Computers and Chemical Engineering 152 (2021) 107379

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Computers and Chemical Engineering

journal homepage: www.elsevier.com/locate/compchemeng

Impact of kinetic models in the prediction accuracy of an industrial

steam methane reforming unit
P.P.S. Quirino a, A. Amaral b, K.V. Pontes a, F. Rossi c, F. Manenti b,∗
a
Industrial Engineering Graduate Program (PEI), Federal University of Bahia (UFBA), Rua Professor Aristides Novis 02, 40210-630, Bahia, Brazil
b
Dipartimento di Chimica, Materiali e Ingegneria Chimica “Giulio Natta” Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
c
Purdue University, Forney Hall of Chemical Engineering, West Lafayette, IN, 47907, United States

a r t i c l e i n f o a b s t r a c t

Article history: Catalyzed chemical reactions inside tubular reactors play a key role in steam methane reforming process,
Received 25 January 2021 as they define the temperature and composition profiles of the reformer and, consequently, its efficiency.
Revised 1 May 2021
Although the kinetic modeling of reforming reactions has already been widely discussed, studies that
Accepted 20 May 2021
carry out the comparison between kinetic reform models are still scarce. In this context, this work aims
Available online 25 May 2021
to evaluate the influence of different kinetics on the stationary behavior of industrial reformer. The ki-
Keywords: netic models of Singh and Saraf (1979), Numaguchi and Kikuchi (1988), Xu and Froment (1989a), and
kinetic model Hou and Hughes (2001) were analyzed and compared with each other. All models describe the reformer
steam methane reforming satisfactorily with relative deviations around 3.5% compared to experimental and literature data. The re-
industrial reformer sults show that the simplest kinetic models present advantages compared to traditional models, such as
Xu and Froment (1989a), which is widely used without any prior analysis.
© 2021 Elsevier Ltd. All rights reserved.

catalyst. Furthermore, more accurate predictions of the maximum

values of the tube wall temperature allow preventive maintenance
1. Introduction and, as a result, an increase in the equipment lifetime. Therefore,
the kinetic modeling is the key to a reliable mathematical model
The steam methane reforming (SMR) is the main catalytic route of the reformer.
for synthesis gas production due to its greater efficiency compared Although the kinetic modeling of reform reactors has been
to other concurrent processes, such as autothermal, dry and par- vastly studied, the literature still lacks a more comprehensive
tial oxidation reforming (Rostrup-Nielsen et al., 2002; Mbodji et al., analysis of the influence of the kinetic model on the over-
2012; Ghouse et al., 2015; Balzarotti et al., 2019; Leonzio, 2019). all performance of the reforming process. The works attempt-
The industrial steam reforming units are composed of different ing to compare kinetic models are scarce and mostly applied
equipment, among which the main and most expensive is the re- to other types of reactors: adiabatic fixed-bed reactors mod-
former, where highly endothermic reforming reactions take place. els, where synthesis gas is produced by partial oxidation of
Such reactions significantly influence the temperature and compo- methane (De Smet et al., 2001); micro-scale catalytic reactor
sition profiles and, consequently, the yield and quality of the syn- (Vaccaro and Malangone, 2012); quartz tubular reactor for CO pro-
thesis gas, as well as the fuel gas consumption. duction (Vidal Vázquez et al., 2017); integrated membrane re-
The mathematical model of the reformer allows a better un- actor for H2 production (Leonzio, 2016). These studies are lim-
derstanding of the behavior of the reforming reactions inside the ited to analyzing only the reactor and do not compare the ki-
catalyst-filled tubes. Further sophistication in modeling the heat netic models investigated herein. In steam industrial reformers,
transfer or diffusion mechanisms, though, might not overcome though, a joint assessment of the catalyst-filled tube and the fur-
model mismatches due to a poor representation of the kinetic nace is essential due to the heat transfer mechanisms between
mechanism. A more precise estimate of the reaction rates allows them. There is still much to know about the influence of reform-
a more accurate prediction of the heat required by combustion, ing reaction kinetics on the tube and on the furnace tempera-
avoiding overheating the reactor and consequently damaging the ture and composition profiles. This work, then, tries to fill this
gap in the literature through comparison of the kinetic models of
∗ Singh and Saraf (1979), Numaguchi and Kikuchi (1988), Xu and Fro-
Corresponding author.
E-mail address: polipastorele@gmail.com (F. Manenti). ment (1989a), and Hou and Hughes (2001), against experimental

https://doi.org/10.1016/j.compchemeng.2021.107379
0098-1354/© 2021 Elsevier Ltd. All rights reserved.
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Nomenclature i control volume chemical species

Superscript
Main Symbols
c catalyst
A absorptivity factor - parameter of the equilibrium
env environment
constant -
f furnace
K adsorption constant - thermal conductivity
g reforming gas
W.m−1 .K−1
r refractory
Z axial coordinate m
t tube
R rate of reaction kg.m−3 .s−1
R chemical reaction rate kmol.m−3 .s−1 ideal gas con-
stant m3 .kPa.kmol−1 .K−1 data from literature and industry. This comparison is performed
h convective coefficient W.m−2 .K−1 with the use of a steady-state model of a steam methane reform-
C pressure conversion factor kPa2 .atm−2 ing unit (SMRU), Our main motivation is to evaluate the quality
D diameter m of fit of different kinetic models, identifying, through simulations,
K equilibrium constant - which one is the most assertive and consistent with the SMRU. Un-
V velocity m. s−1 like the previous studies which focus only on the reactor to com-
Q perimetric heat term J. m−1 pare the kinetic models (De Smet et al., 2001;Vaccaro & Malan-
K kinetic constant - gone, 2012; Vidal Vázquez et al., 2017), the current approach relies
W mass fraction - on a comprehensive model of the SMRU, which considers not only
F molar flow rate kmol. s−1 the catalyst-filled tube (CFT), but also the tube wall, the furnace,
G mass flux kg. s−1 . m−2 and the refractory (Quirino et al., 2020).
y molar fraction - The paper is organized as follows. Section 2 reviews the dif-
M molar weight kg. kmol−1 ferent kinetic models used to describe steam methane reforming
H molar enthalpy J. kmol-1 reactions. The next section describes the case study, which is an
SMRU height m industrial reformer. Section 4 presents the steady-state mathemat-
J. kmol−1 m ical model of this equipment, which will be used for comparison
L SMRU length m purposes of the kinetic models investigated in this work. In ad-
W SMRU width m dition, the section 4 presents the experimental data necessary for
A pre-exponential factor - the model solution and validation. In section 5, the predicted tem-
area m2 perature and composition profiles, considering the different kinetic
B parameter for calculating the equilibrium constant - models, are evaluated against experimental data. This paper ends
P pressure Pa with the final conclusions (section 6).
 perimeter m
R radius m radial coordinate m rate of reaction 2. Kinetic models for the reforming reactions
kg.m−3 .s−1
E activation energy J. kmol−1 Some authors describe the reforming process through two re-
cP specific heat J. kg−1 .K−1 actions, namely the steam reforming reaction of methane (R1) and
H adsorption energy J. kmol−1 reaction enthalpy the water-gas shift reaction (R2), whereby CO can be converted to
J. kmol−1 CO2 (and vice-versa). Other authors further consider reaction (R3),
T temperature K here dubbed ‘shift reforming’, which is the sum of the previous
δ thickness m two reactions. These reactions are shown in Table 1. Other reac-
δij Kronecker delta m tions, such as carbon formation, occur less frequently and are often
neglected, since they would make the kinetic model too complex
Greek Letters for practical purposes.
λ correction factor of the heat radiation - Experimental kinetic data are commonly used to develop
ρ density kg. m−3 new kinetic reaction rate expressions based on simple power
μ dynamic viscosity Pa.s laws or more complex expressions. Such models try to ex-
η effectiveness factor - plicitly represent the interaction of reactant species with the
ε emissivity - catalyst through Langmuir-Hinshelwood (LH) or Hougen-Watson
β heat partition coefficient - (HW) adsorption theories. Different hypotheses are assumed re-
α parameter of the equilibrium constant - garding the controlling steps, the dependence of the reaction
φ porosity - rate on steam and hydrogen concentration as well as on the
σ Stefan–Boltzmann constant s−1 .m−2 . K−4
υ stoichiometric coefficient -
Table 1
Main reactions used to describe the steam methane reforming pro-
Subscripts cesses.
eq equivalent composition temperature after hydroc-
racking Reaction Chemical H298K Reaction
name formula (kJ/mol) number
0 composition/temperature before hydrocracking
out output condition Steam CH 4 + H 2 O  206 (R1)
in inner tube wall reforming CO + 3H2
Water-gas CO + H2 O → ← −41 (R2)
ext outer tube wall shift CO2 + H 2
k reforming reactions number Shift CH 4 + 165 (R3)
j heat transfer phenomena reforming 2H2 O 
CO2 + 4H 2

