A method of improving the accuracy of a variable orifice flow meter
that includes characterizing the flow coefficient of the flow meter
orifice for different orifice openings and for different differential
pressures. The method may be particularly useful with a flow metering
and controlling device that includes a fluid flow conduit having
at least one planar inner wall and an element having a linear edge
configured to mate with the at least one planar inner wall of the
fluid flow conduit. The element is movable relative to the conduit
to define a flow orifice and vary a cross-sectional area of the
orifice. The device also includes a processor configured to calculate
the fluid flow based on the cross-sectional area of the orifice,
the differential pressure, and the flow coefficient.
1. A method of metering fluid flow through a variable orifice,
the fluid having a density and a viscosity, the method comprising
the steps of: determining an orifice geometry defined by the variable
orifice; measuring a pressure differential across the variable orifice;
determining a value of Reynolds number divided by flow coefficient
(Re/K) using the orifice geometry, the pressure differential, the
density and the viscosity; determining a flow coefficient using
the orifice geometry and the determined (Re/K) value; and determining
a fluid flow through the variable orifice using the determined flow
2. The method of claim 1 wherein determining the flow coefficient
includes using a low-order univariate polynomial and a triangulated
surface to interpolate a value for the flow coefficient.
3. The method of claim 1 further comprising determining a temperature
of the fluid and determining the density and the viscosity using
the determined temperature.
4. The method of claim 1 further comprising determining a set
of calibration data points by measuring fluid flow through the variable
orifice opening for a predetermined set of pressure conditions and
orifice opening geometries, wherein the determined orifice geometry
and the measured pressure condition are between the calibration
5. The method of claim 4 further comprising determining an Re/K
value for each calibration data point, and using a low-order univariate
polynomial and a triangulated surface to interpolate a value for
the flow coefficient using the determined Re/K values for the determined
pressure condition and the orifice geometry.
6. A method of determining a flow coefficient in a variable orifice
device having a variable orifice opening, the method comprising
calculating the flow coefficient using a Reynolds number value and
geometry of the variable orifice opening.
7. The method of claim 6 further comprising determining the Reynolds
number value using density and viscosity values of a fluid flowing
through the variable orifice opening.
8. The method of claim 6 further comprising obtaining a set of
calibration data points by measuring fluid flow through the variable
orifice opening for a predetermined set of pressure conditions and
orifice opening geometries.
9. The method of claim 8 further comprising determining an Re/K
value for each of the calibration data points.
10. The method of claim 9 further comprising determining a surface
by fitting a univariate polynomial in orifice geometry to generate
a flow coefficient for each of the calibration data points.
11. The method of claim 10 further comprising determining a residual
surface that is a difference (.DELTA.K) between the surface defined
by the polynomial and the calibration data points.
12. The method of claim 11 further comprising determining a flow
coefficient for a new Reynolds number and a new orifice opening
geometry that are between the calibration data points, the step
of determining the flow coefficient comprising: evaluating the polynomial
at the new orifice opening geometry to obtain an approximate flow
coefficient value; and determining points on the residual surface
representing calibration orifice opening sizes bounding the new
orifice opening geometry.
13. The method of claim 12 wherein the step of determining the
flow coefficient further comprises: interpolating a value of .DELTA.K
using the determined points on the residual surface; and adding
the value of .DELTA.K to the approximate flow coefficient value
to generate the flow coefficient that corresponds to new orifice
opening geometry and new Reynolds number.
14. A device for metering fluid flow, of the type having a variable
orifice, comprising: a variable sized orifice defined by a fluid
flow conduit and an element movable relative to the fluid flow conduit
to vary a size of the orifice; a pressure sensor configured to determine
a pressure differential across the orifice and generate a pressure
signal; a positioning device configured to determine a position
of the element relative to the conduit and generate a position signal;
and a processor configured to determine the fluid flow rate using
the pressure signal, the position signal, and a flow coefficient
that is dependent on the pressure signal, the position signal, and
the viscosity and density of the fluid.
15. The device of claim 14 further comprising a temperature sensor
configured to determine a temperature of the fluid.
16. The device of claim 15 wherein the fluid temperature is used
to determine the viscosity and density of the fluid.
17. The device of claim 14 further comprising nonvolatile memory
accessible by the processor, wherein a plurality of constants used
for determining the flow coefficient are stored in the nonvolatile
CROSS REFERENCE TO RELATED APPLICATIONS
 This application is a continuation-in-part application of
U.S. patent application Ser. No. 10/877377 filed on Sep. 25 2004
and entitled SOFTWARE CORRECTION METHOD AND APPARATUS FOR A VARIABLE
ORIFICE FLOW METER, which application is incorporated herein by
BACKGROUND OF THE INVENTION
 1. Technical Field
 The present invention generally relates to fluid flow metering
and control devices, and more particularly relates to software related
correction methods for such flow devices.
 2. Related Art
 In process control industries, it is common to use small
diameter tubes to carry process fluids at low flow rates when small
amounts of fluids are required for manufacturing processes. The
tubes are almost always of a circular cross-section. Instruments
used to measure a flow rate in the tubes must interface with a fluid
flowing in the tube while minimizing disturbance to the fluid flow.
To minimize disturbance to the fluid flow, the instrument typically
includes a circular cross-section to match the cross-section of
the tubes. The flow rate for a flow meter measuring a change in
pressure across an orifice is defined by the following Equation
1: 1 Q = C A o ( 1 1 - ( A o A p ) 2 ) 1 2 ( 2 ( P hi - P lo ) )
1 2 Equation 1
 Q=volumetric flow rate
 C=orifice discharge coefficient
 A.sub.o=cross-sectional area of the orifice
 A.sub.p=cross-sectional area of the pipe
 P.sub.hi=upstream pressure
 P.sub.lo=downstream pressure
 .rho.=density of the fluid
 The differential pressure measurement (P.sub.hi-P.sub.lo)
could be made using two individual pressure measurements and combining
them to get the pressure difference or pressure drop or using a
single device as represented in FIG. 14.
 When orifices and differential pressure measurements are
used to calculate flow through large pipes it is common for them
to be discrete devices that are bolted or otherwise attached to
the pipe. There are also devices available for measuring the flow
in small tubes that have the orifice and pressure sensors integrated
into the same housing. In almost all cases, the measuring device
orifices are of a fixed size for measuring flow over a fixed flow
range. The flow characteristic or "flow coefficient" of
the orifice is measured, or determined by design, by the manufacturer.
