How Is The Line-Segment Lever Rule Applied To Graphical Material Balance Calculations? A Guide For Extraction Pilot Plants

The line-segment lever rule determines stream ratios from geometry. In a ternary liquid-liquid extraction diagram, the composition of a mixture formed by combining feed stream (R) and solvent stream (S) lies on the straight line joining their composition points. If the mixture composition is (M), the stream mass ratio is found from the opposite line segments:

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

The same principle applies after phase separation: along the equilibrium tie line, the extract-to-raffinate mass ratio is inversely proportional to the corresponding segment lengths.

The lever rule converts graphical distances into material-balance ratios. First use the line between the feed and solvent to determine the mixed-stream composition or feed-to-solvent ratio; then use the tie line between equilibrium extract and raffinate compositions to determine the phase flow rates.

How the Composition Diagram Represents the Process

Ternary coordinates define stream compositions

A ternary diagram represents the relative amounts of three components, commonly a solute, a carrier liquid, and an immiscible extraction solvent. Each stream composition is plotted as a point whose coordinates sum to one, or to 100%.

The diagram may be equilateral or right-triangular. The lever-rule geometry is valid in either form, provided the axes represent a consistent mass or molar basis.

Each stream becomes a composition point

The feed composition is plotted as point (R), and the solvent composition as point (S). Their total flow rates or masses are (W_R) and (W_S), respectively.

After mixing, the combined stream has composition point (M). Because the mixture contains only material from (R) and (S), (M) must lie on the straight line segment connecting those two points.

Applying the Lever Rule to Mixing

Locate the mixture point

Draw a straight line from the feed point (R) to the solvent point (S). The overall mixture composition must be somewhere on this line.

Its exact location depends on the relative amounts of feed and solvent:

(M) closer to (R) means the feed contributes more mass.
(M) closer to (S) means the solvent contributes more mass.
Equal feed and solvent masses place (M) at the midpoint of (RS), assuming the compositions are represented on the same basis.

Use opposite line segments for the ratio

The lever relationship is:

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

The mass of each stream is paired with the segment opposite that stream. Therefore, the solvent mass (W_S) is associated with segment (RM), while the feed mass (W_R) is associated with segment (MS).

This inverse relationship is the central geometric rule: the larger stream lies opposite the shorter segment.

Calculate unknown stream flow rates

If the total mixture flow rate (W_M) is known, then:

[ W_M=W_R+W_S ]

Using the measured or graphically determined segment lengths:

[ W_R=W_M\frac{MS}{RM+MS} ]

[ W_S=W_M\frac{RM}{RM+MS} ]

These equations allow a pilot plant operator or student to calculate individual stream flow rates from the total flow and the location of (M).

Example of the mixing calculation

Suppose the diagram gives:

[ RM=3 \text{ cm}, \qquad MS=2 \text{ cm} ]

Then:

[ \frac{W_S}{W_R}=\frac{3}{2} ]

The solvent flow is therefore 1.5 times the feed flow. If the total mixture flow is (500\ \text{kg/h}):

[ W_R=500\left(\frac{2}{5}\right)=200\ \text{kg/h} ]

[ W_S=500\left(\frac{3}{5}\right)=300\ \text{kg/h} ]

The graphical distances have been converted directly into a material balance.

Applying the Lever Rule After Phase Separation

Plot the equilibrium phases

After the feed and solvent are contacted, the mixture may separate into two liquid phases:

Extract (E): enriched in the transferred solute and extraction solvent.
Raffinate (R_f): depleted in solute and richer in the carrier liquid.

The equilibrium compositions of these phases are located at the two ends of a tie line. The overall mixture point (M) lies somewhere between them:

[ E \longleftrightarrow M \longleftrightarrow R_f ]

The subscript (f) distinguishes the raffinate phase from a feed point that may also be labeled (R).

Determine the phase flow-rate ratio

The phase lever rule is:

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

The extract flow is paired with the segment on the raffinate side, and the raffinate flow is paired with the segment on the extract side.

If the total flow entering the separator is (W_M), then:

[ W_E=W_M\frac{MR_f}{ME+MR_f} ]

[ W_{R_f}=W_M\frac{ME}{ME+MR_f} ]

This gives the extract and raffinate flow rates without separately solving three component-balance equations.

Example of the phase-split calculation

Suppose the tie-line distances are:

[ ME=1.5 \text{ cm}, \qquad MR_f=4.5 \text{ cm} ]

Then:

[ \frac{W_E}{W_{R_f}}=\frac{4.5}{1.5}=3 ]

The extract flow is three times the raffinate flow. For a total mixture flow of (400\ \text{kg/h}):

[ W_E=400\left(\frac{4.5}{6.0}\right)=300\ \text{kg/h} ]

[ W_{R_f}=400\left(\frac{1.5}{6.0}\right)=100\ \text{kg/h} ]

The phase closer to the mixture point receives the larger flow because its opposite segment is longer.

