4.1. Effect of Inlet CO2 Volume Fraction
When the underwater power system operates under different conditions, the mass flow rate of the exhaust gas and the required cooling water alters, resulting in a change in the gas volume fraction. Numerical simulations under different inlet CO
2 volume fractions were performed to study the influence of the gas volume fraction. Here, the inlet CO
2 volume fraction changed from 0.8 to 1 by considering the operating range of the thermal power system. The rotational speed of the Roots pump was 2500 rpm, the pressure ratio was 2, and other parameters are shown in
Table 4.
The corresponding efficiencies are depicted in
Figure 5. The volumetric efficiency increases from 92% to 97%, and then decreases to 80%, when the volume fraction is larger than 0.9. Meanwhile, the flow efficiency shows an increased trend under the investigated conditions. The pump efficiency maintains an upward trend from 48% to 64% and then drops to 59% for pure CO
2.
To explain this variation, the flow characteristics first need to be explored. For three-lobe Roots pumps, the theoretical pulsation period is the time that the rotors take to rotate 120 degrees (0.008 s when the rotational speed is 2500 rpm). The time that the rotors take to rotate 60 degrees (0.004 s when the rotational speed is 2500 rpm) is long enough to show the transient variation of the flow field, taking the conjugacy of the pump into account.
Figure 6 and
Figure 7 depict the pressure streamline contours and the water volume fraction contours when the inlet CO
2 volume fraction is 0.8.
The rotation and the backflow result in the compression of the fluid, as shown in
Figure 6. Due to the suction effect caused by the volume change at the inlet zone, the working fluid flows into the inlet zone. The pressure in the inlet zone remains constant, since the fluid is not compressed in this process. During the rotation, the closed working chamber is generated. Then, the gas–liquid mixture in the inlet zone is transported to the outlet zone. The volume of the chamber remains constant during the transportation process, and the gas–liquid mixture inside is compressible. Therefore, the pressure within the working chamber does not change significantly. When the working chamber continues to move and connects to the outlet zone, the backflow, whose pressure is high, flows into the outlet, increasing the local pressure. The working fluid is finally pressurized and pumped out of the Roots pump together with the backflow, resulting in a decrease in pressure.
To further study the flow characteristics in the outlet zone, as shown in
Figure 6a,b, the upper-left chamber begins to connect to the outlet zone, while the upper-right chamber is about to disappear. Part of the compressed fluid in it flows into the upper-left chamber, mixing with the working fluid; therefore, the local pressure is increased. The rest of the fluid flows out of the outlet. In
Figure 6c, the upper-right chamber gradually disappears and the volume of the upper-left chamber decreases. The pressure in the outlet zone is lower than that of the outlet boundary, leading to the backflow. The pressure in the upper-left chamber continues to increase, due to the backflow and the extrusion of the left rotor. In
Figure 6d, the upper-left chamber is now completely connected to the outlet zone. The compressed working fluid in the upper-left chamber flows into the outlet zone and mixes with the backflow, resulting in the increased outlet zone pressure. As shown in
Figure 6e, the pressure in the outlet zone continues to increase, and a new upper-right chamber begins to connect to the outlet zone. Finally, the flow field nearly conjugates with the initial state, as shown in
Figure 6f.
Comparing
Figure 6 and
Figure 7, the variation in the pressure and water volume fraction is similar. This can be explained by the physical properties of the gas–liquid mixture. The liquid phase is incompressible, while the gas phase is compressible. When the pressure in the domain increases, the gas phase is compressed and the volume decreases, resulting in an increase in the water volume fraction. This phenomenon implies that the main factor affecting the gas–liquid distribution is the compression effect of the pressure in the operation of the gas–liquid mixture Roots pump.
A similar flow pattern is also obtained under the working conditions when the inlet CO
2 volume fraction is 0.9, 0.95, 0.99, and 1, as shown in
Figure 8,
Figure 9,
Figure 10 and
Figure 11, respectively. In the inlet zone, the pressure remains constant, while the pressure in the outlet zone fluctuates periodically. A local pressure higher than the outlet pressure is formed intermittently in the outlet zone and the working chambers, due to the backflow and the extrusion of the rotors.
Figure 12 illustrates the pressure variation at the outlet. It can be seen that both the frequency and the amplitude are varied at different operating conditions.
The increased inlet CO
2 volume fraction tends to reduce the pressure amplitude (see in
Figure 12). The pressure pulsation amplitude is larger than 0.26 MPa at the outlet for the inlet CO
2 volume fraction of 0.8, while the maximum pressure is larger than 2 MPa at the outlet zone. When the inlet CO
2 volume fraction changes to 0.9, the amplitude of the outlet pressure decreases to 0.2 MPa, and the maximum pressure also decreases to 1.8 MPa. As the inlet CO
2 volume fraction continues to increase, the pressure amplitude at the outlet drops to less than 0.1 MPa. The maximum pressure in the Roots pump is 1.5 MPa when the inlet CO
2 volume fraction is 0.95, and 1.26 MPa at an inlet CO
2 volume fraction of 0.99. When pure CO
2 is prescribed, the pressure pulsation disappears almost entirely, and the maximum pressure decreases to 1 MPa.
