A method for calculating leakage flow of straight-through grate teeth with non-uniform gaps

Abstract

The present invention relates to the technical field of analysis and design of aero-engine air systems, and in particular to a method for calculating the leakage flow of non-uniform gap straight-through grate teeth. The method comprises the following steps: obtaining the inlet and outlet pressures of the non-uniform gap straight-through grate teeth; preliminarily selecting the pressures in tooth cavities of each level of grate teeth according to the inlet and outlet pressures; obtaining the flow rates of the tooth top clearances of each level of grate teeth based on the sizes of the tooth top clearances of each level of grate teeth; correcting the preliminarily selected pressures in the tooth cavities of each level of grate teeth according to a flow continuity condition; and determining the leakage flow rates of the non-uniform gap straight-through grate teeth after the tooth top clearance flow rates of each level of grate teeth meet an optimal condition. The method can accurately calculate the leakage flow rates of the non-uniform gap grate teeth.

Description

Technical Field

[0001] The invention relates to the technical field of analysis and design of aero-engine air systems, and in particular to a method for calculating leakage flow of a non-uniform gap straight-through grate. Background Art

[0002] Straight-through sealing grate teeth are commonly used rotary-static sealing devices in aircraft engine air systems. They are responsible for reducing flow leakage and controlling the flow distribution of the air system. Therefore, accurately calculating the leakage flow of the grate teeth is crucial for air system analysis and design.

[0003] The grate tooth tip clearance is recognized as a key factor affecting leakage flow. During the operation of an aircraft engine, the components are subjected to the combined effects of aerodynamic loads, thermal loads, and mechanical loads, causing elastic deformation, resulting in changes in the grate tooth tip sealing clearance. Due to the different structural stiffness of the rotating and stationary components or the uneven axial temperature distribution, the radial deformation of the grate element is not consistent, resulting in a non-uniform distribution of the tooth tip clearance, such as Figure 1 However, existing methods for calculating grate leakage flow are all derived from grate teeth with uniform gaps, and cannot accurately calculate the leakage flow of grate teeth with non-uniform gaps, and cannot support the refined design of aircraft engine air systems. Summary of the Invention

[0004] In view of the above problems, the present invention provides a method for calculating the leakage flow of non-uniform gap straight-through grate teeth, which solves the technical problem in the prior art that it is difficult to accurately calculate the leakage flow of non-uniform gap grate teeth and requires refined design of the aircraft engine air system.

[0005] The present invention provides a method for calculating the leakage flow of a non-uniform gap straight-through grate tooth, comprising the following steps:

[0006] Step S1, obtaining the inlet and outlet pressures of the non-uniform gap straight-through grate teeth;

[0007] Step S2: preliminarily selecting the pressure inside the tooth cavity of each level of grate teeth according to the inlet and outlet pressures;

[0008] Step S3, based on the size of the tooth tip clearance of each level of grate teeth, obtaining the flow rate of the tooth tip clearance of each level of grate teeth;

[0009] Step S4: correcting the pressure in the tooth cavity of each level of the grate teeth initially selected according to the flow continuity condition.

[0010] Step S5, returning to step S2, until the gap flow rate of the tooth tips of each level of grate teeth meets the optimal condition, and then determining the leakage flow rate of the non-uniform gap straight-through grate teeth.

[0011] Preferably, step S2 specifically includes: according to the grate inlet and outlet pressures p0 and p n , preliminarily select the pressure p in the tooth cavity of the i-th stage grate i (i=1,2,…,n-1).

