POPULATION BALANCE IN BALL MILL

Transcription

1 POPULATION BALANCE IN BALL MILL Varun Sogani 1, Pranjal Somani 2, Nalin Singla 3 and Rishabh Mathur 4 1,2,3 Dept. of Chemical Engineering, IIT Delhi 4 Dept. of Biochemical Engineering and Biotechnology, IIT Delhi Abstract Grinding using ball mill is one of the important step in powder processing industry. The ball mill is based on the principle of impact size reduction and due to so many variables and parameters being involved, scale-up and predictive models are difficult to formulate which can give accurate results. Population balance concept provides a useful framework describing and solving powder size distributions and other mill operating parameters. In this work, we have used the breakage function and selection function model approach to simulate the output size distribution using the experimentally determined mill parameters and input size distribution and compared the results to actual output in our pilot plant. The model is descriptive rather than predictive but it is demonstrated that it can be used for designing ball mill within acceptable error limits. Index Terms Wi is the mass fraction of material in i th size class Si is the selection function for ith class bij is the breakage function from j to i class t is the time p is product matrix representing product fractions M is the milling matrix f is feed matrix representing feed size distribution B is max. ball diameter f80 is 80% of mill feed size passing in micron W is bond s work index of f80 S is specific gravity of feed Cs is % critical speed D is effective mill diameter K is a constant U = Ball Loading W = weight of balls ρ = bulk density of balls D = Diameter of mill L = Length of mill particles into smaller ones. It was invented and first used in 1870 s and since has been used extensively for grinding materials like coal, pigments, felspar and many other ores and ceramic materials. Ball mill comes in generally cylindrical design which can rotate about a horizontal axis. There are spherical balls inside (generally made of steel) which rotate with the outer body and fall after a certain height to grind the material inside. The ball mill is based on the principle of impact size reduction and due to so many variables and parameters being involved, scale-up and prediction models are difficult to formulate which can give accurate results. Hence, a simplistic empirical approach is used referred to as population balance over the whole size distribution of inlet grinding material. The ball mills can be of two types based on input conditions of material to be grinded- 1) wet ball millinput is mixed with wetting agent generally water. Generally it produces smaller output size than dry ball mill in which the input is fed without addition of water. On the basis of operation, ball mills can be divided into 2 classes- 1. Batch grinding- (Figure.1) as the name suggests, it is a batch process where the grinding material is kept inside for optimum time for energy efficient size reduction. I. INTRODUCTION Ball mill is a mechanical device (one type out of many grinders) used for efficient conversion of large IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 207

2 Figure 1: Batch Grinding Setup 2. Continuous grinding- (Figure 2) the feed is continuously fed in and output is taken out. The output size distribution depends on average residence time of particles inside the continuous ball mill. The output generated by the mill depends on several parameters like mill dimensions, angular velocity, the mill input rate and characteristics of the material, the work index of the material, shaft power of mill. Mill capacity is defined as a ratio of the mill shaft power and the energy consumed in the grinding process. Classifiers, which separate the different particle sizes, are used after the grinding process to separate the different size fractions, and sometimes to reflux the coarse particles back into the mill for further grinding. III. A. Basic Breakage Mechanism THEORY The size reduction of particles occur through 3 basic mechanisms- Figure 2: Continuous Grinding Setup Abrasion - occurs when low intensity stresses are applied on smaller area of a single big particle and results in formation of many smaller particles. II. DESIGN OF A MILL The design of a mill for optimum output and energy usage is of paramount importance which depends on our needs (load, input, output, material properties, operating parameters etc.). The main physical feature is the length to diameter ratio of the cylindrical portion which generally ranges from.5 to 3.5..The basic characteristics of the mill are mass and size, wear rate, influence on the particle breakage rate and energy efficiency of the milling process. Based on the speed of rotation of the mill three different types of operations can be followed: slow rotation (cascading), fast rotation (contracting) and very fast rotation (centrifugation). Each mode has a specific trajectory of motion of the input in the mill and a different impact of the balls on the ground material. The particle size distribution of output depends on the following basic factors: Characteristics of the material input in the mill (size distribution of the charge, mass, density and volume, hardness); Characteristics of the balls (mass, density, ball size distribution); Rotation velocity of the mill; Slurry density for wet grinding operation Cleavage - slow and intense stresses results in breakage into half the sizes. Fracture - intense stresses applied on a particle as a whole and results in wide particle size distribution. B. Theoretical Approach The theory is based on probabilistic view of fragmentation of particles of a given size into a known distribution. It is basically characterized by 2 functions: Selection Function (also known as probability of breakage) - Si represents the specific breakage rate of particles in class size i, i.e. it represents the fraction of particles that that are going to break in the milling operation. Breakage Function (also called distribution function) - bij represents the distribution of particle sizes obtained after breakage in size class j. That means any given bij is the mass fraction of material j that goes into the size class i after the operation of the breakage. C. Population Balance in a Ball Mill IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 208

