Minimum Amounts of Extracting Solvent of (A&lt;sub&gt;1&lt;/sub&gt;A&lt;sub&gt;2&lt;/sub&gt;&amp;middot;&amp;middot;&amp;middot;A&lt;sub&gt;&lt;i&gt;t&lt;/i&gt;-1&lt;/sub&gt;)/(A&lt;sub&gt;2&lt;/sub&gt;A&lt;sub&gt;3&lt;/sub&gt;&amp;middot;&amp;middot;&amp;middot;A&lt;sub&gt;&lt;i&gt;t&lt;/i&gt;&lt;/sub&gt;) Countercurrent Extraction Separation

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Minimum amounts of extracting solvent and scrubbing agent solution (denoted as Smin and Wmin respectively) mean the theoretical minimal requirements for chemicals to achieve a specified separation duty, and therefore attract more concerns in the process design of countercurrent extraction separation. Over the past decade, hyperlink extraction technology has been widely used in rare earth separation industry in China with the target to decrease chemical consumption as well as pollution discharges. However, the equations of Smin and Wmin derived in the earlier version of the theory of countercurrent extraction can only be correctly applied to two-component separation and become invalid for hyperlink processes. It is significant to develop new aspects of the theory for design of the hyperlink processes. The separation of (A1A2···At－1)/(A2A3···At) is the basic configuration in the hyperlink processes differently from in conventional processes, where A1, A2, ···, At represent different kinds of rare earth ions with the same number of valence. The separations of two and three components have been discussed and the equations of Smin and Wmin are derived in our previous works. Nevertheless, it is still significant, especially for rare earth separations, to investigate the separations with more components. The present article will therefore focus on developing new expressions for Smin and Wmin applied to the (A1A2···At﹣1)/(A2A3···At) separation. Also a five-component separation case is simulated according to the derived equations. This work is an essential part of the new theory to design the hyperlink extraction processes.

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Advances in Materials Physics and Chemistry, 2015, 5, 325-336
[Published Online August 2015 in SciRes. http://www.scirp.org/journal/ampc](http://www.scirp.org/journal/ampc)
http://dx.doi.org/10.4236/ampc.2015.58032

# Minimum Amounts of Extracting Solvent of (A1A2∙∙∙A t −1)/(A2A3∙∙∙A t ) Countercurrent Extraction Separation

Fu-Xiang Cheng 1 , Sheng Wu 1,2 , Song-Ling Wang 1 , Bo Zhang 1 , Yan Liu 1 , Chun-Sheng Liao 1,2* ,
Chun-Hua Yan 2

1China Minmetals (Beijing) Research Institute of RE Co., Ltd., Beijing, China
2Beijing National Laboratory for Molecular Sciences, State Key Laboratory of Rare Earth Materials Chemistry
and Applications, PKU-HKU Joint Lab on Rare Earth Materials and Bioinorganic Chemistry, Peking University,
Beijing, China
Email: * [cfx@cre-ol.com](mailto:cfx@cre-ol.com)

Received 20 July 2015; accepted 18 August 2015; published 21 August 2015

Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
[http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)

Abstract

Minimum amounts of extracting solvent and scrubbing agent solution (denoted as _S_ min and _W_ min
respectively) mean the theoretical minimal requirements for chemicals to achieve a specified separation duty, and therefore attract more concerns in the process design of countercurrent extraction separation. Over the past decade, hyperlink extraction technology has been widely used
in rare earth separation industry in China with the target to decrease chemical consumption as
well as pollution discharges. However, the equations of _S_ min and _W_ min derived in the earlier version
of the theory of countercurrent extraction can only be correctly applied to two-component separation and become invalid for hyperlink processes. It is significant to develop new aspects of the
theory for design of the hyperlink processes. The separation of (A1A2∙∙∙A _t_ −1)/(A2A3∙∙∙A _t_ ) is the basic
configuration in the hyperlink processes differently from in conventional processes, where A1, A2,
∙∙∙, A _t_ represent different kinds of rare earth ions with the same number of valence. The separations of two and three components have been discussed and the equations of _S_ min and _W_ min are derived in our previous works. Nevertheless, it is still significant, especially for rare earth separations, to investigate the separations with more components. The present article will therefore focus on developing new expressions for _S_ min and _W_ min applied to the (A1A2∙∙∙A _t_ −1)/(A2A3∙∙∙A _t_ ) separation. Also a five-component separation case is simulated according to the derived equations. This
work is an essential part of the new theory to design the hyperlink extraction processes.

Keywords

Theory of Countercurrent Extraction, Minimum Amount of Extracting Solvent, Countercurrent

*Corresponding author.

How to cite this paper: Cheng, F.-X., Wu, S., Wang, S.-L., Zhang, B., Liu, Y., Liao, C.-S. and Yan, C.-H. (2015) Minimum
Amounts of Extracting Solvent of (A1A2∙∙∙A _t_ −1)/(A2A3∙∙∙A _t_ ) Countercurrent Extraction Separation. _Advances in Materials Phys-_
_ics and Chemistry_, 5, 325-336. [http://dx.doi.org/10.4236/ampc.2015.58032](http://dx.doi.org/10.4236/ampc.2015.58032)

F.-X. Cheng _et al._

Extraction, Hyperlink Process, Multi-Component Separation

1. Introduction

Rare earth separation processes of solvent extraction may involve as many as fifteen components. As the rare
earths are all so close to each other in terms of extraction chemistry, the techniques by countercurrent extraction
and scrubbing of the extract have been proved essential for the production of pure rare earth products [1]-[3].
Various methods such as McCabe-Thiele method [3]-[6], chemical thermodynamics analysis [7] [8], and mathematical empirical methods [9] [10], have been developed to carry out the calculation and optimization of
countercurrent extraction processes of rare earths. The remarkably rapid progress on separation processes of rare
earths was created by the Theory of Countercurrent Extraction (called hereafter as “the theory” for simplicity
unless otherwise specified) established by Xu _et_ _al._ [11] [12] in 1970s. The theory derived a number of new algebra equations as well as flexible calculation approaches for process optimization. In particular, the theory facilitated the advent of one-step scale-up technique [13], as a result of which the world rare earth industry encountered “China Impact” [14].

Currently due to the increasing environmental concerns, the separation techniques of solvent extraction are
tending to reduce both consumption and emission of chemicals [15]-[17]. Acidic extractants are commonly used
to separate rare earths [1] [18] [19], and saponification by basic substances is usually necessary for the purpose
of increasing and stabilizing the extraction capacity of the acidic extractants. Meanwhile, inorganic acids are
demanded for both scrubbing and stripping. Thus acids and bases are the main consumed chemicals, and the resulted salts are the main pollution during rare earth separation processes. To determine the theoretical minimum
requirement of the base for saponification and the acid for scrubbing, the terms of minimum amounts of extracting solvent and scrubbing agent solution (denoted hereafter as _S_ min and _W_ min respectively) are introduced respectively.

Hyperlink extraction technique has been developed and widely used in Chinese rare earth separation industry
due to its virtue in reducing both consumptions and emissions since the early of last decades [14] [20]. By arranging two separation cascade units to offer each other with the extracting solvent or the scrubbing agent solution containing rare earths, the technique can decrease the consumption of acid or base for both of the two units.
So far, the optimized parameters of a hyperlink process are determined by screening a great deal of simulation
results [21]-[23]. However, the simulation cannot exactly tell the theoretical minimum values of parameters, and
also the simulation task is too technical for common technicians. Therefore we are exploring to develop new aspects of the theory available for design of the hyperlink processes. Differently from the configuration of
(A1A2∙∙∙A _k_ )/(A _k_ +1A _k_ +2∙∙∙A _t_ ) (1 < _k_ < _t_ ) commonly seen in conventional processes, (A1A2∙∙∙A _t_ −1)/(A2A3∙∙∙A _t_ ) is the
basic separation configuration in hyperlink processes [20], where _t_ ( ≥ 3) represents the number of components.
The values of _S_ min and _W_ min are the primary concerns for a given separation duty, and therefore in our previous
works [24]-[27], we discussed the separations with two or three components and derived the equations of _S_ min
and _W_ min. Nevertheless, rare earth separations involve more than 3 components in most cases, and thus in this
article we will derive the equations of _S_ min and _W_ min for multi-component separations with the (A1A2∙∙∙A _t_ −1)/
(A2A3∙∙∙A _t_ ) configuration.

