Validación de la plataforma propuesta

It is generally claimed that the Darcy-Weisbach equation is superior because it is theoretically based, whereas both the Manning equation and the Hazen-Williams expression use empirically determined resistance coefficients. Although it is true that the functional relationship of the

TABLE 2.2 Unit Coefficient C_u for the Hazen-Williams Equation

Darcy-Weisbach formula reflects logical associations implied by the dimensions of the various terms, determination of the equivalent uniform sand-grain size is essentially experimental. Consequently, the relative roughness parameter used in the Moody diagram or the Colebrook-White equations is not theoretically determined. In this section, the Darcy-Weisbach and Hazen-Williams equations are compared briefly using a simple pipe as an example.

In the hydraulic rough range, the increase in ∆hf can be explained easily when the ratio of Eq. (2.16) to Eq. (2.22) is investigated. For hydraulically rough flow, Eq. (2.18) can be simplified by neglecting the second term 2.51 () of the logarithmic argument. This ratio then takes the form of

(2.23) which shows that in most hydraulic rough cases, for the same discharge Q, a larger head loss h_f is predicted using Eq. (2.16) than when using Eq. (2.22). Alternatively, for the same headloss, Eq. (2.22) returns a smaller discharge than does Eq. (2.16).

When comparing headloss relations for the more general case, a great fuss is often made over unimportant issues. For example, it is common to plot various equations on the Moody diagram and comment on their differences. However, such a comparison is of secondary importance. From a hydraulic perspective, the point is this: Different equations should still produce similar head-discharge behavior. That is, the physical relation between headloss and flow for a physical segment of pipe should be predicted well by any practical loss relation. Said even more simply, the issue is how well the hf versus Q curves compare.

To compare the values of h_f determined from Eq. (2.16) and those from Eq. (2.22), consider a pipe for which the parameters D, L, and C are specified. Using the Hazen-Williams relation, it is then possible to calculate hf for a given Q. Then, the Darcy-Weisbach f can be obtained, and with the Colebrook formula Eq. (2.18), the equivalent value of roughness e can be found. Finally, the variation of head with discharge can be plotted for a range of flows.

FIGURE 2.4 Flow in series and parallel pipes.

This analysis is performed for two galvanized iron pipes with e=0.15 mm. One pipe has a diameter of 0.1 m and a length of 100 m; and the dimensions of the other pipe are D=1.0 m and L=1000 m, respectively. The Hazen-Williams C for galvanized iron pipe is approximately 130. Different C values will be used for these two pipes to demonstrate the shift and change of the range within which ∆hf is small. The results of the calculated h_f-Q relation and the difference ∆h^f of the headloss of the two methods for the same discharge are shown in Figs. 2.5 and 2.6.

If h_f,_DW denotes the headloss determined by using Eq. (2.16) and h_f,_HW, then using Eq.

(2.22), ∆h^f (m) can be

(2.24) whereby the Darcy-Weisbach headloss h_f,_DW is used as a reference for comparison.

FIGURE 2.5 Comparison of Hazen-Williams and Darcy-Weisbach loss relations (smaller diameter).

FIGURE 2.6 Comparison of Hazen-Williams and Darcy-Weisbach loss relations (larger diameter).

Figures 2.5 and 2.6 show the existence of three ranges: two ranges, within which hf,DW

>h_f,DW, and the third one for which h_{f ,DW}<h_{f DW}. The first range of h_f,DW>h_f,DW is at a lower headloss and is small. It seems that the difference of ∆h^f in this case is the result of the fact that the Hazen-Williams formula is not valid for the hydraulic smooth and the smooth-to-transitional region. Fortunately, this region is seldom important for design purposes. At high headlosses, the Hazen-Williams formula tends to produce a discharge that is smaller than the one produced by the Darcy-Weisbach equation.

For a considerable part of the curve—primarily the range within which h_f,DW>h_{f ,DW} -∆hf is small compared with the absolute headloss. It can be shown that the range of small ∆h^f changes is shifted when different values of Hazen-Williams’s C are used for the calculation.

Therefore, selecting the proper value of C, which represents an appropriate point on the head-discharge curve, is essential. If such a C value is used, ∆h^f is small, and whether the Hazen-Williams formula or the Darcy-Weisbach equation is used for the design will be of little importance.

This example shows both the strengths and the weaknesses of using Eq. (2.22) as an approximation to Eq. (2.16). Despite its difficulties, the Hazen-Williams formula is often justified because of its conservative results and its simplicity of use. However, choosing a proper value of either the Hazen-Williams C or the relative roughness e/D is often difficult.

In the literature, a range of C values is given for new pipes made of various materials.

Selecting an appropriate C value for an old pipe is even more difficult. However, if an approximate value of C or e is used, the difference between the headloss equations is likely to be inconsequential.

Headloss also is a function of time. As pipes age, they are subject to corrosion, especially if they are made of ferrous materials and develop rust on the inside walls, which increases their relative roughness. Chemical agents, solid particles, or both in the fluid can gradually degrade the smoothness of the pipe wall. Scaling on the inside of pipes can occur if the water is hard. In some instances, biological factors have led to time-dependent headloss. Clams and zebra mussels may grow in some intake pipes and may, in some cases, drastically reduce discharge capacities.