Neural network and CFD-based optimisation of square cavity and curved cavity static labyrinth seals

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Tribology International 40 (2007) 1204–1216 www.elsevier.com/locate/triboint

Neural network and CFD-based optimisation of square cavity and curved cavity static labyrinth seals S.P. Asoka,, K. Sankaranarayanasamyb, T. Sundararajanc, K. Rajeshd, G. Sankar Ganeshane a

Department of Mechanical Engineering, Mepco Schlenk Engineering College (MSEC), Sivakasi 626005, India b Mechanical Engineering, National Institute of Technology, Trichy, India c Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India d Indira Gandhi Centre for Atomic Research, Kalpakkam, India e Department of Mechanical Engineering, MSEC, Sivakasi, India Received 25 July 2005; received in revised form 5 May 2006; accepted 6 January 2007 Available online 20 February 2007

Abstract The pressure drop characteristics for leakage of water through circular grooved, square cavity and curved cavity static labyrinth seals are investigated. A semi-theoretical model employing two new terms named virtual cavity velocity and vortex loss coefficient, to determine the pressure drop across the seal is presented. Five different square cavity labyrinth seals (SCLS) were subjected to flow visualisation tests to observe the leakage flow patterns. Computational fluid dynamic (CFD) analysis was done using Fluent commercial code. The values of the vortex loss coefficient for the SCLS at turbulent flow conditions were obtained experimentally. Using the data pool, an artificial neural network (ANN) simulation model was employed to identify the optimal SCLS configuration. Based on the insights gained, two different curved cavity labyrinth seal (CCLS) geometries were developed and optimised using parametric CFD analysis. They were visualisation tested and experimentally found to have higher pressure drops and vortex loss coefficients as compared to the SCLS configurations. The studies show that the enhanced performance is due to the presence of multiple recirculation zones within their cavities, which dissipate higher amount of leakage flow momentum. r 2007 Published by Elsevier Ltd. Keywords: Labyrinth seal; Vortex loss; Visualisation; CFD; ANN

1. Introduction The functioning of several fluid flow systems is affected by the leakage at particular locations in them. For instance, the piston in an automobile engine has to operate in a leakage-free environment. In multi-stage turbo machinery, leakage flow between stages needs to be reduced for achieving better efficiency. In the turbo pump of a cryogenic rocket engine, the leakage across the oxidiser and fuel streams has to be avoided to prevent accidents. In these applications, by increasing the flow resistance, the leakage can be reduced for a given pressure difference or in other words, the associated pressure drop can be increased for the rated leakage. This can be simply done by reducing Corresponding author. Tel.: +1 91 4574 260352; fax: +1 91 4562 235111. E-mail address: [email protected] (S.P. Asok).

0301-679X/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.triboint.2007.01.003

the clearance in the leakage flow passage. But such reduction in clearance would lead to practical difficulties like removal of accumulated foreign materials, occurrence of thermal and mechanical instabilities and inconvenience in the assembling and disassembling of components. Hence, it is desirable to maintain clearances to lie above a certain minimum value. Compared to the leakage through annular seals, labyrinth seals allow smaller leakage without requiring any drastic decrease in the radial clearance. They can offer interesting possibilities for achieving a higher-pressure drop at rated leakage in the above-mentioned systems, by suitable optimisation of the seal configuration. Apart from dynamic sealing applications, labyrinth seals are also employed in static applications involving the oil/fuel supply systems in heat engines and pressure vessels. In the present paper, the pressure drop characteristics of static labyrinth seals having square cavities or curved

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Nomenclature a b c d D f g he

length of the straight annular portion, m cavity width, m radial clearance, m cavity depth, m maximum OD of the labyrinth, m friction factor acceleration due to gravity, m/s2 sudden expansion loss for flow through pipes, m sudden contraction loss for flow through pipes, m turbulent kinetic energy, m2/s2 coefficient of contraction for flow through pipes

hc k K

cavities machined in a circular fashion on the non-rotating shaft part alone are investigated. Such static seals are also known as straight through, non-rotating labyrinth seals and encountered in fast breeder nuclear reactors. It is possible to have the cavities machined on the shaft in a helical fashion, leading to helical grooved labyrinth seals or screw labyrinth seals. This paper discusses the formulation of a simple theoretical model, CFD predictions and experimental validations for several straight through labyrinth seals. Further, it presents the optimisation of square cavity labyrinth seals (SCLS) using ANN and the development and experimental testing of two better performing curved cavity labyrinth seals (CCLS). 2. Development of theoretical model The general configuration of a labyrinth seal is shown in Fig. 1. The literature available on the type of static, liquid labyrinth seals dealt in this paper are too less compared to the quantum of literature available on dynamic, gas labyrinth seals having their cavities machined on both the shaft and sleeve portions. Mixed views are reported in the Straight Annular Portion Outer Sleeve V1

c

Kv l p Pr Dp Q, q Re V1 V2e V2 Z m o r

vortex loss coefficient total length of the labyrinth seal, m pitch of the labyrinth seal, m pressure drop ratio pressure drop, N/m2 leakage in m3/s and m3/h, respectively Reynolds number average velocity in the annulus, m/s average velocity after sudden expansion in a pipe, m/s virtual cavity velocity, m/s number of cavities in the labyrinth seal dynamic viscosity, Ns/m2 specific weight, N/m3 mass density, kg/m3

