Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
437
The Use of Residence Time and Nonlinear Optimization in Predicting Comminution Parameters in the Swing Hammermilling of Refuse Geoffrey R. Shiflett and George J. Trerek' Depaflment of Mechanical Engineering, University of California, Berkeley, California 94 720
Earlier predictive methods for estimation of the parameters which govern size reduction of solid waste in a swing hammermill have been extended through the use of residence time information. Nonlinear optimization techniques are used to determine values for the selection and breakage functions. These functions are shown to be dependent on the operating conditions of the shredder. The characteristic size of shredded refuse is observed to decrease as the mill throuahDut increases. Within a certain range of oDeratina conditions the model adeauatelv simulates shredder behavior 'and permits the prediction of the particle size distribution of shredded refuse. ~
Introduction In order to provide a means for rigorously evaluating the various schemes for recovering materials and energy from the solid waste stream, increased attention is being focused on characterizing the variables governing individual unit operations. This is particularly true in the case of comminution operations. These are usually a first step common to most processing schemes and often utilize equipment not specifically designed for the size reduction of solid waste. Further, an increasing number of processes require multiple stages of size reduction commonly referred to as fine grinding. Although particle size significantly influences the efficiency of secondary processing, there is a paucity of the design data needed for the development of comminution equipment which will produce specific particle size distributions with a minimum expenditure of energy and machine wear. Initial investigations using the time discrete, size discrete matrix models described by Callcott (19671, Callcott and Lynch (19641, Lynch and Draper (1965) and others (Obeng and Trezek, 1975) as a means of modeling the comminution process in a swing hammermill have been only partially successful. This stems from the fact that these models are essentially black-box representations of the comminution process in which material transport is implicitly assumed to occur by plug flow. Also, because time is not an explicit variable in the governing equations, a detailed description of mill dynamics is not possible. In such models, the mill matrix coefficients are not easily physically interpretable. Although time and size discrete models can be used t.o predict size distributions and illustrate qualitative feedrate and moisture effects, certain basic questions such as: (a) why and how does the feedrate affect the product size distribution; (b) why and how does the moisture affect the energy consumption, etc. cannot be answered in a fundamental manner. Motivated by the successful extension of the size discretized, time-continuous model used for describing mineral batch ball milling to continuous laboratory mills (Herbst et al., 19711, the matrix approach to the analysis of refuse comminution was similarly modified to include a residence time distribution. This distribution, the mean value of which is essentially the mill holdup divided by the feedrate, is a major factor in determining the product size distribution. In the case of the overflow discharge ball or rod mills, the mass of holdup is assumed constant so that the mean residence time is solely a function of the feedrate. Unfortunately, this simplification cannot be extended to the hammermilling of refuse because of its highly com-
pressible nature. Thus the mean residence time will vary with both holdup and feedrate. The model presented here provides a means of simulating the hammermill size reduction of refuse based on the actual physical parameters of the process. It addresses the specific issues of (a) how the feedrate and moisture content affect the mill holdup, (b) how the mill holdup affects the energy consumption and residence time, and (c) how the selection and breakage functions can be estimated for different operating conditions. M a t r i x Analysis As a means of improving previous matrix analyses which were applied to the study of heterogeneous nonbrittle material comminution, the size-discrete, time-continuous approach was implemented. A particulate assembly can be represented by the vector M such that
M = H m = Hlm,m2, ..., mi, ..., mnlT
(1)
If M is the material within the grinder a t any particular time, H represents the mass of material held in the mill (mill holdup) and a mass balance on the ith size class a t time t will yield i-1 d -[Hmi(t)] = -si(Hmi(t)) + Cbijsjsj(Hmj(t)) (2) dt j=l If si and bij are assumed to be independent of the size x i and time, then eq 2 can be linearized allowing the breakage process to be represented by n simultaneous differential equations which, in matrix form are given as d -[[Hm(t)] = -[I - B]SHm(t) (3) dt For a batch process with an initial size distribution mb(0), H is constant and eq 3 becomes (4) whose solution is q ( t ) = exp[-(I - B)Stlq(O) (5) The application of a similarity transform to eq 5 yields q ( t ) = TJT-'q(O) (6) which allows for an expeditious evaluation of the solution to the linearized lumped parameter, size discrete, time continuous equation for batch grinding. By introducing the concept of residence time (the length of time a particle or size interval remains in the mill) the
0019-7882/79/1118-0437$01.00/00 1979 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 kf
Figure 1. Schematic diagram of closed circuit mill operation.
