Hydrodynamic modeling of adsorption -
Application to the separation of xylenes
Master Thesis
ofMiguel Domingos da Silva
Thesis performed in the framework of Dissertation at
Supervisor from IFP: Dr. Frédéric Augier Supervisor from FEUP: Prof. José Carlos Lopes
Department of Chemical Engineering
Acknowledgments
First, I want to thank my supervisor from IFPEN, Dr. Fréderic Augier, for every time spent helping me and advising me during the internship, also for putting me back on track when things seemed confusing and dispersed.
I thank Professor José Carlos Lopes for accepting to be my supervisor from FEUP.
I thank Dr. Aude Royon-Lebeaud for her help in every detail, every problem, for the interest demonstrated about my work and for the help with some issues that I had with Linux and Fluent.
I thank PhD Student Leonel Gomes for the help with every call and every question that I had even before the first day I arrived at IFPEN.
To all my friends, I thank you all for the positive thoughts that you sent me, they helped me a lot to surpass the “being far away from home” feeling and gave me more confidence conclude the project.
Finally, to my family, I thank you for being so supportive and believe that at the end of this little amount of time that was the course I had all the capacity and knowledge to finish it.
Abstract
Xylenes production has been increasing over time and with this there has been a major research in developing and upgrading the separation processes. The main separation process is the Simulated Moving Bed which consists in a chromatographic separation performed in multi fixed-bed adsorption columns.
Within these columns there are several dispersive phenomena that can be related to hydrodynamics or the non-linearity of the adsorption equilibrium, that deviate the flow regime from the desired plug flow. Therefore, it is needed to study the coupling between these dispersive phenomena and adsorption, so that the separation can be accurately modelled. Recently, these interactions were studied by Augier et al (2008).
The main objectives of this project are to study and model hydrodynamics of an adsorption bed of the SMB, couple it with the adsorption equilibrium and study the impact of obstacles (such as pipes and beams) placed inside the porous media, and empty chambers before and after the fixed bed, in the process of separation.
Results from CFD simulations show that hydrodynamics have a huge role in the separation and cannot be separated from the adsorption equilibrium, in the absence of mass transfer limitations. When mass transfer limitations are considered results show that 1D ideal model approximations can be made to predict the separation, in the presence or absence of obstacles in the porous media.
Keywords: Hydrodynamics, Adsorption, Simulated Moving Bed,
Resumo
Ao longo dos ultimos anos a produção de xilenos tem vindo a aumentar, e com isto o aumento do interesse na investigação para melhorar os processos de separação dos mesmos. O processo de separação que mais tem vindo a ser utilizado é o Leito Móvel Simulado, que consiste numa separação cromatográfica feita em colunas de adsorção constituídas de leitos fixos sobrepostos.
No interior destas colunas vários fenómenos dispersivos estão presentes, estes podem estar relacionados com a hidrodinâmica ou devido à não linearidade do fenómeno de adsorção, estes efeitos causam um desvio no regime de escoamento que o faz sair do pretendido fluxo pistão. Assim sendo, é necessário estudar esses efeitos para que a separação possa ser modelisada com precisão. Este caso foi recentemente estudado por Augier et al (2008)
Este trabalho tem como objectivos principais o estudo e modelisação da hidrodinâmica de um leito de adsorção do Leito Móvel Simulado, acoplar a esse estudo o fenómeno de adsorção e estudar o impacto de obstáculos (como tubos e barras) colocados no meio poroso e de zonas livres antes e depois do meio poroso, na separação.
Os resultados das simulações CFD demonstram que a hidrodinâmica tem um papel principal na separação e a mesma não pode ser dissociada da adsorção, aquando da absência de limitações à transferencia de massa. Quando essas limitações são tomadas em conta, os resultados obtidos demonstram que apoximações feitas por modelos ideais podem ser feitas para prever a separação, quer na ausência ou presença de obstaculos dentro do meio poroso. Palavras Chave: Hidrodinâmica, Adsorção, Leito Móvel Simulado, Dinâmica
Declaration
I declare, under honor commitment, that this work is original and every non-original contribution was referenced with the source identification.
i
Index
1 Introduction ... 1
2 Bibliographic study ... 2
2.1 Xylenes ... 2
2.2 Theoretical description and background ... 7
2.3 Flow characterization methods ... 12
3 Technical Description ... 19
3.1 Experimental Setup ... 19
3.2 CFD approach ... 20
3.3 Ideal model approximations ... 21
4 Results and discussion ... 23
4.1 Hydrodynamics ... 23
4.2 Adsorption ... 31
4.3 Ideal model approximations ... 34
5 Conclusions ... 39
References ... 40
Appendix 1 Tables and figures ... 43
ii
List of Figures
Figure 1 – Molecular structure of the mixed xylenes. ...2
Figure 2 – Main utilizations of the mixed xylenes [Fabri et al., 2000]. ...3
Figure 3 – Simplified scheme of a chromatography. ...5
Figure 4 – Simplified scheme of the TMB with its concentration profiles [Ruthven and Ching, 1989]. ...6
Figure 5 – Simplified scheme of the Simulated Moving Bed [Ruthven and Ching, 1989]. ...7
Figure 6 - Example of RTD. ... 13
Figure 7 – Example of three RTD with the same mean times but with distinct variances. ... 14
Figure 8 - Completely segregated system and perfectly mixed system both with the same RTD. ... 15
Figure 9 – Schematic view of the experimental installation, respective dimensions and sensor positions. ... 19
Figure 10 – 2D geometry representative of the experimental installation. ... 20
Figure 11 – Inlet profile injections for experimental and simulated RTD’s ... 25
Figure 12 – Velocity field obtained and zoom of the inlet and outlet. ... 25
Figure 13 – User defined scalar contours for the RTD simulation after injecting the profile displayed in Figure 11. ... 26
Figure 14– RTD obtained for a 2D simulation using an inlet UDF injection (solid lines) and comparison with the experimental result (points). ... 27
Figure 15 – Local mean age and variance obtained through Liu and Tilton’s methodology (2010) ... 29
Figure 16 – Mean age for the three geometries for a flow rate of 6 m3.h-1. ... 31
Figure 17 - Separation of the xylene mixture performed with the 2D geometry without obstacles .... 32
