Research Article Volume 2 Issue 6
1003 SW 142 PL, CP 33184 Miami, Florida, USA
Correspondence: Oscar Jimenez Medina, 1003 SW 142 PL, CP 33184 Miami, Florida, USA, Tel +305 (786) 2963 793
Received: November 14, 2018  Published: December 12, 2018
Citation: Medina OJ. General formula for the evaluation of linear load losses. Int J Hydro. 2018;2(6):726735. DOI: 10.15406/ijh.2018.02.00150
The main objective of this technical article is to unify the diversity of criteria, formulas, tables, diagrams, abacuses, photos, etc. They exist for calculating the coefficients of hydraulic resistances (CCH Chezy, nM Manning, fWD, WeisbachDarcy, CWH, Williams Hazen), and then evaluate the losses of linear load in the lines of any geometric shape, and working without pressure for review and search of the international literature and the Internet, respectively, the formula proposed by the French engineer recognized. A. Chézy in 1769, as the first, which is also considered as a paradigm of hydraulic channels. Until, in 1789, the Irish Engineer R. Manning presented his formula, which is most commonly used today. And the DarcyWeisbach formula which is considered to be of universal application and the Hazen Williams practiced in the case of water conveyance.
The author of this white paper, to conduct an analysis of the above equations and compare them with the general formula of fluid resistance, says that the latter has the attributes of all of them and with the advantage that it is applicable to any laminar or turbulent flow with and without pressure and for all possible cases geometrically duct. In other works, the author has exposed the deduction of the general law of fluid resistance from the fundamental equation of hydrodynamics (Bernoulli). That is the principle of energy applied to the fluid flow.
Keywords: load losses, hydraulic resistance coefficients
As an antecedent to mention, of the here proposed as the general formula for the computation of linear load losses. That is, the general law of fluid resistance (1765). It is the fundamental equation of hydrodynamics, (Bernoulli, 1738), which is the origin of it. It is necessary to clarify that the general formula of fluid resistance is the foundation of the equations of.^{1} The equation proposed here can be used to solve an infinity of the theoreticalpractical problems of the most important that occur in hydraulics in a general way as it is the determination of the linear load losses in the pipes. How often students, designers, researchers, etc. We have seen the need to select a method to calculate the head losses in a given hydraulic problem, sometimes it is more difficult to select the method to be used than to give the solution to the problem. Unbelievably often the situation is solved in such a simple way that we have overlooked it, this case is one of them. The author states that it would be very healthy to use the general formula of fluid resistance to calculate linear load losses, because this provides the results that best represent the real conditions of the problem, because it is a law, that is, it takes into account the relationships between the elements that participate in the phenomenon. The author cites the article. ID (0229NS), "General formulas for the Chezy and Manning coefficients". In which it was demonstrated that these are only particular cases applicable conceptually applicable to the category of full turbulent flow, (rough). That is, when the pair, (Re, ε/Di), is located in the zone of complete turbulence, (quadratic resistance zone in the Moody diagram). On or above the dashed line. This proposal pursues, obtaining the most accurate and accurate results of the problem analyzed in a simple and quick way within the existing limitations in the solution of this problem.
The deductive method is used. The author acknowledges that it is recurrent in relation to the deduction of the general formula of fluid resistance based on the fundamental equation of hydrodynamics, (Bernoulli). The fundamental reasons are, the Bernoulli equation, is the law of conservation of energy and / or conservation of the amount of movement applied to the flow of fluids and because one of the main questions of hydraulics is solved efficiently and correctly, as is the determination of linear load losses. Not by insisting there is unnecessary repetition. The undersigned stresses that, the WeisbachDarcy formula, is a particular case of the general law of fluid resistance, for the calculation of linear load losses in pipes fully filled.