2
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Table 2
Kinetic models proposed in the literature for modeling the steam reforming process.

Reference - Singh and Saraf Numaguchi and Xu and Froment Hou and Hughes
(1979) Kikuchi (1988) (1989a) (2001)

Year - 1979 1988 1989 2001

# of References - 52 117 1203 449
Model Type - PL LH / F LH LH / F
Temperature - Min K 644 674 573 598
Temperature - Max K 1317 1160 848 823
Pressure - Min Bar 14 1.2 3 1.2
Pressure - Max Bar 35 25.5 15 6
S/C ratio - Min % 384 144 300 400
S/C ratio - Max % 673 450 500 700
H2 / CO ratio - Min % - - 50 50
H2 / CO ratio - Max % - - 100 75
Catalyst - Nickel wt % ? 8.7 15.2 11.8 - 13.4
Catalyst - Support - Al2 O3 Al2 O3 MgAl2 O4 α -Al2 O3
Catalyst - SSA m2 / g ? ? 9.3 14.3
Catalyst - BD kg / m3 1,200 - 1,400 ? 1,870 1,790
Catalyst - Type - Multiple Cylindrical Crushed Crushed
Catalyst - Size Mm 6 - 16 6 - 20 0.18 - 0.25 0.12 - 0.18
Catalyst - DL - YES YES NO NO
Ri = αi ki πi Ref. Singh and Saraf Numaguchi and Xu and Froment Hou and Hughes
(1979) Kikuchi (1988) (1989a) (2001)
PCH4 P0H.5O
π1 = PCH4 PH2 O − PCO P3H2 /∗∗∗1 α1 1
P0.5 PH2 O
1
P1H.596
1
P2H.5 D2 P1H.25 D2
2

2O 2 2
PCO P0H.5O
π2 = PCO PH2 O − PCO2 PH2 /∗∗∗2 α2 1
P0.5 PH2 O
1
PH2 O
1
PH2 D2
2
P0H.5 D2
2
PCH4 PH
π3 = PCH4 P2H2 O − PCO2 P4H2 /∗∗∗3 α3 0 0 1
P3H.5 D2 P1H.75 D2
2O

2 2
PL: power-law; LH: Reference D
Langmuir-Hinshelwood; F: Xu and Froment 1 + KCH4 PCH4 +
Freundlich; S/C: steam / carbon (1989a) KCO PCO +
molar ratio (@ reactor inlet); KH2 PH2 +
BD: bulk density; SSA: specific KH2 O PH2 O /PH2
surface area; ?: information not Hou and Hughes 1 + KCO PCO +
available; DL: diffusion limited. (2001) KH2 P0H.25 +
KH2 O PH2 O /PH2

species adsorption on the catalyst. Although a plenty of mod- Studies on reforming reactions show that there is no agreement
els have been proposed to describe the reforming reactions regarding kinetic parameters, such as activation energies, which
(Akers and Camp, 1955; Moe and Gerhard, 1965; Bodrov et al., have quite different values among the kinetic models reported in
1964, 1967,1968; Ross and Steel, 1973; Al-Ubaid, 1984; Allen et al., the literature. This is due to the different catalysts and experi-
1975; Singh and Saraf, 1979; De Deken et al., 1982; Numaguchi and mental conditions used and, above all, it results from the lack of
Kikuchi, 1988; Xu and Froment, 1989a; Hou and Hughes, 2001), a more improved analysis of the heat and mass diffusion limita-
four kinetic models are most commonly applied. Table 2 summa- tions. Numaguchi and Kikuchi (1988), for example, performed sim-
rizes the mathematical expressions, the operating conditions used ulations with a model of an integral fixed-bed reactor to compare
in the experiments and the most important assumptions adopted intrinsic and apparent kinetic rates and to identify their influence
for the formulation and validation of these four kinetic models. from the design viewpoint. Based on mass and heat balances con-
The model proposed by Singh and Saraf (1979) - (SS) is sidering interphase and intraphase diffusions, the kinetic param-
based on reforming experiments performed with hydrocarbons eters of the intrinsic rates of the reactions R1 and R2 were esti-
heavier than methane, such as propane, butane, and hexane. In mated to fit the measured outlet composition. The authors further
these experiments, though, all hydrocarbons are rapidly hydroc- investigated the apparent kinetic rates which do not consider the
racked to methane, as found out by most studies in the literature interphase and intraphase diffusions. The axial profiles of the in-
(Hyman, 1968; Latham et al., 2011). The SS model, then, makes trinsic and apparent reaction rates, facing changes in the reactor
exclusive reference to methane as hydrocarbon. Singh and Saraf inlet temperature, proved to be quite different, although the outlet
(1979) consider that the shift reforming reaction R3 is promptly concentrations only differ by 5%. In addition, the apparent activa-
completed at the reformer inlet, as Numaguchi and Kikuchi (1988), tion energy obtained was approximately half of the intrinsic acti-
then only the reactions R1 and R2 are considered. On the other vation energy, which shows that the diffusion on the catalyst sur-
hand, Xu and Froment (1989a) and Hou and Hughes (2001) fur- face has a significant effect on the values attributed to the kinetic
ther consider the shift reforming reaction R3. Singh and Saraf parameters. The authors concluded that, at high inlet temperatures
(1979) adapted the kinetic model from Haldor Topsoe (1965) and or high steam/carbon ratios, the reactor can be over-designed if the
Hyman (1968), which relies on power laws, assuming that pore dif- diffusion effects are not considered, i.e., if apparent kinetic reaction
fusion resistance for reforming reactions is very high. They addi- rates are used.
tionally considered the influence of the total pressure on the re- The NK model is usually reported in process conditions closer
action rates, according to Ruthven (1969), and adjusted the kinetic to those employed by Numaguchi and Kikuchi (1988), such
parameters to better fit experimental data. The SS kinetic model as autothermal reforming reactors, where there is partial ox-
could reproduce experimental data of reforming processes reason- idation of methane (De Smet et al., 2001; Shi et al., 2008;
ably well, despite its simplified power-law structure (Rydén and Azarhoosh et al., 2015). These reactors require higher pressures, in
Lyngfelt, 2006; Vakhshouri and Hashemi, 2008; Shayegan et al., cases of methanol production, and tend to operate at higher tem-
2008). peratures, due to the exothermic character of the combustion re-

