1. Volume II Companion Presentation Frank R. Miele Pegasus Lectures, Inc.
Ultrasound Physics & Instrumentation 4th Edition
Volume II
Companion Presentation
Frank R. Miele
Pegasus Lectures, Inc.
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3. Volume II Outline Pegasus Lectures, Inc. Chapter 7: Doppler
Chapter 8: Artifacts
Chapter 9: Bioeffects
Chapter 10: Contrast and Harmonics
Chapter 11: Quality Assurance
Chapter 12: Fluid Dynamics
Level 2
Chapter 13: Hemodynamics
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4. Energy Pegasus Lectures, Inc. Potential Energy:
Energy which is stored -- the ability to do work
Kinetic Energy:
Energy which is related to motion – proportional to the velocity squared
Conservation of Energy:
Energy is always conserved – energy is never lost, only converted between forms
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5. Flow Conversion between Potential and Kinetic Energy
Assuming no loss of energy to heat, as the flow accelerates, there is a decrease in potential energy and a compensatory increase in kinetic energy (transitioning from region 1 to region 2) As the velocity decreases (region 2 to region 3) the kinetic energy decreases and the potential energy increases back to the same level as in region 1. PE KE
Region 1
Lower Velocity
Region 2
Higher Velocity
Region 3
Fig. 1: (Pg 742)
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6. Hydrostatic Pressure Pegasus Lectures, Inc.
Hydrostatic pressure is the pressure that results from the force of the fluid which results from a column of fluid. The hydrostatic pressure is proportional to the density of the fluid, the height of the fluid, and gravity. h1 h2 h3
Fig. 2: (Pg 743)
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7. Hydrostatic Pressure Pegasus Lectures, Inc.
By substituting into the equation the average density for blood and the value for gravity, the equation for the hydrostatic pressure simplifies to:
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8. Capacitance Pegasus Lectures, Inc.
Capacitance is defined as a change in volume per time:
Low
Capacitance
Higher
Highest
Fig. 3: (Pg 745)
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9. Developing the Resistance Equation
The principal relationships that constitute the resistance equation are developed throughout the next group of slides. Instead of just writing the equation outright, we will consider different physical situations to gain an intuitive understanding of these relationships.
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10. The Effect of Length on Resistance
For the same radius and the same volumetric flow, if the length increases the resistance increases.
ℓ = 5 m
ℓ = 10 m
Q = 10ℓ / min r
Fig. 4: (Pg 748)
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11. The Effect of Radius (Area) on Resistance
For the same length, if the radius (area) increases the resistance decreases.
Pipeline A
Pipeline B v
AreaA = rA2
AreaB = rB2
Fig. 5: (Pg 748)
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12. The Dominance of Radius on Resistance
Notice that if the radius and the length both change by a factor of 2, the radius still has a greater impact on the resistance. Therefore, the resistance equation is dominated by the radius (highest power). That power happens to be the fourth power.
Pipeline A
Pipeline B
Q = 10ℓ / min
r = 1 m
r = 2 m
ℓ = 5 m
ℓ = 10 m
Fig. 6: (Pg 749)
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13. The Effect of Viscosity on Resistance
The resistance is not just determined by the geometry of the flow conduit, but also by the viscosity of the fluid flowing. A higher viscosity results in a higher resistance to flow.
Pipeline A
Pipeline B
Q = 10ℓ / min
r = 1 m
ℓ = 5 m
Water
Honey
Fig. 7: (Pg 750)
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14. The Resistance Equation
We have just seen that the resistance is proportional to the length, proportional to the viscosity, and inversely proportional to the radius to the fourth power. The equation for the resistance, including a constant therefore becomes:
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15. Developing the Continuity (Volumetric Flow) Equation
The principal relationships that constitute the continuity equation are developed throughout the next group of slides. As with the resistance equation, instead of just writing the equation outright, we will consider different physical situations to gain an intuitive understanding of these relationships.
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16. The Effect of Cross-sectional Area on Volumetric Flow
Notice that for the same velocity, a larger cross-sectional area increases the volumetric flow (Q).
Pipeline A
Pipeline B
v = 1 m/sec
Fig. 8: (Pg 751)
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17. The Effect of Velocity on Volumetric Flow
Notice that for the same cross-sectional area, a higher average velocity increases the volumetric flow (Q).
