TELKOM
NIKA
, Vol. 11, No. 10, Octobe
r 2013, pp. 5
669 ~ 5
674
ISSN: 2302-4
046
5669
Re
cei
v
ed Ap
ril 13, 2013; Revi
sed
Jun
e
29, 2013; Accepted July 1
3
,
2013
A Curve
-fitting Calibration Method applied for
Ultrasonic Flow-meter
Yong Luo
1
, Rang
ding Wang
*
1
, Ling Yao
2
1
School of Infor
m
ation Sci
enc
e and En
gi
neer
ing,
Ni
ngb
o Un
iversit
y
, N
i
n
g
b
o
315
21
1, Chin
a
2
Ning
bo W
a
ter Meter CO. L
T
D, Ningb
o, 315
0
00, Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
a
ngr
ang
di
n
g
@n
bu.ed
u.cn
A
b
st
r
a
ct
As the influe
n
c
e of fluid d
i
stributi
on in th
e
in
terna
l
pi
pe,
the me
asur
e
m
e
n
t characte
ristics of
theory an
d pra
c
tice exist sign
ificant differe
nc
es in
Ultraso
ni
c F
l
ow
-meter (USF
). T
h
rough
analysis
of flui
d
state, the meth
od of curve-fitti
ng is
ap
pli
ed f
o
r the cali
brati
on of USF
.
Experi
m
e
n
tal res
u
lts show
that the
USF
can ach
i
e
v
e leve
l-1 acc
u
racy w
i
th
just a
correction
of 5
flow
points
,
a
nd this
meth
od
perfor
m
s a l
o
w
computati
o
n
a
l
compl
e
xity and
strong practic
a
lity.
Ke
y
w
ords
:
Cu
rve-fitting, Cal
i
b
ratio
n
, Ultraso
nic F
l
ow
-met
er, R
e
y
n
o
l
d
s
n
u
m
b
e
r
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
USF u
s
e
s
the propa
gati
on ch
ar
acte
ristics of ultraso
n
ic
wave
in fluid to get the
informatio
n a
bout fluid velocity. Due to
the ultra
s
oni
c signal i
s
se
n
s
itive to external facto
r
s, a
n
d
the law of diff
erent velo
city in clo
s
ed
co
nduits
i
s
com
p
licate
d
. The
measurement
cha
r
a
c
teri
stics
of theory an
d pra
c
tice
exist signifi
cant
differen
c
es i
n
USF, whi
c
h woul
d se
ri
ously affect the
accuracy
of USF
[1
-2].
At present, based on the IS
O standa
rd o
f
<With the USF of time-transit
method to
measure the
fluid flow i
n
clo
s
e
d
pip
e
>a
nd the
Chin
ese sta
ndard <JJG1
030-
2007
USF
>
, many re
sea
r
che
r
s h
a
ve studied the
problem cau
s
e
d
by the heavy workl
oad
of
calib
ration, b
u
t these meth
ods
still can’t
adapt to the mass produ
ct
ion [3-4].
In this pa
per,
the cu
rve-fitting metho
d
i
s
ad
opted fo
r the
calib
rati
on of USF
after the
analysi
s
of flu
i
d state. Thi
s
method
not o
n
ly can
acqui
re a
better
accuracy
of US
F, but also
ca
n
redu
ce the
workl
oad of cal
i
bration a
nd p
r
omote e
n
terprise produ
cti
on.
2. The Rea
s
on of Calibra
tion in USF
2.1. The Principle of USF
In study of USF, the time
differen
c
e me
t
hod po
ssesses a majo
rity, and who
s
e p
r
inci
ple
of mea
s
u
r
e i
s
sho
w
n i
n
F
i
gure
1. Th
e
ultrasoni
c tra
n
sd
ucer1
(
U
1
)
an
d ultraso
n
ic tran
sdu
c
e
r
2
(
U
2
)
alternat
ely transmit or re
ceive ul
t
r
asoni
c pul
se
s (with a
n
incide
nt angle
). Because the
ultrasoni
c vel
o
city in the d
o
wn
stre
am a
nd up
stre
am
i
s
differe
nt, the time for the
ultrasoni
c si
gnal
to
r
e
ac
h
th
e tw
o
u
l
tr
as
on
ic
tr
an
sd
uce
r
s
is
different, with a time difference
T
. The linea
r
averag
e velo
city
v
L
is obtai
ned by ca
pturing this time differen
c
e
2
2t
a
n
L
C
VT
D
(1)
max
0
1
max
1
)
(
1
)
(
1
V
n
n
dr
R
r
R
V
R
dl
r
V
L
V
R
n
L
L
(2)
Whe
r
e
C
is
the veloc
i
ty of ultras
onic
wav
e
in fluid,
D
is the diameter
of this pipe.
