Abstrict An improved accuracy Coriolis mass flow meter. Signal processing
embodiments provide improved accuracy by accounting for the nonlinear
relationship between phase angle difference of motion sensor signals
and mass flow rate. Compensation is made for asymmetric and viscous
damping effects, as well.
Claims We claim:
1. A flow meter for flowable materials comprising:
a. a support;
b. a continuous conduit which
(1) is free of pressure sensitive joints,
(2) is solidly mounted to said support at an inlet end and an outlet
end of said conduit,
(3) is characterized by the presence of an oscillation axis and
a second axis, substantially transverse to said oscillation axis,
about which Coriolis force acts when fluid is flowing through said
conduit under oscillation conditions;
c. driver means for oscillating said conduit about said oscillation
axis; and
d. sensor means for measuring motion of said conduit as a result
of elastic deformation of said conduit around said second axis thereof
upon oscillation of said conduit with fluid flow therethrough, thereby
providing motion signals;
e. signal processing means, comprising:
(1) means for measuring frequency of oscillation of said conduit;
(2) logic means for converting said frequency of oscillation to
a frequency response function;
(3) logic means for comparing said motion signals to determine
a phase angle difference between said signals;
(4) logic means for converting said phase angle difference to a
halfangle tangent function of said phase angle difference;
(5) logic means for mathematically combining said frequency response
function, said frequency of oscillation and said halfangle tangent
function with at least one calibration constant which comprise
means to calculate mass flow rate m.sub.o according to the formula
where ##EQU63## is a phase angle difference between output signals
developed by said motion sensors
tan (.delta..theta..degree./2) is a halfangle tangent function;
K.sub.1 is a calibration constant proportional to a ratio of a
bending force acting on said flow conduit to a viscous damping force
acting on said flow conduit;
k.sub.c is an elastic spring constant of said flow conduit which
is fixed at constant temperature and is otherwise temperature compensated;
##EQU64## is a frequency response function; .omega..sub.d is a natural
frequency of said flow conduit about said oscillation axis;
.omega..sub.c is a natural frequency of said flow conduit about
said second axis,
.zeta..sub.c is a damping coefficient about said second axis,
whereby a mass flow rate measurement is obtained.
2. A flow meter as in claim 1 wherein
a. said sensor means comprises
first sensor means for measuring motion of said conduit at a first
location along said conduit, thereby providing a first motion signal
and
second sensor means for measuring motion of said conduit at a second
location along said conduit, thereby providing a second motion signal;
and
b. in said signal processing means,
said means for measuring frequency of oscillation of said conduit
comprise
(i) logic means for comparing said first and second motion signals
to determine a phase angle difference between said signals and
(ii) means for converting at least one of said first and second
motion signals to a frequency of oscillation.
3. A flow meter as in claim 2 wherein said means for converting
at least one of said first and second motion signals to a frequency
of oscillation comprises
a. means for converting said first motion signal to a first frequency
of oscillation,
b. means for converting said second motion signal to a second frequency
of oscillation and
c. logic means for averaging said first and second frequencies
of oscillation.
4. A flow meter as in claim 1 wherein
a. said sensor means comprises
first sensor means for measuring motion of said conduit at a first
location along said conduit, thereby providing a first motion signal
and
second sensor means for measuring motion of said conduit at a second
location along said conduit, thereby providing a second motion signal;
b. in said signal processing means
(1) the means for measuring frequency of oscillation of said conduit
comprise
(i) sensor means for detecting a driver motion signal and
(ii) logic means for measuring a frequency of oscillation of said
conduit;
(2) the logic means for comparing said motion signals comprises
logic means for comparing said first and second motion signals to
determine a motion sensor phase angle difference and
(3) the logic means for converting said phase angle difference
comprises
(i) means for dividing said motion sensor phase angle difference
by 2 to produce a motion sensor phase halfangle difference,
(ii) logic means for comparing said first motion signal and said
driver motion signal to determine a first phase angle difference
between said first motion signal and said driver motion signal,
(iii) logic means for converting said first phase angle difference
to a first asymmetrycompensated phase angle difference,
(iv) logic means for comparing said second motion signal and said
driver motion signal to determine a second phase angle difference
between said second motion signal and said driver motion signal,
(v) logic means for converting said second phase angle difference
to a second asymmetrycompensated phase angle difference,
(vi) logic means for comparing said motion sensor phase halfangle
difference, said first asymmetrycompensated angle difference and
said second asymmetrycompensated phase angle difference to produce
a system phase halfangle difference and
(vii) logic means for converting said system phase halfangle difference
to a halfangle tangent function.
5. A flow meter as in claim 4 wherein
a. Said logic means for converting said first phase angle difference
to a first asymmetrycompensated phase angle difference comprises
means for converting according to the formula ##EQU65## b. Said
logic means for converting said second phase angle difference to
a second asymmetrycompensated phase angle difference comprises
means for converting according to the formula ##EQU66## where .delta..theta..sub.R
is said first phase angle difference,
.delta..theta..sub.1 is said first asymmetrycompensated phase
angle difference,
.phi..sub.c is a Coriolis phase angle function,
.delta..theta..sub.L is said second phase angle difference, and
.delta..theta..sub.2 is said second asymmetrycompensated phase
angle difference; and
c. Said logic means for comparing said motion sensor phase halfangle
difference, [1/2.delta..theta..sub.c, ] said first asymmetrycompensated
phase angle difference and said difference and said second asymmetrycompensated
phase angle difference, to produce a system phase halfangle difference
comprises means to do so according to the relationships
whereby lack of agreement with b(1) through b(4) produces a warning
signal.
6. A flow meter as in claim 1 wherein, in said signal processing
means,
a. the means for measuring frequency of oscillation of said conduit
comprises
(i) sensor means for detecting a driver motion signal and
(ii) logic means for measuring a frequency of oscillation of said
conduit;
b. the logic means for comparing said motion signals comprises
means for comparing at least one of said motion signals and said
driver motion signal to determine a phase angle difference between
said motion signal and said driver motion signal;
c. the logic means for converting said phase angle difference comprises
(i) means for converting said phase angle difference to an asymmetrycompensated
halfangle difference and
(ii) logic means for converting said asymmetrycompensated phase
halfangle difference to a halfangle tangent function.
7. A flow meter as in claim 1 further comprising:
a. temperature sensor means for measuring conduit temperature,
thereby providing a temperature signal and
b. logic means for converting said temperature signal to at least
one temperature compensated spring constant signal,
Wherein
said logic means for mathematically combining said frequency response
function, said frequency, and said halfangle tangent function with
at least one calibration constant comprises means which also mathematically
combine said temperature compensated spring constant signal therewith,
thereby producing a mass flow rate measurement.
8. A flow meter as in claim 1 further comprising:
a. temperature sensor means for measuring conduit temperature,
thereby providing a temperature signal;
b. logic means for converting said temperature signal to a temperature
compensated spring constant representing the oscillation degree
of freedom of said conduit and corresponding to conduit motion about
said oscillation axis;
c. logic means for converting said temperature signal to a temperaturecompensated
spring constant representing the Coriolis degree of freedom of said
conduit and corresponding to conduit motion about said second axis;
d. logic means for converting said temperaturecompensated spring
constant representing the oscillation degree of freedom and said
frequency of oscillation of said conduit to a fluid density;
e. logic means for converting said temperaturecompensated spring
constant representing the Coriolis degree of freedom of said conduit
and said fluid density to a Coriolis natural frequency;
f. logic means for converting said Coriolis natural frequency and
said frequency of oscillation to a frequency response function,
thereby correcting for density variations; wherein
g. the logic means for mathematically combining said frequency,
said frequency response function and said halfangle tangent function
with at least one calibration constant comprise means which also
mathematically combine said temperature compensated spring constant
representing the Coriolis degree of freedom therewith,
thereby producing a mass flow rate measurement.
9. A flow meter for flowable materials comprising:
a. a support;
b. a continuous conduit which
(1) is free of pressure sensitive joints,
(2) is solidly mounted to said support at an inlet end and an outlet
end of said conduit,
(3) is characterized by the presence of an oscillation axis and
a second axis, substantially transverse to said oscillation axis,
about which Coriolis force acts when fluid is flowing through said
conduit under oscillation conditions;
c. driver means for oscillating said conduit about said oscillation
axis;
d. sensor means for measuring motion of said driver means, thereby
providing a driver motion signal;
e. sensor means for measuring motion of said conduit as a result
of elastic deformation of said conduit around said second axis thereof
upon oscillation of said conduit with fluid flow therethrough, thereby
providing motion signals; and
f. signal processing means, comprising:
(1) means for measuring frequency of oscillation of said conduit;
(2) logic means for converting said driver motion signal to a damping
measurement;
(3) logic means for converting said frequency of oscillation and
said damping measurement to a frequency response function and a
Coriolis phase angle function;
(4) logic means for comparing said driver motion signal and at
least one of said flow conduit motion signals to determine a phase
angle difference between said signals;
(5) logic means for converting said phase angle difference and
said Coriolis phase angle function to a trigonometric function of
said phase angle difference;
(6) logic means for mathematically combining said frequency response
function, said frequency of oscillation and said trigonometric function
with at least one calibration constant; thereby producing a mass
flow rate measurement.
10. A flow meter as in claim 9 wherein said
logic means for mathematically combining said frequency response
function, said frequency of oscillation, and said trigonometric
function comprises logic means which further mathematically combines
therewith said damping measurement, to calculate mass flow rate
m.sub.o according to the formula ##EQU67##
11. a flow meter as in claim 9 further comprising:
a. temperature means for measuring conduit temperature, thereby
providing a temperature signal and
b. logic means for converting said temperature signal to at least
one temperaturecompensated spring constant, wherein
c. said logic means for mathematically combining said frequency
response function, said frequency, and said trigonometric function,
with at least one calibration constant comprises means which also
mathematically combine said temperaturecompensated spring constant
therewith,
thereby producing a mass flow rate measurement.
