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International Journal of
eISSN: 2576-4454

Hydrology

Research Article Volume 2 Issue 6

General formula for the evaluation of linear load losses

Oscar Jimenez Medina

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):726-735. DOI: 10.15406/ijh.2018.02.00150

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Abstract

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, fW-D, Weisbach-Darcy, 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 Darcy-Weisbach 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

Introduction

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 theoretical-practical 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.

Methodology

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 Weisbach-Darcy 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 * L R h * V 2 2g = f DW * L D i * V 2 2g =4 C R * L 4 R h * V 2 2g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAai aadAgacqGH9aqpcaWGdbWaaSbaaeaajugWaiaadkfaaKqbagqaaiaa cQcadaWcaaqaaiaadYeaaeaacaWGsbWcdaWgaaqcfayaaKqzadGaam iAaaqcfayabaaaaiaacQcadaWcaaqaaiaadAfalmaaCaaajuaGbeqa aKqzadGaaGOmaaaaaKqbagaacaaIYaGaam4zaaaacqGH9aqpcaWGMb WaaSbaaeaacaWGebGaeyOeI0Iaam4vaaqabaGaaiOkamaalaaabaGa amitaaqaaiaadsealmaaBaaajuaGbaqcLbmacaWGPbaajuaGbeaaaa GaaiOkamaalaaabaGaamOvamaaCaaabeqaaKqzadGaaGOmaaaaaKqb agaacaaIYaGaam4zaaaacqGH9aqpcaaI0aGaam4qaSWaaSbaaKqbag aajugWaiaadkfaaKqbagqaaiaacQcadaWcaaqaaiaadYeaaeaacaaI 0aGaamOuamaaBaaabaqcLbmacaWGObaajuaGbeaaaaGaaiOkamaala aabaGaamOvaSWaaWbaaKqbagqabaqcLbmacaaIYaaaaaqcfayaaiaa ikdacaWGNbaaaaaa@6B91@
Observar:
τ 0 = C R ρ V 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3cdaWgaaqcfayaaKqzadGaaGimaaqcfayabaGaeyypa0Jaam4qamaa BaaabaqcLbmacaWGsbaajuaGbeaacqGHxiIkcqaHbpGCcqGHxiIkda WcaaqaaiaadAfalmaaCaaajuaGbeqaaKqzadGaaGOmaaaaaKqbagaa caaIYaaaaaaa@4873@   τ 0 = f WD 4 ρ V 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3cdaWgaaqcfayaaKqzadGaaGimaaqcfayabaGaeyypa0ZaaSaaaeaa caWGMbWaaSbaaeaacaWGxbGaeyOeI0IaamiraaqabaaabaGaaGinaa aacqGHxiIkcqaHbpGCcqGHxiIkdaWcaaqaaiaadAfalmaaCaaajuaG beqaaKqzadGaaGOmaaaaaKqbagaacaaIYaaaaaaa@4963@  & τ 0 =γ R h S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3cdaWgaaqcfayaaKqzadGaaGimaaqcfayabaGaeyypa0Jaeq4SdCMa ey4fIOIaamOuaSWaaSbaaKqbagaajugWaiaadIgaaKqbagqaaiabgE HiQiaadofaaaa@4516@  

C R ρ V 2 2 =ρg R h S= f WD 4 ρ V 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qaS WaaSbaaKqbagaajugWaiaadkfaaKqbagqaaiabgEHiQiabeg8aYjab gEHiQmaalaaabaGaamOvaSWaaWbaaKqbagqabaqcLbmacaaIYaaaaa qcfayaaiaaikdaaaGaeyypa0JaeqyWdiNaey4fIOIaam4zaiabgEHi QiaadkfadaWgaaqaaKqzadGaamiAaaqcfayabaGaey4fIOIaam4uai abg2da9maalaaabaGaamOzamaaBaaabaGaam4vaiabgkHiTiaadsea aeqaaaqaaiaaisdaaaGaey4fIOIaeqyWdiNaey4fIOYaaSaaaeaaca WGwbWaaWbaaeqabaqcLbmacaaIYaaaaaqcfayaaiaaikdaaaaaaa@5B5A@
Por tanto:

