## Introduction and population growth and forecast and Water Demand

## introduction

- Before designing a proper waterworks project, it is necessary to identify the quantity of water that is required daily.
- The water requirement is estimated considering provisions for future population and also considering the requirement for various purposes like industrial or institutional.
- A water supply project is designed to achieve the need of the region up to the designed period, which can be extended up to 2-4 decades.
- There are several methods to forecast the population.
- The final method is decided by the engineer who will be most suitable and applicable to that particular location.
- After the calculation of water demand, a reliable source of water is located.

### estimating water demand involves the determination of the following items:

#### population forecasting:

- The most important factor in planning is population forecasting.
- Generally, a design period of 20-40 years is taken.

#### Rate of demand:

- The water consumption in a city may be conveniently divided into the following categories:
- Domestic Water Demand
- Industrial Water Demand
- Institutional and Commercial
- Demand for Public Use
- Fire Demand
- Losses

- The total quantity of water required divided by the total population gives per capita water demand.

#### Design periods:

- Future periods for which provision is made in designing the capacities or various components of the water supply scheme is known as the design period.
- For designing various components of the water supply project, the design period given by the GOl manual on water supply (CPHEEO-1999) is as given below in table

Sr. No. | Components | Design Period (in Years) |

1 | Storage by dams | 50 |

2 | Infiltration Works | 30 |

3 | Pumping | - |

A) Pump House (civil works) | 30 | |

B) Electric Motors & Pumps | 15 | |

4 | Water Treatment Units | 15 |

5 | Pipe connections to the several treatment units and other small appurtenances | 30 |

6 | Raw water and clear water conveying mains | 30 |

7 | Clear water reservoirs at the headworks balancing tanks and service reservoirs (Overhead or ground level) | 15 |

8 | Distribution system | 30 |

**Design Period for Various Components of a Water Supply System**

## Population growth

**The Three Main Factors Are:**

- Births (Population Gain)
- Deaths (Population Loss)
- Migrations (Population loss or gain depending on whether movement out or movement in occurs in excess).

**Population forecasting is also done graphically**

**Note:** Population forecasting is obtained by making and combining separate but related curves of natural increases and of net migration. The net effect of birth and death on the population is termed as a natural increase (natural decrease, if death exceeds births).

### Growth curve:

**1) **The population would mostly follow the growth curve nature of living things with limited space or with limited economic opportunity.

**2) **The curve having S-Shaped as shown below, is known as a logistic curve.

**3)** The curve represents early growth AB at an increasing rate {i.e., geometric or log growth, (dp/dt)*p} and late growth DE at a decreasing rate order curve as the saturation value P_{s} is approached.

**4) **The Traditional middle curve BD follows an arithmetic increase {i.e., (dt/dt) = constant what the future holds for a given population, depending upon where the point lies on the growth curve at a given time.

### the logistic curve method:

**1) **It was explained earlier that under normal conditions, the population of a city should grow as per the logistic curve, shown in **fig. 1**.

**2)** P. F. Verhulst has put forward a mathematical solution for this logistic curve.

**3)** According to him, the entire curve A-D (**Fig. 1**) can be represented by an auto-catalytic first-order equation, given as:

log_{e} (P_{S}-P)/P - log_{e} (P_{S}-P_{o})/P_{o} = -KP_{S}*t

Where, P_{o} = the population at the start point of the curve A

P_{S} = Saturation population

P = Population at any time t from the origin A

K = Constant

from above equation

log_{e} [ {(P_{S}-P)/P} * {(P_{S}-P_{o})/P_{o}} ] = -KP_{S}*t

On simplifying we get

P = P/ 1 + m* log^{-1}_{e}(nt)

This is the required equation of the logistic curve.

**4) **Mclean further suggested that if only three pairs of characteristic values P_{o} at times t = t_{o} = 0, P_{1} at time and P_{2} at time t_{2} = 2t_{1} extending over the useful range of the census population are chosen, the saturation value P_{s} and the constants m and n can be evaluated from three simultaneous equations, as follows:

P_{S}= 2P_{o}P_{1}P_{2}-P^{2}_{1} (P_{o}+P_{2}) / P_{o}P_{2}-P^{2}_{1}

m = P_{S}- P_{o} /P_{o}

n = (1/t_{1}) log_{e}[ P_{o}(P_{S}-P_{1})/P_{1}(P_{S}-P_{o})]

Knowing P_{o}, P_{1}, and P_{2} from census data and using them in these equations, the values of P_{S}, m, and n can be calculated, and the equation of the logistic curve thus obtained can be used to forecast the population P at any time t.

