Separation of Ions with Similar Properties and Isotopes in an Ion Exchanger–Solution System under Conditions of a Moving Ion Exchanger with an Applied Constant Electric Field
Abstract Generated abstract
This paper develops a theoretical model for separating ions with similar properties, including isotopic ions, in an ion exchanger solution system subjected to a constant electric field while both the solution and granular ion exchanger phases are in motion. Treating the apparatus as a two conductor, two phase countercurrent electromigration system, the authors formulate flux balances including electric migration, medium transport, and diffusion, with equilibrium assumed between phases across the column. The resulting differential equation yields an expression for the stationary concentration distribution of the light component along the column under nonselective operation. The analysis also derives a formula for the height equivalent to a theoretical plate, providing a means to estimate separation efficiency when the ion exchanger phase is mobile.
Full Text
UDC 621.039.3
SEPARATION OF IONS WITH SIMILAR PROPERTIES AND ISOTOPES IN THE ION EXCHANGER—SOLUTION SYSTEM UNDER CONDITIONS OF A MOVING ION EXCHANGER WITH AN APPLIED CONSTANT ELECTRIC FIELD
I. A. Tikhomirov, A. P. Vergun, L. A. Avdeev,
V. V. Larionov
(Presented by the scientific seminar of the Faculty of Physics and Engineering)
The theory of the process of separating ions with similar properties and isotopes during electromigration in the ion exchanger—solution system under conditions of motion of both phases of the system is considered.
The solution of the proposed equation of the process gives an expression for estimating the distribution of ions along the column, and also makes it possible to find the value of the equivalent theoretical plate (ETP).
Illustrations 1, references 4.
A number of papers [1, 2] have been devoted to the study of the process of separating isotopic ions in the ion exchanger—solution system with the application of a constant electric field, where this process is considered as countercurrent electrochromatography. These works also note that, at a concentration of the feed solution \(c_0 < 0.3\), the fraction of ions transferred through the ion exchanger increases, and since counterflow is effected only through the solution, the efficiency of the separation process decreases.
In our work, the process of separating a binary mixture of isotopic ions is considered under conditions of counterflow of both the solution and the ion exchanger; moreover, at the initial moment of time the ion exchanger is in the salt form in equilibrium with the feed solution. Such an organization of the process makes it possible to regard it as ion separation in countercurrent electromigration in a two-phase system. Here the granular ion exchanger performs the following functions: it serves as a packing that reduces convective mixing, and as the medium in which electromigration of the ions being separated takes place. On this basis, the separation device is treated as a two-conductor model. The fraction of solution in a unit cross section \((\gamma)\) depends on the sizes of the granules and on the porosity of the ion exchanger. For the concentration of the feed solution under consideration, \(c_0 > 0.1N\), processes at the ion exchanger—solution boundary may be neglected [3, 4]. Proceeding from the above model (Fig. 1), we formulate the differential equation of this process
\[ I=\gamma I_1+(1-\gamma)I_1', \tag{1} \]
where \(I\) is the flux of the light component through the device, and \(I_1\) and \(I_1'\) are the fluxes of the light component through the solution and ion-exchanger phases, respectively.
The ion flux in each phase is composed of the following quantities: the flux under the action of the electric field \(I_E\); the transport of moving
by the medium \(I_W\) and the flux due to concentration diffusion \(I_D\). The values of the indicated quantities are determined by the relations:
\[ I_{E_1}=c_1v_1,\qquad I'_{E_1}=m_1v'_1\div I_{w_1}=c_1w_1,\qquad I'_{w_1}=m_1w'_1;\qquad I_{D_1}=D_1\frac{dc_1}{dx}; \]
\[ I'_{D_1}=D'_1\frac{dm_1}{dx}, \]
where \(C_1\) and \(m_1\) are the concentrations of the light component in the solution and in the ion exchanger, respectively; \(v_1\) is the velocity of motion of the ion under the action of the electric field, \(w_1\) is the velocity of motion of the medium, \(D_1\) is the diffusion coefficient; subscript 1 indicates that the quantity refers to the light component; primed quantities characterize the ion-exchanger phase.
Fig. 1. Scheme of fluxes in the separation of ions by countercurrent electromigration in the ion exchanger–solution system: hatched area—ion-exchanger phase; unhatched area—solution phase.
