Fluid Mechanics 2 (SCEE08003) laboratory briefing note
F2: Bernoulli (I): pressure variation in a convergent-divergent duct
1. Safety
A copy of the completed Risk Assessment “RA1” form for this experiment is kept with each
set of the apparatus.
The experiment carries very little safety risk, but we require that you wear safety glasses,
safety (steel toe-capped) boots/shoes, and a lab coat while in the laboratory. The apparatus
is set up in the main, open area of the lab in which there can be a variety of other activities
happening at the same time as your work on this experiment. Because there are no physical
partitions within this large space, we must adopt the level of PPE which aligns with the
highest level required anywhere in the lab. You will not be allowed to enter the work area
unless you are wearing your safety glasses, your safety boots, and your lab coat.
The apparatus is located away from the immediate proximity of mains electricity outlets.
Nevertheless, you should check that there are no mains extension cables nearby, which
could be reached by any splash from the apparatus. If you are in any doubt, you should
consult a member of staff.
2. Background
In Chapter 3 of the course, we derive the relationship between flow velocity and pressure,
arriving at the steady flow Bernoulli equation – henceforth just the ‘Bernoulli equation’ (or
lazily, just “Bernoulli”).
In this experiment you will explore the variation in pressure as the flow passes through a
duct whose cross-section is changing; first contracting and then expanding again – a
“convergent – divergent duct”. If we know the flow rate and how the cross-section varies
through the convergent-divergent section, we can work out how the flow speed must
change, and thus, by Bernoulli, predict how the pressure should vary through the same
duct. We can then ask the question “how good are these Bernoulli-based predictions of
p(x)?” and then use the apparatus to check. You can approach this experiment once you
know about the Continuity Equation (u1A1 = u2A2, Section 3.2) and Bernoulli (Section 3.4).
3. Objectives
Objective 1. From experiment, determine the variation in pressure distribution through the
convergent – divergent venturi section, and compare these with expectations from theory.
Seek to quantify and explain any differences.
Objective 2. Explore the influence of flow rate on the findings from objective 1.
Objective 3. Draw clear, concise conclusions relating to Objectives 1 and 2. These should be
quantitative where possible, and supported by the evidence of your measurements.
4. Preparation
Introductory note: The tasks listed here and in the analysis section later are presented in detail
where they are new for F2, but we will not ‘walk you through’ the Excel spreadsheeting aspects
again – your ever-improving skills in Excel are now assumed. The concepts, methods and
approach required to undertake the uncertainty analysis are also not set out in great detail
here, these being the subject of the voluntary ‘homework’ set, exploring the uncertainties in
your F1 data, and how you can deal with these uncertainties in a practical and useful way.
Objectives 1 & 2: pressure variation through convergent – divergent duct
In this experiment, we explore how the pressure in the water flow varies as it passes
through the venturi; first through the contracting section, and then through the expansion.
We wish to predict this variation with knowledge of the variation in the cross section shown
in Fig. F2.1.
Figure F2.1: Cross-section of the venturi section.
Preparation task 1: It is always a good idea to think about your spreadsheet layout now, at
the very beginning. It is often useful to imagine the graph(s) that you want to plot, and
arrange the spreadsheet to make this easy. Here, you will want to plot p vs x, so you will
want the x’s in a column, with a row to do the calculations on each x position.
Enter the x and A data into a new spreadsheet (called e.g. F2 working
v01”). You might want to enter the locations and areas in mm and mm2 respectively,
reading directly from Figure F2.1, AND then add two further columns to give x and A in m
and m2 respectively. This way, you can work with the m and m2 numbers, and you will have
much less chance of any subsequent units error.
We would like to explore the influence of the flow rate on the pressure distribution in the
venturi, so we will use a lower and a higher flow rate. You may choose these in the
experiment, but 8 and 12 litres per minute (l/min) respectively should work well and are
suggested. If you set up your spreadsheet well, it will be very easy to replace these two
nominal flow rates with the actual ones that you measure in the experiment.
Preparation task 2: Assuming two experiments: one at 8 l/min and one at 12 l/min, and
setting the upstream pressure to 0 Pa *, expand your spreadsheet to get predictions for
pressures at each of the six locations, for both flow rates. This is a harder task. Hint: you only
need the continuity equation and the basic ‘Bernoulli’ equation here – not yet the equation
derived in Section 4.4 for flow meters. For each x, you work out the speed u (from Q and A)
and then p using this u and Bernoulli. Take as many columns as you like to work through the
calculation clearly to avoid errors.
x1 =$0$mm
A1 =$338.6$mm2
x3 =$58$mm
A3 =$84.6$mm2
x4 =$95.5$mm
A4 =$170.2$mm2
x5 =$129.5$mm
A5 =$255.2$mm2
x6 =$173$mm
A6 =$338.6$mm2
x2 =$29$mm
A2 =$233.5$mm2
21o 8o
* Because we are interested in the pressure variation through the section, it does not
matter what we choose as the upstream value. We choose 0, and then predict and measure
the differences from this, positive or negative.
