Material Balance in Froth Flotation using Microsoft Excel Solver

MATERIAL BALANCE IN FROTH FLOTATION USING MICROSOFT EXCEL SOLVER Revised – February, 2017

PROCESS DESIGNER JOSEPH KAFUMBILA

Material balance in froth flotation using Microsoft Excel Solver © 2017 Joseph Kafumbila [email protected]

Joseph Kafumbila

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Contents 1.

INTRODUCTION..................................................................................................................... 3

2.

MATERIAL BALANCE.............................................................................................................. 3 2.1. FLOW PARAMETERS ..................................................................................................................... 3 2.1.1. SOLID .................................................................................................................................. 3 2.1.2. WATER ................................................................................................................................ 6 2.1.3. PULP ................................................................................................................................... 6 2.2. UNIT OPERATION OF FROTH FLOTATION CIRCUIT ................................................................................ 7 2.2.1. FROTH FLOTATION UNIT........................................................................................................... 7 2.2.2. HYDROCYCLONE..................................................................................................................... 8 2.2.3. BALL MILL ............................................................................................................................. 9

3.

MATERIAL BALANCE IN AN OPERATING PLANT ................................................................... 11 3.1. OPERATING PLANT DESCRIPTION .................................................................................................. 11 3.2. OPERATING PLANT DATA ............................................................................................................ 13 3.3. MATERIAL BALANCE USING MICROSOFT EXCEL SOLVER .................................................................... 14 3.3.1. MATERIAL BALANCE SIMULATION TABLE ................................................................................... 14 3.3.2. MATERIAL BALANCE SIMULATION PROCEDURE ........................................................................... 16

4.

REFERENCES ........................................................................................................................ 32

Joseph Kafumbila

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1.

Introduction

Material balance calculations define an engineering problem where flow parameters between unit operations are partly known. The purpose of a material balance is to mathematically examine the known flow parameters to solve for the unknown flow parameters. Two main types of material balances are commonly made: design material balance and operating plant material balance. The design material balance is typically faced during plant design when the test work results and a flowsheet diagram are the only known values. Design material balance purpose is to find values for the unknown flow parameters. Operating plant material balance is tried to have a large amount of data from operating plant. Operating plant material balance purpose is to produce a picture of the state of an operating plant. This paper will give a procedure of operating plant material balance using Microsoft Excel Solver on Excel spreadsheet. An example solving a copper concentrator flotation circuit is presented and the process flow diagram is given below.

2.

Material balance

2.1.

Flow parameters

In mineral processing, the flow is the pulp which is a suspension of particles in water. The suspended particles will be called solid. Therefore, the flow will always consist of two components: solid and water. It will be discussed first the characterization of solid before the characterization of pulp. The characterization means, in this paper, designation of parameters and development of equations linking these parameters.

2.1.1. Solid Solid is characterized by a mass (Ms ) expressed in (ton) and a volume (Vs ) expressed in (m3). The specific gravity (SGs ) expressed in (kg/m3) is the ratio of mass on volume of solid. Equation (1) gives the mathematical expression that links mass, volume and specific gravity of solid.

SGs =

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Ms Vs

x 1000

( (1)

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2.1.1.1. Solid specific density There are two methods for obtaining a specific gravity of solid: laboratory method and mineralogical composition method.

A.

Laboratory method

In the case where it is possible to have physically a solid, laboratory method for obtaining a specific gravity of solid consisting of mineral rock finely crushed is as follows:       

Dry crushed solid in an oven at 80 ° C for 24 hours, Weigh crushed solid (kg) (mass between 0.100 and 0.300 kg), Put crushed solid in a test tube of one liter, Add water into the test tube up to 500 ml, Mix crushed solid and water until complete homogenization, Add more water in the test tube to the mark of a liter and, Weigh one liter volume of pulp.

After the practical operations, other data is determined as follows:    

Water mass is the difference between pulp mass and solid mass. Water volume is the ratio of water mass on water specific gravity (1,000 kg/m3). Solid volume is the difference between pulp volume and the water volume. Finally, solid specific gravity is the ratio of mass on volume of solid.

Table 1 shows an example for obtaining a solid specific gravity by the laboratory method. This method seems simple, but it requires great accuracy during weighing and measuring of values. Table 1: Solid specific gravity from the laboratory method Description Solid mass Pulp mass Pulp volume Water mass Water Volume Solid volume Solid SG

B.

unit kg kg l kg l l kg/m3

Equations

Pulp mass – Solid mass Water mass/ Water specific gravity Pulp volume – Water volume (Solid mass / Solid volume) x 1000

values 0.141 1.089 1.000 0.948 0.948 0.052 2,711.54

Mineralogical composition method

In the case where a solid is not provided in order to obtain a specific gravity by the laboratory method, obtaining of solid specific gravity is taken place by using mineralogical composition method. This method is based on the principle that rock is a juxtaposition of minerals. Therefore, the mass of rock is the sum of mineral masses and the volume of rock is the sum of mineral volumes. Based on these assumptions, the method for obtaining the rock specific gravity consists of:  

Knowing mass percent of minerals into a solid. Knowing specific gravity of minerals.

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   

Calculating mineral masses per unit solid mass. Calculating volumes of minerals. Determining solid volume by adding mineral volumes. Calculating the solid specific gravity by using the equation (1)

The weakness of this method is that it ignores porosities or structural defects of solid. Table 2 shows an example for obtaining a specific gravity of solid by using the mineralogical composition method. Results from Table 2 shows that for a total solid weight of 1,000 ton and a total solid volume of 359.77 m3 which is the sum of mineral volumes, the value of solid specific gravity is 2,779.55 kg/m3. Table 2: Solid specific gravity from solid mineralogical composition Minerals Cu2(OH)2(CO3) Cu3(PO4)2.Cu2(OH)4 2CuO.2SiO2.3H2O CuO CuS Cu2S CuFeS2 CoOOH FeO(OH) Ni(OH)2 CaCO3.MgCO3 MnO2 ZnS SiO2 UO3 Mg2SiO4 Ca2SiO4 CaCl2 Al2SiO5 Cr2O3 CdO Total

Mineralogical composition % 1.584 0.146 0.199 0.503 0.005 0.005 0.005 0.507 1.954 0.001 0.900 0.127 0.007 77.139 0.004 5.381 0.011 0.025 11.471 0.026 0.001

Mineral masses t 15.836 1.456 1.989 5.029 0.055 0.055 0.055 5.070 19.543 0.010 9.000 1.266 0.067 771.393 0.040 53.805 0.110 0.250 114.710 0.256 0.006 1000

Specific gravity of mineral t/m3 4.00 4.20 2.20 6.40 4.68 5.65 4.20 4.00 3.65 4.10 2.85 4.85 4.00 2.65 10.97 3.15 2.71 2.15 3.25 5.22 8.15

Mineral volumes m3 3.959 0.347 0.904 0.786 0.012 0.010 0.013 1.268 5.354 0.002 3.158 0.261 0.017 291.092 0.004 17.081 0.041 0.116 35.295 0.049 0.001 359.77

Specific gravity of solid kg/m3

2,779.55

2.1.1.2. Chemical composition of solid Solid is constituted with chemical elements. The index “k” is an identification number of a chemical element in this paper. At this level, two other parameters are defined; a mass of chemical element of index “k” (Mk ) expressed in (kg) into a solid and a grade of chemical element of index “k” (Tk ) expressed in (%) into a solid. Equation (2) gives mathematical expression that links mass of chemical element of index “k”, grade of chemical element of index “k” and solid mass.

