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tec R h A nTcal report section NAVAL POSTGRADUATE SCHOOL Monterey, caufoania 93*40 NPS55TO72071A NAVAL POSTGRADUATE SCHOOL Monterey, California A MATHEMATICAL FORMULATION FOR SELECTING HOLDING-TANK-PROCESSOR REQUIREMENTS FOR SHIPBOARD SEWAGE TREATMENT M. U. Thomas R. D. Rantschler July, 1972 Approved for public release; distribution unlimited. FEDDOCS D 208.14/2. NPS-55TO72071A NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral M. Superintendent . Freeman, USN M. U. Clauser Provost ABSTRACT: This paper describes a formulation of the problem that sys- tems designers face in selecting a combination of holding tank and processor for shipboard sewage treatment systems. Two decision models are discussed within this framework. In one case the generation of sewage, aboard ships, is assumed to consist of deterministic arrival streams. In a second model, sewage generation is assumed to behave in accordance with a Poisson process. Allowances for maintenance and re- liability are discussed. This task was supported by the Naval Ship Systems Command under Work Request 2-5010, 17 August 1971 (Project Director: C. Rowell) . Prepared by: 1. INTRODUCTION Our increasing concern for a clean environment, and recent legislation by the United States Congress in support of this con- cern have prompted the U.S. Navy to expedite their efforts to eliminate the discharge of sewage in inland waters. To date, it appears that there are two alternative directions available to the Navy: (1) to make shore connections with municipal sewage treat- ment systems, and (2) to provide on-board sewage treatment facil- ities. Although the first alternative is appealing because of its simplicity and apparently lower initial investment cost, it has several disadvantages. Since municipal cooperation is required, the Navy stands to lose mobile flexibility unless such cooperation can be sustained indefinitely with each coastal municipality, both domestic and foreign. The actual cost for this is not known. Furthermore, there are other costs that would have to be incurred over and above the "connection costs." Every naval vessel would be required to have an on-board pumping system, plus a holding tank system for collecting sewage generated while in-transit or at an- chorage. The second alternative direction indeed offers sustained mobile flexibility for the Navy, but the initial investment can be very high. Several shipboard treatment facilities have been pro- posed [ 1] . These consist of a chemical or bacteriological treat- ment process coupled with a holding tank system. Unfortunately, to date, none of these systems have proven to be effective. Although cost is an important factor, other reasons that contribute to this defeat is the lack of capability to meet anticipated Environmental Pollution Agency (EPA) standards and a host of interfacing problems in implementing proposed designs aboard ships. The approach to the development of these shipboard treatment systems has been one of building a set of hardware in one phase, followed by an implementation phase where a ship and crew must ac- comodate the new facility. Another approach is to prescribe the needs of the ship and design the hardware accordingly. Since we now have some knowledge of our hardware development capabilities, perhaps this latter approach is in order. This paper describes a mathematical formulation for examin- ing trade-offs between holding tank capacity and processing rates of any proposed facility subject to the restriction that all sewage generated must be processed (i.e., no overflow of unprocessed efflu- ent). Following a general formulation of the problem in section 2, we shall consider the case where the generation of sewage is assumed to be deterministic in section 3. In section 4, we consider the more general case where arrival streams are random but we assume that the distribution is Poisson. Section 5 discusses the effect of interrupted operation of the processor, followed by final com- ments in section 6. 