Figure 18 – Separation of the xylene mixture performed with the 2D geometry with the cylinder as obstacle ... 33
Figure 19 - Separation of the xylene mixture performed with the 2D geometry with the quadrangular prism as obstacle ... 33
Figure 20 – RTD comparison between the three geometries ... 36
Figure 21– Adsorption separation, without mass transfer limitations, results comparison for the 2D geometry in the absence of obstacle ... 36
Figure 22 – Adsorption separation, without mass transfer limitations, results comparison for the 2D geometry with a cylinder as obstacle ... 37
iii Figure 23 – Adsorption separation, with mass transfer limitations, results comparison for the 2D geometry with a cylinder as obstacle ... 38 Figure 24 - Schematic view of the experimental installation with the obstacles (cylinder at left and prism at right), with the respective dimensions. ... 43 Figure 25 - Adsorption separation, with no mass transfer limitations, results comparison for the 2D geometry with a quadrangular prism as an obstacle ... 44
iv
List of Tables
Table 1 – Physical properties of mixed xylenes [Fabri et al., 2000]. ...4 Table 2 – Hydrodynamic simulation parameters ... 23 Table 3 – First two RTD moments obtained with different turbulence models at the outlet of the vessel. ... 24 Table 4 – Mean of the age distribution of the probes for the two turbulence models ... 24 Table 5 –Comparison of the mean ages between a CFD simulation made at a flow rate of 6.63 m3.h-1
and the experimental results at a flow rate of 6 m3/h ... 28
Table 6 – Comparison of internal age distribution and variance for the geometry/installation with a cylinder as obstacle. ... 30 Table 7 - Comparison of internal age distribution and variance for the geometry/installation with a quadrangular prism as obstacle ... 30 Table 8 – Table of moments of age comparison between hydrodynamics and adsorption in the three cases of study ... 34 Table 9 – Simulated results, with Liu and Tilton’s methodology (2010), for the top and bottom free flow zones for the first and second moments, variance and equivalent number of CSTR’s ... 35 Table 10 – Moments distribution for the same residence time and different Schmidt number ... 43 Table 11 – Table of moments of age comparison between the already presented cases and their
v
Notation and Glossary
List of symbols
Terminology
Spatial mean age(s) (s)
Langmuir coefficient of the compound i (m3/mol)
Concentration of the compound i (mol/m3)
k- model parameter (-)
k- model parameter (-)
k- model parameter (-)
CoV
Coefficient of variation (-)Diffusivity coefficient (m2/s)
Flow’s length scale (m)
Particle’s diameter (m)
Turbulent diffusion (m2/s)
Exit age distribution (-)
F
Liquid-solid friction force (N/m3)Spalding’s spatial invariant term (mol/m3)
Turbulent kinetic energy (m2/s2)
Bed permeability (m2)
Overall mass transfer coefficient (s-1)
Outlet’s length (m)
Molar mass (kg/mol)
nth raw moment (sn)
Moment order (-)
Number of CSTR (-)
Pressure (Pa)
Peclet number (-)
Turbulent model equation (-)
Adsorbed concentration of the compound i (kg/m3)
Volumetric flow (m3/s)
Adsorbed concentration of the compound i in equilibrium (kg/m3)
Adsorbent mass capacity for the compound i (kg/kg)
Molar source term (mol/m3.s)
Reynolds number (-) Schmidt number (-) Time (s) a i b i C 1 C 2 C
C
Dd
pd
TD
E Ik
K 1k
L M n m n CSTR N pPe
k P i q Q * iq
i mq
, RRe
Sc
t
vi
Switching time (s)
Fluid interstitial velocity (m/s)
Fluid average superficial velocity (m/s)
Vessel’s volume (m3)
Fluid superficial velocity (m/s)
Position (m)
Greek symbols
Age frequency function (-)
Molecule’s age (s)
Molecules’ mean age (s)
Molecule’s age in a point (s)
* Damping coefficient (-)
Non-Darcy coefficient (m-1)
i Bed porosity (-)
Turbulent kinetic energy dissipation rate (m2/s3)
Volume average age frequency function (-)
Fluid dynamic viscosity (Pa.s)
Fluid effective dynamic viscosity (Pa.s)
nth central moment (sn)
nth normalized central moment (-)
Turbulent dynamic viscosity (Pa.s)
Mean residence time (s)
Fluid density (kg/m3) Particle density (kg/m3) Variance (s2) k- model parameter of k (-) k- model parameter of (-) Residence time (s)
ω
Specific turbulence dissipation rate (s-1)Indexes and exponents
Exit Inlet Longitudinal Outlet Transversal Volume averaged * t u u
V
v x
p
T
e
n
* n
T
1
p
2
k
e inl
out
t
V
vii Radial
Axial
Abbreviations
CFD
Computational fluid dynamicsCSTR
Continuous stirred-tank reactorDPFR
Dispersive plug flow reactorEB
EthylbenzeneMX
m-xyleneMOX
m-xylene and o-xylene mixtureOX
o-xylenePDEB
p-diethylbenzenePET
Polyethylene terephthalatePX
p-xyleneRTD
Residence time distributionSMB
Simulated moving bedTMB
True moving bedUDF
User defined function xIntroduction 1
1 Introduction
Para-xylene (PX) is the main raw material for the manufacture of polyethylene terephthalate (PET), a polymer used for the production of beverage bottles and polyester fibers. The constant growth of the PET consumption has boosted the production of PX.
Since PX is always produced in a mixture with its isomers, there has been a big interest in developing and improving the processes of separation of this mixture. The most used process to perform this separation is the Simulated Moving Bed (SMB).
The objective of the companies that are interested in studying this process is to maximize PX production. To do this it is necessary to evaluate the impact of the different dispersive phenomena such as the intra-particle mass transfer and axial dispersion of the flow in the fixed bed. It is also important to describe hydrodynamics of the system and its interactions with adsorption, when its isotherm is not linear.
Then, for this master thesis project the main objectives proposed are to:
Study and model hydrodynamics of an adsorption bed of a SMB through CFD simulations and experimental tracer injections.
Identify the effect that obstacles provoke in hydrodynamics.
Couple the adsorption with the previous model and observe the effect of hydrodynamics in the xylene separation.
Bibliographic study 2
2 Bibliographic study
2.1 Xylenes
The term xylene designates the dimethylbenzene, an aromatic ring with two methyl groups. There are three different molecular configurations in which the xylenes can appear, ortho (1,2-dimethylbenzene, OX), meta (1,3-dimethylbenzene, MX) and para (1,4-dimethylbenzene, PX), depending on which carbon atoms the methyl groups are attached. The term mixed xylenes also considers the ethylbenzene (EB) which instead of two methyl groups has one group ethyl and also shares the same molecular formula and molar mass as the xylenes, C8H10
and 106.16 g /mol. The molecular configurations of these four molecules are shown in Figure 1.
Figure 1 – Molecular structure of the mixed xylenes.