## IMPORTANT QUESTION

**Question -** A city has the following recorded population;

Year 1971: 60000

Year 1991: 120000

Year 2011: 180000

Estimate a) The saturation population and b) The expected population in the year 2031 by the logistic curve method. **[ESE 2019]**

### factors affecting population growth:

**Economic factors**

Example the discovery of minerals or oils in the vicinity of the city, the development of new Industries etc.

**Development programs**

Development of projects of national Importance, such as river valley projects, etc.

**Social facilities**

Educational, medical, recreational, and other social facilities.

**Tourism**

Tourist facilities, religious places, or historical buildings.

**Community life**

Living habits, social customs, and general education in the community.

**Unforeseen factors**

Epidemics frequent famines Earthquakes, floods, etc.

## population forecast

- The data about the present population of a city can always be obtained from records of the municipality or civic body.
- There are several methods for population forecast, but the Judgment must be exercised by the engineers as to which method is most applicable for a particular location.
- The population density, indicating the number of persons per unit area, and the distribution of population should be studied for efficient design of the distribution system.
- Following are some of the important methods of population forecasts or population projections.
- Arithmetic Increase Method
- Geometric Increase Method
- Incremental Increase Method
- Graphical Method
- Comparative Method
- Zoning Method
- Decreasing Growth Rate Method

### Arithmetic Increase Method:

- This method is most simple and gives lower results.
- In this method, the increase in population is assumed to be constant.
- For each successive future decade, the average increment is added.
- The Future population P
_{n}after n decades is thus given by:

P_{n}= P + nx

Where,

P_{n}= The future population at the end of n decades.

P = Present population.

X = Average increment for a decade. - This method should be used for forecasting the population of those large cities which have reached their saturation population.

### Geometric increase method or uniform percentage growth method:

- In this method, it is assumed that the percentage increase in population from decade to decade is constant.
- From the population data of the previous three or four decades, the percentage increase in population is found, and its average is found.

P_{n}= P {1+(r/100)}^{n}

Where, r = Average percentage increase per decade.

P_{n}= Population after n decades. - This method gives high results since the percentage increase never remains constant but instead decreases when the growth of the city reaches saturation.

### Incremental increase method:

- This method combines both the arithmetic average method and the geometrical overage method.

- The actual increase in each decade is found.

- From these, an average increment of the increases known as incremental increase) is found.

- Thus, the future population at the end of n decade is given by:
- P
_{n}= P+ nx̄ + { n(n+ 1)/2 } Ȳ - P = Present population = Average increase per decade = average incremental increase in population.
- n = number of decades

- P

### Decrease growth rate method:

- This method is applicable only when the rate of growth shows a downward trend.
- Since the rate of increase in population goes on reducing, as the cities reach saturation, a method which makes use of the decrease in the percentage increase is many times used and gives quite rational results.
- In this method, the average decrease in the percentage increase is worked out and is then subtracted from the latest percentage increase for each successive decade, as explained in the example below.

## important question

**Question 1**

The population of a locality as obtained from Census records is as follows:

Years | 1970 | 1980 | 1990 | 2000 | 2010 |

Population | 15000 | 20000 | 24500 | 29500 | 32500 |

Estimate the population of the locality in 2040 by Arithmetic Increase, Geometrical Increase, Incremental Increase, and Rate of Decrease Methods. **[ESE MAINS 2014]**.

**Solution **

### Arithmetic Increase Method

Years | Population | Increase in Population |

1970 | 15,000 | - |

1980 | 20,000 | 5000 |

1990 | 24,500 | 4000 |

2000 | 29,500 | 5000 |

2010 | 32,500 | 3000 |

Average increase per decades,

x̄ = (5000 + 4500 + 5000 + 3000) / 4 = 4375

Population after 3 decade beyond 2040

P_{2040} = P_{2010} + 3 x̄

P_{2040} = 32500 + 3 x 4375

P_{2040} = 45625

### Geometrical Increase Method

Year | Population | Increase in Population | % Increase in Population |

1970 | 15,000 | - | - |

1980 | 20,000 | 5000 | 33.33 |

1990 | 24,500 | 4500 | 22.50 |

2000 | 29,500 | 5000 | 20.40 |

2010 | 32,500 | 3000 | 10.17 |

Constant growth rate assumed for future,

r = geometric mean of past growth rate.