(

T

k Mk = Ms x 100 x 1000

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(2)

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2.1.2.

Water

Water is characterized by a mass (ML ) expressed in (ton) and a volume (VL ) expressed in (m3). The specific gravity (SGL ) expressed in (kg/m3) is the ratio of mass to volume of water. The value of water specific gravity is 1,000 kg/m3.

2.1.3. Pulp The pulp is a mixture of solid and water. The pulp will be characterized by a mass “MP ” expressed in (ton), a volume “VP ” expressed in (m3) and a specific gravity “SGP ” expressed in (kg/m3). Equation (3) gives mathematical expression that links mass, volume and specific gravity of pulp. SGP =

MP VP

x 1000

(3)

(

Pulp mass is a sum of solid and water masses. Mathematical expression of this principle is given by equation (4). MP = M S + ML

(4)

(

Pulp volume is a sum of solid and water volumes. Mathematical expression of this principle is given by equation (5). VP = VS + VL

(5)

(

Solid mass can also be calculated from pulp volume and specific gravities of pulp, solid and water. Equation (6) gives the mathematical expression. (SG −1000)

Ms = (SGP−1000) x SGs x Vp / 1000 S

(6)

(

Volume percent of solid in the pulp expressed in (%) is given by equation (7). V

Cvs = V s x 100 (%)

(7)

p

(

Weight percent of solid in the pulp expressed in (%) is given by equation (8). s Cw =

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MS MP

x 100 =

(

100 x SGS x (SGp − 1000) SGp x (SGS − 1000)

(8)

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2.2.

Unit operation of froth flotation circuit

2.2.1. Froth flotation unit 2.2.1.1. Description Froth flotation is a method for physically separating particles based on differences in the facility of air bubbles to selectively adhere to specific mineral surfaces into a pulp. The particles attached to air bubbles are then carried to the tank surface; however the particles that are not attached to air bubbles stay into the pulp. In the industrial practice, chemical treatments are used to selectively alter mineral surfaces so that they have the necessary properties to adhere or not to air bubbles.

2.2.1.2. Material balance equations Figure 1 gives a flow diagram of froth flotation unit operation. A froth flotation receives a feed flow and produces concentrate flow and tailing flow. Exponent f, c ant t will respectively designate feed, concentrate and tailing.

Figure 1: Flow diagram of froth flotation unit operation In a continuous system at steady state, the principle of conservation of matter gives the following mathematical expression: Mpf = Mpc + Mpt

(9)

Vpf = Vpc + Vpt

(10)

MSf = MSc + MSt

(11)

VSf = VSc + VSt

(12)

MLf = MLc + MLt

(13)

VLf = VLc + VLt

(14)

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MSf x Tkf x 10 = MSc x Tkc x 10 + MSt x Tkt x 10

(15)

Froth flotation metallurgical performances are given by the following mathematical expression: 

Mass pull (%) is the ratio of concentrate solid mass on feed solid mass. Mass pull =



McS MfS

(Tf −Tt )

(16)

x 100 = (Tkc −Tkt ) x 100 k

k

Metal recovery (%) is the ratio of element mass in the concentrate on element mass in the feed. Metal recovery =

Mck Mfk

Tck

x 100 =

Tfk

x

(Tfk −Ttk ) (Tck −Ttk )

x 100

(17)

2.2.2. Hydrocyclone 2.2.2.1. Description A hydrocyclone is a separator mechanism that uses centrifugal force to separate solids from liquids.

2.2.2.2. Material balance equations Figure 2 gives flow diagram of a hydrocyclone. In the mineral processing, a hydrocyclone receives a feed flow and produces underflow and overflow. Pulp specific gravity of underflow is greater than that of overflow. Exponent f, u ant o will designate respectively feed, underflow and overflow. In a continuous system at steady state, the principle of conservation of matter gives the following mathematical expression: Mpf = Mpu + Mpo

(18)

Vpf = Vpu + Vpo

(19)

MSf = MSu + MSo

(20)

VSf = VSu + VSo

(21)

MLf = MLu + MLo

(22)

VLf = VLu + VLo

(23)

MSf x Tkf x 10 = MSu x Tku x 10 + MSo x Tko x 10

(24)

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Figure 2: Flow diagram of a hydrocyclone

2.2.3. Ball mill 2.2.3.1. Description Ball mill is a pulverizing machine consisting of a rotating drum which contains pebbles or metal balls as the grinding implements.

2.2.3.2. Material balance equations Figure 3 and 4 give flow diagrams of ball mill. In mineral processing, ball mill is generally coupled with a hydrocyclone. Figure 3 gives a flow diagram where ball mill receives hydrocyclone underflow and sometimes water and produces a pulp. Figure 4 gives a flow diagram where a ball mill receives new feed pulp, hydrocyclone underflow and sometimes water and produces a pulp. Exponent i and o will designate respectively inlet and outlet pulps of a ball mill. In a continuous system at steady state, the principle of conservation of matter gives the following mathematical expression: Mpi = Mpo

(25)

Vpi = Vpo

(26)

MSi = MSo

(27)

VSi = VSo

(28)

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Figure 3: Ball mill flow diagram (ball mill inlet pulp = hydrocyclone UF + water)

Figure 4: Ball mill flow diagram (ball mill inlet pulp = hydrocyclone UF + feed + water)

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MLi = MLo

(29)

VLi = VLo

(30)

Tki = Tko

(31)

3.

Material balance in an operating plant

3.1.

