Compurers and Chemical Engineering, Vol. 2, pp. 153-159 0 Pergamon Press Ltd., 1978. Printed m Great Britain

OPTIMAL SYNTHESIS OF WASTE TREATMENT PLANTS

M. J. BUSH* Department of Chemical Engineering, University of Calgary, Calgary, Alberta, U.S.A.

and

P. L. SILVEST~N Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada.

(Receioed 17 May 1978)

Ah&a&--It is possible to simulate waste treatment plants with increasing accuracy and reliability. This study examines the application of the SEPSIM steady-state simulation program as the modelling basis of an optimal synthesis program. A modified form of Box’s Complex method coupled with Fan’s structural parameter method is presented and employed in the solution of the multidimensional optimization problem. The computational ditliculties associated with the optimal synthesis of processing systems are discussed in some detail. It is found that an operating arrangement consisting of a trickling filter and an activated sludge system working in parallel is clearly indicated by the models as being the optimal system.

Scope-In the last few years the computer simulation of waste treatment plants has received a great deal of attention. The models developed have become complex, approaching, perhaps, the limit of sophistication permitted by the data available for their development. The objectives of the work described in this study were (1) to identify problems which occur in employing a process simulation program within an optimization program, and (2) to prepare an easily used versatile optimal design program for waste treatment systems. Progress towards both objectives is discussed in the paper. Specifically, the use of the SEPSIM computer program[ 11 in the optimal synthesis of a municipal waste treatment facility is described.

Parameters for the models used in the simulation are taken from the Kitchener. Ontario plant which was recently in the process of expansion. It has been demonstrated that the mean monthly operating characteristics of this municipal plant can be accurately modelled by the SEPSIM program provided that data used to evaluate the model parameters are taken from the actual plant. Accurate simulation provides an excellent and necessary basis for optimizing the design of a plant expansion. In such an application the degree of uncertainty associated with the plant models should be minimal avoiding, perhaps, the need to resort to Monte Carlo methods or the quantification approach of Chen et al. [2] and Berthouex & Polkowski(31. It is certainly a reasonable assumption that the plant will continue to operate in the same way after expansion provided, of course, that the characteristics of the influent remain generally unaltered.

Plant expansion, however, is rarely simply a matter of adding process units in parallel. Often new inter- connections or even new types of units are considered. The optimal design problem, then, is concerned not only with the optimization of a fixed network but simultaneously with the determination of that network itself. Not all networks will be available for consideration. Certain regulatory agencies impose constraints by broadly defining what they consider an acceptable waste treatment system. This often simplifies the optimization problem.

Conclusions and Sigrdfleanee- It is demonstrated that it is quite practical to couple a modular simulation program such as SEPSIM to a crude search technique, such as Box’s complex method, to solve small but still quite difficult network synthesis problems. The complex method appears to be a relatively inefficient search method when compared to many others but with the modifications discussed in this study it has the important advantage that it does not fail on a multimodal function: a characteristic commonly encountered in synthesis problems of this nature and liable to become increasingly common in all optimization studies of real plants using accurate and detailed cost data-without which, we may add, any study is of limited application and significance.

The optimal waste treatment system configuration suggested by this work is largely insensitive to fluctuations in the major variables and costs. In nearly all cases the structure consists of an activated sludge and a trickling filter system operated in parallel. The major part of the biochemical oxygen demand and suspended solids removal is accomplished by the activated sludge system which is capable of greater removal efficiencies than the trickling filter. However, trickling filter costs are low in comparison to activated sludge system costs and the optimal network structure that is indicated is a combination of a low performance but low cost unit and a high performance but high cost one. The poor quality trickling filter effluent is mixed with the high quality activated sludge system effluent to produce a final effluent of the required standard.

It is interesting to note that this conclusion has recently been supported by results from a pilot plant study of combinations of activated sludge and trickling filter units coupled both in series and parallel. Data from this study suggest that there are a number of significant operational advantages to a parallel arrangement of units. According to our work, not only do combined systems have distinct practical advantages. they also realize significant economic benefits

*Author to whom correspondence should be addressed.

CACE Vol. 2. No. 44 153

154 M. J. BUSH and P. L. SILVESMN

INTRODUCIION

The optimal synthesis technique employed in this study is a modified form of Fan’s structural parameter method[4,5]. In this technique a network is constructed which incorporates all possible system networks or at least all networks that one feels it is necessary to consider. At each node or splitting junction there are a number of associated structural parameters, 0 I aij 5 I, which define the stream split. The structure of a network is thus uniquely determined once a certain value has been assigned to each structural parameter since a non- zero value of any ail indicates the physical presence of a stream flow. We therefore have a relatively simple method, in principle at least, of generating an optimal system network.

Mishra et al.161 present a brief survey of some of the more recent work in this field. However, as they them- selves point out, most of the effort unfortunately appears to be more concerned with the demonstration of a parti- cular optimization technique rather than with the ques- tion of whether the design is satisfactory given the uncertainty in the models, in the projected flows, and in the waste characteristics. Berthouex & PolkowskiD] have specifically addressed this problem. The question has been taken up still more recently by Tarrer et al.[7] who present an optimization of the activated sludge process so as to minimize the uncertainty associated with model parameters or parameters describing waste characteristics. However, a fixed network was consi- dered and no attempt was made to examine a different arrangement of process units or to determine the effects of coupling the aeration process to a trickling filter system. Also, allowance for uncertainty may not be necessary when an accurate simulation is available for a plant operating on the waste to be treated.

The problem, formally stated, is

minimize

where

F = g&-N + g*(sY (1)

subject to

@I = @,(a, 4, dj) j = 1, N (2)

OS&, S 1 (3)

o~u;c:I4~c:u~. (4)

Here, a is the matrix of stream splitting factors, Uj is the bounded decision variable associated with each unit (assuming just one per unit), di is the vector of state variables defined a priori, 4 is the cost vector, and se is the exit stream vector which generates a cost penalty if certain of its elements exceed a preset level.

Figures 1 and 2 show the Kitchener waste treatment plant and its corresponding network. Figure 3 shows the network that will be used to determine the optimal system structure. There are four network structural parameters associated with units 2, 7, 9 and 13 and four variables defining the dimensions of units 3, 4, 5 and 8. We thus have an eight-dimensional optimization problem.

The SEPSIM simulation program requires that the

Fig. 1. Kitchener waste treatment plant.

INPUMP

Fig. 2. Network for Kitchener waste treatment plant.

Fig. 3. Initial optimal synthesis network.

Optimal synthesis of waste treatment plants

models for the various process units by written as subroutines. Restrictions on the subroutines are few but one of these is that each subroutine must operate with a stream vector of the same length and order. However, the user may fix both. For this study the vector was: flowrate, MGD; total BOD, mgll; dissolved BOD, mgll; total suspended solids, mg/ 1; volatile suspended solids, mg/l. The mathematical models of the processing units and the cost relationships employed in the calculation of the objective function have been described in an earlier paper 181.

(a33 - 1) E’)~ a23 ., . . a2n s2

$3 .. : 1.1 =-s,

. . . . . (arm - 1) s,

155

Flows of supematant from the secondary digester and filtrate from the vacuum filter (see Figs. 1 and 2) are very small compared to the overflow from the primary clarifier. Simulations show that, even though the BOD of these streams is quite high, the streams do not appreci- ably affect the quality of the treatment plant outflow. Consequently, the optimization problem can be tom into fluid side and solid side problems. We follow this prac- tice so the optimal design procedure discussed in this paper is applied only to the fluid side (Fig. 3).

Although fairly sophisticated models for digesters and vacuum filters exist[l], adequate plant data to evaluate model parameters are rarely available. Optimization of the solids side, therefore, has not been extensively studied.

COMPUTATIONAL ASPECIS

The structural parameter method of Fan was modified in the following manner. We allow no more than two streams to exit from a splitting junction. If a stream is to be split into three or more substreams, then it is neces- sary to have a number of junctions in series. Since a junction with p exiting streams has an associated p - 1 decision variables, this modification has the advantage that only one decision variable is associated with each node; a variable, furthermore, that is constrained only by simple bounds.

