Mathematical Modeling of Gas Oil HDS and Optimization of ...

I NTERNATIONAL J OURNAL OF C HEMICAL R EACTOR E NGINEERING Volume 7

2009

Article A8

Mathematical Modeling of Gas Oil HDS and Optimization of Operational Conditions in Trickle Bed Reactor by Genetic Algorithm Reza Abbasi∗

∗

Shohreh Fatemi†

University of Tehran, [email protected] University of Tehran, [email protected] ISSN 1542-6580 c Copyright 2009 The Berkeley Electronic Press. All rights reserved. †

Mathematical Modeling of Gas Oil HDS and Optimization of Operational Conditions in Trickle Bed Reactor by Genetic Algorithm Reza Abbasi and Shohreh Fatemi

Abstract The present work aims to employ genetic algorithm (GA) to optimize a HDS process, which is difficult to optimize by conventional methods. The considered chemical process is the three phase catalytic trickle-bed reactor in which hydrodesulphurization reaction occurs. Non-linear kinetics coupled with the transitional mathematical model of the gas, liquid and solid phases are used to describe the dynamic behavior of the multivariable process. The model, based on a two-film theory, was tested with regards to hydrodesulphurization of vacuum gas oil in a high-pressure pilot plant operated under isothermal conditions. Due to the high dimensionality and non-linearity of the model, a rigorous one, the solution of the optimization problem through conventional algorithms does not always lead to the convergence. This fact justifies the use of an evolutionary method, based on the GAs, to deal with this process. In this way, in order to optimize the process, the GA code is coupled with the rigorous model of the reactor. The aim of the optimization through GAs was to search for the optimal conditions that minimize the gas make and sulfur content of the outlet oil. Many simulations are conducted in order to find the maximization of the objective function without violating the constraints. The results show that the GA is used successfully in the process optimization. KEYWORDS: mathematical modeling, trickle bed reactor, dynamic simulation, non-linear dynamics, optimization, genetic algorithm

Abbasi and Fatemi: Mathematical Modeling of Gas Oil HDS

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1- Introduction Three phase catalytic reactors are widely used in industrial processes like hydrogenation and oxidation with usually high throughput production. The importance of hydrogenation multiphase reactors is due to its large range of applications, such as hydrotreating reactors. Investments in these types of reactors are high, mainly in the mathematical modeling development, since the dynamic behavior of these reactors is complex. This normally leads to high dimensional non-linear models (Rezende et al., 2008). Development of mathematical models that describe the dynamic behavior of the three phase catalytic reactors is important for their optimization since it is necessary at the same time to operate with high productivity and within products specifications without causing environmental impact. In this way, optimization problems of three phase reactors are postulated considering an objective function motivated by the profit or by another variable of interest to the process, but subject to the environmental and operational constraints. The non-linear high dimensional problems with a complex search space can be solved by two classes of search methods, to know, deterministic and stochastic ones. The deterministic methods, such as successive quadratic programming (SQP), have the characteristic of requiring the computation of first and/or second order derivatives of the objective function and/or constraints, which make these methods not applicable to a non-differentiable or discontinuous problem. Another disadvantage of the deterministic methods is the large dependence on the chosen initial solution in the search for the optimal solution (Costa et al., 2005; Deb, 1999). On the other hand, stochastic methods, such as genetic algorithms (GAs), have the advantages of not requiring manipulation of the mathematical structure of the objective function and/or constraint and not requiring an initial feasible point (Deb, 2000; Leboreiro and Acevedo, 2004). Such characteristics have increased the application of the GAs in the various search and optimization problems. The present work proposes the optimization of a three phase catalytic reactor making use of the GAs coupled with a rigorous mathematical model of the process. The mathematical model is characterized by non-linearity and high dimensionality, which bring difficulties for deterministic optimization methods. In this work, the performance of the GAs to optimize the reactor represented by a rigorous model is studied and the suitability of the GA application in real time implementation is discussed.

