Maximization of Yields of Clean Coal from Coal Preparation Plants

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979

calculated by the proposed correlation, then to determine the actual high pressure thermal conductivity, k, it is necessary to know the low pressure value, k*. This may be determined by a suitable correlation or obtained from experimental data. To use the Stiel and Thodos method you must know, in addition to k*, T,, P,, and M , the critical compressibility factor, z,, or the critical density, p,, and the density, p. The acentric factor is also needed if a generalized method, such as the Pitzer (1955) correlation, is used to calculate the required densities. Conclusions The results of this work show that the effect of pressure on the thermal conductivity of hydrocarbon gases may be accurately described using a three-parameter corresponding states correlation. These three parameters are the critical temperature, critical pressure, and isochoric heat capacity. The isochoric heat capacity was used to effectively separate the translational and internal energy contributions to the thermal conductivity. The form of the thermal conductivity correlation developed in this study is given by eq 6 where (k/k*)' is given by eq 12. The ( k / k * ) " function is given by eq 15 for noncyclic hydrocarbons and by eq 16 for cyclic hydrocarbons. Alternatively, Figures 1, 2, and 3 may be used to obtain the functions defined by eq 12, 15, and 16, respectively. The low pressure thermal conductivity, k*, can be obtained from standard sources such as The American Petroleum Institute's Technical Data Book (1977). Nomenclature A , B = reduced temperature functions b = constant C = heat capacity k = thermal conductivity M = molecular weight p = pressure R = gas constant T = temperature Greek Letters

p =

x

511

density

= function of density p = viscosity of gas w = acentric factor

Superscripts * = low pressure ' = translational energy contribution " = internal energy contributions O = ideal gas (0) = simple fluid (1) = correction term for molecular acentricity = molar

-

Subscripts r = reduced v = constant volume p = constant pressure

L i t e r a t u r e Cited American Petroleum Institute, "Technical Data Book, Petroleum Refining", Third Edition, The American Petroleum Institute, Washington, D.C., 1977. Chapman, S., Cowling, T. G., "The Mathematical Theory of Non-Uniform Gases", 3rd ed, Cambridge University Press, London, 1970. Comings, E. W., Mayland, 6.J., Chem. Met. Eng., 52(3),115 (1945). Comings, E. W.. Nathan, M. F., Ind. Eng. Chem., 39,964 (1947). Crooks, R. G., M.S. Thesis, The Pennsylvania State University, University Park, Pa., 1978. Hanley, H. J. M., McCarty, R. D., Cohen, E. G. D., Physica, 80(2),322 (1972). Hoerl, A. R., Jr., "Chemical Business Handbook", John H. Peny, Ed., McGraw-Hill, New York, N.Y., 1954. Jossi, J. A., Stiel, L. I., Thodos. G., AIChE J., 8,59 (1962). Kramer. F. R., Comings, E. W., J . Chem. Eng. Data, 5, 462 (1960). Lee, 6. I., Kesler, M. G., AIChE J., 21,510 (1975). Leng, D. E.. Comings, E. W.. Ind. Eng. Chem., 49,2042 (1957). Lenoir, J. M., Junk, W. A., Comings, E. W., Chem. Eng. Frog., 49,539 (1953). Mason, E. A., Monchick, L., J . Chem. Phys., 36, 1622 (1962). Pitzer, K. S.,Lippman, D. Z.,Curl, R. F., Jr., Huggins, C. M., Petersen, D. E., J . Am. Chem. SOC.,77. 3433 (1955). Reid, R. C., Prausnitz, J. M., Sherwood, T. K., "The Properties of Gases and Liquids", 3rd ed, McGraw-Hill, New York, N.Y., 1977. Stiel, L. I., Thodos, G., AIChE J . , 10, 26 (1964).

Received for review July 5, 1978 Accepted February 21, 1979

The Department of Refining of the American Petroleum Institute provided partial financial support of this work.

Maximization of Yields of Clean Coal from Coal Preparation Plants Jephthah A. Abara and Byron S. Gottfrled' Department of Industrial Engineering, Systems Management Engineering and Operations Research, University of Pittsburgh, Pittsburgh, Pennsylvania 1526 1

A computer simulation model of coal preparation plants has been developed with provisions for crushers, screens, and washers, and a capability of interconnecting plant components into any desired configuration. The model predicts yield of clean coal, Btu recovery, and ash and sulfur levels. An efficient optimization method utilizes the simulation model for maximizing clean coal yields, within ash and sulfur level restrictions. The method, based on classical nonlinear search techniques, enhances its efficiency by exploiting special characteristics of coal preparation processes. Results obtained with the method compare favorably with theoretical results for ideal washers. Agreement between predicted and actual performance for entire plants is less satisfactory. Nevertheless, use of the optimizer is shown to result in significant improvements in the yield of clean coal. I n one case featuring a crusher, two screens, and three washers, the clean coal yield increased by approximately 2 % , and the annual net revenue by over $500 000 for 600 tons per hour of feed.

Introduction Coal preparation or coal washing involves the removal of ash and pyritic sulfur from raw coal. One of its major objectives is to maximize the yield of the washed product within accepted levels of the impurities such as ash, sulfur 0019-7882/79/1118-0511$01.00/0

(or SO,). For instance, the ash in a typical U.S. Northern Appalachian bituminous coal can be reduced from 14% to 6% and the total (pyritic and organic) sulfur halved from 3% to 1.5% (Carallaro et al., 1976). The heating value per unit weight of coal is also increased as a result 0 1979 American

Chemical Society

512

I


run-of-rn:ne

coal

a g g r e g a t e clean c o a l c!ean

I

I

Cnal

1 I

l I

*

Specific G r a v i t y , p

C : crusher

s:

screen W : washing v e s s e l 1.

