Linearity of Coriolis
Mass Flowmeters
Dean Standiford
Director Global Calibration Quality
Overview
!!
Purpose:
–! To answer the customer’s question, “How do I know
the output from my Coriolis flow meter is valid beyond
the calibrated mass flow range?”
!!
Scope:
–! High Capacity Coriolis flow meters
•! 6”, 8”, 10”
•! >25% of maximum flow range
•! Based on water calibration
March 26, 2014
2
1!
Summary
Coriolis flowmeters are inherently linear devices.
Each Coriolis flowmeter has a unique mass flow
calibration constant (flow calibration factor, FCF),
that when determined correctly, is valid for the
entire operating mass flow range.
March 26, 2014
3
Agenda
!!
Coriolis mass flow equation
!!
Calibration processes
!!
Mass Flow Linearity Results
!!
Validating the Calibration
!!
Conclusion
March 26, 2014
4
2!
Mass Flow w/Coriolis Flowmeters
!!
Basic Measurement
Measuring tubes are forced to oscillate producing a sine wave. At zero flow, the two tubes
vibrate in phase with each other. When flow is introduced, the Coriolis forces cause the
tubes to twist resulting in a phase shift. The time difference between the sine waves is
measured and is directly proportional to mass flow rate.
!"#$%&'()%
!"#$%"&#
&'()$(%*"+,-..%
/"01)2+$#$3(%
43)$(%*"+,-..%
/"01)2+$#$3(%
+,-.$-%/"*01''%
&"23.4*$#$)-%
5).$-%/"*01''%
&"23.4*$#$)-%
!"#$%
!"#$%
!"#$%&"''$($)*$%
March 26, 2014
5
Coriolis Mass Flow Equation
!!
Mathematical Derivation
Force:
"!#$%&
"#!
""#'()
"#$%&'(
"#+!
!
""#)*
(1) Coriolis Force, named
for Gustav Coriolis
(2)
Because the direction of the flow changes between
the inlet and outlet tube, the direction of the force
changes.
As the tube vibrates about axis “o”, an
oscillation moment develops:
%%%&$ '$ ( %%%%&
!"#
#) ') !
(3)
####$& '(')" % )& !
!" % !
Since: ###$
*%,!
! " #$% & " #&'#()
*%+
, combine equations (2) & (3)
'(!
'()
or ! " #$%&
(4)
March 26, 2014
6
3!
Coriolis Mass Flow Equation
!!
Mathematical Derivation cont’d.
'(!
'()
Making the substitution into: ! " #$%&
%&
!$
!
'
!" #
(5)
Equation (4) becomes:
(6)
! " #$
%&'()*!
The moment “M” is being resisted by the mechanical
properties of the sensor tube. The torque on the tube
is given by:
(7)
! " #$ %!
Where: !" ! is the stiffness of the tube & !! is the
amount of twist (in degrees). By definition:
! " #! so: !" # $ %&
'()*+,!
(8)
$% &
!" #
!
'(
)*+,
(9)
March 26, 2014
7
Coriolis Mass Flow Equation
!!
Mathematical Derivation cont’d.
By definition:
!" #
$
!
%
&'
(10)
$ &
%
!
Substituting (10) into (9): !" # '(
)*+,
$%
) *+!
&'(
Coriolis Mass Flowmeter
(11)
All of the values (except !"!) are constants that when
lumped together MMI calls the flow calibration factor:
(12)
!" # $%$ & '(!
!" # $%$ & '() * ()+ ,!
!" # $%$ & '( ) *$%$ & '(+ ,!
30000
25000
Reference Mass Flow, kg/min
!" #
20000
15000
10000
5000
equation of a line: ! " # $ % & '!
! " #$%&' ( )*)!
0
! " #$%&'(&)% * +,-.- / 0%1 2!
0
5000
10000
15000
20000
Coriolis Mass Flow, kg/min
25000
30000
March 26, 2014
8
4!
Calibration Processes
!!
Calibration
operation that, under specified conditions, in a first step, establishes a relation
between the quantity values with measurement uncertainties provided by
measurement standards and corresponding indications with associated
measurement uncertainties and, in a second step, uses this information to
establish a relation for obtaining a measurement result from an indication.
