Basics of fluid flow. Types of flow. Fluid Ideal/Real Compressible/Incompressible

Transcription

1 Basics of fluid flow Types of flow Fluid Ideal/Real Compressible/Incompressible Flow Steady/Unsteady Uniform/Non-uniform Laminar/Turbulent Pressure/Gravity (free surface) 1

2 Basics of fluid flow (Chapter 4) Basics of fluid flow, kinematics Statics Mechanics Kinematics Dynamics Kinetics Kinematics: deals with motion apart from considerations of mass, force or energy 2

3 Basics of fluid flow Path lines, streamlines and streak lines Path line: the trajectory that a fluid particle would make as it moves around with the flow Streamline: line that shows the flow direction, local velocity vector is tangent to the streamline at every point along the line at that instant 3

4 Basics of fluid flow Types of flow Steady flow: all fluid/flow properties at any point in the flow do not change with time; however, conditions may be different at different points. For steady flows: Uniform flow: at every point in the flow, the velocity (in both magnitude and direction) is identical at any given instant. 4

5 Basics of fluid flow One -, two -, and three- dimensional flows This is the most general 3-D flow: The flow is classified as 2-D if: The flow can be viewed as 1-D if: 5

6 Partial derivative Differentiating a function of more than one variable with respect to a particular variable, with the other variables kept constant: the notation f/ t means the partial derivative of the function f with respect to t f/ t : partial derivative df/dt : total derivative For more info: 6

7 Basics of fluid flow Velocity and Acceleration, 4.12 a t a n Convective (spatial) acceleration Local (temporal) acceleration 7

8 Basics of fluid flow Flow rate and Mean velocity Flow rate: the rate at which fluid crosses a known surface volume flow rate mass flow rate The volume flow rate passing through the element of area da (in yz plane) is dq = u(cosθ)da=uda volume flow rate is equal to the magnitude of the mean velocity multiplied by the flow area at right angles to the direction of the mean velocity 8

9 Basics of fluid flow Flow rate and Mean velocity The volume flow rate passing through the element of area da is dq = u da =uda the local time mean velocity, u, will vary across the section for real fluid Q uda A AV m uda AV Q 9 A

10 Basics of fluid flow Reynolds Transport Theorem & Continuity AV 1 1 A2V 2 Q 10

11 FIGURE 5-24 Copyright The McGraw-Hill Companies, Inc. Bernoulli s Equation (Energy per unit weight)

12 Energy in Steady Flow (Chapter 5) Energies of a Flowing Fluid (Euler s Equation) Kinetic Energy 1/2mV 2 V 2 /2g Unit: L Potential Energy Wz z (Energy per unit weight) Pressure Head p = γh p/γ 12

13 Derivation of the Bernoulli Equation Steady flow: The forces acting on a fluid particle along a streamline. Steady, incompressible flow: Bernoulli equation The sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow when compressibility and frictional effects are negligible. The Bernoulli equation between any two points on the same streamline: 13

14 Energy in Steady Flow (Chapter 5) Bernoulli s Equation Unit: L (Energy per unit weight) Piezometric pressure Basic assumptions: Inviscid & incompressible fluid Steady flow Applies along a streamline No energy added or removed from the fluid along the streamline 14

15 FIGURE 5-22 Copyright The McGraw-Hill Companies, Inc.

16 Energy in Steady Flow, Pipe Flow V p/γ V 2 /2g p/γ Pitot Tube (Measures stagnation pressure) Free stream dynamic pressure Free stream static pressure Bring moving water to a halt, and it'll drive a column of water up to exactly the height from which water would flow to gain that velocity. 16

17 Energy in Steady Flow, Free surface flow Pitot Tube (Measures stagnation pressure) V V 2 /2g Bring moving water to a halt, and it'll drive a column of water up to exactly the height from which water would flow to gain that velocity. 17

18 Example: Bernoulli s principle, Pitot Tube 18

19 Example: Water Discharge from a Large Tank Example: Spraying Water into the Air 19

20 Hydraulic grade line (HGL), P/ g + z The line that represents the sum of the static pressure and the elevation heads. Energy grade line (EGL), P/ g + V 2 /2g + z The line that represents the total head of the fluid. Dynamic head, V 2 /2g The difference between the heights of EGL and HGL. The hydraulic grade line (HGL) and the energy grade line (EGL) for free discharge from a reservoir through a horizontal pipe with a diffuser. 20

21 Energy in Steady Flow Stagnation pressure, ideal fluid (5.4) V 1 2 V 1 = V, p 2 is the stagnation pressure 21

22 Energy in Steady Flow General Energy Equation, steady flow, incompressible fluid For an incompressible fluid with γ = const. and α =1: 22

23 Energy in Steady Flow General Energy Equation, steady flow, incompressible fluid Real fluid For an incompressible fluid with γ = const. and α =1: If there is no machine between points 1 and 2: If head loss is neglected: Ideal fluid 23

24 Energy in Steady Flow Power considerations in fluid flow, Derivation of Power Equation Power: P = (Force) x (Velocity) Power: P = Energy / Time P = FV (F =ΔpA) P = (Energy/Weight) x (Weight/Time) P = (ΔpA)V (Δp = γh) P = (γha)v (Q = AV) head (h) γq P = γhq P = h γq P = ΔpQ 24

25 Energy in Steady Flow Power considerations in fluid flow, Units of Power P = γhq Power in BG units Horsepower = P = γhq/550 ( [Q] = cfs, [h] = ft, [γ] = pcf ) Power in SI units Kilowatts = P = γhq/1000 ( [Q] = m 3 /s, [h] = m, [γ] = N/m 3 ) P = γhq P: power put into flow by a pump, then h = h pump P: power lost because of friction, then h = h L Pump efficiency, η = (power output) / (power input) 25