Flow in the waste sludge line to the primary clarifier in Fig. 1 is small and can be neglected for design purposes. In some systems, furthermore, waste sludge goes directly to the digester as shown in Fig. 2. Although the usual practice is to use the trickling filter as a roughing unit in advance of an activated sludge unit, it could in principle operate in parallel or even as a final stage. Ah these possible configurations are allowed for in the network of Fig. 3. Examination of this flowsheet shows two recycle loops. It is necessary to tear both loops. Streams 6 and 11 are the tom streams.

Waste treatment systems have relatively simple networks that converge without undue difficulty using the successive substitution method. It is not usuahy necessary, therefore, to build in a convergence promo- tion algorithm. But the simplicity of the structural parameter method makes it possible to immediately solve for all the stream flows even before we embark on the recycle calculations. If aii is the fraction of stream j passing into stream i, then we may write for each of n streams:

ailsl+ai2s2+ ..fai,s.=+ i=l,n. (5)

Furthermore, if there is just one input stream to the network: st, and no streams in the network flow back into sI then au = 0, j = 1, n. It is possible then to set up the following system of equations.

A standard matrix inversion routine will solve this prob- lem quite rapidly.

This initial calculation results in a significant reduction in computer time. It is subsequently only necessary to force convergence on the stream component concen- trations and these converge very rapidly when the stream flows are already determined.

The stopping criterion at the mth iteration is

%I I I -.%-I <Q S,

where E is a tolerance vector with each element very small.

Each physical unit in the network, apart from the pump and splitting junctions, has a lower bound on its decision variable. When this bound is reached, the unit is suppressed, that is, it disappears from the network. In effect, the unit functions as a simple mixer in order to maintain the overall mass balance and it no longer generates an attendant cost. In addition, some units are coupled so that when one unit is suppressed so is the other. For example, if the activated sludge unit is suppressed then so too is the secondary settler, unit 8. Similarly, if the trickling filter is suppressed, so too is its associated settler, unit 5. Additional, if the flow to a unit fags below a set value (0.001 MGD) that unit is also automatically suppressed and withdrawn from the network.

Two constraints on the operation of the activated sludge system were introduced. To avoid bulking sludge and failure of the secondary clarifier, the sludge age is usually held between 2 and 10 days [l]. This translates roughly into residence times of between 3 and 9 hr duration in the activated sludge tank. If the residence time is calculated to be outside this range the activated sludge unit is suppressed. When this happens the outflow quality greatly exceeds specifications invoking a severe cost penalty for the parameter set chosen by the opti- mization program.

Examination of years of records disclosed that the Kitchener waste treatment plant maintains about a mixed liquor suspended solids level of 3500 mg/l. Plant opera- tors evidently have found that this level gives adequate performance and stable plant operation. This information is available in a plant expansion study and deserves inclusion in an optimal design exercise. A constraint of about 3500 mg/l was introduced. It functioned as follows. The solids level is controlled by manipulating the split fraction in unit 12 which varies the amount of active sludge that returns to the aeration tanks. In essence, we have a simple proportional controller, sensing the suspended solids concentration in stream 11, and acting on the a value for unit 12.

These factors, particularly the constraint on the aera- tion tank residence time, cause the response hypersur- face to be highly irregular-and more importantly- multimodal, characteristics that caused serious and often intractable problems for all of the optimization methods that were employed in our study. The unusual nature of

If6 M. J. BUSH and P. L. SILVESTON

the response hypersurface is illustrated by Figs. 4-7 which represent planes through the surface. All planes examined were multimodal and piecewise continuous.

The eight elements of the vector shown on each figure correspond to the foUo~ng variables: 1, 2, 3, 4-the a values for units 2, 7, 9 and 13 respectively (a is the fraction of flow going to the lowest numbered stream); 5-aeration tank volume (unit 3); btrickhng filter sur- face area (unit 4); ‘I-secondary settler surface area (unit 5); 8-secondary settler surface area (unit 8).