2. PROBLEM FORUMLATION The problem to be studied involves an arrival stream of sewage, aboard a naval vessel, to a collection point where it is stored in a tank for processing and removal from the parent facil- ity. We shall consider this arrival stream as countable numbers of a fixed volume of sewage that are generated at discrete points in time. This problem is essentially the same as the "dam", or storage, problem which has received considerable attention in the literature over the past several years (see Moran [3] and Prabhu [4]). In particular, random inputs accumulate in a finite capac- ity storage facility (provided the capacity is not exceeded) with outputs that depend both on the volume accumulated and the release policy employed. The analytical detail provided by storage theory, however, far exceeds that of our current understanding of the nature of sewage generation aboard ships. For this reason, we limit our present discussion to a formal treatment of the decision problem of selecting combinations of holding tank and processing capabilities From the point of view of a systems designer, his role is that of a decision maker whose objective is to make an appropriate selection of processor and holding tank system. Among his alterna- tives are various combinations of processing units and tank sizes each of which has associated costs. His constraints are determined by cost, space utilization, risk of overflow of effluent sewage, and reliability. Basically, this is a decision problem in which the decision maker is uncertain as to the "future states of nature." In this case nature corresponds to the characteristics of sewage generation by shipboard crews. For each alternative, i.e., combination of pro- cessor and holding tank, and possible state of nature there is a cost value that depends upon a set of constraints. Our problem is to find a decision rule for solving choice among the alternatives. In general, one can treat the problem as a decision under risk by assuming knowledge of the probability structure of the future generation of sewage. Before we take this approach, however, we shall consider a more simplified approach by assuming complete knowledge of the generation of sewage aboard ships. Our motivation for this latter approach is based on the relatively systematic ship- board routine. 3. DETERMINISTIC ARRIVAL STREAMS 3.1 Constant Arrival Rates Consider an arrival stream of sewage to a shipboard holding tank of capacity V. The stream alternates deterministically be- tween two rates, r and r , over a period of time of length x. Thus, the arrival rate is given by (r , if £ t £ yt (1) U(t) = lr 2 , if yj ^ t ^ x where ^ y ^ 1 and r > r . This is shown in Figure 1. fJL(t) r. H f-n ~(S- ► t T 2T Bt T Fig. I. A typical deterministic arrival stream with y = 1/3. Note that ^(t) = r for Br < t < T = Wt. A cycle of length T comprises an integer number W of these per- iods of length t. The first B < W, also an integer, of these periods are repetitions and have arrival rates given by equation (1) . For the remaining time duration (W-B)t, the rate is r„. From the holding tank, sewage is processed at a constant rate a. Thus, the net accumulation in the holding tank during t for any one of the first B periods is given by (2) !YT T ) + / (^-a) dt + / (r 2 -a) dt> o yt ) = T[yr ;L + (l-y)r 2 - a] . There are two cases to consider. First, for a > Y r i + (l~Y) r o» M(t) = and there will be idle processor time during period t in the amount xfyr- + (l-y)r 2 - a] t. = r„ - a The maximum accumulation for this case will occur at x = y t • Thus, M max (YT) = ( r r a )YT , and this will be the minimum tank volume that will insure against overflow. The second case to consider is where M(t) > 0. Here the maximum accumulation in the tank will occur at the earliest time t such that u(t) < a for all t : t ^ t ^ T, which will be y t m m time units into period B. Since the