Bernoulli & Energy Equations – Comprehensive Study Notes

Mechanical Energy of Flowing Fluids

• A fluid element in motion simultaneously possesses three classical forms of mechanical energy.
  • Potential (Elevation) Energy
  • Absolute form: $E_p = mgz$
  • Per–unit–weight (head) form: $z$
  • Kinetic (Velocity) Energy
  • Absolute form: $E_k = \frac12 m v^2$
  • Per–unit–weight head: $\frac{v^2}{2g}$
  • Pressure (Flow) Energy
  • Work done by pressure force as a fluid element is displaced downstream.
  • For a weight element $mg$ that occupies volume $\frac{m}{\rho}$ and crosses a section of area $A$ under pressure $p$:
    \text{Flow\/work} = pA \Big[\frac{m/\rho}{A}\Big] \;\Rightarrow\; \text{head} = \frac{p}{\rho g}
• Total mechanical energy per unit weight (total head)$H = \frac{p}{\rho g} + \frac{v^2}{2g} + z$
  • Components are respectively called pressure head, velocity head, and elevation (potential) head.

Total Head & Bernoulli Equation

• Ideal (inviscid, steady, incompressible, along a streamline, no shaft work)$\frac{p<em>1}{\rho g}+\frac{v</em>1^2}{2g}+z<em>1 = \frac{p</em>2}{\rho g}+\frac{v<em>2^2}{2g}+z</em>2 = H = \text{constant}$
  • Statement of mechanical‐energy conservation between any two points, traditionally known as Bernoulli’s equation.
• Extended form (to include real‐world additions or losses)$\frac{p<em>1}{\rho g}+\frac{v</em>1^2}{2g}+z<em>1\; + h</em>q\; = \;\frac{p<em>2}{\rho g}+\frac{v</em>2^2}{2g}+z<em>2\; + h</em>w\; + h_l$
  • $h_q$ : energy added (e.g. by a pump)
  • $h_w$ : shaft work removed (e.g. turbine output)
  • $h_l$ : irreversible loss (major/minor friction, entrance/exit disturbances, expansions, contractions).

Worked Examples (Chap 6)

Example 6.1 – Free Jet from a Tank (Torricelli)

• 5 m water head above a smooth tap discharging to atmosphere.
• Large tank → $v<em>1 \approx 0$, $p</em>1 = p<em>2 = p</em>{atm}$, $z<em>1 = 5 \text{ m}$, $z</em>2 = 0$.
• Bernoulli: $\frac{v<em>2^2}{2g}=z</em>1\;\Rightarrow\; v<em>2 = \sqrt{2gz</em>1}=9.9\ \text{m\,/s}$.
• Result is identical to velocity acquired by a body in free fall of 5 m.
  • Highlights energy conversion: potential → kinetic.

Example 6.2 – Gasoline Siphon

• Data: $\Delta z = 0.75\ \text{m}\ (1\to2),\ z_3 = 2\ \text{m\ above point 1}, \rho=750\ \text{kg/m}^3$, tube Ø = 5 mm.
• Best‐case (ignore friction): $v_2 = \sqrt{2g\Delta z}=3.84\ \text{m/s}$.
• Flow area $A = \pi D^2/4 = 1.96\times10^{-5}\ \text{m}^2$.
• Volumetric flow $Q = v_2 A = 7.53\times10^{-5}\ \text{m}^3/s =0.0753\ \text{L/s}$.
• Time for 4 L: $t = 4/0.0753 = 53.1\ \text{s}$.
• Pressure at summit (point 3): $p<em>3 = p</em>{atm} - \rho g z_3 = 101.3\ \text{kPa} - 750\,9.81\,2.75 \approx 81.1\ \text{kPa}$
  • Sub‐atmospheric (partial vacuum); explains why vapor pockets may form if $p_3$ drops below vapor pressure.

