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. Pegasus Lectures, Inc.

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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 Pegasus Lectures, Inc.

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 Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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: Pegasus Lectures, Inc.

Capacitance Pegasus Lectures, Inc. Capacitance is defined as a change in volume per time: Low Capacitance Higher Highest Fig. 3: (Pg 745) Pegasus Lectures, Inc.

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. Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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: Pegasus Lectures, Inc.

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. Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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: Pegasus Lectures, Inc.

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. Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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: Pegasus Lectures, Inc.

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: Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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: Pegasus Lectures, Inc.

Developing Bernoulli’s Equation By applying the conservation of energy we arrive at: Pegasus Lectures, Inc.

Developing Bernoulli’s Equation By expressing the given relationship in terms of the change in potential energy (P1-P2) we achieve: Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

Bernoulli’s Equation with Hydrostatic Term Rewriting the equation in terms of the potential energy difference (the pressure gradient) yields: Pegasus Lectures, Inc.

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) Pegasus Lectures, Inc.

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. Pegasus Lectures, Inc.

Recirculation and Turbulence at a Pump Inlet (Animation) (Pg 765 A) Pegasus Lectures, Inc.

Pump Outlet in Cross Section Fig. 19: (Pg 764) Pegasus Lectures, Inc.

Pump Outlet Flow (Animation) (Pg 765 B) Pegasus Lectures, Inc.

Turbulence Fig. 24: (Pg 765) Pegasus Lectures, Inc.

Turbulence (Animation) (Pg 765 C) Pegasus Lectures, Inc.

Flow Over an Obstruction (Animation) (Pg 766 A) Pegasus Lectures, Inc.

Slow Flow Over an Obstruction (Animation) (Pg 766 B) Pegasus Lectures, Inc.

Low Reynold’s Number (Animation) (Pg 769 A) Pegasus Lectures, Inc.

High Reynold’s Number (Animation) (Pg 769 B) Pegasus Lectures, Inc.

Damped High Reynold’s Number (Animation) (Pg 769 C) Pegasus Lectures, Inc.

Distal Disturbance (Two Cylinders) (Animation) (Pg 769 D) Pegasus Lectures, Inc.

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