12. A flow meter as in claim 9 further comprising:
a. temperature sensor means for measuring conduit temperature,
thereby providing a temperature signal,
b. logic means for converting said temperature signal to a temperaturecompensated
spring constant representing the oscillation degree of freedom of
said conduit and corresponding to conduit motion about said oscillation
axis,
c. logic means for converting said temperature signal to a temperaturecompensated
spring constant representing the Coriolis degree of freedom of said
conduit and corresponding to conduit motion about said second axis,
d. logic means for converting said temperaturecompensated spring
constant representing the oscillation degree of freedom of the conduit
and said frequency oscillation to a fluid density,
e. logic means for converting said temperaturecompensated spring
constant representing the Coriolis degree of freedom of said conduit
and said fluid density to a Coriolis natural frequency and
f. logic means for converting said Coriolis natural frequency,
said frequency of oscillation and said damping measurement to a
frequency response function, thereby correcting for density variations;
wherein
g. the logic means for mathematically combining said frequency
response function, said frequency and said trigonometric function
with at least one calibration constant comprises means which also
mathematically combine therewith said temperature compensated spring
constant representing the Coriolis degree of freedom, thereby producing
a mass flow rate measurement.
13. A flow meter for flowable materials comprising:
a. a support;
b. a continuous conduit which
(1) is free of pressure sensitive joints,
(2) is solidly mounted to said support at an inlet end and an outlet
end of said conduit,
(3) is characterised by the presence of an oscillation axis and
a second axis, substantially transverse to said oscillation axis,
about which Coriolis force acts when fluid is flowing through said
conduit under oscillation conditions;
c. driver means for oscillating said conduit about said oscillation
axis;
d. sensor means for measuring motion of said driver means, thereby
providing a driver motion signal;
e. sensor means for measuring motion of said conduit as a result
of elastic deformation of said conduit around said second axis thereof
upon oscillation of said conduit with fluid flow therethrough, thereby
providing motion signals; and
f. signal processing means, comprising:
(1) means for measuring frequency of of oscillation of said conduit;
(2) logic means for converting said driver motion signal to a damping
measurement;
(3) logic means for converting said damping measurement and said
frequency of oscillation to a frequency response function and a
Coriolis phase angle function;
(4) logic means for comparing said driver motion signal and one
of said flow conduit motion signals to determine a phase angle difference
between said signals;
(5) logic means for converting said phase angle difference and
said Coriolis phase angle function to an asymmetrycompensated halfangle
function;
(6) logic means for converting said asymmetrycompensated halfangle
function to a halfangle tangent function;
(7) logic means for mathematically combining said frequency response
function, said frequency of oscillation and said halfangle tangent
function with at least one calibration constant; thereby producing
a mass flow rate measurement.
14. A flow meter as in claim 13 wherein said logic means for mathematically
combining said frequency response function, said frequency of oscillation,
and said halfangle tangent function comprises means to calculate
mass flow rate m.sub.o according to the formula. ##EQU68##
15. A flow meter as in claim 13 further comprising:
(a) temperature sensor means for measuring conduit temperature,
thereby providing a temperature signal and
(b) logic means for converting said temperature signal to at least
temperature compensated spring constant; wherein
(c) said logic means for mathematically combining said frequency
response function, said frequency, and said half angle tangent function
with at least one calibration constant comprises means which also
mathematically combine said temperature compensated spring constant
therewith,
thereby producing a mass flow rate measurement.
16. A flow meter as in claim 13 further comprising:
a. temperature sensor means for measuring conduit temperature,
thereby providing a temperature signal;
b. logic means for converting said temperature signal to a temperature
compensated spring constant representing the oscillation degree
of freedom of said conduit and corresponding to conduit motion about
said oscillation axis;
c. logic means for converting said temperature signal to a temperaturecompensated
spring constant representing the Coriolis degree of freedom of said
conduit and corresponding to conduit motion about said second axis;
d. logic means for converting said frequency and said temperaturecompensated
spring constant representing the oscillation degree of freedom to
a fluid density;
e. logic means for converting said temperature compensated spring
constant representing the Coriolis degree of freedom and said fluid
density to a Coriolis natural frequency; and
f. logic means for converting said coriolis natural frequency,
said frequency of oscillation and said damping measurement to a
frequency response function, thereby correcting for density variations;
wherein
g. said logic means for mathematically combining said frequency
response function, said frequency and said halfangle tangent function
with at least one calibration constant comprises means which also
mathematically combine said coriolis spring constant therewith thereby
producing a mass flow rate measurement.
17. A flow meter for flowable materials comprising:
a. a support;
b. a continuous conduit which,
(1) is free of pressure sensitive joints,
(2) is solidly mounted to said support at an inlet end and an outlet
end of said conduit,
(3) is characterized by the presence of an oscillation axis and
a second axis, substantially transverse to said oscillation axis,
about which Coriolis force acts when fluid is flowing through said
conduit under oscillation conditions;
c. driver means for oscillating said conduit about said oscillation
axis;
d. sensor means for measuring motion of said driver means, thereby
providing a driver motion signal;
e. sensor means for measuring motion of said conduit as a result
of elastic deformation of said conduit around said second axis thereof
upon oscillation of said conduit with fluid flow therethrough, thereby
providing motion signals; and
f. signal processing means, comprising:
(1) means for measuring frequency of oscillation of said conduit;
(2) logic means for converting said driver motion signal to a damping
measurement;
(3) logic means for converting said frequency of oscillation and
said damping measurement to a frequency response function;
(4) logic means for comparing said motion signals to determine
a phase angle difference between said signals;
(5) logic means for converting said phase angle difference to a
halfangle tangent function of said phase angle difference;
(6) logic means for mathematically combining said frequency response
function, said frequency of oscillation and said halfangle tangent
function with at least one calibration constant; thereby producing
a mass flow rate measurement.
18. A flow meter as in claim 17 wherein said logic means for mathematically
combining said frequency response function, said frequency of oscillation,
and said halfangle tangent function with at least one calibration
constant comprises means to calculate mass flow rate m.sub.o according
to the formula ##EQU69##
19. A flow meter as in claim 17 wherein said sensor means for measuring
motion of the conduit comprise
(1) first sensor means for measuring motion of said conduit at
a first location along said conduit, thereby providing a first motion
signal and
(2) second sensor means for measuring motion of said conduit at
a second location along said conduit, thereby providing a second
motion signal.
20. A flow meter as in claim 17 further comprising:
a. temperature sensor means for measuring conduit temperature,
thereby providing a temperature signal and
b. logic means for converting said temperature signal to a at least
one temperature compensated spring constant, wherein
c. said logic means for mathematically combining said frequency
response function, said frequency, and said halfangle tangent function
with at least one calibration constant comprise means for also combining
said temperature compensated spring constant therewith, thereby
producing a mass flow rate measurement.
21. A flow meter as in claim 17 further comprising:
a. temperature sensor means for measuring conduit temperature,
thereby providing a temperature signal;
b. logic means for converting said temperature signal to a temperature
compensated spring constant representing the oscillation degree
of freedom of said conduit and corresponding to conduit motion about
said oscillation axis;
c. Logic means for converting said temperature signal to a temperature
compensated spring constant representing the Coriolis degree of
freedom of said conduit and corresponding to conduit motion about
said second axis;
d. logic means for converting said temperature compensated spring
constant representing the oscillation degree of freedom and said
frequency of oscillation to a fluid density;
e. logic means for converting said temperature compensated spring
constant representing the Coriolis degree of freedom and said fluid
density to a Coriolis natural frequency;
f. logic means for converting said Coriolis natural frequency,
said frequency of oscillation and said damping measurement to a
frequency response function, thereby correcting for density variations
and wherein
g. said logic means for mathematically combining said frequency,
said frequency response function and said halfangle tangent function
with at least one calibration constant comprise means for also combining
said temperature compensated spring constant representing the Coriolis
degree of freedom therewith, thereby producing a mass flow rate
measurement.
Description TECHNICAL FIELD
The present invention relates to Coriolis mass flow rate meters
that include one or more flow conduits which are driven to oscillate
at the resonant frequency associated with the combined mass of the
flow conduit and the fluid flowing therethrough. The drive frequency
is maintained at resonance by a feedback system, which detects a
change in the resonant behavior of the fluidfilled conduit as a
result of the fluid mass change due to the change in fluid density.
The flow conduits used in these Coriolis mass flow rate meters are
mounted so they can oscillate at the resonant frequency associated
with the oscillation axis about which externally applied forces
act to oscillate each flow conduit. The flow conduit also deforms
about another axis which is associated with deflections of each
flow conduit caused by Coriolis forces arising from the combination
of the driven oscillation and flow of fluids through the flow conduit.
The axis associated with Coriolis deflections is substantially transverse
to the oscillation axis. According to the present invention greater
accuracy of measurement at high flow rates is obtained in flow meter
designs which employ signal processing which includes a nonlinear
phase angle difference relationship between the signals of motion
sensors mounted on or adjacent the flow conduit. The present invention
also accounts for asymmetries in flow conduit deflections and for
viscous damping in the flow conduits due to properties of the flow
conduit materials and the flow of fluids through the conduits.
BACKGROUND ART
In the art of measuring mass flow rates of flowing substances it
is known that flowing a fluid through an oscillating flow conduit
induces Coriolis forces to act on the conduit. It is also known
that the magnitudes of such Coriolis forces are related to both
the mass flow rate of the fluid passing through the conduit and
the angular velocity at which the conduit is oscillated.
One of the major technical problems previously associated with
efforts to design and make Coriolis mass flow rate instruments was
the necessity either to measure accurately or control precisely
the angular velocity of an oscillated flow conduit so that the mass
flow rate of the fluid flowing through the flow conduits could be
calculated using measurements of effects caused by Coriolis forces.
Even if the angular velocity of a flow conduit could be accurately
determined or controlled, precise measurement of the magnitude of
effects caused by Coriolis forces was another severe technical problem.
This problem arises in part because the magnitudes of generated
Coriolis forces are very small in comparison to other forces such
as inertia and damping, therefore resulting Coriolis forceinduced
effects are minute. Further, because of the small magnitude of the
Coriolis forces, effects resulting from external sources such as
vibrations induced, for example, by neighboring machinery or pressure
surges in fluid lines, may cause erroneous determinations of mass
flow rates. Such error sources as discontinuities in the flow tube
may even completely mask the effects caused by generated Coriolis
forces rendering a flow meter useless.
