Quantity of Water and Quantity of Water and WastewaterWastewater

CE 547CE 547

ProbabilityProbabilityQuantity of WaterQuantity of WaterTypes of WastewaterTypes of WastewaterSources of WastewaterSources of WastewaterPopulation ProjectionPopulation ProjectionDeriving Design Flows of WastewaterDeriving Design Flows of Wastewater

ContentsContents

ProbabilityProbability

1. Values Equaled or Exceeded1. Values Equaled or ExceededOne element equal to the valueOne element equal to the valueElements exceeding the valueElements exceeding the value

Prob (value equaled or exceeded) = Prob (value Prob (value equaled or exceeded) = Prob (value equaled) + Prob (value exceeded) – Prob (value equaled) + Prob (value exceeded) – Prob (value equaled equaled value exceeded) value exceeded)

Since the intersection probability = zeroSince the intersection probability = zero

Then,Then,

Prob (value exceeded) = Prob (value 1 exceeded) Prob (value exceeded) = Prob (value 1 exceeded) + Prob (value 2 exceeded) +…+ Prob (value + Prob (value 2 exceeded) +…+ Prob (value exceeded)exceeded)

Prob (value equaled or exceeded) = Prob (value Prob (value equaled or exceeded) = Prob (value equaled) + Prob (value 1 exceeded) + Prob equaled) + Prob (value 1 exceeded) + Prob (value 2 exceeded) +…+ Prob (value (value 2 exceeded) +…+ Prob (value exceeded)exceeded)

ProbabilityProbability

2. Derivation of Probability from Recorded 2. Derivation of Probability from Recorded ObservationObservation

(E) = occurrence of the event(E) = occurrence of the event = no of units favorable= no of units favorable

ss = total possible number of events = total possible number of events

s

Eob

)(Pr

Determination of Determination of ss

CostlyCostlyNot availableNot available

So, So, approxapprox is used instead. If is used instead. If approxapprox is small, then is small, then

the probability produced might be wrong. To the probability produced might be wrong. To correct this, 1 is added to the denominatorcorrect this, 1 is added to the denominator

1)(Pr

approxapproxEob

Example 1Example 1

ProbabilityProbability

3. Values Equaled or Not Exceeded3. Values Equaled or Not Exceeded

Values equaled or not exceeded is just the Values equaled or not exceeded is just the reverse of values equaled or exceededreverse of values equaled or exceeded

Prob (value equaled or not exceeded) = Prob Prob (value equaled or not exceeded) = Prob (value equaled ) + Prob (value 1 not exceeded) + (value equaled ) + Prob (value 1 not exceeded) + Prob (value 2 not exceeded) +….+ Prob (value Prob (value 2 not exceeded) +….+ Prob (value not exceeded)not exceeded)

Example 2Example 2

Quantity of WaterQuantity of Water

Quantities of water and wastewater are required by Quantities of water and wastewater are required by designers. Examples:designers. Examples:

I. Maximum daily flow is used to designI. Maximum daily flow is used to design community water suppliescommunity water supplies water intakeswater intakes wellswells treatment plantstreatment plants pumping stationspumping stations transmission linestransmission lines

Hourly variations are handled by storage.Hourly variations are handled by storage.

II. Water distribution systems are designed on II. Water distribution systems are designed on the basis of the MAXIMUM DAY PLUS FLOW the basis of the MAXIMUM DAY PLUS FLOW FOR FIRE FIGHTING or on the basis of the FOR FIRE FIGHTING or on the basis of the MAXIMUM HOURLY MAXIMUM HOURLY whichever is greaterwhichever is greater

Another parameter needed by designers is the Another parameter needed by designers is the DESIGN PERIODDESIGN PERIOD

What is Design Period?What is Design Period?

Time from the initial design years to the time Time from the initial design years to the time that the facility is to receive the final design that the facility is to receive the final design flows. Facilities would be designed at stages. It flows. Facilities would be designed at stages. It starts smaller and it gets bigger with time starts smaller and it gets bigger with time (staging period) due to increase in population.(staging period) due to increase in population.