r = 4 √ 33.33*22.50*20.40*10.17

r = 19.86% per decade

= P_{n} = P_{o} {1 + (r/100)^{n}}

P_{2040} = Population after 3 decades from 2010

P_{2040} = P_{2010} {1 + (19.86/100)^{3}}

P_{2040} = 55933

### incremental increase method

Year | Population | Increase in Population | Increment over the increase,i,e., Incremental Increase |

1970 | 15,000 | - | - |

1980 | 20,000 | 5000 | - |

1990 | 24,500 | 4500 | (-) 500 |

2000 | 29,500 | 5000 | (+) 500 |

2010 | 32,500 | 3000 | (-) 2000 |

Total | 17,500 | (-) 2000 |

The average increase per decade,

x̄ = 4375

**Ȳ** = - 666.67

The expected population at the end of the year 2040, i.e., after 3 decade from 2010

P = P_{o} + 3x̄ + {3(3+1)/2}/**Ȳ**

P = 41625

### question 2

Which of the following method is used to forecast the population of an old and very large city? **[SSC JE 2018]**

(A) Graphical method

(B) Logistic curve method

(C) Arithmetical increase method

(D) Geometric progression method

**Solution [C]**

### question 3

The population for 4 decades from 1951 to 1981 are given in the table. Find out the population by the end of 2011 by using the arithmetic increase method. **[SSC JE 2020]**

Years | 1951 | 1961 | 1971 | 1981 |

Population | 1,00,000 | 1,09,000 | 1,16,000 | 1,28,000 |

A) 1,36,000

B) 1,56,000

C) 1,46,000

D) 1,26,000

**Solution (B)**

### question 4

the census record of a particular town shows the population figures as follows:

Years | 1960 | 1970 | 1980 | 1990 |

Population | 55.500 | 63,700 | 71,300 | 79,500 |

Estimate the population for the year 2020 by decreasing rate growth. **[ESE 1999]**

**Solution 1,00,765**

### question 5

The population of a small town for 5 decades from 1970 to 2010 is given below. Find out the population in the year 2020, 2030 and 2040 by using the arithmetic increase method, geometric increase method and incremental increase method. **[IFS 2013]**

Years | 1970 | 1980 | 1990 | 2000 | 2010 |

Population | 25,000 | 28,000 | 34,000 | 42,000 | 47,000 |

**Solution **

**AIM** = 52,500 , 58,000, 63,000

**GIM** = 53,167, 60,000, 67,500

**IIM **= 54,708, 63,680, 74,124

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## pOINTS TO REMEMBER

Method | Population P_{n} After n Decades | Applicable To |

Arithmetical Increase Method | P= P + nl_{n} | Large and old cities |

Graphical Increase Method | P= P { 1+ (l_{n} _{a}/100)^{n} } | Growing towns and cities having a vast scope of expansion |

Incremental Increase Method | P= P + nl + { n (n+1)/2}r_{n} | - |

Decrease the Rate of Growth | P= P_{n} _{o} + { (r-r')/100} P_{o} | In such cases where the rate of growth shows a downward trends |

**Method**,

**Population P**

_{n}After n Decades**and Applicable To**

### Geometrical Increase Method:

- In this method, a curve is drawn between the population P and time T, with the help of census data of the previous few decades, so that the shape of the population curve is obtained up to the present period.

### Graphical comparison method:

- This method is graphical variation of the previous method.
- It presumes that the city under consideration will develop similar to cities developed in the past.
- This method includes plotting curves of cities that, one or more decades ago, had attained the present population of that city.

### Zoning method:

- A master plan of the city is prepared, dividing it into various zones such as industrial, commercial, residential and other zones.
- Each zone is allowed to develop as per the master plan only.
- The future population of each zone, when fully developed can be easily found.

### Ratio and correlation method:

- In this method, the local to national (or state) population ratio is determined in the previous two to four decades.
- Depending upon conditions or other factors, even changing ratios may be adopted.
- These ratios may be used in predicting the future population.