Carbon-dioxide Emission Analysis in a Convective Cylindrical Pipe
RAMOSWHEU SOLOMON LEBELO
Education Department
Vaal University of Technology
Andries Pogieter Blvd, Vanderbijlpark, 1911
SOUTH AFRICA
SAMUEL OLUMIDE ADESANYA
Department of Mathematics and Statistics
Redeemer’s University
Ede 232101
NIGERIA
MOHANA SUNDARAM MUTHUVALU
Department of Fundamental and Applied Sciences
Universiti Teknologi PETRONAS
32610 Bandar Seri Iskandar, Perak
MALAYSIA
SAHEED OJO AKINDEINDE
Department of Mathematics and Statistical Sciences
Botswana International University of Science and Technology
Private Bag 16, Palapye
BOTSWANA
TUNDE ABDULKADIR YUSUF
Department of Mathematics
Adeleke University
Ede 23210
NIGERIA
ADESHINA TAOFEEQ ADEOSUN
Department of Mathematics
Federal College of Education
Iwo, Osun State
NIGERIA
Abstract: - This article analyzed carbon dioxide (CO2) emission from the combustion of reactive materials
modeled in a cylindrical domain. Reactive materials in this case involve carbon-containing substances that react
spontaneously with the oxygen of the surrounding environment under the influence of an exothermic chemical
reaction. In this analysis, the reactant (oxygen) consumption was neglected. The nonlinear differential equation
governing the problem was solved numerically using the Finite Difference Method embedded within the Maple
software. It was found that there are kinetic parameters that enhance the emission of CO2, like the rate of reaction,
and others, like the heat loss parameter, retard the CO2 emission during the exothermic chemical reaction.
Key-Words: - Carbon-dioxide emission, exothermic chemical reaction, convective heat loss, mass transfer,
combustible stockpile, cylindrical pipe.
Received: March 24, 2022. Revised: November 17, 2022. Accepted: December 15, 2022. Published: December 31, 2022.
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1 Introduction
Carbon dioxide (CO2) emission has drawn the
attention of many researchers because of its
devastating impact on the environment. Continued
CO2 emission in the atmosphere is considered one of
the challenges in the 21st century that threatens
human safety and health because it brings
degradation of the environment and is therefore a
natural disaster, [1,2]. The emission of CO2 is also
considered to be one of the fundamental factors that
breed climate change and is therefore one of the
world’s problems with the environment, [3,4].
Rehman et al. mentioned that it is necessary to limit
CO2 emissions to tackle climate change to sustain
worldwide economic growth, [4]. Fernández-
Amador et al. claim CO2 to be the main greenhouse
gas that is present in the atmosphere with longer
atmospheric life, though its global warming potential
per mole is lower compared to other greenhouse
gases, [5]. According to the study done by Distefano
et al., climate change led to water scarcity, which also
caused the shrinking of economic activities in the
countries under the Organization for Economic
Cooperation and Development (OECD), [6]. Further
studies by Kahia et al., Marques et al., and Mikayilov
et al. have shown that economic growth is to some
extent inhibited and undermined by climate change
to delay social development, [7,8,9]. Though CO2
emissions are detrimental to the environment, studies
show that economic growth goes hand in hand with
CO2 emissions, [4]. This is the reason China, believed
to be one of the CO2 emissions accelerators due to its
economic growth excellency, is determined to peak
CO2 emissions by 2030, and before 2060, it is
determined to achieve the neutrality of carbon, [4].
Since economic growth is accompanied by CO2
emissions, different activities of the economy such as
trade and urbanization, energy consumption,
industrial structure, fossil, and cleaner fuels
consumption structure, technological advancement,
and foreign investment enhance the emissions of
greenhouse gas, [10,11,12,13].
Although CO2 is a large contributor to greenhouse
gases, it is also usable in many instances, for
example, it is turned into a protein-rich powder for
animal feed, [14]. Other applications of CO2 are
found in dry blasters, fire extinguishers, soft drinks,
and as end products of fertilizers, fuels, and
materials, [15]. Lately, the study by Palm, Nilsson,
and Åhman showed that CO2 is also applicable in
drop-in plastics, [16], CO2 is also used for lowering
climate change in concrete mixtures, [17], and it is
also used for enhancing oil recovery, [18].
