Abstract— This paper developed two mathematical models

of a packed tubular reactor loaded with reforming catalysts

which results in a simultaneous, nonlinear, two dimensional

PDE in cylindrical coordinates. It generates 2-D radial and

axial plots of components concentration and temperature.

Here both regimes of steady state and transient flow are

assumed. We also ignored the presence of WGS reaction and

analyzed the consequences by comparing the results with

experimental data.

Keywords—Catalytic reactor, Reformer modeling, methane

Steam reforming, Shazand Arak refinery, Hydrogen

production unit.

I. INTRODUCTION

Ethan steam reforming is one of the most common

methods to produce hydrogen at industrial scale

nowadays. It is a large scale operation carries out in 240

rows of plug reactors pass through a firebox of reformer. In

this process the reformer is charged with a mixture of CH4

and H2O at a molar ratio from 1:3 to 1:4 [i]. Although the

lower steam ratio could lead to a reduction in the life of

reformer tubes, economical restrictions limit our choices. At

unit 17 this ratio is 1:3.9[ii]. Reforming is extremely

endothermic and is limited by thermodynamic equilibrium.

Therefore, the development of catalysts significantly

increases the conversion of the reactants.

The temperature inside the tubes evolves from 545 ̊C to

860 ̊C and the pressure alters from 24.5 barg to 21.30 barg.

Indeed precise simulations requires vast knowledge and

kinetic and thermodynamic data of the process, Xu &

Froment [iii,iv] derived twenty-one sets of three rate

equations. They also examined the accuracy and credit of

Langmuir-Hinshelwood rate equation. Rostrup-Nielsen [v]

investigated the details of catalyst’s deactivation and

principles of coke formation. While taking all of the aspects

of the above methods into account leads us to theoretical

insolvable PDEs, logical results will appear by making

reasonable assumptions to reduce the complexity of

intrinsic kinetics.

In order to halt cracking reaction and to prevent the

formation of hot spot and overheated bands, reformer is

charged with two types of catalysts [i,ii]. While their shapes

are similar and their density values are close, their

MS student at Sharif University of technology, International campus

(E-Mail: arash.nasiris@gmail.com).

MS student at Babol Noshirvani University of Technology

(E-Mail: ahosseini.che.eng@gmail.com).

composition differs. One is a Nickel oxide dispersed on

calcium aluminate ceramic support catalyst, Katalco57-4Q

and the other is a Nickel oxide dispersed on calcium

aluminate ceramic support promoted by alkali catalyst,

Katalco25-4Q.

II. A BRIEF DESCRIPTION OF THE REFORMER SECTION

The treated feed vapor is mixed with a controlled quantity

of superheated process steam and is preheated in the

convection section of the reformer furnace (firebox).

The hot feed plus steam mixture is distributed over the

catalyst tubes of the reformer, where the hydrocarbons in

the feed gas are converted to hydrogen, carbon monoxide

and carbon dioxide in the presence of steam over the nickel

catalyst (Ni/Ca-Al2O3).

The produced gas, which leaves the reformer, is essentially

a mixture of hydrogen, carbon monoxide, carbon dioxide,

methane and steam, which will make the equilibrium to

approach a certain degree. The corresponding (dry gas)

methane concentration is generally referred to as ‘methane

slip’ at the outlet temperature.

(1). CnHm+nH2O nCO + (n+m/2) H2

(2)WGS reaction CO+H2O CO2 +H2

Besides the above-mentioned reactions there are a number

of side reactions, which are not desirable:

(3)Boudouard reaction: 2CO C+CO2

(4)CO reduction: CO+H2 C+H2O

(5)Methane Cracking CH4 C+2H2

These reactions are suppressed by applying an excess of

steam, so that eventually formed carbon will be removed by

the reversed CO reduction reaction (4). A low steam to

carbon ratio can lead to carbon deposition and thus catalyst

damage.

The overall heat effect of the steam reforming reactions is

strongly endothermic i.e. heat has to be supplied externally

to achieve the required conversion. This heat is provided by

combustion of PSA purge gas as the priority fuel and

natural gas as the make-up fuel.

