COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Optimal Design Technique of Spiral Grooved Mechanical seal Based on
Thermo-Hydrodynamic
J. F. Zhou*, B. Q. Gu
College of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, 210009 China
Email: [email protected], [email protected]
Abstract The spiral grooved mechanical seal is a new type of non-contacting mechanical seal, and widely used as
shaft sealing on pumps, compressors and agitators. Based on the frictional heat transfer analysis of the sealing
members and the coupling analysis of their thermal deformation and the frictional heat of the fluid film, the primary
parameters of the sealing members and the shape of the gap between the two end faces can be determined. A finite
element program is developed to calculate the bearing force and the leakage rate of the fluid film in the spiral grooved
mechanical seals. The optimal design technique of the spiral grooved mechanical seal is desired to obtain the
geometrical parameters of the sealing members which correspond to the relative large bearing force, the small leakage
rate and the minimum temperature of the sealing members. By means of this technique, the geometrical parameters of
the sealing members, such as the length L, the inner radius Ri, the end radius of the grooves Rg, the outer radius Ro, the
spiral angle α, the ratio of the groove width to the weir width δ and the number of the grooves Ng are obtained. Because
most of the related factors, such as the thermal deformation of the end faces and the viscosity change of the fluid film,
are considered in the method, the sealing members possess good heat transfer performance and sealing capability.
Key words: spiral groove, mechanical seal, optimum design, bearing force, leakage rate
INTRODUCTION
The non-contacting and non-leaking mechanical seal with multiple lobe grooves on the end face of the rotating ring
was developed by Etsion in 1984 [1]. The shallow grooves can generate hydrodynamic in the end faces of the rotating
ring and the stationary ring. The hydrodynamic of the fluid film can resist the leakage flow driven by the differential
pressure between the outer and inner radiu of the sealing face. The shape of the shallow grooves may be spiral grooves,
radial grooves, rectangular grooves and so on. The configuration of the spiral groove can be log spiral or Archimedes’
spiral. The log spiral is more effective than the Archimedes’ spiral and the spiral angle of the log spiral α is constant.
The spiral grooved sealing ring is illustrated in Fig. 1.
Figure 1: Model of the spiral grooved sealing ring
The log spiral is defined by Eq. (1).
r = Rg eθ tgα
(1)
⎯ 1054 ⎯
The bearing force provided by the fluid film in the spiral grooved mechanical seal is much larger than that in the flat
end face mechanical seal. It changes with the thickness of the fluid film which is adjusted to keep the balance between
the bearing force of the fluid film and the closing force provided by the spring or the bellow. When the mechanical seal
is running, the frictional heat is generated continuously and the sealing performance is different from the design
objective. Some investigations on the thermo-hydrodynamic (THD) in the mechanical seals were carried out in
consideration of the frictional heat, the change of the physical characteristics of the fluid film and the thermal
deformation of the end faces of the sealing members [2-4]. The optimal design technique of the spiral grooved
mechanical seal based on THD is investigated to give the method to select the appropriate material and to determine
the geometrical parameters of the sealing members.
THERMAL-HYDRODYNAMIC IN SPIRAL GROOVED MECHANICA SEAL
A THD analysis is carried out based on the heat transfer model of spiral grooved mechanical seal to investigate the
effect of the frictional heat on the sealing performance of the mechanical seal.
1. Model of the mechanical seals Since the leakage rate of the sealed medium through the gap between the two end
faces is very small, the frictional heat taken away by the leakage can be neglected. The cooling effect of the air is poor
enough to be neglected and therefore, the boundary conditions of the rings at the air side can be regarded as thermal
isolation. Hence the total frictional heat is conducted by the two rings from their end faces to the sealed medium. Some
other assumptions are made as follows:
(1) The model of the sealing member is regarded as axisymmetrical cylinder. The temperature field and deformation
are also axisymmetry in spite of the effect of the spiral grooves.
(2) The sealed medium is a Newtonian fluid, and its dynamic viscosity is constant in the direction of the thickness of
the fluid film but varies along radius.
(3) The temperature of the sealed medium T0 is constant.
(4) The radial outlines of the two deformed end faces are approximately regarded as straight lines because the seal
faces are narrow and the order of magnitude of the deformation of the end face is micro-size.
(5) The heat transfer takes place under a steady state condition.
