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Measuring and optimizing air classifier performance
Measuring and optimizing air classifier performance Ivan V. Klumpar Badger
Engineers,
Cambridge,
MA, USA
Operation of air classifiers, particularly in closed circuit with grinding mills, is described. Performance measures are reviewed and an improvement of the AIChE standard method of calculating the selectivity curve is proposed. Approach to performance optimization is derived from the fundamentals of air classt$cation. The computations of power requirements for classical air separators and for modern high efficiency machines are compared. Trial-and-error calculations for optimizing the complex heat balance of a grinding mill in closed circuit with an air classifier are discussed.
Kevwords: powder/bulk solid classification: circuit grinding, heat balance
particle size; selectivity;
Introduction Air classifiers or air separators are rotary machines that separate a particulate feed into a fine and coarse fraction, using air or other gases to entrain the fine product and a rotor to reject any airborne coarser particles.’ While air classifiers can be used for various purposes, the most common application is in closed circuit grinding as shown in Figure 1. New material, N, (makeup) is fed into mill 1 together with tailings (or “tails”), T, recycled from air classifier 2. The mill receives a power input, I, is swept by air, S, and further cooled by a water spray, W. Mill discharge, D, becomes air classifier feed, F. The classification is primarily done by ambient air, A, and the tailings are returned to the mill. In classical machines, the bulk of the fine product is recovered in the air classifier while in the modern high efficiency equipment the fines are separated from the entraining air in baghouse 3. Mill vent, V, leaves the plant as exhaust, X, and/or is used as stream, U, in the air classifier. The fictitious loss streams, K, M, and the boundary lines will be explained later. In the first part of this paper, the performance measures are reviewed and an improvement of the AIChE standard method of selectivity calculation proposed. Next, the approch to performance optimization is derived from air classification principles. In a separate section, the minimization of power requirements is re-
air classifiers; power requirements;
closed
lated to air classifier performance. Finally, cooling optimization is discussed using an illustrative example.
Performance measures Air classifier performance is measured in different ways.2 Yield is defined as the amount of product per unit amount of feed expressed as a fraction or percent. For example, fine product yield Yp = w,/w,
(1)
Symbols are listed in the Notation section. Recovery (sometimes called efficiency) is the percentage of a selected cumulative fraction of feed that goes to either product. For fines EP = X, WJX,
W,
(2)
where X, and X, are the fractions of fines and feed, respectively, that pass a specific screen size, e.g., 200 mesh. Based on Equation 2, the following more practical formula can be derived by mathematically eliminating the material flow rates, which are sometimes difficult to measure in a plant:3 Ep = X,(X,
- X,)/X,(X,
- X,)
(3)
where subscript T refers to the tailings. Efficiency (or overall efficiency) is the difference between fine and coarse product recoveries: E = EP - E, = (X, W, - X, WT)/XF W,
Based on experimental and engineering work done at Sturtevant, Inc., Boston, MA, USA. Address reprint requests to Dr. Ivan V. Klumpar, Badger Engineers, One Broadway, Cambridge, MA, 02142, USA. Received 31 October 1991; accepted 6 February 1992
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Sep. Technol.,
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(4)
A more practical formula is E = (X, - X,)(X,
- X,)/X,(1
- X,)(X,
- X,) (5)
0 1992 Butterworth-Heinemann
Air classifier performance: Recovery and efficiency, while widely used, are not very accurate because only one screen size specifies the feed or products. Air classifier performance is best characterized by the plot of selectivity vs. particle size. Selectivity can be calculated if the particle size distributions of at least two of the three materials involved (feed, fines, and tailings) are known. Selectivity is the weight percentage of all particles of a given diameter in the feed that go to the tailings.4 As there is an extremely high number of different particle sizes and shapes, the group of particles of a
I. V. Klumpar
given diameter is defined as the narrow fraction passing one screen (e.g., 200 mesh) but retained on the next smaller one (250 mesh). The mass of this narrow fraction is denoted as m. If one of the automatic particle size analyzers based on X rays, light beams, or lasers is used, the corresponding results are obtained in microns , i.e., 74 and 63, respectively. The average diameter of that fraction is defined as the arithmetic mean or 68.5 pm. Selectivity for particles with an average diameter d, is mathematically expressed as
Notation A b C L d E f G
Ez h
i .i
k
L M m N
If, P Q 4
8e
s s
St T t u
u
W
i X
Y YJ
z
Projected area Intercept Centrifugal force Specific heat Drag force (Screen opening) diameter (Classification) efficiency (or recovery) Friction factor Gravity Acceleration due to gravity Enthalpy flow Height Size interval number Sharpness index Drag coefficient Fractional heat loss Mass of a body Mass or mass flow rate of a narrow size fraction Number of rejection elements Number of size intervals Power Pressure Volumetric flow rate Quotient, ratio Radius Reynolds number Selectivity Speed Stokes number Start-up time Temperature Air velocity Particle velocity Mass or mass flow rate of a material stream Rejection element velocity Cumulative mass fraction Mass fraction of narrow size fraction Yield Auxiliary variables Equivalent length
Greeks h Difference operator 2 Sum operator Viscosity p 7T = 3.1416 Density P Stream symbols (also used as subscripts) A Ambient air D Mill discharge E Air classifier exhaust F Air classifier feed Power input to mill Z K Air classifier heat loss M Mill heat loss N New material P Fine product S Sweep air to mill T Tailings U Mill vent used in air classifier V Mill vent W Water to mill X Mill exhaust Subscripts “b : e ; ;, m n P r
s :, 1 2
Average particle Blade, rod, or other rejection element Air-particle cloud Drag Effective Gas, air Hydraulic Size interval Consist based Mass based Normalized Individual particle Rotor Start-up Particle trajectory Reference Direct interception Inertial impaction
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Air classifier performance:
I. V. Klumpar
,._._._.-._._._._._.t_._., A
IE
ii
3
I
&....i i
---
T
i
_. U
i’
!
_,-.-.-a_.-
c
_--,_._._._.
i i P.
i J
---+X --I
f
I ,.......)
M
i
Figure 1 Closed grinding circuit. ----, mass or power stream; ------, alternative stream for classical air classifiers; ......., loss stream; classifier system boundary; -.-.-.-.-r --.----.--., mill system boundary; 1, mill; 2, air classifier; 3, bag house
S = m,lm,
(6) Figure 2 is a typical plot of selectivity vs. particle size, expressed as diameter, for a regular cement (specific surface of 370 m2/kg according to Blaine) produced in a commercial Sturtevant High Efficiency SD Classifier under normal plant operating conditions. In interpreting the selectivity curve, two features are important: the slope of the main portion of the curve and the position of the “fishhook” in the lower portion. The slope reflects the sharpness of cut or how much the actual classifier differs from an ideal machine. Ideal classification would result in a vertical straight line intersecting the actual curve at 50% selectivity. In the case shown in Figure 2, all particles of 40 pm or smaller would go to the fines while larger particles would report to the tailings. The slope is measured with the sharpness index conventionally defined as the ratio of the particle sizes corresponding to the 25% and 75% selectivity, or (7) The lowest point of the selectivity curve is sometimes called the apparent bypass because it is the measure of the amount of particles that behave as if bypassing the separation zone of the air classifier and going directly to the “wrong” product, e.g., fine particles reporting to the coarse fraction. The apparent bypass in Figure 2 is 11%. While some particles actually pass by the separation zone without being exposed to classification, there are also other causes of the fishhook, such as particle agglomeration or inadequate dispersion of agglomerates present in the feed, grinding of material in the air classifier, and entrainment by recycle air of particles already classified. j
=
d~S%ld?S%
a
Selectivity computation In computing selectivity, it is convenient to adjust the experimental data in the following way. The average particle diameter is defined as 126
Sep. Technol.,
1992, vol. 2, July
where i is the interval number or the sequence number of the narrow material fraction passing the larger screen il but retained on the next smaller screen i2, and d is the screen opening diameter. In automatic particle size analyzers, d is the largest particle size diameter of the cumulative material fraction. Let us refer to d as the indicated diameter for simplicity, and d, as the midpoint diameter because it is in the middle of an adjacent pair of ds. In the sample calculations in Table 1, i, d,, and d are recorded in the first three columns. In the next step of selectivity computation, cumulative mass fractions or percentages X,, X, and X, in the feed, tailings, and fines (columns 4, 5, and 6 in Table J), respectively, are converted to the respective mass fractions or percentages +, xr, and xP of the individual particle size intervals (columns 7, 8, and 9) as follows: x = xi, - xi2
(9)
In batch classification tests and many industrial operations, consist and mass of the tailings and fines are known. Consist is used here to denote particle size distribution. Mass is defined either as weight of the batches or as flow rate in continuous operations. For these cases, Equation 6 can be rewritten as S,, = m,l(m, + mP) = xT W,lx, W, = = XTW&x, w, + xp W,)
(10)
based on the following material balance relationships: mF = mP + mT
(11)
w, = w, + w,
(12)
m = XW
(13)
Selectivity S,, is referred to as mass based because the mass of two out of the three materials (feed, tailings, and fines) have to be given. Even if all three are known, it is recommended that Equation 10 is used for consistency. Sample calculations are shown in columns 2-5 of Table 2 and the resulting selectivity curve plotted in Figure 2. In industrial grinding mills operating in closed circuit with air classifiers, the flow rates of classifier feed, tailings, and fines are often not known while the consist of these three materials can be easily determined by sampling and particle size analysis. This is routinely done for at least one of the three materials. In this case, selectivity can be computed as follows: Sk = xr(xp - +Yx,(x,
- x7)
(14)
This can be conveniently rewritten (Braun, R. Canada Lafarge Cement, Ltd., Montreal, private communication, 1987) as Sk = x,ylx,z
(15)
where y = x, - XF
(16)
2 = xp - xr
(17)
Air classifier
performance:
I. V. Klumpar
65
.
