detailed 10- lump model proposed to lump the reaction feed into paraffins, napthenes, and aromatic groups, and two product groups of C1-C4 plus coke and gasoline (Gupta et al., 2010). The accurate prediction of coke formation helps to estimate the endothermic heat needed in the reactor. However, the models developed so far have no advantage for estimating the amount of coke, whereas the 4-lump model is developed to estimate the accurate amount of deposited coke (Ali e et al., 1997).

The microscopic shell balance approach has been used in literature to drive partial differential equations or simultaneous differential equations that can describe the products yields and the reactants conversions in tubular reactors as a function of residence time, position, temperature, pressure, velocity, catalyst feed ratio, gas void fraction, type of catalyst, inlet feed droplet size, reactor height, inlet catalyst temperature, mean boiling point of inlet feed, reactor radius, etc. (Ali et al., 1997; Fogler, 1999; Lee, 1989; Reza, 2000; Vogt and Weckhucysen, 2015). This study employed the microscopic approach for the riser reactor modelling. The approach combines the kinetic models, by considering the reactions occurring at a microscopic scale, and the mass flow rate of the components through the reactor system.

The FCC riser reactor, Figure 1, is used in the production of gasoline form the cracking of heavier petroleum fractions where the catalyst pellets deactivated very rapidly by coking. In the FCC riser reactor, the catalyst and the reactant feed enter together from the bottom point and the gas velocity also increases continuously along the riser height because of decrease in vapor density due to formation of lower molecular weight products on cracking of vacuum gas oil (VGO). The catalyst particles are carried through at the same velocity as the gas velocity (Ali et al., 1997; Gupta et al., 2010). A microscopic approach was used to write a mass balance across the control volume as depicted in Figure 1 (Mahfud, 2016). A mass balance on the component j (j = A, B, C and D) over the riser reactor volume ∆V = A_C∆z is done based on the following assumptions: steady state process, one-dimensional flow process, isothermal process, isobaric process, and ideal-gas behaviour. The species mass balance over the control volume includes gas phase only.

Differential balance equations were performed on an infinitesimally small control volume. These equations enable us to determine how mole fractions of the components vary with the height of the riser reactor.

Dividing by Δz and taking the limit as Δz→0 we obtain

Equation 2 is the design equation of the reactor. The four-lump kinetic model, Figure 2, is based on the following assumptions: Gas oil cracking reaction is second order, gasoline cracking reaction is first order, gas flow in the reactor is an ideal plug flow, and gas does not produce coke. The rate of consumption of reactant j per unit catalyst volume was expressed as (Ali et al., 1997):

where C_j is the concentration of component j, C_jo is the initial concentration of pure component j, n=1 for vacuum gas oil (VGO), and n = 0 for gasoline cracking, ∅ catalyst activity.

Combining Equations 2 and 3 gives the following ordinary differential equations:

where $Km=∅ ACεgLρgFg$⁠.

With the boundary conditions, at z=0: y_A (0)=1, y_i (0)=0; i = B, C and D where: y_i; weight percent of hydrocarbons in the riser (i = A, B, C and D), k_ij; reaction rate constants between species i and j, where i ≠ j, m_r³/(m_cat³.s), AC; cross-sectional area of the riser (m²), Fg; hydrocarbon gases mass flow rate in the riser, kg.sec^-1, ϵ_g; hydrocarbon gases void fraction in the riser, Ф = 0.2 (Gupta and Subba, 2003); catalyst activity, ρ_g; density of gas phase in the riser, kg.m^-3, L; riser height, m.

Each of the first order ordinary differential equations is accompanied by one initial condition. These first order ordinary differential equations are simultaneous in nature but can be solved by the numerical methods such as Runge-Kutta methods for differential-algebraic equations. The four-lump kinetic model was developed based on a number of assumptions, which may not be applicable to all systems. The validity of the model should be checked against experimental data prior to the laborious computations.