A mechanical structure and measurement technique which, among other
advantages: (a) avoids the need to measure or control the magnitude
of the angular velocity of a Coriolis mass flow rate instrument's
oscillating flow conduit; (b) concurrently provides requisite sensitivity
and accuracy for the measurement of effects caused by Coriolis forces;
and, (c) minimizes susceptibility to errors resulting from external
vibration sources, is taught in U.S. Pat. Nos. Re 31450 entitled
"Method and Structure for Flow Measurement" and issued
Nov. 29 1983; U.S. Pat. No. 4422338 entitled "Method and
Apparatus for Mass Flow Measurement" and issued Dec. 27 1983;
and U.S. Pat. No. 4491025 entitled "Parallel Path Coriolis
Mass Flow Rate Meter" and issued Jan. 1 1985. The mechanical
arrangements disclosed in these patents incorporate flow conduits
having no pressure sensitive joints or sections, such as bellows
or other pressure deformable portions. These flow conduits are solidly
mounted in a cantilevered fashion from their inlet and outlet ports.
For example, in U.S. Pat. No. 4491025 but not limited thereto,
the flow conduits can be welded or brazed to a support, so they
can be oscillated in springlike fashion about axes which are located
near the solidly mounted sections of the flow conduits. Additionally
these solidly mounted flow conduits are preferably designed so they
have resonant frequencies about the axes located near the mountings
which are lower than the resonant frequencies about the axes relative
to which Coriolis forces act. By so designing the flow conduits,
a mechanical situation arises whereby, under flow conditions, the
forces opposing generated Coriolis forces are essentially linear
spring forces. The Coriolis forces, opposed by essentially linear
spring forces, deflect the flow conduit containing flowing fluid
about axes located between and essentially equidistant from the
portions of the flow conduits in which Coriolis forces are generated.
The magnitudes of the deflections are a function of the magnitudes
of the generated Coriolis forces and the linear spring forces opposing
the generated Coriolis forces.
As stated above, the flow conduits, in addition to being deflected
by the Coriolis forces, are also driven to oscillate. Accordingly,
under flow conditions, one portion of each flow conduit on which
the Coriolis forces act will be deflected so as to move ahead, in
the direction in which the flow conduit is moving, of the other
portion of the flow conduit on which Coriolis forces are acting.
The time or phase relationship between when the first portion of
the oscillating flow conduit deflected by Coriolis forces has passed
a preselected point on the path of oscillation for the flow conduit
to the instant when the second portion passes a corresponding preselected
point is a function of the mass flow rate of the fluid passing through
the flow conduit.
A number of other Coriolis mass flow meters have been developed
which are governed by similar equations of motion. Among these are
specific embodiments disclosed in U.S. Pat. No. 4127028 (Cox et
al., 1978), U.S. Pat. No. 4559833 (Sipin, 1985), U.S. Pat. No.
4622858 (Mizerak, 1986), PCT Application No. PCT/US85/01046 (Dahlin,
filed 1985) and U.S. Pat. No. 4660421 (Dahlin, et al., 1987).
Prior art mass flow meters have been limited in their accuracy
by the method of processing motion sensor signals and the relationship
used in such processing. This limitation becomes important for phase
angle differences above the range of 3 to 4 degrees (0.0524 to 0.0698
radians). The three Smith patents named above employ a linear relationship
between the time difference of two portions of the flow conduit
passing through a preselected point and mass flow rate. This time
difference measurement may be made by optical sensors as specifically
exemplified in U.S. Pat. No. Re 31450 electromagnetic velocity
sensors as specifically exemplified in U.S. Pat. Nos. 4422338
and 4491025 or position or acceleration sensors as also disclosed
in U.S. Pat. No. 4422338.
A double flow conduit embodiment with sensors for making the preferred
time measurements is described in U.S. Pat. No. 4491025. The double
flow conduit embodiment described in U.S. Pat. No. 4491025 provides
a Coriolis mass flow rate meter structure which is operated in a
tuning forklike manner as is also described in U.S. Pat. No. Re
31450. The tuning fork operation contributes to minimizing effects
of external vibration forces. Minimizing effects of external vibration
forces is important because these forces can induce errors in the
required time measurement.
The approach which has been taken in the prior art has been to
assume that flow conduits exhibit symmetric behavior in the deformations
about the Coriolis axis transverse to the oscillation axis. This
is because an assumed absence of damping permits each portion of
the flow conduit to respond essentially identically to the forces
acting on the tube as would portions located symmetrically about
the transverse axis. As one skilled in the art will recognize from
the disclosures herein, a general solution to phase angle difference
equations, assuming no asymmetric behavior of the flow conduit,
yields the following expression for mass flow: ##EQU1## where the
meaning of the variables and parameters for this and all subsequent
equations is given in Table 1 herein.
The mass flow rate measurement scheme embodied in eq. (1) is identical
with previously published schemes, e.g. U.S. Pat. No. Re 31450.
The correspondence can be shown by eliminating some of the additional
considerations of this analysis.
Assume the phase angle difference .delta..theta. is sufficiently
small that
As shown herein, the phase angle difference and time delay .delta.t
are related by
where
Using eqs. (2) and (3) together with eq. (4) in eq. (1) gives ##EQU2##
Thus, when the phase angle difference is small, which is typically
true for phase angle differences below 3 to 4 degrees, the general
solution to the phase angle difference equations reduces to a time
delay mass flow measurement scheme. The simple change of variables
from time to phase angle (eq. 50 herein) does not alter the physics
of the mass flow measurement scheme. Taking account of the nonlinear
phase angle difference relationship for phase angle differences
above 3 to 4 degrees, does, however, increase the range of mass
flow rate measurement accuracy.
If the viscous damping associated with the Coriolis motions of
the flow tube is sufficiently small, it can be neglected. The critical
damping ratio .zeta..sub.c is set to zero in eq. (6) resulting in
As before, the first term in the parentheses is a constant, as
is the spring constant k.sub.c. Equation (8) states that when damping
can be neglected, the mass flow rate scheme developed from the phase
angle difference equations is identical to a constant times the
frequency response term times the time delay.
It is assumed that the combined inertia of the flow tube, appendages,
and fluid is sufficiently low that it can be neglected. As one skilled
in the art will recognize, as the mass becomes smaller, the natural
frequency .omega..sub.c increases. In the limit, as m goes to zero,
.omega..sub.c tends to infinity, such that the frequency response
term in eq. (8) approaches unity. Thus, when inertia is neglected,
the mass flow rate measurement scheme based on the phase angle difference
equations reduces to ##EQU4## Equation (9) is identical in form
to the mass flow rate equation of U.S. Pat. No. Re 31450. This
equation can be presented as a phase angle difference equation by
substituting the expression ##EQU5##
The nonlinear relationship employed in the Dahlin PCT Appln. No.
PCT/US85/01046 is reached by employing mutually contradictory assumptions
of the existence of damping and the lack of damping to develop the
underlying zero crossing equations and their solutions. In addition,
asymmetric effects are ignored. This results in two equations in
the solution which are reduced to a single solution by assuming
zero damping. The Dahlin nonlinear relationship can be reached
by one skilled in the art by employing the following phase angle
difference equations based on a lumped parameter model shown in
FIG. 2 herein: ##EQU6##
If electronics are coupled to the outputs of velocity sensors such
that the phase angle difference .delta..theta. is measured, then
at a zero crossing either eq. (11) or (12) presented herein can
be solved for the amplitude function H.sub.c. Wellknown mathematical
identities can be used to expand eq. (12), and then dividing by
cos (.delta..theta./2), which never is zero for realistic .delta..theta.,
the following expression for H.sub.c results: ##EQU7## However,
H.sub.c is related to the mass flow rate through eq. (37), presented
herein. Setting eq. (13) equal to eq. (37) and solving for the mass
flow rate yields ##EQU8## An alternate form for eq. (14) is obtained
by setting ##EQU9## Substituting eq. (15) into eq. (14) and using
wellknown mathematical identities yields ##EQU10## Equation (16)
is the mass flow measurement scheme given by PCT Appln. No. PCT/US85/01046.
Note that the first term in parentheses is a nondimensional constant,
since it is assumed that the ratio f.sub.b /u.sub.d is constant
for any particular driver design. Also note that eq. (16) shows
how the measurement of mass flow rate depends on the frequency response
of the flow tube.
Equation (16) is not the only mass flow measurement scheme that
can be generated from the phase angle difference equations (11)
and (12). Equation (16) resulted from manipulations on eq. (12)
only. The same procedures applied to eq. 11) yields. ##EQU11## Note
that eq. (17) is identical with eq. (16) except that the sign of
the last term in the denominator has changed. This is a contradiction,
and at least one of the mass flow measurement schemes embodied in
these equations (16) and (17) must be incorrect. The contradiction
is removed for the case of zero damping. Equation (36) indicates
that .phi..sub.c is zero when .zeta..sub.c is zero. By eq. (15),
the phase angle becomes .pi./2. Making this substitution in eqs.
(16) and (17) makes them identical.
If the simplifications are made in eqs. (16) and (17) that were
made to the general solution, eq. (1), then the result is the linearized
formula of eq. (9).
The source of the error with the PCT Appln. No. PCT/US85/01046
approach is that the solution is not unique. There the mass flow
equation is contradicted by a companion equation developed by identical
methods. As explained above, both equations cannot be correct in
the general case. For the special case of no damping in the Coriolis
motions, the two equations (16) and (17) become identical, but this
violates the assumption of nonzero damping used throughout PCT
Appln. No. PCT/US85/01046. The new solution of the phase angle difference
equations given by equation (71) disclosed herein overcomes this
contradiction.
Prior mass flow measurement schemes have employed assumptions which
are limited in their appropriateness to the materials of which the
flow conduit is made. Both U.S. Pat. No. Re 31450 and PCT Appln.
