Computation of the pressure drop of the refrigerant flowing in a pipe is only one step in the decision process of selecting the pipe size. Ultimately the decision of the size of a vapor pipe is economic, trading off the additional cost of a larger pipe against the saving in compressor energy over the equipment lifetime. For a given situation, the cost trends are as shown in Fig. 9.3 where all costs reflect present worth.
It may seem at first that for a given flow rate and refrigerant condition the optimum diameter of a long pipe will be greater than that of a short one. Richards7 showed, however, that by setting to zero the derivative of the total cost, the length cancels. A summary form of the equation representing the costs shown in Fig. 9.3 is:
where C1 incorporates the material and labor cost of the pipe installation and assumes that these costs are proportional to the pipe diameter. The C2Δp term is the present worth of the lifetime costs to overcome Δp. The constant C2 thus incorporates the number of hours of operation per year, the life of the facility, and the cost of money. When the equation for Δp, Eq. 9.1 or 9.2, is substituted into Eq. 9.5 for the existing flow rate, the new equation is:
To find the optimum diameter, differentiate Eq. 9.6 and set equal to zero:
The length L cancels, which shows that the optimum diameter is independent of length.
In principle, the optimization calculation, subject to such constraints as minimum diameter to achieve a certain velocity or maximum diameter to meet space limitations, could be performed on each design. Such an effort is not practical, and the best that can be hoped for is a periodic check of the optimum to accommodate shifts in cost of materials and energy.