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Abstract

The pressure loads acting on the teeth of a labyrinth seal are usually not uniform. Sometimes, one tooth would take almost half of the total pressure difference, which is detrimental to the teeth’s working life and sealing effect. Therefore, the estimation of maximum pressure load on the teeth is helpful to design a sound structure of labyrinth seal. This article analyzes the influence of teeth number, boundary condition, clearances, and structure parameters on maximum pressure loads of labyrinth seal teeth in low pressure ratio conditions (PR≤ 0.5, PR = P_out/P_in). Both the computational fluid dynamics model and test results show that teeth number and pressure load are main factors influencing the maximum pressure load on teeth. Finally, a fitting equation is given to estimate the maximum pressure load on labyrinth seal teeth and its results are in good agreement with the computational fluid dynamics model.

Keywords

Labyrinth seal pressure loads on the teeth teeth number pressure ratio clearances

Introduction

The labyrinth seals are important annular gas seals in turbomachines.^1–3 As traditional annular gas seals, their major advantages are that they can be used for higher pressure gradients compared with other seals, like brush seals and finger seals.^4,5 Labyrinth seal is most typically used as impeller seals, shaft seals, and balance piston seals in centrifugal compressors. In centrifugal compressors, its working efficiency can be raised by 2% if the seal clearance can be reduced by 50%.⁶

The efficiency of labyrinth seal is determined by the values of its groove geometry, clearance, and teeth number. In terms of theoretical study, the basic principles of operation can still be described by the methods in the study by Martin⁷ and Childs and Scharrer.^8,9 Today, based on them, the computational fluid dynamics (CFD) methods are used to calculate flow behavior of labyrinth seal.^10,11 Numerical calculation can give a clear visual display of the results.^12–14 In experiments, labyrinth seal has been put to more severe conditions, such as greater pressure differential and higher rotational speed in recent years.^15,16

Generally, leakage is the main factor in evaluating the sealing performance of labyrinth seal design. However, insufficient researches have been conducted in the estimation of the maximum pressure load on labyrinth seal teeth in the process of engine design. This problem is particularly serious for labyrinth seals made of polymer materials, which are often used to reduce wear. These polymer materials deform more easily than traditional metals in high pressure difference. Hence, this study on the maximum pressure load of labyrinth seal teeth is helpful to know their stress and deformation conditions.

The earliest research on pressure distribution of labyrinth seal dated back to the early twentieth century.^7,17 Their results showed that the pressure load increased gradually from the first tooth to the last one, and the last tooth carried the maximum pressure load. However, their research did not consider heat exchange and assumed that the carry-over coefficients were the same for all teeth.

Suryanarayanan and Morrison¹⁸ studied the influence of flow parameters on carry-over coefficient of labyrinth seal. Their results showed that the maximum pressure load occurred on the first tooth and remained constant on the other teeth. Their explanation was that there was an entrance effect caused by the difference between the upstream flow of first tooth and its inside cavity.¹⁸ Moreover, their model is only applicable to incompressible flows.

Other researches^19,20 also indicated that the maximum pressure load occurred on the first tooth, but they offered a different explanation which is applicable to high pressure ratio and compressible flows. Their explanation goes like this: the first tooth has a high flow contraction because there is no upstream “carry-over” effect. Meanwhile, fluid turbulence intensity decreases along the flow, therefore the kinetic energy loss decreases as the fluid flows through each cavity.

Dogu et al.²¹ discussed the effect of pressure ratio on the pressure drops on each tooth. Their results show that the highest pressure load was on the first tooth (the first tooth carried about 62% of total pressure load) when the pressure ratio was 0.67. However, the last tooth would carry the maximum pressure load in low pressure ratio conditions. For instance, the first tooth carries about 46% of total pressure load for PR = 0.28 (PR = P_out/P_in). They thought that the large Mach number of the flow in last clearance was the main reason for the last tooth to carry the highest pressure load in low pressure ratio conditions.²¹

J Jiang et al.²² studied the uniformity of pressure load on labyrinth seal teeth. The results showed that the pressure load on the teeth was not completely uniform. The test results showed that the first tooth carried the maximum pressure load for PR > 0.5. However, the maximum pressure load occurred at the last tooth for PR≤ 0.5.