literature about the effect of shaft rotation on leakage. Stocker [1] reports minimal effect on leakage due to rotation speeds up to 235 m/s. El-Gamal et al. [2] feel that shaft rotation has little effect on the leakage from grooved shaft and grooved casing labyrinth seals while it makes a considerable improvement on the performance of the upthe-step seal. The model of Nikitin et al. [3] considered leakage of petroleum-based liquids through labyrinth seals having rectangular/triangular cavities. The model of Idelchik et al. [4] dealt with liquid flow through trapezoidal cavities. The cavities discussed in this paper are different from the above. The present theoretical model is based on the work of Asok et al. [5] where the vortex loss in a labyrinth seal was suggested to be the main contributor to the overall pressure drop. Since the nature of the labyrinth seal problem does not lend itself to complete analytical conceptualisation, all the above theoretical models need to deal with unknown values of resistance coefficients. Consider one pitch length portion of a labyrinth seal comprising of the straight annular passage and cavity chamber portions. Basically, the Bernoulli’s equation accounts for the losses taking place in the seal. The pressure drop occurring in the straight annular passage of the labyrinth seal can be calculated with the help of Eq. (1).

a

Dpannular ¼ V2

d

oflV 21 . 2gð2cÞ

(1)

The value of the Darcy-Weisbach friction factor f is calculated by using the Blasius formula D/2

Cavity Chamber

1205

f ¼ 0:316Re0:25 ,

(2)

b

Axis of the Labyrinth Fig. 1. Labyrinth seal configuration.

where Re ¼ rV 1 ð2cÞ=m. The labyrinth cavity pressure loss is affected by several factors like the viscosity of the fluid, cavity geometry, level of turbulence, flow direction and strength of the cavity vortex, boundary and shear layers developed, presence of vapour bubbles if any, etc. Finding a precise analytical

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expression is impossible because of the complex nature of cavity flow. The complexity is further increased due to transitional effects (Reo10 000) and turbulence exhibited during separation at leading edge, free shear layer and reattachment at trailing edge. The flow condition after the inlet to the seal would not be fully developed. Chochua et al. [6] showed that flow can become fully developed after about 30% of seal length. They demonstrated that when the flow becomes fully developed the profiles of mass flux and non-dimensionalised turbulent kinetic energy and eddy dissipation rate become unchanged. Villasmil et al. [7] have shown that depending on Reynolds number and cavity size, the flow can achieve the fully developed condition after passing over a few cavities. Now, a simplified theoretical approach is given as described below: Consider the loss of head due to a sudden enlargement for flow through a pipe found as

he ¼

ðV 1  V 2e Þ2 , 2g

(3)

where V2e is the average velocity downstream of the sudden expansion. In the case of a static labyrinth seal, subsequently discussed visualisation tests have shown that not all the leakage flow issuing from the straight annular passage enters into the cavity and contributes to sudden expansion loss. Since the mean flow entering a cavity is zero, the flow field can be considered to comprise of a free shear layer, which expands and contracts in the cavity region and a recirculatory flow confined within the cavity. It can be noted from Fig. 2 that the angle of penetration of the dividing streamline corresponding to the free shear layer is small. Hence the head loss occurring due to enlargement at the cavity is negligible. If it is visualised that the whole leakage flow expands and enters into the cavity, then the imaginary or virtual cavity velocity V2 can be calculated using the following equation: V2 ¼

Q . p½D  ðD  2dÞ2 =4

(4)

2

Free Shear Layer

Main Flow Stream

Infusing Stream Line for the Vortex Flow

Stream Lines Accelerated into the Main Flow

Fig. 2. Flow pattern in the SCLS.

hc ¼

KV 21 . 2g

(5)

For flow through pipes, the well-established value of the coefficient of contraction K for valves, elbows, etc. is 0.5. General values of K for labyrinth seals are unavailable, since the fraction of leakage that contracts from the cavity before entering the next groove is unknown. A weak vena contracta is formed at the inlet region of the ensuing straight annular portion. Considering the overall seal geometry, there will also be some inlet and exit losses, which are usually small. Therefore, the losses associated with the vortex developed inside each cavity would be the major source of head loss. In this regard, Chochua [8] has identified a similar phenomenon of head loss, while Arghir et al. [9] have discussed about an ‘enlightened inertial lift’ effect. Villasmil et al. [10] have concluded that wall shear stress at the straight annular portion plays a secondary role and a stable mean flow recirculation zone within each cavity could lead to larger friction factors and higher pressure drops. Based on the above reasoning, a general expression can be formulated to determine the cavity vortex pressure loss in analogy to the Eqs. (3) and (4) and the pressure difference formula for a forced vortex, assuming that small quantities of pressure losses caused by the sudden contraction/expansion are imbibed within it and the effect of gravity is negligible. Hence it is proposed that Dpcavity ¼ K v rðV 21  V 22 Þ,

(6)

where Kv is the vortex loss coefficient. The virtual cavity velocity term V2 has been intended to take care of the cavity/vortex size and speed of the vortex developed inside the cavity. Now, the total pressure drop over length l for leakage flow through a static, liquid labyrinth seal is obtained by combining the Eqs. (1) and (6), leading to the final form Dp ¼ Here;

rfZaV 21 þ K v ZrðV 21  V 22 Þ. 4c Z¼

l l l ¼ ) a ¼  b. p aþb Z

Now, the Eq. (7) is recast as   rfZððl=ZÞ  bÞV 21 Dp ¼ þ K v ZrðV 21  V 22 Þ 4c

(7) (8)

