Indonesi
an
Journa
l
of El
ect
ri
cal Engineer
ing
an
d
Comp
ut
er
S
ci
ence
Vo
l.
23
,
No.
3
,
Septem
ber
2021
, pp.
1590
~
1601
IS
S
N: 25
02
-
4752, DO
I: 10
.11
591/ijeecs
.v
23
.i
3
.
pp
1590
-
1601
1590
Journ
al h
om
e
page
:
http:
//
ij
eecs.i
aesc
or
e.c
om
Math
ematical m
odeling
and
algorithm
fo
r calcula
tion of
therm
ocatalyti
c proce
ss of p
ro
du
cing n
anomate
rial
Bakh
tiy
ar
Is
mailo
v
1
, Z
ha
n
at U
m
ar
ova
2
,
Kha
ir
ull
a Ism
ailov
3
, Aibars
ha
D
os
m
akan
bet
ova
4
,
Sa
ule
Meldeb
ekova
5
1,2,3
Depa
rtment
o
f
Inform
at
ion
S
y
stems
and
M
ode
li
ng,
M.
Aue
zov S
outh
Kaz
akhst
a
n
Univer
sit
y
,
Ka
za
khstan
4
Depa
rtment Technological m
achine
s a
nd
equ
ip
m
ent
,
M.
Auez
o
v
South Ka
z
akhs
ta
n
Univ
ersity
,
Kaz
akhsta
n
5
Depa
rtment of
Com
puti
ng
tech
nolog
y
and
soft
ware
,
M.
Aue
zo
v
South Ka
z
akhs
ta
n
Univ
ersity
,
Kaz
akh
stan
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
A
pr
1
8
,
2021
Re
vised
Ju
l
8
,
2021
Accepte
d
Aug
5
,
2021
At
pre
sent,
whe
n
construc
ti
ng
a
m
at
hemati
c
al
d
esc
ription
of
th
e
p
y
ro
l
y
s
is
rea
c
tor,
par
t
ia
l
d
iffe
ren
ti
a
l
equa
t
i
ons
for
the
components
of
the
g
as
phase
and
the
ca
t
aly
st
phas
e
are
used.
In
the
well
-
known
works
on
m
odel
ing
p
y
rol
y
sis,
the
obt
ai
ned
m
odel
s
are
applic
ab
le
onl
y
fo
r
a
n
ar
row
ran
ge
of
ch
ange
s
in
th
e
proc
ess
par
amet
ers,
the
g
eometr
ic
dimensions
ar
e
conside
r
ed
co
nstant
.
The
art
i
cl
e
poses
th
e
ta
sk
of
cr
eat
ing
a
complex
m
at
hemati
c
al
m
odel
with
addi
ti
on
al
te
rm
s,
ta
king
int
o
a
ccount
nonli
n
ea
r
e
ffe
ct
s,
where
th
e
geometr
ic
dimensions
of
t
he
app
aratus
and
oper
at
ing
cha
r
ac
t
eri
sti
cs
var
y
over
a
wide
ran
ge.
An
an
aly
t
ic
a
l
m
et
hod
has
bee
n
dev
el
oped
for
th
e
impleme
nta
ti
on
of
a
m
at
hemati
c
al
m
odel
of
cata
l
y
t
i
c
p
y
rol
y
sis
of
m
et
hane
for
th
e
pr
oduct
ion
of
nanomate
ri
al
s
in
a
cont
inuous
mode
.
The
diff
erenti
al
equ
at
ion
f
or
gase
ous
components
with
ini
ti
a
l
and
boundar
y
cond
it
ions
of
the
thi
rd
t
y
p
e
is
red
uce
d
to
a
d
im
ensionl
e
ss
form
with
a
sm
al
l
val
u
e
of
the
pec
l
et
criter
ion
with
a
form
fac
tor
.
It
is
sho
wn
tha
t
the
la
pl
ac
e
tra
nsform
m
et
hod
is
m
ai
n
l
y
suita
ble
for
thi
s
ca
se
,
whi
ch
is
applica
b
le
b
oth
for
diff
ere
n
t
ia
l
equations
for
solid
-
phase
components
and
ca
l
culati
on
in
a
per
iodic
m
ode.
The
ade
qua
c
y
of
the
m
odel
result
s with
the
known e
xper
ime
nta
l
data is
ch
ecked
.
Ke
yw
or
ds:
Laplace
tra
nsf
or
m
Ma
them
a
ti
cal
m
od
el
ing
Nanom
at
erial
s
Nu
m
erical
algo
rithm
Perio
dic a
nd c
on
ti
nu
ous m
odes
Ther
m
ocatal
yt
i
c p
yr
olysi
s
This
is an
o
pen
acc
ess arti
cl
e
un
der
the
CC
B
Y
-
SA
l
ic
ense
.
Corres
pond
in
g
Aut
h
or
:
Zha
nat U
m
arova
Dep
a
rtm
ent o
f Info
rm
at
ion
System
s an
d
M
od
el
ing
M. A
uez
ov S
outh
Kaza
khsta
n Un
i
ver
sit
y
Tau
ke kh
a
n
a
ve
nu
e
,
5
, Shym
ken
t,
K
aza
khst
an
Em
a
il
:
zhan
at
.u
m
aro
va@
a
ue
zov.ed
u.kz
1.
INTROD
U
CTION
Ma
them
a
ti
cal
m
od
el
ing
of
c
hem
ic
al
te
chn
olo
gical
process
es
is
one
of
th
e
sci
ence
-
i
nten
sive
areas
of
knowle
dge.
A
la
rg
e
num
ber
of
m
at
he
m
at
ic
a
l
m
od
el
s
known
in
the
sci
entifi
c
li
te
ratur
e
are
associat
ed
wi
th
the
conditi
ons
f
or
perform
ing
the
process
with
f
ixed
par
am
et
ers
-
the
dim
ension
s
a
nd
ge
om
etr
y
of
the
a
ppar
at
us
,
the
val
ues
of
tem
per
at
ure
and
press
ur
e
,
the
c
om
po
sit
ion
of
the
fe
edstoc
k,
an
d
the
ty
pe
of
c
at
al
yst
.
Re
searche
rs
pro
ve
the
so
l
va
bili
ty
of
m
od
el
s
and
the
c
onve
r
gen
ce
of
the
ap
plied
nu
m
erical
m
et
ho
ds
in
diff
e
re
nt
ways,
so
we
can
sta
te
the
s
pread
of
m
e
tho
ds
for
de
sign
i
ng
an
d
i
m
ple
m
enting
m
at
he
m
at
ic
a
l
m
od
el
s.
This
ci
rcu
m
stan
ce
create
s
dif
ficult
ie
s
in
the
app
li
cat
io
n
an
d
com
par
iso
n
o
f
the
res
ults
of
the
i
m
ple
m
e
ntati
on
of
m
od
el
s
ob
ta
ined
by
va
rio
us
researc
her
s
.
I
n
the
fiel
d
of
chem
ic
al
eng
ine
erin
g,
ne
w
cha
ll
eng
es
are
em
erg
i
ng
that
require
qu
ic
k
so
luti
ons
.
The
creati
on
of
hydro
car
bon
-
base
d
na
no
m
ater
ia
ls
(h
e
reina
f
te
r
NM)
is
of
gr
ea
t
inte
rest.
