Analysis of Leakage Model of All-Metal Screw Pump

Traditional rubber screw pumps use an interference method to engage. When the pressure is less than the breakdown pressure of the pump, there is no leakage in the pump. *e all-metal screw pump stator and rotor are made of metal, and the stator and rotor adopt a gap-fitting engagement method, so even when the pump is working normally, the leakage is objective. Based on the concentric annular gap flow, the pressure drop leakage caused by the fluid inertia force is fully considered, and the calculation formula of the leakage of the all-metal screw pump is studied from the three aspects of transverse leakage, longitudinal leakage, and oblique leakage. *is model was experimentally verified by a commercial all-metal screw pump produced by Shihong Petroleum Equipment Company. *e model results show that the leakage of the all-metal screw pump is mainly affected by the structure parameters of the pump itself and the density of the pumped fluid, and the gap height is the main factor affecting the leakage.


Introduction
As a new mechanical oil extraction equipment, the screw pump has many advantages, such as simple structure and stable displacement [1,2]. e stator and rotor of traditional screw pumps are made of rubber, which is inexpensive, but the pressure at the exit of the pump is large, and it is easy to be broken down [3]. It is not suitable for high temperature and high pressure working environments, so its application has certain limitations [4][5][6][7]. e stator and rotor of the allmetal screw pump are made of metal materials and are regarded as a new generation of screw pumps [8][9][10], especially in the fields of heavy oil thermal recovery, which are widely used [11,12].
Due to the gap between the stator and the rotor of the allmetal screw pump, it is very necessary to study the leakage of the pump. e researchers used digital simulation software (CFD) to study the relevant leakage laws [13][14][15][16][17][18], but the digital analog analysis often ignored the complicated working conditions at the site which would cause a large error between the analysis results and the actual situation. e combination of theoretical modeling and experimental methods can avoid the above situation. rough theoretical modeling and experimental methods, Gamboa et al. [19,20] determined that the pressure in the cavity of the all-metal screw pump is linearly distributed. Pessoa et al. [21] analyzed the leakage in the pump by establishing a flow model. Nguyen et al. [22], on the basis of the leakage in the pump, established a new fluid model to simulate the actual flow of the fluid, but they all only considered the influence of the viscous force and did not consider the effect of the inertial force of the fluid.
In view of this, there is currently a lack of a full metal screw pump leakage calculation model that fully considers the inertia of the fluid. e purpose of this article is to establish a new model of the full metal screw pump leakage. Compared with the existing models, the new model not only fully considers the inertial force of the fluid but also conducts research in three aspects: transverse leakage, longitudinal leakage, and oblique leakage. e model can also provide guidance for the design and optimization of field pumps.

Model Simplification.
Based on the structural characteristics and working principle of the clearance fit of the stator and rotor of the all-metal screw pump, the model makes the following assumptions: (1) e fluid in the pump is pure liquid phase with constant temperature (2) e fluid in the pump is Newtonian fluid, and the fluid meets Newton's internal friction law (3) Uniform pressure changes between pumps (4) e fluid properties are stable and do not change with pressure Using the concentric ring gap flow theory [23], the pressure drop formula can be obtained as where Δp 0 is the single-stage pressure difference between the pumps, Pa; μ is the viscosity of the fluid, Pa·s; L is the gap length, m; q is the gap leakage, m 3 /s; r 0 and R 0 are the inner and outer radius of the concentric ring, m; and h is the height of the gap, m. Now consider the pressure drop loss caused by the fluid inertia force. Figure 1, the inner diameter of the ring is r 0 , the outer diameter is R 0 , and the fluid density is ρ, and it flows in at the velocity of u 0 from the inlet. After time dt, the velocity becomes u. In this process, the expression of fluid kinetic energy can be obtained as

Inlet Inertia Effect. As shown in
Assuming the initial velocity of the fluid is 0, equation (2) can be simplified as Further, the change in power can be expressed as By integrating equation (4), the inertial power at the entrance can be obtained as Introduce the N-S equation in the cylindrical coordinate system because f x � f y � f z � 0 and set the liquid flow condition to one-dimensional flow and axisymmetric, and we can get And the following relationship exists: dp dx � − Δp L .
Taking equation (7) into equation (6) and integrating it, we can get where C 1 and C 2 are integral constants. e boundary condition relation is Substituting formula (9) into formula (8), we can get u � Δp 4μL Substituting formula (10) into formula (5), we can get where N is the inertial power at entrance, W; ρ is fluid density, kg/m 3 ; and d is the average diameter of the gap, m. And there is a relation: From equation (11), we can get the inertia pressure loss at the inlet section: where Δp 1 is the inertia pressure loss at the inlet section, Pa.

