## 3 Results and Discussion  
The measurements of pressure drop across the orifice were made at different flow velocities, for different solid volume fractions of (15, 20, 25, 28, and 30)%, and for orifice bores of (10 and 15) mm. Figure 3 exhibits the pressure difference (\(\Delta P_{13}\)) between the pressures measured at 50 mm upstream (\(P_{O1}\)) and 1100 mm downstream (\(P_{O3}\)) from the upstream face of the orifice. For a given solid fraction, \(\Delta P_{13}\) increases almost linearly with the increase of flow velocity for both orifice bores of (10 and 15) mm. Figure 3: _Cont._  
The net pressure loss (\(\Delta P\)) caused by the orifice is calculated by subtracting from \(\Delta P_{13}\) the pressure loss (\(\Delta P_{Ld}\)) caused by the straight pipe having the same length between pressure difference measuring locations:  
\[\Delta P=\Delta P_{13}-\Delta P_{Ld} \tag{5}\]  
The empirical equation reported by [PERSON] et al. [39] is utilized to evaluate \(\Delta P_{Ld}\) for the pipe length of 1150 mm:  
\[\frac{\Delta P_{Ld}}{L_{Ld}}=\frac{4}{D}\bigg{(}\frac{8U}{D}\frac{24.27712 \phi_{s}{}^{2}+0.0861\phi_{s}-4}{32.3616\phi_{s}{}^{2}+0.1148\phi_{s}-4}\bigg{)} ^{(-8.0904\phi_{s}{}^{2}-0.0287\phi_{s}+1)}\exp\left(77.7250\phi_{s}{}^{2}-1.888 7\phi_{s}-5.0546\right) \tag{6}\]  
where, \(L_{Ld}\) is the pipe length, and \(D\) is the pipe diameter. The pressure loss coefficient (\(K\)) is calculated according to the common definition:  
\[K=\frac{\Delta P}{\frac{1}{2}\rho U^{2}} \tag{7}\]  
The Metzner-Reed Reynolds number (\(Re_{MR}\)), frequently employed in non-Newtonian fluid analysis [54], is introduced:  
\[Re_{MR}=\frac{D^{n}U^{2-n}\rho_{HS}}{k\Big{(}\frac{1+3n}{4n}\Big{)}^{n}\beta^{n -1}} \tag{8}\]  
where,\(\rho_{HS}\) is the density of hydrate slurry. Empirical correlations for the consistency index k and behavior index n of Equation (8) were also available [39], which revealed the pseudoplastic non-Newtonian behavior of aqueous R134a hydrate slurry:  
\[k=\exp(77.725\phi_{s}{}^{2}-1.8887\phi_{s}-5.0546) \tag{9}\]  
\[n=-8.0904\phi_{s}{}^{2}-0.0287\phi_{s}+1.0 \tag{10}\]  
Figure 4 presents the variations of the pressure loss coefficients with the Metzner-Reed Reynolds number. Based on the near linear relationship between ln(K) and ln(\(Re_{MR}\)) recognized from plots with both axes scaled logarithmically, regression can be performed with the dimensionless model equation of the following type:  
\[K=C(\phi_{s})Re_{MR}^{m} \tag{11}\]  
Figure 3: Pressure difference between 50 mm upstream and 1100 mm downstream from the upstream face of the orifice with respect to the flow velocity at various solid volume fractions for the orifice bores of (**a**) 10 mm, and (**b**) 15 mm. The exponent \(m\) is a constant having the same value for both orifice bores. A model equation for the multiplier C in Equation (11) can be further constructed as a function of \(\phi_{s}\), which may contain different sets of coefficient estimates for the respective orifice bore. However, the highly nonlinear nature of Equation (11), especially \(Re_{\text{MR}}\) with respect to \(\phi_{s}\)[Equations (8)-(10)], causes a serious disadvantage when used for in situ determination of the solid volume fraction. The solution procedure of Equation (11) for \(\phi_{s}\) from the measured pair of net pressure loss and flow velocity data is inevitably iterative.