Efficiency based non-equilibrium modeling of

Separation Science and Technology

ISSN: 0149-6395 (Print) 1520-5754 (Online) Journal homepage: http://www.tandfonline.com/loi/lsst20

Efficiency based non-equilibrium modeling of industrial-scale multicomponent distillation columns Hadi Poortalari, Javad Karimi Sabet & Farshad Varaminian To cite this article: Hadi Poortalari, Javad Karimi Sabet & Farshad Varaminian (2017): Efficiency based non-equilibrium modeling of industrial-scale multicomponent distillation columns, Separation Science and Technology, DOI: 10.1080/01496395.2017.1301471 To link to this article: http://dx.doi.org/10.1080/01496395.2017.1301471

Accepted author version posted online: 20 Mar 2017.

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Date: 21 March 2017, At: 07:57

Efficiency based non-equilibrium modeling of industrial-scale multicomponent distillation columns Hadi Poortalari1, Javad Karimi Sabet21, Farshad Varaminian1*

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1-Faculty of Chemical & Petroleum and Gas Engineering, Semnan University, Semnan, Iran 2-NFCRS, Nuclear Science and Technology Research Institute, Tehran, Iran

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Abstract

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An efficiency-based non-equilibrium model is proposed to simulate two industrial distillation

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columns. In this model, the behavior of trays is studied by considering the mass transfer at interphase. The interphase mass transfer is simply evaluated by the simultaneous solution of the

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vapor material balance equation and the definition of the Murphree efficiency. A MATLAB code is developed to implement the simulation procedure. For verification, the product compositions

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and plate temperatures are compared with reported experimental data. The discrepancy of the experimental and simulation results is about 0.5% and 20% for the temperature and the products

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compositions for both columns, respectively.

Keywords: Multicomponent distillation, Non-equilibrium modeling, Interphase mass transfer,

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Murphree efficiency,

1

Correspond authors: [email protected], [email protected]

1

1- Introduction In recent years, distillation process is highly considered in various industrial applications due to

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its various advantages. Distillation is a practiced technique for the separation of the chemical species mixtures in the refineries and chemical industries. This process is a cost effective method

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for continuous separation of different mixtures in high volumes to obtain high purity products.

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These advantages cause to develop the application of the distillation process in many industrial fields, and therefore researchers have tried to increase the performance of this process. Due to

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high energy usage and operational difficulties of distillation columns, experimental investigations are significantly expensive and time consuming. Hence, various researchers have

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focused on the analytical and numerical approaches to model and simulate this process. Since control, management and operation of the distillation columns in the industrial scale are

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accompanied with various complexities, computer programs that describe this operation should

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be available throughout the unit operation.

In order to simulate this process, the detailed knowledge of the process theory is paramount importance for design purposes. The mass transfer between vapor-liquid is the main phenomena that occurs during the separation inside the distillation column. Basically, two approaches are

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used to model vapor-liquid mass transfer on a distillation stage: the equilibrium stage approach and the non-equilibrium stage approach. The conventional method to model a distillation column is the so called equilibrium or ideal stage model. The ideal stage model is easy to use because a

detailed equipment design is not required. Indeed, some limited data including only equilibrium relationships and enthalpy data are required in this model. It is conceptually simple and

2

mathematically well-designed. Hence, it has been used to simulate and design various columns in real world. More than a century, chemical engineers have modeled distillation columns by the equilibrium stage approach [1]. In the equilibrium stages, the vapor and liquid phases are

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assumed to mix perfectly and reach complete interaction with each other, so that the equilibrium composition is obtained as streams leave the stage. The equilibrium assumption also means

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streams leaving a stage are at the similar temperature that is reasonable for many industrial

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columns.