The above researches all indicate that the pressure load on labyrinth seals is not uniform. In recent years, labyrinth seals are more frequently used in lower pressure ratio conditions (PR≤ 0.5). Meanwhile, for high pressure ratios, the maximum pressure load on the teeth is too small to have any effect on the deformation of teeth. However, low pressure ratio (0.2 ≤PR≤ 0.5) and great differences of pressure loads will force a certain tooth in the seal to suffer an extreme loading condition. Therefore, in labyrinth seal design, estimation of the maximum pressure load on the teeth is helpful to make a desirable structure of labyrinth seal. Recent researches are based on the CFD method to calculate the maximum pressure load on labyrinth seal teeth. This article studies the influence of teeth number, boundary condition, and clearances on the maximum pressure load of labyrinth teeth for the pressure ratio range 0.2–0.5. And it proposes a fitting equation dependent on the pressure ratio and teeth number to estimate maximum pressure load on labyrinth seal teeth.

Theoretical researches and experiment system

CFD model

As shown in Figure 1(a), a four-teeth straight-through labyrinth seal is adopted. The structural parameters of the seal are given in Table 1. The three-dimensional (3D) CFD model and a structural grid for the labyrinth seal are generated in ANSYS ICEM CFD. The mesh independence has been verified to ensure accuracy, and a fine mesh density is used in the region of clearance and grooves, as shown in Figure 1(b). The average value of y⁺ is $15.6 \leq y^{+} \leq 22.3$ in the range of pressure ratio $(0.2 \leq PR \leq 0.5)$ . The results of calculation show that the leakage error remains at 3%, when the number of cells is larger than 200,000.

Figure 1.

Size of labyrinth seal and mesh of flow field: (a) size and flow field of labyrinth seal, (b) local mesh of flow field, (c) mesh independence (PR = 0.2, P₄ = 0.1 MPaA), and (d) leakages under various PR.

Table 1.

Labyrinth parameters.

Parameters	Symbol	Values
Seal radius (mm)	R	57.25
Number of teeth	N	4
Pitch (mm)	T	8.1
Tooth thickness (mm)	t	1.9
Width of groove (mm)	W	6.2
Tooth height (mm)	H	10.2
Radius clearance (mm)	C	0.15

The mass conservation equation, the momentum conservation equations, and the energy conservation equations (equations (1)–(3)) are numerically solved with the CFD software Fluent

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{v}) = S_{m}

(1)

\frac{\partial}{\partial t} (ρ \vec{v}) + \nabla \cdot (ρ \vec{v} \vec{v}) = - \nabla p + \nabla (\vec{τ}) + ρ \vec{g} + \vec{F}

(2)

\begin{matrix} \frac{\partial}{\partial t} (ρ E) + \nabla \cdot (\vec{v} (ρ E + p)) \\ = - \nabla (k_{ε} \nabla T + \vec{v} (τ_{ε})) + S_{h} \end{matrix}

(3)

where ρ is the density, t is time, $\vec{v}$ is the velocity vector, p is the pressure, S_m is the source term, $\vec{τ}$ is the stress tensor, $\vec{g}$ is the gravitational accelerator vector, $\vec{F}$ is the external forces vector, E is the total energy, k_ε is the effective thermal conductivity, T is the temperature, τ_ε is the deviatoric stress tensor, and S_h is the heat source.

The airflow is assumed to comply with the ideal gas law, and the compressibility of gas is also considered

ρ = p / (R T_{em} / M_{w})

(4)

where p is the pressure, M_w is the molecular weight of the gas, R is the universal gas constant, and T_em is the temperature.

The Reynolds number of the flow in the tip clearance of the four teeth on a straight-through labyrinth seal varies from about 2000 to 7700, so the standard k-ε turbulence model is adopted. In the standard k-ε model, turbulence kinetic energy k and its rate of dissipation ε are obtained from the following transport equations

\begin{matrix} \frac{\partial}{\partial t} (ρ k) + \nabla \cdot (ρ k \vec{v}) = \nabla ((μ + \frac{μ_{t}}{σ_{k}}) \nabla k) \\ + G_{k} + G_{b} - ρ ε - Y_{M} + S_{k} \end{matrix}