2. Description of the Separation System

Figure 1 displays a typical countercurrent extraction cascade unit with the separation configuration of (A1A2∙∙∙A _t_

−1 )/(A 2 A 3 ∙∙∙A _t_ ), where _t_ ( ≥ 3) means the number of components and A 1 , A 2 , ∙∙∙, A _t_ represents _t_ kinds of trivalent
rare earth ions in the feed with a prior extractable sequence of A1 > A2 > ∙∙∙ > A _t_ . The feed flowrates of the _t_
components are _f_ 1, _f_ 2,∙∙∙, and _ft_ (hereafter in mole per unit time) respectively all the time, where _f_ 1 + _f_ 2 + ∙∙∙ + _ft_ = _f_ .
The separation factor of the neighboring A _i_ /A _i_ +1 ( _t_ - _i_ ≥ 1) pair is denoted as _βi_ / _i_ +1; the separation factor of A _i_ /A _i_ + _j_
pair ( _t_ - _i_ ≥ 1; _t_ - _i_ ≥ _j_ ≥ 1), _βi_ / _i_ + _j_, can be calculated according to:

_i_ + _j_

_i_ + _j_ - 1

_i_ + _j_      - 1
#### + j = ∏ β k / k + 1                  (1)

_k_ = _i_

#### β i / i + j = ∏ β k / k

_k_ = _i_

_i_ / _i_ + _j_ _k_ / _k_ + 1

326

Feed:A1,A2, `…` ,A _t_

F.-X. Cheng _et al._

Aqueous

Organic

Extracting

solvent

1 2 _n_ _n_ +1 _n+m-_ ~~1~~ _n+m_

Extract:
A1,A2, `…` ,A _t-_ 1

Scrubbing agent
solution

Raffinate:
A2,A3, `…` ,A _t_

Extraction

stages

Scrubbing

stages

Figure 1. Schematic diagram for the countercurrent extraction cascade of
hyperlink processes.

The separation cascade couples _n_ extraction stages and _m_ scrubbing stages. The partly saponified barren solvent with the flowrate _S_ and hydrochloric acid with the flowrate _W_ enter at opposite ends of the cascade, serving
as extracting solvent and scrubbing agent solution respectively. Here for brevity the term “extracting solvent” is
used to include solvent mixtures. The feed stream enters either stage _n_ or stage ( _n_ + 1) depending on its phase
type, as shown in Figure 1 . The final raffinate and the final extract, which can behave as the feed in subsequent
countercurrent units for further separation, contain no A1 or no A _t_ respectively as required.

It is assumed that the extracting solvent and the aqueous solution are immiscible and remain so at all cases of
the operation. It is assumed that each stage is an equilibrium stage, and therefore all the flowrates are out of each
stage in equilibrium. It is also assumed that only the saponified extractant has the extraction capacity with a single cation-exchange mechanism. The saponified extracting solvent is assumed to be always saturated with components when leaving all stages except ( _n_ _+_ _m_ ) th stage. All the H + ion in the starting scrubbing acid is supposed
to immediately exchange and migrate towards the organic phase after entering the cascade, and therefore the
scrubbing agent solution has contained no acid since leaving ( _n_ _+_ _m_ ) th stage. And also as all the components has
a same number of valence, the mixed extraction factor which denotes the ratio of the total mole amount of all the
components in the organic phase to that in the aqueous phase will keep constant either through extraction stages
except stage 1 or through scrubbing stages except stage ( _n_ _+_ _m_ ).

3. Expressions of Adjacent Stage Impurity

To derive the equation of _S_ min and _W_ min, we need to introduce a term Adjacent Stage Impurity Ratio (briefly as
ASIR) for the case of multi-component separation [24] as we did before [25]-[27]. The separation goal can be
achieved by stepwise impurity removal in the countercurrent extraction cascade, and ASIR is then defined as
Equation (2) which can evaluate the purification efficiency of a certain stage.

impurity flowrate entering stage before purified _k_ ( )

_i_, _k_ = impurity flowrate leaving stage after purified _k_ ( )

impurity flowrate entering stage before purified _k_ ( )

_gi_, _k_ = impurity flowrate leaving stage after purified _k_ ( ) (2)

impurity flowrate entering stage before purified _k_ (

_i_, _k_ impurity flowrate leaving stage after purified _k_ (

where _g_ denotes ASIR. The subscripts _i_ and _k_ represent an impurity and a stage, respectively. The _gi,k_ then
means ASIR of impurity A _i_ at stage _k_, and so hereafter.

In the extraction stages in Figure 1, the extracting solvent should withdraw component A1 stage by stage to
obtain the final raffinate with no A1, and therefore component A1 is regarded as the impurity of the aqueous
phase in the extraction section. Instead, component A _t_ behaves as the impurity in the organic phase to be removed by scrubbing agent solution in the scrubbing stages. Figure 2 shows the equilibrium flows of a continuous three stages, where _x_ and _y_ denote the flowrates of a component in the aqueous and the organic phases
respectively. The subscripts _i_ and _k_ represent a component and a stage again, respectively. The _xi_, _k_ then represent
the flowrates of component A _i_ in the aqueous phase, and so hereafter. If 1 < _k_ < _n_, which means that the three
stages locate in extraction section, extraction equilibrium and mass balance in view of _k_ th and ( _k_ + 1) th stages
give the relations of Equations (3), (4) and (5).

_t_

= ∑ _yi_, _k_ (3)

_i_ = 1

_S_ = ∑ _yi_, _k_
_i_ = 1

_i_,

= 1

_y_ 1, _k_ ⋅ _x_

β 1/ _i_ = _xy_ 1,1, _kk_ ⋅⋅ _yxii_,, _kk_ ( 2 ≤≤ _i_ _t_ ) (4)

_i_ = _x_ 1, _k_ ⋅ _yi_ , _k_

1/ _i_ = _x_ 1,1, _kk_ ⋅ _yii_,,

_k_ _i_, _k_

327

F.-X. Cheng _et al._

## yi,k- 1 xi,k- 1

## _yi,k_ _xi,k_

## yi,k+ 1 xi,k+ 1

Figure 2. Mass flows in a continuous three stages.

_x_ 1,1 = _x_ 1, _k_ + 1                    - _y_ 1, _k_ (5)

where _x_ 1,1 means the flowrates of A1 in the final raffinate, which should be approximate to zero according to the
separation goal. From Equations (3) and (4), we have Equation (6).

_S_
_y_ 1, _k_ = _t_ _xi_

#### + ∑ t xi,

_i_ = 2 _x_ 1, _k_ ⋅

1, _k_ _t_

_i_ , _k_

#### 1, k 1 + ∑ t xi,

(6)

#### 1 + ∑ i, k

_i_ = 2 _x_ 1, _k_ ⋅β 1/ _i_

_i_ = 2 _x_ 1, _k_ ⋅β 1/

_i_ = 2 _x_ 1, _k_ ⋅β 1/ _i_

Substituting the expression for _y_ 1, _k_ as in Equation (6) into Equation (5) gives Equation (7).

