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josephkafumbila

This paper will give a procedure of operating plant material balance using Microsoft Excel Solver on Excel spreadsheet. Operating plant material balance purpose is to produce a picture of the state of an operating plant. An example solving a copper concentrator flotation circuit is presented. This method is simple and easy to make from simple to a complex froth flotation circuit.

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MATERIAL BALANCE IN FROTH FLOTATION USING MICROSOFT EXCEL SOLVER Revised–February, 2017 PROCESS DESIGNER JOSEPH KAFUMBILA

© 2017 Joseph Kafumbila jokafumbila@hotmail.com

Material balance in froth flotation using Microsoft Excel Solver

Joseph Kafumbila

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Contents

1.INTRODUCTION..................................................................................................................... 32.MATERIAL BALANCE.............................................................................................................. 32.1.FLOW PARAMETERS..................................................................................................................... 32.1.1. SOLID.................................................................................................................................. 32.1.2.WATER................................................................................................................................ 62.1.3.PULP................................................................................................................................... 62.2.UNIT OPERATION OF FROTH FLOTATION CIRCUIT................................................................................ 72.2.1.FROTH FLOTATION UNIT........................................................................................................... 72.2.2.HYDROCYCLONE..................................................................................................................... 82.2.3.BALL MILL............................................................................................................................. 93.MATERIAL BALANCE IN AN OPERATING PLANT ................................................................... 113.1.OPERATING PLANT DESCRIPTION.................................................................................................. 113.2.OPERATING PLANT DATA............................................................................................................ 133.3.MATERIAL BALANCE USINGMICROSOFTEXCELSOLVER.................................................................... 143.3.1.MATERIAL BALANCE SIMULATION TABLE................................................................................... 143.3.2.MATERIAL BALANCE SIMULATION PROCEDURE........................................................................... 164.REFERENCES........................................................................................................................ 31

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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 3 Solid is characterized by a mass (M) expressed in (ton) and a volume (V) expressed in (m ). The ୱ ୱ 3 specific gravity (SG) is the ratio of mass on volume of solid. Equation (1) gives the) expressed in (kg/m ୱ mathematical expression that links mass, volume and specific gravity of solid. ౩ SG1000= x ୱ v ౩(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. 3 Water volume is the ratio of water mass on water specific gravity (1,000 kg/m ). 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 unit Equations values Solid mass kg 0.141 Pulp mass kg 1.089 Pulp volume l 1.000 Water mass kg Pulp mass–0.948Solid mass Water Volume l Water mass/ Water specific gravity 0.948 Solid volume l Pulp volume–0.052Water volume 3 Solid SG kg/m (Solid mass / Solid volume) x 1000 2,711.54 B.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 3 3 m which is the sum of mineral volumes, the value of solid specific gravity is 2,779.55 kg/m . Table 2: Solid specific gravity from solid mineralogical composition Minerals Mineralogical Mineral Specific gravity Mineral Specific gravity composition masses of mineral volumes of solid 3 3 % t t/m m3 kg/m Cu2(OH)2(CO33.95915.836 4.00 ) 1.584 Cu3(PO4)2.Cu2(OH)40.3471.456 4.20 0.146 2CuO.2SiO2.3H20.904O 0.199 1.989 2.20 CuO 0.503 5.029 6.40 0.786 CuS 0.005 0.055 4.68 0.012 Cu2S 0.005 0.055 5.65 0.010 CuFeS20.055 4.20 0.005 0.013 CoOOH 0.507 5.070 4.00 1.268 FeO(OH) 1.954 19.543 3.65 5.354 Ni(OH)2 0.001 0.010 4.10 0.002 CaCO3.MgCO33.1589.000 2.85 0.900 MnO2 0.127 1.266 4.85 0.261 ZnS 0.007 0.067 4.00 0.017 SiO2 77.139 291.092771.393 2.65 UO30.004 0.004 0.040 10.97 Mg2SiO417.08153.805 3.15 5.381 Ca2SiO40.0410.110 2.71 0.011 CaCl2 0.025 0.250 2.15 0.116 Al2SiO5114.710 3.25 11.471 35.295 Cr2O3 0.026 0.256 5.22 0.049 CdO 0.001 0.006 8.15 0.001 Total 1000 359.77 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 chemicalelement of index “k” (M) expressed in (kg) into a solid and a grade of chemical elementof index “k” (T) expressed in (%) into a k k solid. Equation (2) gives mathematical expression that links mass ofchemical element of index “k”, grade of chemicalelement of index “k” andsolid mass. K M=M1000x x k ୱ ଵ(2)

