Columns with rotating elements

Columns provided with built-in, rotating elements were first proposed in 1925. In view of the results of Jost et al. [73] the question has arisen whether this tyj)e of column really offers so great advantages over plate and packed columns as well as over columns with stationary elements that further development would seejii promis-iug. 

Their evolution was prompted by the need to improve the separating effect without increasing the hold-up and the pressure drop. At that time an efficient column was lacking for certain classes of work, particularly for charges of 1—5 g. 

The columns belonging to this group can be subdivided into the following types (cf. chap. 5.1.1)  

Spinning hand columns, containing a flat, spiral or cross-shaj)ed rotating metal strip (Fig. 282)  

Rotating concentric-tube columns, having a stationary and a rotating cylinder separated by a gap of 1—2 inin (Fig. 286)  

---

The development of distillation columns with rotating elements, described by Baker et al. (5) and Willingham et al. (70), indicates a trend which probably will be followed in future developments to reduce the time required to reach equilibrium and hence the time for an efficient fractionation. Hickman s (27) type of molecular distillation will certainly acquire increasing importance for analytical uses as one becomes more concerned with the higher boiling constituents. 

Starting from the hydrodynamic model of the liquid film generated mechanically in columns with rotating elements and using simplifying assumptions Dietz et al. [lid] deduced an equation for the calcidation of film thicknesses in the range 0 < film thickness < gap width ... 

Data of columns with rotating elements (of. also Table 37)... 

This table shows that columns with rotating elements are clearly superior as r ards pressure drop, since the values for packed columns (and even more those for plate columns) may lie several orders of magnitude higher. 

Equation (4.11) describes the transformation of coordinates under an anticlockwise rotation by an angle, 0. This coordinate transformation is completely characterized by a square matrix. A, with the elements cos 0 and + sin 0, and the column matrices, r and r, involving the initial and final coordinates, respectively ... 

The first column with a rotating element, constructed by My ers and Jones, was intended for laboratory use and contained conical segments. Urey and Huffniann... 

The other characteristics of rotating columns are comparable to those of concentric-tube and multiple-tube columns, and of columns with wire-gauze or wire helix elements (sections 7.3.1 and 7.3.4). 

In the language of linear algebra, these operations rotate A and B, with T referred to as a transformation matrix or a rotation matrix. Since any T will leave n unaltered, each of the elements of T must be specified to define a unique solution. Normalization of each column determines F elements, leaving F F - 1) parameters undefined. Extra information is necessary to determine these parameters. This is known as the rotation problem. Methods for addressing the rotation problem are discussed in Section IV. 

The elements of structured packing are installed in layers in the column, with cate being taken that the adjoining elements of adjacent layers are otiemed such dut the liquid flow direction is reversed and that gas is redistributed. For the simple pilot plant elenwnt shown in Fig. S.8-4, adjacent layers are usually rotated by 90. ... 

The next step in the whole process is to extract another 2 x 2 to diagonalize. In this case, the 2x2 has to be that in the second and third rows and columns. With this second transformation matrix designated V, the rotation angle and the elements of V are... 

The procedure is schematically shown in Fig. 34.29. Equation (34.10) represents a homogeneous system of equations with a trivial solution r, = 0. Because component / is absent in the concentration vector, this component does not contribute to the matrix T °. As a consequence the rank of T is one less than its number of rows. A non-trivial solution therefore can be calculated. The value of one element of r, is arbitrarily chosen and the other elements are calculated by a simple regression [17]. Because the solution depends on the initially chosen value, the size (scale) of the true factors remains undetermined. By repeating this procedure for all columns c, (t = 1 to p), one obtains all columns of R, the entire rotation matrix. 

It is easy to check that the rows and columns of this matrix are orthogonal and its determinant equals unity. The independent complex matrix elements in eq. (3.45) are known as Cayley-Klein parameters of the rotation group. Also, one can see that for quaternions connected by the relation r = ri o r2 the corresponding 2x2 matrices are connected by the same relation with replacement of the quaternion multiplication by the usual matrix product. This establishes isomorphism between the SU(2) group and the group of normalized quaternions HP which can be continued to the homomorphism on 50(3). 

---