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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:

 

  1. The pressure drop across the stenosis is independent of the stenosis length. (Alternative hypothesis: Pressure drop increases with stenosis length).
  2. The pressure drop across the stenosis is independent of the flow rate.  (Alternative hypothesis: Pressure drop decreases with stenosis diameter).

 

 

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, Dv is diameter of the vein, Ds 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

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 3 curves on the same plot, one for a 25% diameter reduction, one for a 50% diameter reduction, and one for a 66% diameter reduction.

 

Set up the formulas in Excel to do the following:

 

  1. Translate flow rate from the volume of fluid collected and the time over which it was collected.
  2. Calculate theoretical pressure drop for a given measured flow rate value.
  3. Calculate the pressure drop, given the transducer voltage measurement and the transducer calibration constant.

 

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:

 

  1. Graduated cylinder to measure volume.
  2. Stop watch to measure time of volume collection.
  3. Tape measure to measure height of the transducer.
  4. Two large white buckets (fluid reservoirs)
  5. A length of ¼ inch inner diameter tubing
  6. Smaller pieces of tubing to make connections to the pressure transducer
  7. A pressure transducer that is connected to a signal conditioner
  8. A digital multimeter
  9. Stopcocks to control connections to the pressure transducer.
  10. Plastic glue.
  11. Three stenoses: 50% 1 cm long, 50% 2 cm long, 66% 1 cm long.

 

 

 

 

 

 

 

The experimental setup is as shown in Figure 1.

                                                                                                          

                                                              

 

 

 

 

 

 

 

                                                                                     

 

Text Box: Figure 1: Experimental apparatus for the measurement of pressure drops across stenoses.

You will need to set up this flow configuration. with the materials provided.  Note that the pressure transducer has two fluid inlets to it.  However, it is not a differential transducer and only senses the static pressure that is applied to its sensing element.  Connect the transducer to one of the pressure taps at a time.  The other inlet is used to bleed air out of the transducer after each connection is made.  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.

 

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

 

.

 

Take several readings of voltage from the signal conditioner as a function of the height of the reservoir surface above the transducer.  Vary this height by changing the position of the transducer, rather than the position of the reservoir.

 

To obtain the calibration, perform a linear least squares fit of pressure as a function of voltage (V).  This will enable you to translate all of the voltage readings directly to pressure from the slope m.  That is, .

 

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 ().

 

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 2).

 

 

 

 

 

 

 

Connect the pressure transducer to the upstream pressure tap.  Make sure that there are no air bubbles in the line.

 

Flow Rate

Calculate the flow rate that is required to obtain a cross-sectional average velocity of 50 cm/sec  ().  Use a graduated cylinder to set the flow rate to approximately this value.  Measure the flow rate as precisely as possible.  You will need to measure the pressure when the flow is off and when the flow is on.  The difference between these two is the pressure drop caused by fluid flow.

 

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.

 

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 no-flow measurement upstream, but may not be as a result of transducer drift.  This should not be a large problem as long as the flow-on and flow-off 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 1.

 

 

 

 

 

 

 

For Flow Rate

For Pressure Drops

Conditions

Volume

(ml)

Time

(sec)

Voltage Up-stream No Flow

Voltage Up-stream With Flow

Voltage Down-stream No Flow

Voltage Down-stream With Flow

50% Short Stenosis

Flow Rate 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flow Rate 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

50% Long Stenosis

Flow Rate 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flow Rate 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Text Box: Table 1: Required data for this laboratory.Recalibration

To ensure that your calibration has not changed, perform a post-experiment 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/cm2).   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 pre-experiment and post-experiment 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 T-tests to test the two hypotheses.  The first T-test will compare data taken with different flow rates but the same stenosis.  The second T-test will compare the two stenoses of different lengths.  Provide a p-value 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: 157-175, 1979

 

 

 

Biomedical Engineering Senior Laboratory (BIEN 435)

Louisiana Tech University

Steven A. Jones

 

 

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