No. PCT/US85/01046 ignore the asymmetry considerations disclosed
herein, to assume that the time or phase angle difference between
the motion of one portion of the flow conduit and the locus of the
intersection of the conduit with a plane bisecting the conduit into
two equal portions is equal to the time or phase angle difference
between the motion of an opposed symmetricallylocated portion of
the flow conduit and the same locus. The damping of the mechanical
system influences the degree of asymmetry, however. For systems
having little damping, such as when the flow tubes are made from
metal tubing, the amount of asymmetry is too small to measure (on
the order of a few millionths of a degree). For these cases, it
is appropriate to assume symmetry of the motion sensor signals with
respect to the driver signal. If nonmetallic tubes are used, however,
it is expected that the inherent damping of such materials would
create an amount of asymmetry large enough to measure, and hence
of sufficient magnitude for correction in the mass flow rate determination.
Suitable nonmetallic materials include, but are by no means limited
to, hightemperature glass, such as PYREX, manufactured by the Corning
Glass Company, hightemperature ceramics, or fiberreinforced high
mechanical strength, temperatureresistant plastics. Additionally,
the flowing fluids through flow conduits produce viscous damping
which can be accounted for by appropriate modeling, as described
herein. This modeling, which employs lumped parameters, accounts
for both the damping due to the flow conduit properties and the
damping due to the flow of fluids through the conduits.
Prior art mass flow devices have permitted the determination of
fluid density. See, for example, U.S. Pat. Nos. Re 31450 and 4491009
Which disclose such a density determination. Specific density meter
circuitry embodiments are disclosed by Ruesch in U.S. patent application
Nos. 916973 and 916780 both filed Oct. 9 1986. These circuits
can be employed with the embodiments disclosed herein to determine
density as well as mass flow rate.
DISCLOSURE OF INVENTION
Several embodiments of Coriolis mass flow meters are disclosed
which incorporate logic to process motion sensor signals to determine
the frequency at which the flow conduit is driven, a frequency response
function, a halfangle tangent function, and a temperaturecompensated
spring constant. These processed signals are further processed with
calibration constants to provide a measurement of mass flow rate.
In a preferred embodiment, signals are processed from two motion
sensors which are positioned to monitor two points on a flow conduit
essentially equidistant from the center or midpoint of the conduit,
which is assumed to be symmetrical The signal from one motion sensor
is processed by a logic element to provide a measurement of the
drive frequency at which the flow conduit is oscillated. The signal
is further processed by a logic element to provide a frequency response
function. The signal from a second motion sensor is compared with
the signal from the first motion sensor in a phase angle difference
logic element to produce a phase angle difference measurement. The
phase angle difference of the signals at their respective zerocrossings
is used. This phase angle difference measurement is then processed
further by a halfangle tangent function logic element to produce
a halfangle tangent function. A signal from a temperature sensor
which monitors flow conduit temperature is processed by a temperature
compensation logic element to provide a temperaturecompensated
flow conduit spring constant. The resultant frequency response function,
halfangle tangent function and temperaturecompensated spring constant
are then combined with various known calibration constants in a
mathematics logic element. This mathematics logic element combines
the various processed signals to produce a measurement of mass flow
rate.
For the case in which there is significant asymmetric behavior
of the flow conduit under flow conditions, two other preferred mass
flow meter embodiments are disclosed which incorporate logic elements
which similarly process motion sensor signals for the flow conduit
motion and driver motion to produce a conduit drive frequency, a
frequency response function, a halfangle tangent function and a
temperaturecompensated flow conduit spring constant along with
calibration constants to produce mass flow rate.
Finally, three embodiments which account for both asymmetric behavior
and viscous damping are disclosed. These embodiments incorporate
logic elements which similarly process motion sensor signals for
the flow conduit and driver motion to produce a conduit drive frequency,
a frequency response function, a temperaturecompensated flow conduit
spring constant, calibration constants, along with trigonometric
logic elements which produce either a halfangle tangent function
or another trigonometric function. Viscous damping is accounted
for by a damping measurement logic element. One of these embodiments
employs the use of two flow conduit motion sensor signals.
Several techniques are known in the prior art for making damping
measurements. For many applications the damping is constant, and
very accurate modal analysis techniques can be employed. For example,
the Modal 3.0 software produced by Structural Measurement Systems,
Inc. of San Jose, Calif. has been used to measure the damping ratios
of mass flow meters.
When the damping is not constant, a near continuous signal must
be supplied. One technique is to monitor the power delivered by
the drive system. This quantity is directly proportional to the
damping in the system. Small changes in the drive frequency could
also be used to make damping measurements. Another approach is to
use standard decay techniques, such as described in the "Shock
and Vibration Handbook", second edition, 1976 published by
McGraw Hill Book Company. The measurement could be made on an intermittent
basis so as not to interfere with the mass flow measurement.
The advantages of the embodiments disclosed herein are at least
fourfold: (1) accuracy is preserved at high flow rates; (2) account
can be taken for asymmetries between the motion sensor signals;
(3) accurate mass flow measurement is permitted for higher signaltonoise
ratios; and (4) viscous damping is accounted for in three of the
embodiments.
BRIEF DESCRIPTION OF DRAWINGS
FIG. 1 is a perspective view of an oscillating parallel path flow
meter with both motion and temperature sensors mounted on oscillating
tubes;
FIG. 2 is a diagram which illustrates idealized geometry of a lumped
parameter, single tube mass flow meter;
FIG. 3 is a diagram which illustrates the geometric relationship
of the superposition of bending and Coriolis motions of a lumped
parameter mass flow meter;
FIG. 4 is a graphical presentation of sinusoidal waveforms which
illustrates zero crossings of motion sensors for two lumped masses
and a driver;
FIG. 5 is a graphical presentation of a representative asymmetric
zero crossing for motion sensors for two lumped masses and a driver;
FIG. 6 is a block diagram of signal processing logic for an existing
linearized signal processing approach incorporated in a mass flow
meter to measure mass flow rate;
FIG. 7 is a block diagram of signal processing logic for an embodiment
of the invention for a symmetric nonlinear signal processing approach
incorporated in a mass flow meter to measure mass flow rate;
FIG. 8 is a block diagram of signal processing logic for an embodiment
of the invention for an asymmetric nonlinear signal processing approach
incorporated in a mass flow meter to measure mass flow rate;
FIG. 9 is a block diagram of signal processing logic for halfangle
polling, used in logic element 212 of FIG. 8; and
FIG. 10 is a block diagram of signal processing logic for an embodiment
of the invention of an asymmetric nonlinear signal processing approach
incorporated in a mass flow meter using one motion sensor to measure
mass flow rate.
FIG. 11 is a block diagram of signal processing logic for an embodiment
of the invention for a signal processing approach incorporated in
a mass flow meter to measure mass flow rate using one flow conduit
motion sensor which accounts for asymmetries and viscous damping.
FIG. 12 is a block diagram of signal processing logic for an embodiment
of the invention for a signal processing approach incorporated in
a mass flow meter to measure mass flow rate using one flow conduit
motion sensor which accounts for asymmetries and viscous damping.
FIG. 13 is a block diagram of signal processing logic for an embodiment
of the invention for a signal processing approach incorporated in
a mass flow meter to measure mass flow rate using two flow conduit
motion sensors which account for asymmetries and viscous damping.
BEST MODE FOR CARRYING OUT THE INVENTION
A. Signal Processing Application
A Coriolis mass flow rate meter, as generally designated by numeral
10 for which the present invention can be used, is shown in FIG.
1. The flow meter 10 incorporates twin flow conduits 12. The flow
conduits 12 are solidly mounted to the supports 22 with the conduits
being free of pressuresensitive joints. Other arrangements utilizing
a single flow conduit and a spring arm, or a single light weight
flow conduit solidly mounted to a relatively massive support can
also be used with the present invention. In addition, the invention
herein is applicable to various shapes and configurations in which
Coriolis force couples are generated by fluids flowing through solidly
mounted oscillated flow conduits. This includes various full and
partial loop shapes, Sshapes, Bshapes and straight tubes. As one
skilled in the art will recognize, tube shape is not a limitation
of the current invention. The flow meter 10 in addition to the
flow conduits 12 includes a driver 14 such as a permanent magnet
and wire coil combination as is known in the art, to oscillate the
flow conduits 12 about axes BB essentially 180 degrees out of
phase with one another as the prongs of a tuning fork. The flow
meter 10 further includes sensors 16 mounted on the flow conduits
12. The sensors 16 shown in FIG. 1 are velocity sensors which continuously
provide signals linearly representative of the actual movement of
the flow conduits 12 over their entire path of motion. When the
flow conduits 12 are oscillating and fluid is flowing through them,
the flow
conduits 12 are deflected about axes AA by Coriolis forces. The
effects of these deflections are monitored by the sensors 16. A
detailed description of the mechanical operation of flow meter 10
is set forth in the aforementioned U.S. Pat. Nos.: U.S. Pat. No.
Re 31450 4422338 and 4491025.
Flow conduit temperature is monitored by temperature sensor 20
which is mounted on the flow conduit where flow is incoming. Temperature
sensor 20 may be a resistance temperature device (RTD), a thermocouple
or other temperature measurement device known in the art.
The sensors 16 are electromagnetic velocity sensors. Each sensor,
16 consists of a permanent magnet and a coil, with the coil designed
so as always to be moved within the essentially uniform magnetic
field of the magnet. Descriptions of the operation of sensors 16
for single and twin conduit Coriolis mass flow rate meters are set
forth in the aforementioned U.S. Pat. Nos.: 4422338 and 4491025.
The invention disclosed herein relates to the processing of motion
sensor signals and a driver motion signal to determine accurately
the mass flow rate of fluids passing through oscillated conduits.
The flow conduits can be curved or straight and/or single or multiple.
Motion sensor signals and driver motion signals are employed in
various combinations to produce phase angle differences between
the signals. The key feature is that the Coriolis forces produced
by the combination of oscillation and flow yield a net phase angle
difference which is related to mass flow rate.
The motion sensing transducers may be position or displacement,
velocity or acceleration. In what follows, however, velocity sensors
are discussed. This does not limit the use of other motion sensors.
The primary requirements for the motion sensors are that their output
signals have different phases as a result of the deformation of
the flow conduit due to Coriolis force and that their output signals
be linearly representative of the whole range of adjacent conduit
motion. The driver yields a periodic, preferably sinusoidal signal.