Although equilibrium model has several advantages, stages rarely operate at equilibrium state in

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actual operation despite attempts to approach this state by proper design and operating conditions tuning. In fact, the behavior of a real tray is not ideal due to some reasons such as incomplete

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contact between the phases, not enough residence time, vapor and liquid entrainment, weeping, and poor vapor and liquid distribution [2]. In the design and operating of distillation columns,

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deviation from equilibrium state on the stages have important effects. This is often large enough,

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so that it might reduce the effectiveness of the separation unit to 50-70% of ideal operation [3]. Therefore, the designers have always interested in modelling of the trays with real conditions. In general, two common approaches are proposed to explain the non-equilibrium nature of

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distillation stages and overcome the gap between the model and reality: the stage efficiency method and the rate-based method. In the first approach, the concept of efficiency is used to link the performance of practical trays to the ideal equilibrium stages. Indeed, the efficiency is recognized as an important factor in distillation column design. For dealing with deviation from the equilibrium condition, stage efficiencies incorporate into the equilibrium relations. The most

3

common definition for stage efficiency is the vapor phase Murphree efficiency as it is simply jointed with the equilibrium relations. The Murphree tray efficiency is defined as the ratio of vapor concentration change for the actual conditions to equilibrium conditions. On the second

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approach, the interphase mass transfer method known as the rate-based method is used to describe the deviation from equilibrium conditions. Since separation process which is actually

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reached in the real plates mainly depends on the rates of mass transfer from the vapor phase to

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the liquid phase, the actual mass transfer at plates is directly calculated in the rate-based approach. The rates of mass transfer depend on the non-ideality extent of the vapor and liquid

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streams with each other. This method gives more detailed temperature and composition profiles at the expense of heavier computational cost and more equipment design details.

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In recent years, different researchers motivated to apply efficiency calculations along with

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equilibrium modeling approaches to consider the non-ideal nature of distillation process. The efficiencies are conventionally obtained from experiment or empirical methods and usually

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just a same fixed value (overall efficiency) is considered for the whole column. However, it has been experimentally revealed that the plate or even component efficiencies might be different with each other in multicomponent mixtures [4, 5]. Thus, the concept of overall column

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efficiency cannot result in an accurate simulation and a more rigorous definition should consider for individual efficiencies that can differ along the column. Consequently, the prediction of the component or stage efficiencies during the simulation improves the column design by avoiding unnecessary over-sizing, and so savings are obtained in the capital and operation costs.

4

Frequently, these calculations are based on multicomponent mass transfer models and the estimation of binary NTUs (Number of Transfer Units). Aittamaa [7] initially applied the multicomponent plate efficiencies in the modelling of distillation column based on equilibrium

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stage approach. Ilme [8] later used this algorithm and developed it further. The methods of distillation process modeling according to efficiency concept have been extensively examined by

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Klemola [9], Jakobsson [10], and also applied for industrial columns by Ilme et al. [11] and

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Jakobsson et al. [12].

Although various aspects of these methods are comprehensively studied, the concept of the

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efficiency have not been applied in the non-equilibrium modelling approaches. In fact, the researchers often focused on modifying the equilibrium algorithms using the efficiency concept

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to consider the actual condition of the separation units. Furthermore, researchers frequently have been applied multicomponent mass transfer models to predict individual component efficiencies

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at any plate. The rigorous methods for the multicomponent efficiencies calculation are

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accompanied with many complexities that lead to convergence problems in calculations and consequently the difficulty of applications. Therefore, it would be very desirable to simulate a multicomponent distillation process based on pseudo-binary efficiency calculations without significant reduction in the accuracy of results. According to this, the aim of this study is to

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present a rigorously modified method for the modeling of industrial-scale distillation columns by using pseudo-binary efficiency calculations into a simple non-equilibrium approach. In this model, the interphase mass transfer relations are applied instead using of the phase equilibrium

relations. The interphase mass transfer coefficient is determined simply from the simultaneous solution of the vapor phase material balance and the definition of the Murphree efficiency. The

5

main characteristics of this model is to combine the advantages of the both efficiency based equilibrium model with interphase mass transfer non-equilibrium model to obtain the efficient and applicable approach for the modeling of distillation process. In the proposed model, the

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behavior of real plates is initially estimated in form of stage efficiencies, and then column performance for separation of key components is obtained. Finally, the estimated separation is

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compared with actual measurements of the industrial distillation columns. According to this

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procedure, the column design can be analyzed, and it would be modified if needed. The calculated results include the concentration and temperature profiles along the column and

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finally product compositions are compared against reported industrial data.

