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Measurement of
Pressure Drop across a Stenosis
Null Hypothesis
This
experiment models the pressure drop across a stenosis in an artery. You will test two null hypotheses:
Theory for Pressure Drops Across a Stenosis
Young (1979) summarized the results
of a number of studies on arterial stenoses with an equation for pressure loss
(_{}) that is recast here in terms of flow rate (_{}) instead of Reynolds number:
_{}
Eq. (1)
where _{} is dynamic viscosity, _{}is fluid density, D_{v}
is diameter of the vein, D_{s}
is stenosis diameter at its narrowest point, and _{} is the turbulent loss
coefficient (1.52). The expression in square brackets _{} is called the laminar
loss coefficient, and the expression within in angled brackets _{} is the effective
stenosis length. Physically, the first term represents losses caused by
separation and turbulence, where as the second term represents viscous (or
laminar) losses. As a consequence, the
dynamic viscosity appears only in the second term.
For the
laboratory you will be doing, the following values or ranges of parameters are
appropriate:
_{}
_{}
_{}
_{} for 50% diameter
reduction
_{} will range from 2 ml/s
to 20 ml/s
_{} will be 1 to 2 cm
_{}
Pressure Transducers
The
pressure transducers you will be using are known as “strain gauge” type
transducers. The pressure on the sensing
element changes the resistance of a resistor according to the equation _{}. When a wire is
stretched, its length, _{}, increases slightly and its crosssectional area, _{}, decreases. The small
change in resistance can be detected by the use of a Wheatstone bridge
setup. The bridge is shown in Figure 1.
Initially
the bridge is balanced so that all of the resistances are the same. As one resistor is strained, its resistance
changes and the bridge becomes out of balance.
If the positive supply voltage (at the top of the bridge) is equal in
magnitude to the negative supply voltage, the two output voltages at balance
are zero, which is convenient since it is then not necessary to subtract the DC
offset from the output. However, in our
case we will use a positive supply voltage at the top of the bridge and ground
the bottom. If we use a 9 V supply, we
expect that both V_{out} and V_{out} wil be +4.5 V. These two outputs can be input to a
differential amplifier so that the DC offset does not appear in the output.
Before You Come to the Laboratory
You should
have some understanding of the behavior of pressure as a function of stenosis
diameter. Use Excel to plot pressure
drop as a function of flow rate for a few typical cases. Use the typical parameter values given
above. Plot three curves on the same
plot, one for a 25% diameter reduction, one for a 50% diameter reduction, and
one for a 66% diameter reduction. An
example Excel file that sets up the
equation has been generated. The file
uses the following techniques to make the equation easier to create and easier
to verify. You should use these
techniques whenever you set up equations in Excel:
and by
defining _{}, _{}, _{}, and _{} separately I have
eliminated the need to use a large number of confusing parentheses in setting
up the equation. Furthermore, I can now
easily look at the pieces of this equation individually.
Set up the
formulas in Excel to do the following:
If this is
set up properly, you can obtain the expected pressure drop immediately whenever
you take a flow rate measurement. This
pressure can then be immediately compared to the pressure drop you obtained
from the pressure transducer to see if there is an obvious error in either your
procedure or your calculations.
Experimental Setup
You will
need the following materials for this experiment:
You will need the following:
The experimental setup is as shown
in Figure 2.
You will need to set up this flow configuration with the materials
provided. The pressure transducer has
two fluid inlets to it, but it is not a
differential transducer and only senses the pressure that is applied to its
sensing element. One of the ports of the
transducer is to be connected to your experiment, and the other is to be used
to “bleed” air from the transducer. Once
all air has been removed from the transducer and the tubing, shut off the extra
port. Connect the transducer to one of the pressure
taps at a time. If you wish, you can
physically connect the transducer to both of the pressure taps and use the
stopcocks to control which of the taps is open to the transducer. However, if you do this, make sure that one tap is open to the transducer and the other is
closed to the transducer when you make your measurement.
Preliminaries
You will
need two pressure taps, one upstream and one downstream of the stenosis.
Insert the
first stenosis in the tubing. The
easiest way to do this is to cut the tubing at the location where you wish to
place the stenosis. The stenosis is then
inserted halfway into one side and halfway into the other so that the stenosis
itself holds the two pieces of tubing together (Figure 3).
Connect the
pressure transducer to the upstream pressure tap. Make sure that there are no air bubbles in
the line.
Transducer Calibration
Calibrate
the pressure transducer using the weight of water as a standard. The pressure at the transducer is rgh, where r is the density of water, g is the acceleration of gravity, and h is the height of the fluid surface
above the pressure transducer. For
example, if the transducer height changes by 5 cm, the pressure changes by
_{}.
With the
experiment set up as in Figure 1, fill the upstream reservoir to about ½ its
height and stop up the end of the tubing so that there is no flow in the
system. Make sure that there is no air
in the tubing. Now you can change the
pressure to which the transducer is exposed by changing the height of the
transducer relative to the surface of the water in the upper reservoir. This height difference is the _{} in _{}. Take several
readings of voltage from the signal conditioner as a function of the height of
the reservoir surface above the transducer.
Take measurements for heights that are both above and below the level of
the pressure tap up to ±12”.
Read the voltage for each transducer position and obtain about 10 total
data points. Pressure transducer
calibration data should be tabulated in a table similar to Table 1. If you generate this table in Excel, the
linear fit to find the slope of the calibration will be simple. You should calculate the linear fit of the
transducer before you move on to the next stage of the experiment so that you
can find and check any incorrect values in the calibration data.
Before Experiment 
After Experiment 