[0012] Preferably, step S3 specifically includes: considering the n-stage grate teeth as n 1-stage grate teeth, and calculating the kinetic energy residual factor α in the front cavity of the i-th stage grate teeth for each stage of tooth top clearance. i , the exit velocity V of the tooth tip clearance of the i-th stage grate i , total airflow pressure p in the front cavity of the i-th stage grate t,i-1 and the airflow mass flow rate of the tooth tip clearance of the i-th stage grate

[0013] Preferably, the residual kinetic energy factor α in the front cavity of the i-th stage grate teeth is i The calculation method is:

[0014]

[0015] Among them, V i-1 V′ is the velocity of the airflow when it flows out from the gap between the teeth of the i-1th stage grate. i-1 c is the velocity of the airflow from the gap between the teeth tips of the i-1th stage grate to the end of the jet section, i-1 is the tooth tip clearance of the i-1th level grate teeth, c i is the gap between the teeth of the i-th level grate, B is the distance between the teeth of the grate, and t is the width of the teeth of the grate.

[0016] Preferably, the residual kinetic energy factor α in the front cavity of the i-th stage grate teeth is i The calculation method is:

[0017]

[0018] Among them, V i-1 V′ is the velocity of the airflow when it flows out from the gap between the teeth of the i-1th stage grate. i-1 c is the velocity of the airflow from the gap between the teeth tips of the i-1th stage grate to the end of the jet section, i-1 is the tooth tip clearance of the i-1th level grate teeth, c i is the gap between the teeth of the i-th level grate, B is the distance between the teeth of the grate, and t is the width of the teeth of the grate.

[0019] Preferably, the exit velocity V of the tooth tip clearance of the i-th stage grate is i The calculation method is:

[0020]

[0021] Among them, p i-1 The pressure in the front cavity of the i-th stage grate, A iis the tooth tip clearance area of ​​the i-th stage grate, κ is the gas specific heat ratio, T is the temperature, R g is the gas constant, The airflow mass flow rate at the tooth tip clearance of the i-th stage grate teeth.

[0022] Preferably, the total airflow pressure p in the front cavity of the i-th stage grate is t,i-1 The calculation method is:

[0023]

[0024] Among them, p t,i-1 is the total airflow pressure in the front cavity of the i-th stage grate, p i-1 is the static pressure of the airflow in the front cavity of the i-th stage grate, V i-1 is the velocity of the airflow when it flows out from the gap between the teeth tops of the i-1th stage grate, and ρ is the gas density.

[0025] Preferably, the airflow mass flow rate of the i-th stage grate tooth tip clearance is The calculation method is:

[0026]

[0027] Among them, p t,i-1 is the total airflow pressure in the front cavity of the i-th stage grate, p i is the static pressure of the airflow in the back cavity of the grate teeth, R g is the gas constant;

[0028] β cr is the critical pressure ratio, and the calculation formula is:

[0029]

[0030] where κ is the ratio of gas specific heats.

[0031] Preferably, step S4 specifically includes: obtaining Then, p is modified according to the flow continuity condition. i (i=1, 2, ..., n-1); the flow continuity condition is: the mass flow rate passing through the n grate tooth top gaps is the same.

[0032] Preferably, step S5 specifically includes: returning to step S2 until The calculation is completed when the difference between them is less than the limit value, and the grate leakage flow is taken as The average value of .

[0033] Compared with the prior art, the present invention has at least the following beneficial effects:

[0034] When calculating the leakage flow of straight-through grate teeth with non-uniform gaps, the present invention takes into account the non-uniformity of the gaps at the top of each tooth and the non-consistency of the airflow in the tooth chambers at each level. Based on the calculation formula for the influence of the non-uniformity of the gaps at the top of each tooth on the kinetic energy residual factor in the tooth chamber, the kinetic energy residual factor in each tooth chamber is independently corrected and calculated. After multiple iterative calculations, the leakage flow of each single-stage grate tooth is independently calculated using the pressure in the front and rear chambers in each iterative cycle, and then the pressure in each tooth chamber is corrected according to the flow conservation principle until the flow conservation requirement is met. The present invention breaks through the "fixed and uniform" sealing gap assumption of the aircraft engine air system through the above method, improves the flow calculation capability of the air system under the "dynamic and non-uniform" sealing gap, and provides a basis for the precise design of the flow distribution of each branch of the air system during the transition process. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] The drawings are only for purposes of illustrating particular embodiments and are not to be considered limiting of the invention.