Despite the similar molecular configuration of these isomers, they do not share the same industrial interest. The PX is by far the most important aromatic C8, followed by the OX. To
valorise the isomers of PX, they are sent to an isomerization unit so they can be converted into PX.
2.1.1 Mixed Xylenes - production and use
There are four ways to obtain these isomers: the catalytic reforming (80%) [Chem Systems Ltd. 1995], pyrolysis gasoline (11.1%), toluene disproportionation (7.6%) and coke-oven light oil (1.3%). EB is not present when the xylenes are produced via toluene disproportionation [Cannella, 2007].
During 2007-2012 the consumption of mixed xylenes increased by 15%, the capacity also increased in a rate of 31%. So, in the next years, it is expected the operational rates to increase. In terms of consumption and production, the Asian market is the one that has the highest share. For the isomers of xylene, PX accounted for 79% of 2012 global mixed xylenes
Bibliographic study 3 demand, followed by the OX with 7% of the mixed xylenes consumption, after comes the MX with 1.3% [Xylenes report, IHS Chemical, 2012].
Figure 2 shows in a schematic view the main uses for the mixed xylenes.
Figure 2 – Main utilizations of the mixed xylenes [Fabri et al., 2000].
From the mixed xylenes recovered, between 50 and 60% are destined to PX production, from 10 to 15% to OX production, 10 to 25% are sent to gasoline blending and merely 1% to MX production [Cannella, 2007]. Around 98% of the PX is consumed in the polyester chain, mainly to produce PET. For this, the PX must be separated from its isomers, with high level purity, so it can be used in this valuable industrial chain.
Bibliographic study 4 2.1.2 Separation of Mixed Xylenes
The physical properties of the mixed xylenes are shown in Table 1.
Table 1 – Physical properties of mixed xylenes [Fabri et al., 2000].
o-xylene m-xylene p-xylene ethylbenzene
boiling point (°C) 144.4 139.1 138.4 136.2
melting point (°C) -25.2 -47.9 13.3 -95.0
density at 25°C (kg/m3) 876.0 859.9 856.7 862.4
To separate this mixture the process of distillation is not economically viable due to the close boiling points of these isomers. If the EB and the OX were separated from the other two isomers it would be necessary 300 theoretical trays [Cannella, 2007]. This process would have high energy consumption and the cost of the equipment would be impracticable, also the MX would still need to be removed from the resultant mixture.
Crystallization was the first industrial PX purification process. It consists in cooling the xylene mixture below the PX's freezing point, so it can be extracted from the other isomers, still in the liquid phase. Even if it is possible to achieve a purity of 99%, due to the eutectic point the maximal recovery rate of PX is around 60-65%, resulting in a high volume of recycled xylenes to the isomerization unit [Fabri et al., 2000].
Nowadays, around 60% [Minceva and Rodrigues, 2007] of the PX produced comes from chromatographic processes. This kind of industrial processes consists in the exploitation of the difference of affinities of a given adsorbent for the PX and its isomers, and can achieve a recovery rate of around 97% per pass, much higher than the crystallization [Candella, 2007].
2.1.2.1 Chromatography
With this method, a small sample of a mixture (mobile phase) is injected in a column filled with the adsorbent (stationary phase). To better understand this process a scheme of a chromatographic separation is shown in Figure 3.
Bibliographic study 5
Figure 3 – Simplified scheme of a chromatography.
The difference of affinities of the adsorbent for the compounds within the mixture will result in different residence times: compounds with respective higher affinity will remain longer inside the chromatographic column, while those with lower affinity will be less affected by the adsorbent and will have shorter residence times. These laboratory scale chromatographic columns must have small diameters to minimize radial heterogeneities. For industrial scale columns this problem cannot be avoided since their diameters can vary between 5 and 10 meters. To maximize the mass transfer between the mobile phase and the so called “stationary phase”, these are put in counter current, analogously to the heat transfer in heat exchangers.
2.1.2.2 True Moving Bed (TMB) process
The principle of the TMB consists in the motion of the solid phase (adsorbent) and the liquid phase (xylenes) in countercurrent within a column. The PX is the more strongly adsorbed species, and must be separated from its isomers, OX, MX and EB, that are less strongly adsorbed. The desorbent used is the para-diethylbenzene (PDEB), for who the adsorbent has an intermediate affinity, so the order of affinities is PX>PDEB>EB>OX/MX. The process separates the original mixtures into two streams: the extract (PX and PDEB) and the raffinate (OX, MX, EB and PDEB). These streams must then be purified through distillation.
Bibliographic study 6
Figure 4 – Simplified scheme of the TMB with its concentration profiles [Ruthven and Ching, 1989].
The most used desorbent is the p-diethylbenzene (PDEB), which molecular structure is similar to the PX, but instead of the two methyl groups (CH3), PDEB has two ethyl groups (C2H5) in the para position. When the stationary state is achieved, two mixtures may be recovered from the column. The role of the desorbent is to clean the solid phase from the mixed xylenes. The boiling point of the PDEB is different from the mixed xylenes, so that a posterior distillation can be easily performed.
The main technical difficulty of this process is the circulation of the solid. It is hard to control its velocity, resulting in heterogeneous motion of the adsorbent and consequently a great loss of efficiency of the mass transfer between the bulk and the solid. The movement of the solid is expensive, and it causes its corrosion and backmixing, which highly decreases the separation efficiency of the process [Ruthven and Ching, 1989].
2.1.2.3 Simulated Moving Bed process
In this process instead of circulating the solid, the countercurrent is simulated by periodically switching the inlets and outlets of the column towards the fluid flow. This process is called Simulated Moving Bed (SMB) and is schematically represented in Figure 5.
Bibliographic study 7
Figure 5 – Simplified scheme of the Simulated Moving Bed [Ruthven and Ching, 1989].
The adsorbent is kept in several fixed beds. TMB’s respective four sections are visible and distinguishable. Solid arrows show the actual streams at time t and the dashed arrows show the streams in use after the stream switch, at t+t* where t* is the switching time.
While the TMB reaches a stationary state, the SMB leans towards a cyclic steady state that is reached after several cycles of the unit’s operation. When the cyclic steady state is achieved, the concentration profiles change over time in each fixed bed of the adsorber, although, these profiles are stationary in each section of the SMB.
This process is ideal for xylene separation due to its ability to work continuously and consequently to treat industrial quantities of mixed xylenes.
2.2 Theoretical description and background
In this section the equations and models used to describe hydrodynamics in free flow and porous media, the mass transfer and the adsorption thermodynamics equilibrium will be presented. Since in the scope of the project is based in the separation of a mixture of xylenes, and this separation is assumed to be isothermal, the heat transfer equations are not presented, only the mass balance will be studied.
