Optimum pipe diameter, example

Operating line, humidifying towers 778 Optimum pipe diameter, example 371 — water velocity, heat exchanger 505 Orifice meter 244,246, 248... 

This will be possible for only a few practical design problems. The technique is illustrated in Example 1.1, and in the derivation of the formula for optimum pipe diameter in Chapter 5. The determination of the economic reflux ratio for a distillation column, which is discussed in Volume 2, Chapter 11, is an example of the use of a graphical procedure to find the optimum value. 

The sensitivity to the particular property how much will a small error in the property affect the design calculation. For example, it was shown in Chapter 4 that the estimation of the optimum pipe diameter is insensitive to viscosity. The sensitivity of a design method to errors in physical properties, and other data, can be checked by repeating the calculation using slightly altered values. 

Adequate information must be available for installed costs of piping and pumping equipment. Although suppliers quotations are desirable, published correlations may be adequate. Some data and references to other published sources ate given in Chapter 20. A simplification in locating the optimum usually is permissible by ignoring the costs of pumps and drivers since they are essentially insensitive to pipe diameter near the optimum value. This fact is clear in Example 6.8 for instance and in the examples worked out by Happel and Jordan (Chemical Process Economics, Dekkcr, New York, 1975). 

A classic example showing how added refinements can come into an analysis for optimum conditions is involved in the development of methods for determining optimum economic pipe diameter for transportation of fluids. The following analysis, dealing with economic pipe diameters, gives a detailed derivation to illustrate how simplified expressions for optimum conditions can be developed. Further discussion showing the effects of other variables on the sensitivity is also presented. 

The power requirement can, therefore, be calculated as a function of d and the cost obtained. The total cost per annum is then plotted against the diameter of pipe and the optimum conditions are found either graphically or by differentiation as shown in Example 8.9. 

As the process model is made more accurate and complicated, you can lose the possibility of obtaining an analytical solution of the optimization problem. For example, if (1) the pressure losses through the pipe fittings and valves are included in the model, (2) the pump investment costs are included as a separate term with a cost exponent (n) that is not equal to 1.0, (3) elevation changes must be taken into account, (4) contained solids are present in the flow, or (5) significant changes in density occur, the optimum diameter will have to be calculated numerically. 

---