The feed flow rate, solvent flow rate, and feed 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:
overall: \(\dot{F} + \dot{S} = \dot{M}\),
solute A: \(x_A^F\dot{F} = x_A^M\dot{M}\) (because the solvent feed contains no solute),
solvent S: \(x_S^F\dot{F} + \dot{S} = x_S^M\dot{M}\),
carrier C: \(x_A^M + x_S^M + x_C^M = 1\),
where \(\dot{F}\) is the feed flow rate (kg/h); \(\dot{S}\) is the solvent flow rate (kg/h); \(\dot{M}\) is the combined feed and solvent flow rate (kg/h); \(x_A^i\), \(x_S^i\), and \(x_C^i\) refer to the mass fractions of the solute, solvent, and carrier in \(i\) = (F, S, M), where the superscripts F, S, and M refer to the three kinds of stream: feed, solvent, and mixed. The coordinates of the mixing point \(M\) on the ternary diagram are (\(x_A^M\), \(x_A^M\), \(x_A^M\)).
The mixing point \(M\) can also located on the phase diagram using the lever rule:
$$
\dot{F}/\dot{S} = \overline{MF}/\overline{MS},
$$
where \(\overline{MF}\) is the line segment from the mixing point \(M\) to the feed location \(F\), and \(\overline{MS}\) is the line segment from the mixing point \(M\) to the solvent location \(S\).
A line drawn from the desired raffinate composition \(R\)_{N} through the mixing point until it intersects the phase boundary gives the extract composition leaving stage 1, \(E\)_{1}.
The operating point \(P\) is located at the intersection of a line drawn through points \(F\) and \(E\)_{1} and a line drawn through \(R\)_{N} and \(S\) because the overall mass balance for the system is:
$$
F + S = E_1 + R_N.
$$
This equation is rearranged to define the operating point:
$$
P = E_1  F = R_N  S.
$$
A tie line from \(E\)_{1} to the right side of the phase boundary yields the raffinate composition leaving stage 1, \(R\)_{1}; this line represents the first equilbrium stage (orange).
A mass balance in stage 1 is:
$$
E_2 + F = E_1 + R_1,
$$
which can be written as:
$$
E_2  R_1 = E_1  F = P.
$$
Thus, the extract composition \(E_2\) leaving stage 2 is found by drawing a straight line from \(R_1\) to \(P\) (because \(P + R_1 = E_2\) from the previous equation). Where this line intersects the left side of the phase boundary is the composition of \(E_2\).
This procedure is repeated for additional stages until the raffinate composition is nearly equal to the desired value of \(R_N\) without being less than \(R_N\). The number of blue equilibrium lines drawn is the number of equilibrium stages needed to obtain \(R_N\).
See [14] for screencasts that describe the HunterNash method and present examples.

HunterNash Method: LiquidLiquid Extraction [Video]. (July 31, 2017) www.youtube.com/watch?v=e0Yt5RMkyj4.

HunterNash Method: LiquidLiquid Extraction Example [Video]. (July 31, 2017) www.youtube.com/watch?v=FaKIZU3XPR8.

HunterNash Method 1: Mixing and Operating Points [Video]. (July 31, 2017) www.youtube.com/watch?v=yW0jIcH0_E.

HunterNash Method 2: Number of Stages [Video]. (July 31, 2017) www.youtube.com/watch?v=JZuavrt8ksQ.