Figure 2: Heat transfer model of the mechanical seal
The heat transfer model of the sealing members consists of the rotating ring, the stationary ring, the fluid film and the
sealed medium, as illustrated in Fig. 2. x and y are the coordinates of the axial direction and the radial direction
respectively. Ro and Ri are the outer radius and the inner radius of the fluid film respectively. hi is the thickness of the
fluid film at the inner radius and β is the separation angle of the two deformed end faces. m1 and m2 are the lengths of
the rotating ring and the stationary ring respectively. The thickness of the fluid film in the non-grooved area and in the
grooved area can be expressed by Eqs. (2) and (3), respectively.
h1 ( y ) = hi + ( y − Ri ) tan β
(2)
h2 ( y ) = hg + hi + ( y − Ri ) tan β
(3)
2. Frictional heat of the fluid film The frictional heat flux can be calculated by Eq. (4).
q ( y ) = μ(Tm )ω 2
r2
h( y)
(4)
⎯ 1055 ⎯
where μ (Tm ) is the viscosity of the fluid film which is determined by the mean temperature of the fluid film Tm. For
water, the relationship between the viscosity and the temperature can be expressed by Eq. (5)
μ(T ) = 0.001 × ( 0.258 + 2.112e−0.0341T
m
m
)
(5)
Eq. (5) is obtained by fitting the data of the viscosity and the temperature ranging from 0 to 100°C.
3. Temperature field of the sealing rings The model of the seal rings is illustrated in Fig. 3.
In Fig. 3, m and n are the length and the width of the cross-section of the rings, respectively. The heat transfer
governing equation is in the form
Figure 3: Cross-section of the sealing ring
∂ 2T ∂ 2T
+
=0
∂x 2 ∂y 2
(6)
Eq. (7) is obtained by using the boundary condition at y=n.
λ
k = cot nk
α
(7)
Using the thermal insulation conditions at x=0 and y=0, the general solution of Eq. (6) can be obtained, and it is in the
form
∞
θ ( x, y ) = ∑ Bi ( e k x + e − k x ) cos ki y
i
(8)
i
i =1
where ki is the solutions of Eq. (7) and Bi is an undetermined value. According to the heat flux boundary condition
when x = m, Eq. (9) is obtained.
∞
∑ B k (e
i =1
i i
ki m
− e − ki m ) cos ki y =
q
(9)
λ
where q is the heat flux exerted on the end face. For the rotating ring, q = δ1q ( y ) and for the stationary ring,
q = δ 2 q ( y ) , δ1 and δ2 are the heat distribution ratio for the rotating ring and the stationary ring respectively. Because
Figure 4: Discrete heat flux values and linear fit
⎯ 1056 ⎯
only when the distribution of q is linear along radius, can Eq. (9) be solved, Eq. (4) should be simplified to a linear
equation. Some discrete values of yi and qi in the dam area, the weir area and the grooved area are given to fit the linear
equation, as shown in Fig. 4.
Introducing the parameters y for the average value of yi and q for the average value of qi, the linear distribution
equation of q can be obtained by means of least square and it is in the form
ˆ + bˆ
q ( y ) = ay
where aˆ =
S yq
S yy
(10)
n
n
n
ˆ , S yy = ∑ ( yi − y ) , S qq = ∑ ( qi − q ) and S yq = ∑ ( yi − y )( qi − q ) .
, bˆ = q − ay
2
i =1
2
i =1
i =1
Hence, Bi is solved and the temperature distribution equation of the ring is in the form
T ( x, y ) =
where Δ =
2δ
∞
∑ Δ (e
λ
ki x
i =1
)
+ e− ki x cos ki y + T0
(
(11)
)
ˆ + bˆ sin nki
aˆ cos nki − aˆ + ki an
k
2
i
(e
ki m
−e
− ki m
) ( k n + sin k n cos k n )
i
i
.
i
4. Heat distribution ratio The heat transfer coefficients for the rotating ring α1 and for the stationary ring α2 can be
calculated by Eqs. (12) and (13), respectively.
α1 = 0.135λ ⎡⎣( 0.5Rec2 + Rea2 ) Pr ⎤⎦
0.33
(12)
Dr
α 2 = 0.0115λε1Re0.8 Pr 0.4 Ss
(13)
where Re c = ωDr2 ν , Re a = UDr ν and Re = 2VSs ν .
The heat quantity Q1 and Q2 transferred by the two rings can be approximately calculated by Eqs. (14) and (15).
Q1 = A1α f (Tf − T1 )
(14)
Q2 = A2α f (Tf − T2 )
(15)
The heat transfer coefficient from the fluid film to the end faces of the two rings αf can be calculated by Eq. (16).
