Chemical Engineering and Processing 46 (2007) 25–36
Separation of para-xylene from xylene mixture via crystallization
H.A. Mohameed a,∗ , B. Abu Jdayil b , K. Takrouri a
a
Jordan University of Science and Technology, Department of Chemical Engineering, P.O. Box 3030, Irbid 22110, Jordan
b Department of Chemical Engineering, University of Bahrain, Bahrain
Received 21 December 2005; received in revised form 9 April 2006; accepted 10 April 2006
Available online 23 May 2006
Abstract
Crystallization kinetics of para-xylene from xylene isomers mixture using a lab-scale cooling batch crystallizer were determined. The cooling
batch crystallizer type is simple, flexible and requires less process development. Dynamic mass and population balances were used to model
the batch crystallizer. The model equations were solved using the numerical method of lines; a new proposed solution method. The kinetic
parameters of nucleation and growth rates were estimated by measuring the concentration and the total mass of para-xylene suspended crystals
during the process time. A nonlinear optimization technique was then applied to estimate the parameters. The effect of the cooling strategy on
the estimated parameters was studied. It was found that model predictions using the optimum estimated parameters were in good agreement with
the experimental results under various cooling strategies. The optimal kinetic parameters were then used to find the optimum cooling strategy to
maximize the yield of para-xylene crystals which have an average size greater than 0.5 mm. A new objective function was formulated and also,
a nonlinear optimization technique was applied to find the optimum cooling strategy to achieve this product characterization. The optimization
technique converged successfully and the proposed objective function was found to be effective to optimize the para-xylene crystallization process.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Batch crystallization; para-Xylene; Parameters estimation; Optimum cooling; Method of lines; Population balance
1. Introduction
Xylene, which is a colorless liquid with a sweet odor, occurs
mainly in petroleum and coal tar. However, xylene is primarily a
man-made chemical. Xylene is mainly produced from petroleum
and to a smaller extent from coal [1]. There are three isomers of xylene having the same chemical formula C6 H4 (CH3 )2 ,
but with different structural configurations. These isomers are:
meta-xylene, ortho-xylene and para-xylene. In production of
xylene, a mixture of the three isomers is usually formed and
other chemicals may also be present in smaller amounts such
as benzene and ethylbenzene. The term “mixed xylene” is used
to refer to the mixture of xylene isomers that usually contains
6–15% ethylbenzene.
Abbreviations: CSD, crystal size distribution; DMT, di-methyl terephthalate; GC, gas chromatography; MOL, method of lines; ODE, ordinary differential
equation; PBE, population balance equation; PET, poly-ethelene terephthalate;
TPA, tere-phthalic acid; TSS, total suspended solid
∗ Corresponding author.
E-mail address: [email protected] (H.A. Mohameed).
0255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.cep.2006.04.002
Xylene is used in many applications in the modern industry. Solvents and thinners for varnishes and paints often contain
xylene. Xylene is used in airplanes fuel and gasoline. It is also
used in rubber, cleaning agents and in leather industries. However, among the three isomers, para-xylene is in high demand
for conversion to terephthalic acid (TPA) and then to dimethyl
terephthalate (DMT).
Dimethyl terephthalate (DMT) is then reacted with ethylene
glycol to form polyethelene terephthalate (PET).
Polyethelene terephthalate (PET) is the raw material for most
polyesters used in production of fibers, packaging materials and
containers. Worldwide production of para-xylene in year 2001
was near 21.4 million metric tons (approximately 679 kg/s).
In production of para-xylene, ortho-and meta-xylenes and
other substances are either separated from or converted to paraxylene. These substances boil so closely together that separating
them by fractional distillation is not practical. Therefore, other
methods are used to separate para-xylene from the mixture. One
of para-xylene separation methods is the UOP Parex Process [2].
This process depends on adsorption of para-xylene molecules
by a shape-selective zeolite adsorbent, which is selective only to
the para structure. However, the high cost of the para-selective
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H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
Table 1
Thermodynamic properties of xylene isomers and ethylbenzene
Component
Melting point, Tm (◦ C)
Boiling point, Tb (◦ C)
para-Xylene
ortho-Xylene
meta-Xylene
Ethylbenzene
13.3
−25.2
−47.9
−95.0
138.5
144.0
139.3
136.2
catalyst makes this method limited to small scale productions.
One of the recently used methods for para-xylene separation is
the fractional crystallization process. This process depends on
the variation of melting points of the chemicals in the mixed
xylene. Therefore, by lowering the temperature of this mixture,
para-xylene, which has the highest melting point, will crystallize
firstly (as solid crystals) and then can be separated. Table 1 shows
some of the thermodynamic properties of the “mixed xylene”
components, including the boiling and the melting points [3].
