eth

.

matter in refineries and lots of investigations have been made concerning this issue. In this

a boiling temperature between 30 C and 200 C, and consti-

have an improved process its better to revamp naphtha

reforming strategies due to its impact on overall refinery

profits [5].

various hydrocarbons and related isomers. It is too complex to

represent the catalytic reforming reactions. In this regard,

the first significant attempt to model a reforming system has

been made by Smith [7]. He considered naphtha to consist of

three basic components including paraffins, naphthenes, and

* Corresponding author. Tel.: 98 711 2303071; fax: 98 711 6287294.

Avai lab le at www.sc iencedi rect .com

w.

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9E-mail address: rahimpor@shirazu.ac.ir (M.R. Rahimpour).tutes typically 15e30% byweight of the crude oil. The catalytic

naphtha reforming reactions are carried out over commercial

Pt/Re/Al2O3 catalyst (0.3% Platinum, 0.3% Rhenium). Naphtha

and reformate are complex mixtures of paraffins, naph-

thenes, and aromatics in the C5eC12 range [1,2]. Naphtha

reforming plays a major role in improving the aromatic and

hydrogen production in petroleum refineries [3,4]. In order to

consider a detailed kinetic model taking into account all

components and reactions [6]. Studies on the catalytic

naphtha reforming process have been categorized in two basic

groups. The first group involves studies on the kinetics of

catalytic naphtha reforming process. Various attempts have

been made to find better lumped groups of reactions to1. Introduction

Full-range naphtha is the fraction of the crude oil with

2. Literature review

Naphtha as a complex reforming feedstock is composed ofReceived 21 May 2010

Received in revised form

15 July 2010

Accepted 23 August 2010

Keywords:

Axial-flow

Spherical packed-bed reactor

Catalytic naphtha reforming

Dynamic modeling

Hydrogen production

Aromatic production0360-3199/$ e see front matter 2010 Profedoi:10.1016/j.ijhydene.2010.08.124study, an axial-flow spherical packed-bed reactor (AF-SPBR) is considered for naphtha

reforming process in the presence of catalyst deactivation. Model equations are solved by

the orthogonal collocation method. The AF-SPBR results are compared with the plant data

of a conventional tubular packed-bed reactor (TR). The effects of some important param-

eters such as pressure and temperature on aromatic and hydrogen production rates and

catalyst activity have been investigated. Higher production rates of aromatics can

successfully be achieved in this novel reactor. Moreover, results show the capability of flow

augmentation in the proposed configuration in comparison with the TR. This study shows

the superiority of AF-SPBR configuration to the conventional types.

2010 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.Article history: Improving the octane number of the aromatics compounds has always been an importanta r t i c l e i n f o a b s t r a c tDepartment of Chemical Engineering, School of Chemical and Petroleum Engineering, Shiraz University, Shiraz 71345, IranModeling of an axial flow, sphnaphtha reforming process indeactivation

D. Iranshahi, E. Pourazadi, K. Paymooni, A.M

journa l homepage : wwssor T. Nejat Veziroglu. Prical packed-bed reactor fore presence of the catalyst

Bahmanpour, M.R. Rahimpour*, A. Shariati

e lsev ie r . com/ loca te /heublished by Elsevier Ltd. All rights reserved.

Moreover, Brumbaugh has patented a modified novel axial-

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9 12785aromatics industrially known as PNA. Thereafter, more

extensive attempts have been made to model reforming

reactions. Juarez and Macias [3] developed a new kinetic

model in which the most important reactions in terms of

isomers of the same nature (paraffins, naphthenes and

aromatics) were taken into account. The average deviation of

the new kinetic model from experimental data was reported

to be less than 3%. Weifeng et al. [8] developed a new kinetic

model including 20 lumped components and 31 reactions.

Boyas and Froment [9] studied a fundamental chemistry of

naphtha. The current model imposed the equilibriums of

hydrogenation and dehydrogenations. Stijepovic et al. [10]

recommended a semi-empirical kinetic model for catalytic

reforming. Their lumping strategy was based on paraffins,

olefins, naphthenes and aromatics (PONA) analyses. Similar

studies in this field were carried out by Ramange et al. [11,12],

Krane et al. [13], Kmak [14] and Marin et al. [15]. The second

group involves studies on the modeling and improvement of

aromatics and hydrogen yields of the conventional reactors. Li

et al. [16] modeled and optimized a semi-regenerative cata-

lytic naphtha reformer by considering most of its key consti-

tutes. Taskar and Riggs [17] done a similar study as Li et al. But

they have used a reaction network composed of 35 reactions

for 35 pseudo components. Juarez et al. [18] modeled and

simulated four serially catalytic reactors for naphtha reform-

ing. Weifeng et al. [6] considered 18-lumped kinetic models to

simulate and optimize a whole industrial catalytic naphtha

reforming process by Aspen Plus platform. Stijepovic et al. [19]

introduced a new simulation and optimization approach for

CRs. They applied a new proposed objective function in which

economical and environmental performance was taken into

consideration. Khosravanipour and Rahimpour [4] presented

a membrane catalytic bed concept for naphtha reforming in

the presence of catalyst deactivation. Weifeng et al. [20]

examined a multi objective optimization strategy for a CR

process in order to obtain aromatic products. Rahimpour [21]

proposed a novel fluidized-bed membrane reactor (FBMR) for

naphtha reforming in the presence of catalyst deactivation.

He shows that the FBMR increases catalyst activity, aromatic

and hydrogen production rates.

In order to enhance the octane number of aromatic

compounds, the reaction conditions have to be improved.

Aromatics are enhanced by shifting the reactions (using

a hydrogen permeselective membrane) or decreasing the

pressure drop. In industrial plants, the pressure drop is

a serious problem in unit operations such as reactors. The

configurationswhich havemuch lower pressure drop than the

conventional fixed-bed reactors are radial flow spherical

packed-bed reactors (RF-SPBR) and radial flow tubular packed-

bed reactors (RF-TPBR). A complete literature review on the

RF-SPBR and RF-TPBR has been prepared by Iranshahi et al.

[22] have recentlymodeled the RF-SPBR for naphtha reforming

process in the presence of catalyst deactivation. They have

shown that in addition to pressure drop reduction through the

catalytic bed, the amount of aromatic production increases.

The AF-SPBR is an alternative configuration for pressure drop

reduction in high pressure processes such as naphtha

reforming. This novel configuration is not as well known asthe previous ones and little studies have been done on it.

Fogler has modeled the dehydrogenation of paraffins turningflow spherical reactor for naphtha reforming process [25].

The main goal of this work is to investigate the production

yield of aromatics in the AF-SPBR. A comparison between the

AF-SPBR and the TR are carried out in this study. The dynamic

modeling of this novel reactor is done by considering the

catalyst deactivation.

