SIMULA TION AS A RESOURCE RECOVERY
PLANT DESIGN TOOL
H. G. RIGO CSI Resource Systems, Inc.
Boston, Massachusetts
ABSTRACT
After a short general introduction to Monte Carlo simulation techniques and the mathematical formulation of necessary stochastic variables, various waste-to-energy plant configurations are studied using Monte Carlo simulation. A method for predicting long-tenn plant technical performance which considers the statistical nature of waste deliveries and plant operations is presented. The worked examples investigate trade-offs between capacity
and redundancy design parameters. The results can be combined with standard cost engineering techniques to quantify plant economic performance.
INTRODUCTION
Modem resource recovery plants are expensive
facilities to build and maintain. To minimize the cost of these facilities while continuing to provide environmentally sound and fiscally responsible waste management services, it is important to strike the proper balance between plant design capacity and waste stream characteristics.
Traditional engineering calculations for resource recovery plants are deterministic in nature and assume that the design criteria are adequately represented by average or peak processing requirements and properties. However, such static treatment of highly variable waste and processing characteristics can lead to inappropriately sized plants. Fortunately, a number of techniques can be employed to account for the variable nature of waste
characteristics and plant processing capabilities. This paper explores one type of analysis, Monte Carlo simulation. Monte Carlo techniques also allow designers to simulate the long-term perfoIlIlance of a particular plant configuration during the process engineering phase rather than after the plant is built.
Monte Carlo simulation * is performed by repeated analysis of a system during which the inputs and plant status are allowed to vary between each step according to known statistical distribution of the input variables. Repeated sampling using random numbers instead of dice or a roulette wheel to randomly select the value for every stochastic variable at each step of the simulation will produce predicted operating results also having a statistical distribution. After sufficient trials, the resulting plant operating estimates will characterize the plant's performance distribution.
When systems include components like surge bins and storage pits whose ability to deliver or receive waste is dependent on the entire plant history up to that point in time, analysis can indicate that all incoming waste will be processed easily. However, a series of high waste delivery days and/or concurrent forced outages is likely, making it necessary to bypass waste to the landfill.
·In this analysis, the plant status at the end of the pre
ceding time step becomes the initial condition for the cur
rent time step. Since the path followed to arrive at a par
ticular point of time has no bearing on the next estimate
of the stochastic variables, this is a Markov analysis. A more extensive discussion of this analysis can be found in the appendix.
459
When the stochastic variables are solid waste characteristics, furnace scheduled and forced outage rates, and refuse-derived fuel (RDF) processing line up and down time distributions, the effect of varying the plant's receiving, processing line, and boiler capacities can be studied. By analyzing a proposed system for its ability to process the delivered waste and the amount of time the furnaces will be operated, the technical parameters necessary to characterize a plant's output (considering uncertainty) can be predicted. Once the plant's output is determined, revenues from the processed waste and disposal costs for unprocessed waste and residue can be estimated. Conventional cost engineering techniques can be used to determine the net cost of a design alternative.
Such simulations provide a more firm basis for estimating system costs and revenues than do sensitivity analyses. In the case of waste processing plants, the input parameters all vary simultaneously along with plant performance and operating status. Sensitivity analysis varies only one parameter at a time. In a Monte Carlo simulation, the center of the response distribution is the system performance when all input variables are at tl;eir average values, but the range of results is much
\
narrower than what would result from estimates based on the extreme values of the input variables. This is because it is highly unlikely that every variable would be simultaneously at an extreme value. The Monte Carlo simulation recognizes both probability and value. The technique provides a practical means of analyzing very complex stochastic problems.
To provide a greater understanding of the Monte Carlo technique, two example problems are presented in the balance of this paper. Some of the information developed by the simulation is discussed.
A MODULAR COMBUSTION UNIT DESIGN PROBLEM
Displayed in Fig. 1 is a flowchart for a relatively simple waste-to-energy plant composed of a tipping floor with a fixed capacity and three or four identical modular combustion units (MCUs). Each MCU has its own waste heat boiler. For Simplicity, it is assumed that the plant operates 5 days/week, 24 hr/day. Each MCU is brought down on the weekend for routine maintenance and municipal solid waste (MSW) is delivered 5 days/week. A.5 percent forced outage rate, the average weekday waste de-
460
livery (x) with a standard deviation (s) equal to 25 percent of the average daily and Saturday deliveries equal to one quarter of the average daily delivery is specified. The effect on the receiving area and individual unit capacities will both be studied using Monte Carlo simulation techniques.
