Transcript Chapter12_level_2

Ultrasound Physics & Instrumentation
4th Edition
Volume II
Companion Presentation
Frank R. Miele
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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
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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
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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
PE
KE
PE
KE
KE
Fig. 1: (Pg 742)
Region 1
Lower Velocity
Region 2
Higher Velocity
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Region 3
Lower Velocity
Hydrostatic Pressure
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   gh
h1
h2
h3
Fig. 2: (Pg 743)
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Hydrostatic Pressure
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   gh
mmHg
mmHg
mmHg
 gh ( Blood )  0.776
 1.97
2
cm
inch
inch
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Capacitance
Capacitance is defined as a change in volume per time:
V
Capacitance 
t
Low
Capacitance
Higher
Capacitance
Highest
Capacitance
Fig. 3: (Pg 745)
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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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The Effect of Length on Resistance
For the same radius and the same volumetric flow, if the length
increases the resistance increases.
ℓ=5m
Q = 10ℓ / min
r
ℓ = 10 m
r
Q = 10ℓ / min
Fig. 4: (Pg 748)
R (Resistance)  (Length)
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The Effect of Radius (Area) on
Resistance
For the same length, if the radius (area) increases the resistance
decreases.
Pipeline A
AreaA = rA2
v
Pipeline B
v
Fig. 5: (Pg 748)
1
R (Resistance) 
r (radius)
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AreaB = rB2
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.
ℓ=5m
r=1m
Pipeline A
Q = 10ℓ / min
ℓ = 10 m
r=2m
Q = 10ℓ / min
Pipeline B
Fig. 6: (Pg 749)
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1
R 4
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.
ℓ=5m
Water
r=1m
Q = 10ℓ / min
Pipeline A
R    viscosity 
ℓ=5m
Honey
r=1m
Pipeline B
Q = 10ℓ / min
Fig. 7: (Pg 750)
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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:
8 
R 4
r
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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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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
v = 1 m/sec
Pipeline B
v = 1 m/sec
Fig. 8: (Pg 751)
Q  area or Q  r
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2
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
v = 1 m/sec
Q  v (mean spatial velocity)
Pipeline B
v = 2 m/sec
Fig. 9: (Pg 752)
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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:
Q  v  Area
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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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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:
P  R (Resistance)
ℓ=5m
Pipeline A
Q = 10ℓ / min
ℓ=5m
Pipeline B
Q = 10ℓ / min
Fig. 10: (Pg 753)
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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:
P  Q (Volumetric Flow)
Pipeline A
Q = 10ℓ / min
Pipeline B
Q = 1ℓ / min
Fig. 11: (Pg 753)
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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:
P  Q  R
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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,
P
or:
P  Q  R
 Q
R
P P r 4
 Q

8 
8 
 r4
P r
Q
8 
4
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Bernoulli’s Equation
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:
Energy1  Energy2
Point 1
Point 2
Energy2
Energy1
Fig. 12: (Pg 756)
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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:
1
Energy1  PE1  KE1  P1   v12
2
1
Energy2  PE2  KE2  P2   v2 2
2
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Developing Bernoulli’s Equation
By applying the conservation of energy we arrive at:
Energy1  Energy2
1
1
2
P1   v1  P2   v2 2
2
2
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Developing Bernoulli’s Equation
By expressing the given relationship in terms of the change in potential
energy (P1-P2) we achieve:
1
1
2
P1   v1  P2   v2 2
2
2
1
1
2
2
 P1  P2   v2   v1
2
2
1
P  P1  P2    v2 2  v12 
2
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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:
P1 
1
1
 v12   gh1  P2   v2 2   gh2
2
2
Point 1
Height h1
Point 2
Height h2
Energy1
Fig. 13: (Pg 758)
Energy2
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Bernoulli’s Equation with Hydrostatic
Term
Rewriting the equation in terms of the potential energy difference (the
pressure gradient) yields:
1
 1

2
P1  P2    v2   gh2     v12   gh1 
2
 2

1
   v2 2  v12    g  h2  h1 
2
1
2
2
P  P1  P2    v2  v1    g  h2  h1 
2
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Flow Through a Rigid Tube
1: Laminar Flow
5: Entrance Effect: “Plug Flow”
2: Entrance Effect: “Plug Flow”
6: Exit Effect: turbulence
3: Laminar Flow
7: Laminar Flow
4: “Steeper” Parabolic Laminar
Laminar
Plug
“Blunt” “Parabolic”
4
2
Turbulence
6
1
Laminar
3
Laminar
5
Plug
Fig. 16: (Pg 762)
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7
Laminar
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.
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Recirculation and Turbulence at a Pump
Inlet (Animation)
(Pg 765 A)
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Pump Outlet in Cross Section
Fig. 19: (Pg 764)
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Pump Outlet Flow (Animation)
(Pg 765 B)
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Turbulence
Fig. 24: (Pg 765)
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Turbulence (Animation)
(Pg 765 C)
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Flow Over an Obstruction (Animation)
(Pg 766 A)
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Slow Flow Over an Obstruction
(Animation)
(Pg 766 B)
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Low Reynold’s Number (Animation)
(Pg 769 A)
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High Reynold’s Number (Animation)
(Pg 769 B)
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Damped High Reynold’s Number
(Animation)
(Pg 769 C)
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Distal Disturbance (Two Cylinders)
(Animation)
(Pg 769 D)
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