The AF-SPBR configuration is superior to the previous ones

(AF-TPBR and RF-SPBR) briefly as follows:

The AF-SPBR in comparison with the AF-TPBR:

Lower pressure drop is encountered in this configuration. Lower required material thickness (Material thickness fora spherical and a tubular pipe of the same radius, subjected to

action of an internal pressure, P, are tsph P$r=2s andttub P$r=s respectively.Where t is thewall thickness, s is thetensile stressand r is theradiusofthetubeandthesphere [26]).

Consequently the required surface has been decreasedwhich reduces effectively the costs of investment and

maintenance during the operation (this is a wise decision to

use membranes with lower costs of maintenance).

Smaller catalytic pellets with higher effectiveness factor canbe applied owing to the reduction of pressure drop in this

configuration.

Highermolar flow rate can be applied (due to lower pressuredrop) which increases the production rate.

Lower power supply of recompression is needed.

The AF-SPBR in comparison with the RF-SPBR:

The feed distribution has been improved (Utilizing the radialflow pattern to distribute the feed stream is not easily

applicable).

Modifications have been done on the reactor structure toprovide an effective contact between the reactant gas and

the catalytic bedwhen the quantity of the applied catalyst is

only a fraction of the designed quantity of catalyst [25].

Membrane technologies can be easily introduced in the AF-SPBR while it is hard to apply the membrane concept in

radial flow spherical packed-bed reactor (RF-SPBR).

A homogeneous one-dimensional model has been

considered. The basic structure of the model consists of heat

and mass balance equations. These equations must be

coupled with the deactivation model, and also thermody-

namic and kinetic equations, as well as auxiliary correlations

for predicting physical properties.

The effect of various parameters such as reactor length,

time, variable physical properties, and operating conditions

on the performance of the reactor has been investigated.

3. Reaction scheme and kineticsinto olefins and confirmed that the pressure drop in axial

spherical reactor decreases significantly [23]. Zardi et al. have

modeled the ammonia synthesis in an axial-radial reactor [24].Kinetics of multi-component reactions were presented by

Smith [7]. He assumed some pseudo components in order to

used, so the results are:

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 912786simplify the feedstock of catalytic naphtha reforming. Four

dominant idealized reactions can be used to simplify the

catalytic reforming system. The following four reactions are

considered in the model:

(1) Dehydrogenation of naphthenes to aromatics.

(2) Dehydrocyclization of paraffins to naphthenes.

(3) Hydrocracking of naphthenes to lower hydrocarbons.

(4) Hydrocracking of paraffins to lower hydrocarbons.

The related reactions are

Naphthenes (CnH2n)4 Aromatics (CnH2n6) 3H2 (1)

Naphthenes (CnH2n) H24 Paraffins (CnH2n2) (2)

Naphthenes (CnH2n) n/3H2/ Lighter ends (C1eC5) (3)

Paraffins (CnH2n2) (n3)/3H2/ Lighter ends (C1eC5 (4)

Naphtha reforming reactions are limited by equilibrium. In

order to achieve higher aromatics production, the naphtha

reformingprocess should be carried out athigher temperatures.

The rate equations of these reactions are as follows:

r1 kf1Ke1

ke1pn pap3h

(5)

r2 kf2Ke2

ke2pnph pp

(6)

r3 kf3pt

pn (7)

r4 kf4pt

pp (8)

where kf and Ke are the forward rate constant and the

equilibrium constant, respectively. Rase [27] reported the

following equations for these constants.

kf1 9:87exp23:21 E1

1:8T

(9)

kf2 9:87exp35:98 E2

1:8T

(10)

kf3 kf4 exp42:97 E3

1:8T

(11)

Ke1 1:04 103exp46:15 46; 045

1:8T

(12)

Ke2 9:87exp7:12 8000

1:8T

(13)

where E is the activation energy of each reaction. The acti-vation energies depend on the catalyst which is used.

According to the previous works, the activation energy isE1 36,350E2 58,550E3 63,800

4. Process description

4.1. Conventional configuration

Naphtha reforming is amajor process practiced extensively by

petroleum refineries and the petrochemical industry to

convert paraffins and naphthenes into aromatics. A simplified

flow diagram of continuous catalytic reforming process is

shown in Fig. 1. The fresh naphtha feedstock (middle distillate

of atmospheric distillation column) is combined with a recy-

cled gas stream containing 60e90% (by mole) hydrogen.

Hydrogen can adjust the H2/HC molar ratio through the

reactors to prevent coking and also it sweeps the products

through the catalyst pores. The total reactor charges are

heated and passed through the catalytic reformers which are

designedwith three adiabatically operating reactors and three

heat exchangers between the reactors to maintain the reac-

tion temperatures at designed levels. The effluent from the

3rd reactor is cooled, and then it enters the separators. Off

gases and reformates are separated from the top and the

bottom of the separator [28]. Table 1 shows the specific

properties and operating conditions of the conventional

naphtha reactors. Boiling point ranges are determined by

Distillation Petro Test D86.

4.2. Spherical reactor setup

Although, tubular packed-bed reactors are used extensively in

industry [29], due to some disadvantages of this type of reac-

tors the spherical packed-bed reactors attract more atten-

tions. Some potential disadvantages of tubular reactors are

the pressure drop along the reactor, highmanufacturing costs

and low production capacity. In order to avoid serious pres-

sure drop in the TR, the effective diameters of the catalyst

particles are usually considered more than 3 mm which lead

to a certain inner mass transfer resistance. In this study, the

AF-SPBR is proposed for naphtha reforming process.found by minimizing the differences between the calculated

and the observed values of outlet temperature and aromatic

yield of bed simultaneously. A constrained optimization

procedure is used to find the activation energies. The activa-

tion energies are adjustable parameters. The objective func-

tion to be optimized is:

OF Xmi1

(NA cal NA plant

2X3i1

TOut cal TOut plant

2)(14)

where NA is the aromatic molar flow rate, T is the outlet

temperature of each bed and m is the number if data sets areFig. 2(a) shows the schematic diagram of the spherical

configuration setup. In the AF-SPBR, catalysts are situated

between two perforated screens. As depicted in Fig. 2(b), the

naphtha feed enters the top of the reactor and flows steadily

to the bottom of the reactor. Attempts should be made to

have a continuous flow without any channeling in the

reactor. The goal is to achieve a uniform flow distribution

through the catalytic bed, because the flow is mainly occur-

ring in an axial direction. The two screens in upper and lower

parts of the reactor hold the catalyst and act as a mechanical

support. Since the cross-sectional area is smaller near the

inlet and the outlet of the reactor, the presence of catalysts in

these parts would cause substantial pressure drop and

spherical packed-bed reactors [30]. In this study, a homoge-

Compressor

EX

-1

R-1

EX-2 EX-3

R-2

Naphtha Feed

R: Reactor

EX: Heat

Exchanger

S-1: Separator

S-2: Stabilizer

Fig. 1 e A simple process flow diagram for conv

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9 12787Table 1 e Specifications of reactors, feed, product andcatalyst of plant for fresh catalyst.