The frequency of the pressure pulsation at the outlet is also discussed. When the inlet fluid is pure CO2, the outlet pressure pulsates four times in the time it takes for the rotors to rotate 60 degrees. This pulse number decreases to two when the inlet CO2 volume fraction is 0.99. When the inlet CO2 volume fraction decreases to 0.95 or 0.9, the outlet pressure pulsates once per 60-degree rotation of the rotors. The pulse number of the outlet pressure is also one when the inlet CO2 volume fraction is 0.8, while two peaks with different values exist in each pulsation.
The differences in the flow characteristics can be attributed to the fluid properties resulting from different volume fractions. The pressure difference overcomes the inertia and pushes the gas–liquid mixture flow out of the pump. The incompressible liquid phase has the characteristics of high density and large inertia, while the gas phase is compressible and is characterized as having low density and inertia. When the inlet CO2 volume fraction is low, the density and the inertia of the compressible gas–liquid mixture are relatively high. The high pressure difference is needed to push the fluid flow out of the pump, which results in the prolonged and high-intensity backflow. Meanwhile, the local pressure fluctuation is created in the outlet zone. Therefore, the high-amplitude and low-frequency pressure pulsation flow is formed, and multiple pulsation peaks may appear under some working conditions. The density and the inertia of the gas–liquid mixture decrease with the increase in the inlet CO2 volume fraction, resulting in a decrease in the pressure pulsation amplitude and an increase in the frequency at the outlet.
The volumetric efficiency is mainly determined by the leakage, and the leakage is determined by the properties of the fluid and the pressure difference between the inlet and the outlet. To further study the influence of the gas volume fraction and the pressure difference on the leakage, a two-dimensional model of the gap between the rotor and the casing was established, as shown in
Figure 13. The rotor rotation is not considered in the simulation. The node number of the mesh is 18,436 in order to ensure that the wall y+ number is less than 2. The simulation settings were set according to the method described in
Section 3.1. The flow direction of the gas–liquid mixture is from the outlet zone to the inlet zone, so the pressure was set as 0.46 MPa at the pressure outlet, and the temperature was 365 K. The pressure ratio was from 1.5 to 4 and the gas volume fraction was from 0.8 to 1 under different working conditions.
Figure 14 shows the pressure, velocity, and water volume fraction contours when the pressure ratio is 2 and the CO
2 volume fraction at the inlet zone is 0.8. As shown in
Figure 14a, the pressure decreases from the pressure inlet to the pressure outlet, while the lowest pressure occurs at the accessory with the smallest clearance. Meanwhile, the pressure changes are not uniform in the region close to the pressure outlet; this is caused by the variation of the velocity. As shown in
Figure 14b, the fluid gradually accelerates, and reaches a maximum value at the minimum clearance as the gap decreases. The high-velocity fluid then flows into the region close to the pressure outlet and mixes with the low-velocity fluid within it. The increase in the velocity represents an increase in the dynamic pressure, which leads to a decrease in the static pressure. As shown in
Figure 14c, the variation of the water volume fraction is similar to that of the pressure. This further indicates that the change in the volume fraction is mainly caused by the compression of the gas phase.
Figure 15 shows the leakage volume flow rates under different conditions. With the decrease in the inlet zone gas volume fraction, the leakage volume flow rate decreases accordingly, and the rate of decrease becomes smaller. This indicates that the presence of the liquid phase is beneficial in terms of sealing the gaps. A similar phenomenon can be found in the comparison of the cases with different pressure ratios. The high pressure ratio leads to a large amount of leakage, while the leakage increment decreases as the pressure ratio increases. The gas volume fraction presents a larger influence on the leakage than the pressure ratio.
To summarize, the leakage in the Roots pump is jointly influenced by the gas volume fraction in the pump and the pressure difference between the inlet and the outlet zones. With the decrease in the gas volume fraction, the sealing effectiveness of the gaps improves, leading to a decrease in the leakage. Meanwhile, with the increased pressure difference between the inlet and the outlet zones, the driving force of the flow in the gaps increases and the leakage rises. In addition, the relatively low gas volume fraction in the pump leads to the high local pressure in the outlet zone, increasing the pressure difference between the inlet zone and the outlet zone. Hence, the pressure difference is dominant, leading to leakage when the inlet CO
2 volume fraction is less than 0.9. In this range, the high local pressure decreases with the increased inlet CO
2 volume fraction, reducing the pressure difference and the leakage. As a result, the volumetric efficiency increases. When the inlet CO
2 volume fraction is larger than 0.9, the key factor affecting the leakage switches to the gas volume fraction in the pump. In this case, the sealing effectiveness of the gaps is weakened with the increased inlet CO
2 volume fraction, decreasing the volumetric efficiency, as shown in
Figure 5a.
