## 6 Low-Duty Turbo Compressor Mass Flow and Tank Equilibrium Pressure Maintained during a Voyage  
In the mixtures of various gases, equilibrium exists at specific temperatures and pressures. Changes in the temperature or the pressure move the gas mixture from one equilibrium state to another. In this part, we estimated the tank temperature during maneuvering and when sailing with the assumption that the mass flow rate of the main boiler is constant all the way to the discharging port for a 15-day voyage. The equation of the state for the gas component, assuming that the component is at the same volume and temperature, of the mixture is [69; 70; 71]:  
\[p_{i}\cdot v=Z_{i}\cdot R\cdot T. \tag{23}\]  
The specific volume is:  
\[v=\frac{V}{m}. \tag{24}\]  
Substitution of Equation (23) into Equation (24) gives:  
\[p_{i}\cdot V=Z_{i}\cdot m\cdot R\cdot T. \tag{25}\]  
As per an additive pressure rule known as [PERSON]'s model, the gas mixture occupying volume \(V\) at temperature \(T\) can be expressed as the sum of the pressures exerted by the individual components:  
\[p_{m}=\sum_{i=1}^{j}p_{i}(T_{m},V_{m}). \tag{26}\]  
Substitution of Equation (25) into (26) gives:  
\[Z=\sum_{i=1}^{j}y_{i}\cdot Z_{i}|_{T_{m}}V_{m}. \tag{27}\]  
Figure 10: Exergy efficiency with surrounding temperature change. Approximation of compressibility factor \(Z\) and temperature \(T\) by the variation of the gas mixture equilibrium pressure from 0.105 to 0.125 MPa according to Equation (22) gives the following polynomials:  
\[Zf(p)=0.762860\cdot p^{3}-2.191443\cdot p^{2}+0.638980\cdot p+0.877609, \tag{28}\]  
\[R^{2}=0.999317,\]  
\[Tf(p)=-392.45\cdot p^{2}+205.53\cdot p-179.66, \tag{29}\]  
\[R^{2}=1.000000.\]  
Finally, collecting the partial components in Equations (26)-(29) to a real mixture equation of state at equilibrium pressure gives:  
\[v=\frac{V}{m}. \tag{24}\]0  
In an LNG tanker, cargo tanks are filled up to 98% as required according to the Lloyd register, which is the safety margin to delay cargo tank overflowing in the case of fire [72]. However, some administrations will allow for a greater tank volume, typically 98.5%, which considers the expected boil-off of vapors from the cargo tanks during the loaded voyage, [73]. Membrane cargo tanks for a 150,000 m\({}^{3}\) LNG carrier are built for an average daily vaporization rate of 0.15% of the total cargo [74; 75]. The measured cargo tank pressure after loading and before starting the low-duty turbo compressor was 0.1097 MPa. The average time from dead slow ahead to fully maneuvering was divided into three one-hour-long steps. The last measurement of the main boiler was recorded as the constant consumption rate of the main boiler for the next 15 days of the assumed voyage. According to the given assumptions, we calculated 6559.4 kg/h available gas in the cargo tanks with 2% available volume, as shown in \[v=\frac{V}{m}. \tag{24}\]1 As the voyage continues, there is more available space in the cargo tanks due to gas consumption by the main boiler. As the main boilers are consuming more gas than available in the cargo tanks, one part of the gas must be evaporated in the LNG vaporizer, which has to simultaneously compensate for the lack of gas in the tanks and balance cargo tank pressure. In the case of the steady model as presented here, the vaporizer must vaporize an additional 3045 kg/h of LNG to maintain the cargo pressure in the tank. As the cargo tank pressure and liquid temperature are correlated, the cargo temperature is maintained for the entire voyage at about \(-\)161.92 to \(-\)161.85 \({}^{\circ}\)C. The only disturbance in the equilibrium was noticed during maneuvering, where the cargo tank pressure was not steady but dependent on the load change in the boiler. The total consumption of the cargo for the voyage with this setup was 2554 MT of LNG.