Example 6.3 – Rising Pipe with Area Change

• Upward flow, $Q=0.9\ \text{m}^3/s,\ d<em>1=0.5\ \text{m},\ z</em>2-z<em>1=1.5\ \text{m},\ p</em>1=800\ \text{kPa},\ p_2=600\ \text{kPa}$.
• Ignore losses; unknown $d_2$.
• Use continuity $Q = A<em>1v</em>1 = A<em>2v</em>2$ and Bernoulli to solve for $v<em>2\ (\Rightarrow d</em>2)$.

Example 6.4 – Aircraft Wing (Subsonic)

• Flight speed 134 m/s at 4000 m. Over-wing velocity 26 % higher → $v<em>2 = 1.26 v</em>1 = 168.8$ m/s.
• Neglect altitude change & compressibility.
• Bernoulli gives pressure drop: $p<em>2 = p</em>1 - \tfrac12\rho(v<em>2^2 - v</em>1^2).$ Use standard‐air $\rho<em>{4000m}\approx0.819\ \text{kg/m}^3$ to compute $p</em>2$.
  • Demonstrates lift generation through velocity-induced pressure gradient.

Example 6.5 – Fire Engine Pump & Nozzle

• Data: pump head 50 m, suction line d = 150 mm (loss $5u<em>1^2/2g$), delivery line d = 100 mm (loss $12u</em>2^2/2g$), nozzle d = 75 mm, elevation of C 30 m above pump, z₂ = 2 m.
• Part (a): Jet velocity $u_3$.
  • Combine extended Bernoulli from reservoir A (free surface) to nozzle exit C.
  • Use continuity to express $u<em>1, u</em>2$ in terms of $u_3$.
  • Substitute into
    $u<em>3^2 + 5u</em>1^2 + 12u<em>2^2 = 2g(18\,\text{m})$ → $u</em>3 = 8.314\ \text{m/s}$.
• Part (b): Pressure at pump inlet B: $\frac{p<em>B}{\rho g} = -z</em>1 - 6\frac{u<em>1^2}{2g} = -3.32\ \text{m}\;\Rightarrow\; p</em>B = -32.6\ \text{kPa}$ (suction).
  • Signifies potential risk of cavitation if vapor pressure is approached.

Example 6.6 – Closed Tank, Absolute Pressure & Head Loss

• Air space above water: 70 kPa (abs). Barometer: 750 mm Hg ($p_{atm}=100.1\ \text{kPa}$).
• Vacuum relative to atmosphere at point O: $p_O = 70-100.1 = -30.1\ \text{kPa}$.
• Head loss from O→A: $h=0.1\ \text{m}$; elevation change $z<em>O - z</em>A = 4\ \text{m}$.
• Bernoulli →
  $v<em>A = \sqrt{2g\big[z</em>O - z<em>A - h + \frac{p</em>O}{\rho g}\big]} = 4.05\ \text{m/s}.$

Energy Line & Hydraulic Grade Line (HGL)

• Energy Line (EL): Graph of total head $H$ along the flow path.
  • For real fluids EL slopes downward due to $h_l$.
  • Rises at a pump (energy addition), falls sharply across a turbine or major disturbance.
• Hydraulic Grade Line (HGL): Joins points representing $\frac{p}{\rho g}+z$ only.
  • Always lies one velocity-head ($\frac{v^2}{2g}$) below the EL.
  • If a piezometer were tapped, fluid would rise to the HGL.
• Visual interpretation for reservoirs A→D through pump C:
  • Entrance/disturbance → immediate EL drop.
  • Friction → gradual EL decline; steeper where velocity (and pipe loss coefficient) is larger.
  • Pump → vertical jump of $h_p$ in EL (HGL jumps by same amount since velocity unchanged across pump impeller).
  • Delivery reservoir → EL and HGL terminate at free-surface elevation.