α f = 0.664λm Pr
0.33
⎛ uf ⎞
⎜
⎟
⎝ ν Lc ⎠
0.5
(16)
where Lc = π (Ro + Ri ) and uf = (Ro + Ri )ω 4 .
The relationship between the heat quantity conducted by each ring and the average temperature on its end face can be
expressed by Eqs. (17) and (18) according to the reference [5].
Q1 = ζ 1λ1 A1tanh (ζ 1m1 ) T1
(17)
Q2 = ζ 2λ2 A2 tanh (ζ 2 m2 ) T2
(18)
where ζ 1 = α1 L1 / ( A1λ1 ) and ζ 2 = α 2 L2 / ( A2 λ2 ) .
Hence the heat distribution ratio for the rotating ring δ1 and that for the stationary ring δ2 can be calculated by Eqs. (19)
and (20), respectively, which are obtained from Eqs. (14), (15), (17) and (18).
δ1 =
δ2 =
A1 z1 ( z2 + α f A2 )
(19)
( A1 + A2 ) z1 z2 + ( z1 + z2 ) A1 A2α f
A2 z2 ( z1 + α f A1 )
(20)
( A1 + A2 ) z1z2 + ( z1 + z2 ) A1 A2α f
⎯ 1057 ⎯
where z1 = ζ 1λ1 A1 tanh (ζ 1 L1 ) and z2 = ζ 2 λ2 A2 tanh (ζ 2 L2 ) .
5. Temperature distribution equation of the fluid film Based on the assumption that the heat is conducted by the
two rings, the energy equation (21) for the fluid film holds
2
λm
∂ 2T
⎛ ∂u ⎞
+ μ(Tm ) ⎜ ⎟ = 0
2
∂x
⎝ ∂x ⎠
where
(21)
∂u
can be obtained from the velocity boundaries at the end faces of the rotating ring and the stationary ring. T
∂x
can be obtained by integrating Eq. (21) twice and substituting the temperature boundaries at the end faces which can be
acquired from Eq. (11).The local mean temperature is
∞
∞
Tm ( y ) = ∑ f (δ1 , λ1 , k1,i , m1 , n1 ) cos k1,i y + ∑ f (δ 2 , λ2 , k2,i , m2 , n2 ) cos k2,i y +
i =1
where f (δ , λ , ki , m, n ) =
μ(T )ω 2 ( Ri + y )
i =1
(
ˆ
)
m
12λm
2
+ T0
(22)
ˆ + b sin nki
δ aˆ cos nki − aˆ + ki an
( ek m + e− k m ) .
km
−k m
2
λ ki ( e − e ) ( ki n + sin ki n cos ki n )
i
i
i
i
DEFORMATION OF THE END FACES OF THE SEALING RINGS
The heat expansion of the sealing rings caused by the frictional heat is inevitably in mechanical seal. The gap form
which is composed of the two deformed end faces is different from the parallel gap and the characteristics of the fluid
film is also different from that in the parallel gap.
1. Calculating the deformation by FEM The deformation of the end faces of the sealing members can be calculated
by means of FEM. When the pressure of the sealed medium is less than 2MPa, the deformation caused by the pressure
is much less than that did by the frictional heat. Hence the effect of the pressure on the deformation is neglected. The
heat expansion at the inner radius is larger than that at the outer radius. The deformation of the end face can by
expressed by the relative axial displacement Δz between the outer radius and the inner radius.
2. Forecasting the deformation based on BP ANN The calculating of the deformation by FEM is accurate but
time-consuming. By means of back propagation artificial neural nets (BP ANN), an efficient method can be realized to
forecast the deformation magnitude. A three-layer BP ANN is built to express the relationship between the heat flux
exerted on the end face and the value of Δz. The ANN is composed of the input layer, the mid layer and the output
layer.
The input training samples which affect the precision of ANN are related to the viscosity and thickness of the fluid film
and the rotational speed. These factors can be selected according to the orthogonal design [6, 7]. The table of the
orthogonal design for the selection of the three factors is L25(56). Each factor has five levels, as Tab.1 shown.