25
3 4 > 4 G al rl rt
10
_i
I __I I 5
0.1
1
10
102
103
Particle diameter, d, microns Figure 2
Table 1
Interval no. i 1
1 2 3 4 5 6 7 8 9 10 11 12
Mass based selectivity
Sample selectivity calculations,
Indicated diameter km) d 2 2.8 3.9 5.5 7.8 11 16 22 31 44 tf 125 177
part 1
Mid point diameter (;m) 5
3.4 4.7 6.7 9.4 13.5 19 27 38 53 75 107 151
Cumulative mass % of mtl. smaller than d in Feed
Tails
Mass % of material between 2 adjacent ds in Fines
Feed
Tails
Fines
xF
XT
XP
xF
4
XT
5
6
XP
7
8
9
2 5.5 9 10.5
5 12.5 19.5 25.5 38.5 51.5 65.5 79.5 96 100 100 100 100
6.4
3.5 3.5 1.5 4.5
7.5 7 6 13 13 14 14 16.5 4 0 0 0
z.6 10 16.5 20 28.5 51 64 81 89 95 100 100
:z 23.5 33 55 72 86 98 100
1:: 8.5 11.5 11 13 17 8 6 5 0
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127
Air classifier Table 2
performance:
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Sample selectivity calculations,
part 2
Mass* of material between 2 adjacent ds
Interval no. i 1
Tails
Feed
w
mF
1 2 3 4 5 6 7 8 9 10 11 12
Fines m
2
3
4
953 903 687 1,561 1,590 1,603 1,951 2,926 1,386 812 696 116
203 203 87 261 290 203 551 1,276 986 812 696 116
750 700 600 1,300 1,300 1,400 1,400 1,650 400 0 0 0
21 22 13 17 18 13 28 44 71 100 100 100
This selectivity is referred to as consist based. Sample calculations are shown in columns 6, 7, and 8 of Table 2 and results plotted in Figure 3. The data in columns 2, 4, 5, and 6 of Table I have been taken from an actual industrial operation. The low accuracy of plant measurements compared with laboratory tests is reflected in the wide scatter of the points in Figure 3. The points in Figure 2 are less scattered because the W,/W, ratio in Equation 10 is a constant while the corresponding y/z ratio in Equation 15 might vary with particle size. The consist based selectivity curve can be smoothed out by using a constant mean y/z ratio, (18)
It can be determined graphically on plotting y vs. z and measuring the slope (see Figure 4). Alternatively, q can be computed by regression analysis, e.g., using the least squares method: nxyz - Izyzz q =
nCz2- (xz)2
The corresponding
intercept
on the Y-axis
= 10,000
sy
5
8
1.1 0.5 2.5 4.5 1.5 3 1 -0.5 -4 -6 -5 0
4 3.5 4.5 8.5 8.0 10.5 4.5 -5.5 -13 -14 -12 -2
15 8 24 28 8 9 16 12 65 100 100
(20)
Air classification is affected by two basic phenomena: selective entrainment and selective rejection. For the sake of argument, let us look at the particle size distribution of the feed as a mixture of fine, medium, and coarse particles. First, the fine and medium fractions become airborne. Then the medium-sized particles are rejected and fall with the coarse ones to the bottom chamber of the equipment while the fine particles are entrained by the air through another chamber to the outlet. An example is shown in Figure 6. Selective entrainment occurs in a swirl of particles and air. The air rotation is generated by an internal fan or by blowing the air tangentially into the classifier. Unless the solids are also fed tangentially by pneumatic conveying using the classification air, the swirling of particles is caused primarily by a rotating distributor plate and only to a lesser degree by the air movement if it is curved. In any case, for the selective entrainment to operate effectively, the drag force of the air, DP , has to balance gravity, G, and centrifugal force, C. These three forces acting on a particle can be expressed as
G = M,g(p,
- t1)~/2 - P,)
(21)
It is shown in the last column of Table 2 for q = 0.349 and plotted in Figure 5. 128
Sep. Technol.,
1992, vol. 2, July
(22)
(23) (24)
where the particle drag coefficient, k,,, is constant or increases with the Reynolds number. Re = d,(u - v)p,Ipg
s, = qx,lx,
19 19 15 18 15 11 25 45 74 81 84
Principles of air classifier performance
C = MPv21r, should be close to zero. The straight line in Figure 4 was calculated by this method. The “normalized” selectivity can then be computed as
Normalized selectivity
(fines).
DP = k,A,p,(u
b = ZyZz2 - Z,Cyz nCz2 - (Zz)2
Consist based selectivity (%I Sk
Cl 5
l Mass in kilograms per hour here or in grams in batch tests. Mass flow rates (kg/h): kV, = 15,800 (feed), Wr = 5,800 (tails), W, Mean y/z ratio: q= 0.349.
q = y/z = constant
Auxiliary variables (%I
Mass based selectivity (%I
(25)
According to the principles of air conveying, there are of course limits as to how much a given air volume can entrain.
Air classifier
performance:
I. V. Klumpar
65
Particle
diameter,
d,
microns
Figure 3 Consist based selectivity
Rejection happens when a particle collides with a rejection element of the air classifier rotor, such as a blade or rod. Rejection is measured with the efficiency of capture, Eb ,5that occurs either by direct interception or inertial impaction.6 In the former case, the particle is swept to the rejection element by air and the corresponding efficiency is E, = (1 + d,ldb) -
l/( 1 + d,/d,)
(26)
where db is an equivalent diameter unless the rejection element is a rod. In inertial impaction, the collision course is controlled by particle inertia rather than by the air streamlines, and the efficiency, E,, is approximately proportional to the Stokes number. St = W, d;@, - pJ18~
db
(27)
The effective blade or rod velocity, we, is discussed below. The total capture efficiency is the sum of E, and E, . Furthermore, statistics indicate that collision frequency increases with the number of blades or rods per unit circumference of the rotor. Suppose the rotor is a “cage” formed by a circle of vertical rods or blades, and the air swirls around and
inside the cage, as in modern high efficiency equipment, e.g., the Sturtevant SD Classifier or the Fuller O‘SEPA machine2 (Figure 6). In that case, the radial velocity component, ub, of air flowing through a “window” between two adjacent rods or blades can be readily computed from the volumetric air flow rate and the cage geometry,7*8 as will be subsequently explained. If the rotor diameter, d, , is defined by the periphery of the set of rejection elements, the actual rod or blade velocity is w = rrs d,
(28)
The effective rod or blade velocity with respect to the air velocity is then given by the parallelogram of vectors w2e = w2 + L2b
(29)
where the window velocity can be approximated ub = Qg/hb(r d, - N db) Performance
as (30)
optimization
The following parameters control air classifier performance in terms of yield, recovery, efficiency, and selectivity: Sep. Technol.,
1992, vol. 2, July
129
Air classifier performance:
Figure 4 0.95
l l l l
Graphical
selectivity
I. V. Klumpar
normalization.