3 Applying simple mass balance over the ball mill, and assuming perfect mixing (no diffusion term) the following equation holds: dw i (l, t) dt i 1 = S i w i (l, t) + S j b ij w j (l, t) i=1 After the assumption of perfectly mixed model, the dependence on horizontal axis of the mass function is neglected that means the diffusion term and convective term which makes it second order disappears. dw i (t) dt i 1 = S i w i (t) + S j b ij w j (t) i=1 This simple mass balance can be explained as follows; the left hand side term represents the rate of change of mass fraction of material in i ht size class. The right hand side term has two parts; first tells us the fraction going out from i th size class due to fragmentation of particles in this size range; the second term represents the increment in fraction of i th size class due to breakage in j th size class and distribution in i class. This effectively gets reduced to: Rate of change = Rate of formation Rate of decomposition The selection and breakage functions parameters are empirically determined and modeled as follows: D. Selection Function Is modeled as following equation, where α, δ, µ, and λ are experimentally determined parameters. S i = E. Breakage Function B i,j = [ x i 1 x j α[ x i x 1 ] δ [1 + x i μ ]λ ] γ + [1 ][ x i 1 ] β x j Again, Φ, γ and β are experimentally determined parameters. Sometimes these equations are also written in a matrix form [p] = [M][f] The matrix representation is used when we have a classifier which separates the coarse particles and recycles them back to the mill for further milling operation to increase the efficiency of the plant. IV. METHODOLOGY To predict the output size distribution we need the values of several parameters used in the model which were used to define selection and breakage function. This parameter dependence of selection function and breakage function is a lot which makes it descriptive rather than predictive. These parameters are actually determined by experimental methods and then empirically fitting the data for a small pilot plant and then scale up procedures are used to simulate the values for the main plant. After getting the parameters from empirical relations, the differential equation is solved numerically and outlet particle size distribution with respect to time is plotted through a Matlab code. The set of values obtained from the experimental work done in the pilot plant are: Selection function Breakage Function S 1 = /min β = α = γ = µ = 1.4 mm δ = 0.0 λ = Φ = Table 1: Estimated Parameters for ball mill Output size distribution of particles was simulated for a given the input size distribution (used as a boundary condition in the differential equation)assuming that the selection function and breakage function are independent of time which is almost true for batch milling and also the parameters of the selection function and breakage function are assumed independent of the mill parameters. In the following section we provide information related to mill parameters importance: Operating Variables of ball mill- 1. Mill size 2. no. of chambers 3. Diaphragm design details 4. Shape 5. % opening 6. Ball loading 7. material loading 8. Speed of rotation 9. Ball size distribution 10. Classifiers- liners recirculation ratio 11. air flow/ gravity flow IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 209

4 12. concentration of slurry 13. feed size Ball Loading = Ball Volume Mill Volume Generally the value of ball loading varies from 0.45 to 0.5 in ball milling and 0.25 to 0.35 in continuous milling. Outlet size distribution for different average residence times has been simulated and as can be seen the particles in highest size range decrease continuously and lower size range increases with a maxima occurring in midrange as it increases first and then decreases eventually. These also confirm with the actual observations of the pilot plant run. U = W balls πd 2 L 4 Material Loading = ρ bulk Volume Of Powder Volume of voids in the bed Generally the value of material loading varies from 0.5 to 0.2 for both continuous and batch milling. Defining the Ball size for the operation is usually given by the Bond s Formula. OPERATING MILL PARAMETERS Most importantly it is necessary to determine the feed size distribution that is going to be input in the system. Using the Rosin Rammler equation parameters are evaluated. Thus the ball size distribution is evaluated. Figure 3: Number fraction for average residence time = 10s for Set-1 W balls = 0.785ρ BB DLU N balls = 0.785ρ BB DLU W sb Rotational speed of the mill is also an important factor whose value ranges from.7 to.8 of the critical speed of the mill. At critical speed the balls would not fall rather obey complete revolution due to high centrifugal force. Figure 4: Number fraction for average residence time = 5s for Set-1 N C = 42.2 D d max These are the general loading design equations for a ball mill. V. RESULTS IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 210

5 ACKNOWLEDGMENT Figure 5: Number fraction for average residence time = 10s for Set-2 We thank Chemical Department, IIT Delhi for their support in experimental data determination and various professors and colleagues for their valuable inputs. REFERENCES Figure 6: Number fraction for average residence time = 5s for Set-2 Input particle sizes for the two sets are different Particle size Number Frequency (Set 1) Number Frequency (Set2) [1] Joel Ducoste An Introduction to population balance, MBR Training Seminar, Ghent University, 2008 [2] Narni Nageswara Rao Simulations for modelling of population balance equations of particulate processing using discrete particle model (DPM) [3] L.G. Austin, K.Shoji, P.T. Luckie The effect of ball size on ball mill performance Elsevier Powder Technology, vol. 14, Issue 1, pp , May [4] S.Y. Sonaye, Dr. R. N. Baxi Particle size measurement and analysis of flour IJERA, vol.2, Issue 3, pp , May 2012 [5] K. Viswanathan, B.P.Mani A new particle size distribution Ind. Eng. Chem. Process Des. Dev., vol 21, Issue 4, pp , October 1982 [6] Sheryl Ehrman (June 2005). Population balance modeling an application in particle technology [Online].Available: alppt.pdf [7] Mingwei Gao, Eric Forssberg Prediction of product size distributions for a stirred ball mill Powder Technology., vol 1. Issue 84, pp , 1995 [8] K Shoji, P.T. Luckie The effect of ball size on mill performance Powder Technology., vol 1. Issue 14, pp , 1976 [9] J.A. Herbst, D.W. Fuerstenau Scale up procedure for continuous grinding mill IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 211

6 design using population balance models International Journal of Mineral Processing., vol 7., pp. 1-31, 1980 IJIRT INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY 212