3 INTRODUCTION The three equations commonly used in fluid mechanics are: the mass, Bernoulli, and energy equations.The mass equation is an expression of the conservation of mass principle.The Bernoulli equation: Conservation of kinetic, potential, and flow energies ( viscous forces are negligible.)
4 Fluid Motion Two ways to describe fluid motion Lagrangian Eularian Follow particles aroundEularianWatch fluid pass by a point or an entire regionFlow patternStreamlines – velocity is tangent to them
5 STEADY AND UNSTEADY FLOW Steady flow: the flow in which conditions at any point do not change with time is called steady flow.Then, …etc.Unsteady flow: the flow in which conditions at any point change with time, is called unsteady flow.Then, …etc.
6 UNIFORM AND NON-UNIFORM The flow in which the conditions at all points are the same at the same instant is uniform flow.The flow in which the conditions vary from point to point at thesame instant is non-uniform flow.
7 ACCELERATION Acceleration = rate of change of velocity Components: Normal – changing directionTangential – changing speed
8 ACCELERATION Cartesian coordinates In steady flow ∂u/∂t = 0 , local acceleration is zero.In unsteady flow ∂u/∂t ≠ 0 ; local acceleration Occurs.Other terms u ∂u/∂x, v ∂u/∂y,.. are called convective accelerations. Convective acceleration Occurs when the velocity varies with position.Uniform flow: convective acceleration = 0Non-uniform flow: convective acceleration ≠0ConvectiveLocalFluid Mechanics I
9 Example Valve at C is opened slowly The flow at B is non uniform The flow at A is uniformFluid Mechanics I
14 ExamplesDischarge in a 25-cm pipe is 0.03 m3/s. What is the average velocity?A pipe whose diameter is 8 cm transports air with a temp. of 20oC and pressure of 200 kPa abs. At 20 m/s. What is the mass flow rate?
15 Example: The velocity distribution in a circular duct is , where r is the radial location in the duct, R is the duct radius, and Vo is the velocity on the axis. Find the ratio of the mean velocity to the velocity on the axis.Find:
16 Example: Air (ρ =1.2 kg/m^3) enters the duct shown: Find:V/10=y/.5 V=20ydA=1*dy
17 Example: In this flow passage the discharge is varying with time according to the following expression: At time t=0.5 s, it is known that at section A-A the velocity gradient in the S direction is +2m/s per meter. Given that Qo, Q1, and to are constants with values of m^3/s, 0.5 m^3/s, and 1 s, respectively, and assuming that one-dimensional flow, answer the following questions for time t=0.5 s. a. What is the velocity a A-A? b. What is the local acceleration at A-A? c. What is the convective acceleration at A-A?Fluid Mechanics I
19 Systems, Control Volume, and Control Surface System (sys)A fluid system: contains thesame fluid particles.Mass does not cross thesystem boundaries.Thus the mass of the systemis constant.Control Volume (C.V)A control volume is a selected volumetric region in space. It’s shape and position may change with time.(Open System)Control Surface (C.S)The surface enclosing the control volume is called the control surface.
20 Systems, Control Volume, and Control Surface (continued ) Consider the tank shown, assume:the control volume is defined by the tank walls and the top of the liquid.The control surface that encloses the control volume is designated by the dashed line.The liquid in the tank at time t is elected as the system and is indicated by the solid line.At this instant in time, the system completely occupies the control volume and is contained by the control surface. Thus, at this time:During the same period some liquid has entered the control volume from the left, the amount being:After a time some liquid has flowed out of the control volume to the right. The amount that flowed out is:
21 Systems, Control Volume, and Control Surface (continued ) Now the system has been deformed as shown in Fig. (b).Part of the system is the liquid that has flowed out across the control surface.The system remaining in the control volume has been deformed by the mass that has flowed in across the control surface.Also, the height of the control volume has changed to accommodate the net flow into the tank.The mass of the system at time t + Δt can be determined by taking the mass in the control volume, subtracting the mass that entered, and adding the mass that left.Subtracting (1) from (2), we have:Dividing by Δt and taking the limit as Δt 0 yieldsThe equation relates the rate of change of the mass of the system to the rate of change of mass in the control volume plus the net outflow (efflux) across the control surface
22 Systems, Control Volume, and Control Surface (continued ) By definition, the mass of the system is constant soLagrangian statementAnd Eq. (3) becomesThe corresponding Eulerian statement and can be written as:This equation states that there is an increasing mass in the control volume if there is a net mass influx through the control surface and decreasing mass if there is a net mass efflux. This is identified as the continuity equation.