Published by The Berkeley Electronic Press, 2009

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International Journal of Chemical Reactor Engineering

Vol. 7 [2009], Article A8

2- Mathematical modeling Three phase catalytic reactors are systems in which both gas and liquid phases are in contact with a catalyst in the solid phase, which is wet and adequately contacted by reactants (in both liquid and gas phase) to promote the chemical reaction(Rezende et al., 2005). This is clearly a problem of interactions among chemical reactions and heat and mass transfer. In order to represent the main phenomena and be able to describe the real behavior, the mathematical model should take into account the different phases (heterogeneous model).The process considered in this work is the three phase catalytic trickle-bed reactor in which the hydrodesulphurization reaction of vacuum gas oil on NiMo/Al2O3 catalyst occurs. The stoichiometric coefficients of the overall hydrodesulphurization reaction as a characteristic of the feed have to be estimated experimentally because information about the chemical composition of high-boiling petroleum fractions is lacking. The process can be described by the reaction that follows (Korsten et al., 1996) υ S S( liquid ) + υ H 2 H 2( gas ) → υ HC HC( liquid ) + υ H 2 S H 2 S( gas ) The exit sulfur concentration should be as low as possible due to severe environmental specifications. In this way, the exit sulfur concentration is defined as a constraint of the process, limited to 100 ppm or even less. The mathematical model used to describe this process is the one developed by (Fabian S. Mederos, Jorge Ancheyta, 2007). This model includes material balances for the three phases. It neglects the diffusion in the catalyst particles. The following hypotheses were assumed in development of the model: (1) The reactor operates in dynamic condition. (2) Gas and liquid flows are co-current. (3) Gas and liquid velocities are constant through the reactor. (4) Constant density of gas and liquid phases. (5) There are not radial concentration and temperature gradients. (6) Catalyst activity does not change with time. (7) Vaporization and condensation of oil do not take place. (8) Constant pressure. (9) Chemical reactions take place only at the solid catalyst. (10) Mass resistance in the gas side of the gas–liquid interface is assumed to be negligible.

http://www.bepress.com/ijcre/vol7/A8


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2-1- Mass balance Mass-balance equations in the TBR (trickle bed reactor) for co-current operations are described with the following set of partial differential equations (PDEs) and ordinary differential equations (ODEs): (i) Gas phase

εG ∂p iG RT G ∂t

=−

⎛ PG ⎞ u G ∂Pi G − k iL aL ⎜ i − C iL ⎟ RT G ∂z ⎝ Hi ⎠

(1)

Where i = H2, H2S. (ii) Liquid phase

εL

⎛ Pi G ⎞ ∂CiL ∂C L ∂ 2CiL L k a = −u L i + ε L DaL + − CiL ⎟ − kiS aS ( CiL − CiS ) i L⎜ 2 ∂t ∂z ∂z ⎝ Hi ⎠

(2)

Where i = H2, H2S. 2 L ∂C iL ∂C iL L ∂ Ci = −uL + ε L Da − k iS as (C iL − C iS ) εL 2 ∂t ∂z ∂z

(3)

Where i = S, HC. In these equations, axial dispersion coefficient of liquid phase ( D aL ) is needed, which can be determined from the Peclet number: Pe aL,m =

d peu L D aL ε L

(4)

Peclet number can be calculated from different correlations reported in the literature depending on the mode of operation of the reactor. In this work we have used the Hochman–Effron correlation for low interaction regime, pilot plant (M.O. Tarhan1983): 0.003ReG Pe aL,m = 0.034 Re 0.5 L 10

(5)

(iii) Solid phase

ε p (1− ∈)

∂C iS = k iS as (C iL − C iS ) ± ρB ζηHDS rHDS (C iS ,...,T s ) ∂t

(6)


