18.08.2015   NUCLEAR CHEMICAL ENGINEERING

NUMBER OF THEORETICAL STAGES IN EXCHANGE COLUMNS

This section derives a general equation for the dependence of stream compositions in an exchange column on stream flow rates and number of equilibrium stages. In Fig. 13.22, the vapor flow rate is V kg-mol of exchangeable element in unit time, and the liquid flow rate is L in the same units. In the present simplified derivation these flow rates are treated as constant throughout the column. Vapor compositions у and liquid compositions x are expressed as atom fraction of desired isotope of exchangeable element. To keep the derivation simple, atom fractions are to be restricted to values below 0.05, as are found in the large stages of plants to concentrate deuterium, 13 C, 15 N, or 18 О from the natural element.

The equilibrium relation between vapor and liquid leaving stage і is

*, = <W — (13.79)

where a is the separation factor. This convention is used to make a greater than unity for exchange separation of deuterium, the example of greatest practical importance.

The material-balance equation for the section between the top of the column and the top of stage і is

(*i-i — x0)L

Hence

y, xQ Atom fraction

V L Molal flow rote

t

I

I

і 11

Stage I

Stage n

yn+i *„ Atom fraction

V L Molal flow rote Figure 13.22 Flow rates and compositions in exchange

Vapor Liquid column.

xi = ixi-1 -*o) (jr) + «Pi

(13.81)

This is a first-order difference equation in x whose general solution is

xt=ABijrZ

(13.82)

Substitution into (13.81) gives

Aff + Z = (AB1-1 +Z-X o) ^ + ay t

(13.83)

For this to hold, it is necessary that

B- у

(13.84)

„ cix0L/V-yi)

311(1 2= aL/V-l

(13.85)

A is obtained from (13.82) with і = 0,

x0=A + Z

(13.86)

so that

, „ ayi ~Xq 0 aL/V-.

(13.87)

(ayi — x0)(cxL/Vy + ajxoL/V-yi)
aL/V- 1

(ayі -*оХ«£/Ю” +Фо^/У~Уі)

» aL/V-1

_ ~XoYaL/VT +x0]L/V-yi ~ aL/V-1

<*Уп+1 — X» = (ayi ~*o)(y)

In 1(<*Уп+1 — хп)І(аУі — x0)]

П In (aL/V)

From (13.82),

This is the general equation for the number of theoretical stages needed in exchange columns. It is a form of the Kremser [K5, S5] equation, derived originally for gas absorption.