3
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

action, which occurs when oxygen or air is injected into the re- hydrocarbons that are rapidly converted to CO and H2 . In addition,
forming tubular reactor. Although the NK model has proved to be this component is considered necessary in the reactor feed to pro-
reasonable for these processes, it is necessary to be careful with tect the Ni catalyst from oxidation by steam, and/or to prevent car-
its use, as Numaguchi and Kikuchi (1988) performed only five ex- bon formation on the catalyst surface. When the XF kinetic model
periments to obtain the kinetic model. Therefore, the conclusions is used in the SMRU model, some numerical instabilities are ob-
reached by these authors may not be reliable and the application served during integration, despite the presence of hydrogen in the
of the NK model may result in inaccurate responses, even in sit- stream feeding the tubular reactor. This is probably due to its small
uations with diffusional limitations. Despite being frequently used amount in the gas mixture. Thus, we established a minimum tol-
in autothermal reform processes, the NK model was developed for erated value for hydrogen to avoid this problem.
reforming processes in general and will therefore be tested in this Among the kinetic models available in the literature, the model
work. proposed by Xu and Froment (1989a) stands out, being widely
To explain the adsorption of reagents on the catalyst surface, used in both academia and industry. Considered as the most gen-
the kinetic models from Numaguchi and Kikuchi (1988)- (NK), eral and representative of reforming reactions, this model is com-
Xu and Froment (1989a)- (XF) and Hou and Hughes (2001) - (HH) monly used without prior comparison with other kinetic mod-
consider the adsorption model proposed by Langmuir – Hinshel- els. Xu and Froment (1989a), though, performed their experiments
wood (LH). NK and HH further suggest combining LH theory with at much lower temperatures than those practiced in industry, as
Freundlich’s concept of non-ideal adsorption. The NK model sub- listed in Table 2, to avoid reaching the kinetic equilibrium. In SM-
stantially reduces the structure of their kinetic model considering RUs, the typical pressure is 15 to 35 bar (Rostrup-Nielsen, 1975;
that the adsorption equilibrium constant of steam is considerably Twigg, 1989; Latham, 2008; Copenor, 2014; Abid and Jassem, 2014)
larger than that of other components. Xu and Froment (1989a) and and the temperature of the reactants at the inlet of the catalyst-
Hou and Hughes (2001) consider that all reactions occur at the filled tube is 723K to 923 K, and the products leave at a tem-
same catalyst active sites; therefore, the reaction rates of each perature of 1073 K to 1223K, depending on the application of the
model have the same denominator, which acts as an inhibitor of synthesis gas (Rostrup-Nielsen et al., 2002; Ferreira-Aparicio et al.,
the reaction rates. The model from XF considers the adsorption 20 05; Basini, 20 05; Vasconcelos, 20 06). Thus, temperature and
of all species on the catalyst surface, except carbon dioxide since pressure conditions, where the model is validated, may substan-
its adsorption is negligible compared to other species. The model tially influence the model predictions, resulting in misleading con-
from HH neglects not only the adsorption of carbon dioxide, but clusions. It raises the question if another kinetic model, developed
also methane due to the high steam concentration which hinders at temperature and pressure conditions like that of the industrial
the adsorption of methane on the catalyst surface, particularly at reformer, would be more representative of the industrial data than
high temperatures. Hou and Hughes (2001) came to the kinetic the XF model.
model in Table 2 after adjusting the model by trial and error until
all terms have physical meaning. 3. Process Description
The kinetic models in Table 2 also differ on the order of reac-
tion regarding the partial pressure of steam and hydrogen. In the The case study of this work consists of an industrial steam
NK model, the dependence on steam is of negative order, proba- methane reforming unit (SMRU) from Northeastern Petrochemical
bly due to the competitive adsorption between methane and steam Company – COPENOR (Camaçari – Brazil), shown in Figure 1. The
in the active sites of the catalyst. Both XF and HH models have a natural gas, steam and CO2 flows are previously mixed, and the re-
negative order for H2 , consequently the reaction rates tend to in- sulting supply stream is distributed between the tubular reactors.
finite when hydrogen inlet concentration approaches zero, which At the outlet of this reactors, synthesis gas (reformed gas) is ob-
can lead to numerical instabilities. The orders of H2 in the XF tained, whose main components are H2 , CO and CO2 . The energy
model have a higher absolute value than in the HH model, then required for the reforming reactions comes from burning the fuel
the XF model is more sensitive regarding the H2 partial pressure. gas with the combustion air in the furnace burners. Part of the
To avoid numerical instabilities, some authors prefer to use other energy transferred to the tube is irradiated to the furnace and to
models, such as those proposed by SS and NK, or to define a neg- the refractory and the other part is conducted to its interior, by
ligible amount of H2 in the reactor feed, if XF and HH models are convection and radiation. Similarly, part of the energy transferred
used. Xu and Froment (1989a) affirmed, however, that their model to the refractory is lost, by convection, and the other part is ir-
is adequate and may represent most industrial reformers, because radiated to the furnace and to the tube wall. Table 3 summarizes
the reforming tube feed is usually composed of hydrogen or higher the two sets of industrial data (Copenor 1 and Copenor 2) used

Table 3
Tube-side and Furnace-side input conditions used for Steam Methane Reforming simulations (Copenor, 2014). RG: reforming gas.

Case - Copenor 1 Copenor 2

Stream - RG Fuel Air RG Fuel Air

T0 K 646.15 412.88 545.15 645.63 412.86 545.15

P0 bar 21.61 2.16 1.03 21.61 2.16 1.03
m˙ kg/s 7.29 0.95 13.29 7.47 1.03 14.65
Composition (mass fractions)

CH4 % 16.05 60.29 - 16.77 61.85 -

C2 H6 % 2.96 7.60 - 3.09 8.17 -
C3 H8 % 0.21 0.50 - 0.22 0.55 -
CO2 % 12.56 17.10 0.04 12.63 15.97 0. 04
CO % 0.05 4.59 - 0.04 4.23 -
H2 O % 67.87 - 0.6 66.94 - 0.6
H2 % 0.06 5.12 - 0.05 4.72 -
O2 % - - 23.10 - - 23.10
N2 % 0.25 4.80 78.26 0.26 4.51 78.26

4
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Figure 1. Schematic flow diagram of the industrial SMRU (Steam Methane Reforming Unit) and heat transfer mechanisms.

