# Identification of the Wear Margin of a Pipeline-Machine Subsystem
## 3 The Hydrodynamic Model of an Open Pipeline-Machine Subsystem
### Energy Losses and Hydraulic Efficiency of the Suction and Discharge Pipes  
A pipeline section can be composed of straight pipe sections and required built-in elements: elbows, branched pipes, cross-section changes and instruments. Built-in components in terms of hydrodynamics are contractions, flown-round bodies or a combination of both and, due to energy losses, are regarded as obstacles. Obstacles may also be undesired obstructions created due to manufacturing and installation errors (e.g., welding of pipelines) and as a result of wear. Characteristics of a straight section stem from characteristics of elements making up that section and characteristics obtained after inserting them, including inside diameter \(d\) (m), length \(l\) (m), absolute roughness of internal surface \(k\) (peak height of asperities) (m) and relative roughness \(k/d\). Fluid characteristics include kinematic viscosity \(v\) (m\({}^{2}\)/s) and density \(\rho\) (kg/m\({}^{3}\)). Characteristics of fluid load and motion are fluid flow velocity \(w\), mean flow velocity \(w_{m}\) (m\({}^{2}\)/s).and height above reference plane \(h\) (m).  
It follows from the energy equation that energy losses between two pipeline cross-sections can only result in a change of pressure energy \(\Delta p\)[27] (Figure 4). Pressure drops (losses) \(\Delta p\) along a straight section of a pipeline are described by the Darcy-Weisbach equation [27]:  
\[\Delta p\ =\ \lambda\frac{l}{d}\rho\frac{w_{m}^{2}}{2} \tag{10}\]  
The number \(\lambda\) is called the linear resistance coefficient or the friction factor. The coefficient \(\lambda\) depends on the value of Reynolds number Re (Re = \(wl/v\)) and relative roughness \(k/d\)[26; 27].  
Pressure drops along an obstacle are accounted for by the coefficient \(\zeta\), which reflects additional losses compared to losses on a straight section of a pipeline [27].  
\[\Delta p_{p}\ =\ \zeta p\frac{w_{m}^{2}}{2} \tag{11}\]  
The values of coefficient \(\zeta\) are calculated experimentally.  
The resistance coefficients \(\lambda\) and \(\zeta\) can be referred to as hydraulic resistance coefficients. Conduits in machines where a fluid flows also have a specific value of the hydraulic resistance coefficient \(\zeta\). The resistance coefficients \(\lambda\) and \(\zeta\) should be regarded as wear margin measures [8] of a straight section and an obstacle. These coefficients are established at the manufacturing stage and get larger due to wear. Any device built in a pipeline can be regarded as an obstacle.  
If there are no mass losses or changes in cross-section or height above reference plane between two cross-sections of a pipe, X1 and X2, then the efficiency of a pipe section \(\eta_{od}\) can be expressed in the following form:  
\[\eta_{od}\ =\ \frac{E_{\text{X2}}}{E_{\text{X1}}} \tag{12}\]  
Hence, the hydraulic efficiency of the suction pipe is  
\[\eta_{Hsp}\ =\ \frac{p_{s}}{p_{1}}\ =\ \frac{p_{s}}{p_{s}+\Delta p_{sp}} \tag{13}\]  
Figure 4: Pressure loss between cross-sections 1 and 2 [29], where P is an obstacle, \(l_{p}\) is the length of the obstacle, \(\Delta p_{p}\) is the loss at the obstacle (loss focused locally), \((p^{\prime}{}_{X1}-p^{\prime}{}_{X2})\) is the loss at the straight section (loss evenly distributed) and \((p^{\prime}{}_{X1}-p^{\prime}{}_{X2})\) is the total loss at the section \(l_{o}\).  
and hydraulic efficiency of the discharge pipe is  
\[\eta_{Hdp}\ =\ \frac{p_{2}}{p_{d}}\ =\ \frac{p_{d}-\Delta p_{dp}}{p_{d}}\ =\ 1-\frac{\Delta p_{dp}}{p_{d}} \tag{14}\]  
where \(\Delta p_{dp}\) represents the pressure losses in the discharge pipe and \(\Delta p_{sp}\) represents the pressure losses in the suction pipe.