NOTE 1 A calibration may be expressed by a statement, calibration function, calibration
diagram, calibration curve, or calibration table. In some cases, it may consist of an
additive or multiplicative correction of the indication with associated measurement
uncertainty.
NOTE 2 Calibration should not be confused with adjustment of a measuring system, often
mistakenly called “self-calibration”, nor with verification of calibration
VIM – International vocabulary of metrology, Basic and general concepts and terms (JCGM 200:2008)
March 26, 2014
9
Calibration Processes
!!
calibration - The comparison of measuring
equipment of unknown measurement uncertainty
to a reference standard of known measurement
uncertainty to determine an estimate of the error.
–! Such as: comparing the reading of a flow meter to a calibrated
reference to determine the estimate of error (step 1), which can
then be used to adjust the flow meter output if required (step 2).
!!
verification - Evidence by calibration that
specified requirements have been met.
–! A decision made by analyzing the calibration data.
March 26, 2014
10
5!
Calibration Processes
!!
Micro Motion mass flow calibration process for a
Coriolis Mass Flowmeter
–! Ensure stable process conditions
–! Determine Coriolis zero offset
•! Intercept
!" # $%$ & '( ) *$%$ & '(+ ,!
–! Determine mass flow error at the best flowrate for
adjusting FCF
–! Make an adjustment to the FCF
•! Slope
!" # $%$ & '( ) *$%$ & '(+ ,!
–! Verify the adjustment is valid
March 26, 2014
11
Calibration Processes
!!
What is the “best flowrate” for adjusting the FCF?
–! Because Coriolis flowmeters are linear devices:
•! Any flowrate where errors are minimized
–! Zero stability
–! Process repeatability
–! System Limitations
March 26, 2014
12
6!
Calibration Processes
254cm (10”)+ line sizes
186kw (250hp) + pumps
30t (66,139lb) weigh systems
Multiple reference meters
March 26, 2014
13
Calibration Process – FCF Adjustment
Coriolis Mass Flowmeter
0.50
50% of Maximum Flow Rate = Nominal Flow Rate (~1 bar dp)
0.40
<#%'5':".
=>= ?.@/-%5"0%* !$0A"
0.30
Typical Error, %
0.20
0.10
0.00
-0.10
-0.20
!"#"$%$&'('%)*"++,+-*./"*%,*'01+"$-".*2(,3* 0,'-"
$0.*-4,+%"+*&$%14* %'5"-
-0.30
-0.40
-0.50
6++,+- ./"*%,* 789*,2*:"+,*-%$&'('%)*-#"1'2'1$%',0*;$(/"
0
10
20
30
40
50
60
70
Flow Rate, % Maximum
80
90
100
110
March 26, 2014
14
7!
Mass Flow Linearity Results
CMF300 S/N: 11005156
0.50
3.5
0.40
3.0
!"#$%&'()*)(&+,-./012/(3'456&/1(
0.30
Mass Error, % of Reading
0.10
2.0
0.00
1.5
-0.10
-0.20
1.0
-0.30
761..-61(864#
-0.40
-0.50
Pressure Drop, bar
2.5
0.20
0.5
50% of Maximum Flow Rate = Nominal Flow Rate (~1bar dp)
20
30
40
50
60
70
Flow Rate, % Maximum
Test #1
Test #2
80
90
100
0.0
Test #3
March 26, 2014
15
Mass Flow Linearity Results
CMFHC2 S/N: 13053120
0.50
0.40
!"#$%&'()*)(&+,-./012/(3'456&/1(7899(:;<0$2
0.30
Mass Error, % of Reading
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
2,000
4,000
6,000
8,000
Flow Rate, kg/min
Test #1
10,000
12,000
14,000
16,000
Test #2
March 26, 2014
16
8!