in the comparative analysis of the kinetic models. This table shows ances, using the kinetic model available in Tran et al. (2017). In ad-
the operational conditions of the streams fed to the reforming tube dition, for the study of radiation, the authors consider four control
and the furnace burners. volumes in the reformer (tube, tube wall, furnace, and refractory).
The results of this work showed that it is not necessary to subdi-
4. Mathematical Model vide the furnace into more control volumes according to the spatial
position, as proposed by Hottel et al. (1967) and commonly seen
The stationary phenomenological model of the steam reform- in the literature (Olivieri and Vegliò, 2008; Zamaniyan et al., 2008;
ing industrial furnace, used in this work and developed by Latham et al., 2011; Ebrahimi et al., 2013; Kumar et al., 2017). Labo-
Quirino et al. (2020), considers four control volumes: (i) the rious calculations to determine the radiation heat exchange areas
catalyst-filled tube, (ii) the tube wall, (iii) the furnace and (iv) the would be required in this type of approach, which would make the
refractory. The streams fed to the tube and the furnace are com- model by Quirino et al. (2020) too complex. This is not desirable
posed of different components, within which light hydrocarbons, since the model was developed for practical purposes, in order to
such as methane, ethane, and propane, as indicated in Table 3. It help the operator in monitoring the temperature and composition
is then assumed that alkanes higher than methane in the process profiles of the industrial SMRU.
gas are rapidly hydrocracked at the inlet of the tubular reactor, as Quirino et al. (2020) still investigate in detail the heat trans-
suggested by Latham et al. (2011). Table 4 summarizes the equa- fer between the control volumes to understand its effect on the
tions to compute the new equivalent streams after the hydrocrack- model’s prediction. Some phenomena usually despised in the liter-
ing at the tubes inlet. The temperature of the equivalent stream is ature are considered, such as the energy reflected by the refractory
calculated by considering that enthalpy is conserved in the trans- and the tube wall, the radiation heat transfer from the tube wall
formation (which is true for an isobaric process in the absence of to the process gas and the convection heat transfer from the fur-
friction and shaft work). nace to the tube wall and the refractory. This model was validated
The model from Quirino et al. (2020), unlike the literature, pro- against experimental data from the industrial partner and the lit-
vides a phenomenological description of the radiation mechanisms erature (Latham, 2008), using the kinetic expressions proposed by
and combustion models in the furnace. The literature usually em- Xu and Froment (1989a). The results of this validation demonstrate
ploys empirical expressions for the distribution of heat along of the that the conditions at the outputs of the tubular reactor and fur-
furnace and assumes a homogeneous composition in the furnace nace are predicted with reasonable accuracy. New simulations will
(Zamaniyan et al., 2008; Latham et al., 2011; Kumar et al., 2017). be conducted in this work to verify the validity of this model when
On the other hand, Quirino et al. (2020) describes the combustion the different kinetic models of Table 2 are applied, and to identify
reactions in the furnace through rigorous mass and energy bal- which is the most assertive for the description of the SMRU.
For the previously summarized kinetic models in Table 2, the
Table 4
Model of hydrocracking to compute the equivalent reformate stream. kinetic constants (ki ) are computed by the Arrhenius law, and the
adsorption constants (Ki ), by the Van’t Hoff expression, using the
Fi,eq = (1) Cx Hy + υ1 H2 O → υ2 CH4 + ν1 CO
 parameters provided by the corresponding authors, according to:
F i , 0 + υi , n r n υ1 = 15 x + 15 y (3)

( Fi Hi )eq = (2)
 
 −Ei
( Fi Hi )0 υ2 = 45 x − 15 y (4) ki = Ai exp i = 1, 2, 3 (5)
RT

5
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Table 5
Parameters used by Twigg (1989) for the calculation of equilibrium constants.

i 0 1 2 3 4

ai −3.2770 27.1337 −0.58101 −0.3665 0.2513

bi 0.31688 4.1778 0.63508 −0.29353 -

Table 6
Typical system parameters.

M=  1
wi /Mi
(11) ρ= PM
RT
(12) At = π r2in (13)

yi = wi
MM i
(14) G = ρv = ˙
m
A
(15) A = f
(16)
LW
nt
− π r2ext

Table 7 Table 8
Mass balance and reaction rates. Pressure profiles along the reformer.
dwi ri dP
dz
= G
i∈ (17) dz
= Reformate (21)
Reformate
−150 (1−φφ3 ) ·
2
{ }
Flue Gas μv
−
(Dc )2
ri = i ∈ Reformate (18)
1.75 (1φ−3φ ) ·
( 1 − ε )ρ c Ri Gv
ri = Ri i ∈ Flue Gas (19) Dc

Ri = i∈ (20) P = constant Flue Gas (22)


Mi ηk υi,k Rk {
Reformate
}
Flue Gas
Table 9
Energy balance for all control volumes.
  dTi
= A ddzT = β Q [A]ij = (23)
−H0i dz
Ki = Ai exp i = CH4 , CO, H2 O, H2 (6) βi,j Qj δij Gi cP,i
RT
The equilibrium constants (Ki ) are calculated according to Table 10
Twigg (1989), with the temperature in Kelvin: Summary of heat transfer terms.
  
Qj = − Rk Hk if j = 1, 2 (24) i i, Control Volume
K1 = c · exp ai α i (7) 1 Reformate
Qj = −hj j Tj if j = 3 to 6 (25) 2 Tube – Int
  3 Tube – Ext
Qj = σ ε j j T4j if j = 7 to 12
K2 = exp bi α i (8) (26) 4 Furnace
5 Refract – Int
T
Qj = −kj j r j if j = 13, 14 (27)
j
6 Refract – Ext
K3 = K1 K2 (9)

10 0 0
α= −1 (10) expressions can be considered equal to 1 (Hou and Hughes, 2001;
Tg Xiu et al., 2002; Gallucci et al., 2004; Silva et al., 2010;
where the parameter c, in Eq. 7, is a conversion factor for the Kuznetsov and Kozlov, 2011). The reforming gas pressure can be
pressure, from atm to Pa. The parameters ai and bi are shown in calculated through the Ergun equation, while for the flue gas it
Table 5. can be assumed that is has essentially constant pressure. The en-
The mathematical model developed for the steam reforming ergy equation can be used to write the temperature derivative for
process results in a differential algebraic equation (DAE) system. the control volumes (reformate, tubes, flue gas and refractory) in
The control volumes considered in this model have different dy- a compact matrix form. The above expression consists in a system
namics since the reforming reactions occur much more slowly on of coupled differential algebraic equations, with tube and refrac-
the tubes side than the combustion reactions on the furnace side. tory temperatures appearing in the Qj terms. Such terms may be
This results in highly steep temperature and concentration profiles. calculated with the following logic presented in Table 10, account-
Thus, the model is represented by a system of stiff ordinary differ- ing for the different heat transfer phenomena in the system. The
ential equations coupled with algebraic equations. The system is order of the heat terms in the Table 11 structures the β matrix
solved using a smart implementation of the Adams-Moulton and in a block diagonal form. The λ terms are correction factors that
the Gear methods using BzzMath, a sophisticated and performant that take into account reabsorption of the emitted radiation by the
numerical library written in C++ which exploits the features of same control volume. This may be due to reflection or shading ef-
object-oriented programming. The BzzDae method is especially ro- fects caused by the furnace geometry and tubes arrangement.
bust and efficient due to the automatic selection control of rela- The Beek’s correlation (1962) is used to compute the convec-
tive and absolute tolerances and a better Jacobian evaluation policy tive coefficient on the tube side (Murty and Krishna Murthy, 1988;
(Buzzi-Ferraris and Manenti, 2012, 2015). Acuña et al., 1999; Rajesh et al., 20 0 0; Nandasana et al., 2003;
Table 6-Table 9 present the reformer model equations. ri and Ghouse and Adams, 2013) and the Dittus and Boelter’s equa-
Ri are the net mass and volumetric generation rates of the i-th tion, (1930), to calculate the convective coefficient on the furnace
component. The system is modelled under a simplifying hypothesis side (Yu et al., 2006; Latham et al., 2011; Darvishi and Zareie-
that the mass transport resistances are insignificant. Consequently, Kordshouli, 2017). The convective coefficient to the external envi-
the effectiveness factors (η ) applied in the reforming kinetic ronment is a constant.