Bibliographic study 8 2.2.1 Hydrodynamics
In this work, hydrodynamics are studied while assuming the flow as steady and incompressible. Different models were chosen for two different media: free flow and porous media.
2.2.1.1 Free flow
For the free flow media, if the turbulence effects are non-existent, the Navier-Stokes equations for an incompressible Newtonian fluid will be solved:
u
u
p
2u
(2.1)where is the fluid density (kg/m3), u the fluid velocity (m/s), p the pressure (Pa), the
fluid dynamic viscosity (Pa.s) and 2 is the Laplacian operator. Since the Reynolds number is
high, , the models chosen to characterize the flow are the standard k-ε turbulence model proposed by [Launder and Spalding (1972)] and standard k-ω turbulence model developed by [Wilcox (1998)].
For the k-ε model, the turbulent viscosity is related to the local values of turbulent kinetic energy (m2/s2) and its dissipation rate
T (m2/s3) as shown in equation (2.2).
(2.2)
is a k-ε model parameter, determined experimentally and equals to 0,09. This introduces two new transport equations, (2.3) and (2.4).
⃗ [( ) ] (2.3)
⃗ [( ) ] (2.4)
, , , are constant parameters of the model determined experimentally and is the production term.
Bibliographic study 9 The standard k-ω model is an empirical based model with transport equations for turbulent kinetic energy and its specific dissipation rate . This model has been modified several times to improve its accuracy and as result the turbulent viscosity is defined using a damping coefficient , defined and characterized by [Wilcox (1998)], as shown in equation (2.5).
(2.5)
The transport equations (2.6) and (2.7) are used in this model.
( ) ( ) [( ) ] (2.6) ( ) ( ) [( ) ] (2.7)
These two turbulence models were used in a comparative study in order to perceive which one is more suitable for this study.
2.2.1.2 Porous Media
The Brinkman-Forchheimer model, is a modification of the laminar Navier-Stokes equations where the diffusion of momentum due to viscosity effects and the inertia due to friction between the fluid and particles are modeled. This model has been used to simulate hydrodynamics in porous media, [Chan et al. (2000)]. The fluids are considered to be incompressible since they are in liquid state, and the equations to be solved are:
(2.8)
for continuity, where represents the fluid interstitial velocity (m/s).
( ) (2.9) for momentum, is the inter-particle porosity, is the pressure (Pa), is the apparent liquid viscosity (Pa.s), is the liquid density (kg/m3) and is the liquid-solid friction force (N/m3)
Bibliographic study 10
(2.10)
v is the superficial velocity (m/s), is the permeability of the porous media (m2) and is the non-Darcy term (m-1). In the case of fixed bed the values of and are expressed by the
Ergun’s law [Ergun (1952)] and their values can be obtained through equation (2.11) and (2.12).
( ) (2.11)
(2.12)
These empirical expressions were originally developed for one–dimensional flows within isotropic media, but when the first term of the equation (2.9) / viscous term, is dominant the use of the friction term coupled in the Navier-Stokes equation is acceptable, [Zeng and Crigg (2006)].
2.2.2 Mass Transfer
2.2.2.1 Inter-particle phase
The mass transfer equations that are presented in this section were coupled with the solution previously obtained for hydrodynamics. In the free flow regions there is no adsorption, the concentration of a compound is transported by diffusion and convection.
( ⃗ ) (2.13)
If the regime is turbulent, the turbulent diffusion must be added to the molecular. This turbulent diffusion , can be obtained by assuming a turbulent Schmidt number of 0.7 and using the turbulent viscosity obtained by turbulent model in use, either k-ε or k-ω, as shown in equation (2.14) .
Bibliographic study 11 This turbulent diffusion, in opposite of the molecular diffusion that is assumed constant, suffers a variation in space, this happens because it is a function of the turbulent viscosity and consequently of the local velocity.
In the porous media the transport of species can be described by equation (2.15),
( )
( ⃗ ) (2.15)
i is the bed porosity, is the particle density (kg/m3), the molar mass of the compound
(kg/mol) and u is the interstitial velocity (m/s). The diffusivity coefficient in the equation (2.15) comprises the molecular diffusion and the mechanical dispersion. Contrarily to the molecular and turbulent diffusion that are isotropic, the mechanical dispersion is described by a diagonal tensor comprising the radial ( ) and axial dispersion ( ), relatively to the flow direction. These two vectors are calculated through the definition of the Peclet number that is the product of the Reynolds and Schmidt numbers and quantifies the ratio between convective and diffusive transport.
(2.16)
The longitudinal ( ) and transversal dispersion ( ) coefficients can be calculated by using a constant Peclet number.
(2.17)
(2.18)
Since they are dependent to the local fluid velocity, the tensor can be calculated for a two-dimensional domain such as,
| | | | (| | | | ) (2.19) | | | | (| | | | ) (2.20)
The longitudinal and transversal Peclet numbers are assumed to take the value of 2 and 11 respectively, following the results obtained by [Founemy et al (1992)].
Bibliographic study 12 2.2.2.2 Intra-particle phase
Two possible ways to define the adsorption term are, through direct equilibria between both phases using partial derivative equations (PDE), equation (2.21), or through a linear driving force (LDF), equation (2.22), to simulate the mass transfer resistance between the bulk and the adsorbed phase.
∑ (2.21) ( ) (2.22)
is the overall mas transfer coefficient for the fluid film resistance (s-1). The equation (2.20)
can be applied when molecules migrate between the bulk and the adsorbed phases almost instantaneously, so the mass transfer limitations are not important. When the mass transfer limitations are substantial, the adsorbent must be discretized and the mass balance must be solved locally taking into account a linear driving force and by adjusting the overall mass transfer coefficient.
2.2.3 Adsorption Equilibria
The adsorbed concentration is obtained through the Langmuir multicomponent adsorption model [Ruthven (1984)].
∑
(2.23)
is the adsorbent mass capacity (kg/kg), is the adsorbent affinity (m3/mol) for each
compound on a given system, is the concentration (mol/m3) of a compound in the mobile
phase. This model is generally used for gas/solid adsorption systems. However, Daems et al. (2006) found a good agreement between this model and experimental results for liquid/solid adsorption for alkanes, alkenes and aromatics mixtures.
2.3 Flow characterization methods
2.3.1 Residence Time Distribution (RTD)A classical approach to provide some basic information of the macro-mixing state and hydrodynamics of a given system is the Residence Time Distribution (RTD) theory. To
Bibliographic study 13 determine the RTD of a system normally transient experiments with inert tracers are made. These experiments consist in injecting an inert tracer at the inlet of a vessel (reactor, column), and the measurement of its concentration at the outlet. These injections must be close to the ideal, such as perfect step or Dirac. The RTD is represented by an external residence time distribution or an exit age distribution, E(t). The function E(t) has the units of time-1 and one example shown in the Figure 6.