(9)

Stagnation Zone

or Dp ¼ C 1  C 2 Z þ C 3 K v Z, Clockwise Rotating Vortex

Stagnation Zone

Consider the head loss due to sudden contraction for flow through a pipe expressed in the form

Stagnation Zone

rflV 21 =4c,

rfbV 21 =4c

(10)

where C 1 ¼ C2 ¼ and C 3 ¼ rðV 21  2 V 2Þ The vortex loss coefficient Kv in conjunction with the virtual cavity velocity V2 forms the set of core parameters for describing the present theoretical model. Since the values of Kv for a particular labyrinth seal have to be

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determined using extensive experimental/computational pressure drops, the above model essentially constitutes a semi-theoretical approach. Also, it can be observed that instead of the well-known two-coefficient correlation, a single-coefficient correlation for predicting the leakage has been formulated using Kv. The values of the clearance c and length of the straight annular portion a for all the seals discussed in this paper are 0.5 and 1 mm, respectively which results in a (a/c) ratio of 2. Since the value of this ratio is larger than 1.6, the jet trailing vena contracta would reattach to the walls of the straight annular portion. Now, the associated losses are fairly constant and no additional coefficient might be needed. 3. Pressure drop characteristics of SCLS A SCLS has its cavity width b equal to the cavity depth d. An optimal geometry yielding the maximum pressure drop would have to strike a balance between making the cavity width as large as possible to reduce carry over of kinetic energy and making the pitch as small as possible to get the maximum number of throttlings over a given length. Morrison et al. [11] noted that the number of throttlings is the most important factor in designing a labyrinth seal. Since the size and shape of the labyrinth seal cavity affect the strengths of the vortex as well as those of the turbulent eddies, effective labyrinth seal geometries should exhibit increased internal cavity turbulence. As an aid to optimisation of seal configuration, visualisation of flow pattern inside the cavities of five different SCLS geometries specified under Table 1 was carried out. The water tunnel flow visualisation test rig used glittering aluminium powders of 20–60 mm size as tracer while the air tunnel flow visualisation test rig employed the vapour arising out of solid carbon dioxide viz. dry ice added with a few drops of water, as tracer. Fig. 2 presents an artistic view of the observed flow pattern in the SCLS. Under turbulent flow conditions, it was noted that the momentum of the main flow emerging from the straight annular passage and the recirculatory pattern developed inside the cavity did not permit any appreciable inflow/outflow of fluid into/from the cavity. All square cavities had one distinct, clockwise vortex whose average rotational speed was qualitatively observed to Table 1 Specifications of five SCLS Sl. no.

Name of the seal

a

b¼d

p ¼ a+b

2 2 3 4 5

4 3 4 5 6

(mm) 1 2 3 4 5

LS1 LS2 LS3 LS4 LS5

2 1 1 1 1

1207

reduce with increase in cavity size. However, it was noted that the recirculation flow rate increases with cavity size. Using an order of magnitude analysis it can be shown that the rotational speed of vortex varies as (2V1/b) and the maximum cavity stream function value i.e. recirculation flow rate varies as (V1b/2). The stagnation zones were symbolised by poor tracer particle densities. Subsequent CFD analysis predicted the same flow pattern observed in the above visualisation studies. Increasing the size of cavities can yield larger sized vortices, which can promote higher pressure drop per cavity and also higher overall pressure drop of the seal, until a maximum value is reached. But, when the cavity size is increased beyond the optimal value, the combined effects of reduced rotational speed of vortex in the cavity and the reduction in the number of grooves per unit length will bring down the total pressure drop across the seal. 3.1. CFD analysis of SCLS Rhode et al. [12] and Schramm et al. [13] have applied CFD to analyse leakage flow in labyrinth seals. A CFD model validated by selective experimentation can be used as an optimisation tool also. This paper employs the commercial, finite volume CFD code Fluent to solve the incompressible, Navier–Stokes equations. A structured finite volume grid employing quadrilateral mesh is used to discretise the axisymmetric flow domains of all the seals studied. The solutions were obtained using the secondorder upwind, SIMPLEC algorithm [14]. The five SCLS geometries analysed are listed in Table 1. The cavity pitches lie in the range of 3–6 mm corresponding to 40–20 cavities, respectively over the whole length of the seal. Normally, about 25–15 cavities were considered for the simulation. Literature suggests that most pressure drop curves are linear, indicating a constant pressure gradient in the stream-wise direction. A uniform velocity of 5.1 m/s at inlet corresponding to the rated water leakage flow rate of 2.5 m3/h and a static pressure of 1 bar at the exit of the labyrinth seal were prescribed as the boundary conditions. The standard k–e model was used to simulate the turbulence characteristics of flow. The near-wall features were defined using logarithmic wall functions. Wherever required, the solution-adaptive grid feature of Fluent was invoked which refines the original grid. Most of the values of y+ for the grid points near the wall were kept between 30 and 100 in order to prevent the wall-adjacent cells from being placed in the buffer layer. Schramm et al. [13] have shown that such a numerical scheme is capable of simulating labyrinth flow even when low values of y+ existed in the stagnation zones. Villasmil et al. [7] have expressed surprise at the comparable values of mass flow rate predictions obtained using a similar k–e model and a more elaborate Reynolds-stress model with a two-layer zonal approach. Incidentally, labyrinth seal leakage is mostly pressure driven and the losses are not too sensitive to wall friction as compared to loss in the vena contracta,

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1.8 1.6 1.4

Pr

1.2 1 LS1

0.8

LS2 LS3

0.6

LS4 LS5

0.4 20000 30000 40000 50000 60000 70000 80000 90000 100000 Number of nodes Fig. 4. Grid independence test for the SCLS.