(5)

\begin{matrix} \frac{\partial}{\partial t} (ρ ε) + \nabla \cdot (ρ ε \vec{v}) = \nabla ((μ + \frac{μ_{t}}{σ_{ε}}) \nabla ε) \\ + C_{1 ε} \frac{ε}{k} (G_{k} + C_{3 ε} G_{b}) - C_{2 ε} ρ \frac{ε^{2}}{k} + S_{ε} \end{matrix}

(6)

In these equations, G_k represents the generation of turbulence kinetic energy due to the mean velocity gradients, G_b is the generation of turbulence kinetic energy due to buoyancy, and Y_M represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. μ_t (= ρC_μ k²/ε) is the turbulent viscosity. C_1ε(= 1.44), C_2ε (= 1.92), C_3ε (= 1.3), and C_μ (= 0.09) are constants. σ_k (= 1.0) and σ_ε (= 1.3) are the turbulent Prandtl numbers for k and ε. S_k and S_ε are source terms.

Maximum pressure load on the tooth

As shown in Figure 1, pressure of the ith groove is defined as $P_{i}$ , which can be obtained by CFD model and tests. The axial pressure load over each tooth can be defined as

Δ P_{i} = P_{i - 1} - P_{i}

(7)

Substituting the relative pressure $α_{i}$ for the pressure load on each tooth, we have equation (8)

α_{i} = \frac{Δ P_{i}}{Δ P}

(8)

where the total pressure difference ΔP and pressure ratio PR can be obtained by equations (9) and (10)

Δ P = P_{0} - P_{4}

(9)

PR = \frac{P_{4}}{P_{0}}

(10)

According to formulas (7)–(10), the pressure on each tooth can be obtained by

Δ P_{i} = α_{i} \times P_{0} \times (1 - PR)

(11)

Equation (11) indicates that pressure load on the tooth is dependent on $α_{i}$ , $P_{0}$ , and PR. Because the sum of $α_{i}$ is 1, $\bar{α_{i}}$ can be obtained by equation (12). The maximum relative pressure $α_{\max}$ can be expressed as equation (13)

\bar{α_{i}} = 1 / N

(12)

α_{max} = \bar{α_{i}} + x σ

(13)

where $σ$ is the standard deviation of $α_{i}$ ; x is a unknown coefficient.

Consequently, the maximum pressure load on the tooth can be written as

Δ P_{max} = (1 + Nx σ) \times \frac{Δ P}{N} = (1 + Nx σ) \times \frac{P_{0} \times (1 - PR)}{N}

(14)

where the coefficient $Nx σ$ is substituted by the coefficient $γ$ , which describes the uniformity of pressure load on teeth. Equation (14) can be written as

Δ P_{max} = (1 + γ) \times \frac{Δ P}{N} = (1 + γ) \times \frac{P_{0} \times (1 - PR)}{N}

(15)

Equation (15) shows that the max pressure load on the teeth relates to the coefficient $γ$ , inlet pressure, and the pressure ratio.

Experiment testing system

Our experiment testing system can measure the outlet leakage and the pressure at each groove. Measurable leakage ranges and the maximum upstream pressure are 0–150 Nm³/h and 0.5 MPa, respectively. To measure the pressure of each groove, three grooves are connected to pressure sensors by tubes illustrated in Figure 2(a). Therefore, the pressure difference between the two sides of a groove represents the pressure load on the tooth. As shown in Figure 2(b), the test end consists of the following parts: test cavity, pressure sensors, and temperature sensor. The experimental measurement error of labyrinth seal pressure is 1% of the measured value. High pressure gas passes into cavity from two sides of the test cavity at the same time, which is beneficial for the stability of inlet pressure.

Figure 2.

Labyrinth seal test system: (a) labyrinth seal and (b) photograph of test end.

Results and discussion

In order to obtain coefficient $γ$ , the results of CFD and experiment are used in equation (15). The coefficient $γ$ can be obtained by

γ = \frac{N \times Δ P_{max} (CFD, EXP)}{P_{0} \times (1 - PR)} - 1

(16)

Moreover, the coefficient $γ$ is dependent on teeth number, clearances, and boundary conditions. Here, the influence of teeth number, clearances, and boundary condition on coefficient $γ$ is discussed.

