10.1 Metallurgical Plant Flowsheets & Pulp Mass Balances

10.1 Metallurgical Plant Flowsheets & Pulp Mass Balances

Extractive metallurgy and mineral processing rely on flowsheets and mass balance calculations to design, monitor, and optimize plant performance. Metallurgical plant flowsheets delineate unit operations—such as comminution, classification, flotation, leaching, and dewatering—providing the engineering framework for material, water, and metal routing.

Metallurgical Plant Flowsheets: Block Diagrams & PFDs

Flowsheets are categorized based on their level of technical detail:

1. Block Flow Diagrams (BFDs): Conceptual schematics showing major processing circuits (e.g., Primary Crushing, Grinding, Flotation, Thickening) as simple blocks connected by directional material flow lines. BFDs establish overall plant architecture during preliminary scoping studies.
2. Process Flow Diagrams (PFDs): Detailed engineering drawings depicting major equipment items (e.g., SAG mills, jaw crushers, hydrocyclones, flotation banks, high-rate thickeners), stream identification tags, operating temperatures, pressures, volumetric flow rates, and pulp densities under nominal design conditions.

Flowsheet symbols conform to standardized conventions (such as ISO or ANSI standards) to ensure consistent communication across engineering disciplines.

Two-Product Metallurgical Mass Balances

A steady-state two-product separation circuit splits a single feed stream ($F$) into a concentrate stream ($C$) and a tailings stream ($T$). Mass conservation applies to both total wet/dry material mass and individual metal contents.

Fundamental Mass Conservation Equations

1. Total Mass Conservation: $F = C + T$

2. Metal (Assay) Conservation: $F f = C c + T t$

Where:

• $F$ = Mass flow rate of dry feed (tonnes/hour or kg/s)
• $C$ = Mass flow rate of dry concentrate (tonnes/hour or kg/s)
• $T$ = Mass flow rate of dry tailings (tonnes/hour or kg/s)
• $f$ = Grade or assay of valuable element in feed (decimal fraction or %)
• $c$ = Grade or assay of valuable element in concentrate (decimal fraction or %)
• $t$ = Grade or assay of valuable element in tailings (decimal fraction or %)

Derivation of Mass Flow Ratios

Substituting $T = F - C$ into the metal balance equation yields:

$F f = C c + (F - C) t = C c + F t - C t$

$F (f - t) = C (c - t)$

Rearranging gives the weight yield ratio of concentrate to feed:

$\frac{C}{F} = \frac{f - t}{c - t}$

Similarly, substituting $C = F - T$ yields the tailings weight yield ratio:

$\frac{T}{F} = \frac{c - f}{c - t}$

Ratio of Concentration ($K$) and Enrichment Ratio ($E$)

• Ratio of Concentration ($K$): The mass of feed required to produce one unit mass of concentrate: $K = \frac{F}{C} = \frac{c - t}{f - t}$

• Enrichment Ratio ($E$): The factor by which the feed grade is upgraded in the concentrate: $E = \frac{c}{f}$

Metallurgical Recovery ($R$)

Metallurgical recovery ($R$) represents the percentage of total valuable metal in the feed that is successfully recovered into the concentrate:

$R = \left( \frac{C c}{F f} \right) \times 100\%$

Substituting the mass ratio $\frac{C}{F} = \frac{f - t}{c - t}$ into the recovery equation gives the Two-Product Recovery Formula:

$R = \frac{c (f - t)}{f (c - t)} \times 100\%$

This formula allows calculation of metallurgical recovery directly from chemical assays ($f$, $c$, $t$) without requiring direct mass flow measurements.

Pulp Density & Slurry Calculations

In hydrometallurgical and mineral processing plants, solids are transported as aqueous suspensions (slurries or pulps). Accurately determining pulp density, specific gravity, and percent solids is vital for volumetric equipment sizing and mass balance accounting.

Key Pulp Parameters & Formulas

1. Percent Solids by Weight ($%S$): $\%S = \left( \frac{W_s}{W_p} \right) \times 100\% = \left( \frac{W_s}{W_s + W_w} \right) \times 100\%$ Where $W_s$ is dry solid weight, $W_p$ is total pulp weight, and $W_w$ is water weight.

2. Pulp Specific Gravity ($S_p$): $S_p = \frac{100}{\frac{\%S}{S_s} + \frac{100 - \%S}{S_w}}$ Where $S_s$ is solid specific gravity (e.g., $2.70$ for quartz/chalcopyrite mix) and $S_w$ is liquid specific gravity ($1.00$ for water).

3. Percent Solids by Volume ($%V$): $\%V = \%S \times \left( \frac{S_p}{S_s} \right)$

4. Volumetric Flow Rate of Pulp ($Q_p$): $Q_p = \frac{M_s}{\%S / 100 \times S_p \times \rho_w}$ Where $M_s$ is dry solid throughput rate (t/h) and $\rho_w$ is water density ($1.0\text{ t/m}^3$).

Slurry Pumping Hydraulics & Pipeline Design

Pulp slurry transport requires specialized centrifugal slurry pumps and lined pipelines. Key design parameters include:

• Critical Velocity ($V_c$): The minimum line velocity required to prevent solid particles from settling out of suspension and forming a stagnant bed (Durand formula): $V_c = F_L \sqrt{2 g D (S_s - 1)}$ Where $F_L$ is a particle size coefficient, $g$ is gravitational acceleration ($9.81\text{ m/s}^2$), $D$ is pipe internal diameter (m), and $S_s$ is solid specific gravity.
• Total Dynamic Head (TDH): Sum of static head, friction head losses in pipes/fittings, and pressure head. Friction losses increase with slurry viscosity and particle concentration.
• Pump Head and Efficiency Derating: Heavy slurries decrease centrifugal pump performance. Correction factors ($C_h$ for head, $C_e$ for efficiency) are applied to clear water performance curves: $H_m = H_w \times C_h$.

Equipment Sizing & Plant Water Balance Circuits

Balancing water and solids across closed-loop grinding and flotation circuits ensures steady-state operation.

Grinding Circuit Hydrocyclone Water Balance

Hydrocyclones separate coarse mill discharge (underflow, recycled to SAG/ball mill) from fine particles (overflow, routed to flotation).

Mill Feed + Recycled Water -> Ball Mill -> Sump + Dilution Water -> Hydrocyclone Bank
 hydrocyclone underflow (65-75% solids) -> Recycled to Mill
 hydrocyclone overflow (30-40% solids) -> Sent to Flotation

Thickeners consolidate tailings or concentrates and return overflow water when quality and inventory permit. The recovered fraction follows the mass balance, underflow density, evaporation, entrainment, leakage, and downstream demand rather than a universal percentage.

Summary Table of Flowsheet Formulas