Rectifying Section Operating Line (ROL)

Consider the rectifying section as shown in the Figure below.
(System shows a total condenser and the reflux is at bubble point)

Material balance around the envelope shown:

IN = OUT

 Overall Balance (Total): Vn+1 = Ln + D Component Balance (MVC): Vn+1 yn+1 = Ln xn + D xD

Thus, we have

(Ln + D) yn+1 = Ln xn + D xD

Under constant molal overflow assumption:

L1 = L2 = .......... Ln-1 = Ln = Ln+1 = L = constant

V1 = V2 = .......... Vn-1 = Vn = Vn+1 = V = constant

The subscripts can be dropped. Thus, the equation simplifies to:

(L + D) yn+1 = L xn + D xD

Re-arranging in the form y = f(x), we have

Introducing Reflux Ratio R = L / D,

This is the Operating Line Equation for the rectifying section, or ROL in short.

Characteristics: Straight Line Equation

slope R/(R+1), constant for given value of R
Intercept (1/R+1) xD, constant for given R and purity of distillate xD

In addition, if xn = xD, then yn+1 = xD .

Thus, the operating line passed through the point (xD , xD) on the 45o diagonal line.

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The operating line connects the concentrations of the more volatile component in the vapour and liquid between 2 adjacent phases. By plotting the operating line on the equilibrium curve, we can graphically construct using the "staircase" construction to determine the number of theoretical stages required for the separation in the rectifying section. See the Figure below.

The method starts off by identifying the point xD on the diagonal. Starting from (xD, y1) on the diagonal (note: y1 = xD), draw a horizontal line to the left until it touches the equilibrium curve: this gives the point (x1, y1).

From this point (x1, y1) draw a vertical line down to the ROL: this gives the point (x1, y2). In this manner we had obtained one triangle (no.1) where the horizontal distance is (xD - x1) and the vertical distance is (y1 - y2). One triangle is equivalent to one theoretical tray. See the Figure below.

The difference (xD - x1) represents the decrease in the concentration of the more volatile component in the liquid phase as its moves down one tray, i.e. from tray 1 to tray 2. The difference (y1 - y2) represents the increase in the concentration of the more volatile component in the vapour phase as its moves up one tray, i.e. from tray 2 to tray 1. See the Figure below:

Usually the subscripts "n+1" and "n" are not shown: