# Identification of the Wear Margin of a Pipeline-Machine Subsystem
## 4 The Thermodynamic Model of an Open Pipeline-Machine Subsystem  
If a difference in temperatures, \(\Delta T\ =\ T_{1}-T_{2}\), exists in a fluid flow in a pipeline, it means that heat is transported along with fluid itself. Heat flow rate \(\dot{Q}\) in a fluid flow rate is a product of fluid flow rate \(\dot{m}\), specific heat capacity of the fluid \(c_{w}\) and temperature difference \(\Delta T\). \[\dot{Q}\ =\ \dot{m}c_{W}\Delta T \tag{19}\]  
Heat flow rate can change in a fluid flow rate, as heat flow can be perpendicular to the direction of fluid motion and it is possible to convert mechanical energy of the fluid (hydraulic energy) into heat energy. Heat Flow Perpendicular to the Direction of Fluid Flow: Determination of the Heat Penetration Coefficient  
In the case of heat flow perpendicular to the direction of pipeline fluid motion, the pipe wall is a barrier. Figure 5 presents a drop in temperature in a plane perpendicular to the pipeline axis (the ordinate \(T\) of the coordinate system overlaps the pipeline section axis). The process of heat transfer from the fluid flux of higher temperature through a barrier to the flux of lower temperature is called heat penetration. Heat flow through a pipe wall can be desired or undesired. In the case of undesired heat flow, the energy of undesired flow is energy loss. Desired heat flow perpendicular to the fluid flow is implemented in pipeline devices called heat exchangers. In heat exchangers, the process of heat exchange takes place at a certain pipeline section (along a specific length), which is related with temperature change along the heat exchanging section. Therefore, mean temperature must be determined. The logarithmic mean temperature difference \(\Delta T_{m}\) is a universal measure of the mean temperature \(\Delta T\). The method of determining the mean logarithmic difference of temperatures based on an anticurrent heat exchanger is taken from [38] and shown in Figure 6.  
\[\Delta T_{m}\ =\ \frac{\Delta T_{\max}-\Delta T_{\min}}{\ln\Bigl{(}\frac{ \Delta T_{\max}}{\Delta T_{\min}}\Bigr{)}} \tag{20}\]  
Figure 5: Temperature drop from hot fluid, through the barrier, to cold fluid [37], where \(d\) is the path of heat transport/distance from the pipeline axis, \(d_{p}\) is the barrier thickness/pipe wall thickness, \(T_{G}\) is the temperature of hot fluid (flowing in the pipeline) in the observation plane, \(T_{Z}\) is the temperature of cold fluid (flowing on pipeline outside surface) in the observation plane, \(T_{1}\) and \(T_{2}\) are temperatures at fluid–solid interface corresponding to the \(T_{p1}\) and \(T_{p2}\) temperatures at the internal and external surfaces of the barrier (pipeline wall), 1 and 3 indicate convection and 2 indicates heat conduction. The flow rate of heat penetrating barriers with surface area \(A\) is described in the following form [38,39]:  
\[\dot{Q}\ =\ \Lambda\ k\ \Delta T \tag{21}\]  
where \(k\) is the heat penetration coefficient and \(\Delta T\ =\ T_{G}-T_{Z}\). Heat penetration coefficient \(k\)[37,40] is a substitute measure of wear margin of the heat exchanger because it depends on properties of materials, shape of built-in elements, working fluids and quantities describing motion and load. A possible experimental determination of the heat penetration coefficient \(k\) is indicated in [41]. It follows from the heat exchanger's heat/energy balance (Figure 5) that  
\[\dot{Q}_{1}\ =\ \dot{Q}_{2}\ =\ \dot{Q}_{3}\ =\ A\ k\ \Delta T_{m} \tag{22}\]  
because heat flow rate in a fluid flow rate  
\[\dot{Q}_{1}\ =\ \dot{m}_{G}\ c_{vG}\ \Delta T_{G} \tag{23}\]  
\[\dot{Q}_{3}\ =\ \dot{m}_{Z}\ c_{vZ}\ \Delta T_{Z} \tag{24}\]  
then  
\[k\ =\ \frac{\dot{m}_{G}\ c_{vG}\ \Delta T_{G}}{A\ \Delta T_{m}}\ =\ \frac{\dot{m}_{Z}\ c_{vZ}\ \Delta T_{Z}}{A\ \Delta T_{m}} \tag{25}\]  
where \(\dot{m}_{G}\) and \(\dot{m}_{Z}\) are hot and cold fluid flow rates, respectively, and \(c_{vG}\) and \(c_{vZ}\) represent specific heat capacities at constant volume of hot and cold fluids, respectively.