This study focuses on CO2 emission analysis in a
stockpile of reactive materials modeled in a
cylindrical pipe. This study aims to investigate the
factors (kinetic parameters) that enhance or reduce
the emission of CO2 during a spontaneous
combustion process due to an exothermic chemical
reaction of reactive materials in a stockpile. The CO2
emission analysis in a reactive stockpile modeled in
a rectangular slap was studied by Lebelo and
Makinde, [19]. Their study considered a steady-state
combustion process with heat loss to the ambient by
radiation. In, [20], the study was considered in a
cylindrical domain with heat loss to the environment
by both convection and radiation. The study in a
spherical domain was conducted in, [21], where the
heat loss to the surrounding environment was due to
convection. The study considered simultaneous CO2
emission, O2 consumption, and transient heat
stability during the spontaneous combustion of
reactive materials. The studies done above
considered a combustion process in a one-step
scenario, in this study, a two-step combustion process
analogous to the one taking place in fuel combustion
in an automobile exhaust system.
2 Problem Formulation
A one-dimensional energy and mass transfer partial
differential equations with constant thermal
conductivity are used to model the problem. The
reactant concentration is neglected. The problem is
modeled in a long cylindrical pipe illustrated in Fig.
1 below.
Fig. 1: Geometry of the Problem
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Coupled equations describing the heat and mass
transfer are given as follows:
(1)
(2)
The initial and boundary conditions for equations (1)
and (2) are respectively expressed as:
(3)
and
(4)
Convective heat loss to the surrounding
environment follows Newton’s law of cooling,
expressed as
, where is the heat
transfer coefficient and is the thermal conductivity,
is the ambient temperature, is the initial
temperature, and is the cylinder’s thermodynamic
temperature. The mass transfer at the cylinder’s
surface is given by
, where is the CO2
transfer coefficient and is the diffusivity of CO2 in
the cylinder. is the CO2 emission concentration,
is the concentration of CO2 in the environment,
, and are the density and the specific heat at
constant pressure. and are, respectively, the
heat of the reaction, and the Arrhenius constant, for
the first step of the reaction, and and represent
the heat of the reaction and the Arrhenius constant,
respectively, for the second step. In addition, is the
Boltzmann’s constant, is the vibration frequency,
is Planck’s number, and are, respectively, the
activation energy for the first step and the second
step, is the universal gas number, represents the
type of chemical kinetics, where is for the
sensitized (light-induced is for the Arrhenius
and
is for the bimolecular kinetics and lastly,
is the heat loss parameter, where is the
inner surface.
The introduction of dimensionless quantities on
equations (1) – (4) is done as follows:
(5)
Equations (1) – (4) then become
(6)
(7)
The corresponding initial and boundary conditions
are respectively expressed as follows:
and
(8)
The dimensionless parameters are described as
follows:
and are the dimensionless temperature and
dimensionless initial temperature respectively, is
the Frank-Kamenetskii parameter also called the
reaction rate parameter, are, respectively
parameters for, the dimensionless activation energy,
the dimensionless radial distance, the activation
energy ratio, and the two-step low-temperature
oxidation parameter. are, respectively,
the Biot and CO2 Biot numbers, CO2 emission rate
parameter, CO2 two-step emission rate parameter,
and dimensionless time. is the dimensionless heat
loss parameter
2.1 Steady-state combustion formulation
As time , the combustion process attains a
steady state situation, where the energy and mass
transfer equations are temperature independent. The
expressions for the energy and mass transfer
equations, and their boundary conditions are
respectively expressed as follows:
(9)
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(10)
The boundary conditions are:
(11)
The Runge-Kutta (RK-4) coupled with Shooting
technique is applied to numerically solve the steady
state problem.
3 Problem Solution
The problem to be solved in this case concentrates on
equation (7) and its corresponding boundary and
initial conditions. The focus is on CO2 analysis in a
spontaneous combustible process of reactive
materials. Following, [22], equation (7) was solved
numerically using the Finite Difference Method
(FDM) due to the nonlinearity of the equation. The
following FDM expression describes the solution
algorithm.
(12)
The temperature at current time step is expressed
by , and describes the new temperature. The
space position in the direction is denoted by . The
(j + ) describes the intermediate time level, where
. To allow larger time steps that can
accommodate any value, we let , to determine
the implicit terms.