[0036] Figure 1 A schematic diagram of a non-uniform gap straight-through grate meridian flow channel provided by an embodiment of the present invention;

[0037] Figure 2 A schematic diagram of airflow parameter changes before and after the airflow passes through the grate tooth tip gap under ideal conditions provided by an embodiment of the present invention;

[0038] Figure 3 A schematic diagram of the change in airflow parameters before and after the airflow passes through the grate tooth tip gap, taking into account the non-uniform tooth tip gap and the residual kinetic energy in the tooth cavity, provided by an embodiment of the present invention;

[0039] Figure 4 This is a flow chart of the method for calculating the leakage flow of non-uniform gap straight-through grate teeth provided by the present invention. DETAILED DESCRIPTION

[0040] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited by the specific embodiments disclosed below.

[0041] In order to illustrate the effectiveness of the method proposed in the present invention, the above technical solution of the present invention is described in detail below through a specific embodiment.

[0042] The present invention equates multi-stage straight-through grate teeth to a form of multiple single-stage grate teeth connected in series, and considers the situation that the gaps at the tops of each stage are uneven and the kinetic energy of the airflow in the tooth cavity is not completely dissipated, and proposes a method for calculating the leakage flow of non-uniform gap straight-through grate teeth.

[0043] (1) Theoretical model

[0044] When the speed is not high, the grate leakage flow is less affected and the effect of wall rotation on the grate leakage flow can be ignored. Assume that the airflow flows isothermally at the tooth tip gap and the airflow energy is completely dissipated in the tooth cavity. The aerodynamic parameters before and after the airflow flows through the grate tooth tip gap are as follows: Figure 2 As shown, if friction is not considered, when the air flow passes through the grate teeth gap, the pressure change is very small, and the relationship between the pressure change and the tooth top gap outlet velocity can be obtained as shown in the following formula.

[0045] V 2 =-2δp(v+δv)

[0046] Where p is the pressure, v is the specific volume, V is the air velocity at the outlet of the tooth tip clearance, δp is the pressure change after the airflow passes through the tooth tip clearance, and δv is the specific volume change after the airflow passes through the tooth tip clearance.

[0047] The calculation formula for the flow rate of air flowing through the grate teeth gap is:

[0048]

[0049] in, is the air mass flow rate, and A is the tooth tip clearance flow area.

[0050] From the above formulas for airflow outlet velocity and flow rate, we can get:

[0051]

[0052] When the airflow flows isothermally, the formula is satisfied:

[0053]

[0054] The state equation of the airflow satisfies the formula:

[0055] pv=R g T

[0056] Where T is the temperature, R g is the gas constant. For air, R g =287J / (kg·K).

[0057] From the above three formulas we can get:

[0058]

[0059] The frictional work done by the rotating wall on the airflow will slightly increase the total temperature of the airflow, but when the frictional work is small, the temperature rise can be ignored, so it can be approximately assumed that the total temperature T of the airflow remains unchanged. The airflow satisfies the flow continuity law, so the mass flow rate through the tooth tip gap of n grate teeth is the same. For n grate teeth, by summing the above formula, we can get:

[0060]

[0061] When the pressure change δp is very small, the above formula can be expressed in the form of an integral:

[0062]

[0063] The mathematical relationship between the mass flow rate and the pressure in the front and rear tooth cavities when the airflow passes through the grate tooth top gap can be obtained:

[0064]

[0065] Among them, p0 and p n are the grate inlet and outlet pressures, respectively.

