Section 1b Basic Fluid Flow in Pipes (2024).pdf

Section 1b: Review of Basic Concepts of Fluid Flow  Flowing Fluids There are three regimes of flow: - Laminar flow - Turbulent flow - Potential flow (or flows that behave like potential flow) Laminar and turbulent flow are the main flow regimes in pipe...

Section 1b: Review of Basic Concepts of Fluid Flow  Flowing Fluids There are three regimes of flow: - Laminar flow - Turbulent flow - Potential flow (or flows that behave like potential flow) Laminar and turbulent flow are the main flow regimes in pipe flow. Laminar Flow Viscous forces (viscosity) dominate in this flow. Little mixing except for molecular diffusion. Smooth, streaming flow. Tends to occur for a very viscous, small depths, and/or low velocity Source: https://www.iahrmedialibrary.net/the-library/methods-in-hydraulics/fluid- mechanics/hele-shaw-cell-experiments-2/293, Accessed September 3, 2023. flow. Turbulent Flow Inertial forces dominate this flow. Made up of eddies from very large scales to very small scales. Lots of mixing. Turbulence is three-dimensional in nature. Turbulence appears to be random.  Turbulent flow occurs for a higher velocity, large sized (large depths), and/or low viscosity fluid flows. Whitewater is turbulent, but what it is indicating is entrainment of air by turbulence at the water surface not whether the flow is turbulent or not.  For a turbulent flow, the water surface can look Turbulence in a Boundary Layer on a Flat Plate smooth - turbulence is (Ref: https://www.iahrmedialibrary.net/the-library/methods-in-hydraulics/fluid-mechanics/turbulent- flow-large-eddies-in-a-turbulent-boundary-layer/333, Accessed Sept. 3, 2023) about what is happening inside the flow (are there eddies or not). Eddies in a Turbulent Jet (Ref: Van Dyke, M. (1982) An Album of Fluid Motion, The Parabolic Press, Stanford, California, 176 pp.)  Examples - Laminar Flow: slow flow in a small diameter pipe (glycerine in tubing) blood flow in veins and arteries very shallow overland flow pouring of engine oil into car engine pouring of cold honey from one jar to another groundwater flow  Examples - Turbulent Flow: flow in rivers and streams flow in water distribution systems flow in sewers atmosphere (wind) high velocity flow in large diameter pipes your breath as you breathe out flow in pipelines flow from smoke stacks at factories  How can you tell if a flow is turbulent? Inject dye into the flow. Measure the velocity at some point in the fluid flow. Using tufts that follow the local flow. LAMINAR FLOW TURBULENT FLOW u What causes the transition between laminar and turbulent flow? In a laminar flow, “perturbations”, or disturbances, in the flow are damped out by viscous forces and the flow remains stable (as a laminar flow). At higher velocities, disturbances are not damped out by viscous forces and turbulence occurs. The criteria used for determining whether a flow is laminar or turbulent is the Reynolds number, R. In general form, the Reynolds number has the form: ρVcLc VcLc R= = μ ν Vc = a characteristic velocity of the flow Lc = a characteristic dimension of flow ρ = density of the fluid μ = dynamic viscosity of fluid ν = kinematic viscosity of fluid The Reynolds number is the ratio of the inertial forces to the viscous forces in a flow. If R is large, inertial forces dominate the flow and viscosity is not very important (turbulent flow). If R is small, viscous forces dominate the flow and viscosity is important (laminar flow). For flow in a circular pipe, the Reynolds number has the form: ρVd Vd R= = μ ν V = the average velocity of flow in the pipe d = the diameter of the pipe Typical range of Reynolds number for a pipe flow of 0.01 to 108.  For a flow in a circular pipe: If R < 2000, have laminar flow If R > 2000, have turbulent flow. R=2000 for transition from laminar to turbulent for pipe flow is not an exact value. Average Velocity of Flow, V V u = velocity of flow in x-direction (along length of pipe) r = perpendicular distance from centre of pipe towards pipe wall  Q Flow Rate V= = A C ross − S ectional Area of Flow volume of fluid collected in time t Q= time t Mathematical Definition of Q - Find Q from integrating the velocity profile through the depth of flow. By definition: Q = ∫ u dA Q for a Circular Pipe: ro = radius of pipe Analysis of Fluid Flows Conservation of Mass (Control Volume) from Reynolds Transport Theorem: Rate of Net Creation of change of transport of = + mass in mass inside mass into control control control volume volume volume Control volume = An arbitrary region within a flow that may be fixed in size or one that changes in size and shape; it may or may not be moving. For flows of only water, mass is not created or destroyed: 0 Rate of Net Creation of change of transport of = + mass in mass inside mass into control control control volume volume volume  Rate of Net change of transport of mass inside = mass into control control volume volume Mass transport (mass flow rate) = fluid density (ρ) x volumetric flow rate (Q) ∑ Mass transport into control volume = ρQin ∑ Mass transport out of control volume = ρQout  Rate of Net change of transport of mass inside = mass into control control volume volume Rate of Change of Mass Inside Control Volume of Volume ∀ ∂ρ ∀ = ∂t  Conservation of mass becomes: ∂ρ ∀ ∑ ∑ = ρQin − ρQout ∂t For an incompressible flow, for which the fluid’s density is constant: ∂∀ ∑ ∑ = Qin − Qout ∂t For a control volume that is not changing in size and for a steady flow (not changing in time): ∂∀ ∑ ∑ =0= Qin − Qout ∂t ∑ ∑ Qin = Qout Sum of the flows in = Sum of the flows out Steady Flow through a pipe of constant diameter (one inflow, one outflow): Pipe itself Qout is control Qin volume ∑ ∑ Qin = Qout Qin = Qout Flow rate has to be constant through the pipe.  Qout Qin Qin = Qout = VA Q is constant and pipe’s Average velocity V cross-sectional A is must be constant constant (diameter is through the pipe. constant). For pipes in series (steady flow, control volume fixed): Pipe 2 Pipe 1 (diameter d2) (diameter d1) Qout Qin ∑ ∑ Qin = Qout Qin = Qout Velocity in pipe 1, Q = constant = V1A1 = V2 A2 V1, has to be larger than velocity in pipe 2, V2.  2 1 h1 h2 datum g= gravitational acceleration For flow in pipes, regardless of whether the flow is laminar or turbulent, the energy equation can be used to predict the velocity and pressure at any point along the pipe.  Losses in Energy Head between Total Energy Head Total Energy Head Points 1 and 2 at Section 1 at Section 2 (upstream (downstream location) location) Head loss is made up of two components: friction and minor losses. hL = h f + hm Total Head Losses Minor Head Losses Frictional Head Losses  Friction Losses: - Losses due to shear stresses on the pipe wall and, for a turbulent flow, transfer of energy to heat by turbulence. Minor Losses - Losses due to turbulence created in fittings in pipes (e.g., entrances, exits, valves, contractions, expansions, bends, T’s). - Minor losses are applied for turbulent flow only. Laminar Flow: We will consider frictional losses only. The energy equation then becomes: 2 2 p1 V p2 V + h1 + = 1 + h2 + + hf 2 γ 2g γ 2g For a laminar flow, frictional losses in a circular pipe are given by Poiseulle’s Law (can derive from conservation of momentum): 32μVL hf = Poiseulle’s Law γd 2 μ = dynamic viscosity of the fluid V = average velocity of flow in pipe L = length of flow between sections of interest γ = specific weight of the fluid d = pipe’s diameter Next: Section 1c - Applications of energy equation in laminar flow problems.