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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 

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and Administrative Sciences 
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of Naval Laboratories 
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Washington, D. C. 20350 

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USS Springfield (CLG-7) 
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Professor E. A. Brill 1 

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Professor M. U. Thomas 10 

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Department of Material Science and Chemistry 
Naval Postgraduate School 
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LT R. D. Rantschler 10 

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and Administrative Sciences 
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Rl.FOR! riTLK 



A Mathematical Formulation for Selecting Holding-Tank-Processor Requirements for 
Shipboard Sewage Treatment 



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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 



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key wo ROi 



Sewage System 

Navy, Shipboard Sewage Processing 

Water Pollution 

Sewage Treatment 

Decision Analysis 

Ship Systems Design 



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S/N 0101 -807-6821 



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DUDLEY KNOX LIBRARY ■ RESEARCH REPORTS 



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