Becau
s
e vel
o
city from E
q
.(1) i
s
line
a
r
avera
ge ve
locity, the volume flow
sh
ould be
comp
uted u
s
i
ng profile av
erag
e velocit
y
. The fl
ow of fluid in the
pipe present
s a non-unifo
r
m
distrib
u
tion,
so, the mea
s
u
r
ed lin
ea
r vel
o
city diffe
re
ntiates the
profile velocity.
Some sch
o
la
rs
have rai
s
ed
method of measure by mu
lti-cha
nnel
so
that the measured data
ca
n be better cl
ose
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 566
9 –
5674
5670
to the real values. But it still needs to be
amende
d by software co
mpen
sation,
and will b
r
ing
on
the com
p
lexity of design a
nd incre
a
se the co
s
t. To fully unde
rsta
nd the chara
c
teri
stics of the
media me
asured i
s
the
premi
s
e of
accurate
me
asu
r
em
ent. Hen
c
e, the
analysi
s
of fluid
characteri
stics is an essent
ial link in
imp
r
oving the accura
cy of USF [5-6].
2.2. The Ana
l
y
s
is of Flo
w
Fluid Chara
c
teris
t
ics
Und
e
r th
e ide
a
l co
ndition
s
of that the pi
pe wa
ll i
s
sm
ooth an
d the
fluid viscosity
is
zero,
the flow
rate
of fluid in pi
pe is in u
n
ifo
r
m di
strib
u
tio
n
, as
sh
own
in Figu
re 2
(
a
)
. The
adja
c
e
n
t
particl
e of fluid has
sh
ear
stre
ss be
cau
s
e of the ex
i
s
ten
c
e of viscou
s effe
ct in the actual fl
uid.
Whe
n
the fluid flows, the velocity
is ze
ro
at the wall,
which in
crea
se
s while a
w
ay
from the wall
to
the axis, and
is shown in
Figure 2(b).
Whe
n
the fl
ui
d is in the
st
ate of full de
velopment of
flow,
the velocity
distrib
u
tion
o
f
fluid ch
ang
es al
ong
wit
h
the in
crea
se of velo
city, mean
whil
e, the
Reynol
ds numbers
incr
ease. Velocity will go
through three stat
es
－
l
a
mina
r, transie
nt a
nd
turbule
n
t, whi
c
h will di
re
ctly lead to the diffe
ren
c
e
s
b
e
twee
n profil
e-line
a
r velo
city [7-8]
Figure 1. Flow mea
s
u
r
em
ent prin
ciple
of the USF
(a
)
T
he
id
ea
l
state
(b)
T
he
a
c
tu
al
state
Figu
re 2. T
he
Di
stri
buti
on
of Flui
d Vel
o
city
Reyn
ol
d
s
n
u
m
b
e
r
i
s
a
unit to
de
scri
be t
h
e f
l
ui
d flo
w
by
the th
eo
ry
of flui
d
mechani
c
s.
R
s
e
VD
, where
v
S
is the profile velocity,
D
is the diamete
r
of the pipe,
is the
kinem
atic visco
s
ity. From the Reyn
old
s
numbe
r
expe
riment
s, it can be co
ncl
u
d
ed that [9-12]
:
Whe
n
Re
<
230
0
－
lamin
a
r,
the parti
cle
has a li
nea
r
motion sm
oot
hly along the
pipe axis in
parall
e
l. Rela
tionshi
p of v
e
locity bet
we
en the
par
ti
cle and
axis i
s
sho
w
ed
in
the follo
win
g
:
2
ma
x
[1
(
)
]
r
r
VV
R
, the c
o
rrec
tion c
oeffic
i
ent is
a co
nsta
nt at this situatio
n.
Whe
n
Re
>3
750
－
turbule
n
t, the fluid d
oesn’t remai
n
stratified flow any mo
re, but in all
dire
ction
s
. Relation
ship
o
f
velocity be
tween
the
p
a
rticle
an
d
axis i
s
sho
w
ed
a
s
follo
w:
1
ma
x
(1
)
n
r
r
VV
R
,
n
is a function relate
d to Reynol
ds nu
mber, r is the
distance fro
m
axis of
the pipe, R is the diamet
er of the pip
e
. T
he linear average vel
o
city and the
profile average
veloc
i
ty c
an be c
a
lc
ulated res
p
ec
tively as
follow:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Curve-fittin
g
Calib
ration
Method ap
pli
ed for Ultraso
n
ic Flo
w
-m
eter (Ran
gding
Wan
g
)
5671
max
2
1
max
)
1
)(
1
2
(
2
2
)
(
1
)
(
1
)
(
V
n
n
n
rdr
R
r
R
V
s
ds
r
V
s
t
V
s
n
s
s
(3)
()
2
21
()
s
L
Vt
n
K
n
Vt
(4)
So the coefficient of corre
c
tion ca
n be ob
tained from th
e above two
equatio
n sh
o
w
n in Eq.(4
)
10
()
1
1.