Fig. 3. Adaptive grid for the SCLS.

isobaric kinetic energy loss in cavities and kinetic energy carry-over. One of the meshes generated for a SCLS is shown in Fig. 3. To establish the residual level for convergence, five different, trial residual values including the software’s default value of 103 were used during the analysis of the profiles. For all the seals, it was found that the parameters like mass flow rate, pressures, velocity components and turbulence quantities were generally found to converge at a residual value lying between 103 and 104. Following [7], reaching a residual value of 105 was fixed as the criterion for the convergence of solution. Grid independence test should be conducted until the predictions are essentially independent of the grid employed [15]. While carrying out this test, successive grid adaptations were done. Convergence of mass flow rate and stabilisation of different flow properties and turbulence quantities were verified. The dimensionless pressure drop ratio Pr is obtained by dividing the pressure drop with a reference pressure. The variations in the prediction of Pr with number of mesh points at the rated leakage flow are shown in Fig. 4 for the SCLS. Similar grid independence tests were carried out for the remaining seals of the present study also. For the highest number of nodes considered, the convergence of solution required around three hours of CPU time on a Pentium-IV, 2.8 GHz, 1 GB RAM computer. With reference to Table 2, in comparison to the subsequently obtained experimental results, the CFD analysis has slightly over predicted the Pr values for all the SCLS considered and under predicted for the two curved cavity labyrinth seal geometries LS6 and LS7 tested later. 3.2. Experimental investigations The labyrinth seals chosen for testing in this work along with the experimental labyrinth seal test section shown in

Table 2 Comparison of Pr for SCLS and CCLS Sl. no.

1 2 3 4 5 6 7

Name of the labyrinth seal

LS1 LS2 LS3 LS4 LS5 LS6 LS7

Pressure ratio Pr CFD

Experimental

1.24 1.37 1.54 1.27 1.17 2.28 2.52

1.21 1.35 1.51 1.24 1.14 2.36 2.69

% deviation

2.48 1.48 1.98 2.42 2.63 3.38 6.32

sectional elevation on Fig. 5, were fabricated using stainless steel (SS304) on computer numerical controlled machines. The experimental facility consists of a sump, water pump delivering water to pass through the labyrinth seal along the vertical upward direction, 10 mm filter, water heater capable of maintaining the temperature of water at 70 1C, by pass and flow control valves and a collection tank with micrometre level measurement facility. The test section includes the outer sleeve, a 40 mm long straight annular seal passage, 121 mm long labyrinth seal, piezometric rings, Rosemount differential pressure transducers connected across the pressure tapings provided on the piezometric rings along with scanner facility for high accuracy in measuring the pressure differences and a RTD thermometer. The symbols DP1, DP2 and DP3 in the Fig. 5 indicate differential pressure transducers for measuring the pressure drop across the annular seal passage and the first and second halves of the labyrinth seal, respectively. For all the seals tested the DP2 and DP3 values were found to be virtually the same for Re values up to around 30,000 which are well above the rated Re of 12,300. The maximum increase in the DP2 value over that of DP3 was observed to be 7% at the highest values of Re.

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Direction of Water Flow

Ø87

Piezometric Ring

Details at 'A'

61

DP3

Square Cavity Labyrinth Seal

DP2

40

318

60

'A'

DP1

Positioning Ring

Ø86 g

Sleeve

Fig. 5. Experimental labyrinth seal test section.

q

q 0

1

2

3

4

5

6

7

8

9

0

16

2

4

6

8

10

0.25 LS1 LS2 LS3

12

LS4 LS5

8 LS1

Kv

Pr

0.2

LS2

4

LS3 LS4

0.15

LS5

0 0

10000

20000

30000

40000

50000

Re Fig. 6. Pressure drop ratio vs. Re and q.

0.1 0

10000

20000

30000

40000

50000

Re Fig. 7. Vortex loss coefficient vs. Re and q for SCLS.

Fig. 6 shows the variation of Pr against Re as well as the leakage flow q for all the SCLS configurations. Among LS1 and LS2 having the same cavity size of b ¼ 2 mm, Pr is higher for LS2 having Z of 40, which is 10 more than that for LS1. Though LS1 and LS3 have the same pitch of 4 mm and Z of 30, LS3 has exhibited higher Pr values over LS1. In fact, the LS3 square cavity configuration with 3 mm cavity depth and 4 mm pitch gives the highest pressure drop at any flow rate among the experimentally tested SCLS configurations. Thus, the interwoven effects of cavity size and number of cavities Z on Pr are confirmed. Also, it can be inferred that there is little possibility for a SCLS geometry having b values above 5 mm and below 2 mm to have higher pressure drops than LS3.