Using the Rule in Pilot Plant Material Balances

Verify measured flow rates

Pilot plant measurements often include feed, solvent, extract, and raffinate flow rates together with laboratory composition data. Plotting those compositions allows the lever-rule prediction to be compared with measured flow ratios.

A significant discrepancy may indicate inaccurate flow measurement, incomplete phase separation, non-equilibrium operation, sampling error, or an inconsistent composition basis.

Evaluate solvent-to-feed requirements

The mixing line between feed and solvent shows how changing the solvent flow moves the mixture point. Increasing the solvent-to-feed ratio shifts (M) toward the solvent composition and changes the available phase split.

This provides a visual way to examine solvent usage before performing more detailed stage calculations.

Analyze multistage extraction

For multistage countercurrent extraction, each stage involves a combination of material balances and equilibrium relationships. Tie lines connect equilibrium extract and raffinate compositions, while operating lines represent the stage-to-stage material balance.

In the Hunter-Nash graphical method, operating lines pass through a common difference point (O). Alternating between an operating line and an equilibrium tie line permits the intermediate compositions and approximate number of theoretical stages to be determined graphically.

Cross-check component balances

The lever rule is a geometric expression of component conservation. It should therefore be checked against the component balances:

[ W_M z_i=W_E y_i+W_{R_f}x_i ]

where (z_i) is the overall mixture composition, (y_i) is the extract composition, and (x_i) is the raffinate composition for component (i).

Agreement between the graphical result and the component balances confirms that the plotted points, tie line, and flow basis have been used correctly.

Understanding the Trade-offs

Graphical accuracy depends on the diagram

A hand-drawn diagram introduces uncertainty when line segments are short, points are crowded, or the phase envelope and tie lines are poorly resolved. A small position error can produce a large flow-ratio error, particularly when one phase fraction is very small.

For design-quality calculations, use accurate equilibrium data and numerical verification rather than relying solely on measurements from a printed diagram.

The basis must remain consistent

The lever rule applies to mass flows when the diagram uses mass fractions and to molar flows when it uses mole fractions. Mixing a mass-based diagram with molar flow rates produces an invalid result.

Density-based volumetric flow measurements must be converted appropriately before being compared with mass-based graphical calculations.

Equilibrium is an assumption

The tie-line lever rule describes a phase split at the specified equilibrium condition. Real pilot equipment may not reach equilibrium because of inadequate mixing, insufficient settling time, entrainment, temperature variation, or stage inefficiency.

The graphical result is therefore an equilibrium benchmark, not automatically the measured performance of the equipment.

A tie line is not the same as a mixing line

The feed-solvent line describes mixing two incoming streams. A tie line describes equilibrium separation into two phases.

Using the wrong line for a lever calculation is a common error. The mixing ratio is obtained from (RM/MS), whereas the phase-split ratio is obtained from (MR_f/ME).

Making the Right Choice for Your Goal

Use the diagram as a material-balance tool, then validate the result with measured flows and component balances.

If your primary focus is feed-to-solvent ratio: Plot the feed and solvent compositions, locate the mixture point on their connecting line, and use (W_S/W_R=RM/MS).
If your primary focus is extract and raffinate flow rates: Locate the mixture point on the equilibrium tie line and use (W_E/W_{R_f}=MR_f/ME).
If your primary focus is pilot plant verification: Compare the graphical flow predictions with measured phase flows and component balances on the same mass or molar basis.
If your primary focus is multistage design: Combine tie-line lever calculations with operating lines and the common difference point used in the Hunter-Nash method.

The line-segment lever rule makes ternary extraction balances visible: stream ratios are encoded in the distances between composition points.

Summary Table:

Application	Diagram elements	Lever-rule relationship	Main use
Feed-solvent mixing	Feed point, solvent point, mixture point	W_S/W_R = RM/MS	Calculate solvent-to-feed ratios or individual stream flows
Phase separation	Extract, mixture, and raffinate points on a tie line	W_E/W_Rf = MRf/ME	Determine equilibrium extract and raffinate flow rates
Pilot plant verification	Graphical predictions and measured flows	Compare predicted and actual ratios	Identify measurement, sampling, separation, or equilibrium issues
Multistage extraction	Tie lines, operating lines, and difference point	Hunter-Nash graphical construction	Estimate intermediate compositions and theoretical stages
Balance validation	Overall and phase compositions	W_M z_i = W_E y_i + W_Rf x_i	Confirm conservation and consistent mass or molar bases

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