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2- Model description

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2-1- Interphase mass transfer

In our model, the interphase mass transfer approach is used to describe the deviation from

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equilibrium stage model in two industrial scale distillation columns, a C4 splitter and a column for separating benzene, toluene, and xylene (BTX). A specific equation is required to calculate the interphase mass transfer coefficient. This relation obtained from the simultaneous solution of

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the vapor phase material balance equation and the definition of the Murphree efficiency factor (

) as follow [3]:

=

(1)

1−

6

represents the vapor mass transfer coefficient,

where and

is the effective area for mass transfer,

is vapor flow rate, all for stage j. In derivation of this expression, it is assumed that the

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bulk vapor rate does not change within the liquid and/or froth on the stage. To consider the non-equilibrium behavior of real distillation trays, it is necessary to include the

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interface mass transfer effect in the component-material balances written for any stage in distillation column. Mass transfer on a two-phase contact stage may be described by two material

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balance equations, one written for the liquid phase and one for the vapor phase. The balance

=

,

,

.

−

,

−

,

+

∗ ,

−

,

M

,

∗ ,

−

,

+

,

=0

(2)

+

,

=0

(3)

ed

∗ ,

−

, represents

the K-value of component i at stage j.

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where

,

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equation for any component in the liquid phase and the vapor phase are stated as:

The term related to feed, the last term, should be considered in both expressions, since feed may enter either phase. These expressions for the phase material balance are derived based on the

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relation of the mass transfer driving force and equilibrium state. According to this, the rate of mass transfer is decreased as conditions proceed to equilibrium. Figure 1 schematically depicts the flow pattern of stages according to the interface mass transfer effect.

7

In distillation modeling, the condenser and the reboiler are always considered as equilibrium stages even though equilibrium is not obtained on the trays within the column. So, the balance

condenser (stage 1) :



−

,

+

,

=−

,



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equations for condenser as first stage and reboiler as last stage are as follows [13]:

,

,

−

+

=−

,

,



,

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reboiler (stage N) :

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(4)

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2-2- Efficiency model

The Murphree vapor phase tray efficiency [14] is the most popular efficiency definition, and it is

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the only one with practical aspects [15, 16]. It is defined as the ratio of the actual change in a component vapor mole fraction to the change in mole fraction of the vapor while it is in

ed

equilibrium with liquid at the tray output. For each tray, the Murphree vapor efficiency is defined

=

∗

– −

(5)

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by:

where

represents the vapor average composition above the froth and

vapor composition in equilibrium with the liquid composition,



∗

corresponds to the

, leaving the tray. Since the

compositions often vary across the interface, the point efficiency is defined at any specified position on the tray in terms of compositions. The Murphree vapor point efficiency can be calculated as:

8

=

(6)

represents the point composition in the vapor phase.

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where

– −

∗

Point efficiency describes the degree of tendency to equilibrium condition at any point on the

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tray. This factor is not constant on the tray because there are concentration gradients in the liquid

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flowing on the tray. The tray efficiency is more useful factor than the point efficiency for the

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design purpose.

The Murphree point efficiency is often calculated from a binary data correlation based on type of

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the tray. This correlation provides the number of liquid and vapor phase transfer units (NTUs). Then, the number of overall vapor phase transfer units is calculated from the number of liquid

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and vapor NTUs as below [17]:

=

1

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1

+

,

=

(7)

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This is used with:

where

= 1 − exp(−

is the slope of the equilibrium line and

rates through the tray, respectively.