The rearrangement of equation (12) after
multiplying through by and writing
gave
the following equation:
(13)
The initial and boundary conditions are as follows:
(14)
The FDM is embedded withing Maple software,
this method was used to solve equations (13) to (14)
using the software.
The Shooting‐Runge–Kutta Method, also
embedded within Maple software, is applied to solve
the steady-state equation (10), the algorithm is as
follows:
Let
Then, equation (10) is transformed into a first order
differential equation as follows:
(15)
The corresponding boundary conditions are:
(16)
4 Results and Discussion
The results obtained in the CO2 analysis in a
combustible stockpile modelled in a cylindrical
domain, are presented, and discussed in this section.
The steady-state combustion process is looked at
first, then followed by the unsteady-state condition.
In both processes, the following dimensionless
parameters’ settings were applied:
4.1 Steady-state combustion process
The following Figs. 2 and 3 depict the steady-state
attainment as
.
Fig. 2: 2-D steady-state illustration
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It is clearly observable that the steady-state condition
is attained as time increases from zero to infinity.
Table 1 indicates that the temperature values reach a
constant number (0.15123) over time .
This shows that the CO2 constantly keeps on being
emitted when the combustion process continues even
if the time component is ignored. Fig. 2 illustrates too
that the CO2 is highly concentrated at the center of
the stockpile and declines towards the surface of the
combusting material, due to its high rate of emission
to the surrounding environment.
Table 1: Steady-state temperatures as at
Time
Temperature
4.2 Unsteady-state combustion process
Each kinetic parameter mentioned above was varied
to study the process of CO2 emission under its
influence. The effects of the following parameters on
CO2 emission are depicted in Figs. 4-14.
Figs. 4-11 illustrate the effects of
and . It is observed that an increase in the
parameters mentioned brings a corresponding
increase in the emission of CO2. The CO2 emission
corresponds to the temperature levels increments, as
heat is released to the environment, during the
exothermic chemical reaction. Both heat release and
CO2 emission are detrimental to the environment.
What is more interesting is to observe that the CO2
emission is least in sensitized kinetics () as
compared to bimolecular kinetics , as
indicated by Fig. 10. Figs. 4 and 7, show that the
reaction rate and the CO2 emission rate parameters
contribute much to the emission of the greenhouse
gas. From Figs. 5, 8 and 11, we observe that the two-
step low-temperature oxidation parameter, CO2 two-
step emission rate parameter, and dimensionless
initial temperature, respectively, have almost the
same effect on the CO2 emission. It is observed also
from Figs. 6 and 9 that the activation energy, and the
activation energy ratio also show the same effect on
the CO2 emission. Tables 2-9 confirm the CO2
emission increment as the magnitudes of the
mentioned parameters are increased. The Sherwood
number
which is the CO2 rate transfer
at the cylinder’s wall to the surrounding environment,
also shows that the CO2 emission rate increases with
increasing values of the parameters under the study.
For example, Table 2 shows that as is increased
from 0.1 to 0.5, the CO2 concentration increases from
1.06474 to 1.61785. The table also indicates the
increases from 0.06474 to 0.61785. The negative sign
means that the CO2 is lost to the environment at the
surface of the cylinder.