Based on consist;
Air velocity Feed rate Particle velocity Rate of particle rejection by the rotor, or rejection for short
Among these four parameters, rejection is unique in that it directly affects another important performance factor, the fineness of the fine product, because particle diameter, dP, does not appear in Equations 22, 23, or 24, and Equation 25 has only a minor effect. Rejection, in turn, is controlled by l l l
Rotor speed Number of rejection elements of the rotor such as blades or rods Type, i.e., size and shape of these elements
Additional parameters that control performance are air density and viscosity. The latter has a less significant effect via the Reynolds number (see Equation 25). In special applications, density (and viscosity for that matter) can be varied by using gases other than air. This is expensive unless one of the modern air classifiers with gas recycle is used. Moreover, in close circuit grinding of large volume materials such as cement in air-cooled mills, the mill exhaust is often passed through the air classifier. The use of another gas would 130
Sep. Technol.,
1992, vol. 2, July
slope q = 0.349; Y-axis intercept
b = -0.03;
correlation
coefficient
=
require putting the entire system under a non-air atmosphere and provide expensive coolers of dust-laden gas. There are other factors that affect performance, such as overall equipment size, proportions of the internal parts, and the distances between them. However, these are design parameters that usually cannot be varied once a piece of equipment is made. Design optimization is discussed elsewhere.9 In the classical air classifiers, only feed rate and rotational speed can be changed independently, because the air velocity is controlled by an internal fan that is mounted on the same shaft as the rotor. Changing the number and type of rejection elements usually requires a partial disassembly of the equipment. Only in the Sturtevant SD Classifier can the rods be readily taken out or inserted through a manhole, though the machine still has to be shut down temporarily. Air velocity improves performance if it is compensated for by increasing rejection, because otherwise drag prevails over gravity and centrifugal force (see Equations 22-24), shifting the force equilibrium toward the entrainment of some coarse particles. On the other hand, with more rejection, product fineness rises but performance deteriorates unless air velocity is increased at the same time to maintain a balance between entrainment and rejection.
Air classifier performance:
I. V. Klumpar
65
25
III
Particle
I
diameter,
Iill
d,
microns
Figure 5 Normalized selectivity
Analysis of Equations 26-30 and the statistical findings show that rejection increases with increasing particle diameter, rotational speed, particle-air density difference, number of rejection elements, and to a certain extent with window air velocity. Rejection is smaller with bigger rejection elements and greater gas viscosity. The most significant factors are particle and blade or rod size, because the other parameters appear only in one of Equations 26 and 27, and in some cases are adversely affected by the rotor geometry. However, particle size is a given parameter in most applications. Blade or rod size is not a convenient rejection control parameter. First, changing these parts requires partial dismounting of the air classifier in most cases. Second, rods, which are the easiest to change, can be reduced only to a certain degree because they have to be robust to support the rotor structure and withstand erosion. While rotor speed has only a secondary effect on performance, it can be varied in a wide range. Consequently, it is the best rejection controlling parameter. Particle velocity is not an important optimization parameter because it can be independently controlled only in the rare air classifiers with the rotating internal
feed distributor mounted on a separate shaft. In all other mechanically fed classifiers, particle velocity is a function of rotor speed and thus tied to rejection. In pneumatically fed machines, particle velocity depends on air velocity. Decreasing the feed rate improves performance because the feeding of fewer solids effectively undersizes the air classifier, thus relatively increasing entrainment capacity and rejection.
Power minimization From the discussion in the previous section on performance optimization, it appears advantageous to run air classifiers at high air flow rate and high rotor speed, at least if a very fine product is called for and a modern machine is used. Let us look at how this conforms to the general requirement of keeping plant power consumption low. An air classifier needs power to pump the air and run the feed distribution and rejection rotor. In the classical machines, the fan and the rotor are mounted on the same shaft, and power is computed by start-up requirements, i.e., to bring the shaft from standstill to operating speed, s, expressed as Sep. Technol.,
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131
Air classifier performance:
I. V. Klumpar I
PARTICLE FEED
exceeds the start-up power requirements. This resistance has two components. One is the friction of the rotor with the particle-air cloud that can be modeled as the power to counteract the drag of the rejection elements passing through that cloud. P, = Nw,D,
(34)
where the rod or blade velocity is given by Equation 29 and drag by D, = k,,A,p,w;l2
(35)
The drag coefficient, k,,, is constant or increases with a Reynolds number that depends on the shape and size of the rejection element. The effective rod or blade velocity for some air classifiers is defined by Equation 29. The second resistance applies only to machines with rotating feed distributors. The power, PF , to overcome this resistance corresponds to the kinetic energy to accelerate the feed from zero horizontal velocity, when the material drops on the distributor, to the velocity, u, of the particles as they slide off the distributor rim (Figure 6): P, = w,v2/2
As the particle velocity, u, cannot exceed rim velocity, it can be conservatively approximated as u = w. The total rotor power requirements are the sum of the two components
\ TO COARSE PARTICLE CHAMBER TO FINE PARTICLE CHAMBER TANGENTIAL
0
AIR
@
P = P, + PF
FLOW
RADIAL
AIR
FLOW
Figure 6 Operation of a high efficiency air classifier C, centrifugal force; D, drag; G, gravity. Based on reference 2 by special permission of Chemical Engineering. Copyright 0 1986 by McGraw-Hill, Inc., New York, NY 10020
(31)
where My; is the rotor inertia. Modern high efficiency classifiers have a separate fan, the required power for which is proportional to air flow rate, Qg, and pressure drop. Ap = 2fu2Zld,,
(32)
The friction factor, f, is constant or increases with the Reynolds number. Re = d,,up,Ip,
(33)
Minimizing fan power consumption requires cutting air velocity, which runs against the needs of the performance optimization of the air classifier proper. It is obvious that there is a trade-off between saving electricity and improving performance. Power requirements of the air classifier rotor can be minimized by extending the start-up time, T, using invertors. This can be done only to the point where the resistance of the rotor under operating conditions 132
Sep. Technol.,
1992, vol. 2, July
(37)
Power computations are illustrated in the Appendix. Substituting the previously discussed relationships into Equation 37 leads to a very complex mathematical model. It was reduced by simplifying assumptions to the following approximate formula:‘,* P = constant
P s = h=M rr?s=IT I
(36)
s2 d2r WF
(38)
again, trying to minimize the rotor power requirements by reducing speed runs against performance optimization. Accordingly, the price to be paid for better performance is often higher power costs. Cooling optimization Modern air classifiers in closed circuit with grinding mills contribute significantly to the cooling of the fine product. The thermal conditions of the circuit can be optimized based on a heat balance that sometimes requires a trial-and-error solution because of the complexity of the problem. Using the system boundaries indicated in Figure I, the following heat balances can be written: Classifier:
Ho + HA + H,
= HP + HT + HE + HK
(39) Mill: HN + H, + H, - H, + H, = Ho + H, + HM (40)
Air classifier performance: Table 3
Input data for illustrative
example
Production in thousands of lbsih (Wp = W,) Air in thousands of actual ft3/min Ambient to classifier CQ, = Q,) Mill vent (0” = 0s = QX) Circulating load ( WTl W,) Mill power input (P,) Water to mill (Owl, postulated Temperatures (“F) New feed (t) Air to classifier and mill (f,,= ts) Water (tw) Product (t,J, assumed Temperature differences (“F), selected Tailings and product (tr - tp) Exhaust air and product (tp - tE) Mill vent and discharge (to - tv) Fractional heat losses selected Classifier (LK) Mill (Lnn)
Enthalpies
100 k PPH 50 k ACFM 12 k ACFM 150% 4,000 HP e.g., 5.2 GPM 160 80 70 e.g., 160”
of most process streams are computed
15 2” 20” 12% 20%
as
H = Wc(t - to)
(41) with the reference temperature, c,, conveniently set at zero. The mill cooling water enthalpy, H,, is approximated as the heat of vaporization and taken from steam tables. Power input to the mill is also considered a stream. Expressed in enthalpy units, it is denoted as HI. To compare heat losses HK and HM of air classifiers and mills, respectively, of various types, sizes, and configurations, it was propsoed to define the fractional heat loss, L, as the ratio of the absolute heat loss to the enthalpy of the heat source.” For the classifier system, the heat source enthalpy is the difference between the inlet and outlet enthalpies of the warmer streams. For the mill system, the heat source is the power input to the mill proper since power to run other equipment such as elevators can be neglected. The two fractional heat losses are then L, = HKI(HD - HP - HT) L,
(42)
= H,IH,
(43) In trial-and-error calculations, the “internal” streams D and T of the circuit are often not available, while the “external” streams A, E, and U are (or can be more readily approximated). On combining Equations 39 and 42, a more useful relationship is obtained: HK = (HE - HA - H,)L,I(l
- L,) (44) To demonstrate the application of Equations 39-44 to the assessment of thermal operating conditions of closed grinding circuits, let us use the following illustrative example. Calculate the effect on product temperature of the amount of water sprayed into the mill for various scenarios such as seasonal changes of ambient temperature and humidity, maximum product temperature tolerated by industrial customers or wholesalers, expected water cost increases, etc. Input data are presented in Table 3. The mass and
I. V. Klumpar
volumetric flow rate equations at the beginning of the table neglect material losses for the sake of simplicity. No mill vent goes to the classifier. For each scenario, one or more series of trial-and-error computations are required. A flow rate of the water spray is postulated for each series. A product temperature is assumed for, and firmed up by, each trial and error. The data denoted as selected are based on actual experience that, of course, can vary from plant to plant and also when operating conditions at the same plane change. The last of a series of trial-and-error runs is summarized in Table 4 The computations are done in the following way: 1. Compute tailings flow rate from circulating load. 2. Postulate water flow rate. 3. Compute flow rates of streams N, S, and E according to Table 3. 4. Assume product temperature. 5. Temperatures t, and tE are higher and lower by 15°F and 2”F, respectively, than tp (see Table 3). 6. Compute enthalpies of all streams except D and Vusing Equation 41, steam tables, and conversion from HP to BTU/h. 7. Compute classifier and mill heat losses on substituting L, and L, from Table 3 into Equations 44 and 43, respectively. Also note that U = 0. 8. Compute mill discharge enthalpy from Equation 39. 9. Compute mill discharge temperature from Equation 41. 10. Temperature t, is 20°F less than to. Il. Compute mill vent enthalpy from Equation 41. 12. Insert all enthalpies into Equation 40 to check assumed t,. Other problems of evaluating and optimizing the cooling of closed grinding circuits can be solved in a similar way. For repetitive calculations, the development of a spreadsheet computer program is recommended.
