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.