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sc
i
IS
S
N:
25
02
-
4752
Mathe
m
atical
modeli
ng
and alg
or
it
hm for
c
alculatio
n of t
he
rmocat
alyti
c…
(
Bakh
ti
yar
Isma
il
ov
)
1591
In
this
a
rtic
le
,
to
co
ntinu
e
t
he
ideas
of
w
orks
[
1
]
-
[
5],
m
at
he
m
at
ic
a
l
m
od
el
ing
of
low
-
te
m
per
at
ure
synthesis
of
ca
rbo
n
na
nostr
uc
tures
is
ca
rr
ie
d
out.
T
he
re
su
l
ts
of
a
com
pr
e
hensi
ve
st
ud
y
of
t
he
str
uctu
r
e
an
d
m
or
phology
of
m
et
a
l
nan
opowde
rs
synth
esi
ze
d
by
the
m
e
t
hod
of
the
el
ect
ric
exp
losi
on
of
c
onduct
ors,
wh
i
c
h
wer
e
us
ed
as
c
at
al
yst
s,
are
presented
.
I
n
the
course
of
the
exp
e
rim
ents,
t
he
te
ch
no
l
og
ic
al
par
am
et
ers
op
ti
m
al
for
the
lo
w
-
te
m
per
at
ur
e
gro
wth
of
ca
rbo
n
nanostr
uctu
re
s
wer
e
determ
ined.
I
n
pa
rtic
u
la
r,
t
he
ex
pe
r
i
m
ental
reg
im
es
of
lo
w
-
te
m
per
at
ur
e
chem
ic
a
l
vap
or
de
posit
io
n
of
car
bon
nanotu
bes
us
i
ng
ir
on
an
d
nickel
nan
opow
der
s
as
a
cat
al
yst
wer
e
fou
nd
for
the
first
ti
m
e.
Mod
e
r
n
sci
entifi
c
de
velo
pm
ents
in
the
fiel
d
of
synthesis
of
NM
m
at
he
m
a
ti
cal
m
od
el
i
ng
us
in
g
the
cat
al
ytic
pyro
ly
sis
m
et
ho
d
are
descr
ibe
d
by
tw
o
appr
oach
es:
M
od
el
in
g
t
he
f
or
m
at
ion
of
st
r
uctu
ral
bl
ocks
and
the
proce
s
s
of
NM
grow
t
h,
w
hich
pro
vi
des
for
t
rack
i
ng
the str
uctu
re
of N
M
(m
or
phol
og
y
)
at
the
m
icr
o
-
le
vel;
M
od
el
in
g
th
e
process
of
f
orm
ing
str
uctu
ra
l
blo
c
ks
at
the
m
acro
le
vel,
pro
vid
in
g
a
n
a
ssessm
ent
of
t
he
perform
ance o
f
the a
pp
a
ratus
.
These
a
ppr
oac
hes
are
well
know
n,
a
nd
t
he
y
are
the
m
ain
m
et
ho
ds
of
a
syst
e
m
at
ic
a
ppr
oach
t
o
m
od
el
ing
the
m
ai
n
processe
s
of
chem
ic
al
pro
du
ct
io
n.
T
he
work
s
[6
]
-
[
7]
desc
ribe
a
new
te
ch
no
l
ogy
f
or
pro
du
ci
ng
ca
r
bon
nanotu
bes
and
pure
hydro
ge
n
by
cat
al
yt
ic
pyro
ly
sis
of
hydroca
r
bon
ra
w
m
at
erials,
th
e
introd
uction
of
wh
ic
h
opens
up
wide
oppor
tun
it
ie
s
du
e
t
o
it
s
cost
-
eff
ect
i
ven
e
ss.
I
n
ou
r
op
i
nion,
on
e
s
houl
d
disti
ngu
ish
bet
ween
"
fast"
an
d
"sl
ow"
(
dif
fusion
ty
pe
)
te
ch
no
l
og
ic
al
pr
oc
esses.
F
or
ty
pe
1,
rat
her
high
flo
w
rates
are
cha
r
act
erist
ic
,
as
a
res
ult
of
wh
i
ch
vortex
an
d
tur
bule
nt
fl
ows
a
re
reali
zed.
M
odel
ing
of
s
uch
pro
blem
s
is
giv
en
,
for
e
xam
ple,
in
w
orks
[
8
]
-
[
11]
.
T
o
si
m
ula
te
te
chnolog
ic
al
process
es,
the
m
et
ho
dolo
gy
of
works
[
12
]
-
[
14]
can
be
ap
pl
ie
d.
Di
ff
e
ren
c
e
schem
es
for
pa
raboli
c
eq
ua
ti
on
s
with
no
n
-
sm
oo
th
bo
unda
ry
conditi
ons
a
nd
rig
ht
-
ha
nd
sid
es
are
m
od
ifie
d
in
them
.
The
pro
blem
s
of
t
he
c
orrectnes
s
of
the
i
de
nti
ficat
io
n
pro
blem
fo
r
th
e
rig
ht
-
ha
nd
side
of
a
par
a
bo
li
c
equ
at
io
n
w
ere
s
olv
e
d
i
n
[
15
]
-
[
20
]
.
Re
ce
ntly
,
ne
w
w
orks
ha
ve
app
ea
re
d
on
th
e
m
od
ern
iz
at
io
n
of
Sm
olu
ch
owski'
s
theor
y
[
21
]
-
[
22
]
,
as
ap
plied
to
the
prob
le
m
s
of
chem
ic
al
te
chnolo
gy
f
or
the
sy
nth
esi
s
of
pa
r
ti
cl
es.
I
n
these
w
orks,
t
he
pro
blem
s
of
fl
ows
m
od
el
ing
in
te
ch
no
l
ogic
al
dev
ic
es
a
nd
s
cal
ing
w
he
n
c
al
culat
ing
the
sp
eed
of
pa
r
ti
cl
es
in
the
bin
a
ry
ag
gr
e
ga
ti
on
are
c
ons
idere
d.
Diff
e
re
ntial
eq
uations
of
var
i
ou
s
f
or
m
s
are
widely
us
e
d
t
o
desc
ribe
t
he
py
ro
ly
sis
proces
s
an
d
they
i
nclud
e
,
a
m
on
g
oth
e
r
thin
gs
,
e
xpressi
on
s
t
hat
ta
ke
into
acc
ount
th
e
influ
e
nce
of
the
co
ncen
trat
i
on
s
of
inte
rm
e
diate
rad
ic
al
s,
acco
r
ding
to
the
sta
ges
of
tra
ns
f
orm
at
io
n
of
m
at
e
rial
s
to f
inis
he
d
NM.
C
urren
t
ly
,
there
a
re
m
od
el
s
of
cat
al
yt
ic
p
yroly
sis
of
natu
ral
gas
tog
et
her
with
the
f
or
m
at
ion
of
NM
in
a
con
ti
nuous
and
va
riable
tubular
reactor
.
H
ow
e
ver,
these
m
od
el
s
are
no
t
rig
oro
us
ly
checke
d
f
or
their
a
de
qu
acy
,
si
nce
they
char
act
eri
ze
the
process
unde
r
stud
y
on
ly
f
or
certai
n
co
ndi
ti
on
s
an
d
ini
ti
al
con
t
ro
l
data.