2. 3.

c o a r s e coal washer intermediate c o a l w a s h e r fine c o a l w a s h e r

Y

Figure 2. Typical distribution curve illustrating specific gravity of separation, p'.

refuse

Figure 1. Simplifiedschematic of a coal preparation plant (with Baum jig).

of such beneficiation. The importance of coal preparation has grown as environmental and emission standards associated with coal utilization have become stricter. A coal preparation plant consists basically of interconnections of crushers, screens, and washers. Typically, the feed is crushed in a crusher, then segregated into two or more streams by size using screens. These streams are next passed through appropriate washers whose essential outputs are clean coal, with reduced concentrations of ash and sulfur, and refuse. Different washers are used for different sizes of coal. The individual clean coal streams are blended into an aggregate product. Figure 1 is a simplified schematic designed to illustrate the sequence of operations and flowstreams. Dewatering and drying units reduce surface moisture of the product streams to necessary limits. The characteristics of the coal feed and desired product purity determine the appropriate combination of crushers, screens and washers to be utilized. On the other hand, the intended use of the final product and its mode of transportation establish the degree of dewatering and prescribe the requisite drying and dewatering units. Therefore, these latter units are absent from the figure since the primary consideration here is the purity of the clean coal. How to establish settings of a plant which result in a prescribed quality while improving the yield of clean coal has been of recurring interest to the coal industry in the past two decades. Some of the studies (Porshnev, 1964; Wright, 1961) have applied linear programming to blending and transportation problems. Coal preparation processes, however, are not linear functions of the decision variables. Consequently, other efforts (Absil et al., 1966; Brookes and Whitmore, 1966; Ivanov and Yampolsky, 1973; Kindig et al., 1966; Laverick, 1973; Sarkar et al., 1960; Venkatesan, 1976; Walters et al., 1976; Weyher, 1971) have used simulation and graphical techniques for carrying out optimal washability studies. The models involved in these studies are limited in that they include only washing vessels and are mostly simple. They have tended not to consider the effects of size and specific gravity distributions of the coal feeds. The optimization attempts, in the main, have been variations of explicit enumeration.

Model of the Process The objective of this study is the development of an efficient computer-oriented method to maximize the yields of clean coal from coal preparation plants subject to ash and sulfur level quality constraints. The overall preparation process is best described by examining mathematical representations of the process of each kind of unit. These processes are defined by many

N o r m a h z e d S p e c i f i c Gravity, p i p '

Figure 3. Typical generalized distribution curve.

basic equations representing particular units and their settings (decision variables). However, no explicit analytical functions of the decision variables exist for the unit product streams because the functional relationships are determined also by the distributions of sizes and specific gravity fractions present in the coal feed to each unit. But only the distributions of the initial raw feed are given at the outset, whereas those of intermediate feed streams result from the preparation processes and are not known in advance. A computer program (Gottfried, 1977; Gottfried and Abara, 1978) has been written to simulate the principal preparation plant units under a given operating policy, that is, a set of decision variables for a specified plant configuration. The simulator introduces the influences of feed specific gravity and size without unduly cumbersome input requirements. To use it one only needs to specify the appearance of the plant configuration, the operating conditions, and a description of the coal feed. Thus the user can interconnect the units into any desired configuration. Since the distributions of size and specific gravity for the intermediate feeds are not known in advance, their effects on the respective processes can be determined only during the simulation. Therefore, it is sufficient here to present general expressions of the individual unit product streams using only the decision variables or unit settings as parameters. Washers. To simulate the performance of a washer, the feed is considered as comprising several size increments, and each size increment as subdivided into specific gravity fractions. Therefore, a different set of generalized distribution data (Gottfried, 1978; Gottfried and Jacobsen, 1977) is obtained for each of the size increments. The distribution data indicate what proportion of each specific gravity fraction reports to the top stream (clean coal) at a given specific gravity of separation or separation gravity, p', that is the specific gravity of the material that is divided equally between clean coal and refuse (see Figures 2 and 3). The washer's effect on each size and specific gravity is determined, and then the overall clean coal and refuse products are reconstituted. Mathematically, this can be expressed as

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 513

FC(p’)= CCdijb’)fij i l

P(P’) =

CC[1 - dij(p’)lfij i l

(1)

(2)

where Fc and P represent the overall clean coal and refuse flowrates, respectively, from a washer, f i j represents the flowrate of feed in the ith size fraction and the j t h specific gravity fraction, and di,(p’) is the distribution factor for the ith size fraction and j t h specific gravity fraction at a separation gravity of p’. Crushers. The simulator uses two selection functions in the calculations. They are p i ( c ) , the fraction of feed particles in the ith size increment that will be crushed at a crusher setting c, and a breakage function for crushed particles, bim(c),the fraction of crushed particles originally in the ith size fraction that ends up in the mth size fraction. Both pi(c) and bi,(c) are assumed to depend on the initial particle size, but to be independent of specific gravity. It is also assumed that the breakage of a particle creates fragments that are unchanged in specific gravity analysis. Accordingly, the flowrate of crushed product in the mth size fraction, fmP(c),is given by f m P ( c ) = C C pi(c)bim(c)fij + C[1 - pm(c)Ifmj (3) I iEBM

J

where f i j represents the flowrate of feed in the ith size fraction and j t h specific gravity fraction, and BM is the set of size increments larger than the mth size fraction. The flowrate of the product from the crusher, i.e., F ,then is FP(c) = C f m P ( c ) (4)

size fractions and specific gravity fractions, respectively, in the feed. Blenders and Splitters. A “blender” is a point within the plant where two different flowstreams are combined. The flowrate of the product stream is the sum of the flowrates of the two feed streams. The product stream’s attributes are weighted averages, based on flowrates, of attributes of the feed streams. Conversely, a “splitter” represents a point within the plant where a portion of a flowstream is diverted, thereby creating two flow streams whose attributes are identical with those of the original flowstream. Further details on all the units are given by Gottfried (1977). Product Streams. Only the washer product streams are final streams; the others are necessarily intermediate, and are ultimate feeds to washers. Therefore, the objective function, the aggregate yield of clean coal (Y), is given by N

Y(X,W,Q)=

P ( s )= C o i ( S ) JE f i j

(5)

1

and the flowrate of underflow (fine) material, F“,can be expressed as JYs) = [1- O i ( S ) l Cfij (6)

x i

I

where f i , again represents the flowrate of feed in the ith size fraction and j t h specific gravity fraction. The equation for the selection function depends on the type of screen in question and several exponential-type equations are included in the simulator. Rotary Breaker. The rotary breaker is considered a sequence of crusher and screen operations, with breakage and screening alternating as the feed material undergoes successive falls in a rotating, perforated drum. The three input parameters are drum length (A), drum diameter ( T ) , and size openings (s) in the drum (screen size). These are the “external” decision variables for the rotary breaker. Two streams are produced: the rock stream, P , which is discarded, and the coal stream, FP. The flowrate of the coal stream is given by FP(A,T,s) = + ( A , ~ , ~ , f i j ;i = 1, ..., I ; j = 1, ..., J ) (7) where f i j is the flowrate of feed in the ith size fraction and j t h specific gravity fraction; I and J are the numbers of

(8)

where X is the set of decision variables for all the units, W and Q represent the distributions of size fractions and specific gravity fractions, respectively, in the initial raw coal feed, and N is the number of units. The additional subscript, n, refers to the position of a given unit in the sequence of units in the plant. Obviously, FnC= 0 unless the nth unit is a washer. Constraints. Constraints placed on the ash and total sulfur levels of the aggregate clean coal product can be represented algebraically as

kFkC

m

As with d i j ( p ’ ) and washers, the explicit forms of pi(c), and bim(c)depend on the type of crusher and the crusher setting. In general, however, they are expressed as exponential functions. Screens. Like the washer, the screen has two product streams. A selection function, oi(s), similar to the dij(p’) of washers, is used to determine the fraction of feed particles in the ith size increment that passes over the screen into the overflow (coarse) product, given a screen opening size, s. The flowrate of overflow material, P,is then given by

