BE 310 Project - Penn Engineering



BE 310 Project

Pulsatile Flow

May 11, 2000

Group T1

Walter Aughenbaugh

Barry Lee

Ameesha Patel

Matthew Samuelowitz

Adrianne Wenisch

Introduction

Can a modified, one-dimensional Navier-Stokes equation be used to model pulsatile flow? Can the large vessels of the mammalian arterial system be biomatched with rigid Pygon tubing connected to a syringe pump filled with water or sucrose solution to simulate blood flow? Which parameters affect flow rate, and how? These questions were addressed by sending water and sucrose solutions through single tubes of varying diameter at varying flow rates. From the recorded pressure change, a theoretical flow rate was calculated using a modified version of Navier-Stokes. Comparing this theoretical flow rate to the set flow rate reveals whether pulsatile flow can be applied to the Navier-Stokes equation. Also, the parameters of tube length, diameter, and liquid viscosity were varied and their effect on flow rate was investigated.

A. Biomedical Relevance and Importance

The cardiovascular system starts by pumping oxygenated blood from the left ventricle of the heart into the aorta. From the aorta, the blood enters the arteries. Both the aorta and the arteries have walls that expand and contract with the pumping of the blood; the aorta and arteries act as auxiliary pumps and pressure reservoirs. The branching of the arteries is demonstrated in the diagram below:

[pic]

Figure 1. A general diagram of the arterial system

From any given artery, the blood flows into arterioles. The arterioles are smaller than the arteries, and in addition, the body uses the arterioles to regulate the flow of blood in various parts of the body. For instance, when the body is cold, the arterioles leading to the skin will narrow, lessening the flow of blood and therefore the amount of heat to the skin. Likewise, the arterioles leading to the digestive system expand while digestion is taking place, to increase the efficiency of that system.

From an arteriole, the blood enters one or more capillaries. The capillaries are extremely small; so small that the red blood cells slide through one at a time, and the blood can no longer be considered to be a homogeneous fluid with a well-defined viscosity. In the capillaries, the body uses the oxygen carried by the blood, and the blood that leaves the capillaries is called "deoxygenated". Blood flow in the capillaries is smooth; that is, the pulses of the heartbeat have been smoothed out and the blood flows continuously with a constant pressure:

[pic]

Figure 2. The pulses and velocities of the heart in different parts of the arterial system

[pic]

Figure 3. The velocity of blood in different parts of the arterial system

Many capillaries feed into one venule. A venule is larger than a capillary, and smaller than a vein. Many venules feed into one vein. At this point, the blood has lost so much pressure to dissipation (below) that it can barely make its way back to the heart. The veins have one-way valves, which prevent the flow of blood away from the heart, and the movements of the body’s muscles help to pump the blood toward the heart. All of the veins connect to the vena cava, which is the entry for deoxygenated blood into the heart.

As the heart beats, the blood pressure in the aorta varies from the "systolic" (maximum) pressure, which occurs as the ventricle pumps blood into the aorta, to the "diastolic" (minimum) pressure, which occurs as the ventricle is filling. The average blood pressure of a healthy person of average build is referred to as "120 over 80", which means that the systolic pressure is 120 mmHg (millimeters of mercury, that is, the increase in height of a tube of mercury due to the pressure) and the diastolic pressure is 80 mmHg. Expansion and contraction of the arterial walls, as well as dissipation from viscosity, reduce the pressure variations as the blood moves away from the heart.

B. Scientific Background

Navier-Stokes Equation is the principle behind this experiment. For pulsatile flow in our experiment, the Equation takes the form

[pic]

Equation 2

for incompressible, flow with no body force. Density ( is assumed to be constant. The symbol ( is the dynamic viscosity, P is the pressure, u is the velocity of the fluid. A different parameter can then be applied to fit our model. In the case of low Reynolds number (< 2000), the inertia term is smaller than the viscous term and can therefore be ignored, leaving the equation of creeping motion.

[pic]

Equation 3

Certain assumptions can be made based on the experimental procedure to simplify Equation 3: the velocity between the pressure transducers does not change, (u/(x = 0; viscosity of the fluid is negligible, ( = 0; the velocity is not a function of radius (u/(r = 0.

For the purpose of this experiment, only flow in the x-direction (parallel to the tube) will be considered, and both the head loss and the viscosity factor will be ignored. Therefore, (P is the partial derivative of pressure with respect to x, along the length of the tube. Rearrangement of this equation yields

(u(x)/(t = ((1/() (P

Equation 8

Integration of equation 8 yields equation 9

u = ( (1/( ) (P t

Equation 9

where t is the time period that is set to empty the syringe. To yield the flow rate equation, equation 9 is multiplied by area. This then yields equation 10.

Q = ( (1/( ) (P t A

Equation 10

Due to the fact that in this experiment we are only considering flow in the along the x-direction, along the length of the tube, (P may be represented as (P, the change in pressure along the length of the tube.

C. Experimental Background

In order to simulate pulsatile flow, a KDS200 series infusion/withdrawal pump was used. Linear motion of the pusher block delivered a known volume of fluid. After entering the syringe diameter, a dispense volume and flow rate, the pump automatically performed all calibration and control functions.

The mode of operation for the pump was set at infusion; the pump infused at the set rate and stopped automatically when the target volume was reached.

Methods

Materials and Apparatus

KDS200 series infusion/withdrawal pump

60 mL syringe

Tubes with diameters of: .03048mm, .5588mm, 1.068mm, 1.5875mm, 3.175mm

LabView Software

Figure 4. Experimental setup.

Procedure

The pressure transducers were calibrated; tubes of water at various heights gave voltage readings that were related to the pressure calculated using Bernoulli’s Equation. The calibration equations for both transducers are given below.