For discrete systems, the end user may calculate the flow based
upon the parameters in Equation 1 including a manufacturer provided
discharge coefficient. In integrated systems, the discharge coefficient
can simply be accounted for as part of a total device calibration
performed by the manufacturer and maintains a constant value.
 Differential pressure orifice flow metering is most accurate
when the flow rate is near the upper end of the flow range that
the meter is designed for; that is, where the pressure change is
relatively large for a given change in flow rate. As the flow rate
decreases, the accuracy of the device decreases because there is
a relatively small pressure change for a given change in flow rate.
This phenomena can also be described as a decrease in the differential
pressure to flow rate ratio, which ratio is shown in the graph of
FIG. 15. Since the pressure differential must be accurately known
to calculate the flow rate, any error in the differential pressure
measurement causes an error in the flow calculation. As the slope
of the curve gets steeper at low flow rates (see FIG. 15), any pressure
measurement error causes a larger flow calculation error.
 In order to make more accurate flow measurements over a
larger range of flow rates using an orifice and differential pressure
measurement, it may be advantageous to use a variable-sized orifice.
A variable-sized orifice can be used to improve the flow measurement
accuracy over the range of orifice openings by providing a relatively
high pressure differential for each flow rate. However, even though
computational fluid dynamics (CFD) software can be used to optimize
the design of a variable-sized orifice, there is still a small change
in the discharge coefficient as the size of the orifice is varied.
This change is due to the range of flows that the device is designed
to measure, and the physical factors that contribute to the discharge
coefficient of an orifice.
 Some variable-sized orifice devices are designed to cover
flow ranges that begin in the laminar flow region and end in the
turbulent flow region, which make it likely that the discharge coefficient
will vary in the different flow ranges. Also, it is known that the
discharge coefficient of an orifice is comprised of a combination
of physical effects relating to the fluid and the shape of the orifice.
When the orifice is set for a very small opening, the surface area
of the walls of the flow path are large relative to the cross-sectional
area of the flow path. This is because a "slit" type opening
results. In a slit type opening, the viscous force of the liquid
against the walls in the orifice region of the flow path becomes
much more significant than when a larger opening is present. A larger
ratio of the wall surface area to the flow path cross-sectional
area has the effect of lowering the discharge coefficient of the
 Although a variable orifice flow meter may have the advantage
of extending the range of a flow meter by as much as a factor of
10 or more, it may have the inherent drawback of decreased accuracy
due to slight changes in the discharge coefficient at different
openings, and for different flow rates at any given opening size.
 In addition to the above noted disadvantages related to
discharge coefficients, known variable orifice devices are ineffective
for several other reasons. First, known variable orifice devices
typically use circular or curved members that are moved with respect
to the fluid flow to change the size of the orifice. Because of
the curved nature of these members, the shape of the orifice changes
as the size of the orifice changes, which results in significant
errors when calculating the fluid flow over a range of orifice sizes.
Second, the changed shape of the orifice leads to non-ideal orifice
shapes for at least a portion of the flow range. This leads to inconsistent
flow characteristics for any given opening as flow rate changes,
again leading to errors in the calculation of fluid flow.
 A flow device that addresses these and other shortcomings
of known flow control and metering devices would be an important
advance in the art.
SUMMARY OF THE INVENTION
 The present invention generally relates to software related
correction methods for flow devices such as differential pressure
flow metering and controlling devices.
 These and further objects of the present invention will
become clearer in light of the following detailed description of
illustrative embodiments of this invention described in connection
with the drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
 The illustrative embodiments may best be described by reference
to the accompanying drawings where:
 FIG. 1 is a top perspective view of a flow device according
to principles of the invention;
 FIG. 2 is a top plan view of the flow device shown in FIG.
 FIG. 3 is a cross-sectional view of one example configuration
of the flow device shown in FIG. 2 taken along cross-section indicators
 FIG. 4 is a cross-sectional view of the example flow device
shown in FIG. 3 taken along cross-section indicators 4-4;
 FIG. 5 is an enlarged view of the orifice and movable element
portion of the device shown in FIG. 3;
 FIG. 6 is a cross-sectional view of the example flow device
shown in FIG. 2 taken along cross-section indicators 6-6;
 FIG. 7 is a cross-sectional view of the example flow device
shown in FIG. 2 taken along cross-section indicators 7-7 the example
device having a rectangular inlet to the orifice;
 FIG. 8 is a cross-sectional view of the example flow device
shown in FIG. 2 taken along cross-section indicators 8-8;
 FIG. 9 is schematic process diagram of an example flow device
according to principles of the present invention;
 FIG. 10 is an example array of discharge coefficients based
on orifice size and pressure differential for an example variable
orifice flow device according to principles of the present invention;
 FIG. 11 is a schematic representation of a fluid flow control
 FIG. 12 is a flow diagram representing an example method
of determining fluid flow through a flow device according to principles
of the present invention;
 FIG. 13 is a flow diagram representing another example method
of determining fluid flow through a flow device according to principles
of the present invention;
 FIG. 14 is a schematic representation of a pressure differential
measuring device configured to measure a pressure differential across
 FIG. 15 is a chart representing flow rate verses pressure
differential for an example flow device having a fixed orifice size;
 FIG. 16 is a graph representing discharge coefficient verses
an orifice size for a simulated flow device;
 FIG. 17 is a graph representing discharge coefficient verses
flow rate for a simulated flow device having a fixed orifice size;
 FIG. 18 is a graph representing known values of flow coefficient
verses Reynolds number for an orifice;
 FIG. 19 is a three dimensional graph illustrating a typical
set of data points to be approximated according to principles of
 FIG. 20 is an example array of flow coefficients based on
orifice size and a value of Reynolds number divided by flow coefficient
for an example variable orifice flow device according to principles
of the present invention;
 FIG. 21 is a graph illustrating errors that occur from triangulation;
 FIGS. 22A and 22B are two and three dimensional graphs illustrating
a polynomial curve fit to the set of data points shown in FIG. 19;
 FIG. 23 is a graph illustrating residual values or the difference
between the original data and the polynomial;
 FIGS. 24A and 24B illustrate the difference between scattered
and semi-gridded data;
 FIGS. 25A and 25B are graphs illustrating a top view of
the sample data set of FIGS. 22A and 22B and the resulting triangulation;
 FIG. 26 is a three dimensional graph illustrating the triangulated
residual surface of FIGS. 22A and 22B;
 FIG. 27 illustrates construction of the triangulated region
between two parallel grid lines; and
 FIG. 28 is a flow diagram representing another example method
of determining fluid flow through a flow device according to principles
of the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
 The invention generally relates to fluid flow metering and
control devices, and more particularly relates to variable-sized
orifice flow devices and software related correction methods for
such flow devices. The variable-sized orifice may be particularly
suited for use in a differential pressure flow meter as will be
described herein with reference to the several drawings, although
such an application is only exemplary of the many applications to
which principles of the present invention may be applied.