_S_
_x_ 1, _k_ + 1 = _t_ _xi_ , _k_ + _x_

1, _k_ + 1 _t_

_i_ , _k_

#### 1, k + 1 1 + ∑ t xi, k 1,1

(7)

= +
#### 1 + ∑ t xi, k

_i_ = 2 _x_ 1, _k_ ⋅β 1/ _i_

#### 1 + ∑ i, k

_i_ = 2 _x_ 1, _k_ ⋅β 1/ _i_

_x_ 1, _k_ ⋅β 1/

Thus the ASIR of component A1 at stage _k_ of the extraction section, denoted by _g_ 1, _k_, can be expressed as the
following equation:

#### g 1, k = xx 1,1, kk + 1 = x 1, k + S ∑ t xi, k + xx

_i_ = 2 β 1/ _i_

_k_ = _x_ 1, _k_ + 1 = _St_

#### 1, k = x 1,1, kk + 1 = x 1, k + S ∑ t xi, k + x 1,1,1 k

_i_ = 2 β 1/ _i_

+

= 1, _k_ + 1 = +
#### x 1, k x 1, k + ∑ t xi, k

(8)

#### 1, k x 1, k + ∑ i, k x 1, k

_i_ = 2 β 1/ _i_

_i_ =

_t_

_x_

_t_

_xi_, _k_, should have an unchanged value through the extraction stages with the
#### ∑ i = 1

As the total aqueous flowrate, _xi_, _k_
#### ∑ i = 1

_i_, _k_
#### ∑ i = 1

exception of stage 1 according to the constant-mixed-extraction-factor assumption mentioned before, and also
the cation exchange reaction takes place always between a more extractable ion in the aqueous phase and a less

extractable ion in the organic phase,

_t_ _xi_ , _k_ as well as _x_ 1, _k_ will decrease with stage from the central feed stage
### ∑ i = 2 β 1/ i

_i_ 2 β 1/ _i_

_t_ _xi_ , _k_
### ∑ = 2 β 1/ i

_x_

_i_ ,

towards the left end of the cascade, and consequently the ASIR of impurity A1 will exhibit an increasing tendency towards this direction.

In the scrubbing stages, the same analysis can give the expression for ASIR of impurity A _t_ as Equation (9) on
the basis of extraction equilibrium and mass balance in view of ( _k_ - 1) th stage and _k_ th stage where ( _n_ _+_ 1) _<_ _k_ ≤
( _n_ _+_ _m_ ) in Figure 2 .

#### gt, k = yyt, tk, k − 1 = ∑ t − 1 yiW, k + yt, k + yyt,

_i_ = 1 β _i_ / _t_

_t_, _k_ = _yt_ , _k_ − 1 = _t_ - 1 _y_ _W_ + _yt_ , _n_ + _m_

#### t, k = yt, tk, k − 1 = ∑ t − 1 yiW, k + yt, k + yt, nt, + k

_i_ = 1 β _i_ / _t_

− 1 = _W_ + _t_ , _n_ +

(9)

= _t_ , _k_ − 1 = +
#### yt, k ∑ t − 1 yi, k + yt, k

#### t, k ∑ i, k + yt, k yt, k

_i_ = 1 β _i_ / _t_

in which _yt_, _n_ + _m_ represents the flowrate of impurity A _t_ leaving the organic outlet in stage ( _n_ _+_ _m_ ). From the same
consideration as above, ASIR of impurity A _t_ should also present an increasing tendency with stage from the feed
stage to stage ( _n_ _+_ _m_ ) according to Equation (9).

ASIR presents an intuitive view of the status to remove an impurity in a separation cascade. Generally ASIR
should have a value larger than 1 in order to perform purification stage by stage. Nevertheless when the extracting solvent or scrubbing agent solution loses their concentration driving force to withdraw the impurity from the
opposite liquid phase at a certain stage, the value of ASIR will drop to 1 there. However, ASIR is always not

328

smaller than 1.

4. Derivation of the Equations of _S_ min and _W_ min

4.1. Principle Procedures of Deviation

F.-X. Cheng _et al._

For more accessible, the derivation procedures are outlined as the following before the details are presented.

Firstly, we need to know the conditions where both _S_ and _W_ have their theoretical minimum values to achieve
a specified separation duty as shown in Figure 1 . ASIR analysis can help to find out such conditions. When an _S_
lower than _S_ min is given to the cascade, the two following events will occur in extraction section if this section is
long enough: (i) more impurity A1 than required exists in the final raffinate; (ii) ASIR at some continuous extraction stages next to scrubbing section drops to 1. In this case, adding more extraction stages cannot decrease
A1 in the raffinate any longer. Then if _S_ is gradually increased, impurity A1 will become less and less in the final
raffinate, while the event (ii) will remain until impurity A1 completely disappears in the final raffinate. Therefore
it can be addressed that _S_ min should correspond to the _S_ causing impurity A1 in the final raffinate just removed to
the required level while the lowest value of ASIR in the extraction section still remaining at 1. Similarly, _W_ min
should be the _W_ that can just reduce impurity A _t_ in final extract to the required level, and meanwhile there are
some continuous stages with ASIR of 1 next to extraction section.

Secondly based on the above conditions, flowrate analysis is taken on the extraction and the scrubbing sections separately. We can deduct the expressions for both the aqueous and the organic flowrates of all components leaving stage _n_ except for impurity A1. The flowrates of each intermediate component are temporarily expressed with its flowrate in final raffinate which still remains unknown now. Similarly we will also deduct the
expressions for both the aqueous and organic flowrates of each component leaving stage ( _n_ + 1) except for impurity A _t_, and the flowrate expressions of each intermediate component here involve its unknown flowrate in the
final extract.

In the third step, we will consider the mass balance between stage _n_ and stage ( _n_ + 1) which connect the extraction and the scrubbing sections. As a result, we can obtain the quantitative relationships between the two
flowrate expressions of each intermediate component leaving stage _n_ and stage ( _n_ + 1) as well as the ratio of the
flowrates of each intermediate component leaving the two opposite ends consequently. Taking the mass balance
of each component in the two ends into account, the exiting flowrates of each component in the two ends can
then become known.

Finally the equations of _S_ min and _W_ min can be derived according to Equations (8) and (9) with the use of the
flowrate expressions obtained in the steps above.

4.2. Conditions of _S_ min and _W_ min

Mass balance of the (A1A2∙∙∙A _t_ −1)/(A2A3∙∙∙A _t_ ) separation duty suggests the quantitative relations as the following
expressions.

_xt_,1 ≈ _ft_

 _xt_,1 ≈ _ft_

 _yt_, _n_ + _m_ ≈
 _x_ 1,1 ≈ 0

 _y_ 1, _n_ + _m_ ≈

_t_,1 _t_

(10)

0 (11)

_t_,1

_y_

_x_

_t_, _n_ + _m_

_t_, _n_

1,1

0 (12)

_y_ 1, _n_ + _m_ ≈ _f_ 1

1, _n_ + _m_

1, _n_ + _m_ 1

(13)

_xi_,1 + _yi_, _n_ + _m_ = _fi_ ( 1 < _i_ < _t_

 _xi_,1 + _yi_, _n_ + _m_ = _fi_ ( 1 < _i_ <

_i_,1 _i_, _n_ + _m_ _i_

( 1 < _i_ < _t_ )

1 < _i_ < _t_ ) (14)

_i_,1 _i_,

where _xi_,1 and _yi_, _n_ + _m_ represent two exiting flowrates of A _i_ (1 ≤ _i_ ≤ _t_ ) in the aqueous and the organic streams, respectively. If stage _k_ is an extraction stage and has the ASIR value of 1, according to the definition of ASIR, we
know that

_x_ 1, _k_ = _x_ 1, _k_ + 1 (15)

And taking the mass balance of A1 represented by Equation (16) into account, we can obtain Equation (17).

