Lecture 8 Bernoulli s Equation: Limitations and Applications Last time, we derived the steady form of Bernoulli s Equation along a streamline p + ρv + ρgz = P t static hydrostatic total pressure q = dynamic pressure pressure pressure o p = static pressure is the thermodynamic pressure o if there is no flow, we have a static fluid and p at a point in a static fluid total pressure form () + ρ gz is the absolute pressure one would measure o q = dynamic pressure is the kinetic energy per unit volume of a fluid element in motion divide () by ρ to get the energy/mass form p + V + gz = C ρ PE/mass flow energy/mass KE/mass divide () by ρ g density to get the form different constant for each streamline p V + + z = H elevation total ρ g g Pressure velocity Latter is often plotted in graphical form (civil engineering)
o line that represents p ρ g + z is called the hydraulic grade line ( HGL ) o line that represents H is called the energy grade line ( EGL ), which is always above the HGL by a distance Restrictions are. steady flow V g o there is an unsteady form that has an extra term in the equation advanced fluid dynamics o Don't use during transient start-up and shut-down periods. frictionless (no losses)
o Every flow involves some friction, just a question of whether frictional effects may be negligible o Frictional effects are generally negligible for short flow sections with large cross sections especially at low flow velocities. o Frictional effects are usually significant in long and narrow flow passages, wake region downstream of an object, separated flows, and near solid walls where friction effects are dominant due to no slip BC. o Not applicable in a component that disturbs the streamlines and causes considerable mixing and backflow such as a sudden expansion, or a valve 3. No shaft work o not applicable across a pump, turbine, fan, or any other turbomachine o use the energy equation instead to account for the shaft work o Can usually still use the Bernoulli equation before or after the machine, but the Bernoulli constant changes across the device 4. Incompressible flow o We assumed ρ was constant usually true for liquid and low-speed gas flows o There is a compressible form compressible fluid dynamics 5. No heat transfer
o not applicable in flows where there is a significant temperature change in flow o use energy equation instead 6. Flow along a streamline o Different Bernoulli constant for each streamline (recall normal direction equation tells us that pressure increases in outward normal direction along radius of curvature) o When the flow is irrotational, the vorticity ζ = V is zero in the flow and the Bernoulli constant is the same everywhere Example V = 0 Is the equation V ( P P ) stag = correct if we ρ use a U-tube manometer with a fluid density V h ρ man? The density of flowing fluid is ρ. h ρ man Δh ρ man gδ h= ρv h = distance from stagnation point to static pressure tap h = height of left fluid column Δ h = difference in height of each fluid column
Solution Assume incompressible, steady, frictionless flow along a straight streamline from to stagnation point Bernoulli equation Ptotal = P + ρv + ρgz = Pstag + ρ Vstag ρ gzstag + so V ( stag ) P P q = =. ρ ρ Along the stagnation streamline, the flow decelerates from V to 0! Occurs over a very short distance. What is Pstag P? o The figure shows that the Δ h in the U-tube corresponds to q or ρ g h man Δ = ρv o The fluid in the pitot tube is at rest, so the pressure at the top of the right fluid column is stag ( ) P + ρ g h + h +Δ h, (assuming gravity acts downward).. Is this true? o Jumping across the manometer to the other side and applying fluid statics to obtain the pressure at the interface between the fluids is ρ ( ) P = P + g h + h +Δh ρ gδ h interface stag p o Since the streamlines are straight, = 0 (no change in static pressure), there is only a change in the n hydrostatic pressure so ρ ( ) ρ ρ ( ρ ρ) ( ) P P = ρ ρ gδh ρ gδ h if ρman stag man man P P g h h P g h g h P g h = + = + Δ Δ = Δ or interface stag man stag man man ρ (valid in gases but not liquids).
Example Recall Example 5. stated that V = gh. Show that this is true using Bernoulli s equation. jet Solution We will follow a streamline from the free surface at the top of the tank to the jet issuing from the tank. Assume that the flow is steady, incompressible, and frictionless. P + ρv + ρgz = P + ρv + ρgz jet jet jet But P = Pjet = Patm and V 0 so ρvjet = ρg( z zjet ) = ρgh Vjet = gh