A Gibbs energy-driving force method for the optimal design of non-reactive and reactive distillation columns

Computers and Chemical Engineering 128 (2019) 53–68

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Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng

A Gibbs energy-driving force method for the optimal design of non-reactive and reactive distillation columns Teresa Lopez-Arenas a, Seyed Soheil Mansouri d, Mauricio Sales-Cruz a,∗, Rafiqul Gani b,e, Eduardo S. Pérez-Cisneros c a

Departamento de Procesos y Tecnología, Universidad Autónoma Metropolitana Cuajimalpa, Avenida Vasco de Quiroga 4871, Ciudad de México C.P. 05348, México PSE for SPEED Company Ltd., Skyttemosen 6, DK-3450 Allerod, Denmark c Departamento de Ingeniería de Procesos e Hidráulica, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Ciudad de México C.P. 09340, México d Process and Systems Engineering Center, Department of Chemical and Biochemical Engineering, Technical University of Denmark, Kgs. Lyngby DK-2800, Denmark e College of Control Science and Engineering, Zhejiang University, Hangzhou, China b

a r t i c l e

i n f o

Article history: Received 11 February 2019 Revised 29 April 2019 Accepted 14 May 2019 Available online 23 May 2019 Keywords: Driving force Gibbs energy Distillation columns design Reactive distillation design

a b s t r a c t A simple Gibbs energy-driving force method for the design of non-reactive and reactive distillation columns has been developed. Based on the binary driving force concept and the equilibrium Gibbs energy computation, a systematic procedure for the design of non-reactive distillation columns (NRDC) and reactive distillation columns (RDC) is proposed. The design method exploits the connection between the driving force values and the equilibrium Gibbs energy to determine the number of stages, the optimal feed location, and the heat required at the top and bottom of the column and the minimum reflux ratio. The final design guarantees an optimal operation since maximum thermodynamic efficiency is achieved. The maximum thermodynamic efficiency criterion is equivalent to the minimum entropy condition required for a stable operation of the distillation columns. The method is applied for the design of two non-reactive systems: a) Benzene-Toluene ideal system and b) Ethanol-Water non-ideal system. A reactive distillation column considering the isomerization of n-butane in the presence of an inert compound is designed. The optimal thermal feed condition obtained through the maximum separation efficiency guarantees that the final designs obtained correspond to the minimum energy requirements for the design target of separation. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Distillation is the most widely used industrial separation technology and distillation units consume a significant part of the total heating energy in the world’s process industry. Distillation has been in use for a very long time and is often regarded as mature technology. However, demands to reduce capital costs, energy consumption and operation and maintenance costs, have led to rethinking of how the separation should be carried out. As CO2 emissions are directly related to energy consumption (when a nongreen energy source is utilized), a more energy efficient distillation would also directly contribute to reduction in CO2 emissions. To date, computers have superseded graphical techniques as the main tool for distillation design and performance evaluation.

∗

Corresponding author. E-mail address: [email protected] (M. Sales-Cruz).

https://doi.org/10.1016/j.compchemeng.2019.05.024 0098-1354/© 2019 Elsevier Ltd. All rights reserved.

Nevertheless, graphical techniques are still widely used in modern distillation technology. They provide the means for visualizing the process characteristics and enable spotting pinch conditions, excessive reflux, incorrect feed points, and a sub-optimal thermal condition of the feed. They are powerful for setting up and analyzing computer-aided solution strategies. Other applications are the screening and optimization of design options and providing initial estimates for computer calculations (Beneke et al., 2013). When such graphical techniques and the computational procedures are combined for the design of distillation columns, the visualization and fast computation become a powerful hybrid method for the design and evaluation of many reactive and non-reactive separation processes (Sánchez-Daza et al., 2003). On the other hand, it is well-known that the thermodynamic analysis of a distillation column is important for synthesizing and developing energy efficient distillation processes (Bandyopadhyay, 2002). It allows identification of thermodynamic

54

T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

efficiency of the process, regions with poor energy efficiency, and the thermodynamic limits. Several studies show the application of exergy analysis to improve the thermodynamic efficiency of a distillation column (Atkinson, 1987; Ratkje et al., 1995; Taprap and Ishida, 1996; Agrawal and Herron, 1997). Atkinson (1987) has developed a graphical representation of exergy loss in a distillation column. Ratkje et al. (1995) have analytically shown that entropy generation for a distillation column is at a minimum when the driving force for separation is distributed uniformly along the column. Taprap and Ishida (1996) have presented different exergy losses in a distillation column on energy-utilization diagrams. These diagrams identify the amount of energy transformation and exergy loss of individual process steps. Agrawal and Herron (1997) have given equations to quantify thermodynamic efficiency of a distillation column that separates an ideal binary mixture, with constant relative volatility, into pure components. Also, Blahusiak et al. (2016) proposed a quantitative efficiency analysis based on exergy analysis. They analyzed distillation of near-ideal binary systems from a heat engine perspective (considering distillation as a process creating separation work) and investigated the influence of the feed composition and thermal properties of separated compounds on the internal efficiency of the heat engine. In a recent work, Pérez-Cisneros and Sales-Cruz (2017), through a thermodynamic analysis of the driving force approach, showed that the condition of maximum driving force is explicitly related to the energy involved in the separation of a binary mixture. That is, the maximum driving force corresponds to a unique energy value were the largest difference of the vapor composition to the liquid composition is attained. The link between the maximum driving force and the energy value leads to the concerns about the influence of the thermal feed condition on the total energy consumption of reactive or non-reactive distillation columns. Thus, the main objective of this paper is to propose a systematic method to design an energy efficient distillation process using the driving force approach. The objective is to design the distillation column system at the maximum driving force where the energy required for the separation is minimum. Despite the interesting research around the thermodynamic analysis of distillation columns, there is no evidence in the open literature that proposes a systematic procedure for the design of non-reactive and reactive distillation columns incorporating the influence of the thermal feed condition and its impact on the separation efficiency, the real number of stages and the total energy required. Therefore, in the present work, a systematic method for the design of reactive and non-reactive distillation columns considering binary mixtures (elements or components), is proposed. The novelty of the method is that, once the design variables are defined (i.e., operating pressure, feed flow, feed composition, purity targets and operating to minimum reflux ratio), the number of stages (reactive or non-reactive), the feed location and heat requirements are determined by incorporating an embedded procedure to maximize the thermodynamic efficiency of the column through the optimal thermal feed condition evaluation. It should be pointed out that there exist several thermodynamic efficiency definitions (Blahusiak et al., 2016), and in the present work the thermodynamic efficiency definition proposed by Fonyo and Rev (1999) is used. The design method is applied to three different mixtures: 1) a non-reactive ideal distillation column design for benzene-toluene separation; 2) a non-reactive non-ideal distillation column design for ethanol-water separation and; 3) an ideal reactive distillation column for the isomerization of the n-butane reaction in the presence of an inert compound. 2. Driving force concept A brief description of the fundamental equations of the nonreactive driving force approach is given below (Pérez-Cisneros and

