M093 · Fluid mechanics

Continuity & Bernoulli Flow Bench

Where does the pressure go when liquid flows through a narrow part of a pipe? Into speed. The same volume has to pass every second, so the liquid speeds up where the pipe narrows: half the area, twice the speed. Bernoulli's equation says the pressure drops to pay for that speed. Pressure recovers downstream as the pipe widens, but what friction takes along the way is never paid back.

Shape a transparent Venturi tube, move a pressure probe and compare the energy grade line with the hydraulic grade line. Record raw pressure and speed readings, then use them to work out the flow and a Darcy friction factor.

Inference status $\text{Ready}$

Accepted transport flow $\text{Ready}$

Candidate probe speed $\text{Ready}$

Candidate probe gauge pressure $\text{Ready}$

Candidate probe absolute pressure $\text{Ready}$

Minimum candidate absolute pressure $\text{Ready}$

Minimum margin above vapor pressure $\text{Ready}$

Probe centreline elevation $\text{Ready}$

Tube centreline length $\text{Ready}$

Candidate probe energy head $\text{Ready}$

Candidate probe hydraulic head $\text{Ready}$

Candidate total loss head $\text{Ready}$

Candidate energy balance residual $\text{Ready}$

Acquired stations $\text{Ready}$

Flow inferred from raw readings $\text{Ready}$

Fit coordinate residual $\text{Ready}$

Single-phase status $\text{Ready}$

Inlet gauge pressure $\text{Ready}$

Throat gauge pressure $\text{Ready}$

Outlet gauge pressure $\text{Ready}$

Darcy factor inferred from data $\text{Ready}$

Shape the pipe, then measure the pressure

Can pressure recover after the throat?

Prescribed flow

Q

4.50\,L/s

Throat diameter ratio

D_t/D

0.50\,

Outlet elevation change

\Delta z

0.00\,m

Inlet gauge pressure

p_1

14.00\,kPa

Tube inner diameter

D

70.00\,mm

Reference Darcy friction factor

f_D

0.00\,

Known diffuser loss coefficient

K

0.00\,

Pressure noise bound

\epsilon_p

0.00\,kPa

Probe path fraction

\xi_p

0.50\,

Liquid density

\rho

1000.00\,kg/m^3

Gravity

g

9.81\,m/s^2

Ambient absolute pressure

p_a

101.33\,kPa

Specified vapor pressure

p_v

2.34\,kPa

The flow is steady and incompressible and is described by its mean speed, with a kinetic-energy correction factor of one. The upstream source sets the flow rate and gauge pressure, and the outlet pressure follows from the model. The Darcy friction factor is set directly; it does not come from a Reynolds-number correlation. The known diffuser loss is based on the throat speed. Flow separation and pressure changes across the pipe are left out.

The electronic probes report the pressure as if it were measured on the centreline; fixtures and sensors are schematic. Pressure and speed come from the same pipe geometry. Tube radii are enlarged fourfold; reference standpipe levels retain the height scale. If the absolute pressure falls below the set vapor pressure, single-phase flow and acquisition are both rejected; the dashed curves stay only as candidate references.

An acquisition records raw pressure and speed readings. Weighting the speeds by area gives the flow rate. A straight-line fit of the energy along the pipe, with the known diffuser loss, then gives the Darcy friction factor. Moving the probe keeps the data; changing the apparatus or noise clears it. The CSV leaves out the reference flow and friction. A new reading at the same station replaces the old one, and the noise is bounded and reproducible.

The flow is steady, incompressible and one-dimensional, with a prescribed Darcy factor and a known diffuser loss. If the pressure anywhere drops below the vapor threshold, that single-phase solution is not used for flow or for readings.

Start with the essentials

Focus question: Can the pressure recover while energy is still lost?
One-sentence intuition: In the throat the liquid trades pressure head for speed, and dissipative losses lower the total head.

Core mathematical model

Continuity: the same flow through every section

Q=A(s)v(s),\qquad A(s)=\pi r(s)^2

Every cross-section carries the same prescribed volume flow, so a smaller area means a higher speed.

Energy and hydraulic grade lines

\begin{aligned}H(s)&=z(s)+\frac{p(s)}{\rho g}+\frac{v(s)^2}{2g}\\H_{\mathrm{hyd}}(s)&=H(s)-\frac{v(s)^2}{2g}\end{aligned}

Pressure here is gauge pressure, and the source sets the head at the inlet.

Head loss from local speed

h_L(s)=f_D\!\int_0^s\!\frac{v(u)^2}{2gD(u)}\,du+K\chi(s)\frac{v_t^2}{2g}

The known diffuser loss builds up after the throat. The Darcy loss uses the local diameter and speed at each point.

Estimating flow and friction from the readings

\begin{aligned}\widehat Q&=\frac1n\sum_i A_i v_i\\\widetilde H_i&=b-f_D X_i\end{aligned}

First correct the measured total head for the known diffuser loss. Then fit it against the wall-loss coordinate, which is computed from the estimated flow.

Check that the pressure stays above vapor pressure

\min_s\!\left[p_a+p(s)\right]>p_v

If the pressure crosses the threshold, the solution is rejected. The model does not compute cavitating flow.

Common difficulties

Gauge and absolute pressure

Typical misconceptionA negative gauge pressure always means the liquid cannot flow.

Better mental modelAdd the ambient pressure first, then compare the result with the specified vapor pressure. The check covers the whole path.

The fitted friction factor is an estimate

Typical misconceptionThe fitted coefficient is just the reference slider under another name.

Better mental modelClear the record and the estimate disappears. Noise can even give an unphysical negative coefficient: the fit is unconstrained, so the result is not clipped, and it is not a safety certificate.

Run the experiment

01
Find the throat
Move the gold probe handle through the narrow section.
What to observe: The speed rises and the pressure drops. Without losses, the energy head stays flat.
02
Separate recovery from loss
Choose With losses and compare the inlet, throat and outlet.
What to observe: Pressure recovers downstream, but the energy grade line stays below its inlet value.
03
Record readings and estimate
Set a Pressure noise bound, press Acquire pressure profile, then press Export raw readings.
What to observe: The pressure and speed readings give estimates of flow and friction; the reference flow is never used in the fit.
04
Push past the limit of the model
Choose Pressure boundary or lower the ambient pressure.
What to observe: Even a smooth solution can drop below the vapor threshold, and then acquisition is disabled.