C Fn‘b’) n = l

-

a=-

k

cFkc

I a

(9)

k

CPkFkC (10) k

where uk and Pk are the ash and total sulfur percent levels, respectively, for the hth flowstream, and a and /3 are the corresponding upper limits for the aggregate clean coal product. Note that FkC,the flowrate of clean coal in the hth flowstream is zero if the hth flowstream is not a clean coal stream from a washer. The sulfur restriction may also be given in terms of pounds of SO2 per, say, lo6 Btu of energy. The decision variables, X, typically have upper and lower bounds determined by the characteristics of the respective vessels, the coal feeds, or other factors such as the availability of particular screen sizes. Thus XI+I I 1 = 1, ..., L (11)

x1 x1++

where X+ and X++are the lower and upper bounds, respectively, and L is the number of decision variables. Thus, we have

subject to g,(X, W, Q) = ii 5 a

(13)

g,(X, W, Q) = P 5 P x,+I IX,++ 1 = 1, ..., L

(14)

x,

(15) Characteristics of the Objective Function and Constraints Dominance of Washers on Clean Coal Yield. The effect of separation gravities has been observed to be the

514


dominant factor in determining the aggregate yield of clean coal in a plant in which all the original raw coal feed (except the rock stream from a rotary breaker) is washed. Crusher settings and screen sizes have relatively small effects by themselves, except for the hole size in a rotary breaker. But even here the variation of the size is, in practice, over a narrow range and so has limited impact. The above is true primarily because crushers and screens affect feed volumes to respective washers and, thereby, the yields of clean coal from individual washers. However, if the separation gravities of the washers are fixed, quite different individual yields of clean coal result in little change in the aggregate yield. I t is only when the separation gravities themselves are varied that substantial changes occur in the aggregate yield. Several cases were investigated to illustrate this point. In one set of results, using typical ranges of the values of the different unit control parameters, the average improvement in yield of clean coal from varying only the washer separation gravities was almost nine times the gain from varying either a crusher setting or a screen size only. Monotonicity of the Objective Function. Dominance of washers on the value of the aggregate yield of clean coal prompted an analysis of the behavior of the objective function as washer separation gravities are changed. The impetus was to discover any properties that would expedite the development of an efficient optimization method. It can be demonstrated that for one (and any) washer, yield of clean coal is monotonic nondecreasing with respect to separation gravity. Consider the case of a composite feed of coal (one size). Let

Y = total yield (% of feed) = Cy(p,,p’) I

(16)

where y(p,,p’) = weight of the j t h specific gravity fraction, pI, in the clean coal as a percent of feed, for a separation gravity of p’ Y(P],P’) = d ( p j , p ’ ) f ( p j ) (17) with d(p,,p’) = weight percent of the j t h specific gravity fraction reporting to clean coal, for a separation gravity of p‘, and f ( p l ) = weight of the jth specific gravity fraction as a percent of feed. Equations 16 and 17 give y = Cd(P,,P’)f(P,) (18) I

If the p are considered continuous variables, (18) becomes an integrand rather than a sum; then

Y = j-P-d(P,P’)f(P)dP

(19)

P@U

where d ( p , p ’ ) is the yield distribution curve, f ( p ) d p is the percent weight of feed contained within a differential specific gravity increment, d p , and d ( p , p ’ ) is a nonincreasing function of p. Now, by normalizing the independent variable, p , the function d ( p , p ’ ) can be transformed into the general form, d * ( p / p ’ ) , which holds for all values of p’. Thus d * ( p / p ’ ) , the generalized curve, like d ( p , p ’ ) , is a nonincreasing function of p l p ‘ . (See Figure 3.) Now PIP1’ > p / p i if pz’ > pl’. Therefore d*(p/p2’) 2 d*(p/pl’) if ~ 2 > ’ PI’ (20) Hence, from (20) d(P,Pz’) 2 d ( P , Q l ’ )

P*’

> P*’

(21)

Since f ( p ) is obviously nonnegative, (20) can be modified to give d(P,Pi)f(P) 2 d(P,Pl’)f(P)

P2’

> P1’

Table I. Typical Yield of Clean Coal for a Washer separation gravity, p

yield, % of feed

1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60

21.7 36.6 45.6 50.8 54.1 56.7 58.3 60.4 60.4 62.2 65.8 80.2 81.2 100.0 100.0 100.0 100.0

-

.

I

Screen

Washer (Dense ,Medium Vessel) ~

Washer i c o n c en trat1ng Tables1

Refuse

Figure 4. Circuit with two washers.

Consequently, by one of the properties of integrals on rectangles

or more exactly, since excluded regions give trivial integrands

From (19) and (22), we then get y2 2 Yl P2’ > P1’ (23) Thus Y is a monotonic nondecreasing function of p’. Empirical illustration of this is provided in Table I with data for a dense-medium vessel. There is also empirical evidence that for small sequential increases in the separation gravities of two or more washers, the yield is monotonic nondecreasing. Table I1 offers an example of this using the configuration of Figure 4 consisting of a dense-medium vessel and concentrating tables. Unlike the well-behaved nature of the objective function when only the separation gravities of washers are varied, the nature of the resultant surface is not easily defined when crusher settings and screen sizes are altered. A major reason for this is that the performance characteristics of screens and crushers depend on a feed’s size consist (Le., the kinds of sizes present) which varies from flowstream to flowstream for the same raw coal feed. In contrast, the performance of washers is largely influenced by the specific gravity fraction consist (Le., the types of specific gravity fractions present) which, unlike the size consist are the same for all flowstreams. The specific gravity consist defines which segments of the distribution curves are to


515

Table 111. Typical Association of Clean Coal Yield with Its Ash and Total Sulfur Levels yield, % of feed 21.7 36.6 45.6 50.8 54.1 56.7 58.3 60.4 62.2 65.8 80.2 81.2