S= C R 1 R h V 2 2g = f WD L Di V 2 2g , f WD =4 C R ,y,Di=4 R h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai abg2da9iaadoealmaaBaaajuaGbaqcLbmacaWGsbaajuaGbeaacqGH xiIkdaWcaaqaaiaaigdaaeaacaWGsbWcdaWgaaqcfayaaKqzadGaam iAaaqcfayabaaaaiabgEHiQmaalaaabaGaamOvaSWaaWbaaKqbagqa baqcLbmacaaIYaaaaaqcfayaaiaaikdacaWGNbaaaiabg2da9iaadA gadaWgaaqaaiaadEfacqGHsislcaWGebaabeaacqGHxiIkdaWcaaqa aiaadYeaaeaacaWGebGaamyAaaaacqGHxiIkdaWcaaqaaiaadAfalm aaCaaajuaGbeqaaKqzadGaaGOmaaaaaKqbagaacaaIYaGaam4zaaaa caGGSaGaeyOeI0IaamOzaSWaaSbaaKqbagaajugWaiaadEfacqGHsi slcaWGebaajuaGbeaacqGH9aqpcaaI0aGaam4qaSWaaSbaaKqbagaa jugWaiaadkfaaKqbagqaaiaacYcacaWG5bGaaiilaiabgkHiTiaads eacaWGPbGaeyypa0JaaGinaiaadkfalmaaBaaajuaGbaqcLbmacaWG ObaajuaGbeaaaaa@7069@

Deduction of the general form of fluid resistance shows in Figure 1

P 1 A P 2 AγALSenα= τ 0 PL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiuam aaBaaabaqcLbmacaaIXaaajuaGbeaacaWGbbGaeyOeI0Iaamiuamaa BaaabaqcLbmacaaIYaaajuaGbeaacaWGbbGaeyOeI0Iaeq4SdCMaam yqaiaadYeacaWGtbGaamyzaiaad6gacaaMc8UaaGPaVlaaykW7cqaH XoqycqGH9aqpcqaHepaDdaWgaaqaaKqzadGaaGimaaqcfayabaGaam iuaiaadYeaaaa@5400@
÷γA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaey49aG Raeq4SdCMaamyqaaaa@3B2C@
P 1 A γA P 2 A γA γALSen γA α= τ 0 PL γA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGqbWaaSbaaeaajugWaiaaigdaaKqbagqaaiaadgeaaeaacqaH ZoWzcaWGbbaaaiabgkHiTmaalaaabaGaamiuamaaBaaabaqcLbmaca aIYaaajuaGbeaacaWGbbaabaGaeq4SdCMaamyqaaaacqGHsisldaWc aaqaaiabeo7aNjaadgeacaWGmbGaam4uaiaadwgacaWGUbaabaGaeq 4SdCMaamyqaaaacaaMc8UaaGPaVlaaykW7cqaHXoqycqGH9aqpdaWc aaqaaiabes8a0naaBaaabaqcLbmacaaIWaaajuaGbeaacaWGqbGaam itaaqaaiabeo7aNjaadgeaaaaaaa@5DF4@

P 1 A γA P 2 A γA LSenα= τ 0 PL γA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGqbWaaSbaaeaajugWaiaaigdaaKqbagqaaiaadgeaaeaacqaH ZoWzcaWGbbaaaiabgkHiTmaalaaabaGaamiuamaaBaaabaqcLbmaca aIYaaajuaGbeaacaWGbbaabaGaeq4SdCMaamyqaaaacqGHsislcaWG mbGaam4uaiaadwgacaWGUbGaaGPaVlabeg7aHjabg2da9maalaaaba GaeqiXdq3aaSbaaeaajugWaiaaicdaaKqbagqaaiaadcfacaWGmbaa baGaeq4SdCMaamyqaaaaaaa@55F4@
Senα=( h 2 h 1 L ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aadwgacaWGUbGaaGPaVlabeg7aHjabg2da9maabmaabaWaaSaaaeaa caWGObWaaSbaaeaajugWaiaaikdaaKqbagqaaiabgkHiTiaadIgalm aaBaaajuaGbaqcLbmacaaIXaaajuaGbeaaaeaacaWGmbaaaaGaayjk aiaawMcaaaaa@4864@
[ ( P 1 γ P 2 γ )( h 2 h 1 ) ]=hf MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaae aadaqadaqaamaalaaabaGaamiuamaaBaaabaqcLbmacaaIXaaajuaG beaaaeaacqaHZoWzaaGaeyOeI0YaaSaaaeaacaWGqbWaaSbaaeaaju gWaiaaikdaaKqbagqaaaqaaiabeo7aNbaaaiaawIcacaGLPaaacqGH sislcaGGOaGaaiiAamaaBaaabaqcLbmacaaIYaaajuaGbeaacqGHsi slcaGGObWaaSbaaeaajugWaiaaigdaaKqbagqaaiaacMcaaiaawUfa caGLDbaacqGH9aqpcaWGObGaamOzaaaa@524F@