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Where i = H2, H2S, S, and HC. ‘‘_’’ sign is for the reactants, and the ‘‘+’’ sign is for the products. Because the concentration of hydrocarbon is the main component of the feedstock and it does not change significantly during HDS of VGO, Eqs. (3) and (6) for i = HC will not be considered in the simulations. The reactor of this case study is designed as a tube with an inside diameter of 3 cm and a length of 125 cm. The length of the reactor is subdivided into three sections. The first section, having a length of 33.5 cm, was packed with inert SiC particles. This entrance section was used to heat up the mixture to a maximum temperature of 500°C, and to provide a uniform distribution of gas and liquid .The following section with a length of 66.5 cm (this is the active length of the reactor in which the chemical reaction occurs and this length is used as the reactor length in the model calculations) contained a packing of 160-g catalyst, and inert particles. The catalysts used were commercial NiMo/Al2O3, extrudates with a trilobe shape and an equivalent diameter of 1.72 mm. For all the experiments reported here a single charge of the presulfided catalysts was used. The inert exit section was packed with SiC particles of nearly the same size as the catalyst and the particles in the other reactor sections. At the center line of the reactor there is a thermo well containing ten thermocouples used to measure the axial temperature profile within the reactor. The reactor temperature was maintained at the desired level by five electric heaters, which provided an isothermal temperature along the active reactor section. The greatest deviation from the desired value was about 1°C. . More details about reactor parameters, feed specifications, operating conditions and properties of catalyst can be found in Table 1 and 2. (Hans Korsten, 1996). Table 1.Operational conditions of the reactor Conditions Gas superficial velocity (cm/s) Oil superficial velocity (cm/s) Catalytic bed length (cm) Internal diameter (cm) Temperature ( °C) Pressure (MPa) WHSV (h-1) Gas composition (mol %) H2 H2S Light hydrocarbons N2


Reactor 0.1606 7.522×10-3 66.5 3 370 10 0.85 100 -


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Table 2.Main properties of HDT feedstock and catalyst Property Feedstock Density at 15.6 °C(g/cm3) Density at 50°C(g/cm3) Molecular weight Mean average boiling point (°C) Simulated distillation: IBP 10 vol. % 30 vol. % 50 vol. % 70 vol. % 90 vol. % FBP Sulfur( wt %) Mean average boiling point (°C) Catalyst Equivalent diameter(mm) Pore volume (cm3/g) Mean pore diameter(A°) Molybdenum content(wt %) Nickel content (wt%) Bulk density(g/cm3)

Value 0.915 0.894 420 451 245 353 456 524 551 583 617 2 451 1.72 0.56 127 10.7 2.9 0.816

3- Reaction kinetics The oil fraction feedstock contains a great amount of organic sulfur compounds, and the overall HDS reaction is usually represented by the practical and widely accepted generalized stoichiometric equation, that lumps the HDS reaction of all of the sulfur compounds into one expression. Since hydrodesulphurization reactions at the usual process conditions are irreversible (Girgis and Gates, 1991; Vrinat, 1983), reverse reactions need not be considered. Assuming the sulfur content of the oil and the concentration of hydrogen to have a positive effect, and hydrogen sulfide to adsorb at the active catalyst sites, the following kinetic equation of the Langmuir-Hinshelwood type is used:

rHDS = k app .

(c SS )m1 .(c HS 2 )m 2 (1 + K 4 .c HS 2S )2

(7)

Where the rate of reaction per unit mass of the catalyst is correlated with the concentrations at the outer catalyst surface; m1 and m2 represent the reaction order concerning the sulfur compound and hydrogen, respectively; kapp is the



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apparent rate constant, as discussed below; and the adsorption-equilibrium constant of hydrogen sulfide at the catalyst surface is represented by K4. It has often been observed that the rate of chemical reaction decreases with increasing particle size. Especially in the hydrotreating of high-boiling mineral-oil fractions, a great intra-pellet mass-transfer resistance must be expected. In the literature effectiveness factors in the range of η = 0.0057-1 are mentioned (Li et al., 1995; Scamangas and Marangozis, 1982; Weiss et al., 1987). In the calculations described below we assumed that the catalyst effectiveness factor is constant down through the reactor. Since the experiments reported in this article were carried out using a single catalyst charge, the value of the catalyst effectiveness factor is included in the chemical reaction rate constant. (Hans Korsten, 1996). The validity of assuming a constant effectiveness factor in the field of hydroprocessing is confirmed by the good agreement with the experimental results. This simplification makes it much easier to solve the system of equations than it would be if the concentration profile inside the catalyst were included, as recently noted (Froment et al, 1994). The effect of temperature on HDS reaction rate parameters are experimentally estimated in the literature. The intrinsic rate constant on Alumina supported Ni-Mo catalyst has been reported by the activation energy and preexponential term as the following (Korsten et al., 1996): 3 cm3 0.45 J 6 cm )( ) , K 0 = 0.545 × 10 ( E A = 72,500 mol gr.s mol