Fig. 4: effect on CO2 emission
Fig. 5: effect on CO2 emission
Fig. 3: 3-D steady-state illustration
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Fig. 6: effect on CO2 emission
Fig. 7: effect on CO2 emission
Fig. 8: effect on CO2 emission
Fig. 9: effect on CO2 emission
Fig. 10: effect on CO2 emission
Fig. 11: effect on CO2 emission
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Table 2: Effects of on CO2 emissions (Fig. 4)
variation
CO2 emissions
values
0.1
1.06474
-0.06474
0.2
1.15079
-0.15079
0.3
1.26387
-0.26387
0.4
1.41344
-0.41344
0.5
1.61785
-0.61785
Table 3: on CO2 emissions (Fig. 5)
variation
CO2 emissions
values
0.1
1.06474
-0.06474
0.2
1.06483
-0.06483
0.3
1.06493
-0.06493
0.4
1.06502
-0.06502
0.5
1.06512
-0.06512
Table 4: on CO2 emissions (Fig. 6)
variation
CO2 emissions
values
0.1
1.06474
-0.06474
0.2
1.06514
-0.06514
0.3
1.06553
-0.06553
0.4
1.06593
-0.06593
0.5
1.06632
-0.06632
Table 5: on CO2 emissions (Fig. 7)
variation
CO2 emissions
values
1
1.06474
-0.06474
2
1.12948
-0.12948
3
1.19425
-0.19425
4
1.25896
-0.25896
5
1.32369
-0.32369
Table 6: on CO2 emissions (Fig. 8)
variation
CO2 emissions
values
1
1.06474
-0.06474
2
1.06987
-0.06987
3
1.07499
-0.07499
4
1.08012
-0.08012
5
1.08525
-0.08525
Table 7: Effects of on CO2 emissions (Fig. 9)
Table 8: on CO2 emissions (Fig. 10)
variation
CO2 emissions
values
-2
1.06194
-0.06194
0
1.06416
-0.06416
0.5
1.06474
-0.06474
Table 9: on CO2 emissions (Fig. 11)
variation
CO2 emissions
values
0.1
1.06474
-0.06474
0.2
1.07123
-0.07123
0.3
1.07832
-0.07832
0.4
1.08605
-0.08605
0.5
1.09448
-0.09448
We now consider the kinetic parameters that were
observed to lower the CO2 emission during an
exothermic chemical reaction. The parameters are
and . An increase in the values of these
parameters shows a decline in CO2 emissions. The
scenario is depicted in Figs. 12-14. This observation
also confirms that the emissions of CO2 are very high
in materials that combust spontaneously. Figs. 12 and
14 show that the heat loss and the Biot number
parameters have a lesser effect on the retardation of
CO2 emission as compared to the CO2 Biot number,
illustrated in Fig. 13. Tables 10-12 confirm also what
Figs. 12-14 demonstrate. These parameters are
helpful to the environment because the
discouragement of CO2 emission also means a lesser
heat emission to the environment to reduce the fast-
growing climate change that has adversely affected
the weather globally.
variation
CO2 emissions
values
1
1.06474
-0.06474
2
1.06514
-0.06514
3
1.06554
-0.06554
4
1.06595
-0.06595
5
1.06636
-0.06636
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Fig. 12: effect on CO2 emission
Fig. 13: effect on CO2 emission
Fig. 14: effect on CO2 emission
Table 10: on CO2 emissions (Fig. 12)
variation
CO2 emissions
values
1
1.06474
-0.06474
2
1.06411
-0.06411
3
1.06356
-0.06356
4
1.06308
-0.06308
5
1.06264
-0.06264
Table 11: on CO2 emissions (Fig. 13)
variation
CO2 emissions
values
0.1
1.06474
-0.06474
0.2
1.03237
-0.06474
0.3
1.02158
-0.06474
0.4
1.01618
-0.06474
0.5
1.01295
-0.06474
Table 12: on CO2 emissions (Fig. 14)
variation
CO2 emissions
values
1
1.06474
-0.06474
2
1.06302
-0.06302
3
1.06247
-0.06247
4
1.06219
-0.06219
5
1.06202
-0.06202
5 Conclusion
In this study, the analysis of CO2 emission in a
stockpile of reactive materials modeled in a
cylindrical domain was considered. The kinetic
parameters embedded within the differential equation
governing the problem were used to bring an
understanding of the CO2 analysis process. Maple
software was used to give the numerical solutions. It
was found that the parameters and
enhance the emission of CO2 during an
exothermic chemical reaction. These parameters are
not friendly to the environment because the CO2
emissions have contributed much to climate change.
On the other hand, the parameters such as
and
were found to reduce the CO2 emission during a
combustion process. The lessening of CO2 emission
is helpful to reduce climate change. The
mathematical approach to this study allowed the CO2
analysis in a quicker and cheaper manner as
compared to the experimental one. This study can be
extended to a situation where the stockpile is
modeled in a spherical domain.
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
RS Lebelo, SO Adesanya carried out the
conceptualization, simulation, the overall writing,
language editing and plagiarism checking.
MS Muthuvalu, SO Akindeinde, numerical methods
presentation and Algorithms implementation.
TA Yusuf, AT Adeosun, responsible discussion of
results and conclusive remarks.
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