Conclusions After reviewing the fundamentals of air classification and traditional techniques for evaluating and optimizing air classifiers, the following new methods are presented. For industrial measurements that are not as accurate as those in research laboratories, it is proposed to substantially improve the standard method of selectivity computation by first determining the mean value of ratios that tie together the concentration in feed, product, and tailings of each particle size. For modern energy-efficient air classifiers that use inverters, the classical rotor inertia-based formula for computing power requirements is obsolete. Instead, a new method is discussed based on feed particle acceleration and air resistance to the machine rotation. The trial-and-error computation of the complex heat balance of a grinding mill in closed circuit with an Sep. Technol.,
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133
Air classifier performance: Table 4
Summary
I. V. Klumpar
of illustrative example C*
Stream New material Mill discharge Tailings Fine product Water to mill Air to mill Mill vent Ambient air to classifier Classifier exhaust Power to mill Classifier loss Mill loss
Stream symbol
BTU lb (“F)
N D T P W S V A E I K M
0.19 0.19 0.19 0.19 -t 0.25 0.25 0.25 0.25 -
200 500 300 c 200 2.6 p 53 53 221 221 -
* Quantity given unless indicated otherwise. t Not applicable, heat of vaporization taken from steam tables. Abbreviations: a, assumption firmed up by trial and error; c, straight calculation;
air classifier can be greatly simplified of-thumb temperature differences streams and by defining fractional terms of those stream thermal data obtained.
by using rulebetween key heat losses in that are easily
References 1. 2.
3. 4.
5.
6. I. 8. 9. 10.
Sgaslik, F. New approach to air separator design. ZKG Intern&/. 1985, 38(l), 22 Klumpar, IV., Currier, F.N. and T.A. Ring, Air classifiers. Encyclopediu of Fluid Mechanics, N.P. Cheremisinoff, ed.
vol. 4, Solids and gas-solids flow, Houston: Gulf Publishing, 1986, p. 1361. Also Chem Eng. 1986 93(5), 77 Air Separators. Boston, MA: Sturtevant, Bulletin B-087-N, 1987 Particle Size ClassiJiers. A Guide to Performance Evaluation, AIChE Equipment Testing Procedure. New York: American Institute Chemical Engineers, 1980 Wong, J.B., Ranz, W.E. and Johnstone, H.F. Collection efficiency of aerosol particles and resistance to flow through fiber mats. J. Appl. Phys. 1956, 27(2), 161 Licht, W. Removal of Particulate Matter from Gaseous Wastes-Filtration. American Petroleum Institute, 1961 Klumpar, I.V. Verifying air classifier design. ZKG Inr. 1991, 44(3), 119 Klumpar, I.V. Control and scale-up of air classifiers. Eng. Progr. 1992, 88(4), 50
Chem.
Klumpar, I.V., et al. Air classifiers with optimum design and operation. ZKG Int. 1986, 39(6), 305 Klumpar, I.V. An efficiency boost for size reduction. Chem.
150g 204 t 165t 150 a
6.08 19.38 9.41 5.70 2.87 1.06 2.44 4.42 8.18 10.18 0.51 2.04
70 g 80 g 184t 80 g 148t -
g, given; p, postulation;
t, trial-and-error
Perry, R.H., Green, D.W. and Maloney, J.O. Perry’s Chemical Engineers’ Handbook, 6th ed. New York: McGraw-Hill, 1984, Fig. 5-76
calculation.
Rod height, hb = 0.872 m Air flow rate, Q, = 19.5 m3/s Air density at atmospheric pressure and 0°C p0 = 1.293 kg/m3 Actual temperature, t = 50°C Air viscosity at atmospheric pressure and 50°C p = 1.9 (lo-‘) kg/m s Feed rate, W, = 27 kg/s Startup power requirements
Inertia computation, based on the configuration of the rotor and the size, shape and weight of its structural elements, is a standard procedure and is not shown here. The result is M,rS = 1626 kg m2
For a conventional
startup time T = 8s, the power is
P, = 4 (3.142) 1626 (32)/8 (1,000) = 72.2 kW Power requirements
to overcome
(31)
drag
Projected rod area, A, = hb db = 0.872 (0.0127) = 0.0111 An analysis of field data indicates that the particle cloud density is approximately 10% higher than the air density. pc = 1.1 p,, 273/(t + 273) = 1.1 (1.293) 2731323 = 1.20 kg/m3
Eng. 1992, 99(3), 106
11.
c t t t c c t c t g t t
I!4 b = 19.YO.872 [3.14 (2.17) - 196 (0.0127)1 = 5.17 m/s
(30)
Appendix: Power computation example
w = 3.14 (3) 2.17 = 20.5 m/s
(28)
Given data
w, = (20.52 - 5.172)o.5 = 19.8 m/s
(29)
Actual rotor diameter, d = 2.23 m Rotor speed, s = 3s~’ No. of rejection elements (rods), N = 196 Diameter of the circle of rods, d, = 2.17 m Rod diameter, db = 0.0127 m 134
Sep. Technol.,
1992, vol. 2, July
The effect on the drag coefficient of particle cloud viscosity, pc, is minor because, for most modern air classifiers, the applicable portion of the kb vs. Re curve” is in the flat, turbulent region. Let us assume
Air classifier
(see statement
= 0.0127 (1.20) 19.8/3.8 (10w5) = 7,940
k, = 1.2
following Equation 36)
PF = 27 (20S2)/2 (1,000) = 5.67 kW
(36)
Total operating power requirements
D, = 1.2 (0.0111) 1.20 (19.g2)/2 = 3.13 N
from chart” P, = 196 (19.8) 3.13/1000 = 12.15 kW Power requirements
I. V. Klumpar
v = w = 20.5 m/s
pC = 2~ = 2 (1.9) 10T5 = 3.8 (10m5) kg/m s Re = db~r~J~,
performance:
to accelerate
feed
P = 12.15 + 5.67 = 17.8 kW
(35) (34)
The startup power can be reduced start-up time is extended to T(P,IP)
= 8 (72.207.8)
Sep. Technol.,
(37) to 9.11 kW if the
= 32s
1992, vol. 2, July
135
Engineers,
Cambridge,
MA, USA
Operation of air classifiers, particularly in closed circuit with grinding mills, is described. Performance measures are reviewed and an improvement of the AIChE standard method of calculating the selectivity curve is proposed. Approach to performance optimization is derived from the fundamentals of air classt$cation. The computations of power requirements for classical air separators and for modern high efficiency machines are compared. Trial-and-error calculations for optimizing the complex heat balance of a grinding mill in closed circuit with an air classifier are discussed.