above model can be extended to continuous grinding. Assuming that all size intervals are characterized by a single residence time distribution, then the open circuit mill is the average of the batch response weighted with respect to residence time, i.e. mcp=
Sorn
mb(t)R(t) dt
(7)
Substituting eq 6 for mb in eq 7 yields the solution for continuous grinding, namely
mcp= T[
SornJ(t)R(t) dt]T-lmCf
(8)
When eq 8 is expressed in matrix form it becomes
mCp= TJC(~)T-'mcf
(9)
where
N O R M A LI Z E D TIME,@ Figure 2. Dimensionless residence time distribution.
=O
i#j
The integral expression for J c i i ( ~can ) be evaluated by Simpson's approximation. A schematic diagram of a closed circuit mill operation is shown in Figure 1. Here the classifier matrix C is an n X n diagonal matrix whose entries represent the fraction within each size class which is recycled to the mill. Applying standard matrix operations to the system of Figure 1 gives the solution to the continuous closed circuit mill product equation under steady-state conditions as (Herbst e t al., 1973) mp = (I - C)D(I - CD)-'mf
(10)
since M = Mf under steady state conditions. The mill matrix h is given by
D = TJC(7)T-'
(11)
P a r a m e t e r Estimation The fundamental assumptions involved in the parameter estimation are the consideration of the size reduction process as linear and the subsequent normalizability in the sense of Epstein (1947) of the size discretized breakage function. This means that the breakage function depends only on the difference, i - j, not independently on i and j and it also implies that if Bij is plotted against ( x i z / xjxj+l)llza single curve will result for all ij. Following the approach of Herbst and Fuerstenau (19681, the breakage functions can be given by
where K* and CY are time invariant constants derived from the zero-order production rate. The selection functions are given by (Herbst and Fuerstenau, 1968) (13)
An evaluation of the parameters K*, a , and s1 follows from the nonlinear optimization techniques which follow. Finally the individual breakage functions may be obtained from the difference from the normalized equation j = 1, ..., n - 1; j b.. = bL-l+l,l .
+ 1, ..., n - 1
(14b)
and from " Residence Time Distribution Previous investigations into the comminution of solid waste assumed that the residence time was negligible and hence could be approximated by a once through process (Obeng, 1974). To both test this assumption and provide the residence time distribution used in eq 9, a series of residence time experiments were done in which distinctively colored paper was used as a tracer material. The characteristic size of the tracer was near that of raw residential refuse to minimize any size effects and the tracer was introduced in the feed as a close approximation to a unit impulse. From the experimental data the dimensionless residence time distribution shown in Figure 2 was obtained for a mean residence time of 51.4 s. In a study of a rotary cutting mill similar to the horizontal hammermill of the present work, Klimpel and Austin (1972) assumed that material finer than the internal grate spacing of the mill will be rapidly removed from the mill and that the product size distribution is thus independent of the feed rate and holdup. However, the holdup involved in the hammermilling of refuse has been shown to increase more rapidly than the flow rate (Shiflett, 1978). The mean residence time, 7 , as defined by r = HIM,, thus increases with flow rate and should lead to a smaller particle size as the flow rate increases. This result is confirmed by the experimental observation that the characteristic particle size of shredded refuse does not
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