$hf={C}_{R}*\frac{L}{{R}_{h}}*\frac{{V}^{2}}{2g}={f}_{DW}*\frac{L}{{D}_{i}}*\frac{{V}^{2}}{2g}=4{C}_{R}*\frac{L}{4{R}_{h}}*\frac{{V}^{2}}{2g}$
Observar:
${\tau}_{0}={C}_{R}\ast \rho \ast \frac{{V}^{2}}{2}$
${\tau}_{0}=\frac{{f}_{WD}}{4}\ast \rho \ast \frac{{V}^{2}}{2}$
& ${\tau}_{0}=\gamma \ast {R}_{h}\ast S$
${C}_{R}\ast \rho \ast \frac{{V}^{2}}{2}=\rho \ast g\ast {R}_{h}\ast S=\frac{{f}_{WD}}{4}\ast \rho \ast \frac{{V}^{2}}{2}$
Por tanto:
$S={C}_{R}\ast \frac{1}{{R}_{h}}\ast \frac{{V}^{2}}{2g}={f}_{WD}\ast \frac{L}{Di}\ast \frac{{V}^{2}}{2g},{f}_{WD}=4{C}_{R},y,Di=4{R}_{h}$
Deduction of the general form of fluid resistance shows in Figure 1
${P}_{1}A{P}_{2}A\gamma ALSen\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha ={\tau}_{0}PL$
$\xf7\gamma A$
$\frac{{P}_{1}A}{\gamma A}\frac{{P}_{2}A}{\gamma A}\frac{\gamma ALSen}{\gamma A}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\frac{{\tau}_{0}PL}{\gamma A}$
$\frac{{P}_{1}A}{\gamma A}\frac{{P}_{2}A}{\gamma A}LSen\text{\hspace{0.17em}}\alpha =\frac{{\tau}_{0}PL}{\gamma A}$
$Sen\text{\hspace{0.17em}}\alpha =\left(\frac{{h}_{2}{h}_{1}}{L}\right)$
$\left[\left(\frac{{P}_{1}}{\gamma}\frac{{P}_{2}}{\gamma}\right)({h}_{2}{h}_{1})\right]=hf$
$hf=\frac{{\tau}_{0}*P*L}{\gamma *A}$
${\tau}_{0}={C}_{R}\ast \rho \ast \frac{{V}^{2}}{2}$
$\frac{P}{A}=\frac{1}{R}$
$\frac{\rho}{\gamma}=\frac{1}{g}$
$hf={C}_{R}*\frac{L}{R}*\frac{{V}^{2}}{2g}$
That is the general law of fluid resistance.
By means of calculations in Excel, using the general formula of fluid resistance, in order to evaluate linear load losses later, we will demonstrate the veracity of the foregoing.
Before proceeding with the examples, we want to specify the scope and limitation of the formulas discussed above.
Considered as a paradigm of channel hydraulics, it is a particular case, conceptually valid for the category of full turbulent flow, (rough). Coincides with the zone of complete turbulence in the Moody diagram, are the points that are located on or above the dashed line, (the influence of the Reynolds number is ignored).
It has the same scope and limitation as Chezy's. But it has been the most used in recent times in free conductions. The caveat is made, that if in 1 and 2, the formulas proposed by the author in the article are used. ID (0229NS), "General formulas for the Chezy and Manning coefficients". The results are correct, that is, they coincide with those of the formula proposed here.
It is the general equation of the fluid resistance, but to be used specifically in pipes working under pressure, it is valid for the three possible categories of turbulent flow, (full, transitional and smooth), that is to say for the three zones of the Moody diagram, (quadratic resistance, transition and curve for smooth tubes).
It is the law for the evaluation of linear load losses. That is, valid for all possible cases of hydraulic problems of linear load losses. Calculation by trial and error of the dimensions of the sections, triangular, rectangular, trapezoidal and partially circular and completely filled respectively. Data and results of the conductions. Q, Ks, γ, S. Same for all examples. (for the rectangular case, the channel is real).
Ex.1: Triangular cannel (Table 1).
Q_{d} 
Ks 
n 
h 
m 
0.0297 
0.00025 
0.000001 
0.1634 
1.5 
0.0327 
0.00025 
0.000001 
0.16947 
1.5 
0.0483 
0.00025 
0.000001 
0.19645 
1.5 
0.0511 
0.00025 
0.000001 
0.2007 
1.5 
0.0628 
0.00025 
0.000001 
0.21703 
1.5 
0.0655 
0.00025 
0.000001 
0.22052 
1.5 
0.0722 
0.00025 
0.000001 
0.22882 
1.5 
0.0874 
0.00025 
0.000001 
0.24605 
1.5 
0.1024 
0.00025 
0.000001 
0.2613 
1.5 
0.1075 
0.00025 
0.000001 
0.26618 
1.5 
A 
P 
R 
V_{r} 
Re_{m} 
C_{Rm} 
C_{CHm} 
n_{m} 
0.04005 
0.58915 
0.06798 
0.742 
201647 
0.00521 
61.341 
0.01041 
0.04308 
0.61103 
0.0705 
0.759 
214064 
0.00516 
61.659 
0.01042 
0.05789 
0.70831 
0.08173 
0.834 
272762 
0.00495 
62.941 
0.01047 
0.06042 
0.72363 
0.0835 
0.846 
282463 
0.00492 
63.126 
0.01047 
0.07065 
0.78251 
0.09029 
0.889 
321017 
0.00482 
63.801 
0.0105 
0.07294 
0.7951 
0.09174 
0.898 
329520 
0.0048 
63.939 
0.0105 
0.07854 
0.82502 
0.09519 
0.919 
350051 
0.00475 
64.257 
0.01052 
0.09081 
0.88715 
0.10236 
0.962 
394073 
0.00466 
64.88 
0.01054 
0.10242 
0.94213 
0.10871 
1 
434759 
0.00459 
65.395 
0.01056 
0.10628 
0.95973 
0.11074 
1.012 
448045 
0.00457 
65.553 
0.01057 