OFWMZATiON METHODS l-ECl%D

The optimization method o~~n~ly employed to tackle this problem is due to Best & RitterlP, 101. The technique is a feasible conjugate direction method for linearly constrained optimization problems and it requires an evaluation of the local gradient. On multivariable poly- nomial problems with exact derivatives the method ap- pears extremely effective but on the type of problem we

have here it did not have the same success. The method

109,09,09,09,u~, 2000,15000,15000) I~~ , ,

I I 1 IO 20 30

Aerct~on tank volume, UJ

Fig. 4.

ti 5 = ‘0

p” 1 (0 9.0 9. I 0, 0 96,u,, 20000, 15000,150001

s It

I I I I IO 20 20

Aeration tank volume. us

Fig. 5.

2.100

~

(Qi, IO, l0,0,20000,15000,15ooo,45

I I / 02 cl4 06 08

Stream splat. 0,

Fig. 6.

u)

G _ .,o

5 200. =

5 0; 3

9 r look-

0

2

I= (a,,I0,I0.07.20000,15000,15000,45)

I I I 02 04 06 08

Stream spilt, a,

Fig. 7.

repeatedly failed to find an extremum, either stopping prematurely at different points, or terminating with a logic error. The failure of the method is probably due both to the difficult nature of the surface and to the fact that the estimation of the gradient at any point is sub- jected to error-a factor that causes additional problems for this type of optimization method. The source of error is that the solution to the mainly non-linear material balance equations through successive substitution is never exact. As can be seen in Figs. 4, 5 and 7, portions of the hypersurface are virtuahy flat, Smaller tolerance vectors were applied, but the Best & fitter algorithm still failed to locate an extremum with any consistency.

The complex method of Box[ll] was implemented after the failure described above. Although this method requires much more time on well behaved hy~rsu~a~es, it does not require the estimation of derivatives and has been used on the type of problem we have here with some success.

Box’s complex method is a development of the Simplex method of Spendley et al.[12], the latter being applicable only to unconstrained optimization problems. In the Complex method, for an n-dimensional problem, k z n t 1 points are chosen and these form the vertices of what is referred to here as the complex. The function is evaluated at each vertex and the vertex with the highest function value or cost (we seek a ~~murn) is replaced by a point collinear with the reflection of the worst point through the centroid of the complex, such that an over-reflection takes place. If the point is feasible its value is calculated; if not, it is moved a small distance into the feasible region in the direction of the centroid and there evaluated. If its value is acceptable, which we take to mean that if its cost is less than the second highest value present in the complex, it is accepted as a vertex. If not, the new point is moved towards the centroid until an acceptable point is found. When this occurs the second highest point becomes the worst point and the process is repeated. In this way the complex moves in a rather desultory fashion into the region of the minimum point. The slowness of the method would be less important if it could be shown to be exceptionally reliable, but u~o~unately this has not proved to be the case, at least insofar as the present application is concerned. The procedure has a tendency to halt when the local surface is relatively flat or when the complex moves into a boundary since this increases the tendency of the complex to contract. When first applied to this problem it was unable to locate the minimum unless started in the neighbourhood-not a very satisfactory characteristic.

A MOD~D COMPLEX =OD 7. Repeat 6 no more than t, times or until 4 holds. If All complex methods usually stop when the complex no acceptable point is found then

becomes small in some sense. Most methods actually 8. x, = /3x, t (1-/3)x, j= 1,k encourage the contraction of the complex when the sur- xj f xmax face is nearly flat. In fact, all the methods fail on a /3 *2. perfectly flat surface since any new experimental point 9. Repeat from 4. will be unacceptable because it obviously cannot have a value less than the existing points. A simple solution to here contraction and premature termination is to develop a method that stops only when it begins to repeat its search cycle. This may be accomplished by preventing the complex from contracting below a certain size. It is then forced either to move if it finds a better point, or to or

remain station~y but continue to examine points in the immediate neighbourhood if it in fact lies at the optimum.

t 5

= In tP - 1) -In y

el < 9 5 e2.