hf= τ 0 *P*L γ*A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAai aadAgacqGH9aqpdaWcaaqaaiabes8a0naaBaaabaqcLbmacaaIWaaa juaGbeaacaGGQaGaamiuaiaacQcacaWGmbaabaGaeq4SdCMaaiOkai aadgeaaaaaaa@43EB@
τ 0 = C R ρ V 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq 3cdaWgaaqcfayaaKqzadGaaGimaaqcfayabaGaeyypa0Jaam4qaSWa aSbaaKqbagaajugWaiaadkfaaKqbagqaaiabgEHiQiabeg8aYjabgE HiQmaalaaabaGaamOvaSWaaWbaaKqbagqabaqcLbmacaaIYaaaaaqc fayaaiaaikdaaaaaaa@490C@  
P A = 1 R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGqbaabaGaamyqaaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWG sbaaaaaa@3AD7@   ρ γ = 1 g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaHbpGCaeaacqaHZoWzaaGaeyypa0ZaaSaaaeaacaaIXaaabaGa am4zaaaaaaa@3CB8@
hf= C R * L R * V 2 2g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAai aadAgacqGH9aqpcaWGdbWaaSbaaeaajugWaiaadkfaaKqbagqaaiaa cQcadaWcaaqaaiaadYeaaeaacaWGsbaaaiaacQcadaWcaaqaaiaadA falmaaCaaajuaGbeqaaKqzadGaaGOmaaaaaKqbagaacaaIYaGaam4z aaaaaaa@45B8@

That is the general law of fluid resistance.

Figure 1 Deduction of the general form of fluid resistance.

Results and discussion

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.

  1. Formula of A. Chezy,1

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).

  1. Formula of R. Manning,1

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.

  1. Formula of Weisbach-Darcy, (1855).

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).

  1. General formula of the fluid resistance, (1765).

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).

Qd

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

Vr

Rem

CRm

CCHm

nm

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

 

CRm

 

 

fw-d

 

 

fw-d

Sm

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

Rem

CRm

CCHm

nm

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

 

CRm

 

 

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

Rem

CRm

CCHm

nm

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

 

CRm

 

 

Fw-d

 

 

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).

Qd

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

CR

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).

 

CRm

 

 

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). > fW-D, 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).

Qd

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

CR

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

 

 

CRm

 

 

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).

Qd

 

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

 

 

CRm

 

 

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)

Observe

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 fw-d), 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

Rem

CRm

CCHm

nm

0.04005

0.58915

0.06798

0.742

201647

0.00521

61.341

0.01041

fw-d

Sm

a

Suα

Su

α

Suα

 

0.02086

0.00215

1.04472

0.002246

0.00215

1.04472

0.002246

CRm

CCHmm

nm

fw-d

 

 

 

 

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

Rem

CRm

CCHm

nm

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

CRm

CCHm

nm

fc

 

 

 

 

 

0.00523

61.227

0.01041

0.02093

 

 

 

 

Table 7 Canal rectangular

Data

Qd

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

Rem

CRm

CCHm

nm

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

CCHm

fc

 

 

 

 

 

0.00526

0.01041

61.07531

0.02104

 

 

 

 

Table 8 Canal trapezoidal

Data

Qd

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

Rem

CR

n

154.1581

0.04084

0.61934

0.06594

0.727

191817

0.00526

0.01041

CCHm

fc

Su

α

Suα

Su

α

Suα

61.07531

0.02104

0.00215

1.0451

0.002247

0.00215

1.0451

0.002247

CR

n

CCHm

fc

 

 

 

 

 

0.00526

0.01041

61.07531

0.02104

 

 

 

 

Table 9 Circular canal parallely filled, (h/Di=0.95)

Data

Qd

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

 

 

 

 

Vr

Rem

CRm

CCHm

nm

fw-d

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 fw-d/d), is the same, therefore, the condition of hydrodynamic similarity in this case consists of the equal value of the coefficient, (λ or fw-d), 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

Conclusion

  1. The general formula of the fluid resistance, (law), is valid for the calculation of all possible cases of linear load losses in the pipes, the hydraulic concept being more efficient for this purpose, because with it the more accurate and accurate results.
  2. The general law of fluid resistance is the origin of the coefficients of Chezy, Manning and Weisbach-Darcy, it is also the first formula of the uniform regime and the general formula for the calculation of linear load losses.

Acknowledgements

None.

Conflict of interest

The author declares that there are no conflicts of interest.

References

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  2. Nekrasov B. Hydraulics. 1968;84‒85.
  3. Agroskin II. Hydraulics. Volume I, Ministry of Higher Education. ISC. TO. Havana. 1960;285‒336.
  4. Chow Come Te. Open-Chanel Hydraulics. Revolutionary Edition, Instituto del Libro. 1959;98‒197.
  5. Basic documentation of the Master of Hydraulic and Environmental Engineering. University of Pinar del Rio. 2003‒2006.
  6. King HW. Manual of Hydraulics. Revolutionary Edition. Book Institute. 1959;254:336‒358.
  7. Leon, MJFA. Hydraulics of Free Conductions. Volume I and Volume II, Havana. 2000;87‒676.
  8. Montes JS. Hydraulics of Open Channel Flow. Asce press. 2000;147‒207.
  9. Sotelo AG. Hydraulics of Channels. UNAM. 2002;79‒89.
  10. Sturm. Open-Channel Hydraulics. 2001;4:97‒150.
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