Figure 6 - Example of RTD.
This function is defined as shown in equations (2.25) and (2.26)
∫ ( ) (2.24)
( )
∫ ( ) ( )̅ ∫ ∫ ( )
(2.25)
is the outlet tracer concentration (mol/m3), and ̅ are the fluid superficial and its
average at the outlet and is the outlet’s length. Residence time distributions can be described by using an infinite set of parameters known as moments.
∫ ( ) (2.26)
is the nth raw moment of the distribution. For a better comprehension of these moments,
Bibliographic study 14
∫ ( ) ( ) (2.27)
Where is the nth central moment. The first moment is the mean of the distribution or mean
residence time and can be obtained through equation (2.28).
̅ ∫ ( ) (2.28)
It is equal to the average residence time of a system and it is defined as the expected value of a molecule’s age at the outlet of a system. The second central moment measures the variance and it is the degree of dispersion around the mean, equation (2.29).
∫ ( ) ( ) ∫ ( ) (2.29) If the value of the variance is closer to zero, it means that all the values are close to the mean age, and the system is not very dispersive. As this value tends to grow, the dispersion also grows and the points tend to spread out from each other and from the mean age.
One example of this variation is demonstrated in the Figure 7.
Figure 7 – Example of three RTD with the same mean times but with distinct variances.
The main limitation of the RTD calculations is that they just give information about the system hydrodynamics at its outlet, leaving its internal information unknown. With this limitation the RTD theory can be insufficient when non-linear phenomena is present, with this in mind it is possible to conclude that this theory is incomplete to characterize hydrodynamics. It is then needed to adopt a supplementary method that can enable one to access the internal information of a vessel.
Bibliographic study 15 2.3.2 Degree of mixing
In the design of the continuous-flow reactors the characterization of the system made by the RTD could be, in most of the cases, insufficient. This happens because the RTD characterization is focused in demonstrating the macro mixing of a vessel. This problem could be demonstrated by comparing the RTD’s of a perfectly mixed system, described by a continuous stirred-tank reactor, and a system composed by parallel plug flow reactors. These systems and their RTD are presented in the Figure 8, and also is their RTD.
Figure 8 - Completely segregated system and perfectly mixed system both with the same RTD.
To better describe the internal information of these two distinct systems, Danckwerts (1958) and Zweetering (1959) proposed a criterion that describes the quality of the mixing of a vessel. Danckwerts uses the concept of “age of the fluid at a point”, which implies the age averaged over a region, small compared to the whole system and even smaller compared to the scale of segregation. If , the age of a molecule, is defined as the time passed since the molecule entered the system, it is possible to calculate the variance of the ages of all the molecules in the system (equation (2.30)).
( ̅̅̅̅̅̅̅̅̅̅)̅̅̅̅̅̅̅̅̅̅̅ (2.30) ̅ is the mean age of all molecules in the system at a given time and the upper bar indicates that the values are averaged over all the molecules. For each “point” (Danckwerts concept) the variance of the mean age of the molecules is given by the equation (2.31).
( ̅̅̅̅̅̅̅̅̅̅)̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ (2.31) Where is the mean age of the molecules at a given point, and the upper bar indicates that the values are averaged over all the “points” of the vessel.
Bibliographic study 16 For the cases described previously, in the “well-mixed” stirred tank system, the at each point is the same and equal to the mean age of every molecule in the system, which gives a variance for all the points the value of 0. In the other system the values for and are the same for every molecule in the system. Danckwerts define the degree of mixing, J, as
a ratio between the two variances (equation (2.32)). ( ̅ ̅̅̅̅̅̅̅̅̅̅) ̅̅̅̅̅̅̅̅̅̅̅̅̅ ( ̅ ̅̅̅̅̅̅̅̅̅) (2.32)
This quantity lies between 0, for the systems mixed at a molecular scale, and 1, for a completely segregated system. The degree of mixing helps to study and characterize the internal information of a system, the impact of hydrodynamics on a system that is at a degree of mixing between 0 and 1.
Recently, Liu (2012), developed a CFD methodology to transport the moments of internal age distribution that allows the computation of the degree of mixing.
2.3.3 Transport of moments of internal age distribution
As shown previously, the raw moments of the RTD can be calculated through equation (2.33). ∫ ( ) ∫
∫
(2.33)
The previous equation can be applied at any point in a system, as Danckwerts (1958) has shown. The resulting equation can be solved in function of the spatial position, and can be used to obtain local measurements of the tracer concentration in every point of the system.
( ) ∫ ( )
∫ ( ) ∫ ( ) (2.34)
( ) is the nth moment frequency function.
Then when a pulse injection of tracer is performed in a steady and incompressible flow, the transport of molecules is made by convection and diffusion, and the variation of concentration, as function of time and position, is given by equation (2.35).
( )
( ) ( ) (2.35)
D comprises all the dispersive phenomena in the system (molecular diffusion, turbulent mixing, mechanical dispersion, etc). If this expression is multiplied by tn and integrated over
Bibliographic study 17
∫ ( )
∫ ( ) ( ) (2.36) To simplify the equation, the term on the left-hand side can be integrated by parts, resulting in equation (2.37).
∫ ( )
( )| ∫
( ) (2.37)
Adding to this, Spalding (1958) considered a steady and incompressible flow in a closed system and inferred the quantity I, as shown in equation (2.38).
∫ ( ) (2.38)
I is spatially invariant. With these two equations, (2.36) and (2.37), and considering that in
the equation (2.36) the term ( )| will be zero when , because C tends to be zero faster than t goes to infinity [Liu and Tilton (2010)]. With ( )| , replacing equation (36) in equation (35) and dividing everything by I, results in equation (2.39)
∫ ( ) ∫ ( ) [ ( ∫ ( ) ∫ ( ) ) ( ∫ ( ) ∫ ( ) )] (2.39)
Where the invariance of u and D, and spatial invariance of I have been used. The term on the
left-hand side corresponds to n times the nth-1 moment of the RTD and the transport equation
of the nth moment, for an incompressible flow, is then shown in equation (2.40).
(2.40)
With the aid of this equation it is understandable that all the RTD moments are diffused and convected. The equation (2.39) is generally solved by CFD solvers and can be easily used to provide the spatial distribution of ages by simply adding a source term in steady state simulations.
