Figure 8: Linear least square error reduction for training data.
Results give \(\beta_{o}\) = 1.2082, and \(\beta_{1}\)= -0.3241. Eq (16)
is then modified to yield \(C_{\text{LLS}}\):
\(C_{\text{LLS}}=\beta_{o}+\beta_{1}{\overset{\overline{}}{v}}_{T\text{est}}\)… (19)
where \({\overset{\overline{}}{v}}_{T\text{est}}\) is the test data
given by transit time ultrasonic flow meter.
Next, the revised flow coefficient \(C_{\text{LLS}}\) was applied to the
correction equation, which corrects the value of ultrasonic flow
measurement.
\(v_{\text{Rev}}=\frac{v_{T\text{est}}}{C_{\text{LLS}}}\) …
(20)
By applying the correction equation, the mean flow coefficient\(\overset{\overline{}}{C_{\text{\ Rev}}}\) has been revised from 0.917
to 0.957. As shown in Table 2, the mean relative error has reduced from
-0.0830 (-8.3%) to -0.0422 (-4.2%). However, the revised standard
deviation of relative error \({\sigma_{\text{Rev}}}_{\text{\ \ }}\)is
about the same at ±0.2004 (20.04%). This indicates the LLS regression
method has reduced both the systematic error and the relative error only
slightly.