The first step in cooling crystallization technique is to reach
supersaturation. Supersaturation, which is the driving force for
crystallization, occurs as a result of the reduction in solute solubility when the temperature is reduced. After that, formation of
nuclei occurs and finally the subsequent growth of these nuclei
to form large crystals. Thus, continuous cooling is necessary and
the conventional way for cooling is by using jacketed crystallizers or similar forms of heat exchange devices.
In any crystallization process, the crystal size distribution
(CSD) is a key issue since most quality requirements and enduse properties of the crystals are strongly dependent on the CSD
[4]. One of the separation methods mainly used to separate the
solid crystals from the solution after crystallization is filtration.
If the obtained crystals have relatively small sizes, the filtration process is expected to be difficult because small crystals
will cause filter clogging. Thus, it is of great importance to control the crystallizer conditions to obtain the desired CSD. The
CSD depends mainly on the degree of supersaturation, as well
as on crystals nucleation and growth rates. All these three processes (i.e. supersaturation, nucleation and growth of crystals)
may occur simultaneously in the crystallizer [5]. Controlling
the CSD is usually achieved by cooling the solution following
a pre-determined cooling strategy, or a cooling profile. In general, the temperature of the solution should be reduced slowly
in the early stages of the process, and more rapidly in the final
stages to get relatively large crystals [6]. To determine the suitable cooling profile, the behavior of the crystals under cooling
should be known and this is provided from the study of crystallization kinetics, i.e. nucleation and growth rates. Quantitatively,
each system has its own crystallization kinetics which should be
investigated experimentally.
In order to get a complete description of the CSD, it is necessary to apply the conservation laws of mass, energy and crystals
population in addition to nucleation and growth rates kinetics
[4,7]. For crystals population, the concept of population balance has been a major contribution to crystallizers analysis and
design. The CSD is affected by generation and destruction of
crystals by breakage and agglomeration processes as well as by
nucleation and growth processes [4]. Thus, all crystals present in
the crystallizer must be accounted for, and this is accomplished
by the Population Balance Equation (PBE).
The aim of this study is to present a methodological framework for optimization of para-xylene crystallization process
using a developed rigorous model and model parameters estimation. The kinetic model parameters were estimated by reconciling experimental data of para-xylene concentration and total
mass of para-xylene suspended crystals with model predictions
using a nonlinear optimization technique. For this purpose, the
numerical method of lines (MOL) was used to solve the model’s
mathematical equations. The effect of cooling strategy on the
estimated parameters was studied. To find the optimum cooling profile, a new objective function was proposed and also,
a nonlinear optimization technique was applied. The new proposed objective function maximizes the yield of crystals which
have a certain required average size or greater. This objective
function requires the calculation of mass crystals along the size
range.
In this study, the homogenous well-mixed cooling batch crystallizer was used for para-xylene crystallization in order to
investigate its advantages. In addition, batch crystallizer is suitable equipment for the study of nucleation and growth rates.
1.1. para-Xylene crystallization
de Goede [1] studied the crystallization of para-xylene with
scraped surface heat exchangers. He studied the structure and
growth phenomena of para-xylene crystals as well as the heat
transfer properties of the scraped surface crystallizer under
both crystallizing and non-crystallizing conditions on pilot-plant
scale. He concluded that decreasing the wall temperature below
−30 ◦ C did not result in further cooling of the bulk, because
the effect of the increasing temperature difference is countered
by a decrease in heat transfer coefficient due to an increasing
thickness of the crystals layer.
Patience et al. [8] determined crystallization kinetics of paraxylene from xylenes mixture containing about 25% para-xylene
produced in Amoco plants using a batch pilot-scale scraped surface crystallizer. They found that the xylene mixture was always
saturated during all runs of crystallization and thus developed
a reduced two-parameter model to describe these crystallizers.
They assumed that nucleation of crystals occurs at the walls of
the crystallizer and growth of the crystals occurs in the bulk.
de Goede and de Jong [9] studied heat transfer properties
of a scraped surface heat exchanger for para-xylene separation
in the turbulent flow regime. They found that the heat transfer
coefficient depends more strongly on the rotational frequency
of the scrapers than the theoretical model predicts. This was
because of the generation of vortices due to the scraper action.
They corrected the crystallizer model to take into consideration
the occurrence of these vortices and get satisfactory agreement
between the heat transfer coefficients predicted by the corrected
model and those determined experimentally. However, the disadvantages of the scraped crystallizers arise from the fact that
the scraping blades cause crystals damage and thus producing
slurries which are difficult to filter. Also these devices are often
costly and require considerable maintenance.