_x_ 1, _k_ + 1 − _y_ 1, _k_ = _x_ 1,1 ≈ 0 (16)

_x_ 1, _k_ = _y_ 1, _k_ (17)

329

F.-X. Cheng _et al._

Equation (17) reveals that the aqueous and the organic flowrates of impurity A1 leaving one equilibrium extraction stage with ASIR of 1 should be equal to each other. In this case the extraction equilibrium relations between A1 and the other components in the _k_ th stage can be reduced as

β 1/ _i_ = _xy_ 1,1, _kk_ ⋅⋅ _yxii_,, _kk_ = _xyii_ ,, _kk_ ( 2 ≤≤ _i_ _t_ ) (18)

β 1/ _i_ = _xy_ 1,1, _kk_ ⋅⋅ _yxii_,, _kk_ = _xyii_ ,, _kk_ ( 2 ≤≤ _i_ _t_ )

_i_ = _x_ 1, _k_ ⋅ _yi_ , _k_ = _yi_ , _k_ ( 2

_k_ _i_, _k_ _i_, _k_

Moreover according to the constant-mixed-extraction-factor assumption, when an aqueous stream is used as
feed, the overall flowrates of all components in the aqueous and the organic phases should always be ( _f_ + _W_ ) and
_S_ respectively when leaving an extraction stage with the exception of 1 st stage. If it is also assumed that the extraction section of the cascade is long enough and there are at least ( _t_ - 2) continuous extraction stages including
the feed stage (stage _n_ ) with the same ASIR value of 1, a series of relationships can be noted as below in view of
mass balance within these stages.

 _t_
∑(β 1/ _i_ _yi_, _n_ ) = _f_ + _W_
 _i_ = 1
 _t_
∑ _yi_, _u_ + _j_ = _S_ 0 ( ≤ _j_ ≤− _t_ 3 )
 _i_ = 1
β 1/ _i_ _yi_, _u_ + _j_ - _yi_, _u_ + _j_ - 1 = _xi_,1 2 ( ≤≤ _i_ _t_ ;1 ≤ _j_ ≤− _t_

_t_

(β 1/ _i_ _yi_, _n_ )

β 1/

_t_
#### ∑

(19)

_yi_, _n_ ) = _f_ + _W_

1/ _i_ _i_, _n_

_i_ = 1

1/ _i_ _i_,

= 1

_i_ = 1

_t_

_yi_, _u_ + _j_ = _S_ 0 ( ≤ _j_ ≤− _t_

_t_
#### ∑

_i_, _u_ + _j_
_i_ = 1

0 ( ≤ _j_ ≤− _t_ 3 ) (20)

( ≤ _j_ ≤− _t_ 3 )

_i_,

= 1

_i_, _u_ +
_i_ = 1

β 1/

_yi_, _u_ + _j_ - _yi_, _u_ + _j_ - 1 = _xi_,1 2 ( ≤≤ _i_ _t_ ;1 ≤ _j_ ≤− _t_

+ _j_ _i_, _u_ + _j_

_i_ _i_, _u_ + _j_ _i_, _u_ + _j_ - 1 _i_

1/ _i_ _i_, _u_ + _j_ _i_, _u_ + _j_ - 1 _i_,1

2 ( ≤≤ _i_ _t_ ;1 ≤ _j_ ≤− _t_ 3 ) (21)

2 ( ≤≤ _i_ _t_ ;1 ≤ _j_ ≤− _t_ 3 )

where _u_ _=_ _n_ - _t_ + 3; expressions (20) and (21) are two collectives of ( _t_ - 2) and ( _t_ 2 - 4 _t_ + 3) equations respectively, and therefore expressions (19), (20) and (21) consist of a system involving ( _t_ 2 - 3 _t_ + 2) independent equations. Then we analyze the number of the unknowns involved in this equation system. As all of these ( _t_ - 2)
stages have ASIR value of 1, the aqueous flowrates of impurity A1 leaving these stages ( _i.e._, _x_ 1, _u_ + _j_ ) remain unvaried, and thus _y_ 1, _u_ + _j_ is also unvaried as it always equals _x_ 1, _u_ + _j_ in these ( _t_ - 2) stages. Moreover, the aqueous exiting flowrates of each component, _xi,_ 1 (1 ≤ _i_ ≤ _t_ ), will not change with time as the cascade is in a steady state, and
they can therefore be treated as constant terms. Thus in the equation system, only _yi,u_ + _j_ (2 ≤ _i_ ≤ _t_ ; 0 ≤ _j_ ≤ _t_ -3) are
regarded as unknown quantities, with the number of ( _t_ 2 - 3 _t_ + 2) which is just same as that of the independent
equations. Solving the equation system enables us to express all _yi,u_ + _j_ with the involved constant terms _W_, _S_, _β_ 1/ _i_
(2 ≤ _i_ ≤ _t_ ), _xi_,1(2 ≤ _i_ ≤ _t_ ) and _y_ 1, _n_ . Furthermore, when more continuous extraction stages than ( _t_ - 2) have ASIR
of 1, any continuous ( _t_ - 2) extraction stages among them should have a same group of solutions. To satisfy this
condition, it is necessary for the organic flowrate of each component to have the same value when leaving any
extraction stage involved in these stages with ASIR of 1, and accordingly the aqueous flowrate of each component also remains constant when leaving these stages, which means _yi,n_ = _yi,n_ −1 = ∙∙∙ = _yi,u_ −1 and _xi,n_ = _xi,n_ −1 = ∙∙∙ =
_xi,u_ −1 (1 ≤ _i_ ≤ _t_ ). Working the case using an organic stream as starting feed in the same manner, we can address
the same conclusion.

4.3. Separate Analysis of Extraction and Scrubbing Sections

Figure 3 displays the material flows of four continuous stages, where the aqueous or the organic feed will flow
into _n_ th stage or ( _n_ + 1) th stage in Figure 3, respectively. According to the theory of countercurrent extraction

[12], the stage receiving an aqueous feed is generally treated as an extraction stage, whereas the stage that an
organic feed enters is regarded as a scrubbing stage instead, as illustrated in Figure 1 .

When there are ( _t_ - 1) or more extraction stages with ASIR of 1 next to scrubbing section, the mass balance
on Interface I between ( _n −_ 1) th and _n_ th stages in Figure 3 can be expressed as the following three equations.

 _x_ 1, _n_ - _y_ 1, _n_ - 1 = _x_ 1, _n_ - _y_ 1, _n_ ≈ 0
 _xi_, _n_ - _yi_, _n_ - 1 = _xi_, _n_ - _yi_, _n_ = _xi_,1 ( 2 ≤≤− _i_ _t_

 _xt_, _n_ - _yt_, _n_ - 1 = _xt_, _n_ - _yt_, _n_ = _ft_

_x_ 1, _n_ - _y_ 1, _n_ - 1 = _x_ 1, _n_ - _y_

_n_ 1, _n_ - 1 1, _n_ 1, _n_

0 (22)

1, _n_ 1, _n_ - 1 1, _n_ 1,

- 1

_xi_, _n_ - _yi_, _n_ - 1 = _xi_, _n_ - _yi_, _n_ = _xi_,1 ( 2 ≤≤− _i_ _t_

_i_, _n_ _i_, _n_ - 1 _i_, _n_ _i_, _n_ _i_

_i_, _n_ _i_, _n_ - 1 _i_, _n_ _i_, _n_ _i_,1

( 2 ≤≤− _i_ _t_ 1 )

2 ≤≤− _i_ _t_ 1 ) (23)

_xt_, _n_ - _yt_, _n_ - 1 = _xt_, _n_ - _yt_, _n_ = _ft_

_t_, _n_ _t_, _n_ - 1 _t_, _n_ _t_, _n_ _t_

(24)

, _n_ _t_, _n_ - 1 _t_, _n_ _t_,

Considering that _xi,n_ and _yi,n_ in Equation (23) as well as _xt,n_ and _yt,n_ in Equation (24) should satisfy Equation
(18) by letting _k_ _=_ _n_, we can obtain ( _t_ - 1) equation systems, and each of them involves two unknowns _xi,n_ and
_y_ i, _n_ (2 ≤ _i_ ≤ _t_ ) as well as two independent equations. Solving these equation systems gives the expressions of the
aqueous and the organic flowrates of A2, A3, ∙∙∙, A _t_ leaving stage _n_ as follows.