indeed decrease as the flow rate increases (Trezek and Savage, 1975). The work of Kelsall et al. (1970) provides some justification for the assumptions in the present work that the form of the dimensionless residence time distribution is invariant with the flow rate and applies equally well to all particle sizes, thus permitting extension of the model to flow rates other than those used in the residence time distribution experiments. Determination of Selection and Breakage Functions A. Optimization Technique. Values for the initial selection function and cumulative breakage functions are customarily estimated from data obtained with narrowly sized initial feed and short grind times. Unfortunately, due to the facts that narrowly sized feed is not a characteristic of municipal solid waste and short grind time experiments with the test hammermill were not possible t o perform, an alternative method for evaluating the selection and breakage functions was required. Austin and Klimpel (1970) and Klimpel and Austin (1977) used a conjugate gradient optimization technique for back calculating values for the selection and breakage functions from batch grinding data. Although the present study also uses a nonlinear optimization technique, the mill under investigation differs in that continuous feeding is assumed and batch grinding data cannot be obtained. The assumption that the selection and breakage functions are related by eq 1 2 and 13 simplifies the search for the n selection functions and n(n - 1 ) / 2 breakage functions considerably since both sets of parameters are dependent on only three unknowns-the initial selection function, sl, the zero-order production rate coefficient, K*, and the size modulus, a. The optimization technique used in the search process was a modified form of Powell's nonderivative search technique (Powell, 1964; Fiacco and McCormick, 1968; Suh and Radcliffe, 1978). Since a complete description of this method is outside the scope of this paper the reader is referred to the references for details. As in all optimization routines there is an objective function to be minimized. For this routine the function to be minimized has the form
439
Table I. Summary of Optimization Results
Q, s,
0.405 0.297 0.295 0.300 0.106 0.407 0.336 0.514 0.306 0.332 0.071 0.073 0.287 0.286 0.285 0.289 0.290 0.290
K*
K*/S,
0.190 0.347 0.347 0.336 0.311 0.411 0.253 0.396 0.294 0.252 0.292 0.300 0.314 0.308 0.308 0.290 0.284 0.281
0.469 1.168 1.176 1.120 2.934 1.010 0.753 0.770 0.961 0.759 4.113 4.110 1.094 1.077 1.081 1.003 0.979 0.969
a
F(x)
0.826 0.804 0.804 0.804 0.809 0.977 0.805 1.057 0.804 0.814 0.841 0.898 0.804 0.804 0.804 0.804 0.804 0.804
0.0119 0.0271 0.0614 0.0504 0.00597 0.00458 0.0161 0.00166 0.0215 0.00745 0.00964 0.00277 0.0196 0.0549 0.0249 0.0168 0.0135 0.0123
B,,
t/h
0.351 0.884 0.890 0.849 2.220 0.720 0.569 0.553 0.726 0.573 3.090 3.010 0.829 0.815 0.817 0.759 0.742 0.932
1.0
0.9 1.4 1.5 1.5 2.1 2.4 2.4 2.4 3.0 3.4 3.8 3.8 3.9 4.6 5.0 6.2 6.6 7.0
I
.2-
A
A
2.0
where F(x) = unconstrained objective function, gi(x) = inequality constraint, hi(x) = equality constraint, and rg, rh = inequality and equality weighting factors. The inequality constraints are all of the univariant type, that is of the form a < f(xJ < b (16) On the other hand, the equality constraints are all multivariant and, since each is a function of all three variables, could equally well be modified for use as unconstrained objective functions. The possible equality constraints are
B*1 = 1.0
(17)
n i=l
[mi,exptl - mi,pred.l' = 0.0
(18)
n
i= 1
mi,pred. = 1.0
(19)
It can be noted, however, that satisfaction of eq 18 will automatically satisfy eq 19 and that inclusion of eq 19 is therefore redundant. For a purely random set of initial guesses, the use of eq 17 a t F(x) led to rapid convergence
3.0
FLOWRATE,O
6.0
I . 0
10.0
(TONNES/HRl
Figure 3. Initial selection function, s1 and zero-order production rate constant, K*.