 The software related correction methods may utilize a matrix
or array of stored discharge or flow coefficients that correlate
to specific pressure differential and orifice size characteristics
of the flow device. Other software related correction methods may
utilize equations or algorithms to calculate an exact discharge
or flow coefficient for each determined pressure differential and
orifice size of the flow device. The arrays of discharge and flow
coefficients and the equations/algorithms used to calculate discharge
and flow coefficients may be stored in memory and used by a controller,
such as a processor, to determined fluid flow.
 I. Example Flow Device
 An example flow device 10 constructed in accordance with
the principles of the present invention for controlling and metering
fluid flow is shown in FIGS. 1-9. The device includes a housing
12 a moveable element 14 first and second pressure sensors 16
18 and inlet and outlet conduit connectors 22 20. A conduit 30
is formed through the housing and includes first, second and third
segments 50 52 54. The housing also includes first and second
sensor bores 36 38 that intersect with the conduit 30 in a direction
transverse to the conduit 30 and an element bore 40 that also intersects
with conduit 30 in a direction transverse to conduit 30. In this
example, element bore 40 and sensor bores 36 38 extend parallel
to each other, but may be aligned perpendicular to each other in
other embodiments. Housing 12 may be divided into separate pieces
or halves 13 15 (see FIG. 1) to facilitate precise formation of
intricate features within the housing, or may be integrally formed
as a single piece.
 Moveable element 14 includes a base 42 and a contact member
44 and is positioned in element bore 40 so as to extend into second
segment 52 of the conduit 30. Contact member 44 includes a leading
edge 46 a tapered trailing edge 48 and a planar contact surface
49 (see FIG. 5) configured to mate with a planar surface (for example,
fixed wall 90 described below and shown in FIG. 5) of second segment
52. The movable element 14 is moveably adjustable along a linear
axis through a range of positions between an open (retracted) position
and a closed position, with movement of the movable element 14 being
limited to the linear axis. The open position allows a maximum fluid
flow through the conduit 30. The fluid flow through the conduit
30 decreases as the movable element 14 is moved toward the closed
position due to contact with the fluid. Adjustment of the movable
element 14 in element bore 40 may be performed using, for example,
a linear actuator, a stepper motor, a hydraulic or pneumatic actuator,
a solenoid, a servo motor, or a manual device such as a threaded
shaft with a thumb turn button. The position of the movable element
14 may be determined using, for example, a Hall effect sensor, magnetostrictive
devices, linear variable differential transformers (LVDTs), optical
encoder, and other position determining technologies.
 Limiting movement of element 14 to linear motion within
element bore 40 may simplify positioning of movable element 14.
Other methods may "infer" a position of the moveable element
14 based on incremental movement related to the moveable element.
In one example method, the movable element 14 may be moveable a
certain number of steps from a reference position such as a fully
open or fully closed position. Software controlling the device 10
may be programmed to convert the number of steps traveled into the
distance traveled. An independent position measuring device would
not be needed in such a configuration, which may reduce the amount
and complexity of hardware used for device 10. A possible drawback
of this method is the potential for inaccurate position measurements
if the element becomes locked in a single position and the processor
thinks that the element is moving a certain number of steps when
the element is actually stationary. An encoder used with a stepper
motor or with a linear actuator, or other devices that "infer"
a linear position from related incremental movement may have similar
issues of potential inaccuracy.
 Second segment 52 includes an inlet portion 60 an outlet
portion 62 and an orifice portion 64 positioned between the inlet
and outlet portions 60 62. The inlet portion 60 is in fluid communication
with sensor chamber 32 at one end, and includes a plurality of tapered
surfaces at a second end adjacent to the orifice portion 64. Similarly,
outlet portion 62 is in fluid communication with sensor chamber
34 at one end, and includes a plurality of tapered surfaces at an
opposing end adjacent to orifice portion 64.
 The inlet and outlet portions of the orifice segment of
the device include a plurality of fixed sidewalls that define a
noncircular cross-section in this embodiment. Other embodiments
may include inlet and outlet portions of the orifice segment that
have a circular cross-section, which configuration may be preferred
in some instances. The example first and third portions 60 62 include
four fixed walls substantially in the shape of a square (see example
cross-section of inlet portion 60 in FIG. 7). As used throughout
this document, rectangular is defined as a four-walled shape and
a square is defined as a rectangle that has four walls of the same
length. The walls of a rectangle are substantially flat or linear
and the intersection of two walls provides an angle of about 90.degree..
In some applications, the corners of the rectangle may be tapered
slightly with a round, fillet, chamfer or like feature as a result
of manufacturing limitations. Further, a portion of one or more
of the walls may be slanted or chamfered slightly to create sealing
points or to meet other design goals and/or address manufacturing
limitations. In embodiments that include a combination of linear
and curved walls (not shown), the intersection of these walls may
also include features such as rounds, fillets, chamfers, etc. Finally,
a portion of one or more of the walls may be formed by the exposed
face of a gasket or seal.