6
 P.P.S. Quirino, A. Amaral, K.V. Pontes et al.
Table 11
Heat partition coefficients.

j Type Source Sink i, Control Volume

1 2 3 4 5 6

βi,j , heat partition coefficients

1 Reaction - Reformate +1 0 0 0 0 0
2 - Furnace 0 0 0 +1 0 0
3 Convection Reformate Tube – Int. −1 +1 0 0 0 0
4 Tube – Ext. Furnace 0 0 −1 +1 0 0
5 Furnace – Refract. – Int. 0 0 0 −1 +1 0
Ext.
7

6 Refract. – Environment 0 0 0 0 0 −1
Ext.
7 Radiation Reformate Tube – Int. −1 +1 0 0 0 0
8 Tube – Int. Reformate +1 −1 0 0 0 0
9 Furnace – Tube – Ext. 0 0 +1 −1 0 0
Int.
10 Tube – Ext. Multiple 0 0 −λt + λt a f λt (1 − af ) 0
11 Refract. – Int. Multiple 0 0 λr ( 1 − a f ) + λr a f −λr 0
12 Furnace – Refract. – Int. 0 0 0 −1 +1 0
Ext.
13 Conduction Tube – Int. Tube. – Ext. 0 −1 +1 0 0 0
14 Refract. – Int. Refract. – 0 0 0 0 −1 +1

Computers and Chemical Engineering 152 (2021) 107379

Ext.
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Table 12
Parameters used in the SMRU model.

Symbol Name Units Value Reference

H SMRU - Height m 13.4 Copenor (2014)

L SMRU - Length m 9.4
W SMRU - Width m 4.9
nt Tube - Number − 72
rin Tube - Inner radius mm 53
rext Tube - outer radius mm 63
kt Tube - Conduct. W.m−1 .K−1 28.5 Ghouse and Adams (2013)
εt Tube - Emissivity − 0.85 Latham (2008) Kumar et al.
(2017) Lao et al. (2016)
Darvishi and
Zareie-Kordshouli, (2017)
δr Refractory - Thickness m 0.305 Latham (2008) Kumar et al.
(2017)
−1 −1
k r
Refractory - Conduct. W.m .K 2.6 Tran et al. (2017)
εr Refractory - Emissivity − 0.60 Latham (2008), Kumar et al.
(2017), Darvishi and
Zareie-Kordshouli, (2017)
Dc Catalyst - Particle diam. m 0.0054 Latham (2008)
ρc Catalyst - Density (bulk) kg. m−3 2355.2 Xu and Froment (1989b),
Zamaniyan et al. (2008),
Ghouse and Adams (2013)
φc Catalyst - Porosity − 0.519 Copenor (2014)
εf Flue gases - Emissivity − 0.3758 Darvishi and
Zareie-Kordshouli, (2017)
henv Env. - Conv. Coeff. W.m−2 .K−1 11.63 Olivieri and Vegliò (2008)
Tenv Env. - Temperature K 298 Copenor (2014)
Data Fitting Parameters

af Furnace Gas - Abs. Factor % 70 Quirino et al. (2020)

λt Tubes - Rad. Factor % 100 Quirino et al. (2020)
λr Refract. - Rad. Factor % 58 Quirino et al. (2020)

The ideal mixing rules are considered to compute the physical Hou and Hughes (2001), to verify the model’s ability to predict the
properties of the process gases in the catalytic tubular reactor and real behavior of the system and to discriminate, among these four
in the furnace. The thermal conductivity and viscosity of the chem- kinetic models, the one that best fits the experimental measure-
ical species are determined according to Yaws, (1999). The Wassil- ments. The only distinguishing factor between the different com-
jewa equation and the Wilke method are used to evaluate these parisons is the kinetic models of the reforming reactions, all the
properties in the gas mixture, respectively (Poling et al., 2001). The rest of the steam reformer model remaining as it was presented
specific heat of the pure components is obtained from NASA cor- in section 1. Table 13 shows the relative deviations (RD) obtained
g
relations (Burcat and Ruscic, 2005), and used to calculate the reac- for the flue gas temperature (Tfout ) and for the temperature (Tout ),
g g
tion enthalpy, based on the specific enthalpies of the products and pressure (Pout ) and composition (yi,out ) of the reformed gas at the
reagents. outlet of the SMRU, when the model predictions are compared
The SMRU model is validated against experimental data from against the data from (Latham, 2008). All kinetic models simu-
literature and from our industrial partner, comparing the predic- late with acceptable accuracy the experimental data and presented
tion of the four kinetic models presented in Table 2. In general, very similar relative deviations (RDs) with a maximum of 3.3%
only experimental data from the output of the industrial reformer for all variables. No significant differences have been observed be-
are available for the validation since temperature and composi- tween the computational times of the different reaction schemes.
tion measurements are not performed along the length of the Figure 2 compares the composition, temperature, and methane
industrial reformers. Five data sets are used for the validation: conversion profiles, predicted by the four reforming kinetics used
Latham 1 to Latham 3 refer to the experimental data provided by in the SMRU model, with the experimental data from Latham 3.
(Latham, 2008), and Copenor 1 and Copenor 2 refer to data from The results for Latham 1 and Latham 2 are similar to Latham
our industrial partner, according to Table 3. Latham’s (2008) work 3, therefore only Latham 3, which presented smallest deviations
was selected because the author provides information on the re- (Table 13), is illustrated in Figure 2 for the sake of simplicity. A fo-
former’s output variables, as well as reports the experimental tem- cus is given in the initial reformer section, where the differences
perature at two points along the tubular reactor: one located at between the kinetic models can be better visualized. Figure 2 also
3.57 meters and the other at 8.56 meters from the top of the re- shows the experimental data reported by Latham (2008), namely:
former. The parameters used in model validation are summarized the composition (b) and the temperature at tubular reactor outlet
in Table 12, in which the equipment geometry data was provided (d), the temperature of the outer wall of the reform tube at two in-
by our industrial partner (Copenor, 2014), some parameters were termediate points along the axial coordinate and the flue gas tem-
estimated by Quirino et al. (2020) and other parameters were ob- perature at the furnace outlet (f).
tained from the literature. The XF and HH kinetic models showed very similar behavior,
especially the temperature profiles, which were practically coin-
5. Results and Discussion cident. Figure 2 (c and e) allows only the visualization of the XF
curve, which is overlapped on the HH curve. This is due to the
The model of the stationary SMRU was simulated us- great similarity between the basic mechanistic premises and the
ing the kinetic models proposed by Singh and Saraf (1979), final forms of these two models, despite the differences in the cat-
Numaguchi and Kikuchi (1988), Xu and Froment (1989a) and alyst support (refer to Table 2). Such results agree with the study

8
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Figure 2. Influence of the kinetic models on composition (a and b) and temperature (c to f) profiles and methane conversion (g and h) along the length of the reformer.
Validation against data from Latham 3 (Latham, 2008).

