APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION


IIUM Engineering Journal, Vol. 22, No. 1, 2021 Suharsono et al. 
https://doi.org/10.31436/iiumej.v22i1.1398 

SOLUTION OF THE REVERSE FLOW REACTOR MODEL 

USING HOMOTOPY ANALYSIS METHOD 

SUHARSONO1, SRI WULANDARI1, AANG NURYAMAN1, MUSTOFA USMAN1,

WAMILIANA1 AND JAMAL IBRAHIM DAOUD2

 
1
Department of Mathematics, FMIPA Universitas Lampung,  

Jl. Prof. Dr. Soemantri Brojonegoro No. 1, Bandarlampung 35145, Indonesia
2
Department of Science in Engineering, Kulliyyah of Engineering, 

International Islamic University Malaysia,  

Jalan Gombak, 53100 Kuala Lumpur, Malaysia 

*
Corresponding author: aang.nuryaman@fmipa.unila.ac.id 

(Received: 13th March 2020; Accepted: 21st July 2020; Published on-line: 4th January 2021) 

ABSTRACT:   Methane (CH4) is one of the most dangerous greenhouse gases in the 

atmosphere. A reverse flow reactor is utilized to convert CH4 to carbon dioxide (CO2) as 

a means of reducing the effect of global warming. The dynamics of its dependent variables 

can be stated by a set of convective-diffusion equations. In this article, we examined 

analytical solutions of temperature dynamics and methane conversion for a 1-D pseudo 

homogeneous model without refrigeration by using the homotopy analysis method. The 

results show that temperature and conversion of methane will go to constant when time 

goes to infinity.  

ABSTRAK: Metana (CH4) merupakan salah satu gas rumah hijau paling berbahaya di 

atmosfera. Reaktor aliran balik telah dipakai bagi menukar CH4 kepada CO2 bagi 

mengurangkan kesan pemanasan global. Dinamik pemboleh ubah bersandar ini dapat 

diterangkan melalui satu set persamaan konvektif-difusi. Artikel ini akan mengkaji 

penyelesaian analisis dinamik suhu dan penukaran metana bagi model 1-D pseudo-

homogen tanpa penyejukan dengan menggunakan kaedah analisis homotopi. Hasil kajian 

menunjukkan bahawa suhu dan penukaran metana akan berterusan dengan masa tak 

terhingga. 

KEYWORDS: analytical solution; 1-D pseudo homogeneous model; reverse flow reactor; 

homotopy analysis method 

1. INTRODUCTION

There are several mathematical models in differential equation form that are difficult

to solve using ordinary analytic partial differential equations. Hence, various methods have 

been developed to solve these equations, such as the Laplace transformation method, the 

perturbation method, the finite difference method, etc. 

One natural phenomena that gets very intensive attention is global warming caused by 

greenhouse gas emissions. One of the dangerous and numerous greenhouse gases in the 

atmosphere is methane (CH4). Reducing the global warming effect can be achieved by 

converting CH4 into carbon dioxide (CO2) according to the oxidation (combustion) 

equation: 

CH4 + 2O2 → CO2 + 2H2O  ΔH298  = −802,7 kj/mol  (1) 

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Every one mole of oxidized CH4 gas will release as much heat energy of 802.7 kJ. 

Hence, converting CH4 gas to CO2 gas will reduce the heating effect by 87%. The presence 

of fairly small amounts of methane gas in the air (0.1–1% by volume) causes the conversion 

of methane gas to CO2 gas but needs a catalyst so that the reaction can take place. On the 

other hand, low methane temperature (around 303 K), so far from the reaction temperature, 

requires preheating of the feed gas. 