Kevwords: powder/bulk solid classification: circuit grinding, heat balance
particle size; selectivity;
Introduction Air classifiers or air separators are rotary machines that separate a particulate feed into a fine and coarse fraction, using air or other gases to entrain the fine product and a rotor to reject any airborne coarser particles.’ While air classifiers can be used for various purposes, the most common application is in closed circuit grinding as shown in Figure 1. New material, N, (makeup) is fed into mill 1 together with tailings (or “tails”), T, recycled from air classifier 2. The mill receives a power input, I, is swept by air, S, and further cooled by a water spray, W. Mill discharge, D, becomes air classifier feed, F. The classification is primarily done by ambient air, A, and the tailings are returned to the mill. In classical machines, the bulk of the fine product is recovered in the air classifier while in the modern high efficiency equipment the fines are separated from the entraining air in baghouse 3. Mill vent, V, leaves the plant as exhaust, X, and/or is used as stream, U, in the air classifier. The fictitious loss streams, K, M, and the boundary lines will be explained later. In the first part of this paper, the performance measures are reviewed and an improvement of the AIChE standard method of selectivity calculation proposed. Next, the approch to performance optimization is derived from air classification principles. In a separate section, the minimization of power requirements is re-
air classifiers; power requirements;
closed
lated to air classifier performance. Finally, cooling optimization is discussed using an illustrative example.
Performance measures Air classifier performance is measured in different ways.2 Yield is defined as the amount of product per unit amount of feed expressed as a fraction or percent. For example, fine product yield Yp = w,/w,
(1)
Symbols are listed in the Notation section. Recovery (sometimes called efficiency) is the percentage of a selected cumulative fraction of feed that goes to either product. For fines EP = X, WJX,
W,
(2)
where X, and X, are the fractions of fines and feed, respectively, that pass a specific screen size, e.g., 200 mesh. Based on Equation 2, the following more practical formula can be derived by mathematically eliminating the material flow rates, which are sometimes difficult to measure in a plant:3 Ep = X,(X,
- X,)/X,(X,
- X,)
(3)
where subscript T refers to the tailings. Efficiency (or overall efficiency) is the difference between fine and coarse product recoveries: E = EP - E, = (X, W, - X, WT)/XF W,
Based on experimental and engineering work done at Sturtevant, Inc., Boston, MA, USA. Address reprint requests to Dr. Ivan V. Klumpar, Badger Engineers, One Broadway, Cambridge, MA, 02142, USA. Received 31 October 1991; accepted 6 February 1992
124
Sep. Technol.,
1992, vol. 2, July
(4)
A more practical formula is E = (X, - X,)(X,
- X,)/X,(1
- X,)(X,
- X,) (5)
0 1992 Butterworth-Heinemann
Air classifier performance: Recovery and efficiency, while widely used, are not very accurate because only one screen size specifies the feed or products. Air classifier performance is best characterized by the plot of selectivity vs. particle size. Selectivity can be calculated if the particle size distributions of at least two of the three materials involved (feed, fines, and tailings) are known. Selectivity is the weight percentage of all particles of a given diameter in the feed that go to the tailings.4 As there is an extremely high number of different particle sizes and shapes, the group of particles of a
I. V. Klumpar
given diameter is defined as the narrow fraction passing one screen (e.g., 200 mesh) but retained on the next smaller one (250 mesh). The mass of this narrow fraction is denoted as m. If one of the automatic particle size analyzers based on X rays, light beams, or lasers is used, the corresponding results are obtained in microns , i.e., 74 and 63, respectively. The average diameter of that fraction is defined as the arithmetic mean or 68.5 pm. Selectivity for particles with an average diameter d, is mathematically expressed as
Notation A b C L d E f G
Ez h
i .i
k
L M m N
If, P Q 4
8e
s s
St T t u
u
W
i X
Y YJ
z
Projected area Intercept Centrifugal force Specific heat Drag force (Screen opening) diameter (Classification) efficiency (or recovery) Friction factor Gravity Acceleration due to gravity Enthalpy flow Height Size interval number Sharpness index Drag coefficient Fractional heat loss Mass of a body Mass or mass flow rate of a narrow size fraction Number of rejection elements Number of size intervals Power Pressure Volumetric flow rate Quotient, ratio Radius Reynolds number Selectivity Speed Stokes number Start-up time Temperature Air velocity Particle velocity Mass or mass flow rate of a material stream Rejection element velocity Cumulative mass fraction Mass fraction of narrow size fraction Yield Auxiliary variables Equivalent length
Greeks h Difference operator 2 Sum operator Viscosity p 7T = 3.1416 Density P Stream symbols (also used as subscripts) A Ambient air D Mill discharge E Air classifier exhaust F Air classifier feed Power input to mill Z K Air classifier heat loss M Mill heat loss N New material P Fine product S Sweep air to mill T Tailings U Mill vent used in air classifier V Mill vent W Water to mill X Mill exhaust Subscripts “b : e ; ;, m n P r
s :, 1 2
Average particle Blade, rod, or other rejection element Air-particle cloud Drag Effective Gas, air Hydraulic Size interval Consist based Mass based Normalized Individual particle Rotor Start-up Particle trajectory Reference Direct interception Inertial impaction
Sep. Technol.,
1992, vol. 2, July
125
Air classifier performance:
I. V. Klumpar
,._._._.-._._._._._.t_._., A
IE
ii
3
I
&....i i
---
T
i
_. U
i’
!
_,-.-.-a_.-
c
_--,_._._._.
i i P.
i J
---+X --I
f
I ,.......)
M
i
Figure 1 Closed grinding circuit. ----, mass or power stream; ------, alternative stream for classical air classifiers; ......., loss stream; classifier system boundary; -.-.-.-.-r --.----.--., mill system boundary; 1, mill; 2, air classifier; 3, bag house
S = m,lm,
(6) Figure 2 is a typical plot of selectivity vs. particle size, expressed as diameter, for a regular cement (specific surface of 370 m2/kg according to Blaine) produced in a commercial Sturtevant High Efficiency SD Classifier under normal plant operating conditions. In interpreting the selectivity curve, two features are important: the slope of the main portion of the curve and the position of the “fishhook” in the lower portion. The slope reflects the sharpness of cut or how much the actual classifier differs from an ideal machine. Ideal classification would result in a vertical straight line intersecting the actual curve at 50% selectivity. In the case shown in Figure 2, all particles of 40 pm or smaller would go to the fines while larger particles would report to the tailings. The slope is measured with the sharpness index conventionally defined as the ratio of the particle sizes corresponding to the 25% and 75% selectivity, or (7) The lowest point of the selectivity curve is sometimes called the apparent bypass because it is the measure of the amount of particles that behave as if bypassing the separation zone of the air classifier and going directly to the “wrong” product, e.g., fine particles reporting to the coarse fraction. The apparent bypass in Figure 2 is 11%. While some particles actually pass by the separation zone without being exposed to classification, there are also other causes of the fishhook, such as particle agglomeration or inadequate dispersion of agglomerates present in the feed, grinding of material in the air classifier, and entrainment by recycle air of particles already classified. j
=
d~S%ld?S%
a
Selectivity computation In computing selectivity, it is convenient to adjust the experimental data in the following way. The average particle diameter is defined as 126
Sep. Technol.,
1992, vol. 2, July
where i is the interval number or the sequence number of the narrow material fraction passing the larger screen il but retained on the next smaller screen i2, and d is the screen opening diameter. In automatic particle size analyzers, d is the largest particle size diameter of the cumulative material fraction. Let us refer to d as the indicated diameter for simplicity, and d, as the midpoint diameter because it is in the middle of an adjacent pair of ds. In the sample calculations in Table 1, i, d,, and d are recorded in the first three columns. In the next step of selectivity computation, cumulative mass fractions or percentages X,, X, and X, in the feed, tailings, and fines (columns 4, 5, and 6 in Table J), respectively, are converted to the respective mass fractions or percentages +, xr, and xP of the individual particle size intervals (columns 7, 8, and 9) as follows: x = xi, - xi2
(9)
In batch classification tests and many industrial operations, consist and mass of the tailings and fines are known. Consist is used here to denote particle size distribution. Mass is defined either as weight of the batches or as flow rate in continuous operations. For these cases, Equation 6 can be rewritten as S,, = m,l(m, + mP) = xT W,lx, W, = = XTW&x, w, + xp W,)
(10)
based on the following material balance relationships: mF = mP + mT
(11)
w, = w, + w,
(12)
m = XW
(13)
Selectivity S,, is referred to as mass based because the mass of two out of the three materials (feed, tailings, and fines) have to be given. Even if all three are known, it is recommended that Equation 10 is used for consistency. Sample calculations are shown in columns 2-5 of Table 2 and the resulting selectivity curve plotted in Figure 2. In industrial grinding mills operating in closed circuit with air classifiers, the flow rates of classifier feed, tailings, and fines are often not known while the consist of these three materials can be easily determined by sampling and particle size analysis. This is routinely done for at least one of the three materials. In this case, selectivity can be computed as follows: Sk = xr(xp - +Yx,(x,
- x7)
(14)
This can be conveniently rewritten (Braun, R. Canada Lafarge Cement, Ltd., Montreal, private communication, 1987) as Sk = x,ylx,z
(15)
where y = x, - XF
(16)
2 = xp - xr
(17)
Air classifier
performance:
I. V. Klumpar
65
.