Sales-Cruz, 2017). It is well-known (Thompson, 20 0 0) that by using departure functions it is possible to write the chemical potential for two species in the vapor and liquid phases as:

 ideal  v nonideal μv1 = μv1 + μ1

(1)

 ideal  l nonideal μl1 = μl1 + μ1

(2)

 ideal  v nonideal μv2 = μv2 + μ2

(3)

 ideal  l nonideal μl2 = μl2 + μ2

(4)

In addition, the general phase equilibrium equation in terms of the chemical potentials for species i is:

μvi = μli

(5)

By introducing Eqs. (1)–(4) into Eq. (5) and subtracting such equations for a binary mixture, we obtain as follows:



ideal  l ideal  v nonideal  l nonideal  − μ1 + μ1 − μ1 μv1  ideal  ideal  nonideal  nonideal  − μv2 − μl2 + μv2 − μl2 =0

(6)

Eq. (6) could be considered as an extended phase equilibrium condition. The ideal contributions, assuming ideal gas and an ideal solution, to the chemical potentials are:



ideal



ideal



ideal



 l ideal

μv1 μl1 μv2 μ2

= μv10 + RT ln(y1 P ) s = μl0 1 + RT ln (x1 P1 )

= μv20 + RT ln(y2 P ) s = μl0 2 + RT ln (x2 P2 )

(7)

By incorporating Eqs. (7) into Eq. (6), considering the equality of the reference chemical potential for the two species, we can write:

RT ln(y1 P ) − RT ln(x1 P1s ) − RT ln(y2 P ) + RT ln(x2 P2s ) +



μv1 − μl1

nonideal

−



μv2 − μl2

Or, in condensed form as:

RT ln

 y  1 1 − y1



+ RT ln

where nonideal RT ln(ε12 )=



(1 − x1 )P2s

nonideal

nonideal

(8)

nonideal + RT ln(ε12 )=0

x1 P1s

μv1 − μl1

=0

−



μv2 − μl2

nonideal

(9)

(10)

Re-arranging Eq. (9) we obtain the generalized equilibrium vapor-liquid composition difference [y1 -x1 ] equation as a function of the liquid composition, the ideal relative volatility and accounting for the non-ideality of the phases as:

[y 1 − x 1 ] =

ideal x1 α12

nonideal ideal nonideal ε12 + x1 (α12 − ε12 )

− x1

(11)

ideal = kideal /kideal and kideal = P S /P , kideal = P S /P . It Where α12 1 2 1 1 2 2 should be pointed out that Eq. (11) is valid for ideal as well nonideal as non-ideal systems. Considering ideal phases (ε12 = 1), that is, the vapor phase behaves as an ideal gas (PV=nRT) and the liquid phase is represented by an ideal solution (Raoult’s law), Eq. (11) renders the driving force equation given by Gani and BekPedersen (20 0 0) for constant volatility:

FD = [y1 − x1 ] =

ideal x1 α12

ideal 1 + x1 (α12 − 1)

− x1

(12)

T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

It is important to note that Eq. (12) is valid only when the temperature and pressure conditions of the vapor-liquid equilibrium are far from the critical conditions and assuming weak intermolecular forces in the liquid phase. Also, it should be recognized that the non-ideal volatility is related to the ideal volatility for a γ - ϕ approach by the following relationship: nonideal α12 =

ideal α12 P1s γ1 ϕˆ 2 = = α12 nonideal P2s γ2 ϕˆ 1 ε12

(13)

Moreover, we have the generalized non-ideal driving force expression as:

FD = [y1 − x1 ] =

nonideal x1 α12

1 + x1 ( α

nonideal 12

− 1)

− x1 =

x1 α12 − x1 1 + x1 (α12 − 1 )

55

It should be pointed out that Eq. (23) is valid for ideal and nonideal binary systems, since the residual and excess enthalpies of the binary mixture are considered. It is worthwhile to observe the terms in Eq. (23) compared to the terms of the driving force approach obtained through the phase equilibrium analysis Eq. (14)). Clearly, both Equations of the driving force approach must render the same values of FD , considering the ideality or not of the binary mixture. It is evident from observing Eqs. (14) and ((23) that the driving force definition has a direct connection to the energy involved in the phase separation. In addition, it should be pointed out that at the maximum driving force point, the ideal energy condition holds, this is the heat of vaporization of the mixture and the difference of the ideal vapor and liquid mixture enthalpies remain constant.

(14) 3. The driving force approach and the phase change energy

4. The driving force design equations for non-reactive distillation columns (NRDC)

It is known that the enthalpy for a non-ideal gas-liquid mixture can be represented by:

The equilibrium condition for an ideal or non-ideal binary mixture can be represented as:

hV,nonideal = hV,ideal + hR

(15)

y1 =

hL,nonideal = hL,ideal + hE

(16)

and the driving force relationship is given by Eq. (14):

hR

hE

where and are the residual and excess non-ideal contributions. The ideal enthalpies can be obtained with the well-known relationships (Seader and Henley, 1998):

hV,ideal =

NC

yi h0i V =

NC

i=1

hL,ideal =

NC

 yi

T T0

i=1

C pig dT i

xi (h0i V − Hivap )

FD = y1 − x1 =

V,ideal

L,ideal

R

−h =h −h +h −h NC NC



= yi h0i V − xi (h0i V − Hivap ) + hR − hE

h

i=1

(18)

(FD )RL =

E

(19)

i=1

and for a binary mixture we have

+ x1 H1

vap

+ x1 H1

where

+

hR12

−

hE12

x1 x1D − R+1 R+1

(20)

hR12 = y1 hR1 + y2 hR2 ;

  hR1 = −RT 2 ∂ ln∂(Tϕˆ1 ) ;

  hR2 = −RT 2 ∂ ln∂(Tϕˆ2 )

hE12 = x1 hE1 + x2 hE2 ;

  hE1 = −RT 2 ∂ ln∂(Tγ1 ) ;

  hE2 = −RT 2 ∂ ln∂(Tγ2 ) (21)

x1B x1 − VB VB

(FD )SL =

h

−h

L,nonideal

vap

+ x2 H2

+

= (y1 − x1 )(

hR12

−

h01V

−

h02V

) + x1 H1

hE12

x1 |q=0 =



FD =

hV,nonideal − hL,nonideal

 −

h01V − h02V hR12 − hE12

h01V − h02V







−

x1 H1vap + x2 H2vap



h01V − h02V

zF zF + α12 (1 − zF )