Flow‐Measurement Devices

Four classical differential‐pressure instruments rely on Bernoulli and continuity:

1. Venturi meter
2. Orifice meter (plate)
3. Flow nozzle
4. Pitot tube

Venturi Meter

• Structural parts: entry section, converging cone, throat, diverging cone.
• Operates on induced pressure drop between entry (1) and throat (2).
• Ideal discharge:
  • Continuity: $A<em>1 v</em>1 = A<em>2 v</em>2$
  • Bernoulli → $v<em>2 = \sqrt{\frac{2gH}{1-(A</em>2/A<em>1)^2}}$ where $H = \frac{p</em>1-p<em>2}{\rho g} + (z</em>1 - z_2)$.
  • Volume flow (theoretical)
    $Q<em>t = A</em>2 v<em>2 = A</em>2 \sqrt{\frac{2gH}{1-(A<em>2/A</em>1)^2}} = \frac{A<em>1}{\sqrt{m^2-1}}\sqrt{2gH}\,,\quad m=\frac{A</em>1}{A_2}.$
• Coefficient of discharge $C<em>d \,(0.95!\text{–}!0.98)$ accounts for small losses: $Q</em>{act} = C<em>d Q</em>t.$
• Pressure difference $H$ usually obtained from U-tube manometer:
  $H = h\Big(\frac{\rho_{man}}{\rho}-1\Big).$ Independent of vertical inclination if measurement planes are horizontal.
• Example 5.4 (gas flow): Given $d<em>1=0.3\,\text{m},\ d</em>2=0.2\,\text{m},\ C<em>d=0.96,\ \gamma</em>{gas}=19.62\,\text{N/m}^3, \Delta h =0.06\,\text{m water}$ ⇒ $Q=0.158\,\text{m}^3!\,/s$ (worked algebra included in slide).

Orifice Meter

• Flat plate with sharp-edged hole; produces a vena contracta just downstream of plate.
• Much cheaper than Venturi, but:
  • Larger permanent pressure loss.
  • Lower accuracy, therefore $C_d\approx0.60!\text{–}!0.65.$
• Same basic equations as Venturi, but use appropriate $C_d$ and account for recovery factor if pressure taps are at standard locations.

Flow Nozzle

• Converging section ending at sharp exit; divergent recovery cone omitted.
• Cost and performance intermediate between Venturi and orifice.
  • $C_d\approx0.70!\text{–}!0.80.$ Increases slightly with Reynolds number.
  • Higher head loss than Venturi (due to flow separation in abrupt expansion) yet lower than orifice.

Pitot Tube

• L-shaped tube; open mouth faces upstream, stagnates the flow (point B), while static opening (point A) senses local static pressure.
• For an incompressible fluid:
  • Bernoulli between A (moving stream) and B (stagnation, $v<em>0=0$): $\frac{p</em>0}{\rho g} = \frac{p}{\rho g}+\frac{v^2}{2g}.$
  • Use manometer to read difference $h$ between stagnation and static ports:
    $v = C\sqrt{2gh}$ where $C$ is instrument coefficient (≈1.00 for well-designed probe, <1 if alignment errors, finite diameter effects, etc.).
• Widely used for field measurements, wind-tunnel testing, and as velocity probe in pipe flows.

Comparison of Differential Flowmeters

• Bullet summary:
  • Venturi: highest accuracy, highest cost, lowest permanent loss, $C_d\approx0.95–0.98$.
  • Orifice: lowest cost, low accuracy, highest loss, $C_d\approx0.60–0.65$.
  • Flow nozzle: middle ground in all three respects, $C_d\approx0.70–0.80$.
• Selection therefore depends on acceptable error, energy penalty, and installation/maintenance budget.

Practical & Conceptual Notes

• Bernoulli’s equation is an energy statement, not a momentum statement; real-fluid limitations stem from viscous dissipation.
• When velocity cannot increase (e.g., constant-diameter pipe carrying incompressible flow), gravitational head loss manifests as pressure rise, not kinetic energy gain.
• Negative (gauge) pressures predicted by Bernoulli (examples 6.2 & 6.5) highlight cavitation risk; ensure $p<em>{abs} > p</em>{vap}$ in design.
• Energy and hydraulic grade lines are indispensable tools for diagnosing pump placement, estimating head losses, and explaining piezometer readings.
• Flow-measuring devices fundamentally trade off cost versus accuracy versus head loss — a design balance engineers must evaluate for every installation.