25
3 4 > 4 G al rl rt
10
_i
I __I I 5
0.1
1
10
102
103
Particle diameter, d, microns Figure 2
Table 1
Interval no. i 1
1 2 3 4 5 6 7 8 9 10 11 12
Mass based selectivity
Sample selectivity calculations,
Indicated diameter km) d 2 2.8 3.9 5.5 7.8 11 16 22 31 44 tf 125 177
part 1
Mid point diameter (;m) 5
3.4 4.7 6.7 9.4 13.5 19 27 38 53 75 107 151
Cumulative mass % of mtl. smaller than d in Feed
Tails
Mass % of material between 2 adjacent ds in Fines
Feed
Tails
Fines
xF
XT
XP
xF
4
XT
5
6
XP
7
8
9
2 5.5 9 10.5
5 12.5 19.5 25.5 38.5 51.5 65.5 79.5 96 100 100 100 100
6.4
3.5 3.5 1.5 4.5
7.5 7 6 13 13 14 14 16.5 4 0 0 0
z.6 10 16.5 20 28.5 51 64 81 89 95 100 100
:z 23.5 33 55 72 86 98 100
1:: 8.5 11.5 11 13 17 8 6 5 0
Sep. Technol.,
g.5 9.5 22 17 14 12 2
1992, vol. 2, July
127
Air classifier Table 2
performance:
I. V. Klumpar
Sample selectivity calculations,
part 2
Mass* of material between 2 adjacent ds
Interval no. i 1
Tails
Feed
w
mF
1 2 3 4 5 6 7 8 9 10 11 12
Fines m
2
3
4
953 903 687 1,561 1,590 1,603 1,951 2,926 1,386 812 696 116
203 203 87 261 290 203 551 1,276 986 812 696 116
750 700 600 1,300 1,300 1,400 1,400 1,650 400 0 0 0
21 22 13 17 18 13 28 44 71 100 100 100
This selectivity is referred to as consist based. Sample calculations are shown in columns 6, 7, and 8 of Table 2 and results plotted in Figure 3. The data in columns 2, 4, 5, and 6 of Table I have been taken from an actual industrial operation. The low accuracy of plant measurements compared with laboratory tests is reflected in the wide scatter of the points in Figure 3. The points in Figure 2 are less scattered because the W,/W, ratio in Equation 10 is a constant while the corresponding y/z ratio in Equation 15 might vary with particle size. The consist based selectivity curve can be smoothed out by using a constant mean y/z ratio, (18)
It can be determined graphically on plotting y vs. z and measuring the slope (see Figure 4). Alternatively, q can be computed by regression analysis, e.g., using the least squares method: nxyz - Izyzz q =
nCz2- (xz)2
The corresponding
intercept
on the Y-axis
= 10,000
sy
5
8
1.1 0.5 2.5 4.5 1.5 3 1 -0.5 -4 -6 -5 0
4 3.5 4.5 8.5 8.0 10.5 4.5 -5.5 -13 -14 -12 -2
15 8 24 28 8 9 16 12 65 100 100
(20)
Air classification is affected by two basic phenomena: selective entrainment and selective rejection. For the sake of argument, let us look at the particle size distribution of the feed as a mixture of fine, medium, and coarse particles. First, the fine and medium fractions become airborne. Then the medium-sized particles are rejected and fall with the coarse ones to the bottom chamber of the equipment while the fine particles are entrained by the air through another chamber to the outlet. An example is shown in Figure 6. Selective entrainment occurs in a swirl of particles and air. The air rotation is generated by an internal fan or by blowing the air tangentially into the classifier. Unless the solids are also fed tangentially by pneumatic conveying using the classification air, the swirling of particles is caused primarily by a rotating distributor plate and only to a lesser degree by the air movement if it is curved. In any case, for the selective entrainment to operate effectively, the drag force of the air, DP , has to balance gravity, G, and centrifugal force, C. These three forces acting on a particle can be expressed as
G = M,g(p,
- t1)~/2 - P,)
(21)
It is shown in the last column of Table 2 for q = 0.349 and plotted in Figure 5. 128
Sep. Technol.,
1992, vol. 2, July
(22)
(23) (24)
where the particle drag coefficient, k,,, is constant or increases with the Reynolds number. Re = d,(u - v)p,Ipg
s, = qx,lx,
19 19 15 18 15 11 25 45 74 81 84
Principles of air classifier performance
C = MPv21r, should be close to zero. The straight line in Figure 4 was calculated by this method. The “normalized” selectivity can then be computed as
Normalized selectivity
(fines).
DP = k,A,p,(u
b = ZyZz2 - Z,Cyz nCz2 - (Zz)2
Consist based selectivity (%I Sk
Cl 5
l Mass in kilograms per hour here or in grams in batch tests. Mass flow rates (kg/h): kV, = 15,800 (feed), Wr = 5,800 (tails), W, Mean y/z ratio: q= 0.349.
q = y/z = constant
Auxiliary variables (%I
Mass based selectivity (%I
(25)
According to the principles of air conveying, there are of course limits as to how much a given air volume can entrain.
Air classifier
performance:
I. V. Klumpar
65
Particle
diameter,
d,
microns
Figure 3 Consist based selectivity
Rejection happens when a particle collides with a rejection element of the air classifier rotor, such as a blade or rod. Rejection is measured with the efficiency of capture, Eb ,5that occurs either by direct interception or inertial impaction.6 In the former case, the particle is swept to the rejection element by air and the corresponding efficiency is E, = (1 + d,ldb) -
l/( 1 + d,/d,)
(26)
where db is an equivalent diameter unless the rejection element is a rod. In inertial impaction, the collision course is controlled by particle inertia rather than by the air streamlines, and the efficiency, E,, is approximately proportional to the Stokes number. St = W, d;@, - pJ18~
db
(27)
The effective blade or rod velocity, we, is discussed below. The total capture efficiency is the sum of E, and E, . Furthermore, statistics indicate that collision frequency increases with the number of blades or rods per unit circumference of the rotor. Suppose the rotor is a “cage” formed by a circle of vertical rods or blades, and the air swirls around and
inside the cage, as in modern high efficiency equipment, e.g., the Sturtevant SD Classifier or the Fuller O‘SEPA machine2 (Figure 6). In that case, the radial velocity component, ub, of air flowing through a “window” between two adjacent rods or blades can be readily computed from the volumetric air flow rate and the cage geometry,7*8 as will be subsequently explained. If the rotor diameter, d, , is defined by the periphery of the set of rejection elements, the actual rod or blade velocity is w = rrs d,
(28)
The effective rod or blade velocity with respect to the air velocity is then given by the parallelogram of vectors w2e = w2 + L2b
(29)
where the window velocity can be approximated ub = Qg/hb(r d, - N db) Performance
as (30)
optimization
The following parameters control air classifier performance in terms of yield, recovery, efficiency, and selectivity: Sep. Technol.,
1992, vol. 2, July
129
Air classifier performance:
Figure 4 0.95
l l l l
Graphical
selectivity
I. V. Klumpar
normalization.