The simulation proceeded as follows. The first step was to estimate the amount of waste to be delivered to the plant using a random number and the mean and standard deviation for the day of the week as explained in the appendix. That estimated amount was then added to the waste already in the plant. Because most waste is delivered during a relatively short period each day, any tonnage in excess of remaining plant storage capacity was accumulated as bypassed waste to be sent to a landfill.
Next, the status of the plant (the number of units scheduled to be operating) and the actual number of units that were up, were determined by using a random number and a "look-up" table (discussed under Combustion Unit Availability in the appendix). The waste on the tipping floor was then reduced by the daily "up" processing capacity. If there was insufficient waste to operate the units at their rated capacity, all unit capacities were fust reduced to no less than 60 percent of their rating, an assumed turn-down ratio. If there was still insufficient waste, units were sequentially taken offline until only those units necessary to process the available waste were utilized. As units were down because of either scheduled or forced outage or lack of waste, unit-hours of down time were accumulated.
The waste bypassed from the plant was accumulated and divided by the total tons delivered to calculate the bypass ratio. The total number of unit-hours of down time were divided by the unithours simulated (number of units times hours simulated) to compute the estimated boiler down time ratio. The simulation was repeated day by day until the bypass and unit down time ratios stabilized.
SOLID WASTE x & s KNOWN
WEEKDAYS - 5(, s SATURDAY - 15(, Is SUNDAY - 0, °
RECEIVING
MCUs CAPACITY KNOWN
T CAPACITY 1-----1�-I KNOWN
, ' , ' , I L_ •• J
FIG. 1 PROCESS FLOW DIAGRAM FOR THE THREE OR FOUR MCU PROBLEM
'" '"
@ � co
if ->- <D '" -
.W .. t; -� '" -w
51 -' '"
� co
g <D a: a.
u. ..
0
2R '"
100 150
3-49 TPO UNITS Wi 500 T PIT
4-36 TPD UNITS Wi 500 T PIT
I I •
I I I I
I I I ,
I I I I
I .L4-49 TPO ,. " UNITS wI
" I 300 T PIT I I
J I I I
I "4-49 TPO UNITS ,/ / WI 500 T PIT
I I I ,
I / / /
, I -' / - /
� .. ; " /
200
T OF PROCESSIBLE WASTE DELIVERED PER WEEKDAY
FIG. 2 COMPARISON OF ALTERNATIVE PLANT
CONFIGURATIONS
Shown in Fig. 2 are the results of Markov-type Monte Carlo simulations for four different plant . configurations that use commercially available equipment. A plant with three 40-TPD * units behaves almost like a plant with four 36-TPD units with an equivalently sized pit or storage/receiving area. Increasing the size of the pit increases the average daily arrival rate at which waste begins to bypass the plant.
Figure 2 can be used to compare the relative merits of different plant configurations on both seasonally varying waste streams and the future requirements of growing waste sheds. A worked example problem is presented in Table 1, estimating the waste bypassed for a specific plant (4 by 36 TPD) and tipping floor (500 T) configuration and the equivalent receiving days per month. The equivalent working days are the sum of week days plus one quarter of the Saturdays to account for different weekday and weekend deliveries. The estimated bypass is 8.9 percent of the waste delivered. Similar calculations for other plant configurations result in an estimated 0.2 percent bypass for the 4 by 49 TPD case with a 500 T pit and a 1.1 percent bypass for the 4 by 49 TPD unit case with a 300 T pit. Hence, the number of units, their capacity, and the receiving area/storage capacity of the plant are all important variables to be optimized during system development.
The usefulness of the results of a few simulations can be greatly enhanced by non dimensionalizing results like those presented in Fig. 1. This can be accomplished by using the average weekly processing capacity (hours/week x TPH/unit x num-
• Because all inputs have been normalized to T of MSW
input, these results are equally valid if TPD is read as
either metric or English tons. A metric ton is approxi
mately 1.1 English ton.
