Energy-Mass-Size balance model for dynamic control of a wet open circuit grinding mill: Part I - Model development

Transcription

1 ISSN (e): Volume, 06 Issue, 02 Feruary 2016 International Journal o Computational Engineering Research (IJCER) Energy-Mass-Size alance model or dynamic control o a wet open circuit grinding mill: Part I - Model development Augustine B. Makokha Moi University, School o Engineering, Astract Majority o traditional mill simulation models have een successully applied in mill circuit design and optimization. However, only limited success has een reported with respect to dynamic mill control. This is partly attriutale to the act that these models are ased on steady state analysis and ail to account or energy alance inside the mill. In this paper, an integrated model has een developed ased on energy-mass-size alance or dynamic simulation o open circuit wet grinding mills under MATLAB environment. The model comines inormation rom energy-mass alance, material reakage mechanisms, undamental material properties and the milling conditions in a simple and clear representation o the physics and thermodynamics o the wet milling process, therey addressing the limitations o traditional model ormulations. Simulation test results depicting the dynamic response o the model are presented in Part II. Successul validation o this modelling approach would go alongside improving the current methods o continuous control o grinding mills. Keywords: Energy alance, Modelling, Simulation, Mixing, Grinding mill, Mill power, Mill temperature I. Introduction Eective mill control is elieved to e a key element in the drive towards lower energy and material costs in grinding circuits ut such a control system would require good understanding o the dynamic ehaviour o mills when the operating variales are changed and the interactions etween external process variales (open to manipulations) and perormance variales o the process. In deed grinding mills in mineral processing plants never really operate under steady state conditions as variations in eed size distriutions, eed rate, eed slurry density and ore characteristics continuously upset the smooth running process. Dynamic simulation (on-line or o-line) can help to gain etter understanding o the dynamic ehaviour o the mill y assessing the what i situations. Hence, prior insights into the ehaviour o the milling circuit can e made availale such as response to grinding process disturances during transitions etween various steady-states. Also, the complex interactions within the mill circuit can e explored and evaluated with minimum cost and changes can then e made to the circuit to realise high eiciency and throughput. The key elements required or accurate dynamic simulation are good dynamic models, precise measurements o process variales and accurate estimation procedures or process parameters [1]. A range o simulation models exist or mill simulation (JKSimMet, METSIM, MODSIM etc) and have een widely used or mill circuit design and optimization with varying levels o success. However these models do not provide or analysis o energy alance inside the mill and are mainly ased on steady state analysis. Energy alance is a key aspect in control and optimization o a comminution process apart rom mass and size alance. Energy alance data which is related to the mill temperature proile is elieved to e a key indicator o the process dynamics as shown in the work y Van Drunick and Moys [2]. Interpretation o the temperature signature may yield some useul inormation o the mill operation characteristics such as variations in eed rate, slurry properties and holdup mass inside the mill which are directly linked to grinding eiciency. Having the knowledge o these dynamics enales eective control o the grinding circuit which in turn results in stale operation with optimum production capacity and energy eiciency. This research roadly aims to develop dynamic models ased on energy, mass and size alance or simulation o open circuit wet all mills under MATLAB environment. It integrates the energy alance with the population alance model in a simple and clear representation o the physics and thermodynamics o the wet all milling process. Easily measurale parameters such as mill temperature and product size distriution would serve as control parameters. The success o this technique would go alongside improving the current methods o continuous control o grinding mills. Open Access Journal Page 36