9

)

and

(8)

are the molar vapor and liquid flow

The Murphree plate efficiency is determined based on the point efficiency when the degree of phase mixing on the plates has been determined. For trayed columns, the concentration varies across the tray as a result of non-ideal mixing. If the liquid on the tray is perfectly mixed, the

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concentration is constant across the tray and consequently the Murphree tray efficiency is equal to the point efficiency. However, liquid is never perfectly mixed on a real tray and concentration

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gradients exist. If the concentration gradient is maximized, it leads to the extreme limit of plug

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flow across the tray and the difference between tray and point efficiencies will be maximized. The degree of liquid mixing is characterized by the Peclet number known as the ratio of liquid

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=

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transport and longitudinal liquid mixing and is defined as follow [11]:

is the liquid flow path length,

is the liquid phase residence time and

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where

(9)

is the eddy

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diffusion coefficient. A small Peclet number corresponds to a high degree of mixing and vice versa.

Welch et al. [18] presented a correlation for estimation of the eddy diffusion coefficient as

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follow:

= 0.088

ℎ

10

(10)

where

is the liquid flow path width and ℎ is the clear

is the volumetric liquid flow rate,

liquid height on the plate. In conventional industrial scale columns, the liquid on the plate is partially mixed while the

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vapor is often assumed to be perfectly mixed. For modeling the liquid mixing on the tray,

(11)

is the number of perfectly mixed pools that can be determined from the equation given

ed

by Ashley and Haselden [20]:

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where

−1

1+

an

=

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the following relation between the tray and point efficiencies:

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Gautreaux and O'Connell [19] divided a tray into n perfectly mixed liquid pools and proposed

+2 2

(12)

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=

The advantage of the mixed pool model is that the Murphree plate efficiencies are calculated in

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the whole liquid mixing range. The number of perfectly mixed pools depends on the liquid flow path on the tray.

2-3- Efficiency prediction in the multicomponent mixtures In a binary mixture, the component efficiencies are equal, but multicomponent mixtures do not have similar component efficiencies. However, individual component efficiencies are not often

11

used in design practice because of the complexity of rigorous methods for the multicomponent efficiencies calculation, the difficulty of applying for design purposes, and unknown reliabilities. The most common and the simplest procedure used for multicomponent efficiency calculations is

The selection of light and heavy key components and calculation of the pseudo mixture

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1-

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pseudo binary method, and it proceeds in the following steps [21]:

=

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composition as:

an

+

=

3)

M

+

refer to light and heavy key, respectively.

The prediction of the key components binary diffusion coefficients in each phase at the

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2-

and

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where the subscripts

mixture temperature and pressure.

The determination of the equilibrium curve slope:

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3-

4-

=

, ,

− −

, ,

4)

Using the binary correlations to predict the Murphree point and plate efficiencies.

12

5-

Assuming the same Murphree efficiency for all components at the any stage and apply

this value in the simulation procedure. It is worthy to note that some judgments are required to select the pseudo keys, and two

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components selected can be different for various parts of a column. Naturally the light and heavy

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key approach is more appropriate for distillation columns in which two main components are present in the feed. However, this method can be extended to allow for multi-pair efficiencies

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that may be different. The choice of binary pairs depends on feed and product compositions,

majority fraction of the mixture [21].

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volatility differences, components of major interest for design, and the components which are the

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2-4- Efficiency prediction in the binary mixtures

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Several methods have been proposed for the prediction of the efficiencies in a binary mixture. In the present study, two well-known methods, AIChE (1958) [22] and Chan-Fair (1984) [23,24],

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are used.

Based on the AIChE correlation, the vapor phase number of transfer units on both bubble caps

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and sieve trays is:

where

=



=

(0.776 + 4.57 ℎ − 0.238

is the superficial f-factor and

number.

13

+ 104.8

=

/ )



(15)

is the vapor-phase Schmidt

Chan and Fair developed an alternative correlation for



(16)

ℎ

is the fractional approach to flooding defined as the ratio of the superficial velocity ( ) ).

