1. Conceptual Framework & Infrastructure Design Periods

A. Purpose of Population Forecasting in Environmental Engineering

Definition

Population forecasting is the mathematical estimation of the future population of a town, city, or region over a specified planning period.

Why is it important?

Environmental infrastructure is generally designed for future demand, not only for the present population.

Population forecasting is therefore required for designing:

• Water Treatment Plants (WTPs)

• Sewage Treatment Plants (STPs)

• Water distribution networks

• Sewerage networks

• Drainage systems

• Pumping stations

• Sanitary landfills

• Solid waste management facilities

Basic Design Logic

Present Population
        ↓
Population Forecast
        ↓
Future Water / Wastewater / Waste Load
        ↓
Infrastructure Capacity
        ↓
Design Period

B. Over-Design vs Under-Design

1. Under-Estimation of Population

If future population is underestimated:

• Water supply becomes inadequate

• Sewage systems become overloaded

• Sewer overflows may occur

• Solid waste collection capacity becomes insufficient

• Treatment plants may become overloaded

• Environmental pollution increases

• Public health risks increase

2. Over-Estimation of Population

If future population is overestimated:

• Excess infrastructure capacity is constructed

• Capital investment is unnecessarily high

• Infrastructure may remain underutilized

• Water resources may be unnecessarily over-extracted

• Oversized sewer systems may operate at low flow and velocity

• Low velocity can contribute to deposition and septic conditions

Key Engineering Principle

Under-design → inadequate service

Over-design → unnecessary capital and operational cost

Therefore, the population forecasting method should be selected carefully.

C. Typical Design Periods of Environmental Infrastructure

Exam idea: Different infrastructure components have different economic and functional lifetimes. Therefore, the design period is not the same for every component.

2. Demographic Drivers of Population Change

Population changes because of two major processes:

1. Natural increase

2. Migration

General Population Balance

Pₜ = P₀ + (Births − Deaths) + (Immigration − Emigration)

Where:

• Pₜ = population at time t

• P₀ = initial population

A. Natural Increase

Natural increase is:

Births − Deaths

It depends on:

• Birth rate / natality

• Death rate / mortality

If births exceed deaths:

Population increases

If deaths exceed births:

Population decreases

B. Net Migration

Net migration is:

Immigration − Emigration

Immigration

People entering the city or region.

Emigration

People leaving the city or region.

Factors affecting migration

Urban Pull Factors

• Employment

• Education

• Healthcare

• Better infrastructure

• Business opportunities

Rural Push Factors

• Lack of employment

• Poor infrastructure

• Limited educational opportunities

• Limited healthcare

• Agricultural uncertainty

3. Mathematical Population Forecasting Methods

              POPULATION FORECASTING METHODS
                         │
        ┌────────────────┼────────────────┐
        ▼                ▼                ▼
   ARITHMETIC        GEOMETRIC       INCREMENTAL
    INCREASE          INCREASE          INCREASE
        │                │                │
 Constant absolute   Constant %       Average increase
    increase          increase         + incremental
                                         change
        │                │                │
   Mature cities    Rapidly growing   Moderately
                     cities            growing cities

Additional methods include:

• Decreasing Rate of Growth / Logistic Curve Method

• Master Plan / Graphical Method

• Ratio Method

3.1 Arithmetic Increase Method

Growth Premise

The population increases by approximately the same absolute amount during each period.

Mathematically:

dP/dt = constant

Therefore, population growth is approximately linear.

Formula

Pₙ = P₀ + n × x̄

Where:

• Pₙ = projected population after n periods

• P₀ = present population

• n = number of periods

• x̄ = average population increase per period

If census data are available at 10-year intervals:

n = (Target Year − Base Year) / 10

Suitability

Generally suitable for:

• Old cities

• Mature cities

• Large established cities

• Cities approaching development limits

Limitation

It may produce relatively low estimates when applied to rapidly growing cities.

Memory Trick

Arithmetic → Same Amount

3.2 Geometric Increase Method

Growth Premise

The population grows at approximately the same percentage rate during each period.

Mathematically:

dP/dt = kP

This produces compound/exponential growth.

Formula

Pₙ = P₀ × (1 + rᵍ/100)ⁿ

Where:

• Pₙ = projected population

• P₀ = present population

• rᵍ = geometric mean growth rate (% per period)

• n = number of periods

Geometric Mean Growth Rate

If the percentage growth rates for different periods are r₁, r₂, r₃, etc.:

rᵍ = (r₁ × r₂ × r₃ × ... × rₘ)^(1/m)

Where:

• rᵢ = percentage growth rate during period i

• m = number of growth periods

For each period:

rᵢ = [(Population increase during period) / (Population at beginning of period)] × 100

Suitability

Generally suitable for:

• Young cities

• Rapidly growing cities

• Industrial boomtowns

• Areas experiencing strong economic migration

Limitation

It can give very high projections if a high growth rate is assumed to continue indefinitely.