330

Interface

III

F.-X. Cheng _et al._

Stage _n_ -1 Stage _n_ Stage _n_ +1 Stage _n_ +2

_n_ - 1

_n_ + 2

_y_



_y_

,1 _n_ + 1

,2 _n_ + 1

_n_ +

,2 _n_ +

,1 _n_ + 2

,2 _n_ + 2

_n_ +

,2 _n_ +

_n_ + 2

_y_,1 _n_

_n_ + 1

_n_ - 1

Extracting

solvent

_y_



_y_

,1 _n_ - 1

,2 _n_ - 1

_n_

,2 _n_

+ 2

- 1

_y_,1

_y_,2



_yt_

,2 _n_

_n_

,2 _n_

+ 1

_t_, _n_ +

_t_, _n_ + 1

_t_, _n_ +

_t_, _n_ + 2

_n_

_n_ + 1

_y_,1

_y_,2



_yt_,

_t_, _n_

_t_, _n_ - 1

_t_, _n_

_n_ + 1

_x_

_n_ + _x_ 1

_n_ + 1

_x_



_x_

_n_ - 1

_n_ + 2

_x_



_x_

,1 _n_ + 2

,2 _n_ + 2

_n_ +

,2 _n_ +

_n_ + 2

A, _n_ + 1

B, _n_ + 1

C, _n_ + 1

_x_



_x_

_n_ +

,2 _n_ +

_n_ +

+ 1

_n_

,2 _n_

,1 _n_ - 1

,2 _n_ - 1

_x_,1 _n_

水
相

_n_ - 1

Scrubbing
agent solution

,1 _n_ + _x_ 1

,2 _n_ + 1

_n_

_n_ + 1

- 1

_x_,2



_xt_

,2 _n_

_n_

,2 _n_

_t_, _n_

_t_, _n_ + 1

_t_, _n_

_t_, _n_ + 2

_t_, _n_

_t_, _n_ - 1

_t_, _n_

Extraction section Scrubbing section

Figure 3. Mass flows near feed stage.

⋅β 1/ _i_ (β 1/ _i_ - 1 )

β 1/ _t_ (β 1/ _t_ - 1 )

(β 1/ _i_ - 1 )
(β 1/ _t_ - 1 )

_xi_, _n_ = _xi_,1 ⋅β 1/ _i_ (β 1/ _i_ - 1 )
_xt_, _n_ = _ft_ ⋅β 1/ _t_ (β 1/ _t_ - 1 ) ( 2 ≤≤− _i_ _t_
_yi_, _n_ = _xi_,1 (β 1/ _i_ - 1 )
_yt_, _n_ = _ft_ (β 1/ _t_ - 1 )

 _xi_, _n_ = _xi_,1 ⋅β 1/ _i_ (β 1/ _i_ - 1 )
 _xt_, _n_ = _ft_ ⋅β 1/ _t_ (β 1/ _t_ - 1 ) ( 2 ≤≤− _i_ _t_
 _yi_, _n_ = _xi_,1 (β 1/ _i_ - 1 )
 _yt_, _n_ = _ft_ (β 1/ _t_ - 1 )

⋅β 1/ _i_ (β 1/
⋅β 1/ _t_ (β 1/ _t_

(β 1/ _i_ - 1 )
(β 1/ _t_ - 1 )

_i_, _n_ _i_,1 1/ _i_ 1/ _i_

_xt_, _n_ = _ft_ ⋅β 1/ _t_ (β 1/ _t_

( 2 ≤≤− _i_ _t_ 1 )

_i_, _n_ _i_,1 1/ _i_ 1/ _i_

_t_, _n_ _t_ 1/ _t_ 1/

_i_, _n_ _i_,1 1/ _i_

β 1/ _i_ - 1 ) (25)
β 1/ _t_ - 1 ) ( 2 ≤≤− _i_ _t_ 1 ) (26)

1 ) (27)
1 ) (28)

_i_, _n_ _i_,1 1/

_t_, _n_ _t_ 1/ _t_

_t_, _n_ _t_ 1/

Equations (25) and (26) specify the lowest requirement on the aqueous flowrate of each component leaving
the feed stage in order to reach its exiting aqueous flowrate as required respectively by Equations (23) or (24).

To derive the equation of _S_ min from Equation (8), we should turn to analyze the scrubbing section to obtain the
expressions for the flowrates of A1 as well as _xi_,1 (2 ≤ _i_ ≤ _t_ - 1). In scrubbing section, when there are more stages
than ( _t_ - 2) involving stage ( _n_ + 1) with ASIR of 1, the analysis in exactly the same manner as in extraction section can reveal that the aqueous and the organic flowrates of each component do not vary with stage in these
stages. Therefore the mass balance on interface III in Figure 3 can provide the relations below, similarly to Equations (22)-(24).

 _y_ 1, _n_ + 1 - _x_ 1, _n_ + 2 = _y_ 1, _n_ + 1 - _x_ 1, _n_ + 1 = _f_ 1
 _yi_, _n_ + 1 - _xi_, _n_ + 2 = _yi_, _n_ + 1 - _xi_, _n_ + 1 = _yi_, _n_ + _m_ ( 2 ≤≤− _i_ _t_

 _yt_, _n_ + 1 - _xt_, _n_ + 2 = _yt_, _n_ + 1 - _xt_, _n_ + 1 ≈ 0

_y_ 1, _n_ + 1 - _x_ 1, _n_ + 2 = _y_ 1, _n_ + 1 - _x_ 1, _n_ + 1 = _f_ 1

1, _n_ + 1 1, _n_ + 2 1, _n_ + 1 1, _n_

1, _n_ + 1 1, _n_ + 2 1, _n_ + 1 1, _n_ + 1 1

+ 1 1, _n_ + 2 1, _n_ + 1 1, _n_ + 1

(29)

_yi_, _n_ + 1 - _xi_, _n_ + 2 = _yi_, _n_ + 1 - _xi_, _n_ + 1 = _yi_, _n_ + _m_ ( 2 ≤≤− _i_ _t_

_i_, _n_ + 1 _i_, _n_ + 2 _i_, _n_ + 1 _i_, _n_ + 1 _i_, _n_ + _m_

( 2 ≤≤− _i_ _t_ 1 )

2 ≤≤− _i_ _t_ 1 ) (30)

_i_, _n_ + 1 _i_, _n_ + 2 _i_, _n_ + 1 _i_, _n_ + 1 _i_, _n_

+ 1 _i_, _n_ + 2 _i_, _n_ + 1 _i_, _n_ + 1 _i_, _n_ +

_yt_, _n_ + 1 - _xt_, _n_ + 2 = _yt_, _n_ + 1 - _xt_

_t_, _n_ + 1 _t_, _n_ + 2 _t_, _n_ + 1 _t_, _n_

_t_, _n_ + 1 _t_, _n_ + 2 _t_, _n_ + 1 _t_, _n_ + 1

+ 1 _t_, _n_ + 2 _t_, _n_ + 1 _t_, _n_ + 1

0 (31)

The expressions of extraction equilibrium between A _t_ and other components according to Equation (31) can
be simplified to result in

β _i_ / _t_ = _xyii_,, _nn_ ++ 11 ⋅⋅ _yxtt_ ,, _nn_ ++ 11 = _xyii_,, _nn_ ++ 11 ( 1 ≤≤− _i_ _t_

= _yi_ , _n_ + 1 ⋅ _xt_ , _n_ + 1 = _yi_ , _n_ + 1 ( 1 ≤≤− _i_ _t_ 1 ) (32)

_xi_, _n_ + 1 ⋅ _yt_, _n_ + 1 _xi_, _n_ + 1

_i_ / _t_ = _xyi_ , _n_ + 1 ⋅⋅ _yxt_ , _n_ + 1 = _xyi_ , _n_ + 1 ( 1 ≤≤− _i_ _t_ 1 )

β _i_ / _t_ = _xii_,, _nn_ ++ 11 ⋅ _ytt_ ,, _nn_ ++ 11 = _xii_,, _nn_ ++ 11 ( 1 ≤≤− _i_ _t_ 1

+ 1 _t_ , _n_ + 1 = _i_ , _n_ +

+ 1 ⋅ _yt_, _n_ + 1 _xi_, _n_ +

_i_, _n_ + 1 _t_, _n_ + 1 _i_, _n_

Furthermore, by constructing and then solving ( _t_ - 1) equation systems involving two independent equations
and two unknowns _xi,n_ +1, _yi_, _n_ +1 (1 ≤ _i_ ≤ _t_ - 1) respectively according to Equations (29), (30) and (32), we can obtain the expressions for the aqueous and the organic flowrates of A1, A2, ∙∙∙, A _t_ −1 leaving ( _n_ + 1) th stage as presented in Equations (33)-(36), similarly to Equations (25)-(28).