100.0

ash, % of yield 3.3 3.8 4.4 5.2

5.9 6.5 7.1 7.9 8.8 12.7 25.3 26.0 36.7

total S, % ' of yield

0.45 0.45 0.46 0.47 0.48 0.48 0.48 0.49 0.50 0.54 0.61 0.62 0.64

be used in determining yield of clean coal. Besides, the distribution curves of a given type of washer do not differ substantially for different sizes. Positive Association between Yield and Quality. It also has been observed that, in general, the product ash and sulfur levels both rise as the yield of clean coal increases, provided the cumulative ash and sulfur levels of specific gravity fractions of the coal feed are nondecreasing while the average specific gravity goes up. Violation of the above condition is a rarity and can be assumed to be highly unlikely. Table I11 illustrates this point for both ash and sulfur using data for a dense-medium vessel. In other words, at lower yields, feasibility is generally present. And as yield increases, the system approaches the upper limits on ash and sulfur levels, i.e., the constraint barriers. The Solution Algorithm The dichotomy of characteristics of the objective function suggests the possibility of optimizing in two stages: in one, vary the washer separation gravities; in the other, vary crushers and screens as well as washers. Because the dominant effect on the aggregate yield of clean coal is from washers, a variation in the decision variables of the latter, with crushers and screens fixed, should not result in any serious deviation from a superior result, but could lead to fewer computations. In developing a solution method, a major effort was made to balance computational efficiency with analytic effectiveness. Based on the observations of the preceding paragraph, the optimization procedure is divided into two main stages, viz. (1)vary decision variables (separation gravities) of washers only; (2) vary all decision variables. The functions involved and the dimensionality of the problem preclude the use of indirect optimization methods. Of the direct methods, linear programming techniques would require that the effects of intermediate feed composition either be neglected or be grossly generalized. Therefore, those techniques are not considered. A simple direct search technique is considered adequate. The pattern method of Hooke and Jeeves (1961) is used as the nucleus of the solution algorithm because it is relatively simple to program. Besides, it can be modified to become self-bounding on the decision (independent) variables, thereby precluding additional explicit constraints to depict bounds on the variables. Specifically, whenever the value of a decision variable is changed, the new value is checked to ensure that it lies within the proper bounds before functions of the model are evaluated. Because it has proved effective in many previous studies and has continued to be recommended by several authors and experimenters (Beveridge and Schechter, 1970; Gaver and Thompson, 1973; Gottfried et al., 1970; Pierre, 1969; Reklaitis and Phillips, 1975; Zahradnik, 1971), the internal

516


U

I

Aggregate Clean C o a l

f

Clean Coal

t ’ Clean

Clean Coal

Coal

4 23I

D P reiyrm sU Pa-prM py eeSrdi Sinucgm rl ee V eRneosl sl eCl r u s h e r

5

Concentrati-g Tables Froth Flotation Cells

1L-b Unit

0

6

Refuse

Figure 5. Configuration of example case.

penalty function treatment of constraints has been applied here where appropriate. Since feasilbilty is present a t lower yields and the constraint barriers are approached as yield of clean coal increases, the interior penalty approach has been adopted. During the first stage of the search, all screen sizes, crusher settings, and rotary breaker decision variables are fixed at their initial values. Only washer separation gravities are varied. Given that the aggregate clean coal yield surface is monotonic nondecreasing for increasing washer separation gravities, the search step sizes are relatively large and do not pose a danger of bypassing local peaks. In the second stage, all the decision variables may be varied. Also, the objective function now is modified by the introduction of an interior penalty function. Details on all aspects of the algorithm, including its associated computer program, are provided by Abara (1978). The program is written in standard Fortran IV. It has been tested on a DEC System/lO computer. Computations, otherwise called functional evaluations, giving the yield of clean coal and its associated attributes, including ash and sulfur levels, were found to take about 3 seconds per unit in a plant. The core requirement of the model is 72K of memory for a maximum of 12 units and 20 flowstreams. Applications of t h e Method Yield Improvement. Several feeds and plant configurations were used in testing the algorithm. One of these is discussed below. The configuration is illustrated in Figure 5. It includes a crusher, two screens, and three washers for a total of six units. There are only five decision variables because the froth flotation process has no explicit control parameter. The raw feed has a size range of 18 in. X 0 consisting of 8 sizes with the bulk of the coal concentrated in the more coarse sizes of 6 X 2 in. or bigger. The feed has low sulfur content with an overall pyritic sulfur level of only 0.26% and a total sulfur level of 0.65%. It is more fully described in Table IV. The results of this example are summarized in Table V. The performance criteria chosen in assessing the results are computational efficiency and analytic effectiveness. The two must be considered together since the philosophy in developing the algorithm has been to balance them. A good measure of efficiency is an indication of the number and types of computations involved. In this context, the number of functional evaluations is the indicator chosen, where a functional evaluation includes all

the calculations needed to obtain a point consisting of yield of clean coal, product ash level, and sulfur level. A practical effectiveness is defined as follows: given a set of decision variable values and the resultant yield of clean coal as well as product ash and sulfur levels, does the method provide a different set of decision variable values with higher or equal yield at the same product ash and sulfur levels? In the former case, increased yield, the benefits are obvious-increased revenue with little or no increase in cost. In the latter case, same yield with different values for the decision variables, the decision maker is afforded the flexibility of alternative policies. The clean coal ash and total sulfur levels corresponding to the starting point (that is, initial set of decision variables) are used as the respective upper limits on ash and sulfur for this example. The search terminates after 82 functional evaluations. This number of functional evaluations is considered quite reasonable since it allows about 16 values for each of the five decision variables. With regard to the effectiveness, the final yield of 53.2% represents almost a 2% relative increase. The significance of this increase is, perhaps, better appreciated in net revenue terms where it would result in an increase of $560 000 per annum, assuming a sale price of $24 per ton of clean coal and a feed rate of 600 tons per hour. Also, the results indicate that more of the coal (83% vs. 53%) is channeled to the dense-medium vessel for cleaning in the final solution operating policy than occurs with the policy of the starting point. Thus the results reflect real-life preferences since the dense-medium vessel is closer to perfect washers than are either the concentrating tables or froth flotation cells. Sensitivity Analysis. To investigate the possible sensitivity of the final result to the starting point, the example case is run with several starting points. Consistent with the defined effectiveness, the clean coal ash and total sulfur levels at the original starting point, also called the reference point, serve as the ash and total sulfur level barriers for all the runs with the different starting points. The results are summarized in Table VI and indicate that there are differences both in the final operating policies and in the final yields of clean coal for the different starting points, with the yields of clean coal going from 52.7% to 53.6%. The range of 0.9% is significant. However, it represents the difference between case C (or case D) and case B. In the latter case, as well as in cases E and H, the starting point includes the minimum feasible specific gravities of separation for all the washing vessels. This is not a very practical starting point and is used in the algorithm only when a given starting point is found infeasible with respect to the ash and sulfur constraints. When cases B, E, and H are not considered, the remaining cases produce more comparable yields of clean coal (between 53.1 and 53.6%). On the basis of the above discussion, it is recommended that, in using the algorithm, solutions be obtained for a small number of judiciously selected starting points. The best solution from this set can then be adopted as the optimal operating policy. The effect of eliminating the stagewise search, that is, allowing all the decision variables to vary throughout the search, was also examined. Both approaches, stagewise and nonstagewise, produced essentially equal clean coal yields as demonstrated by the six cases given in Table VII. However, the stagewise approach required fewer functional evaluations than the nonstagewise in all six cases. Because both approaches have comparable average computation time per functional evaluation, the stagewise algorithm is