Other periodic wave forms can be used, but they have an increased
harmonic content.
THEORY OF OPERATION
FIG. 2 depicts a single loop flow meter idealized as a lumped parameter
mass flow meter. The flow meter is modeled as two uncoupled masses
with lumped elastic and damping characteristics. The model is based
on an approach developed by Dahlin (1985 "Apparatus for Mass
Flow Rate and Density Measurement," Patent Cooperation Treaty
Application Number PCT/US85/01046). The flow tube is fixed at each
end along the e.sub.1 axis of an inertial reference frame having
mutually orthogonal axes e.sub.1 e.sub.2 and e.sub.3. The flow
tube is symmetric about the e.sub.2 axis and lies entirely in the
e.sub.1  e.sub.2 plane. The elastic, damping and inertial properties
of the flow tube are lumped at points on both the right and left
sides of the flow tube as shown. Hence, in the discussion that follows,
the subscripts "r" and "l" will be used to designate
right and left. Of course, other orientations of the flow conduit
are contemplated. For example, a Utube is obtained from the geometry
of FIG. 2 by straightening the inlet and outlet flow tubes such
that they are tangent to the upper curve. A sinusoidal drive force,
f.sub.d (t), is applied at the top of the loop in transverse direction
e.sub.3. Coriolis forces, f.sub.c (t), are illustrated acting on
the two lumped mass points. The analysis as presented herein is
directed to curved tubes for the sake of simplicity. However, the
equations of motion are applicable to any flow conduit configuration
for which Coriolis force couples are generated by flow through an
oscillated conduit. FIG. 2 illustrates a curved conduit. As one
skilled in the art will recognize, because the equations of motion
are the same, the model derived is applicable to straight tubes
as well as curved tubes.
FIG. 3 illustrates the characteristic motion of the idealized system
of FIG. 2. As shown in FIG. 3 the flow conduit undergoes a displacement,
u, along the e.sub.3 axis which has two components. The displacement
component u.sub.b is the bending displacement along e.sub.3 that
results from the drive force f.sub.d (t). The displacement component
u.sub.c is the Coriolis displacement along e.sub.3. The total displacement
is given by superposition of the solutions to the equations of motion
for bending and Coriolis forces.
In the discussion which follows, the symbols used and their definitions
are given in Table 1.
TABLE 1 ______________________________________ System Parameters
and Variables English SI Symbol Description Units Units ______________________________________
.m.sub.o mass flow rate lb.sub.m /min kg/min m mass of fluidfilled
flow lb.sub.m kg tube f.sub.b drive force lb.sub.f N f.sub.c Coriolis
force lb.sub.f N f.sub.o viscous damping force lb.sub.f N c.sub.b
bending damping con lb.sub.f  sec/in Nsec/cm stant c.sub.c Coriolis
damping con lb.sub.i  sec/in Nsec/cm stant u displacement in
cm u.sub.b bending displacement in cm u.sub.c Coriolis displacement
in cm u.sub.d drive displacement in cm u.sub.l left mass displacement
in cm u.sub.r right mass displacement in cm .u.sub.b bending velocity
in/sec cm/sec .u.sub.c Coriolis velocity in/sec cm/sec .u.sub.d
drive velocity in/sec cm/sec .u.sub.l left mass velocity in/sec
cm/sec .u.sub.r right mass velocity in/sec cm/sec u.sub.b bending
acceleration in/sec.sup.2 cm/sec.sup.2 u.sub.c Coriolis acceleration
in/sec.sup.2 cm/sec.sup.2 .omega. angular velocity hz hz .omega..sub.b
natural frequency hz hz (bending) .omega..sub.c natural frequency
hz hz (Coriolis) .omega..sub.d drive frequency hz hz .theta. phase
angle radians radians .theta..sub.r phase angle (right mass) radians
radians .theta..sub.l phase angle (left mass) radians radians .delta..theta.
phase angle difference radians radians .delta..theta..sub.r phase
angle difference radians radians (right mass) .delta..theta..sub.l
phase angle difference radians radians (left mass) t time sec sec
.delta..theta..sup.o phase angle difference radians radians (undamped)
t.sub.r time (right mass) sec sec t.sub.l time (left mass) sec sec
.delta.t time delay sec sec .alpha. rotation radians radians r moment
arm in cm ..alpha. rotational velocity radians/sec radians/sec ...alpha.
rotational acceleration radians/sec.sup.2 radians/sec.sup.2 M.sub.e
elastic restoring moment inlb.sub.f Ncm M.sub.v viscous damping
moment inlb.sub.f Ncm I.sub.c moment of inertia lb.sub.m  in.sup.2
kgcm.sup.2 .phi. phase angle function radians radians .phi..sub.b
bending phase angle radians radians function .phi..sub.c Coriolis
phase angle radians radians function H.sub.b bending amplitude in
cm H.sub.c Coriolis amplitude in cm .zeta..sub.b bending damping
ratio % % .zeta. .sub.c Coriolis damping ratio % % k.sub.b bending
elastic spring lb.sub.f /in N/cm constant k.sub.c Coriolis elastic
spring lb.sub.f /in N/cm constant .epsilon. phase angle lag radians
radians .theta..sub.d driver phase angle radians radians .rho. fluid
density lb.sub.m /ft.sup.3 kg/m.sup.3 P oscillation period sec sec
m.sub.T conduit and appendage lb.sub.m kg mass m.sub.F fluid mass
lb.sub.m kg V.sub.o conduit volume cu. ft. m.sup.3 ______________________________________
The equation of motion for the bending motion due to the driver
force is ##EQU12##
The steadystate solution for the idealized bending motion of the
flow tube is
where H.sub.b is the amplitude function and .phi..sub.b is the
phase angle function. These functions are given by ##EQU13##
By similar means, a second equation of motion is obtained for the
case when fluid of constant density flows through the tube at constant
speed. FIG. 3 shows the geometry of the situation. At time t, the
idealized masses have been displaced an amount u.sub.b (t) due to
the driver. In addition, the flowing fluid gives rise to a Coriolis
force f.sub.c (t) acting on each mass of the form
where m.sub.o is the constant mass flow rate, and u.sub.d is the
amplitude of the driver velocity, which is also assumed constant.
Due to the curvature of the flow tube, the Coriolis force on the
left mass is 180 degrees out of phase with the Coriolis force acting
on the right mass. The net result is that the flowing fluid produces
a pure moment causing the rotation .alpha. of the masses about the
e.sub.2 axis which is out of the e.sub.1 e.sub.3 plane as shown
in FIG. 3.
Since all displacements are small, it is assumed that the resulting
torsional motions induced by the flowing fluid can be superposed
on the bending motions by neglecting any coupling between the degrees
of freedom. The equation of motion for the uncoupled torsional motions
is obtained as follows. An elastic restoring moment M.sub.e is assumed
to act on each mass of the form
where k.sub.c is the elastic spring constant for this degree of
freedom and r is the distance between the e.sub.2 axis and the mass
point. A viscous damping moment is also assumed of the form
where c.sub.c is the viscous damping constant. Then the rotational
form of Newton's Second Law of Motion gives for the motion of either
mass
where I.sub.c is the moment of inertia of one mass about the e.sub.2
axis, i.e.,
From the geometry of FIG. 3 the component of displacement in the
e.sub.3 direction due to the rotation .alpha. is
which for small angles reduces to
Substituting eqs. (24), (28) and (30) into eq. (27) and dividing
by r produces the equation of motion for the Coriolis displacements
or in canonical form ##EQU14## where the natural frequency .omega..sub.c
and critical damping ratio .zeta..sub.c of the Coriolis motions
are given by ##EQU15## The solution to the equation of motion for
the Coriolis displacements is
where the amplitude function H.sub.c and phase angle function .phi..sub.c
are given by ##EQU16##
The total motion of the two mass system is composed of the Coriolis
motions superimposed on the bending motions as shown in FIG. 3.
The displacements of the right mass u.sub.r and the left mass u.sub.l
are given by
This completes the description of the idealized lumped parameter
model of the flow tube. Given the parameters of the system
.omega..sub.b, .zeta..sub.b, k.sub.b . . . bending motions
.omega..sub.c, .zeta..sub.c, k.sub.e . . . Coriolis motions
.omega..sub.d, u.sub.d, f.sub.b . . . driver parameters
m.sub.o . . . mass flow rate
the complete steady state motion of the flow tube can be calculated.
FIG. 4 illustrates zero crossings of sine curves generated by velocity
transducers, which can be used to derive mass flow rate as explained
below. The two solid sinusoids represent the velocities of the two
sides of the flow conduit. The broken sinusoid represents the velocity
of the driver element.
The analysis above permits the development of various schemes for
measuring the steady flow of mass through the flow tube. The fundamental
concept is, that by making measurements of the motions of the flow
tube, the mass flow rate can be determined. Only velocity measurements
will be considered here, but similar schemes based on the use of
displacement or acceleration transducers can be developed. Similarly,
only schemes based on time delays or phase angle differences will
be discussed. Only zero crossings will be used, but it is understood
that by similar analysis arbitrary level crossings could be used.
It is assumed that the idealized flow tube in FIG. 2 is equipped
with velocity transducers such that continuous time measurements
are made of the velocity of the right and left masses. Using eqs.
(21), (35), (38) and (39), and differentiating with respect to time,
yields these velocities:
Further, assume that the system is driven at the resonance of the
bending motions, i.e. set
From eqs. (19), (20), (22), and (23) the amplitude function H.sub.b
and phase angle .phi..sub.b reduce to ##EQU17## Substituting eqs.
(43) and (44) into the velocity equations and using wellknown mathematical
identities gives ##EQU18##
FIG. 4 shows a plot of these velocities as functions of time, where
it is assumed that H.sub.c is small in comparison to H.sub.b, and
that .phi..sub.c is a small angle. The two sine waves are offset
such that u.sub.l crosses zero before u.sub.r. The times when these
zero crossings occur are t.sub.l and t.sub.r, and they can be found
by setting eqs. (45) and (46) to zero: ##EQU19## Of course, if level
crossings were used, the right hand side of equations 30 and 31
would be set equal to the level value rather than zero and the analysis
would proceed.