Based on consist;
Air velocity Feed rate Particle velocity Rate of particle rejection by the rotor, or rejection for short
Among these four parameters, rejection is unique in that it directly affects another important performance factor, the fineness of the fine product, because particle diameter, dP, does not appear in Equations 22, 23, or 24, and Equation 25 has only a minor effect. Rejection, in turn, is controlled by l l l
Rotor speed Number of rejection elements of the rotor such as blades or rods Type, i.e., size and shape of these elements
Additional parameters that control performance are air density and viscosity. The latter has a less significant effect via the Reynolds number (see Equation 25). In special applications, density (and viscosity for that matter) can be varied by using gases other than air. This is expensive unless one of the modern air classifiers with gas recycle is used. Moreover, in close circuit grinding of large volume materials such as cement in air-cooled mills, the mill exhaust is often passed through the air classifier. The use of another gas would 130
Sep. Technol.,
1992, vol. 2, July
slope q = 0.349; Y-axis intercept
b = -0.03;
correlation
coefficient
=
require putting the entire system under a non-air atmosphere and provide expensive coolers of dust-laden gas. There are other factors that affect performance, such as overall equipment size, proportions of the internal parts, and the distances between them. However, these are design parameters that usually cannot be varied once a piece of equipment is made. Design optimization is discussed elsewhere.9 In the classical air classifiers, only feed rate and rotational speed can be changed independently, because the air velocity is controlled by an internal fan that is mounted on the same shaft as the rotor. Changing the number and type of rejection elements usually requires a partial disassembly of the equipment. Only in the Sturtevant SD Classifier can the rods be readily taken out or inserted through a manhole, though the machine still has to be shut down temporarily. Air velocity improves performance if it is compensated for by increasing rejection, because otherwise drag prevails over gravity and centrifugal force (see Equations 22-24), shifting the force equilibrium toward the entrainment of some coarse particles. On the other hand, with more rejection, product fineness rises but performance deteriorates unless air velocity is increased at the same time to maintain a balance between entrainment and rejection.
Air classifier performance:
I. V. Klumpar
65
25
III
Particle
I
diameter,
Iill
d,
microns
Figure 5 Normalized selectivity
Analysis of Equations 26-30 and the statistical findings show that rejection increases with increasing particle diameter, rotational speed, particle-air density difference, number of rejection elements, and to a certain extent with window air velocity. Rejection is smaller with bigger rejection elements and greater gas viscosity. The most significant factors are particle and blade or rod size, because the other parameters appear only in one of Equations 26 and 27, and in some cases are adversely affected by the rotor geometry. However, particle size is a given parameter in most applications. Blade or rod size is not a convenient rejection control parameter. First, changing these parts requires partial dismounting of the air classifier in most cases. Second, rods, which are the easiest to change, can be reduced only to a certain degree because they have to be robust to support the rotor structure and withstand erosion. While rotor speed has only a secondary effect on performance, it can be varied in a wide range. Consequently, it is the best rejection controlling parameter. Particle velocity is not an important optimization parameter because it can be independently controlled only in the rare air classifiers with the rotating internal
feed distributor mounted on a separate shaft. In all other mechanically fed classifiers, particle velocity is a function of rotor speed and thus tied to rejection. In pneumatically fed machines, particle velocity depends on air velocity. Decreasing the feed rate improves performance because the feeding of fewer solids effectively undersizes the air classifier, thus relatively increasing entrainment capacity and rejection.
Power minimization From the discussion in the previous section on performance optimization, it appears advantageous to run air classifiers at high air flow rate and high rotor speed, at least if a very fine product is called for and a modern machine is used. Let us look at how this conforms to the general requirement of keeping plant power consumption low. An air classifier needs power to pump the air and run the feed distribution and rejection rotor. In the classical machines, the fan and the rotor are mounted on the same shaft, and power is computed by start-up requirements, i.e., to bring the shaft from standstill to operating speed, s, expressed as Sep. Technol.,
1992, vol. 2, July
131
Air classifier performance:
I. V. Klumpar I
PARTICLE FEED
exceeds the start-up power requirements. This resistance has two components. One is the friction of the rotor with the particle-air cloud that can be modeled as the power to counteract the drag of the rejection elements passing through that cloud. P, = Nw,D,
(34)
where the rod or blade velocity is given by Equation 29 and drag by D, = k,,A,p,w;l2
(35)
The drag coefficient, k,,, is constant or increases with a Reynolds number that depends on the shape and size of the rejection element. The effective rod or blade velocity for some air classifiers is defined by Equation 29. The second resistance applies only to machines with rotating feed distributors. The power, PF , to overcome this resistance corresponds to the kinetic energy to accelerate the feed from zero horizontal velocity, when the material drops on the distributor, to the velocity, u, of the particles as they slide off the distributor rim (Figure 6): P, = w,v2/2
As the particle velocity, u, cannot exceed rim velocity, it can be conservatively approximated as u = w. The total rotor power requirements are the sum of the two components
\ TO COARSE PARTICLE CHAMBER TO FINE PARTICLE CHAMBER TANGENTIAL
0
AIR
@
P = P, + PF
FLOW
RADIAL
AIR
FLOW
Figure 6 Operation of a high efficiency air classifier C, centrifugal force; D, drag; G, gravity. Based on reference 2 by special permission of Chemical Engineering. Copyright 0 1986 by McGraw-Hill, Inc., New York, NY 10020
(31)
where My; is the rotor inertia. Modern high efficiency classifiers have a separate fan, the required power for which is proportional to air flow rate, Qg, and pressure drop. Ap = 2fu2Zld,,
(32)
The friction factor, f, is constant or increases with the Reynolds number. Re = d,,up,Ip,
(33)
Minimizing fan power consumption requires cutting air velocity, which runs against the needs of the performance optimization of the air classifier proper. It is obvious that there is a trade-off between saving electricity and improving performance. Power requirements of the air classifier rotor can be minimized by extending the start-up time, T, using invertors. This can be done only to the point where the resistance of the rotor under operating conditions 132
Sep. Technol.,
1992, vol. 2, July
(37)
Power computations are illustrated in the Appendix. Substituting the previously discussed relationships into Equation 37 leads to a very complex mathematical model. It was reduced by simplifying assumptions to the following approximate formula:‘,* P = constant
P s = h=M rr?s=IT I
(36)
s2 d2r WF
(38)
again, trying to minimize the rotor power requirements by reducing speed runs against performance optimization. Accordingly, the price to be paid for better performance is often higher power costs. Cooling optimization Modern air classifiers in closed circuit with grinding mills contribute significantly to the cooling of the fine product. The thermal conditions of the circuit can be optimized based on a heat balance that sometimes requires a trial-and-error solution because of the complexity of the problem. Using the system boundaries indicated in Figure I, the following heat balances can be written: Classifier:
Ho + HA + H,
= HP + HT + HE + HK
(39) Mill: HN + H, + H, - H, + H, = Ho + H, + HM (40)
Air classifier performance: Table 3
Input data for illustrative
example
Production in thousands of lbsih (Wp = W,) Air in thousands of actual ft3/min Ambient to classifier CQ, = Q,) Mill vent (0” = 0s = QX) Circulating load ( WTl W,) Mill power input (P,) Water to mill (Owl, postulated Temperatures (“F) New feed (t) Air to classifier and mill (f,,= ts) Water (tw) Product (t,J, assumed Temperature differences (“F), selected Tailings and product (tr - tp) Exhaust air and product (tp - tE) Mill vent and discharge (to - tv) Fractional heat losses selected Classifier (LK) Mill (Lnn)
Enthalpies
100 k PPH 50 k ACFM 12 k ACFM 150% 4,000 HP e.g., 5.2 GPM 160 80 70 e.g., 160”
of most process streams are computed
15 2” 20” 12% 20%
as
H = Wc(t - to)
(41) with the reference temperature, c,, conveniently set at zero. The mill cooling water enthalpy, H,, is approximated as the heat of vaporization and taken from steam tables. Power input to the mill is also considered a stream. Expressed in enthalpy units, it is denoted as HI. To compare heat losses HK and HM of air classifiers and mills, respectively, of various types, sizes, and configurations, it was propsoed to define the fractional heat loss, L, as the ratio of the absolute heat loss to the enthalpy of the heat source.” For the classifier system, the heat source enthalpy is the difference between the inlet and outlet enthalpies of the warmer streams. For the mill system, the heat source is the power input to the mill proper since power to run other equipment such as elevators can be neglected. The two fractional heat losses are then L, = HKI(HD - HP - HT) L,
(42)
= H,IH,
(43) In trial-and-error calculations, the “internal” streams D and T of the circuit are often not available, while the “external” streams A, E, and U are (or can be more readily approximated). On combining Equations 39 and 42, a more useful relationship is obtained: HK = (HE - HA - H,)L,I(l
- L,) (44) To demonstrate the application of Equations 39-44 to the assessment of thermal operating conditions of closed grinding circuits, let us use the following illustrative example. Calculate the effect on product temperature of the amount of water sprayed into the mill for various scenarios such as seasonal changes of ambient temperature and humidity, maximum product temperature tolerated by industrial customers or wholesalers, expected water cost increases, etc. Input data are presented in Table 3. The mass and
I. V. Klumpar
volumetric flow rate equations at the beginning of the table neglect material losses for the sake of simplicity. No mill vent goes to the classifier. For each scenario, one or more series of trial-and-error computations are required. A flow rate of the water spray is postulated for each series. A product temperature is assumed for, and firmed up by, each trial and error. The data denoted as selected are based on actual experience that, of course, can vary from plant to plant and also when operating conditions at the same plane change. The last of a series of trial-and-error runs is summarized in Table 4 The computations are done in the following way: 1. Compute tailings flow rate from circulating load. 2. Postulate water flow rate. 3. Compute flow rates of streams N, S, and E according to Table 3. 4. Assume product temperature. 5. Temperatures t, and tE are higher and lower by 15°F and 2”F, respectively, than tp (see Table 3). 6. Compute enthalpies of all streams except D and Vusing Equation 41, steam tables, and conversion from HP to BTU/h. 7. Compute classifier and mill heat losses on substituting L, and L, from Table 3 into Equations 44 and 43, respectively. Also note that U = 0. 8. Compute mill discharge enthalpy from Equation 39. 9. Compute mill discharge temperature from Equation 41. 10. Temperature t, is 20°F less than to. Il. Compute mill vent enthalpy from Equation 41. 12. Insert all enthalpies into Equation 40 to check assumed t,. Other problems of evaluating and optimizing the cooling of closed grinding circuits can be solved in a similar way. For repetitive calculations, the development of a spreadsheet computer program is recommended.