Fig. 7 presents the variation in the experimental values of Kv, evaluated using Eq. (10) for all the five SCLS tested. Nikitin et al. [3] reported that the local resistance coefficient decreased steeply for increasing Re in the turbulent flow regime above the critical Re value of around 2000. The vortex loss coefficient Kv of the present work in contrast, is found to gently and nearly linearly drop against Re for all the labyrinth seals tested. Its values are predominantly affected by the geometry and flow pattern within the cavity. The increased cavity size of LS3 over that of LS1 having b ¼ 2 mm, has also led to higher values of Kv as seen in Fig. 7. Generally, the values of Kv have increased as the

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cavity size increases. A detailed examination of the cavity stream function values showed that the circulation strength increases with cavity size since as discussed earlier, the cavity stream function values are proportional to (V1b/2). Yet, increased cavity sizes have not guaranteed enhanced values of pressure drop across the seal. Though LS3 and LS4 have nearly the same values of Kv, the performance of LS3 is better. In fact, the values of Pr are the least for LS5 having the biggest cavity size of b ¼ 5 mm and highest values of Kv. Higher pressure drop in a single cavity does not imply larger overall pressure drop across the seal since the number of cavities per unit length also plays an important role. Hence, it is again seen that the pressure drop occurring over a given length of labyrinth seal is a complex function of the geometrical and flow variables. Repeatability of experimental results was verified by conducting the experiments at least twice. The tolerances maintained for D and other dimensions were +0 mm/ 0.02 mm and 70.01 to 70.02 mm, respectively. Because of such close tolerances, any uncertainty in pressure drop ratio arising due to variation of dimensions was neglected. However, the flow rate was observed to have a maximum variation of 1%. The maximum uncertainty in the pressure drop ratio due to variations in these parameters did not exceed 2%. 4. Optimisation of SCLS configuration using artificial neural network (ANN) modelling In the previous section, it was shown that the pressure drop depends on the seal dimensions in a complex manner. Hence the LS3 geometry need not be the ultimate optimal square cavity configuration, although among the experimentally tested seals it gives the maximum pressure drop. It will be of interest to evaluate in a systematic way, the set of dimensions which correspond to the optimal seal geometry. Table 3 shows the configurations of 55 candidate SCLS considered for the optimisation studies, inclusive of the five already tested SCLS. The Pr values of the remaining 50 intermediate sized seals have to be known for the purpose of comparison. Fabrication and experimental testing of these seals would consume a lot of time and huge expenses. Though CFD analysis can be resorted to, large amount of CPU time would be needed. Because of the coupled nature of the governing equations, high number of geometrical variables, complexly interrelated turbulence parameters and nonlinear nature of pressure drops, any conventional optimisation method will be cumbersome to implement. Of late, ANN applications are rapidly growing in a wide variety of fields. Rajkumar et al. [16] have applied the ANN technique for optimisation of rocket engine components. Marko et al. [17] have used ANN for modelling a fluid dynamics problem. ANN employs artificial neurons similar to the biological ones, to perform complex operations using already gathered knowledge. It can perform nonlinear mapping using randomly assigned weights. With the ANN approach, it is easier to

incorporate all the optimisation inputs. In view of the above considerations, this paper employs the ANN model as an optimisation tool to quickly identify the better SCLS configurations. No such previous work is cited in the literature dealing with the optimisation of labyrinth seals. The ANN tool with a feed forward back propagation (FFBP) learning algorithm available in the Matlab, version 6.5 software is used as the network model. Readers can refer to [16] for a detailed glossary of ANN terms found in this paper. Different trial networks were trained with five input parameters viz. a, b, d, Z and q with the target being either or both the vortex loss coefficient and pressure drop ratio. Different training procedures using 288 and 336 experimental data derived from the four geometries of LS1, LS2, LS3 and LS4, respectively for a single target and twin target ANN models were employed. The initial weights were chosen at random and training was continued with different networks in a trial and error fashion. The weights were modified until an ANN model would predict the targets to an accuracy level of within 0.5% of the experimental results for LS5. Networks not meeting the desired accuracies were discarded. It was found that only when the FFBP ANN model was employed to predict the vortex loss coefficients, the predictions could reach the targeted accuracy level. This situation occurred for a three layer FFBP ANN model shown in Fig. 8 employing the Trainlm training option. Besides an input layer, there is a hidden layer having 20 neurons with a Tansig transfer function and an output layer with a Purelin transfer function. The number of epochs specified was 20,000 for a set performance goal of 10e-15. The CPU time required for the training was around 15 min in the same computer that performed the CFD analysis. The Kv values at the rated leakage obtained from the simulation results of this ANN model for different candidate seals are listed in Table 3. Using these values of Kv in Eq. (10), the respective Pr values have been calculated and listed in Table 3. It is seen from the Table 3 that ANN has quickly identified the candidate seals LS116, LS120, LS324 and LS328 as four potential optimal seal candidates. Their performances can be validated either by CFD analysis or experimental testing. CFD analysis was performed on all of these four seals. The percentage over prediction of Pr by ANN over CFD for these seals were found to be 1.3%, 2.5%, 4.2% and 5.7%, respectively. Since the minimum percentage deviation of 1.3% occurred for the candidate seal LS116, having a ¼ 1 mm and b ¼ 2.75 mm, the same is identified as the optimal square cavity labyrinth seal. Fig. 9 shows the variation of axial velocity along a radial line drawn through the middle of a cavity of the optimal seal LS116 at the rated leakage flow condition. Fig. 10 shows the contours of turbulent kinetic energy within the cavity geometry of LS116. These figures show that the maximum values of axial velocity and turbulent kinetic energy occur only in the free shear layer region at the top portion of the cavity, implying that the flow penetration

