Pharos University ME 259 Fluid Mechanics II Review of Previous Fluid Mechanics Dr. Shibl

Fluid Properties: Liquid or Gas Liquids are: Incompressible, DV ≠ f(DP) Viscous (high viscosity) Viscosity decreases with temperature Gases are: Compressible, DV = f(DP) Low viscosity Viscosity increases with temperature

Equations for Fluid Property Circular Area: Weight: w = m*g Newton Density: r = m/V Kg/m3 Specific Weight: g = w/V N/m3 Specific gravity: SG=r/rwater Area = p/4*D2

Viscosity = Shear Stress/Slope of velocity profile Dynamic Viscosity = Shear Stress/Slope of velocity profile Kinematic Viscosity cS (centistokes) or m2/Sec. cP (centipoise) or Pa-sec n v F Slope = v/y y

Pressure and Elevation Change in pressure in homogeneous liquid at rest due to a change in elevation DP = gh Where, DP = change in pressure, kPa = specific weight, N/m3 h = change in elevation, m

Pressure-Elevation Relationship Valid for homogeneous fluids at rest (static) P2 = Patm + rgh Free Surface Free Surface P2 P1 P1 > P2

Example: Manometer Calculate pressure (psig) or kPa (gage) at Point A. Open end is at atmospheric pressure. A 0.15 m Water 0.4 m Hg: SG = 13.54

Forces due to Static Fluids Pressure =Force/Area (definition) Force = Pressure*Area Example: If a cylinder has an internal diameter of 50 mm and operates at a pressure of 20 bar, calculate the force on the ends of the cylinder.

Force-Pressure: Rectangular Walls Patm Vertical wall DP = g*h d

Flow Classification Classification of Fluid Dynamics Laminar Inviscid µ = 0 Viscous Turbulent Compressible Incompressible ϱ = constant Internal External 22-Apr-17

Q = Area*Distance/Unit Time Definitions Volume (Volumetric) Flow Rate Q = Cross Sectional Area*Average Velocity of the fluid Q = A*v Weight Flow Rate W = g*Q Mass Flow Rate M = r*Q v Volume Q = Volume/Unit time Q = Area*Distance/Unit Time

Key Principles in Fluid Flow Continuity for any fluid (gas or liquid) Mass flow rate In = Mass Flow Rate out M1 = M2 r1*A1*v1 = r2*A2*v2 Continuity for liquids Q1 = Q2 A1*v1 = A2*v2 M1 M2

Newton’s Laws Newton’s laws are relations between motions of bodies and the forces acting on them. First law: a body at rest remains at rest, and a body in motion remains in motion at the same velocity in a straight path when the net force acting on it is zero. Second law: the acceleration of a body is proportional to the net force acting on it and is inversely proportional to its mass. Third law: when a body exerts a force on a second body, the second body exerts an equal and opposite force on the first.

Momentum Equation Steady Flow Average velocities Approximate momentum flow rate

Total Energy and Conservation of Energy Principle E = FE + PE + KE Two points along the same pipe: E1 = E2 Bernoulli’s Equation:

Conservation of Energy

Bernoulli’s Equation

piezometric head kinetic head

Flow through a contraction 1 2 head Total Energy Kinetic Head Piezometric Head position

head Energy Grade Line Kinetic Head Hydraulic Grade Line Piezometric Head position

Bernoulli’s Equation No heat transfer No shear work (frictionless) Single uniform inlet and single uniform outlet No shaft work Steady state Constant temperature Incompressible

Frictional Effects Pipes are NOT frictionless Add a loss due to friction to Bernoulli’s eq. Head loss due to friction

head EGL (ideal) hL EGL (actual) HGL (actual) position

Friction Losses hL = head losses due to friction Fittings (valves, elbows, etc) Pipe friction

Minor Losses Km - minor loss coefficient

Pipe Friction Losses Darcy-Weisbach equation Kinetic pressure Length/diameter ratio Darcy friction factor - pipe roughness (Table 6.1) - Reynolds number

Reynolds Number Dimensionless Ratio of inertial forces to viscous forces Used to characterize the flow regime 27

Reynolds Number Describes if the flow is: Laminar - smooth and steady Turbulent - agitated, irregular Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 28

Osborne Reynolds Tests Laminar Turbulent Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 29

Friction Factor For circular pipe Re ≈ 2300 for transition L = Diameter of pipe Laminar flow Turbulent flow Colebrook Equation 30

Piping Systems Three examples of piping systems Pipes in series 32

Home work Two reservoirs are connected by a pipe as shown. The volume flow rate in pipe A is 2.2 L/s. Find the difference in elevation between the two surfaces. r = 1000 kg/m3 m = 0.001 kg/(m·s) e = 0.05 mm Dz B A D = 2 cm L = 5 m D = 4 cm L = 5 m 33 33

Centrifugal Pumps Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Head increase over pump

Fixed system pressure Variable system pressure

Nature of Dimensional Analysis Example: Drag on a Sphere Drag depends on FOUR parameters: sphere size (D); speed (V); fluid density (r); fluid viscosity (m) Difficult to know how to set up experiments to determine dependencies Difficult to know how to present results (four graphs?)

Nature of Dimensional Analysis Example: Drag on a Sphere Only one dependent and one independent variable Easy to set up experiments to determine dependency Easy to present results (one graph)

Buckingham Pi Theorem Step 1: List all the dimensional parameters involved Let n be the number of parameters Example: For drag on a sphere, F, V, D, r, m, and n = 5

Buckingham Pi Theorem Step 5 Set up dimensional equations, combining the parameters selected in Step 4 with each of the other parameters in turn, to form dimensionless groups There will be n – m equations Example: For drag on a sphere

Dimensional Analysis and Similarity Geometric Similarity - the model must be the same shape as the prototype. Each dimension must be scaled by the same factor. Kinematic Similarity - velocity as any point in the model must be proportional Dynamic Similarity - all forces in the model flow scale by a constant factor to corresponding forces in the prototype flow. Complete Similarity is achieved only if all 3 conditions are met.

Flow Similarity and Model Studies Example: Drag on a Sphere For dynamic similarity … … then …

Flow Similarity and Model Studies Scaling with Multiple Dependent Parameters Example: Centrifugal Pump (Negligible Viscous Effects) If … … then …

System Curve Static head

Pump Curve

System pressure Pump pressure

Operating point

changing

changing

Pump Power Recall that Power provided to fluid Power required by pump

Home Work Water is pumped between two reservoirs at 0.2 ft3/s (5.6 L/s) through 400 ft (124m) of 2-in (50mm) -diameter pipe and several minor losses, as shown. The roughness ratio is e/d = 0.001. Compute the pump horsepower required