T
he
sci
entifi
c
novelty
of
this
a
rt
ic
le
is
the
m
et
ho
d
of
ob
ta
ini
ng
a
n
a
naly
ti
cal
rep
re
sentat
ion
f
or
the
ca
rbo
n
c
oncentrati
on
with
crit
erio
n
par
a
m
et
ers,
wh
ic
h
al
lo
ws
c
om
par
ing
the
obta
ined
m
od
el
ing
res
ults
with
the
data
of
othe
r
aut
hors,
w
it
h
the
justi
ficat
ion
of
the
existe
nce
and
uniq
uen
e
s
s
of
t
he
so
l
ution
of
the
f
orm
ula
te
d
init
ia
l
-
bo
undar
y
value
pro
blem
fo
r
th
e
diff
e
re
ntial
eq
ua
ti
on
.
2.
THE
PROPO
SED
RESEA
RCH
METH
OD
2.1.
Dev
el
opme
nt
of
a
m
athema
tical m
odel
of therm
oc
ataly
tic p
yro
l
ys
is
Let
'
s
wr
it
e
th
e
m
at
he
m
a
ti
ca
l
m
od
el
of
th
e
therm
ocatal
yt
ic
pyro
ly
sis
proces
s
in
t
he
fo
ll
owin
g
form
[7
]
:
∂
∂
+
1
∂
∂
=
∂
2
∂
2
+
∂
2
∂
2
+
∂
∂
+
(1)
(
=
0
,
,
)
=
0
(
,
)
(2)
(
=
0
,
)
=
(
)
(3)
∂
∂
|
=
=
0
(4)
∂
∂
|
−
0
=
∑
=
1
2
, if
=
/
2
(5)
∂
∂
|
−
0
=
0
, if
≠
/
2
(6)
∂
∂
|
=
/
2
=
0
(7)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
23
, N
o.
3
,
Se
ptem
ber
2
02
1
:
15
90
-
16
01
1592
Wh
e
re
,
D
ap
-
le
ng
t
h
an
d
dia
m
et
er
of
t
he
a
pp
a
ratus
,
m
;
D
i
-
m
olecul
ar
diffusi
on
c
oeff
ic
ie
nts
f
or
eac
h
i
,
m
2
/se
c;
-
init
ia
l
con
ce
ntrati
on
of
the
gas
c
ompone
nt
at
the
r
eact
or
inlet
,
m
ol/
m
3
;
∑
=
1
2
-
the
sum
of
the
rates
of
form
a
ti
on
an
d
co
nsu
m
pt
ion
of
the
i
-
com
po
ne
nt
of
the
gas
ph
as
e
by
re
act
ion
s
ta
kin
g
place
on
th
e
cat
al
yst
su
rf
ac
e,
m
ol/(
m
3
*s
ec);
-
st
oichiom
et
ric
coeffic
ie
nt
of
the
i
-
com
po
ne
nt
of
t
he
ga
s
phase
i
n
th
e
j
-
su
r
face
reacti
on;
W
j
-
j
-
s
urfa
ce
reacti
on
ra
te
,
m
ol/
(m
3
*
s
ec);
(
1)
-
a
m
od
ifie
d
syst
e
m
of
e
quat
ions
f
or
t
he
c
o
n
c
e
n
t
r
a
t
i
o
n
s
o
f
g
a
s
c
o
m
p
o
n
e
n
t
s
w
i
t
h
a
n
a
d
d
i
t
i
o
n
a
l
t
e
r
m
t
ha
t
t
a
k
e
s
i
nt
o
a
c
c
o
u
n
t
t
h
e
e
f
f
e
c
t
o
f
t
h
e
c
o
n
c
e
n
t
r
a
t
i
o
n
s
o
f
i
n
t
e
r
m
e
di
a
t
e
r
a
d
i
c
a
l
s
;
(
2
)
i
n
i
t
i
a
l
c
o
n
d
i
t
i
o
ns
;
(
3
)
,
(
4
)
,
(
5
)
,
(
6
)
,
(
7
)
-
b
o
u
n
d
a
r
y
c
o
n
d
i
t
i
o
n
s
f
o
r
(
1
)
,
ta
king
into
account t
he
in
f
lux
of
reag
e
nts
int
o
the a
pp
a
r
at
us
to
gether
w
it
h
the init
ia
l
m
ixtur
e, the
a
r
rival
-
dep
a
rtu
re
o
f gas
-
ph
a
se com
pone
nts as a res
ult of
reacti
ons
on a cata
ly
st l
ocated in
t
he
ce
nter
of th
e ap
pa
ratus.
The
equati
on
f
or
the ch
a
nge i
n
t
he
c
on
ce
ntrati
ons
of the
dis
persed
phase c
ompone
nts:
∂
,
∂
=
,
(8)
W
he
re
C
k,i
-
s
urface
c
on
ce
ntr
at
ion
of
a
so
li
d
phase
c
om
po
ne
nt,
re
duce
d
to
un
it
y
cat
al
yst
m
ass,
m
ol/
kg
;
J
k,i
-
the
rate
of
for
m
at
ion
or
co
nsum
ption
of
the
i
-
com
po
ne
nt
of
t
he
dis
perse
d
phase
by
rea
ct
ion
s
procee
di
ng
on
t
he
cat
al
yst
su
r
face, m
ol
/(k
g*sec).
The
i
niti
al
co
ndit
ion
f
or
(
8)
ha
s the
form
:
,
(
=
0
)
=
,
0
(9)
Wh
e
re
,
0
-
the
c
oncent
rati
on
of
the
i
-
c
om
po
ne
nt
of
the
dis
per
s
ed
phase
at
t
he
init
ia
l
m
o
m
ent
of
ti
m
e,
m
ol/kg
(at
the
init
ia
l
m
o
m
ent
of
ti
m
e,
the
c
on
c
entr
at
ion
of
al
l
c
om
po
nen
ts
of
t
he
disp
e
rse
d
phase
is
zer
o,
the
cat
al
yst
act
ivit
y
is
m
a
xim
u
m
).
For
t
he
c
onve
nienc
e
of
s
olv
i
ng
a
nd
com
par
in
g
the
res
ults
with
the
re
su
lt
s
of
oth
e
r
auth
or
s
,
we
tu
rn
t
o
dim
ension
le
ss
va
riabl
es:
le
t
x
*
-
the
char
a
ct
e
risti
c
li
near
siz
e
of
t
he
ap
pa
ratus;
∗
-
the
aver
a
ge
val
ue
of
the
c
on
ce
ntr
at
ion
of
the
co
m
po
nen
t;
t
*
-
the
aver
a
ge
tim
e
of
the p
r
ocess;
r
*
-
the
char
act
e
risti
c
value
of
the
ra
dial
directi
on.
Dim
ension
le
ss
values
are
m
ark
e
d
with
a
da
sh
.
