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. 4
Solution of Pipe Flow Problems Single Path – Find p for a given L, D, and Q Use energy equation directly – Find L for a given p, D, and Q Use energy equation directly
Solution of Pipe Flow Problems Single Path (Continued) – Find Q for a given p, L, and D 1.Manually iterate energy equation and friction factor formula to find V (or Q), or 2.Directly solve, simultaneously, energy equation and friction factor formula using (for example) Excel – Find D for a given p, L, and Q 1.Manually iterate energy equation and friction factor formula to find D, or 2.Directly solve, simultaneously, energy equation and friction factor formula using (for example) Excel
7 Example 1 Water at 10 C 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/m 3 and =1.30x10 -3 N.s/m 2 V = Q/A andA= R 2 A = (0.15/2) 2 = 0.01767 m 2 V = Q/A =0.03/.0.01767 =1.7 m/s Re = (1000x1.7x0.15)/(1.30x10 -3 ) = 1.96x10 5 > 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= gh f = 1000 x 9.81 x 8.84 ΔP = 8.67 x 104 Pa
8 Example 2 Determine the energy loss that will occur as 0.06 m 3 /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; D a /D b = 40/100 = 0.4 From Table 6.3: K = 0.70 Thus, the head loss is
9 Ex ample 3 Calculate the head added by the pump when the water system shown below carries a discharge of 0.27 m 3 /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) P 1 = P 2 = P atm = 0 (atm) and V 1 =V 2 0 Thus equation (1) reduces to: (2) H L1-2 = h f + h entrance + h bend + h exit From (2):
The velocity can be calculated using the continuity equation: Thus, the head added by the pump: H p = 39.3 m P in = 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
30 The measured stagnation pressure cannot of itself be used to determine the fluid velocity (airspeed in aviation). However, Bernoulli's equation states:Bernoulli's equation Stagnation pressure = static pressure + dynamic pressure Which can also be written
31 Solving that for velocity we get: Note: The above equation applies only to incompressible fluid. where: V is fluid velocity; p t is stagnation or total pressure; p s is static pressure; and ρ is fluid density.
32 The value for the pressure drop p 2 – p 1 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
Momentum Integral Equation Provides Approximate Alternative to Exact (Blassius) Solution
Momentum Integral Equation Equation is used to estimate the boundary-layer thickness as a function of x: 1.Obtain a first approximation to the freestream velocity distribution, U(x). The pressure in the boundary layer is related to the freestream velocity, U(x), using the Bernoulli equation 2.Assume a reasonable velocity-profile shape inside the boundary layer 3.Derive an expression for w using the results obtained from item 2
Use of the Momentum Equation for Flow with Zero Pressure Gradient Simplify Momentum Integral Equation (Item 1) The Momentum Integral Equation becomes
Use of the Momentum Equation for Flow with Zero Pressure Gradient Laminar Flow – Example: Assume a Polynomial Velocity Profile (Item 2) The wall shear stress w is then (Item 3)
Use of the Momentum Equation for Flow with Zero Pressure Gradient Laminar Flow Results (Polynomial Velocity Profile) Compare to Exact (Blassius) results!
Use of the Momentum Equation for Flow with Zero Pressure Gradient Turbulent Flow – Example: 1/7-Power Law Profile (Item 2)
Use of the Momentum Equation for Flow with Zero Pressure Gradient Turbulent Flow Results (1/7-Power Law Profile)
Drag Pure Friction Drag: Flat Plate Parallel to the Flow Pure Pressure Drag: Flat Plate Perpendicular to the Flow Friction and Pressure Drag: Flow over a Sphere and Cylinder Streamlining
Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available
Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag (Continued) Laminar BL: Turbulent BL: … plus others for transitional flow