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to the correspond value under flooding conditions (

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where

10,300 − 8670

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=

as follow:

Both AIChE and Chan-Fair methods proposed a similar correlation for the liquid-phase number

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of transfer units of sieve trays only as follow:

(0.4

+ 0.17)

(17)

M

= 19,700

and

are Fick diffusivities (

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In above equations

the weir length ( ), and

2

/ ), ℎ is the exit weir height ( ),

is the volumetric liquid flow rate (

3

/ ).

and

are the vapor

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phase and the liquid phase residence time, respectively that are given by:

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=

(1 −

)ℎ

(18)



=

ℎ

(19)

The clear liquid height, ℎ is calculated by the correlation of Bennett et al. [24]:

14

is

.

ℎ =

ℎ +

.

ip t

(20)

−12.55

−

(−137.8 ℎ )

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= 0.50 + 0.438

cr

=

an

3- Simulation procedure

A distillation column is considered as a set of N separation stages and includes a condenser at the

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top as stage 1 and a reboiler at the bottom as stage N. A schematic view of a single separation stage is shown in Figure 2 where vi,j , li,j and fi,j are component flow rates of vapor, liquid and

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feed streams, respectively, and Sj and sj are dimensionless side-stream flow rates of vapor and

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liquid streams, respectively.

The modified Newton-Raphson method [26] was used to solve the equations of present model. This approach has been widely applied to solve the equations of both the equilibrium and non-

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equilibrium models. Based on Naphtali and Sandholm [27] method, whole functions consist of material balances for each phases in addition to energy balance are categorized according to stage locations. Details of this method are presented in the appendix A. According to Newton-Raphson method, the set of governing equations for each stage is as follows:

15

: 1 +



,

+ 1+

−

,

,

− (21)

: 1 +

,

−

+

,

,

,

,

−

∑

,

−

,

,

∑

,

,

∑

−

−

,

,

∑

ip t

,

−

,

= 0 (1 equation)

= 0 (c equations) (22)

,

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: 1 +

+

,

,

−

,

= 0 (c equations) (23)

an

,

−

,

us



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The entire model equations, N (2c+1), are solved simultaneously based on the modified Newton's method using an in-built program in MATLAB. The framework of the developed code

ed

is represented in details in Figure 3. The simulation is carried out with non-equilibrium stage model according to interphase mass transfer method, but the reboiler and the condenser are

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described as equilibrium stages. The sequence of simulation steps is depicted as a flowchart in Figure 4. The simulation model is specified by defining various specifications such as the thermodynamics of the system, the feed rate, condition and composition of the feed, the column

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and plate geometry. Since the characteristics of the components are similar for both phases, the well-known Soave Redlich Kwong (SRK) equation of state [28] is used as the VLE-method. The simulation procedure needs to estimation of several physical properties at any calculation iteration. The methods used for estimation of necessary properties are presented in Table 1.

16

The numerical algorithm consists of two computational loops: outer loop and inner loop. In each iteration, a set of values for the interphase mass transfer coefficients (

) is initially calculated

from the estimated tray efficiencies during outer loop, and then these values are used to

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determine the compositions and temperature profiles based on Newton-Raphson method through

inner loop. The point efficiencies in each computation step are calculated using the NTU

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correlations of AIChE and Chan-Fair, and these values are then related to the Murphree tray

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efficiencies using the mixed pool model of Gautreaux and O'Connell. The calculation sequence

4- Columns specifications

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for the prediction of each tray efficiency has been shown in Figure 5.

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To evaluate and validate performed simulation, the results of two industrial-scale distillation columns are used. The first column is a C4 splitter that was introduced by Ilme et al. [11, 12]. In

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this column, i-butane and n-butane are separated from a liquid feed containing about 29.4 w% ibutane, 67.7 w% n-butane, 1.5 w% propane and 1.0 w% pentanes. In the mixture, i-butane and n-

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butane are considered as light and heavy key components, respectively. The second column (BTX column) introduced by Biddulph [29] separates benzene from a liquid feed containing approximately 37.2 mol% benzene, 57.3 mol% toluene, and 5.5 mol% C8 aromatics. The C8

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aromatics will be lumped together and treated as m-xylene. In this column benzene and toluene are chosen as light and heavy key components, respectively. A simplified flow sheet of these two columns are shown in Figure 6. As mentioned, in each columns, samples were taken from the feed, top and bottom products when the operation is

