Mathematical modeling of two-phase flow leakage
The scroll compressor is the focus of the international compressor industry since the 1980s. It belongs to the volumetric rotary compressor.
It has the characteristics of high efficiency, low vibration, low noise, simple structure and good reliability, and is an ideal model for small displacement compressors. In recent years, researchers have done a lot of work in the research of scroll compressors, and have achieved certain results, but the leakage problem is still the main obstacle to the development of scroll compressors. Vortex compressors generally use gap seals, which increase the reliability of the compressor and create problems that are difficult to control. The internal leakage of the scroll compressor increases its power consumption, and the external leakage increases the power consumption and reduces the displacement. In order to reduce leakage and reduce power consumption, it is necessary to improve the processing precision of the profile, thereby increasing the cost of the whole machine. Therefore, it is necessary to study the leakage mechanism in depth and try to reduce the leakage amount to further improve the efficiency of the compressor and reduce the manufacturing cost.
According to the special structure and unique working principle of the scroll compressor, the sealing gap is divided into two types: radial clearance and axial clearance. The leakage through the radial clearance is called tangential leakage, and the leakage through the axial clearance is called For radial leakage. Since the radial leakage line of the scroll compressor is much longer than the axial leakage line, the tangential leakage of the scroll compressor is the focus of this paper, and the leakage model is shown in the figure. For the amount of gas leaking through such gaps, most of the literature is based on the flow formula of gas flowing through the nozzle. The effect of viscous friction on the flow is ignored. In a small gap, the viscous friction of the gas is comparable to the inertial force and cannot be ignored. According to this feature, this paper establishes a mathematical model that considers both the viscous term and the inertia term.
1 Mathematical model of single-phase leakage When the compressed gas leaks through the small gap, when the leakage occurs, the gas with higher pressure enters the flow channel from the 1-1 section, passes through the small-gap passage with variable cross-section, and flows out from the 2- 2 section. Since the channel widths are opposite and far greater than the height, the flow field can be considered to be uniform in the Y direction and can be processed in binary flow. In order to mathematically describe the leakage, the convection action is assumed as follows (1) pxpz( 2)z = 0( 3) The flow process is adiabatic. Based on the above assumptions, taking into account the common effects of viscous and inertial forces on gas flow, based on Momentum equations, continuity equations, actual gas state equations and process equations, etc., derive a more accurate calculation model of leakage, the specific process is as follows: momentum equation ux + uz = - px + zuz ( 1) continuous Equation x( u ) + z( ) = 0( 2) Equation of state p = RT( 3) Process equation p / k = C(4) Boundary condition u (x, 0) = 0 u (x, h) = 0 From the assumption (2), the equation (1) can be transformed into ux + uz = - px + 2 uz 2 in the small gap flow channel, the gas flow film is very thin, so the inertial term in the above formula can be averaged in the z direction, and the result is 2 uz 2 = 1 px + h
h 0 ux + uz dz( 5) It can be seen from the above equation that the right side of the equation is only a function of x, that is, 2 uz 2 = f (x) (6) using the boundary condition to quadraticize the above equation to obtain u = z( z - h) 2 f (x ) ( 7) The leakage mass flow of the gas through the unit width channel is Q =
h 0 udz( 8) Bring ( 7) into (8) and get Q = - h 3 f (x )12( 9) Obviously, the leakage amount Q must be obtained, and the deviation of x and z is obtained for equation (7). Derivative ux = z( z - h )2 f (x )x - zf (x)2 hx( 10)uz =(2z - h)f (x )2( 11) Equation ( 2) Expand ux + ux +
z = 0( 12) (4) Find the partial derivative of xpx = kp x( 13) Combine the equations ( 3), ( 10), ( 12 ), ( 13) and use the boundary conditions (x, 0) = 0 integral = z 2 f (x) 4 hx - 1 2 z 3 - z 2 h 2 f(x)x + f (x)kp px( 14) will be (7) ( 9) (11) ( 14) Bring in (5), and integrate it to get 2 uz 2 = px + h 2 f (x ) h 60 f (x) x + f (x ) 24 hx + hf (x) 120kp x ( 15) from ( 6) ( 12) ( 9) Eliminate f (x) in (15) and sort out px = 120 Q h - 6Q 2 5h 3 hx 6Q 2 5k 2 p 2 h 2 - 1 ( 16) The above equation is the flow field The differential relationship between internal pressure changes and leakage, pressure, and channel geometry parameters.
For the parallel plate gap, considering that the height of the leakage channel is a certain value, ie hx = 0, the equation (16) can be simplified as px = 60k 2 p 2 Qh 6Q 2 2 - 5k 2 p 2 h 2 ( 17) 2 two phases Mathematical model of flow leakage For the injection scroll compressor, the leakage gap is generally a mixed flow of oil and gas. Due to the blocking effect of the oil, the leakage of the compressor gas is reduced. For the calculation of gas phase leakage in this two-phase flow, a phase separation flow model is used in this paper. Since the lubricating oil has a certain adhesive force, in the leakage process, there is always a certain thickness of the oil film on both sides of the axial gap, and the gas phase leaks from the center. In this paper, the gas phase leakage is calculated according to the method of calculating the gas fraction of the two-phase flow section. 1 + 1 - XX g 1 S - 1(18)S = K + (1 - K )g 1 + K 1 - XX 1 + K 1 - XX 1 2(19)A g = A(20)bg = h (21) By substituting the formula (21) into the equations (16) and (17), the amount of gas leakage under the radial gap and the axial gap of the two-phase flow through the scroll compressor can be obtained. Px = 120 Q(ah )- 6Q 2 5(ah )3(ah )x 6Q 2 5k 2 p 2(ah)2 - 1(22)px = 60k 2 p 2 Qah 6Q 2 - 5k 2 p 2(ah 2(23)3 Calculation examples and analysis In order to make the calculation results more comparable with the existing experimental results, this paper only validates the single-phase leakage model of tangential leakage.
Using the leakage mathematical calculation model of this paper, we have compiled a corresponding program to calculate the leakage Q when the refrigerant is R22. It shows the relationship between the leakage amount Q and the minimum clearance hm in, indicating the leakage Q and the pressure of the high pressure chamber. relationship. Curves A, B, and C represent the nozzle model, the viscous flow model, and the calculation model considering both the viscous force and the inertial force. The 0 point in the figure represents the actual measured value. Obviously, the model calculation results established in this paper agree well with the experimental results. P l = 0 8M Pa P b = 0 098MPa
It can be seen that when the gap h and the high and low pressure difference are small, the curves B and C are closer to the experimental results, and the curve A is different. This is because the flow velocity in the leakage flow field changes little at this time, and the corresponding inertial force is also small, which can be neglected, that is, the viscous force is dominant, so it is appropriate to consider only the viscous force. When the gap h and the high and low pressure difference are large, the airflow velocity changes greatly, and the effect of the inertial force is greater than the viscous force. Therefore, the calculation results of the model C and the model A are more accurate than the results of the model B. P b = 0 098MPa hm in = 10 m
4 Conclusions (1) The mathematical model established in this paper, considering the influence of viscous force, the accuracy of the calculation results is much higher than when using the nozzle flow model and the viscous flow model, so the mathematical model established in this paper is feasible; 2) When the gap and the difference between high and low pressure are small, the viscous force predominates; on the contrary, the influence of inertial force predominates.
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