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Table 3 Configurations of the candidate SCLS, the ANN predicted Kv and calculated Pr at rated leakage Seal name

a (mm)

b, d (mm)

Z

Kv predicted by ANN

Pr from Eq. (10)

LS2 LS22 LS23 LS24 LS1 LS16 LS17 LS18 LS19 LS110 LS111 LS112 LS113 LS114 LS115 LS116 LS117 LS118 LS119 LS120 LS3 LS322 LS323 LS324 LS325 LS326 LS327 LS328 LS329 LS330 LS331 LS332 LS333 LS334 LS335 LS336 LS337 LS338 LS339 LS340 LS4 LS442 LS443 LS444 LS445 LS446 LS447 LS448 LS449 LS450 LS451 LS452 LS453 LS454 LS5

1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 1 1.25 1.5 1 1.25 1

2 2 2 2 2 2.25 2.25 2.25 2.25 2.25 2.5 2.5 2.5 2.5 2.5 2.75 2.75 2.75 2.75 2.75 3 3 3 3 3 3.25 3.25 3.25 3.25 3.25 3.5 3.5 3.5 3.5 3.5 3.75 3.75 3.75 3.75 3.75 4 4 4 4 4 4.25 4.25 4.25 4.25 4.5 4.5 4.5 4.75 4.75 5

40 37 34 32 30 37 34 32 30 28 34 32 30 28 27 32 30 28 27 25 30 28 27 25 24 28 27 25 24 23 27 25 24 23 22 25 24 23 22 21 24 23 22 21 20 23 22 21 20 22 21 20 21 20 20

0.12544 0.12467 0.12647 0.12826 0.13602 0.12790 0.13166 0.13576 0.14390 0.15049 0.13896 0.14063 0.14880 0.15864 0.16451 0.15149 0.17106 0.18419 0.18808 0.19169 0.18760 0.18922 0.18960 0.19253 0.19076 0.19017 0.18894 0.19204 0.19103 0.18829 0.18969 0.19045 0.19117 0.18872 0.18497 0.19026 0.19059 0.18914 0.18555 0.18085 0.18997 0.18941 0.18616 0.18154 0.17618 0.18974 0.18683 0.18232 0.17693 0.18848 0.18347 0.17816 0.18801 0.18249 0.21016

1.35137 1.38234 1.43784 1.49326 1.20427 1.37487 1.44910 1.52657 1.23198 1.30792 1.48051 1.53477 1.23834 1.33757 1.40835 1.60018 1.36906 1.49186 1.54846 1.60306 1.51078 1.55157 1.58313 1.63367 1.31939 1.52990 1.54948 1.60128 1.29801 1.30469 1.52633 1.53198 1.27585 1.28426 1.28492 1.53057 1.24941 1.26377 1.26539 1.26040 1.23763 1.25724 1.26063 1.25576 1.03869 1.23624 1.24168 1.23747 1.02330 1.22865 1.22142 1.01032 1.22581 1.01293 1.13926

into the cavity is not appreciable. Also, the maximum turbulent kinetic energy value exists near the stagnation region where the flow decelerates. These features are in conformity with the flow pattern depicted in Fig. 2.

The largest deviations of 5.7% between the ANN and CFD predictions for the LS328 configuration can be attributed to the fact that both of its dimensions a and b are experimentally untested. This indicates that availability

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1

a

1

2

b

2 3

d

1

Kv

4

Z

3 5

q

20 Input Layer

Hidden Layer

Output Layer

Fig. 8. Schematic of the ANN model used. Fig. 10. Contours of turbulent kinetic energy for the optimal SCLS–LS116.

5. Development and testing of two CCLS

Radial distance in m

0.0435

0.0425

0.0415

0.0405

0.0395 -2

0

2 Axial Velocity in m/s

4

6

Fig. 9. Variation of axial velocity with radial distance at the mid-plane of the cavity for the optimal SCLS viz. LS116.

of additional training data from at least one experimentally tested seal having intermediate values of a and b would have made the ANN predictions even more precise for the untested seals. The parameter Kv is found to be ideally suited for performing optimisation studies in association with an ANN model. Further investigations into the vortex loss coefficient Kv can provide a vital key to develop better labyrinth seal geometries. In the absence of the ANN approach, extensive experiments/computational studies would have been required to generate the values of Kv. It is possible that different candidate seals like LS116, LS120, LS324 and LS328 could vie for the optimal configuration status. Such an outcome of identifying more than one optimal seal configuration is conveniently possible only with the ANN optimisation scheme.