 Tapers 70 72 74 76 are formed in the sidewalls of inlet
portion 60 to reduce the cross-sectional area at the point where
inlet portion 60 abuts to orifice portion 64. The tapers 70 72
74 76 are aligned at a single axial position so as to create a
reduction in cross-sectional area of portion 60 in a single step
(see FIG. 3-5). Outlet portion 62 also includes a square shaped
cross-section with tapered surfaces 78 80 (see FIG. 4) on opposing
sidewalls so as to reduce the cross-sectional area of outlet portion
62 at the transition point between orifice portion 64 and outlet
 Orifice portion 64 includes three fixed walls 90 92 94
with fixed wall 90 including a tapered trailing edge 96 and a leading
edge 98 (see FIG. 5). As a result, the cross-sectional area of orifice
portion 64 tapers out to the larger cross-sectional area of portion
62 in two steps with sets of tapers 96 48 and 78 80. As shown
in the cross-sectional view of FIG. 8 orifice portion 64 has a
relatively small cross-sectional area as compared to the cross-sectional
area of inlet portion 60 shown in FIG. 7.
 The leading edges 46 98 and trailing edges 96 48 of respective
moving element 44 and orifice portion 64 provide consistent flow
characteristics into and out of the orifice portion 64. A cross-sectional
size of the orifice portion 64 is determined by the location of
the movable element 14 in relation to the fixed walls 90 92 94
of the orifice portion 64. The orifice portion 64 is void of sensor
openings and dead volume spaces to avoid disruptions to the fluid
flow and potential accumulation of process material or sediment.
 A linear actuator (not shown) such as those discussed above
(e.g., stepper motor, servo motor, etc.) may be used to affect movement
of the movable element 14. By moving along a single linear axis,
the movable element 14 linearly changes the cross-sectional size
of the orifice portion 64 while maintaining a generally uniform
shape to provide a relatively consistent set of flow characteristics
through the range of movable element positions. The cross-sectional
shape of orifice portion 64 allows repeatable regulation of the
fluid flow in accordance with the position in the range of positions
of the movable element 14. In one example wherein the uniform shape
is a rectangle, the height of the cross-sectional area of the orifice
portion 64 is reduced in size as the movable element 14 moves between
the open and closed positions. Maintaining a rectangular shape,
or at least a shape having at least one planar or linear sidewall,
minimizes variations in flow characteristics (variable "C"
in the flow rate equation in the Background section), thus reducing
errors when determining the flow rate for each orifice size.
 In use, fluid first enters flow device 10 (which example
will be used for the remainder of the description of various aspects
of the invention) through first segment 50 of conduit 30. The flow
through segment 50 has flow characteristics that match the circular
cross-section of first segment 50. The flow then enters the open
sensor chamber 32 where a transition volume is provided prior to
the fluid flow entering the non-circular inlet portion 60 of second
segment 52. The flow is then reduced in cross-sectional area by
the several tapers formed in inlet portion 60 just before orifice
portion 64. As mentioned above, a higher pressure is generated at
the inlet to orifice portion 64 due to the very small cross-sectional
area of orifice portion 64 and the wall-like structure created by
leading edges 46 98. The cross-sectional area of orifice portion
64 is dependent on the position of moveable element 14 in the direction
A. Each position along the direction A corresponds to a different
cross-sectional area of the orifice portion 64 for use in determining
the volumetric flow through the flow device 10.
 As the fluid exits orifice portion 64 the cross-sectional
area of the fluid flow increases due to tapers 78 and 80 and trailing
edges 48 and 96 of the moveable element 14 and orifice portion 64
as the flow enters portion 62. The cross-sectional area of outlet
portion 62 preferably has the same size and shape as the cross-section
of inlet portion 60 (which is a square cross-section in the example
flow device in flow device 10--see FIGS. 2 and 6-9). Flow exiting
outlet portion 62 enters sensor chamber 34 where another transition
volume is provided before the fluid flow enters the third segment
54 and takes on a flow pattern for the circular cross-section of
third segment 54.
 The first and second pressure sensors 16 18 are positioned
at opposing sides of orifice portion 64 so as to be able to determine
a difference in pressure at the inlet and outlet sides of second
segment 52 of conduit 30. The first and second pressure sensors
16 18 may be mounted proximate the process liquid to minimize the
amount of dead volume of the fluid and reduce crystallization and
particle buildup between the first and second pressure sensors 16
18 and the fluid in conduit 30. In other aspects of the present
invention, a single differential pressure sensor may be used to
communicate with both the first and second sensor chambers 32 34
to determine the pressure difference. Furthermore, only a single
pressure sensor may be required in applications where one of the
first or second sensor chamber 32 34 has a fixed pressure. For
example, if the second sensor chamber 34 is downstream of the orifice
and empties into an open tank at atmospheric pressure, a downstream
pressure measurement is not required and the pressure measurement
from the first sensor 16 may be used singly with atmospheric pressure
to determine the pressure differential. Likewise, if the first sensor
chamber 32 is upstream of the orifice portion 64 and is accepting
liquid from a pressurized tank where pressure is tightly controlled
to a fixed value, an upstream pressure is not required and the pressure
measurement from the second sensor 18 may be used singly with the
fixed upstream pressure value to determine the pressure differential.
 Other example embodiments may use a single differential
pressure sensor that takes pressure readings from the inlet and
outlet sides of the orifice portion of the device and determines
a differential pressure across the orifice portion. This and other
types of sensors do not necessarily have to be mounted in a sensor
bore, nor does the sensor bore being used require a larger cross-sectional
area than the cross-sectional area of the conduit. For example,
a sensor may be configured to obtain pressure readings using a small
probe that requires a very small entrance opening into the conduit
relative to the conduit size, and the sensor can be mounted at a
different location within or adjacent to the device housing.
 Yet further embodiments may not include any sensors associated
directly with the device, but may be configured to use pressure
signals provided by outside sources. Such pressure readings from
an outside source may include, for example, a pressure reading from
a pressure sensor positioned up or down stream from the device,
or a pressure signal representative of a known static pressure condition
for the system either up or down stream of the device. Thus, although
the device does not require a pressure sensor, the device is preferable
configured to use a pressure signal for purposes of metering and
controlling fluid flowing through the device.
 A pressure signal representing a pressure differential across
an orifice may be used with the cross-sectional area of the orifice,
the cross-sectional area of the inlet and outlet portions just before
and after the orifice, and the density of the fluid to determine
the volumetric flow rate (discussed in the Background section above).
 An advantage of the present invention is that the pressure
signal (.DELTA.P) may be optimized at each flow rate by varying
the orifice size. For example, the pressure signal may be set at
a minimum value for a given flow rate by varying the orifice size.
Furthermore, the pressure signal may be optimized for a desired
flow rate or inlet pressure by varying the orifice size.