One technology which can be used to anticipate the negative impact and characteristic 

of methane is the use of a reverse flow reactor (RFR) to oxidize CH4 into CO2. Further 

explanation about RFR can be seen in [1,2].  A mathematical model illustrating the 

dynamics of temperature and concentration on oxidation of CH4 through RFR has been 

revealed by Khinast et al. [3] and van Norden [4]. In those articles, a one-dimensional (1-

D) pseudo homogeneous model was used to describe the dynamic of dependent variables in 

a cooled-reverse flow reactor. Previous studies [5,6] used this model to investigate operating 

parameter sensitivities of  RFR on the behavior of dependent variables for periodic feed gas 

by using a numerical approach. Whereas in [7,8], this model was used to construct singular 

perturbation problems by considering certain assumptions for steady state conditions and 

solved them using asymptotic methods. While for the unsteady state case, Nuryaman [9] 

reported an analytical solution for conversion equation that was derived from the 1-D 

pseudo homogeneous model which assumed that the reaction took place spontaneously at a 

certain reaction rate. The homotopy perturbation method was used to get its analytical 

solution.  

In this article, we consider the 1-D pseudo homogeneous model in [4] and we assume 

that the reactor is in the condition without cooling such that the equations become as 

follows: 

𝑢𝑡 =  𝑎𝑢𝑥𝑥 − 𝑏𝑢𝑥 + 𝑐𝑔(𝑢)(1 − 𝑣),          𝑥 ∈ [0,1]         (2) 

𝑣𝑡 =  𝑒𝑣𝑥𝑥 − 𝑓𝑣𝑥 + 𝑙 𝑔(𝑢)(1 − 𝑣),             𝑡 ≥ 0, (3) 

𝑔(𝑢) =
1,6656 ×10−5exp (

25,785(𝑢−1)

𝑢
)

1,6656 ×10−5+exp (−
25,785

𝑢
)

                                          (4) 

where 𝑢 = 𝑢(𝑥, 𝑡), 𝑣 = 𝑣(𝑥, 𝑡) are dimensionless variables for temperature and conversion 
variables. Here 𝑎, 𝑏, 𝑐, 𝑒, 𝑓,and 𝑙  are dimensionless parameters which values are given in 
Table 1, and 𝑔(𝑢)  is a nonlinear function that corresponds to the rate of reaction in the RFR. 

 

Table 1: The dimensionless parameter values of RFR [4] 

No Parameter Values 

1 𝑎 6.9393 x 10−4 

2 𝑏 0.1749 

3 𝑐 1.5577 x 10−6 

4 𝑒 2.4038 x 10−3 

5 𝑓 174.06 

6 𝑙 0.01 

Based on equations (2)-(3) and under the certain assumptions, in this article, we 

investigate an analytical solution by applying the homotopy analysis method (HAM). In 

recent years, HAM can be applied for solving various linear and nonlinear systems, and 

homogeneous and nonhomogeneous equations [10]. The HAM is used to solve problems 

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using the determination of series convergence with respect to an embedded parameter [11]. 

In fact, the homotopy method is easier to use in solving difficult problems. Therefore, the 

HAM method will be applied herein solving the RFR model.  

2.   METHODOLOGY 

The Homotopy Analysis Method (HAM) was designed firstly in 1992 by Liao [12] and 

was then modified in 1997 [13]. This is a semi-analytics technique for solving ordinary 

nonlinear problems or partial differential equations. Homotopy can be defined as a link 

between two different objects in mathematics that have the same characteristics in several 

aspects [13]. 

The HAM is based on concepts in topology and differential geometry to produce a 

series convergence of a nonlinear system. The concept of homotopy is then traced back to 

Jules Henri Poincare, a French mathematician.  Homotopy explains a kind of deformation 

variation in mathematics. For example, a circle can be deformed continuously into an 

ellipse, and the shape of a coffee cup can be deformed continuously into a donut shape. 

