International Journal of Engineering Trends and Technology (IJETT) – Volume 36 Number 8 - June 2016
Application of Laplace Transform to Ethane
Cracking in Furnace Reactor
Wordu, A. A, Ojong, O.E
Department of Chemical/Petrochemical Engineering,
University of Science and Technology,
Port Harcourt, Nigeria.
ABSTRACT
kinetic
A research applying Laplace transform as basically
the mathematical technique to resolve the steady
state plug flow reactor model for ethane cracking
into ethylene has been developed. The process
chemistry was kinetically applied to derive the
required rate law. Steady state models expressing
the ethane depleting process were evaluated using
the industrial plant data of Indorama Eleme
Petrochemical as boundary conditions to solve the
models analytically. The variation of the feed along
the length of the reactor shows an exponential
function of the feed cracking process to a
maximum. While, temperature effects showed a
progression of 1000 k to 1100 k is being the reactor
design operating temperature for optimum yield of
ethylene exhibiting complete cracking action.
for
the
writing
and
/or
development of the rate laws on all the species
including the intermediates species rate laws
contributing to formation products by free-radical
mechanisms (Wordu, 2014). The simulations of the
models give a clear flat plateau where PSSH is
valid.
The process chemistry adopted on the pyrolysis of
ethane is irreversible first order chemical reaction
with reactor temperature range of 900k-1200k
given as:
k1
C2 H 6  g  
C2 H 4 g   H 2  g 
(1)
Keywords: Furnace reactor, Laplace-transform,
radiation zone, rate Laws, ethylene.
1.
modeling
The yield of ethylene can be as high as 65-70%.
The thermodynamic properties such as standard
INTRODUCTION
According to [Wordu and Akinola, 2013] on the
heat
of
numerical design models for ethane cracking was
based on predicting a computer-based model for
capacity

formation H f
C 
p
298
,
specific
heat
and related kinetics data for
ethane cracking in a furnace reactor. The kinetics of
chemical species associated with the reaction
radiation and the convective zone of the furnace
process given in table 1.
reactor were derived as well as the temperature
2. MATERIALS
effects and Pressure drop. The method of resolution
2.1 Rate law for ethane cracking process in
of the model equation were numerical and mat lab
radiation zone
ode-45 to generate the plots profiles in respect to
The writing of the rate law is constrained by the
fractional conversion,  i along the length of the
fact that K >>1, and the equilibrium shifts much
reactor, pressure drop effects and energy balance
model. The comparison of the work with plant data
of
Indorama
Eleme
petrochemicals
was
in
agreement with what occurs operationally in the
reactor. Also, an attempt have been made to
applying pseudo-steady-state hypothesis (PSSH)
ISSN: 2231-5381
more towards right hand side of the reaction
process. When this holds for the cracking process,
more yield of ethylene is achieved at the operating
reactor temperature of 1113 K and a low pressure
of 0.4kgcm-2 with feed rate of 55,000kghr-1. When
the preceding conditions are adopted, the PSSH
kinetics
http://www.ijettjournal.org
where
free-radical
mechanism
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International Journal of Engineering Trends and Technology (IJETT) – Volume 36 Number 8 - June 2016
predominates is therefore neglected for the purpose
[Coulson and Richardson, 2005].This then cracked
of this research.
the ethane to ethylene and hydrogen as the major
Pure ethane is fed in the furnace at temperatures of
products obtained. Conversion of ethane to
about
of
ethylene is as high as 68-75%. The thermodynamic
. Heat is supplied by the
properties for the process are shown in table 1
and
about
pressure
2
furnace to the reactor of about 30-50KW/m
below.
Table 1 Physical properties and data for ethane cracking.
Component
Cp(KJ/KmolK)
298
-83820
52510
0.00
Reaction rate constant:
The furnace reactor is modeled as a tubular flow reactor with no radial mixing along the flow reactor.
3. METHOD
Using figure 1 to illustrate a component balance for a differential volume element.
Formation by chemical
Reaction
ri V
U plug flow velocity
Figure Differential volume in an unsteady plug reactor
Taking a species i balance for 1 mole of feed C2H6 over a differential volume element dv gives;
(2)
Diving equation (2) by
yields;
(3)
But,
rearranging equation (3)
(4)
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International Journal of Engineering Trends and Technology (IJETT) – Volume 36 Number 8 - June 2016
Assuming pssh-model mechanism yields;
Equation (4) becomes;
(5)
Equation [5] is plug flow reactor model for pyrolysis of ethane which can be evaluated using Laplace transform
analytically with boundary conditions
, rearranging equation (5) yields where,
.