[0066] When the pressure in front of the tooth tip clearance is constant, as the back pressure of the cavity behind the tooth tip clearance continues to decrease, the tooth tip airflow rate does not always increase. When the tooth tip clearance enters the critical state, the tooth tip leakage flow rate no longer changes. Critical pressure ratio β cr The calculation formula is:

[0067]

[0068] where κ is the ratio of gas specific heats.

[0069] The above formula for the mathematical relationship between the mass flow rate and the pressure in the front and rear tooth cavities when the airflow flows through the tooth top gap of the comb teeth is derived on the basis of assuming that the pressure change δp after the airflow flows through the tooth top gap is very small and the multi-stage comb tooth structure. It can describe the mathematical relationship between the mass flow rate and the pressure in the front and rear tooth cavities when the airflow flows through the tooth top gap of the comb teeth. The present invention extends it to the case of "independent calculation of a single tooth and flow conservation of multiple teeth in series" in the following way.

[0070] Let n = 1, then the calculation formula for the airflow mass flow rate of the tooth tip clearance of the i-th stage grate is:

[0071]

[0072] Among them, p t,i-1 is the total airflow pressure in the front cavity of the i-th stage grate, p i is the static pressure of the airflow in the back cavity of the grate teeth, R g is the gas constant.

[0073] like Figure 2As shown in the figure, the front chamber of the i-th stage grate is a static chamber, and the total airflow pressure is the same as the static pressure. After the throttling and acceleration of the grate tooth tip clearance, the exit velocity V of the tooth tip clearance is usually larger, and the total pressure of the rear chamber of the grate is not equal to the static pressure.

[0074] In the above manner, the present invention describes the relationship between the pressure change of the airflow and the speed and flow rate under the conditions of low rotation speed, isothermal flow of the airflow at the tooth top gap, and all kinetic energy dissipated inside the tooth cavity, and derives a theoretical model of the airflow mass flow rate in the comb tooth top gap, which can be used to describe the flow characteristics of the airflow in the comb tooth gaps at various levels.

[0075] (2) Correction of total pressure in tooth cavity

[0076] Inside the grate element, the kinetic energy of the airflow is not completely dissipated during the flow process in the tooth cavity. The flow process in the tooth cavity can be simplified into the jet section and the contraction section, such as Figure 3 As shown. The airflow velocity when it flows out from the gap between the teeth of the i-1th stage is V i-1 , the velocity after the jet section ends is V′ i-1 Under the assumption that static fluid flows from an infinite container, the starting point of the contraction section is 2% of the dynamic pressure head of the grate tooth tip clearance, which is located 2ac away from the i-th level grate tooth. i The position (parameters a = 1.26, c i is the tooth top clearance of the i-th level grate teeth).

[0077] Previous researchers derived the kinetic energy dissipation process of the jet section based on the free jet theory and gave a calculation method for the kinetic energy residual factor α under the condition of uniform gap:

[0078]

[0079] Among them, B is the grate tooth spacing, t is the grate tooth top width, and c is the grate tooth top clearance.

[0080] Based on the above formula, when the tooth top clearances of different levels of grate teeth are different, the residual kinetic energy factor α in the front cavity of the i-th level grate teeth is i The calculation formula can be expressed as follows:

[0081]

[0082] Among them, V i-1 V is the velocity of the airflow flowing out from the gap between the teeth tips of the i-1th stage grate, i ' -1 c is the velocity of the airflow from the gap between the teeth tips of the i-1th stage grate to the end of the jet section, i-1 is the tooth tip clearance of the i-1th level grate teeth, c i is the tooth tip clearance of the i-th stage comb teeth.

[0083] Considering that the kinetic energy in the tooth cavity is not completely dissipated, the total pressure in the tooth cavity is corrected using the fluid dynamic pressure head. It is approximately considered that the total pressure of the airflow in the front cavity of the i-th stage grate tooth is p t,i-1 Equal to the static pressure of the airflow p i-1 The sum of the dynamic pressure head and the total pressure in the tooth cavity of each level of the grate can be obtained:

[0084]

[0085] Among them, p i-1 is the static pressure of the airflow in the front cavity of the i-th stage grate teeth, and ρ is the gas density.