461
TABLE 1 ESTIMATED ANNUAL BYPASS FOR A
4 BY 36 TPD MCU PLANT WITH A 500-T PIT
Equivalent Average Work Weekly •
Daysl Delivery Percent T T Month Months (TPW) Bypass Received Bypassed
Oct. 24 145 1.6 3,480 56
Nov. 23 143 1.0 3,289 33
Dec. 22% 143 1.0 3,182 32
Jan. 24 188 21.8 4,512 984
Feb. 22 181 18.5 3,982 737
March 22% 173 13.7 3,849 527
April 23 175 15.0 4,025 604
May 23% 158 5.3 3,674 195
June 22 164 8.5 3,608 307
July 23 158 5.3 3,634 193
August 22% 155 4.5 3,449 155
Sept. 23 150 3.0 3,450 104
44,133 3,924
Annual T Processed = 40,209
Annual Bypass = 8.9 percent
ber of units x availability) as the denominator, and by using variables which describe average week-ly MSW deliveries and receiving/storage area capac-ity as the numerator, as shown below.
_ average weekly MSW delivery '" = average weekly processing capacity
For: 0.5 < '" < 1.5
d - tons of pit capacity an 1/>-- average weekly finll plant capacity
For: 0.1 < I/> < 1.0, •
Using Eqs. (10) and (11) along with other simu-lations, Fig. 3 was developed for a plant with four installed combustion units scheduled to operate 5 days per week and receiving waste each weekday and 1/4 day on Saturday. Two major results of this simulation are presented in Fig. 3. First, the average unit down time is plotted. The percent of boiler down time becomes asymptotic to the unit's (and plant's) forced outage rate as waste delivery to the plant is increased. This is due to routine mainte
nance being scheduled during weekend shutdowns. Second, waste bypass is plotted. Waste bypasses the plant when the receiving area is full; this can be the result of either a series of high delivery days or a series of coincidental forced outages.
Curves like Fig. 3 can be developed for many general plant configurations. Each curve is applicable to specific numbers of units, availability and operating schedule assumptions, and a specific statistical deSCription of the waste arrival rate. Such
general curves can be very useful for initially screening alternatives. However, once a configuration has
been selected for further investigation, a site-specific simulation which uses local waste delivery characteristics and equipment-specific operating data is needed to provide an accurate estimate of long-term plant performance. These technical performance predictions are coupled with energy balances, cost engineering outputs, and locally determined bypassed waste disposal costs and product
o 0.25 0.50 lOO 1.25 l50 1'. AVG. WEEKLY DELIVERY/ AVG. WEEKLY
PROCESS CAPACIT Y
FIG.3 GENERALIZED FOUR UNIT MCU PERFORMANCE CURVE, FIVE DAY PER WEEK OPERATION
sales prices to establish the probable cost of disposal for a proposed plant. Of course, the fmal configuration will be governed by the engineer's judgment and local preferences for balancing plant costs with an acceptable degree of waste bypass.
AN RDF/DEDICATED BOILER DESIGN PROBLEM
A more complex problem is presented by an RDFjdedicated boiler plant. The plant displayed
PROCESSING LINE
SOLID WASTE
PROCESSING LINE
Solid Waste Delivery
5 J days/week 4 hours/day
i & 5 specified
Receivino Pit
F1x� tonnage specified • �ste is delivered - Was te is processed out
Process 1"9 Pi t
Fhed capacity specified (single line) Run time distribution Xo & b specified
Down time distribution xo' b specified
- Process at maxlm� capacity that does not: o exceed line rat 1ng o overflow sludge bin o exceed bol1er rating
Beilers
C\XfIu:ative probability spec.iried for 2. then I or more bol1e:-s ur
Run at maximlln capacity if line capacfty • boiler caoacity
If 1 ines + surge > 0.6 bo1 leI" up, run 'at 0.6 MCR
r�e)(t. shut down 1 line and check prey lous criteria
Reduce boiler to 0.6 HeR
Then shut down the las.t boile:".
FIG.4 GENERAL PROCESS ARRANGEMENT DIAGRAM FOR THE RDF PROBLEM
TABLE 2 TABULATED RESULTS OF THE RDF PLANT SIMULATION
Pit Surge Line MCR T T TPH TPH
5400 400 40 37.5 Variable Line Capacity problem
• � 30 50
Variable MCR problem
1 1 40
1 Variable Surge problem
• 1 • 1 000 Variable Pit problem 3600 400 t 7200 t
,
25 30 35 35 40 50
37.5
t
t
MSW xIs
1 964/490
,
1 t t
Variable MSW Delivery, Std. Deviation problem 5400 t t • 1969/245
t 1964/980 Variable average run time problem
t t t t 1969/490
Run Down Boiler Time Time Bypass Outage to/b-1 xo/b-1 Ratib Ratio
1 6.68/0.879 1 .1 1 9/1 .726 0.020±0.012 0.1 48±0.01 5
, , 0.1 85 0.1 20 0.005 0.1 03
1 1 0.273 0.043 0.1 47 0.045 0.016 0.148 0.056 0.159
0 0.194 0.003 0.1 69
+ t 0.01 1 0.1 57 0.005 0.227
• + 0.056 0.1 58 0.Q1 5 0.1 03
• + 0.021 0.141 0.075 0.170
33/0.879 t 0 0.131
462
in Fig. 4 was simulated, and the results are displayed in Table 2 and Fig. 5-1 0.