C_{Rm} 


fwd 


fwd 
S_{m} 
a 
Suα 
Su 
α 
Suα 
0.02086 
0.00215 
1.04472 
0.002246 
0.00215 
1.04472 
0.002246 
0.02064 
0.00215 
1.04429 
0.002245 
0.00215 
1.04429 
0.002245 
0.01981 
0.00215 
1.04258 
0.002242 
0.00215 
1.04258 
0.002242 
0.01969 
0.00215 
1.04234 
0.002241 
0.00215 
1.04234 
0.002241 
0.01928 
0.00215 
1.04149 
0.002239 
0.00215 
1.04149 
0.002239 
0.0192 
0.00215 
1.04132 
0.002239 
0.00215 
1.04132 
0.002239 
0.01901 
0.00215 
1.04093 
0.002238 
0.00215 
1.04093 
0.002238 
0.01864 
0.00215 
1.04018 
0.002236 
0.00215 
1.04018 
0.002236 
0.01835 
0.00215 
1.03958 
0.002235 
0.00215 
1.03958 
0.002235 
0.01826 
0.00215 
1.0394 
0.002235 
0.00215 
1.0394 
0.002235 
Table 1 Triangular channel
Ex.2: Canal rectangular (Table 2).
Qd 
Ks 
n 
g 
b 
h 
m 
0.0297 
0.00025 
0.000001 
9.81 
0.4 
0.10097 
0 
0.0327 
0.00025 
0.000001 
9.81 
0.4 
0.10803 
0 
0.0483 
0.00025 
0.000001 
9.81 
0.4 
0.14293 
0 
0.0511 
0.00025 
0.000001 
9.81 
0.4 
0.14894 
0 
0.0628 
0.00025 
0.000001 
9.81 
0.4 
0.17345 
0 
0.0655 
0.00025 
0.000001 
9.81 
0.4 
0.179 
0 
0.0722 
0.00025 
0.000001 
9.81 
0.4 
0.19263 
0 
0.0874 
0.00025 
0.000001 
9.81 
0.4 
0.22285 
0 
0.1024 
0.00025 
0.000001 
9.81 
0.4 
0.252 
0 
0.1075 
0.00025 
0.000001 
9.81 
0.4 
0.2618 
0 
A 
P 
R 
V 
Re_{m} 
C_{Rm} 
C_{CHm} 
n_{m} 
0.04039 
0.60194 
0.0671 
0.73537 
197362 
0.00523 
61.227 
0.01041 
0.04321 
0.61606 
0.07014 
0.75673 
212317 
0.00517 
61.615 
0.01042 
0.05717 
0.68586 
0.08336 
0.84482 
281690 
0.00493 
63.112 
0.01047 
0.05958 
0.69788 
0.08537 
0.85773 
292887 
0.00489 
63.318 
0.01048 
0.06938 
0.7469 
0.09289 
0.90516 
336323 
0.00478 
64.046 
0.01051 
0.0716 
0.758 
0.09446 
0.9148 
345646 
0.00476 
64.19 
0.01051 
0.07705 
0.78526 
0.09812 
0.93703 
367776 
0.00471 
64.517 
0.01053 
0.08914 
0.8457 
0.1054 
0.98048 
413385 
0.00463 
65.131 
0.01055 
0.1008 
0.904 
0.1115 
1.01587 
453097 
0.00456 
65.612 
0.01057 
0.10472 
0.9236 
0.11338 
1.02655 
465570 
0.00454 
65.754 
0.01058 

