Modeling of Fluid Flow in the Cone Seal

Available online at www.sciencedirect.com

Procedia Engineering 39 (2012) 132 – 139

XIIIth International Scientific and Engineering Conference “HERVICON-2011”

Modeling of Fluid Flow in the Cone Seal Leonid Savina, Elena Kornaevab, b* a

State university – education-research-and- production complex, Naugorskoye Shosse 29, Orel 302020,Russia b Stary Oskol state institute, Makarenko micro-district 42, Stary Oskol 309516, Russia

Abstract The fluid flow in cylinder-cone rotor-seal system is investigated. Mathematical model of three dimensional enforced and shear flow of viscosity incompressible fluid in the gap between cylinder rotor and cone stator is produced. The dimensional analysis of model is carried out. The velocity and pressure fields, leakage are presented as result, which are calculated by control volume approach.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Sumy State University Keywords: Hydrodynamic theory of lubrication; hydrodynamic seal; dimensional analysis; enforced and shear flow; finite elements approach; control volume approach.

1. Introduction Modeling of the fluid flow in the gap of rotor-stator system is the applied problem of hydrodynamics. In this paper we study hydrodynamic groove seals, which are assigned for rotor of compression pump closure, for example. Groove seals advantages are simplicity, dependability and reliability. Seals construction is resemble with journal bearings, that’s why they may be used like complementary bearings [1]. As a result it is necessary to calculate the velocity and pressure fields, and theirs integral characteristic such as leakage through the gap, load-carrying capability and shear torque. In spite of construction simplicity, the groove seal modeling is very intricate problem. Basically, the three-dimensional fluid flow in the gap between axle and groove seal can be described by Navier-Stokes equation solved together with continuity equation. These equations have no analytical solution in case of

* Corresponding author. Tel.: +7-903-886-83-56. E-mail address: [email protected].

1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.07.017

Leonid Savin and Elena Kornaeva / Procedia Engineering 39 (2012) 132 – 139

nonsymmetrical flow region. Also, the Navier-Stokes equation is nonlinear because of quadratic inertia components. Using dimensional analysis, it is possible some times to simplify basic model as it is done in squeeze film analysis for example. As the result of stated below dimensional analysis research of fluid flow in the cone seal, basic model cant be simplified, except cases of very low conicity. See the for a more complete discussion of dimensional analysis research. The literature review shows that theoretical analysis of fluid flow in cone seal is insufficient in opposite to squeeze film flow in regions with small constant gap, such as cylindrical and conical bearings. 2. Definition of problem Three-dimensional flow of viscosity incompressible fluid in the gap, which is represented on figure 1, is investigated. The gap is formed by cylinder rotor and cone stator.

Fig.1. Rated operating conditions

Cylinder with radius r, cone with radiuses R1 and R2 respectively, L is length of seal. h01 and h02 are middle value of radial gap, ȕ is a conicity parameter. Cylinder is rotates with constant tangential velocity Ȧ, cone is fixed. The fluid is fill up the gap fully and flows at the expense of the pressure drop ǻP along axial direction in confuser. Navier-Stokes and continuity are base equations, which are described investigated process. In cylindrical coordinate system dimensionless equations have the appearance:

133

134

Leonid Savin and Elena Kornaeva / Procedia Engineering 39 (2012) 132 – 139

~ V~ϕ ∂V~ρ V~ϕ ∂V~ρ ­ 2 ~ ∂V~ρ ∂P ~ 2 + A Vz − = − Eu + ° A Vρ ~ + A ~ ρ + B ∂ϕ ρ~ + B ∂ρ ∂~z ∂ρ~ ° ° ~ ~ 2~ 2~ ∂ 2V~ρ ∂V~ϕ ·¸ ° A §¨ ∂ V ρ 1 ∂V ρ 1 2 2 ∂ Vρ − Vρ C + + − + + ° ¸; ¨ ρ~ + B ∂ρ~ (ρ~ + B )2 ∂ϕ 2 ~ + B )2 A(ρ~ + B )2 ∂ϕ ¸ ∂~z 2 ° Re ¨ ∂ρ~ 2 ( ρ © ¹ ° ~ ~ ∂V~ ~ ~ V~ ~ ° ∂ ∂ V V V V Eu ∂P ϕ ϕ ρ ϕ ° AV~ρ ϕ + ϕ + AV~z +A =− + ρ~ + B ∂ϕ ρ~ + B ρ~ + B ∂ϕ ∂ρ~ ∂~ z ° ° ~ ~ · °° 1 §¨ ∂ 2V~ϕ ∂ 2V~ϕ ∂ 2V~ϕ V~ϕ 1 ∂Vϕ 1 2 A ∂V ρ ¸ 2 (1) + + +C − + ®+ ¸; ¨ 2 ρ~ + B ∂ρ~ (ρ~ + B )2 ∂ϕ 2 ∂~z 2 (ρ~ + B )2 (ρ~ + B )2 ∂ϕ ¸¹ ° Re ¨© ∂ρ~ ° ~ ~ ° V~ϕ ∂V~ ∂V~z ~ z + AV~ ∂V z = − EuγC ∂P + ° AV ρ + z ∂~z ∂ρ~ ρ~ + B ∂ϕ ∂~z ° ° ~ ° 1 §¨ ∂ 2V~z ∂ 2V~z ∂ 2V~z ·¸ 1 ∂V z 1 + + + C2 °+ ¸; ¨ 2 ρ~ + B ∂ρ~ (ρ~ + B )2 ∂ϕ 2 z 2 ¸¹ ∂~ ° Re © ∂ρ~ ° ~ ~ ~ V~ρ ° ∂V ρ 1 1 ∂Vϕ ∂V z + + + = 0; ° ~ ρ~ + B A ρ~ + B ∂ϕ ∂~z °¯ ∂ρ z ρ − ( r − e) where: ~ , ~ - dimensionless radial and axial coordinates respectively; z= ρ= h02 + e L Vρ Vϕ V ~ P − P0 ~ ~ ~ , Vϕ = , Vz = z - dimensionless components of velocity; P = Vρ = ωL ΔP ω(h02 + e) ωr ω r ( h + e) ΔP 02 dimensionless pressure; Re = , Eu = - Reynolds and Euler numbers respectively; 2 ν ρ (ωr ) w

ΔR

h r e β= , γ = , γ = 01 , ξ = h01 L L r

- geometric similarity parameters;

A = η(1 + ξ) + βγ −1 ,

B = A −1 (1 − ηξ) , C = Aγ - dimensionless coefficients formed by similarity parameters. Boundary conditions for velocity on cylinder and cone surface and on seal butts are presented: ~

­Vρ ( ~r (ϕ), ϕ, ~z ) = 0; °° ~ ~ ~ ®Vϕ ( r (ϕ), ϕ, z ) = 1; , °~ ~ ~ ¯°V z ( r (ϕ), ϕ, z ) = 0;

~ ~

­Vρ ( R (ϕ), ϕ, ~z ) = 0; °° ~ ~ ~ ®Vϕ ( R (ϕ), ϕ, z ) = 0; , °~ ~ ~ ¯°V z ( R (ϕ), ϕ, z ) = 0;

­ ∂V~ ° ~i °° ∂z ® ~ ° ∂Vi ° ~ °¯ ∂z

~ z =0

~ z =1

= 0; (2)

= 0;

135

Leonid Savin and Elena Kornaeva / Procedia Engineering 39 (2012) 132 – 139

where functions of outer and inner boundary are:

β ~ R (~ z) = − ~ z + 1,

C

γ ­~ § · 2 2 2 ¨ ηξ( γ + cos(ϕ + α )) + 1 − η ξ sin (ϕ + α) − 1¸; °r (ϕ) = ηγ ( 1 + ξ ) + β © ¹ ® °ϕ ∈ [0;2π]; ¯ Boundary conditions for pressure:

~ ∂P ~ ~ ~ P (~ ρ, ϕ,0) = 1 , P ( ρ , ϕ,1) = 0 , & & ~ ~ = 0 ∂n n ⊥ R ( z )

(3)

There is a periodic condition for each unknown function:

~ ~ Fi ( ~ ρ,0, ~ z ) = Fi ( ~ ρ,2π, ~z )

(4)

As the result, mathematical model, which is described investigated process, are presented by (1)-(4). The dimensional analysis was carried out, order of equations system terms are presented in table 1. Sequent summaries were obtained on the basis of the dimensional analysis: For small conicity (10-3): - with such value of Re>100 inertia terms provides a considerable contribution; - inertia terms rank over viscosity terms with such fully developed turbulence, what are confirmed in article [1]; - Reynolds assumption [2] about negligible of normal velocity component and partial derivatives of velocity by tangential and axel coordinates are reasonable. For value of conicity more than 10-3: - inertia terms is negligible with small Reynolds number (Re<<100) like small conicity; - normal velocity component and derivate of pressure by fluid layer thickness provide a considerable contribution. Table 1. Order of base equations terms

Num. of projection Oȡ Oij Oz

Inertia & terms, & ( V ⊗ ∇ ) ⋅V į2 į2 į2 į į į į į2 į į į -

1

į

* ∇ ⋅V 1

Navier-Stokes equation Pressure,

∇P

Eu į/Re į2/Re įEu 1/Re į/Re įEu 1/Re į/Re Continuity equation

1

where: δ = ( h02 + e ) L− 1 - size of small order.

