Contents

- Introduction
- Objectives
- Reading
- Mass and Volume Balancing
- Mass and Component Balance
- Mass Yield or Mass Recovery
- Two-Product Formula
- Component Recovery
- Component Recovery Example
- Concentration Ratio, CR
- Complex Sulfide Ore Metallurgical Balance
- Mass/Volume Flow Determination
- Pulp Density
- Marcy Scale
- Mass Flow Determination Example
- Volumetric Balancing Using Pulp Density
- Circuit Balancing
- Material Balancing
- What to Do with Unbalanced Data

## Introduction

Introductory narrative or video

## Objectives

Upon completion of this lesson students should be able to:

- Differentiate between different type metallurgical accounting methods and benefit of metallurgical accounting.
- Recall the definitions of grade, recovery, yield and other characteristics used in mass balancing.
- Recognize the application of two product formula for balancing.
- Provide a brief idea about multi component balancing for complex ores.

## Reading

### Mass and Volume Balancing

**The most important rule governing the accounting of a processing plant or unit operation is that mass can be neither created nor destroyed.**

**Mass In = Mass Out**

**Volume In = Volume Out**

### Mass and Component Balance

**The component balance is true for any assay type such as any element content, % finer than, water-to-solids ratio etc.**

**Ff = Cc + Rr**

### Mass Yield or Mass Recovery

### Two-Product Formula

Given the mass and component balance equations :

F = C + R

Ff = Cc + Rr

Mass yield (Y) can be solved as a function of the stream, assays which yields the *two-product formula*:

R = F- C

Ff = Cc + r (F – C)

#### Mass Yield Formula Example

### Component Recovery

In many mineral processing applications, the focus is to maintain maximum recovery of the valued mineral or element while also maximizing upgrade or concentrate assay.

### Component Recovery Example

### Complex Sulfide Ore Metallurgical Balance

**Brunswick No.1 Concentrator**

Process Stream | Weight (%) | Assays | |||
---|---|---|---|---|---|

Pb% | Zn% | Cu% | Ag% | ||

Mill Feed | 100.00 | 3.21 | 7.93 | 0.33 | 2.26 |

Copper Concentrate | 0.74 | 4.69 | 4.65 | 22.64 | 50.39 |

Lead Concentrate | 4.80 | 42.20 | 9.62 | 0.33 | 15.48 |

Zinc Concentrate | 10.52 | 1.12 | 57.76 | 0.18 | 1.75 |

Tailings | 83.94 | 1.22 | 1.61 | 0.15 | 1.14 |

Metal Recovery (%) |
63.1 |
76.6 |
50.9 |

Section | Process Stream | Mass Flow (tph) | Copper Assay (%) | Cu Recovery (%) |
---|---|---|---|---|

Complete Plant |
Mill Feed | 709 | 0.53 | 100 |

Mill Concentrate | 10 | 33.00 | 88.3 | |

Mill Tailings | 699 | 0.06 | 11.7 | |

Rougher |
Feed | 716 | 0.55 | |

Concentrate | 28 | 12.00 | 83.6 | |

Tailings | 688 | 0.09 | ||

Scavenger |
Feed | 253 | 0.13 | |

Concentrate | 6 | 2.00 | 2.4 | |

Tailings | 247 | 0.09 | ||

Cleaner |
Feed | 29 | 12.00 | |

Concentrate | 15 | 22.00 | ||

Tailings | 14 | 1.30 | ||

Re-Cleaner |
Feed | 18 | 20.00 | |

Concentrate | 11 | 31.00 | ||

Tailings | 7 | 3.00 | ||

Final Cleaner |
Feed | 11 | 31.00 | |

Concentrate | 10 | 33.00 | 88.3 | |

Tailings | 1 | 5.00 |

### Mass/Volume Flow Determination

- The relationship between the mass (M) and the volume flow (Q rates in a given stream is defined as:

= the pulp density (solids and water) in lbs/ft^{3 }X = solids concentration in % by weight - Volumetric flow rate can be measured by flow meters, P-Q curve relationships for pumps or directly measured.
- Pulp density can be measured using a Marcy Density of nuclear density gauge.
- The solids concentration can be estimated using a common expression or directly measured.

### Pulp Density

- Density (ρ) is the ratio of the mass weight (M) of a substance and the total volume (V):

- Water has a density of:
- 1.0gm/ml or 1.0gm/cm
^{3} - 1000kg/m
^{3}– 1tonne/m^{3} - 62.4lbs/ft
^{3}

- 1.0gm/ml or 1.0gm/cm
- Specific gravity is the ratio of the material density over the density of water:

### Marcy Scale

- The Marcy scale is a common tool used in preparation plants to measure pulp density.
- The scale uses a container that allows a volume of exactly 1000 ml.
- After collecting the sample, the container is hung on a weight scale which measures in grams.
- The printed scale reads the pulp density directly in grams/ 1ml.
- Since water density is 1.0 gm/ml, the scale also indicates the specific gravity of the pulp.