Suppose there are zeroth-order differential equations:  

𝑁𝑘[𝑧𝑘(𝜔, 𝜏)] = 0,     𝑘 = 1, 2, … , 𝑚      (5) 

where 𝑁𝑘 are nonlinear operators that represent the whole equations, 𝜔 and  𝜏 denote the 
independent variables, and 𝑧𝑘(𝜔, 𝜏) are unknown functions. Liao constructed the 
deformation equations as 

(1 − 𝑞)𝐿[𝜙𝑘(𝜔, 𝜏; 𝑞) − 𝑧𝑘,0(𝜔, 𝜏)] = 𝑞ℏ𝑘𝑁𝑘[𝜙𝑘(𝜔, 𝜏; 𝑞)]   (6) 

where 𝑞 is an embedding parameter, 𝑞 ∈ [0,1], ℏ𝑘are nonzero auxiliary functions, 𝐿 is an 
auxiliary linear operator, 𝑧𝑘,0(𝜔, 𝜏) are initial guesses of 𝑧𝑘(𝜔, 𝜏), and 𝜙𝑘(𝜔, 𝜏; 𝑞) are 
unknown functions. One has great freedom to choose auxiliary objects such as ℏ𝑘 and 𝐿. 
Obviously, when 𝑞 = 0  and  𝑞 = 1, 𝜙𝑘 hold: 

𝜙𝑘(𝜔, 𝜏; 0) = 𝑧𝑘,0(𝜔, 𝜏)and 𝜙𝑘(𝜔, 𝜏; 1) = 𝑧𝑘(𝜔, 𝜏)    (7) 

Thus, if  𝑞   increases from 0 to 1, then the solutions 𝜙𝑘(𝜔, 𝜏; 𝑞) move from 
𝑧𝑘,0(𝜔, 𝜏)  to 𝑧𝑘(𝜔, 𝜏).  𝜙𝑘(𝜔, 𝜏; 𝑞) are then expanded in Taylor series with respect to 𝑞, and 
then becomes 

𝜙𝑘(𝜔, 𝜏; 𝑞) = 𝑧𝑘,0(𝜔, 𝜏) + ∑ 𝑧𝑘,𝑛(𝜔, 𝜏)𝑞
𝑛+∞

𝑛=1      (8) 

where  

𝑧𝑘,𝑛 =
1

𝑛!

𝜕𝑛𝜙𝑘(𝜔,𝜏;𝑞)

𝜕𝑞𝑛
|𝑞=0.        (9) 

When ℏ𝑘,  𝐿, 𝑧𝑘,0(𝜔, 𝜏), and 𝜙𝑘(𝜔, 𝜏; 𝑞)are properly chosen, then Equation (8) converges 
at  𝑞 = 1 and 

𝜙𝑘(𝜔, 𝜏; 1) = 𝑧𝑘,0(𝜔, 𝜏) + ∑ 𝑧𝑘,𝑛(𝜔, 𝜏)
+∞
𝑛=1      (10) 

which has to be one of the solutions. As  ℏ𝑘 = −1, Equation (6) becomes 

(1 − 𝑞)𝐿[𝜙𝑘(𝜔, 𝜏; 𝑞) − 𝑧𝑘,0(𝜔, 𝜏)] + 𝑞𝑁𝑘[𝜙𝑘(𝜔, 𝜏; 𝑞)] = 0   (11) 

The governing equations can be deduced from the zeroth-order deformation Equation 

(6). Define the vectors 

𝑧𝑘,𝑚⃗⃗⃗⃗⃗⃗⃗⃗ = {𝑧𝑘,0(𝜔, 𝜏), 𝑧𝑘,1(𝜔, 𝜏), … , 𝑧𝑘,𝑚(𝜔, 𝜏)}     (12) 

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The nth order deformation can be found by differentiating (6) n times with respect 

to 𝑞 and then putting 𝑞 = 0. After that divide it by 𝑛! such that 

𝐿[𝑧𝑘,𝑛(𝜔, 𝜏) − 𝜒𝑛𝑧𝑘,𝑛−1(𝜔, 𝜏)] = ℏ𝑘𝑅𝑘,𝑛(𝑧𝑖,𝑛−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )    (13) 

where 

𝑅𝑘,𝑛(𝑧𝑘,𝑛−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) =
1

(𝑛−1)!