9
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Table 13
Relative deviations (in %) between model predictions and experimental results from Latham (2008) considering different kinetic models. All values refer to outlet
conditions. Composition values refer to the reformed gas and are taken on a molar basis.

Case # Latham 1 Latham 2 Latham 3

Model SS NK XF HH SS NK XF HH SS NK XF HH
Tfout (K ) -1.91 -1.89 -1.87 -1.87 -1.40 -1.37 -1.36 -1.36 -1.55 -1.51 -1.50 -1.50
Tgout (K ) -0.56 -1.31 -0.56 -0.54 -0.60 -0.57 -0.59 -0.58 -0.30 -0.26 -0.27 -0.27
-2.98 -2.97 -2.95 -2.95 -2.13 -2.12 -2.11 -2.11 -2.43 -2.46 -2.42 -2.42
Pgout (bar )
CH4 2.47 2.84 2.94 2.94 2.72 3.16 3.25 3.25 -0.60 -0.08 -0.02 0.01
CO2 1.04 0.97 1.00 1.00 1.31 1.25 1.28 1.28 0.48 0.41 0.44 0.44
CO -1.62 -1.72 -1.79 -1.78 -1.77 -1.90 -1.97 -1.97 -0.02 -0.17 -0.23 -0.22
H2 O 0.32 0.42 0.44 0.44 0.42 0.54 0.55 0.56 -0.37 -0.24 -0.22 -0.22
H2 -0.33 -0.42 -0.44 -0.44 -0.44 -0.54 -0.56 -0.56 0.26 0.14 0.12 0.12

developed by Vidal Vázquez et al. (2017), who investigated the in- The difference in profiles obtained by the SS and NK model,
fluence of kinetic models in a quartz tubular reactor for Fischer- when compared with the XF and HH kinetic models, may be asso-
Tropsch applications. In their study, four kinetic models were se- ciated with the use of higher temperature and pressure conditions
lected: a power law model and three models based on assumed for the formulation of their models. Furthermore, unlike Xu and
reaction mechanisms on the active sites of a metal catalyst, which Froment (1989a) and Hou and Hughes (2001), Singh and Saraf
are: XF, HH and Hakeem, Alstrup and Weatherbee (HAW) models. (1979a) and Numaguchi and Kikuchi (1988) disregard the R3 reac-
The last model is a different approach where the kinetic equations tion, which dominates at the lower temperatures. In the NK model,
have different denominator for the rate equations due to the re- a lower nickel content and metal surface area are used than in the
action mechanisms being developed separately for each reaction. XF and HH models, which must also have contributed to this dif-
The comparative analysis of these models showed that the kinetic ference seen at reactor inlet. Singh and Saraf (1979a) did not pro-
models XF and HH were the ones that provided the best agree- vide this information, thus making this comparative analysis im-
ment of the model and among them, the XF model was chosen for possible. The greater similarity of the SS and NK models, especially
use in future works. in the composition profiles (a and b), is associated with the exper-
Just after the reactor inlet, HH and XF provide a more abrupt imental conditions used and their simplified structures. However,
drop in methane and water molar fractions, compared to the SS Singh and Saraf (1979a) additionally incorporates the influence of
and NK models (a), which agrees with the temperature profiles the total operating pressure in the kinetic reaction rates. Studies
presented in the (c) and (e) and with the methane conversion pro- that consider the kinetic expressions suggested by Singh and Saraf
file (g). This drop is more pronounced in the HH model than in (1979a) report that this kinetic model reproduces well the exper-
the XF model. The process gas temperature (Tg ) decreases about imental data from reformers, even in atypical operational condi-
110K and the methane reaches a conversion of approximately 10.7% tions, as seen in Vakhshouri and Hashemi (2008). These authors
in the XF and HH models, while in the SS and NK models, this developed a rigorous two-dimensional model of a stationary re-
temperature decrease is around 90K and 60K, with methane con- former that operated under pressure conditions of 2 to 3 bar and
version of 7.38% and 5.1%, respectively. This drop in the process mass velocities of 2 to 5 kg.m−2 . s−1 . The reformer model was val-
gas temperature and tube wall profiles (c and e) is due to the idated with data from an industrial plant, which allowed to satis-
endothermic character of the steam methane reforming (R1) and factorily predict the reactor’s output variables, as well as the radial
of shift reforming (R3) reactions. In the case of XF and HH, the transfer of heat and mass inside the reforming tubes.
negative order in relation to hydrogen further favors these reac- Similar results are observed when the experimental data pro-
tion rates at the reactor inlet. Low concentrations of this compo- vided by the industrial partner are used in the SMRU model.
nent in the stream fed to the tube result in higher rates. On the Table 14 shows the deviations from the model’s predictions in re-
other hand, in the presence of a significant amount of H2 , i.e., as lation to these industrial data, Copenor 1 and Copenor 2, for the
methane is converted in this component, the reaction rates become four kinetic models considered in the Table 2. All kinetic mod-
lower and the differences between the kinetic models are less pro- els simulate with acceptable accuracy the experimental data and
nounced. presented very similar relative deviations (RDs) with a maximum
The rates of reforming reactions are slower between 0.15m and of approximately 6.4% for the flue gas temperature. Despite these
1.6m, due to the net endothermic effect of parallel reforming re- higher deviations, they are considered acceptable, as the reliability
actions. Consequently, there is a slight decrease of 20 K in Tg (d) of the experimental data cannot be guaranteed. The flue gas tem-
for the four kinetic models considered. From 1.6m, flue gas tem- perature measuring instruments require periodic calibrations and
perature (Tf ) abruptly increases, providing the necessary heat for are considered critical in the industrial SMRU.
increasing the reaction rates and the temperature inside the tube. The comparative analysis of the kinetic models presented in
Around 2.75m, the temperature in the outer wall of the tube (Ttext ) Table 2 allowed to infer that all four models were able to predict
reaches a maximum value of 1176K for the SS model, of 1173K for the temperature, composition, and conversion profiles with good
NK model and of approximately 1171K, for the XF and HH mod- accuracy. As seen in Figure 2, the greatest differences among the
els. These temperature values are consistent with what is seen kinetic models are in the reformer’s initial section. However, as it
in industrial reformers, which usually have an outer reforming advances along the tubular reactor length, this difference becomes
tube wall temperature around 1100K to 1200K (Nandasana et al., practically insignificant. Such results support the simplifying hy-
20 03; Latham, 20 08; Ghouse and Adams, 2013, Lao et al., 2016; pothesis that the effectiveness factor can be considered 1. As seen
Aguirre, 2017). From then on, the difference between these four in Table 2, the SS model intrinsically incorporates diffusion limita-
kinetic models decreases and at around 5.4 m the curves get over- tions, however all the kinetic models are equivalent (i.e., produce
lapped. The water-gas shift reaction (R2), which is exothermic, is the same results) under the above assumption.
reversed at high temperatures, explaining the higher amount of CO The results also suggest that the reforming reaction rates are
than that of CO2 at the reformer outlet (3-b). very fast and, consequently, tend to achieve chemical equilibrium

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P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Table 14
Relative deviations (in %) between the results predicted by the model and by the industrial MSRU, for the different kinetic
models used. Experimental data from Copenor (2014). All values refer to outlet conditions. Composition values refer to the
reformed gas and are taken on a molar basis.

