KULIAH VIII - IX MEKANIKA FLUIDA II NazaruddinSinaga
Shear stress and velocity distribution in pipe for laminar flow
Typical velocity and shear distributions in turbulent flow near a wall: (a) shear; (b) velocity.
Solution of Pipe Flow Problems • Single Path • Find Dp for a given L, D, and Q • Use energy equation directly • Find L for a given Dp, D, and Q • Use energy equation directly
Solution of Pipe Flow Problems • Single Path (Continued) • Find Q for a given Dp, L, and D • Manually iterate energy equation and friction factor formula to find V (or Q), or • Directly solve, simultaneously, energy equation and friction factor formula using (for example) Excel • Find D for a given Dp, L, and Q • Manually iterate energy equation and friction factor formula to find D, or • Directly solve, simultaneously, energy equation and friction factor formula using (for example) Excel
Example 1 • Water at 10C is flowing at a rate of 0.03 m3/s through a pipe. The pipe has 150-mm diameter, 500 m long, and the surface roughness is estimated at 0.06 mm. Find the head loss and the pressure drop throughout the length of the pipe. Solution: • From Table 1.3 (for water): = 1000 kg/m3 and =1.30x10-3 N.s/m2 V = Q/A and A=R2 A = (0.15/2)2 = 0.01767 m2 V = Q/A =0.03/.0.01767 =1.7 m/s Re = (1000x1.7x0.15)/(1.30x10-3) = 1.96x105 > 2000 turbulent flow To find , use Moody Diagram with Re and relative roughness (k/D). k/D = 0.06x10-3/0.15 = 4x10-4 From Moody diagram, 0.018 The head loss may be computed using the Darcy-Weisbach equation. The pressure drop along the pipe can be calculated using the relationship: ΔP=ghf = 1000 x 9.81 x 8.84 ΔP = 8.67 x 104 Pa
Example 2 • Determine the energy loss that will occur as 0.06 m3/s water flows from a 40-mm pipe diameter into a 100-mm pipe diameter through a sudden expansion. Solution: • The head loss through a sudden enlargement is given by; Da/Db = 40/100 = 0.4 From Table 6.3: K = 0.70 Thus, the head loss is
Example 3 • Calculate the head added by the pump when the water system shown below carries a discharge of 0.27 m3/s. If the efficiency of the pump is 80%, calculate the power input required by the pump to maintain the flow.
Solution: Applying Bernoulli equation between section 1 and 2 (1) P1 = P2 = Patm = 0 (atm) and V1=V2 0 Thus equation (1) reduces to: (2) HL1-2 = hf + hentrance + hbend + hexit From (2):
The velocity can be calculated using the continuity equation: Thus, the head added by the pump: Hp = 39.3 m Pin = 130.117 Watt ≈ 130 kW.
EGL & HGL for a Pipe System • Energy equation • All terms are in dimension of length (head, or energy per unit weight) • HGL – Hydraulic Grade Line • EGL – Energy Grade Line • EGL=HGL when V=0 (reservoir surface, etc.) • EGL slopes in the direction of flow
EGL & HGL for a Pipe System • A pump causes an abrupt rise in EGL (and HGL) since energy is introduced here
EGL & HGL for a Pipe System • A turbine causes an abrupt drop in EGL (and HGL) as energy is taken out • Gradual expansion increases turbine efficiency
EGL & HGL for a Pipe System • When the flow passage changes diameter, the velocity changes so that the distance between the EGL and HGL changes • When the pressure becomes 0, the HGL coincides with the system
EGL & HGL for a Pipe System • Abrupt expansion into reservoir causes a complete loss of kinetic energy there
EGL & HGL for a Pipe System • When HGL falls below the pipe the pressure is below atmospheric pressure
FLOW MEASUREMENT • Direct Methods • Examples: Accumulation in a Container; Positive Displacement Flowmeter • Restriction Flow Meters for Internal Flows • Examples: Orifice Plate; Flow Nozzle; Venturi; Laminar Flow Element
Flow Measurement • Linear Flow Meters • Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic Float-type variable-area flow meter
Flow Measurement • Linear Flow Meters • Examples: Float Meter (Rotameter); Turbine; Vortex; Electromagnetic; Magnetic; Ultrasonic Turbine flow meter
Flow Measurement • Traversing Methods • Examples: Pitot (or Pitot Static) Tube; Laser Doppler Anemometer
The measured stagnation pressure cannot of itself be used to determine the fluid velocity (airspeed in aviation). • However, Bernoulli's equation states: • Stagnation pressure = static pressure + dynamic pressure • Which can also be written
Solving that for velocity we get: • Note: The above equation applies only to incompressible fluid. • where: • V is fluid velocity; • pt is stagnation or total pressure; • ps is static pressure; • and ρ is fluid density.
The value for the pressure drop p2 – p1 or Δp to Δh, the reading on the manometer: Δp = Δh(ρA-ρ)g • Where: • ρA is the density of the fluid in the manometer • Δh is the manometer reading
Main Topics • The Boundary-Layer Concept • Boundary-Layer Thickness • Laminar Flat-Plate Boundary Layer: Exact Solution • Momentum Integral Equation • Use of the Momentum Equation for Flow with Zero Pressure Gradient • Pressure Gradients in Boundary-Layer Flow • Drag • Lift
Boundary Layer Thickness • Disturbance Thickness, d where • Displacement Thickness, d* • Momentum Thickness, q
Laminar Flat-PlateBoundary Layer: Exact Solution • Governing Equations
Laminar Flat-PlateBoundary Layer: Exact Solution • Boundary Conditions
Laminar Flat-PlateBoundary Layer: Exact Solution • Equations are Coupled, Nonlinear, Partial Differential Equations • Blassius Solution: • Transform to single, higher-order, nonlinear, ordinary differential equation
Laminar Flat-PlateBoundary Layer: Exact Solution • Results of Numerical Analysis
Momentum Integral Equation • Provides Approximate Alternative to Exact (Blassius) Solution