Part I Characteristics of Water and Wastewater Before any water or wastewater can be treated, it must first be characterized. Thus, characterization needs to be addressed. Waters and wastewaters may be characterized according to their quantities and according to their constituent physical, chemical, and microbiological characteristics. Therefore, Part I is composed of two chapters: “Quantity of Water and Wastewater,” and “Constituents of Water and Wastewater.” Tx249_Frame_C01.fm Page 73 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero Quantity of Water and Wastewater Related to and integral with the discussion on quantity are the important knowledge and background on the types of wastewater, sources of water and wastewater, and methods of population projection. The various categories of quantities in the form of design flow rates are also very important. These topics are discussed in this chapter. Because of various factors that have influenced the rate of wastewater generation in recent times, including water conservation and the expanded use of onsite systems, it is critical that designers have more than just typical wastewater generation statistics to project future usage. Thus, a method of determining accurate design flow rates calculated through use of probability concepts are also discussed in this chapter. This method is called probability distribution analysis ; it is used in the determination of the quantities of water and wastewater, so it will be discussed first. 1.1 PROBABILITY DISTRIBUTION ANALYSIS Figure 1.1 shows a typical daily variation for municipal sewage, indicating two maxima and two minima during the day. Discharge flows of industrial wastewaters will also show variability; they are, in general, extremely variable and “explosive” in nature, however. They can show variation by the hour, day, or even by the minute. Despite these seemingly uncorrelated variability of flows from municipal and indus- trial wastewaters, some form of pattern will emerge. For municipal wastewaters, these patterns are well-behaved. For industrial discharges, these patterns are constituted with erratic behavior, but they are patterns nonetheless and are amenable to analysis. Observe Figure 1.2. This figure definitely shows some form of pattern, but is not of such a character that meaningful values can be obtained directly for design purposes. If enough data of this pattern is available, however, they may be subjected to a statistical analysis to predict design values, or probability distribution analysis, which uses the tools of probability. Only two rules of probability apply to our present problem: the addition rule and the multiplication rule. 1.1.1 A DDITION AND M ULTIPLICATION R ULES OF P ROBABILITY Before proceeding with the discussion of these rules, we must define the terms events, favorable event, and events not favorable to another event. An event is an occurrence, or a happening. For example, consider Figure 1.3, which defines Z as “Going from A to B .” As shown, if the traveler goes through path E , he or she arrives at the destination point B . The arrival at B is an event. The travel through path E that causes event Z to occur is also an event. 1 Tx249_Frame_C01.fm Page 75 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero FIGURE 1.1 A typical variation of sewage flow. FIGURE 1.2 A three-day variation of sewage flow. FIGURE 1.3 Definition of event Z as “Going from A to B .” 70 60 50 40 30 20 10 0 Cubic meters per hour 12 4 8 12 4 8 12 Midnight Noon Midnight Average Flow 70 60 50 40 30 20 10 0 Midnight Midnight Midnight Midnight First day Second day Third day Average flow Cubic meters per hour A B D J K C E F G H I Tx249_Frame_C01.fm Page 76 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero 77 The path through E is an event or happening favorable to the occurrence of event Z . The other paths that a traveler could take to reach B are C , F , G , H , and I . Thus, the occurrence of any of these event paths will cause the occurrence of event Z ; the occurrence of Z does not, however, mean that all of the event paths E , C , F , G , H , and I have occurred, but that at least one of them has occurred. These events are all said to be favorable to the occurrence of event Z . The paths D , J , and K are events unrelated to Z ; if the traveller chooses these paths she or he would never reach the destination point B . The events are not favorable to the occurrence of Z . All the events both favorable and not favorable to the occurrence of a given