III. MATHEMATICAL MODEL AND EQUATIONS DERIVATION

I. ASSUMPTIONS AND VARIABLES

The following assumptions were considered:

1. Unsteady State operation

Simultaneous, nonlinear, 2D modeling of tubular reactor of CH4

reforming in H2 production unit of Shazand Arak refinery

Arash Nasiri Savadkouhi, Seyed Adel Hoseini

M

2. Non-iso thermal condition.

3. Plug flow on reaction zone.

4. Reaction progress in circumferential direction is

ignored.

Since the maximum ratio of the lateral perimeter to

height is 2(π) (5.65)/ (1200) =0.0295

5. All the reactants are gaseous and follow the ideal gas

equation.

Since CH4 is non-polar and the temperature is also

high enough.

6. The First law of Fick is utilized with the constant

diffusion coefficient.

7. The conduction heat rates evaluated from Fourier’s

law

8. Heat transfer by radiation is neglected.

Since the temperature inside is less than 1000 ̊C [vi]

9. Pressure drop through the tubes is ignored.

While ∆Preal=3.2 bar

10. We considered that the reformer is loaded by a

uniform catalyst.

11. WGS reaction is ignored.

12. Although the true shape of catalyst is shown below

to simplify the calculations we consider one catalyst

as four simple cylindrical catalysts.

Fig 1. True shape of catalyst and the assumed shape.

II. Governing Equations

We start with the derivation of the PDEs. The physical

system is a convection–diffusion–reaction system, for

example, a tubular reactor, which is modeled in cylindrical

coordinates, (r, z) (we assume angular symmetry so that the

third cylindrical coordinate, θ, is not required). The PDE

system has two dependent variables, ca and Tk, which

physically are the reacting fluid concentration, ca, and the

fluid temperature, Tk. Thus we require two PDEs.

These two dependent variables are functions of three

independent variables: r, the radial coordinate; z, the axial

coordinate; and t, time, as illustrated by Figure 2. The

solution will be ca(r, z, t) and Tk(r, z, t) in numerical form as

a function of r, z, t.

Fig 2.Represention of a convection-diffusion-reaction system in a plug

reactor

Fig 3. Precise view of the reactor charged with catalysts.

The mass balance for an incremental element of length

∆z is:

(1)

Division by 2πr∆r∆z and minor rearrangement gives:

(2)

Or in the limit r →0, ∆z→ 0,

We may briefly discuss the physical meaning of each

term in this PDE. For the differential volume in r and z,

As mentioned before we now assume Fick’s first law for

the flux with a mass diffusivity, Dc

(1)

We obtain

Or expanding the radial group,

(2)

Equation (2) is the required material balance for ca(r, z, t)

and would be sufficient for the calculation of ca(r, z, t)

except we consider the case of an exothermic reaction that

liberates heat so that system is not isothermal. That is, the

reacting fluid temperature varies as a function of r, z, and t

so that a second PDE is required, an energy balance, for the

computation of the temperature Tk (r, z, t). Further, Eq. (2)

is coupled to the temperature through the reaction rate

coefficient kr as it changes dramatically by changes in

temperature.

Also note that the volumetric rate of reaction in Eq.

(2),kr.ca2, is second order; that is, the rate of reaction is

proportional to the square of the concentration ca .This

kinetic term is therefore a source of non-linearity in the

model.

The thermal energy balance for the incremental section is

Divided by 2πr.∆r.∆z.ρ gives

Or in the limit r →0, ∆z→ 0,

Briefly, the physical meaning of each term in this PDE for

the differential volume in r and z is:

Also, there is one technical detail about this energy

balance which we may consider. The derivative in t is based

on the specific heat Cp reflecting the fluid enthalpy. More

generally, an energy balance is based on the specific heat

Cv reflecting the fluid internal energy. For a liquid, for

which pressure effects are negligible, the two are essentially

the same, although our system is gaseous, we assumed that

P is constant therefore we use Cp instead.

If we now assume Fourier’s first law for the flux with a

thermal conductivity, k,

(3)

We obtain (after its division by Cp)

Or expanding the radial group, (4)

Where Dt =k/ρCp. Equation (4) is the required energy

balance for T(r, z, t).