In this example, waste is received at the plant 4 hr/day, 5% days per week. When the receiving pit is full, waste is bypassed to a sanitary landfill. This waste is processed out of storage by two RDF lines running 24 hours per day, 7 days per week. * Whenever a line is available, waste is processed at a rate limited by the amount of waste available, the capacity remaining in intermediate storage, or line capacity. The waste is then removed from intermediate storage for use in two boilers whose availabilities are governed by forced outage rates. **
Plotted in Fig. 5 are the results from seven repeated 2,000 hour simulations of the base case [5,400 T pit, 400 T surge bin, 40 TPD individual line capacity, 37.5 TPH MSW equivalent individual boiler capacity, MSW receiving average (x) = 1 964 TPD and standard deviation(s) = 490 TPD 5 days per week and x = 491 TPD and s = 1 22 TPD on Saturday; the processing line statistical characteristics are described by the 2 parameter Weibul coefficients discussed in the appendix: to = 1 6.68 and l/b = 0.879 (up time) and to = 1 .11 9 and l/b = 1.726 (down time); forced outage rate (FOR) = 0.04]. The base case was selected because it results in some bypass and some boiler down time. Thus, it is not an optimum situation. The shaded area in Fig. 5 contains the 90 percent confidence interval for the true value of the bypass ratio and boiler down time - it is not an interval that contains 90 percent of the data. This ellipse is repeated on all subsequent figures to provide an indication of whether or not a specific result is likely to be different from the base case.
Shown in Fig. 6 is the effect of changing the boiler rating on the bypass ratio and the boiler down time ratio. At a combined fum weekly boiler capacity which is less than the average we�kly delivery, the boilers operate during all of the time available and waste is b�ing bypassed because the boilers cannot burn the material. As average boiler
·Thls somewhat artificial constraint can be relaxed; however, the Impacts of some design alternatives are not as easy to see in a plant with standard two-shift, 5 day/week processing.
··Insurance Inspections and annual turn-around require
ments Indicate one 2-week and one 1-week outage per boiler. Separate simulations, rather than programming the
outage schedule Into the simulation, are easier because re
peated runs with the program Indicate that at least 2,000 simulated hours are required to develop representative
predictions without considering the annual turn-arounds. The Inclusion of scheduled outages would extend the required simulation over 100,000 cycle simulations, a pro
hibitive simulation time.
463
"1 ��
'!1 en en � >-m :!: sa en :::<
*-.,.,
0
Ll) '"
en Ll) en -� fu :s: en 0 :::< -
�
Ll)
0
:� •
5 10 15 20 25 % BOILER DOWN TIME
FIG.5 RELATED RUNS FOR BASE CASE
.25 TPH
.30 TPH
35 TPH �37.5 TPH 50 TPH 4 TPH 5 10 15 20 25
% BOILER DOWN TIME
FIG.6 VARIABLE BOILER MCR
available capacity approaches the average weekly delivery, bypass and down time are similar to those of the base case. As boiler capacity is increased, the amount of waste bypassed approaches zero, but the frequency with which units are taken off-line begins to increase rapidly. Apparently, there is a hyperbolic relationship between boiler down time (which jeopardizes unit reliability and life, while increasing plant cost) and increasing waste bypass (which uses more landfill while reducing product revenues).
'" '"
�
C/) C/) 'f1 � >-CD 3: C/) ::;; Q *
'"
o 5 10
.30 TPH
�40TPH
15 20
% BOILER DOWN TIME
FIG.7 VARIABLE LINE CAPACITY
25
and availability characteristics, it appears that boiler down time and bypass can both be significantly affected by RDF line capacity. This is a comparatively inexpensive part of a plant to enlarge and a prime candidate for optimization.