The positive direction is taken to be from the anode to the cathode. Then, according to the flux scheme shown in Fig. 1, the expression for \(I\) is written as:
\[ I=\gamma\left[(v_1-w_1)c_1-D_1\frac{dc_1}{dx}\right]+ \]
\[ +(1-\gamma)\left[(v'_1-w'_1)m_1-D'_1\frac{dm_1}{dx}\right]. \tag{2} \]
When considering the equation, we assume that the ion concentration in the solution phase \(c_0=c_1+c_2\) and in the ion-exchanger phase \(m_0=m_1+m_2\) does not vary along the length of the separation device. It follows from this that the electric-field strength in the device is also a constant quantity. On the basis of the assumption of constant concentration, we obtain \(I_1+I_2=0\) and \(I'_1+I'_2=0\). Expanding these expressions and solving them with respect to \(w\) and \(w'\), we obtain
\[ w=\frac{1}{c_0}(v_1c_1+v_2c_2)-\frac{1}{c_0}\left(D_1\frac{dc_1}{dx}-D_2\frac{dc_2}{dx}\right), \tag{3} \]
\[ w'=\frac{1}{m_0}(m_1v'_1+m_2v'_2)-\frac{1}{c_0}\left(D'_1\frac{dm_1}{dx}-D'_2\frac{dm_2}{dx}\right). \tag{4} \]
Substituting (3) and (4) into (2) and introducing the notation
\[ v_1-v_2=\Delta v;\qquad v'_1-v'_2=\Delta v';\qquad \frac{\Delta v'}{\Delta v}=p;\qquad \frac{m_1}{m_0}=\mu; \]
\[ \frac{c_2}{c_0}D_1-\frac{c_1}{c_0}D_2=D;\qquad \frac{m_2}{m_1}D'_1-\frac{m_1}{m_0}D'_2=D', \]
we obtain
\[ I=\gamma \Delta v\,\frac{c_1(c_0-c_1)}{c_0} -\gamma \frac{1}{c_0}D\frac{dc_1}{dx} -(1-\gamma)p\Delta v(\mu-\mu^2) -(1-\gamma)D'\frac{d\mu}{dx}. \tag{5} \]
Proceeding from the condition of equilibrium between the ion exchanger and the solution in the column cross section, we can write
\[ (1+\varepsilon_2)\frac{c_1}{c_0-c_1}=\frac{\mu}{1-\mu}, \tag{6} \]
where \(\varepsilon_2\) is the enrichment coefficient in ion exchange.
Since \(\varepsilon_2\ll 1\), expanding expression (6) in a series and neglecting quantities of second order of smallness, we obtain
\[ c_1=c_0\bigl[(1-\varepsilon+\varepsilon^2)\mu+(\varepsilon-2\varepsilon^2)\mu^2\bigr]. \tag{7} \]
Substituting (7) and introducing the notation
\[ \gamma c_0\Delta v(1-\varepsilon_2)+(1-\gamma)m_0\Delta v p=A, \]
\[ \gamma c_0\Delta v(1-3\varepsilon_2)+(1-\gamma)m_0\Delta v p=B. \]
\[ \gamma c_0D+(1-\gamma)m_0D'=K, \]
we obtain the expression for \(I\) in the form
\[ I=A\mu-B\mu^2-K\frac{d\mu}{dx}. \tag{8} \]
We shall seek the solution of equation (8) for the case of a nonselective mode of operation of the separating device. The boundary conditions in this case may be written in the form
\[ \mu\big|_{x=0}=\mu_0,\qquad I\big|_{x=L}=0. \]
For the stationary case \((I=0)\), under the given boundary conditions the solution of equation (8) is written as
\[ \mu=\mu_0\frac{A}{B\mu_0+(A-B\mu_0)\exp\left(-\frac{A}{K}x\right)}. \tag{9} \]
One of the principal characteristics of a separating device is the height equivalent to a theoretical plate (HETP). In the stationary state the separation \((Q)\) achieved in the column is determined as
\[ Q=(1+\varepsilon)\frac{L}{\text{HETP}}, \]
where
\[ Q=\frac{\mu/(1-\mu)}{\mu_0/(1-\mu_0)},\qquad \varepsilon=\gamma\varepsilon_1+(1-\gamma)\varepsilon_1', \]
\(L\) is the column length. Hence
\[ \text{HETP}=L\frac{\ln[1+\gamma\varepsilon_1+(1-\gamma)\varepsilon_1']}{\ln\left(\dfrac{\mu}{1-\mu}\bigg/\dfrac{\mu_0}{1-\mu_0}\right)}. \tag{10} \]
The expressions obtained, (9) and (10), make it possible to estimate the efficiency of separation of ions with similar properties and isotopes under conditions of electromigration with motion of the ion-exchanger phase.
REFERENCES
- M. N. Kurin, I. A. Tikhomirov, G. S. Tikhonov, ZhFKh, 40, 1966, No. 9.
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- N. P. Gnusin, V. D. Grebenyuk, ZhFKh, 39, 1965, No. 12.
- G. V. Samsonov, E. B. Trostyanskaya, G. E. Elkin, Ion Exchange. Sorption of Organic Substances. “Nauka,” 1969.