Operating plant description

Figure 5 gives the flow diagram of an operating plant. This operating plant is processed copper sulfide ore. The unit operations of the operating plant are the following: A. – Rougher B. – Scavenger C. – Regrind Ball Mill D. – Hydrocyclone E. – Cleaner F. – Cleaner scavenger G. – Re-cleaner Flow designations of the operation plant are the following: 1. – Feed 2. – Cleaner scavenger tailing 3. – Rougher concentrate 4. – Rougher concentrate launder water addition 5. – Rougher concentrate pulp 6. – Rougher tailing 7. – Scavenger concentrate 8. – Scavenger concentrate launder water addition 9. – Scavenger concentrate pulp 10.– Scavenger tailing 11.– Outlet regrind ball mill 12.– Cleaner scavenger concentrate 13.– Cleaner scavenger concentrate launder water addition 14.– Cleaner scavenger concentrate pulp

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Figure 5: Operating plant flow diagram

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15.– water 16. – Hydrocyclone feed 17. – Hydrocyclone underflow 18. – Hydrocyclone overflow 19. – Re-cleaner tailing 20.– Cleaner concentrate 21.– Cleaner concentrate launder water addition 22.– Cleaner concentrate pulp 23.– Cleaner tailings 24.– Re-cleaner concentrate

3.2.

Operating plant data Sampling campaign has been done and Table 3 gives data of the operating plant for each flow. Table 3: Sampling data of the operating plant Designation Feed Scavenger cleaner tailing Rougher concentrate Rougher concentrate launder water addition Rougher tailing Scavenger concentrate Scavenger concentrate launder water addition Scavenger tailing Outlet regrind ball mill Scavenger cleaner concentrate Scavenger cleaner concentrate launder water addition water Hydrocyclone feed Hydroclyclone underflow Hydrocyclone overflow Re-cleaner tailing Cleaner concentrate Cleaner concentrate launder water addition Cleaner tailing Re-cleaner concentrate

SGp Kg/m3 1,239 1,104 1,234

Solid mass t/m3 1,243.01

Pulp volume m3/h

Water m3/h

Cu grade % 1.93 0.88 14.57

2 1,256 1,152

0.84 4.28 4

1,264

0.67

1,070

9.49 5

1,301 1,740 1,123 1,072 1,203

1,275 7.44 11.29 1.18 19.15 4

1,095 1,275

2.57 23.78

The hydrocyclone feed data come from the control room. In this case, mathematical expression (32) gives the relationship between solid specific gravity and copper grade. In general for complex ore, solid specific gravity must be found for each flow by using method explained in chapter 2. SGs (t/m3) = 0.0005 x (%Cu)2 +0.0259 x (%Cu) + 2.6728

Joseph Kafumbila

(32)

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3.3.

Material balance using Microsoft Excel Solver

3.3.1. Material balance simulation table The simulation table of material balance of froth flotation of the operating plant is given in Table 4 as it appears on the Excel spreadsheet. Table 4 is divided into two small tables. Table 4A takes flows from 1 to 10 and Table 4B take flow from 11 to 20. In Table 4A and 4B, lines from 6 to 20 and from 28 to 42 give the values of flow parameters of each flow. Below table 4B, there is simulation constraint table that will contain excel solver constraints. The abbreviation into Table 4A and 4B means: -

SCT: scavenger cleaner tailing RC: Rougher concentrate RCW: Rougher concentrate launder water RT: Rougher tailing SC: Scavenger concentrate SCW: Scavenger concentrate launder water ST: Scavenger tailing ORBM: Out regrind ball mill SCC: Scavenger cleaner concentrate SCCW: Scavenger cleaner concentrate launder water W: Water HF: Hydrocyclone feed HUF: Hydrocyclone underflow HOF: Hydrocyclone overflow ReCT: Re-cleaner tailing CC: Cleaner concentrate CCW: Cleaner concentrate launder water CT: Cleaner tailing ReCC: Re-cleaner concentrate

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A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

B

C

D

E

Table 4A

F

G

H

I

J

K

L

8 ST

9 ORBM

10 SCC

18 CCW

19 CT

20 ReCC

Material balance simulation table 1 Feed

2 SCT

3 RC

4 RCW

5 RT

6 SC

7 SCW

Solid Mass t/h Vol. m3/h kg/m3 SGS Cu % Cu kg/h Water Mass t/h Vol. m3/h kg/m3 SGS Pulp Mass t/h Vol. m3/h kg/m3 SGS S % Cw

Table 4B

Material balance simulation table 11 SCCW

12 W

13 HF

14 HUF

15 HOF

16 ReCT

17 CC


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Simulation solver constraint table Constraint 1 Constraint 2 Constraint 3 Constraint 4

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3.3.2. Material balance simulation procedure Material balance of froth flotation circuit of the operating plant will be done step by step. The procedure is as follow:

1. Calculation of unknown parameters of flow 1 (plant Feed). - Known parameters: o o o o -

Solid mass flowrate. Copper grade. Pulp specific gravity. Water specific gravity.

In the excel cell “C7” (solid mass flowrate), type number “1243.01”. In the excel cell “C10” (copper grade), type number “1.93”. In the excel cell “C19”(pulp specific gravity), type number “1239”. In the excel cell “C15” (water specific gravity), type number “1000”. In the excel cell “C9” (solid specific gravity), type “=(0.0005*C10^2+0.0259*C10+2.6728)*1000” (equation 32). In the excel cell “C8” (solid volume flowrate), type “=C7/C9*1000”. In the excel cell “C11” (copper mass flowrate), type “=C7*C10*10” In the excel cell “C20” (solid percent), type “=100*C9*(C19-C15)/(C19*(C9-C15))” (equation 8). In the excel cell “C17” (pulp mass flowrate), type “=C7/C20*100”. In the excel cell “C13” (water mass flowrate), type “=C17-C7”. In the excel cell “C14” (water volume flowrate), type “=C13/C15*1000”. In the excel cell “C18” (pulp volume flowrate), type “=C8+C14”.

2. Calculation of unknown parameters of flow 2 (scavenger cleaner tailing) - Known parameters: o Copper grade. o Pulp specific gravity. o Water specific gravity. - In the excel cell “D10” (copper grade), type number “0.88”. - In the excel cell “D19” (pulp specific gravity), type number “1104”. - In the excel cell “D15” (water specific gravity), type number “1000”. The solid mass flowrate of the scavenger cleaner tailing is an unknown value (recycle flow) and it becomes the Excel solver variable. The starting value of the solid mass flowrate of the scavenger cleaner tailing is “100”. Joseph Kafumbila

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- In the excel cell “D7” (solid mass flowrate), type number “100” (blue color). - In the excel cell “D9” (solid specific gravity), type “=(0.0005*D10^2+0.0259*D10+2.6728)*1000” (equation 32). - In the excel cell “D8” (solid volume flowrate), type “=D7/D9*1000”. - In the excel cell “D11” (copper mass flowrate), type “=D7*D10*10” - In the excel cell “D20” (solid percent), type “=100*D9*(D19-D15)/(D19*(D9-D15))” (equation 8). - In the excel cell “D17” (pulp mass flowrate), type “=D7/D20*100”. - In the excel cell “D13” (water mass flowrate), type “=D17-D7”. - In the excel cell “D14” (water volume flowrate), type “=D13/D15*1000”. - In the excel cell “D18” (pulp volume flowrate), type “=D8+D14”.