2.3.4 Degree of mixing calculation using the internal age distribution
The concept of internal age distribution can be used to calculate the degree of mixing [Liu (2012)]. By taking a case of a steady incompressible flow where the tracer concentration is constant at the inlet, and the age of every molecule at the inlet is zero, if the age can be identified at any point of the system, then the age at any point of the system can be sampled with the equation (2.40).
Bibliographic study 18 ( ) ( )
∫ ( ) (2.41)
With this equation, concentration stands as a function of the molecular age. Repeating all the steps made by Liu and Tilton (2010) and doing the integrations in molecular age, an equation similar to the equation (2.39) can be obtained. The age frequency function of all molecules in a system can be obtained by integrating the equation of a determined point, equation (2.41), for all the points of the system.
By using the definition of the average age of all the molecules,
̅ it is easy to obtain a
relation that equals this average to the volume averaged distribution moments.̅̅̅̅ ∫ ( ) ∫ ( ) ̅ (2.42)
With this equation, and knowing the concept of variance, Liu (2012) showed that Zwietering’s (1959) degree of mixing was easily computed using the volume averaged first and second raw moments of distribution.
̅̅̅̅̅ ̅̅̅̅
̅̅̅̅
(2.43)
This methodology can only be applied for steady incompressible flows. In CFD simulations this can be a pragmatic way to obtain results about de mixing state of a given system and compare them with others that produce the same RTD but have unrelated hydrodynamics.
Technical Description 19
3 Technical Description
3.1 Experimental Setup
In order to validate the CFD models and the chosen assumptions, concerning hydrodynamic models, boundary conditions and the Peclet and Schmidt numbers, tracer tests were performed using an experimental installation, represented in Figure 9.
Figure 9 – Schematic view of the experimental installation, respective dimensions and sensor positions.
This installation was filled with glass spheres of a diameter of 1 mm with a resulting average porosity of 0.357. The inlet volumetric flow varied between 2, 4, 6 and 8 m3/h. When
hydrodynamics inside the installation were stable, a saline solution was injected in a pulse shape. The conductivity was measured throughout time with the aid of 13 conductivity sensors with a frequency of 8 Hz. These 13 localizations of the sensors are also used for the CFD simulations as probes in order to compare the experimental results with the numerical results. The obtained RTD and its moments are compared with those obtained through CFD simulations. 1 3 4 5 6 2 7 8 9 10 11 12 13
Technical Description 20
3.2 CFD approach
CFD simulations were performed to characterize hydrodynamics of the SMB adsorbent beds. With such simulations it was possible to obtain the local moments of the internal age distribution and the RTD resulting from the injection of inert tracer or obtain the outlet curve concentrations of mixed xylenes in the presence of adsorption. In view of this knowledge, a criterion is defined to develop new simple models that remain coherent in the presence of non-linear phenomena, such as adsorption. This need to simplification also surges because the real time of CFD simulations for these non-linear phenomena can take up to 72 hours, which is a huge amount of time regarding to what is normal in simpler RTD simulations (2/3 hours). When one needs to optimize or to perform a parameter fitting of this process, simulations of 72 hours renders such task are infeasible. It is then interesting to study the simplification of the processes, such as 1D models which can take to converge.
3.2.1 2D CFD
In order to compare the hydrodynamic results for the experimental setup, a two-dimensional geometry was drawn as shown in Figure 10.
Figure 10 – 2D geometry representative of the experimental installation.
To this geometry some modifications were made, such as the addition of macro scale obstacles within the porous media, or the displacement of the inlet and outlet of the vessel.
Technical Description 21 To perform all the simulations the software ANSYS Fluent 15.0 was used. The mesh for this geometry was optimized and adapted to all the changes made in the geometry. Hydrodynamics of this volume were characterized through the mapping of the local moments of internal age distribution obtained with Liu’s methodology (2012) and through RTD resulting from pulse injections of inert tracer. These dynamic simulations performed in order to obtain the RTD of the 2D domain can take up to 3 hours. Thus, it is interesting to study the accuracy of traditional and simple reactor models that will be introduced in the next section.
3.3 Ideal model approximations
Generally, to ease the modelling of industrial processes, these are assumed to behave as ideal vessels. The two main simple models are the Continuous Stirred-tank Reactor and the Plug Flow Reactor. The first assumes that the vessel is perfectly mixed and that the concentration is the same in every point of the vessel at any given time.
3.3.1 Continuous Stirred Tank Reactor
In ideal Continuous stirred tank reactors, the assumption that the concentration is uniform throughout the vessel is made. Thus, the outlet concentration is equal to the concentration inside the reactor, and its temporal variation can be calculated by:
iin i
i
iin i
i iR
C
C
R
C
C
V
Q
t
C
, ,1
(3.1)Q is the volumetric flow (m3/s) fed to the vessel, C
i,in the inlet concentration (mol/m3), τ is
the vessel's residence time (s) and Ri the molar source term (mol/m3.s). It must be assumed
that the fluid's density is constant, which is a decent assumption since xylenes are in liquid state when separated.
As said before, the degree of mixing of a CSTR is equal to zero. The central moments of the RTD of a CSTR were deduced with the help of an equation similar to (2.39) knowing that the inlet moment mn,in is null equation (3.2),
m
n
m
n,in
m
n1
n
m
n
m
n1
n
1
(3.2)The resulting equation for the distribution central moments is equation (3.3),
nn
n
1
!
(3.3)where (n-1)! is the factorial of the order of the moment minus 1. When the process deviates from perfect mixing, a cascade of CSTR is generally used for the modelling. It consists in several consecutive CSTR linked between themselves. The equations of the distribution
Technical Description 22 moments of an enchainment of CSTR can be deduced with the help of the equation (3.2) but where mn,in is not null, (except for the first reactor of the enchainment) since the moments of
a given reactor i depend on the moments of the previous reactor (equation (3.4)).
m
nm
n,in
m
n 1n
m
nm
n 1n
m
n,in1
(3.4)where mn is the distribution raw moment of order n of the reactor i, mn-1 is the moment of
order n-1 of the reactor i and mn,in is the moment of order n of the reactor i-1.
Using this equation, the distribution central moments of a cascade of NCSTR reactors can be
deduced, if they share their residence times, and is equal to equation (3.3) multiplied by NCSTR:
1
N
CSTR
(3.5) 2 2
N
CSTR
(3.6) 32
N
CSTR
(3.7)The fourth distribution moment does not follow the logic order of the previous ones, and is equal to 3(2+NCSTR)τ4. The degree of mixing for a homogeneous (constant τ) cascade of CSTR
was deduced. It only depends on the number of consecutive reactors:
5
1
CSTR CSTRN
N
J
(3.8)By changing the two variables of a cascade of CSTR, NCSTR and τ, it is possible to fit two from
Results and discussion 23
4 Results and discussion
4.1 Hydrodynamics
With the geometry shown in the previous chapter it is possible to study hydrodynamics of the system by CFD simulations. The set of parameters used for these simulations are shown in Table 2.