event, such as Z , constitute an event space of a particular domain. This particular domain space is called a probability space . Addition rule of probability . Now, what is the probability that one event or the other will occur? The answer is best illustrated with the help of the Venn diagram , an example of which is shown in Figure 1.4, for the events A and B . There is D , which contains events from A and B ; it is called the intersection of A and B , designated as A ʝ B . This intersection means that D has events or results coming from both A and B . C has all its events coming from A , while E has all its events coming from B . The sum of the events in A and B constitutes the union of A and B. This is written as A ʜ B . From the figure, A ʜ B = A + B − D = A + B − A ʝ B (1.1) where the subtraction comes from the fact that when A and B “unite,” they each contribute to the events at the intersection part of the union ( D ). This part counted the intersection events twice; thus, the other “half” must be subtracted. The union of A and B is the occurrence of the event: event A or event B has occurred—not event A and event B have occurred. The event, event A and event B , have occurred is the intersection mentioned previously. From Equation (1.1), the probability that one event or the other will occur can now be answered. Specifically, what is the probability that the one event A or the other FIGURE 1.4 A Venn diagram for the union and intersection. A B C D E Tx249_Frame_C01.fm Page 77 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero event B will occur? Because the right side of the previous equation is equal to the left side, the probability of the right side must be equal to the probability of the left side. Or, Prob(A ʜ B) = Prob(A) + Prob(B) − Prob(A ʝ B) (1.2) where Prob stands for probability. Equation (1.2) is the probability that one event or the other will occur—the addition rule of probability. Multiplication rule of probability. The intersection of A and B means that it contains events favorable to A as well as events favorable to B. Let the number of these events be designated as N(A ʝ B). Also, let the number of events of B be designated as N(B). Then the expression (1.3) is the probability of the intersection with respect to the event B. In the previous formula, event is synonymous with unit event. Unit events are also called outcomes. Probability values are referred to the total number of unit events or outcomes in the probability space, which would be the denominator of the above equation. As shown, however, the denominator of the above probability is referred to N(B). N(B) is smaller than the total number of unit events in the domain space; thus, it is called a reduced probability space. Because the reference probability space is that of B and because N(A ʝ B) is equal to the number of unit events of A in the intersection, the previous equation is called the conditional probability of A with respect to B designated as , or (1.4) Let ζ designate the total number of unit events in the domain probability space in which event A is a part as well as event B is a part. Divide the numerator and the denominator of the above equation by ζ . Thus, (1.5) The numerator of the previous equation is the intersection probability Prob(A ʝ B) and the denominator is Prob(B). Substituting and performing the algebra, the fol- lowing equation is produced: (1.6) Equation (1.6) is the multiplication rule of probability. If the reduced space is referred to A, then the intersection probability would be (1.7) In the multiplication rule, if one event precludes the occurrence of the other, the intersection does not exist and the probability is zero. These events are mutually exclusive. NA ʝ B() NB() Prob( AB) Prob( AB) NA ʝ B() NB() = Prob( AB) NA ʝ B()/ ζ NB()/ ζ = Prob A ʝ B()Prob A B()Prob B()= Prob A ʝ B()Prob B A()Prob A()= Tx249_Frame_C01.fm Page 78 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero If the occurrence of one event does not affect the occurrence of the other, the events are independent of each other. They are independent, so their probabilities are also independent of each other and Prob( ) becomes Prob(A) and Prob( ) becomes Prob(B). Using the intersection probabilities, the addition rule of probability becomes (1.8) 1.1.2 VALUES EQUALED OR EXCEEDED One of the values that is often determined is the value equaled or exceeded. The probability of a value equaled or exceeded may be calculated by the application of the addition rule of probability. The