tank capacity V must ex- ceed M (t) in order to prevent overflows, max r M (t) = (B-l)M(x) + YT(r,-a) £ V max 1 from which it follows that (3) V + T(B-H-y)a ^ xtBy^ + (B-l) (1- Y ) r^ . We shall impose the restriction that the system processes all ar- rivals in each cycle, i.e., M(T) = 0, therefore, the processing capacity must exceed the volume generated. Thus, Wxa ^ t [Byr + (W-yB)r „ or Byr + (W-yB)r (4) a * - -^ 2 - This is equivalent to requiring that the amount of sewage processed in [t , Tl be greater than M (t) plus the amount of sewage m b max generated in [t ,T] . It is noted that if a > r , then the only holding tank requirement would be to accomodate the accumulation during maintenance periods. Hence, the range of interest for a is Byr 1 + (W-yBr 2 ) (5) ■ £ a <; r x Let us now consider the decision problem of selecting an ap- propriate system for a particular ship. Any processing system has associated costs which we shall assume can be related to processing rate, holding tank capacity, and spatial requirements. From the above we can determine the minimum holding tank size required to satisfy the system requirements. For each of i = 1, ... , n can- didate systems let C. . (a) be the cost to process at rate a , and C_ . (V) the cost associated with a capacity V. For simplicity, we are including the spacial requirements for the processor with the holding tank volume. Our problem then is the following mathe- matical programming problem: min C = C (a) + C (V) s.t.: V + i(B-l+y)a ;> x[Byr + (B-l)(l-y)r ] (P) BYr 1 + (W-yB)r 2 a ^ _ a ^ , V ;> . The solution to (P) determines a set of optimal values (a ,V) ; for each candidate system i = l,...,n. The minimum cost then is * * *, C = min C. , and our decision rule is to select i ={i: C . = C ) . i i l Example - 1 To illustrate the approach described in this section, consider the situation where sewage generation is relatively constant at 216 gal./hr. for the first 40 percent of a workday. At the end of this duration, the rate instantaneously drops to 1Q8 gal. /day and reamins at that level for the remaining 60 percent. Our cycle is 1 week in duration which is comprised of 5 workdays by a 2 day weekend. Thus we have, t = 24 hours W = 7 days B = 5 days Y = 0.4 r- = 216 gal./hr r 2 = 108 gal./hr. The net accumulation during a workday, from equation (2), is M(t) = 24[151.2 - a] + , and from equation (5) the range of interest for a is 138y £ a ^ 216 gal./hr. Therefore; 138y <; a <: 151.2 gal./hr. determines a critical area in which there is a gradual accumulation of stored sewage during the week, which is to be processed over the 10 weekend. For this range, on applying the inequality of equation (3) , we have V + 105a ^ 16,588.8 gal. Problem (P) then is min C . l s.t. : V + 105a ^ 16,588.8 a :> 138.86 V, a ^ . In this section we have assumed that the generation of sew- age was both deterministic and constant at fixed known durations of time. The systems designer will indeed have to make allowances for these assumptions. In general; he will solve problem (P) for many representative values of B, W, x, r.. , r_ , and y, and de- termine a range of values for a subset of the candidate systems. He then, of course, must apply judgment. In some cases, however, he cannot justify assuming that the rates are constant even though he might be able to assume that they are deterministic. 3.2 Variable Arrival Rate-Fourier Series Approach Let us now consider deterministic arrival streams that are variable, but identical for B periods each of length x. 11 Any cyclic arrival stream may be represented by a sum of cosine and sine terms of increasing harmonics. For illustration, a typical arrival flow in terms of an incremental volume may be represented by the Fourier series: (6) v(t) = A - A n cos 2tt — - A_ cos 4it — . O 1 T 2 T The total volume of sewage generated in a period is (7) A = / v(t) dt and the net accumulation of sewage over time t t (8) M(t) = / [v(u) - a] du , o for suitable values of