Chanson H. Jean-Baptiste Charles Joseph Belanger (1790-1874), the Backwater Equation and the Belanger Equation (796976), страница 6
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A-2 - Details of the engraved names of scientists on the Eiffel Tower on 25 July 200821Appendix C - The Prony flow resistance formulaGaspard Clair François Marie Riche de PRONY (1755-1839) introduced a flow resistance formula for pipesand rivers composed of a linear and a quadratic velocity term (PRONY 1804).
The Prony formula wasoriginally presented as :1g S f D H = a ' V + b' V 24(B-1)where g is the gravity acceleration, DH is the hydraulic diameter: DH = 4 A/Pw, A is the cross-section area, Pwis the wetted perimeter, Sf is the friction slope: Sf = -∂H/∂x, H is the total head, x is the longitudinal directionpositive downstream, V is the cross-sectional averaged velocity, and a' and b' are constant. The Pronyformula was typically used in the form:Sf =(4a V + b V2DH)(B-2)Several values were proposed for the coefficients a and b (Table B-1). However the differences wererelatively small as shown in Figure B-1, with discrepancies between the formulae of less than 3% for V = 1m/s.Despite its crude nature, the Prony formula gave results close to modern estimates.
Its success may beexplained by its simple form at a time when all the calculations were performed manually (BROWN 2002).Table B-1 - Coefficients a and b of the Prony formula (Eq. (B-2))Reference / ResearcherabOpen channel flowsPRONY (1804, p. xxix)31 experiments (0.03 < V < 2.3 m/s) 4.4444 10-5 3.0928 10-4BÉLANGER (1828, p. 6)Gaspard Clair François Marie Riche de PRONY (1755-1839) 4.445 10-5 3.09314 10-4Johann EYTELWEIN (1764-1848) 2.4265 10-5 3.655410-4BÉLANGER (1841, pp.
66-67; 1849, p. 67)Gaspard Clair François Marie Riche de PRONY (1755-1839) 4.44 10-53.09 10-4Johann EYTELWEIN (1764-1848)2.4 10-53.66 10-4Circular pipe flowsPRONY (1804, p. xxix)-53.4822 10-451 experiments (0.03 < ∅ < 0.5 m) 1.7329 10BÉLANGER (1849, p. 49)Gaspard Clair François Marie Riche de PRONY (1755-1839) 1.73 10-53.48 10-4-5Johann EYTELWEIN (1764-1848) 2.22 102.80 10-4-5Jean-François D'AUBUISSON (1769-1841), 1.884 103.425 10-4Note: SI units.221.4E-3a=4.445 10-5, b=3.093 10-4a=2.426 10-5, b=3.655 10-4a=1.73 10-5, b=3.48 10-4a=2.22 10-5, b=2.80 10-4a=1.884 10-5, b=3.425 10-41.2E-3-∂H/∂x DH/4 (m)1E-38E-46E-44E-42E-4000.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9V (m/s)2Fig.
B-1 - Prony flow resistance formula: comparative results between various values of the coefficients aand b23ReferencesBAKHMETEFF, B.A. (1932). "Hydraulics of Open Channels." McGraw-Hill, New York, USA, 1st ed., 329pages.BARRÉ de SAINT-VENANT, A.J.C. (1871). "Théorie du Mouvement Non Permanent des Eaux, avecApplication aux Crues des Rivières et à l'Introduction des Marées dans leur Lit." Comptes Rendus desséances de l'Académie des Sciences, Paris, France, Vol.
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