Table 2 – Hydrodynamic simulation parameters
Value Units μ 1E-3 Pa.s ρ 998.2 kg/m3 dp 1E-3 m Pel 2 Pet 11
The bed porosity εi was calculated experimentally and the average value for this parameter
was 0.357. The diffusivity of the species is given by an user defined function (Appendix 2) and has two different calculations, one for the porous media and the other for the free flow zones.
The inlet flow rate of the simulations was defined as being 6 m3/h since it gives a superficial
velocity of the fixed bed close to the one passing through the SMB beds. 4.1.1 Comparison with experimental results
The goal of such simulations is to obtain the spatial distribution of ages at every point and the RTD of the experimental installation, and validate de CFD results with the aid of experimental ones.
4.1.1.1 Turbulent models comparison
With the geometry drawn, it was necessary to study which hydrodynamic turbulent model was best suited for this study, and their results must be compared to experimental data. These comparisons were made by converging hydrodynamics of the system in steady state and then use the equation (2.39), to calculate the internal age distribution for the probes shown in Figure 9. The moments of the resulting simulations are displayed in Table 3.
Results and discussion 24
Table 3 – First two RTD moments obtained with different turbulence models at the outlet of the vessel.
Turbulent Model μ1 (s) σ2 (s2)
Standard k-ε model 29.06 3.23
Standard k-ω model 29.04 3.10
This table only includes the first two moments in the outlet. The moments of the age distribution at the internal probes are displayed in the Table 4.
Table 4 – Mean of the age distribution of the probes for the two turbulence models
Top probes Middle probes Bottom probes Outlet
(s)
4.85 1.44 0.99 16.33 14.68 13.98 13.80 13.94 27.00 27.70 29.06
(s)
4.63 1.56 1.04 16.25 14.78 14.09 13.78 14.03 27.07 27.80 29.04
Since the experimental residence time obtained was 27.2 seconds and the variance at the outlet was equal to 2.5 s2 the best method to do the simulations and represent hydrodynamics is the Standard k-ω model since the residence time and the variance at the outlet are almost the same, the criteria to choose was based in the first layer of probes and taking it into account the k-ω model was chosen to be the model used in all the simulations of this project.
4.1.1.2 RTD and Internal age distribution
After choosing the best turbulence model for the system it was possible to do RTD simulations. The simulations were made in transient state and the input was an injection profile developed to represent the experimental injections of tracer, the profile was then compiled to the solver as an UDF (Appendix 2). The profile result and comparison with the experimental injection is presented in the Figure 11. This injection profile has this shape because it is difficult to represent experimentally the perfect Dirac injection, since it is manually controlled. Thus, for better accuracy of the results comparison, an approximated profile was developed.
Results and discussion 25
Figure 11 – Inlet profile injections for experimental and simulated RTD’s
The experimental values (red squares) are mean values calculated with the information of all the injections made experimentally. With this profile and the velocity field obtained for a steady state simulation of hydrodynamics, (Figure 12), RTD simulations were made.
Figure 12 – Velocity field obtained and zoom of the inlet and outlet.
Snapshots at different times, of a RTD simulation, were taken and are shown in Figure 13.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 E (t ) ( 1/s ) t(s) Experimental Simulated
Results and discussion 26
Figure 13 – User defined scalar contours for the RTD simulation after injecting the profile displayed in Figure 11.
Figure 13 shows the behavior of the tracer injected along the column. It is possible to see that exist dispersion inside the bed since the area covered by the tracer increases. To study the behavior of the tracer inside the geometry and also to compare it with the experimental ones, the concentration curves were obtained for each probe of the experimental installation, and compared to the curves obtained experimentally. This data is divided in 3 sections depending on the zone of the bed where the probes are, Figure 14.
Results and discussion 27
Figure 14– RTD obtained for a 2D simulation using an inlet UDF injection (solid lines) and comparison with the experimental result (points).
For better comprehension of the figure, the graphic (a) (top) presents the results of the probes 2 (blue), 3 (green) and 5 (red) of the Figure 9, the graphic (b) (middle) the probes 6 (dark blue), 7 (red), 8 (green), 9 (violet) and 10 (light blue) and the graphic (c) (bottom and outlet) the probes 11 (green), 12 (red) and 13 (blue).
As it has been demonstrated before the probes inside the column are distributed by three zones. The results of the first zone, in the beginning of the porous media, are represented in the graphic (a) of the Figure 14, the second zone, in the middle of the porous media and the representation of results appears in the graphic (b) and the third zone is at the end of the experimental set-up, graphic (c).After analyzing these graphics some conclusions could be made. In the graphic (a) it is shown that the numeric tracer takes more time since it enters the column in the opposite side, this would be a problem because it could cause the concentration profile shown in the snapshot of the Figure 13, to solve this problem other simulations were made doing some changes in the diffusive term for the free zones, by decreasing the Schmidt number which increases the degree of mixing. This turned out to be discarded after showing that even for a very small Schmidt number the solution would be better for the point that is far from the inlet, but for the other two it would be worse (Table
(a) top
(b) middle
(c) bottom
Results and discussion 28 10 in Appendix 1). In the middle zone (graphic (b)) results show that the first half of the installation in the experiments was more dispersive that in the simulations, one option of solution was to decrease the axial Peclet ( ). After one simulation this option was dropped because it influenced the dispersion at all zones and the variance at the outlet would increase, turning it far away from the experimental value.
Observing the graphic (c) it was easy to conclude that the variance at the outlet was almost the same but the mean residence time had a delay of 1.5 seconds. to solve this delay without changing the dispersion of the result, the solution proposed was to make simulations considering a bigger flow rate to equalize the mean residence time of the experiments. The best result for this solution was at a flow rate of 6.63 m3/h and it is shown in the next table.
This delay could be related to the flow meter error of 5% to highest flow rate (5%*12m3/h=0.6m3/h).
This representation of the results was made graphically in order to avoid the integration errors provoked by the noise read by the experimental sensors, which can result in an overestimation of the moments.