phrase “equaled or exceeded” denotes an element equaling a value and elements exceeding the value. Therefore, the probability that a value is equaled or exceeded is by the addition rule, (1.9) But there are 1, 2, 3,… ψ of the elements exceeding the value. Also, for mutually exclusive events, the intersection probability is equal to zero. Thus, (1.10) Substituting Equation (1.10) into Equation (1.9), assuming mutually exclusive events, (1.11) 1.1.3 DERIVATION OF PROBABILITY FROM RECORDED OBSERVATION In principle, to determine the probability of occurrence of a certain event, the experiment to determine the total number of unit events or outcomes for the prob- ability space should be performed. Then the probability of occurrence of the event is equal to the number of unit events favorable to the event divided by the total possible number of unit events. If the number of unit events favorable to the given event is η and the total possible number of unit events in the probability space is ζ , the probability of the event, Prob(E), is (1.12) AB AB Prob A ʜ B()Prob A() Prob B() Prob A B()Prob B()–+= Prob value equaled or exceeded() Prob value equaled()Prob value exceeded()+= Prob value equaled ʝ value exceeded()– Prob value exceeded() Prob value1 exceeding()Prob value2 exceeding()+= … Prob value ψ exceeding()++ Prob value equaled or exceeded() Prob value equaled()Prob value1 exceeding()+= Prob value2 exceeding() … Prob value ψ exceeding()+++ Prob E() η ζ = Tx249_Frame_C01.fm Page 79 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero In practical situations, either the determination of the total number of ζ s is very costly or the total number is just not available. Assume that the available ζ is ζ avail , then the approximate probability, Prob(E) approx , is (1.13) The use of the previous equation, however, may result in fallacy, especially if ζ avail is very small. η can become equal to ζ total making the probability equal to 1 and claiming that the event is certain to occur. Of course, the event is not certain to occur, that is the reason why we are using probability. What can be claimed with correctness is that there is a high degree of probability that the event will occur (or a high degree of probability that the event will not occur). Because there is no absolute certainty, in practice, a correction of 1 is applied to the denominator of Equation (1.13) resulting in (1.14) Take note that for large values of ζ avail the correction 1 in the denominator becomes negligible. To apply Equation (1.14), recorded data are arranged into arrays either from the highest to the lowest or from the lowest to the highest. The number of values above a given element and including the element is counted and the probability equation applied to each individual element of the array. Because the number of values above and at a particular element is a sum, this application of the equation is, in effect, an application of the probability of the union of events. The probability is called cumulative, or union probability. After all union probabilities are calculated, an array of probability distribution results. This method is therefore called probability distri- bution analysis. This method will be illustrated in the next example. Example 1.1 In a facility plan survey, data for Sewer A were obtained as follows: (a) What is the probability that the flow is 3700 m 3 /wk? (b) What is the probability that the flow is equal to or greater than 3700 m 3 /wk? (c) What is the flow that will never be exceeded? Week No. Flow (m 3 /wk) Week No. Flow (m 3 /wk) 1 2900 8 4020 2 3028 9 3675 3 3540 10 3785 4 3300 11 3459 5 3700 12 3200 6 4000 13 3180 7 3135 14 3644 Prob E() approx η ζ avail = Prob E() approx η ζ avail 1+ = Tx249_Frame_C01.fm Page 80 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero Solution: (a) = = = 0.07 Ans (b) From the problem, the values greater than 3700 are 3785, 4000, and 4020. Thus, The problem may also be solved by arranging the data into an array. Because the problem is asking for the probability that is “equal or greater,” arrange the data in descending order. This is the probability distribution analysis. The analysis is indicated in the following table. The values under the column cumulative represent the total number of values above and including the element at a given serial number. For example, consider the serial no. 4. The flow rate at this serial number is indicated as 3700 m 3 /wk and, under the column of , the value is 4. This 4 represents the sum of the