E1C10 09/14/2010
13:4:41 Page 458
As a fluid particle of elemental mass, dm, flows through a section of the flow meter, it
experiences the Coriolis acceleration, 2v  _
r S=O
0 , and an inertial Coriolis force
dF ¼ 2v  _
r S=O
0
dm
ð10:34Þ
acting in the z direction. For the flow meter design of Figure 10.22 and as depicted in Figure 10.24,
as the particle travels along the meter the direction of the velocity vector changes. This results,
using the right-hand rule, in a change in direction of vector force, dF, between the left and right
sides. The resultant forces experienced by the tube are of equal magnitude but the opposite sign of
those experienced by the particle. Each tube segment senses a corresponding differential torque,
dT, a rotation about the y axis at a frequency v c ,
dT ¼ x
0
 dF ¼ x
0
 2v  _
r S=O
0
dm
ð10:35Þ
where x
0 refers to the x distance between the elemental mass and the y axis, x
0
¼ x
0_
e x . The total
magnitude of torque experienced by each tube is found by integration along the total path length of
the tube, L,
T ¼
Z L
0
dT
ð10:36Þ
A differential element of fluid has mass dm ¼ rA dl, for an elemental cross section of fluid A, of
differential length dl, and of density r. If this mass moves with an average velocity, U , then the
differential mass can be written as
dm ¼ rAdl ¼ _
m dl=U
À
Á
ð10:37Þ
The Coriolis cross-product can be expressed as
2v  _
r S=O
0 ¼ 2v  _
r S=O
0 sin u
_
e z ¼ ð2 v c Usin uÞ
_
e z
ð10:38Þ
where u is the angle between the Coriolis rotation and the velocity vector. Then,
_
m ¼
T
2
R L
0 rx 0 v c sin u
ð
Þ dl
_
e y
ð10:39Þ
Since the velocity direction changes by 180 degrees, the Coriolis-developed torque acts in
opposite directions on each side of the tube.
3 The meter tubes twist about the y axis of the tube (i.e.,
wobbles) at an angle d. For small angles of rotation the twist angle is related to torque by
d ¼ k s T ¼ constant  _
m
ð10:40Þ
where k s is related to the stiffness of the tube. The objective becomes to measure the twist angle,
which can be accomplished in many ways. For example, by driving the two meter tubes 180 degrees
out of phase, the relative phase at any time between the two tubes is directly related to the mass flow
rate. The exact relationship is linear over a wide flow range and determined by calibration with any
fluid (14).
3 We can envision a design in which the velocity does not reverse direction along the flow path but v does. In fact, after the
first publication of this footnote, one vendor adopted this approach in a commercial product.
458 Chapter 10 Flow Measurements
13:4:41 Page 458
As a fluid particle of elemental mass, dm, flows through a section of the flow meter, it
experiences the Coriolis acceleration, 2v  _
r S=O
0 , and an inertial Coriolis force
dF ¼ 2v  _
r S=O
0
dm
ð10:34Þ
acting in the z direction. For the flow meter design of Figure 10.22 and as depicted in Figure 10.24,
as the particle travels along the meter the direction of the velocity vector changes. This results,
using the right-hand rule, in a change in direction of vector force, dF, between the left and right
sides. The resultant forces experienced by the tube are of equal magnitude but the opposite sign of
those experienced by the particle. Each tube segment senses a corresponding differential torque,
dT, a rotation about the y axis at a frequency v c ,
dT ¼ x
0
 dF ¼ x
0
 2v  _
r S=O
0
dm
ð10:35Þ
where x
0 refers to the x distance between the elemental mass and the y axis, x
0
¼ x
0_
e x . The total
magnitude of torque experienced by each tube is found by integration along the total path length of
the tube, L,
T ¼
Z L
0
dT
ð10:36Þ
A differential element of fluid has mass dm ¼ rA dl, for an elemental cross section of fluid A, of
differential length dl, and of density r. If this mass moves with an average velocity, U , then the
differential mass can be written as
dm ¼ rAdl ¼ _
m dl=U
À
Á
ð10:37Þ
The Coriolis cross-product can be expressed as
2v  _
r S=O
0 ¼ 2v  _
r S=O
0 sin u
_
e z ¼ ð2 v c Usin uÞ
_
e z
ð10:38Þ
where u is the angle between the Coriolis rotation and the velocity vector. Then,
_
m ¼
T
2
R L
0 rx 0 v c sin u
ð
Þ dl
_
e y
ð10:39Þ
Since the velocity direction changes by 180 degrees, the Coriolis-developed torque acts in
opposite directions on each side of the tube.
3 The meter tubes twist about the y axis of the tube (i.e.,
wobbles) at an angle d. For small angles of rotation the twist angle is related to torque by
d ¼ k s T ¼ constant  _
m
ð10:40Þ
where k s is related to the stiffness of the tube. The objective becomes to measure the twist angle,
which can be accomplished in many ways. For example, by driving the two meter tubes 180 degrees
out of phase, the relative phase at any time between the two tubes is directly related to the mass flow
rate. The exact relationship is linear over a wide flow range and determined by calibration with any
fluid (14).
3 We can envision a design in which the velocity does not reverse direction along the flow path but v does. In fact, after the
first publication of this footnote, one vendor adopted this approach in a commercial product.
458 Chapter 10 Flow Measurements