Transducer 1: Pressure (Pa) = 4873 * Voltage (V) – 7431

Transducer 2: Pressure (Pa) = 4906 * Voltage (V) – 7338

Equations 11 and 12

The syringe pump expelled a set volume of liquid at a preset flow rate. Pressure transducers recorded the change in pressure, (P. The recorded (P was then used to calculate the theoretical flow rate, Q, from the modified Navier-Stokes Equation. The theoretical Q was compared to the preset flow rate.

The tube length, tube diameter, and the viscosity of the liquid were varied and are given below in Table 1.

|Length (cm) |Diameter (cm) |Viscosity (%sucrose) |

|139.7 |0.31 |10 |

|111.76 |0.16 |15 |

|83.82 |0.10 |20 |

Table 1. Variable Parameters

Results

A plot of pressure vs. time was constructed from raw data and is given below in Figure 5a. Since there were no negative pressures, a correction factor was used to adjust the (P. The resulting plot of pressure vs. time is given in Figure 5b.

The instantaneous flow rates calculated from the pressure vs. time graphs, both with and without the correction factor, were plotted and are given below in Figures 6a and 6b, respectively.

The flow rates for varying the tube length are given below in Figure 7. The plot suggests that varying tube length does not affect flow rate.

The flow rates when the tube diameters were varied are given below in Figure 8. According to the plot below, as tube diameter decreases, flow rate increases.

The flow rates for varying liquid viscosities are given in Figure 9. Increasing the viscosity appears to decrease flow rate.

Discussion

The experimental flow of each trial was not consistent with the actual flow. This is mainly because the experimental system was not closed. The human arterial system is a closed system where a constant pressure is applied. To accurately model the human arterial system, the liquid needs to be pumped into a raised reservoir above the transducers to simulate a closed system.

Since the system was an open one, the second pressure transducer was open to atmospheric pressure for the entire length of each of the trials. As a result, the second transducer made readings of nearly one atmosphere at all times; there was no increase in pressure measured in the second transducer as the syringe started to pump. Therefore, all the values of ΔP for the system are the pressure measured in the first transducer minus approximately one atmosphere (the pressure measured at the second transducer).

To compensate for not having a closed system, a correction factor was introduced to shift the ΔP curve. The correction factor represents a raised reservoir at the end of the system. This correction factor was found using the preset influx volume, and working backwards to calculate new values of ΔP. This new corrected curve contained the necessary positive and negative ΔP values, which produced reasonable fluid velocities and flow rates.

The correction factor is the mathematical equivalent of a reservoir raised a specific height. This height can be calculated directly from the correction factor using the simplified form of Bernoulli’s Equation, (P = (gh. For example, in the trials where the viscosity of the pumped liquid was varied, the correcting pressure shift was found to be between 2940-3430 Pascals. The necessary height of the reservoir was found to be around 30-35cm above the transducers for the viscosity trials.

One observation made was that flow appeared to be independent of tube length. This is unpredicted by Navier-Stokes equation, which says there is an inverse relationship between length and flow. Increasing length should almost certainly decrease flow, but as Figure 7 illustrates, this was not the case.

These unusual results were caused by the system not being pressurized. Since length does not affect the pressure measured at the first transducer, varying tube length did not change the values of ΔP and therefore did not affect calculated flow. In order to make accurate measurements of flow as length varies, the system must be pressurized.

The effects of increasing either tube diameter or liquid viscosity causes a decrease in flow rate. Although the Navier-Stokes Equation predicts these results, they cannot be relied upon. The flow rates appear to continuously increase in Figures 8,9, and 10. This is a physical impossibility for pulsatile flow.

Another error in the experimental set up resulted from the diameter differences between the tubing from the pulsatile syringe pump to the first pressure transducer and the tubing throughout the rest of the system. The most significant errors occurred when the smallest diameter tubing was used throughout the system while the first segment of tubing between the syringe pump and first pressure transducer remained constant for all trials. The first section had the largest diameter of 0.31 cm and the rest of the system had tubing with a 0.10 cm diameter. This diameter difference would cause turbulence at the first pressure transducer and therefore the resulting pressure reading would be inaccurate at that transducer.

Other errors include the possible backflow within the system, leaks in the system, and inaccurate pressure recordings due to air bubbles trapped in the transducers. Although these errors may have affected the measured (P, the much greater magnitude of error from using an open system would decrease the significance of these other errors.

Conclusions

The affect of trying to model the human arterial system with an unpressurized, open system produced physically impossible results. Therefore, future experimenters should either use closed systems for their experimental set up, or use a raised reservoir to pressurize the system.

Due to the inaccurate experimental set up, the results from varying tube length, diameter and liquid viscosity are not reliable. Without accurately modeling the human arterial system with a closed system, the true effects of changing these parameters cannot be determined.

The outcome of this experiment also suggests that the syringe pump is not a good device to use to model the closed human arterial system. The motor pump, which is always used in a closed system, would be preferable for modeling the arterial system.

References

kdScientific Model 200 Series Manual

© 1996-2000 Eric W. Weisstein

2000-03-18

physics/Navier-StokesEquation.html

©1996, Kenneth R. Koehler.



Appendix

Corrected Q

[pic]

Acquired Q

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Figure 5b. Pressure as a function of time with correction factor.

Figure 5a. Recorded pressure as a function of time

.

Figure 6a. Measured flow rate as a function of time.

Figure 6b. Measured flow rate as a function of time with correction factor.

Figure 7. Effect of tube length on flow rate.

Figure 8. Effect of tube diameter on flow rate.

Figure 9. Effect of viscosity on flow rate.

Dð

Dð

mð

D

L

Pump

P2

P1

fect of viscosity on flow rate.

Δ

Δ

μ

D

L

Pump

P2

P1

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