Memory Trick

Geometric → Same Percentage

3.3 Incremental Increase Method

Growth Premise

This method considers:

1. Average population increase

2. Change in that increase from one period to the next

Therefore, it provides a correction to the simple arithmetic method.

Formula

Pₙ = P₀ + n × x̄ + [n(n + 1)/2] × ȳ

Where:

• x̄ = average population increase per period

• ȳ = average incremental increase

• n = number of periods

Incremental Increase

If successive population increases are:

x₁, x₂, x₃...

Then:

y₁ = x₂ − x₁

y₂ = x₃ − x₂

and so on.

Average incremental increase:

ȳ = Sum of incremental increases / Number of increments

Suitability

Generally suitable for:

• Moderately growing cities

• Cities showing changing growth rates

• Situations where neither purely arithmetic nor purely geometric growth is appropriate

Memory Trick

Incremental → Increase + Change in Increase

3.4 Decreasing Rate of Growth / Logistic Curve Method

Real urban population growth does not necessarily continue exponentially forever.

As land, resources, infrastructure and economic opportunities become limited, the growth rate may decline.

This can be represented by an S-shaped logistic curve.

Population
   │
   │                         ───────── Saturation Population (Pₛ)
   │                    .────
   │                 .´
   │              .´       Deceleration
   │           .´
   │        .´
   │     .´      Exponential Growth
   │  .´
   │.´  Early Growth
   └──────────────────────────────────── Time

Stages

1. Early/Lag Phase

   • Slow population growth

2. Exponential Growth Phase

   • Rapid population increase

3. Deceleration Phase

   • Growth begins to slow

4. Saturation Phase

   • Population approaches a limiting value

The limiting population is called the:

Saturation Population (Pₛ)

Verhulst-Pearl Logistic Equation

P = Pₛ / [1 + m × e^(−kt)]

Where:

• P = population at time t

• Pₛ = saturation population

• m = empirical constant

• k = growth constant

• t = time

Saturation Population Formula

If three population values P₀, P₁ and P₂ are available at equal time intervals:

Pₛ = [2P₀P₁P₂ − P₁²(P₀ + P₂)] / [P₀P₂ − P₁²]

Important Condition

The three population observations should correspond to equal time intervals.

Concept

Logistic method → Growth cannot continue indefinitely → population approaches Pₛ

3.5 Master Plan / Graphical Method

Population is estimated using the city's master plan and zoning pattern.

The city may be divided into:

• Residential zones

• Commercial zones

• Industrial zones

• Institutional areas

• Recreational areas

Population can then be estimated using:

Population = Area × Permitted Population Density

For example:

If:

• Residential area = 500 hectares

• Permitted density = 200 persons/hectare

Then:

Population = 500 × 200

= 100,000 persons

Advantage

It incorporates the planned physical development of the city.

3.6 Ratio / Comparative Method

This method assumes that the population of a city changes approximately in proportion to the population growth of a larger region.

The comparison may involve:

• State population

• District population

• National population

• Other similar cities

Basic Concept

City growth is related proportionally to the growth of the reference population.

4. Comparative Selection & Hierarchy

A commonly used conceptual hierarchy is:

FORECASTED POPULATION

Geometric
    ↓
Highest estimate

Incremental
    ↓
Intermediate estimate

Arithmetic
    ↓
Lowest estimate

Therefore, under the same historical trend:

P_geometric > P_incremental > P_arithmetic

However, this is a general comparative tendency, not an absolute rule for every dataset.