 Furthermore, although the cross-sections of the inlet, outlet
and orifice portions 60 62 64 of second segment 52 are shown having
a rectangular shape, it may be appreciated that the cross-sections
may be cross-sections of different shapes, such as, but not limited
to, rectangles, isosceles triangles or the like. Furthermore, different
portions of the second segment 52 may have dissimilar cross-sectional
shapes and sizes, and may have varying shapes or sizes along a length
of each portion of the second segment 52. Additionally, although
the orifice portion 64 has a rectangular cross-section, the leading
and trailing portions of the orifice portion 64 defined by the leading
and trailing edges 46 48 of the contact member 44 of the movable
element 14 and the leading and trailing edges 98 96 of the fixed
walls 90 92 94 may be of different sizes, shapes and orientations
than those shown in the Figures.
 Other example flow devices and further aspects of the flow
device 10 are shown and described in U.S. patent application Ser.
No. 10/728594 filed on Dec. 3 2003 and entitled APPARATUS FOR
CONTROLLING AND METERING FLUID FLOW, which patent application is
incorporated by reference herein in its entirety
 Features of the preferred embodiment flow device 10 shown
in FIGS. 1-8 are shown schematically as part of a flow device assembly
100 in FIG. 9. Assembly 100 includes a microcontroller 102 that
controls and communicates with most of the other assembly features.
Assembly 100 includes a actuator drive circuit 104 a linear actuator
106 a position sensor reference 108 a position sensor 110 and
an analog-to-digital converter (ADC) 112 that relate to the flow
device variable sized orifice 113 and a switch 114 regulator 116
regulator 150 and linear regulator 118 that control power to the
features 106 108 110 112. Microprocessor 102 may be any suitable
processor or controller such as, for example, the HD64F3062 16-bit
microprocessor manufactured by RENESAS of San Jose, Calif.
 The assembly 100 also includes a pressure sensor reference
120 a high pressure sensor 122 a low pressure sensor 124 and
difference amplifiers 126 128 and an ADC 129 that together are
used to determine a pressure differential in the flow device. The
assembly 100 also includes a temperature sensor 121 and a temperature
amplifier 127 that are used to determine a temperature of the fluid
in the flow device. Different memory devices such as RAM 130 NVROM
132 and program memory 134 may be used by the microprocessor 102
to store data, such as the example array of FIG. 10 (and/or the
polynomial equations below), instructions, code, algorithms, etc.
 The microprocessor 102 may receive inputs in the form of
current signals having a magnitude of, for example, 4-20 mA that
are converted to digital signals using ADC 136 and voltage isolation
137 and may communicate with direct digital signals through a UART
138 and a digital interface 140. Microprocessor 102 may also generate
output signals that are converted to analog signals with the voltage
reference 142 digital-to-analog converter (DAC) 144 voltage isolation
145 and an output circuit 146 that generates signals having a magnitude
of, for example, 4-20 mA. Assembly 100 may use a power source that
includes a negative regulator 148 and the switching regulator 150
for powering various features of the assembly 100.
 II. Software Correction Methods
A. EXAMPLE #1
 A variable orifice flow meter has an inherent drawback of
losing some accuracy due to slight changes in the discharge coefficient
at different orifice openings, and for different flow rates at any
given opening. The example flow metering and controlling devices
disclosed herein provides a means of overcoming these two drawbacks
using software correction for the discharge coefficient. Rather
than using a single discharge coefficient for the device for all
flow calculations, a discharge coefficient that is dependent upon
the orifice opening and the differential pressure measured may be
used in each flow calculation. An array (see example array in FIG.
10) of discharge coefficient values for the range of differential
pressures and orifice openings to be used may be stored in memory
of the flow meter and the appropriate value can be accessed and
used by the flow meter for each individual flow calculation.
 As discussed above, a Hall effect sensor may be used to
measure a linear position of a magnet contained in the movable element
of the flow device that varies the orifice opening of the flow device.
Since the orifice opening in the flow device 10 shown in FIGS. 1-9
has at least one planar wall, the orifice cross-sectional area is
linearly proportional to the position of this sliding element. By
monitoring the Hall effect sensor output, the microprocessor 102
in flow meter assembly can determine the orifice opening area, which
is one index of the example array shown in FIG. 10. The microprocessor
102 is configured to read the pressure sensors 122 124 each time
it performs a flow calculation. By reading the pressure sensors
122 124 and calculating the differential pressure, the microprocessor
102 determines the pressure value for the second index of the array
shown in FIG. 10. For values between the differential pressure and
position points listed on the array, simple linear interpolation
can be used to determine an exact discharge coefficient value between
values in the array.
 The two dimensional array of values for discharge coefficient
shown in FIG. 10 may be determined by design and stored in the program
memory 134 of each flow meter. The discharge coefficient values
in the array may also be determined by testing for each individual
flow meter manufactured, which would provide a unique and more accurate
array for each flow meter. Array values may be stored in the nonvolatile
memory (NVROM) 132 or other memory associated with the flow meter
assembly 100. Other embodiments may include arrays having indices
that extend in range beyond the range shown in FIG. 10 and may
include more or less resolution depending upon the level of accuracy
desired. Also, since the change in discharge coefficient versus
orifice opening and flow rate is not linear (see simulated results
of FIGS. 16 and 17), the values on each axis of the array need not
be linear. In this way, the array can be kept as small as possible
to reduce memory requirements while maintaining the desired accuracy.
Further, although the position of the orifice is linearly proportional
to the cross-sectional area of the orifice in the flow device 10
other embodiments may not include a linear relationship between
the area and linear position of the movable element. In such embodiments,
the index of the array could be either a position of the movable
element or the cross-sectional area of the orifice.