517

Table IV. Feed of Example Case. Specific Gravity Analysis of Feeda direct, % size fraction 18 x 12

wt 9.0%

12x 6

28.1%

6x 2

26.9%

2x 1

17.0%

1x

3 i 8

8.2%

28

5.0%

28 X 48

2.3%

48X 0

3.4%

3/aX

composite

100.0%

specific gravity float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.7 0 1.7 0-1 .BO sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.5 0-1.60 1.60-1.70 1.70-1.80 sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.70 1.70-1.80 sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.7 0 1.70-1.80 sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.70 1.70-1.80 sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.70 1.70-1.80 sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.70 1.70-1.80 sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.70 1.70-1.80 sink 1.80 float 1.30 1.30-1.35 1.35-1.40 1.40-1.50 1.50-1.60 1.60-1.70 1.70-1.80 sink 1.80

pyritic total S S

wt

ash

19.7 11.0 2.3 11.2 2.1 1.9 0.8 51.0 16.7 14.3 5.3 5.3 4.1 2.1 1.2 51.0 23.3 21.0 7.3 7.9 4.7 2.4 1.7 31.7 26.8 21.0 8.0 7.4 3.9 2.3 1.8 28.9 31.6 21.9 7.2 6.8 3.2 1.9 1.7 25.8 40.8 20.7 6.8 6.1 2.9 1.8 1.5 19.5 45.5 18.8 5.8 5.1 2.5 1.6 1.2 19.5 50.0 20.0 5.0 5.0 2.0 2.0 1.0 15.0 24.7 18.2 6.3 7.0 3.8 2.2 1.4 36.4

2.8 0.08 4.4 0.09 7.8 0.03 15.4 0.13 25.0 0.10 31.5 0.18 41.0 0.17 84.6 0.32 2.9 0.14 0.12 4.7 9.4 0.19 16.7 0.26 0.23 26.7 35.1 0.25 0.54 45.7 82.9 0.26 2.4 0.14 0.25 4.7 9.7 0.22 16.2 0.27 0.30 25.9 34.2 0.38 0.52 42.5 82.2 0.33 0.14 2.4 0.19 4.7 9.6 0.19 15.9 0.29 24.8 0.36 33.0 0.39 0.66 42.0 82.9 0.44 2.2 0.12 4.6 0.12 9.2 0.23 15.4 0.30 23.8 0.39 32.7 0.51 41.0 0.78 82.2 0.65 2.2 0.09 4.6 0.14 9.2 0.27 15.4 0.30 23.8 0.37 32.7 0.54 41.0 0.77 82.2 0.98 1.8 0.12 4.3 0.18 8.9 0.23 15.5 0.31 24.6 0.43 33.6 0.54 41.9 0.77 83.0 0.98 21.0 0.33 21.0 0.33 21.0 0.33 21.0 0.33 21.0 0.33 21.0 0.33 21.0 0.33 21.0 0.33 3.7 0.14 5.3 0.18 9.8 0.21 16.1 0.26 25.6 0.29 33.4 0.35 42.3 0.58 82.0 0.36

0.32 0.35 0.39 0.45 0.46 0.51 0.55 0.58 0.27 0.30 0.54 0.54 0.55 0.55 0.68 0.70 0.35 0.58 0.62 0.63 0.63 0.73 0.73 0.73 0.45 0.50 0.60 0.62 0.65 0.77 1.67 1.84 0.59 0.60 0.63 0.63 0.64 1.00 1.07 1.07 0.59 0.60 0.61 0.63 0.67 0.80 0.97 1.01 0.59 0.60 0.63 0.63 0.64 0.79 0.88 0.98 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.43 0.50 0.59 0.58 0.60 0.69 0.95 0.88

cumulative, % Btu/lb 12500.0 12500.0 12500.0 125 00.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0

wt 19.7 30.6 32.9 44.2 46.3 48.2 49.0 100.0 16.7 31.0 36.3 41.6 45.7 47.8 49.0 100.0 23.2 44.3 51.6 59.5 64.2 66.6 68.3 100.0 26.8 47.8 55.7 63.1 67.0 69.3 71.1 100.0 31.6 53.4 60.6 67.4 70.6 72.5 74.2 100.0 40.8 61.4 68.2 74.3 77.2 79.0 80.5 100.0 45.5 64.3 70.1 75.2 77.7 79.3 80.5 100.0 50.0 70.0 75.0 80.0 82.0 84.0 85.0 100.0 24.7 42.9 49.1 56.2 60.0 62.1 63.6 100.0

ash 2.8 3.4 3.7 6.7 7.5 8.4 9.0 47.5 2.9 3.7 4.6 6.1 8.0 9.1 10.0 47.2 2.4 3.5 4.4 5.9 7.4 8.4 9.2 32.4 2.4 3.4 4.3 5.7 6.8 7.6 8.5 30.0 2.2 3.2 3.9 5.1 5.9 6.6 7.4 26.7 2.2 3.0 3.6 4.6 5.3 5.9 6.6 21.3 1.8 2.5 3.1 3.9 4.6 5.2 5.7 20.8 21.0 21.0 21.0 21.0 21.0 21.0 21.0 21.0 3.7 4.4 5.1 6.5 7.7 8.6 9.3 35.8

pyritic total S S 0.08 0.08 0.08 0.09 0.09 0.10 0.10 0.21 0.14 0.13 0.14 0.15 0.16 0.17 0.17 0.22 0.14 0.19 0.20 0.21 0.21 0.22 0.23 0.26 0.14 0.16 0.17 0.18 0.19 0.20 0.21 0.28 0.12 0.12 0.13 0.15 0.16 0.17 0.18 0.30 0.09 0.11 0.12 0.14 0.15 0.16 0.17 0.33 0.12 0.14 0.15 0.16 0.17 0.17 0.18 0.34 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.14 0.16 0.17 0.18 0.18 0.19 0.20 0.26

0.32 0.33 0.33 0.36 0.37 0.37 0.38 0.48 0.27 0.28 0.32 0.35 0.37 0.38 0.38 0.54 0.35 0.46 0.48 0.50 0.51 0.52 0.52 0.59 0.45 0.47 0.49 0.51 0.51 0.52 0.55 0.92 0.59 0.59 0.60 0.60 0.60 0.61 0.62 0.74 0.59 0.59 0.60 0.60 0.60 0.61 0.61 0.69 0.59 0.59 0.60 0.60 0.60 0.60 0.61 0.68 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.43 0.46 0.47 0.49 0.50 0.50 0.51 0.65

Btu/lb 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0 12500.0

Flowstream summary: flowrate = 100.0% of feed; ash = 35 .8%, PYritic sulfur = 0.26%; total s#ulfur= 0.65%; Btu content = 12500.0 Btu/lb; SO, con tent = 1.03 lb of SO,/million Btu.

a

518


Table V. Results for Example Case (Upper Limit on Clean Coal Ash: 6.2%; Upper Limit o n Clean Coal Total Sulfur: 0.48%) clean coal attributes yield, %of (x,P feed .

decision variables x ,... in.

x,. in.

feasible values (range)

(1.00, 4.00)

starting point final point

3.00 2.75

x.. in.

x.

