# Pressure Study on Pipe Transportation Associated with Cement Coal Gangue Fly-Ash Backfill Slurry
## 5 Mechanisms  
The effect of the feeding velocity on the resistance loss of the CGFB slurry pipe transportation was analyzed and an attempt has been made to develop a correlation to estimate the resistance loss as a function associated with the conveying velocity in the straight pipe and 90\({}^{\circ}\) elbow pipe based on the mathematical analysis. The function can be used to predict the CGFB slurry pressure loss in the pipe transportation regarding the corresponding conveying velocity and provide the possibility of estimating the minimum pump power that delivers the slurry to the gob. The CGFB slurry was investigated as a non-Newtonian fluid and which conforms to Bingham plastic body characteristics. The relation [34] of the shear stress to shear rate for the Bingham plastic body can be expressed as follows:  
\[\tau=\tau_{0}+\gamma\mu \tag{4}\]  
where \(\tau_{0}\) is the yield stress, Pa, \(\mu\) is the viscosity coefficient, Pa\(\cdot\)s and \(\gamma\) is the typical shear rate, 1/s. Based on the Bingham plastic body linen relationship, [PERSON] created an equation that can be employed to determine the slurry transmission pipe diameter [1]. The Buckingham equation was shown as the following:  
\[\frac{8v}{D}=\frac{\tau_{w}}{\mu}[1-\frac{4}{3}\frac{\tau_{0}}{\tau_{w}}+ \frac{1}{3}\bigg{(}\frac{\tau_{0}}{\tau_{w}}\bigg{)}^{4}] \tag{5}\]  
where \(v\) is the slurry conveying velocity, m/s, \(D\) is the diameter of the transportation pipeline, m, \(\tau_{w}\) is the shear stress at the wall, 1/s. Considering that the magnitude of the \((\tau_{0}/\tau_{w})^{4}\) is so small that it can be ignored because of the CGFB slurry having a high solid concentration up to 76.05% by weight, Equation (5) was modified to Equation (6):  
\[\frac{8v}{D}=\frac{\tau_{w}}{\mu}-\frac{4}{3}\frac{\tau_{0}}{\mu} \tag{6}\]  
Figure 9 provides the overview of the slurry conveyance in the horizontal pipes (straight pipe) along the slurry's advance direction. The slurry with a length of \(dL\) was intercepted as the research object to make the static analysis. The pressure loss \(\Delta p\) at the length of \(dL\) is calculated. The static balance available for the objected slurry at the length of \(dL\) was shown as follows:  
\[\pi R^{2}\Delta p=2\pi R\tau_{w}l \tag{7}\]  
According to Equation (7), the shear stress occurring at the pipeline wall can be expressed as follows:  
\[\tau_{w}=\frac{R\Delta p}{2l} \tag{8}\]  
In China's engineering field, the resistance loss (\(i_{\mathrm{s}}\)) in the straight pipe is introduced to describe the value of the pressure loss over unit length in practice [29]. \(i_{\mathrm{s}}\) was defined as follows:  
\[i_{\mathrm{s}}=\frac{\Delta p}{l} \tag{9}\]  
Combining Equations (6) and (8), Equation (9) is created as follows:  
\[i_{\mathrm{s}}=\frac{16}{3D}\tau_{0}+\frac{32v}{D^{2}}\mu \tag{10}\]  
where \(l\) is the length of the fluid in the pipeline, \(\Delta p\) is the delivery distance of the pressure drop and \(i_{\mathrm{s}}\) is the resistance in straight pipe. Compared with the data from the CFD simulation result and the theoretical data calculated from Equation (10), Figure 10 was made. From the statistical analysis, the predicted data is well in agreement with the theoretical data. Based on the CFD simulation data, the equation that can be applied in the 90\({}^{\circ}\) elbow pipe was derived in combination with Equation (10). Initially, a log-log graph was plotted for the local resistance in the 90\({}^{\circ}\) elbow pipe with the resistance loss of the straight pipe to obtain the function between them. Then, all the data from the variation of local resistance collected in the 90\({}^{\circ}\) elbow pipe was plotted  
Figure 9: The overview of the static analysis for the CGFB slurry. against the slurry conveying velocity. Thus, the functional relationship of the pipe diameter (\(D\)), the CGFB slurry yield stress (\(\tau_{0}\)) and the CGFB slurry relative viscosity (\(\mu\)) is given in Equation (11) as  
\[