 FIG. 18 further illustrates the relationship between discharge
coefficient and flow rate as a plot of know values that is presented
in the technical book: JOHN A. ROBERSON AND LAYTON T. CROWE, ENGINEERING
FLUID MECHANICS, at 612 (1993). FIG. 18 plots the flow coefficient
(K) as the y-axis and the Reynolds Number (Re) as the x-axis, wherein
the flow coefficient and Reynolds Number relate to the discharge
coefficient (C) and the flow rate (Q), respectively, as follows
in Equations 2 and 3: 2 K = C [ 1 - ( A o A p ) 2 ] 1 2 Equation
2 Re d = 4 Q d Equation 3
 FIG. 18 also plots across the top axis a relationship between
the Reynolds Number and flow coefficient as follows in Equation
4: 3 Re d K = ( 2 g h ) 1 2 d v = d v ( 2 P ) 1 2 Equation 4
 Re.sub.d=Reynolds Number in the orifice section
 D=diameter of the pipe
 d=diameter of the orifice (rectangle orifice related to
d using hydraulic radius)
 .nu.=kinematic viscosity of the fluid
 .rho.=density of the fluid
 For fixed values of the orifice diameter and the kinematic
viscosity, the Reynolds Number changes only with a change in flow
rate. The coefficient relationships provided in equations 2-4 result
in FIG. 18 essentially representing the change in discharge coefficient
versus flow rate.
 The individual curves shown in FIG. 18 illustrate the significant
changes in discharge coefficient that result as flow rate (represented
as Reynolds Number) increases from low flow rates to high flow rates
for a given orifice-to-pipe size ratio. The difference from one
curve to the next in FIG. 18 represents the change in discharge
coefficient that occurs when the size of the orifice is changed
with respect to the pipe size.
 The flow chart in FIG. 12 illustrates the steps involved
in performing a flow calculation and updating input/output (I/O)
for a given flow device. Some steps in the flow calculation include
determining discharge or flow coefficients. These steps may be implemented
in programming stored in local memory (e.g., program memory 134)
or may be downloaded or transmitted to the microcontroller 102.
The process may begin by reading a position sensor to determine
a position of the flow meter movable element thereby determining
an area of the variable orifice. Reading the pressure sensors may
be performed before, after, or concurrently with reading the position
sensor. The pressure and position readings are used as indices to
look up a discharge coefficient in a stored array of discharge coefficients.
The discharge coefficient is interpolated if necessary to determine
an exact discharge coefficient value. The discharge coefficient
is then input into a flow equation and a fluid flow is calculated
from the flow equation. The analog and digital output variables
can then be updated using the fluid flow and sensor readings. If
there are no changes in the orifice position or any interrupts to
service of the flow device, the cycle repeats with a new pressure
reading. If there are interrupts to service or a change in the position
sensor, the cycle repeats from the beginning.
 The two dimensional array of discharge coefficient values
shown in FIG. 10 may also be determined by the microprocessor 102
using a polynomial that inputs the orifice opening size and the
differential pressure as variables. An example polynomial is listed
below as Equation 5.
ltidot.(A.sub.o).sup.2+E.multidot.(A.sub.o)+F Equation 5
 A, B, D, E and F=constants
 C=orifice discharge coefficient
 A.sub.o=cross-sectional area of the orifice
 P.sub.hi=upstream pressure
 P.sub.lo=downstream pressure
 The constants A, B, D, E and F are typically determined
during manufacture when characterizing differential pressure and
the discharge coefficient versus orifice opening. The constants
rather than array values may be stored in the memory of the flow
meter assembly 100. The polynomial coefficients could be determined
by design and be the same for each flow meter manufactured and stored
in program memory 134 or the coefficients could be determined by
calibration and be unique for each flow meter manufactured and then
stored in nonvolatile memory 132.
 The flow chart in FIG. 13 illustrates the steps for determining
an optimum discharge coefficient and performing a flow calculation
using a polynomial. The process may begin by reading a position
sensor to determine a position of the flow meter movable element
thereby determining an area of the variable orifice. Reading the
pressure sensors may be performed before, after, or concurrently
with reading the position sensor. The pressure and position readings
are input into a polynomial and an exact discharge coefficient is
calculated. The calculated discharge coefficient is input into a
flow equation and a fluid flow is calculated from the flow equation.
The analog and digital output variables can then be updated using
the fluid flow and sensor readings. If there are no changes in the
orifice position or any interrupts to service of the flow device,
the cycle repeats by taking new pressure sensor readings. If there
are interrupts to service or a change in the position sensor reading,
the cycle repeats from the beginning.
 Principles of the present invention also encompass a device
that can function either as a variable orifice flow meter or as
a flow controller. The electrical hardware for a variable orifice
flow meter and a flow controller may be similar. One difference
between a flow metering and a flow control device involves additional
software functionality required for the flow controller. FIG. 11
is a block diagram showing the basic features of a flow device 200.
Device 200 includes a controller 210 a control valve 212 and a
flow meter 214. The controller 210 may include software that compares
the desired flow set point to a measured flow rate measured by the
flow meter 214. The controller 210 then sends a signal to the control
valve 212 to vary an orifice opening to increase or decrease flow
rate as required to meet the flow set point. Flow metering is performed
in the same or a similar way for the flow controller as performed
for the variable orifice flow meter 10 described above. Therefore,
the two-dimensional correction for the discharge coefficient of
the orifice via the array (e.g., see FIG. 10) or a polynomial (e.g.,
Equation 5) can be used both in the variable orifice flow meter
10 and in the flow meter function in the flow controller 200.
B. EXAMPLE #2
 Compensation for Viscosity and Density Changes in the Process
 Typical flow calculations based on the measured differential
pressure (.DELTA.P) across an orifice contain a process liquid density
term, but do not contain a viscosity term (see Equation 1). As a
result, the calculations are only accurate for the single viscosity
at which the device was calibrated.
 For a fixed orifice size/geometry and a limited range of
differential pressures, the discharge coefficient C in Equation
1 is a constant determined experimentally. For a device with a variable
orifice size the discharge coefficient C is no longer a constant
but must now be a function of the orifice size. In the simplest
case, the discharge coefficient is a function only of orifice size
C=.function.(H.sub.o) Equation 6
 In this case an implementation for function .function. must
be determined empirically by measuring Q at a number of orifice
sizes and then fitting an interpolant or approximant to the values
of C calculated from the measured Q values in Equation 1. Methods
for this might include fitting a polynomial or spline curve to the
data or piecewise linear interpolation between data points. In any
method, the goal is to fit the curve defined by .function. to a
set of two dimensional (2D) data points that are measured experimentally
 To provide increased accuracy over a wide range of pressures,
the above procedure can be expanded to include characterization
of C over a range of pressures (.DELTA.P) as well as a range of
orifice sizes (H.sub.o). The discharge coefficient is then a function
of both orifice size and differential pressure, and .function. is
now an interpolant or approximant that is fit to a set of points
in three dimensions rather than two:
C=.function.(H.sub.o,.DELTA.P) Equation 7
 In the previous examples, a constant viscosity is assumed.