Th
us
,
we
obta
in
the
f
ollo
wing
syst
e
m
o
f
e
qu
a
ti
on
s
for
(1):
∂
‾
∂
‾
+
∂
‾
∂
‾
=
∂
2
‾
∂
‾
2
+
(
∂
2
‾
∂
2
+
1
∂
‾
∂
)
+
‾
−
1
‾
−
1
+
‾
+
1
‾
+
1
+
‾
(10)
Wh
e
re
=
∗
1
∗
-
dim
ensio
nl
ess
nu
m
ber
char
act
e
rizi
ng
the
le
ngth
of
t
he
a
pparat
us
;
K
xi
,
K
ri
–
a
re
t
he
diffusi
on
par
a
m
et
ers
of
t
he
com
po
ne
nts
over
x
,
r
;
‾
−
1
=
−
1
∗
−
1
∗
,
‾
+
1
=
−
1
∗
+
1
∗
-
dim
ension
le
ss
c
on
ce
ntrati
on
factors;
‾
=
∗
‾
∗
-
the
re
la
ti
ve
rate
of
f
or
m
at
ion
of
t
he
i
-
com
ponen
t
in
the
reacti
on
in
the
gas
ph
as
e;
within
the
m
eaning
of
t
he
n
otati
on
-
‾
0
=
0
,
+
1
=
0
. Ini
ti
al
an
d
bo
unda
ry co
nd
it
io
ns
(2)
-
(7),
e
quat
io
ns
a
nd
c
onditi
ons
for
the
so
li
d
phase
rem
ai
n
pract
ic
al
ly
un
cha
ng
e
d.
The
m
ath
e
m
atical
m
od
el
-
(2)
-
(
9)
-
(
10)
can
be
reali
zed
us
in
g
analy
ti
cal
o
r n
um
erical
m
e
tho
ds.
At the
f
irs
t st
age, we
will
conside
r
a
sim
pler
m
od
e
-
c
onti
nuous.
In
t
his
w
ork,
a
m
at
he
m
at
ic
a
l
m
od
el
of
the
c
at
al
yt
ic
pyro
lysis
p
r
ocess
will
be
i
m
ple
m
ented
ba
sed
on
the
la
place
tran
sform
(
s
m
al
l
init
ia
l con
centrat
ion
of
the
gas
c
om
po
ne
nt)
a
nd a n
um
erical
m
et
hod
(lar
ge va
lues
of
the
init
ia
l
con
ce
ntrati
on
).
The
or
et
ic
al
an
d
pr
act
ic
al
quest
ion
s
of
t
he
so
lva
bili
ty
and
uniq
ue
ness
of
t
he
so
luti
on
of
dif
fer
e
ntial
equ
at
ion
s
,
wh
ic
h
ar
e
m
os
t
of
te
n
use
d
in
the
pro
blem
con
side
r
ed
in
this
paper,
are
descr
i
bed
i
n
suffici
ent
detai
l
in
[
23
]
-
[
24
]
.
T
he
eq
uatio
n
f
or
the
cha
nge
in
the
co
ncen
trat
ion
of
the
gas
ph
a
se
com
po
ne
nt in
the
on
e
-
dim
ensi
on
al
f
or
m
ula
tio
n t
akes
the
form
:
∂
∂
+
1
∂
∂
=
D
md
∂
2
∂
2
+
∗
,
0
<
<
,
0
<
<
(11)
In
it
ia
l co
nd
it
io
n:
(
,
0
)
=
0
(
)
(12)
Border
con
diti
on
s:
(
0
,
)
=
-
That is
, th
e
concent
rati
on at the inlet
of the
app
a
ratus
(13)
∂
∂
|
−
=
0
-
T
he
c
oncent
r
at
ion
at t
he
e
nd of the
apparat
us
does
no
t c
ha
ng
e
(14)
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sc
i
IS
S
N:
25
02
-
4752
Mathe
m
atical
modeli
ng
and alg
or
it
hm for
c
alculatio
n of t
he
rmocat
alyti
c…
(
Bakh
ti
yar
Isma
il
ov
)
1593
Her
e
l
is t
he
le
ng
t
h
of the app
aratus,
m
;
1
v
-
gas
f
low
rate
, m/
s
ec;
D
md
-
m
olec
ular
dif
f
us
io
n
c
oeffici
ent, m
2
/se
c;
с
-
gas
c
oncent
r
at
ion
in
t
he
re
act
or
,
m
ol/
m
3
;
J
*
-
the
rate
of
form
ation
or
consum
ption
of
a
com
ponen
t
in
the
gas p
hase,
m
ol
/(
m
3
*s
ec).
Fo
r
the
c
onve
nience
of
cal
culat
ion
s
,
le
t
us
pass
t
o
dim
en
sion
le
ss
va
riab
le
s,
ch
oosin
g
c
har
act
erist
i
c
values
for ea
ch
of them
. A
fte
r
tran
s
f
or
m
at
ion
s,
we
get
(
the
dashes
over
the
var
ia
bles are
om
itted):
∂
∂
+
∂
∂
=
−
1
∗
∗
∂
2
∂
2
+
(15)
Wh
e
re
Pe
=
1
v
l
/
D
md
-
Pecl
et
criter
io
n;
F=L/D
-
geo
m
et
ric
factor
of
the
ap
pa
r
at
us
sh
a
pe;
b=
F/Pe=c
onst,
J
is
the
dim
ension
le
ss
rate
of
form
ati
on
of
the
ga
s
ph
a
se
of
t
he
c
om
po
ne
nts
f
o
r
the
r
eact
ion
ta
king
place
on
th
e
cat
al
yst
su
rf
ac
e (c
on
sta
nt v
al
ue
in
the is
oth
e
rm
al
f
or
m
ulatio
n o
f
t
he pr
ob
l
e
m
).
In
it
ia
l co
nd
it
io
n:
(
,
0
)
=
0
(
)
(16)
In a
par
ti
cular
case, in
(16) c
ould
be
ta
ken
:
0
(
)
=
0
=
(17)
Border
con
dit
ion
s:
(
0
,
)
=
-
I
nlet co
n
ce
ntr
at
ion
(18)
∂
∂
|
=
1
=
0
-
The
conce
ntra
ti
on
at th
e e
nd
of the a
pparat
us d
oes n
ot ch
a
nge
(19)
If
0
(
)
≅
0
, th
e
n
the
pr
oble
m
is fo
rm
ulate
d
as
fo
ll
ows
:
∂
2
∂
2
−
∂
∂
−
∂
∂
=
−
(20)
0
(
,
0
)
=
0
(
)
(21)
(
0
,
)
=
0
(22)
∂
∂
|
=
1
=
0
(23)
b=
F/Pe
=
c
on
st
(24)
No
te
:
Pro
blem
(20)
-
(
23)
can
be
f
or
m
ulate
d
and
s
olv
e
d
acc
ordin
g
to
the
m
et
ho
dolo
gy
presente
d
in
[12]
-
[
20]
. Nam
el
y, in a m
or
e
gen
e
ral sett
in
g, we re
pr
ese
nt t
his
pro
blem
in
the form
:
∂
∂
−
∂
2
∂
2
+
∂
∂
=
(
,
)
,
0
<
<
,
0
<
<
1
(25)
(
,
0
)
=
(
)
,
0
≤
≤
1
(26)
(
0
,
)
=
(
)
,
0
≤
≤
(27)
∂
(
1
,
)
∂
=
0
(28)
We
i
ntr
oduce
a
ne
w
functi
on
v(
x,t)
,
so
t
hat
ho
m
og
e
ne
ou
s
bounda
ry
co
ndit
ion
s
ar
e
s
at
isfie
d:
(
,
)
=
(
,
)
+
(
,
)
, whe
re
(
,
)
=
2
−
2
.
The
n prob
le
m
(
25)
-
(
28)
takes
the
fo
ll
owin
g form
:
∂
∂
−
∂
2
∂
2
+
∂
∂
=
(
,
)
,
0
<
<
,
0
<
<
1
(29)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
23
, N
o.