(8)

The influence of temperature on the adsorption- constant described by the van't Hoff equation as: K 4 (T ) = K 0 . exp(

∆ H ads ) R .T

(9)

In which the enthalpy of adsorption has been obtained as: (Frye and Mosby, 1967) ∆H ads = 2, 761

J mol

(10)

The enthalpy of the HDS reaction at standard temperature is reported as: (M.O. Tarhan1983) ∆H R = −251, 000 (

J ) mol

(11)



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4- Reactor Performance

The three-phase reactor model contains a number of parameters. Most of these, the properties of gases and liquids, solubilities, diffusivities, and mass-transfer coefficients, can be calculated by using the correlations which are presented in Table 3 (F.S. Mederos, M.A. Rodrı´guez, and J. Ancheyta, 2006). Table 3.Correlation to estimate heat- and mass-transfer coefficients, gas solubilites and properties of oil and gases Parameter

Correlation

Oil density

ρ L ( P , TL ) = ρ o + ∆ ρ P − ∆ ρ T 2

⎛ p ⎞ −0.0603ρ0 ⎛ p ⎞ ⎤⎦ ⎜ ∆ρP = ⎡⎣0.167 + (16.181×10−0.0425ρ0 )⎤⎦ ⎜ ⎟ − 0.01⎡⎣0.299 + (293×10 ⎟ ⎝ 1000 ⎠ ⎝ 1000 ⎠ −0.764(ρ0+∆ρp )

∆ρT = ⎡⎣0.0133+152.4(ρ0 +∆ρp )−2.45 ⎤⎦(TL −520) −⎡⎣8.1×10−6 −0.0622×10 Henry coefficient

Hi =

⎤(TL −250) 2 ⎦

υN λiρ

L

⎛ TL ⎞ −6 2 0.835783 ⎟ +1.94593×10 TL + ρ ρ202 20 ⎝ ⎠

Solubility of hydrogen

λH =−0.559729 − 0.42947×10−3TL + 3.07539×10−3 ⎜

Solubility of H2S

λ H2S = exp(3.367 − 0.00847TL )

Gas- liquid mass-transfer

2

k iL aL DiL

1/ 2

0.4

⎛G ⎞ ⎛ µ ⎞ = 7⎜ L ⎟ ⎜ L L ⎟ ⎝ µ L ⎠ ⎝ ρ L Di ⎠

Coefficient Dynamic liquid viscosity

µL = 3.141× 1010 (TL − 460)−3.444 [ log10 ( API )]

a

a = 10.313 [ log10 (T L − 460) ] − 36.447 Diffusivity

⎛ υ 0.267 ⎞⎛ T ⎞ DiL = 8.93 × 10−8 ⎜ L0.433 ⎟⎜ L ⎟ ⎝ υi ⎠ ⎝ µ L ⎠

Molar volume

v = 0.285vc1.048 0.2896 −0.7666 vcm = 7.5214 × 10 −3 TMeABP .d15.6 1/ 2

Liquid-solid mass-transfer

⎛ G ⎞ k si = 1.8 ⎜ L ⎟ DiL a S ⎝ a Sµ L ⎠

1/ 3

⎛ µL ⎞ ⎜ L ⎟ ⎝ ρL Di ⎠

Coefficient Specific surface area

as =

6 (1 − ε ) d pe

The rate constant, order of reaction, and adsorption equilibrium constant are the result of experimental determination (Hans Korsten, 1996). Orders of reaction are m1=1 and m2 = 0.45. The value of m2 appears to be due to a dissociation of H, at the catalyst sites, where the theoretical value would be m2 = 0.5. These parameters and the adsorption-equilibrium constant of



