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.

Figure 1 displays a typical countercurrent extraction cascade unit with the separation configuration of (A₁A₂∙∙∙A_t−1)/(A₂A₃∙∙∙A_t), where t ( ≥ 3) means the number of components and A₁, A₂, ∙∙∙, 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 f₁, f₂,∙∙∙, and f_t (hereafter in mole per unit time) respectively all the time, where f₁ + f₂ + ∙∙∙ + f_t = f. The separation factor of the neighboring A_i/A_i+1 (t > i ≥ 1) pair is denoted as β_i/i+1; the separation factor of A_i/A_i+j pair (t > i ≥ 1; t - i ≥ j ≥ 1), β_i/i+j, can be calculated according to:

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).

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.

where g denotes ASIR. The subscripts i and k represent an impurity and a stage, respectively. The g_i,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 x_i,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).

(4)

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).

Substituting the expression for y_1,k as in Equation (6) into Equation (5) gives Equation (7).

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:

As the total aqueous flowrate, , should have an unchanged value through the extraction stages with the

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, as well as x_1,k will decrease with stage from the central feed stage

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.

in which y_t,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 smaller than 1.

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 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.

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

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.

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

Fu-XiangCheng,ShengWu,Song-LingWang,BoZhang,YanLiu,Chun-ShengLiao,Chun-HuaYan, (2015) Minimum Amounts of Extracting Solvent of (A₁A₂&middot;&middot;&middot;A_t-1)/(A₂A₃&middot;&middot;&middot;A_t) Countercurrent Extraction Separation. Advances in Materials Physics and Chemistry,05,325-336. doi: 10.4236/ampc.2015.58032