A flow of hot air is cooled in a heat exchanger with chilled water in a closed-loop system. The water is chilled in a refrigeration system as shown in Figure P6.20. The total length of pipe transporting the water is 4000 ft. The friction factor in the pipe is 0.018.

The heat transfer rate at the evaporator of the refrigeration system is 350 tons. In order to maintain this heat transfer duty, the average temperature between the chilled water supply and the warm return water must be,

In order to overcome the friction effects in the pipe, the pump must provide a head of 70 ft, which can be considered independent of the mass flow rate of the water.
It is desired to design this system such that the cost is minimized. The cost can be considered to be made up of three components. The initial cost of the pipe is,

In this equation, b1 = 91,450 $/ft and D is the inside diameter of the pipe. The annual cost of the pumping power required to circulate the water is,

In this equation, b2 = 0.06 $-hr/lbm and m is the mass flow rate of the water. The annual cost of operating the refrigeration system used to produce the chilled water is,

Here, b3 = 41,300 $, b4 = −800 $/°F, and t1 is the chilled water temperature leaving the refrigeration system. This thermal energy system is expected to last 40 years and provide a rate of return of 18%.

a. Formulate the optimization problem and constraint(s).

b. Use the method of Lagrange multipliers to solve the optimization problem formulated in Part (a) and determine the inside diameter of the pipe that minimizes the total cost of the thermal energy system. What is the minimum present worth of the thermal energy system cost at the calculated diameter?

c. How does the velocity of the water in the pipe compare to the economic velocity computed in Table 4.18?
Without resolving the optimization problem, use the Lagrange multipliers to estimate the value of the objective function if the

d. Pump head was 71 ft instead of 70 ft

e. Heat transfer rate at the evaporator of the chiller was 351 kW instead of 350 kW It is unlikely that the diameter specified in Part (b) is a standard pipe size. If sch 40 commercial steel pipe is being used in this system,

f. Specify the nominal pipe size for this application. Justify your selection.

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T avg t+t. 2 = 50F