Suitable geometrical modifications were tried on the basic square cavity geometry to get better profiles. With reference to the Fig. 2, any attempt to weaken the stagnation pocket existing near the exit of the cavity of a SCLS would result in more infusion of fluid into the cavity thereby aiding better viscous dissipation of momentum of the leakage flow. In other words, the boundary layer formed mainly on the right side vertical wall of the cavity in the SCLS has to be narrowed. This is expected to help the development of a fast rotating vortex inside the cavity. Accordingly, the right side vertical wall i.e. the knife face of the cavity is made convex in the axial flow direction. This CCLS geometry was subjected to parametric CFD analysis towards maximising the pressure drop, culminating in labyrinth seal-6 (LS6) geometry shown in Fig. 11. The flow visualisation plot for LS6 obtained by CFD analysis is seen in Fig. 12. Fig. 13 shows the axial velocity of leakage flow along a radial line shown in Fig. 12 as plane A and drawn at 1.125 mm from the left knife face of the cavity. Fig. 14 shows the contours of turbulent kinetic energy within the cavity. It is evident from Figs. 12–14 that a double vortex structure exists. A clockwise vortex fills the top and right potions and a thin anti-clockwise vortex prevails in the bottom and left portions of the cavity. The numbers placed on the contour lines of turbulent kinetic energy in Fig. 14 indicate the values in m2/s2. Due to this twin vortex pattern, maximum turbulent kinetic energy values are observed to occur at the top right and bottom left portions. Comparing Figs. 9 and 10 with Figs. 13 and 14, the dissipation of main flow momentum through turbulent kinetic energy production, is higher for LS6, which plays a significant role in increasing the pressure drop across the seal.

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1.5

1.5

1213

0.75

1.5

2.25

Fig. 11. The LS6 curved cavity labyrinth seal (all dimensions in mm).

Fig. 14. Contours of turbulent kinetic energy for LS6.

0.75

3

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0.75

1

2±0.1

2.5±0.1

Fig. 15. The LS7 curved cavity labyrinth seal (all dimensions in mm).

Fig. 12. CFD flow visualisation for LS6.

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0.0435

0.0425

0.0415

0.0405

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0

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6

Fig. 13. Axial velocity variation with radial distance on plane A of the cavity of LS6.

Another CCLS shape is proposed by modifying LS6 in such a way that two large vortices would be created inside the cavity. This change in cavity flow structure is contemplated to further increase the pressure drop. The dimensions of this profile were also optimised through a parametric CFD analysis yielding labyrinth seal-7 (LS7) shown in Fig. 15. The parametric CFD investigations conducted on the CCLS geometries have illustrated that the curvature of the cavity is an important parameter. The adaptive grid employed for the cavity of LS7 is shown on Fig. 16. Nodes in excess of 1.2 million were required to obtain grid independent CFD results for LS6 and LS7. The number of nodes and CPU time required at the grid independent condition for the CCLS were nearly 10 times of those required for the SCLS. The CFD flow visualisation plot for LS7 is shown in Fig. 17. As expected, the LS7 geometry entertains two counter-rotating vortices inside every cavity. The arrows indicate the directions of the leakage flow and vortices inside the cavities. Figs. 18 and 19 show the axial velocity of leakage flow along two radial planes, Planes B and C shown on Fig. 17 and drawn at the distances of 1.375 and

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Radial Distance in m

0.0435

0.0425

0.0415

0.0405

0.0395 -2

0

2 Axial Velocity in m/s

4

6

Fig. 18. Axial velocity variation with radial distance on plane B of the cavity of LS7. Fig. 16. Adaptive grid for LS7.

Radial distance in m

0.0435 0.0427 0.0419 0.0411 0.0403 0.0395 -2

0

2 Velocity in m/s

4

6

Fig. 19. Axial velocity variation with radial distance on plane C of the cavity of LS7.

Fig. 17. CFD Flow visualisation view for LS7.

3.25 mm, respectively from the left knife face. Fig. 20 shows the contours of turbulent kinetic energy in the cavity of LS7. Existence of two strong counter-rotating vortices is evident from the velocity vector plots and the velocity profiles shown in Figs. 17–19. The turbulent kinetic energy contours for the cavity of LS7 shown in Fig. 20, indicate that higher turbulent kinetic energy values occur throughout the cavity of LS7; on the other hand, only weaker pockets of turbulent kinetic energy were observed with the optimal SCLS and LS6 geometries. Thus, the highest level of internal cavity turbulence brought on by the large sized, faster and counter rotating vortices of LS7 dissipate more

Fig. 20. Contours of turbulent kinetic energy for LS7.

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momentum of leakage flow. The corresponding energy is transferred to eddies by the recirculating flow for the generation of turbulent kinetic energy which is eventually dissipated to thermal form at the scale of very small eddies. Visualisation tests on planar geometries of LS6 and LS7 were conducted which impressively confirmed the same flow patterns portrayed by CFD analysis. These two CCLS were fabricated and experimentally tested. The variations of Pr and Kv for LS6 and LS7 are plotted against Re and q in Figs. 21and 22, respectively. Figs. 21and 22 illustrate that the CCLS geometries of LS6 and LS7 have higher values of pressure drop ratio Pr and vortex loss coefficient Kv than the SCLS. The highest values of Pr and Kv have been observed in the case of LS7. Demko et al. [18] have noticed the highly desirable secondary recirculation zone in a dynamic labyrinth seal q 0

2

4

6

8

10

1215

only beyond an extremely high shaft speed. For the LS7 geometry, CFD analysis and visualisation tests have confirmed the existence of two counter rotating vortices even at fairly low values of Re, mainly due to the geometry of the seal. With reference to Table 2, the minimum and maximum deviations between the experimental and the CFD predicted values of Pr are found to be 1.48% and 6.32%, respectively. There is good agreement between the predicted and experimental values for all the square cavity labyrinth seals and LS6. Though the k–e model with the wall functions approach is very stable, the presence of significant mean streamline curvature and multiple separations and recirculations has resulted in a high difference of 6.32% for LS7. Chochua et al. [6] have shown that the low Reynolds number model performs better than the wall function treatment. 6. Conclusions

40

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10 LS6 LS7

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Re Fig. 21. Pr vs. Re and q for the CCLS.