number of values equal to or greater than 3700; these values being 3700, 3785, 4000, and 4020. The values 3785, 4000, and 4020 are the values above the element 3700 which numbers 3. Adding 3 to the count of element 3700, itself, which is 1, gives 4, the number under the column . The column = is the cumulative probability of the item at a given serial number. For example, the element 3700 has a cumulative probability of 0.27. This cumulative probability is the same Prob(value equaled or exceeded) = Prob(value equaled) + Prob(value1 exceeding) + … used previously. Serial No. Flow (m 3 /wk) 1 4020 1 2 4000 2 3 3785 3 4 3700 4 5 3675 5 6 3644 6 7 3540 7 8 3459 8 (continued) Prob(3700) Prob(E) approx = η ζ avail 1+ 1 14 1+ Prob value equaled or exceeded() = Prob value equaled()Prob value1 exceeding() … ++ Prob flow 3700≥() = Prob 3700()Prob 3785()Prob 4000()Prob 4020()+++ 1 15 1 15 1 15 1 15 +++ 0.27 Ans== η ∑ η = ∑ η ∑ η ∑ η /(ζ avail 1)+∑ η /(14 1)+ ∑∑ ∑∑ ηη ηη ∑∑ ∑∑ ηη ηη ζζ ζζ total 1++ ++ 1 14 1+ 0.07= 2 14 1+ 0.13= 3 14 1+ 0.20= 4 14 1+ 0.27= 5 14 1+ 0.33= 6 14 1+ 0.40= 7 14 1+ 0.47= 8 14 1+ 0.53= Tx249_Frame_C01.fm Page 81 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero (c) The value that will never be exceeded is the largest value. Thus, there will be no value above it and the cumulative count for this element is 1. The data for flows in the previous table are simply for values obtained from a field survey. To answer the question of what is the value that will never be exceeded, we have to obtain this value from an exhaustive length of record and read the value that is never exceeded on that particular record. Of course, the count for this largest value would be 1, as mentioned. Now, what has the field data to do with the determination of the largest value? The use of the field data is to develop a probability distribution. The resulting distri- bution is then assumed to model the probability distribution of all the possible data obtainable from the problem domain. The larger the number of data and the more representative they are, the more accurate this model will be. Obtaining the largest value means that the amount of data used to obtain the probability distribution model must be infinitely large; and, in this infinitely large amount of data, there is only one value that is equaled or exceeded. This means that the probability of this one value is 1/infinity = 0. From the probability distribution, the peak weekly flow rate can be extrapolated at probability 0. This is done as follows (with x representing the weekly flow rate): Therefore, 1.1.4 VALUES EQUALED OR NOT EXCEEDED The probability of values equaled or not exceeded is just the reverse of values equaled or exceeded. In the previous example, the values were arranged in descending order. For the case of value equaled or not exceeded, the values are arranged in ascending order. Serial No. Flow (m 3 /wk) 9 3300 9 10 3200 10 11 3180 11 12 3135 12 13 3028 13 14 2900 14 x 0 4020 0.07 4000 0.13 ∑∑ ∑∑ ηη ηη ∑∑ ∑∑ ηη ηη ζζ ζζ total 1++ ++ 9 14 1+ 0.60= 10 14 1+ 0.67= 11 14 1+ 0.73= 12 14 1+ 0.80= 13 14 1+ 0.87= 14 14 1+ 0.93= x 4020– 4020 4000– 0 0.07– 0.07 0.13– x 4043 m 3 /wk Ans== Tx249_Frame_C01.fm Page 82 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero Deducing from Equation (1.11), the probability of a value equaled or not exceeded is (1.15) Example 1.2 In Example 1.1, what is the probability that the flow is equal to or less than 2800? 3000? Solution: The probability distribution arranged in descending order is as follows: Example 1.3 In Example 1.1, calculate the probability that the flow is equal to less than 3700 m 3 /wk. Serial No. Flow (m 3 /wk) ∑∑ ∑∑ ηη ηη 1 2900 1 2 3028 2 0.13 3 3135 3 0.2 4 3180 4 5 3200 5 6 3300 6 7 3459 7 8 3540 8 9 3644 9 10 3675 10 11 3700 11 12 3785 12 13 4000 13 14 4020 14 2800 x 2900 0.07 3000 y 3028 0.13 Prob value equaled or not exceeded() Prob value equaled()Prob value1 not exceeding()+= + Prob value2 not exceeding() … + ∑∑ ∑∑ ηη ηη ζζ ζζ total 1++ ++ ∑∑ ∑∑ ηη ηη 14 1++ ++ == == 1 14 1+ 0.07= 4 14 1+ 0.27= 11 14 1+ 0.73= x 0.07– 0.07 0.13– 2800 2900– 2900 3028– = y 0.07– 0.13 0.07– 3000 2900– 3028 2900– = x 0.023 Prob flow 2800≤()Ans== y 0.12 Prob flow 3000≤()Ans== Tx249_Frame_C01.fm Page 83 Friday, June 14, 2002 1:49 PM © 2003 by A. P. Sincero and G. A. Sincero [...]