3
,
Se
ptem
ber
2
02
1
:
15
90
-
16
01
1594
(
,
0
)
=
1
(
)
,
0
≤
≤
1
(30)
(
0
,
)
=
0
,
0
≤
≤
(31)
∂
(
1
,
)
∂
=
0
,
0
≤
≤
(32)
In
it
ia
l co
nd
it
io
n:
(
,
0
)
=
(
,
0
)
−
2
+
2
(33)
The fu
nctio
ns
v(
x
,
t)
and
c(
x,t)
are
relat
ed by
the r
el
at
io
n:
(
,
)
=
(
,
)
−
2
+
2
(34)
Hen
ce
, fo
r (1)
we get
:
(
,
)
=
∂
∂
−
∂
2
∂
2
+
∂
∂
=
+
2
+
2
(
1
−
)
(35)
The
ta
s
k
ta
ke
s
the foll
owin
g
f
or
m
:
∂
∂
−
∂
2
∂
2
+
∂
∂
=
+
2
+
2
(
1
−
)
,
0
<
<
,
0
<
<
1
(36)
(
,
0
)
=
0
−
2
+
2
,
0
≤
≤
1
(37)
(
0
,
)
=
0
,
0
≤
≤
(38)
∂
(
1
,
)
∂
=
0
,
0
≤
≤
(39)
Th
us
,
t
he
ori
gi
nal
pro
blem
i
s
reduce
d
to
the
prob
le
m
(36)
-
(
39),
f
or
w
hich
t
he
co
ndi
ti
on
s
for
the
existe
nce
an
d
un
i
qu
e
ness
of
the
so
luti
on
wer
e
prov
e
d
in
[12]
-
[14].
T
her
e
fore,
we
can
assum
e
that
the
equ
i
valent
pro
blem
(
20
)
-
(
23)
h
as
a
un
i
qu
e
s
olu
ti
on.
2.2.
S
olu
tion
of t
he
pr
ob
le
m by t
he L
ap
l
ace
transf
orm
method
We s
olv
e
the i
niti
al
b
ounda
ry v
al
ue
pr
ob
le
m
b
y a
pp
ly
in
g
th
e Laplace t
ran
s
form
:
(
(
,
)
,
,
)
=
=
∫
0
∞
−
(
,
)
(40)
Wh
e
re
p
is
a
num
ber
with
a
la
rg
e
e
nough
posit
ive
real
pa
r
t
fo
r
the
inte
gral
to
converge.
Applyi
ng
the Laplace
trans
form
to
(20)
-
(
23), we
obt
ai
n
the
f
ollow
i
ng bo
unda
ry va
lue pr
ob
le
m
f
or an
ord
i
nar
y
diff
e
re
ntial
eq
ua
ti
on
:
2
2
−
−
=
−
(41)
(
0
)
=
0
(42)
∂
∂
|
=
1
=
0
(43)
Let
’s write
the
char
act
e
risti
c equ
at
io
n f
or
t
he ho
m
og
e
neous
in (4
1) as
fo
ll
ows:
2
−
−
=
0
(
44
)
W
her
e
1
,
2
=
1
±
√
1
+
4
2
Let
:
α
=
1
+
√
1
+
4
2
,
=
1
−
√
1
+
4
2
(
45
)
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i
IS
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02
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Mathe
m
atical
modeli
ng
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or
it
hm for
c
alculatio
n of t
he
rmocat
alyti
c…
(
Bakh
ti
yar
Isma
il
ov
)
1595
The
n:
(
)
=
1
+
2
(46)
The parti
c
ular soluti
on
of the
inhom
og
ene
ous equati
on c
ou
l
d be
fou
nd from
the (41).
Let
’s
‾
=
, th
en
=
−
,
=
2
(
,
)
=
1
+
2
+
2
(47)
1
+
2
=
−
2
,
1
+
2
=
0
(48)
The
s
olu
ti
ons
of syst
em
(
48
)
are the
foll
ow
i
ng num
ber
s:
1
=
−
2
(
−
)
,
2
=
2
(
−
)
(49)
The
n
the
r
e
qu
i
red f
un
ct
io
n
is
equ
al
t
o:
(
,
)
=
2
(
−
)
(
−
)
+
2
(50)
Fo
r
fur
t
her cal
culat
ion
s
, we t
ake in
t
o
acc
ou
nt the f
ollow
i
ng
rati
o:
−
=
−
√
1
+
4
(51)
The
n
it
can
b
e
wr
it
te
n
as:
(
,
)
=
2
√
1
+
4
(
−
)
+
2
(52)
Let
'
s p
erfor
m
the inve
rse
La
pl
ace t
ran
s
form
:
(
,
)
=
1
2
∫
−
∞
∞
(
,
)
(53)
To
cal
c
ulate
th
e integ
ral in
(
53)
, th
e
re is c
ou
ld
be
used
the t
heory
of
resid
ue
s:
1
2
∫
−
∞
∞
(
,
)
=
∑
=
1
(
)
(54)
Fo
r
fu
nction
v
(
x, t
):
re
z
(
)
=
(
1
2
(
,
)
)
=
(
+
−
+
+
√
1
+
4
)
(55)
Der
i
vatives
for
α
a
nd
β
a
re cal
culat
ed by f
orm
ulas (
55):
=
1
√
1
+
4
,
=
−
1
√
1
+
4
(56)
At the
pole
poi
nt, we
ob
ta
in
t
he follo
wing s
olu
ti
on:
(
,
)
=
(
1
−
/
−
−
)
(57)
If
th
e
ass
um
pt
ion
a
bout
the
sm
a
ll
ness
of
t
he
init
ia
l
conc
entrati
on
0
(
)
≅
0
is
inappr
opriat
e,
th
en
te
chn
ic
al
diff
ic
ulti
es arise f
or
so
lvi
ng equati
on:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
23
, N
o.
3
,
Se
ptem
ber
2
02
1
:
15
90
-
16
01
1596
∂
∂
+
∂
∂
=
−
1
∗
∗
∂
2
∂
2
+
(58)
By
the
La
place
trans
f
or
m
m
eth
od.
T
her
e
fore
,
it
is
necessa
r
y
to
a
pp
ly
finite
dif
fer
e
nce
m
et
hods
.
T
he
iss
ues
of
nu
m
erical
m
o
deling
of
he
at
a
nd
m
ass
tra
ns
fe
r
proces
se
s,
conve
rg
e
nc
e,
and
sta
bili
ty
of
finite
-
di
f
fer
e
nce
schem
es
(F
our
ie
r
m
et
ho
d)
from
a
pr
act
ic
al
po
i
nt
of
view
are
prese
nted
in
[
9].
T
he
n
we
f
ollo
w
ed
t
he
reco
m
m
end
at
ion
s
of t
his
wor
k.
2.2.
1
.
E
xp
li
ci
t
s
cheme
Fo
r
c
onve
nien
ce, w
e
d
e
note
:
/
=
(59)
Inde
xing
of
gr
i
d
po
i
nts:
i
-
by
х
,
j
-
by
t
.
We
a
ppr
oxim
a
te
the
first
-
order
de
rivati
ves
for
x
a
nd
t
acco
rd
i
ng
to
th
e
“forw
a
r
d”
sch
e
m
e,
and
the
second
-
orde
r
de
rivati
ve
f
or
х
by
the
central
diff
e
re
nce.