_x_ 1, _n_ + 1 = _f_ 1 (β 1/ _t_ - 1 )
_xi_, _n_ + 1 = _yi_, _n_ + _m_ (β _i_ / _t_ - 1 ) ( 2 ≤≤− _i_ _t_
_y_ 1, _n_ + 1 = _f_ 1 ⋅β 1/ _t_ (β 1/ _t_ - 1 )
_yi_, _n_ + 1 = _yi_, _n_ + _m_ ⋅β _i_ / _t_ (β _i_ / _t_ - 1 )

 _x_ 1, _n_ + 1 = _f_ 1 (β 1/ _t_ - 1 )
 _xi_, _n_ + 1 = _yi_, _n_ + _m_ (β _i_ / _t_ - 1 ) ( 2 ≤≤− _i_ _t_
 _y_ 1, _n_ + 1 = _f_ 1 ⋅β 1/ _t_ (β 1/ _t_ - 1 )
 _yi_, _n_ + 1 = _yi_, _n_ + _m_ ⋅β _i_ / _t_ (β _i_ / _t_ - 1 )

_n_ + 1 1 1/ _t_

1 ) (33)

_i_ / _t_ - 1 ) ( 2 ≤≤− _i_ _t_ 1 ) (34)

β 1/ _t_ - 1 ) (35)

/ _t_ (β _i_ / _t_ - 1 ) (36)

1, _n_ + 1 1 1/ _t_

_n_ +

β 1/

_i_, _n_ + 1 _i_, _n_ + _m_ _i_ / _t_

( 2 ≤≤− _i_ _t_ 1 )

_n_ + 1 _i_, _n_ +

1, _n_ + 1 1 1/ _t_ 1/ _t_

(β 1/ _t_ - 1 )

+ _m_ (β _i_ / _t_ - 1 )

β 1/ _t_ (β 1/ _t_ - 1 )

+ _m_ ⋅β _i_ / _t_ (β _i_ / _t_ - 1 )

+ _m_ (β _i_ / _t_ - 1 )

β 1/ _t_ (β 1/ _t_
+ _m_ ⋅β _i_ / _t_ (β _i_ /

_i_, _n_ + 1 _i_, _n_ + _m_ _i_ / _t_ _i_ / _t_

, _n_ + 1 _i_, _n_ + _m_ _i_ / _t_ _i_ / _t_

+ 1 _i_, _n_ +

331

F.-X. Cheng _et al._

4.4. Combining Analysis of Extraction and Scrubbing Sections

The relations of Equations (22)-(36) can apply to the cases of using either the aqueous feed or the organic feed.
Nevertheless, the different mass balance of all components on interface II in Figure 3 will occur if different feed
is used, as expressed in Equations (37)-(39) for aqueous feed and Equations (40)-(42) for organic feed, where 2
≤ _i_ ≤ _t_    - 1.

 _y_ 1, _n_ - _x_ A, _n_ + 1 = _f_ 1
 _yi_, _n_ - _xi_, _n_ + 1 = _yi_, _n_ + _m_

 _yt_, _n_ - _xt_, _n_ + 1 = _yt_, _n_ + _m_ ≈

_y_ 1, _n_ - _x_ A, _n_ + 1 = _f_ 1

_yi_, _n_ - _xi_, _n_ + 1 = _yi_,

_yt_, _n_ - _xt_, _n_ + 1 = _yt_,

1, _n_ A, _n_ +

1, _n_ A, _n_ + 1 1

_n_ +

(37)

_i_, _n_ _i_, _n_ + 1 _i_, _n_ + _m_

_yt_, _n_ - _xt_, _n_ + 1 = _yt_, _n_ + _m_

( for aqueous feed )

for aqueous feed ) (38)

_i_, _n_ _i_, _n_ + 1 _i_,

_t_, _n_ - _xt_, _n_ + 1 = _yt_,

+ 1 _i_, _n_ + _m_

+ 1 = _yt_, _n_ +

0 (39)

 _x_ 1, _n_ + 1 - _y_ 1, _n_ = _x_ 1,1 ≈
 _xi_, _n_ + 1 - _yi_, _n_ = _xi_,1

 _xt_, _n_ + 1 - _yt_, _n_ = _ft_

_x_ 1, _n_ + 1 - _y_ 1, _n_ = _x_ 1,1

_xi_, _n_ + 1 - _yi_, _n_ = _xi_

_xt_, _n_ + 1 - _yt_, _n_ = _ft_

(42)

_n_ + 1 1, _n_

1, _n_ + 1 1, _n_ 1,1

0 (40)

_n_ +

_i_, _n_ + 1 _i_, _n_ _i_,1

_xt_, _n_ + 1 - _yt_, _n_ = _ft_

_i_, _n_ + 1 _i_, _n_ _i_,1

( for organic feed )

for organic feed ) (41)

_t_, _n_ + 1 _t_,

In the case of using an aqueous stream as feed, the comparison of Equation (30) and Equation (38) gives

_yi_, _n_ = _yi_, _n_ + 1 (43)

and thus by substituting Equation (27) for _yi,n_ as well as Equation (36) for _yi,n_ +1 and rearrangement, we have

_xi_ ,1 = β 1/ _t_ − β _i_ / _t_ ( _i_ = 2,3, , _t_
_yi_, _n_ + _m_ β _i_ / _t_ - 1

_i_ ,1 = β 1/ _t_ − β _i_ / _t_

_n_ + _m_ β _i_ / _t_ - 1

= β 1/ _t_ − β _i_ / _t_ ( _i_ = 2,3, , _t_ - 1 ) (44)

β _i_ / _t_ - 1

_i_ ,1 = β 1/ _t_ − β _i_ / _t_ ( _i_ = 2,3, , _t_ - 1 )

_i_ ,1 = β 1/ _t_ − β _i_ / _t_

_i_, _n_ + _m_ _i_ / _t_

_i_ / _t_ ( _i_ = 2,3, , _t_ - 1

Equation (44), termed by us as optimal flowrate ratio equation of intermediate components, gives the optimal
ratio between its flowrates leaving two opposite outlets when both of _S_ min and _W_ min are applied to the
(A1∙∙∙A _t_ −1)/(A2∙∙∙A _t_ ) separation system using an aqueous feed. It can be found from Equation (44) that the optimal flowrate ratio of an intermediate component, such as A _i_, is determined only by _β_ 1 _/i_ and _βi/t_ (Note: _β_ 1 _/t_ = _β_ 1 _/I_ ×
_βi/t_ ), but independent of the separation factors of other pairs as well as the composition of the initial feed.

Further, by combining Equation (14) with Equation (44) and simultaneously solving them, we can immediately obtain the optimal flowrate equations of intermediate components as expressed by Equations (45) and
(46). The equations permit us to calculate the exiting flowrates of each intermediate component when both _S_ min
and _W_ min are given to the specified separation cascade.