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A semi-theoretical model developed to determine the pressure drop for leakage through static, liquid labyrinth seals is found useful in the analysis and optimisation of such seals. The visualisation tests employed were handy in observing the fluid flow pattern and qualitative assessment of the internal cavity turbulence, providing useful information for the formulation of the theoretical model. Presence of large sized, fast rotating and multiple vortices inside the cavities is an essential ingredient to achieve better sealing characteristics. This is in view of the fact that the flow penetration into the cavity is small and the cavity vortex loss is the major contributor to the overall pressure drop. Among the five SCLS tested, the LS3 seal has the highest pressure drop measured experimentally. The concept of vortex loss coefficient in association with an ANN model serves as a reliable tool for identifying the optimal SCLS configuration. The CFD predictions are in very close agreement with the pressure drop obtained experimentally. The two newer curved cavity labyrinth seals outperformed the square cavity seals. The optimal LS7 seal obtained through parametric CFD analysis exhibited the highest level of internal cavity resistance and was the best among all the seals tested. Evidently, the curvature of the impinging wall of the cavity plays a crucial role in deciding the size and strength of the recirculation pattern. The LS7 seal is found to have the maximum vortex loss coefficient. Its pressure drop ratio increased by more than 75%, at the rated leakage flow over the optimal square cavity seal. The reason for the improvement is identified to be the counter rotating, double vortex structure within the cavity, which promotes higher turbulent dissipation of leakage flow momentum.

LS7

0.2 0

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30000 40000 Re

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Fig. 22. Vortex loss coefficient vs. Re and q for CCLS.

70000

Acknowledgements This paper is a partial outcome of a research project sponsored by the Indira Gandhi Centre for Atomic

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Research, Government of India and supported by the Mepco Schlenk Engineering College. The assistance of Mr. Srinivasarao Aryasomayajula, Mr. M. Regupathy Ragavan and the research team members on either side are gratefully appreciated. References [1] Stocker HL. Advanced labyrinth seal design performance for high pressure ratio gas turbines. In: Winter annual meeting of the Gas Turbine Division/ASME, Houston, TX, 1975; 75-WA/GT-22. [2] El-Gamal HA, Awad TH, Saber E. Leakage from labyrinth seals under stationary and rotating conditions. Tribol Int 1996;29(4):291–7. [3] Nikitin GA, Ipatov AM. Design of labyrinth seals in hydraulic equipment. Russ Eng J 1973;LIII:26–31. [4] Idelchik IE, Fried E. Flow resistance: a design guide for engineers. New York, USA: Hemisphere Publishing Corporation; 1989. p. 284–85. [5] Asok SP, Rajesh K, Padmakumar G, Sundararajan T, Sankaranarayanasamy K, Govindarajan S, et al. Theoretical and experimental investigations on incompressible flow through labyrinth seal. In: Proceedings of the 10th international topical meeting on nuclear reactor thermal hydraulics (NURETH-10), Seoul, South Korea, October 5–9 2003. Paper number K00110. [6] Chochua G, Shyy W, Moore J. Computational modeling for honeycomb stator gas annular seal. Int J Heat Mass Transfer 2002;45:1849–963. [7] Villasmil LA, Chen HC, Childs DW. Evaluation of near-wall turbulence models for liquid annular seals with roughened walls. AIAA J 2005;43(10):2137–46.

[8] Chochua G. Computations of gas annular damper seal flows. PhD. dissertation, University of Florida, Gainesville, 2002. [9] Arghir M, Roucou N, Helene M, Frene J. Theoretical analysis of the incompressible laminar flow in a macro-roughness cell. J Tribol 2002;125:309–18. [10] Villasmil LA, Childs DW, Chen HC. Understanding friction factor behavior in liquid annular seals with deliberately roughened surfaces. ASME J Tribol 2005;27:213–22. [11] Morrison GL, Daesung Chi. Incompressible flow in stepped labyrinth seals. In: Jt.ASME/ACSE applied mechanics, bioengineering and fluids engineering conference, Albuquerque, NM, 24–26 June 1985. ASME Paper-85-FE-4. [12] Rhode DL, Demko JA, Traegner UK, Morrison GL, Sobolik SR. Prediction of incompressible flow in labyrinth seals. J Fluid Eng 1986;108:19–25. [13] Schramm V, Denecke J, Kim S, Wittig S. Shape optimization of a labyrinth seal applying the simulated annealing method. In: Proceedings of the 9th of international symposium on transport phenomena and dynamics of rotating machinery, Honolulu, Hawaii, 10–14 February 2002. p. 1–6. [14] FLUENT 6 User’s guide, Fluent incorporated, Lebanon. [15] Roache PJ. Perspective: a method for uniform reporting of grid refinement studies. ASME J Fluids Eng 1994;116:405–13. [16] Rajkumar V, Nilay P, Wei S, Kevin TP, Lisa WG, Raphael TH, et al. Neural network and response surface methodology for rocket engine component optimization. AIAA 2000 AIAA-2000-4880. [17] Marko H, Brane S, Igor G. Experimental turbulent field modeling by visualization and neural networks. Trans ASME 2004;126(May): 316–22. [18] Demko JA, Morrison GL, Rhode DL. Effect of shaft rotation on the incompressible flow in a labyrinth seal. J Propuls 1990;6(2):171–6.