12
0.
011
*
l
o
g
R
e
()
s
L
Vt
K
Vt
(5)
max
0
2
2
max
3
2
)
1
(
1
)
(
1
V
dr
R
r
V
R
dl
r
V
L
V
R
L
L
(6)
w
h
en
23
0
0
<
Re
<37
5
0
－
transi
ent, th
e rel
a
tion
shi
p
of p
r
ofile
-linear velo
city has not
con
c
lu
ded up
to now. The chief re
ason i
s
that this
sta
t
e isn’t stable
and the velocity distributio
n
is extremely compli
cate
d. The relatio
n
ship of prof
ile-linear velo
cit
y
is difficult to expre
ss u
s
i
ng
definite functi
ons, which ca
n only be obt
ained from
ex
perim
ental eq
uation. One
commonly u
s
e
d
equatio
n is a
s
Eq.(5
)
.
For the turb
u
l
ent and tran
sient flow, the theor
eti
c
al
equatio
n of correctio
n
is h
a
rd to
impleme
n
t du
e to its
com
p
l
e
xity. In this pape
r, t
he
cu
rve-fitting of l
east-sq
ua
re i
s
ap
plied t
o
the
calibration.
3. The Calibr
a
tion Me
tho
d
We all
kn
ow
that the fluid
exists three
stat
es, lamina
r, transie
nt an
d turbul
ent flow by
previou
s
an
al
ysis. Beca
use the law of fluid move
me
nt varies wid
e
ly in each fluid state, this
pape
r intend
s to calibrate t
he USF in ea
ch fluid state
respe
c
tively.
Lamina
r
, the distrib
u
tion of
fluid velocity likes
a pa
ra
bolic
curve
shown in Figu
re 3. As is
dedu
ce
d fro
m
[9], the rel
a
tionship
of p
r
ofile ave
r
a
g
e
velocity a
nd
maximum vel
o
city is a
s
foll
ow:
ma
x
1
2
s
VV
. The linear a
v
erage velo
ci
ty can be cal
c
ulated by inte
gral a
s
follow.
ma
x
m
a
x
12
3
/
23
4
S
L
V
KV
V
V
(7)
2
1
1
[(
,
,
)
]
n
Li
n
S
i
i
D
fx
a
a
x
……
，
(8)
So the corre
c
tion coeffici
en
t can be com
puted a
s
(7
)
Turb
ulent
an
d tra
n
si
ent,
who
s
e
velo
ci
ty distrib
u
tion
is so
compli
cated
that
n
o
con
c
lu
sion
coul
d be d
r
a
w
n ab
out the
relation
shi
p
of profile-l
i
n
e
a
r velo
city, and the velo
ci
ty distribution
is
s
h
ow
n in
F
i
gu
r
e
3
.
T
o
th
is p
r
ob
le
m, th
e p
o
l
yn
o
m
ia
l i
s
used to
cali
b
r
ate the
e
rro
r of profile-lin
e
a
r
veloc
i
ty.
If to c
a
librate n+
1 flow points
, s
u
c
h
as (x
Li
，
x
Si
), where i
=
0
，
1
，
2
，
…
…
n, and
x
L
is
linear velo
cit
y
, x
S
is profile velocity. The functio
n
f(x
Li
) is deman
ded to match
the real dat
a of
flow poi
nts a
s
cl
ose a
s
po
ssi
ble. In ind
u
strial
appli
c
ation, the fun
c
tional
relatio
n
shi
p
bet
wee
n
x
L
and x
S
i
s
ofte
n obtai
ned
th
roug
h a
lot of
experi
m
ental
data
su
ch
a
s
（
x
L1
, x
S1
）
,……,(x
Ln
, x
Sn
).
Thro
ugh p
r
o
c
essing of ex
perim
ental d
a
ta, the f
unctional eq
uatio
n can b
e
de
scribe
d like t
h
is:
x
S
=f
(x
L
,a
1
,a
2
,……
，
a
n
). T
hes
e p
a
ra
met
e
rs
(i.e.
a
1
,a
2
,……
，
a
n
) det
ermin
ed by
the
lea
s
t squa
res
method
are
sele
cted
to
make
the
su
m of
squ
a
re
s of
deviatio
n
to mi
nimu
m, as de
scri
bed i
n
Eq.(8):
The nu
mbe
r
D
can reflect
the deg
ree
of fitting betwe
en fun
c
tion
x
S
=f
(
x
L
,
a
1
,
a
2
,……
，
a
n
)
and experi
m
e
n
tal
data
－
(
x
Li
，
x
Si
). Th
e v
a
lue
of
D
is smaller, th
e
co
nsi
s
ten
c
y bet
wee
n
the
val
u
e
of function an
d the data of
experim
ent is better.
These pa
ram
e
ters i
n
fitting cu
rve are
obtai
ne
d by the followi
ng
mean
s: assu
ming the
got data of these m
sampli
ng point
s are
sho
w
n Ta
ble
1
.
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ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 566
9 –
5674
5672
Table 1. sa
m
p
le point
s
x
L
x
L1
x
L2
x
L3
……
x
Lm
f(x
L
) x
S1
x
S2
x
S3
……
x
Sm
Figure 3. The
distributio
n o
f
velocity in pipe
The polyno
m
ial is used to
interpolate, i
f
p
n
(
x
L
)=
a
0
+
a
1
x
L
+…
…+
a
n
x
L
n
, and enabl
ing the
equatio
n p
n
(x
Li
)=f(x
Li
)(i=
1
,2,……
，
m), then,
1
01
1
1
01
2
2
2
01
L
m
n
Ln
S
n
L
nL
S
n
L
mn
L
S
m
aa
x
a
x
x
aa
x
a
x
x
aa
x
a
x
x
……
……
……
……
(9)
Setting an auxiliary function calle
d error
function as Eq.(10),
whi
c
h is a mu
ltivariate function related to
a
0
,
a
1
,……
，
a
n
. To get the minimum of
, method
s of
extreme valu
e of multivariate function a
r
e
ado
pted. Partial de
rivative to each in
depe
ndent
variable i
s
set
to zero a
s
sh
own in e
q
.(10
)
01
1
0
01
1
1
01
1
2[
]
0
2[
]
0
2[
]
0
i
m
n
Li
n
L
i
S
i
i
m
n
Li
n
L
i
S
i
L
i
i
m
nn
Li
n
L
S
i
Li
i
n
aa
x
a
x
x
a
aa
x
a
x
x
x
a
aa
x
a
x
x
x
a
……
……
……
……
(10
)
Eq.(10) i
s
written a
s
lin
ear e
quation
gro
up
relating
to the
unkno
wn
c
h
ar
ac
te
r(
a
0
,a
1
,……
，
a
n
).
01
21
01
12
01
1
n
Li
n
L
i
L
i
n
L
i
L
i
n
Li
Li
Si
nn
n
n
L
i
L
i
n
Li
Li
S
i
aa
x
a
x
x
ax
a
x
a
x
x
x
ax
a
x
a
x
x
x
……
……
……
……
(11
)
2
23
1
12
2
1
n
L
i
Li
Li
n
Li
Li
L
i
Li
nn
n
n
Li
Li
Li
Li
xx
x
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TELKOM
NIKA
ISSN:
2302-4
046
A Curve-fittin
g
Calib
ration
Method ap
pli
ed for Ultraso
n
ic Flo
w
-m
eter (Ran
gding
Wan
g
)
5673
Whi
c
h can be
written a
s
u=
C
,
wh
e
r
e
22
01
0
1
11
(,
,
,
)
[
]
[
(
)
]
mm
n
nL
i
n
L
i
S
i
n
L
i
S
i
ii
aa
a
a
a
x
a
x
x
p
x
x
(10)
02
[,
,
,
]
T
n
ua
a
a
,
[,
,
,
]
nT
S
i
Li
Si
Li
Si
C
x
xx
xx
(11
)
It is easy to get the solutio
n
of equation,
from
whi
c
h the cu
rve equ
ation fitted ca
n be acquired
.
If taking the
worklo
ad of
calibratio
n
an
d
accuracy int
o
acco
unt, he
re we take
m
=3,
n
=2,
and the
n
we
have
p
n
(
x
L
)=
a
0
+
a
1
x
L
+
a
2
x
L
2
. There exist
s
the com
m
on
flow point
s b
e
twee
n lamin
a
r
flow and tran
sient flow, turbulent flow
a
nd tran
sie
n
t flow, su
ch
as
sampli
ng poi
nt1 and p
o
int
3
.
So, the entire
pro
c
e
s
s of
calibratio
n
onl
y need
s
5
sa
mpling
points. Based
on t
he expe
rime
ntal
platform of DN10
0 USF, the field samp
li
ng data obtai
ned are sh
own in TABLE 2.
Figure 4. Sch
e
me of calib
ration
Table 2. Data
Sheet of Field Sampling
sampling
points
point1 point2
point3
point4
point5
v
L
0.0447
0.0581
0.0727
0.1299
0.1823
V
S
0.0335
0.0512
0.0683
0.1123
0.1745
By sampling
flow point1, p
o
int2 and
poi
nt3,
the fittin
g
cu
rve obtai
ned is
as foll
ow:
v
S
=-
0.0394
+1.8
7
0
3
v
L
-5.353
4
v
L
2
. By sampl
i
ng flow poi
nt3, poi
nt4 and poi
nt5, the fitting curve
obtaine
d is a
s
follow:
v
S
=0
.0116-0.285
1
v
L
+3
4.717
4
v
L
2
.
4. Analy
s
is o
f
Experimen
t
al Results
To test the p
e
rform
a
n
c
e o
f
the sche
m
e
,
t
he samplin
g data of three addition
al
point of
flow is
obtain
ed in e
a
ch fluid state. T
h
e re
sult
s are
sho
w
e
d
a
s
T
able 3
-
5. By the cu
rve fitting
method for
USF, calibratio
n
results fro
m
the above t
ables
can p
r
op
erly refle
c
t the true situatio
n.
Table 3. Lami
nar
Flow-points
(m/s
)
0.02010
0.02470
0.03650
Average
accur
a
c
y
Repeat-a
bility
Measured value
v
S
(m/s
)
0.02016
0.02485
0.03668
accurac
y
0.3%
0.64%
0.51%
0.48%
0.17%
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 566
9 –
5674
5674
Table 4. Tran
sient
Flow-points
(m/s
)
0.04860
0.06260
0.07140
Average
accur
a
c
y
Repeat-a
bility
Measured value
v
S
(m/s
)
0.04855
0.06277
0.07152
accurac
y
-0.09%
0.28%
0.17%
0.12%
0.19%
Table 5.Tu
rb
ulent
Flow-points
(m/s
)
0.09520
0.15830
0.17970
Average
accur
a
c
y
Repeat-a
bility
Measured value
v
S
(m/s
)
0.09551
0.15823
0.18005
accurac
y
0.33%
-0.04%
0.20%
0.16%
0.18%
Accuracy, repeatability,
respectively within
the range
of ±1%
、
±0.2
%, can meet
the requi
rem
ent
of Level-1 set by the National Metrol
ogi
cal Bureau.
5. Conclusio
n
In this pa
pe
r, the nonli
nea
r
model i
s
g
e
n
e
rated
by an
alysis
of fluid
state an
d test
of field
calib
ration,
a
nd the
cu
rve-fitting method
is a
pplie
d
to
the process
of calib
ratio
n
. As the
re
sul
t
sho
w
s, this
method n
o
t only can
gua
rantee th
e a
c
cura
cy of USF, but also
can
redu
ce
the
worklo
ad of
calibratio
n
an
d
perfo
rm lo
w
comp
utationa
l compl
e
xity.
Also, the a
c
tual ap
plicatio
n
ability is improved.
Ackn
o
w
l
e
dg
ments
This work is supp
orted
by the key indu
stri
al proj
ect of priorit
y
theme in Zhejian
g
Province (2
0
10C110
25), t
he majo
r sci
entific and
t
e
ch
nolo
g
ical
resea
r
ch proj
ect in Zh
ejia
ng
Province (Z
D20
090
12),
the
sche
me of
“ten
der
grass” talents
i
n
Zhejian
g
Province
(201
1R405
05
6).
Referen
ces
[1]
Q Liu, RD W
ang, Y Z
hu and
CT
Du.
An algorith
m
to el
i
m
inate stoch
a
sti
c
jump meas
ur
ement of USF
with tim
e
difference
m
e
thod
. Manufactur
i
ng S
y
stems
a
nd
Industr
y App
lic
ation. 20
10: 41
4-42
1.
[2]
MW Li, GS Lv
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n
i
m
pr
ovin
g the accu
racy of the ultr
ason
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-me
t
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i
th time
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e met
hod
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0
1
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rnatio
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onfe
r
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259.
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he Metro
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a, 20
07.
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h
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a
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aso
n
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w
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e
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