At this stage, it is assumed that all the system parameters are
known, such that eq. (47) has only the one unknown quantity t.sub.r.
Similarly eq. (48) has only the one unknown t.sub.l. If each of
these equations are solved separately, t.sub.r and t.sub.l are obtained,
i.e. the time when the velocity of the respective mass crosses the
zero level.
There are several methods for solving eqs. (47) and (48). One approach
is to make a simple change of variables as follows. The general
relationship between phase angle and time is
where .theta. is phase angle in radians, and .omega. is an arbitrary
angular frequency. For the steady state motions of the flow tube
under consideration, all motions occur at the frequency of the driver
.omega..sub.d. Thus, the proper change of variables for this problem
is
The phase angle .theta. is measured with respect to time zero,
and increases as time increases. The previously defined phase angles
.phi..sub.b and .phi..sub.c are of different type in that they are
restricted to the interval (.pi., .pi.).
From the sinusoidal driver force, the driver phase has zero crossing
at
where n is an integer in the set (0 1 2 . . . ). The even values
of n correspond to zero crossings with positive slope, and the odd
values of n correspond to zero crossings with negative slope. This
is plotted in FIG. 4. Now define the difference in phase between
the right and left velocity traces as .delta..theta., and consider
only the zero crossings with positive slope. Then, assuming the
symmetry shown in FIG. 4 the zero crossings of the two velocity
traces occur at
Using eqs. (50), (52) and (53) and wellknown mathematical identities,
the time equations (47) and (48) are transformed to the following
phase angle difference equations: ##EQU20##
A general solution to the phase angle difference equations (54)
and (55) can be obtained. Restated they are ##EQU21## There are
two equations in the one unknown quantity H.sub.c. Adding the squares
of eqs. (56) and (57) yields ##EQU22## Employing wellknown mathematical
identities and taking the negative square root yields ##EQU23##
Setting eq. (59) equal to eq. (37) and solving for the mass flow
rate yields ##EQU24## Equation (60) is the general solution to the
phase angle difference equations.
The formula in eq. (60) is the general solution for the prior art
formulas discussed previously, all of which are based on an assumption
of no asymmetric behavior. FIG. 5 is a detailed view of the zero
crossings. Instead of the zero crossing times t.sub.r and t.sub.l
being symmetric with respect to the driver, both times lag the symmetric
position by the same small amount. For the asymmetric case, eqs.
(52) and (53) are replaced by
Proceeding as before, eqs. (61) and (62) are substituted into the
zero crossing equations (47) and (48), and eq. (50) is used to make
the change of variables. These equations are then solved separately
for the phase angle difference .delta..theta..sub.r and .delta..theta..sub.l
: ##EQU25##
Consider the special case when the Coriolis motions are undamped.
The phase angle .phi..sub.c is zero, and eqs. (63) and (64) reduce
to the identical expression ##EQU26## But from FIG. 5 this is just
the symmetric case previously discussed, i.e. ##EQU27## For nonzero
damping, define ##EQU28## where .epsilon. is the nonzero lag shown
in FIG. 5. Then substituting eq. (65) into eqs. (63) and (64), and
employing wellknown mathematical identities yields the following
expressions for the zero crossing asymmetry: ##EQU29## As is readily
apparent, eq. 69 can be solved for ##EQU30## to yield ##EQU31##
Similarly, eq. 70 can be solved for tan to yield ##EQU32## If the
total phase angle difference .delta..theta..degree.is measured,
then the right side of eq. (69) is known, and it can be set equal
to the right hand side of eq. (63). This results in a simple expression
for the amplitude function H.sub.c, which again can be used in eq.
(37). Solving for the mass flow rate yields ##EQU33## This is the
correct solution to the zero crossing equations. The same manipulations
on eqs. (64) and (70) yield identical results. It should be noted
that had level crossing equations been employed, the same result
would be obtained as for zero crossing equations.
For the case of nonmetallic conduits or for the case of damping
due to fluid flow, such as that due to air entrained in liquid,
it can be demonstrated by one skilled in the art that equations
(6971) are modified by a cos .phi..sub.c term as follows: ##EQU34##
For all cases wherein damping is unimportant, cos .phi..sub.c is
essentially unity and hence does not need to be accounted for. For
materials, such as, but not limited to, nonmetallic tubes, wherein
damping is important, or when there is two phase flow which results
in damping, account should be taken of cos .phi..sub.c. The derivation
of Equations 7274 is as follows. For zero damping, .zeta..sub.c
and .phi..sub.c are zero. Thus, the amplitude function (eq. 37)
becomes ##EQU35## where the zero superscript indicates zero damping.
From equations (63) and (64) ##EQU36## i.e., there is no asymmetry
and
Thus eq. (76) can be rewritten ##EQU37## For a lightly damped
system with .omega..sub.d not near .omega..sub.c, the frequency
response amplitude function has little dependence on .zeta..sub.c.
Thus, the following approximation is made.
and eq. (78) becomes ##EQU38## Substituting eq. (80) into eq. (63)
with some manipulations produces eq. (69): ##EQU39## The same approximation
is used to derive eq. (70). Now suppose the flow meter is not lightly
damped such that eq. (79) is not valid. Equation (37) can be rewritten
##EQU40## where use has been made of eq. (78). Multiplying eq. (81)
by (H.sup.o.sub.c).sup.2 and simplifying yields ##EQU41## But making
use of eq. (36), eq. (82) becomes ##EQU42## Substituting eq. (84)
into equation (78) yields ##EQU43## Finally, substituting eq. (85)
into eq. (63) and simplifying yields ##EQU44## which is eq. (72).
Equation (73) is similarly derived by substituting eq. (85) into
eq. (64). To derive eq. (74), set eq. (63) equal to eq. (72) and
solve for the frequency response amplitude function. The result
is ##EQU45## Now set eq. (86) equal to eq. (37) and solve for the
mass flow rate m.sub.o. This yields eq. (74).
Some additional mass flow measurement schemes can be developed
from this analysis. The phase angle difference equation (54) can
be solved for H.sub.c : ##EQU46## Then setting eq. (87) equal to
eq. (37) and solving for the mass flow rate yields ##EQU47## The
same equation can be derived from eq. (55), but the sign of .delta..theta..sub.l
must be taken into account. Equation (88) can be used to determine
mass flow rate by direct measurement of either .delta..theta..sub.r
or .delta..theta..sub.l.
As asymmetric correction scheme can be derived from the previous
analysis. Equations (72) can be solved for ##EQU48## i.e., ##EQU49##
Similarly from eq. (73) ##EQU50## Either eq. (89) or eq. (90) can
be used to correct for asymmetry, when the flow meter is not lightly
damped. To determine the mass flow rate, the value of .delta..theta..degree.given
by these equation is used in eq. (74).
Use of dual sensors requires some additional analysis. The trigonometric
identity ##EQU51## can be used to combine eqs. (72) and (73). After
simplification the result is ##EQU52## Equation (92) can be solved
for ##EQU53## Let ##EQU54## Equation (93) can be substituted into
eq. (92) and expanded to yield
Using the quadratic formula and taking the positive root gives
##EQU55## This is the dual sensor asymmetric compensation scheme
for nonlightly damped flow meters. The value of resulting from
eq. (95) is used in eq. (74).
C. Embodiments of Invention
Several embodiments of mass flow meters which measure and process
motion sensor signals are disclosed which provide more accurate
determination of mass flow rates in Coriolis mass flow meters than
have heretofore been achieved. It is envisioned that the signal
processing elements of each of the embodiments may be incorporated
into a microchip or other integrated circuit device. Alternatively,
the signal processing portions of each embodiment may be incorporated
into discrete elements which are connected to form a signal processing
system.
Three embodiments disclosed herein relate to detecting motion sensor
signals and processing such signals according to equation 71: ##EQU56##
The terms of the equation can be grouped as follows:
______________________________________ m.sub.o is mass flow rate
##STR1## where K.sub.1 represents a group of calibration constants
f.sub.b is the force due to the driver acting on half of the flow
tube. c.sub.b is the combined damping constant for the flow tube
for bending motions u.sub.d is the driver velocity c.sub.b u.sub.d
can be recast as a viscous damping force f.sub.o resisting oscillations
of the flow tube Therefore ##STR2## where the respective forces
can be measured by techniques known to one skilled in the art, k.sub.c
is the elastic spring constant of the flow tube which interacts
against Coriolis forces. For known constant temperature operation,
this is a constant and would be a second calibration constant K.sub.2.
For unknown or varying temperature operation, this parameter must
be temperaturecompensated. .omega..sub.d is the frequency at which
the flow tube is driven to oscillate. ##STR3## Is a frequency response
function, where .omega..sub.d is the drive frequency, above; .omega..sub.c
is the natural frequency of the flow tube about the axis where the
Coriolis force acts; ##STR4## is the damping ratio of the flow tube
about the axis where the Coriolis force acts and can be determined
from techniques known to those skilled in the art; c.sub.c is the
combined damping constant for motion of the flow tube about the
axis where the Coriolis force acts, and can be determined by dividing
the viscous damping force acting about the Coriolis axis by the
velocity of the flow tube about the same axis, k.sub.c is the spring
constant above, m is the mass of the flow tube, flow tube appendages,
and the fluid filling the tube whose flow is to be measured. tan
(.delta..theta..degree./2) is a halfangle tangent function, where
.delta..theta..degree. is the phase angle difference between the
velocities mea sured by two motion sensors located essentially
equidistant from the midpoint of the flow tube. ______________________________________
The result of this equation is that mass flow rate can be determined
by determining the phase angle difference of motion signals from
two locations on a flow tube, measuring the drive frequency of the
flow tube, and compensating for temperature, as needed. The advantages
of this equation, which is employed in the following embodiments,
are that it allows for greater phase angles and increased accuracy
at full flow rates. Greater phase angles will improve the signaltonoise
ratio in noisy environments, such as gas flow environments.
