A DAPTACIÓN E IMPLANTACIÓN DEL P ORTAL N OVA LNX

Screening cost estimates are often based on a single variable representing the capacity or some physical measure of the design such as floor area in buildings, length of highways, volume of storage bins and production volumes of processing plants. Costs

do not always vary linearly with respect to different facility sizes. Typically, scale economies or diseconomies exist. If the average cost per unit of capacity is declining,

then scale economies exist. Conversely, scale diseconomies exist if average costs increase with greater size. Empirical data are sought to establish the economies of scale for various types of facility, if they exist, in order to take advantage of lower

costs per unit of capacity.

Let x be a variable representing the facility capacity, and y be the resulting construction cost. Then, a linear cost relationship can be expressed in the form:

(5.1)

where a and b are positive constants to be determined on the basis of historical data.

Note that in Equation (5.1), a fixed cost of y = a at x = 0 is implied as shown in Figure 5-2. In general, this relationship is applicable only in a certain range of the variable x,

such as between x = c and x = d. If the values of y corresponding to x = c and x = d are known, then the cost of a facility corresponding to any x within the specified range may be obtained by linear interpolation. For example, the construction cost of a school building can be estimated on the basis of a linear relationship between cost and floor area if the unit cost per square foot of floor area is known for school buildings

within certain limits of size.

Figure 5-2: Linear Cost Relationship with Economies of Scale

A nonlinear cost relationship between the facility capacity x and construction cost y can often be represented in the form:

(5.2)

where a and b are positive constants to be determined on the basis of historical data.

For 0 < b < 1, Equation (5.2) represents the case of increasing returns to scale, and for b ;gt 1, the relationship becomes the case of decreasing returns to scale, as shown in Figure 5-3. Taking the logarithm of both sides this equation, a linear relationship can

be obtained as follows:

Figure 5-3: Nonlinear Cost Relationship with increasing or Decreasing Economies of Scale

(5.3)

Although no fixed cost is implied in Eq.(5.2), the equation is usually applicable only for a certain range of x. The same limitation applies to Eq.(5.3). A nonlinear cost relationship often used in estimating the cost of a new industrial processing plant from

the known cost of an existing facility of a different size is known as the exponential rule. Let y_n be the known cost of an existing facility with capacity Q_n, and y be the estimated cost of the new facility which has a capacity Q. Then, from the empirical

data, it can be assumed that:

(5.4)

where m usually varies from 0.5 to 0.9, depending on a specific type of facility. A value of m = 0.6 is often used for chemical processing plants. The exponential rule

can be reduced to a linear relationship if the logarithm of Equation (5.4) is used:

(5.5)

or (5.6)

The exponential rule can be applied to estimate the total cost of a complete facility or the cost of some particular component of a facility.

Example 5-4: Determination of m for the exponential rule

Figure 5-4: Log-Log Scale Graph of Exponential Rule Example

The empirical cost data from a number of sewage treatment plants are plotted on a log-log scale for ln(Q/Qn) and ln(y/yn) and a linear relationship between these logarithmic ratios is shown in Figure 5-4.

For (Q/Q_n) = 1 or ln(Q/Q_n) = 0, ln(y/y_n) = 0; and for Q/Q_n = 2 or ln(Q/Qn) = 0.301, ln(y/yn) = 0.1765. Since m is the slope of the line in the figure, it can be determined from the geometric relation as follows:

For ln(y/yn) = 0.1765, y/yn = 1.5, while the corresponding value of Q/Q_n is 2. In words, for m = 0.585, the cost of a plant increases only

1.5 times when the capacity is doubled.

Example 5-5: Cost exponents for water and wastewater treatment plants[4]

The magnitude of the cost exponent m in the exponential rule provides a simple measure of the economy of scale associated with building extra capacity for future growth and system reliability for the present in the design of treatment plants. When

m is small, there is considerable incentive to provide extra capacity since scale economies exist as illustrated in Figure 5-3. When m is close to 1, the cost is directly proportional to the design capacity. The value of m tends to increase as the number of

duplicate units in a system increases. The values of m for several types of treatment plants with different plant components derived from statistical correlation of actual

construction costs are shown in Table 5-3.

TABLE 5-3 Estimated Values of Cost Exponents for Water Treatment Plants

Treatment plant type

Exponent m

Capacity range (millions of gallons per day)

1. Water treatment 0.67 1-100

2. Waste treatment

Primary with digestion (small) 0.55 0.1-10 Primary with digestion (large) 0.75 0.7-100

Trickling filter 0.60 0.1-20

Activated sludge 0.77 0.1-100

Stabilization ponds 0.57 0.1-100

Source: Data are collected from various sources by P.M. Berthouex. See the references in his article for the primary sources.

Example 5-6: Some Historical Cost Data for the Exponential Rule The exponential rule as represented by Equation (5.4) can be expressed

in a different form as:

where

If m and K are known for a given type of facility, then the cost y for a proposed new facility of specified capacity Q can be readily computed.

TABLE 5-4 Cost Factors of Processing Units for Treatment Plants Processing

unit

Unit of capacity

K Value (1968 $)

m value 1. Liquid processing

Oil separation mgd 58,000 0.84

Hydroclone degritter mgd 3,820 0.35

Primary sedimentation ft² 399 0.60

Furial clarifier ft² 700 0.57

Sludge aeration basin mil. gal. 170,000 0.50

Tickling filter ft² 21,000 0.71

Aerated lagoon basin mil. gal. 46,000 0.67

Equalization mil. gal. 72,000 0.52

Neutralization mgd 60,000 0.70

2. Sludge handling

Digestion ft³ 67,500 0.59

Vacuum filter ft² 9,360 0.84

Centrifuge lb dry

solids/hr 318 0.81

Source: Data are collected from various sources by P.M. Berthouex. See the references in his article for the primary sources.

The estimated values of K and m for various water and sewage treatment plant components are shown in Table 5-4. The K values are based on 1968 dollars. The

range of data from which the K and m values are derived in the primary sources should be observed in order to use them in making cost estimates.

As an example, take K = $399 and m = 0.60 for a primary sedimentation component in Table 5-4. For a proposed new plant with the primary sedimentation process having

a capacity of 15,000 sq. ft., the estimated cost (in 1968 dollars) is:

y = ($399)(15,000)^0.60 = $128,000.

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