Staging Periods (Table page 87)Staging Periods (Table page 87)

Design Periods (Table page 87)Design Periods (Table page 87)

Average Rates of Water Use (Tables Average Rates of Water Use (Tables page 88 and 89)page 88 and 89)

Types and Sources of WastewaterTypes and Sources of Wastewater

Types of Wastewater (two main types)Types of Wastewater (two main types)

Sanitary (from human activities)Sanitary (from human activities) Residential (domestic wastewater)Residential (domestic wastewater) Industries (industrial sanitary wastewater)Industries (industrial sanitary wastewater)

Industrial (from manufacturing processes)Industrial (from manufacturing processes)

Infiltration: Infiltration: water entering the sewer through cracks or water entering the sewer through cracks or imperfect connectionsimperfect connections

Inflow: Inflow: water entering the sewer through openings that water entering the sewer through openings that were not meant for that purposewere not meant for that purpose

Sources of Wastewater (Tables page 91 to 93)Sources of Wastewater (Tables page 91 to 93) ResidentialResidential CommercialCommercial InstitutionalInstitutional RecreationalRecreational IndustrialIndustrial

Population PredictionPopulation Prediction

Why is it Needed?Why is it Needed?

To determine the design flows for a communityTo determine the design flows for a community

Several methods are usedSeveral methods are used Arithmetic methodArithmetic method Geometric methodGeometric method Declining rate of increase methodDeclining rate of increase method Logistic methodLogistic method Graphical comparison methodGraphical comparison method

Arithmetic MethodArithmetic Method

The population at present increase at a The population at present increase at a constant rateconstant rate

The method is applicable for short-term The method is applicable for short-term projections (projections ( 30 years) 30 years)

12

12

1212 )(

int

tan

YY

PPk

or

YYkPP

egrate

ratetconsk

yeargivenanyatpopulationdY

dP

kdY

dP

a

a

a

a

ExampleExample

The population for City A is as follows:The population for City A is as follows:

19801980 15,00015,000

19901990 18,00018,000

What will be the population in What will be the population in 20002000??

SolutionSolution

personskPP

yeark

a

a

210001030018000)19902000(

/30010

3000

19801990

1500018000

19902000

Geometric MethodGeometric Method

The population at present increase in The population at present increase in proportion to the number at presentproportion to the number at present

Used for short-term projectionsUsed for short-term projections

12

12

1212

lnln

)(lnln

int

tan

YY

PPk

or

YYkPP

egrate

tconsrategeometrick

PkdY

dP

g

g

g

g

ExampleExample

Repeat the previous example using the geometric Repeat the previous example using the geometric method.method.

SolutionSolution

personsP

kPP

yearYY

PPk

g

g

21593

98013.9100182.018000ln)19902000(lnln

/0182.019801990

15000ln18000lnlnln

2000

19902000

12

12

Declining-Rate-of-Increase MethodDeclining-Rate-of-Increase Method

The population will reach The population will reach a saturation valuea saturation value

The rate of increase will The rate of increase will decline until it becomes decline until it becomes zero at saturationzero at saturation

)(3

231

2231

2

3

23

1

2

12

2323

1212

3)(

2

ln1

ln1

)()ln()ln(

)()ln()ln(

int

tan

)(

YYearkssyear

s

s

sd

s

sd

dss

dss

d

s

sd

dePPPP

PPP

PPPP

PP

PP

YYk

PP

PP

YYk

YYkPPPP

YYkPPPP

twiceegrating

tconsratek

populationSaturationP

PPkdY

dP

ExampleExample

If the population of City A is as follows:If the population of City A is as follows:

19801980 15,00015,000

19901990 18,00018,000

20002000 20,00020,000

What will be the population in What will be the population in 20202020??

SolutionSolution

personseP

ePPPP

yearPP

PP

YYk

PPP

PPPP

Ykss

s

sd

s

d

22203)2000024000(24000

)(

/04.01800024000

2000024000ln

10

1ln

1

24000)18000(22000015000

)18000()20000(15000

2

)20002020(04.02020

)2020(32020

2

3

23

2

231

2231

3

Logistic MethodLogistic Method

If environmental conditions are optimum, If environmental conditions are optimum, population will increase at geometric rate. In population will increase at geometric rate. In reality, this will be slowed down due to reality, this will be slowed down due to environmental constraints such as:environmental constraints such as:

Decreasing rate of food suppliesDecreasing rate of food suppliesOver-crowdingOver-crowdingDeathDeath

According to the geometric method:According to the geometric method:

To enforce the environmental constraints, kTo enforce the environmental constraints, kggP P

should be multiplied by a factor less than 1. In should be multiplied by a factor less than 1. In logistic method, the factor 1 is reduced by P/K.logistic method, the factor 1 is reduced by P/K.