𝜕𝑛−1𝑁𝑘[𝜙𝑘(𝜔,𝜏;𝑞)

𝜕𝑞𝑛−1
|𝑞=0     (14) 

and 

𝜒𝑛 = {
0, 𝑛 ≤ 1
1, 𝑛 > 1

  (15) 

Note that 𝑧𝑘,𝑛(𝜔, 𝜏) (𝑛 ≥ 1) are governed by (13) with boundary conditions coming from 
the original problem. 

3.   RESULT AND DISCUSSION 

Consider the 1-D pseudo homogeneous model in Equations (2)-(3). By using a 

rescaling process and the assumption that the reaction rate takes place at certain temperature 

such that the nonlinear term approach to one. We obtain a dimensionless equation set that 

illustrates the dynamics of temperature and conversion of methane gas to methane oxidation 

using RFR without cooling as follows: 

𝑢𝑡 − 𝑎𝑢𝑥𝑥 + 𝑏𝑢𝑥 + 𝑐𝑣 − 𝑐 = 0  (16) 

𝑣𝑡 − 𝑒𝑣𝑥𝑥 + 𝑓𝑣𝑥 + 𝑙𝑣 − 𝑙 = 0  (17) 

where 𝑢 = 𝑢(𝑥, 𝑡), 𝑣 = 𝑣(𝑥, 𝑡) are dimensionless variables for temperature and conversion 
and   𝑎, 𝑏, 𝑐, 𝑒, 𝑓, and  𝑙 are dimensionless parameters which values given in Table 1. In this 
case, initial conditions are 

𝑢(𝑥, 0) = 𝛽, 𝛽 > 1    (18) 

where 𝛽 is constant and  

𝑣(𝑥, 0) = 0  (19) 

The linear operator  

𝐿[𝜑𝑘(𝑥, 𝑡; 𝑞)] =   
𝜕𝜑𝑘(𝑥,𝑡;𝑞)

𝜕𝑡
, 𝑘 = 1,2  (20) 

with  𝐿[𝑝𝑘] = 0, where  𝑝𝑘(𝑘 = 1,2) are integral constants. 

The nonlinear operator 

𝑁1[𝜑1(𝑥, 𝑡; 𝑞)] =
𝜕𝜑1(𝑥,𝑡;𝑞)

𝜕𝑡
− 𝑎

𝜕2𝜑1(𝑥,𝑡;𝑞)

𝜕𝑥2
+ 𝑏

𝜕𝜑1(𝑥,𝑡;𝑞)

𝜕𝑥
+ 𝑐𝜑2(𝑥, 𝑡; 𝑞) − 𝑐

  

(21) 

𝑁2[𝜑2(𝑥, 𝑡; 𝑞)] =
𝜕𝜑2(𝑥,𝑡;𝑞)

𝜕𝑡
− 𝑒

𝜕2𝜑2(𝑥,𝑡;𝑞)

𝜕𝑥 2
+ 𝑓

𝜕𝜑2(𝑥,𝑡;𝑞)

𝜕𝑥
+

𝑙𝜑2(𝑥, 𝑡; 𝑞) − 𝑙   

(22) 

Using the above definition, we construct the zeroth-order deformation equations 

(1 − 𝑞)𝐿[𝜑𝑘(𝑥, 𝑡; 𝑞) − 𝑧𝑘,0(𝑥, 𝑡)] = 𝑞ℎ𝑘𝑁𝑘[𝜑𝑘(𝑥, 𝑡; 𝑞)], 𝑘 = 1, 2 

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When 𝑞 = 0 and  𝑞 = 1, respectively, yields 

𝜑1(𝑥, 𝑡; 0) = 𝑧1,0(𝑥, 𝑡) = 𝑢0(𝑥, 𝑡) 

𝜑2(𝑥, 𝑡; 0) = 𝑧2,0(𝑥, 𝑡) = 𝑣0(𝑥, 𝑡) 

𝜑1(𝑥, 𝑡; 1) = 𝑢(𝑥, 𝑡) 

𝜑2(𝑥, 𝑡; 1) = 𝑣(𝑥, 𝑡) 

After expanding  𝜑𝑘(𝑥, 𝑡; 𝑞)in Taylor series with respect to 𝑞, it yields 

𝜑𝑘(𝑥, 𝑡; 𝑞) =  𝑧𝑘,0(𝑥, 𝑡) + ∑ 𝑧𝑘,𝑛(𝑥, 𝑡)𝑞
𝑛

+∞

𝑛=1

 

where 

𝑧𝑘,𝑛(𝑥, 𝑡) =
1

𝑛!