 _xi_,1 = β 1/ _t_ − β _i_ / _t_ ⋅ _fi_
 β 1/ _t_ - 1 ( 2 ≤≤− _i_ _t_
 β _i_ / _t_ − 1
 _yi_, _n_ + _m_ = β 1/ _t_ - 1 ⋅ _fi_

β 1/ _t_ − β _i_ / _t_
β 1/ _t_ - 1

( 2 ≤≤− _i_ _t_ 1 )

_i_,1 = β 1/ _t_ 1/ _t_ - 1 _i_ / _t_ ⋅ _fi_

_yi_, _n_ + _m_ = ββ 1/ _i_ / _tt_ −− 11 ⋅ _fi_

_xi_,1 = 1/ _t_ _i_ / _t_ ⋅ _fi_

_i_,1 = β 1/ _t_ - _i_ / _t_

1/ _t_ ( 2 ≤≤− _i_ _t_

_yi_, _n_ + _m_ = β _i_ / _t_ − 1 ⋅ _fi_

+ _m_ = ββ 1/ _i_ /

, _n_ + _m_ = β _i_ /

1 _i_ / _t_ ⋅ _fi_ (45)

( 2 ≤≤− _i_ _t_ 1 )

− 1

- 1 ⋅ _fi_ (46)

From Equations (17) and (43), the relations _x_ 1, _n_ = _y_ 1, _n_ = _y_ 1, _n_ +1 can be easily found when using the aqueous feed,
and then taking Equation (35) into account can lead to Equation (47); introduction of the expression for _xi_,1 as in
Equation (45) into Equation (25) results in Equation (48); and for the convenience of comparison, Equation (26)
is listed after Equation (48) as below. It is worthwhile to notice from Equations (47), (48) and (26) that the
aqueous stream leaving the feed stage has a composition exactly the same as the initial aqueous feed.

 _x_ 1, _n_ = _f_ 1 ⋅β 1/ _t_ (β 1/ _t_ - 1 )
 _xi_, _n_ = _fi_ ⋅β 1/ _t_ (β 1/ _t_ - 1 ) ( 2 ≤≤− _i_ _t_

 _xt_, _n_ = _ft_ ⋅β 1/ _t_ (β 1/ _t_ - 1 )

_x_ 1, _n_ = _f_

_n_ 1 1/ _t_ 1/ _t_

1 ) (47)

1 ) ( 2 ≤≤− _i_ _t_ 1 ) (48)

1 ) (26)

( 2 ≤≤− _i_ _t_ 1 )

1, _n_ 1 1/ _t_ 1/ _t_

_xi_, _n_ = _fi_ ⋅β 1/ _t_ (β 1/ _t_ - 1 ) ( 2 ≤≤− _i_ _t_

_i_, _n_ _i_ 1/ _t_ 1/ _t_

_t_, _n_ = _ft_ ⋅β 1/ _t_ (β 1/ _t_

_i_, _n_ _i_ 1/ _t_ 1/ _t_

_t_, _n_ = _ft_ ⋅β 1/ _t_ (β 1/ _t_

_xt_, _n_ = _f_

β 1/ _t_ (β 1/ _t_

(β 1/ _t_ - 1 )
(β 1/ _t_ - 1 )
(β 1/ _t_ - 1 )

The analysis in the same manner will be applied when an organic feed is used. The combination of Equation
(23) with Equation (41) leads to

_xi_, _n_ = _xi_, _n_ + 1 ( 2 ≤≤− _i_ _t_ 1 ) (49)

Then the optimal flowrate ratio equation as well as the optimal flowrate equations for the separation system
using organic feed can be derived respectively as Equation (50) and Equations (51), (52) respectively, and the

332

F.-X. Cheng _et al._

organic flowrates of all components leaving the feed stage can be obtained and summarized as Equations (35),
(53), (54).

_xi_ ,1 = β 1/ _i_ − 1 ( 2 ≤≤− _i_ _t_
_yi_, _n_ + _m_ β 1/ _t_ - β 1/ _i_

_xi_ ,1 = β 1/ _i_ − 1 ( 2 ≤≤− _i_ _t_ 1

_i_ ,1 = β 1/ _i_ − 1

_n_ + _m_ β 1/ _t_ - β 1/

= β 1/ _i_ − 1 ( 2 ≤≤− _i_ _t_ 1 ) (50)

β 1/ _t_ - β 1/ _i_

_i_ ,1 = β 1/ _i_ − 1 ( 2 ≤≤− _i_ _t_ 1 )

_i_, _n_ + _m_ 1/ _t_ 1/ _i_

 _xi_,1 = β 1/ _i_ − 1 ⋅ _fi_
 β 1/ _t_ - 1 ( 2 ≤≤− _i_ _t_
 β 1/ _t_ − β 1/ _i_
 _yi_, _n_ + _m_ = β 1/ _t_ - 1 ⋅ _fi_

β 1/ _i_
β 1/ _t_

_xi_,1 = β 1/1/ _ti_ - 1 ⋅ _fi_

_xi_,1 = 1/ _i_ ⋅ _f_

_i_,1 = β 1/ _i_

1/ _t_

1
1 ⋅ _fi_ (51)

( 2 ≤≤− _i_ _t_ 1 )

1/ _t_ ( 2 ≤≤− _i_ _t_

_yi_, _n_ + _m_ = β 1/ _t_ − β 1/ _i_ ⋅ _fi_

+ _m_ = ββ 1/ _t_ 1/ − _t_ −β 11/

( 2 ≤≤− _i_ _t_ 1 )

_yi_, _n_ + _m_ = β 1/ _t_ 1/ _t_ - 11/ _i_ ⋅ _fi_

_i_, _n_ + _m_ = β 1/ _t_ - 1/

11/ _i_ ⋅ _fi_ (52)

 _y_ 1, _n_ + 1 = _f_ 1 ⋅β 1/ _t_ (β 1/ _t_ - 1 )
 _yi_, _n_ + 1 = _fi_ ⋅β 1/ _t_ (β 1/ _t_ - 1 ) ( _i_ = 2,3, , _t_ 
 _yt_, _n_ + 1 = _ft_ ⋅β 1/ _t_ (β 1/ _t_ - 1 )

_y_ 1, _n_ + 1 = _f_

_n_ + 1 1 1/ _t_ 1/ _t_

1 ) (35)

1 ) ( _i_ = 2,3, , _t_ - 1 ) (53)

1 ) (54)

( _i_ = 2,3, , _t_ - 1 )

1, _n_ + 1 1 1/ _t_ 1/ _t_

_yi_, _n_ + 1 = _fi_ ⋅β 1/ _t_ (β 1/ _t_

_yt_, _n_ + 1 = _ft_ ⋅β 1/ _t_ (β 1/ _t_

β 1/ _t_ (β 1/ _t_

_yi_, _n_ + 1 = _fi_ ⋅β 1/ _t_ (β 1/ _t_ - 1 ) ( _i_ = 2,3, , _t_

⋅β 1/ _t_ (β 1/

_i_, _n_ + 1 _i_ 1/ _t_ 1/ _t_

_t_, _n_ + 1 = _ft_ ⋅β 1/ _t_ (β 1/ _t_



_yt_, _n_ + 1 = _f_

(β 1/ _t_ - 1 )
(β 1/ _t_ - 1 )
(β 1/ _t_ - 1 )

β 1/ _t_ (β 1/ _t_

Equations (35), (53) and (54) clearly indicate that the composition of the organic stream leaving the feed stage
is exactly the same as that of the initial organic feed.

4.5. Derivation of _S_ min and _W_ min Equations

The equation of _S_ min for the (A1A2∙∙∙A _k_ )/(A _k_ +1A _k_ +2∙∙∙A _t_ ) separation using aqueous feed can be derived as Equation
(55) by introducing Equations (47), (48), (26) into Equation (8) and using the conditions of _g_ 1, _n_ = 1 and _x_ 1,1 = 0.