3. Calculation of unknown parameters of flows 3 (rougher concentrate) - Known parameters: o Copper grade. o Water specific gravity. o Pulp specific gravity. - In the excel cell “E10” (copper grade), type number “14.57”. - In the excel cell “E15” (water specific gravity), type number “1000”. - In the excel cell “E19” (pulp specific gravity), type number “1234”. The solid mass flowrate of rougher concentrate will be estimate by using the mathematical expression (16). The mathematical expression (16) needs known values of copper grades of rougher feed and rougher tailing. The rougher feed copper grade is the copper grade of the mixture of plant feed (flow 1) and scavenger cleaner (flow2). -

In the excel cell “D22” (rougher feed copper grade designation), type “RT%Cu”. In the excel cell “E22” (rougher feed copper grade), type “=(C11+D11)/(C7+D7)*10)”. In the excel cell “G10” (rougher tailing copper grade), type number “0.84”. In the excel cell “E7” (solid mass flowrate), type “=(E22-G10)/(E10-G10)*(C7+D7)” ( equation 16). In the excel cell “E9” (solid specific gravity), type “=(0.0005*E10^2+0.0259*E10+2.6728)*1000” (equation 32). In the excel cell “E8” (solid volume flowrate), type “=E7/E9*1000”. In the excel cell “E11” (copper mass flowrate), type “=E7*E10*10” In the excel cell “E20” (solid percent), type “=100*E9*(E19-E15)/(E19*(E9-E15))” (equation 8). In the excel cell “E17” (pulp mass flowrate), type “=E7/E20*100”. In the excel cell “E13” (water mass flowrate), type “=E17-E7”. In the excel cell “E14” (water volume flowrate), type “=E13/E15*1000”. In the excel cell “E18” (pulp volume flowrate), type “=E8+E14”.

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4. Calculation of unknown parameters of flows 4 (rougher concentrate launder water addition) - Known parameters: o o o o -

Solid mass flowrate. Copper grade. Water volume flowrate. Water specific gravity.

In the excel cell “F7” (solid mass flowrate), type number “0”. In the excel cell “F10” (copper grade), type number “0” In the excel cell “F14” (water volume flowrate), type number “2”. In the excel cell “F15” (water specific gravity), type number “1000”. In the excel cell “F9” (solid specific gravity), type number “0”. In the excel cell “F8” (solid volume flowrate), type number “0”. In the excel cell “F11” (copper mass flowrate), type number “0”. In the excel cell “F13” (water mass flowrate), type “=F14*F15/1000”. In the excel cell “F17” (pulp mass flowrate), type “=F7+F13”. In the excel cell “F18” (pulp volume flowrate), type “=F8+F14”. In the excel cell “F19” (pulp specific gravity), type “=F17/F18*1000”. In the excel cell “F20” (solid percent), type “=F7/F17*100”.

5. Calculation of unknown parameters of flows 5 (rougher tailing) - Known parameters: o Copper grade. o Water specific gravity. o Pulp specific gravity. To close the water balance into the rougher cells, the rougher tailing specific gravity will be calculated. - In the excel cell “G15” (water specific gravity), type number “1000”. - In the excel cell “G7” (solid mass flowrate), type “=C7+D7-E7”. - In the excel cell “G9” (solid specific gravity), type “=(0.0005*G10^2+0.0259*G10+2.6728)*1000” (equation 32). - In the excel cell “G8” (solid volume flowrate), type “=G7/G9*1000”. - In the excel cell “G11” (copper mass flowrate), type “=G7*G10*10” - In the excel cell “G13” (water mass flowrate), type “=C13+D13-E13”. - In the excel cell “G14” (water volume flowrate), type “=G13/G15*1000”. - In the excel cell “G17” (pulp mass flowrate), type “=G7+G13”. - In the excel cell “G18” (pulp volume flowrate), type “=G8+G14”. - In the excel cell “G19” (pulp specific gravity), type “=G17/G18*1000”. - In the excel cell “G20” (solid percent), type “=100*G9*(G19-G15)/(G19*(G9-G15))” (equation 8). Joseph Kafumbila

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6. Calculation of unknown parameters of flows 6 (scavenger concentrate) - Known parameters: o Copper grade. o Water specific gravity. o Pulp specific gravity. - In the excel cell “H10” (copper grade), type number “4.28”. - In the excel cell “H15” (water specific gravity), type number “1000”. - In the excel cell “H19” (pulp specific gravity), type number “1152”. The solid mass flowrate of scavenger concentrate will be estimate by using the mathematical expression (16). The mathematical expression (16) needs known value of copper grade of scavenger tailing. - In the excel cell “J10” (copper grade of scavenger tailing), type number “0.67”. - In the excel cell “H7” (solid mass), type “=(G10-J10)/(H10-J10)*G7” ( equation 16). - In the excel cell “H9” (solid specific gravity), type “=(0.0005*H10^2+0.0259*H10+2.6728)*1000” (equation 32). - In the excel cell “H8” (solid volume flowrate), type “=H7/H9*1000”. - In the excel cell “H11” (copper mass flowrate), type “=H7*H10*10” - In the excel cell “H20” (solid percent), type “=100*H9*(H19-H15)/(H19*(H9-H15))” (equation 8). - In the excel cell “H17” (pulp mass flowrate), type “=H7/H20*100”. - In the excel cell “H13” (water mass flowrate), type “=H17-H7”. - In the excel cell “H14” (water volume flowrate), type “=H13/H15*1000”. - In the excel cell “H18” (pulp volume flowrate), type “=H8+H14”.

7. Calculation of unknown parameters of flows 7 (scavenger concentrate launder water addition) - Known parameters: o o o o -


In the excel cell “I7” (solid mass flowrate), type number “0”. In the excel cell “I10” (copper grade), type number “0” In the excel cell “I14” (water volume flowrate), type number “4”. In the excel cell “I15” (water specific gravity), type number “1000”. In the excel cell “I9” (solid specific gravity), type number “0”. In the excel cell “I8” (solid volume flowrate), type number “0”. In the excel cell “I11” (copper mass flowrate), type number “0”.

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-

In the excel cell “I13” (water mass flowrate), type “=I14*I15/1000”. In the excel cell “I17” (pulp mass flowrate), type “=I7+I13”. In the excel cell “I18” (pulp volume flowrate), type “=I8+I14”. In the excel cell “I19” (pulp specific gravity), type “=I17/I18*1000”. In the excel cell “I20” (solid percent), type “=I7/I17*100”.