As shown in Fig. 8, for the conditions under study, plant performance is relatively insensitive to surge capacity. This apparently counter-intuitive finding is a result of the 24 hr/day operating schedule for the two front-end processing lines and the fact that one base case RDF processing line has sufficient capacity to keep both boilers operating at 60 percent of their maximum continuous rating. For other configurations and operating schedules, the amount of surge capacity is expected to be a significant design variable.
The effects of varying the receiving pit's size on plant performance are shown in Fig. 9. With both boilers scheduled up, a 7,200 T pit reduces bypass and down time, while a 3,600 T pit increases these characteristic parameters. During I-week outages, the larger the pit, the less by bypass. Obviously, for a 4-week scheduled outage, the pit would have to be able to hold at least 2 week's worth of accumulated waste as well as buffer daily surges unless there was a significant amount of reserve capacity in the boilers. The total scheduled outage time is such a small fraction of the total operating year that the infrequency of occurrences mitigates against the almost certain 50 percent bypass the last week of each 2-week outage. Hence, careful
The effect of variable line capacity is shown in Fig. 7. Under the. assumed plant operating schedule
464
o
� 400T
1000T.�O T
% BOILER DOWN TIME
FIG.8 VARIABLE SURGE
1:>5400 T
D 7200 T
7200 T ..
5 10
• NORMAL RUN TIME D ONE BOILER SCHEDULED
DOWN
.3600 T
� 5400T
15 20
% BOILER DOWN TIME
FIG. 9 VARIABLE PIT
25
planning of scheduled boiler down time throughout the year to minimize sending waste to the landfill during long scheduled outages is indicated.
The effect of changes in waste delivery on plant performance is shown in Fig. 1 0. Clearly, local waste delivery variability can have a Significant impact on plant performance. The careful determination of this parameter is warranted.
til N
til
o
50%�� 25%
10 15 % BOILER DOWN TIME
, .
20 25
FIG. 10 VARIABLE DELIVERY COEFFICIENT OF
VARIANCE
Unlike the simpler pit/MCU plant case, the number of physical variables is too great in RDF /dedicated boiler plants to permit the generation of simple trade-off curves. However, this analysis has indicated the direction of changes to be made in major system parameters in order to develop plant designs which balance waste bypass, boiler down time, and system economics. A system simulation routine can be coupled with good engineering judgment and some "experimental design" procedures, such as response surface methodologies (Box, Hunter and Hunter, 1980), to lead rapidly to an appropriate solution to a local community's waste disposal problem.
CONCLUSIONS
Monte Carlo simulation can be used to study the long-term appropriateness of a plant design and to trade off investment alternatives while preparing fmal system configurations. Future applications of Monte Carlo simulation include the statistical sizing of processing components and conveyors, including variations in mass flow, waste composition, and material density .
The methodologies employed in this paper are well known in reliability, traffic, and utility load engineering circles. They also seem to be emerging in power plant design practice, but are novel in waste-to-energy plant design. As a result, this paper relied heavily on sparce data in order to begin to use Monte Carlo simulation as a tool for waste-toenergy plant configuration evaluation.
465
The author is convinced that the tool is powerful and will enable designers to perform better as more relevant data are collected and analyzed. The data collection should include not only additional plant and component performance information, but also information on the statistiCal nature of
•
waste. Because plant operators frequently consider their facility performance data proprietary and many communities do not weigh their waste, establishing the requisite data base may be difficult and not available in the immediate future. However, the simulation results are certainly informative and indicate that such a data collection effort is warranted.
APPENDIX:- FORMULATION OF A MARKOV MODEL
STATISTICAL DESCRIPTION OF MAJOR PLANT VARIABLES
When developing a Markov model of a resource recovery plant, a statistical description of stochastic variables and the availability of a pseudo-random number generator are essential. The latter is employed to create a set of random numbers between o and 1. These random numbers can then be used in conjunction with known parameter statistics to develop a random series of values which have the same statistical distribution as the parent population for the stochastic variables. Repeated engineering analysis of the system using the randomly selected stochastic variable values will yield a set of results. These results defme the statistical distribution of plant performance. Using the output distribution, the engineer can decide whether the chance or risk of some outcome is acceptable and alter his design as appropriate.