𝜕𝑛𝜑𝑖(𝑥, 𝑡; 𝑞)

𝜕𝑞𝑛
|𝑞=0 

The above series will converge at 𝑞 = 1, so that 

𝑢(𝑥, 𝑡) =  𝑧1,0(𝑥, 𝑡) + ∑ 𝑧1,𝑛(𝑥, 𝑡)

+∞

𝑛=1

 

𝑣(𝑥, 𝑡) =  𝑧2,0(𝑥, 𝑡) + ∑ 𝑧2,𝑛(𝑥, 𝑡)

+∞

𝑛=1

 

These are the solution of the nonlinear equation systems (16, 17). Now, we define the vector 

𝑧𝑘,𝑚⃗⃗⃗⃗⃗⃗⃗⃗ = {𝑧𝑘,0(𝑥, 𝑡), 𝑧𝑘,1(𝑥, 𝑡), … , 𝑧𝑘,𝑚(𝑥, 𝑡)} 

So, the nth-order deformation equations is 

𝐿[𝑧𝑘,𝑛(𝑥, 𝑡) − 𝜒𝑛𝑧𝑘,𝑛−1(𝑥, 𝑡)] = ℏ𝑘𝑅𝑘,𝑛(𝑧 𝑘,𝑛−1) 

where 

𝑅1,𝑛(𝑧 𝑘,𝑛−1) = (𝑧1,𝑛−1)𝑡 − 𝑎(𝑧1,𝑛−1)𝑥𝑥
+ 𝑏(𝑧1,𝑛−1)𝑥

+ 𝑐(𝑧2,𝑛−1) − 𝑐 + 𝑐𝑋𝑛 

𝑅2,𝑛(𝑧 𝑘,𝑛−1) = (𝑧2,𝑛−1)𝑡 − 𝑒(𝑧2,𝑛−1)𝑥𝑥
+ 𝑓(𝑧2,𝑛−1)𝑥

+ 𝑙(𝑧2,𝑛−1) − 𝑙 + 𝑙𝑋𝑛 

Now, the solution of the nth-order deformation equation for  𝑛 ≥ 1  becomes 

𝑧𝑘,𝑛(𝑥, 𝑡) = 𝜒𝑛𝑧𝑘,𝑛−1(𝑥, 𝑡) + ℏ𝑘 ∫ 𝑅𝑘,𝑛(

𝑡

0

𝑧 𝑘,𝑛−1) 𝑑𝜏 + 𝑝𝑘 

where the integration constants  𝑝𝑘 = 0. 

We, now successively have 

𝑧1,0(𝑥, 𝑡) = 𝛽 

𝑧1,1(𝑥, 𝑡) = −𝑐ℎ𝑡 

𝑧1,2(𝑥, 𝑡) = −𝑐ℎ𝑡 − 𝑐ℎ
2𝑡 −

𝑐𝑙ℎ2𝑡2

2
 

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z1,3(x, t) = −𝑐ℎ𝑡 − 2𝑐ℎ
2𝑡 − 𝑐𝑙ℎ2𝑡2 − 𝑐ℎ3𝑡 − 𝑐𝑙ℎ3𝑡2 −