An embodiment which accounts for asymmetry and viscous damping
employs equation (88): ##EQU57## The sign for this equation is as
shown if .delta..theta..sub.l >O. If .delta..theta..sub.1 <O.sub.l
then the sign is changed. The symbols used in the above equation
are the same as discussed for equation 71 with the exception that
______________________________________ ##STR5## is the trigonometric
function which replaces the halfangle tangent function of equation
(71) and .phi..sub.c is the Coriolis phase angle function. ______________________________________
The embodiment employing this equation is shown in FIG. 11.
Another embodiment which accounts for both asymmetry and viscous
damping employs equation (74): ##EQU58## where the symbols are as
before in equation 71 but where .phi..sub.c is the Coriolis phase
angle function and tan (.delta..theta..degree./2) is the halfangle
tangent function for undamped oscillations given by equation (89):
with the positive sign in the denominator being used for phase lag
and negative for phase lead. The embodiment employing this approach
is shown in FIG. 12.
A final embodiment, shown in FIG. 13 also employs the signal combination
approach of equation (74). This embodiment accounts for asymmetry
and viscous damping and employs the signals from two motion sensors.
The halfangle tangent function that is employed uses equation (95):
##EQU59## The symbols are the same as described above.
The mass flow rate equation (71) permits a compensation scheme
for the changes in the resonant frequency of oscillation and the
Coriolis natural frequency which result from changes in density
of the fluid flowing through the conduit. In the embodiments which
follow, temperaturecompensated spring constants about the oscillation
axis and the transverse Coriolis axis are obtained by measuring
the flow conduit temperature. As one skilled in the art will recognize,
the density of the fluid in the conduit can be determined according
to the formula: ##EQU60## A derivation and discussion of this formula
can be found in U.S. Pat. No. 4491009. Specific density meter
circuitry embodiments employing this technique are found in U.S.
patent application Ser. Nos. 916973 and 916780 both filed Oct.
9 1986.
Since the total mass of the fluidfilled conduit is given by /
where
m=total mass
m.sub.T =tube mass and appendages
m.sub.F =fluid mass
and
where V.sub.o is the tube volume,
then
Therefore, the Coriolis natural frequency is given by ##EQU61##
The Coriolis natural frequency can then be determined by employing
the temperaturecompensated spring constant about the Coriolis axis
transverse to the oscillation axis, the known tube mass and volume
and the measured density.
Equation (37) demonstrates how the amplitude of the Coriolis motions
H.sub.c, depends on the frequency response function. The frequency
response function contains terms which account for the density of
the process fluid. If the density changes, both the resonant frequency
of oscillation and the Coriolis natural frequency change. The feedback
system of the meter works to keep the system at resonance, i.e.
the drive frequency moves to the new bending frequency. However,
unless the ratio of the new frequencies is unchanged, the amplitude
of the Coriolis motions will change without any change in the mass
flow rate. This causes an error in the measured mass flow rate.
The embodiments herein contain a compensation scheme for this problem.
The drive frequency is continuously measured. From the drive frequency
it is possible to determine the density of the fluid in the conduit.
Equation (100) above is used to correct the Coriolis frequency.
Then according to eq. (71) with these frequencies plus the damping
ratio of the Coriolis motions, the frequency response effect is
cancelled. This correction is carried out in frequency response
function logic elements in the embodiments disclosed herein.
An existing linearized embodiment of the signal processing logic
for a mass flow meter is depicted in a block diagram in FIG. 6.
In this embodiment, the signals of two motion sensors are employed
to determine a phase angle difference which is processed further
with calibration constants to produce a mass flow measurement. Motion
sensors 1 and 2 are mounted to monitor motion of the flow conduit
at locations which are essentially equidistant from the midpoint
of the flow conduit, which is assumed symmetrical in shape. In this
embodiment, the motion sensors are assumed to be velocity sensors
which provide signals u.sub.1 and u.sub.2 which have associated
phase angles .theta..sub.1 and .theta..sub.2 respectively, but
could also be displacement or acceleration sensors. The signals
from motion sensors 1 and 2 are each input to phase detector element
4. Phase detector element 4 performs a comparison of the signals
from motion sensors 1 and 2 and processes the two signals to produce
a phase angle difference measurement between the two signals. Temperature
sensor 3 measures the temperature of the flow conduit. Its signal
is processed by temperaturecompensation logic element 5. The processed
temperature signal provides a temperaturecompensated elastic spring
constant k.sub.c (T) for the degree of freedom corresponding to
the Coriolis force. The phase difference measurement and temperaturecompensated
spring constant are then further processed by mathematics logic
element 6 where they are combined with calibration constants according
to the formula ##EQU62## As one skilled in the art will recognize,
this linear approach is limited by the inherent nonlinear relationship
between mass flow rate and phase angle.
In the first, second, and third embodiments disclosed below, damping
is either zero (undamped), or damping is very slight. In such a
case, the damping ratio, .zeta..sub.c, is assumed a constant, based
on characteristics of the flow conduits. The damping ratio is measured
beforehand for the flow meters and a constant is employed based
on that predetermined value.
FIG. 7 depicts a first preferred embodiment of the signal processing
logic for a mass flow meter which includes two motion sensors 101
and 102 positioned essentially equidistant from the center of the
flow conduit to monitor the motion of two sides of the oscillating
flow conduit. One of the motion sensor signals, here shown as a
velocity signal from sensor 101 is input to frequency measurement
element 104 to determine the frequency at which the oscillating
flow conduit is driven. This drive frequency is input to frequency
response function element 106. The drive frequency is also input
to mathematics logic element 109 for further processing. Temperature
sensor 103 is employed to provide a temperature of the oscillating
flow conduit. This temperature measurement is input to temperaturecompensation
logic element 108 which processes the signal to produce two temperaturecompensated
spring constant signals. One spring constant signal k.sub.b (T),
is for the oscillation degree of freedom. The other, k.sub.c (T),
is for the Coriolis degree of freedom. Both temperaturecompensated
spring constant signals are input to frequency response function
logic element 106 which combines these signals with the drive frequency
to determine a frequency response function which accounts for any
change in fluid density. The Coriolis spring constant is also input
to mathematics logic element 109. The signals from each of the motion
sensors u.sub.1 and u.sub.2 which have associated respective phase
angles .theta..sub.1 and .theta..sub.2 are input to phase detector
element 105 which determines the phase angle difference between
the two signals. The phase angle difference is then processed by
halfangle tangent function logic element 107. The result of this
processing is input to mathematics logic element 109 for further
processing. The measured drive frequency, processed frequency response
function, processed halfangle tangent function and temperaturecompensated
Coriolis spring constant are combined with calibration constants
in mathematics logic element 109 according to the formula of equation
(71) to produce the mass flow rate as an output. As one skilled
in the art will recognize, this embodiment takes into account inherent
nonlinear effects in measuring mass flow rates, which are not accounted
for in the existing linear embodiments. Such inherent nonlinear
effects are reflected in the relationship between mass flow rate
and phase angle in equation (71) and are embodied in the system
of FIG. 7.
A second preferred embodiment of the signal processing logic for
a mass flow meter, illustrated in FIG. 8 also processes velocity
signals u.sub.1 and u.sub.2 having respective associated phase angles
.theta..sub.1 and .theta..sub.2 from two motion sensors 201 and
202 again positioned essentially equidistant from the midpoint
of the flow conduits. The voltage signal V.sub.d, having associated
phase angle .theta..sub.d, controlling driver 204 is sensed and
is input to frequency measurement logic element 205. The frequency
measurement logic element 205 processes the driver signal and provides
as output the driver frequency. This output signal is input both
to mathematics logic element 216 and to a frequency response function
logic element 213. Temperature sensor 203 is employed to provide
a temperature of the oscillating flow conduit. This temperature
measurement is input to temperaturecompensation logic element 215
which processes the signal to produce two temperaturecompensated
spring constant signals. One spring constant signal k.sub.b (T),
is for the oscillation degree of freedom. The other, k.sub.c (T),
is for the Coriolis degree of freedom. Both temperaturecompensated
spring constant signals are input to frequency response function
logic element 213 which combines these signals with the drive frequency
to determine a frequency response function which accounts for any
change in fluid density. The Coriolis spring constant is also input
to mathematics logic element 216.
The signal from motion sensor 201 is input to phase detector element
206 along with the signal from driver 204. Phase detector element
206 compares the two signals and produces a phase angle difference
measurement between the two signals. This phase angle difference
is then input to asymmetry compensation logic element 209 which
processes the signal to produce an asymmetrycompensated phase angle
difference between motion sensor 201 and driver 204. Similarly,
the signals of motion sensor 202 and driver 204 are input to phase
detector element 208 where they are processed to produce a phase
angle difference, which is in turn processed by asymmetry compensation
logic element 211 to produce an asymmetrycompensated phase angle
difference between motion sensor 202 and driver 204. The two motion
sensor 201 and 202 signals are also processed by phase detector
element 207 to produce a phase angle difference measurement, which
is then input to divider element 210 where the phase angle difference
signal is divided by 2. The three signals from asymmetry compensation
logic elements 209 and 211 and divider element 210 which are each
halfangle measurements, are then input to halfangle logic element
212 where they are compared according to a polling scheme, shown
in FIG. 9. The purpose of the polling scheme is to determine if
there is a zero fluctuation of the flow conduit, which would be
indicative of unbalanced operation.
A halfangle difference is output by halfangle logic element 212
which is then input to halfangle tangent function logic element
214 which processes the signal to produce a halfangle tangent
function signal. The measured drive frequency, processed frequency
response function, processed halfangle tangent function, and temperaturecompensated
Coriolis spring constant are combined with calibration constants
in mathematics logic element 216. The output signal of mathematics
logic element 216 is the mass flow rate, also according to the formula
of equation (71). This embodiment has several advantages. It takes
into account both the nonlinearities accounted for in the previous
embodiment and asymmetries of the zerocrossings of the motion sensor
signals. In addition, logic element 212 provides a warning when
the device experiences unbalanced operation such as might be caused
by ambient vibrations.
This second embodiment offers yet another advantage in determining
the proper operation of a mass flow meter measuring the asymmetry,
.epsilon., of motion signals versus the driver signal of two motion
sensors located essentially equidistant from the driver, which should
result in the same value of .epsilon.. If the values are not the
same, then there is a problem with either the sensors or the conduit
and a warning signal can be provided.