[0086] From the above formula, it can be concluded that the premise for calculating the total pressure of the airflow is to know the velocity of the airflow when it flows out from the tooth tip gap of the previous level grate. The velocity can be calculated using the flow calculation formula, the total static temperature relationship formula and the gas state equation. The three formulas are:

[0087]

[0088]

[0089] p i v i =R g T i

[0090] Among them, i represents the i-th level grate, A i is the tooth tip clearance area of ​​the i-th level grate teeth, is the flow rate flowing through the tooth tip clearance of the i-th stage grate teeth, V i 、T i 、p i 、v i are respectively the air flow velocity, temperature, pressure and specific volume in the front cavity of the i-th stage grate. p is the constant-pressure specific heat capacity of the airflow, which is calculated as:

[0091]

[0092] According to the above formula, the expression of the exit velocity of the tooth tip clearance of the i-th stage grate can be obtained:

[0093]

[0094] The total pressure of the airflow in the tooth cavity can be corrected by applying the outlet velocity of the tooth tip clearance of the i-th stage grate tooth to the calculation formula of the total pressure in the tooth cavity of each stage of the grate tooth.

[0095] The method proposed in this paper considers the kinetic energy dissipation of the airflow after exiting the gaps between the grate teeth. It simplifies the airflow into a jet phase and a contraction phase, and introduces a residual kinetic energy factor to describe the partial dissipation of kinetic energy. By incorporating parameters such as airflow velocity, gap size, and tooth spacing, this modified model allows for a more accurate estimation of the total pressure within the front chamber of each grate tooth.

[0096] (3) Calculation process

[0097] After knowing the total airflow pressure in each grate cavity, the multi-stage grate structure is decomposed into multiple single-stage grate teeth connected in series for calculation. The pressure in the middle tooth cavity is obtained through multiple iterative calculations. The specific iterative process is as follows:

[0098] (1) According to the grate inlet and outlet pressures p0 and p n , preliminarily select the pressure p in the tooth cavity of the i-th stage grate i (i=1,2,…,n-1);

[0099] (2) Considering the n-stage grate teeth as n 1-stage grate teeth, for each stage of tooth top clearance, calculate the kinetic energy residual factor α in the front cavity of the i-stage grate teeth in turn. i , the exit velocity V of the tooth tip clearance of the i-th stage grate i , total pressure p in the tooth cavity of each level of the grate teeth t,i-1 and the airflow mass flow rate of the tooth tip clearance of the i-th stage grate

[0100] (3) Get Then, p is modified according to the flow continuity condition. i (i=1,2,…,n-1);

[0101] The flow continuity condition indicates that during the flow process, the mass flow rate flowing into a certain control volume is equal to the mass flow rate flowing out of the control volume. Specifically, in the present invention, the mass flow rate through the gaps between the n teeth should be the same. By adjusting the pressure p in each middle tooth cavity i To ensure consistent.

[0102] (4) Repeat the above three steps until The calculation is completed when the difference between them is less than a certain limit value, and the grate leakage flow is taken as The average value of .

[0103] This method, combined with calculations of the residual kinetic energy factor, velocity, total pressure, and mass flow, progressively corrects the pressure within each tooth chamber until the mass flow rate at all levels is consistent, thus ensuring flow continuity. This method effectively evaluates and optimizes the performance of grate elements, improving their accuracy and reliability in practical applications.

[0104] Although the specific embodiments of the present invention have been described in a particular order, it should be understood that such actions or steps are required to be performed in the particular order shown or in a sequential order, or that all illustrated actions or steps are required to be performed to obtain the desired result. Under certain circumstances, multitasking and parallel processing may be advantageous. Similarly, although some specific implementation details have been included in the above discussion, these should not be construed as limiting the scope of the present disclosure. Some features described in the context of a separate embodiment can also be implemented in a single implementation in combination. On the contrary, the various features described in the context of a single implementation can also be implemented in multiple implementations individually or in any suitable sub-combination.