... Quantity of water has already been addressed in the previous treatments, so it is now time to address quantity of wastewaters Before discussing quantity of wastewaters, however, it is important that the various types of wastewaters be discussed first The two general types of wastewaters are sanitary and non-sanitary The non-sanitary wastewaters are normally industrial wastewaters Sanitary wastewaters... chapter: the quantity of water and the quantity of wastewater The quantity of water is discussed first To design water treatment units, the engineer, among other things, may need to know the average flow, the maximum daily flow, and maximum hourly flow The following information are examples of the use of the design flows: 1 Community water supplies, water intakes, wells, treatment plants, pumping, and transmission... rains and slow entries of water from ponded areas into openings of manholes Infiltration and inflows are collectively called infiltration-inflow 1.4 SOURCES AND QUANTITIES OF WASTEWATER The types of wastewaters mentioned above come from various sources Sanitary wastewaters may come from residential, commercial, institutional, and recreational areas Infiltration-inflow, of course, comes from rainfall and groundwater,... groundwater, and industrial wastewaters come from manufacturing industries The quantities of these wastewaters as they come from various sources are varied and, sometimes, one portion of the literature would report a value for a quantity of a parameter that conflicts on information of the quantity of the same parameter reported in another portion of the literature For example, many designers often assume... for, some form of projection must be made For industrial facilities, production may need to be projected into the future, since the use of water and the production of wastewater are directly related to industrial production Design of recreational facilities, resort communities, commercial establishments, and the like all need some form of projection of the quantities of water use and wastewater produced... shown by the dashed line and is approximately 24,800 1.6 DERIVATION OF DESIGN FLOWS OF WASTEWATERS The accurate determination of wastewater flows is the first fundamental step in the design of wastewater facilities To ensure proper design, accurate and reliable data must be available This entails proper selection of design period, accurate population projection, and the determination of the various flow rates... because a myriad of manufacturing processes are used, a myriad of industrial wastewaters are also produced Sanitary wastewaters produced in industries may be called industrial sanitary wastewaters To these wastewaters may also be added infiltration and inflow Wastewaters are conveyed through sewers Various incidental flows can be mixed with them as they flow For example, infiltration refers to the water that... facility is smaller, and it gets bigger as it is being expanded during the staging period corresponding to the increase in population until finally reaching the end of the planning period Table 1.1 shows staging periods for expansion of water and wastewater plants, and Table 1.2 shows design periods for various water supply and sewerage components Tables 1.3 through 1.6 show average rates of water use for... particulars of the community; however, parallel concerns should also be directed to the determination of industrial wastewaters and other types of wastewaters 1.6.1 DESIGN FLOWS Different types of flow rates are used in design of wastewater facilities: average daily flow rate, maximum daily flow rate, peak hourly flow rate, minimum daily flow rate, minimum hourly flow rate, sustained high flow rate, and sustained... literature For example, many designers often assume that the amount of wastewater produced is equal to the amount of water consumed, including the allowance for infiltration-inflow, although one report indicates that 60 to 130% of the water consumed ends up as wastewater, and still another report indicates that 60 to 85% ends up as wastewater For this reason, quantities provided below should not be used . address quantity of wastewaters. Before discussing quantity of wastewaters, however, it is important that the various types of wastewaters be discussed first. The two general types of wastewaters. discussion on quantity are the important knowledge and background on the types of wastewater, sources of water and wastewater, and methods of population projection. The various categories of quantities. P. Sincero and G. A. Sincero 1.3 TYPES OF WASTEWATER As mentioned earlier, basically two quantities are addressed in this chapter: those of water and those of wastewater. Quantity of water has