STOCHASTIC VARIABLES
WASTE DELIVERY
All of the solid waste delivery records the author has used over the last 10 years can be characterized by normal distribution. Therefore, an adequate representation of Monday through Friday deliveries can be made if the average and standard deviation for the daily tonnage deliveries are known. Usually, the coefficient of variance for weekday deliveries (the ratio of standard deviation to average delivery) is the same as the coefficient of variation for Saturday deliveries. Saturday deliveries, however, are usually some fraction less than one of the weekday
deliveries. Additional subtleties can be introduced here by allowing the average daily delivery to vary throughout the simulation in order to represent seasonal fluctuations in waste collection.
The amount of waste delivered on any given day is determined by first generating a random number (r) and then computing the associated normal deviate, or z-score, as follows (Abramowitz and Stegur, 1 970):
(1 )
1R if 0 < r " 0.5
rl t =
jin (1 � r)l
(2) if 0.5 < r < 1
Co = 2.51551 7 d1 = 1.432788
Cl = 0.802853 d2 = 0.1 89269
C2 = 0.01 0328 d3 = 0.001308
if 0 < r" 0.5 with error Then x = , y
( -y if 0.5 < r < I Ic(r)1 < 4.5 X---,1 0-4 (3)
The amount of waste delivered on a given day is then:
Delivery = F. [x + ZS]
where: F is 1 for weekdays
< 1 for Saturdays
o for Sundays
x is the "average" daily delivery such that
(5 + FSAT) x = average TPW
S is the standard deviation of daily deliveries
RDF PRODUCTION LINE AVAILABI LlTY
The status of RDF processing lines can be estimated once the characteristic distributions of the processing line's up and down times are known. 'The author's analysis of a limited number of processing line operating and maintenance duration records indicates that these mechanical systems can be described by the two-parameter Weibul distribution. This distribution, which is used extensively in machinery failure analysis (Lipson and Sheth, 1973), is:
466
(5)
where: P (r) is the probability associated with time t
to is the center describing statistic
b is the slope describing statistic
O<P(r)<1
The distribution can be inverted and the up- or down time estimated by:
1 t=to[l n( l-r)]b (6)
The status of an operating line can be continuously determined by calculating an operating period and the line to run for t hours and then computing a down time and keeping the line unavailable during repairs. Of course, if a line is undergoing repair and not scheduled to run, repair would continue so that it might be available for operation at the next scheduled run.
COMBUSTION UNIT AVAILABILITY
Assuming that a maintenance schedule is available for a system's boilers, the forced outage rates of individual units and a· system unit operating schedule can be developed. Estimates of overall system availability and the recommended preventative maintenance schedule can be estimated from the literature on each system. Using this information, the forced outage rate can be estimated as follows:
A = (1 - FOR) (I - SOR) (7)
where: FOR is the forced outage rate (hr down/hr scheduled to run),
SOR is the scheduled outage rate (hr scheduled down/hr per year),
and, A is the overall availability.
Thus, FOR = 1 -A/(1 - SOR) (8)
Now, the probability profUe for the number of units which are actually up can be estimated by considering the number of units scheduled to be operational and the probability profUe for that number of units. A convenient method for tracking the plant's status is to develop operating probability profUes for all combinations of operating units. A scheduled outage clock or timer can then be introduced into the computer code to track the
deliveries. Additional subtleties can be introduced here by allowing the average daily delivery to vary throughout the simulation in order to represent seasonal fluctuations in waste collection.
The amount of waste delivered on any given day is determined by first generating a random number (r) and then computing the associated normal deviate, or z-score, as follows (Abramowitz and Stegur, 1 970):
�A t = l jln
( l �r)Z
Co = 2.51 551 7
Cl = 0.802853
Cz = 0.01 0328
if 0 < r .;; 0.5
if 0.5 < r < 1
d1 = 1 .432788
dz = 0.1 89269
d3 = 0.001 308
Th
1 y if 0 < r .;; 0.5 with error en x =
_y if 0.5 < r < 1 le(r)1 < 4.5 x
(1 )
(2)
1 0-4 (3)
The amount of waste delivered on a given day is then:
Delivery = F. [x + ZS]
where: F is 1 for weekdays
< 1 for Saturdays
o for Sundays
x is the "average" daily delivery such that
Per) = (5)
where: P (r) is the probability associated with time t
to is the center describing statistic
b is the slope describing statistic
O.;;P(r).;; 1
The distribution can be inverted and the up- or down time estimated by:
1 _ ]b t - to [In ( l-r) (6)
The status of an operating line can be continuously determined by calculating an operating period and the line to run for t hours and then computing a down time and keeping the line unavailable during repairs. Of course, if a line is undergoing repair and not scheduled to run, repair would continue s"o that it might be available for operation at the next scheduled run.
