# Identification of the Wear Margin of a Pipeline-Machine Subsystem
## 4 The Thermodynamic Model of an Open Pipeline-Machine Subsystem
### Identification of a Subsystem Delivering Gas to a Constant Volume Vessel  
If a vessel has a constant volume \(V_{z}\), the pressure inside increases from \(p_{1z}\) to \(p_{2z}\) and temperature rises from \(T_{1z}\) to \(T_{2z}\) due to the gas filling process. The mass \(n_{zd}\) of the medium discharged into a vessel [28] is the difference between the mass in vessel \(n_{2z}\) after discharge and \(n_{1z}\) before discharge:  
\[n_{zd}\ =\ n_{2z}-n_{1z}\ =\ \frac{V_{z}\cdot p_{2z}}{R\cdot T_{2z}}-\frac{V_{z} \cdot p_{1z}}{R\cdot T_{1z}}\ =\ \frac{V_{z}}{R}\left(\frac{p_{2z}}{T_{2z}}-\frac{p_{1z}}{T_{1z}}\right) \tag{36}\]  
The values of gas temperatures in the vessel are difficult to determine. According to [28], the most accurate measurement of the input mass is obtained when pressure \(p_{2}\) is measured after temperature \(T_{2}\) becomes equal to ambient temperature \(T_{2}\ =\ T_{1}\ =\ T_{o}\), so  
\[n_{zd}\ =\ \frac{V_{z}}{RT_{o}}(p_{2z}-p_{1z}) \tag{37}\]  
The mass defined by Equation (36) is expressed in moles. Mass \(m\) in grams can be calculated by multiplying the number of moles \(n\) by molar mass of a specific gas \(\mu\) (g/mol), \(m\) (g) = \(n\) (mol) \(\cdot\mu\) (g/mol), and  
\[m_{zd}\ =\ \mu\frac{V_{z}}{R}\left(\frac{p_{2z}}{T_{2z}}-\frac{p_{1z}}{T_{1z}}\right) \tag{38}\]  
However, the temperature after the discharge is completed can be estimated by measuring \(T_{d}\) of gas in a conduit connecting the compressor with the vessel, assuming that the heat exchange with the surroundings is negligibly small during discharge. Heat delivered together with the gas discharged into the vessel heats the gas contained in the vessel before filling it up to a temperature \(T_{2}\). The quantity of heat \(Q_{d}\) delivered from the compressor to the air in the vessel is expressed as follows:  
\[Q_{d}\ =\ m_{zd}\cdot c_{V}\cdot(T_{d}-T_{2z}) \tag{39}\]Heat \(Q_{1}\) required to heat gas contained in the vessel before the vessel is filled up is expressed as follows:  
\[Q_{1}\ =\ m_{1z}\cdot c_{0^{*}}(T_{2z}-T_{1z}) \tag{40}\]  
\[Q_{1}\ =\ Q_{d} \tag{41}\]  
\[m_{1z}\cdot c_{0^{*}}(T_{2z}-T_{1z})\ =\ m_{zd}\cdot c_{V^{*}}(T_{d}-T_{2z}) \tag{42}\]  
\[T_{2z}\ =\ \frac{m_{zd}\cdot T_{d}+m_{1z}\cdot T_{1z}}{m_{1z}+m_{zd}} \tag{43}\]  
\[m_{1z}+m_{zd}\ =\ m_{2z} \tag{44}\]  
\[T_{2z}\ =\ \frac{\frac{V_{2}}{R}\left(\frac{p_{2z}}{T_{2z}}-\frac{p_{1z}}{T_{1z}} \right)T_{d}+\frac{V_{2}\cdot p_{1z}}{R\cdot T_{1z}}\cdot T_{1z}}{\frac{V_{2} \cdot p_{2z}}{R\cdot T_{2z}}} \tag{45}\]  
\[T_{2z}\ =\ \frac{\left(\frac{p_{2z}}{T_{2z}}-\frac{p_{1z}}{T_{1z}}\right)T_{d }+p_{1z}}{\frac{p_{2z}}{T_{2z}}} \tag{46}\]  
\[T_{2z}\ =\ T_{d}\cdot\frac{p_{2z}}{p_{2z}+p_{1z}\left(\frac{T_{d}}{T_{1z}}-1\right)} \tag{47}\]  
hence  
\[m_{zd}\ =\ \mu\cdot\frac{V_{z}}{R}\Bigg{(}\frac{p_{2z}}{T_{d}\frac{p_{2z}}{p _{2z}+p_{1z}\left(\frac{T_{d}}{T_{1z}}-1\right)}}-\frac{p_{1z}}{T_{1z}}\Bigg{)} \ =\ \mu\cdot\frac{V_{z}}{R}\left(\frac{p_{2z}-p_{1z}}{T_{d}}\right) \tag{48}\]  
The capacity of vessel filling can be defined as the quotient of discharged mass \(m_{zd}\) and the filling time of discharge \(t\).  
\[\dot{m}_{zd}\ =\ \mu\cdot\frac{1}{\Delta t_{N}}\cdot\frac{V_{z}}{R}\left(\frac{p _{2z}-p_{1z}}{T_{d}}\right) \tag{49}\]  
Efficiency \(\eta_{zz}\) of filling the vessel can be referred to as the quotient of energy accumulated in the vessel and electric energy absorbed for its accumulation.  
\[\eta_{ez}\ =\ \frac{V_{z}\cdot(p_{2z}-p_{1z})}{E_{el}} \tag{50}\]  
The efficiency of a subsystem filling a pressure vessel (machine set-vessel subsystem) \(\eta_{azz}\) can be calculated from Equations (7) and (52) for the efficiency of an open subsystem, substituting  
\[m\ =\ m_{zd}\ \ \ \text{and}\ \ \ \ p_{2}\ =\ p_{1z}+\frac{p_{2z}-p_{1z}}{2}\ =\ \frac{p_{1z}+p_{2z}}{2} \tag{51}\]  
\[\eta_{e}\ =\ \frac{m\cdot\frac{(p_{2z}-p_{1})}{\rho}}{E_{el}} \tag{52}\]  
\[\eta_{azz}\ =\ \frac{\mu\cdot V_{z}}{R\cdot\rho\cdot E_{d}}\Bigg{(}\frac{p_{2z}-p_{1z }}{T_{d}}\Bigg{)}\Bigg{(}\frac{p_{1z}+p_{2z}}{2}-p_{1}\Bigg{)} \tag{53}\]