{fi;;)

(1.35, 1.75)

{~~(l~~~ f 0.0058,

(1.35, 1.75)

1.00 0.25

1.45 1.46

0.023 0.0058

1.50 1.50

_ I

X,

I

I

I ,

ash, %

functtional total evaluaS, % tions

-

-

-

-

-

-

52.2 53.2

6.2 6.2

0.48 0.48

1 82

-

a Variables enclosed in parentheses are not decision variables, but are included here for completeness based on both the configuration and the algorithm.

Table VI. Effect of Starting Points o n Final Results (Upper Limit on Clean Coal Ash: 6.2%; Upper Limit on Clean Coal Total Sulfur: 0.48%) clean coal attributes f,lncdecision variables

description of starting point

-

A, Reference point

B, Start at minimum separation gravities C,

Increase feed t o dense-medium vessel D, Larger increase in feed to densemedium vessel E, Decrease feed t o dense-medium vessel

F,

Increase feed t o concentrating table G, Decrease feed t o concentrating tables

H,

A variation of B

x,, in.

x2, in.

x3

x4, in.

x,

(1.35, 1.75)

0.0058

(1.35, 1.75)

1.00

1.45

0.023

2.75

0.25

1.46

3.00

1.00

2.50

feasible values (range)

(1.00, 4.00) 0.0058

starting point

3.00

final point starting point final point starting point

( X

yield, %of ~ )feed ~

ash, %

tional total evaluaS, % tions

-

-

-

1.50

52.2

6.2

0.48

1

0.0058

1.50

53.2

6.2

0.48

82

1.35

0.023

1.35

35.1

5.1

0.46

1

0.25

1.46

0.25

1.38

52.7

6.2

0.48

179

4.00

1.00

1.45

0.023

1.50

51.3

5.8

0.48

1

final point starting point

4.00

0.25

1.48

0.0058

1.50

53.6

6.2

0.48

68

4.00

0.25

1.45

0.023

1.50

51.3

5.8

0.48

1


4.00

0.25

1.48

0.023

1.50

53.6

6.2

0.48

56

3.00

1.50

1.35

0.023

1.35

34.0

5.3

0.46

1


1.90

0.25

1.46

0.25

1.42

52.9

6.2

0.48

184

3.00

1.00

1.45

0.0195

1.50

52.2

6.2

0.48

1


2.75

0.25

1.46

0.0058

1.50

53.2

6.2

0.48

82

3.00

1.00

1.45

0.25

1.50

52.4

6.2

0.48

1


2.00

0.25

1.46

0.25

1.52

53.1

6.2

0.48

124

3.00

1.50

1.35

0.023

1.35

34.0

5.3

0.46

1

final point

1.90

0.25

1.46

0.25

1.42

52.9

6.2

0.48

184

-

Variables enclosed in parentheses are not decision variables, but are included here for completeness based on both the configuration and the algorithm.

considered more efficient and superior in the problem of this study.

Comparison of Results with Theory Results obtained with the algorithm have also been analyzed to assess their consistency with results developed

theoretically for simple circuits. Cierpisz and Gottfried (1977) have presented a theoretical analysis of the performance of float-and-sink-type coal washing devices. They arrive at some conclusions regarding the performance of perfect washers. Two of the examples they studied are investigated here.

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 3, 1979 519

Table VII. Comparison of Stagewise and Nonstagewise Algorithms clean coal attributes functional yield, % case evaluations of feed

total S, ash, %

%

Stagewise Algorithm 1

82 92 148 236 107 99

2 3 4 5 6

53.2 53.0 56.3 56.4 69.9 69.6

6.2 6.2 7.2 7.2 10.2 10.5

0.48 0.48 1.49 1.49 0.56 0.57

Nonstagewise Algorithm 1

114 105 227 349 137 114

2 3 4 5 6

53.2 53.0 56.3 56.3 69.9 69.6

6.2 6.2 7.2 7.2 10.2 10.5

F1

Feed

I-

0.48 0.48 1.49 1.49 0.56 0.57 FC

U

1I

b

Table VIII. Results for One Washer with Bypass (DenseMedium Vessel) (Upper Limit o n Product Ash Level for Grouu 1: 8.1%: GrouD 2: 32.0%) product attributes startine noint final point FC ash F, F2 F, F2 % of level, % o f feed p' % o f feed p' feed %

-.

~

Group 1 0.0 100.0 1.50 100.0 0.0 25.0 50.0 75.0 100.0

75.0 50.0 25.0 0.0

1.50 1.50 1.50 1.50

0.0 100.0 75.0 25.0 50.0 50.0 75.0 25.0 100.0 0.0

1.50 1.50 1.50 1.50 1.50

100.0 100.0 100.0 100.0

Comparison Example I. The circuit for this example is illustrated in Figure 6. It involves a single washer and part of the feed, F,, is washed while the other part, F2, bypasses the washer. The clean coal stream and F2 are then blended to give the final product F". Both F1and F2 may vary but their sum must remain constant. Thus F1 and F2 can be considered as percentages of a constant feed, where F1 F2 = 100. This makes the example compatible with the structure of the algorithm in which all the flowstreams are fractions of a constant feed. The purpose is to determine the values of F1,and F2, and the washer separation gravity, p', that will maximize the flowrate of the product and at the same time satisfy some quality restrictions. From their theoretical analysis, Cierpisz and Gottfried conclude that, for a perfect washer, the flowrate of the product is maximized by choosing a separation gravity such that the cumulative quality level (ash or sulfur) of the washed coal is equal to the desired cumulative quality level of the blended product. This requires that all of the feed material be washed; that is, F1 = 100 and F2 = 0. The results, for a dense-medium vessel and concentrating tables, respectively, are summarized in Tables VI11 and IX. The results are consistent with theory when the ash (quality) restriction is relatively low (8.1%);in other words, all of the feed is washed. For higher desired product ash levels (32.070),it appears as if all of the feed need not be washed. The apparent discrepancy occurs because the washers used here are not perfect. Their distribution curves are not the ideal step functions of perfect washers. Therefore, for such real washers, two or more unique combinations of F1, F2, and p' can produce identical results in terms of yield and quality of clean coal. As the quality standards become less restrictive, the combinations of F1,F2, and p' with identical results grow more numerous. In other words, there are more local optima, including the case in which F1is loo%, with the higher ash level of 32.0% than with 8.170.And since the algorithm is primarily designed to identify a local maximum in yield of clean coal, it is more likely to detect more locally optimal combinations of F1,