A third preferred embodiment of the signal processing logic for
a mass flow meter, illustrated in FIG. 10 processes the signals
from motion sensor 301 driver 302 and temperature sensor 303 to
produce mass flow rate. The voltage sign V.sub.d, having associated
phase angle .theta..sub.d, controlling driver 302 is input to a
frequency measurement element 305 which processes the signal and
outputs a driver frequency. Alternatively, a driver motion sensor
can be used and such motion signal be converted to a frequency signal.
This driver frequency output is input to mathematics logic element
310 and to a frequency response function logic element 308. Temperature
sensor 303 is employed to provide a temperature of the oscillating
flow conduit. This temperature measurement is input to temperaturecompensation
logic element 309 which processes the signal to produce two temperaturecompensated
spring constant signals. One spring constant signal, k.sub.b (T),
is for the oscillation degree of freedom. The other, k.sub.c (T),
is for the Coriolis degree of freedom. Both temperaturecompensated
spring constant signals are input to frequency response function
element 308 which combines these signals. The driver 302 signal
is also input to phase detector element 304 which compares this
signal with the motion signal u.sub.1 having associated phase angle
.theta..sub.1 from the flow conduit motion sensor 301. The output
from phase detector element 304 is then input to asymmetry compensation
logic element 306. The resulting processed signal is input to a
halfangle tangent function logic element 307. The outputs of the
halfangle tangent function logic element 307 the frequency response
function logic element 308 and the temperaturecompensated spring
constant are combined with calibration constants in mathematics
logic element 310. The resulting output of the processed signals
from the driver and flow conduit motion sensor is the mass flow
rate, according to the formula of equation (71). This embodiment
takes account of both nonlinearity and asymmetry, as in the previous
embodiment, but has the further advantage of requiring only one
motion sensor signal, in addition to the driver signal.
In the fourth, fifth and sixth embodiments which follow, a damping
measurement is employed. It is contemplated that several techniques
can be used for making damping measurements. For many applications
the damping is constant, and very accurate modal analysis techniques
can be employed. For example, the Modal 3.0 software produced by
Structural Measurement Systems, Inc. of San Jose, Calif. has been
used to measure the damping ratios of mass flow meters.
When the damping is not constant, a near continuous signal must
be supplied. One technique is to monitor the power delivered by
the drive system. This quantity is directly proportional to the
damping in the system. Small changes in the drive frequency could
also be used to make damping measurements. Another approach is to
use standard decay techniques, such as described in the "Shock
and Vibration Handbook", second edition, 1976 published by
McGraw Hill Book Company. The measurement could be made on an intermittent
basis so as not to interfere with the mass flow measurement.
A fourth preferred embodiment of the signal processing logic for
a mass flow meter, illustrated in FIG. 11 processes the signals
from motion sensor 401 driver 402 and temperature sensor 403 to
produce mass flow rate. The voltage signal V.sub.d, having associated
phase angle .theta..sub.d, controlling driver 402 is input to a
frequency measurement element 405 which processes the signal and
outputs a driver frequency. Alternatively, a driver motion sensor
can be used and such motion signal be converted to a frequency signal.
This driver frequency output is input to mathematics logic element
410 and to a frequency response function logic element 408. Temperature
sensor 403 is employed to provide a temperature of the oscillating
flow conduit. This temperature measurement is input to temperaturecompensation
logic element 409 which processes the signal to produce two temperaturecompensated
spring constant signals. One spring constant signal k.sub.b (T),
is for the oscillation degree of freedom. The other, k.sub.c (T),
is for the Coriolis degree of freedom. The Coriolis spring constant
k.sub.c (T) is also input to mathematics logic element 410. The
driver 402 signal is also input to damping measurement element 406.
The output from damping measurement element 406 is then input to
the frequency response element 408. Both temperaturecompensated
spring constant signals are input to frequency response logic element
408 which combines these signals with the driver frequency and
damping measurement to determine a frequency response function which
accounts for any change in fluid density. Frequency response element
408 supplies a Coriolis phase angle function .phi..sub.c to trigonometric
function element 407 and a frequency response function to mathematics
logic element 410. The driver 402 signal is also input to phase
detector element 404 which compares this signal with the motion
signal u.sub.1 having associated phase angle .phi..sub.1 from
the flow conduit motion sensor 401. The output from phase detector
element 404 is then input to a trigonometric function logic element
407. The outputs of the trigonometric function logic element 407
the frequency response function logic element 408 and the temperaturecompensated
spring constant are combined with calibration constants in mathematics
logic element 410. The resulting output of the processed signals
from the driver and flow conduit motion sensor is the mass flow
rate, according to the formula of equation (88). This embodiment
takes account of both nonlinearity and viscous damping as in the
previous embodiment and has the advantage of requiring only one
motion sensor signal, in addition to the driver signal.
A fifth preferred embodiment of the signal processing logic for
a mass flow meter, illustrated in FIG. 12 processes the signals
from motion sensor 501 driver 502 and temperature sensor 503 to
produce mass flow rate according to equation (74). The voltage signal
V.sub.d, having associated phase angle .phi..sub.d, controlling
driver 502 is input to a frequency measurement element 505 which
processes the signal and outputs a driver frequency .omega..sub.d.
Alternatively, a driver motion sensor can be used and such motion
signal be converted to a frequency signal. This driver frequency
output is input to mathematics logic element 510 and to a frequency
response function logic element 508. Temperature sensor 503 is employed
to provide a temperature of the oscillating flow conduit. This temperature
measurement is input to temperaturecompensation logic element 509
which processes the signal to produce two temperaturecompensated
spring constant signals. One spring constant signal, k.sub.b (T),
is for the oscillation degree of freedom. The other k.sub.c (T),
is for the Coriolis degree of freedom. The driver voltage signal
V.sub.d is also input to a damping measurement element 511. The
output of the damping measurement element 511 is fed to the frequency
response function element 508. Both temperaturecompensated spring
constant signals are input to frequency response logic element 508
which combines these signals with the driver frequency and damping
measurement to determine a frequency response function which accounts
for any change in fluid density. Frequency response function element
508 generates Coriolis phase angle function .phi..sub.c which is
input to asymmetry compensation logic element 506 and also to mathematics
element 510 along with a frequency response function R. The driver
502 signal is also input to phase detector element 504 which compares
this signal with the motion signal u.sub.1 having associated phase
angle .phi..sub.1 from the flow conduit motion sensor 501. The
output from phase detector element 504 is then input to asymmetry
compensation logic element 506. Asymmetry compensation logic element
506 employs eqs. 89 or 90. The resulting processed signal is input
to a halfangle tangent function logic element 507. The outputs
of the halfangle tangent function logic element 507 the frequency
response function logic element 508 and the temperaturecompensated
spring constant are combined with calibration constants in mathematics
logic element 510. The resulting output of the processed signals
from the driver and flow conduit motion sensor is the mass flow
rate, according to the formula of equation (74). This embodiment
takes account of asymmetry and viscous damping effects.
FIG. 13 depicts a sixth preferred embodiment of the signal processing
logic for a mass flow meter which includes two motion sensors 601
and 602 positioned essentially equidistant from the center of the
flow conduit to monitor the motion of two sides of the oscillating
flow conduit. Driver 604 voltage signal V.sub.d is input to frequency
measurement element 605 to determine the frequency at which the
oscillating flow conduit is driven. This drive frequency is input
to frequency response function element 609. The drive frequency
is also input to mathematics logic element 611 for further processing.
The driver 604 voltage signal is also input to damping measurement
element 606 which generates a damping ratio .zeta..sub.c which
is input to frequency response function element 609. Frequency response
function element 609 generates a frequency response function R and
a Coriolis phase angle function .phi..sub.c which are input to mathematics
logic element 611. The Coriolis phase angle function .phi..sub.c
is also input to asymmetry compensation logic element 610. Temperature
sensor 603 is employed to provide a temperature of the oscillating
flow conduit. This temperature measurement is input to temperaturecompensation
logic element 612 which processes the signal to produce two temperaturecompensated
spring constant signals. One spring constant signal k.sub.b (T),
is for the oscillation degree of freedom. The other, k.sub.c (T),
is for the Coriolis degree of freedom. Both temperaturecompensated
spring constant signals are input to frequency response function
logic element 609 which combines these signals with the drive frequency
and damping measurement to determine a frequency response function
which accounts for any change in fluid density and for viscous damping.
The Coriolis spring constant is also input to mathematics logic
element 611. The signals u.sub.1 and u.sub.2 from each of the motion
sensors 601 and 602 which have associated respective phase angles
.phi..sub.1 and .phi..sub.2 are input to phase detector element
607 which determines the phase angle difference between the two
signals. The phase angle difference is then processed by asymmetry
compensation logic element 610 which combines this signal with the
Coriolis phase angle function. Asymmetry compensation logic element
610 employs eq. 95. The result of this processing is input to mathematics
logic element 611 for further processing. The measured drive frequency,
processed frequency response function, processed halfangle tangent
function and temperaturecompensated Coriolis spring constant are
combined with calibration constants in mathematics logic element
611 according to the formula of equation (74) to produce the mass
flow rate as an output. As one skilled in the art will recognize,
this embodiment takes into account inherent nonlinear effects in
measuring mass flow rates along with asymmetry and viscous damping
effects, which are not accounted for in the existing embodiments.
Such inherent nonlinear, asymmetric and viscous damping effects
are reflected in the relationship between mass flow rate and phase
angle in equation (74) and are embodied in the system of FIG. 13.
It should be noted that for multiple motion sensors, the frequency
of oscillation can be extracted and averaged to give a systemaveraged
frequency of oscillation. Similarly, for multiple tube mass flow
meters, the frequency signals from the multiple tubes can be compared
and if the frequencies are not within predetermined limits, a warning
signal can be given.
The above discussion and related illustrations of the present invention
are directed primarily to preferred embodiments and practices of
the invention. However, it is believed that numerous changes and
modifications in the actual implementation of the concepts described
herein will be apparent to those skilled in the art, and it is contemplated
that such changes and modifications may be made without departing
from the scope of the invention as defined by the following claims.