Table 1 Factors and values of input samples
Factors
values of different levels
1
2
3
4
5
ω/rad·s
100
300
500
700
900
hi/μm
1.0
1.5
2.0
2.5
3.0
β/rad
0.0001
-1
0.0002 0.0003 0.0004 0.0005
Because the input samples are the frictional heat flux exerted on each node in the radial direction on the end face, the
number of the input elements is equal to the number of the nodes of the end face in the radial direction. The output
samples are the values of the displacement Δz and the output layer has only one element. The number of the elements
of mid layer is not definite, but it should satisfy the requirement of the precision of the BP ANN.
Some random values of heat flux and the corresponding value of Δz calculated by FEM are used to examine the
reliability of the BP ANN. The forecasting results illustrate that the BP ANN can calculate the deformation accurately.
The BP ANN is the more efficient method to calculate the deformation of the end faces than FEM. The deformation
⎯ 1058 ⎯
can be calculated by the BP ANN when the heat flux changes. It is conveniently to carry out the noupling analysis of
Frictional heat and thermal deformation by means of the BP ANN.
CHARACTERISTICS OF THE FLUID FILM IN THE END FACES
The flow of the fluid film between the end faces of the sealing members can be simplified as laminar flow of the ideal
liquid, which can be described by the Reynolds equation for incompressible fluid [8].
∂ ⎛ h3r ∂p ⎞
∂ ⎛ h3 ∂p ⎞
∂h
+
⎜
⎟
⎜
⎟ = 6ω r
r∂r ⎝ μ ∂r ⎠ r∂θ ⎝ μ r∂θ ⎠
r∂θ
(23)
Eq. (23) is a convection-diffusion transport equation, and the pressure distribution of the fluid film can be calculated
by solving Eq. (23). By using the Galerkin weighted residual method with second-order shape functions, a finite
element program is developed to solve this equation numerically. Based on the calculation results, the main sealing
performance parameters can be obtained, including the bearing force of the fluid film and the leakage rate and the
frictional torque.
OPTIMAL DESIGN TECHNIQUE
The thermal deformation of the end faces and the viscosity change of the fluid film is considered in the design
technique. The primary parameters of the sealing rings can be determined by the requirement of reducing the
temperature of the sealing rings. The geometrical parameters of the spiral grooves, such as the width of the groove, the
spiral angle, the number of the grooves and so on, can be obtained by optimal design in consideration of the maximum
bearing force as design objective. Some methods are also carried out to minimize the effect of the deformation on the
bearing force of the fluid film and to reduce the leakage rate.
The sealed medium is water and its coefficient of heat conductivity λ is 0.06 W·m−1·K−1，its dynamic viscosity μ is
0.001 Pa·s, its kinematic viscosity ν is 1×10−6 m2·s−1. Other parameters are as: Pr=7.02, U=V=0.5 m·s−1, Ss＝0.15 m.
The sealed medium pressure is 1.013×106 Pa.
1. Determination of the primary parameters of the sealing rings The width of the end face n (n = Ro−Ri) heavily
influences the quantity of the frictional heat. The larger the width is, the larger the quantity of the frictional heat is.
Similarly, the larger the area of the sealing face, the larger the heat transfer area is. The calculated tempreture field of the
sealing members and the fluid film illustrates that the highest temperature Tmax locates at the inner radius of the fluid film
next to the rotating ring. Exerting the heat fux q=2000W·m−1 on the end face of the sealing ring to investigate the change
of Tmax with the length of the rings m and the width of the end face n, as illustrated in Fig. 5. When n is less than 8mm, the
increase of the cooling area lowers the maximum temperature Tmax. Once n is larger than 8mm, the temperature gradient
in the radial direction of the end face increases and the maximum temperature Tmax rises accordingly. Therefore, there
exists the optimal width of the seal face in terms of reducing the maximum temperature of the end face.
Figure 5: Variation of Tmax with m and n
The length of the rotating ring m has less evident influence on the maximum temperature than the width of the end face.
Tmax always decreases with the increase of m because the increase of m enlarges the heat transfer area but not changes the
frictional heat, as shown in Fig. 5. The effect of m on the maximum temperature is almost negligible when m>11mm.
⎯ 1059 ⎯
Another important parameter is the coefficient of heat conductivity of the material of the rotating ring λ. The
coefficient λ of the frequently used materials for mechanical seals ranges from 15 to 100 W·m−1·K−1. The maximum
temperature of the fluid film Tmax can be obtained using Eq. (22). For example, when λ=15 W·m−1·K−1, Tmax is about 46
℃, and when λ=100 W·m−1·K−1, Tmax=26 ℃ under the conditions that ω =500 rad·s−1, hi=1.0 μm, m=8 mm and n=11
mm. It can be seen that the large coefficient of heat conductivity of the material of the seal rings can effectively lower
the maximum temperature of the end faces of mechanical seals.
