## Constant Optimum Zone Length in Each Pass

For this case, we have dYn/dZ = 0 but Yn_1 O Yn. Eqns (1)-(7) are applicable if Yn(Z) is substituted by Yn. The optimal zone lengths Y% for each pass in n-passes were obtained by following the same calculation procedure described in previous sections. The results are shown in Figure 4.

Figure 2 Numerical values of the optimal variable zone length for each pass.

burettes, of equal internal area a, each with a stopcock, connected in parallel to a common horizontal tube leading to the zone tube. The solute concentration at any point is represented by the height of the liquid level in the tube. Initially, the zone tube is Zone refining region, 0 ) x ) (L — l) Figure 5 emptied and ingot tubes are filled to height h0. The shows a zone refining simulator for k < 1. The ingot number of open tubes, m, corresponds to the zone is operated by an array of vertical tubes, such as length.

Separation Theory in Analogue Simulators

Zone Refining Simulator for k< 1

Figure 3 Numerical values of the constant optimum zone length for all passes.
Figure 4 Numerical values of the constant optimal zone length for each pass.

The area az of the zone tube determines the value of k that is operated, in accordance with: ma(1 — k)

az" K

The liquid level in tube i after the nth pass is: (m — 1)ahn-1 # ahm~#i-x # a^-i

It can be seen that the horizontal tube in conjunction with the zone tube performs the essential operations of a travelling molten zone: namely, taking in solute at the leading end; remixing it with the solute in the zone; and freezing out. At the trailing end, a solute concentration is k times that in the zone. In the analogue simulator, as the molten zone advances, it opens the next tube in line, produces the same liquid height in the zone tube and the empty tube connected to it, and then leaves this height in the closed off tube.

Figure 5 Hand-operated liquid-level zone refining simulator for k< 1

Normal freezing region, (L — l) ) x )L Since after the front of the zone reaches the end of the ingot, at (x = L — l), the zone length l as well as the number of ingot tubes m, operating the zone length, is no longer constant. In order to keep the distribution coefficient k constant in this section, the cross-sectional area aZii of the zone tube must be readjusted continuously according to eqn (12) as:

A zone pass is performed by continuously readjusting the cross-sectional area of the zone tube according to eqn (14) after closing each rearmost tube. At the end of each pass the total amount of the liquid in the zone tube can readily be forced into the last tube, which is then closed as the zone leaves the ingot.

The liquid level in tube i after the nth pass is:

hn = (N — i)a + az> i ' (N — i)a + az, i i = N — m + 1, N — m + 2,..., N — 1

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