If xmaxr xIn.x2, x0 x., xmin are the points with the highest value, the next highest value, the centroid, the 10. Stop after a set time or when the complex is new reflected point and the best point respectively, then oscillating. Box’s method can be represented by the following statements: Choosing 8, y and t, to satisfy yfs = (B - I)-’ where t, is

integer ensures that the only way that the complex I. Calculate F(xi) at .all vertices. contracts is when it runs into a boundary. However, the 2. Find xmaxr xmex2. above modification to Box’s method prevents the 3. x,=px,+(1-/3)x,n,X P>l complex from contracting prematurely and forces it to

thoroughly search the local hyperspace. where

OFWMAL. SY!XJ%

xc =

if

4. F(G) < F(x,,,z)

then

k-l

Because of the nature of the surface it was necessary to make a number of runs starting at different points in the search space. A sixteen point complex was used. Four runs of 60 s duration were found to be sufficient to determine the optimum system configuration and design. The formulation of the objective function deserves some comment since its nature has a very real effect on the solution to the optimal synthesis problem. The cost of the system is taken as the net present value which is calculated as follows

5. x, repiaces xmax xmax replaces xmax2

find xmax2, xmln and continue from 3,

if not then

where C, is the yearly cash flow, i is the annual rate of discount, and n is the life of the project in years. Assuming for simplicity that design and construction are

6. x, = (I - Y)XC + yx, O<y<l. completed in the first year of the project and if the yearly

7. Repeat 6 until 4 holds. If no acceptable point is cash flows are equal, we have

found halt

8. Stop when

B = 2 [F(xi) - ~(xmin)l

F(xmn) 22 e,.

This procedure is modified in the following manner:

3. x, = Bx, + (1 - P)X,.X p+2

The cash flows are conventionally positive for an inflow and negative for an outflow and one usually selects that project with the largest net present value. In this case where we have only costs, that is, negative cash flows, we reverse this convention, take all costs as positive, and seek the system with the minimum value. If the suepended solids or BOD level in the exit stream from the system are excessive, i.e. greater than 30 or 20 mgll respectively, then a penalty is added to the cost as indicated by Eq. 1. The penalty is calculated as

if gz(s’) = 10”. 5 c (sr’ - sX) for skc > st. 110) =z

4. F(x,) < Fbmz)

then 5.6 as above. The formulation indicated by Eq. (9) means that a unit ._ .

with low operating costs, such as tnckhng tilter, will

Optimal synthesis of waste treatment plants 157

Case

M. J. BUSH and P. L. SILVE~~ON

Table 1. Sensitivity studies

Optimal vector e2 a3 a4 u3 84 System

Base Trickling filter cost doubled Activated sludge cost doubled k, doubled y halved Trickling filter capacity reduced

p halved 0.24 0 0.73 0 3.51 2%5 403 BODri, = 30 mg/l 0.17 0 0.70 0 3.86 1248 401 BODri, = 50 mgfl 0 0 0 0 4.35 2402 4%

BODi, = 30 mg/l ACS cost doubled BOD,, = 30 mgll ACS cost tripled

0.19 0 0.99 0 0.21 0 0.99 0

0.22 0 0.93 0 3.54 2906 931 12980 Parallel

0.22 0 0.97 0 3.40 2727 860 0.23 0 0.94 0 3.53 2951 938 0.08 0 0.92 0 4.05 410 430

0.51 0 0.85 0 2.45 5428 17320

0.51 0 1.0 0 2.16 5683 18790 14830 Parallel

ar fraction of flow to trickling biter. a2 fraction of flow recycled around trickling filter. a3 fraction of trickling filter effluent passed to activated sludge system. a4 fraction of activated sludge settler overtlow passed to trickling filter. u3 aeration vessel volume, million gal. u4 trickling filter surface area, ft’. US trickling filter settler surface area, ft’. UE activated sludge settler surface area, ft’.

appear less costly over a period of many years than a unit with a comparable capital cost but much higher operating costs such as the activated sludge system. If the project is sufficiently long, it is annual operating costs which will dictate the optimal system.

Using the cost relationships [8] we can find an apparent global minimum ~o~es~nding to the network shown in Pig. 8. The optima1 vector is (0.19, 0, 0.99, 0, 3.59, 547, 13970,2297). The system is basically parallel with a small fraction of the trickling fiber efIluent returned to the activated sludge system inlet. This flow (stream IO) is often almost negIigible but, as the sensitivity studies show, it is generally present. The recycle stream on the trickling filter is suppressed and the primary settler is also not present in the optimal system structure. The incoming stream, stream 1, is defined as follows: flow- 20 MGD; TROD-242 mg/l; DBOD-73 mgll; TSS-275 mgfl; VSS-234 mgll. A trickling filtered system is deemed incapable of reducing the dissolved BOD level below 25 mg/l. Thus a trickling filter system operating in

Fig. 8. Optimal structure.