Therefore, the primary parameters of the sealing members are determined, as shown in Table 2.
Table 2 Parameters of the sealing members
λ
C
/W·m−1·K−1 /J·kg−1·K−1
ε
m
/mm
Ri
/mm
Ro
/mm
E
/GPa
τ
/K−1
Rotating ring
15
20
30
200
1.6×10-5
16
502
0.3
Stationary ring
15
20
30
25
5×10-6
10
174
0.15
The coefficients α1 and α2 is calculated by using the parameters in Table 2, as illustrated in Fig. 6. The coefficient α1
increases obviously with the increase of the rotational speed ω. For the medium is circled and its temperature T0
remains constant, the axial velocity of the medium around the stationary ring increases slightly with ω and results in
the small variation of the coefficient α2. The heat distribution ratios δ1 and δ2 are mainly related to the heat transfer
coefficient of the rotating ring α1 and that of the stationary ring α2. Because α1 is much larger than α2, most of the
frictional heat is conducted by the rotating ring.
Figure 6: Values of α and δ corresponding to ω
2. Optimal design of the parameters of the spiral grooves The operation parameters of the mechanical seal consist
of the rotational speed ω, the medium pressure po and the viscosity μ of the medium. The parameters of the spiral
groove consist of the spiral angle α, the number of the grooves Ng, the groove depth hg, and the width of the groove
width b and the end radius of the groove Rg.
The fluid film in the spiral grooved end faces should provide the bearing force to open the two end faces, and the
leakage rate should also be less than the allowable value. Therefore, the larger bearing force and the smaller leakage
rate are the design objective. Under the given operation conditions, the variables which affect the design objective
include the values of β, Ng, hg, b and Rg and the fluid film thickness h1. The complex method is employed to solve the
multivariable optimization problem. The complex method is a mathematical programming technique, suitable for
obtaining an optimal solution to a nonlinear, constrained optimization problem.
The optimal design process consists of two modules. One is to improve the geometrical parameters of the spiral groove
by using the complex method. The other is to evaluate the value of the objective function by FEM. The optimized
parameters of the spiral groove are shown in Table 3. In this example, h1 is assumed to be 3 μm.
The parameter h1 should be controlled in consideration of the two requirements. One is to ensure the fluid lubrication
of the two end faces, and the other is to reduce the leakage rate, but the parameter h1 can not meet the both of the two
requirements simultaneously. The relationship between the groove depth hg and h1 can be expressed by the
non-dimensional parameter H which holds
⎯ 1060 ⎯
Table 3 Parameters of the spiral grooves
α
Ng
/rad
0.14
H = 1+
hg
h1
12
hg
b
Rg
/μm
/mm
/mm
6.2
0.72
25
=3
(23)
The parameter H determines the hydrodynamic effect of the spiral grooves. By using the parameters shown in Table 3,
when ω=500 rad·s-1, the bearing force is 2873 N the volume leakage rate is 2.2×10−8 m3·s−1.
3. Coupling analysis of Frictional heat and thermal deformation In order to determine the relationship between the
frictional heat and the thermal deformation, the coupling analysis is carried out in consideration of the viscosity change
with the fluid film temperature. Because the frictional heat decreases with the increase of the separation angle β, and β
increases with the increase of the frictional heat, there exists the only separation angle β for the given values of hi and
ω. The iterative computation is to adjust the value of β to obtain the frictional heat which results in the same separation
angle β.
For a certain β, the viscosity, which affects the frictional heat, is calculated according to the mean temperature of the fluid
film, and the calculating of viscosity is included in the coupling iteration. The viscosity increases with the increase of β,
namely if β is enlarged, the viscosity is enlarged too, and the frictional heat is enlarged. Therefore, the viscosity change
slows up the frictional heat change with β, and the iteration step of β should be a very small value in order to get the
required value of β. The values of β corresponding to different hi and ω are obtained and illustrated in Fig. 7.
Figure 7: Values of β corresponding to given hi and ω
4. Reshaping of the depth of the spiral groove Based on the coupling analysis of Frictional heat and thermal
deformation, the separation angle of the end faces is determined. When hi=3 μm and ω=500 rad·s-1, the value of β is
equal to 0.00035 rad. The bearing force is 2312 N, and the volume leakage rate is equal to 2.4×10-8 m3·s-1. It is obvious
that the deformation decreases the bearing force but increases the leakage rate.