Case # Copenor 1 Copenor 2

Model SS NK XF HH SS NK XF HH

Tfout (K ) 6.13 6.11 6.13 6.13 6.42 6.41 6.43 6.43

Tgout (K ) 0.48 0.44 0.44 0.44 0.35 0.32 0.32 0.32
Pgout (bar ) 1.29 1.30 1.29 1.28 1.46 1.48 1.46 1.46
CH4 3.08 3.51 3.48 3.45 0.85 1.21 1.16 1.13
CO2 1.01 0.96 0.96 0.96 0.75 0.71 0.70 0.70
CO -1.09 -1.12 -1.14 -1.11 -0.58 -0.60 -0.59 -0.59
H2 O 0.16 0.23 0.22 0.22 0.02 0.07 0.07 0.06
H2 -0.18 -0.23 -0.23 -0.23 -0.02 -0.07 -0.06 -0.06

Figure 3. Deviations of chemical equilibrium along the reformer length for different kinetic models - Latham data 3. The red dotted line refers to a 1% deviation from
equilibrium.

Table 15 nificantly from the equilibrium condition. The results of this anal-
Operating range applied in the simulations for the evaluation of
ysis can be found in Appendix A. More extreme conditions, lower
chemical equilibrium.
temperature, and pressure, imply greater deviations from equilib-
Parameter Unit Range rium at the reactor outlet, where maximum value reached approx-
Tgin K 723 - 923 imately 5.5% for the reaction R2 at a pressure of 5 bar. The NK
Pgin bar 5 - 15 model has the smallest relative deviations for this reaction and re-
SCR − 2-5 sults in rapid reactions, such as the HH model. The SS model, on
the other hand, is the slowest among the evaluated kinetic models.
The simpler structures and the reduced number of parameters
give the SS and NK models an advantage over the XF and HH mod-
in the catalytic bed. Figure 3 illustrates the deviations from chemi-
els. The first models have only 4 parameters (pre-exponential fac-
cal equilibrium along the reformer length to different kinetic mod-
tors and activation energies), while the last two have 14 and 12
els, using the Latham 3 data. Such deviations were calculated by
parameters, respectively, due to the additional consideration of the
comparing the equilibrium constants Eq. 7 to (9) with the reaction
effects of reagent adsorption on the catalyst surface. Moreover, the
quotients, which are the ratios between the relative amounts of
operating range used by XF and HH is below that practiced in the
products and reagents. If the equilibrium constant is equal to the
industry, while SS and NK apply operating conditions more like
reaction quotient, the chemical equilibrium is achieved. The model
those of the industrial unit reported by Quirino et al. (2020) and
HH results in faster reforming reactions (R1 and R3), which reach
by Latham (2008).
equilibrium almost instantly upon entering the tubular reactor. The
It is appropriate to emphasize that the parameters used in the
R2 reaction is faster than the R1 reaction in the NK and XF models,
SMRU model were reported from Quirino et al. (2020), who esti-
while in the SS model the opposite effect occurs. These reactions
mated their parameters based on the XF kinetic model. Thus, the
show equilibrium deviations of less than 1% from 5.4 m for all ki-
SS and NK models will be able to predict the reformer’s station-
netic models, except the SS model, whose deviation at the reactor
ary behavior more accurately if a new parameter estimation of the
output is around 2% for reaction R1. These results agree with the
proposed model by Quirino et al. (2020) is performed. However,
profiles presented in Figure 2 (b), in which the SS model is pre-
the rigorous estimation of kinetic parameters is difficult due to the
sented as the slowest compared to the other kinetic models.
lack of experimental data and literature that allow to quantitatively
A sensitivity analysis is performed to investigate the feasibil-
infer the temperature and composition profiles along the length of
ity of these kinetic models, varying the three operating condi-
the reformer. Thus, the authors consider that the use of the pa-
tions that affect the chemical equilibrium: temperature, pressure,
rameters estimated by Quirino et al. (2020) is sufficient, since they
and molar steam to carbon ratio (SCR) of the stream fed to the
already provide an excellent prediction of the reformer model.
tubular reactor. Table 15 illustrates the variation range applied to
Due to the great similarity between the results obtained with
each of these variables, based on industrial and literature data
the use of the XF and HH models, as shown in Table 13, Table 14
(Twigg, 1989; Rostrup-Nielsen, 1993; Ferreira-Aparicio et al., 2005;
and Figure 2, a more detailed study is conducted to evaluate the
Basini, 20 05; Vasconcelos, 20 06; Latham, 20 08; Copenor, 2014;
contribution of the methane adsorption term in the denominator
Abid and Jassem, 2014). Lower pressures than those commonly ap-
of the kinetic expressions suggested by Xu and Froment (1989a).
plied to industrial reformers (15 to 35 bar) were used in order to
Therefore, we consider that the adsorption of methane on the cat-
identify the lower limit, from which the kinetic models diverge sig-