2 Energy-Mass-Size alance model or dynamic control II. Development o dynamic model A dynamic model is developed or a wet overlow all mill ased on a set o mass and energy alances to simulate the mill. The energy alance relies on temperature and mass low data. The key control parameters to e measured are mill temperature (eed and discharge streams), mill power draw and the mass low rate in the eed stream. Manipulated variales are eed size distriution, eed % solids, eed dilution water and ore hardness. Figure 1 is a schematic representation o the energy and mass low streams around the mill operating in an open circuit coniguration. Dilution water Q Loss Feed y i, F T, T d y pi P Figure 1: Representation o mass and energy streams around the mill and the modelling structure in MATLAB Where, P is the power input (kw) to the mill (corrected or motor losses), F is the eed low rate into the mill (resh eed + dilution water), T is the temperature o the mill eed, T d is the temperature o mill discharge, Q Loss is the rate o energy loss to the environment (kj/s), y i is the mass raction o mill eed solids in size class i while y pi is the mass raction o mill product solids in size class i. The mill is depicted as system o N perectly mixed cells o equal volume, V with a net volumetric low rate, F and recirculation rate, F. The Figure 2 is the conceptual ramework o the model. The total low rate through stages 1 and N is F + F while the low rate through stages 2 to N-1 is F + 2F. The set o equations which descries the dynamics o the system can e otained y perorming a material alance. The temperature o the mill discharge is assumed to e equal to the temperature in the last mixer and so is the solids concentration in the slurry. Cell 1 Cell k = 2 : N -1 Cell N Q L1 Q Lk Q LN F, T V, T 1 F + F V, T k F + F V, T N F d, T d y, k y p, m F F P 1 P k P N Figure 2: Depiction o in-mill mass and energy streams represented y N equally sized and ully mixed segments with ack-mixing. The model inputs are the reakage and selection unction parameters, mill eed rate, mill dilution water, eed solids concentration, eed size distriution, ore characteristics, mill rotational speed and mill ill level. The outputs (target variales) are the mill discharge temperature, mill power and the product size distriution. 2.1 Energy alance From the undamental laws o physics, the energy proile within a ody depends upon the rate o energy input including internal generation, its capacity to store some o the energy and the rate o energy transer to the surrounding environment i.e. Energy input = Energy output + Energy accumulation. Considering a wet all mill operating in a continuous mode, then this equation can e expanded as ollows: E in (Power, mill eed) = E out (mill discharge) + E asored (particle racture) + E loss (Sound, Viration, Evaporation, conduction, convection, radiation) + accumulation. d E ( t ) Q ( t ) P ( t ) Q ( t ) Q ( t ) Q ( t ) (1) d p L o ss Where the suscripts and d denote eed and discharge respectively while the symol P represents the mill power. The term Q p represents the actual energy asored per unit time y the particles to cause racture, which is a unction o material properties and mill conditions [3]. Typically this is so small relative to the total energy input to the mill that it cannot e relialy measured. Since the model will e tested and validated against data otained on a single mill, some simpliying assumptions will e considered in the energy alance. The assumptions are as ollows: Open Access Journal Page 37

3 Energy-Mass-Size alance model or dynamic control That all the net power supplied to the mill, is transmitted to the mill charge as rictional energy where part o it is converted to heat and lost to the environment through mill discharge product and vapour with the rest eing lost in orm o sound and viration. That the mill contents are well mixed oth in the radial and axial directions. That the slurry and grinding media in the mill are at thermal equilirium and that the reerence enthalpy o water and solids is zero at 0 0 C. The overall dynamic energy alance around the mill can then e presented as in equation 2 where Q Loss represents the energy loss per unit time which comprises losses through conduction, convection, vapour, sound and viration while M is the slurry holdup mass inside the mill approximated as, ( M F m ). d C M ( t ) T ( t ) m C T F ( t ) C T ( t ) F ( t ) P ( t ) Q ( t ) Q ( t ) (2) d d p L o ss The value o C (speciic heat capacity o slurry) varies with the proportion o solids (χ) in the slurry and can e estimated y equation 3 in which the suscripts s, w,, m denote solids, water, eed and mill respectively. C C (1 ) C m m s m w m C C (1 ) C s w (3) The rate o energy loss through sound and viration is typically small relative to the net energy input into the mill, hence it could e neglected. Also, or mills itted with ruer liners, energy loss through convection and conduction is minimal and hence can e ignored (note that measuring liner temperature could equally e a challenging task). A signiicant proportion o the energy loss to the environment will occur through evaporation (i.e. heat loss