Height Difference (in) 
Voltage Out (mV) 
Height Difference (in) 
Voltage Out (mV) 








































To obtain
the calibration factor that converts volts to
pressure, perform a linear least squares fit of pressure as a function of
voltage (V). The slope of the derived
function (_{}) allows voltage to be converted to pressure. We will not be concerned with the
yintercept, only the slope because all calculations will be done in terms of a
voltage difference (voltage upstream – voltage downstream). Thus, to obtain the pressure drop
you will use _{}.
You will
need to know the true diameter of the tubing used in
this experiment. This may not be the
same as what is written on the tubing.
Fill a length of the tubing with water, and use the volume of water
required to fill the tubing to calculate back to tubing diameter (_{}, hence _{}).
Flow Rate
You do not
have a valve to adjust the flow rate.
However, you can adjust the flow rate by changing the vertical distance
from the fluid level in the upper reservoir to the outlet of the tubing. You can also adjust the flow rate by
inserting the second stenosis into the end of the tubing. Calculate the flow rate that is required to
obtain a crosssectional average velocity (_{}) of 50 cm/sec (_{}). Set up the system
so that the surface of the upper reservoir is 2 to 4 feet above the outlet of the
tubing into the lower reservoir, and use a graduated cylinder to measure the
flow rate. If the flow rate is
significantly different from the target of 50 cm/s (say, by a factor of 2),
then readjust the height difference and remeasure. This time, measure the flow rate as precisely
as possible. You can reasonably expect
to obtain a flow rate that varies by no more than 5% from measurement to
measurement.
The flow
rate will change somewhat as a result of changes in the height of the fluid in
the reservoir. Be sure to maintain this
height constant as much as possible.
Also, determine how much a given change in height changes the flow
rate. To do this, take repeated flow
rate measurements as the fluid reservoir height decreases.
Pressure
Record the voltage
on the digital multimeter both when flow is turned on and when flow is turned
off. You can insert a plug (made from
the same material used to make the stenosis) to turn off the flow. When you remove the plug, the flow rate
should return to its original value, assuming that you have made no other
changes in the configuration (such as changing the vertical position of the end
of the tubing).
Now connect
the pressure transducer to the downstream pressure tap. Again, measure pressure with the flow on and
the flow off. The pressure with the flow
off should be the same as the noflow measurement upstream, but may not be as a
result of (1) transducer drift or (2) a significant change in the height of
fluid in the reservoir. You should
doublecheck the height of the fluid in the reservoir before doing the
downstream measurement and add water if the height has changed significantly. The drift should not be a large problem as
long as the flowon and flowoff measurements are taken close enough to one
another in time. Repeat these measurements five times.
Reduce the flow rate by a factor of
2, and repeat the set of 5 measurements again. In the end
you should have enough data to fill Table 2.

For Flow Rate 
For Pressure Drops 

Conditions 
Volume (ml) 
Time (sec) 
Voltage Upstream No Flow 
Voltage Upstream With Flow 
Voltage Downstream No Flow 
Voltage Downstream With Flow 

50% Short Stenosis 
Flow Rate 1 


































Flow Rate 2 



































50% Long Stenosis 
Flow Rate 1 


































Flow Rate 2 


































Recalibration
To ensure
that your calibration has not changed, perform a postexperiment calibration,
again using the weight of water as your standard.
Data analysis
1. Plot the measurements of output
voltage as a function of pressure for the transducer calibration. Perform a least square fit of pressure as a
function of voltage. The slope is the
transducer sensitivity, in volts/(dynes/cm^{2}). Show the raw data for the calibration on a
plot as individual points without connecting lines. Show the least square fit model as a solid
line. If the preexperiment and
postexperiment calibrations are significantly different, perform least squares
fits for these data sets separately.
2. For each data set value of (Volume
Collected, Time of Collection), translate to flow rate (Q).
3. For each data value of (Change in
voltage upstream, Change in voltage downstream), translate to pressure drop.
4. Plot Pressure as a function of Q
(use symbols for these data).
5. Plot Young’s equation for the same
flow rate range (use a solid line for these data).
6. Discuss reasons for any differences
(theory vs experiment).
7. Perform Student’s Ttests to test
the two hypotheses. The first Ttest
will compare data taken with different flow rates but the same stenosis. The second Ttest will compare the two
stenoses of different lengths. Provide a
pvalue in both cases. Are the results
significant.
8. As part of your discussion, determine
whether the overall pressure drop in the system (_{}) is accounted for by the pressure drop across the stenosis
and the Poiseuille flow pressure drop in the tube. Do you expect the measured flow rate to be
higher or lower than what would be predicted by the combination of Young’s
equation and Poiseuille’s equation?
References
Young DF:
Fluid mechanics of arterial stenosis. J Biomech Eng 101: 157175, 1979
Biomedical Engineering
Senior Laboratory (BIEN 435)
Steven A. Jones