A , A n and A. . Letting t 1 and t o 1 2 1 I be the first and last times that v(t) = a over length T, it is convenient to translate the origin to t,. Thus, (9) M (t) = max f m [v(t) - a]dt , max J t , t 1 m and for the simple case shown in Figure 2, 1 w provided / [v(u) - a ]du :> for all w <. t. We shall spare o the details here. 12 Fig. 2. A variable arrival stream showing periods of of over - production (v(t) > a; hatched) and under production. M max (t) = M( V = lA o " aJ 'VV A 1 t 2it Ax 2-nt —^— sin 1- — r— sin 2tt t 2tt t A 2 T 4TTt 2 A T 4TTt —7 — sin 1- — — sin 4tt t 4tt t This is the hatched area in Figure 2. 13 We have ignored the remaining (W-B)t time units of the cycle of length T. In practice, of course, one would apply the above technique to this interval of time and determine an approx- imation for the overall arrival behavior. Again, the system de- signer will have to apply judgment. We shall next consider the situation where arrivals of sewage to the holding tank are random. 4. RANDOM ARRIVAL STREAMS 4.1 Poisson Arrivals Assume that the counting process, N(t), for quantities of sewage is Poisson with rate A. Each i arrival consists of a volume of amount denoted by random Variable Y.. Thus, the amount of sewage generated prior to some time t is given by N(t) (10) X(t) = I Y , i=l and (X(t) ; t ^ 0} is a compound Poisson process. For the special case where each arrival is a fixed volume Q, i.e., Y. = Q for all i, (11) X(t) = Q N(t) and (12) E[X(t)] = Q E[N(t)] - Q At 14 We shall now consider the stochastic analog to the arrival pattern described in section 3.1. For time period of length t, the rate \ alternates between two constant values, at a fixed point y t j given by |r 1 , <; t <: Y t c«>- (r 2 , yt (13) £ t ^ T Thus, the arrival process is described by the two processes {N..(t) ; t ^ 0} and {N (t) ; t ^ 0} with parameters r and r_ respectively. Assuming X(0) > 0, the amount of accumulation in a tank at time t is (14) M(t) - X(t) - ta = Q N(t) - t a provided X(u) - ua > for all ue(0,t). Here we assume that X(0) and r are both sufficiently large, so that the probability of remaining at a boundary, i.e. V or 0, is small. This is reasonable aboard most ships particularly since if the input rate is very small, then it is impractical to process. Thus, for a period (15) M(t) = M(yx) + M((1-v)t) = Q[N 1 ( Y t) + N 2 ((1-y)t)J - xa and for M(x) > , 15 (16a) E[M(t)] = Q[r lY x + r^l-y^] - ai and (16b) VarlM(x)] = Q 2 Ir yt + r 2 (l- Y )x] . As with the case of deterministic arrival streams, described in section 3, we restrict the tank capacity V by the maximum accumulation over a cycle of length T. Hence, P{M (t) £ V> i £. , (0 J! t i T , i 5. i 1) , max 1 1 or P((B-l)(QlN 1 (yT) + N 2 ((l-Y)x)j - to) + Q N (yt) - ytcx ^ V> :> 5 , from which it follows that (17) p{N al ( T M V+ ( B - 1+Y)TO ^l where N ..(t) is Poisson distributed with parameter al A al = tByr l + ( B - 1 H 1 - Y ) r 2 jT ' Similarly, the requirement that all sewage generated is processed during each cycle becomes 16 P{B M(t) + (W-B)Q N (t) - (W-B)ia ^ 0} ^ ? 2 , (0 £ ^ 2 ^ D This leads to (18) P{N a2 ( T ) S ^), 52 where N „(t) is Poisson distributed with parameter A a2 = ByTr l + ( w ~YB)xr 2 . The decision problem for the case of Poisson arrival streams may now be formulated, similar to problem (P) in section 3. Let- ting, 7 V + (B-l+y)T0t z i " m Wta '2 Q we wish to min C. = C (a) + C 2 (V) <Z > i " X al 1 A ., e s.t. (P') i=l I "If" y 2 A a2 e * e , iii i! a :> , V^O 17 where the symbol < >" denotes greatest integer. 4.2 Normal Approximation It is well known that for large values of A, Poisson dis- tributed N(t) can be approximated by a Normal density function. Therefore; since M(t ) = Q N (yt) - YTa , (19) M(t ) ~ N[E(M(t )), Var(M(t ))] m m m Thus, for M(t) ^ 0, the tank capacity V must be such that V :> E(M(t )) + $ 1 (l-g 1 )[Var(M(t ))] 1/2 m 1 m or (20) V :> QYTr L - Y Ta + $ 1 (1-? 