Exit Inertia Effect.
For the exit inertia effect of the annular gap flow model, the pressure drop can be calculated by the following formula: where Δp 2 is the inertia pressure loss at the outlet section, Pa, and ξ is the sudden expansion pressure loss coefficient of the annular gap flow channel, and the value is 1. e total pressure drop of the fluid through the model is obtained: Mathematical Problems in Engineering Simplify equation (15) and solve the one-variable quadratic equation: Equation (16) is the formula for calculating total leakage in the case of clearance fit of the all-metal screw pump. Analyze the effect of gap length on the amount of leakage. When L � 0, equation (16) can be simplified as Equation (17) considers only the leakage under the influence of inertial force, and the amount of leakage is proportional to the gap height and inversely proportional to the square root of the fluid density.
When L approaches positive infinity, formula (15) is simplified as It is further simplified to Because L ⟶ ∞, erefore, equation (19) becomes It can be seen that formula (21) is the same as formula (1), which shows that if the inertial force of the fluid is not considered, the leakage of the all-metal screw pump is inversely proportional to the pressure difference and decreases with the increase of the viscosity of the lifting fluid.

Analysis of Leakage Process.
According to the special meshing method and movement mechanism between the stator and rotor of the all-metal screw pump, the leakage process mainly includes horizontal and vertical leakage, as well as a small amount of oblique leakage, as shown in Figure 2, where the red a-cavity, c-cavity, and e-cavity are relatively high-pressure cavities, while the green b-cavity, d-cavity, and f-cavity are relatively low-pressure cavities.
In summary, the leakage process mainly includes the following three types: (1) Transverse leakage: as shown in the green and red parts in Figure 2, the corresponding liquid will leak into the relatively low-pressure cavity along the relatively high-pressure cavity, that is, from a to b, c to d, and e to f. (2) Longitudinal leakage: after different low-pressure and high-pressure zones are formed in the radial direction, the corresponding low-pressure and highpressure zones continue to increase in pressure from the suction end to the discharge end in the axial direction. at is to say, the pressures in the b, d, and f chambers are constantly increasing, and the pressures in the a, c, and e chambers are also increasing, so there is a leakage in the longitudinal direction from the high pressure region to the low pressure region, that is, f to d and d to b and e to c and c to a. (3) Oblique leakage: when the fluid in the upper-stage high-pressure chamber flows in the axial direction to the lower-stage chamber, most of the fluid flows to the relatively high-pressure area on the same side, but a small amount of fluid flows into the relatively low-pressure chamber, so oblique leakage occurs, that is, fluid leakage occurs between the high-pressure chambers and low-pressure chambers in the axial direction. Because the oblique leakage fluid must not only overcome the pressure loss in the longitudinal direction but also overcome the pressure loss in the transverse direction, compared to the transverse and longitudinal leakage, the amount of oblique leakage is small, which is the secondary leakage of the entire pump.

Calculation of Transverse Leakage.
Transverse leakage is one of the main leakages. e calculation model is shown in Figure 3, where the determination of d is complicated. It can be determined by the spiral method on the surface of the rotor [19]: where e is the eccentricity of the screw pump, m, and T is the stator lead, m. Substituting equation (22) into equation (17), the transverse leakage is Equation (23) is the calculation formula for transverse leakage of all-metal screw pumps.

Calculation of Longitudinal Leakage.
Longitudinal leakage is one of the main leakages, and its calculation model is shown in Figure 4. ere is a relationship: where D r is the rotor diameter, m, and D s is the stator diameter, m. Combined with formula (17), the longitudinal leakage can be obtained as

Calculation of Oblique
Leakage. e oblique leakage should overcome not only the resistance of the transverse leakage but also the resistance of the longitudinal leakage. erefore, the amount of the oblique leakage is relatively small, which is the secondary leakage of the screw pump.
It can be known from the screw pump leakage mechanism that the oblique leakage, the transverse leakage, and longitudinal leakage have the following relationship: So, we can get Combining formulas (23), (25), and (27), we can calculate the oblique leakage formula as Combining transverse leakage, longitudinal leakage, and oblique leakage, the total leakage is Equation (29) is the formula for calculating the total leakage of the all-metal screw pump.

Performance Experiments
In order to verify the accuracy of the leakage model, laboratory experiments were conducted by using a special experimental platform, as shown in Figure 5. e experimental system includes the control system, fluid system, and data collection system. e control system is composed of a computer and a console, which is responsible for sending instructions to the fluid system; the fluid system is composed of a screw pump and a motor, which is responsible for the fluid transmission; the data collection system is composed of a flow meter and a pressure gauge, which is responsible for collecting data during the experiment. e systems are tightly coordinated and safe. e experiment uses a single-head screw pump, as shown in Figure 6. First, the control system transmits instructions to the motor, and then the motor drives the screw pump to rotate. In order to conveniently record the input torque of the screw pump, a torque meter is installed between the motor and the screw pump. After measurement, such as pressure gauge and flow meter, it finally flows into the liquid pool. e whole system is a circulation system. e experiment was performed under safe and reliable conditions. By adjusting the valve, the pressure difference between the inlet and outlet was changed. e experimental fluid was hydraulic oil, viscosity was 60 mPa·s, and the ambient temperature was 20°C. e experimental pumps were two commercial pumps with different clearances, and the calculation formula of the gap between the stator and the rotor is shown in formula (30). e specific models are shown in Table 1.
where δ is the gap value, m.