3.59 2297 547 3.60 2055 966

13970 Paraliel 13110 Parallel

13410 Parallel 13420 Parallel 13860 Activated

sludge only

15340 Parallel 14020 Parallel 13830 Activated

sludge only

5416 Parallel

isolation will always generate a penalty cost. If this minimum allowable dissolved BOD level is reduced to say lOmg/l then a simple filter system is the optimal system configuration.

A number of sensitivity studies were made. The results are shown in Table 1 which indicate that the optimal system configuration is largely insensitive to fluctuations in the major variables and costs. It should be noted that the location of the pump modules in the initial optimal synthesis network (Fig. 3) is of critical irn~~ce. Pumping costs appear to make a significant contribution to the total system costs. This accounts for the supres- sion of streams 7 and 13. The position of these units should therefore properly reflect the arrangement of pumps in the physical plant.

A recent paper by Roskopf et ai.[131 discusses the results of a pilot plant study of activated sludge and trickling filter systems in series and parallel. They point out that a parallel system has obvious advantages for expansion of an existing trickling filter plant which is heavily loaded hydrauii~~y. They note that the neces- sary aeration tanks would be smaller than those required if the trickling filter were abandoned, that less aeratioan capacity would be required, that secondary sludge hand- ling problems might be reduced and that, in general, combined systems provide better operating stability than single-stage systems.

BOD C cap

C, di

DBOD el, e2

F gi,g2

NOMENCLATURE

biochemical oxygen demand, mg/i capital cost, 16 annual cash flow, $/year state variable vector for unit j dissolved or soluble BOD, mg/l error tolerances, e2 > et objective function, S cost functions

Optimal synthesis of waste treatment plants 159

N P

PV S

S' SBOD

T& UI

u;, u; vss

x (I

all E e

a,

number of plant units number of exit streams from a node net present value, S stream vector vector of maximum allowable effluent concentrations suspended BOD, mgjl number of contraction steps in optimization routine total suspended solids, mg/l decision variable for unit j lower and upper bounds on ui volatile suspended solids, mg/l point in search space matrix of stream splitting factors fraction of stream j passing into stream i tolerance vector stopping criterion cost vector for unit j

Subscripts e effluent i influent

lim limiting value

Superscripts e exit

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13.

REFERENClZS

Report No. EPS 6-WP-74-1, Environmental Protection Service, Environment Canada (1974). M. S. K. Chen, L. E. Erickson & L. T. Fan, I/EC Process Des. Dev. 9(4), 514 (1970). P. M. Berthouex & L. B. Polkowski, J. War. Pollut. Control Fed. 42(9), 1589 (1970). P. N. Mishra, L. T. Fan L L. F. Erickson, Can. 1. Chem. Engng 31(6), 694 (1973). L. T. Fan & K. Osakada, Report 38, Institute for Systems Design and Optimization, Kansas State University (1972). P. N. Mishra, L. T. Fan & L. E. Erickson, A.1.Ch.E. fl Symp. Series, 71(145), 136 (1976). A. R. Tarrer, C. P. L. Grady, Jr., H. C. Lim & L. B. Koppel, J. Enu. engng Dia. Am. Sot. Chem. Engrs, 102(3), 657 (1976). M. J. Bush & P. L. Silveston, Cornput. Chem. Engng 2 (1978). M. J. Best & K. Ritter, Research Report CORR 74-12, University of Waterloo (1974). M. J. Best, J. Proc. Oil Technol. Ass. India. 16 (l/2), 25 (1975). M. J. Box, Compuf. fl, 8, 42 (1%5). W. Spendley, G. R. Hext & F. R. Himsworth, Technometrics 4, 441 (1%2). R. F. Roskopf, J. C. Young, E. R. Baumann, .I. Env. Engng Die. Am. Sot. Chem. Engrs 102(5), 1005 (1976).