Elementary Fluid Dynamics:The Bernoulli Equation : Elementary Fluid Dynamics:The Bernoulli Equation CEE 331
29 March 2010 ???
BernoulliAlong a Streamline : BernoulliAlong a Streamline Separate acceleration due to gravity. Coordinate system may be in any orientation!
k is vertical, s is in direction of flow, n is normal. Component of g in s direction Note: No shear forces! Therefore flow must be frictionless.
BernoulliAlong a Streamline : BernoulliAlong a Streamline 0 (n is constant along streamline, dn=0) Write acceleration as derivative wrt s Chain rule and Can we eliminate the partial derivative?
Integrate F=ma Along a Streamline : Integrate F=ma Along a Streamline If density is constant… But density is a function of ________. pressure Eliminate ds Now let’s integrate…
Bernoulli Equation : Assumptions needed for Bernoulli Equation
Eliminate the constant in the Bernoulli equation? _______________________________________
Bernoulli equation does not include
___________________________
___________________________ Bernoulli Equation Apply at two points along a streamline. Mechanical energy to thermal energy Heat transfer, Shaft Work Frictionless Steady Constant density (incompressible) Along a streamline
Bernoulli Equation : Bernoulli Equation The Bernoulli Equation is a statement of the conservation of ____________________ Mechanical Energy p.e. k.e. Pressure head Elevation head Velocity head Piezometric head Total head Energy Grade Line Hydraulic Grade Line
Bernoulli Equation: Simple Case (V = 0) : Bernoulli Equation: Simple Case (V = 0) Reservoir (V = 0)
Put one point on the surface, one point anywhere else z Elevation datum Pressure datum Same as we found using statics We didn’t cross any streamlines so this analysis is okay!
Hydraulic and Energy Grade Lines (neglecting losses for now) : Mechanical Energy Conserved Hydraulic and Energy Grade Lines (neglecting losses for now) The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)? Elevation datum Pressure datum? __________________ Atmospheric pressure Teams Mechanical energy
Bernoulli Equation: Simple Case (p = 0 or constant) : Bernoulli Equation: Simple Case (p = 0 or constant) What is an example of a fluid experiencing a change in elevation, but remaining at a constant pressure? ________ Free jet
Bernoulli Equation Application:Stagnation Tube : Bernoulli Equation Application:Stagnation Tube What happens when the water starts flowing in the channel?
Does the orientation of the tube matter? _______
How high does the water rise in the stagnation tube?
How do we choose the points on the streamline? Stagnation point Yes!
Bernoulli Equation Application:Stagnation Tube : Bernoulli Equation Application:Stagnation Tube 1a-2a
_______________
1b-2a
_______________
1a-2b
____________________________ Same streamline Crosses || streamlines Doesn’t cross streamlines V = f(?p) V = f(z2) V = f(?p) In all cases we don’t know p1
Stagnation Tube : Stagnation Tube Great for measuring __________________
How could you measure Q?
Could you use a stagnation tube in a pipeline?
What problem might you encounter?
How could you modify the stagnation tube to solve the problem? EGL (defined for a point)
Pitot Tubes : Pitot Tubes Used to measure air speed on airplanes
Can connect a differential pressure transducer to directly measure V2/2g
Can be used to measure the flow of water in pipelines Point measurement!
Pitot Tube : Pitot Tube V1 = 1 2 Connect two ports to differential pressure transducer. Make sure Pitot tube is completely filled with the fluid that is being measured.
Solve for velocity as function of pressure difference z1 = z2 Static pressure tap Stagnation pressure tap 0
Relaxed Assumptions for Bernoulli Equation : Relaxed Assumptions for Bernoulli Equation Frictionless (velocity not influenced by viscosity)
Steady
Constant density (incompressible)
Along a streamline Small energy loss (accelerating flow, short distances) Or gradually varying Small changes in density Don’t cross streamlines
Bernoulli Normal to the Streamlines : Bernoulli Normal to the Streamlines Separate acceleration due to gravity. Coordinate system may be in any orientation! Component of g in n direction
Bernoulli Normal to the Streamlines : Bernoulli Normal to the Streamlines 0 (s is constant normal to streamline) R is local radius of curvature n is toward the center of the radius of curvature
Integrate F=ma Normal to the Streamlines : Integrate F=ma Normal to the Streamlines (If density is constant) Multiply by dn Integrate
Pressure Change Across Streamlines : Pressure Change Across Streamlines If you cross streamlines that are straight and parallel, then ___________ and the pressure is ____________. hydrostatic r As r decreases p ______________ decreases
End of pipeline? : End of pipeline? What must be happening when a horizontal pipe discharges to the atmosphere? Try applying statics… Streamlines must be curved! (assume straight streamlines)
Nozzle Flow Rate: Find Q : Nozzle Flow Rate: Find Q D1=30 cm D2=10 cm Q 90 cm Coordinate system Pressure datum____________ Crossing streamlines Along streamline h h=105 cm gage pressure
Solution to Nozzle Flow : Solution to Nozzle Flow Now along the streamline h Two unknowns…
_______________ Mass conservation
Solution to Nozzle Flow (continued) : Solution to Nozzle Flow (continued)
Incorrect technique… : Incorrect technique… The constants of integration are not equal!
Bernoulli Equation Applications : Bernoulli Equation Applications Stagnation tube
Pitot tube
Free Jets
Orifice
Venturi
Sluice gate
Sharp-crested weir Applicable to contracting streamlines (accelerating flow).
Ping Pong Ball : Ping Pong Ball Why does the ping pong ball try to return to the center of the jet?
What forces are acting on the ball when it is not centered on the jet? How does the ball choose the distance above the source of the jet? Teams
Summary : Summary By integrating F=ma along a streamline we found…
That energy can be converted between pressure, elevation, and velocity
That we can understand many simple flows by applying the Bernoulli equation
However, the Bernoulli equation can not be applied to flows where viscosity is large, where mechanical energy is converted into thermal energy, or where there is shaft work. mechanical
Jet Problem : Jet Problem How could you choose your elevation datum to help simplify the problem?
How can you pick 2 locations where you know enough of the parameters to solve for the velocity?
You have one equation (so one unknown!)
Jet Solution : Jet Solution The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)? z Elevation datum Are the 2 points on the same streamline?
What is the radius of curvature at the end of the pipe? : What is the radius of curvature at the end of the pipe? R Uniform velocity Assume R>>D2
Example: Venturi : Example: Venturi
Example: Venturi : Example: Venturi How would you find the flow (Q) given the pressure drop between point 1 and 2 and the diameters of the two sections? You may assume the head loss is negligible. Draw the EGL and the HGL over the contracting section of the Venturi. 1 2 ?h How many unknowns?
What equations will you use?
Example Venturi : Example Venturi