CIVL2611 Chap.6 Bernoulli Energy Analysis
Bernoulli Energy Analysis
Bernoulli analysis relates pressure, velocity and elevation heads between carefully selected points under a restricted ideal model. This chapter focuses on point selection, transparent cancellation, continuity coupling and physical admissibility. The worked discharge problem shows why stating atmospheric boundaries and negligible reservoir velocity is part of the solution rather than optional commentary.
In head form, every term is a length representing mechanical energy per unit weight, so pressure cannot be added directly in pascals without conversion. A useful point table records pressure reference, velocity and elevation before the full equation is written.
Equal atmospheric pressures cancel only when both points share that boundary, and a large free surface has negligible velocity only when continuity makes its area ratio convincing. Contractions require continuity and Bernoulli on separate lines: the first links speeds, while the second links the associated pressure and elevation states.
The simple form assumes steady, incompressible, inviscid flow along a streamline without unrepresented machinery. Pump head, turbine head and dissipative loss must be added when those effects matter. Original reservoir and contraction examples include numerical cross-checks between pressure head and velocity-head change.
The chapter closes by testing direction, absolute-pressure plausibility and whether the ideal calculation is an upper estimate for a real lossy discharge.
What this chapter covers
- 01
Pressure, velocity and elevation head
- 02
Point selection along a streamline
- 03
Gauge-pressure cancellation
- 04
Continuity coupled with Bernoulli
- 05
Steady incompressible inviscid restrictions
- 06
Pump, turbine and loss terms outside the simple form
Worked example · free
Reservoir discharge through a nozzle
- +1Choose the free surface and free jet as points with equal atmospheric pressure.
- +1Take free-surface velocity as negligible and set the nozzle elevation datum.
- +1Reduce Bernoulli to 8.0 metres of elevation head becoming velocity head.
- +1Find jet speed 12.53 metres per second.
- +1Multiply area by speed to obtain 0.0501 cubic metres per second.
- +1Label the answer as an ideal estimate without losses.
Key terms
- Pressure head
- Pressure divided by specific weight, expressed as an equivalent fluid-column length.
- Velocity head
- Speed squared divided by twice gravity, expressing kinetic energy per unit weight.
- Total head
- The sum of pressure, velocity and elevation heads within the selected energy model.
Bernoulli Energy Analysis FAQ
When can atmospheric pressures cancel?
They cancel when both selected points are exposed to the same atmosphere and every pressure term uses the same reference. A sealed or pressurised surface cannot be cancelled against a free jet without including the pressure difference.
Why is reservoir velocity often neglected?
Continuity makes the free-surface velocity small when reservoir area is much larger than outlet area. It is an approximation based on the area ratio, not a universal property of reservoirs.
When is simple Bernoulli insufficient?
Add or change the model when pumps supply energy, turbines remove it, viscous losses matter, density changes significantly or the chosen points do not meet streamline conditions. A correct formula substitution cannot repair invalid assumptions.
Exam move
For every problem, circle two points and make a table of pressure, velocity and elevation information. State the pressure reference and datum, then write the full head equation before cancelling. Beside each removed term, give a reason: equal atmosphere, negligible reservoir velocity, equal elevation or another explicit boundary condition. This habit turns cancellation into visible modelling evidence.
Rebuild pressure, velocity and elevation heads from energy per unit weight and confirm that all three have length units. Convert one pressure difference into metres of water head so the connection becomes numerical rather than verbal. Use continuity separately when areas change.
Solve for the section velocity first, then place its square into Bernoulli; do not mix area ratios directly into the head equation without showing the intervening relation. Prepare a restriction checklist containing steady flow, incompressibility, negligible viscosity in the simple form, streamline selection and absence of unrepresented machinery.
For a pump or long-pipe question, write the extended balance with pump head, turbine head and nonnegative loss head. Check their signs using the device purpose rather than memory. Compare an ideal reservoir discharge with a version containing a stated loss to see why real speed is lower. For a horizontal contraction, predict higher speed and lower ideal static pressure before calculating.
If the algebra reverses that direction, inspect the velocity-square difference. Also test physical admissibility: an impossible negative absolute pressure may signal cavitation or breakdown of the assumed state. During closed-book practice, use four lines: point table, full equation, cancellation reasons and computed result. End with the pressure reference, units and idealisation.
If the large-reservoir approximation is used, quantify the area ratio and estimate the discarded surface velocity instead of calling it zero automatically. That comparison makes the assumption transferable to unfamiliar geometry and prevents a convenient simplification from becoming a rule applied everywhere. Re-solve one example with a nonzero surface speed to see precisely where the discarded head returns.
Restate every cancellation reason.