Experimental catalytic cracking data were utilized at three temperatures 482, 549, and 616°C (Ali et al., 1997; Gupta et al., 2010). The dependence of the rate constant K_ij of chemical reactions on the temperature T and activation energy E_a are given by Arrhenius equation (Gupta et al., 2010):

K_o represents the frequency factor (second), and R represents the gas constant.

Table 1 shows the values of the reaction rate constant at different temperatures that are calculated from the following equation:

The specific heat capacity of each lump in the reactor was not considered as function of temperature, which was not introduced into the model equations. The model equations were only adopted to investigate the influence of the riser temperature on the rate of reaction as a consequence of increasing the reaction rate constant that described by Arrhenius equation. However, the rate of reaction (Equation 3) could probably be doubled or tripled as a result of increase in the reactor temperature, inside the riser, especially at the catalyst surface, where the reaction rates are controlled by reaction kinetics which affects the conversion or yields of products.

More details can be found in the included tables. In this result, the kinetic constants reported by Gupta and Subba (2003) were used. For the riser, 0.8 m is the inside diameter, 33 m is the riser height, 20 kg/s is the feed flow rate, 4.5 kg/m³ is the gas phase density. The kinetic constants, the molecular weights of the components, the activation energy for cracking reaction, and the industrial riser data are summarized in Tables 2–5, respectively. The reaction rate constants at different temperatures are calculated by using Equation 9 and listed in Table 1, were used in all the calculations. It should be noted that production data for VGO conversion and for product yields were predicted by Ali et al. (1997) with a predictive four lumps kinetic model, where lumps represented VGO, gasoline, gas and coke.

Equations 4 to 7 are initial value ordinary differential equations (ODEs) and solved using MATLAB ODE45 solvers (see Figure 3). Figure 4 illustrates the effect of temperature on the yields (wt %) of Gas, Gasoline, Coke and conversion of vacuum gas oil (wt %). The mean average boiling point of the vacuum gas oil is equal 690 K (Ali et al., 1997), and the cracking reactions are endothermic reactions for that reasons the reactor temperature were selected to be higher than 690 K. Figure 5 compares the predicted data and the actual plant data. Obviously, the derived models indicate reasonable predictions for the outlet data from the riser reactor. The continuous cracking of VGO and Gasoline leads to increase the Gas and Coke, to avoid increasing the yield of the unwanted products such as Gas and Coke the contact time has to be short (less than 0.5 seconds) (Gupta et al., 2010).

Increasing reactor temperature leads to increase both liquefied petroleum gas yield and coke yield, and to decrease the gasoline yield. Table 1 gives the products yields of the FCC unit at three different operating temperatures 756 K, 822 K and 889 K. 756 K gives the highest gasoline yield, and higher than this riser temperature leads in over-cracking of the feed gas oil and resulting in more light hydrocarbons (Gas) and more deposited coke on the catalyst surface.

There are a number of chemical kinetic models available in the literature and used to improve and predict the FCC process. Detailed production data such as compositional distributions along a riser is necessary to estimate the appropriate length for maximum gasoline yield. However, using a simplified four-lump kinetic model is quite useful if the validity of the model is verified with comparing to actual plant data. A shell balance was used to derive simultaneous ordinary differential equations (ODEs) that employed to determine the products yield from the riser. The best gasoline yield was being obtained (64%) for conditions specified above. Operating temperature was varied and its effect on the gasoline yield, coke yield and total conversion for a riser FCC unit was studied. The cracking product distribution is measured at three different temperatures 756 K, 822 K, and 889 K. Results of simulation showed good agreement with actual plant data. These ODEs are perfect in predicting the amount of precipitated coke at the catalyst surface. The obtained results of the products yield from the riser show that the four lump kinetic model of Ali et al. (1997) is predictive as it is coupled with the predicted simultaneous ODEs, that is, the reaction rate constants are independent of the feed type.