H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
de Goede and Van Rosmalen [10] studied the growth rate
phenomena of para-xylene crystals in a small cell. They studied
the growth rate of the crystals for two systems: pure para-xylene
melt and a binary mixture of 70% para-xylene and 30% other
hydrocarbons (xylene isomers, ethylene, benzene and toluene).
They concluded that the growth rate depends on surface integration and diffusion kinetics. Moreover, they indicated that the
growth rate appears to be length-independent in the size range
of their experiments.
Duncan and West [11] studied the effect of ultrasonic vibration in prevention or removal of crystals incrustation. They chose
two chemical systems in their study; water-sodium chloride
system and meta-xylene/para-xylene system in a tank crystallizer. They chose these systems because of their industrial
importance. For the second system they used a mixture of 85%
para-xylene in meta-xylene which has a saturation temperature
of about 6 ◦ C. This mixture was firstly maintained at 3 ◦ C and
no ultrasonic vibration was applied. It was found that the heat
exchanger surface became incrusted within a few minutes. This
experiment was repeated under the same conditions of cooling but with applying ultrasonic excitation at full power. It was
found that nucleation occurred spontaneously just below the
saturation temperature. As a result, slurry of much finer crystals was obtained and it could be cooled to a temperature of
−2 ◦ C before incrustation was observed. When the ultrasonic
excitation was reduced (by halving the voltage of the ultrasonic generator) they found that no incrustation took place and
the obtained crystals were similar in shape and size to those
obtained when no ultrasonic excitation was applied (i.e. larger
crystals were obtained). They also indicated that the required
energy for ultrasonic excitation was about 1 or 2% of the
overall energy requirement for the process. However, this technique has not been scaled up for use in large-scale industrial
plants.
Another cooling method is the cooling by direct contact of
the refrigerant with the crystallizing solution. This technique was
utilized in para-xylene crystallization by using an immiscible,
non-boiling coolant [12]. By this method, no crystals incrustation occurs since there is no heat transfer surface. This method
has often been used in the field of direct heat exchangers and
emergency cooling systems. However, it requires using considerable amounts of the refrigerant for practical production rates.
Also, experience had shown that complete separation between
the crystallizing liquor and the refrigerant was difficult to be
achieved within the short time-scale required for the process.
As a result, this process is not used now on an industrial scale.
Another technique utilized in para-xylene crystallization is
by using the miscible refrigerant CO2 . This refrigerant is first
liquefied in a refrigeration cycle and then introduced to the
crystallizer under conditions of temperature and pressure which
should be above the triple point. The pressure of the crystallizer
is then reduced causing evaporation of CO2 and this will provide
cooling necessary for the crystallization process. However, one
disadvantage of this process is its requirement of using highly
expensive compressors [12].
Duncan and Philips [12] had proposed a new miscible refrigerant process for para-xylene crystallization from its isomers.
27
They used liquid natural gas (LNG), liquefied in a simple, ambient pressure condenser cycle as a cooler. They investigated
this process under different operating conditions of feedstock
composition, residence time in the crystallizer and degree of agitation. They indicated convenient operating pressures and easy
control of the operation as well as a wider choice of the refrigerant and easier separation of this refrigerant from the crystallizing
liquor.
However, no study in the literature tried to estimate the
crystallization kinetic parameters of para-xylene except the
study of Patience et al. [8]. Also, most of the previous studies focused on heat transfer efficiency and mass transfer issues
related to the crystallization process of para-xylene using layer
crystallizers.
In this study, the cooling batch crystallizer for para-xylene
crystallization from xylene isomers mixture was introduced
because it is suitable to study most of the affecting variables
of the crystallization process, such as the effect of the applied
cooling strategy.
2. Mathematical model of the cooling batch crystallizer
2.1. Population density, concentration and temperature
For a well-mixed constant volume batch cooling crystallizer
in which crystal aggregation and breakage are neglected, the
population balance equation is a partial differential equation in
time t and crystal size L. This equation takes the form:
∂n(L, t)
∂n(L, t)
= −G
∂L
∂t
(1)
subject to the spatial boundary condition:
n(0, t) =
B0
G(L = 0)
(2)
and to the initial condition:
n(L, t = 0) = ns
(3)
where n(L, t) is the population density of crystals, G the sizeindependent crystals growth rate, B0 the time-dependent nucleation rate at the initial crystal size L0 , provided that L0 goes to
zero and ns is the initial size distribution of the seeding crystals.
In this study, unseeded crystallizer was used and hence ns = 0.