C_{Rm} 


fc 


fc 
Su 
α 
Suα 
Su 
α 
Suα 
0.02093 
0.00215 
1.04488 
0.002246 
0.00215 
1.04488 
0.002246 
0.02067 
0.00215 
1.04435 
0.002246 
0.00215 
1.04435 
0.002246 
0.0197 
0.00215 
1.04236 
0.002241 
0.00215 
1.04236 
0.002241 
0.01958 
0.00215 
1.0421 
0.00224 
0.00215 
1.0421 
0.00224 
0.01913 
0.00215 
1.04119 
0.002239 
0.00215 
1.04119 
0.002239 
0.01905 
0.00215 
1.04101 
0.002238 
0.00215 
1.04101 
0.002238 
0.01885 
0.00215 
1.04061 
0.002237 
0.00215 
1.04061 
0.002237 
0.0185 
0.00215 
1.03989 
0.002236 
0.00215 
1.03989 
0.002236 
0.01823 
0.00215 
1.03933 
0.002234 
0.00215 
1.03933 
0.002234 
0.01815 
0.00215 
1.03917 
0.002234 
0.00215 
1.03917 
0.002234 
Table 2 Triangular channel
Ex.3: Canal trapezoidal (Table 3).

Ks 
n 
g 
b 
h 
m 
0.0297 
0.00025 
0.000001 
9.81 
0.4 
0.08174 
1.5 
0.0327 
0.00025 
0.000001 
9.81 
0.4 
0.08634 
1.5 
0.0483 
0.00025 
0.000001 
9.81 
0.4 
0.1075 
1.5 
0.0511 
0.00025 
0.000001 
9.81 
0.4 
0.11092 
1.5 
0.0628 
0.00025 
0.000001 
9.81 
0.4 
0.1243 
1.5 
0.0655 
0.00025 
0.000001 
9.81 
0.4 
0.1272 
1.5 
0.0722 
0.00025 
0.000001 
9.81 
0.4 
0.13416 
1.5 
0.0874 
0.00025 
0.000001 
9.81 
0.4 
0.1488 
1.5 
0.1024 
0.00025 
0.000001 
9.81 
0.4 
0.162 
1.5 
0.1075 
0.00025 
0.000001 
9.81 
0.4 
0.16625 
1.5 

P 
R 
V 
Re_{m} 
C_{Rm} 
C_{CHm} 
n_{m} 
0.04272 
0.69472 
0.06149 
0.69525 
171005 
0.00537 
60.464 
0.01039 
0.04572 
0.7113 
0.06427 
0.71526 
183888 
0.0053 
60.851 
0.0104 
0.06033 
0.7876 
0.07661 
0.80054 
245303 
0.00504 
62.381 
0.01045 
0.06282 
0.79993 
0.07854 
0.8134 
255523 
0.00501 
62.596 
0.01045 
0.0729 
0.84817 
0.08594 
0.8615 
296167 
0.00488 
63.376 
0.01048 
0.07515 
0.85863 
0.08752 
0.87159 
305139 
0.00486 
63.533 
0.01049 
0.08066 
0.88372 
0.09128 
0.89509 
326800 
0.00481 
63.895 
0.0105 
0.09273 
0.93651 
0.09902 
0.9425 
373302 
0.0047 
64.595 
0.01053 
0.10417 
0.9841 
0.10585 
0.98305 
416218 
0.00462 
65.167 
0.01055 
0.10796 
0.99942 
0.10802 
0.99575 
430248 
0.0046 
65.34 
0.01056 

