This paper develops two mathematical models of a packed tubular reactor for methane steam reforming to produce hydrogen. The models generate 2D radial and axial plots of component concentrations and temperature over time. Both steady state and transient flow regimes are considered. The models consist of two coupled partial differential equations, one for material balance and one for energy balance, with initial and boundary conditions. The equations include terms for convection, diffusion, reaction, and heat transfer. Simplifying assumptions are made to reduce complexity, such as ignoring pressure drop and the water-gas shift reaction.

1. 
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. 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)

3. 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.ca
2
, 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,

4. 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)

5. (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

6. 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

7. 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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i
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