The solution phase concentration is described by the following mass balance:
∞
dĈ
= −3ρc kv hG
n(L, t)L2 dL
(4)
dt
0
where Ĉ is the solute concentration (mass solute/mass solvent),
ρc the density of crystals, kv a volume shape factor and h is a
conversion factor to convert solvent mass to slurry volume. This
equation is subject to the following initial condition:
Ĉ(t = 0) = Ĉ0
(5)
The temperature T of the cooling batch crystallizer was
known from the applied cooling profile. The required cooling
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H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
profile was applied to the crystallizer using a thermostat with a
controller such that the required cooling rate at any time interval
can be applied, and from this pre-determined cooling profile,
the crystallizer temperature is known at any instant during the
experiment time.
2.2. Birth and growth rates
In the model equations, nucleation and growth rates are
assumed to take the following power forms:
B = kb (S)b
(6)
G = kg (S)g
(7)
where kb , b, kg , and g are the kinetic parameters and S is the
relative supersaturation defined as (C − Csat )/Csat where Csat is
the equilibrium concentration.
Fig. 1. Solubility of para-xylene in ortho-xylene/para-xylene mixture.
2.4. Kinetic parameters
2.3. para-Xylene equilibrium concentration
para-Xylene phase equilibria is an important information in
crystallization processes to determine para-xylene maximum
possible recovery and to determine the temperature level at
which the separation equipment, such as a filter or a centrifuge,
must be operated.
Theoretically, several equations can be used to calculate
solid–liquid equilibrium concentration. One of these equations
is Van’t Hoff equation.
ln xi,L =
HS→L (Tm − T )
RTm T
(8)
where xi is the mole fraction of component i, HS→L is the heat
of fusion of material i at the melting temperature Tm , and R is
the gas constant.
The values of the thermodynamic properties appearing in
this equation are shown in Table 1 for xylene isomers and for
ethylbenzene [3].
However, some experimental data for para-xylene solubility
are available in the literature. Porter and Johnson [13] obtained
solubility data for para-xylene in different xylene mixtures in
the temperature ranges of −29 to 10 and −85 to −57 ◦ C. Haddon
and Johnson [14] determined the solubility of para-xylene in a
mixture of C6 –C8 aromatics in the temperature range of −78 to
−29 ◦ C.
de Goede [1] determined the solubility of para-xylene in
a binary mixture with toluene, ortho-xylene, meta-xylene and
ethylbenzene and in a multicomponent mixture containing
xylene isomers and ethylbenzene.
A plot of para-xylene fraction in ortho-xylene/para-xylene
mixture versus temperature as predicted by Eq. (8) as well as
the experimental data of de Goede [1] are shown in Fig. 1.
Van’t Hoff equation gives good prediction of para-xylene
solubility and was used in this study to predict the equilibrium
concentration Csat .
In this study, kb and kg were assumed to be functions of temperature and take the form of Arrhenius expression as follows:
Eb
(9)
kb = kb0 exp −
RT
Eg
kg = kg0 exp −
(10)
RT
where kb0 , kg0 are constants and Eb and Eg may be defined as the
activation energies for nucleation and growth, respectively. This
form is selected based upon analogy with chemical reaction studies which show that the rate constant usually takes the form of
Arrhenius expression. The effect of temperature on birth rate was
indicated by Miller [15]. Nucleation can actually decrease with
temperature, which corresponds to negative activation energy.
This may be due to the fact that at higher temperatures the efficiency of molecular diffusion towards crystal surfaces increases
and hence decreasing the number of nuclei necessary for secondary nucleation to occur and thus decreasing the nucleation
rate. Also, when molecular diffusion towards crystals surfaces
increases, the growth rate increases.
3. Solution of the mathematical model
General solution methods to solve the batch crystallizer mathematical model are summarized in the study of Mohameed et
al. [16], where it was indicated that several solution methods
have been proposed to solve the model equations. In selecting a
method of solution, its accuracy and simplicity is of great importance. This is because the solution procedure will be later used
in tasks such as optimization and control techniques.
In this study, the numerical method of lines (MOL) was used
to solve the model’s mathematical equations. This method was
described in details in the study of Mohameed et al. [16] and
here it is summarized briefly.
H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
3.1. The method of lines (MOL)
Using this method, the partial derivative ∂n/∂L in Eq. (1) is
discretized in the internal spatial direction L into a set of ordinary
differential equations in time domain and this set can then be
solved numerically.
In this study, the L domain in Eq. (1) is represented by grid
points Li , such that:
Li = (i)L,
i = 2, 3, . . . , N
(11)
where L is the grid spacing, calculated as L = (LN − L0 )/
(N − 1), where L0 and LN are the initial and final points in the L
domain and N is the number of grid points.