8. Calculation of unknown parameters of flows 8 (scavenger tailing) - Known parameters: o Copper grade. o Water specific gravity. o Pulp specific gravity. To close the water balance into the scavenger cells, the scavenger tailing specific gravity will be calculated. - In the excel cell “J15” (water specific gravity), type number “1000”. - In the excel cell “J7” (solid mass flowrate), type “=G7-H7” - In the excel cell “J9” (solid specific gravity), type “=(0.0005*J10^2+0.0259*J10+2.6728)*1000” (equation 32). - In the excel cell “J8” (solid volume flowrate), type “=J7/J9*1000”. - In the excel cell “J11” (copper mass flowrate), type “=J7*J10*10”. - In the excel cell “J13” (water mass flowrate), type “=G13-H13”. - In the excel cell “J14” (water volume flowrate), type “=J13/J15*1000”. - In the excel cell “J17” (pulp mass flowrate), type “=J7+J13”. - In the excel cell “J18” (pulp volume flowrate), type “=J8+J14”. - In the excel cell “J19” (pulp specific gravity), type “=J17/J18*1000”. - In the excel cell “J20” (solid percent), type “=100*J9*(J19-J15)/(J19*(J9-J15))” (equation 8).

9. Calculation of unknown parameters of flow 9 (outlet regrind ball mill) - Known parameter: o Water specific gravity. -

In the excel cell “K15” (water specific gravity), type number “1000”.

The solid mass flowrate of the outlet regrind ball mill is an unknown value (recycle flow) and it becomes the excel solver variable). The starting value of the solid mass flowrate is “100”. - In the excel cell “K7” (solid mass flowrate), type number “100” (blue color).

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The copper grade of the outlet regrind ball mill is the same as the copper grade of hydrocyclone underflow. - In the excel cell “K10” (copper grade), type number “7.44”. - In the excel cell “K9” (solid specific gravity), type “=(0.0005*K10^2+0.0259*K10+2.6728)*1000” (equation 32). - In the excel cell “K8” (solid volume flowrate), type “=K7/K9*1000”. - In the excel cell “K11” (copper mass flowrate), type “=K7*K10*10”. The pulp specific gravity of the outlet regrind ball mill is the same as the pulp specific gravity of hydrocyclone underflow. -

In the excel cell “K19” (pulp specific gravity), type number “1740”. In the excel cell “K20” (solid percent), type “=100*K9*(K19-K15)/(K19*(K9-K15))” (equation 8). In the excel cell “K17” (pulp mass flowrate), type “=K7/K20*100”. In the excel cell “K13” (water mass flowrate), type “=K17-K7”. In the excel cell “K14” (water volume flowrate), type “=K13/K15*1000”. In the excel cell “K18” (pulp volume flowrate), type “=K8+K14”.

10. Calculation of unknown parameters of flow 10 (scavenger cleaner concentrate) - Known parameters: o Copper grade. o Water Specific gravity. o Pulp specific gravity. - In the excel cell “L10” (copper grade), type number “9.49”. - In the excel cell “L15” (water specific gravity), type number “1000”. - In the excel cell “L19” (pulp specific gravity), type number “1070”. The solid mass flowrate of scavenger cleaner concentrate is an unknown value (recycle flow) and it becomes the Excel solver variable. The staring value of the solid mass flowrate of the scavenger concentrate is “100”. - In the excel cell “L7” (solid mass flowrate), type number “100”. - In the excel cell “L9” (solid specific gravity), type “=(0.0005*L10^2+0.0259*L10+2.6728)*1000” (equation 32). - In the excel cell “L8” (solid volume flowrate), type “=L7/L9*1000”. - In the excel cell “L11” (copper mass flowrate), type “=L7*L10*10” - In the excel cell “L20” (solid percent), type “=100*L9*(L19-L15)/(L19*(L9-L15))” (equation 8). - In the excel cell “L17” (pulp mass flowrate), type “=L7/L20*100”. - In the excel cell “L13” (water mass flowrate), type “=L17-L7”. - In the excel cell “L14” (water volume flowrate), type “=L13/L15*1000”. - In the excel cell “L18” (pulp volume flowrate), type “=L8+L14”. Joseph Kafumbila

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11. Calculation of unknown parameters of flows 11 (scavenger cleaner concentrate launder water addition) - Known parameters: o o o o -


In the excel cell “C29” (solid mass flowrate), type number “0”. In the excel cell “C32” (copper grade), type number “0” In the excel cell “C36” (water volume flowrate), type number “5”. In the excel cell “C37” (water specific gravity), type number “1000”. In the excel cell “C31” (solid specific gravity), type number “0”. In the excel cell “C30” (solid volume flowrate), type number “0”. In the excel cell “C33” (copper mass flowrate), type number “0”. In the excel cell “C35” (water mass flowrate), type “=C36*C37/1000”. In the excel cell “C39” (pulp mass flowrate), type “=C29+C35”. In the excel cell “C40” (pulp volume flowrate), type “=C30+C36”. In the excel cell “C41” (pulp specific gravity), type “=C39/C40*1000”. In the excel cell “C42” (solid percent), type “=C29/C39*100”.

12. Calculation of unknown parameters of flow 13 (hydrocyclone feed) - Known parameters: o Water Specific gravity. o Pulp volume. o Pulp specific gravity. -

In the excel cell “E37” (water specific gravity), type number “1000”. In the excel cell “E40” (pulp volume flowrate), type number “1275”. In the excel cell “E41” (pulp specific gravity), type number “1301”. In the excel cell “E29” (solid mass flowrate), type “=E7+F7+H7+I7+K7+C29+L7”. In the excel cell “E33” (copper mass flowrate), type “=E11+F11+H11+I11+K11+C33+L11” In the excel cell “E32” (copper grade), type “=E33/E29/10”. In the excel cell “E31” (solid specific gravity), type “=(0.0005*E32^2+0.0259*E32+2.6728)*1000” (equation 32). - In the excel cell “E30” (solid volume flowrate), type “=E29/E31*1000”. - In the excel cell “E36” (water volume flowrate), type “=E40-E30”. - In the excel cell “E35” (water mass flowrate), type “=E36*E37/1000”.

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- In the excel cell “E39” (pulp mass flowrate), type “=E29+E35”. - In the excel cell “E42” (solid percent), type “=100*E31*(E41-E37)/(E41*(E31-E37))” (equation 8).