(28)



x1

α12 q=0 − x1 |q=0 1 + x1 (α12 − 1 ) q=0

(29)

If 0 < q < 1 (two phases) then zF is split in two phases (x1 ,y1 ), thus, from Eq. (14) is obtained for the corresponding liquid composition

x1 |0
 (23)

(27)

and the driving force at this condition is

(22)

and introducing the driving force (FD) definition into Eq. (22), the following relationship between the driving force and the phase change enthalpy is obtained:

zF α12 − zF 1 + zF (α12 − 1 )

If q = 0 (saturated vapor) then y1 = zF , thus, the following expression is obtained from Eq. (14) for the driving force corresponding to liquid composition

FD |q=0 =

vap

(26)

where R is the reflux ratio and VB is the boil-up ratio and the composition subscripts D and B refer to the distillate and bottom sections, respectively. The q-Line (Thermal feed condition): If q = 1 (saturated liquid) then x1 = zF , thus, the following expression is obtained from Eq. (14) for the driving force

Reorganizing Eq. (20) we obtain: V,nonideal

(25)

Stripping section:

FD |q=1 =

hV,nonideal − hL,nonideal = y1 h01V + y2 h02V − x1 h01V − x2 h02V vap

x1 α12 − x1 1 + x1 (α12 − 1 )

(17)

Where hi 0V and Hi vap are the ideal gas enthalpy and heat of vaporization of species i at temperature T, respectively. Subtracting Eqs. (15) and (16) and introducing Eqs. (17) and (18), the following expression is obtained: L,nonideal

(24)

Based on the equilibrium relationship and the driving force definition, the design equations of the driving force for NRDC can be written as: Rectifying section:

i=1

V,nonideal

x1 α12 1 + x1 (α12 − 1 )

−

2



+ zF

q − α12 (q − 1 ) − zF (α12 − 1 ) q(α12 − 1 )

q(α12 − 1 )



x1 |0




=0

(30)

56

T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

And defining



κ= and

ξ=

q − α12 (q − 1 ) − zF (α12 − 1 ) q(α12 − 1 )



(32)

q(α12 − 1 )



x1 |0
(31)



zF

−κ +



κ + 4ξ

2

(33)

x1

α 0
− x1 |0
x1D − x1 min −1 (FD )RL min

(34)

(36)

And the corresponding driving force at the operating condition is:

x1D − x1 |op Rop + 1

=

For element B

x1 |op − zF q−1

(37)

Also, the boil-up ratio VB can be calculated as

(FD )op 1 = VB x1 |op − x1B

(38)

Based on the above equations, Appendix A gives the detailed procedure to determine the minimum number of stages and the real number of stages for a defined non-reactive distillation column design using the driving force concept. 5. The driving force design equations for reactive distillation columns (RDC) In this section, the design of RDC is based on the reactive element driving force concept (Pérez-Cisneros et al., 1997) and the basic design equations for a binary element reactive system are developed. However, unlike what happens with non-reactive mixtures, recently, Lopez-Arenas et al. (2019) showed that the explicit mathematical expression of the driving force FDW in terms of the element liquid mole fraction Wl A can be established only after the reactive system is defined. That is, the formula matrix must be known. Thus, the general reactive driving force equations based on the element concept are:

FDW = WAv − WAl



RT ln PSB = RT SB,1 ln(P1S ) + SB,2 ln(P2S )

(39)



    XB WAl PsB YA (FDW )   RT ln + RT ln + RT ln( nonideal )=0 AB YB (FDW ) XA WAl PsA (40) where YA (FDW ) and YB (FDW ) are functions of the element driving force FDW and XA (Wl A ) and XB (Wl A ) are functions of the element

(42)

and the non-ideality term is given as:



M

(35)

xD1 (q − 1 ) + zF (Rop + 1 ) Rop + q

(41)

RT ln YB = RT [SB,1 ln(y1 ) + SB,2 ln(y2 )]

Where (FD )RLmin and x1min are obtained from the thermal condition of the feed or q value. Operating Reflux ratio (intersection of the rectifying and q lines): The liquid composition at the intersection of the q-line and the rectifying line, that is, at the operating reflux ratio condition Rop can be obtained as:

(FD )op =



RT ln XB = RT [SB,1 ln(x1 ) + SB,2 ln(x2 )]

Minimum Reflux ratio:

x1 |op =

RT ln XA = RT [SA,1 ln(x1 ) + SA,2 ln(x2 )]

2



Rmin =

RT ln YA = RT [SA,1 ln(y1 ) + SA,2 ln(y2 )] RT ln PSA = RT SA,1 ln(P1S ) + SA,2 ln(P2S )

and the corresponding driving force at this condition is

FD |0
liquid mole fraction Wl A . The term RT ln ( AB nonideal ) represents the non-ideality of the mixture. The relationships that connect the above functions for a binary element reactive system with the species compositions present in the mixture are as follows: For element A

It is obtained



ϕˆ i RT ln ( AB ) = RT SA,i ln γi i=1



M



ϕˆ i − RT SB,i ln γi i=1

(43)

The matrix S is a sub-matrix obtained from the formula matrix. In general, the formula matrix A can be partitioned into two submatrices as follows:



. A = AI .. AII



(44)

where the leading M × M block AI is assumed to be nonsingular. Since A is of full rank, a non-singular leading submatrix can be always constructed by permuting the columns of A. From the two sub-matrices AI and AII we can construct two matrices S and Z, given by



−1

AI S= 0





− A−1 I AII Z= I

,



(45)

where the dimension of S is NC × M and that of Z is NC × (NCM=NR), and NC is the number of components present in the reactive system and M is the number of elements. Thus, the application of the Gibbs Energy-Driving Force method is highlighted through a simple isomerization reaction with the presence of an inert compound. That is, the ideal isomerization reaction of n-butane to iso-butane with an inert component such as:

n - butane (1 ) ↔ iso − butane (3 ) Inert component: iso - pentane (2 )

(46)

In terms of elements A and B the above reaction can be written as:

(inert element : B )

n−A ↔i−A

(47)

Here, butane (1) is element A, iso-pentane (2) is element B. Therefore, a reduced formula matrix A (consisting of two partitions, AI and AII ) is written as Component Element A B

n-Butane (1)