Model Validation.
According to the model, the theoretical leakage amount can be calculated, and the actual leakage amount can be calculated according to the experimental results, and the two are compared and analyzed, as shown in Figure 7. e calculation method of the leakage amount in the experiment can be calculated by the following formula: where Q is the actual displacement of the pump, m 3 /d, and q r is the theoretical displacement of a single revolution, ml/r. Figures 7(a) and 7(b), respectively, show the experimental results of two different types of pumps. It can be seen from the figure that no matter what type of pump, as the pressure difference increases, the amount of leakage gradually increases, but the increasing trend is gradually decreasing. It is worth noting that compared to the JDGLB160 screw pump with a smaller clearance value, the leakage of the screw pump of the model JDGLB350 is greater.  It can be seen from Figure 7 that the experimental results agree well with the model results. e comparison of the relative error results is shown in Figure 8. e overall error is less than 5%. e results show that the accuracy of the model is well verified.
It is worth noting that the experimental results also show that although increasing the rotational speed is beneficial to increasing the output of the pump, it has no obvious effect on reducing the amount of leakage, which is consistent with the model results.

Influencing Factor Analysis and Model Verification with
Variable Parameters. By further analyzing the derived model, it can be seen from equation (29) that the pump leakage is not only related to the pressure difference but also related to the fluid density and the parameters of the pump itself.
At the same time, in order to verify whether the calculation results of the model are consistent with the theoretical results when other parameters are changed, we change fluid density values and change the gap values of the pump to continue the experiments. We also compare the experimental results with the theoretical calculation results (Figures 9 and 10).

Gap Height.
e height of the gap between the stator and rotor of the all-metal screw pump is a very important structural parameter of the pump. In order to analyze the effect of the gap height on the amount of leakage, a sensitivity analysis is performed on the gap height h, and the theoretical calculation results are compared with the experimental results, as shown in Figure 9.
It can be seen from Figure 9 that the experimental results are in good agreement with the theoretical calculations even after changing the h values, which further verifies the accuracy of the model. It can also be seen that as the pressure difference increases, the amount of leakage gradually increases, but the rate of growth is declining; at the same time, it can be seen that different gap values have a great impact on the amount of leakage, that is, at the same pressure, the larger the gap value, the greater the amount of leakage, and this disparity is more obvious in the case of high pressure difference.

Fluid Density.
In order to investigate the effect of fluid density on the amount of leakage and verify the accuracy of the model when changing the fluid density values, the experiments of changing the fluid density values were carried out, as shown in Figure 10.
It can be seen from Figure 10 that the experimental data are in good agreement with the model calculation results, under the condition of varying fluid density values, which further verifies the accuracy of the model. It can also be seen that under certain other conditions, the amount of leakage decreases with increasing density, but compared to the structure parameters of the pump itself, the density of the fluid has a smaller effect on the amount of leakage, which is a secondary influencing factor.

Conclusions
In this paper, a new all-metal screw pump leakage model is established with full consideration of inertial forces, which will help to better understand the performance of all-metal screw pumps. e following conclusions are drawn: it can be known that the gap height is the main factor affecting the pump leakage, and the fluid density has an effect on the leakage but is a secondary factor.

Nomenclature
Δp 0 : Single-stage pressure difference between the pumps, Pa Δp 1 : Inertia pressure loss at the inlet section, Pa Δp 2 : Inertia pressure loss at the outlet section, Pa R 0 : Outer radius of the concentric ring, m r 0 : Inner radius of the concentric ring, m h: Clearance height, m u: Fluid velocity, m/s E: Fluid kinetic energy, J N: Inertial power at entrance, W d: e average diameter of the gap, m μ: Fluid viscosity, Pa·s L: Gap length, m Δp: Total pressure loss, Pa q 1 : Transverse leakage, m 3 /s q 2 : Longitudinal leakage, m 3 /s q 3 : Oblique leakage, m 3 /s q: Total leakage, m 3 /s D r : Rotor diameter, m D s : Stator diameter, m Q: eoretical flow rate, m 3 /d q r : eoretical displacement of a single revolution, ml/r.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Mathematical Problems in Engineering 7