CR_{m} 


Fwd 


fc 
Su 
α 
Suα 
Su 
α 
Suα 
0.02147 
0.00215 
1.04597 
0.002249 
0.00215 
1.04597 
0.002249 
0.02119 
0.00215 
1.04541 
0.002247 
0.00215 
1.04541 
0.002247 
0.02017 
0.00215 
1.04331 
0.002243 
0.00215 
1.04331 
0.002243 
0.02003 
0.00215 
1.04303 
0.002243 
0.00215 
1.04303 
0.002243 
0.01954 
0.00215 
1.04202 
0.00224 
0.00215 
1.04202 
0.00224 
0.01944 
0.00215 
1.04182 
0.00224 
0.00215 
1.04182 
0.00224 
0.01922 
0.00215 
1.04137 
0.002239 
0.00215 
1.04137 
0.002239 
0.01881 
0.00215 
1.04052 
0.002237 
0.00215 
1.04052 
0.002237 
0.01848 
0.00215 
1.03984 
0.002236 
0.00215 
1.03984 
0.002236 
0.01838 
0.00215 
1.03964 
0.002235 
0.00215 
1.03964 
0.002235 
Table 3 Canal trapezoidal
Ex.4: Circular canal. (Partially filled pipe) (Table 4).
Q_{d} 
Ks 
n 
g 
Di 
h/Di 
h 
0.0297 
0.00025 
0.000001 
9.81 
0.23019 
0.93 
0.21408 
0.0327 
0.00025 
0.000001 
9.81 
0.23872 
0.93 
0.22201 
0.0483 
0.00025 
0.000001 
9.81 
0.27674 
0.93 
0.25737 
0.0511 
0.00025 
0.000001 
9.81 
0.28272 
0.93 
0.26293 
0.0628 
0.00025 
0.000001 
9.81 
0.3057 
0.93 
0.2843 
0.0655 
0.00025 
0.000001 
9.81 
0.31062 
0.93 
0.28888 
0.0722 
0.00025 
0.000001 
9.81 
0.32234 
0.93 
0.29978 
0.0874 
0.00025 
0.000001 
9.81 
0.3466 
0.93 
0.32234 
0.1024 
0.00025 
0.000001 
9.81 
0.3681 
0.93 
0.34233 
0.1075 
0.00025 
0.000001 
9.81 
0.37494 
0.93 
0.34869 
Table 4 Canal circular
Observe:
For the maximum expense, (h/Di=0.95). <Say, that: For the maximum speed, (h/Di=0.813) and that: For the pipeline occupied halfway, (h/Di=0.50).
Ex.4.1: For the maximum expense, (h/Di=0.95).
b

A 
P 
R 
V 
Re 
C_{R} 
n 
149.3166 
0.04034 
0.59989 
0.06724 
0.736 
198036 
0.00523 
0.01041 
149.3166 
0.04338 
0.62212 
0.06973 
0.754 
210249 
0.00518 
0.01042 
149.3166 
0.0583 
0.7212 
0.08084 
0.828 
267886 
0.00497 
0.01046 
149.3166 
0.06085 
0.73679 
0.08258 
0.84 
277421 
0.00494 
0.01047 
149.3166 
0.07114 
0.79667 
0.08929 
0.883 
315311 
0.00483 
0.01049 
149.3166 
0.07345 
0.8095 
0.09073 
0.892 
323658 
0.00481 
0.0105 
149.3166 
0.07909 
0.84004 
0.09416 
0.913 
343793 
0.00477 
0.01051 
149.3166 
0.09145 
0.90326 
0.10124 
0.956 
387041 
0.00467 
0.01054 
149.3166 
0.10314 
0.95929 
0.10752 
0.993 
426981 
0.0046 
0.01056 
149.3166 
0.10701 
0.97712 
0.10952 
1.005 
440070 
0.00458 
0.01057 
Table 4.1 For the maximum expense, (h/Di=0.95)
For the maximum expense, (h/Di=0.95). >CR and <nM, that: For the maximum speed, (h/Di=0.813) and that: For the pipeline occupied halfway, (h/Di=0.50).
Ex.4.2: For the maximum speed, (h/Di=0.813).

C_{Rm} 


fc 


fc 
Su 
α 
Suα 
Su 
α 
Suα 
0.02092 
0.00215 
1.04486 
0.002246 
0.00215 
1.04486 
0.002246 
0.02071 
0.00215 
1.04442 
0.002246 
0.00215 
1.04442 
0.002246 
0.01987 
0.00215 
1.0427 
0.002242 
0.00215 
1.0427 
0.002242 
0.01975 
0.00215 
1.04246 
0.002241 
0.00215 
1.04246 
0.002241 
0.01934 
0.00215 
1.04161 
0.00224 
0.00215 
1.04161 
0.00224 
0.01925 
0.00215 
1.04144 
0.00224 
0.00215 
1.04144 
0.00224 
0.01906 
0.00215 
1.04104 
0.002238 
0.00215 
1.04104 
0.002238 
0.0187 
0.00215 
1.04029 
0.002236 
0.00215 
1.04029 
0.002236 
0.0184 
0.00215 
1.03969 
0.002235 
0.00215 
1.03969 
0.002235 
0.01832 
0.00215 
1.0395 
0.002235 
0.00215 
1.0395 
0.002235 
Table 4.2 For the maximum speed, (h / Di = 0.813)
For the maximum expense, (h/Di=0.95). > fWD, that: For the maximum speed, (h/Di=0.813). and that: For the pipeline occupied halfway, (h/Di=0.50).
Ex.4.3: For the pipeline occupied halfway, (h/Di=0.50).
Q_{d} 
Ks 
n 
g 
Di 
h/Di 
h 
0.0297 
0.00025 
0.000001 
9.81 
0.23019 
0.95 
0.21868 
0.0297 
0.00025 
0.000001 
9.81 
0.23765 
0.813 
0.1925 
0.0297 
0.00025 
0.000001 
9.81 
0.30714 
0.5 
0.15357 
β 
A 
P 
R 
V 
Re 
C_{R} 
n 
154.1581 
0.04084 
0.61934 
0.06594 
0.727 
191817 
0.00526 
0.01041 
128.3161 
0.03849 
0.53223 
0.07232 
0.772 
223213 
0.00512 
0.01043 
90 
0.03705 
0.48245 
0.07679 
0.802 
246241 
0.00504 
0.01045 