Then, the population balance equation is transformed to a set
of ordinary differential equations by replacing the ∂n/∂L term
in Eq. (1) with a second-order approximation. This approximation was adopted because it is less oscillatory than higher-order
approximations and can provide adequate accuracy with minimal computational effort [16]. The population balance is then
transformed to the following set of ordinary differential equations:
dn(L0 )
−3n(L0 ) + 4n(L1 ) − n(L2 )
=
+ O(L)2 ,
dL
2L
i=0
(12)
dn(Li )
n(Li+1 ) − n(Li−1 )
=
,
dL
2L
i = 1, 2, . . . , N − 1
(13)
4n(LN ) − 7n(LN−1 ) + 4n(LN−2 ) − n(LN−3 )
dn(LN )
=
,
dL
2L
i=N
(14)
The resulting system of nonlinear ordinary differential equations combined with the mass balance equation was solved using
Rung-Kutta Method implemented in an ODE solver. In this
study, MATLAB software was used to solve the above system
of differential equations.
Numerically, it is known that as the discretization increases
the accuracy of the solution increases. In this study, the model
equations were solved for N = 5, 10, 20 and 100. It was found
that there were very small differences between model outputs
for N equals 20 and for N equals 100. However, the program
execution time required for solution when N equals 20 (about
5 s) was about one fourth of the execution time when N equals
100. Thus, N was selected to be 20.
After solving the model by this method, the population density of crystals at any time will be known and hence, it is easy to
find the mass of crystals, Mc , of any size L. This can be calculated
as follows:
Mc =
P
ni Li ρc (Li )3 hMs
(15)
i
where P is the number of the node point in the size domain
at which the mass of crystals is to be calculated. The maxi-
29
mum value of P is N and this corresponds to crystals of LN
size.
3.2. Estimation of model parameters
In order to solve the above model, the unknown parameters of
nucleation and growth rates must be determined. These parameters are estimated so that the model is allowed to describe a set
of data in an optimal way by matching the model predictions
with the measured experimental data. In crystallization studies, variable such as concentration is usually measured. In this
study two measurements will be used; one is the concentration
of para-xylene in the liquid phase and the other measurement for
the solid phase which is the crystal suspension density. Kinetic
parameters estimation using both of these measurements gives
more reliable parameters than the estimation using only one
measurement. This is because the estimation will be based on
data from both liquid and solid phases and not from just one
phase.
In the estimation process, an objective function is usually
defined by calculating the deviations between the model predictions and the experimentally collected data. Then, a nonlinear
optimization technique tries to find the optimum parameters
that minimize these deviations. Quantitatively, the optimization
problem can be written as:
min
Φ(y, ŷ; θ); θL ≤ θ ≤ θU
θ
subject to model constraints
(16)
where y and ŷ are model prediction and experimental measurement, respectively. θ is the set of kinetic parameters to be
estimated; θ = [kb0 , Eb , b, kg0 , Eg , g]. Φ is an objective function represents the deviations between model predictions and
the experimental data. θ L and θ U are the possible lower and
upper values of θ, respectively. In this study Φ was formulated
as:
2
nc
nsc Mc (ti ) − Mci 2
C(ti ) − Ĉi
Φ = Φc + Φs =
+
C0 − C f
Mcf
i=1
i=1
(17)
where Φc and Φs are the concentration-part and the suspended crystals-part of the objective function, respectively. The
concentration-part term was divided by the maximum concentration change (the difference between the initial concentration
C0 and the final concentration Cf ) and the suspended crystalspart was divided by the maximum crystals mass change (the
difference between the mass of crystals at final time Mcf and
the initial mass of crystals which equals zero). This is useful in
obtaining a dimensionless objective function. nc and nsc are the
number of measurements of concentration and suspended mass
of crystals, respectively.
The optimization technique is based on the sequential
quadratic programming (SQP) method and a quasi-Newton
method for evaluation and adapting of the Hessian Matrix (H␪ )
which is implemented in the optimization routine. In optimization techniques, it is common to assign minimum and maximum
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H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
values of the parameters to be estimated. This assignment is necessary when the physical meaning of the estimated parameter
restricts its value or its sign.
4. Optimum cooling profile
After determining the model kinetic parameters, the model
was utilized to optimize the performance of the crystallization
process by finding the optimum cooling profile that achieves certain desired criterion of the final product. The optimum cooling
profile is usually found by solving a nonlinear optimization problem to minimize or maximize certain objective function. Most
of the previous studies in crystallization have dealt with finding the temperature trajectory that optimizes some performance
criterion derived from the final CSD [15,17]. The optimization
problem to find the optimum cooling profile was formulated as:
In this study, T was discretized to six piecewise linear functions and the objective function was formulated as:
Θ=
Mcf5
Mcf
(19)
where Mcf5 is the mass of crystals at final time that posses a
size larger than 0.5 mm (this is the size of the fifth mesh after
discretizing the size domain, L, to ten mesh sizes), Mcf is the
total mass of crystals of all sizes obtained at final time. This
objective function could be modified to maximize the yield of
crystals between any two meshes [16].