13. Calculation of unknown parameters of flow 12 (water) - Known parameters: o Solid mass flowrate. o Copper grade. o Water specific gravity. -

In the excel cell “D29” (solid mass flowrate), type number “0”. In the excel cell “D32” (copper grade), type number “0”. In the excel cell “D37” (water specific gravity), type number “1000”. In the excel cell “D36” (water volume flowrate), type number “=E36-C36-L14-K14-I14-H14-F14-E14”. In the excel cell “D31” (solid specific gravity), type number “0”. In the excel cell “D30” (solid volume flowrate), type number “0”. In the excel cell “D33” (copper mass flowrate), type number “0”. In the excel cell “D35” (water mass flowrate), type “=D36*D37/1000”. In the excel cell “D39” (pulp mass flowrate), type “=D29+D35”. In the excel cell “D40” (pulp volume flowrate), type “=D30+D36”. In the excel cell “D41” (pulp specific gravity), type “=D39/D40*1000”. In the excel cell “D42” (solid percent), type “=D29/D39*100”.

14. Calculation of unknown parameters of flows 14 (hydrocyclone underflow) - Known parameters: o o o o -

Solid mass. Copper grade. Pulp specific gravity. Water specific gravity.

In the excel cell “F29” (solid mass flowrate), type “=K7”. In the excel cell “F32” (copper grade), type “=K10”. In the excel cell “F41” (pulp specific gravity), type “=K19”. In the excel cell “F37” (water specific gravity), type number “1000”. In the excel cell “F31” (solid specific gravity), type “=(0.0005*F32^2+0.0259*F32+2.6728)*1000” (equation 32). In the excel cell “F30” (solid volume flowrate), type “=F29/F31*1000”. In the excel cell “F33” (copper mass flowrate), type “=F29*F32*10”. In the excel cell “F42” (Solid percent), type “=100*F31*(F41-F37)/(F41*(F31-F37))” (equation 8). In the excel cell “F39” (pulp mass flowrate), type “=F29/F42*100”. In the excel cell “F35” (water mass flowrate), type “=F39-F29”.

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- In the excel cell “F36” (water volume flowrate), type “=F35/F37*1000”. - In the excel cell “F40” (pulp volume flowrate), type “=F30+F36”.

15. Calculation of unknown parameters of flows 15 (hydrocyclone overflow) - Known parameters: o Water specific gravity. - In the excel cell “G37” (water specific gravity), type number “1000”. To close the copper and water balance into the hydrocyclone, the copper grade and the pulp specific gravity will be calculated. -

In the excel cell “G29” (solid mass flowrate), type “=E29-F29”. In the excel cell “G33” (copper mass flowrate), type “=E33-F33”. In the excel cell “G32” (copper grade), type “=G33/G29/10”. In the excel cell “G31” (solid specific gravity), type “=(0.0005*G32^2+0.0259*G32+2.6728)*1000” (equation 32). In the excel cell “G30” (solid volume flowrate), type “=G29/G31*1000”. In the excel cell “G35” (water mass flowrate), type “=E35-F35”. In the excel cell “G36” (water volume flowrate), type “=G35/G37*1000”. In the excel cell “G39” (pulp mass flowrate), type “=G29+G35”. In the excel cell “G40” (pulp volume flowrate), type “=G30+G36”. In the excel cell “G41” (pulp specific gravity), type “=G39/G40*1000”. In the excel cell “G42” (solid percent), type “=100*G31*(G41-G37)/(G41*(G31-G37))” (equation 8).

16. Calculation of unknown parameters of flow 16 (Re-cleaner tailing) - Known parameters: o Copper grade. o Water specific gravity. o Pulp specific gravity. - In the excel cell “H32” (copper grade), type number “1.18”. - In the excel cell “H37” (water specific gravity), type number “1000”. - In the excel cell “H41” (pulp specific gravity), type number “1072”. The solid mass flowrate of Re-cleaner tailing is an unknown value (recycle flow) and it becomes the solver variable. The starting value of the solid mass flowrate is “100”. - In the excel cell “H29” (solid mass flowrate), type number “100”.

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- In the excel cell “H31” (solid specific gravity), type “=(0.0005*H32^2+0.0259*H32+2.6728)*1000” (equation 32). - In the excel cell “H30” (solid volume flowrate), type “=H29/H31*1000”. - In the excel cell “H33” (copper mass flowrate), type “=H29*H32*10”. - In the excel cell “H42” (solid percent), type “=100*H31*(H41-H37)/(H41*(H31-H37))” (equation 8). - In the excel cell “H39” (pulp mass flowrate), type “=H29/H42*100”. - In the excel cell “H35” (water mass flowrate), type “=H39-H29”. - In the excel cell “H36” (water volume), type “=H35/H37*1000”. - In the excel cell “H40” (pulp volume flowrate), type “=H30+H36”.

17. Calculation of unknown parameters of flows 17 (cleaner concentrate) - Known parameters: o Copper grade. o Water specific gravity. To close the water balance into the cleaner cells, the pulp specific gravity of cleaner concentrate will be calculated from the known value of cleaner tailing pulp specific gravity. - In the excel cell “I32” (copper grade), type number “19.15”. - In the excel cell “I37” (water specific gravity), type number “1000”. The cleaner flotation feed is the mixture of the hydrocyclone overflow and the re-cleaner tailing. The copper grade of cleaner flotation feed is calculated. - In the excel cell “H44” (cleaner copper grade designation), type “CF %Cu”. - In the excel cell “I44” (cleaner copper grade), type “=(G33+H33)/((G29+H29)*10)”. The estimation of the solid mass flowrate of cleaner concentrate needs the value of copper grade of cleaner tailing. - In the excel cell “K32” (cleaner tailing copper grade), type number “2.57”. - In the excel cell “I29” (solid mass flowrate), type “=(I44-K32)/(I32-K32)*(G29+H29)” ( equation 16). - In the excel cell “I31” (solid specific gravity), type “=(0.0005*I32^2+0.0259*I32+2.6728)*1000” (equation 32). - In the excel cell “I30” (solid volume flowrate), type “=I29/I31*1000”. - In the excel cell “I33” (copper mass flowrate), type “=I29*I32*10” - In the excel cell “I35” (water mass flowrate), type “=G35+H35-K35”. - In the excel cell “I36” (water volume flowrate), type “=I35/I37*1000”. - In the excel cell “I39” (pulp mass flowrate), type “=I29+I35”. - In the excel cell “I40” (pulp volume flowrate), type “=I30+I36”. - In the excel cell “I41” (pulp specific gravity), type “=I39/I40*1000”. - In the excel cell “I42” (solid percent), type “=100*I31*(I41-I37)/(I41*(I31-I37))” (equation 8). Joseph Kafumbila

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18. Calculation of unknown parameters of flows 18 (cleaner concentrate launder water addition) - Known parameters: o o o o -


In the excel cell “J29” (solid mass flowrate), type number “0”. In the excel cell “J32” (copper grade), type number “0” In the excel cell “J36” (water volume flowrate), type number “4”. In the excel cell “J37” (water specific gravity), type number “1000”. In the excel cell “J31” (solid specific gravity), type number “0”. In the excel cell “J30” (solid volume flowrate), type number “0”. In the excel cell “J33” (copper mass flowrate), type number “0”. In the excel cell “J35” (water mass flowrate), type “=J36*J37/1000”. In the excel cell “J39” (pulp mass flowrate), type “=J29+J35”. In the excel cell “J40” (pulp volume flowrate), type “=J30+J36”. In the excel cell “J41” (pulp specific gravity), type “=J39/J40*1000”. In the excel cell “J42” (solid percent), type “=J29/J39*100”.

