The feed (

) flow rate, solvent (

) flow rate and feed composition (select one of three compositions) are given. The desired raffinate composition is specified, and the extract composition and the number of stages to obtain the desired raffinate composition are determined by mass balances. First, the mixing point composition is calculated and located on the ternary phase diagram. The mixing point corresponds to the composition that would be obtained if the feed and the solvent flows were mixed together:

,

solute

:

(because the solvent feed contains no solute),

solvent

:

,

carrier

:

.

Here

is the combined feed and solvent flow rate (kg/hr) and

,

and

refer to the mass fractions of solute, solvent and carrier in

(feed

, solvent

and mixed

), which is the stream fed to the cascade.

The mixing point

is located on the phase diagram using the lever rule:

,

where

is the line segment from the mixing point

to the feed location

, and

is the line segment from the mixing point

to the solvent location

.

A line drawn from the desired raffinate composition

through the mixing point until it intersects the phase boundary gives the extract composition leaving stage 1,

.

The operating point

is located at the intersection of a line drawn through points

and

and a line drawn through

and

because the overall mass balance for the system is:

.

This equation is rearranged to define the operating point:

.

A tie line from

to the right side of the phase boundary yields the raffinate composition leaving stage 1,

; this line represents the first equilibrium stage (orange).

A mass balance on stage 1 (total feed in = total feed out) is:

,

.

Thus, the extract composition leaving stage 2,

, is found by drawing a straight line from

to

(because

from the previous equation). Where this line intersects the left side of the phase boundary is the composition of

.

This procedure is repeated for additional stages until the raffinate composition is nearly equal to the desired value

. The number of orange equilibrium lines drawn is the number of equilibrium stages needed to obtain

.

See [1–4] for screencasts that describe the Hunter–Nash method and present examples.