After the optimum cooling profile was found, it was inserted
into the simulator to find the resulting CSD of the product and
this CSD was compared with that obtained under natural cooling.
It was found that the objective function was effective to optimize
para-xylene crystallization process.
5. Experimental setup and methodology
max
Θ(T, x(tf ); θ)
T
subject to the constraints
[Tmin , . . . , Tmin ]T ≤ AI T ≤ [Tmax , . . . , Tmax ]T
5.1. Materials
(18)
where AI is the identity matrix, T is a vector of temperature
values at discrete points in time between t = 0 and t = tf (the
operation time). T will be taken as discrete time linear piecewise
functions with a constant cooling rate (slope) within each time
interval. Θ defines a final state objective function that describes
the desired performance which could be a function of crystallizer temperature T, final model state x(tf ), and the model
parameters θ.
The crystallizing mixture was prepared by mixing paraxylene and ortho-xylene isomers in two different ratios. paraXylene was purchased from ACROS Company/USA with a
purity of 99% and ortho-xylene was purchased from Scharlau
Chemie/Spain also with a purity of 99%.
5.2. Crystallizer assembly
A sketch of the crystallizer used in this study is shown in
Fig. 2. It was a jacketed, glass vessel with a flat bottom. The
Fig. 2. Crystallizer assembly sketch.
H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
total volume of the crystallizer was 1.20 L and the volume of the
used mixture was 0.80 L. The volume of the jacket surrounding the crystallizer was about 1.45 L. The crystallizer standed
on a magnetic stirrer base to rotate a magnetic stirrer inside the
crystallizer for mixing purposes. From the top of the crystallizer, a sensor to measure the Total Suspended Solid (TSS) was
introduced. This sensor is described below. Also, a thermometer
with measuring range from −10 to 52 ◦ C and with 0.1 ◦ C accuracy was introduced from the top of the crystallizer to measure
the mixture temperature. The glass crystallizer is manufactured
locally by the University workshops.
A cooling liquid circulator from Haake company (Phoenix
P1 Circulator) was used to circulate the cooling liquid (water
with ethanol as an anti-freezing agent) through the jacket of the
crystallizer to cool the mixture following the desired cooling
profile.
5.3. Methodology
The mixture was prepared by mixing para-xylene and orthoxylene isomers in a ratio of 69/31. This ratio was selected to
give a starting melting point of −2 ◦ C. Initially, the crystallizer
was charged with 0.80 L of the mixture at the ambient temperature. A sample for determining the initial concentration was
taken out. Cooling was started by circulating the cooling liquid
through the jacket. Three cooling profiles were applied, namely,
natural, linear and optimum-like profiles. These cooling profiles
are shown in Fig. 3.
When the first crystals appear to form, and using 5-mL
syringes, samples of less than 1 mL of the clear solution were
withdrawn periodically (every 5 min) to measure para-xylene
concentration in the mixture. On the tip of the syringe needle, a
cotton plug was fitted to prevent para-xylene solid crystals from
being sucked in the syringe.
The mass of the suspended solids as well as the temperature
inside the crystallizer was recorded every time the clear samples were withdrawn using the ViSolid 700 IQ Sensor and the
Fig. 3. Experimental natural, linear and optimum-like cooling profiles used to
cool the xylenes mixture.
31
thermometer, respectively. At the end of each run, the concentrations of the collected samples were determined using the Gas
Chromatography analyzer (GC).
5.4. Concentration measurement
As indicated above, samples concentration was determined
using the Gas Chromatography analyzer (GC). For the mixture
under study (ortho-xylene/para-xylene mixture), a 6 feet stainless steel column with 1/8 in. diameter was used for analysis.
This column is packed with 5% SP-1200/1.75% Bentone. The
column temperature was 75 ◦ C and the flow rate of the carrier
gas (N2 ) was 20 mL/min. A sample output of the GC analyzer
is shown in Fig. 4 where each peak represents a component.
The GC output represents the area percentage under each
peak. This requires preparing a calibration line to get the concentration that corresponds to the area percentage under the peak of
the given species. Twelve samples of pre-known concentrations
were analyzed using the GC and these known concentrations
were plotted versus areas under the peaks obtained by the GC.
The resulting calibration line is shown in Fig. 5.
5.5. TSS measurement
From the top of the crystallizer, a sensor for measuring the
mass of Total Suspended Solids (TSS) was introduced. This is
from WTW company ‘IQ Net TSS Sensor’ (ViSolid 700 IQ).
To assure accurate measurement by this sensor, it should be
vertically inserted into the crystallizer. This was assured by using
‘air-bubble water balance’.