𝑐𝑙2ℎ3𝑡3

6
 

z1,4(x, t) = −𝑐ℎ𝑡 − 3𝑐ℎ
2𝑡 − 𝑐𝑙ℎ2𝑡2 − 3𝑐ℎ3𝑡 − 3𝑐𝑙ℎ3𝑡2 −

𝑐𝑙2ℎ3𝑡3

2
− 𝑐ℎ4𝑡 −

3𝑐𝑙ℎ4𝑡2

2

−
𝑐𝑙2ℎ4𝑡3

2
−

𝑐𝑙ℎ2𝑡2

2
−

𝑐[3ℎ4𝑡4

24
 

z1,5(x, t) = −𝑐ℎ𝑡 − 4𝑐ℎ
2𝑡 − 2𝑐𝑙ℎ2𝑡2 − 6𝑐ℎ3𝑡 − 6𝑐𝑙ℎ3𝑡2 − 𝑐𝑙2ℎ3𝑡3 − 4𝑐ℎ4𝑡 − 6𝑐𝑙ℎ4𝑡2

− 2𝑐𝑙2ℎ4𝑡3 −
𝑐𝑙3ℎ4𝑡4

6
− 𝑐ℎ5𝑡 − 2𝑐𝑙ℎ5𝑡2 − 𝑐𝑙2ℎ5𝑡3 −

𝑐𝑙3ℎ5𝑡4

6
−

𝑐𝑙4ℎ5𝑡5

120
 

z2,0(x, t) = 0 

z2,1(x, t) = −𝑙ℎ𝑡 

z2,2(x, t) = −𝑙ℎ𝑡 − 𝑙ℎ
2𝑡 −

𝑙2ℎ2𝑡2

2
 

z2,3(x, t) = −𝑙ℎ𝑡 − 2𝑙ℎ
2𝑡 − 𝑙2ℎ2𝑡2 − 𝑙ℎ3𝑡 − 𝑙2ℎ3𝑡2 −

𝑙3ℎ3𝑡3

6
 

z2,4(x, t) = −𝑙ℎ𝑡 − 3𝑙ℎ
2𝑡 − 3𝑙ℎ3𝑡 − 3𝑙2ℎ3𝑡2 −

𝑙3ℎ3𝑡3

2
− 𝑙ℎ4𝑡 −

𝑙3ℎ4𝑡3

2
−

3𝑙2ℎ2𝑡2

2

−
3𝑙2ℎ4𝑡2

2
−

𝑙4ℎ4𝑡4

24
 

z2,5(x, t) = −𝑙ℎ𝑡 − 4𝑙ℎ
2𝑡 − 6𝑙ℎ3𝑡 − 𝑙3ℎ3𝑡3 − 4𝑙ℎ4𝑡 − 2𝑙3ℎ4𝑡3 − 2𝑙2ℎ2𝑡2 − 6𝑙2ℎ4𝑡2

−
𝑙4ℎ4𝑡4

6
− 𝑙ℎ5𝑡 − 𝑙3ℎ5𝑡3 − 6𝑙2ℎ3𝑡2 −

𝑙4ℎ5𝑡4

6
− 2𝑙2ℎ5𝑡2 −

𝑙5ℎ5𝑡5

120
 

The solutions then have the form 

𝑢(𝑥, 𝑡) = 𝑧1,0(𝑥, 𝑡) + 𝑧1,1(𝑥, 𝑡) + 𝑧1,2(𝑥, 𝑡) + 𝑧1,3(𝑥, 𝑡) + 𝑧1,4(𝑥, 𝑡) + 𝑧1,5(𝑥, 𝑡) + ⋯ 

𝑣(𝑥, 𝑡) = 𝑧2,0(𝑥, 𝑡) + 𝑧2,1(𝑥, 𝑡) + 𝑧2,2(𝑥, 𝑡) + 𝑧2,3(𝑥, 𝑡) + 𝑧2,4(𝑥, 𝑡) + 𝑧2,5(𝑥, 𝑡) + ⋯ 

By putting ℎ = −1, yields 

𝑢(𝑥, 𝑡) = 𝛽 + 𝑐𝑡 −
𝑐𝑙𝑡2

2
+

𝑐𝑙2𝑡3

6
−

𝑐𝑙3𝑡4

24
+

𝑐𝑙4𝑡5

120
− ⋯ 

              = 𝛽 + 𝑐𝑡 +
𝑐

𝑙
(−

𝑙2𝑡2

2
+

𝑙3𝑡3

6
−

𝑙4𝑡4

24
+

𝑙5𝑡5

120
− ⋯ ) 

