Pegasus Lectures, Inc. Volume II Companion Presentation Frank R. Miele Pegasus Lectures, Inc. Ultrasound Physics & Instrumentation 4 th Edition

Pegasus Lectures, Inc.

Volume II

Companion Presentation

Frank R. MielePegasus Lectures, Inc.

Ultrasound Physics & Instrumentation4th Edition


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Volume II Outline

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


Energy

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


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.

Flow Conversion between Potential and Kinetic Energy

Fig. 1: (Pg 742)

PE

KE PE

KE

PE

KE

Region 1Lower Velocity

Region 2Higher Velocity

Region 3Lower Velocity


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.

Hydrostatic Pressure

Fig. 2: (Pg 743)

h1

h2

h3

Hydrostatic Pressure gh


By substituting into the equation the average density for blood and the value for gravity, the equation for the hydrostatic pressure simplifies to:

Hydrostatic Pressure

( ) 0.776 1.97 2

Hydrostatic Pressure gh

mmHg mmHg mmHggh Blood

cm inch inch


Capacitance is defined as a change in volume per time:

Capacitance

Fig. 3: (Pg 745)

LowCapacitance

HigherCapacitance

HighestCapacitance

CapacitanceV

t


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.

Developing the Resistance Equation


For the same radius and the same volumetric flow, if the length increases the resistance increases.

The Effect of Length on Resistance

Fig. 4: (Pg 748)

ℓ = 5 m

ℓ = 10 m

Q = 10ℓ / min

Q = 10ℓ / min

r

r

R (Resistance) (Length)


The Effect of Radius (Area) on Resistance

For the same length, if the radius (area) increases the resistance decreases.

Fig. 5: (Pg 748)

Pipeline A

Pipeline B

v

v

AreaA = rA2

AreaB = rB2

1R (Resistance)

r (radius)


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

Q = 10ℓ / min

r = 1 m

r = 2 m

ℓ = 5 m

ℓ = 10 m

Fig. 6: (Pg 749)

4

1R

r


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.

Fig. 7: (Pg 750)

Pipeline A

Pipeline B

Q = 10ℓ / min

Q = 10ℓ / min

r = 1 m

r = 1 m

ℓ = 5 m

ℓ = 5 m

Water

Honey

viscosityR


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:

The Resistance Equation

4

8R

r


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.

Developing the Continuity (Volumetric Flow) Equation


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).

Fig. 8: (Pg 751)

Pipeline A

Pipeline B

v = 1 m/sec

v = 1 m/sec

2 or Q area Q r


The Effect of Velocity on Volumetric Flow

Notice that for the same cross-sectional area, a higher average velocity increases the volumetric flow (Q).

Fig. 9: (Pg 752)

Pipeline A

Pipeline B

v = 1 m/sec

v = 2 m/sec

(mean spatial velocity)Q v


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:

The Continuity (Volumetric Flow) Equation

Q v Area


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.

Developing the Simplified Law of Hemodynamics


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:

Fig. 10: (Pg 753)

Pipeline A

Pipeline B

Q = 10ℓ / min

Q = 10ℓ / min

ℓ = 5 m

ℓ = 5 m

P R (Resistance)


Effect of Volumetric Flow on the Pressure Gradient

Fig. 11: (Pg 753)

Pipeline A

Pipeline B

Q = 10ℓ / min

Q = 1ℓ / min

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:

P Q (Volumetric Flow)


We have just seen that the pressure gradient is proportional to both the resistance, R, and the volumetric flow, Q, or:

The Simplified Law of Hemodynamics

P Q R


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:

4

4

8 8

PP Q R Q

R

P P rQ

r

4

8

P rQ


Bernoulli’s Equation

Point 2

Point 1

Energy1

Energy2

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:

1 2Energy Energy

Fig. 12: (Pg 756)


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:

21 1 1 1 1

22 2 2 2 2

1

2

1

2

Energy PE KE P v

Energy PE KE P v



By applying the conservation of energy we arrive at:

1 2

2 21 1 2 2

1 1

2 2

Energy Energy

P v P v



By expressing the given relationship in terms of the change in potential energy (P1-P2) we achieve:

2 21 1 2 2

2 21 2 2 1

1 1

2 21 1

2 2

P v P v

P P v v

2 21 2 2 1

1

2P P P v v


Bernoulli’s Equation with Hydrostatic Term

Height h1

Height h2

Energy1

Energy2

Point 2

Point 1

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:

2 21 1 1 2 2 2

1 1

2 2P v gh P v gh

Fig. 13: (Pg 758)


Bernoulli’s Equation with Hydrostatic Term

Rewriting the equation in terms of the potential energy difference (the pressure gradient) yields:

2 21 2 2 2 1 1

2 22 1 2 1

1 1

2 2

1

2

P P v gh v gh

v v g h h

2 21 2 2 1 2 1

1

2P P P v v g h h


1: Laminar Flow

2: Entrance Effect: “Plug Flow”

3: Laminar Flow

4: “Steeper” Parabolic Laminar

Flow Through a Rigid Tube

Fig. 16: (Pg 762)

1Laminar

Plug“Blunt”

2

Laminar“Parabolic”

4Turbulence

6

3Laminar 7

Laminar

5Plug

5: Entrance Effect: “Plug Flow”

6: Exit Effect: turbulence

7: Laminar Flow


Flow Examples

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.


Recirculation and Turbulence at a Pump Inlet (Animation)

(Pg 765 A)


Pump Outlet in Cross Section

Fig. 19: (Pg 764)


Pump Outlet Flow (Animation)

(Pg 765 B)


Turbulence

Fig. 24: (Pg 765)


Turbulence (Animation)

(Pg 765 C)


Flow Over an Obstruction (Animation)

(Pg 766 A)


Slow Flow Over an Obstruction (Animation)

(Pg 766 B)


Low Reynold’s Number (Animation)

(Pg 769 A)


High Reynold’s Number (Animation)

(Pg 769 B)


Damped High Reynold’s Number (Animation)

(Pg 769 C)


Distal Disturbance (Two Cylinders) (Animation)

(Pg 769 D)


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Pegasus Lectures, Inc. Volume II Companion Presentation Frank R. Miele Pegasus Lectures, Inc. Ultrasound Physics & Instrumentation 4 th Edition