The optical measurement head of this sensor measures the
light scattered and reflected by the suspended particles (crystals)
in the measuring medium. The level of total suspended solids
Fig. 4. GC sample analysis output for ortho-xylene/para-xylene mixture.
32
H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
Table 2
Optimum parameters under natural cooling profile
Parameter
(number/m3
kb0
Eb (J/mol)
b
kg0 (m/s)
Eg (J/mol)
g
Fig. 5. GC calibration line.
Value
s)
768.10
−43.21
1.08
20.58
695.25
3.70
which the first crystals started to appear. This freezing temperature is suitable to work at using the described apparatus. The
time of each run was 80 min. The runs were stopped after this
time because a layer of crystals started to incrust on the crystallizer inside wall and the magnetic stirrer was noticed not to be
able to keep a homogenous suspension of the mixture.
In this study, the numerical method of lines (MOL) was
used to find the optimum kinetic parameters under each cooling
profile using the objective function described in Eq. (17). The
validity of the parameters was investigated and then the optimum
cooling profile to optimize para-xylene crystallization process
was determined.
6.1. Effect of cooling profile on the estimated parameters
Fig. 6. ViSolid 700 IQ Sensor calibration curve.
that corresponds to the amount of reflected light was displayed
on a screen attached to the sensor.
To find the mass of suspended crystals in (g crystals/L mixture) units, calibration was needed. To make this, the temperature
of the crystallizer was firstly reduced to −2 ◦ C and set at this
point for 45 min. The reading on the sensor’s display during this
time was recorded (every 1 min) and the average value of these
measurements was calculated. Then, sample from the crystallizer was taken and analyzed using the GC analyzer. The temperature was then reduced to a lower point (−0.5 ◦ C decrease)
and the same procedure was repeated (for six points). Finally,
and using the resulting data, a calibration curve was obtained
by plotting the mass of suspended crystals per mixture volume
(which is actually the third moment) versus the sensor’s displaced measurement. This curve is shown in Fig. 6.
6. Results and discussion
The parameters to be estimated are kb0 , Eb , b, kg0 , Eg and g. To
investigate the effect of cooling strategy on the estimates, three
cooling profiles were applied to cool the 69% para-xylene mixture. These cooling profiles were natural, linear and optimumlike cooling profiles shown in Fig. 3. All of these profiles started
at a temperature near to −2 ◦ C, which was the temperature at
Most of the complicated optimization problems, like the one
in this study, suffer from the fact that the estimated parameters
are highly dependent on their initial guesses. In this case, local
minimization rather than global minimization of the objective
function occurs. To overcome this problem, a wide set of initial
guesses was tried until the global minimization was probably
reached. The estimated parameters are shown below for each of
the applied cooling profile.
The three cooling profiles applied to the crystallizer; natural,
linear and the optimum-like profiles are shown in Fig. 3. Liquid
concentration and mass of suspended crystals were measured
for each cooling profile. The optimum kinetic parameters are
shown in Tables 2–4, respectively. The optimum parameters in
the three tables were used to compare the model prediction y with
the experimental data ŷ of concentration and mass of suspended
crystals. The results for the concentration data are shown in
Fig. 7 for natural cooling profile while the results for suspension
density are shown in Figs. 8 and 9 for linear and optimumlike cooling profile, respectively. To see the accuracy of model
predictions, the percentages of deviation between the predictions
Table 3
Optimum parameters under linear cooling profile
Parameter
MOL
kb0 (number/m3 s)
Eb (J/mol)
b
kg0 (m/s)
Eg (J/mol)
g
812.56
−56.19
0.89
28.45
785.25
3.95
H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
33
Table 4
Optimum parameters under optimum-like cooling profile
Parameter
(number/m3
kb0
Eb (J/mol)
b
kg0 (m/s)
Eg (J/mol)
g
MOL
s)
354.59
−63.37
0.71
16.25
370.23
4.74
Fig. 9. Measured and predicted mass of suspended crystals under optimum-like
cooling.
6.2. Global set of optimum parameters
Fig. 7. Measured and predicted concentrations under natural cooling.
and the experimental measurements were calculated as follows:
deviation percentage =
y − ŷ
× 100
ŷ
(20)
Good agreement between model predictions and the experimental data was noticed in all applied cooling profiles as shown
in Figs. 7–9. The deviations in concentration measurements were
less than ±5%, while those in suspension density were nearly
within ±20%.
It was noticed from the previous results that the estimated
parameters were affected by the applied cooling strategy. However, in order to find the optimum cooling profile, it is better
to find a global set of parameters which minimizes a global
objective function formulated by summing the three objective
functions of the three cooling profiles together. This will give a
set of optimum parameters that predicts the experimental measurements whatever the applied cooling profile is. The resulting
global set of parameters is shown in Table 5.