### Mass Flow Determination Example

A classifying cyclone is being fed slurry at a volumetric flow rate of 1000 gpm. The specific gravity of the slurry is 1.1 and the solids concentration was determined to be 15.0°/o by weight. Determine the mass feed flow rate in tons per hour:

**Solution:**

### Volumetric Balancing Using Pulp Density

- Pulp density measurements around unit operations such as dense medium separators and classifying, cyclones can be used to assess volume yield.
- Consider the balance around a classifying cyclone:

- Thus, the volumetric yield to the underflow stream can be obtained from the following expression:

#### Slurry Solids Concentration by Weight

A classifying cyclone is treating 150 gallons/min of slurry. A plant technician has measured the pulp densities of the process streams using a Marcy density gauge and reported the following: ρf = 1.08 gm/ml, ρo = 1.03 gm/ml and ρu = 1.20 gm/ml. Determine the volumetric flow rates to each stream.

**Solution:**

_________%

_________ = _________ gallons/min

_________ = 150 X (1 – _________) = __________ gallons/min

- The solids concentration
*X*of a slurry by weight can be measured directly by:- Collecting a sample,
- Measuring the weight of the total slurry or pulp M
_{p} - Drying the sample and
- Measuring the dry weight of solids M
_{s }

M_{w}= water weight

- This equation is often used to determine the amount of water in a process stream1 knowing the solids concentration by weight and the tons/hr of solids in a process stream.

#### Solids Concentration Example

The underflow stream from the classifying cyclone bank contains 100 tons/hour of ore which represents 45% of the total slurry mass.

The downstream concentrators require a feed solids concentration of 30% by weight. How much water is required to be added to dilute the stream to the required solids concentration?

**Solution:**

________ tph

________ tph

________ tph

= _______ gpm

#### % Solids & Pulp Density Relationship

- It often occurs that the knowledge of the solids content is needed within a time frame less than the sample preparation and analysis time.
- When this situation arises, the solids content by weight can be estimated knowing the pulp density (ρ
_{p}=) as measured with the Marcy scale and using the following equation:

X% = ρ_{s}= solids density or specific gravity

≈ 2.65 for most host rock minerals.ρ_{w}= water density (62.4 lbs/ft^{3}or 1gm/ml) or specific gravity (=1).

#### Estimation of Solid Density

- The density of solid can be estimated if you know the composition by weight or volume of each component in the solid and the respective solid densities.
- For example, assume that raw ore is a two component system comprised of a valued mineral and host rock having relative densities of 6.30 and 2.65, respectively. The amount of valued mineral is 30% of the total ore.Total Solid Mass, M: Mineral + Host Rock – 30 + 70 = 100

Total Volume, Vs:

Total Solid Relative Density

### Circuit Balancing

- The data generated from process units and circuits are used to make important decisions.
- Plant design considerations
- Operational efficiencies
- Potential upgrades

- As such, reliable data is very important for an operating plant.
- All data must be checked for proper balancing based on the ‘laws of conservation’.
- ‘What goes in’ = ‘What goes out’

### Material Balancing

Balance Summary

Overall: F = C+R

Pb: Ff_{1} = Cc_{1} + Rr_{1}

Zn: Ff_{2} = Cc_{2} + Rr_{2}

L/S Ratio: Ff_{3} = Cc_{3} + Rr_{3}

**Questions:**

- Is this a good set of data?
- Which values are not reliable?

#### Slurry Balance

Mass balance equations must be satisfied!

- Must use 1/Solids%

Product Stream | Mass (tph) | Solids (%) | Inverse Solids | Slurry (tph) |
---|---|---|---|---|

Clean | 10 | 50 | 1/0.5 | 20 |

Reject | 90 | 20 | 1/0.2 | 450 |

Feed | 100 | 25 | 1/0.25 | 400 |

#### Two-Product Formula

**Balances:**

F = C + R

Ff = Cc + Rr

**Multiply by r:**

Ff = Cc + Rr

-(Ff = Cc + Rr)

______________

Ff- Ff = Cr – Cc

**Rearrange:**

F (r-f) = C (c-r)

**or**

Yield = C/F = (r-f) / (r-c)

* Only assays needed to get product-to- feed ratio

#### Two-Product Formula (Pb assays)

**Overall:**

C/F = 10/100

= 10%

Pb:

C/F=(5.20-0.20)/(50.20-0.20)

= 10%

#### Two-Product Formula Characteristics

- Formula only applies under steady-state conditions

- Formula very sensitive to variations in the reject assay, “r”

- Formula inaccurate when component is not separated
- Formula may calculate different yields for each assay (due to experimental errors)

#### Two-Product Formula (various assays)

**Pb:**

C/F = (5.2-0.2)/ (50.2-0.2)

= 10%

**Zn:**

C/F=(2.3-0.3)/(22.3-0.3)

= 9%

**1/Solids:**

C/F = (1/25-1/20) / ( 1/50-1/20)

= 33%

### What to Do with Unbalanced Data

#### Eliminate Data

- avoid collecting or ignore data that conflicts
- most common approach in industry
- arbitrary and highly subject to user biases
- not getting full value out of your data

#### Adjust Data

- create consistent balances by adjusting data
- use method of “weighted least squares”
- adjustments should (i) satisfy all mass balances equations and (ii) be as small as possible