Representation on the x-y plane
Prof. Dr. Marco Mazzotti - Institut für Verfahrenstechnik
Introduction
We have seen that the mass balance equation of a process like absorption is called operating line and can
be represented on the x-y plane. Such a representation can be of great interest for the understanding and
resolution of the problem.
The operating line is usually used in combination with the equilibrium line, in order to calculate the number
of ideal stages that a certain operation requires. The equilibrium line is the representation of the
equilibrium relation for a given temperature and pressure.
y
x
Types of representations
When we talk about x-y plane, we mean gas composition against liquid composition. Because all of them
describe the equilibrium between liquid and gas phase, we choose the one which is the best for our case.
x-y plane
X-Y plane
x-p plane
Molar fractions
Molar ratios
Molar fraction &
partial pressure
y ,x 
mol i
Y,X 
n
 mol
mol i
mol n
n : inert
i 1
X
x
1 x
Y
y
1 y
Y
y
x
p
X
x
Operating line equations
The operating line equation expressed in terms of the molar fractions can be derived as we have seen before,
from the mass balance to the column. For instance, in an absorption column:
Lo xo  Gn1 y n1  L x  G y

y   (x )
In the general case, the operating line is a curve. Nevertheless a straight line can be obtained when the gas
and liquid molar flow-rates are constant or can be considered constant.
y
This is a straight
operating line
This is a curve
operating line
x
Working with molar fractions
When the absorption involves very low concentrations in both phases, the gas and liquid molar flow-rates can
be taken as constant along the column. In this case, the mass balance and operating line become:
L xo  G y n1  L x  G y

L  L

y   y 0  x0   x
G  G

This equation corresponds now to a straight line of slope (L/G). When working with molar fraction, the operating
line is only a straight line when the molar flow-rates are approximately constant.
Working with molar ratios
When the assumption of constant molar flow-rates cannot be taken, it is convenient to use the molar ratios.
Doing so, the mass balance appears as:
l Xo  g Yn1  l X  g Y


 l
l
Y  Y0  X 0   X
g

 g
Where l and g are the molar flow-rates of inert in the liquid phase (solvent) and in the gas phase (carrier gas all components but the solute-). Because the molar flow rate of inert is constant for both phases by definition,
when working with molar ratios, the operating line is always a straight line.
The equilibrium line
The equilibrium line can be expressed in general as:
y  f (x )
For a range of low concentrations (small values of x and y), the equilibrium line can be often represented by a
straight line:
Of course, other ways of expressing the composition may be used, as molar ratios and partial pressure. When
the equilibrium line is a straight line in the x-y coordinates, it will happen that it will be a curve in the X-Y plane.
y
y
Y
y = f (x)
y=mx
x
Y= f (X)
x
X
Resume. Example: absorption case
x-y
X-Y
x-p
O.L: Straight line
O.L: (Always) straight line
O.L: Straight line
L xo  G y n1  L x  G y
l Xo  g Yn1  l X  g Y
E. L: a curve, but often a
straight line
E. L: if a straight line in x-y
coordinates, the a curve in
the X-Y plane
E. L: a curve, but often a
straight line
y  mx
Y  f ( X )
p  mx
O.L: A curve
O.L: (Always) straight line
O.L: A curve
Lo xo  Gn1 y n1  L x  G y
l Xo  g Yn1  l X  g Y
L xo  pT ,n1H pn1  L x  pT H p
L xo  pT H pn1  L x  pT H p
x, y small
G, L  cte
x, y small
G, L  cte
y   (x )
p    (x )
E. L: a curve
E. L: a curve
E. L: a curve
y  f (x )
Y  f ( X )
p  mx