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**California State University, Chico**

CE 150 Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico CE 150

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**Elementary Fluid Dynamics**

Reading: Munson, et al., Chapter 3 CE 150

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Inviscid Flow In this chapter we consider “ideal” fluid motion known as inviscid flow; this type of flow occurs when either 1) 0 (only valid for He near 0 K), or 2) viscous shearing stresses are negligible The inviscid flow assumption is often valid in regions removed from solid surfaces; it can be applied to many problems involving flow through pipes and flow over aerodynamic shapes CE 150

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**Newton’s 2nd Law for a Fluid Particle**

CE 150

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**Newton’s 2nd Law for a Fluid Particle**

The equation of motion for a fluid particle in a steady inviscid flow: We consider force components in two directions: along a streamline (s) and normal to a streamline (n): CE 150

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**Newton’s 2nd Law Along a Streamline**

Noting that we have: CE 150

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**Newton’s 2nd Law Along a Streamline**

Integrating along the streamline: If the fluid density remains constant This is the Bernoulli equation CE 150

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**Newton’s 2nd Law Across a Streamline**

A similar analysis applied normal to the streamline for a fluid of constant density yields This equation is not as useful as the Bernoulli equation because the radius of curvature of the streamline is seldom known CE 150

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**Physical Interpretation of the Bernoulli Equation**

Acceleration of a fluid particle is due to an imbalance of pressure forces and fluid weight Conservation equation involving three energy processes: kinetic energy potential energy pressure work CE 150

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**Alternate Form of the Bernoulli Equation**

Pressure head (p/g) - height of fluid column needed to produce a pressure p Velocity head (V2/2g) - vertical distance required for fluid to fall from rest and reach velocity V Elevation head (z) - actual elevation of the fluid w.r.t. a datum CE 150

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**Bernoulli Equation Restrictions**

The following restrictions apply to the use of the (simple) Bernoulli equation: 1) fluid flow must be inviscid 2) fluid flow must be steady (i.e., flow properties are not f(t) at a given location) 3) fluid density must be constant 4) equation must be applied along a streamline (unless flow is irrotational) 5) no energy sources or sinks may exist along streamline (e.g., pumps, turbines, compressors, fans, etc.) CE 150

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**Using the Bernoulli Equation**

The Bernoulli equation can be applied between any two points, (1) and (2), along a streamline: Free jets - pressure at the surface is atmospheric, or gage pressure is zero; pressure inside jet is also zero if streamlines are straight Confined flows - pressures cannot be prescribed unless velocities and elevations are known CE 150

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**Mass and Volumetric Flow Rates**

Mass flow rate: fluid mass conveyed per unit time [kg/s] where Vn = velocity normal to area [m/s] = fluid density [kg/m3] A = cross-sectional area [m2] if is uniform over the area A and the average velocity V is used, then Volumetric flow rate [m3/s]: CE 150

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**Conservation of Mass “Mass can neither be created nor destroyed”**

For a control volume undergoing steady fluid flow, the rate of mass entering must equal the rate of mass exiting: If = constant, then CE 150

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**The Bernoulli Equation in Terms of Pressure**

Each term of the Bernoulli equation can be written to represent a pressure: pgh : this is known as the hydrostatic pressure; while not a real pressure, it represents the possible pressure in the fluid due to changes in elevation CE 150

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**The Bernoulli Equation in Terms of Pressure**

p : this is known as the static pressure and represents the actual thermodynamic pressure of the fluid CE 150

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**The Bernoulli Equation in Terms of Pressure**

The static pressure at (1) in Figure 3.4 can be measured from the liquid level in the open tube as pgh : this is known as the dynamic pressure; it is the pressure measured by the fluid level (pgH) in the stagnation tube shown in Figure 3.4 minus the static pressure; thus, it is the pressure due to the fluid velocity CE 150

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**The Bernoulli Equation in Terms of Pressure**

The stagnation pressure is the sum of the static and dynamic pressures: the stagnation pressure exists at a stagnation point, where a fluid streamline abruptly terminates at the surface of a stationary body; here, the velocity of the fluid must be zero Total pressure (pT) is the sum of the static, dynamic, and hydrostatic pressures CE 150

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**Velocity and Flow Measurement**

Pitot-static tube - utilizes the static and stagnation pressures to measure the velocity of a fluid flow (usually gases): CE 150

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**Velocity and Flow Measurement**

Orifice, Nozzle, and Venturi meters - restriction devices that allow measurement of flow rate in pipes: CE 150

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**Velocity and Flow Measurement**

Bernoulli equation analysis yields the following equation for orifice, nozzle, and venturi meters: Theoretical flowrate: Actual flowrate: CE 150

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**Velocity and Flow Measurement**

Sluice gates and weirs - restriction devices that allow flow rate measurement of open-channel flows: CE 150

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**Velocity and Flow Measurement**

CE 150

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