Conclusions After reviewing the fundamentals of air classification and traditional techniques for evaluating and optimizing air classifiers, the following new methods are presented. For industrial measurements that are not as accurate as those in research laboratories, it is proposed to substantially improve the standard method of selectivity computation by first determining the mean value of ratios that tie together the concentration in feed, product, and tailings of each particle size. For modern energy-efficient air classifiers that use inverters, the classical rotor inertia-based formula for computing power requirements is obsolete. Instead, a new method is discussed based on feed particle acceleration and air resistance to the machine rotation. The trial-and-error computation of the complex heat balance of a grinding mill in closed circuit with an Sep. Technol.,
1992, vol. 2, July
133
Air classifier performance: Table 4
Summary
I. V. Klumpar
of illustrative example C*
Stream New material Mill discharge Tailings Fine product Water to mill Air to mill Mill vent Ambient air to classifier Classifier exhaust Power to mill Classifier loss Mill loss
Stream symbol
BTU lb (“F)
N D T P W S V A E I K M
0.19 0.19 0.19 0.19 -t 0.25 0.25 0.25 0.25 -
200 500 300 c 200 2.6 p 53 53 221 221 -
* Quantity given unless indicated otherwise. t Not applicable, heat of vaporization taken from steam tables. Abbreviations: a, assumption firmed up by trial and error; c, straight calculation;
air classifier can be greatly simplified of-thumb temperature differences streams and by defining fractional terms of those stream thermal data obtained.
by using rulebetween key heat losses in that are easily
References 1. 2.
3. 4.
5.
6. I. 8. 9. 10.
Sgaslik, F. New approach to air separator design. ZKG Intern&/. 1985, 38(l), 22 Klumpar, IV., Currier, F.N. and T.A. Ring, Air classifiers. Encyclopediu of Fluid Mechanics, N.P. Cheremisinoff, ed.
vol. 4, Solids and gas-solids flow, Houston: Gulf Publishing, 1986, p. 1361. Also Chem Eng. 1986 93(5), 77 Air Separators. Boston, MA: Sturtevant, Bulletin B-087-N, 1987 Particle Size ClassiJiers. A Guide to Performance Evaluation, AIChE Equipment Testing Procedure. New York: American Institute Chemical Engineers, 1980 Wong, J.B., Ranz, W.E. and Johnstone, H.F. Collection efficiency of aerosol particles and resistance to flow through fiber mats. J. Appl. Phys. 1956, 27(2), 161 Licht, W. Removal of Particulate Matter from Gaseous Wastes-Filtration. American Petroleum Institute, 1961 Klumpar, I.V. Verifying air classifier design. ZKG Inr. 1991, 44(3), 119 Klumpar, I.V. Control and scale-up of air classifiers. Eng. Progr. 1992, 88(4), 50
Chem.
Klumpar, I.V., et al. Air classifiers with optimum design and operation. ZKG Int. 1986, 39(6), 305 Klumpar, I.V. An efficiency boost for size reduction. Chem.
150g 204 t 165t 150 a
6.08 19.38 9.41 5.70 2.87 1.06 2.44 4.42 8.18 10.18 0.51 2.04
70 g 80 g 184t 80 g 148t -
g, given; p, postulation;
t, trial-and-error
Perry, R.H., Green, D.W. and Maloney, J.O. Perry’s Chemical Engineers’ Handbook, 6th ed. New York: McGraw-Hill, 1984, Fig. 5-76
calculation.
Rod height, hb = 0.872 m Air flow rate, Q, = 19.5 m3/s Air density at atmospheric pressure and 0°C p0 = 1.293 kg/m3 Actual temperature, t = 50°C Air viscosity at atmospheric pressure and 50°C p = 1.9 (lo-‘) kg/m s Feed rate, W, = 27 kg/s Startup power requirements
Inertia computation, based on the configuration of the rotor and the size, shape and weight of its structural elements, is a standard procedure and is not shown here. The result is M,rS = 1626 kg m2
For a conventional
startup time T = 8s, the power is
P, = 4 (3.142) 1626 (32)/8 (1,000) = 72.2 kW Power requirements
to overcome
(31)
drag
Projected rod area, A, = hb db = 0.872 (0.0127) = 0.0111 An analysis of field data indicates that the particle cloud density is approximately 10% higher than the air density. pc = 1.1 p,, 273/(t + 273) = 1.1 (1.293) 2731323 = 1.20 kg/m3
Eng. 1992, 99(3), 106
11.
c t t t c c t c t g t t
I!4 b = 19.YO.872 [3.14 (2.17) - 196 (0.0127)1 = 5.17 m/s
(30)
Appendix: Power computation example
w = 3.14 (3) 2.17 = 20.5 m/s
(28)
Given data
w, = (20.52 - 5.172)o.5 = 19.8 m/s
(29)
Actual rotor diameter, d = 2.23 m Rotor speed, s = 3s~’ No. of rejection elements (rods), N = 196 Diameter of the circle of rods, d, = 2.17 m Rod diameter, db = 0.0127 m 134
Sep. Technol.,
1992, vol. 2, July
The effect on the drag coefficient of particle cloud viscosity, pc, is minor because, for most modern air classifiers, the applicable portion of the kb vs. Re curve” is in the flat, turbulent region. Let us assume
Air classifier
(see statement
= 0.0127 (1.20) 19.8/3.8 (10w5) = 7,940
k, = 1.2
following Equation 36)
PF = 27 (20S2)/2 (1,000) = 5.67 kW
(36)
Total operating power requirements
D, = 1.2 (0.0111) 1.20 (19.g2)/2 = 3.13 N
from chart” P, = 196 (19.8) 3.13/1000 = 12.15 kW Power requirements
I. V. Klumpar
v = w = 20.5 m/s
pC = 2~ = 2 (1.9) 10T5 = 3.8 (10m5) kg/m s Re = db~r~J~,
performance:
to accelerate
feed
P = 12.15 + 5.67 = 17.8 kW
(35) (34)
The startup power can be reduced start-up time is extended to T(P,IP)
= 8 (72.207.8)
Sep. Technol.,
(37) to 9.11 kW if the
= 32s
1992, vol. 2, July
135