Mass Flow Linearity Results
CMFHC3 S/N: 12074130
0.50
0.40
0.30
Typical FCF adjustment flowrate = 12,000 kg/min
Mass Error, % of Reading
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
2000
4000
6000
8000
10000
12000
Mass Flow Rate, kg/min
USA
14000
Netherlands
16000
18000
20000
China
March 26, 2014
17
Mass Flow Linearity Results
CMFHC4 S/N: 12074922
0.50
0.40
0.30
Typical FCF adjustment flowrate = 12,000 kg/min
Mass Error, % of Reading
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
5000
10000
USA
15000
Flow Rate, kg/min
China Test #1
20000
25000
30000
China Test #2
March 26, 2014
18
9!
Validating the Calibration
Coriolis mass flow meters are linear, but how does
MMI provide confidence to its customers about the
data?
!! Gather Historical Data
–! 30+ production meter calibration samples
–! FCF calibrated at a flow rate less than maximum
!!
Analyze data to a 95% confidence level for
conclusions
March 26, 2014
19
Validating the Calibration
CMFHC3
FCF Calibration at Boulder and then Verified at Higher Flowrate at Ede or AFTC
0.50
0.40
0.30
Batch Error, %
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
5000
USA- FCF Calibration
10000
15000
Flow Rate, kg/min
Netherlands- FCF Verify
20000
25000
China- FCF Verify
March 26, 2014
20
10!
Validating the Calibration
Summary CMFHC3 FCF Verify Error %
A nderson-Darling N ormality Test
-0.04
0.00
0.04
0.08
A -S quared
P -V alue
0.46
0.240
M ean
S tDev
V ariance
S kew ness
Kurtosis
N
0.008767
0.031685
0.001004
-0.154286
0.245871
30
M inimum
1st Q uartile
M edian
3rd Q uartile
M aximum
-0.061000
-0.007925
0.009600
0.024375
0.083300
95% C onfidence Interv al for M ean
-0.003065
0.020598
95% C onfidence Interv al for M edian
0.003675
0.021654
95% C onfidence Interv al for S tDev
9 5 % C onfidence Inter vals
0.025234
Mean
0.042594
RSS of lab uncertainties = 0.042%
Median
0.000
0.005
0.010
0.015
0.020
0.025
March 26, 2014
21
Validating the Calibration
CMFHC4
FCF Calibration at Boulder and then Verified at Higher Flowrate at Ede or AFTC
0.50
0.40
0.30
Batch Error, %
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
5000
10000
USA- FCF Calibration
15000
20000
Flow Rate, kg/min
Netherlands- FCF Verify
25000
30000
35000
China- FCF Verify
March 26, 2014
22
11!
Validating the Calibration
Summary CMFHC4 FCF Verify Error %
A nderson-Darling N ormality Test
-0.10
-0.05
0.00
0.05
A -S quared
P -V alue
0.27
0.659
M ean
S tDev
V ariance
S kew ness
Kurtosis
N
-0.022345
0.040204
0.001616
0.404551
0.565730
33
M inimum
1st Q uartile
M edian
3rd Q uartile
M aximum
-0.100360
-0.047830
-0.019780
-0.001220
0.076900
95% C onfidence Interv al for M ean
-0.036600
-0.008089
95% C onfidence Interv al for M edian
-0.037828
-0.009015
95% C onfidence Interv al for S tDev
9 5 % C onfidence Inter vals
0.032331
Mean
0.053177
RSS of lab uncertainties = 0.042%
Median
-0.040
-0.035
-0.030
-0.025
-0.020
-0.015
-0.010
March 26, 2014
23
Validating the Calibration - Conclusions
!!
Gather Historical Data
–! 30+ production meter calibration samples
–! FCF calibrated at a flow rate less than maximum
!!
Analyze data to a 95% confidence level for
conclusions
–! Sample data is normally distributed
–! Mean Error for Both meters is less than 0.042%
•! (RSS of flow stand uncertainty)
March 26, 2014
24
12!
Conclusion
Coriolis flowmeters are inherently linear devices.
Each Coriolis flowmeter has a unique mass flow
calibration constant (flow calibration factor, FCF),
that when determined correctly, is valid for the
entire operating mass flow range.
Mass Flow Error is within the meter specification
when FCF is calibrated at less than maximum flow rate.
March 26, 2014
25
13!
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