Comparison Table

5. Solved Numerical Examples

Problem 1: Comparison of Arithmetic, Geometric and Incremental Methods

Question

Census data for a town:

Estimate the population in 2040, using:

1. Arithmetic Increase Method

2. Incremental Increase Method

3. Geometric Increase Method

Step 1: Census Analysis

Step 2: Average Values

Average decadal increase

x̄ = (40,000 + 50,000 + 60,000) / 3

= 50,000 per decade

Average incremental increase

ȳ = (10,000 + 10,000) / 2

= 10,000 per decade

Step 3: Geometric Mean Growth Rate

rᵍ = (50.00 × 41.67 × 35.29)^(1/3)

≈ 41.89% per decade

Step 4: Number of Decades

From 2020 to 2040:

n = (2040 − 2020) / 10

= 2 decades

1. Arithmetic Increase Method

P₂₀₄₀ = P₂₀₂₀ + n × x̄

= 230,000 + (2 × 50,000)

= 230,000 + 100,000

= 330,000 persons

2. Incremental Increase Method

P₂₀₄₀ = P₂₀₂₀ + n × x̄ + [n(n + 1)/2] × ȳ

= 230,000 + (2 × 50,000) + [2(3)/2 × 10,000]

= 230,000 + 100,000 + 30,000

= 360,000 persons

3. Geometric Increase Method

P₂₀₄₀ = P₂₀₂₀ × (1 + rᵍ/100)ⁿ

= 230,000 × (1 + 41.89/100)²

= 230,000 × (1.4189)²

≈ 463,059 persons

Answer Summary

Interpretation

The three methods give significantly different results because they make different assumptions about future growth.

Arithmetic → lowest

Incremental → intermediate

Geometric → highest

Problem 2: Saturation Population Using Logistic Method

Question

Population data:

• 1980: P₀ = 100,000

• 2000: P₁ = 160,000

• 2020: P₂ = 205,000

The intervals are equal.

Calculate the saturation population Pₛ.

Step 1: Formula

Pₛ = [2P₀P₁P₂ − P₁²(P₀ + P₂)] / [P₀P₂ − P₁²]

Step 2: Substitute Values

Pₛ = [2(100,000)(160,000)(205,000) − (160,000)²(100,000 + 205,000)]

÷ [(100,000)(205,000) − (160,000)²]

Numerator — First Term

2 × 100,000 × 160,000 × 205,000

= 6,560,000,000,000,000

Numerator — Second Term

(160,000)² × 305,000

= 7,808,000,000,000,000

Therefore:

Numerator = −1,248,000,000,000,000

Denominator

(100,000 × 205,000) − (160,000)²

= 20,500,000,000 − 25,600,000,000

= −5,100,000,000

Step 3: Saturation Population

Pₛ = (−1,248,000,000,000,000) / (−5,100,000,000)

≈ 244,706 persons

Answer

Saturation population ≈ 244,706 persons

This represents the estimated population limit under the assumed logistic-growth model.

Problem 3: Infrastructure Sizing — Water & Sewage Load

Question

A city has:

• Present population = 250,000

• Base year = 2020

• Design year = 2050

• Average decadal increase = 40,000

• Average incremental increase = 5,000

• Water demand = 135 LPCD

• Sewage generation = 80% of water supplied

Calculate:

1. Population in 2050

2. Water requirement in MLD

3. Sewage flow in MLD

Step 1: Population Projection

Number of decades:

n = (2050 − 2020) / 10

= 3 decades

Formula:

Pₙ = P₀ + n × x̄ + [n(n + 1)/2] × ȳ

P₂₀₅₀ = 250,000 + (3 × 40,000) + [3(4)/2 × 5,000]

= 250,000 + 120,000 + 30,000

= 400,000 persons

Step 2: Water Requirement

Water demand:

Q_water = Population × Per capita demand

= 400,000 × 135

= 54,000,000 litres/day

Since:

1 MLD = 1,000,000 litres/day

Therefore:

Q_water = 54,000,000 / 1,000,000

= 54 MLD

Step 3: Sewage Generation

Assuming 80% of supplied water becomes wastewater:

Q_sewage = 0.80 × Q_water

= 0.80 × 54

= 43.2 MLD

Answer

Projected 2050 population = 400,000 persons

Water requirement = 54 MLD

Sewage flow = 43.2 MLD

6. Formula Summary & Quick Reference Sheet

7. Final Memory Map

POPULATION FORECASTING
        │
        ├── Arithmetic
        │      ↓
        │   Same AMOUNT
        │   → Linear
        │   → Mature city
        │
        ├── Geometric
        │      ↓
        │   Same %
        │   → Exponential
        │   → Fast-growing city
        │
        ├── Incremental
        │      ↓
        │   Increase + change in increase
        │   → Intermediate estimate
        │
        ├── Logistic
        │      ↓
        │   Growth → Deceleration → Saturation
        │   → Pₛ
        │
        └── Master Plan / Ratio
               ↓
            Planned / comparative growth

One-Line Exam Recall

Arithmetic = constant amount

Geometric = constant percentage

Incremental = average increase + incremental increase

Logistic = growth limited by saturation population

Population forecast → future demand → infrastructure capacity