Parameter NumericalValue

Unit

Naphtha feed stock 30.41 103 Kg/hReformate 24.66 103 Kg/hH2/HC mole ratio 4.73 e

LHSV 1.25 h1

Mole percent of hydrogen in recycle 69.5 eDiameter and length of 1st reactor 1.25, 6.29 m

Diameter and length of 2nd reactor 1.67, 7.13 m

Diameter and length of 3rd reactor 1.98, 7.89 m

Distillation fraction of naphtha feed and reformate

TBP Naphtha feed

(C)Reformate

(C)IBP 106 44

10% 113 73

30% 119 105

50% 125 123

70% 133 136

90% 144 153

FBP 173 181

Typical properties of catalyst in use

dp 1.2 mm

Pt 0.3 wt%

Re 0.3 wt%

sa 220 m2/g

rB 0.3 Kg/l

3 0.36 eneous one-dimensional model has been considered. The flow

pattern in the AF-SPBR is assumed to be axial. The basic

structure of the model consists of heat and mass balance

equations. These equations must be coupled with the deac-

tivationmodel, and also thermodynamic and kinetic relations

as well as auxiliary correlations for predicting physical prop-

erties. In order to solve the equations, an element with theconsequently, reduce the efficiency of the spherical reactors.

The other advantage of these screens is to balance the free

zones (free catalyst zones) to find a desirable pressure drop

during the process. The radial flow is assumed to be negli-

gible in comparison with the axial flow. As a result, the

equations in the axial coordinate are being taken into

account exclusively.

5. Reactor modeling

5.1. Axial- flow spherical packed-bed reactor model (AF-SPBR model)

Rahimpour et al. have established the dynamic modeling of

R-3

S-1S-2

Hydrogen Rich Gas

Reformate to Storage

Off Gas

entional catalytic naphtha reforming (TR).length dz (as shown in Fig. 3) has been considered and the

material and energy balances are written upon this element.

Themass and energy balance equations for fluid phase can be

formulated as follows:

Dej1Ac

v

vz

AcvCjvz

1Ac

v

vz

AcuzCj

rBaXmi1

nijri 3vCjvt

j 1;2;.;n 15

keff1Ac

v

vz

AcvTvz

1Ac

v

vz

rAcuzcp

T Tref

rBaXmi1

DHiri

3vrcps

T Tref

vt

(16)

where a is the catalyst activity, i represents the reaction

number and j represents the component number, Dej is the

effective diffusivity of component j, C is the gas phase

concentration, rB is the catalyst bulk density, vij is the

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 912788R-1 R-2

R: Reactor

EX: Heat Exchanger

S-1: Separator

S-2: Stabilizer

astoichiometry coefficient of the reactant j in the reaction i, ri is

the reaction rate, keff is the thermal conductivity of the gas

phase, T is the temperature, r is the density of gas phase, CP is

the heat capacity of the gas phase, 3 is the void fraction andDH

is the heat of the reaction.

Its worthmentioning that the cross-section area, Ac, in the

AF-SPBR is a function of reactor length. Therefore it must

remain between the parentheses of the derivative equation of

mass and energy balances. The formulation of cross-section

area, Ac, is described as follows [23]:

Ac phR2 z L12

i(17)

The boundary and initial conditions are as follows:

Compressor

EX

-1

EX-2 EX-3

Naphtha Feed

z+dz

z

dz

F1

Naphtha Feed

Product

F1

b

Fig. 2 e (a) Spherical axial-flow configuration for catalytic naph

specifications for spherical reactor in the catalytic naphtha refoR-3

S-1S-2

Off Gasz 0; Cj Cjo; T To (18)

z L1 L2 : vCjvz

0; vTvz

0 (19)

t 0; Cj Cssj ; T Tss; Ts Tsss ; a 1; (20)

where L1 and L2 represent the vertical distance from the center

of the reactor to the top and bottom screens in the axial

coordinate. The superscript ss represents the steady state

condition. The steady state mass and energy balance equa-

tions are the same when the accumulation term is set to zero.

Mass and heat transfer coefficients are estimated by several

Hydrogen Rich Gas

Reformate to Storage

Screen z=0

Screen z=L1+L2

L1

L2

R

Z

tha reforming and (b) conceptual Flow pattern and

rming process.

where TR, Ed and Kd are the reference temperature, the acti-

formula of order two. It is noted that this method is well

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9 12789auxiliary correlations. These auxiliary correlations are pre-

sented in Appendix B. The heat of reactions is reported as

follows (kJ/kmol of H2):

DH1 71;038:06DH2 36;953:33DH3 51;939:31DH4 56;597:54

5.2. Conventional tubular reactor model (CR model)

Themodeling assumptions of TR andAF-SPBR are similar. The

general fluid-phase balance is a model with the balances

typically accounting for accumulation, convection, and reac-

tion. The energy and mass balances for the gas phase can be

written as Eqs. (21) and (22):

Dejv2Cjvz2

vuCj

vz

rBaXmi1

nijri 3vCjvt

j 1;2;.;n

i 1;2;.;m 21

keffv2Tvz2

vvz

rucp

T Tref

rBaXmi1

DHiri 3 vvt

rcpT Tref

(22)

The boundary and initial conditions are as follows:

z 0 : Cj Cj0; T T0 (23)

Z L : vCjvz

0; vTvz

0 (24)

t 0 : Cj Cssj ; T Tss; a 1 (25)

Fz

dz

Fz+dz

Fig. 3 e Schematic for differential element along the flow

direction.5.3. Pressure drop (Ergun equation)

The pressure drop throughout the catalyst bed is calculated

based on the Ergun equation. This equation covers the entire

range of flow rates by assuming that the viscous losses and

the kinetic energy losses are additive [23]. This equation for

Cartesian systems is derived as below:

dPdz

150mf2s d

2p

1 3233

QAc

1:75rfsdp

1 333

Q2

Ac(26)

where dP is the pressure gradient,Q is the volumetric flow rate,

dp is the particle diameter, fs is the sphericity (for spherical

particles fs equals one), m is the fluid viscosity and r is the fluid

density.suited for the systems of stiff equations [34]. In addition,

Gears method can be used for solving the set of such stiff

ODEs.

8. Model validation

8.1. Model validation with the presence of catalystdeactivation

Model validation is carried out by comparing the TR results

[28] with the proposed configuration over 800 operating days.