_t_

β _i_ / _t_

_t_ - 1

_t_ 1
#### = x 1, n + ∑ t xi, n = ∑ i = 1 β i / t fi + ft                (55)

_i_ = 2 β 1/ _i_ β 1/ _t_          - 1

#### S min = x 1, n + ∑ t xi, n = ∑ i = 1 β i / t fi + ft

#### 1, n + ∑ t xi, n = i = 1 i / t i t

β β

#### min = x 1, n + ∑ t xi, n = i = 1 i / t i

_i_ = 2 β 1/ _i_ β 1/ _t_              - 1

_i_ = 1

_i_ = 2 β 1/ _i_ β 1/ _t_

_i_ =

β 1/ _i_ β 1/ _t_

Then taking the mass balance in the organic outlet as given by Equation (56) into account, we will have the
expression for _W_ min as in Equation (57) by replacing _S_ min and _yi,n_ + _m_ in Equation (56) with Equation (55) and Equation (46) respectively and rearranging it, if the total feed flowrate is unity, that is, _f_ = _f_ 1 + _f_ 2 + ∙∙∙ + _ft_ = 1.

_t_

_t_ - 1

_t_ 1
#### − W min = f 1 + ∑ yi, n + m                 (56)

_i_ = 2

#### S min − W min = f 1 + ∑ yi, n + m

_i_ = 2

min min 1 _i_, _n_

_i_ = 2

_t_

_t_
#### ∑ fi

= _i_ = 1 = 1

β 1/ _t_ - 1 β 1/ _t_

#### ∑ fi

_W_ min = β _i_ 1/ = 1 _t_ - 1 = β 1/

_i_
_i_ = 1

(57)

min = β _i_ = 1

1

1 β 1/ _t_ - 1

_t_ 1/ _t_

1/ _t_ 1/ _t_

Also, Equation (57) can be derived by introducing the organic flowrates of all components leaving feed stage,
as expressed by Equation (35), (53) and (54) into Equation (9) and making _gt_, _n_ +1 = 1 as well as neglecting the
second item. Similarly the expressions of _S_ min and _W_ min for organic feed system can be derived as follows:

_t_

_t_
#### ∑ fi

= _i_ = 1 = 1

β 1/ _t_ - 1 β 1/ _t_

#### ∑ fi

_S_ min = β _i_ 1/ = 1 _t_ - 1 = β 1/

_i_
_i_ = 1

(58)

min = β _i_ = 1

_i_ 1

1 β 1/ _t_ - 1

_t_ 1/ _t_

1/ _t_ 1/ _t_

_t_

β 1/

_t_
#### f 1 + ∑β

= _i_ = 2

β 1/ _t_

#### f 1 + ∑β 1/ i fi

_W_ min = _i_ = 2

1/ _i_ _i_

_i_ = 2

(59)

1 1/ _i_
min = β _i_ = 2

1/ _t_ - 1

β 1/

_t_

5. Case Study

Calculation of Parameters

The results above were used to design a separation of an aqueous solution containing Ho, Er, Tm, Yb and Lu
with the (HoErTmYb)/(ErTmYbLu) separation configuration. Equal volumes of HEH(EHP) and refined kerosene, partly saponified by NaOH, consists of the starting organic phase, and 4.0 mol∙L −1 hydrochloric acid is

333

F.-X. Cheng _et al._

used as a starting scrubbing agent solution, as the common cases in industry. The separation factors [12] [19] between the neighboring components involved in the feed together with their feeding flowrates are listed in Table 1 .
The exiting aqueous flowrate of Lu and organic flowrate of Ho, _x_ Lu,1 and _y_ Ho, _n_ + _m_, were determined on the basis of
the total required purity (99.99%) of (HoErTmYb) in the exiting aqueous stream and that of (ErTmYbLu) in the
exiting organic stream, respectively. The exiting flowrates of all intermediate components could then be calculated
according to Equations (45) and (46) with the use of the data in Table 1 and summarized in Table 2 . Then _S_ min and
_W_ min could be calculated as 0.2639 and 0.0176 according to Equation (55) and Equation (57), respectively.

Self-developed software [22] [23] was used to conduct static simulation of the five-component separation described above. Based on mass balance and extraction equilibrium, the calculation was started from two opposite
ends stagewise towards centre simultaneously until there were enough stages where ASIR dropped to 1 for each
side. Figure 4 displays the varied compositions of the aqueous and organic streams in addition to ASIR with stage
through the cascade according to the calculation results. It can be seen from Figure 4(c) that ASIR in some stages
on both sides of the feed stage (95 th stage) has approached to 1 and accordingly the flowrates of all components do
not change within these stages as shown in both Figure 4(a) and Figure 4(b), which implies that the driving force
of both extracting solvent and scrubbing agent solution for separation has been used up here. Moreover also from
Figure 4(a) the rare earth composition in the aqueous stream near the feed stage is exactly same as that in the initial aqueous feed, which is also in accordance with the conclusion deducted earlier in the present article.

6. Conclusions

As the basic separation configuration of hyperlink extraction process, (A1A2∙∙∙A _t_ −1)/(A2A3∙∙∙A _t_ ) separation was

100

2.5

1.5

(a)

(b)

(c)

_xLu_
_x_ _~~Yb~~_
_xTm_
_~~x~~_ _Er_
_xHo_

_yLu_
_yYb_
_yTm_
_yEr_
_yHo_

1
0 20 40 60 80 100 120 140 160 180 200

stage

Figure 4. Variations with stage of (1) composition in aqueous phase; (2) composition in organic phase; (3)
ASIR.

Table 1. Parameters for a five-component separation.

_f_ Lu _f_ Yb _f_ Tm _f_ Er _f_ Ho

0.045 0.325 0.06 0.415 0.155

_β_ Lu/Yb _β_ Yb/Tm _β_ Tm/Er _β_ Er/Ho

1.78 3.56 3.34 2.73

334

F.-X. Cheng _et al._

Table 2. Calculated exiting flowrates of all components.

stage 1 flowrates stage ( _n + m_ ) flowrates

_x_ Lu,1 7.537 × 10 −5 _y_ Lu, _n+m_ 0.0450

_x_ Yb,1 0.1449 _y_ Yb, _n+m_ 0.1801

_x_ Tm,1 0.0514 _y_ Tm, _n+m_ 0.0086

_x_ Er,1 0.4024 _y_ Er, _n+m_ 0.0126

_x_ Ho,1 0.1550 _y_ Ho, _n+m_ 2.463 × 10 −5

_W_ min 0.0176 _S_ min 0.2639

discussed with regard to the determination of the minimum amounts of extracting solvent and scrubbing agent
solution ( _S_ min and _W_ min), and some conclusions had been drawn as follows:

(1) Both _S_ min in the case of using an aqueous feed and _W_ min in the case of using an organic feed can be expressed by feed composition together with the separation factors between the components involved, whereas
both _W_ min and _S_ min in the corresponding cases are only determined by the separation factor between the most and
the least extractable components, but independent of either separation factors of other pairs or feed composition.

(2) For any intermediate component, when _S_ min and _W_ min are applied to the cascade with enough stages, the
ratio of its flowrates leaving two opposite outlets is decided by an optimal flowrate ratio equation. This ratio can
be expressed only in terms of the separation factors between the intermediate component and the two end components, and do not involve separation factors of other pairs as well as the feed composition, in spite of the different forms when different kinds of feed are used.

(3) Whichever kind of feed is used, when _S_ min and _W_ min are applied to a cascade with enough stages, the
stream of same kind as feed should have the same composition as the initial feed when leaving the feed stage,
which provides a proof for the feed-stage-composition assumption proposed in the early version of the theory of
countercurrent extraction.

Acknowledgements

This research was supported by the National Basic Research Program of China (973) (2012CBA01200).

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