The line-segment principle provides a graphical material balance for ternary extraction. When feed and solvent are mixed, the overall composition lies on the straight line joining their composition points, and the mass ratio is determined by the inverse ratio of the line segments. When the mixture separates into extract and raffinate, the overall composition lies on the tie line joining the two equilibrium-phase compositions, and the same lever rule determines the mass of each phase.

The triangular diagram determines composition and relative stream amounts through geometry; an independent total mass or flow rate is required to convert those relative amounts into absolute pilot-plant flow rates.

How Ternary Diagrams Represent the Material Balance

Each Point Represents a Composition

A ternary diagram represents three components, such as solute, carrier liquid, and extraction solvent. The three apexes correspond to pure components, while every interior point represents a composition satisfying:

[ x_A+x_B+x_C=1 ]

The composition may be expressed using mass fractions, weight percentages, or mole fractions. The balance calculation must use one basis consistently.

Straight Lines Represent Two-Stream Mixing

Suppose feed stream (R) and solvent stream (S) are combined. Their mixture point (M) must lie on the straight line joining (R) and (S):

[ \mathbf{x}_M= \frac{W_R\mathbf{x}_R+W_S\mathbf{x}_S} {W_R+W_S} ]

Here, (W_R) and (W_S) are the stream masses or mass flow rates, and (\mathbf{x}) represents the three-component composition vector.

The point (M) is therefore a weighted average of the two inlet compositions. It cannot lie beyond either endpoint unless another stream or source of material is involved.

Applying the Lever Rule to Feed and Solvent

Locating the Mixture Point

First plot the feed composition (R) and solvent composition (S) on the triangular diagram. Draw the straight mixing line between them.

The overall mixture composition (M) is located according to the feed-to-solvent ratio. If the flow rates are known, calculate (M) from the component balances and plot it. If the desired mixture composition is known, locate (M) on the line and infer the required ratio.

Calculating the Stream-Mass Ratio

The line-segment relationship is:

[ \frac{W_S}{W_R}=\frac{RM}{MS} ]

where (RM) is the distance from the feed point to the mixture point, and (MS) is the distance from the mixture point to the solvent point.

The ratio is inverse to the segment opposite each stream:

A larger (W_S) moves (M) closer to (S).
A larger (W_R) moves (M) closer to (R).
Equal stream masses place (M) at the midpoint of (RS).

This gives a graphical method for estimating solvent-to-feed ratios in a mixer-settler or similar pilot unit.

Converting the Ratio into Flow Rates

The diagram provides only a ratio unless one total flow is known. If the total mixture flow is (W_M), then:

[ W_M=W_R+W_S ]

Using:

[ r=\frac{W_S}{W_R} ]

the individual flows are:

[ W_R=\frac{W_M}{1+r} ]

[ W_S=\frac{rW_M}{1+r} ]

Thus, the graphical result must be combined with a measured or specified total flow.

Applying the Lever Rule to Phase Separation

Identifying the Two-Phase Region

The binodal curve separates the homogeneous single-phase region from the two-phase liquid-liquid region. If the mixture point (M) lies outside the two-phase envelope, it remains one liquid phase under the stated temperature and pressure.

If (M) lies inside the envelope, it separates into an extract phase (E) and a raffinate phase (R_f).

Using the Tie Line

A tie line connects the equilibrium compositions of the two phases produced from the mixture. Its endpoints are:

(E): the extract composition, usually richer in solvent and extracted solute.
(R_f): the raffinate composition, usually richer in carrier liquid.

For equilibrium separation, the overall mixture point (M) lies on the tie line connecting (E) and (R_f):

[ \mathbf{x}M= \frac{W_E\mathbf{x}E+W{R_f}\mathbf{x}{R_f}} {W_E+W_{R_f}} ]

The tie line supplies the phase compositions; the lever rule supplies their relative quantities.

Calculating Extract and Raffinate Masses

The phase-mass ratio is:

[ \frac{W_E}{W_{R_f}}= \frac{MR_f}{ME} ]

Equivalently:

[ \frac{W_{R_f}}{W_E}= \frac{ME}{MR_f} ]

The phase with the longer opposite segment has the larger mass. For example, if (M) is closer to (E), then (ME) is shorter and the raffinate mass is larger.

If the total two-phase mixture mass is (W_M):

[ W_M=W_E+W_{R_f} ]

and, defining:

[ r_p=\frac{W_E}{W_{R_f}}=\frac{MR_f}{ME} ]

the phase masses are:

[ W_{R_f}=\frac{W_M}{1+r_p} ]

[ W_E=\frac{r_pW_M}{1+r_p} ]

Reading Composition Information Correctly

Projection Lines Give Component Fractions

Component fractions are read by drawing or following gridlines parallel to the appropriate triangle side. The three fractions at any point must sum to unity, subject to reading and plotting error.