C_{Rm} 


fc 


C 
fc 
Su 
α 
Suα 
Su 
α 
Suα 
61.07531 
0.02104 
0.00215 
1.0451 
0.002247 
0.00215 
1.0451 
0.002247 
61.88031 
0.0205 
0.00215 
1.04398 
0.002245 
0.00215 
1.04398 
0.002245 
62.40084 
0.02015 
0.00215 
1.04329 
0.002243 
0.00215 
1.04329 
0.002243 
Table 4.3 For the pipeline occupied halfway, (h/Di=0.50)
Ex.5: Circular pipe. (Pipe completely filled) (Table 5).
Q_{d} 

Ks 

n 

g 

Di 

h/Di 

h 


0.0297 
0.00025 
0.000001 
9.81 
0.23626 
1 
0.23626 

0.0327 
0.00025 
0.000001 
9.81 
0.24503 
1 
0.24503 

0.0483 
0.00025 
0.000001 
9.81 
0.28398 
1 
0.28398 

0.0511 
0.00025 
0.000001 
9.81 
0.2901 
1 
0.2901 

0.0628 
0.00025 
0.000001 
9.81 
0.31367 
1 
0.31367 

0.0655 
0.00025 
0.000001 
9.81 
0.31872 
1 
0.31872 

0.0722 
0.00025 
0.000001 
9.81 
0.3307 
1 
0.3307 

0.0874 
0.00025 
0.000001 
9.81 
0.35558 
1 
0.35558 

0.1024 
0.00025 
0.000001 
9.81 
0.3776 
1 
0.3776 

0.1075 
0.00025 
0.000001 
9.81 
0.38465 
1 
0.38465 

β 
A 

P 

R 

V 

Re 

CR 


n 
180 
0.04384 
0.74223 
0.05907 
0.677 
160058 
0.00543 
0.01038 

180 
0.04716 
0.76978 
0.06126 
0.693 
169918 
0.00537 
0.01039 

180 
0.06334 
0.89215 
0.071 
0.763 
216556 
0.00515 
0.01042 

180 
0.0661 
0.91138 
0.07253 
0.773 
224276 
0.00512 
0.01043 

180 
0.07727 
0.98542 
0.07842 
0.813 
254916 
0.00501 
0.01045 

180 
0.07978 
1.00129 
0.07968 
0.821 
261663 
0.00499 
0.01046 

180 
0.08589 
1.03892 
0.08268 
0.841 
277980 
0.00494 
0.01047 

180 
0.0993 
1.11709 
0.0889 
0.88 
312957 
0.00484 
0.01049 

180 
0.11198 
1.18627 
0.0944 
0.914 
345285 
0.00476 
0.01051 

180 
0.1162 

1.20841 

0.09616 

0.925 

355838 

0.00474 


0.01052 


C_{Rm} 


fc 


C 
fc 
Su 
α 
Suα 
Su 
α 
Suα 
60.11113 
0.02172 
0.00215 
1.04648 
0.00225 
0.00215 
1.04648 
0.00225 
60.43056 
0.02149 
0.00215 
1.04602 
0.002249 
0.00215 
1.04602 
0.002249 
61.71968 
0.0206 
0.00215 
1.0442 
0.002245 
0.00215 
1.0442 
0.002245 
61.90532 
0.02048 
0.00215 
1.04395 
0.002245 
0.00215 
1.04395 
0.002245 
62.58354 
0.02004 
0.00215 
1.04305 
0.002243 
0.00215 
1.04305 
0.002243 
62.72184 
0.01995 
0.00215 
1.04286 
0.002242 
0.00215 
1.04286 
0.002242 
63.04129 
0.01975 
0.00215 
1.04245 
0.002242 
0.00215 
1.04245 
0.002242 
63.66725 
0.01936 
0.00215 
1.04166 
0.002239 
0.00215 
1.04166 
0.002239 
64.18469 
0.01905 
0.00215 
1.04102 
0.002238 
0.00215 
1.04102 
0.002238 
64.34348 
0.01896 
0.00215 
1.04082 
0.002237 
0.00215 
1.04082 
0.002237 
Table 5 Circular pipe (Pipe completely filled)
In the examples above, the veracity of everything expressed in relation to this equation is proved, confirming that it is sufficient for the purpose stated here. That is, to be general, (law), gives all and the best solutions. The general formula of fluid resistance, (law). It is the ideal equation that responds to one of the main questions of hydraulics, as is the correct evaluation of linear loa d losses in the pipes. Taking advantage of the space still available, the author wants to present something interesting in relation to the calculation examples made using Excel and the trial and error method. Observe in the table that follows the similarity of the results of the hydraulic resistance coefficients, (Cr, Cch, nM and fwd), for the different geometric shapes of the sections, (triangular, rectangular and circular, the latter working as channels and pipes). Read from left to right consecutively. Data and results of the conductions. Q, Ks, γ, S. Same for all examples.
