McCabe-thiele For Rectification Section Operating Line which include derivation and steps

1. McCabe Thiele Method (Relevance of
Rectification Section Operating Line)
SIMULTANEOUS HEAT AND MASS TRANSFER
Presented By Haroon Zulqernain

2. RECTIFYING SECTION OPERATING LINE (ROL)
Consider the rectifying section as shown in the Figure below.
Material balance:
Overall Balance (Total): Vn+1 = Ln + D
Component Balance (MVC): Vn+1 yn+1 = Ln xn + D Xd
By Substitution,
(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

3. The Equation is simplifies to:
(L + D) yn+1 = L xn + D xD
Re-arranging in the form y = f(x), we have
yn+1 = (
𝐿
𝐿+𝐷
) 𝑥𝑛 + (
𝐷
𝐿+𝐷
) xD
As we know Reflux ratio = R = L/D , we are Multiplying and Dividing “D” on
Right-side of the above Equation
yn+1 = (
𝑅
𝑅+1
) 𝑥𝑛 + (
1
𝑅+1
) xD
This is the Operating Line Equation for the rectifying section, or ROL in short.
By comparing with line equation ( y = mx + c) we will get:
Slope of ROL = (
𝑅
𝑅+1
)
Intercept of ROL = (
1
𝑅+1
) xD

4. McCabe-Thiele Method
This method uses the equilibrium curve diagram to determine the number
of theoretical stages (trays) required to achieve a desired degree of
separation. It is a simplified method of analysis making use of several
assumptions, but none the less a very useful tool for the understanding of
distillation operation.
We will illustrate it clearly by applying the McCabe-Thiele method to
design a binary column.

5. McCabe Thiele Method assumes constant molar overflow and this implies that:
1) molal heats of vaporisation of the components are roughly the same.
2) heat effects (heats of solution, heat losses to and from column, etc.) are
negligible.
3) for every mole of vapour condensed, 1 mole of liquid is vaporised.

6. THERMAL CONDITIONS OF FEED
The parameter q is the mole fraction of liquid in the feed and the slope of the
q-line is q / (q – 1).
Superheated Vapour, q < 0
Saturated Vapour, q = 0
Partially Vaporised, 0 < q > 1
Saturated Liquid, q = 1
Cold Liquid, q > 1

7. GRAPHICAL METHOD OF McCabe
 The Point intersection of diagonal (y=x) and rectifying section operating line is (
xD , xD ).
 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.
 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.

8. (
1
𝑅+1
) xD