Iso-Pentane (2)

Iso-Butane (3)

1 0 AI

0 1

1 0 AII

Thus, the formula matrix A for this reactive system is



1 A= 0

0 1

1 0



(48)

And the partition matrices are:



1 AI = 0



0 1

 

;

1 AII = 0

(49)

T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

57

and for element B

RT ln YB = RT ln(y2 ) RT ln XB = RT ln(x2 ) RT ln PSB = RT ln(P2S )

(53)

By substituting the above equalities into Eq. (40) we obtain:

RT ln

Y  A

YB

= RT ln



+ RT ln

y  1

y2

XB PsB XA PsA



+ RT ln



x2 P2s x1 P1s

+ RT ln( nonideal ) AB

+ RT ln(1 ) = 0

(54)

Also, from the formula matrix, we know that the element mole fractions are given as:

WAv =

nv1 + nv3 = y1 + y3 nv1 + nv2 + nv3

(55)

WBv =

nv2 = y2 v n1 + nv2 + nv3

(56)

WAl = WBl =

nl1 + nl3 nl1 + nl2 + nl3 nl2 nl1

+ nl2 + nl3

= x1 + x3

(57)

= x2

(58)

From the chemical equilibrium condition:



−μ01v + μ03v y1 = exp y3 RT





= Ky

Ps −μ01l + μ03l x1 = 3s exp x3 P1 RT

(59)

 = Kx

(60)

It should be noted that in this ideal case, the chemical equilibrium constants Ky and Kx are only functions of temperature. By substituting the component mole fractions into the element fractions Eqs. (55)–(58) we obtain:

YA = y1 =

WAv 1 + Ky

WAl 1 + Kx YB = y2 = WBv = 1 − WAv XA = x1 =

XB = x2 = WBl = 1 − WAl Fig. 1. Graphical representation of the optimal thermal feed condition for BenzeneToluene system at P = 1 atm. a) Driving force diagram, b) enthalpy-composition diagram, c) thermodynamic efficiency diagram.

FDW = the matrix product [-AI



− A−1 I AII =

−1 0



−1

AII ] is

S=

1 0 0

0 1 0



(50)



,

(51)

Using the values of matrix S to obtain the different terms of Eqs. (41) and (42) we obtain for element A:

RT ln YA = RT ln(y1 ) RT ln XA = RT ln(x1 ) RT ln PSA = RT ln(P1S )

1 + WAl



ideal −1 αAB

 − WAl

(62)

ideal αAB = Kyx

kideal 1 kideal 2

;

s

kideal = P1/P; 1

s

kideal = P2/P 2

(63)

and



−1 Z 0 1

ideal WAl αAB

Where

and the working matrices S, Z and n are



(61)

Performing some algebra with Eq. (54), the explicit relationship of the reactive driving force with respect to the element liquid mole fraction is obtained as:

(52)

Kyx =

(1 + Ky ) (1 + Kx )

(64)

Once the reactive system is defined and using Eqs. (39) and (62), the reactive distillation design equations for a binary element reactive system can be obtained. For the rectifying section of a simple reactive distillation column, an element mass balance around such section for element A is given by v l l bvp+1WA,p+1 − blpWA,p = bT DWAD

(65)

58

T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

Fig. 2. Flow diagram of the global Gibbs energy – Driving force method.

where bTD represents the total element amount distillated at the top of the reactive column. If bl p /bv p +1 is constant from plate to plate, then Eq. (65) can be written as

bl bT D l WAv = v WAl + v WAD b b

(66)

or in terms of the element reflux ratio Rb

WAv =

Rb 1 Wl + Wl Rb + 1 A Rb + 1 AD

W l − WAl = AD Rb + 1

(68)

l ideal WAF αAB l 1 + WAF

q=0

l W l − WAB = A VbB

=

(FDW )op =

l − WAop

Rbop + 1

(FDW )qL

(73)

WAF



ideal WAF + αAB 1 − WAF



(74)

and the driving force at this condition is

=

l WAop

l − WAF



q−1

(70)

The reactive q-line is given as: l W l − WAF = A q−1

 − WAFl ideal −1 αAB

(69)

Intersection of the operating lines l WAD



If q = 0 (saturated vapor) then Wv A = WAF , thus, from the following equation is obtained for the corresponding liquid composition

WAl

For the stripping section:

(72)

The reactive q-Line (Thermal feed condition): If q = 1 (saturated liquid) then Wl A = WAF , thus, from Eq. (62) is obtained for the reactive driving force

FDW =

where the element reflux ratio is defined as Rb TD . In terms of the reactive driving force, we have for the rectifying section:

(FDW )SL

b∗l − bl bT F

q=

(67) =bl /b

(FDW )RL

where

(71)

FDW |q=0 =

WAl

ideal αAB  ideal  − WAl q=0 αAB − 1 q=0

1 + Wl A

q=0

(75)

If 0 < q < 1 (two phase) then WAF is split into two phases (Wl A , Wv A ), thus, using Eq. (62) the corresponding liquid composition is

T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

obtained from the following second order equation:





WAl 0


2



+



× WAl

0
ideal ideal q − αAB (q − 1 ) − WAF (αAB − 1)



q (α

ideal AB

 −

q (α

WAF

ideal AB

− 1)





=0

and

ξb =

ideal ideal q − αAB (q − 1 ) − WAF (αAB − 1) ideal q(αAB − 1)



WAF

ideal q(αAB − 1)



− 1)

And defining

κb =

We obtain





WAl 0


=

−κb +



59

κb2 + 4ξb

(79)

2

and the corresponding driving force at this condition is



(76) FDW |0


WAl

1 + Wl

0
A 0
(77)

(78)

(α

ideal AB



− 1)

− WAl

0
(80)

Minimum Reflux ratio:

Rb min =



ideal αAB

l WAD − WAl min −1 (FDW )RL min

(81)

Were (FDW )RLmin and Wl Amin are obtained from the thermal condition of the feed. 6. Maximum driving force and thermodynamic separation efficiency (Gibbs-energy approach) It is well-known that operating costs, i.e., the heating and cooling charges, are usually the major cost of a distillation process. In such cases, it would be desirable to decrease the heating and cooling requirements at the expense of additional equipment. From a thermodynamic viewpoint, the inefficiencies of a distillation operation can be grouped into two main categories: (1) those that are a function of the distillation process itself and; (2) those that are related to supplying the necessary energy to the materials being separated. The minimum isothermal thermodynamic work required for separating 1 mol of a liquid binary mixture into its pure liquid constituents is given by the following equation:

Wmin = GF = −RT

NC

zi ln(γi zi )

(82)

i=1

where GF is the minimum work of separation (free Gibbs-energy change) per mol of mixture and zi is the liquid composition of species i in the feed. For an ideal solution, this expression has a maximum value at a mol fraction of 0.5, and for this condition, the minimum work required is equal to 0.693RT. Thus, to separate 1 mol of an ideal mixture of this composition at a temperature of 298 K would require about 1717 (J/mol). In case the mixture is not completely separated, the minimum work required per mol is obviously less and can be calculated from the following equation:



Wmin = GF = −RT

NC

zi ln(γi zi ) − θ D

i=1

−θ

NC

i=1

B

NC

xi,D ln(γi,D xi,D )



xi,B ln(γi,B xi,B )

(83)

i=1

Fig. 3. Ideal, non-reactive benzene-toluene mixture. (a) Driving force diagram, (b) Thermal feed condition q plot, (c) Thermodynamic separation efficiency ξ .

where D refers to distillate and B refers to bottoms and θ D and θ B are the vapor and liquid separation fractions, respectively. Mixtures with positive deviations from Raoult’s law, i.e., solutions with activity coefficients greater than 1, require lower minimum work for separation than do ideal solutions; the reverse is true for those solutions with negative deviations. For example, consider the minimum work for separating a 3 mol percent solution of ethanol and water at its normal boiling point into pure water and 87 mol percent ethanol. The boiling point of such a solution is 351.5 K, and the activity coefficients are approximately 4.4 and 1.01 for the 3 mol percent solution for ethanol and water, respectively; the corresponding activity coefficient values for the 87 percent solution are 1.01 and 2.2. On the basis of Eq. (83), the minimum work of separating 1 mol of this mixture into the desired product would be about 111.76 (kJ). Separating an ideal solution of the

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Fig. 4. Ideal, non-reactive benzene-toluene mixture. (a) NRDC design for saturated liquid feed condition (q = 1), (b) NRDC design for saturated vapor feed condition (q = 0), (c) Minimum Stage computation, (d) NRDC design for optimal thermal feed condition (q = 0.0553).

same composition into the same products would require 161.31 (kJ). The actual (non-ideal) energy requirement is lower than the theoretical (ideal) because ethyl alcohol and water have positive deviations from Raoult’s law, indicating a tendency to immiscibility. An immiscible system requires essentially no work for separation. On the other hand, systems with negative deviations from Raoult’s law, i.e., those that are maximum boiling azeotropes, would need more work for separation than an ideal solution. In the case that the feed splits into two phases the following expression considering the thermal condition of the feed (q) is:



Wmin = GF = −RT (1 − q )

NC

yiF ln(ϕi yiF ) + q

i=1

− θD

NC

xi,D ln(γi,D xi,D ) − θ B

i=1

NC

xiF ln(γi xiF )

i=1 NC

(84)

i=1

The energy required for separating a mixture is supplied by adding heat to the fluid in the still and removing heat at a lower temperature level in the condenser. The available work energy, based on an isentropic process, in the heat supplied to the liquid in the still can be calculated on the basis of the following equation:

WB = available work = QB

TB − T0 TB

WC = available work = QC

(85)

TD − T0 TD

(86)

Where WC is the available work at the condenser, QC is the heat removed from condenser and TD is the temperature of condensation of distillate. The net available work supplied in the heat to the distillation process itself is equal to:

W = QB

 xi,B ln(γi,B xi,B )

where WB is the available work at the reboiler, QB is the heat added in the still, TB is the absolute temperature of liquid in still and T0 is the absolute temperature at which heat can be discharged, i.e., temperature of cooling water. The available work equivalent to the heat removed from the condenser can be calculated by the following equation:

TB − T0 TD − T0 − Qc = QB τB − QC τC TB TD

(87)

Where τ B and τ C are the dimensionless temperature ratios at the bottom and top of the distillation column, respectively. The minimum heat that can be utilized in a distillation corresponds to the minimum reflux condition and, for a binary mixture, can easily be determined from the enthalpy-composition diagram. It is interesting to compare the thermodynamic minimum work with that required by the actual distillation process at the minimum reflux condition. However, it is more instructive to study the ratio of Eq. (84) to Eq. (87) for actual cases. The ratio between the minimum work of separation to the minimum heat that can be used in a distillation considering the effect of the thermal feed

T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

Fig. 5. Non-Ideal, non-reactive ethanol-water mixture. (a) Driving force diagram, (b) Thermal feed condition q plot, (c) Thermodynamic separation efficiency ξ .

Fig. 6. Non-Ideal, non-reactive ethanol-water mixture. (a) NRDC design for saturated liquid feed condition (q = 1), (b) NRDC design for saturated vapor feed condition (q = 0), (c) NRDC design for optimal thermal feed condition (qmax =0.5385).

condition q, is named in the present work as the thermodynamic separation efficiency and is given by the following expression:



ξ=

Wmin = W

−RT (1 − q )

NC  i=1

yiF ln(ϕi yiF ) + q

NC  i=1

xiF ln(γi xiF ) − θ

NC 

D

i=1

QB τB − QC τC

On the other hand, Gani and Bek-Pedersen (20 0 0) showed that at the maximum driving force point the minimum reflux ratio condition is reached. Thus, at this composition, one should have the minimum heat that can be utilized in a distillation column. Also, based on the thermodynamic relationships between the maximum driving force and the phase change energy (PerezCisneros and Sales-Cruz, 2017), it is expected that at this point,

61

xi,D ln(γi,D xi,D ) − θ

B

NC  i=1

 xi,B ln(γi,B xi,B ) (88)

the minimum heat of separation is required and the maximum thermodynamic separation efficiency is achieved. Fig. 1 shows the connection between the maximum driving force diagram with the optimal thermal feed condition (attained with the enthalpy-composition diagram) and the maximum thermodynamic efficiency. Fig. 1a shows for the benzene-toluene system (P = 1

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Fig. 7. Reactive, ideal system: isomerization of n-butane in the presence of iso-pentane (inert). (a) Reactive driving force diagram, (b) Variation of the reactive relative volatility and equilibrium constants.

atm) the different values of the driving force with a fixed feed composition (zF =0.6) considering three different thermal conditions: i) saturated vapor (q = 0) FD = 0.2209; ii) saturated liquid (q = 1) FD = 0.1898 and iii) with a feed split into two phases (qmax =0.0553) FDmax = 0.2223. Fig. 1b shows on an enthalpycomposition diagram the corresponding energy requirements at the condenser (QC ) and reboiler (QB ) for the three different thermal feed conditions. It can be observed through the slopes of the enthalpy tie-lines that the minimum heating (QB ) is attained with the thermal condition qmax =0.0553 and the minimum cooling (QC ) is attained with the thermal condition q = 1. Fig. 1c shows the thermodynamic separation efficiency evaluated considering the thermal condition qmax =0.0553 and varying the feed composition. It should be noted in Fig. 1c that, when the feed composition is zF =0.6, the corresponding liquid and vapor equilibrium feed compositions are xmax =0.390 and ymax = 0.6123, respectively. If the mathematical relationships for the evaluation of the enthalpies are available, a simple computational procedure to obtain the energy requirements can be developed. Thus, considering a total condenser, and assuming that there is sufficient cooling water

and vapor for the condenser and reboiler, respectively, the cooling supplied in the condenser can be calculated as:



vap QC = D(Rop + 1 ) Hmix



TD



= D(Rop + 1 ) x1D H1vap

TD



+ x2D H2vap

(89)

TD

and for a partial reboiler the heating supplied can be calculated as:



vap QB = BVB Hmix



TB



= BVB x1B H1vap

TB



+ x2B H2vap

TB

(90)

Appendix B shows the computational procedure to obtain the optimal thermal feed condition (qmax ) based on the separation efficiency given by Eqs. (88)–(90) for the design of non-reactive and reactive distillation columns. With the above design equations the Gibbs energy – Driving force method can be stated in five steps: 1) with the required thermodynamic data compute the driving force diagram and determine the maximum driving force and the corresponding liquid composition; 2) with the target product compositions xD and xB determine the minimum number of stages Nmin ; 3) determine the optimal thermal feed condition qmax through the computation of the energy supplied (heating and

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63

Fig. 8. Reactive, ideal system: isomerization of n-butane in the presence of iso-pentane (inert). (a) Reactive driving force diagram, (b) Variation of the dimensionless element reactive enthalpies of the vapor and liquid phases.

cooling) and the thermodynamic separation efficiency; 4) at the optimal thermal feed condition qmax , determine the minimum reflux ratio Rmin and the operating reflux ratio Rop ; and 5) determine the intersection of the rectifying and stripping operating lines and obtain the real number of equilibrium stages Nreal and the optimal feed location NF . Fig. 2 shows the flow diagram of the global Gibbs energy-Driving force method. To show the application of the above design method three binary mixtures are considered: 1) the non-reactive ideal system benzene-toluene; 2) the non-reactive non-ideal ethanol-water system and 3) the ideal isomerization reaction of n-butane in the presence of an inert component. 7. Application examples of the Gibbs energy-driving force design method 7.1. Non-reactive, ideal benzene-toluene mixture separation A staged column is to be designed to continuously distill 204 kmol/h of a binary mixture of 60 mol% benzene and 40 mol% toluene. A liquid distillate and a liquid bottoms product of 95

mol% and 5 mol% benzene, respectively, are to be produced. The total operating pressure is 1 atm, considering ideal phases. The design variables (i.e., P, zF , xD , xB , R/Rmin ) and the assumption of ideal phases for this case are taken following the McCabe-Thiele example (7.1) in Seader and Henley book (Seader and Henley, 1998). It should be pointed out that in that example the thermal feed condition, q, of the feed is arbitrarily specified. Using the Gibbs-energy - Driving force method determine: (a) The optimal thermal feed condition q; (b) Minimum number of theoretical stages; (c) Minimum reflux ratio; (d) Number of equilibrium stages for a reflux ratio Rop /Rmin =1.3 and (e) The optimal feed location. Step 1. With the required thermodynamic data compute the driving force diagram and determine the maximum driving force and the corresponding liquid composition (FDmax = 0.223, xmax = 0.390). Step 2. With xD and xB (xD =0.95, xB =0.05) determine the minimum number of stages Nmin (Nmin =7) (see Appendix A). Step 3. Determine the optimal thermal feed condition qmax computing the thermodynamic separation efficiency (qmax = 0.0553), see Appendix B.

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of q (qmax = 0.0553) corresponds to the maximum separation efficiency achieved (ξ =0.8). Fig. 4 shows the design of three distillation columns considering different thermal conditions of the feed. Fig. 4a and b show the design for a saturated liquid and saturated vapor feed conditions, respectively. It can be noted from these figures that for a saturated liquid feed the number of real equilibrium stages (Nreal =14) is superior that for saturated vapor (Nreal =12), however, the operating reflux is the opposite requiring more cooling at the top of the column. Fig. 4c shows the minimum number of stages for the product composition targets. Fig. 4d shows the optimal column design with 12 equilibrium stages and an operating reflux ratio of 1.9748. It should be noted in Fig. 4d that the q-value is close to zero (saturated vapor condition) obtaining the same number of equilibrium stages for q = 0. However, the operating reflux is smaller for the optimal q value (q = 0.0553) leading to less cooling at the top of the column. 7.2. Non-reactive, non-ideal ethanol-water mixture separation A staged column is to be designed to continuously distill 100 kmol/h of a binary mixture of 30 mol% ethanol and 70 mol% water. A liquid distillate and a liquid bottoms product of 75 mol% and 1 mol% ethanol, respectively, are to be produced. The total operating pressure is 1 atm, considering ideal vapor phase and non-ideal liquid phase. For the non-ideal phase, the Wilson GE model is used to predict the activity coefficients and the excess enthalpy of the liquid mixture. The design variables (i.e., P, zF , xD , xB , R/Rmin ) and the assumption of ideal vapor phase and Wilson model for the liquid phases for this case are taken following the example (Chapter 7: Fractional Distillation. Heat Economy Section) in Robinson and Gilliland book (Robinson and Gilliland, 1950). It should be pointed out that in this example, the thermal feed condition of the feed is specified as q = 1 (saturated liquid). The Wilson model for calculating activity coefficients has shown to be a reliable model for the non-ideal ethanol-water system. This is, the azeotropic behavior of the mixture is properly determined (Voutsas et al., 2011). Using the Gibbs-energy - Driving force method determine: (a) The optimal thermal feed condition q; (b) Minimum number of theoretical stages; (c) Minimum reflux ratio; (d) Number of equilibrium stages for a reflux ratio Rop /Rmin =1.3 and (e) The optimal feed location.

Fig. 9. Reactive, ideal system: isomerization of n-butane in the presence of isopentane (inert). (a) Reactive driving force diagram, (b) Thermal feed condition q plot, (c) Thermodynamic separation efficiency ξ .