through vapour at the eed and discharge openings) which has to e accounted or. Thereore esides the temperature measurements, the prediction accuracy o this model will depend on correct estimation o Q Loss and Q p where latter represents the actual energy expended per unit time to cause particle racture (i.e. to create new suraces), which is proportional to the new surace area created. Using standard ond work index, Q p is estimated to lie etween 1-3% o the net energy input to the mill [4, 5]. Thereore, in this study, Q p is modeled as a unction o the net mill power draw in the orm Q p p P where p is a constant that is dependent on ore characteristics (hardness and size) and could e estimated y the semi-empirical 3.5 relation in which, is the Poisson ratio o the ore material while is the size coeicient (= 6 p or spherical particles). The rate o energy loss rom the mill through water vapour is a unction o in-mill temperature, the humidity o the air overlaying the mill load and the slurry-air interacial area where the latter is related to the size o the slurry pool. Thus, Q Loss can e estimated y the ollowing relation. Q ( t ) K L D 1 J q q ; o r J J L oss v L a m m s a m ax Where L (kj/kg) is the latent heat o vapourization [= (T-273)], ρ a is the density o overlaying air, L m is the mill length, J is the load volumetric illing (J max is dependent on the size o the discharge trunnion), D m is the mill diameter while K v is a parameter whose value is to e estimated and is related to the volume o vapour generated. The parameters q s and q a represent the speciic humidity o air overlaying the load at saturated and unsaturated states respectively which are determined y the ollowing expressions in which, (= 0.622) is the ratio o molecular weight o water vapour to that o dry air and RH is the relative humidity (%) at temperature, T [6]. (4) q p p 1 p ; q p p ; p p R H s s a t a tm s a t a v ( d r y a ir ) v s a t w h e r e p e x p ( T 2 7 3) ( T ) in k P a, s a t (5) Open Access Journal Page 38

4 Energy-Mass-Size alance model or dynamic control To urther assess the impact o change in the proportion o solids in slurry on the in-mill temperature, the residence time has een correlated to the eed solids concentration in the orm, ( k k J k / F ) [7]. Here, k , K 2, K 3 are empirical constants to e determined y regression while J is the ractional illing o the mill with charge. Now, perorming an energy alance around the mill, we otain the ollowing equations that descrie the temperature variation with time or the n mixers representing the mill. The symols and represent the length and residence time o a single mixer respectively while dispersion coeicient (see Figure 2). is the ack-mixing coeicient F F F which is related to the axial C T ( t ) C T ( t ) 1 P ( t) p 1 C T ( t )... d T ( t ) Mixer 1: d t C 1 K D 1 J v a L m q q s 1 a 1 (6a) Mixer k = 2 to N-1: C T ( t ) C T ( t ) C T ( t ) 1 P ( t) k k k k k k p k... d T ( t ) 1 1 k (6) d t C k K D 1 J v a L m q q s k a k Mixer N: (6c) C T ( t ) C T ( t ) 1 P ( t) n 1 n 1 n n p n... d T ( t ) 1 1 n d t C n K D 1 J v a L m q q s n a n The values o the ack-mixing coeicient ( ) lie etween 0 (no mixing) and 1 (perectly mixed state). This parameter is a strong unction o solids raction in slurry and could e estimated y the relation, ( k k / ) proposed y Makokha and Moys [7], where k A = and k B = are otained y A B regression. 2.2 Mill power The mill power is expected to vary with changes in mill holdup mass, eed size distriution and slurry properties. The equation used here to predict the power drawn y the mill is ased on modiication o Hogg and Fuerstenau power model. The model gives the relationship o mill power draw with respect to mill dimensions and operating conditions and treats the contriution o grinding media, ore and slurry as independent components in charge. The gross mill power is expressed as, / (7) L p c a p P K D L D N J J S in Where K p is a itting parameter that corrects or the inaccuracies in estimation o the dynamic load shape as well as the eect o slurry pool volume on mill power. The symols L and D represent mill dimensions in meters, J is the mill illing as a raction o the total mill volume, N c is the mill rotational speed as a raction o the critical speed, is the load angle o repose (in radians) while ap is the apparent charge density (media and ore) given y the ollowing equation [9]. Open Access Journal Page 39