W
e
ob
ta
in
the
f
ollow
i
ng
exp
li
ci
t schem
e f
or
t
he
e
qu
at
i
on
:
,
+
1
−
,
+
+
1
,
−
,
ℎ
=
+
1
,
−
2
,
+
−
1
,
ℎ
2
+
,
(60)
Wh
e
n
ap
pro
xi
m
at
ing
the
der
i
vatives
of
the
1
-
orde
r,
the
er
r
or
is
(
)
+
(
ℎ
)
,
an
d
f
or
t
he
de
rivati
ve
of
t
he
2
-
order
−
(
ℎ
2
)
. After
tra
ns
f
or
m
at
ion
s
we get
:
,
+
1
=
1
−
1
,
+
2
,
+
3
+
1
,
+
(61)
Wh
e
re:
i=
1
,…,
n
-
1
;
j=
0
,…,
m
-
1;
1
=
ℎ
,
2
=
1
−
2
ℎ
2
+
ℎ
,
3
=
ℎ
2
−
ℎ
+
,
ℎ
=
1
/
,
=
1
/
.
(62)
Discre
te
a
nalo
gu
e
s
of
c
onditi
on
s
(2
1)
,
(2
2),
and (
23) ha
ve
t
he follo
wing
f
or
m
:
(
,
0
)
=
,
0
,
(
0
,
)
=
0
,
.
,
=
−
1
,
(63)
In the c
onside
r
ed
is
oth
erm
al
r
egim
e,
J =
co
nst
.
The
c
onditi
ons
f
or
no
n
-
ne
gativit
y
1
,
2
,
3
an
d
sta
bi
li
t
y
of
sc
hem
e
(
61)
ta
ke
n
t
og
et
her
ha
ve
th
e
form
:
/
ℎ
2
≤
1
(64)
At
la
rg
e
Pecl
et
nu
m
ber
s
=
1
/
,
wh
e
n
c
onvecti
ve
tra
ns
fe
r
prevail
s
over
th
e
m
olecular
transf
e
r,
conditi
on (6
3)
ta
kes
the
foll
owin
g form
:
≤
ℎ
2
(65)
Whe
re t
he
c
onsta
nt c
decr
eas
es w
it
h i
ncr
ea
s
ing
Pecl
et
num
ber. F
or exam
ple, at
≈
1000
:
c
≈
0
,
1
.
Su
c
h
a sm
oo
th
ing
pro
ce
dure
is
oft
en
us
e
d
a
nd
m
akes
it
pos
sible
to
e
nsure
a
slo
w
a
nd
sm
oo
t
h
c
hange
in
the
cal
culat
ed
dynam
ic
functi
on.
In
our
work,
t
he
Pecl
et
num
ber
is
s
m
al
l
du
e
to
th
e
pred
om
inance
of
th
e
m
olecular
co
m
po
nen
t
(i.e.,
on
the
orde
r
of
10
);
there
fore,
cal
c
ulati
on
s
usi
ng
c
onditi
on
(
65)
w
ere
no
t
perform
ed.
Takin
g
into
acc
ount
the
fact
tha
t
the
synthesis
process
is
lo
ng
,
the
re
qu
i
rem
e
nt
that
co
nd
it
io
n
(
64)
b
e
sat
isfie
d
co
ncernin
g
the
ti
m
e
ste
p
le
ads
t
o
a
la
rg
e
num
ber
of
it
erati
ons
,
wh
ic
h
res
ults
in
the
accum
u
la
ti
on
of roun
d
-
off
e
r
rors.
2.2.
2.
I
mpli
ci
t schem
e
Fo
r
la
rg
e
P
ecl
et
nu
m
ber
s,
w
hen
c
onvecti
ve
m
at
te
r
transf
e
r
prevail
s,
the
us
e
of
an
e
xpli
ci
t
sche
m
e
al
s
o
le
ads
to
c
om
pu
ta
ti
on
al
instabil
it
y.
Fo
r
la
rg
e
peclet
nu
m
ber
s,
wh
e
n
c
onvecti
ve
m
at
t
er
trans
fer
pr
e
vails,
the
us
e
of
a
n
exp
li
ci
t
schem
e
al
so
le
ads
to
com
pu
ta
ti
on
al
instabil
it
y.
Ap
plica
ti
on
to
e
qu
at
io
n,
t
he
im
pl
ic
it
schem
e "backwar
d"
i
n
ti
m
e l
eads t
o
the
fo
ll
ow
i
ng pr
ob
le
m
:
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sc
i
IS
S
N:
25
02
-
4752
Mathe
m
atical
modeli
ng
and alg
or
it
hm for
c
alculatio
n of t
he
rmocat
alyti
c…
(
Bakh
ti
yar
Isma
il
ov
)
1597
,
−
,
−
1
+
+
1
,
−
,
ℎ
=
+
1
,
−
2
,
+
−
1
,
ℎ
2
+
,
(66)
(
,
0
)
=
,
0
,
(
0
,
)
=
0
,
.
,
=
−
1
,
(67)
The
rate of
f
orm
at
ion
or
c
ons
um
ption
o
f
a
c
om
po
ne
nt
in
th
e
gas
phase
at
all
po
i
nts o
f
t
he
reacto
r
ca
n
be
c
onsidere
d
t
he
sam
e, i.e
J
ij
=
J.
Let
us
redu
ce the
pro
blem
(66)
-
(67
)
to
m
at
rix
f
orm
:
−
−
1
,
+
(
ℎ
2
−
ℎ
+
2
)
,
+
(
ℎ
−
)
+
1
,
=
ℎ
2
,
−
1
+
ℎ
2
(68)
i.e., in m
at
rix
f
or
m
:
−
1
−
1
+
+
+
1
+
1
=
(69)
Wh
e
re:
−
1
=
−
,
=
ℎ
2
−
ℎ
+
2
,
+
1
=
ℎ
−
,
=
ℎ
2
,
−
1
+
ℎ
2
(70)
To
s
olv
e
syst
em
(6
9)
Ax
=
b
by
a
tridia
gona
l
m
a
trix,
we
use
the
s
weep
m
et
hod.
For
the
app
li
cabil
it
y
of
t
he
f
orm
ula
s
of
the
s
wee
p
m
e
tho
d,
w
e
c
heck
the
c
ondi
ti
on
s
for
the
in
equ
al
it
y
of
de
nom
inators
to
z
ero
i
n
the
form
ulas
fo
r
cal
culat
in
g
the
swee
p
c
oeff
ic
ie
nts
and
the
diag
on
al
dom
i
nan
ce
of
t
he
m
at
rix
A
acc
ordi
ng
t
o
the form
ula
:
|
ℎ
2
−
ℎ
+
2
|
≥
|
−
|
+
|
ℎ
−
|
(71)
Fo
r
la
rg
e
Pecl
et
nu
m
ber
s,
i.e
.
f
or
sm
al
l
b
,
we
ch
oose
the
ste
p
in
x
f
ro
m
the
co
nd
it
io
n
2b
>
h>
b,
the
ste
p
i
n
tim
e fr
om
the co
ndit
ion
h.