with eq 18 added as equality constraints. Once operating in the feasible region, however, the use of eq 18 (modified to become the sum of the squared errors) as an objective function and no constraints a t all converged rapidly to reasonably good answers. B. Results. The results of the optimization work are summarized in Table I. The exponent, a , is noted to remain relatively constant (average value = 0.835) while both K* and s1tend to decrease with flowrate through the lower half of the range before settling a t a relatively constant value (sl = 0.29, K* = 0.29). It can be noted that the ratio of K* to s1 is approximately unity in all cases leading to a value of Bzl which is dependent on a only and is thus not a function of flowrate. Values of sl,K*, a, and Bzl are plotted against flowrate, Q (tonnes per hour) in Figures 3 and 4. A failing of this model is its inability to predict a value of BZ1equal to unity; however, this failing can be understood when it is considered that eq 12 and 13 are forced
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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979
describe the swing hammermill size reduction of refuse is shown to be a close approximation to the real system. Significant differences between this system and typical ball mill applications are that the mean residence time increases with refuse flowrate rather than decreasing as in overflow discharge ball mills and the fact that it is not possible to determine experimentally any of the selection or breakage functions. The optimization technique used has been found to be an effective means of finding values for the selection and a compromise breakage function. Acknowledgment This work was partially supported by the EPA Project No. R-804034, "Refuse Processing for Material and Energy Recovery", Mr. Carlton Wiles, Project Officer. Nomenclature b , = fraction of material which, once broken, reports to the ith interval from the jth interval; called the breakage function Bi, = cumulative breakage function B = the n X n lower triangular matrix of individual breakage functions H = total mass of the assembly I = the n X n identity matrix J, = the n x n diagonal matrix with elements-f,," exp[(sir)B]R(B)dB K* = zero order production rate constant m, = fraction within the xixifl size class mCf= continuous feed size distribution m , = continuous product size distribution R(t7 = the residence time distribution si = fractional rate at which material is broken out of the ith interval; called the selection function s1 = initial selection function S = the n x n diagonal matrix of selection functions t = time T = model matrix x i = some size usually associated with a set of standard screen sizes
A0 i i 0
0
A
A
0 0 %
008
00
A
4 4
A
A
4.0
2 . 0
1
1.0
6 . 0
CTONNES,HRI Figure 4. a and cummulative breakage function Bzl. FLOWRATE,O
I
P o
Greek Letters 8 = t/r 7 = mean residence time = ratio of holdup to feedrate, H I M d 1
0
> P
$
I O
0
S I Z E 7 . 4
I
O
i
o
I
C
s o
I O 0
C L A S S , ] TONNES/HR
Figure 5. Experimental and predicted size distributions
to hold for all size classes despite evidence that the zero-order production rate constants for the coarse size classes cannot be characterized by the same equation as the fine sizes (Herbst and Fuerstenau, 1968). In view of this, the cumulative breakage functions predicted by this model are compromise values between the actual coarse and fine functions. Despite these compromise values the predicted product still compares favorably with the experimental product as evidenced by Figure 5 in which the experimental product for a flowrate of 7.4 t / h is compared to that predicted. For this comparison s1 = 0.29, K* = 0.29, and a = 0.804, and the sum of the squared errors is found to be 0.0168. Conclusions Despite the fact that Bzl is not predicted to equal unity, the size discretized, time continuous model adapted to
Literature Cited Austin, L. G., Klimpel, R. R., Ind. Eng. Chem. Fundarn., 9,230 (1970). Cailcott, T. G., Lynch, A. J., Proc. Aust. Inst. Min. Met., 209, 109 (1964). Callcott, T. G., Inst. Min. Metall. Trans., Sec. C , 7 6 , 1 (1967). Epstein, B., J. Franklin Inst., 224, 471 (1947). Fiacco, A. V., McCormick, G. P., "Non-Linear Programming-Sequential Unconstrained Minimization Technique", Wiley, New York, 1968. Herbst, J. A,, Fuerstenau, D. W., Trans. AIM€, 241, 538 (1968). Herbst, J. A,, Grandy. G. A,, Mika, T. S., Inst. Min. Metali. Trans., Sec. C , 80, 193 (1971). Herbst, J. A,, Grandy, G. A,, Fuerstenau, D. W., "Population Balance Models for the Design of Continuous Grinding Mills", 10th IMPC, London, 1973. Keisall, D. F., Reid, K. J., Restarick, C. J., Powder Techno/., 3, 170 (1970). Klimpel, R. R., Austin, L. G., DECHEMA-Monogr., 69, 449 (1972). Klimpel, R. R., Austin, L. G., Int. J . Miner. Process., 4, 7 (1977). Lynch, A. J., Draper, N., Proc. Aust. Inst. Min. Met., 213, 89 (1965). Obeng, D. M., Ph.D. Thesis, University of California, 1974. Obeng, D. M., Trezek, G. J., Ind. Eng. Chem. Process. Des. Dev., 14, 113 (1975). Powell, M. J. D., Cornput. J., 7, 303 (1964). Shiflett, G. R., D. Eng. Dissertation, Universlty of California, 1978. Suh, C. H., Radcliffe, C. W., "Kinematics and Mechanism Design", Chapter 12, Wiley, New York. 1978. Trezek, G. J., Savage, G. M., Waste Age, 49 (July 1975).
Received for review March 16, 1978 Accepted January 9, 1979