5 1 1 ap v ore v sl v Energy-Mass-Size alance model or dynamic control J J J U J (8) Where J is the alls illing as a raction o mill volume, v is the voidage within the load while U is the powder / Slurry illing in the load interstices. The load angle o repose, (radians) is dependent on the load dynamic ehaviour and is mathematically computed rom the load toe angle, T (radians) and shoulder angle, S (radians) as, /2. The load toe and shoulder angles can e easily measured y online instruments, T S ut or purpose o modelling, the load angles ( T and S ) could e estimated y the ollowing relations that are slightly modiied rom those presented y Apelt [8] and Morrell [9]. T N 1 a J 1 e x p c a 1 m N / J S T 1 2 m c (9) In which, a, a,, are parameters determined y regression analysis. In this study, data collected rom an industrial all was used to estimate the parameters and the optimal values otained a ; a ; ; are: Mass-Size alance The size-mass alance is developed with reerence to the mill system in Figure 1. Applying the principle o mass conservation to the milling process, the rate o accumulation o material o size i = rate o entry o material o size i rom the eed + entry o material o size i rom reakage o larger sizes - material destroyed in size i y racture disappearance o material o size i through discharge. This can e mathematically represented y the ollowing steady state equation. i 1 F y F y F y M S l ( t ) S l m m p i m i d p i m ij j j i i (10) j1; i1 w h e r e : B B, n i j ; B, j n i, j i, j i 1, j n, j n, j The parameters and S are the reakage distriution unction and the selection unction respectively (that can e otained using equations proposed y Austin et al [10], while F m, χ m and χ denote the mass low rate through the mill, the slurry solids raction inside the mill and in the mill eed respectively. The rest o the parameters retain their earlier deinitions. For a well-mixed mill, the size o particles inside the mill would e equal to the mill product size ( l i y ) and the average residence time, / p i M F. m Austin et al [10] presented an equation or determining the selection unction which characterizes particles only y their sizes. However, It is well recognized that the mechanical properties o rittle materials such as mineral earing ore rocks strongly depend on deormation rate and strain rate. Thereore in population alance modelling o particle reakage process, the selection unction should allow or particles to e characterized simultaneously y their size and racture energy. The unction should comprise the proaility o a particle eing selected or reakage and the proaility o the energies generated y the impacts eing suicient to reak the particles. According to the work y Crespo [11], the proaility o the energy applied eing suicient to reak the particle is related to the raction o the asored impact energies in a given time interval that are higher than the racture energy. Similar oservations were made y Tugcan and Rajamani [12] in a separate study using ultra ast load cell data who urther indicated the dependence o this proaility on particle size and material speciic properties. For instance, a material with a higher hardness index would store more strain energy during deormation hence depending on the level o applied strain, it may require repeated impacts to racture a hard material. This oservation is supported y the data rom Discrete Element Method (DEM) presented y Rajamani et al [13] and Datta et al [14], which estalished that the impact requency inside the mill depends on the level o energy applied. In a continuous all mill, the applied impact energy (E p ) could e equated to the ratio o the actual energy expended to cause particle racture (Q p ) to the mass eed rate o material into the mill as ollows: E Q F P F (11) p P m p m Open Access Journal Page 40

6 Energy-Mass-Size alance model or dynamic control Where P is the net mill power (kw), F m is the mass low rate through the mill (kg/s) while the parameter p (discussed earlier in section 2) represents the proportion o energy input to the mill that is actually utilized in reakage o the particles which lies etween 1 3% [4, 5]. Due to lack o appropriate data on racture energies o various mineral ore rocks, here we suggest the use o hardness numer (), which is an easily measurale variale to simulate the energy-dependent rate o particle racture on single impact. Based on the Mohs hardness scale ( the values o ore hardness numer () range rom 1 to 10 corresponding respectively to the sotest and hardest ores. The hardness numer could e standardized to a hardness index { 1 lo g 2 1 } with values o < 1, >1 or 1 corresponding to soter, harder and normal ores respectively. Using this simpliying assumption, then the dynamic mass-size alance model can e re-written as ollows: d M ( t ) y ( t ) i 1 pi F ( t ) y ( t ) F ( t ) y ( t ) S M ( t ) y ( t ) S M ( t ) y ( t ) s i d s p i ij j p j i p i... (12) j 1 i 1 The vectors, y pi (t) = [y p1 (t), y p2 (t), y pn (t)] T and y i = [y 1, y 2, y m ] T represent the mass raction o mill product solids and mill eed solids in discrete size classes respectively while F s (= χ F ) and F ds = (χ m F d ) are the mass low rates o the solids in the eed and