The predicted results of the production rate, the correspond-vation energy and the deactivation constant of the catalyst,

respectively. The numerical values of TR, Ed and Kd, are 770 K,

1.642 105 J mol1 and 5.926 105 h1, respectively.

7. Numerical solution

In order to solve the set of coupled partial differential-

algebraic equations a two steps procedure is used. Firstly,

the steady state simulation is carried out to obtain the

initial conditions of the dynamic simulation. In the steady

state simulation all the time variations are considered to be

zero and activity will be considered to be one. Secondly,

the results of the steady state simulation are used as the

initial conditions for the time-integration of the dynamic

state equations in each node through the reactor. The

deactivation model and the conservation rules are ordinary

and partial differential equations, respectively. Moreover,

there would be algebraic equations due to the auxiliary

correlations, kinetics and thermodynamics of the reaction

system. These equations constitute a set of dynamic

equations.

The set of aforementioned equations of the model of

spherical and tubular reactors are solved by means of the

orthogonal collocation method [32,33]. Inner collocation

points are chosen as a root of a Jacobi polynomial. The

orthogonal collocation method is discussed further in

Appendix A. The system of PDEs is transformed into an

ordinary differential equation (ODE) system by means of this

numerical method. An energy balance is developed for each

collocation point, as well as a mass balance for each species.

The energy and mass balance equations of the spherical

reactor are systems of ODEs with initial conditions. The

system of ODEs is integrated by a modified Rosenbrock6. Catalyst deactivation model

The catalyst deactivation model is used from the previous

work which was presented by Rahimpour [31].

dadt

Kdexp Ed

R

1T 1TR

a7 (27)ing observed data and the residual error are presented in

Table 2. Good agreements between the daily observed plant

data and simulation results are achieved, so this model can

Table 2 e Comparison between predicted production rate and plant data.

Time (day) Naphtha feed (ton/h) Plant (kmol/h) AF-TPBR (kmol/h) AF-SPBR (kmol/h) Devi % (Tubular- Plant)

0 30.41 225.90 221.5802 221.4080 1.9134 30.41 224.25 222.5122 222.3432 0.7762 31.00 229.65 227.9313 227.7559 0.7597 30.78 229.65 226.7749 226.6038 1.25125 31.22 229.65 230.7985 230.6220 0.50160 31.22 229.65 231.2730 231.0971 0.71188 28.55 211.60 209.8377 209.6965 0.83223 30.33 222.75 224.7291 224.5558 0.88243 31.22 233.05 232.1821 232.0072 0.37298 30.67 228.65 228.1752 228.0081 0.21321 30.76 227.64 229.0932 228.9246 0.64

3

3

3

3

3

3

3

2

2

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 912790satisfy successfully the industrial conditions. The related

deviations from the plant reported values are due to the fact

that the kinetics and models used for reaction system of

naphtha reforming underestimated the true reaction rate.

Also using the average molecular weights and such other

physical properties for the three groups of pseudo compo-

nents (PNA), heat transfer coefficients maybe increase the

uncertainties.398 42.35 317.30

425 42.32 317.94

461 42.32 317.94

490 42.32 317.94

524 42.32 313.09

567 42.54 317.94

610 42.54 313.90

717 37.86 286.15

771 38.51 282.108.2. Steady state model validation

A comparison between the proposed model and TR has been

demonstrated at the steady state condition in Table 3. As seen,

there is a good agreement between the plant data and pre-

dicted mole fractions of components at the output of the

system. Analyses of components (paraffin, naphthene and

aromatic) are performed by PONA Test in Stan Hop Seta

apparatus. The aromatic is tested especially by ASTM 2159

equivalent to UOP 273 method [28].

Table 3 e Comparison between model prediction and plant da

Reactor number Inlet temperature (k) Inlet pressure (kPa)

1 777 3703

2 777 3537

3 775 3401

Reactor number Outlet temperature (k)

Plant NPBR SPFR

1 722 727.38 727.4

2 753 751.03 751.6

3 770 770.54 770.79. Results and discussion

In order to have a desirable prediction, the appropriate

number of interior collocation points has been evaluated.

According to Fig. 4, the aromatic molar flow rate is plotted

versus the mass of catalysts with different number of interior

collocation points. In Fig. 4(a) one, two and three and in Fig. 4

(b) one, five, and seven interior collocation points have been

used. As it is shown in Fig. 4(a), when the number of interior

collocation points is not enough, the diversity appears

24.5555 324.2907 2.2824.4826 324.2216 2.0524.6987 324.4426 2.1224.8622 324.6099 2.1725.0433 324.7952 3.8127.0586 326.8164 2.8627.2581 327.0210 4.2589.3742 289.1475 1.1294.9026 294.6767 4.53between the curves. But by increasing the number of interior

collocation points as shown in Fig. 4(b), the discrepancy

disappears between the curves. The curves with 5 and 7

interior points cover each other thoroughly. Therefore, the

N 5 is the most accurate choice.

Results of modeling involve the following main issues:

9.1 Changes of parameters along the spherical reactors.

9.2 Changes of parameters as a function of time.

ta for fresh catalyst.

Catalyst distribution (wt %) Input feedstock (mole %)

20 Paraffin 49.3

30 Naphthene 36.0

50 Aromatic 14.7

Aromatic in reformate (mole %)

Plant NPBR SPFR

5 e 34.78 34.77

6 e 47.28 47.12

5 57.7 56.26 56.18

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3300

3350

3400

3450

3500

3550

3600

3650

3700

3750

Mass of catalyst (Dimensionless)

Pres

sure

(kPa

)

TR

SR

L=0.95R

L=0.70R

L=0.50R

L=0.90R

L=0.10R L=0.30R

Fig. 5 e Pressure profile along the mass of catalyst (solid

line: tubular reactor, dotted line: spherical reactor).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140

45

50

55

60

65

70

75

80

85

90

Mass of catalyst (Dimensionless)

Aro

matic

m

ola

r flo

w ra

te (k

mole

/hr)

one interior collocation pointNinterior=1

two interior collocation pointNinterior=2

three interior collocation pointNinterior=3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140

45

50

55

60

65

70

75

80

85

90

Mass of catalyst (Dimensionless)

Aro

ma

tic m

ola

r flo

w ra

te (k

mole

/hr)

One interior collocation pointNinterior=1

Five interior collocation pointNinterior=5

Seven interior collocation pointNinterior=7

a

b

Fig. 4 e Aromatic molar flow rate along themass of catalyst

(a) with1, 2 and 3 and (b) and 1, 5 and 7 interior orthogonal

collocation points.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

20

40

60

80

100

120

140

Mass of catalyst (Dimensionless)

Mola

r flo

w ra

te (k

mole

/hr)

800th

Day

1st

Day

Paraffin

Naphthene

Aromatic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1250

1300

1350

1400

1450

1500

Mass of catalyst (Dimensionless)

Hyd

roge

n m

ola

r flo

w ra

te (k

mole

/hr)

800th

Day

1st

Day

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

720

730

740

750

760

770

780

Mass of catalyst (Dimensionless)

Tem

pera

ture

(K

)

TR

SR

a

b

C

Fig. 6 e (a) Production and consumption rates versus mass

of catalyst (solid line: the 1st day of production, dotted line:

the 800th day of production), (b) hydrogen molar flow rate

versus mass of catalyst (solid line: the 1st day of

production, dotted line: the 800th day of production) and (c)

the temperature profiles of TR and SR.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9 12791

9.3 Effect of operating conditions and a comparison between

tubular and spherical reactors in this regard.