19. Calculation of unknown parameters of flows 19 (cleaner tailing) - Known parameters: o Water specific gravity. To close the water balance into the scavenger cleaner cells, the pulp specific gravity of cleaner tailing will be calculated. - In the excel cell “K37” (water specific gravity), type “1000”. - In the excel cell “K29” (solid mass flowrate), type “=G29+H29-I29” - In the excel cell “K31” (solid specific gravity), type “=(0.0005*K32^2+0.0259*K32+2.6728)*1000” (equation 32). - In the excel cell “K30” (solid volume flowrate), type “=K29/K31*1000”. - In the excel cell “K33” (copper mass flowrate), type “=K29*K32*10” - In the excel cell “K35” (water mass flowrate), type “=L13+D13”. - In the excel cell “K36” (water volume flowrate), type “=K35/K37*1000”. - In the excel cell “K39” (pulp mass flowrate), type “=K29+K35”. - In the excel cell “K40” (pulp volume flowrate), type “=K30+K36”. - In the excel cell “K41” (pulp specific gravity), type “=K39/K40*1000”. - In the excel cell “K42” (solid percent), type “=100*K31*(K41-K37)/(K41*(K31-K37))” (equation 8). Joseph Kafumbila

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20. Calculation of unknown parameters of flows 20 (Re-cleaner concentrate) - Known parameters: o Copper grade. o Water specific gravity. To close the water balance into the re-cleaner cells, the pulp specific gravity of the re-cleaner concentrate will be calculated. -

In the excel cell “L32” (copper grade), type number “23.78”. In the excel cell “L37” (water specific gravity), type number “1000”. In the excel cell “L29” (solid mass flowrate), type “=(I32-H32)/(L32-H32)*I29” ( equation 16). In the excel cell “L31” (solid specific gravity), type “=(0.0005*L32^2+0.0259*L32+2.6728)*1000” (equation 32). In the excel cell “L30” (solid volume flowrate), type “=L29/L31*1000”. In the excel cell “L33” (copper mass flowrate), type “=L29*L32*10”. In the excel cell “L35” (water mass flowrate), type “=I35+J35-H35”. In the excel cell “L36” (water volume flowrate), type “=L35/L37*1000”. In the excel cell “L39” (pulp mass flowrate), type “=L29+L35”. In the excel cell “L40” (pulp volume flowrate), type “=L30+L36”. In the excel cell “L41” (pulp specific gravity), type “=L39/L40*1000”. In the excel cell “K42” (solid percent), type “=100*L31*(L41-L37)/(L41*(L31-L37))” (equation 8).

21. Calculation of solver constraints -

In the excel cell “E46” (constraint 1), type “=E39-E40*E41/1000”. In the excel cell “E47” (constraint 2), type “=K29-L7-D7”. In the excel cell “E48” (constraint 3), type “=L7-(K32-D10)/(L10-D10)*K29”. In the excel cell “E49” (constraint 4), type “=I29-L29-H29”. At this level, the non-optimized results are given in Table 5 as it appears on Excel spreadsheet.

22. Excel solver program Excel solver program execution is as follows: 1) On the ‘Data’, in the ‘Analysis group’ click solver (if the solver command is not available, you must activate the solver add-in). 2) In the ‘Set objective’ box, enter the cell reference “E46” of simulation constraint table. 3) Click “Value of” and then type the number “0” in the box.

Joseph Kafumbila

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A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

B

C

D

E

Table 5A


Joseph Kafumbila

G

H

I

J

K

L


2 SCT

3 RC

4 RCW

5 RT

6 SC

7 SCW

8 ST

9 ORBM

10 SCC

1243.0 456.2 2724.6 1.93 23990.1

100.0 37.1 2696.0 0.88 880.0

99.0 31.4 3156.3 14.57 14420.2

0.0 0.0 0.0 0.00 0.0

1244.0 461.6 2694.9 0.84 10449.9

58.6 21.0 2792.8 4.28 2507.4

0.0 0.0 0.0 0.00 0.0

1185.5 440.6 2690.4 0.67 7942.5

100.0 34.6 2893.2 7.44 7440.0

100.0 33.7 2963.6 9.49 9490.0

2835.8 2835.8 1000.0

567.8 567.8 1000.0

257.6 257.6 1000.0

2.0 2.0 1000.0

3146.0 3146.0 1000.0

226.4 226.4 1000.0

4.0 4.0 1000.0

2919.6 2919.6 1000.0

53.9 53.9 1000.0

912.8 912.8 1000.0

4078.9 3292.1 1239.0 30.47

667.8 604.9 1104.0 14.97

356.6 289.0 1234.0 27.76

2.0 2.0 1000.0 0.00

4390.1 3607.7 1216.9 28.34

285.0 247.4 1152.0 20.55

4.0 4.0 1000.0 0.00

4105.1 3360.2 1221.7 28.88

153.9 88.4 1740.0 64.99

1012.8 946.5 1070.5 9.87

RF%Cu

1.85

Table 5B


F


12 W

13 HF

14 HUF

15 HOF

16 ReCT

17 CC

18 CCW

19 CT

20 ReCC

0.0 0.0 0.0 0.00 0.0

0.0 0.0 0.0 0.00 0.0

357.6 120.7 2962.9 9.47 33857.5

100.0 34.6 2893.2 7.44 7440.0

257.6 86.1 2991.1 10.26 26417.5

100.0 37.0 2704.1 1.18 1180.0

111.0 33.1 3352.1 19.15 21261.8

0.0 0.0 0.0 0.00 0.0

246.5 89.9 2742.7 2.57 6335.8

88.3 24.7 3571.4 23.78 20993.4

5.0 5.0 1000.0

-307.4 -3.07.4 1000.0

1154.3 1154.3 1000.0

53.9 53.9 1000.0

1100.5 1100.5 1000.0

838.3 838.3 1000.0

458.2 458.2 1000.0

4.0 4.0 1000.0

1480.6 1480.6 1000.0

-376.1 -376.1 1000.0

5.0 5.0 1000.0 0.00

-307.4 -307.4 1000.0 0.00

1511.9 1275.0 1301.0 34.92

153.9 88.4 1740.0 64.99

1358.0 1186.6 1144.5 18.97

938.3 875.3 1072.0 10.66

569.2 491.3 1158.6 19.51

4.0 4.0 1000.0 0.00

1727.1 1570.5 1099.7 14.27

-287.8 -287.8 819.1 -30.67

CFCu%

7.72

Simulation solver constraint table Constraint 1 -146.818 Constraint 2 46.528 Constraint 3 51.611 Constraint 4 -77.254