Table 5 –Comparison of the mean ages between a CFD simulation made at a flow rate of 6.63 m3.h-1 and the experimental results at a flow rate of 6 m3/h
Top probes Middle probes Bottom probes +Outlet (s)
(6.63 m3/h) 4.58 2.30 1.96 15.38 14.23 13.64 13.31 13.75 25.53 26.03 27.00
(s)
(6 m3/h) 4.17 2.73 1.87 18.90 16.31 15.72 15.52 15.10 25.83 27.19 27.26
The result brought another point of study, since the mean age is almost equal at the outlet. The increase of flow rate just shows that the porosity inside the experimental installation is not constant. This conclusion could also be validated based in the difference of residence times between the top half of the installation and the bottom half. For the first half this difference is around 12 seconds but for the experiments it is about 13~14 seconds, this result shows that the top half of the installation could have a higher value of porosity than the medium value of 0.357. In the bottom half the opposite situation is verified, 11 seconds for the simulations and 9~11 seconds for the experiments. After some calculations it was possible to have some approximation for the porosity of these two situations, those values were 0.37 in the top part and 0.32 in the bottom. This conclusion can also explain the difference
Results and discussion 29 between the mean residence time obtained experimentally and through CFD simulations for equal flow rate, and showed that if the porosity in the bed would be described by a constant value, there would be no possibilities to approximate the simulated values to the experimental ones. The transport of the moments of the internal age distribution was also simulated. The local mean age and variance are shown in Figure 15, for the case where no obstacle is placed inside the porous media.
Figure 15 – Local mean age and variance obtained through Liu and Tilton’s methodology (2010)
Figure 15 demonstrates the result of a simulation through Liu’s methodology (2012). The result for the mean age shows that the flow at the left side of the column is higher, as expected, also there are highly delayed regions in the bottom corners and finally the mean residence time is near the one obtained by a RTD simulation, 26.80 seconds. In the case of the variance, the left side of the column has higher values derived from the delay observed in the mean age distribution. . The result at the outlet for the variance is also near the values obtained before, 2.04 s2.
The next step was to simulate the flow perturbations provoked by the presence of obstacles inside the porous media and evaluate the resulting dispersion. The obstacles chosen and added to the experimental setup were a cylinder and a quadrangular prism with the same volume, the representation of these geometries is displayed in Figure 24 of Appendix 1, with
Results and discussion 30 their respective dimensions. The results of those simulations and the comparison with the experimental ones are displayed in the next tables.
Table 6 – Comparison of internal age distribution and variance for the geometry/installation with a cylinder as obstacle.
Cylinder Top probes Middle probes Bottom probes +
Outlet
(s) 4.65 2.49 1.66 12.79 12.08 20.17 11.16 10.51 25.56 23.77 27.03
(s2) 3.59 0.31 0.58 2.31 0.58 6.13 0.60 0.99 1.08 1.52 5.69
(s) 3.73 2.96 1.64 14.08 13.84 22.12 12.85 12.07 23.87 23.34 24.99
(s2) 0.27 0.25 0.07 1.78 1.78 6.78 1.72 1.46 2.85 1.51 3.84
Table 7 - Comparison of internal age distribution and variance for the geometry/installation with a quadrangular prism as obstacle
Prism Top probes Middle probes Bottom probes
+ Outlet (s) 4.23 2.56 1.70 12.06 11.66 27.69 10.60 10.37 24.95 23.39 26.88 (s2) 1.98 0.27 0.65 0.64 0.43 18.25 0.40 1.11 0.68 1.73 9.26 (s) 3.74 3.21 2.05 13.77 13.65 26.25 12.38 11.84 23.18 23.11 24.71 (s2) 0.34 0.32 0.14 1.78 1.67 10.43 1.62 1.62 1.51 3.21 4.79
After analyzing the results for the two tables it is possible to take some conclusions about the influence of the obstacles in hydrodynamics of the system in the experiments and in the simulations.
For the first topic, the presence of obstacles shows that the residence time inside the column decreases, as expected, because the obstacles occupy a portion of the bed’s total volume. Its
Results and discussion 31 presence increases the variance at the outlet which was also expected because with the inclusion of obstacles causes some perturbations in the system (deviations of the streamlines), the changes on the results are well represented by the middle probe of the middle probes line, where, comparing to the case without obstacles, the difference of the mean age was about 1.74 seconds between the probe 8 and the other four and in the presence of the cylinder this difference raises to 9.66 seconds and it is even higher in the presence of the prism, 17.32 seconds.
The transport of moments of the age distribution was simulated, and the mean age, for the three geometries, is shown in the Figure 16.
Figure 16 – Mean age for the three geometries for a flow rate of 6 m3.h-1.
The effect of the obstacles in the internal age distribution is well verified by this figure where it is demonstrated that the geometry with no obstacles tend to have high delay regions in the bottom corners of the geometry and the below the obstacle which can influence the adsorption separation that will be discussed in the next chapter.
4.2 Adsorption
After studying hydrodynamics with the aid of an inert tracer injection, the adsorption was studied. The parameters used to describe the adsorption equilibria were inserted in a UDF presented in the Appendix 2.
Results and discussion 32 The procedure of simulating this equilibrium was the same as the one of a RTD simulation. The compounds for these simulations were the desorbent (PDEB), the PX and the mixture of MX and OX, or MOX (the abbreviation used in the simulations).
For these simulations the mass transfer limitations were considered negligible, so the values given to the mass transfer coefficients were much higher than the real ones, with this it will be possible to compare the simplified models with the CFD for the most unfavorable case. For this case, the geometries simulated were the ones presented and studied in the previous chapter, but some changes were made, the inlet was changed to the same abscissa point as the outlet.
The simulations were performed with the k- turbulence model and a flow rate of 6 m3/h.
The outlet concentration profiles of PX and MOX for the three geometries are in the Figures 17, 18 and 19.
Figure 17 - Separation of the xylene mixture performed with the 2D geometry without obstacles 0 0.05 0.1 0.15 0.2 0.25 30 35 40 45 50 55 60 65 70 E ( -) Time (s) PX MOX
Results and discussion 33
Figure 18 – Separation of the xylene mixture performed with the 2D geometry with the cylinder as obstacle
Figure 19 - Separation of the xylene mixture performed with the 2D geometry with the quadrangular prism as obstacle
For the three cases the chromatographic separation is evident, although it is possible to see the long tails of both curves (PX and MOX) leaning to the right, when the obstacles are placed inside the porous media. The adsorbent has the highest affinity for the PX which is retained longer and has higher values of residence time than the MOX mixture. Residence times increase comparing to the result of hydrodynamics, this happens because there are mass transfer interactions between liquid and solid phases. The differences between the hydrodynamic and adsorption results in terms of moments are in the Table 8.
0 0.05 0.1 0.15 0.2 0.25 30 35 40 45 50 55 60 65 70 E ( -) Time (s) PX MOX 0 0.05 0.1 0.15 0.2 0.25 30 35 40 45 50 55 60 65 70 E ( -) Time (s) PX MOX
