📈 Validity of Bernoulli's Theorem
A virtual F1-15 apparatus: steady flow through a tapered circular duct with six manometers reading the pressure head, and a traversing hypodermic total-head probe. Watch pressure trade for velocity through the contraction, compare the calculated total head (h + v²/2g) with the measured one, and see where — and why — Bernoulli's ideal picture leaks energy.
Objective
To investigate the validity of the Bernoulli equation applied to steady flow of water through a tapered duct.
Theory
h + v²/2g + z = constant (ideal) • v = Q/A • ht,calc = h + v²/2g • probe: ht,meas = stagnation head
The duct is horizontal so z drops out. As the section contracts, continuity accelerates the flow and the pressure head falls; in the diverging part the process reverses — but not perfectly, because friction and (in the steeper diffuser) separation dissipate energy. The measured total head therefore falls slowly along the duct, most of all through the diffuser.
Equipment
F1-10 hydraulics bench; F1-15 Bernoulli apparatus (venturi test section with tappings h₁–h₆, manometer bank, traversing total-head probe); stopwatch.
Tappings: 25.0 / 13.9 / 11.8 / 10.7 / 10.0 / 25.0 mm Position 1: 14° converging · Position 2: 21° converging Collect ≥ 60 s per flow
Procedure — perform it here
Head distribution along the duct — live
Pressure head (manometers), velocity head from continuity, their sum (calculated total) and the probe's measured total. The gap between the two totals is the energy loss.
Observations & Computations
| Set | Pos. | Q (L/s) | Tap | d (mm) | h (mm) | v = Q/A (m/s) | v²/2g (mm) | ht calc (mm) | ht probe (mm) | loss (mm) |
|---|---|---|---|---|---|---|---|---|---|---|
| No sets yet — set a flow, collect, then Record. | ||||||||||
Report Questions
Is Bernoulli's equation valid through the converging part? The diverging part?
Through the smooth contraction the calculated and measured totals agree within a few millimetres — acceleration is nearly loss-free, so Bernoulli holds well. Through the diffuser the measured total drops noticeably below the ideal: the decelerating flow thickens its boundary layer and (especially at the steeper 21° divergence of Position 1) tends to separate, dissipating energy as turbulence.
Where are maximum velocity and minimum pressure?
Both at the throat (tapping h₅, d = 10 mm): continuity puts the highest velocity at the smallest area, and Bernoulli then demands the lowest pressure exactly there.
How does the energy loss show in the results?
As the growing gap between the probe's measured total head and the upstream value — small through the contraction, largest across the diffuser — and in h₆ failing to recover to h₁ even though the areas are equal.
Precautions: level the apparatus; purge all air from manometers and probe tubing; retract the probe while reading the manometers (it blocks part of the throat); collect for at least 60 s; take readings only when levels are steady.