17

steady state. At the same time, operating data including flow rates, temperature profiles and pressure profiles was gathered from the plants instrumentation through automation systems. Due to presence of inaccuracies in instruments measurement and analysis, the material balance are

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not often satisfied in plant data. Hence, it is needed to reconcile errors in measurements and the compositions should be adjusted to satisfy material balance constraints. In Figure 6, the

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reconciled values for compositions and flow rates are presented. The adjusted performance data

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for each columns are given in Table 2. The C4 column diameter is 2.9 m and its working pressure is about 650 kPa. It has 74 Glitsch Ballast two-pass type V-l valve trays. The BTX column

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consists of 54 plates with 2.43 m in diameter, having Koch type Kx-9-u valves. The column has a sidestream taken from plate 5 which provides the pure benzene product. Details of the columns

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specifications are given in Table 3.

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5- Results and discussion

To assess the quality of the separation, a pseudo-binary McCabe-Thiele diagram for considered

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multi-component system at C4 splitter is presented in Figure 7. The data for equilibrium and operating curves are obtained from the results of the converged simulation. In the Figure, the axes are defined by the relative mole fractions of light key component correspond to equation 13.

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As it is noticed, the staircase of visible triangles is extended to diagram corners. This fact shows that the separation of key components has well done. In order to clearly inspect the variation, a portion of the McCabe-Thiele diagram is enlarged to illustrate the stages extended beyond the equilibrium curve. This means that the efficiencies are greater than 100%. However, this problem is not unusual. In the multicomponent mixtures, the component diffusion rates are

18

affected by the concentration of all species presented in the mixture. According to Kister [21], this leads to unbound efficiencies in the range [-∞, +∞]. The predicted values for the number of transfer units of each tray for both phases are given in

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Figure 8. Figure 8a shows the results of two NTU calculation methods (AIChE and Chan-Fair)

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for C4 splitter. It can be seen that the number of liquid transfer units (NTUL) in the stripping

section is almost constant and equals to 3.5 while values of NTUL in the enriching section

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oscillate within value of 4. Also, it can be observed that the number of vapor transfer units (NTUV) across the column obtained by Chan-Fair method is nearly more uniform rather than

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AIChE method. The average value for NTUV according to Chan-Fair method is 2.8 while AIChE method predicts a mean value of 2.6 and 3 for the enriching section and the stripping section of

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column, respectively. Figure 8b displays the variation of NTUs for each phase based on AIChE method regarding to BTX column. Alike C4 splitter, the values of NTUV value in enriching

ed

section are lower than stripping section while NTUL values in enriching section is higher than

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stripping section of column. It is found that NTUV value is approximately 1.6 and 2.7 in enriching and stripping sections, respectively. However, the value of NTUL is around 3.4 and 2.9 in enriching and stripping sections, respectively. So it can be concluded that in both columns the liquid mass transfer in top section is more than bottom section while this trend for vapor mass

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transfer is vice versa.

Figure 9 presents the estimated vapor phase Murphree point and tray efficiencies. The tray efficiencies were achieved from point efficiencies using the partial liquid mixing method. As shown in Figure 9a, the point efficiencies do not vary significantly from plate to plate along the

19

C4 splitter column, so that the mean value for point efficiencies is equal to 80%. Alike the point efficiencies, the calculated tray efficiencies also do not change considerably along the C4 splitter column, and the average value is equal to 110%. On the other side, Figure 9b shows that in BTX

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column both efficiencies are almost unchanged in stripping section, with 66% for point efficiency and 101% for tray efficiency, while efficiency values oscillate in the enriching section.

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It is clearly observed that the Murphree tray efficiencies considerably differ from point

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efficiencies in both columns. Since the liquid mixing on trays is incomplete, the tray efficiencies appreciably deviate from the point efficiencies. This will enhance the tray efficiency compared to

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point efficiency and consequently improve the separation.