The variables that affect the performance of the catalyst,

the yield and quality of reformate are time, feedstock prop-

erties, reaction temperature, space velocity, reaction pressure,

and hydrogen per hydrocarbon ratio [35].

0 100 200 300 400 500 600 700 800

0.7

0.75

0.8

0.85

0.9

0.95

1

Time (Day)

Catalyst activity

1st

reactor

2nd

reactor

3rd

reactor

Fig. 7 e (a) The catalyst activity during 800 days of

operation for the 1st, the 2nd and the 3rd reactors.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1000

1500

2000

2500

3000

L/R

Pressu

re (kP

a) FR=1.0

FR=2.0

FR=2.5

FR=3.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

L/R

Aro

matic yield

FR=1.0

FR=2.0

FR=2.5

FR=3.0

b

c

0.85

0.86

0.87

0.88

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 912792Fig. 8 e (a) Aromatic and paraffin and (b) light end and

hydrogen mole productions during 800 days of operation.3500

4000a9.1. Changes of parameters along the spherical reactors

The changes of parameters along the length of the spherical

reactors in the 1st and the 800th days of operation will be

investigated in the first section.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.78

0.79

0.8

0.81

0.82

0.83

0.84

L/R

Hyd

ro

gen

yield

FR=1.0

FR=2.0

FR=2.5

FR=3.0

Fig. 9 e The effect of different FR ratios on (a) the pressure

versus L/R, (b) the aromatic yield versus L/R and (c) the

hydrogen yield versus L/R.

2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

1480

1485

1490

1495

1500

1505

Operating pressure (kPa)

Hyd

ro

gen

p

ro

du

ctio

n (km

ole/h

r)

TR

SR

P=3070 kPa

(B)(A)

Fig. 11 e Hydrogen production versus operating pressure

for spherical and tubular reactor.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9 127930.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.26

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

Mass of catalyst (Dimensionless)

Aro

ma

tic yi

eld

TR

SR

P=1703 kPa

P=2703 kPa

0.36

0.37

a

bAs shown in Fig. 5, the pressure is depicted versus the

dimensionless mass of catalysts of three reactors for various

L/Rs. it can be seen, by increasing L/Rs the pressure drop

increases. Thus, for L/R 0.95 the greatest pressure drop isachieved. At higher L/Rs, a huge bulk of feed encounters less

empty zones at the inlet and the outlet of the reactor. As

a result, a sudden pressure drop happens in the pressure

profile. The steps indicate the pressure drop in each reactor.

The dimensionless mass of catalyst for the 1st, the 2nd and

the 3rd reactors is in the range of 0e0.2, 0.2e0.5 and 0.5e1,

respectively. The dimensionless mass of catalyst 0 showsthe inlet of the 1st spherical reactor and the dimensionless

mass of catalyst 1 implies the outlet of the 3rd reactor. Onthe other hand, if L/R decreases to lower than 0.7, the pressure

drop would decrease too. However, working with the L/R 0.5is useless because most of the reactor space is empty without

any catalyst. Thus, the L/R 0.7 is chosen not only to reducethe useless fabrication of materials for free zone but also to

have a little pressure drop. The reason which makes this ratio

a proper choice will be discussed later.

In previous works, the total molar flow rate was assumed

to be constant while it increases throughout the reactor in

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.29

0.3

0.31

0.32

0.33

0.34

0.35

Mass of catalyst (Dimensionless)

Aro

ma

tic yi

eld

TR

SR

P=3703 kPa

P=4703 kPa

Fig. 10 e Aromatic yield versus mass of catalyst in third

reactor (a) for operating pressures 1703 and 2703 kPa and

(b) for operating pressures 3703 and 4703.

0 1 2 3 4 5 6 7

0

500

1000

1500

2000

2500

3000

3500

4000

Feed flow rate scale up ratio

Pressu

re (kP

a)

SR

TR

0 1 2 3 4 5 6 7

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

Feed flow rate scale up ratio

Aro

matic yield

SR

TR

a

b

Fig. 12 e (a) Pressure versus feed flow rate scale up ratio

and (b) aromatic yield versus feed flow rate scale up ratio at

3703 kPa for tubular and spherical reactors(solid line:

tubular reactor, dotted line: spherical reactor).

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9127941

2

3

4

5

6

0

5

10

15

1500

2000

2500

3000

3500

4000

Flow scale up ratioCatalyst scale up ratio

Pre

ss

ure

(k

Pa

)

0.8

0.9

ie

ld

a

cplants. In this study, the total molar flow and also all physical

properties such as heat capacity and thermal conductivity are

assumed to be variable along the reactors.

A scheme of components molar flow rates versus the

mass of catalyst along the reactor for the 1st and the 800th

days of operation is depicted in Fig. 6(a). As seen, the aim of

naphtha reforming is satisfied, because the molar flow rate

of aromatics increases along the reactor and fortunately

the molar flow rates of naphtha and paraffin decrease.

Each break point in this figure indicates the inlet temper-

ature of the following reactor. Because the temperature

increases by using pre heaters at the inlet of each reactor,

the reaction rates change and discontinuities appear in the

figure.

Also, Fig. 6(b) illustrates hydrogen molar flow rate versus

the mass of catalyst along the reactor. Fig. 6(c) presents

the temperature distribution through the TR and SR. As the

naphtha reforming is predominantly endothermic, the

temperature decreases in the reactors.

9.2. Changes of parameters as a function of time

In the second part, the variation of parameters as a function of

time will be discussed. Fig. 7 reveals the average catalyst

10

5

10

15

0.4

0.5

0.6

0.7

Catalyst scale up ratio

Hy

dro

ge

n y

Fig. 13 e 3-D figure which shows the effect of flow scale up ratio

yield.1

2

3

4

5

6

0

5

10

15

0.2

0.25

0.3

0.35

0.4

0.45

Flow scale up ratioCatalyst scale up ratio

Aro

ma

tic

y

ie

ld

bactivity as a function of time in three reactors. As the reactions

proceed, the catalyst activity decreases. The catalyst activity

has an inverse relationship with the temperature, in other

words, as the temperature decreases the catalyst activity

increases.