The
n
the
left
-
ha
nd side
of ine
qu
al
it
y (
71) ha
s the estim
at
e:
|
ℎ
2
−
ℎ
+
2
|
>
ℎ
(72)
The rig
ht
-
hand
side
for
h>b
is
:
|
−
|
+
|
ℎ
−
|
=
+
(
ℎ
−
)
=
ℎ
(73)
Th
us
,
w
hen
s
uffici
ently
wea
k
co
ndit
ion
s
are
sat
isfie
d
f
or
the
ste
ps
of
the
m
esh
do
m
ai
n,
the
s
wee
p
m
eth
od
i
s
conve
rg
e
nt.
W
e
app
li
ed
t
he
pro
po
se
d
a
ppr
oach
t
o
the
pro
blem
of
determ
ining
the
con
ce
ntrati
on
of
th
e
com
po
ne
nts
of a syst
em
w
it
h
a N
i/
Mg
O
cat
a
ly
st al
on
g t
he
l
eng
t
h of t
he
a
pparat
us
.
2.3.
Sele
c
tion o
f
c
atalysts
and c
ompari
son o
f their
p
ar
amet
er
s
Var
i
ou
s
cat
al
yst
s
are
us
ed
to
acce
le
rate
the
synthesis
of
na
nostruct
ur
es
.
Her
e
tw
o
inter
relat
ed
ta
sk
s
arise
-
the
sel
ec
ti
on
an
d
m
an
uf
act
ur
e
of
th
e
cat
al
yst
.
In
[25]
-
[
27]
,
m
on
olit
hic
cat
al
yst
s
based
on
Co,
Ni,
order
e
d
m
eso
por
ou
s
car
bons,
a
nd
Ni/M
gO
cat
al
yst
s
on
str
uctu
re
d
m
et
al
su
pport
s
are
us
e
d
t
o
pro
du
c
e
hydro
ge
n.
Kiri
ll
ov
et
al.
[
27]
pro
posed
t
o
a
pp
ly
Ni/M
gO
on
a
m
et
al
su
pp
ort
,
bu
t
t
his
proce
dure
is
e
xpen
siv
e
and
j
ust
ifie
s it
s
el
f
for
high
-
te
m
per
at
ur
e synthesis in work [27
]
. T
her
e
fore,
f
or
us
e in our
exp
e
rim
ental
st
ud
ie
s
,
wh
e
n
t
he
te
m
per
at
ur
e
does
no
t
excee
d
70
00
C
,
we
hav
e
d
e
velo
pe
d
th
e
f
ollow
i
ng
ca
ta
ly
st
m
anu
fac
turing
te
chn
iq
ue:
f
or
the
synthesis
of
a
Ni
-
Mg
cat
a
ly
st,
we
i
m
pr
e
gn
at
e
d
al
um
ina
extr
ud
at
es
wi
th
so
l
utions
of
act
ive
com
po
ne
nts
-
m
ulti
ple
nic
kels
and
m
agn
esi
um
nitrat
e.
Fo
r
this,
al
um
inu
m
hydro
xid
e
was
m
ixed
with
di
sti
l
le
d
water.
T
he
res
ulti
ng
pa
sty
m
ass
was
passe
d
throu
gh
a
la
borat
or
y
scre
w
pr
ess
-
e
xtr
ud
e
r.
Then
the
e
xtr
ud
at
e
s
wer
e
dr
ie
d
at
roo
m
te
m
per
at
ure, the
n dr
ie
d
a
nd
c
al
ci
ned
at
600
0
С
in atm
os
pheric ai
r. Th
e resu
lt
in
g
cy
li
ndrical
gr
a
nule
s h
a
ve a
d
ia
m
et
er o
f 2
.5
-
3 m
m
an
d
a
le
ng
th
of
5
-
6 m
m
.
The
cat
al
yst
s
a
m
ples
synthe
siz
ed
in
this
w
ay
con
ta
in
the
act
ive
com
po
nen
ts
NiO
an
d
MgO
up
to
10%
eac
h.
We
al
so
stu
died
the
physi
coc
hem
ic
al
ind
ic
at
or
s
of
t
he
sa
m
ples:
sp
eci
fic
an
d
act
i
ve
s
urface;
con
ce
ntrati
on
of
act
ive
cente
rs;
st
rength
fac
tor;
total
aci
dit
y.
The
data
ob
t
ai
ned
are
in
sa
ti
sfactor
y
agr
e
e
m
ent
with the
d
at
a
of wo
rk
s
[2
3]
-
[
24]
.
3.
RESU
LT
S
AND DI
SCUS
S
ION
Figure
1
sho
w
s
the
distri
bu
ti
on
of
the
c
on
c
entrati
on
of
m
et
han
e
c
oncent
rati
on
at
the
outl
et
fr
om
a
batch
react
or
f
or
sm
al
l
Pecl
e
t
nu
m
ber
s,
i.e.
wh
e
n
m
olecular
diffusi
on
pred
om
inate
s.
The
cal
culat
io
n
was
m
ade
for
m
et
h
ane,
the
physi
coch
em
ic
al
par
am
et
ers
of
w
hic
h
ha
ve
been
st
ud
ie
d
in
s
uffici
ent
detai
l
.
As
c
an
be
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
23
, N
o.
3
,
Se
ptem
ber
2
02
1
:
15
90
-
16
01
1598
seen fr
om
Figu
re 2, at large v
al
ues
of the Pe
cl
et
crite
rion
(
i
.e
.,
the
rati
o of
dynam
ic
an
d
di
ff
usi
on p
a
ram
et
ers),
the
m
et
han
e
c
on
ce
ntrati
on
c
hanges
slo
wly.
This
is
du
e
to
the
pr
e
dom
inance
of
the
li
ne
ar
velocit
y
of
t
he
gas
ov
e
r
t
he
rate
of
the
diffusi
on pro
ces
s.
Figure
1. Distri
bu
ti
on
of m
et
han
e c
on
ce
ntrati
on at d
i
ff
e
ren
t
values
of the
P
ecl
et
crite
rion
ov
e
r Ni/
MgO
cat
al
yst
in
tim
e
at tem
per
at
ure
600
0
С
:
1
–
Ре =
10, 2
–
Ре =
20, 3
–
Ре =
30
Figure
2. Distri
bu
ti
on
of m
et
han
e c
on
ce
ntrati
on at di
ff
e
ren
t
values
of the
P
ecl
et
crite
rion
ov
e
r Ni/
MgO
cat
al
yst
in
tim
e
at tem
p
eratu
re
600
0
С:
1
–
Ре =
200,
2
–
Ре =
300,
3
–
Ре =
400
As
ca
n
be
see
n
f
r
om
Figu
re
2,
at
la
r
ge
val
ues
of
t
he
Pecl
et
crit
erion
(i.e
.,
the
rati
o
of
dynam
ic
and
diffusi
on
par
a
m
et
ers)
,
the
m
et
han
e
c
on
ce
nt
rati
on
c
hanges
slow
ly
.
This
is
du
e
to
the
pr
e
dom
inance
of
th
e
li
near
vel
ocity
of
t
he
gas
over
t
he
rate
of
the
diffusi
on
process
.
W
e
carried
out
a
syst
e
m
atic
num
erical
exp
e
rim
ent
us
ing
our
s
wee
p
pro
gr
am
,
wh
ic
h
im
ple
m
e
nts
the
above
sweep
al
gori
thm
,
to
cal
culat
e
the
distrib
ution
of
the
sp
eci
fic
c
arbo
n
co
ntent
on
th
e
c
at
al
yst.