As shown in Figure 3, the vertical error stands for pressure ratio error, and it is calculated by inlet pressure in the experiment. In high pressure ratio conditions, the inlet pressure is very low, close to the lowest pressure to control the valve, which leads to a relatively large error for its value. However, gas valve can be readily controlled at the low pressure ratio, and the inlet pressure error is small. The horizontal error stands for the pressure load error, and it is calculated by the pressure value in each cavity in the experiment. For low pressure ratio conditions, the larger gas velocity results in greater sensor measurement error.

Figure 3.

CFD and EXP results $(P_{4} = 0.1 MPaA)$ : (a) various coefficients $α_{i}$ of pressure load on each tooth and (b) maximum pressure load on the tooth at various PR.

When the outlet pressure is constant $(P_{4} = 0.1 MPaA)$ , the change of relative pressure $α_{i}$ on each tooth with pressure ratios is given in Figure 3(a). The highest pressure load acts on the last tooth, and with decreasing pressure ratio, the last tooth carries increasingly larger pressure load. For instance, the last tooth carries 51% of total pressure load for PR = 0.2. The flow resistance at last clearance will gradually increase with the decreasing pressure ratios. As pressure ratio decreases, velocity of the flow increases in the last clearance. If the velocity of the flow in the tip clearance is as high as the critical speed (the local sound speed), choking phenomena will occur in the labyrinth seal. The velocity of the flow will not increase as the pressure ratio decreases. This phenomenon occurred in labyrinth seal. When the choking occurs, the ratio of the pressure drop acting on the last tooth approaches the maximum value, which indicates that the pressure load is increasingly concentrated on the last tooth. This increasing flow resistance pushes the pressure at the last groove even higher, hence changing the pressure at other cavities, which is shown in Figure 3(a).

The maximum pressure load on the teeth is shown in Figure 3(b), and the maximum errors about CFD and test results are 11%. Therefore, with decreasing pressure ratio, the maximum pressure load on teeth is generally increasing. This means that the last tooth will carry more pressure load as the pressure ratio decreases.

Teeth number

The coefficient $γ$ can be calculated by equation (16), and the relationship between pressure ratio PR and coefficient $γ$ for four-teeth straight-through labyrinth seal is shown in Figure 4. The pressure ratio PR is inversely proportional to coefficient $γ$ . Moreover, the curves indicate that the uniformity of pressure load on the teeth becomes more unbalanced with decreasing pressure ratio. In Figure 4, coefficient γ is calculated by the maximum pressure load based on teeth number, pressure ratio, and inlet pressure. Although pressure ratio and maximum pressure load are consistent in Figures 3(b) and 4, the inlet pressure is different in different pressure ratios. Therefore, the errors of CFD and the experiment are different in Figures 3(b) and 4.

Figure 4.

PR versus coefficient $γ$ .

Figure 4 illustrates a function relation between pressure ratio and coefficient $γ$ . According to the CFD calculation, with varying teeth number, the coefficient $γ$ is related to both pressure ratio and teeth number

{\begin{matrix} γ = k_{1} (N) (PR)^{2} + k_{2} (N) (PR) + k_{3} (N); 2 \leq N \leq 9 \\ γ = K_{1} (PR)^{2} + K_{2} (PR) + K_{3}; N \geq 10 \end{matrix}

(17)

where the coefficients k1, k2, and k3 are dependent on teeth number N. For 2 ≤N≤ 9, the coefficient k is

{\begin{matrix} k_{1} (N) = 0.0198 N^{3} - 0.2575 N^{2} + 2.4117 N - 3.4298 \\ k_{2} (N) = - 0.018 N^{3} + 0.2832 N^{2} - 2.884 N + 3.5845 \\ k_{3} (N) = 0.0056 N^{3} - 0.1127 N^{2} + 1.1219 N - 1.334 \end{matrix}

(18)

For N≥ 10, the teeth number has no effect on coefficient $γ$ . Thus, the coefficient K is constant

K_{1} = 12.661; K_{2} = - 13.246; K_{3} = 3.8866

(19)

In Figure 5, the CFD results show that the coefficient $γ$ is proportional to teeth number and is inversely proportional to pressure ratio PR. For the pressure ratio 0.2–0.5, the highest pressure load occurs on the last tooth. When the teeth number is two, the first tooth also carries a large pressure load. As the teeth number increases, the pressure in all grooves decreases. Teeth behind the first tooth will share more of the first tooth’s pressure load. However, the downstream pressure of the last tooth does not change, so the pressure load on the last tooth is still high. Therefore, γ increases when the teeth number increases. It should be noted that the change in coefficient $γ$ becomes smaller as the teeth number increases and it almost ceases to change when the teeth number is greater than 9. The reason is that when teeth number is greater than 9, further increasing teeth number will have little influence on the uniformity of pressure load on each tooth. The curves are calculated by equations (17)–(19), and they are in good agreement with the CFD results. In addition, another advantage of our fitting method is that coefficient $γ$ can be calculated by pressure ratio and teeth number alone.