discharge streams respectively. For overlow mills, the type used in this study, the volume o slurry present in the mill is reasonaly constant over a wide range o operating conditions, hence the volumetric eed rate to the mill equals volumetric discharge rate at all times. Utilizing this act, the variation in proportion o solids in slurry, deined here as the mass o solids per unit mass o slurry in the mill, can e descried y the ollowing equation. d ( t) 1 m ( m ) (13) The variation o water holdup inside the mill that results rom changes in the water low rates in the eed and discharge streams as the slurry solids concentrations changes could e represented as ollows: d m w ( t ) w m F F (14) Similarly to the energy alance, the size-mass alance is perormed with reerence to the mill system presented in Figure 2. The superscripts in the rackets denote the mixing cell numer i.e. k = 1 to n where n is the total numer o mixing cells considered. Mixer 1: (1 ) ( 2 ) (1 ) d m ( t ) y ( t ) i 1 1 pi y ( t ) y ( t ) 2 p i 1 p i (1 ) (1 ) F ( t ) ( ) ( ) ( ) ( ) ( ) y t m t t S y t S y t i 1 ij j p j i p i 1 1 j 1 i 1 (15a) Mixer k = 2 to n-1: ( k ) d m ( t ) y ( t ) k p i ( k 1 ) ( k 1 ) ( k ) y ( t ) y ( t ) y ( t )... k p i k p i k p i 1 i 1 ( k ) ( k ) m ( t ) ( t ) ( ) ( ) k S y t S y t ij i p j i p i j 1 i 1 Mixer n: (15) Open Access Journal Page 41

7 Energy-Mass-Size alance model or dynamic control ( n ) d m ( t ) y ( t ) i 1 n p i ( n 1 ) ( n ) ( n ) ( n ) y ( t ) y ( t ) m ( t ) ( t ) S y ( t ) S y ( t ) n 1 p i n p i n ij j p j i p i 1 j 1 i 1 (15c) The mass holdup o solids in each individual mixer at any time instant is dependent on the level o solids concentration in the mixer, the mass eed rate and the residence time o the individual mixer as ollows: m ( t ) F ( t ) ; m ( t ) F ( t ) ; m ( t ) F ( t ) (16) 1 1 k k n n It is noteworthy to mention that while all the n mixers may e equal in volume, due to slight variations in slurry solids concentration etween the mixers, the mass o solids m 1, m k and m n may not e equal. Perorming the solids alance around the n mixers, we otain the ollowing equations representing the dynamic variation o solids concentration in each respective mixer. d Mixer 1: ( t ) (17a) Mixer k = 2 to N-1: (17) d ( t) 1 k 1 Mixer N: d ( t) 1 n n 1 n (17c) 1 k 1 k 1 k 1 The task o mass and energy alancing was accomplished numerically using MATLAB ode45 solver. III. Conclusions A dynamic energy-mass-size alance model has een developed that can e utilized or dynamic simulation o a wet open circuit grinding mill. This model may serve as predictive tools or providing insights o the process and thus it would lay a ground on which sound control schemes can e created or improved mill product quality and process perormance. Simulation test results using this model are presented in Part II o this research. Reerences [1] Herst, J., Pate, W and Olad, A., Model ased control o mineral processing operations, Powder Techn, 69: [2] Van Drunick, W.I. and Moys, M.H.,2003. Online process modelling or SAG mill control using temperature measurements, energy alancing and inerential parameter estimation, In Proceedings o IMPC. [3] Stamoliadis, E. Th., A contriution to the relationship o energy and particle size in the comminution o rittle particulate materials, Minerals Engineering, Vol. 15, [4] Fuerstenau, D. W and Aouzeid, A,-Z.M., The energy eiciency o all milling in comminution, Int. J. Mineral Processing, 67: [5] Tromans, D., Mineral comminution: Energy eiciency considerations, Minerals Engineering, 21 (8): [6] Jensen, M.E., R.D. Burman, and R.G. Allen., Evapotranspiration and Irrigation Water Requirements. Amer. Soc. o Civil Eng., New York. [7] Makokha, A. B., Measuring, characterisation and modelling o the load dynamic ehaviour in a wet overlow-discharge all mill, Ph.D Thesis, University o the Witwatersrand, Johannesurg [8] Apelt, T.A., Asprey,S.P., and Thornhill, N.F., Inerential measurement o sag mill parameters, Minerals engineering 6 (14): [9] Morrell, S., The prediction o power draw in wet tumling mills. PhD Thesis, University o Queensland. [10] Austin, L., Klimpel, R., and Luckie, P., 1984, Process engineering o size reduction: Ball milling. Society o mining engineers, pp [11] Crespo, E. F., Application o particle racture energy distriutions to all milling kinetics. Powder Technology, 210 (3): [12] Tugcan, T.E. and Rajamani, R.K., Modeling reakage rates in mills with impact energy spectra and ultra ast load cell data, Minerals Engineering, 24 (3-4): [13] Rajamani, R,K., Songack, P. and Mishra, B.K., 2000, Impact energy spectra o tumling mills, Powder Technology, 109: [14] Datta, A., Mishra, B.K. and Rajamani, R.K., Analysis o power draw in all mills y discrete element method, Canadian Metallurgical Quarterly, 38: Open Access Journal Page 42