Fig. 8(a) illustrates the effect of catalyst deactivation on the

product and reactant production rates. By decreasing the

catalyst activity, the aromatic production rate decreases and

hence the amount of unreacted paraffin increases due to

lower reaction rate. As a result of catalyst deactivation, both

the reaction rate and the production rate decrease. The light

end (off gas) production rate as a function of time is depicted

in Fig. 8(b). As seen, this product also decreases in the course

of time due to the catalyst deactivation. An unpredictable

behavior is observed in Fig. 8(b) for hydrogen production. The

catalyst aging acts in away that hydrogen producesmore after

the 800th day of operation compared with the 1st day.

9.3. Effect of operating conditions and a comparisonbetween tubular and spherical reactors in this regard

The effect of operating conditions such as flow scale up ratio

(FR), operating pressures and other parameters are studied in

the following section.

2

3

4

5

6

Flow scale up ratio

and catalyst scale up ratio on (a) pressure and (b) aromatic

and high catalyst loading, the sufficient time exists for con-

verting the feed to aromatics. Therefore the peak is seen in

low feed flow rates (less than 1). When the feed flow rate scale

up ratio equals to 1, the aromatic yield in the tubular reactor is

slightly higher than the yield in the spherical reactor. By

increasing the scale up ratio in two reactors the aromatic yield

decreases in both reactors but the slope is sharper in TR. The

pressure drop has a strongly negative effect on the aromatic

production in tubular reactors due to the FR. It should be

mentioned that Fig. 12(a, b) are depicted in the constant inlet

pressure of 3070 kPa for tubular and spherical reactors.

The simultaneous effect of catalyst and FRs on pressure,

aromatic and hydrogen yields is investigated in the following

3-D plots. The various catalyst distributions in SR in compar-

ison with the TR are identified by catalyst scale up ratio.

Fig. 13(a) illustrates the simultaneous effect of catalyst and

FR on pressure. For all catalyst scale up ratios the pressure

decreases by increasing the FR. Especially for lower catalyst

scale up ratios, the pressure drop is higher due to lower

reactor diameter and high viscose loss in higher velocities.

When, the catalyst scale up ratio increases to 5, the flux

decreases due to the increase in the reactor diameter. There-

fore the pressure becomes approximately constant. In low

0.3

0.338

0.35

0.4

0.45

Aro

matic yield

CMR=1

CMR=2

CMR=3

CMR=4

a

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9 127959.3.1. The effect of flow scale up ratio (FR)The effect of FR on pressure is illustrated first. The pressure

versus L/R for different FRs is shown in Fig. 9(a). FR indicates

that the flow rate of fresh naphtha feed in spherical reactor is

more compared with the feed stream in TR. As seen,

increasing the FR increases the pressure drop and this will be

more significant for higher L/Rs. The slope of curves starts to

change at the L/R 0.7 and the variation of pressure dropexperiences a new trend in spherical reactors. At the L/R 0.7,the pressure drop increases by increasing the FR. But, the

pressure drop augmentation at this ratio is approximately less

than higher values of L/Rs. Therefore, the L/R 0.7 is anappropriate choice, due to observation of considerable pres-

sure drop after this point. This is an evidence to show that the

L/R 0.7 is a proper choice. The aromatic yield versus L/R fordifferent FRs is illustrated in Fig. 9(b). According to the Le

Chateliers principle, for FRs equal to 1e2 the pressure drop at

L/R 0.7e0.9 shifts the reaction to the aromatics andhydrogen production. The residence time decreases in higher

FRs owing to providing a larger quantity of reactants per unit

of catalyst. Consequently, it decreases the products yield (see

Fig. 9(b)e(c)). Higher pressure drop at FRs of 2.5e3 reduces the

reactants partial pressures and this affects the reaction rates.

It should bementioned that all the previous results in this part

have been achieved at the constant inlet compressor pressure

of 3703 kPa.

9.3.2. Effect of operating pressureFig. 10(a, b) shows the aromatic production in spherical

reactor (SR) and tubular reactor (TR) based on the dimen-

sionless mass of catalyst for various operation pressures. As

seen from Fig. 10(a), when the operating pressure equals

1703 kPa, the aromatic yield in the SR is considerably higher

than the TR. As the operating pressure increases, the differ-

ence between the aromatic yields in two reactor configura-

tions decreases (see the graph for Pressure of 2703 kPa). The

intersection of tubular and spherical curves is named

the turning junction point. However, the arrangement of the

curves will change at the turning junction point and the

aromatic yield in TR becomes more than the one in SR (see

Fig. 10(b)). In other words, the TR unlike the SR is incapable of

operating satisfactorily at low pressures. In spite of SR

advantages (lower pressure drop, lower constructionmaterial,

lower surface area and lower related maintenance costs), the

operating pressure should be wisely determined when

compared with the TR. The same trend is observed for

hydrogen production rate versus the operating pressure for SR

and TR configurations in Fig. 11.

9.3.3. The effect of feed flow rate scale up ratioThe capability of spherical reactors in using higher flow rates

in comparison with the tubular ones is clearly represented by

Fig. 12(a). If the feed flow ratio in TR is doubled, the pressure

decreases drastically. On the other hand if the feed flow ratio

in spherical reactor equals to 7, the pressure declines.

Therefore, if the feed flow ratio in spherical reactor increases

more products would be achieved. The effect of flow rate

augmentation on quality is investigated by considering theproduct yields. According to Fig. 12(b), the peak is observed for

low feed flow rate in both reactors. Due to low feed flow rate0 1 2 3 4 5 6

0.2

0.25

Flow scale up ratio

0 1 2 3 4 5 6 7

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.876

0.9

1

Flow scale up ratio

Hyd

ro

gen

yield

CMR=1

CMR=2

CMR=3

CMR=4

bFig. 14 e The effect of CMR and the flow scale up ratio on (a)

aromatic yield and (b) hydrogen yield.

catalyst scale up ratios, the pressure drop increases by

increasing the FR. Fig. 13(b) illustrates the simultaneous effect

of catalyst and FRs on the aromatic yield. In each FR, if the

catalyst scale up ratio increases, the aromatic yield will

increase. In general, the catalyst scale up ratio causes higher

aromatic yield for each FR. However, increasing FR decreases

the aromatic yield in each catalyst scale up ratio. The simul-

taneous effect of catalyst and FR is considered on hydrogen

yield in Fig. 13(c).