The global set of optimum parameters in Table 5 were used
to compare model predictions with the experimental data of
concentration and mass of suspended crystals under the three
cooling profiles.
The results are shown in Figs. 10 and 11 for concentration and
the deviations from the measurements, respectively. It is noticed
that the global optimum parameters predict the experimental
concentrations within almost 0.5% deviations. Depending on
this result, these parameters were used to find the optimum
cooling profile that optimizes the crystallization process of paraxylene.
6.3. Optimum cooling profile
After finding the global set of optimum parameters, the optimum cooling profile that maximizes the yield of crystals that
have an average size greater than 0.5 mm was determined. This
Table 5
Global optimum kinetic parameters
Fig. 8. Measured and predicted mass of suspended crystals under linear cooling.
Parameter
Value
kb0 (number/m3 s)
Eb (J/mol)
b
kg0 (m/s)
Eg (J/mol)
g
734.60
−61.12
0.94
24.55
578.36
3.82
34
H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
Fig. 10. Measured and predicted concentration profiles using the global optimum parameters.
profile was found by solving the optimization problem described
in Eq. (18) and using the objective function in Eq. (19). The
cooling profile was discretized to six linear piecewises (or six
slopes). This number of piecewises was selected since it was
found to be adequate; and for higher numbers, the time required
to solve the optimization problem became longer. The resulting
optimum cooling profile is shown in Fig. 12.
To investigate the resulting cooling profile, the CSD under the
optimum cooling profile was compared with that under natural
cooling profile. The comparison is shown in Fig. 13.
It is clear from Fig. 13 that the mass of crystals that have
sizes larger than the size of the fifth mesh (0.5 mm) under the
optimal cooling profile is greater than the mass obtained under
natural cooling for the same size range. Thus, the optimization
technique converged successfully and para-xylene crystallization was optimized.
Regarding the optimizer execution time to find the optimum
cooling profile, it has been noted that the time was synchronized
with the batch process time. Thus, this optimizer could be used
Fig. 12. Optimum cooling profile.
Fig. 13. CSD obtained under natural and optimum cooling profiles.
off-line to find the set point trajectory over each time horizon.
Then, the resulting cooling profile could be passed to a regulatory
control system at the cooling liquid circulator.
7. Conclusions
Fig. 11. Deviation percentages of predicted concentration from measurement
using the global optimum parameters.
In this study, crystallization of para-xylene from orthoxylene/para-xylene mixture in a lab-scale cooling batch crystallizer was studied. The mathematical model used in this study
was found to be effective in describing para-xylene crystallization process. The MOL, as a new method of solution, was found
to be effective in solving the model and accurate results were
noticed. A nonlinear optimization technique was used to estimate the kinetic parameters and it was successfully converged.
The parameters kb and kg were found to be weak functions
of temperature as the small values of Eb and Eg suggest. The
model was then utilized in a model-based optimization strategy
to calculate an optimum cooling profile for the crystallization
process. The new proposed objective function which maximizes
H.A. Mohameed et al. / Chemical Engineering and Processing 46 (2007) 25–36
the yield of para-xylene crystals that have an average size larger
than the desired size gave good results and thus can be considered as an effective tool to optimize para-xylene crystallization
processes.
Acknowledgment
The workers at the Glass Workshop at Jordan University of
Science and Technology are acknowledged for manufacturing
the glass crystallizers used in this study.
Appendix A. Constants and factors
Constants and factors are shown in Table A.1.
Table A.1
Constants and factors
Constant/factor
Value
Unit
ρc
kv
h
R
HS→L
Tm
N
L0
LN
1096
1
0.0012
8.314
17096
286.4
20
0
0.001
kg/m3
Dimensionless
m3 slurry/kg solvent
J/mol K
J/mol
K
Dimensionless
m
m
Mc
Mcf
n(L, t)
nk
ns
R
S
t
tf
T
Tj
Tm
T0
V
xi,L
x(tf )
y
ŷ
35
mass of crystals
mass of crystals at final time
population density function
number of measurements of the kth type
seed size distribution
gas constant
relative supersaturation
time
final time
slurry temperature
jacket-cooling liquid temperature
melting temperature
initial slurry temperature
slurry volume
component i mole fraction in liquid
final model state
model prediction
experimental measurement
Greek letters
θ
estimated parameters set
θL
lower values of estimated parameters
θU
upper values of estimated parameters
Θ
optimum cooling profile objective function
μi
ith moment
ρc
crystals density
Φ
optimum parameters objective function
Appendix B. Nomenclature
References
AI
b
B
B0
C
C0
Cf
Cp
Csat
Ĉ
Eb
Eg
g
G
h
Hc
HS→L
kb
kb0
kg
kg0
kv
L
L0
L
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