x = 0.023 = Prob (flow < 2800) Ans y = 0.12 = Prob (flow < 3000) Ans
Example 1.3 In Example 1.1, calculate the probability that the flow is equal to less than 3700 m3/wk.
Solution: The values less than 3700 are 3675, 3644, 3540, 3459, 3300, 3200, 3180, 3135, 3028, and 2900. Thus,
Prob(value equaled or not exceeded)
= Prob(value equaled) + Prob(value1 not exceeding) + Prob(value2 not exceeding) + ••• Prob (flow < 3700)
= Prob( 3700) + Prob (3675) + Prob( 3644) + Prob( 3540) + Prob( 3459) + Prob (3300) + Prob (3200) Prob (3180) + Prob(3135) + Prob(3028) + Prob(2900)
Again, probability distribution analysis may also be applied. The procedure is similar to that of the previous one, except that the data values are arranged in ascending order instead of descending order. Thus,
Any discussion on physical-chemical treatment of water and wastewater is incomplete without knowledge of the quantities involved. How large a volume is being treated? The answer to this question will enable the designer to size the units involved in the treatment.
Two quantities are addressed in this 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 lines are normally designed using the maximum daily flow with hourly variations handled by storage.
2. Water distribution systems are designed on the basis of the maximum day plus flow for fighting fires or on the basis of the maximum hourly, whichever is greater. For emergency purposes, standby units are installed.
3. For industrial plants, resort sites, and so on, special studies may be made to determine the various design values of water usage.
In the case of communities, the actual flows that are used in design are affected by the design period. Designs are not normally made on the basis of flow at the end of this period but are spread over the duration. The design period, also called the planning period, is discussed later.
Of the various design flows, we will discuss the average flow first. The average flow in a community is normally taken as the average daily flow computed over a year as follows:
where x - consumption in units of volume per capita per day y - total cubic units of water delivered to the distribution system P - the midyear population served by the distribution system
Note that in the previous equation, the consumption has been normalized against population. In industrial plants, commercial, institutional, and other facilities, the normalization is done in some other ways such as per tonnes of product per customer, per student, and so on.
From the previous definition of average flow, it is evident that there would be a number of values depending upon the number of years that the averages are computed. For purposes of design, the engineer must decide which particular value to use. The highest value may not be arbitrarily used because this may result in over design; on the other hand, the lowest value may also not be used for a similar but reverse reasoning.
In reality, only one average value exists, and this value is the long-term value. The reason why we have so many average values is that averages have been taken each year when there should only be one. What should be done is to take all the daily values in the record, sum them up, and divide this sum by the number of daily values. The problem with this approach, however, is that once this one single average is obtained, it is still not certain whether or not this particular one value is the average value. In concept, the averaging may be extended for one more additional value and the average recomputed. If the current recomputed average value is equal to or close to the previous value, then it may be concluded that the correct average value has been obtained. If not, then the recomputing of the average may be further extended until the correct average is obtained.
The other way to obtain the average value is to use the probability distribution analysis. From the theory of probability, the long-term average value (which is the average value) is the value that corresponds to 50% probability in the probability distribution. Thus, if sufficient data have been gathered, the average can be obtained from the distribution without going through the trial-and-error method of recomputing the average addressed in the previous paragraph.
Depending upon the source of information, average values are vastly different. The following are average usages in cubic meters per capita per day from two communities obtained from two different sources of information:
Cubic meters per cap Cubic meters per cap
User per day per day
Domestic 0.13 0.24
Commercial and industrial 0.11 0.25
The previous table shows that you are bound to obtain widely differing answers to the same question from different sources. You, therefore, have to gather your own.
These values may be used for preliminary calculations only. For more accurate values on specific situations, a field determination must have to be made.
As mentioned previously, the maximum day and the maximum hour may also be needed in design. To get the maximum day and the maximum hour, the proper average daily flow is multiplied by the ratios of the maximum day and maximum hour to the average day, respectively. The question of what maximum ratios to use is not easy to answer. For any community, industrial plant, commercial establishment, and the like, literally hundreds of maxima are in the record; however, there should only be one maximum.
The following treatment will discuss a method of obtaining the maximum such as the maximum hourly. To obtain these quantities, the record may be scanned for the occurrence of the daily maximum hourly and divided by the corresponding daily average. The results obtained are then arranged serially in decreasing order and probability distribution calculated. The probability obtained from this calculation is a "daily probability" as distinguished from the probabilities of occurrence of storms which are based on years and, hence, are "yearly probabilities." A weekly, monthly, or yearly probability of the maximum hourly may also be calculated. In theory, all these computed probabilities will be equivalent if the records are long enough. Now, what ratio should be obtained from the probability distribution? If the distribution is derived from a very long record, the ratio may be obtained by extrapolating to the probability that the ratio will never be exceeded. This probability would be zero. The definition of the maximum hourly flow of water use is the highest of the hourly flows that will ever occur.
The maximum daily flow may be obtained in a similar manner as used for the maximum hourly flow, only the daily values are used rather than the hourly values. Then, in an analogous manner as used to define the maximum hourly flow, we make the following definition of the maximum daily flow of water use: the highest of the daily flows that will ever occur.
For an expansion to an existing system, the records that already exist may be analyzed in order to obtain the design values. For an entirely new system, the records of a nearby and similar community, industrial plant, commercial establishment, and the like may be utilized in order to obtain the design values. Of course, in selecting the final design value, there are other factors to be considered. For example, in the case of a community water supply, the practice of metering the consumption inhibits the consumer from wasting water. Therefore, if the record analyzed contains time when meter was used and time when not used, the conclusion drawn from statistical analysis must take this fact into consideration. The record might also contain years when use of water was heavily curtailed because of drought and years when use was unrestricted. This condition tends to make the data "inhomogeneous." A considerable engineering judgment must therefore be exercised to arrive at the final value to use.
Generally, it takes years to plan, design, and construct a community water and wastewater facility. Even at the planning stage, the population continues to grow and, along with it, comes the increase of flows during the period. This condition requires that, in addition to determining the ultimate design population, the population must also be predicted during the initial years that the project is put into operation.
The flows that the facility is to be designed for are design flows. The time from the initial design years to the time that the facility is to receive the final design flows is called the design or planning period. The facility would not be sized for the initial years nor for the final year; the design must be staged. At the initial years, the 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 various types of facilities.
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