The effect of FR on aromatic yield for different catalyst

mass ratios (CMRs) is depicted in Fig. 14(a) for SR. The

aromatic yield is considered to be 0.338 in industry and the

industry data for CMR 1 is depicted as solid line. If the FRequals 2, in order to achieve the same (0.338) aromatic yield

the CMR should be a little less than 2 (1.9 times). Similarly, if

the FR equals 3, the CMR should be a little less than 3(2.85

times) in fixed aromatic yield. In high CMRs, as FR increases,

the difference between the fixed aromatic yield (0.338) and the

obtained aromatic yield increases. In general, in order to have

a specific flow (FR) a little less than the specified value is

needed for catalyst loading. The difference between these two

values is larger for higher FRs. The effect of FR on hydrogen

yield in different CMRs is depicted in Fig. 14(b). In CMR 1, as

changes to examine the effect of temperature. Fig.15(a, b)

finite difference method. Results show that the AF-SPBR can

be properly applied instead of TR. This study shows that for

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Hyd

roge

n yi

eld

T1=777, T

2=777

T1=777, T

3=775

T2=777, T

3=775

a

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 912796650 700 750 800 850

0.45

0.5

0.55

Inlet temperature (K)

650 700 750 800 850

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Inlet temperature (K)

Aro

mat

ic y

ield

T1=777, T

2=777

T1=777, T

3=775

T2=777, T

3=775

b

Fig. 15 e The effect of inlet temperatures on (a) hydrogenand (b) aromatic yields.naphtha reforming process the AF-SPBR configuration is more

efficient than the previous ones. The practical operating

conditions data together with the mathematical model can be

used to develop the AF-SPBR model for new plant setups and

revamps of process in future.

Acknowledgement

The authors would like to thanks gratefully to Samira Hafe-

ziyeh for her helpful recommendation to improve the English

language of the manuscript.

Appendix A. Orthogonal Collocation method

Jacobi polynomials

a;billustrates the effect of temperature on hydrogen and

aromatic yield, respectively. As the temperature exceeds,

hydrogen is consumed and the yield decreases. The results

show that increasing the 1st reactor temperature (dotted lines

in Fig.15(a, (b)) is more efficient in order to hydrogen and

aromatic production.

10. Conclusion

In any process, a significant incentive to minimize the pres-

sure drop has led to developing a number of alternatives on

the flow configurations. According to this concept, one

potentially interesting idea for catalytic naphtha reforming

process is the application of AF-SPBR. The pressure drop

problem is successfully overcome in this new configuration. In

this study, the AF-SPBR is proposed for catalytic naphtha

reforming process. The effect of several parameters on

aromatic and hydrogen yields is investigated. Most of the

parameters of the systemare considered to be variable such as

heat capacity, viscosity, molecular weight, pressure, density

and the total molar flow. Optimum pressure and L/R are

determined. The dynamic model is solved by the orthogonal

collocation method and the results are much better in

comparison with the other conventional methods such asFR increases hydrogen yield decreases and a peak is observed

in the figure. Hydrogen yield becomes constant for the CMRs

higher than 3. If higher FRs is desired, the CMR should bemore

than 3. In order to achieve the fixed hydrogen yield (0.87),

when the FR is less than 3, it is satisfactory to work with

minimum CMR which equals to 2.

9.3.4. The effect of temperature on the products yieldsIn order to consider the effect of temperature, the inlet

temperatures of two reactors out of three is considered to be

constant and the inlet temperature of the other reactorThe Jacobi function, JN x, is a polynomial of degreeN that is,orthogonal with respect to the weighting function xb1 xa.

2C

32 2C

32

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 9 12797The Jacobi polynomial of degree N has the power series as

follow:

Ja;bN x XNi0

1NigN;ixi (A e1)

The domain of x is in the range [0, 1].

The evaluation of coefficients is done by using the

following recurrence formula

gN; i

gN; i1 N i 1

i$N i a b

i b (A e2)

Starting with

gN;0 1 (A e3)

gN,i are constant coefficients, and a and b are parameters

characterizing the polynomials.

Lagrange Interpolation Polynomials

For a given set of data points (x1, y1), (x2, y2),.,(xN, yN) and

(xN1, yN1) an interpolation formula passing through all(N 1) points is an Nth degree polynomial. A suitable inter-polation polynomial for the orthogonal collocation method is

Lagrange interpolation polynomial, which passes through the

interior collocation points, roots of Jacobi polynomials, and it

is expressed as

yNx XN1j1

yjljx (A e4)

where yN is the Nth degree polynomial, yi is the value of y at

the point xi, and li(x) is defined as

lix YN1j 1jsi

x xj

xi xj

(A e5)

Furthermore,

lixj 0 isj

1 i j (A e6)

The first and second derivative at the interpolation points

are:

dyNxidx

XN1j1

dljxidx

yj (A e7)

d2yNxidx2

XN1j1

d2ljxidx2

yj (A e8)

For i 1;2.;N;N 1:The first derivative vector, composed of (N 1) first deriv-

atives at the (N 1) interpolation points is:

y0N dyNx1

dx;dyNx2

dx;.;

dyNxNdx

;dyNxN1

dx

T(A e9)

Similarly, the second derivative vector is defined as

y00N "d2yNx1

dx2;d2yNx2

dx2;.;

d2yNxNdx2

;d2yNxN1

dx2

#T(A e10)The function vector is defined as values of y at (N 1)collocation points asCp C1 C26664

3

T

sin h

C3T

7775 C46664

5

T

cos h

C5T

7775 (B e2)

where cp is in J/(kmol K) and T is in K [37].

To complete the simulation, extra correlations should be

added to the model. In the case of heterogeneous model,

because of transfer phenomena, the correlations for estima-

tion of heat and mass transfer between two phases should be

considered. It isbecauseof theconcentrationandheatgradient

betweenbulkof thegasphaseandthefilmofgasonthecatalyst

surface, which caused by the resistance of the film layer.

B.3. Mass transfer correlations

To flow through a packed bed, the correlation is given by the

following equation [38]:

kcidpDim

3

1 31g

um1 3g

1=2m

rDim

(B e3)y y1; y2; y3;.; yN; yN1T (A e11)By means of these definitions of vectors y and derivative

vectors, the first and second derivative vectors can be written

in terms of the function vector y using matrix notation

y0 A$yy00 B$y (A e12)

Where the matrices A and B are defined as

A aij

dljxidx

; i; j 1; 2;.;N;N 1

(A e13)

B (bij

d2ljxidx2

; i; j 1;2;.;N; N 1)

(A e14)

The matrices A and B are (N 1, N 1) square matrices.Once the (N 1) interpolation points are chosen, then all theLagrangian building blocks, li(xi), are completely known, and

thus the matrices A and B are also known [36].

