## 3 Results and Discussion
In order to determine the solid volume fraction of slurry without iteration, the net pressure loss (\(\Delta P\)) is regressed as the simplest polynomial function of \(\phi_{s}\):  
\[\Delta P=A\;\phi_{s}+B \tag{12}\]  
The results of regression shown in Figure 5 imply that Equation (12) well characterizes the linear relation between \(\Delta P\) and \(\phi_{s}\) for the given pair of \(U\) and orifice bore (\(d\)), despite the average value of slope \(m\) in Equation (11) being \(-0.616\), which is noticeably different from \(-1\), as shown in \[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]0 The minimum R2 value is larger than (0.99 and 0.97), when the orifice bore is (10 and 15) mm, respectively. The linear relation between \(\Delta P\) and \(\phi_{s}\) is better validated for the smaller orifice bore. \[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]1 Pressure loss coefficient as a function of the Metzner-Reed Reynolds number for the orifice bores of (**a**) 10 mm, and (**b**) 15 mm. Figures 6 and 7 present the results of regression for the slope \"_A_\" and intercept \"_B_\" of Equation (12) for the orifice bore of (10 and 15) mm, respectively. The minimum R2 values are larger than 0.99 for the slope, and 0.97 for the intercept, which values are not notable, because only four data are curve-fitted with the second order polynomial. \[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]2 Regression of net pressure loss drop as a linear function of solid volume fraction for the orifice bores of (**a**) 10 mm, and (**b**) 15 mm. \[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]3 Regression results of coefficients A (**a**) and B (**b**) in Equation (12) for the 10 mm orifice. \[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]4 _Cont._  
Empirical correlations to determine the solid volume fraction of slurry from the measured values of net pressure loss and velocity are developed separately for the (10 and 15) mm orifice bores from Equation (12) and the regression results:  
\[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]5  
\[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]6  
Figure 8 presents the reproduced solid volume fractions (\(\phi_{s}^{\prime}\)) using Equation (13) for the 10 mm bore orifice. The maximum error is 2.47 vol.%, and the standard error is 0.73 vol.%. Figure 9 presents the reproduced solid volume fractions using Equation (14) for the 15 mm bore orifice. The maximum error is 8.45 vol.%, and the standard error is 2.6 vol.%. The relatively larger errors with the 15 mm bore orifice originate from the smaller R\({}^{2}\) values (Figure 5). \[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]7 Reproduced solid volume fractions using Equation (13) for the 10 mm bore orifice. \[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]8 Regression results of coefficients A (**a**) and B (**b**) in Equation (12) for the 15 mm orifice bore, using the net pressure loss. The application of Equations (13) and (14) still requires iterative solution procedure, because the evaluation of \(\Delta P_{Ld}\) using Equation (6) needs a priori knowledge of \(\phi_{s}\). Moreover, the pressure measurement location at far downstream distance of 1100 mm may not be acceptable for the compact desalination facility design. Figure 10 presents the short span pressure difference (\(\Delta P_{12}\)) between the pressures measured at 50 mm upstream (\(P_{O1}\)) and 100 mm downstream (\(P_{O2}\)) from the upstream face of the orifice.