11
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

alyst surface is hampered because of the high steam concentra- as performed in this work. The results demonstrate that there was
tion in the reforming system, analogously to the approach used by no significant loss in the accuracy of the reformer model with the
Hou and Hughes (2001). A simulation is performed, disregarding use of simpler kinetics, such as SS and NK. These models do not
the methane adsorption term (KCH4 = 0 ) in the XF model. The re- have a negative order for H2 , which minimizes problems of nu-
sults demonstrate that this parameter does not significantly affect merical instability when H2 very small concentrations are fed to
the accuracy of the XF model. The total reaction rate of methane the tubular reactors. In the cases evaluated in this work, the re-
(R1 + R3) was practically equivalent, which agrees with the find- former operates close to equilibrium, in addition, the current fed
ings of Vaccaro and Malangone (2012). to the tubular reactor contains H2 . Consequently, significant dif-
Vaccaroand Malangone (2012) investigated the influence of the ferences in the performance of the tubular reactor were not ob-
XF and HH kinetic models in the prediction of a stationary model served, unlike Vaccaro and Malangone (2012) and Leonzio (2016),
of a micro-scale catalytic reactor. In this reactor, the methane who faced problems arising from the use of the XF model.
steam reforming occurs coupled with the methane catalytic com- Since the system operates close to equilibrium over a broad
bustion, which can be considered as a variant of an autothermal range of conditions, the operation primary concern is heat man-
reforming (ATR). Both kinetics compared in this work were able to agement and bringing the reformer up to temperature as rapidly
simulate the temperature profiles satisfactorily. However, only the as possible. It is therefore essential to understand the heat trans-
HH model allowed to predict the reactor performance with good fer processes that take place in such a system and its practical
accuracy, with respect to the compositions at the reactor outlet implications to properly design and operate a reformer. To max-
and methane conversion. XF kinetics overestimated this conversion imize the equipment lifetime, it is important to ensure proper
by 33.14%. The authors state that the presence or absence of hydro- monitoring and control of the temperature in the tubes and re-
gen in the reactor feed significantly contributes to the differences fractories. The severe exposure of the tube wall to temperatures
between the responses predicted by the two kinetic models. Their around 1373 - 1473 K can lead to a tube life of about minutes
findings allowed to infer that negligible amounts of this compo- (Cromarty, 2004). In addition, such useful life can be reduced
nent in the reactor feed imply widely different reaction rates for by half if the tube wall temperature exceeds the design tem-
XF and HH. The pressure used by them is closer to that used by perature by 20 K (Farnell, 2003; Cromarty, 20 04; Latham, 20 08;
Hou and Hughes (2001), which, according to the authors, may jus- Pantoleontos et al., 2012; Tran et al., 2017; Zecevic and Bolf, 2020).
tify the better accuracy of this kinetic model in predicting the per- If the industrial practice cannot provide experimental data to de-
formance of the micro-reactor. termine if the control of internal temperatures is adequate, math-
The SS and NK models, despite its simplified structure, pro- ematical modeling can fill this gap. A clear application of this
vided predictions like those obtained by the XF and HH models. useful resource derives from the observation of Figures 2 (c-
Thus, we investigate whether it is possible to reduce the complex- f), in which a marked temperature drop at the reactor in-
ity of the XF and HH models, without significant loss of accuracy, let is foreseen from all the kinetic models. Different studies
considering the denominators of the kinetic expressions of XF and have foreseen the same behavior along the reformer length and
HH equal to 1. The XF e HH models acquire a format like the SS thus, it is these authors’ belief that such profiles are physically
model, which is based on power law. This change in the structure consistent.
of these models had practically no effect on the prediction of the
SMRU model. The deviations (RD) presented in Table 13 remained 6. Conclusions
nearly the same. Therefore, in addition to the SS and NK models,
the simplified versions of the XF and HH models, which neglect This paper investigated the influence of kinetic models of re-
the adsorption terms of the reactive species, may also be applied forming reactions on the prediction of a stationary model of
to the SMRU model used in this work. The main difference of these an industrial MSRU. Most of the works reported in the liter-
kinetic models is at the reactor inlet. Due to the lack of experimen- ature on the kinetics of reforming reactions focus on analyz-
tal data, it is not possible to state which of these models best de- ing only the reactor, which may result in the choice of less-
scribes the composition and temperature profiles along the equip- accurate models, since the control volumes are closely intercon-
ment. However, these models resulted in profiles in agreement nected. This model considers the catalyst-filled tube, as well as
with those observed in the literature (Latham, 2008; Kumar et al., the tube wall, the furnace, and the refractory. Thus, the MSRU
2017; Darvishi and Zareie-Kordshouli, 2017). The maximum tem- model can predict the temperature and/or composition profiles of
perature in external tube wall was also consistent with what is these four control volumes, using a more detailed approach to de-
seen in industrial reform units. If there is only interest in the re- scribe the heat transfer phenomena and the combustion kinetics
former’s output data, these kinetic models prove to be reliable and in the furnace. The present study compares four kinetic models,
can be used. SS, NK, XF and HH, to certify its validity and choose, among them,
Most of the works available in the literature on indus- which is the most consistent with experimental data and stud-
trial reformers models use the kinetic model of Xu and Fro- ies by other authors. Simulations were performed using experi-
ment (1989a) without any previous analysis. Other models with a mental data obtained from the literature and the industrial part-
more simplified structure could be used, resulting in precision as ner. The results showed that the kinetic models evaluated repro-
good or even better than that provided by Xu and Froment (1989a), duced the outlet conditions of the catalytic tubular reactor and the
as Leonzio (2016) proved. This author modeled an integrated mem- furnace with reasonable accuracy. The deviations were relatively
brane reactor at steady state to study the steam reforming reac- low and at most 3.5% in relation to experimental and literature
tion using the kinetic models of NK and XF. The results of the data.
simulations indicated that Xu and Froment’s kinetics is very fast, The reforming reactions proved to be fast enough and tend to
implying a unitary methane conversion along the whole reactor reach equilibrium as they move along the reformer length. A sen-
length. On the other hand, the NK model provided better results, sitivity analysis was carried out to evaluate the influence of tem-
which were consistent with the literature (Kyriakides et al., 2014, perature, pressure, and the SCR of the reforming gas on the system
Vásquez Castillo et al., 2015, Patrascu and Sheintuch, 2015). There- operation. For the whole operating range of these variables, the ki-
fore, although the XF model is considered to be the most gen- netics had little influence on the reformer performance. Since the
eral and representative of the reforming reactions, it is essential industrial reformer operates close to equilibrium, any of the evalu-
to compare it with other kinetic models existing in the literature, ated kinetic models can be used to study the equipment, including

12
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

an equilibrium model. The results show that the adsorption term Acknowledgements
in the denominator of the XF and HH models might be neglected
without loss of accuracy. We investigated whether it is possible to The authors would like to thank COPENOR, especially John
reduce the complexity of the XF and HH models, without signifi- Kennedy Fernandes and Alan Rocha dos Santos Pinho Costa for
cant loss of accuracy, considering the denominators of the kinetic providing data and knowledge about the industrial plant under
expressions of XF and HH equal to 1. The SS, NK, XF, HH model study and for authorizing the publication of this information in
and its simplified versions may be good options for future MSRU this paper; the Coordination for the Improvement of Higher Edu-
studies. The SS and NK models, which until then were not widely cation Personnel - Brasil (CAPES) - Finance Code 001, the National
used in steam methane reform studies, may have more advantages Council for Scientific and Technological Development (CNPQ) and
over the XF model, due to their greater simplicity, with fewer pa- the CARIPLO Foundation for the financial support.
rameters and with less instability problems numeric.
Appendix A
Declaration of Competing Interest
The Figure A1, Figure A2, and Figure A3 present the sensitivity
The authors declare that they have no known competing finan- analysis performed to evaluate the influence of temperature, pres-
cial interests or personal relationships that could have appeared to sure and SCR of the stream fed to the tubular reactor on chemical
influence the work reported in this paper. equilibrium, respectively.

Figure A1. Relative deviations from the chemical equilibrium along the reactor length for the four kinetic models considered, under different temperature conditions -
Latham 3. The red dotted line refers to a 1% deviation from equilibrium.

13
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Figure A2. Relative deviations from the chemical equilibrium along the reactor length for the four kinetic models considered, under different pressure conditions - Latham
3. The red dotted line refers to a 1% deviation from equilibrium.

14
P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

Figure A3. Relative deviations from the chemical equilibrium along the reactor length for the four kinetic models considered, under different SCR conditions - Latham 3.
The red dotted line refers to a 1% deviation from equilibrium.

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P.P.S. Quirino, A. Amaral, K.V. Pontes et al. Computers and Chemical Engineering 152 (2021) 107379

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