+

1.69 1.69 1.69 1.69 1.69

60.9 60.9 60.9 60.9 60.9

8.1 8.1 8.1 8.1 8.1

2.23 1.96 2.22 2.21 2.23

90.8 90.7 90.7 90.8 90.8

32.0 32.0 32.0 32.0 32.0

Group 2 73.0 25.0 60.0 55.0 73.0

27.0 75.0 40.0 45.0 27.0

Table IX. Results for One Washer with Bypass (Concentrating Tables) Upper Limit on Product Ash Level for Group 1: 8.1%; Group 2: 32.0%) starting point

Figure 6. A single washer with feed bypass.

0.0 0.0 0.0 0.0

F, F2 % o f feed 100.0 75.0 50.0 25.0 100.0 0.0

0.0 25.0 50.0 75.0

0.0 25.0 50.0 75.0 100.0

100.0 75.0 50.0 25.0 0.0

product attributes

final point

F2

F, P'

% o f feed

P'

Group 1 0.0 0.0 0.0 0.0 0.0

1.64 1.64 1.64 1.64 1.64

59.2 59.2 59.2 59.2 59.2

8.1 8.1 8.1 8.1 8.1

12.0 12.0 54.0 35.0 12.0

2.31 2.31 2.21 2.27 2.31

90.7 90.7 90.7 90.7 90.7

32.0 32.0 32.0 32.0 32.0

100.0 100.0 100.0 100.0

100.0 Group 2 88.0 88.0 46.0 65.0 88.0

FC, ash % o f level, % feed

Clean

Refuse

Blended

Product

7-

Reiuse

Figure 7. Two washers in parallel.

F2, and p' at the higher ash level than at the lower one. Note, however, that the values of F" are practically the same for all the final points obtained for the higher ash level. This suggests that although all of the feed is not washed, the flowrate of the product is still maximized. Indeed, when F1 is loo%, F2 is zero and p' is 2.246 and 2.319, respectively, for the dense-medium vessel and concentrating tables, the corresponding yields of clean coal are 90.8 and 90.770,each with an ash level of 32.0%. These are identical with the results in Tables VI11 and IX. for group 2 (ash level of 32.0%)! Comparison Example 2. As illustrated in Figure 7 , the circuit here consists of two washers in parallel, with half of the feed treated in each washer. Cierpisz and Gottfried have demonstrated analytically for perfect washers that,

520


Table X. Results for Two Washers in Parallel (Upper Limit o n Ash Level for Blended Clean Coal: 8.1%) starting point combi- sepanaragravition tion ties

A

B C

final point separation

gravities

DMVa

DMV

DMV

DMV

1.45 1.50

1.50 1.50

1.675 1.719

1.706 1.667

CT

CT

CT

CT

1.45 1.50

1.50 1.50

1.616 1.636

1.655 1.636

DMV

CT

DMV

CT

1.45 1.50 1.50

1.50 1.50 1.45

1.665 1.690 1.660

1.655 1.635 1.659

blended clean coal attributes yield, ash % of level, feed % 60.9 60.9

8.1 8.1

59.2 59.2

8.1 8.1

60.1 60.0 60.1

8.1 8.1 8.1

DMV = dense-medium vessel; CT = concentrating tables. Table XI. Jig Process-Simple

weight recovery, % percent ash percent sulfur Btu recovery, % Btu/lb

Table XII. Jig Process-Intermediate simulator performance weight recovery, % percent ash percent sulfur Btu recovery, % Btu/lb

59.7 9.1 3.77 88.5 13 300

actual plant performance 56.6

10.0 4.16 83.0 13 100

Table XIII. Heavy Media ProcessComplex A


simulator performance

actual plant performance

72.7 6.5 2.05 87.3 14 300

73.3 1.5 2.21 89.2 14 300

Table XIV. Heavy Media Process--Complex B



57.8 11.2 0.77 86.0 12 800

59.0 7.7 0.79 91.6 13 200

to maximize the flowrate of the blended clean coal subject to given quality restrictions, the separation gravities in both washers should be the same. The results of the current investigation are summarized in Table X for three combinations: two dense-medium vessels, two concentrating tables, and a combination of a dense-medium vessel in parallel with concentrating tables. The results show again that for real washers, which are not perfect, the theory can at best only be approximated. It is safe to assume that behavior similar to the theory should be expected and that an exhaustive enumeration of possible combinations of separation gravities should reveal a combination of equal values which maximizes the flowrate of the blended clean coal. The separation gravity combinations (1.636, 1.636) and (1.660, 1.659) in combinations B and C in Table X produce what are the likely corresponding maximum product flowrates. In both cases these maximum product flowrates are matched when the separation gravities are not equal. This merely underscores the existence of multiple maximum points for real washers. The case in which the two separation gravities are equal is only one of these, not the unique maximum point as with perfect washers, and as observed under Example 1, the algorithm locates local optima. Thus in this instance, with multiple optima, it determines a pair of separation gravities which produce the maximum flowrate of blended clean coal but may or may not be equal. Note, for instance, that the yields in combinations A and B are the same as for the optimal results in Example 1for the dense-medium vessel and concentrating tables, respectively, a t 8.15% ash. Comparison with Actual Plant Performance The performance of several existing plants has been simulated and the results compared with actual operating data. The plants examined range in complexity from a relatively simple jig plant to sophisticated preparation circuits featuring heavy media and froth flotation. The results are summarized in Tables XI-XIV. Details of the control settings are not included in these tables because, as explained below, they are very imprecise for the actual plants.