The reaction rate constant, kr, is given by

(5)

Note from Eq. (5) that kr is also nonlinearly related to the

temperature Tk as e−E/RTk. This Arrhenius temperature

dependency is a strong nonlinearity, and explains why

chemical reaction rates are generally strongly dependent on

temperature; of course, this depends, to some extent, on the

magnitude of the activation energy, E. The variables and

parameters of equations (1)–(5) and the subsequent

auxiliary conditions are summarized in Table 1 (in cgs

units).

Equation (2) requires one initial condition (IC) in t (since it

is the first order in t), two boundary conditions (BCs) in r

(since it is the second order in r), and one BC in z (since it

is the first order in z).

(6)

(7)

(8)

(9)

Equation (4) also requires one IC in t, two BCs in r, and one

BC in z.

(10)

(11)

(12)

(13)

The solution to equations. (1) – (13) gives ca(r, z, t), Tk(r, z,

t) as a function of the independent variables r, z, t.

Now assuming steady state flow, the requisite equations for

the model can be gained presuming that the reaction is the

first order one with respect to CH4.

One of the difficulties of simplifications is the presence

of rate coefficient of reaction in equations since it follows

LHHW model which is Langmuir–Hinshelwood–Hougen–

Watson (LHHW) kinetic; it was proposed with nine

parameters, where the surface decomposition of methane is

assumed as the rate determining step (RDS), and while all

other reaction steps are set as reversible [vii]. Fortunately

kinetic scheme of Hougan-Watson can be easily replaced by

the first-order kinetics. Besides while k is generally

considered as a function of temperature, here it is assumed

to be constant. Although Agnelli et.al [viii] disclosed three

different values of k, KA, KB at various temperatures for

diverse rate models, they stipulated that regression

techniques can be utilized to yield values of k. They also

approved the accuracy of the first order reactions with

respect to CH4.

The overall reactions considering the enthalpies are:

CH4+H2O CO + 3 H2 ∆H298 k=206 KJ/mol

CO+H2O CO2 + H2 ∆H298 k= -41 KJ/mol

CH4+2H2O CO2 + 4 H2 ∆H298 k=165 KJ/mol

Rate equations utilized here are gained from the results of

Xu & Froment researches. The mechanism of adsorption

and desorption contains thirteen steps while only three of

them limit the rate of reaction.

(14)

(15)

(16)

(17)

Table 1. variables and parameters of the 2D PDE

model, unsteady state

Variable/Parameter Symbol

Reactant concentration Ca (r, z, t)

Temperature Tk (r, z, t)

Reaction rate constant ki

Time t

Radial position r

Axial position z

Mass diffusion flux qm

Energy conduction flux qh

Reactants and Products

Entering concentration Cae

Entering temperature Tke

Initial concentration Cao

Initial temperature Tko

average wall temperature Tw

Reactor radius R0

Reactor length Z1

Linear fluid velocity υ

Mass diffusivity Dc

Thermal diffusivity Dt=k/ρCp

Fluid density ρ

Fluid specific heat Cp

Heat of reaction (948 K) ∆H

Specific rate constant K0

Activation energy(948 K) E

Gas constant R

Thermal conductivity k

Heat transfer coefficient h

Table 1. continue

Units/Value None

gmol/cm3 None

K None

cm3/(gmol.s) None

s None

cm None

cm None

gmol/(cm2.s) None

cal/(cm2.s) None

References CH4 H2 None

Calculated based on [ii] 0.3 0.3 None

[ii] 818.15K

- 0 gmol/cm3

[ii] 818.15K

Calculated based on [ii] 1123K

[ii] 5.65cm

[ii] 1200cm

Calculated based on [ii] cm2/s 2.88 0.881

Calculated based on[ix] cm2/s 11.8 4379.4

[x] None

[ii] 0.00676 g/cm3

[ii] 0.62 cal/g.K

[vi] 53536 cal/(gmol)

calculated based on[xi] 4.798 cm3/(gmol.s)

[vi] 57383.9 cal/(gmol)

- 1.987 cal/(gmol.K)

calculated based on[x] 0.000408(cal.cm)/(s.cm2.K)

calculated based on[x] 0300..(cal)/(s.cm2.K)

(18)

(19)

IV. RESULTS

Down below the results of solving the PDEs with using

MATLAB for both the steady and unsteady flow is

illustrated.

