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Elementary Mechanics of Fluids. CE 319 F Daene McKinney. Bernoulli Equation. Euler Equation. Fluid element accelerating in l direction & acted on by pressure and weight forces only (no friction) Newton’s 2 nd Law. Flow. l. 30 o. Ex (5.1).

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Elementary mechanics of fluids l.jpg

Elementary Mechanics of Fluids

CE 319 F

Daene McKinney

Bernoulli

Equation


Euler equation l.jpg
Euler Equation

  • Fluid element accelerating in l direction & acted on by pressure and weight forces only (no friction)

  • Newton’s 2nd Law


Ex 5 1 l.jpg

Flow

l

30o

Ex (5.1)

  • Given: Steady flow. Liquid is decelerating at a rate of 0.3g.

  • Find: Pressure gradient in flow direction in terms of specific weight.


Ex 5 3 l.jpg

vertical

A

1 m

B

EX (5.3)

  • Given: g = 10 kN/m3, pB-pA=12 kPa.

  • Find: Direction of fluid acceleration.


Hw 5 7 l.jpg
HW (5.7)

  • Ex (5.6) What pressure is needed to accelerate water in a horizontal pipe at a rate of 6 m/s2?


Ex 5 10 l.jpg
Ex (5.10)

V1/2=(80+30)/2 ft/s = 55 ft/s

  • Given: Steady flow. Velocity varies linearly with distance through the nozzle.

  • Find: Pressure gradient ½-way through the nozzle

dV/dx =(80-30) ft/s /1 ft = 50 ft/s/ft



Bernoulli equation l.jpg
Bernoulli Equation

  • Consider steady flow along streamline

  • s is along streamline, and t is tangent to streamline


Ex 5 47 l.jpg
Ex (5.47)

Point 1

  • Given: Velocity in outlet pipe from reservoir is 6 m/s and h = 15 m.

  • Find: Pressure at A.

  • Solution: Bernoulli equation

Point A


Example l.jpg

Point 1

Point A

Example

  • Given: D=30 in, d=1 in, h=4 ft

  • Find: VA

  • Solution: Bernoulli equation


Example venturi tube l.jpg

D

D

d

2

1

3

Example – Venturi Tube

  • Given: Water 20oC, V1=2 m/s, p1=50 kPa, D=6 cm, d=3 cm

  • Find: p2 and p3

  • Solution: Continuity Eq.

  • Bernoulli Eq.

Nozzle: velocity increases, pressure decreases

Diffuser: velocity decreases, pressure increases

Similarly for 2  3, or 1  3

Pressure drop is fully recovered, since we assumed no frictional losses

Knowing the pressure drop 1  2 and d/D, we can calculate the velocity and flow rate


Ex 5 48 l.jpg
Ex (5.48)

  • Given: Velocity in circular duct = 100 ft/s, air density = 0.075 lbm/ft3.

  • Find: Pressure change between circular and square section.

  • Solution: Continuity equation

  • Bernoulli equation

Air conditioning (~ 60 oF)


Ex 5 49 l.jpg
Ex (5.49)

  • Given: r = 0.0644 lbm/ft3V1= 100 ft/s, and A2/A1=0.5, gm=120 lbf/ft3

  • Find: Dh

  • Solution: Continuity equation

  • Bernoulli equation

Heating (~ 170 oF)

  • Manometer equation




Stagnation tube in a pipe l.jpg

Pipe

2

Flow

1

Stagnation Tube in a Pipe



Pitot tube application p 170 l.jpg

V

1

z1-z2

2

l

y

Pitot Tube Application (p.170)






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