& Viscous terms, ∇ 2V

į3/Re į2/Re į2/Re

į3/Re į2/Re -

į3/Re į2/Re -

į2/Re į/Re į2/Re

136

Leonid Savin and Elena Kornaeva / Procedia Engineering 39 (2012) 132 – 139

As theresult, Reynolds assumption for investigated object is not reasonable and base equations are getting difficult to calculate. Nonlinear equation system (1) was linearized by Newton approach:

(

) ( ) ~ F (X is, j , k ) -

~ ~ ~ F X is, j , k + F ' X is, j , k his, +j ,1k = 0 , where:

equations

(5)

in

mesh

~ ~ X is, j , k = [[Vρs

node

(i,j,k)

on

s

iteration;

~ ~ ~ Vϕs Vzs i, j , k Pis, j , k ]] - vector of unknown function on S iteration in (i,j,k) i, j , k i, j , k ~ s +1 ~ s +1 ~s ~s mesh node; h i, j, k = X i, j , k − X i , j , k - vector of function increments on s+1 iteration; F ' X i , j , k matrix of coefficients in front of unknown increments. The discrete analog of system (1) was obtained by integration of each equation by control volumes.

(

)

3. Example of two-dimensional flow calculation Two-dimensional enforced flow between two stationary planes (fig.2) is discussed as example. The top plane is obliquity by angle ș to bottom plane.

Fig. 2. Flow between planes

In that case equations have the appearance:

~ 2~ · ~ ­ ∂V~ § 2~ Eu ∂P 1 ¨ ∂ Vρ ρ ~ ∂Vρ 2 ∂ Vρ ¸; °V~ + + γ = − + V z ~ ~ ¨ ~2 ¸ ° ρ ∂~ ρ ∂z ∂~z 2 ¸¹ γ 2 ∂ρ Re γ ¨© ∂ρ ° ° ~ ~ ~ ~ 2~ · ∂P 1 §¨ ∂ 2V z ° ~ ∂V z ~ ∂V z 2 ∂ V z ¸; + + = − + γ V V Eu ® ρ ~ z ~ ~2 ∂~z Re γ ¨ ∂ρ ∂ρ ∂z ∂~ z 2 ¸¹ ° © ° ~ ~ ° ∂Vρ ∂V z + = 0; ° ∂~ ∂~ z ° ρ ¯

(6)

137

Leonid Savin and Elena Kornaeva / Procedia Engineering 39 (2012) 132 – 139

& & P − P0 * * Vx Vx y & & x where: x = , y = , Vx = , Vy = , Pˆ = - dimensionless coordinates and * L ΔP h02 γV * V ΔPh02 h - velocity for dimensionless. functions; γ = 02 - geometrical parameter; V * = V x max = 8μL L Boundary conditions for pressure:

∂Pˆ ∂Pˆ ~ ~~ P (~ ρ,0) = 1, P (ρ ,1) = 0 , = 0, & =0 ∂n * ˆ ∂ρˆ ρˆ = 0 n ⊥ h( zˆ )

(7)

Discrete analog of system (6) was obtained by integration of equations by finite respectively volumes (fig.3).

a) Fig. 3. Finite volume: (a)– for (c) – for continuity equation

b)

Oˆz

projection of Navier-Stokes equation; (b) – for

c)

Oρˆ projection of Navier-Stokes equation;

As the result, discrete analog of system (6) for calculate increments have the appearance:

~ ~ ~ ~ ~ ~ ~ Eu ­ °− an hρ n + aen hρ ne + aωn hρ nω + a N hρ q + a P hρ s − γ 2 Δzˆ (hPN − hPP ) = 0; ° ~ ~ ~ ~ ~ ~ ~ ° ®− be hze + bE hzee + bP hzω + ben hznn + bes hz ss − EuΔρˆ (hPE − hPP ) = 0; ° ~ ~ ~ ~ s s s s °Δzˆhρ n − Δzˆhρ s + Δρˆγhze − Δρˆγhzω = Δzˆ(Vˆρ s − Vˆρ n ) + Δρˆγ (Vˆz ω − Vˆz e ); °¯ where: · § ~ ¸ ¨ Vzs en Δ ~ ρ ~ 1 ¸, a = , a ωn = Vzs ωn Δ~ ρ ¨1 + en ¨ Re ~ s Re ~ s ~z ) − 1 ~z ) − 1 ¸ exp( V Δ exp( V Δ z en ¸ ¨ z ωn γ γ ¹ ©

(8)

138

Leonid Savin and Elena Kornaeva / Procedia Engineering 39 (2012) 132 – 139

~ · § Vρs Δzˆ ~ s ~¨ 1 ¸ N , aN = = Δ + a V z 1 ρ ¨ ¸ P ~s ~ ~s ~ P ¨ exp(Re γVρ Δρ ) − 1 exp(Re γVρ Δρ) − 1 ¸ N P © ¹ · § ¸ ¨ ~ ~ ~ 1 Vzs E Δρ ¸, , b P = Vzs P Δ~ ρ ¨1 + bE = ¸ ¨ Re ~ s ~ Re ~ s ~ exp( Vz E Δ z ) − 1 ¨ exp( Vz P Δ z ) − 1 ¸ γ γ ¹ © ~s § · Vρ Δ~z ¸ ~ s ~¨ 1 en , = Δ + b V z 1 b en = ρ es ¨ ¸, es ~s ~s ~ ~ ¨ exp(Re γVρ Δρ) − 1 ¸ exp(Re γVρ Δρ) − 1 es en © ¹ a = a + a + a + a , b = b +b +b +b .

n en ωn N P e E P en es Velocity and pressure field are presented as a result. On fig.4 vector field of velocity are presented. In case ȕ=8ǜ10-6 velocity have a parabolic law along thickness of the channel and constant along length, that case conforms to Poiseuille flow. With increasing of conicity ȕ normal velocity is being significant. There is a maximum value of velocity on the end of channel.

a)

b)

Fig. 4. Vector field of velocity for different conicity: a – ȕ=8ǜ10-6; b – ȕ=0,08

Fig.5a is presented the relation of pressure from length of channel in dimensional form, which was calculated by finite volume approach and by finite elements (Ansys). Deviate of result amounts to less than 2%. FEM

P_exact solution 150000

140000

140000 P(z), Paskal

P(z), Paskal

P_CVM 150000

130000 120000

P_CVM

P_FEM

130000 120000 110000

110000

100000

100000 0

0,02

0,04

0,06

0,08

z, mm

a) Fig. 5. Relation of pressure from length of channel

0,1

0,12

0

0,02

0,04

0,06 z,mm

b)

0,08

0,1

0,12

Leonid Savin and Elena Kornaeva / Procedia Engineering 39 (2012) 132 – 139

a – ȕ=0,08; b – ȕ=0 On fig.5b pressure is calculated by control volume method (CVM) and finite elements method (FEM) and compared with exact solution in case ȕ=0 [4]. It was developed that results by CVM have a best conform that results by FEM, Therewith, sufficient result can be received on crude mesh by control volume method.

Fig.6. Leakage versus conicity parameter

Leakage (is to width of channel along axle Oij) is presented on fig.6 versus conicity parameter. Leakage recedes with increase angle of angle of inclination of top plane. Leakage in case ȕĺ0 (parallel plane) was confirmed by analytical solution [4]:

Q=

ΔPh 3 12μL

As the result, computing error dimension is no more than 2%. Studying fluid flow in the cone seal with conicity ȕ>10-3 and Reynolds number Re>100, the following conclusions were obtained: it is necessary to account inertia forces; in cases of flat fluid flow between two nonparallel walls the velocity vector normal component is significant, besides the velocity value maximum fit on confusor’s end; derived results was compare with asymptotic analytical solution, the result of comparison has prove the model adequacy.

References [1] Martsinkovsky V.A. Rotor vibrations in centrifugal machines. Hydrodynamic of choked stream orifices. Sumy: SumSU, (2002), 337 p. [2] Hori Yukio. Hydrodynamic Lubrication. Hardcover, (2006), 250 p. [3] S. Patankar. Numerical heat transfer and fluid flow. New York: Hemisphere Publishing Corporation, (1980), 148p. [4] Slezkin N. A. Dynamics of Viscous Incompressible Fluid. Moscow: Gostekhizdat, (1955), 520 p.

139