23 Systems Conservation of Mass Momentum Energy Laws of Mechanics Written for systemsSystem = arbitrary quantity of mass of fixed identityFixed quantity of mass, mConservation of MassMass is conserved and does not changeMomentumIf surroundings exert force on system, mass will accelerateEnergyIf heat is added to system or work is done by system, energy will change
24 Control Volumes Solid Mechanics Fluid Mechanics Follow the system, determine what happens to itFluid MechanicsConsider the behavior in a specific region or Control VolumeConvert System approach to CV approachLook at specific regions, rather than specific massesReynolds Transport TheoremRelates time derivative of system properties to rate of change of property in CV
26 Continuity EquationIn the case of the continuity equation, the extensive property in the control volume equation is the mass of the system, Msys, and the corresponding intensive variable, b, is the mass per unit mass, orSubstituting b equal to unity in the control volume equation yields the general form of the continuity equation.This is sometimes called the integral form of the continuity equation.
27 Continuity EquationThe term on the left is the rate of change of the mass of the system. However, by definition, the mass of a system is constant. Therefore the left-hand side of the equation is zero, and the equation can be written asThis is the general form of the continuity equation. It states that the net rate of the outflow of mass from the control volume is equal to the rate of decrease of mass within the control volume.The continuity equation involving flow streams having a uniform velocity across the flow section is given as
28 Example: at a certain time rate of rising is 0.1 cm/s: Continuity equation
29 Select a CV that moves up and down with the water surface Example: Both pistons are moving to the left, but piston A has a speed twice as great as that of piston B. Then the water level in the tank is: a) rising, b) not moving up or down, c) falling.Select a CV that moves up and down with the water surfaceContinuity EquationCSVA=2VBVBhFluid Mechanics I
30 Euler EquationFluid element accelerating in l direction & acted on by pressure and weight forces only (no friction)Newton’s 2nd Law
31 Example 1:Given: Steady flow. Liquid is decelerating at a rate of 0.3g.Find: Pressure gradient in flow direction in terms of specific weight.Flow30ol
32 Example 2: Given: g = 10 kN/m3, pB-pA=12 kPa. verticalGiven: g = 10 kN/m3, pB-pA=12 kPa.Find: Direction of fluid acceleration.
33 dV/dx = (80-30) ft/s /1 ft = 50 ft/s/ft Example 3:Given: Steady flow. Velocity varies linearly with distance through the nozzle.Find: Pressure gradient ½-way through the nozzleV1/2=(80+30)/2 ft/s = 55 ft/sdV/dx = (80-30) ft/s /1 ft = 50 ft/s/ft
34 Bernoulli Equation Consider steady flow along streamline s is along streamline, and t is tangent to streamline
35 The Bernoulli equation states that the sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow.An alternative form of the Bernoulli equation is expressed in terms of heads as: The sum of the pressure, velocity, and elevation heads is constant along a streamline.
36 Example 4:Given: Velocity in outlet pipe from reservoir is 6 m/s and h = 15 m.Find: Pressure at A.Solution: Bernoulli equationPoint 1Point A
37 Example 5: Given: D=30 in, d=1 in, h=4 ft Find: VA Point APoint 1Given: D=30 in, d=1 in, h=4 ftFind: VASolution: Bernoulli equation
38 Static, Stagnation, Dynamic, and Total Pressure: Bernoulli Equation HydrostaticPressureStatic Pressure: moves along the fluid “static” to the motion.Dynamic Pressure: due to the mean flow going to forced stagnation.Hydrostatic Pressure: potential energy due to elevation changes.Following a streamline:Follow a Streamline from point 1 to 20, no elevation0, no elevationNote:“Total Pressure = Dynamic Pressure + Static Pressure”H > hIn this way we obtain a measurement of the centerline flow with piezometer tube.
39 Bernoulli equation: The sum of flow, kinetic, and potential energies of a fluid particle along a streamline is constant.Each term in this equation has pressure units, and thus each term represents Energy per unit voluneP is the static pressure.ϱV2/2 is the dynamic pressure.ϱ gz is the hydrostatic pressureThe sum of the static, dynamic, and hydrostatic pressures is called the total pressure. Bernoulli equation states that the total pressure along a streamline is constant.The sum of the static and dynamic pressures is called the stagnation pressure, and it is expressed as
40 The static, dynamic, and stagnation pressures are shown. When static and stagnation pressures are measured at a specified location, the fluid velocity at that location can be calculated from
44 Example – Venturi TubeGiven: Water 20oC, V1=2 m/s, p1=50 kPa, D=6 cm, d=3 cmFind: p2 and p3Solution: Continuity Eq.Bernoulli Eq.Dd123Nozzle: velocity increases, pressure decreasesDiffuser: velocity decreases, pressure increasesSimilarly for 2 3, or 1 3Pressure drop is fully recovered, since we assumed no frictional lossesKnowing the pressure drop 1 2 and d/D, we can calculate the velocity and flow rateFluid Mechanics I
45 Air conditioning (~ 60 oF) ExGiven: Velocity in circular duct = 30 m/s, air density = 1.2 kg/m3.Find: Pressure change between circular and square section.Solution: Continuity equationBernoulli equationAir conditioning (~ 60 oF)