If the viscosity of the process liquid is different than that which
was used to characterize C according to Equations 6 and 7 then
errors in the calculated flow rate will result. The following characterization
methods have been developed according to principles of the invention
to provide a more general solution for liquids of different viscosities.
 One solution to the problem of accounting for viscosity
is to add another dimension to the domain of .function. and make
it also a function of viscosity .nu.:
C=.function.(H.sub.o,.DELTA.P,.nu.) Equation 8
 While this works fine in theory, it introduces at least
the following difficult problems in practice:
 The number of points where Q must be measured and C determined
increases by an order of magnitude. Characterizing C for a set of
10 pressures and 10 orifice sizes requires 100 test runs. Adding
a set of 10 viscosities requires a total of 1000 test runs.
 Changing and verifying pressure and orifice size are simple
operations that take a few seconds. Changing viscosity requires
emptying the liquid from the test system and refilling it with a
different liquid. After it is filled, the viscosity must then be
verified. This is a time consuming and labor-intensive process that
would increase the time required to calibrate each unit produced
from hours to days.
 In order to avoid the need to characterize each device over
a range of viscosities, a different flow equation is used [Roberson
and Crowe, 1993 p 612]: 4 Q = K d 2 4 ( 2 P ) 1 2 Equation 9
 Q=volumetric flow rate
 K=flow coefficient
 d=hydraulic diameter of the orifice
 .DELTA.P=differential pressure across the orifice
 .sigma.=density of the fluid
 In this equation, the hydraulic diameter of the orifice
(d) is calculated from the height and width of the orifice (H.sub.o
and W.sub.o). The hydraulic radius of a rectangular orifice is a
function of area and perimeter, otherwise known as the orifice geometry
[Roberson and Crowe, 1993 equations 10.3 10.35]: 5 r 2 = A P Equation
 The diameter, which is twice the radius, is then calculated
from the orifice height and width: 6 d = 2 H o W o H o + W o Equation
 One disadvantage with flow calculations using Equation 8
is that the discharge coefficient C needed to be characterized over
three independent variables: orifice size, differential pressure,
and viscosity. Characterizing over viscosity is a difficult process,
so a way is needed to incorporate viscosity into the equation in
a way that eliminates the need for viscosity characterization. While
Equation 9 above does not appear to take viscosity into account,
it can easily be made to do so using the characteristics of K, the
flow coefficient [Roberson and Crowe, 1993 p. 612], as shown in
Equation 4 above:
 1. The flow coefficient K is known to be a function of orifice
size and Reynolds number R.sub.e within the orifice.
 2. K can be treated as a function of the value R.sub.e/K.
 3. The value of R.sub.e/K can be calculated based on orifice
size, differential pressure, density, and viscosity (see Equation
 Since the calculation of R.sub.e/K includes the effects
of viscosity, K need only be characterized as a function of orifice
size and R.sub.e/K: 7 K = f ( H o , R e K ) Equation 12
 The characterization of the function K=.function.(H.sub.o,
R.sub.e/K) is achieved by empirical measurement of the volumetric
flow rate Q at a range of values of orifice size H.sub.o and differential
pressure .DELTA.P. For each set of measured values for (Q, H.sub.o,
.DELTA.P) determined during calibration (e.g., calibration data
points), the values of R.sub.e/K and K are calculated. R.sub.e/K
is calculated using Equation 4 and K is calculated by substituting
Equation 11 into Equation 9 and solving for K in terms of the measured
values of Q, .DELTA.P, H.sub.o: 8 K = Q ( H o W o H o + W o ) 2
( 2 P ) 1 2 Equation 13
 This gives a set of values for K, H.sub.o, and R.sub.e/K
to which an approximant for the function .function. from Equation
12 can be fitted. A typical set of data points to be approximated
is show in FIG. 19.
 Another important step is to determine an implementation
for the function .function.(H.sub.o, R.sub.e/K). The general form
of the function needs be the same from one unit to the next so that
the same version of firmware can be installed in all units. However,
the exact shape of the surface may vary slightly due to mechanical
manufacturing tolerances. This can be accomplished by downloading
a set of constants (e.g., the table of values shown in FIG. 20 representing
some measured data taken during calibration of the device) that
are used by the firmware to calculate the exact shape of the surface.
To avoid contributing any significant error to the flow calculation,
the method chosen to approximate the surface in one preferred embodiment
needs to reproduce the K values with an error of less than 0.1%.
 A number of implementation approaches are available, but
each results in some significant drawbacks:
 Bivariate Polynomials--In order to obtain a fit within the
desired error bounds, high order polynomials were required. These
polynomials were usually ill-behaved outside the bounds of the measured
data set and often were ill-behaved between data points within the
data set. High order polynomials were also computationally intensive
and the time required to evaluate them would not allow other desirable
features to be implemented.
 Bicubic Spline Surfaces--Spline surfaces require gridded
data (where the data points all lie on a intersections of a rectangular
grid). The practical limitations of gathering calibration data in
a production environment produce data points that are scattered
in one of the independent variable's axis. Adapting spline surfaces
to work with scattered data resulted in poor fits and surfaces that
were not well behaved. Additionally, spline surfaces are also computationally
 Triangulation--A triangulated surface is simple to evaluate,
works well with scattered data, is well-behaved between data points,
and can be made to be well behaved outside the data set. However,
errors for highly curved surfaces can become large, and for a uniformly
concave or convex surface the errors are all in one direction (the
average of the errors does not tend towards zero). FIG. 21 is a
two dimensional (2D) demonstration of the errors that occur from
triangulation. Another disadvantage of a triangulation approach
is that a large amount of data is required to represent a triangulated
surface with scattered data points.
 The combination of a low-order univariate polynomial with
a special case of a triangulated surface was found to meet implementation
objectives for the function .function.(H.sub.o, R.sub.e/K). This
combination provides a good fit, is well-behaved, and requires a
minimal computation time.
 The errors produced with a triangulated surface are proportional
to the curvature of the surface and the distance between the measured
data points which are the vertexes of the triangles. The error can
be reduced by flattening the surface or increasing the number of
measured data points. Since increasing the number of measured data
points increases test time (and therefore manufacturing costs),
a polynomial is used to "flatten" the surface.