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hydrogen sulfide at the catalyst, K4 = 70000 (cm3 mol-1), were estimated as well as the reaction rate constant by applying a univariable method at reactor operational temperature (370°C) (Hans Korsten, 1996). A comparison of experimental data, with the several reactor simulations have showed that the apparent rate constant kapp increases with WHSV (Satterfield, 1975). This effect can be correlated with the wetting efficiency, defined as the ratio of the apparent rate constant kapp and the intrinsic rate constant kin : fw =

kapp kin

≤1

(12)

The trickling-flow regime is subdivided into a complete and a partial wetting region, depending on the liquid flow rate (Ng, 1986; Ng and Chu, 1987). Because of the small superficial mass-flow velocities in pilot trickle-bed reactors, the wetting efficiency is in the range fw = 0.12-0.6 (Satterfield, 1975; Gates et al., 1979). The wetting efficiency of industrial reactors can be expected to be fw = 0.7-1.0. There are a number of correlations in the literature that predict the wetting efficiency of packing materials. Since the wetting efficiency is a function of the initial liquid distribution, the geometry of the packing and of the catalyst, and the mass flow throughout the reactor, the predicted values are scattered over a wide range (Satterfield, 1975). Using an empirical correlation, as recommended by (Satterfield, 1975), the measured values of the apparent rate constant as a function of the superficial mass-flow velocity G, in (kg m-2 s-1) can be described: 1 kapp

−

1 A = B kin GL

(13)

With the intrinsic rate constant at reactor operational temperature (370°C) kin = 0.67 (cm3 g-1 s-1). (cm3 /mol) 0.45 and the constants A=0.21 and B=1.4. It should be noted that the conversion in trickle-bed reactors depends strongly on the wetting efficiency. The strong influence of the wetting efficiency on the conversion is due to small catalyst effectiveness factors, as observed in hydroprocessing heavy fractions. In the case of partial wetting we have to distinguish between an incomplete-contact reactor scale and an incomplete-contact particle scale (Mills and Dudukovic, 1979). The former describes a situation where parts of the catalyst bed are completely dry, so that the total reactor efficiency is proportional to the wetting efficiency. If the catalyst particles are touched by the



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liquid without wetting the whole surface, we call it incomplete-contact particle scale. Because of capillary forces, we assume the pellet to be completely liquidfilled if it comes in contact with the liquid. In pilot plant reactors we consider the incomplete-contact particle scale. In the case of large Thiele modulus, that is, very fast reaction, the total catalyst effectiveness is strongly affected by the external wetting efficiency. This is because the liquid reactant cannot get into the catalyst rapidly enough if the wetting efficiency is small. The reaction takes place only in a thin shell near the wetted catalyst surface. Considering a slow reaction, where the intraparticle mass transfer can be neglected, that is, small Thiele modulus, the problem of liquidreactant depletion is much less, and therefore the total catalyst effectiveness is relatively unaffected by external wetting efficiency (Mills and Dudukovic, 1979; Harold and Ng, 1987). 5- Model boundary conditions and solution

Since the mathematical model is a system of PDEs and ODEs with time and spatial coordinate as independent variables, it is necessary to define the following initial and boundary conditions for the liquid and gas phases. Table 4. Boundary and Initial conditions for the liquid and gas phases (i)Boundary condition For t>0, At z=0 i C ocurrent operation p iG = ( p iG ) 0 , i=H 2 , H 2 S

C iL = (C iL ) 0 , i=H 2 , H 2 S i C ocurrent operation

C iL = (C iL ) 0 , i=S C Si = 0, i=H 2 , H 2 S, S TL = (TL ) 0 ; TS = (TS ) 0 At z=LB i Cocurrent operation with liquid axial dispertion

∂C iL =0 ∂z


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Initial condition for the liquid and gas phases: (ii)Initial condition For t=0, At z=0 i C o c u rre n t o p e ra tio n G i

p

C

L i

= ( p iG ) 0 , i = H = (C

L i

, H 2S

2

) 0 , i= H

2

, H 2S

i C o c u rre n t o p e ra tio n

C

L i

= (C

C

S i

= 0 , i= H

L i

) 0 , i= S 2

, H 2S , S

TG = TL = TS = T0 At 0