For a point (P):

[ x_A+x_B+x_C=1 ]

This consistency check is useful when compositions are read from an imprecise printed diagram.

The Diagram Geometry Is Affine

The lever rule works on equilateral, isosceles, or right-triangular composition diagrams because it depends on linear interpolation, not on the visual shape of the triangle. However, measured distances must be taken along the relevant straight line, and the diagram must use a valid linear composition scale.

Mass and Mole Bases Must Not Be Mixed

A mass-fraction diagram supports mass balances. A mole-fraction diagram supports molar balances. A flow ratio calculated from one basis cannot be applied directly to the other without converting the compositions and flow rates.

This distinction matters particularly when the solvent, carrier, and solute have substantially different molecular weights.

Connecting Mixing and Separation in a Pilot Plant

The Overall Balance Comes First

For a single-stage extraction operation, combine the feed and solvent streams conceptually to obtain the overall mixture point (M). The overall balance is:

[ W_F\mathbf{x}_F+W_S\mathbf{x}_S

W_E\mathbf{x}E+W{R_f}\mathbf{x}_{R_f} ]

with:

[ W_F+W_S=W_E+W_{R_f} ]

The mixture point determined from the inlet streams must be the same point obtained from the weighted extract and raffinate streams.

The Two Geometric Steps Are Distinct

The first line segment, (FS), describes mixing of the inlet streams. The second line segment, (ER_f), describes splitting of the equilibrium mixture into two phases.

These lines generally have different directions because the feed-solvent mixing path and the equilibrium tie-line direction are controlled by different information.

Extending the Method to Multiple Stages

In multistage extraction, each stage has its own inlet mixture, phase split, and potentially different outlet compositions. The same principles apply stage by stage, but the graphical construction must account for the streams entering and leaving each stage.

The diagram does not replace stage balances or equilibrium data. It provides a visual framework for applying them.

Understanding the Trade-offs

Graphical Accuracy Is Limited

The lever rule is exact in principle, but distances and compositions read from a printed or low-resolution diagram are approximate. Small plotting errors can produce noticeable errors when (M) lies close to an endpoint.

For design or scale-up decisions, use measured equilibrium data and numerical balances rather than relying solely on ruler measurements.

Tie Lines Are Not Arbitrary

A tie line must represent an experimentally established or otherwise reliable equilibrium relationship at the operating temperature and pressure. Drawing an arbitrary line through the mixture point does not correctly predict the extract and raffinate compositions.

Near the plait point, the two phase compositions become nearly identical, making both the tie-line direction and phase split especially sensitive to measurement error.

The Lever Rule Does Not Predict Equilibrium

The line-segment principle determines relative amounts once the relevant compositions are known. It does not determine the binodal curve, tie lines, distribution coefficients, mixing quality, residence time, or whether the pilot plant has actually reached equilibrium.

Those require phase-equilibrium data and appropriate operating analysis.

Pilot-Plant Results May Differ from the Diagram

Real equipment can exhibit entrainment, incomplete phase disengagement, non-equilibrium contact, temperature variation, and sampling error. Consequently, the measured outlet streams may not fall exactly on the predicted tie line or satisfy the ideal graphical balance without accounting for experimental uncertainty.

Making the Right Choice for Your Goal

Use the line-segment principle as a structured graphical balance, then validate the result with component balances and pilot-plant measurements.

If your primary focus is feed-to-solvent ratio: Plot the feed and solvent points, locate the desired mixture point on their connecting line, and calculate (W_S/W_F) from the inverse segment ratio.
If your primary focus is extract and raffinate flow rates: Locate the overall mixture on the appropriate tie line and use (W_E/W_{R_f}=MR_f/ME), followed by the total-flow balance.
If your primary focus is phase compositions: Use the binodal endpoints and experimentally valid tie lines; the lever rule determines phase quantities but not equilibrium compositions.
If your primary focus is pilot-plant validation: Compare measured stream compositions and flow rates with the graphical balance while checking temperature, sampling, entrainment, and equilibrium assumptions.

With consistent composition bases and reliable tie-line data, the triangular diagram turns ternary extraction balances into a direct geometric calculation.

Summary Table:

Balance step	Diagram feature	Key result
Feed and solvent mixing	Line joining feed and solvent points	Determines the overall mixture composition and solvent-to-feed ratio
Phase separation	Tie line through the mixture point	Identifies equilibrium extract and raffinate compositions
Phase quantities	Lever rule using tie-line segments	Calculates the relative extract and raffinate masses
Flow-rate conversion	Total mass or flow basis	Converts graphical ratios into absolute pilot-plant flow rates
Validation	Component balances and equilibrium data	Checks diagram accuracy against operating results

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