Hot water is being transported in an insulated type K copper tube as shown in Figure P6.19. Due to space restrictions in the installation, the outside diameter of the insulation is to be minimized. The annual cost of the insulated pipe is limited to $80,000. The annual cost of the insulated pipe has two components; the cost of the heat loss, Cheat, and the cost of the pumping energy required to move the fluid through the tube, Cpump. The expressions for these costs are,

In these equations, a1 = 3000 $-m and a2 = 16 $-m5.

a. Write the formal optimization statement in the form of Equation set 6.26.

b. Solve the optimization problem using the method of Lagrange multipliers and determine the tube diameter (m) and insulation thickness (m) that results in the minimum outside diameter of the insulation.

The solution of the optimization problem from Part

(b) results in the inside tube diameter and insulation thickness that minimize the outside diameter of the insulation. However, the copper tube inside diameter calculated in Part

(b) is likely not a standard size. The next part of this problem deals with specifying the standard copper tube diameter for this application. One possibility is to consider the optimum outside insulation diameter calculated in Part

(b) as a condition that must be met. For this case,

c. Use the copper tube data in Appendix D to specify the standard tubing size for this case.

d. What is the corresponding insulation thickness (m) associated with the copper tube diameter specified in Part (c)?

e. What is the annual cost associated with the tubing size specified in Part (c)
and the insulation thickness specified in Part (d)?
A second possibility is to specify that the annual cost constraint must be met. For this case,

f. Use the copper tube data in Appendix D to specify the standard tubing size for this case.
g. What is the corresponding insulation thickness associated with the copper tube diameter specified in Part (f)?
h. What is the outside diameter of the insulation with the tubing size specified in part (f) and the insulation thickness specified in Part (g)?
From these exercises, it should be clear that using standard size tubing the optimum solution determined in Part (b) cannot be achieved. However, the fact remains that a standard tube size must be specified.
i. Which solution seems more viable, Parts (c) –(e) or Parts (f) – (h)? Explain your reasoning.

Transcribed Image Text:

Cheat t; ins and pump ID