Figure 5.

PR versus coefficient γ (point-CFD; curve-fitting).

Boundary conditions

The boundary conditions include inlet pressure, outlet pressure, and pressure ratio. Equation (10) shows that if two of these parameters are known, the third can be obtained. It means that there are only three cases in operating conditions. It should be noted that the case in which P0 is a constant and PR is a variable is less used in practical situations, thus it is not discussed here.

The case in which P4 is a constant and PR is a variable is shown in Figure 4. Therefore, there is only one case that should be considered, that is, the pressure ratio PR is constant (PR = 0.2, 0.3, 0.4). The relationship between outlet pressure P4 and coefficient $γ$ is shown in Figure 6, which shows that outlet pressure P4 has almost no effect on coefficient $γ$ .

Figure 6.

Outlet pressure versus $γ$ .

Tooth clearance

Tooth clearance is one important parameter of the labyrinth seals.^23,24 Some researches show that clearance is the main reason for leakages. Therefore, the effect of clearance on the coefficient $γ$ should be studied. As shown in Figure 7, The CFD results indicate that the clearance has no obvious effect on coefficient $γ$ .

Figure 7.

PR versus $γ$ in various clearance.

Structural parameters

Figure 8 shows the influence of various structural parameters on coefficient $γ$ . The structural parameters shown in Table 1 are the original parameters. With the change of structural parameters, such as tooth height, tooth thickness, and groove width, coefficient γ has no obvious change and the maximum error is 1.5%. Therefore, for different structural parameters of straight-line labyrinth seal, the linear formulas are still applicable to calculating the maximum pressure load of labyrinth seal teeth.

Figure 8.

Various structure parameters versus coefficient $γ$ .

Fitting method calculation of maximum pressure load

From the above analysis, the maximum pressure load on the teeth is only dependent on inlet pressure, pressure ratio, and teeth number. The detailed flow diagram of the proposed method is illustrated in Figure 9, and the results of calculation are shown in Figure 10. In addition, the maximum pressure loads on labyrinth seal’s teeth can be obtained by Figure 10.

Figure 9.

The flow diagram of the method.

Figure 10.

Fitting method calculation of maximum pressure loads.

Conclusion

The pressure loads on the labyrinth seal teeth are not uniform and the last tooth carries the maximum pressure load in low pressure ratio conditions (PR≤ 0.5). Based on the CFD and tests, this study proposes a mathematical analysis formula, which can calculate the maximum pressure load on each tooth of labyrinth seal. The results show that the maximum pressure load on the teeth is dependent on boundary condition, teeth number, and coefficient $γ$ , which describes the uniformity of pressure loads.

Furthermore, the effect of teeth number, boundary conditions, clearances, and structure parameters on the coefficient $γ$ is analyzed. The results show that coefficient $γ$ is only determined by pressure ratio and teeth number. Coefficient $γ$ is proportional to teeth number and inversely proportional to pressure ratio, but it does not change when teeth number is larger than 9. Moreover, a fitting method is given to estimate the maximum pressure load on labyrinth seal teeth and it is in good agreement with the CFD results. For different teeth number, boundary conditions, clearances, and structure parameters, this fitting method can be used to calculate the maximum pressure load on the teeth of labyrinth seal.

Footnotes

Handling Editor: Kai Bao

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful to National Basic Research Program of China (973) (Grant No. 2012CB026003) and National Science and Technology Major Project (No. 2011ZX02403-4-3).

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Theoretical and experimental investigation on maximum pressure loads of labyrinth seal’s teeth

Abstract

Keywords

Introduction

Theoretical researches and experiment system

CFD model

Maximum pressure load on the tooth

Experiment testing system

Results and discussion

Teeth number

Boundary conditions

Tooth clearance

Structural parameters

Fitting method calculation of maximum pressure load

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References