Figure 10 displays the liquid composition profiles in C4 splitter, resulted in AIChE method. The

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profiles of each eight components presented separately in Figure 10a where the mole fractions are shown on a logarithmic axis so that all composition profiles can simply be seen. It is found

ed

that each component exhibits a different composition profile curve. It is obvious that the

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composition of other components in comparison with key components composition is negligible, so it can be concluded that this process is essentially a binary separation and the separation solely occurs between i-butane (LK) and n-butane (HK). Figure 10a also shows that i-butane (LK) mole fraction decrease smoothly and continuously from the top to the bottom of the column except at

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the feed zone and top end of the column while the trend is contrary for n-butane (HK). i-Butane is more concentrated below the feed stage while the main separation of n-butane occurs above the feed stage. This trend has more precisely plotted in Figure 9b, where the liquid mole fractions are presented in the normal scale. The results for both mass transfer models are also plotted in

20

this Figure. It can be seen that both AIChE and Chan-Fair methods predict approximately the same profiles along the C4 splitter column. The liquid composition profiles along the BTX column is shown in Figure 11. It is clear that the

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column has been able to completely separate benzene at stage 7. As it is mentioned in the

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reference work, the function of further stages above this stage is the removal of some small quantities of non-aromatics components. In contrast, toluene is concentrated at bottom section of

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column. Since the main goal is not the complete separation of toluene, the concentration of this

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component reaches to about 90% and the rest is xylene in bottom product.

A comparison between simulated results obtained by both mass transfer methods and the

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reconciled experimental data related to C4 splitter column is provided in Figure 12. This Figure contains the results of distillate and bottom product compositions. The agreement between the

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predicted column separation based on the efficiency calculation method and the reconciled compositions is fairly good.

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The quantitative comparison of products composition between simulation results and correspond experimental data for both C4 splitter and BTX column are presented in Table 4 and Table 5, respectively. Since the results of two efficiency calculation methods are almost similar,

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compositions resulted from AIChE method are only reported. The comparison of simulated and reconciled experimental compositions of the products shows that the presented model fairly predicts the compositions of all components. The comparisons also illustrates that in C4 splitter

the maximum deviation of the top product is occurred in the nButane while the iButene composition presents the highest discrepancy in the bottom product. On the other hand, in BTX

21

column all compositions of the products except composition of toluene at top product are predicted accurately by our model. In the both columns, deviations in the components compositions of products are so small that it can be confidently said that the accuracy of

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predicted values is acceptable for design purposes.

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The predicted temperature profiles along the columns is compared with measured values at some

trays in Figure 13. As it is observed, the maximum deviations between predicted and measured

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temperatures occurs on the plate 9 (0.3%) for C4 splitter and the plate 16 (0.5%) for BTX column. As expected, the temperature decreases from the bottom to the top of the column. A

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dramatic decrease in temperature occurs from bottom (reboiler) to top (condenser) in both columns. This is caused by the subcooled condenser of columns, so that the reflux stream is

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returned to the column with temperature lower than its bubble point.

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The experimental reboiler heat duty for C4 splitter column is 10.24MW, while the simulation gives the value of 10.27MW. The difference of duties is about 0.3% which sounds reasonable,

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because our simulation model does not consider heat losses of the real column. Finally it worthy to notice that in this work we tried to simulate distillation columns based on estimated stage efficiency profiles with a proposed non-equilibrium approach. These information

Ac

about stage efficiencies can be applied in efficiency-based modified equilibrium approach. According to this, distillation columns simulated once more and calculations repeated. The obtained results in this condition did not show noticeable difference, so that both efficiency modeling approaches led to almost same results. This means that the accurate estimation of stage efficiencies has central importance in distillation modeling procedure.

