Objjjectives

Chapter 4 . Bernoulli 's Theorem Experiment 457.204 Elementary Fluid Mechanics and Lab. Elementary Test To investigate the validity of Bernoulli's Theorem as applied to the flow of water in a tapering circular duct .

VP22 VP Bernoulli's Theorem Experiment 11++=ZZhH 21 +++= 22ggγγ12L

Instructor: Seo, Il Won (35-310) Web: www.ehlab.snu.ac.kr

Bernoulli Theorem Bernoulli Theorem

Born in Netherland

Mathematician, physicist

Hydrodynamique (1738)

Conservation of Energy

Expositi on of a N ew Th eory on th e MtMeasuremet ofRikf Risk (1738)

Dan ie l B ernoulli St. Petersburg Paradox

(1700-1782) Beam theory

ddw22⎛⎞ 22⎜⎟EI= q dx⎝⎠ dx Bernoulli Theorem Bernoulli Theorem

Considering flow at two sections in a pipe VP22 VP Bernoulli’s equation 11++=Z 22 ++ZH = 22ggγγ12 VP22 VP 11++=ZZH 22 ++ = 22ggγγ12

V (m/s)2 V = velocity = vellihdocity head = m 2g m/s g = gravitational acceleration P 22 = pressure hhdead kg⋅⋅ m/s kg m/s P = pressure Piezometer 23 = m γ mm γ = specific weight Z = potential(elevation) head Z = height h = head loss H = total head f H = total head

Piezometer Derivation of Theorem App ly N ew ton ’s 2n d law to the mo tion o f flu id parti cl es Small diameter observation well to measure the hydraulic head Consider a streamline and select a small cylindrical fluid system

Constant

p +=z constant γ P Derivation of Theorem Derivation of Theorem Apply Newton’s 2nd law to the motion of fluid particles dV dpdA+⋅ρρ ds ⋅ dA ⋅ V + g ⋅ dA ⋅ dz =0 ds ∑ F = ma Divide by ρdA dF=−+− pdA() p dp dA dW sinθ dz dp γ = ρg =−dp ⋅ dA =−ρ gdA ⋅ ds ⋅ ++=VdV gdz 0 ds ρ dp 1 =−dp ⋅ dA −ρ g ⋅ dA ⋅ dz ++=VdV dz 0 γ g

(density Ⅹ volume) 2 dddAdm =⋅ρds dA dp⎛⎞ V 2 ++=ddz0 dV()2=⋅ V dV γ ⎜⎟2g dV dV ds dV ⎝⎠ aV==⋅= dt ds dt ds ⎛⎞pV2 dz⎜⎟++=0 Euler’s equation dV ⎝⎠γ 2g ∴∴=−−⋅⋅dpdAρρ g dA dz= ⋅⋅ ds dA ⋅ V ds

Derivation of Theorem Application of Theorem

Bernoulli’s equation? Air lift? Not the main reason ⎛⎞p V 2 Integrate Euler’s equation dz⎜⎟++=0 ⎝⎠γ 2g pV2 ++=z const γ 2g

VP22 VP 11++=Z 22 ++ZH = 22ggγγ12 Experimental Apparatus Procedure

1. Obtain the area of cross sections of the duct point connected to the manometer.

2. Calculate the flowrate with a stopwatch and the volumetric tank level .

3. Calculate mean velocity of each cross section with flowrate and area of cross sections.

4. Compute Reynolds number and velocity head using mean velocity.

5. Measure the ppygressure head by reading Manometer level.

6. The sum of velocity head(4) and pressure head(5) and potential head is the total head. (Potential head is zero, we assumed that the centerline of the duct is datum)

7. Measure the total head of each cross section using Pitot tube.

8. Compare the computed total head(6) with measured total head(7).

9. Repeat process (2-5) 5 times with each other flowrate.

Results Results

Point 1 2 3 4 5 6

Diameter (mm) 25 13.9 11.8 10.7 10 25

Area (mm2) 490.874 1. Using Bernoulli Theorem, discuss relations of diameter of Sd(/)Speed (mm/s) duct, mean velocity, pressure.

Velocity Head (mm)

Pressure Head (mm) 2. Compare the computed total head with measured total head, Potential Head (()mm) discuss why does the difference occurs.

Calculated Total Head

Measured Total Head

Difference