Fig 3. T gradient at first second. Qnormal operation=8977

Fig 4. T gradient through the length of reformer Qnormal operation=8977

Table 2. variables and parameters of the steady state model

Names Variables

Methane A

Water B

Carbon Monoxide C

Hydrogen D

Entrance Temperature T

Cross sectional area of reactor S

Specific heat of component i Cpi

Reaction rate expression rA

Catalyst bulk density Ρb

Heating inside reactor q

Partial pressure of component i Pi

Total pressure PT

Molar flow rate component i ni

Total molar flow rate nT

Equilibrium constant K(T)

Constant [viii] KA

Constant [viii] KB

Reactor length z

Table 2. continue

Values Units

1 Molar rate of x/molar rate of methane

3.51 “

0 “

.00983 “

818 Kelvin

380 cm2

None cal/(g mole. kelvin)

None g mole/ (volume of catalyst.second)

0.860 gram catalyst/ volume of catalyst

None cal/(second. cm)

None KPa

2449.4 KPa

None g mole/ sec

None g mole/ sec

0.45 -

None -

None KPa-1

1200 cm

Fig 5. T gradient through the reformer Qmin=175 J.m-1.min-1

Fig 6. Products production rates

Fig 7. Reactants consumption rates

Fig 8.surface plot of methane concentration at t= 200s

Fig 9. Concentration and temperature gradient at t =150s and t=200s

Fig 9. Concentration and temperature changes at t=200s and t=150s

Fig 10. Surface plot of methane temperature at t=200s

Fig 11. Concentration and temperature gradient at t =50s and t=100s

V. RESULTS AND CONCLUSION

We can perceive the following notes from the plots:

Since the reaction is extremely endothermic, at the entrance

of the reactor the temperature may decrease instantly but it

will rise radically since the heat flux is extremely high. This

can be understood by looking into figure 3.

Figure 4 predicts the outlet temperature of the reactor. The

model estimates its order value 917 ̊C while the true outlet

temperature is 860 ̊C. This 57 ̊C difference highlights the

truth that ignoring WGS reaction, was a reasonable

assumption since it only affects the models results from less

than ten percent in comparison with real data.

Figure 5 illustrates the outlet temperature of the reactor

when the furnaces are working by its 2% of normal

operational load. As it is illustrated, if the reformer works at

this ratio, the outlet temperature will reach to 818 ̊C which

means that all the generated heat by furnace is used to

proceed the reaction, while the reformer works normally by

a ratio fifty one times larger in order to maximize the

production. Moreover, the increase of heat flux helps

halting WGS reaction. Figure 6 estimates the outlet

methane to be approximately zero while based on unit PFD,

the ratio of (rate of methane in outlet stream)/ (rate of

methane in inlet stream) is 0.19, the difference however can

be related to omitting the probable reactions in the first

assumptions.

Figures 8 and 10 show clearly the radial variation of

concentration and temperature along the axial length of the

reactor at time t =200, when a steady-state profile is going

to be reached. The radial variation of concentration is

relatively small however the temperature Tk (figure 10)

shows dramatic increase through the length of reactor.

There is a significant difference in the results of

temperature gradient graphs based on two different steady

state and transient flow models. While the steady state

model predicts that the components may reach the final

temperature of 980 ̊C when it leaves reformer, the transient

flow model predicts that the components reach the final

temperature less than 50 seconds. The reason for this

noticeable difference is related to the ignorance of

convection mechanism of heat transfer in steady state flow.

The same conclusion can be derived for concentration plots.

We can assume that the concentration would be slightly

different from those of being plotted. Since the term

diffusivity may not always be appropriate (particularly for a

high-Reynolds-number situation) in the sense that the radial

dispersion in a reactor or similar physical system is

probably not due solely to molecular diffusion (which is

generally a small effect) as much as stream splitting and

flow around the internal obstructions known as eddy

diffusion. Thus the term dispersion coefficient might be

more appropriate. Also, dispersion coefficients are

frequently measured experimentally since they typically

reflect the effects of complex internal flow patterns.

Eventually it seems that the presented models can generate

reasonable results. Although some magnificent reactions

ignored on purpose, the results approved that through the

reaction conditions, prepared by reforming, the main

progressive reaction is methane reforming.

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