Title 9 Flow in Closed Conduits.pdf ✅

HGE 9: Flow in Closed Conduits 1. Major Head Losses. 1.1. Darcy-Weisbach Formula The general formula for head loss given the friction factor f is h = fL D ∙ v 2 2g For the velocity in terms of flow, h = fL D ∙ ( Q A ) 2 2g = fL D ∙ ( Q π 4 D2 ) 2 2g = 8fLQ 2 πgD5 h = 0. 0826fLQ 2 D5 1.2. Manning’s Formula From HGE 11: Flow in Open Channels, v = 1 n R 2 3S 1 2 Where n is the coefficient of roughness. For the slope of the energy grade line, S = h L Also, from HGE 8: Fundamentals of Fluid Flow, D = 4R R = D 4 Therefore, v = 1 n ( D 4 ) 2 3 ( h L ) 1 2 Q π 4 D2 = 1 n ( D 4 ) 2 3 ( h L ) 1 2 4096Q 6 π 6D12 = 1 n 6 ( D 4 256) ( h 3 L 3 ) h 3 = 1048576n 6L 3Q 6 π 6D16 h = 10. 29n 2LQ 2 D 16 3
1.3. Hazen-Williams Formula An empirical formula for the head lost when pipes are larger than 50 mm and the velocity is slower than 3 m/s. h = 10. 67LQ 1.85 C1.85D4.87 where C is the Hazen-Williams coefficient. 2. Minor Head Losses While the major head losses are due to the friction of the pipe, minor head losses are due to changes in direction. It occurs at the pipe fittings and curvature of the pipe. If there is sudden enlargement, the head lost is equal to the velocity head of the velocity change. HL = (v1 − v2 ) 2 2g If the enlargement is more gradual, the head lost is multiplied by a factor of gradual enlargement. HL = K (v1 − v2 ) 2 2g For contraction of pipes, and for flow passing through fittings, the head lost is equal to the velocity head at the outlet multiplied by a factor. HL = K ( v 2 2g) 3. Pipes in Series and Parallel • Pipes in Series – For pipes in series, flow rate is constant, and head losses are additive. H. L.= h1 + h2 + h3 + ⋯ Q = Q1 = Q2 = Q3 = ⋯ • Pipes in Parallel – For pipes in parallel, head loss is constant, and flow rates are additive. H. L.= h1 = h2 = h3 = ⋯ Q = Q1 + Q2 + Q3 + ⋯ The general steps when solving problems with pipe systems are: 1. Find the relation for the flow rates. The inflow at a node is always equal to the outflow. 2. Find the relation for the head losses. 3. Solve for the unknowns using the major head loss formulas and algebraic procedure (by elimination or substitution). Always remember that the number of required equations is equal to the number of unknowns.

3.3. Sample Reservoir Systems and Relations hA + hB = Elev. A − Elev. B hB − hC = Elev. C − Elev. B hA +hC = Elev. A − Elev. C hA + hB = Elev. B − Elev. A hC + hD = Elev. C − Elev.D hA +hP1P2 +hC = Elev. C − Elev. A hB − hP1P2 + hD = Elev. B − Elev.D 4. Water Hammer. Celerity is the velocity of a pressure wave in a fluid. Pressure waves include sound waves or water hammer. c = √ EB ρ (rigid pipes) c = √ EB ρ (1+ EB E ∙ D t ) (nonrigid pipes) The effective bulk modulus is the value that can replace the value of the liquid in a nonrigid pipe as if it were in a rigid one. EB′ = EB (1+ EB E ∙ D t ) For a pipe with length L, the time it would take for the pressure wave to travel back and forth is t = 2L c