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2.1.2.Water3 Water is characterized by a mass (M) expressed in (ton) and a volume (V). The) expressed in (m 3 specific gravity (SG) expressed in (kg/m ) is the ratio of mass to volume of water. The value of water specific 3 gravity is 1,000 kg/m . 2.1.3.Pulp The pulp is a mixture of solid and water. The pulp will be characterized by a mass “M” expressed in P 3 3 (ton), a volume “V” expressed in (m ) and a specific gravity “SG” expressed in (kg/m). Equation (3) gives P P mathematical expression that links mass, volume and specific gravity of pulp. p SG1000= x P v p(3) Pulp mass is a sum of solid and water masses. Mathematical expression of this principle is given by equation (4). M=M+MP ୗ (4) Pulp volume is a sum of solid and water volumes. Mathematical expression of this principle is given by equation (5). V=V+VP ୗ (5) Solid mass can also be calculated from pulp volume and specific gravities of pulp, solid and water. Equation (6) gives the mathematical expression. ሺୗg−ଵሻ p M= xSGxV/ 1000 ୱ ୱ ୮ s(6) ሺୗg−ଵሻ Volume percent of solid in the pulp expressed in (%) is given by equation (7). v ౩ ୱ C100 (%)= x ୴ v (7) Weight percent of solid in the pulp expressed in (%) is given by equation (8). g୶ ሺୗg ଵ ୶ ୗs − ଵሻ s ୱ C100 == x ୵ p s(8) ୗg ୶ ሺୗg− ଵሻ

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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: f c ୲ (9) M=M+M୮ ୮ ୮ f c ୲ (10) V=V+V୮ ୮ ୮ f c ୲ (11) M=M+Mୗ ୗ ୗ f c ୲ (12) V=V+Vୗ ୗ ୗ f c ୲ (13) M=M+M f c ୲ (14) V=V+V

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f f c c ୲ ୲ (15) MxTx 10 =MxTx 10 +MxTx 10 ୗ k ୗ k ୗ k Froth flotation metallurgical performances are given by the following mathematical expression: Mass pull (%) is the ratio of concentrate solid mass on feed solid mass. ç F ౪ ሺ − ሻ s K K ss pull = MaFx 100 =ç ౪(16)x 100 ሺ − ሻ s K K Metal recovery (%) is the ratio of element mass in the concentrate on element mass in the feed. ç ç F ౪ ሺ − ሻ K K K K Metal recovery = x 100 = x x 100 (17) F F ç ౪ ሺ − ሻ K K K K 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: f ୳ ୭ (18) M=M+M୮ ୮ ୮ f ୳ ୭ (19) V=V+V୮ ୮ ୮ f ୳ ୭ (20) M=M+Mୗ ୗ ୗ f ୳ ୭ (21) V=V+Vୗ ୗ ୗ f ୳ ୭ (22) M=M+M f ୳ ୭ (23) V=V+V f f ୳ ୳ ୭ ୭ (24) MxTx 10 =MxTx 10 +MxTx 10 ୗ k ୗ k ୗ k

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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: i ୭ (25) M=M୮ ୮ i ୭ (26) V=V୮ ୮ i ୭ (27) M=Mୗ ୗ i ୭ (28) V=Vୗ ୗ

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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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