Mathematical programming states that any optimization problem can be defined as a process of minimization or maximization of an objective function in a space restricted by constraints. [Fox, 1971]
The objective function for duct optimization is the life-cycle cost, which is given by:
Life-cycle cost = (First year energy cost) (PWEF) + (Initial cost)
where,
PWEF is the present worth escalation factor.
The PWEF is:
PWEF = x 100%
where,
AER= annual escalation rate
AIR = annual interest rate
a= amortization period, years.
First year energy cost is determined by:
Energy cost = x (Fan Pressure)
Laboratory electrical unit energy cost depends on local industrial retail prices of electricity, including demand charges and consumption costs. The unit energy cost or electrical energy retail prices for all U.S. electric utilities can be obtained from Electric Sales and Revenue [EIA, 1995]. The costs are adjusted for 500-kW demand for industrial consumers, which includes laboratories. Data for the annual escalation rate (AER) are predicted by Utility Costs Forecasting [EIA, 1985 ref176] and Data Research Utility Costs Forecasting [Data Resources, Inc., 1985 ref170]. The accuracy of any calculation cannot be greater than the accuracy of the input data. Economic data are good only for current periods and cannot predict situations such as oil embargoes or Persian Gulf crises. Therefore, precise economic data are not needed for duct design. If the annual interest rate (AIR) is unknown, an interest rate of 6% can be used. If the amortization period (a) is unknown, 10 years can be used.
The initial cost includes the cost of ducts and HVAC equipment. The duct cost is presented as a function of the cost per unit area of duct surface, adjusted for straight ducts and fittings.
Installed duct prices are available from "Mechanical Cost Data" [Means, 1997]. These values are based upon a typical system layout, 25% of which is fittings. Duct costs include material, shop labor, field labor, shop drawings, shipping, and a 35% markup on costs for overhead and profit. Labor is figured at $26.50 per hour. More accurate optimization can be obtained by separating the cost of straight ducts from the cost of fittings. Cost data for fittings are also available from "Mechanical Cost Data" [Means, 1997].
The main equipment included in the objective function is a system's central air-handling unit. The pressure loss of duct-mounted equipment (coils, silencers, terminal control boxes) is included in the duct sections where this equipment is located.
An important factor in duct optimization is the cost of space required by ducts and equipment. This cost can be ignored if the space cannot be otherwise utilized. However, if saved space could be utilized, its cost must be included in the objective function. Including this additional cost could lead to reducing the size of ducts and thus increasing energy consumption.
Electrical energy retail prices vary widely. The maximum difference in electric energy costs between industrial customers in Saint Paul City, Alaska (50.66 c/kWh) and Douglas County, Washington state (1.62 c/kWh) is a factor of 31 to 1. Costs for ductwork range from $12.02 per square foot for 10-gauge galvanized iron to $3.10 per square foot for 26-gauge spiral ducts, a ratio of 3.9 to 1 (Wendes, 1986). Combining the two ratios yields a potential factor of 122 to 1 depending on locale and type of ductwork. Because of the electrical energy and ductwork price variations, there is a great potential for reducing the life-cycle cost of different duct systems.
It is important in duct optimization to satisfy all necessary constraints. A detailed explanation of each constraint can be found in Tsal and Adler (1987). The constraints are:
