IEEE Instrumentation & Measurement - September 2023 - 37

The fitted values of sand content for the online sand content
collection equipment were calculated according to (7), and the
relative errors were calculated according to:
E
ri =
SS
S
−
ni i
i
∗100%
(8)
where Eri is the relative error of sand online collection equipment
in the ith sand-bearing standard field; Sni is the fitted
value of sand content in the ith sand-bearing standard field,
kg/m³; and Si is the standard value of sand content in the ith
standard suspended sand concentration field (Table 3).
Based on the analysis of the measured data in Table 3, it is
concluded that the relative errors of the standard and fitted values
of sand content are larger around the very small and very
large values, as shown in Fig. 6, and the maximum relative error
can reach −3.44%, but the results meet the requirement of
10% error of the technical index of the instrument.
To verify the uncertainty of the sand content online collection
equipment, 1.0 kg/m³ was selected as an example for
calculation, and a standard field of sand content in this range
was built and tested 30 times, respectively. The measurement
results (Table 4) were analyzed according to the measurement
model:
∆∆S kX c
m
= ∗ + − +SJTS


 V


 −
(9)
where ∆S is the error in the indicated value of sand content
measurement by the sand content online collection equipment
in kg/m³; k is the slope; and X is the output current value or
turbidity value of the sand content online collecting equipment
measured in a sand content standard field. By definition,
X nxn
= 1/ 1Σ , where xi is the output current or turbidity value,
i
mA or NTU, measured by the sand content online collection
device in a sand content standard field for the ith time. Other
parameters of (9) include c, which is the intercept distance;
m is the mass of the dry sediment sample acquired in the field
in kg; V is the volume of pure water and dry sediment sample
to be weighed to configure the sand content standard field, m³;
SJ is the deviation from the indicated value introduced by the
uniformity measurement of the sand content standard field in
kg/m³; and ∆ST is the error in the value of the online sand collection
equipment introduced by the temperature variation
in kg/m³.
Since the slope k and the intercept c are correlated in a
straight line fit, the corresponding uncertainty is calculated
from (10), according to the propagation law of uncertainty, in
the following steps:
u S ck u k cX u X cc u c c(k)
∗ c c r k c u k u c c m u m c V
c
2
(∆ ) = ∗ + ∗
+
2
∗+ ∗
2( )( )( )( )( )
2
u V c S u S c ∆S u ∆ST
() ( , ) ( ) () ( )∗
22 2
( )( )( )( )( )
JJ T ∗ 2
+
The sensitivity coefficient is:
September 2023
2
2 22 2
22 ( )
2
+ ∗ +
( ) +
( )
(10)
u c uS X X X n X X
= 3.63248 10 11
( )
= −+

−
∗
kgm/ .
( ni n ii n
Σ
) () / ()2
3
2 1 22 2 1
Σ ΣΣ i
−




ck =
() =
cm
()
cSJ
() =
∂ ∆ ∆∆S
= k cc, () =
()
()
S
∂ k
∂∂
()
()
∂mV
S
∂
∂
()
()
SJ
∆∆
∆
= X cX =
S
=−=, ()
,( )
1
cV
∂ X
∂
= −1, c ((∆ST ) =
∂
()
()
S
()
()
S
= −
∂ V
∂ ∆ST
() according to:
()
()
∂ ∆S
V
2 ,
= −1
◗ The uncertainty component introduced by the slope k in
the linear fit uk
LXS =
L XX Xi
from which: kL /LXS XX =
=
XX = −= −
∑∑
∑ ∑∑
− −= −
()22 1
)( n n)
(
n (
u S( ) =
ni S kX c n
n
Σ 1(
=
i
= −= ∗
(
():
ni − − −= ∗
2 1 78 10 kg m . Then the un−14
/
3
(
) ( ) 1/[ΣΣn ii)2/ ] .8 07255 10−12
2
(kg·m mA
−3/ ).
(X X S S Xi XS
X )
i
0.6560 (kg· m mA
−3/ ), c S kXni
−2.7634 kg/m3.
Calculate the fit result Sn and the uncertainty of uSn():
)2/( ) .
certainty component of the slope k:
u k uS X X n
◗ The uncertainty component introduced by the intercept c
in the linear fit uc


) ni
2
m
∂
∂ c
()
()
= 1,
(11)
= −=
i
◗ Correlation of coefficient straight lines rk ( ,)c :
Σ 2 0.999999249.
rk c X n Xii(,) /
= Σ
=
◗ Uncertainty components introduced by the measurement
of current values from sand content online
collection equipment ( )uX . Using the measurement
uncertainty type A evaluation method to evaluate, then:
ux
( ) = ()− −=
j
Σ1
n ( /( ) .
xj X nn 1 0 001024302 mA.
2
◗ Measurement uncertainty components introduced by
volume measurements uV
(). The same 2000 mL cylinder
was used for 30 separate weighings, the inside of the
cylinder was wiped clean of moisture after each weighing
session, and the 30 weighing sessions were positively
correlated. The configuration of a standard field of 1 kg/m³
requires nearly 66 L of pure water, and the maximum allowable
error of the 2000 mL cylinder is known to be ± 10 mL,

IEEE Instrumentation & Measurement - September 2023

Table of Contents for the Digital Edition of IEEE Instrumentation & Measurement - September 2023

Contents
IEEE Instrumentation & Measurement - September 2023 - Cover1
IEEE Instrumentation & Measurement - September 2023 - Cover2
IEEE Instrumentation & Measurement - September 2023 - Contents
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