COMBUSTION UNIT AVAILABILITY
Assuming that a maintenance schedule is available for a system's boilers, the forced outage rates of individual units and a system unit operating schedule can be developed. Estimates of overall system availability and the recommended preventative maintenance schedule can be estimated from the literature on each system. Using this information, the forced outage rate can be estimated as follows:
A = (1 - FOR) (1 - SOR)
where: FOR is the forced outage rate (hr down/hr scheduled to run),
(7)
(5 + FSAT) x = average TPW SOR is the scheduled outage rate
S is the standard deviation of daily deliveries (hr scheduled down/hr per year),
RDF PRODUCTION LINE AVAILABILITY
The status of RDF processing lines can be estimated once the characteristic distributions of the processing line's up and down times are known. The author's analysis of a limited number of processing line operating and maintenance duration records indicates that these mechanical systems can be described by the two-parameter Weibul distribution. This distribution, which is used extensively in machinery failure analysis (Lipson and Sheth, 1 973), is:
467
and, A is the overall availability.
Thus, FOR = 1 - A/(l - SOR) (8)
Now, the probability profile for the number of units which are actually up can be estimated by considering the number of units scheduled to be operational and the probability profile for that number of units. A convenient method for tracking the plant's status is to develop operating probability profiles for all combinations of operating
units. A scheduled outage clock or timer can then be introduced into the computer code to track the
number of units which are scheduled up and the duration of scheduled repair time. By oscillating between all units scheduled up and one unit scheduled down conditions, short duration routine maintenance outage can be incorporated explicitly into the computer code.
The mathematical formulation for the probability of a given number of units being available for operation is as follows:
P = N! (1- FOR)P (FOR) N -P (9)
(N-p)!p!
where: N is the number. of units installed,
p is the number of units up, and
! is the factional (e.g. 3! = 1 x 2x 3)
Consider the situation of two boilers and a forced outage rate of 4 percent. One combination of boilers and outages gives two boilers up, two combinations yield one boiler up, and one combination results in no boilers up. As shown in tabular form:
Cumula-State tive
No. Combina- Individual Proba- Proba-Up tions Probability bility bility
2 X ( 1 -0.04)' (0'.04)° = 0.921 6 0.921 6 2 X ( 1 -0.04)' (0.04)' = 0.0768 0.9984
0 X ( 1 -0.04)' (0.04)' = 0.001 6 1.0000
1 .0000
Mter generating a random number, r, the cumulative probability column can be used as a "look-up" table to determine the number of boilers that are up. For example, r = 0.4 indicates that 2 units are up but r = 0.96 indicates that only 1 unit is available for operation.
Based on published reports and experience, it appears that a satisfactory simulation can be obtained by assuming that the duration of a forced
outage that Significantly reduces unit capacity or requires an outage for repair will be 24 hr. Thus, at the beginning of each simulated day, the operating status of the boilers must be determined.
COMPLETING THE MODEL
Procedures similar to those described above can be used to characterize any other stochastic variable. These variables include the operating status of turbine generators and the amount of process steam demanded by a customer.
Once the values of the random variables have been estimated for a single simulation, deterministic engineering equations are used to estimate plant performance for that hour (RDF line) or day (massburning facility) and to alter the amount of waste and fuel retained in the plant at the end of the cycle. The cycle is then repeated using new random numbers and estimates of random variables until sufficient time has been simulated to yield stable results.
This repeated calculation is best performed by computer, as 1,000 to 2,000 cycles are usually required to produce stable results for the systems studied by the author.
BI BLiOG RAPHY
1. Abramowitz and Stegun, Handbook of Mathemati
cal Functions, National Bureau of Standards, 1970. .2. Box, G., Hunter, W. G., and Hunter, J. S., Statistics
for Experimentation, John Wiley, New York, 1 978. 3. Dhillon, B. S. and Singh, C., Engineering Reliability:
New Techniques and Applications, John Wiley, New York, 1 98 1 .
4. Lipson, C. and Sheth, N. J., Statistical Design and
Analysis of Engineering Experiments, McGraw Hill, New york,1 973.
5. Rubinstein, R. Y., Simulation and the Monte Carlo
Method, John Wiley, New York, 1 98 1 .
Key Words Mass Burning Boi)er Process Refuse Derived Fuel Simulation System
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