Download - fluid flow and bernoulli
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Fluid Mechanics
Flow Rate & Bernoulli’s Equation
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Volumetric Flow Rate
• volumetric flow rate: the volume of fluid that passes a particular point per unit of time– example: liters/minute coming out of a faucet– metric units: m3/s– note: not the same as flow velocity (m/s)
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Flow Rate
volumetric flow rate = Q = Sv
S
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Continuity Equation
• If the fluid in the pipe is incompressible (density remains constant) then the flow rate must be the same everywhere in the pipe.
• Therefore: Q1 = Q2
• …and S1v1 = S2v2 This is known as the continuity equation.
S
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Sample Problem 1A pipe of non-uniform diameter carries water.
At one point on the pipe, the radius is 2 cm and the flow speed is 6 m/s.
a. What is the volumetric flow rate?
b.What is the flow velocity at a point where the pipe constricts to a radius of 1 cm?
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Sample Problem 2
If the diameter of a pipe increases from 4 cm to 12 cm, what will happen to the flow velocity?
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Bernoulli’s Equation
• One of the most important idea in fluid mechanics.
• It is the statement of the law of conservation of energy for ideal fluid flow (mechanical energy balance equation).
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Conditions for Ideal Fluid Flow
1. The fluid is incompressible. This works well for liquids and also applies to gases if the pressure changes are small.
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Conditions for Ideal Fluid Flow
2. The fluid’s viscosity is negligible or zero.
Viscosity is the force of cohesion between the molecules of a fluid. It can be thought of as internal friction. Syrup has a higher viscosity than water – there’s more resistance to the flow of syrup. Bernoulli’s equation gives good results when applied to water, but not to syrup.
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Conditions for Ideal Fluid Flow
3. The flow is streamline (laminar).
The fluid moves smoothly through the tube. The opposite is turbulent flow.
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Turbulent Flow
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Bernoulli’s Equation
• If the conditions for ideal fluid flow are met and the volumetric flow rate, Q, is steady, Bernoulli’s equation can be applied to any pair of points along a streamline with the flow.
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Bernoulli’s Equation
• P1 & P2 = pressure at points 1 and 2• v1 & v2 = flow velocity• z1 & z2 = elevation above a reference level
Bernoulli’s Equation (each term has a unit of energy per unit mass):
z1
z2
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Bernoulli’s Equation
Bernoulli’s Equation:
or…
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Implications of Bernoulli’s Equation
Bernoulli’s Equation:
• Where the flow speed is high the pressure is low, and where flow speed is low, pressure is high.
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Applications of Bernoulli’s Equation
• Close streamlines above wing indicate high velocity
(continuity equation). Therefore the pressure above the wing is lower which creates a loft force that balances that of gravity.
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Applications of Bernoulli’s Equation
A person with constricted arteries will find that they may experience a temporary lack of blood to the brain (TIA – transient ischemic attack) as blood speeds up to get past the constriction, thereby reducing the pressure.
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Torricelli’s Theorem
Bernoulli’s equation can be usedto determine the efflux speed
(how fast the liquid flows out of the hole)
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Torricelli’s Theoremz2
z1
z2-z1
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Torricelli’s Theoremz2
z1
z2-z1
Points 1 and 2 are open to air, so P1 = P2 = Patm
Also,
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Torricelli’s Theoremz2
z1
z2-z1
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Torricelli’s Theoremz2
z1
z2-z1
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Torricelli’s Theoremz2
z1
z2-z1
v2 0 (when compared to v1)
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Torricelli’s Theoremz2
z1
z2-z1
v2 0 (when compared to v1)
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Torricelli’s Theoremz2
z1
z2-z1
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Torricelli’s Theoremz2
z1
z2-z1
Solving for v1, we have
2gh)2g(zv 121 z
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SiphoningThe figure below shows a siphon that is used to draw water from a swimming
pool. The pipe that makes up the siphon has an inside diameter of 40mm and terminates with a 25-mm diameter nozzle. Assuming that there are no energy losses in the system, calculate the volume flow rate through the siphon. Calculate also the pressure at points b, c, d and e.
1.2m1
.8m1.2m
40-mm inside diameter
25-mm inside diameter
a
f
b
c
d
e
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For Your Information
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Other Examples
How high will the jet of water shown at the left be? (neglecting energy losses) 6m
a
b
c
h