1 ) [-prr.] 1/2 Q in order to have the probability of overflow less than 1 -"£-,• In a similar manner, for M(t) > 0, it can be shown that (21) V + (B-1+y)toi :> BytQ^ + (B-l)(l-y)TQr 2 + $" 1 (l-? 1 )(YTr 1 ] 1/2 Q . For the additional requirement that all sewage generated in a cycle be processed, in the same cycle, with probability £-, 18 (22) E(M(T)) + $ _1 (l-C 2 )[Var(M(T))] 1/2 ^ where (23a) E(M(T)) = QlByrr + (W-yB)xr 2 J - Wxa (23b) VarCM(T)) = Q^Byrr + (W- Y B)xr 2 ] . Hence, it follows that (24) a :> ^- {[B Y Tr 1 + (W- Y B)xr 2 ] + $" 1 (l-e 2 )lB Y Tr 1 + (W- Y B)Tr 2 J 1/2 The decision problem (P*), using the normal approximation for Poisson arrivals of sewage, is to min C. subject to the con- i straints given by equations (20), (21) and (24). Example - 2 For illustration of the above; consider again the data of Example - 1, but with r 1 and r» now in units of arrivals /hour, Letting Q = 4.5 gallons per arrival, Qr ± = 216 gal./hr. Qr = 108 gal./hr. Uft , , „ V + 105.6a Aa., = 3686.4 Z- , = ; — ? 1 al 4.5 A - = 5184 Z _ = 37.33a a2 a2 19 Since A and A are large, we shall use the Normal aDproxi- al az - v mation. For E, = g = 0.975, the constraint of equation (20) is V + 105.6a >. 16,778.1 gal. and from equation (24) a ^ 142.6 gal./hr. 5. INTERRUPTED OPERATION OF PROCESSOR The fact that no system is infallible makes it necessary for systems designers to consider some form of allowance for "down time", or interrupted service time of the processor. This down time comprises planned periods for preventive maintenance as well as unplanned periods that arise due to electrical and mechanical malfunctions. In either case, the holding tank must be of suffi- cient capacity to accomodate the additional sewage that accumulates during these interruptions. One way in which the designer can make allowances for down time is to apply a safety factor to the tank capacity determined by solving program P or P'. Alternatively, he may incorporate this allowance in the analysis by modifying the constraints in the decision problem. For example, the worst situation arises when a down time period of length D commences at the time, t , of m maximum accumulation in the tank. For the case of Poisson arrival 20 streams, discussed in section 4, one can allow for this by examin- ing (25) M (t ) = M (t ) + Q N (D) . max m max ra ^2 Since D depends upon the reliability and maintainability require- ments for a particular system, it will vary in length among differ- ent processing systems. 6. FINAL COMMENTS The foregoing sections provide a descriptive framework for the decision problem of selecting combined holding-tank-processor systems for shipboard sewage treatment. Like any decision problem; the decision maker, systems designer in this case, must tradeoff between simplicity and reality through assumptions, and supplement his final analysis with judgment. In general; the more uncertain he is of the distribution of sewage generated, the more judgment he will be required to make. There are, of course, situations where the generation of sewage cannot be treated deterministic, as in section 3, nor can it be described by a Poisson process (section 4) . The general approach presented in this study, however, is not limited to these two models. Given more complete knowledge of a particular sewage generation process, one can arrive at a better choice of facilities Miner [2] has recently developed an empirical distribution of the 21 generation of shipboard sewage based on known data available to date and subjective ratings by shipboard personnel. He also exam- ined the sensitivity of holding tank capacity to various shipboard operating policies. Only the case of a single holding tank and processor have been considered in both this study and the study by Miner. For large ships it may be necessary to combine one or more holding tanks with one or more processors. It is suggested that future studies address this problem. 