Now plot the graphs; you will want to plot p(x) vs. x for both flow rates. In order for the
graph to enable very easy, visual comparison between the pressure distributions p(x) for
lower and higher Q, you should plot both these graphs on the same set of axes (i.e. on the
same plot).
Preparation task 3: Plot a graph of pressure (relative to its upstream value) vs. position
along convergent-divergent duct, for both flow rates. As per comment above, if you do this
well, then the graph will change automatically when you replace the notional flow rates of 8
and 12 l/min with the actual ones.
Recalling that 33% of the assessment is based upon the quality of your graph, now is the
ideal time to do the detailing on the graph to get it looking good according to conventions of
best practice.
Preparation task 4: Refer to the “Plotting effective graphs” document, and adjust your plot
accordingly. You should pay attention to axis titles; axis labels; font sizes; legend; line / no
line; line style; marker / no markers; …
5. Experiment
Please be certain to bring your PPE (lab coat, safety glasses, safety boots/shoes) with you.
These will NOT be loaned to you in the lab. Please don your PPE prior to arrival.
DO NOT COMMENCE any work with the apparatus until your safety and technical briefing
has been completed by the demonstrator.
During the experiment, if you have any concerns about your safe or correct operation of the
equipment, or if you feel unsafe, STOP THE WORK immediately and seek advice.
In pursuit of objectives 1 & 2: first, you need to make the six manometers associated with
the convergent-divergent ‘venturi’ section ready for use. To do this, follow the steps below.
Don’t worry of you get these steps wrong the first time; practice makes perfect. Or if it
doesn’t, then seek the assistance of the demonstrator.
1. Close the air valve on the top of the manometer array (the white, left-hand valve).
2. Open the through-flow valve on the top of the manometer array (the grey, right-hand
valve)
3. Open the gate valve at the very end of the pipe system.
4. Switch on the pump.
5. Gently close the gate valve, fully.
6. Watch the water flush through the manometers until all air bubbles have left the
system.
7. In quick succession: (why? … )
a. close the through-flow valve on top of the standpipe array (the grey one), and
b. switch off the pump.
8. Check that the manometers are now all totally full of water. If not, go back to (1)!
9. Now, you need to get the manometer water levels into a low-to-middle position to
measure pressure differences. To do this, very gently open the white air valve on the top
of the manometer array. You will see the levels fall. Allow the levels to fall t a bit below
the middle before gently re-closing the air valve.
Finally, you are good to go, unless the six manometers are not even, in which case… back to
step (1)!
Next – for the lower and higher flowrate measurements, you will need to switch on the
pump and adjust the gate (exit) valve to achieve these. It doesn’t matter too much the exact
values, as long as they are quite different. Digital flow meter readings of about 8 l/min and
12 l/min should work well.
While this digital meter reading is fairly accurate, it is not very precise (no decimal places
are displayed. Take a moment to reassure yourself that you understand that ‘accurate’ and
‘precise’ are different things. Now measure the actual flowrate by allowing the discharge to
be collected in the measuring jug and timing this with the stopwatch. Think about the best
strategy.
For each flow rate, set the gate valve and record the six manometer readings.
Now switch off the pump and allow the system, including the manometers, to drain.
The experimental measurements are now complete.
Switch off the pump and open all valves allowing the system to drain and return to its
starting state.
6. Analysis (after lab visit)
If you have done your preparation tasks 1 – 4 well, then the analysis should be relatively
straightforward.
For objectives 1 & 2:
Analysis task 1: For the tests on the pressure distributions, add your measured data for
pressure distributions to the chart with the predictions lines. Remember to replace the
nominal flow rates (from preparation task 3) with the actual Qs that you measured.
Analysis task 2: Work through your “uncertainty treatment” for all measured and derived
quantities. This is probably most easily and elegantly done by adding additional columns in
the spreadsheet for the measurement uncertainties, and the for the consequent (fed-
though) uncertainties in the derived quantities. There is no need to consider the uncertainty
in the x-position because it has no bearing on the conclusions.
Analysis task 3: Add the uncertainty estimates in the pressures (predicted and measured)
onto your graphs as “error bars”.
Analysis task 4: Meriting a separate task number… now take a little time to ensure that your
graphs are looking fully professional according to the conventions set out in the graphs
document.
Finally, 33% of the mark rests with the “quality of conclusions” drawn from the experiment.
You will need to read, carefully, the guidance notes on Writing Conclusions.
Analysis task 8: Following that guidance, and looking carefully back at the experiment’s
objectives for further inspiration, draft 2 to 4 conclusions as text in a new worksheet (called
“conclusions”) in the same Excel file as your analysis. Check these conclusions against the
guidance and redraft as necessary.
You should now check the work, and then submit the Excel file (only) to the “Experiment F2”
drop box on Learn. The only acceptable file formats are .xlsx and .xls. The deadline for
submission is 1600 on Thursday 21st October.
Because this is probably the tightest of turnarounds for any of the FM2 coursework AND
there is quite a lot of work associated to do this well (the exercise is weighted at 10% of
FM2), you will need to prioritise this activity over FM2 lecture work during the latter part of
week 4 and into week 5. The GANTT chart reflects this schedule and prioritisation.
This lab is weighted at 10% of the Fluid Mechanics 2 mark.