[0105] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A method for calculating leakage flow of non-uniform gap straight-through grate teeth, characterized in that: The following steps are involved: Step S1, measuring the inlet and outlet pressures of the non-uniform gap straight-through grate teeth; Step S2: preliminarily selecting the pressure inside the tooth cavity of each level of grate teeth according to the measured inlet and outlet pressures; Step S3, based on the size of the tooth tip clearance of each level of grate teeth, obtaining the flow rate of the tooth tip clearance of each level of grate teeth; Step S4: correcting the pressure in the tooth cavity of each level of the grate teeth initially selected according to the flow continuity condition. Step S5, returning to step S2, until the gap flow rate of the tooth tips of each level of grate teeth meets the optimal condition, determining the leakage flow rate of the non-uniform gap straight-through grate teeth; Step S2 specifically includes: according to the grate inlet and outlet pressure and , preliminarily selected Pressure in the tooth cavity of the stage grate ; Step S3 specifically includes: Grade grate teeth are considered For each level of tooth top clearance, calculate the first level of grate teeth. Residual kinetic energy factor in the front cavity of the stage grate teeth , No. Level grate tooth tip clearance outlet speed , No. Total airflow pressure in the front cavity of the stage grate teeth Hedi Mass flow rate of airflow at the tooth tip clearance of stage grate teeth .

2. The method for calculating leakage flow of non-uniform gap straight-through grate teeth according to claim 1, characterized in that: The said Residual kinetic energy factor in the front cavity of the stage grate teeth The calculation method is: in, For airflow from The outflow velocity of the first-stage grate tooth top clearance is For airflow from The velocity of the flow from the gap between the first-stage grate teeth and the end of the jet section is It is Level 1 grate tooth top clearance, It is Level grate tooth top clearance, is the grate tooth spacing, is the width of the comb teeth top.

3. The method for calculating leakage flow of non-uniform gap straight-through grate teeth according to claim 2, characterized in that: The said Level grate tooth tip clearance outlet speed The calculation method is: in, No. The pressure in the front cavity of the level grate teeth, It is Level grate tooth tip clearance area, is the ratio of specific heats of gases, T is the temperature, is the gas constant, No. The mass flow rate of the airflow at the tooth tip clearance of the stage grate teeth.

4. The method for calculating leakage flow of non-uniform gap straight-through grate teeth according to claim 3, characterized in that: The said Total airflow pressure in the front cavity of the stage grate teeth The calculation method is: in, For the The total airflow pressure in the front cavity of the stage grate teeth, For the The static pressure of the airflow in the front cavity of the stage grate teeth, For airflow from The outflow velocity of the first-stage grate tooth top clearance is is the gas density.

5. The method for calculating leakage flow of non-uniform gap straight-through grate teeth according to claim 4, characterized in that: The said Mass flow rate of airflow at the tooth tip clearance of stage grate teeth The calculation method is: in, For the Total airflow pressure in the front cavity of the stage grate teeth, is the static pressure of the airflow in the back cavity of the grate teeth, is the gas constant; is the critical pressure ratio, and the calculation formula is: in, is the ratio of specific heats of gases.

6. The method for calculating leakage flow of non-uniform gap straight-through grate teeth according to claim 5, characterized in that: Step S4 specifically includes: obtaining Then, according to the flow continuity condition, ; The flow continuity condition is: n The mass flow rate of the tooth tip clearance of each grate tooth is the same.

7. The method for calculating leakage flow of non-uniform gap straight-through grate teeth according to claim 6, characterized in that: Step S5 specifically includes: returning to step S2 until The calculation is completed when the difference between them is less than the limit value, and the grate leakage flow is taken as The average value of .

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