Pipeline A
Pipeline B
v = 1 m/sec
v = 2 m/sec
Fig. 9: (Pg 752)
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18. The Continuity (Volumetric Flow) Equation
We have just seen that the volumetric flow is proportional to mean spatial velocity and proportional to the cross-sectional area. Therefore, the equation can be written:
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19. Developing the Simplified Law of Hemodynamics
The principal relationships that constitute the equation for the simplified law of hemodynamics are developed throughout the next group of slides. As with the resistance equation and the flow equation (continuity equation), instead of just writing the equation outright, we will consider different physical situations to gain an intuitive understanding of these relationships.
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20. The Effect of Resistance on the Pressure Gradient
The smaller radius of pipeline B results in a higher pressure gradient for the same volumetric flow as for pipeline A. Therefore, it is clear that the pressure gradient is proportional to the resistance R, or:
Pipeline A
Pipeline B
Q = 10ℓ / min
ℓ = 5 m
Fig. 10: (Pg 753)
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21. Effect of Volumetric Flow on the Pressure Gradient
The higher volumetric flow of pipeline A results in a higher pressure gradient for the same resistance as for pipeline B. Therefore, it is clear that the pressure gradient is proportional to the volumetric flow (Q), or:
Pipeline A
Pipeline B
Q = 10ℓ / min
Q = 1ℓ / min
Fig. 11: (Pg 753)
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22. The Simplified Law of Hemodynamics
We have just seen that the pressure gradient is proportional to both the resistance, R, and the volumetric flow, Q, or:
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23. Combining the Laws to Create Poiseuille’s Law
Poiseuille’s Law can now be created by rewriting the simplified law in terms of the volumetric flow Q and then substituting for the resistance, R, or:
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24. Bernoulli’s Equation Pegasus Lectures, Inc.
Assuming there is no energy lost to heat, in a closed system, the energy at point 1 must equal the energy at point 2 or:
Point 2
Point 1
Energy1
Energy2
Fig. 12: (Pg 756)
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25. Developing Bernoulli’s Equation
The energy at each point is comprised of a kinetic energy term, related to the square of the velocity, and a potential energy term, or:
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26. Developing Bernoulli’s Equation
By applying the conservation of energy we arrive at:
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27. Developing Bernoulli’s Equation
By expressing the given relationship in terms of the change in potential energy (P1-P2) we achieve:
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28. Bernoulli’s Equation with Hydrostatic Term
If there is a difference in height in the flow conduit, there is another “energy” term that must be considered related to the force produced by the mass and gravity called hydrostatic pressure as follows:
Height h1
Height h2
Energy1
Energy2
Point 2
Point 1
Fig. 13: (Pg 758)
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29. Bernoulli’s Equation with Hydrostatic Term
Rewriting the equation in terms of the potential energy difference (the pressure gradient) yields:
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30. Flow Through a Rigid Tube
1: Laminar Flow
2: Entrance Effect: “Plug Flow”
3: Laminar Flow
4: “Steeper” Parabolic Laminar
5: Entrance Effect: “Plug Flow”
6: Exit Effect: turbulence
7: Laminar Flow
“Blunt”
Plug 2
“Parabolic”
Laminar 4
Turbulence 6 3 7 5
Laminar 1
Fig. 16: (Pg 762)
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31. Flow Examples Pegasus Lectures, Inc.
The following slides are taken from the animation CD demonstrating various flow conditions and states (videos courtesy of Flometrics of Solana Beach California). It is very beneficial to review the animation CD for more in depth descriptions.
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32. Recirculation and Turbulence at a Pump Inlet (Animation)
(Pg 765 A)
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33. Pump Outlet in Cross Section
Fig. 19: (Pg 764)
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34. Pump Outlet Flow (Animation)
(Pg 765 B)
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35. Turbulence
Fig. 24: (Pg 765)
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36. Turbulence (Animation)
(Pg 765 C)
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37. Flow Over an Obstruction (Animation)
(Pg 766 A)
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38. Slow Flow Over an Obstruction (Animation)
(Pg 766 B)
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39. Low Reynold’s Number (Animation)
(Pg 769 A)
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40. High Reynold’s Number (Animation)
(Pg 769 B)
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41. Damped High Reynold’s Number (Animation)
(Pg 769 C)
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42. Distal Disturbance (Two Cylinders) (Animation)
(Pg 769 D)
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