Observe the similarity of, (V, Re, CR, CCH and nM), for the different geometric shapes of the sections, (triangular, rectangular, trapezoidal and circular). The difference between them is in the dimensions of the sections. As expected the most efficient is the circular (Table 6‒10). Observe the dimensions for the geometric shapes of the sections, (triangular, rectangular, trapezoidal and circular, the latter partially and completely filled). The difference between them is in the dimensions of the sections. The examples: 1, 2, 3 and 4, are (Table 10) conduits working without pressure, ie free channels or gravity, and example 5, is working with pressure, which we know as forced pipes. To conclude this article, the author as always humbly asks that they face all the problems and proposals that do not exist, they stop seeing its true dimension in its application, sometimes not perceived by us. That is, they are reviewed with an open mind, without prejudices, because all we pursue the same goal, take our profession to a higher level, to achieve better results, which leads to full satisfaction. As a general information, we present what was exposed by B Nekrasov2 in his book Hidráulica.
Mir Moscow 1968.
Data 
Qd 
Ks 
n 
Di 
h/Di 
b 
h 
m 
0.0297 
0.00025 
0.000001 


0 
0.1634 
1.5 

Results 
A 
P 
R 
Vr 
Re_{m} 
C_{Rm} 
C_{CHm} 
n_{m} 
0.04005 
0.58915 
0.06798 
0.742 
201647 
0.00521 
61.341 
0.01041 

fwd 
Sm 
a 
Suα 
Su 
α 
Suα 


0.02086 
0.00215 
1.04472 
0.002246 
0.00215 
1.04472 
0.002246 

C_{Rm} 
C_{CHmm} 
n_{m} 
fwd 





0.00521 
61.341 
0.01041 
0.02086 




Table 6 Canal triangular
Data 
Qd 
Ks 
n 
g 
b 
h 
m 

0.0297 
0.00025 
0.000001 
9.81 
0.4 
0.10097 
0 

Results 
A 
P 
R 
V 
Re_{m} 
C_{Rm} 
C_{CHm} 
n_{m} 
0.04039 
0.60194 
0.0671 
0.73537 
197362 
0.00523 
61.227 
0.01041 

fc 
Su 
α 
Suα 
Su 
α 
Suα 


0.02093 
0.00215 
1.04488 
0.002246 
0.00215 
1.04488 
0.002246 

C_{Rm} 
C_{CHm} 
n_{m} 
fc 






0.00523 
61.227 
0.01041 
0.02093 




Table 7 Canal rectangular
Data 
Q_{d} 
Ks 
n 
g 
b 
h 
m 

0.0297 
0.00025 
0.000001 
9.81 
0.4 
0.08174 
1.5 

Results 
A 
P 
R 
V 
Re_{m} 
C_{Rm} 
C_{CHm} 
n_{m} 
0.04272 
0.69472 
0.06149 
0.69525 
171005 
0.00537 
60.464 
0.01039 

fc 
Su 
α 
Suα 
Su 
α 
Suα 


0.021 4 
0.00215 
1.0451 
0.002247 
0.00215 
1.0451 
0.002247 

CR 
n 
C_{CHm} 
fc 






0.00526 
0.01041 
61.07531 
0.02104 




Table 8 Canal trapezoidal
Data 
Q_{d} 
Ks 
n 
g 
Di 
h/Di 
h 

0.0297 
0.00025 
0.000001 
9.81 
0.23019 
0.95 
0.21868 

Results 
β 
A 
P 
R 
V 
Re_{m} 
C_{R} 
n 
154.1581 
0.04084 
0.61934 
0.06594 
0.727 
191817 
0.00526 
0.01041 