~




80.2 10.9 1.52 94.0 14 000

86.0 14.7 1.8 94.3 13 100

There appears to be reasonable agreement between the predicted and actual values, especially in the case of the simple jig process. Unfortunately, because of both the limited nature of the effort and the state of the art in the industry, the sensitivity of the results to variations in the control settings was not explored. In practice, control settings of the different plant units often are not established exactly. Instead they are somewhat gross values based on an operator’s past successes and experience. Therefore, in using the simulator, it has been possible only to approximate, rather than duplicate, the control settings of the actual plants. Besides, the degree of approximation is not even precise and could be partially responsible for disparities between the actual and predicted results. Another source of disparity would be the set of functions which represent the screening process in the simulator. These functions, while broadly adequate, are not completely satisfactory and are to be refined. Perhaps the handicap which the gross nature of plant settings poses to any rigorous treatment of yield results and the like underscores the need for a move toward establishing more precise procedures for determining accurate values of control settings. Conclusions The following conclusions are possible from the study. (1)Components of the coal preparation process can be simulated and the units interconnected into any desirable plant configuration. Current state-of-the-art nonlinear optimization methods can then be applied in the analysis of the plant; for instance, maximizing the yield of clean coal. (2) These optimization methods can be made more efficient by exploiting the special characteristics of coal preparation processes, such as the dominance of washers in determining the aggregate yield of clean coal. (3) Computed results compare well with those developed theoretically for ideal washers. Also, significant improvements in yields of clean coal have been achieved for

Ind. Eng. Chem. Process Des. Dev., Vol. 18, No.

complete plant designs. For example, a relative increase of approximately 2% in yield or an additional net revenue of over $500000 has been realized for a feed rate of 600 tons per hour. (4) The performance of several existing plants has been simulated and compared with actual operating data. The agreement is only fair. Disparities are mainly due to difficulties in duplicating gross actual plant control settings and some possible shortcomings in the simulator's screening model. Nomenclature c i = ash level of the aggregate clean coal product, 70 dk = ash level of the kth flowstream, % b,, = fraction of crushed particles originally in the ith size fraction that ends up in the mth size fraction c = crusher setting, in. d,,,d(p,,-) = distribution factor for a washer f , ,f(p,),F,,F, = flowrate of feed, % fi = flowrate of clean coal from a washer, % of feed P = flowrate of overflow material from a screen, % of feed F'= flowrate of crushed product from a crusher or coal stream from a rotary breaker, 70of feed F = flowrate of refuse from a washer, % of feed Ft = flowrate of rock stream from a rotary breaker, % of feed P = flowrate of underflow material from a screen, % of feed g, = constraint on the ash level of the aggregate clean coal product g, = constraint on the sulfur level of the aggregate clean coal product I = number of size fractions in a given coal feed J = number of specific gravity fractions in a given coal feed L = number of decision variables N = number of plant units 0,= fraction of feed particles in the ith size fraction that passes over a screen p L= fraction of feed particles in the ith size fraction that is crushed in a crusher Q = set of specific gravity fractions in the initial raw coal feed F = total sulfur level of the aggregate clean coal product, % i$= total sulfur level of the kth flowstream, 70 s = screen size or opening size of rotary breaker drum, in. W = set of size fractions in the initial raw coal feed X I= decision variable XI+= minimum value of XI X1++= maximum value of Xi Greek Letters a = upper limit on the ash level of the aggregate clean coal product, % p = upper limit on the total sulfur level of the aggregate clean coal product, % X = rotary breaker drum length, ft

p' pj 7

3, 1979 521

= washer separation gravity = average specific gravity of the jth specific gravity fraction = rotary breaker drum diameter, ft

Subscripts

i, m = size fraction

j = specific gravity fraction k = flowstream 1 = decision variable n = position of a unit (in a sequence of units in a plant)

Literature Cited Abara, J. A,, Ph.D. Dissertation, School of Englneering, University of Pittsburgh, 1978. Absil, J. H., Keuning, W., Meerman. P. G., "Prediction of Washery Plant Performance by Data Processing," p 381, 5th Internatlonal Coal Preparation Congress, Pittsburgh, Pa., 1966. Beveridge, G. S.G., Schechter, R. S.,"Optimization: Theory and Practice", Chapter VIII, McGraw-Hill, New York, N.Y., 1970. Brookes, G. F.. Whitmore, R. L., Coaifrep., 95, (May/June 1966). Carallaro, J. A,, Johnston, M. T., Deurbrouck, A. W., U . S . Bur. Mines Rep. Invest., No. 8118 (1976). Cierpisz, S.,Gottfried. B. S.,Int. J. Miner. Process., 4, 261-278 (1977). Gaver, D. P.. Thompson, G. L.. "Rcgramming and ProbabiUty Models in Operatkns Research", p 275, Wadsworth Publishing Co., Belmont, 1973. Gottfried, B. S.,"Computer Simulation of Coal Preparation Plants", U S . Bur. Mines Grant No. GO-155030, Final Report, Aug 1977. Gottfried, B. S., Int. J. Miner. Process, 5, 1-20 (1978). Gottfried, B. S.,Abara, J., Comput. Chem. Eng., 2, No. 2 (1978). Gottfrii, B. S..Bruggink, P. R., Harwocd, E. R., Id.Eng. Chem. Process Des. Dev., 9, 581 (1970). Gottfrii, B. S.,Jacobsen, P. S., U.S. Bw. M h s Rep. Invest., No. 8238 (1977). Hooke, R., Jeeves, T. A., J. Assoc. Comput. Mach., 8, 212 (1961). Ivanov, P. N., Yampolsky, M. N., "An Economical and Mathematical Model of a Coal Preparation Plant", 6th International Coal Preparation Congress, Park, 1973. Kindig. J. K., Luckie, P. T.. Lovell, H. L.. "Mineral Preparation Evaluation by Digital Computer of Washability and Distribution Data", Proceedingsof the Symposium and Short Course in Computers and Operations Research in Mineral Indussbles, April 1966. Laverick, M. K., "Optimization of Coal Washefy Fiowsheet Design and Operation with the Aid of a Computer", 6th International Preparation Congress, Paris, 1973. Pierre, D. A., "Optimization Theory with Applications", Chapter 6, Wiley, New York. N.Y.. 1969. Porshnev, V. M., Coke Chem. USSR, (1964). Reklaitis, G. V., Phillips, D. T., AIIE Trans., 7(3), 235 (1975). Sarkar. G. G., Sakha, S.,Lahiri, A., J. Inst. Fuei, 33,No. 230, 145-150 (1960). Venkatesan, S., Coal Age, 81(6), 108-111 (1976). Wakers, A. D., Ramani, R. V., Stefanko, R., "A Computer Simuhtion Model for Coal Preparation Plant Design and Contrd", Special Research R e m SR-102, Penn State University, Feb 1976. Weyher. L. H. E., Min. Congr. J., 57, 38 (Jan 1971). Wright, F. D., Trans. AIM€, 220,82 (1961). Zahradnik, R. L., "Theory and Techniques of Optimization", p 112, Barnes & Noble, New York, N.Y., 1971.

Received for review July 13, 1978 Accepted February 12, 1979

This study was funded jointly by the US.Bureau of Mines (Department of Energy) and the US.Environmental Protection Agency under U S . Bureau of Mines Grant No. GO-155030.

Maximization of Yields of Clean Coal from Coal Preparation Plants

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