## Table 111

Industrial Wastewater Production in Some Industries

Industry

Cattle

Canning

Dairy

Chicken

Pulp and paper

Meat packaging

Tanning

Steel

Production

50 L/head-day 40 m3/metric ton 80 L/head-day 0.5 L/head-day 700 m3/metric ton 20 m3/metric ton 80 m3/metric ton raw hides processed 290 m3/metric ton

### 1.5 population projection

To determine the flows that water and wastewater treatment facilities need to be designed 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. In determining the design flows for a community, the population in the future needs to be predicted. Knowledge of the population, then enables the determination of the flows. This section, therefore, deals with the various methods of predicting population.

Several methods are used for predicting population: arithmetic method, geometric method, declining rate of increase method, logistic method, and graphical comparison method. Each of these methods is discussed.

### 1.5.1 Arithmetic Method

This method assumes that the population at the present time increases at a constant rate. Whether or not this assumption is true in the past is, of course, subject to question. Thus, this method is applicable only for population projections a short term into the future such as up to thirty years from the present.

Let P be the population at any given year Y and ka be the constant rate. Therefore, dP

TY - ka (U7) Integrating from limits P - Pj to P - P2 and from Y - Yj to Y - Y2,

With ka known, the population to be predicted in any future year can be calculated using Equation (1.18). Call this population as Population and the corresponding year as Year. To project the population into the future, the most current values P2 and Y2 must be used as the basis for the projection. Thus,

Example 1.4 The population data for Anytown is as follows: 1980 - 15,000, and 1990 - 18,000. What will be the population in the year 2000? What is the value of ka?

Solution:

P2- P1 18000-15000 Q„„ a - ^ - 1990-1980 - 300 per year Ans

1.5.2 Geometric Method

In this method, the population at the present time is assumed to increase in proportion to the number at present. As in the case of the arithmetic method, whether or not this assumption held in the past is uncertain. Thus, the geometric method is also simply used for population projection purposes a short term into the future. Using the same symbols as before, the differential equation is dP dY

where kg is the geometric rate constant. Integrating the above equation from P = P1 to P = P2 and from Y = Y1 to Y = Y2, the following equation is obtained:

As was the case of the arithmetic method, this equation also needs two data points. Solving for kg,

Example 1.5 Repeat Example 1.4 using the geometric method. Solution:

lnP2-lnPx ln18,000 - ln15,000 _mo. kg = y2-Yx = kg = -19)9)0-19)880- = 0.0182 per year Ans ln (Population) = lnP2 + kg (Year - Y2) = ln (Population) = ln18,000 + 0.0182 (2000 - 1990)

Population = 21,593 people Ans 1.5.3 Declining-Rate-of-Increase Method

In this method, the community population is assumed to approach a saturation value. Thus, reckoned from the present time, the rate of increase will decline until it becomes zero at saturation. Letting Ps be the saturation population and kd be the rate constant (analogous to ka and kg), the differential equation is dP = kd (Ps - P) (1.25)

where P and Y are the same variable as before. This equation may be integrated twice: the first one, from P = P1 to P = P2 and Y = Y1 to Y = Y2 and the second one, from P = P2 to P = P3 and Y = Y2 to Y = Y3 Thus, ln (Ps - P2) = ln (Ps - P1) - kd (Y2- Y1) ln (Ps - P3) = ln (Ps - P2) - kd (Y3- Y2)

In the previous equations, F2 - F1 may be made equal to F3 - F2, whereupon the value of Ps may be solved for. The final equations including Population, are as follows:

Example 1.6 The population data for Anytown is as follows: 1980 - 15,000, 1990 - 18,000, and 2000 -20,000. What will be the population in the year 2020? What is the value of kd?

Solution:

Ps - Pl + P3>-2P1 - 15000 + 20000-2(18000) - 24,000 people

kd - 'Y^-Y PT-P = "10 24000- 18000 = 0.04 per year Ans

Population - Ps - (Ps - P3)e~k"{Ymr~Y,) = 24000- (24000 - 20000)e-004(2020—2000) = 22,203 people Ans

### 1.5.4 Logistic Method

If food and environmental conditions are at the optimum, organisms, including humans, will reproduce at the geometric rate. In reality, however, the geometric rate is slowed down by environmental constraints such as decreasing rate of food supply, overcrowding, death, and so on. In concept, the factor for the environmental constraints can take several forms, provided, it, in fact, slows down the growth. Let us write the geometric rate of growth again, dP dY

This equation states that without environmental constraints, the population grows unchecked, that is, geometric. kgP is the same as kgP( 1). To enforce the environmental constraint, kgP should be multiplied by a factor less than 1. This means that the growth rate is no longer geometric but is retarded somewhat. In the logistic method the factor 1 is reduced by P/K, where K is called the carrying capacity of the environment and the whole factor, 1 - P/K, is called environmental resistance. The logistic differential equation is therefore dY

where kg has changed to k.. Rearranging, dP

Substituting Equation (1.32) in Equation (1.31) and integrating twice: first, between the limits of P = P1 to P = P2 and Y = Yj to Y = Y2 and second, between the limits of P = P2 to P = P3 and Y = Y2 to Y = Y3 produce the respective equations,

Solving for kh

21K-P3)

As in the declining-rate-of-increase method, Y2 - Y1 may be made equal to Y3 - Y2, whereupon the value of K may be solved for. The final equations, including Population, are as follows:

p2- P1P3

K - P3 [ 1- e 3 ] Example 1.7 Repeat Example 1.6 using the logistic method.

Solution:

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