Step 4. At the optimal thermal feed condition (qmax = 0.0553), determine the minimum reflux ratio Rmin and the operating reflux ratio Rop (Rmin = 1.5191, Rop = 1.9748). Step 5. Determine the intersection of the rectifying and stripping operating lines and obtain the real number of equilibrium stages Nreal (Nreal =12) and the optimal feed location NF (NF =6). Fig. 3a shows the driving force diagram indicating the values of the maximum driving force and the corresponding liquid composition (FDmax = 0.223, xmax = 0.390). Fig. 3b shows the variation of the q-value (thermal condition of the feed) as a function of the liquid composition for different feed composition (zF = 0.3, 0.6, 0.8). It can be noted (see Fig. 3c) that the optimal value

Step 1. With the required thermodynamic data compute the driving force diagram and determine the maximum driving force and the corresponding liquid composition (FDmax = 0.3467, xmax = 0.14). Step 2. With xD and xB (xD =0.75, xB =0.01) determine the minimum number of stages Nmin (Nmin =7) (see Appendix A). Step 3. Determine the optimal thermal feed condition qmax computing the thermodynamic separation efficiency (qmax = 0.5385), see Appendix B. Step 4. At the optimal thermal feed condition (qmax = 0.5385), determine the minimum reflux ratio Rmin and the operating reflux ratio Rop (Rmin = 0.7595, Rop = 0.9873). Step 5. Determine the intersection of the rectifying and stripping operating lines and obtain the real number of equilibrium stages Nreal (Nreal =10) and the optimal feed location NF (NF =7). Fig. 5a shows the driving force diagram indicating the values of the maximum driving force and the corresponding liquid composition (FDmax = 0.3467, xmax = 0.140). Fig. 5b shows the variation of the q-value (thermal condition of the feed) as a function of the liquid composition for different feed compositions (zF = 0.1, 0.3, 0.5). It can be noted (see Fig. 5c) that the optimal value of q (qmax = 0.5385) corresponds to the maximum separation efficiency achieved (ξ =0.85). Fig. 6 shows the design of three

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65

Fig. 10. Reactive, ideal system: isomerization of n-butane in the presence of iso-pentane (inert). Minimum number of reactive stages and component composition at the top of the column.

distillation columns considering different thermal conditions of the feed. Fig. 6a and b show the design for a saturated liquid and saturated vapor feed condition, respectively. It can be noted from these figures that for a saturated liquid feed the number of real equilibrium stages (Nreal =11) is superior that for saturated vapor (Nreal =7), however, the operating reflux is the opposite, requiring more cooling at the top of the column. Fig. 6c shows the optimal column design with 10 equilibrium stages and an operating reflux ratio of 0.9873. It should be noted in Fig. 5c that the q-value (q = 0.5385) is almost in the middle of limiting saturation conditions (0
7.3. Ideal isomerization reaction of n-butane in the presence of an inert component A reactive trayed tower is to be designed to continuously distill 100 kmol/h of a binary element mixture of 60 mol% n-butane 40 mol% iso-pentane (inert): WAF =0.6. An element liquid distillate and a liquid bottoms product WAD =0.99 and WAB =0.01, respectively, are to be produced. The total operating pressure is 1 atm, considering ideal phases. The design variables (i.e., P, WAF , WAD , WAB , R/Rmin ) and the assumption of ideal phases for this case are taken following the process based on oxide catalyst (JSC SIE Neftehim) (SIE Neftehim, LLC 2019). It should be pointed out that this oxide catalyst process is a sequential isomerization reactor/non-reactive distillation column and that in the present work an intensified reactive distillation column design is being proposed. Using the Gibbs energy - Driving force method for reactive distillation column design determine: (a) The optimal thermal feed condition q; (b) Minimum number of reactive theoretical stages; (c) Minimum element reflux ratio; (d) Number of reactive equilib-

rium stages for an element reflux ratio Rbop /Rbmin =1.3 and (e) The optimal feed location. For this reactive system we know that the relationships between the component compositions n-butane (1), iso-pentane (2) and iso-butane (3) and the element A and B composition are:

WAv = y1 + y3 ;

WBv = y2 ;

WAl = x1 + x3 ;

WBl = x2

Thus, a value of WAB =0.01 at the bottom of the reactive distillation column means that we have almost pure iso-pentane at this location and a mixture of n-butane and iso-butane at the top of the column. Step 1. With the required thermodynamic and thermochemical data compute the reactive driving force diagram and determine the maximum reactive driving force and the corresponding element liquid composition (FDWmax = 0.336, Wl Amax = 0.355). Step 2. With WAD =0.99 and WAB =0.01 determine the minimum number of stages Nmin (Nmin =7) (see Appendix A with the corresponding design equations for reactive systems). Step 3. Determine the optimal thermal feed condition qmax computing the thermodynamic reactive separation efficiency (qmax = 0.1872), see Appendix B. Step 4. At the optimal thermal feed condition (qmax = 0.1872), determine the minimum reactive reflux ratio Rbmin and the operating reflux ratio Rbop (Rbmin = 0.987, Rbop = 1.283). Step 5. Determine the intersection of the reactive rectifying and stripping operating lines and obtain the real number of reactive equilibrium stages Nbreal (Nbreal =7) and the optimal feed location NF (NF =4). Fig. 7a shows the reactive driving force diagram with the maximum reactive driving force FDWmax =0.336 located at Wl Amax =0.355. Fig. 7b shows the variation of the chemical equilibrium constants Ky and Kx that are only functions of temperature, with respect to Wl A . It can be noted that these equilibrium relationships increase along the whole Wl A range and that Ky is greater than Kx indicating that the composition of iso-butane (product) is higher in the vapor phase. The reactive relative volatility value goes from α AB ideal =2.5 to α AB ideal =2.85 and, when Wl A is close to 1, the higher reactive relative volatility indicates that the composition of