Page 28

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

B

C

D

E

Table 6A


Joseph Kafumbila

G

H

I

J

K

L


2 SCT

3 RC

4 RCW

5 RT

6 SC

7 SCW

8 ST

9 ORBM

10 SCC

1243.0 456.2 2724.6 1.93 23990.1

89.2 33.1 2696.0 0.88 785.4

98.9 31.3 3156.3 14.57 14415.6

0.0 0.0 0.0 0.00 0.0

1233.3 457.6 2694.9 0.84 10359.9

58.1 20.8 2792.8 4.28 2485.8

0.0 0.0 0.0 0.00 0.0

1175.2 436.8 2690.4 0.67 7874.1

404.2 139.7 2893.2 7.44 30073.6

21.8 7.4 2963.6 9.49 2068.4

2835.8 2835.8 1000.0

506.7 506.7 1000.0

257.5 257.5 1000.0

2.0 2.0 1000.0

3085.1 3085.1 1000.0

224.5 224.5 1000.0

4.0 4.0 1000.0

2860.6 2860.6 1000.0

217.7 217.7 1000.0

199.0 199.0 1000.0

4078.9 3292.1 1239.0 30.47

596.0 539.8 1104.0 14.97

356.5 288.9 1234.0 27.76

2.0 2.0 1000.0 0.00

4318.4 3542.7 1218.9 28.56

282.6 245.3 1152.0 20.55

4.0 4.0 1000.0 0.00

4035.8 3297.4 1223.9 29.12

621.9 357.4 1740.0 64.99

220.7 206.3 1070.0 9.87

RF%Cu

1.86

Table 6B


F


12 W

13 HF

14 HUF

15 HOF

16 ReCT

17 CC

18 CCW

19 CT

20 ReCC

0.0 0.0 0.0 0.00 0.0

0.0 0.0 0.0 0.00 0.0

583.0 199.3 2926.0 8.41 49043.4

404.2 139.7 2893.2 7.44 30073.6

178.8 59.5 3003.8 10.61 18969.8

17.5 6.5 2704.1 1.18 206.0

85.2 25.4 3352.1 19.15 16322.0

0.0 0.0 0.0 0.00 0.0

111.0 40.5 2742.7 2.57 2853.8

67.8 19.0 3571.4 23.78 16116.0

5.0 5.0 1000.0

166.1 166.1 1000.0

1075.7 1075.7 1000.0

217.7 217.7 1000.0

858.0 858.0 1000.0

146.4 146.4 1000.0

298.7 298.7 1000.0

4.0 4.0 1000.0

705.7 705.7 1000.0

156.3 156.3 1000.0

5.0 5.0 1000.0 0.00

166.1 166.1 1000.0 0.00

1658.8 1275.0 1301.0 35.15

621.9 357.4 1740.0 64.99

1036.8 917.6 1130.0 17.25

163.8 152.8 1072.0 10.66

383.9 324.1 1184.5 22.20

4.0 4.0 1000.0 0.00

816.7 746.2 1094.6 13.60

224.1 175.3 1278.3 30.24

CFCu%

9.77

Simulation solver constraint table Constraint 1 0.000 Constraint 2 0.000 Constraint 3 0.000 Constraint 4 0.000

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4) In the “By Changing Variable Cells” box, enter the reference for each solver variable (blue color in Table 5). Separate the references with commas (English version). 5) In the ‘Subject to the constraints’ box, enter solver constraints by doing the following: a) In the ‘Solver Parameters’ dialog box, click ‘Add’. b) In the ‘Cell Reference’ box, enter the cell reference “E47” of simulation constraint table c) Click the ‘relationship’ ‘=‘, in the ‘Constraint’ box, type the number ‘0’. d) Click ‘Add’ for the solver constraint 3. When the last solver constraint 4 is added (excel cell ‘E49’), click ‘OK’ to return to ‘Solver Parameters’ dialog box. 6) Click ‘Solve’. To keep the solution values on the worksheet, in the ‘Solver Results’ dialog box, click ‘Keep solver solution’. At this level, the optimized results are given in Table 6 as it appears on Excel Microsoft spreadsheet. Figure 6 gives the summary of results. Table 7 gives comparison between sampling data and material balance results for the sampling data Table 7: Sampling data vs material balance results Number

4 6 12 14 15 16

Joseph Kafumbila

Designation

Rougher tailing Scavenger tailing Hydrocyclone overflow Cleaner concentrate Cleaner tailing Re-cleaner concentrate

Sampling Data Cu grade SGp 3 Kg/m % 1,256 1,264 1,123 11.29 1,203 1,095 1,275

Material balance results Cu grade SGp 3 Kg/m % 1,218.9 1,223.9 1,130.0 10.61 1,184.5 1,094.6 1,278.3

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Figure 6: Summary o simulation results

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4.

References 1. Luis A. Cistermas, David A. Mendez, Edelmira D. Galvez, Rodrigo E. Jorquera, A MILP model for design of flotation circuits with bank/column and regrind/no regrind selection, International journal of mineral processing, 79 (2006) 253-263 2. Metso (2006), basics in Minerals Processing, 5th Edition. Click on the link. Pre-feasibility studies of copper hydro-metallurgical plants

Pre-feasibility study is conducted first to sort out applicable scenarios. It is desirable to do some prefeasibility studies of your own or with help of consultant before evaluation feasibility study done by engineering offices. If you find out early that the proposed plant design project idea is not feasible, it will save your money. The paper is the metallurgical engineering paper that gives the opportunity to do pre-feasibility study of copper hydrometallurgical plants of your own. The paper gives the update of the design criteria and metallurgical constraints of copper solvent extraction and the models of preliminary flow diagram mass balance of heap leaching, conventional, split circuits. The models of preliminary flow diagram mass balance are executable on Microsoft Excel spreadsheet. https://fr.scribd.com/document/358056400/Pre-feasibility-Studies-of-Copper-HydrometallurgicalPlants

Joseph Kafumbila

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Material Balance in Froth Flotation using Microsoft Excel Solver

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