At
the
sam
e
tim
e,
to
ensure
the
possibil
it
y
of
com
par
ing
t
he
ob
ta
ine
d
nu
m
erical
resu
lt
s
with
kn
own
e
xperim
ental
data
[6
]
-
[7
]
,
t
he
a
ct
ual
ty
pe
of
c
at
al
yst,
tem
per
at
ur
e
va
lues,
a
nd
a
set
of
values
of
the
physi
coc
he
m
ic
al
par
a
m
eter
s
of
the
t
herm
ocatalyt
ic
py
ro
ly
sis
process
w
ere
s
el
ect
ed.
Fig
ur
e
s
3
-
4
show
the
distribu
ti
on
of
the
sp
eci
fic
carbo
n
co
ntent
cal
culat
ed
us
in
g
th
e
above
al
gorith
m
.
The
relat
ive
err
or
in
cal
cul
at
ing
the
sp
eci
fic
carbon
c
on
t
ent
us
in
g
a
Ni/M
gO
cat
al
yst
i
s
5%.
Chan
ges
in
th
e
con
ce
ntrati
on
of
m
et
han
e
and
an
inc
re
ase
in
the
con
cent
rati
on
of
carb
on
na
no
par
ti
cl
es
(F
ig
ur
e
s
1
-
4) a
re in ag
reem
ent w
it
h
the
c
on
c
lusio
ns
of [
6
]
-
[
7].
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sc
i
IS
S
N:
25
02
-
4752
Mathe
m
atical
modeli
ng
and alg
or
it
hm for
c
alculatio
n of t
he
rmocat
alyti
c…
(
Bakh
ti
yar
Isma
il
ov
)
1599
Figure
3. Ca
lc
ulate
d dist
rib
ut
ion
of the
sp
eci
fic car
bon
c
ont
ent on t
he Ni/
La
2
O
3
cat
al
yst
over
tim
e at a
tem
per
at
u
re
of
600
0
С
, (c
urve
1),
650
0
С
, (cu
r
ve 2)
, 700
0
С
, (
curve
3),
po
i
nts
-
e
xperim
ental
d
at
a [
6
]
-
[
7]
Figure
4. The
c
al
culat
ed dist
ribu
ti
on
of the
s
pecific ca
rbo
n con
te
nt on
t
he Ni /
Mg
O
cat
al
yst
o
ve
r
ti
m
e a
t a
tem
per
at
ur
e
of
560
0
C,
(c
urve
1),
580
0
C,
(cur
ve
2),
680
0
C,
(
curve
3),
po
i
nts ar
e e
xperim
e
ntal data
[
6
]
-
[7]
4.
CONCL
US
I
O
N
A
one
-
dim
ensi
on
al
pro
blem
of
m
at
he
m
atic
al
m
od
el
ing
of
the
t
her
m
ocatal
yt
ic
pr
ocess
of
obta
ini
ng
nanom
at
erial
s
is
form
ulate
d,
m
et
ho
ds
of
im
plem
enting
a
m
at
he
m
at
ic
a
l
m
od
el
by
the
nu
m
erica
l
m
eth
od
a
nd
the
Laplace
t
ra
ns
f
or
m
are
de
ve
lop
e
d.
Su
c
h
a
sta
tem
ent
of
t
he
prob
le
m
was
carrie
d
ou
t
f
or
t
he
fi
rst
tim
e
an
d
can
be
de
velop
e
d
f
or
us
e
in
researc
h
and
pr
oductio
n
pur
po
se
s
in
the
pro
du
ct
i
on
of
na
no
m
at
erial
s.
The
distrib
utio
ns
of
m
et
han
e
con
ce
ntr
at
io
n
wer
e
obta
ine
d
for
diff
e
re
nt
va
lues
of
t
he
Pec
le
t
diffusio
n
num
ber
.
The
c
onditi
ons
of
sta
bili
ty
of
num
erical
al
go
rithm
s
for
t
he
eq
uatio
n
of
c
on
ce
ntrati
on
of
gas
c
om
po
ne
nts
ar
e
est
ablished
.
T
he
pr
ese
nted
al
gorithm
fo
r
t
he
nu
m
erical
si
m
ulati
on
of
t
he
pr
ocess
of
t
herm
ocatalyt
ic
s
yn
thesis
of
car
bon
nanom
at
erial
s,
in
ou
r
op
i
nion,
can
be
ge
ne
rali
zed
for
th
e
case
of
the
m
utu
al
influe
nce
of
interm
ediat
e
rad
ic
al
s
on
the prod
uct
yi
el
d.
T
o
do
this
,
it
is necessa
ry
to
de
velo
p
a
ki
netic
schem
e
of
rea
ct
ions
,
so
lve
t
he
diff
e
ren
ti
al
eq
uatio
ns
for
the
tra
nsfo
rm
at
ion
s
of
CH4
-
CH
3
-
CH2
-
CH
-
CH
wi
th
init
ia
l
and
boun
dary
conditi
ons,
i
nc
lud
e
a
ddit
iona
l
te
rm
s
in
the
syst
e
m
,
(i
=
1,
...,
n
-
1,),
ta
king
int
o
a
ccount
the
in
f
luence
con
ce
ntrati
ons
of
inte
rm
edia
te
rad
ic
al
s
on
the
rate
o
f
c
at
al
ysi
s.
Con
sideri
ng
th
e
fac
t
that
the
proc
ess
of
therm
al
ca
ta
lysis
of
car
bon
pyr
olysi
s
is
a
slow
a
nd,
on
aver
a
ge,
is
rea
li
zed
in
sever
a
l
te
ns
of
ho
urs,
the
m
et
ho
d
of
m
a
them
a
ti
cal
m
o
deling
f
ollo
we
d
by
i
m
ple
m
e
ntati
on
by
nu
m
erical
al
go
rithm
s
can
sign
ific
antly
reduce
ti
m
e
and
m
at
erial
costs,
as
well
as
i
m
pr
ov
e
t
he
qual
it
y
of
the
res
ulti
ng
na
nom
ater
ia
ls.
T
he
a
na
ly
ti
cal
form
ula
fo
r
t
he
con
c
entrati
on
distri
bu
ti
on
in
the
f
orm
(5
7)
was
ob
ta
i
ne
d
f
or
t
he
fi
rst
tim
e
and
it
is
m
or
e
accurate
t
han
the
resu
lt
s
of
num
e
rical
so
l
ution
s
us
ing
e
xisti
ng
te
chn
iq
ues
.
Wh
e
n
c
al
cula
ti
ng
th
e
con
ce
ntrati
on,
the
val
ue
of
th
e
crit
erio
n
rati
o
b=
F/Pe
is
in
vo
l
ved
as
a
pa
ram
et
er,
wh
ic
h
m
akes
it
po
ssi
ble
to
carry
ou
t
a
syst
e
m
at
ic
nu
m
erical
exp
e
rim
ent
to
f
or
m
ulate
and
so
l
ve
the
pr
ob
le
m
s
of
opti
m
iz
ing
the
op
e
rati
ng
char
act
e
risti
cs o
f
the catal
yt
ic
p
yr
olysi
s pro
c
ess and t
he geo
m
et
ric d
i
m
ension
s
of t
he
te
ch
no
l
og
ic
al
a
ppa
ratus.
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