22

6- Conclusion In this study, a rigorous efficiency based non-equilibrium model is presented to investigate and

ip t

model two industrial-scale multicomponent fractionators. This approach conducts a more realistic distillation simulation without imposing a large computation cost. This model is based

cr

on the interphase mass transfer evaluation at each separation stage. In present model, the mass

us

transfer at liquid-vapor interphase was considered simply by the simultaneous solution of the vapor phase material balance equation and the definition of the Murphree efficiency. In order to

an

estimate the stage efficiencies, two popular methods (AIChE and Chan-Fair methods) were used. Estimated efficiencies were applied to predict the mass transfer coefficient in each tray. A

M

MATLAB home-code was used to implement the sequence of efficiency based non-equilibrium model for obtaining the profile of compositions and temperature along the columns. Our results

ed

show that the average point efficiency for trays is about 80% for C4 splitter and about 66% for BTX column, while the mean value of the plate efficiency is almost 110% for C4 splitter and

ce pt

almost 94% for BTX column. The results for the product composition and temperature profiles along the columns were also compared with the reported experimental data. The simulation results gave an excellent agreement (deviation less than 1%) with the measured values for

Ac

temperature profile. Furthermore, comparisons display a fairly consistence (average deviation about 20%) between results for products composition. According to these results, it can be concluded that the accuracy of predicted values form the proposed efficiency based nonequilibrium model is acceptable for the accurate modeling of industrial-scale distillation columns.

23

Appendix A. Newton-Raphson method Equations (24)-(26) are solved simultaneously by the Newton-Raphson iterative method in

ip t

which successive sets of the output variables are produced until the values of the ML, MV, and H functions are driven to within some tolerance of zero. During the iterations, nonzero values of

cr

the functions are called discrepancies or errors. The functions and output variables be grouped by

=

,…,

,…,

(A.1)

an

,

us

stage in order from top to bottom as:

M

and

=

T

,…, ,…,

(A.2)

ed

,

ce pt

where X, is the vector of output variables for stage j arranged in the order:

Ac

=

,

,

,…,

,

,

,…,

,

, ,

,

,

,

,…,

,

,…,

(A.3)

,

and Fj is the vector of functions for stage j arranged in the order:

=

,

,

,

,

,…,

,

,…,

,

,

,

24

,

,

,…,

,

,…,

,

(A.4)

The Newton-Raphson iteration is performed by solving for the corrections ΔX to the output variables, which in matrix form becomes:

( )

(A.5)

ip t

( )

=−

cr

( )

)

( )

=

( )

+

(A.6)

an

(

us

These corrections are used to compute the next approximation to the set of output variables from:

M

where t is a nonnegative scalar step factor. At each iteration, a single value of t is applied to all output variables. For each iteration, an optimal value of t is sought to minimize the sum of the ∂F ∂X

is the Jacobian matrix that is a (N×N) matrix of blocks of partial

ed

squares. The quantity

ce pt

derivatives of all functions with respect to all output variables as:



∂F

=



0 . . . . .. .. 0 .. 0 ..

Ac

∂X

0

0 .. 0 .. .. 0 0





0 0 0 . . ..



0

(A.7)

Jacobian has a block tridiagonal form because functions for stage j are only dependent on output variables for stages j-1, j, and j+1. Each A , B , or C block represents a

25

(2C+1) by (2C+1) submatrix of partial derivatives, where the arrangements of output variables and functions are given by (A.3) and (A.4), respectively. Blocks Aj , Bj , and Cj correspond to submatrices of partial derivatives of the functions on stage j with respect

ip t

to the output variables on stages j-1, j, and j+1, respectively. Thus, using (24), (25), and (26) and

denoting only the nonzero partial derivatives by +, we find that the blocks have the following

> Aj

=

+ 1

⋮

M

ed

⋮

ce pt

=−

,

=−

=−

,

−

, = 1:

, = 1:

Ac ,

(A.8)

⋱

=

,

+

an

+

us


Bj =

+ +

⋮

⋱

(A.9)

⋱ +

+

⋮

⋮

+

+

= 1+

+

,

= 1+

,

,



Ac

= 1+

,

,

,

= 1+

, = 1:

−

1

,



,

,



,

, = 1:

,



,

+

,

,

+

,

=

,

+

−

,

,

,

+

,

ce pt

,

=

+ 1+

ed

,

= 1:

M

= 1+

,

,

an

,

+

us

⋮

+

ip t

=

+ +

cr


Cj

us