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tie-lines are different. Fig. 9b shows the q-values for different feed compositions (WAF = 0.4, 0.6, 0.8). It can be noted (see Fig. 9c) that the optimal value of q (qmax = 0.1872) corresponds to the maximum reactive separation efficiency achieved (ξ =0.97). Fig. 10 shows the minimum reactive stages obtained considering the element product specification (NRmin =7). It should be noted that the procedure given in Appendix A is used but with the corresponding reactive driving force equations and the element liquid composition. Fig. 11 shows the design of three reactive distillation columns considering different thermal conditions of the feed. Fig. 11a and b show the design for a saturated liquid and saturated vapor feed condition, respectively. It can be noted from these figures that for both saturated feed conditions the number of real reactive equilibrium stages (NRreal =7) is the same. However, the operating reactive reflux ratio is greater with saturated vapor feed condition requiring more cooling at the top of the column and for the saturated liquid feed condition the reactive boil up ratio is greater, requiring more heating at the reboiler. Fig. 11c shows the optimal reactive distillation column design with 7 reactive equilibrium stages and an operating reactive reflux ratio of 1.283. It should be noted in Fig. 10c that the q-value (qmax =0.1872) is between the limiting saturation conditions (0
Fig. 11. Reactive, ideal system: isomerization of n-butane in the presence of isopentane (inert). (a) RDC design for saturated liquid feed condition (q = 1), (b) RDC design for saturated vapor feed condition (q = 0), (c) RDC design for optimal thermal feed condition (q = 0.1872).

the light components (n-butane and iso-butane) is higher. Fig. 8b shows the dimensionless reactive element-based enthalpies for the vapor and liquid phases HW Videal /RT, HW Lideal /RT, respectively. From Fig. 8, it should be pointed out that the connection between the reactive enthalpies and the reactive driving force gives the possibility to determine the optimal thermal feed condition qW as well as the energy required for any reactive distillation process producing iso-butane. In Fig. 8b it can be noted that, if the element feed composition is set as WAF =[Wl A ]max , the energy required for the reactive separation depends on the thermal condition of the feed. This is, if qW =1 (saturated liquid feed) or qW =0 (saturated vapor feed) two energy consumptions of a reactive distillation column are obtained since the slopes of the element enthalpy

A simple systematic method for designing reactive and nonreactive distillation columns has been developed. The method is based on the driving force concept and the thermodynamic separation efficiency evaluation. The driving force design equations for reactive and non-reactive distillation columns are linked with the thermodynamic separation efficiency equation to obtain the optimal thermal feed condition leading to the minimum energy required for the reactive or non-reactive separation. The design method considers five steps and it has been applied for the design of two no-reactive distillation columns (benzene-toluene and ethanol-water mixtures) and one reactive distillation column (isomerization reaction of n-butane in the presence of iso-pentane as inert component). Results show that for the benzene-toluene mixture the optimal feed condition (qmax = 0.0553) is close to the saturated vapor condition (q = 0) rendering less stages and lower energy consumption compared with the saturated liquid condition (q = 1). On the other hand, for the non-ideal ethanolwater mixture, an optimal thermal condition (qmax = 0.5385) near to the middle of the limiting saturated conditions (0
T. Lopez-Arenas, S.S. Mansouri and M. Sales-Cruz et al. / Computers and Chemical Engineering 128 (2019) 53–68

control of reactive distillation columns (Mansouri et al., 2016). It is important to note that in the present work the method has been applied only to binary elements or components, reactive or non-reactive mixtures and one feed distillation columns. Nevertheless, the method can be extended to deal with multicomponent mixtures or multi-element reactive systems by using the heavy key –light key species concept (Jantharasuk et al., 2011). Besides, when the mixture shows low driving force values, this is, yi ∼xi (azeotropic condition) a careful evaluation of the thermodynamic efficiency and the real number of stages should be considered.

GLB =

NC

67

xi,B ln(γi,B xi,B )

(B.2)

i=1

and



L QB∗ = θ B τB HBL − HZF − m ( xB − zF )



L QD∗ = θ D τD HZF − HDL + m(xD − zF )

m=

HBL −

QB B

L − Heq

xB − xeq

;

τB =

T − T  B 0 TB

;

τD =

T − T  D 0 TD

(B.3)

Step 1. Compute θ D and θ B

θD =

Appendix A. Procedure to obtain Nmin for non-reactive distillation columns (1) First stage For q = 0 (saturated vapor) then y1 = xD , thus from Eq. (57) compute

x1 |q=0

 k=1

=

xD xD + α12 (1 − xD )

(A.1)

And from Eq. (18)

  x1 α12



 q=0 k=1   FD |q=0 = − x1 |q=0 k=1 k=1 1+ x (α − 1 ) 1 q=0

(A.2)

12

k=1

(2) Repeat the computation of the liquid composition and driving force for k = 2 to Nmin using

x1 |q=0

 k

=

 x



1 q=0

k−1

 x1



q=0





k−1

+ α12 1 − x1

q=0





k−1

  x1 α12



 q=0 k   FD |q=0 = − x1 |q=0 k k 1+ x (α − 1 ) 1 q=0

k

(A.3)

(A.4)

12

Where the final Nmin is obtained with the k value when the following condition is satisfied:

x1 |q=0



k

≤ x1B

(A.5)

Appendix B. Procedure to determine the optimal thermal feed condition (q) that maximize the thermodynamic separation efficiency Given P, zF , xD and xB , determine de thermal fed condition (q) that maximize the thermodynamic separation efficiency. The separation efficiency can be written as:

ξ=

−RT (1 − q )GVF + qGLF − θ D GLD − θ B GLB QB∗ − QD∗

Where

GVF =

NC

yiF ln(ϕi yiF )

i=1

GLF =

NC

xiF ln(γi xiF )

i=1

GLD =

NC

i=1

xi,D ln(γi,D xi,D )



(B.1)

zF −xB xD −xB

;

θB = 1 − θD

Step 2. With P, xD obtain TD (bubble point calculation) With P, xB obtain TB and compute τ D and τ B Step 3. Compute γ iD , γ iB Step 4. Compute GL D , GL B (Eq. B.2) Step 5. Compute HL D , HL B (From the enthalpy-composition diagram or relationship) Step 6. Compute for q = 1; x1 = zF calculate Teq (bubble point) Compute GF = GL F ; Compute FD = y1 - x1 ; Compute HL eq Compute Q∗ B and Q∗ D and obtain ξ |q=1 Step 7. Compute for q = 0; y1 = zF calculate x1 and Teq (dew point) Compute GF = GV F ; Compute HV eq Compute Q∗ B and Q∗ D and obtain ξ |q=0 Step 8. Compare ξ |q=1 and ξ |q=0 and set ξmax Step 9. Compute ξ for 0 < q < 1 LOOP q = 0.01 to 0.99, step q = 0.01 with P, zF and q calculate: Teq , xeq , yeq ,Rmin , Rop Calculate GV F and GL F (Eq. B.2) Compute Q∗ B and Q∗ D (Eq. B.3) and obtain ξ |q (Eq. B.1) Compare ξ |q with ξmax If ξ |q > ξmax then ξmax =ξ |q qnew = qold +q END LOOP q qoptimal and ξmax are obtained

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