22 REFERENCES [1] Kinney, E. T. and Constant, A., "Control of Shipboard Wastes," Slav/zt Engiw2.su> Jousinal, June 1971, pp. 118-24. [2] Miner, J. 0., Jr., "Descriptive Model of a Shipboard Ecologi- cal System," Masters Thesis, Naval Postgraduate School, Monterey, California, June 1972. [3] Moran, P. A. P., Thz ThzoHy o& Storage,, Wiley, New York, 1959. [4] Prabhu, N. u. , Time. Vtp2.nd2.nt RzAultA In Storage. Theory, Methuen, London, 1964. 23 INITIAL DISTRIBUTION LIST No. Copies Defense Documentation Center 12 Cameron Station Alexandria, Virginia 22314 Dean of Research Administration 1 Code 023 Naval Postgraduate School Monterey, California 93940 Library, Code 0212 2 Naval Postgraduate School Monterey, California 93940 Library, Code 55 5 Department of Operations Research and Administrative Sciences Naval Postgraduate School Monterey, California 93940 Dr. John Huth , Code 034 1 Special Assistant to the Director of Naval Laboratories Naval Ship Systems Command Washington, D. C. 20350 LCDR Jay 0. Miner, Jr., USN 1 USS Springfield (CLG-7) FPO, New York 09501 Professor J. R. Borsting 1 Department of Operations Research and Administrative Sciences Naval Postgraduate School Monterey, California 93940 Professor D. P. Gaver 1 Department of Operations Research and Administrative Sciences Naval Postgraduate School Monterey, California 93940 Professor C. R. Jones 1 Department of Operations Research and Administrative Sciences Naval Postgraduate School Monterey, California 93940 24 No. Copies Professor E. A. Brill 1 Department of Operations Research and Administrative Sciences Naval Postgraduate School Monterey, California 93940 Professor M. U. Thomas 10 Department of Operations Research and Administrative Sciences Naval Postgraduate School Monterey, California 93940 Professor C. F. Rowell, Code 5413 1 Department of Material Science and Chemistry Naval Postgraduate School Monterey, California 93940 LT R. D. Rantschler 10 c/o Department of Operations Research and Administrative Sciences Naval Postgraduate School Monterey, California 93940 Miss Beatrice Orleans, Code 0311 1 Naval Ship Systems Command Washington, D. C. 20350 UNCLASSIFIED 25 Si'. llMtV I 'I.ISSI til .it HMI DOCUMENT CONTROL DATA R&D >'•■ " ri " ■ ' " ''" .'fi.in .'I titlr. h,„l\ ,.l ., U-.lt,,, I „n,l inif.-xm,; ,»,n.,/;ifi,„i mu-.t hr utitrml iv/irr. >/n- i.v.r.,11 rrfi.trt I ,].,■■ if,.-./, Okigina iing A c T I VI 1 v ( i'i> r/inr/i/p <iu II, or) Naval Postgraduate School Monterey, California to. ML I' OUT 5LCUHITY ClA'.MI ic a lie Unclassified 2b. GMOUP Rl.FOR! riTLK A Mathematical Formulation for Selecting Holding-Tank-Processor Requirements for Shipboard Sewage Treatment 4 DESCRIPTIVE NOT ES (Type o( report and.inclusive dates) Technical Report ?> au THORlS) (First name, middle initial, last name) M. U. Thomas 6 REPOR T D A TE July 1972 7a. TOTAL NO OF PAGES 28 7b. NO OF BEFS CONTRACT OR GRANT NO. b. PROJ EC T NO 9a. ORIGINATOR'S REPORT NUMBER(S) 9b. OTHER REPORT NOIS) (Any other numbers that may be assigned this report) IC DISTRIBUTION STATEMENT Approved for public release; distribution unlimited, II SUPPLEMENTARY NOTES 12- SPONSORING Ml LI TAR Y ACTIVITY Naval Ship Systems Command 13 ABSTRACT This paper describes a formulation of the problem that systems designers face in selecting a combination of holding tank and processor for shipboard sewage treatment systems. Two decision models are discussed within this framework. In one case the generation of sewage, aboard ships, is assumed to consist of deterministic arrival streams. In a second model, sewage gen- eration is assumed to behave in accordance with a Poisson process Allowances for maintenance and reliability are discussed. DD , F r:.,1473 . (PAGE '» S/N 01 01 -807-681 1 UNCLASSIFIED Security Classification UNCLASSIFIED 26 Security Classification key wo ROi Sewage System Navy, Shipboard Sewage Processing Water Pollution Sewage Treatment Decision Analysis Ship Systems Design LINK A DD ,?r..1473 'back UNCLASSIFIED S/N 0101 -807-6821 Security Classification 31 DUDLEY KNOX LIBRARY ■ RESEARCH REPORTS 5 6853 01058039 2