Hi,

I need report for this lab (Static Calibration of Instruments), you will find file for experiment and results lab on the Excel.

Based on the results taken in the lab (Exp_instrument_calibration.pdf), write and submit the Results section of your report for Session 1 experiment. Your submissions should also include the equations used to obtain any calculated results (list at the end of your Word document and Excel).

Your submission should have at least two documents: a Word document with the Results section (all measured and analyzed results clearly organized and presented), and one or more Excel spreadsheets with data calculations that are easy to follow (well annotated, including variable names, units, etc.).

For the report:

1- Analysis Correctness: For full marks, all calculations and analysis must be correct.

2- Results Presentation: For full marks, must include: Properly formatted tables and figures; appropriate text describing results

3- Excel Presentation: For full marks, Excel work must be easy to follow including file name, content annotation, units, results emphasized, etc.

Thank you

Widener University
School of Engineering
ME304 Mechanical Measurements II
Experiment: Static Calibration of Instruments
Objective
To perform two calibrations of a measuring instrument, including determination of calibration
curves and quantifying the observed errors.
Introduction
Measuring instruments can be viewed as systems converting a measured unknown input x to an
easily quantifiable output y (which is commonly an electric signal). For any instrument to be
useful, it must first be calibrated, so that the calibration curve y(x) is known, and the
measurement errors can be estimated. For linear instruments, the calibration curve can be
described by the instrument’s sensitivity K and zero offset. For non-linear sensors, a functional
relation of y(x) is first assumed (usually guided by the understanding of the underlying physics)
and then a best fit regression to the measured data is performed. The instrument errors are then
identified and quantified based on how far the measured results deviate from the calibration
curve. For example, linearity error is evaluated as:
???? = max(|????(????) − ????(????)|)
??????
× 100%
where yL(x) is the linear calibration curve determined by best fit to the calibration measurements
yi. Similar definition can be applied to instruments with non-linear calibration curves (with yL(x) no
longer linear). See lecture notes for definitions of other errors and for other details.
In this experiment, an unconventional measuring system is calibrated, which measures the rate of
linear momentum and the volumetric flow rate of water jet exiting a nozzle by determining the
impact force of the jet on selected vanes.
Theoretical Background
In principle, the flow rate of a liquid exiting a nozzle can be measured by directing the liquid jet at
an instrumented vane and measuring the resulting impact force. When the vane obstructs the
liquid jet, it reduces its linear momentum (a vector quantity). Per Newton’s second law, the rate
of change of this momentum is equal to the force exerted on the fluid by the vane.
Consider a vane, which is axi-symmetrical about the
vertical x-axis, as shown in Figure1. A jet of fluid
flowing at the mass flow rate of along the x-axis
with the velocity u0 strikes the vane, is deflected and
leaves the vane with the velocity u1 directed at an
angle β to the x-axis. Changes in elevation from the
striking the vane to leaving it are neglected and so
are the changes in jet velocity (u1 = u0 ).
The rate of x-momentum entering the system is:
??0 = ??̇??0 (kg m/s2 )
while the rate of x-momentum leaving the system is:
??1 = ??̇??0 cos ??
m
Figure 1
u0
u1
F
β
x
u0
u1
F
β
x
The force on the vane in the x -direction is equal to the rate of change of the x -momentum:
?? = ??0 − ??1 = ??0(1 − cos ??) = ??̇??0(1 − cos ??) (Eq. 1)
The mass flow rate the of water exiting the nozzle can also be expressed as
??̇ = ????̇ = ?????? (Eq. 2)
where ρ is the water density, ??̇ is the volumetric flow rate, u is the average velocity at the nozzle
exit and A is the nozzle opening area.
Finally, if the vane is located at height h above the nozzle exit, velocities u and u0 can be related
as: (Eq. 3)
Equations (1) through (3) can be combined and manipulated to yield the theoretical relation for
??0(??) and ??̇
(??). This way, force measurement can be used to measure rate of momentum and
the volumetric flow rate leaving the nozzle.
Experiment Equipment
The measuring system being calibrated consists of a vertical nozzle (diameter of 0.01 m), which
discharges a jet of water at a vane positioned 0.035 m above the nozzle exit. The vane is
supported by a balance system, which allows for the vertical force acting on the vane to be