Appendix B. Auxiliary Correlations

B.1. Gas phase viscosity

Viscosity of reactants and products is obtained from the

following formula:

m C1TC2

1 C3T C4T2

(B e1)

where m is the viscosity in Pa.s and T is the temperature in K.

Viscosities are at 1 atm [37].

B.2. Gas phase Heat capacity

Heat Capacity of reactants and products at Constant Pressure

is obtained from the following formula:where dp is particle diameter (m), 3b is void fraction of packed

bed, is shape factor of pellet, u is superficial velocity through

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 5 ( 2 0 1 0 ) 1 2 7 8 4e1 2 7 9 912798packed bed (m/s), is viscosity of gas fluid phase (kg/m s) and is

fluid density (kg/m3).

Diffusivity of component i in the gas mixture is given by

Ref. [39].

Dim 1 yi

P

yi=Dij (B e4)

The binary diffusivities are calculated using the Fullere

SchetterGiddins equation which is reported by Reid et al.[40]. In the following FullerSchetterGiddins correlation, vci,Mi are the critical volume andmolecular weight of component

i which are reported in.

Dij 107T3=2

1=Mi 1=Mj

1=2Ptv3=2ci v3=2cj

2 (B e5)

B.4. Heat transfer correlation

The heat transfer coefficient between the gas phase and solid

phase is obtained by the following correlation [41]:

hfcprm

cpmK

2=3 0:458

3b

rudpm

0:407(B e6)

where in the above equation, u is superficial velocity of gas

and the other parameters are those of bulk gas phase, dp is the

equivalent catalyst diameter, K is the thermal conductivity of

gas, r, m are density and viscosity of gas, respectively and 3 is

void fraction of catalyst bed. To see the constants which are

used in these equations please see reference [22].

Appendix C. Nomenclature

Parameter description

a catalyst activity, e

aij element of matrix A, e

A moles of aromatic formed, kmol h1

A matrix defined in Eq. (Ae12), e

Ac cross-section area of reactor, m2

B matrix is defined in Eq. (Ae12), e

bij element of matrix B, e

C concentration, kmol m3

Ci coefficient of Eqs. (B-1) to (B-2), e

Cj0 inlet concentration of component j, kmol m3

cp specific heat capacity, kJ kmol1 K1

dp particle diameter, m

De effective diffusivity, m2s1

Dim diffusivity of component i in the gas mixture, m2 s1

Ed activation energy of catalyst, J mol1

Ei activation energy for ith reaction, kJ kmol1

hf heat transfer coefficient, W m2 K1

HC hydrocarbon, kmol h1

H2 hydrogen, kmol h1

J jacobi function, e

keff effective thermal conductivity, W m1 s1

kci mass transfer coefficient for component i, m h1kf1 forward rate constant for reaction (1), kmol h1

kg cat1 M Pa1kf2 forward rate constant for reaction (2), kmol h1

kgcat1 MPa2

kf3 forward rate constant for reactions (3), kmol h1

kg cat1

kf4 forward rate constant for reactions (4), kmol h1

kg cat1

Ke1 equilibrium constant, MPa3

Ke2 equilibrium constant, MPa1

Kd deactivation constant of the catalyst, h1

L length of reactor, m

lj building block of jacobi polynomial, e

m number of data sets used, e

m number of reaction, e

mc mass of catalyst, kg

Mi molecular weight of component i, kg kmol1

Mw average molecular weight of the feedstock, kg kmol1

n average carbon number for naphtha, e

n number of component, e

N degree of Jacobi function, e

NA molar flow rate of aromatic, kmol h1

Ni molar flow rate of component i, kmol h1

p moles of paraffin formed, kmol h1

Pi partial pressure of ith component, kPa

P total pressure, kPa

Q volumetric flow rate, m3 s1

r radius, m

ri rate of reaction for ith reaction, kmol kg cat1 h1

R gas constant, kJ kmol1 K1

Ri inner radius of spherical reactor, m

Ro outer radius of spherical reactor, m

sa specific surface area of catalyst pellet, m2 kg1

t time, h

T temperature of gas phase, KTref reference temperature, KTR reference temperature, Kur radial velocity, m s

1

x variable represent length of reactor, m

yN jacobi polynomial of degree N, e

yi mole fraction for ith component in gas phase, e

yis mole fraction for ith component on solid phase, e

vc critical volume, cm3 kmol1

y0N first derivative of jacobi Equation, ey00N second derivative of jacobi Equation, ea characteristic parameter of Jacobin equation, e

b characteristic parameter of Jacobin equation, e

g coefficient of equation (Ae1), e

3 void fraction of catalyst bed, e

m viscosity of gas phase, kg m1 s1

vij stoichiometric coefficient of component i in

reaction j, e

r density of gas phase, kg m3

rB reactor bulk density, kg m3

s tensile stress, Nm2

fs sphericity, e

DH heat of reaction, kJ kmol1

a aromatic, e

cal calculated, e

h hydrogen, ei numerator for reaction, e

j numerator for component, e

naphtha reforming process on Aspen Plus platform. Chin J

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out outlet, e

p paraffin, e

ss steady state, e

T transpose, e

AF-SPBR axial-flow spherical packed-bed reactor, e

AF-TPBR axial-flow tubular packed-bed reactor, e

FBP final boiling pint, CIBP initial boiling pint, CLHSV liquid hourly space velocity, hr1

OF objective function, e

Pt platinum, e

RF-SPBR radial flow spherical packed-bed reactor, e

RF-TPBR radial flow tubular packed-bed reactor, e

Re rhenium, e

RON research octane number, e

Sph. spherical reactor, e

tub tubular reactor, e

TBP true boiling point, K

WHSV weight hourly space velocity, h1

r e f e r e n c e s

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Modeling of an axial flow, spherical packed-bed reactor for naphtha reforming process in the presence of the catalyst deact ...IntroductionLiterature reviewReaction scheme and kineticsProcess descriptionConventional configurationSpherical reactor setup

Reactor modelingAxial- flow spherical packed-bed reactor model (AF-SPBR model)Conventional tubular reactor model (CR model)Pressure drop (Ergun equation)

Catalyst deactivation modelNumerical solutionModel validationModel validation with the presence of catalyst deactivationSteady state model validation

Results and discussionChanges of parameters along the spherical reactorsChanges of parameters as a function of timeEffect of operating conditions and a comparison between tubular and spherical reactors in this regardThe effect of flow scale up ratio (FR)Effect of operating pressureThe effect of feed flow rate scale up ratioThe effect of temperature on the products yields

ConclusionAcknowledgementOrthogonal Collocation methodJacobi polynomialsLagrange Interpolation Polynomials

Auxiliary CorrelationsGas phase viscosityGas phase Heat capacityMass transfer correlationsHeat transfer correlation

NomenclatureReferences