Hydraulics ecture # CWR 40 age () ecture # Outline: Review of terminology in fluid mechanics: Energy or work Hydraulic head Bernoulli s aw, Conductivity (examle) ransient & turbulent Friction head loss Pum head Design considerations nnouncements: Dimensions, units: (F) (M) (British) (SI) British Metric [] [a] M [F] F [Work] M F [Power] F M Energy (or work) and total hydraulic head: In Fluid Mechanics: What are the tyes of energy that a fluid can have? Pressure, otential energy, and kinetic energy First, in fluid mechanics, it is customary to exress the energy as head of water. he head is the energy er unit weight of fluid. Remember, work or energy (force * dislacement), divided by the weight (force) equals the head. So the head is exressed in units of length ().
Hydraulics ecture # CWR 40 age () yes of energy head: (a) Elevation head, z, [] otential energy, set from a datum. he energy to lift a weight to elevation z weight x z. So elevation head exressed as units of water head z. (b) (c) Pressure head,, [] reresents the height of a column of fluid to roduce ressure. elocity head,, [] g kinetic energy, vertical distance traveled by a fluid article before it reaches velocity,. otal Energy Head of a fluid: α H + z + g [] Hydraulic or iezometric head of a fluid: Hydraulic head + z (does not include velocity head) Bernoulli s aw: If there is no energy loss along a streamline, the total energy of a fluid article is the same (conserved). fluid article H H H H + z + + z g Q Q + g steady state flow (If there is no energy loss) Bernoulli s aw along a streamline continuity, same discharge
Hydraulics ecture # CWR 40 age () α /g α /g iezometric tube ilot tube P / P / decreasing cross-section z z H H Bernoulli Eq between () and () α α If no energy loss. + z + + z + g g α reflects the velocity distribution in the ie H H constant total head along a streamline Energy Grade ine (EG) + z + z Piezometric or hydraulic head forms the Hydraulic Grade ine (HG). series of ilot tubes indicate the locus of the Energy Grade ine (EG). series of iezometers indicate the locus of the Hydraulic Grade ine (HG). Note: is the mean velocity across the ie cross section α is a kinetic energy correction factor due to the fact that velocity is not uniform across the cross section. elocity is deendent on distance from the center of the ie cross section. u(r) u(r 0) u max
Q uda Q α u da uda Hydraulics ecture # CWR 40 age (4) α:.04-.06 for turbulent flow (uniform velocity distribution) α.0 for laminar flow (arabolic velocity distribution) α for most flow conditions aminar & urbulent flow: Reynolds #, Re Reynolds # is a measure of the ratio of inertial force on an element of fluid to the viscous force on the same liquid element. In a ie: ρ density of fluid ρd Re D ie diameter μ μ dynamic viscosity of fluid Remember the viscosity is a measure of fluid resistance to shearing deformation. (oil and glycerin are more viscous than water) Re 00 aminar flow (viscous effects dominate) 00 < Re 4000 Re > 4000 ransient flow (can be either laminar of turbulent) urbulent flow (inertia effects dominate over viscous effects) aminar Flow Re 00 urbulent Flow Re > 4000 In the transitional zone, flow can be either laminar or turbulent (unstable flow).
Hydraulics ecture # CWR 40 age (5) Examles on using the energy equation, Bernoulli s aw, and continuity equation. Given the deth of water ustream and downstream of a sluice gate, determine the discharge through the gate. ssume no energy loss. /g sluice gate E.. /g otal head is the same, so the energy grade line is a horizontal line. ly Bernoulli s law between oint and. What is the ressure at and? 0 0 0 α + z + + z g α + g From continuity (discharge er unit width) Solving for : Discharge er unit width (for m width) HG coincides with the water surface level. EG is horizontal because of no energy loss. + g 0.8 + g 0.8. 5 + 0.8 + 6.5 9.8 9.8. m s.5. 5. s 0 m Q. 4.4 m s
Hydraulics ecture # CWR 40 age (6) Examle: Water at 60 F is sihoned from a large tank through a constant diameter ie. Determine the maximum height of hill, h, over which the water can be sihoned without causing cavitaion in the ie. otal head at: z 5 h -5 0? 0 g 0?? otal head 5 In able.5 vaor ressure of water is 0.56 sia (absolute gage ressure). P P + P ab gage at 0.56 si 4.7 si P gage 0.56 4.7 4. 4 si
P 4.4 lb 44 in 6.4 lb ft in ft. ft ly Bernoulli between () and (). 5 5 + g ft g. s 5. 9 ft s ly Bernoulli between () and (). 5.9 5 + h.. h 8. ft Hydraulics ecture # CWR 40 age (7) Friction Head oss: Energy loss due to friction in ressurized ie with steady, conserved flow. Head loss is due to viscous resistance to flow (friction) along walls of the ie. h l P P hl f D g ε f,re MoodyDiagram D f D Q Q f D Q 4 g g π 6 D
Head loss, h l has a relationshi to: Hydraulics ecture # CWR 40 age (8) h l kq Pum Head: the amount of energy rovided by the um to the system. diameter ie length Reynolds number ie α g α g + z + + h + z + + h l for section between ints () and (). total head at () total head at () he ower of the um necessary to rovide head h for the discharge Q is: head sulied by the um P Q h h Q flow rate secific weight of water kg m N ρg m s m ( Design Considerations: Design of a ieline conduit system that delivers a certain discharge Q with a head H. In designing your ieline: Initial or caital cost
Hydraulics ecture # CWR 40 age (9) o installing the ieline, the smaller the cross-section the more economical Oerational and maintenance cost o head loss in your system can be higher for smaller ie cross-sections o energy cost for oerating a um that delivers flow Q at head H for a ie with small cross section is higher because a larger force is alied for water to get umed through bends. Objective: to design the ieline (size of your conduit) with the least cost for a articular discharge Q.