 The flattening or unrolling of the curved surface is done
by fitting a polynomial surface to the measured data using a least-squares
algorithm. The surface to be triangulated is then defined by the
residual values (the differences between the data points and the
polynomial surface). Since the change in K is more dependent on
H.sub.o than R.sub.e/K, a univariate polynomial in H.sub.o is used.
The function .function.(H.sub.o, R.sub.e/K) now consists of the
sum of two terms: 9 K = p ( H o ) + T ( H o , R e K ) Equation 14
 p(H.sub.o)=value of the polynomial at H.sub.o
 T(H.sub.o, R.sub.e/K=value of the triangulated surface at
 FIGS. 22A (two dimensional) and 22B (three dimensional)
illustrate a polynomial curve fit to the set of data points shown
in FIG. 19. The polynomial is a third order polynomial with an additional
reciprocal term of the form: 10 ax 3 + bx 2 + cx + d + e x + f Equation
 The triangulated surface is then defined by the differences
between the original data and the polynomial. Some example residual
values are shown in FIG. 23.
 The use of a polynomial's residual values for a triangulated
surface rather than the raw data values provides at least the following
 The surface to be triangulated is no longer concave, resulting
in interpolation errors that are both positive and negative and
will have an average near zero.
 The slope of the surface is significantly reduced in one
direction so that the magnitude of the interpolation errors is reduced.
 The combined value for the triangle surfaces are no longer
flat, but have the same characteristic curve as the data set. This
further reduces the magnitude of the interpolation errors.
 Preliminary tests indicate that the interpolation errors
in a combined approach can be less than half the magnitude of those
when triangulation is used alone.
 The triangulation of the residual surface may be done on-the-fly
at run time or it may be done external to the device and the resulting
list of triangles downloaded into non-volatile memory. Using the
former approach can be advantageous in that it limits the amount
of non-volatile storage needed in the device. To further reduce
storage requirements and simplify the triangulation algorithm, two
additional constraints were placed on the data set and triangulation
 The data points may only be scattered in the R.sub.e/K axis:
the H.sub.o values will be limited to a set of discrete values.
In other words, the data will be "semi-gridded" where
points lie on grid lines in one axis (the grid lines may be irregularly
spaced), but are scattered in the other axis. FIGS. 24A and 24B
illustrate the difference between scattered and semi-gridded data.
This approach reduces the storage space required by approximately
30% since only one copy of each of the unique H.sub.o values need
 The triangulation algorithm will be limited to using vertexes
for a triangle that are either on the same or adjacent grid lines.
This results in an execution time for a Delaunay triangulation that
is O(N) with respect to the number of points per grid line rather
than O(NlogN) with respect to the total number of points, which
is the case for a more general Delaunay triangulation algorithm.
 FIGS. 25A and 25B illustrate a "top view" of the
sample data set and the resulting triangulation. FIG. 26 illustrates
the triangulated residual surface that is summed with the polynomial
surface of FIGS. 22A and 22B to obtain the final value for K as
a function of H.sub.o and R.sub.e/K.
 With the two previously mentioned constraints in place,
the on-the-fly triangulation algorithm is more simple than a general
case Delaunay triangulation algorithm. Since the purpose of triangulating
the surface is to evaluate the surface at a particular value of
(H.sub.o, R.sub.e/K), it is sufficient to be able to find the triangle
containing that point.
 Finding the triangle containing a given point consists of
two parts. First, find the values of H.sub.o in the data set (the
vertical "grid lines" seen in FIGS. 25A and 25B) that
surround the H.sub.o value where the surface is to be evaluated.
This gives two sets of data points that lie on two parallel lines.
Second, find the triangle that contains the point (H.sub.o, R.sub.e/K)
in the triangulation of the region between the two parallels lines.
Each triangle will comprise one line segment defined by two adjacent
points on one of the two parallel lines (drawn vertically in FIG.
27), and two line segments (called "rungs" in the description
below) that connect the endpoints of the vertical segment with one
of the data points on the opposite parallel line (drawn generally
horizontal in FIG. 27).
 The process for triangulating the region includes creating
the first rung using the bottom two points on each of the two parallels
(see FIG. 27). The algorithm may be started at either the top or
bottom as long as the starting end is chosen consistently. The next
rung is determined by first determining the distance from each end
of the most recently created rung to the next point on the opposite
parallel (shown as double-arrowed lines in FIG. 27), and then creating
the next rung from the shortest of the two segments from the previous
step. Finally, the triangle containing the point (H.sub.o, R.sub.e/K)
is defined by the two most recent rungs. There may be error handling
the cases where the point is outside of the triangulated region.
 Once the triangle is found, the .DELTA.K value is determined
by the Z coordinate of the point obtained by projecting (H.sub.o,
R.sub.e/K) onto the plane defined by the vertexes of the triangle.
The .DELTA.K value is added to the approximate K value generated
by the polynomial shown in Equation 15 thus yielding a final flow
coefficient value K. The final flow coefficient K is used to calculate
volumetric flow rate using Equation 9 according to the basic process
steps set forth in the flow diagram of FIG. 28. The process represented
in FIG. 28 may be especially useful as steps of a computer program
that is used to operate a variable orifice flow metering device.
Thus, volumetric flow rate can be determined according to this example
system and method for a range of orifice geometries, Reynolds numbers,
and fluid densities and viscosities.
 The example flow devices and software correction systems
described herein are exemplary of apparatuses and methods for improving
the accuracy of flow measurements in a variable orifice flow meter.
The method includes simultaneously characterizing the discharge
or flow coefficient of the orifice for different orifice openings
and flow rates, while accounting for properties of the fluid such
as viscosity and density. By characterizing the discharge or flow
coefficient of the orifice for these parameters and correcting for
them in the flow calculation, the flow meter is able to maintain
flow measurement accuracy over a broad range of flow rates. In this
way, the flow meter may be useful for flow ranges of up to or exceeding
10 times the flow range of conventional differential pressure flow
meters and perform accurately over that entire flow range.
 The above specification, examples and data provide a complete
description of the manufacture and use of the composition of the
invention. Since many embodiments of the invention may be made without
departing from the spirit and scope of the invention, the invention
resides in the claims hereinafter appended.