(6)
Where K is the rate of ethane feed depletion 1000-1100K
(7)
Lap-lacing the mole fraction gives;
(8)
(9)
Re-arranging the terms of above equation (9) yield;
(10)
(11)
Inverse Laplace transform in z-plane yields
(12)
(13)
3.1
Heat balance equation
Taking a heat balance for 1 mole of feed C2H6 over a differential element dv gives;
(14)
Divide both sides of equation (14) by
(15)
But
Then equation (15) becomes;
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International Journal of Engineering Trends and Technology (IJETT) – Volume 36 Number 8 - June 2016
(16)
Again
The term
in equation (16) is non-linear and can be linearized by Taylor’s series expansion theorem and
neglecting terms of order 2 and above to give;
(17)
Substituting equation (17) into (16) gives;
(18)
After simplification and taking:
, then equation (18) becomes;
(19)
But
The temperature model for the cracking process is given in equation (20) below,
(20)
Applying laplace transform as the solution technique to get the Temperature model in z-domain using equations
(20) and (13) yields;
(21)
(22)
(23)
(24)
(25)
Inverse LT equation (25) gives;
(26)
Where:
SOLUTION TECHNIQUE
Equations (13) and (26) developed were evaluated analytically using laplace procedures to obtain the profiles in
figures 1, 2, 3, and 4. The profiles perfectly represent the dynamics of the reactor operation industrially.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 36 Number 8 - June 2016
RESULTS AND DISCUSSIONS
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05 0
50
100
150
200
Figure 1: Plot of mole fraction ethane depletion with length of the reactor.
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05 0
50
100
150
200
Figure 2: Plot of mole fraction ethylene product with length of reactor
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International Journal of Engineering Trends and Technology (IJETT) – Volume 36 Number 8 - June 2016
1200
1000
800
600
400
200
0
0
50
100
150
200
Figure 3: Plot of Temperature effect of ethane depletion with length of reactor.
3500
3000
2500
2000
1500
1000
500
0
0
50
100
150
200
Figure 4: Plot of Temperature effect of ethane formation with length of Reactor.
effects of ethane depletion along the length of plug
4. DISCUSSION
flow reactor which decreases below 200K at
Figure 1 shows a plot of feed depletion from initial
value 0.37 to a minimum as the length of the
reactor increases. The profile shows a gradual drop
from 0.37 at L0 to L75m, becomes steady as the
length increases to L150m i.e steady state process.
corresponding length of L20m and suddenly became
constant as the reaction rate continues and
maintains steady state process. Figure 4 profile of
temperature of ethylene formation with reactor
length shows that the temperature effects increases
Figure 2 principally shows the ethylene formation
from 1100K at L0 continuously as the length of the
from L0 to approximately L90m when ethylene
reactor increases.
formation commenced exponentially to a maximum
length L150m which can be taken as 60-75 % of the
feed conversion. Figure 3 shows temperature
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International Journal of Engineering Trends and Technology (IJETT) – Volume 36 Number 8 - June 2016
Reactor length (z) = 152m
5. CONCLUSION
Laplace
transform
applied
as
an
analytical
Tube thickness (Δd) = 0.01m
technique to solve the developed models was able
to predict the dynamics of the reaction process i.e
feed depleting actions 100% conversion of ethane
Thermal conductivity of reactor tube (K) =
11.8W/MK
to ethylene at a maximum reactor length.
NOMENCLATURE
REFERENCES
INDUSRIAL DATA FROM INDORAMA,
ELEME PETROCHEMICAL, NIGERIA
1)
Derek, L.J. (1971).Computer model for reactor design of
an ethane cracking unit. M.Sc. thesis, Network College of
Engineering Jersey.
2)
Fogler, H.S, (1998). Elements of chemical reaction
engineering, 3rd edition; Prentice Hall, India.
3)
Green, Don. W & Perry, Robert. H., (2008). Perry’s
Chemical Engineering Handbook, 8th edition pp2-14-184,
7.1-38, 8.1 -8.39 RRDonnelleyShenzhen, McGraw Hill.
4)
George, S.(2008). Chemical
Process Control. An
introduction to Theory & Practice, Prentice-Hall of India,
New Delhi.
5)
Hatch, L.F. & Mater, S. (1981). From hydrocarbon to
Petrochemical Gulf Publishing Co.
6)
7)
Industrial plant data from Indorama, Eleme Petrochemical,
Nigeria
Kim, D. K; Cha, C.Y.; Lee, W.T & Kim, J.H., (2001).
Microwave Dehydrogenation of ethane to ethylene Journal
of Engineering chemistry, 7(6).
8)
Rase, H.F, (1997). Chemical reactor design for process
plant (1) Principles and Techniques, John Wiley and sons,
New York.
9)
Wordu, Animia . A, (2014) Predictive PSSH-mechanisticmodel for ethane cracking in furnace reactor, Journal of
Current
Research in Engineering, Publication in
progress
10) Wordu, A. A, & Akinola, B.S. (2013). Numerical Design
models of Furnace reactor for ethane cracking. B.Tech
degree Univedsity of Science and Technology, Port
Harcourt. Nigeria. Publication in progress
GF = 55,000 Kghr-1
11) Wordu, A. A., & Atunshi, O (2012). Investigating the
Applicability of PSSH-mechanistic-approach to thermal
cracking of ethane to ethylene, B.Tech degree University
of Technology, Port Harcourt, Nigeria.
TOTAL PRESSURE (PT) = 560KPa
Mass flux G = 112.4kg/m2s
Reactor temperature (TR) = 1100K
Pressure drop (ΔP) = 0.45KPa
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