measured indirectly by measuring the position of the balance slider on a balance beam:
?? = ???? ??
??
.
Here ?? is the mass of the balance slider (?? = 0.6 kg), y is its position along the beam and l is the
distance from the vane axis to the beam pivot (l =0.15 m).
Four available vanes and their deflection angles are listed in Table 1.
Table 1: Available vanes and their deflection angles.
Shape β
Angled Plate 60o
Flat Plate 90o
Conical Plate 120o
Cup 180o
The calibration system has a water pump which moves water from a water tank through a
regulating valve and a flow meter, and supplies it to the nozzle. The flow meter readings are
considered true values of the flow rates. The rates of linear momentum calculated from the flow
meter readings are considered true values of the rates of linear momentum leaving the nozzle.
Experimental Goal
To perform a static calibration of the proposed measuring systems, including determination of their
calibration curves and quantifying the observed errors (deviation from the calibration curve).
For this measurement system calibration, the input variable ‘x’ being set as a standard is
either the rate of linear momentum ???? (in kg m/s2) or the volumetric flow rate ??̇out of the
nozzle (in gpm) and the output variable ‘y’ is the balance slider position y (in mm). The two
calibration curves are ????(??) and ??̇(??).
u u 2gh 2
0 = −
Testing
Preliminary: Weigh the balance slider used in the experiment and measure the water
temperature in the tank.
Main: Perform all procedures below for either the flat or the cup vane (you will need to prepare
your own tables)
(a) Determination of the range of calibration and the corresponding maximum output (FSO)
(1) Place the balance slider in zero position and balance the beam by adjusting the spring
knob.
(2) Adjust the flow rate carefully until the water jet just touches the vane. This setting will
determine the lower limits of the calibration range.
(3) Open the valve fully and re-balance the beam by adjusting the slider. This setting will
determine the upper limits and the FSO.
(4) Select and record five or six calibration values of ??̇ spanning entire range for the
calibration tests (the lower limit, the upper limit and three or four points in between).
(b) Estimation of precision error (repeatability) for a single operating point – each student must
perform the procedure below three times – total of 12 or 15 repeated measurements
(1) Decide on the flow rate setting to be tested (in the upper half of the range)
(2) Place the balance slider in zero position and balance the beam by adjusting the spring
knob.
(3) Turn on the water jet and adjust the flow to one selected value.
(4) Bring the beam back to balance. Record the slider position.
(5) Turn off the water jet.
(6) Turn the spring knob to throw the beam zero position off balance.
(c) Determination of the calibration curve and the ‘linearity’ error (for lack of a better term,
meaning the maximum deviation from the determined calibration curve).
(1) Decide on the sequence of flow rate values to be tested by randomizing the flow rate
values selected in procedure (a)
(2) Set the first value of flow rate
(3) Determine the slider position y
(4) Repeat for other values of the flow rate (in random order)
This completes calibration procedures for first vane. Repeat procedures (a) and (c) only for the
other vane (flat plate or cup).
Presentation of Results:
Evaluate all measured results critically and present most important findings as tables and/or
graphs. Each table and each graph should have its main ‘message’ reflected in its title. Your
main goal is to present in a very clear way the obtained calibration results for each of the two
vanes.
At the minimum, the following must be reported in each calibration (not necessary in this order):
• Instrument range
• Full-scale output (FSO)
• Calibration curve equation
• Calibration curve plot with experimental points
• Calibration curve plot compared with the theoretical curve
• ‘Linearity’ error estimate (as % of FSO)
• Precision error estimate (as % of FSO)
For linearity error establish least square fit calibration curve using an appropriate functional form
(as suggested by theoretical analysis). There will be two different functional relations for ??0(??)
and ??̇
(??).
For precision error, calculate the precision index (standard deviation of sample population), and
95% confidence interval on slider position y, using the appropriate t value.
References
(1) J. P. Holman, Experimental Methods for Engineers, 7th Edition, McGraw Hill, New York,
2001
(2) R. S. Figliola and D.E. Beasley, Theory and Design for Mechanical Measurements, 5th
Edition, Wiley, 2011
(3) Instructions of “Impact of a jet” by TQ Intelligent Solutions for Education and Training