Where K = carrying capacity of the environmentWhere K = carrying capacity of the environment

(1-P/K) = environmental resistance(1-P/K) = environmental resistance

PkdY

dPg

Therefore, the logistic equation becomes:Therefore, the logistic equation becomes:

Note that kNote that kgg changed to k changed to kll. Re-arrange:. Re-arrange:

)1( KPPkdY

dPl

)2(11

)/1(

1

sin

)1()/1(

PKPKPP

ce

dYkKPP

dPl

Substitute in (1) and integrate twice:Substitute in (1) and integrate twice:

)(ln

)(ln

233

2

2

3

122

1

1

2

YYkPK

PK

P

P

YYkPK

PK

P

P

l

l

Solve for kSolve for kll

)(3

)(3

312

2

3132212

2312

3

2

2

3

23

2

1

1

2

12

3

3

1

)2(

sin

ln1

ln1

YYeark

YYeark

year

l

l

l

l

ePK

eKPP

PPP

PPPPPPPK

YYYY

ce

PK

PK

P

P

YYk

PK

PK

P

P

YYk

ExampleExample

If the population of City A is as follows:If the population of City A is as follows:

19801980 15,00015,000

19901990 18,00018,000

20002000 20,00020,000

What will be the population in What will be the population in 20202020??

persone

e

ePK

eKPP

yeark

PK

PK

P

P

YYk

personsK

PPP

PPPPPPPK

Yk

Yk

l

l

l

l

2182712000022500

2000022500

1

/07.02000022500

1800022500

18000

20000ln

10

1

ln1

22500200001500018000

200001500022000018000180001500018000

)2(

)20002020(07.0

)20002020(007

)2020(3

)2020(3

2020

3

2

2

3

23

2

312

2

3132212

3

3

Graphical Comparison MethodGraphical Comparison Method

Plot the population of the given City along with other cities Plot the population of the given City along with other cities which are larger in size but have similar characteristics. This which are larger in size but have similar characteristics. This method extends a line reflecting the slope of each ten (10) year method extends a line reflecting the slope of each ten (10) year interval between censuses.  An average line is then determined interval between censuses.  An average line is then determined to reflect the population estimate of future years. to reflect the population estimate of future years. This method involves extension of the population-time curve This method involves extension of the population-time curve of the city C (under consideration) based on comparison with of the city C (under consideration) based on comparison with population-time curves of similar but larger cities A and B. population-time curves of similar but larger cities A and B. These larger cities A and B must have reached the present These larger cities A and B must have reached the present population of the city C one or more decades ago. population of the city C one or more decades ago. Starting from the point on curve C representing the present Starting from the point on curve C representing the present population, the curves corresponding to the growths of A and population, the curves corresponding to the growths of A and B after their reaching that population are plotted. The B after their reaching that population are plotted. The extension of the curve C is modified keeping in view the extension of the curve C is modified keeping in view the projections offered by A and B as well as other related projections offered by A and B as well as other related conditions.conditions.

Deriving Design Flows of WastewaterDeriving Design Flows of Wastewater

Types of flow rates used in the design:Types of flow rates used in the design: Average daily flow rateAverage daily flow rate Maximum daily flow rateMaximum daily flow rate Peak hourly flow ratePeak hourly flow rate Minimum daily flow rateMinimum daily flow rate Minimum hourly flow rateMinimum hourly flow rate Sustained high flow rateSustained high flow rate Sustained low flow rateSustained low flow rate

What is needed to determine these flow rates?What is needed to determine these flow rates? Statistically sufficient amount of data should be Statistically sufficient amount of data should be

availableavailable

Where to get data from?Where to get data from?1.1. Literature (least desirable)Literature (least desirable)

2.2. Treatment plant recordsTreatment plant records

3.3. Field surveys (costly and short-term)Field surveys (costly and short-term)

1 and 2 are not recommended because1 and 2 are not recommended because Habits of people are differentHabits of people are different Characteristics of land area are different (precipitation Characteristics of land area are different (precipitation

which affects infiltration into the sewer)which affects infiltration into the sewer)

Average Daily Flow RateAverage Daily Flow Rate

Average flow rate corresponds to 50% probabilityAverage flow rate corresponds to 50% probability

If we have the data shown in Figures 1.6 and 1.7 for If we have the data shown in Figures 1.6 and 1.7 for June and July, then the daily average can be calculated June and July, then the daily average can be calculated using the graphical method shown in Figure 1.8.using the graphical method shown in Figure 1.8.