              = 𝛽 + 𝑐𝑡 +
𝑐

𝑙
(1 − 𝑙𝑡 − 𝑒−𝑙𝑡) 

              = 𝛽 + 𝑐𝑡 +
𝑐

𝑙
− 𝑐𝑡 −

𝑐𝑒−𝑙𝑡

𝑙
 

             = 𝛽 +
𝑐

𝑙
−

𝑐𝑒−𝑙𝑡

𝑙
 

             =
𝛽𝑙 + 𝑐 − 𝑐𝑒−𝑙𝑡

𝑙
 

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https://doi.org/10.31436/iiumej.v22i1.1398 

 

𝑣(𝑥, 𝑡) = 0 + 𝑙𝑡 −
𝑙2𝑡2

2
+

𝑙3𝑡3

6
−

𝑙4𝑡4

24
+

𝑙5𝑡5

120
− ⋯ 

              = 1 − 𝑒−𝑙𝑡 

Using the physical data available in Table 1, the solution graph for 𝑢(𝑥, 𝑡)  and  𝑣(𝑥, 𝑡), as 
shown in Fig. 1 and Fig. 2. 

 

 

 

 

 

 

 

 

Fig. 1: The solution graph for  𝑢(𝑥, 𝑡) at certain position. 

 

 

 

 

 

 

 

 

Fig. 2: The solution graph for 𝑣(𝑥, 𝑡) at certain position. 

In RFR, the heat that is stored in the reactor can be used to preheat the feed. If the 

reaction temperature has been reached, the reactor system no longer needs a preheater for 

preheating the feed so that the process has high energy efficiency. This condition is 

illustrated by the graph 𝑢(𝑥, 𝑡).As shown in Figure 1, there is an increase in temperature 
within a certain time interval, after which the temperature does not increase or decrease but 

moves constantly. This condition has reached the steady state; thus, no preheater is needed. 

The 𝑣(𝑥, 𝑡)graph in Figure 2 illustrates the amount of concentration that reacts. After the 
concentration reacts entirely within a certain time, then the graph will move constantly. 

When this condition is reached, it means that the feed gas has completely reacted. Thus no 

more heat is released so that the temperature of the reactor becomes constant. 

4. CONCLUSION 

In this article, we consider the 1-D pseudo homogeneous model that describes the 

dynamics of temperature and conversion variabls in RFR without the cooling process. Here, 

we consider only the feed gas flow from left to the right end of RFR. Then, we solve this 

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https://doi.org/10.31436/iiumej.v22i1.1398 

 

model using the homotopy  analysis method. Based on the description above, it can be 

concluded that the solution of the dimensionless equation system describing the dynamics 

of temperature and conversion to methane oxidation using RFR without refrigeration is 

obtained as 𝑢(𝑥, 𝑡) =
𝛽𝑙+𝑐−𝑐𝑒−𝑙𝑡

𝑘
 and 𝑣(𝑥, 𝑡) = 1 − 𝑒−𝑙𝑡. The solution graph of 

𝑢(𝑥, 𝑡) illustrates an increase in temperature within a certain time interval, after which the 
temperature does not increase or decrease but it moves constantly. This condition has 

reached the steady state; thus, no preheater is needed. The solution graph of 𝑣(𝑥, 𝑡) describes 
the amount of concentration that reacts. After the concentration reacts entirely within a 

certain time, then the graph will move constantly. Future studies can be extended by 

considering the cooling process term in the 1-D pseudohomogeneous model. In the real 

problem, the heat energy expended during the methane oxidation should be controlled so 

that reactor overheating will not occur. 

ACKNOWLEDGEMENT  

The authors would like to thank the reviewers for invaluable suggestions and corrections. 

Also, the authors would like to thank Universitas Lampung for the facilities, funding 

through BLU Research Grants No. 2280/UN.21/PN/2019, and support during the 

research. 

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