The flow efficiency is directly determined by the moments of the rotors.
Figure 16 shows the simulated total moments and the theoretical total moments in different cases, in which the theoretical total moment can be achieved by dividing the theoretical compression power of the pump by the angular velocity of the rotors. The theoretical total moment decreases slightly with the increase in the inlet CO
2 volume fraction, which is caused by the difference in compression power between the gas phase and the liquid phase under the same volume. The simulated total moment is larger than the theoretical total moment, and also decreases with the increase in the inlet CO
2 volume fraction. The reduction in the simulated total moment is larger than that of the theoretical total moment.
The simulated total moments are determined by the difference between the pressure in the inlet zone and the outlet zone. Considering that the pressure in the inlet zone is almost constant, the pressure in the outlet zone is the main factor affecting the rotor moment. The local pressure in the outlet zone is relatively high when the inlet CO2 volume fraction is low, leading to the average pressure being significantly higher than the outlet pressure. This increases the total moment, resulting in a low flow efficiency. With the increase in the inlet CO2 volume fraction, the high local pressure in the outlet zone gradually disappears because of the change in the fluid properties, reducing the average pressure. Therefore, the rotor moment decreases, and the flow efficiency increases.
The pump efficiency is the combination of the volumetric efficiency and the flow efficiency. Therefore, it presents a trend of first rising and then falling with the increased inlet CO2 volume fraction.
4.3. Effect of Pressure Ratio
It can be seen that the liquid phase in the fluid is helpful in sealing the Roots pump, and reduces the temperature increase during the compression process [
7]. This indicates that the gas–liquid mixture Roots pump has the potential to achieve a high pressure ratio. To study the flow characteristics of the Roots pump under various pressure ratios, numerical simulations were also conducted for pressure ratios from 4 to 10. The inlet pressure was set as 0.46 MPa, the inlet CO
2 volume fraction was 0.8, and the rotational speed was 2500 rpm. The relevant parameters are shown in
Table 6.
The efficiencies at different pressure ratios were obtained, and are illustrated in
Figure 20.
The flow efficiency gradually declines from 52% to 42% with the increase in the pressure ratio, while the volumetric efficiency decreases from 92% to 76%. The pump efficiency presents a downward trend from 48% to 33% when the pressure ratio increases from 2 to 10. The pressure contours of the Roots pump at the pressure ratios of 6 and 10 are shown in
Figure 21 and
Figure 22, respectively.
The pressure field in the Roots pump is similar at different pressure ratio conditions, except for the differences in the pressure pulsation frequency and the pressure values.
For the volumetric efficiency, the pressure difference between the outlet and inlet zones increases gradually with the increased pressure ratio, leading to an increase in the leakage in the gaps. However, as the pressure difference continues to increase, the increment of the leakage is insensitive to the pressure difference. In addition, when the pressure in the outlet zone is high, the gas volume fraction becomes lower, which contributes to the sealing of the gaps. Therefore, the leakage does not increase and the volume efficiency tends to remain constant when the pressure ratio reaches a certain degree.
For the flow efficiency, the compression of the fluid in the pump depends on the joint action of the rotors’ rotation and the backflow of the high-pressure fluid outside the pump. When the pressure ratio is low, the gas volume fraction of the backflow is high, which has the characteristics of small inertia and high compressibility. Therefore, the maximum pressure in the outlet zone is relatively low, and the flow efficiency is high. When the pressure ratio increases, the pressure of the backflow is high and the gas volume fraction is low, leading to an intense mixing process with the fluid in the outlet zone. Therefore, the maximum pressure in the outlet zone is much higher than that of the outlet boundary, and the flow efficiency is low.
As the volumetric efficiency and the flow efficiency decrease with the increase in the pressure ratio, the value of the pump efficiency becomes smaller as well.
Through the analysis of the characteristics of gas–liquid mixture Roots pumps under various working conditions, the density, compressibility, and specific heat capacity of the mixture fluid were found to be the main factors leading to the performance variations of Roots pumps. These fluid properties are jointly governed by the gas and liquid phases. Therefore, the change in the gas volume fraction of the fluid affects the flow characteristics in Roots pumps, leading to the variation in the backflow rate and the outlet zone pressure and temperature. As a result, the flow efficiency changes with the working conditions. In addition, the sealing of Roots pumps, which determines their volumetric efficiency, is also affected by the fluid properties and the outlet zone pressure.