C_{CHm} 
fc 
Su 
α 
Suα 
Su 
α 
Suα 

61.07531 
0.02104 
0.00215 
1.0451 
0.002247 
0.00215 
1.0451 
0.002247 

C_{R} 
n 
C_{CHm} 
fc 






0.00526 
0.01041 
61.07531 
0.02104 




Table 9 Circular canal parallely filled, (h/Di=0.95)
Data 
Q_{d} 
Ks 
n 
g 
Di 
h/Di 
h 

0.0297 
0.00025 
0.000001 
9.81 
0.23626 
1 
0.23626 

Results 
b 
A 
P 
R 
V 
Re 
CR 
n 
180 
0.04384 
0.74223 
0.05907 
0.677 
160058 
0.00543 
0.01038 

C 
fc 
Su 
α 
Suα 
Su 
α 
Suα 

60.11113 
0.02172 
0.00215 
1.04648 
0.00225 
0.00215 
1.04648 
0.00225 

CR 
n 
C 
fc 






0.00543 
0.01038 
60.11113 
0.02172 




V_{r} 
Re_{m} 
C_{Rm} 
C_{CHm} 
n_{m} 
fwd 
Sectión 
0.742 
201647 
0.00521 
61.341 
0.01041 
0.02086 
Triangular 
0.735 
197362 
0.00523 
61.227 
0.01041 
0.02093 
Rectangular 
0.695 
171005 
0.00537 
60.464 
0.01039 
0.02147 
Trapezoidal 
0.727 
191817 
0.00526 
61.075 
0.01041 
0.02104 
Circular no llena 
0.677 
160058 
0.00543 
60.111 
0.01038 
0.02172 
Circular llena 
1 
b 
h 
m 
b 
A 
P 

Sectión 
0 
0.1634 
1.5 
0.04005 
0.58915 
Triangular 

2 
b 
h 
m 
A 
P 
R 


0.4 
0.10097 
0 
0.04039 
0.60194 
0.0671 
Rectangular 

3 
b 
h 
m 
A 
P 
R 


0.4 
0.08174 
1.5 
0.04272 
0.69472 
0.06149 
Trapezoidal 

4 
Di 
h/Di 
h 
b 
A 
P 
R 

0.23019 
0.95 
0.21868 
154.1581 
0.04084 
0.61934 
0.06594 
Circ. does not fill 

5 
Di 
h/Di 
h 
b 
A 
P 
R 


0.23626 
1 
0.23626 
180 
0.04384 
0.74223 
0.05907 
Circular filled 
Table 10 Circular pipe completely filled
Textual quotation, pages, (84 and 85). "Hydraulic head losses in pressurized currents take place on account of the decrease in the potential specific energy of the liquid, (Z+P/ɣ) along the flow. In this case, if the specific kinetic energy of the liquid, (V2/2g), varies along the flow, it is not due to the load losses, but due to the channel, because the energy depends only on the speed and this it is determined by the expense and the area of the section, (V=Q/A). Therefore, in a constant section tube the average speed and the specific kinetic energy remain unchanged, despite the presence of hydraulic resistance and load height losses. The magnitude of the loss of height of load is determined by in this case by the difference in the indications of two piezometers". "The calculation of the losses of load for several concrete cases comes to be one of the main questions of the hydraulics". "The kinematic similarity is the similarity of the streamlines and the proportionality of the similar speeds. It is evident that for the kinematic similarity of the flows the geometric resemblance of the channels is indispensable". "The equality of the coefficients, α1 and α2, for similar sections of two flows derives from their kinematic similarity". "For the flows with geometric similarity the relation, (λ/do fwd/d), is the same, therefore, the condition of hydrodynamic similarity in this case consists of the equal value of the coefficient, (λ or fwd), for said flows ". "The hydraulic slope, (piezometric), is invariable along a straight tube of constant diameter". End of appointment. The application of the general law of fluid resistance to various problems of hydraulics is very convenient, because it has a solid and proven foundation.^{3‒10}
None.
The author declares that there are no conflicts of interest.
©2018 Medina. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work noncommercially.