This method should be used for each individual day. This method should be used for each individual day. Tabulate the results (Table 1.12):Tabulate the results (Table 1.12):

Arrange the data in ascending or descending orderArrange the data in ascending or descending order Calculate probability distributionCalculate probability distribution Average flow is at 50% probabilityAverage flow is at 50% probability If 50% is not available, then interpolate (Table 1.13)If 50% is not available, then interpolate (Table 1.13)

Peak Hourly Flow RatePeak Hourly Flow Rate

Corresponds to probability of zeroCorresponds to probability of zero Find peak hourly flow rate for each day from Find peak hourly flow rate for each day from

original dataoriginal data Tabulate the results (Table 1.14)Tabulate the results (Table 1.14) Arrange in descending orderArrange in descending order Read or extrapolate flow rate with probability Read or extrapolate flow rate with probability

of zero (Table 1.15)of zero (Table 1.15)

Maximum Daily Flow RateMaximum Daily Flow Rate

Same data used for average daily flow rate Same data used for average daily flow rate (Table 1.13)(Table 1.13)

Corresponds to zero probabilityCorresponds to zero probability Read or extrapolate flow rate at probability of Read or extrapolate flow rate at probability of

zerozero

Minimum Hourly Flow RateMinimum Hourly Flow Rate

Obtain the lowest flow each day from original Obtain the lowest flow each day from original data (Figures 1.6 and 1.7)data (Figures 1.6 and 1.7)

Tabulate the daily minimum hourly flow rates Tabulate the daily minimum hourly flow rates (Table 1.16)(Table 1.16)

Arrange in ascending orderArrange in ascending order Read or extrapolate flow rate at probability of Read or extrapolate flow rate at probability of

zerozero

Minimum Daily Flow RateMinimum Daily Flow Rate

Use data on average flow rates (Table 1.12)Use data on average flow rates (Table 1.12) Rearrange in ascending order (Table 1.18)Rearrange in ascending order (Table 1.18) Read or extrapolate flow rate at probability of Read or extrapolate flow rate at probability of

zerozero

Sustained Peak Flow rate and Sustained Peak Flow rate and Sustained Minimum Flow RateSustained Minimum Flow Rate

Define a parameter called “moving average”Define a parameter called “moving average”

Element No 1 2 3 4 5 6

Element Value 23 21 43 32 34 26

1st moving average = (23+21+43)/3 = 29 23 21 43

2nd moving average = (21+43+32)/3 = 32 21 43 32

3rd moving average = (43+32+34)/3 = 36.3 43 32 34

4th moving average = (32+34+26)/3 = 30.7 32 34 26

No of moving averagesNo of moving averages

If If

ee = no of elements (6) = no of elements (6)

meme = no of moving elements to be averaged = no of moving elements to be averaged

mavmav = = e - e - meme +1 +1

wherewhere

mavmav = no of moving averages = no of moving averages

How to find Sustained Peak Flow How to find Sustained Peak Flow Rate?Rate?

Tabulate data (Table 1.12)Tabulate data (Table 1.12) Find moving average (Table 1.20)Find moving average (Table 1.20) Arrange moving averages in descending order Arrange moving averages in descending order

(Table 1.21)(Table 1.21) Read or extrapolate flow rate at probability of Read or extrapolate flow rate at probability of

zerozero

So, if:So, if:

XX 00

38.2438.24 0.020.02

38.1238.12 0.040.04

Then,Then,

hourmX

X

/36.38

04.002.0

02.00

12.3824.38

24.38

3

How to find Sustained Minimum Flow How to find Sustained Minimum Flow Rate?Rate?

Tabulate data (Table 1.12)Tabulate data (Table 1.12) Find moving average (Table 1.20)Find moving average (Table 1.20) Arrange moving averages in ascending order Arrange moving averages in ascending order

(Table 1.22)(Table 1.22) Read or extrapolate flow rate at probability of Read or extrapolate flow rate at probability of

zerozero

So, if:So, if:

XX 00

35.3535.35 0.020.02

30.0630.06 0.040.04

Then,Then,

hourmX

X

/64.34

04.002.0

02.00

06.3635.35

35.35

3