Once the value of h1 varies along radius, the value of H is not constant and the bearing force of the fluid film decreases.
Therefore, the depth of the spiral groove hg should be the function of the radius r, as Eq. (24) expresses.
hg =
H
⎡ hi + ( r − Ri ) tan β ⎤⎦
1− H ⎣
(24)
The thickness of the groove increases continuously from the end of the groove to the outer radius of the sealing face.
5. Reshaping of sealing dam Since the leakage rate of the convergent gap in larger than the parallel gap, it is
suggested that the sealing faces should compose the parallel gap. The dam area on the end face of the rotational ring
and the corresponding area on the stationary ring can be shaped to compose a divergent gap, as Fig. 8(a) illustrates. β1
is the divergent angle. The divergent gap changes to the parallel gap when the thermal deformation takes place, as
Fig.8(b) illustrates. β2 is the convergent angle of the end faces. Because of the assumption that the axial displacement
of the end face is linear, the divergent angle of the sealing dam β1 is equal to the convergent angle of the end face β2.
⎯ 1061 ⎯
(a)
(b)
Figure 8: Gap models in the end faces (1—Rotating ring; 2—Stationary ring)
The leakage rate is related to the reshaping location Rs. The parameter Rs should be samller than Rg because end faces
of the mechanical seal should contact closely to seal the medium when the mechanical seal rests. The value of Rs is
suggested to be 22mm in this example, and the separation angle β1 is equal to 0.0008 rad. During the course of solving
the value of β1, the linear fit of the heat flux of the fluid film should be done repeatedly according to the model of the
fluid film in the end faces.
After reshaping, the bearing force is equal to 2612 N, and the volume leakage rate is equal to 1.6×10−8 m3·s−1. The
reshaping of the spiral grooves and the sealing dam is beneficial for maintain the bearing force of fluid film and reduce
the leakage rate.
CONCLUSIONS
(1) The temperature distribution equations of the sealing members and the fluid film are derived. The heat flux
distribution equation and the linear fit method of the equation are given. The THD in the spiral grooved mechanical
seal is investigated in consideration of the effect of the frictional heat of the fluid film and the thermal deformation of
the end faces. Corresponding to the certain values of hi and ω, there exists the compatible relationship among the
separation angle of the two deformed end faces, the frictional heat, the viscosity and the temperature of the fluid film.
(2) The Reynolds equation for the flow in the end faces of the spiral grooved mechanical seal is solved by the finite
element program which can calculate the pressure distribution of the fluid film. The parameters of the spiral grooves
are optimized to obtain the relative large bearing force of the fluid film and small leakage rate by means of complex
method.
(3) The deformation of end faces and the viscosity change of the fluid film affect the sealing performance of the spiral
grooved mechanical seal. Therefore, the THD effect is considered in the design of the spiral grooved mechanical seal.
Based on the optimized parameters of the spiral grooves, the depth of the grooves is re-designed to keep the ratio of the
thickness of the fluid film in the grooved area to that in the non-grooved area H as constant and to ensure that the
grooves can generate sufficient hydrodynamic pressure. The dam areas of the end faces are reshaped to compose a
divergent gap so that the parallel gap can be formed to reduce the leakage rate of the mechanical seal when the
mechanical seal is running.
(4) The optimal design technique is to determine the geometrical parameters of the spiral grooves , and to calculate the
deformation of the end faces of the sealing members. The reshaping of the end face and the spiral grooves can increase
the bearing force of the fluid film and decrease the leakage rate. The designed mechanical seal possesses the good heat
transfer performance and meets the design requirements of the bearing force and the leakage rate.
Nomenclature
A
area of the end face
C
specific heat
Dr
diameter of the rotating ring
E
Young’s modulus
h
thickness of the fluid film
hi
thickness of the fluid film at the inner radius
L
circumference of the seal rings
m
length of the seal rings
N
width of the seal rings
p
pressure of the fluid film
Pr
Prandtl number
q
frictional heat flux
r
Ro
Ri
Ss
T
T0
Tm
T1
T2
u
⎯ 1062 ⎯
radius of the fluid film
outer radius of the fluid film
inner radius of the fluid film
distance from the outer surface of the stationary
ring to the inner surface of the sealing chamber
temperature
temperature of the sealed medium
average temperature of the fluid film
average temperature of the end face of the rotating
ring
average temperature of the end face of the
stationary ring
tangential velocity of the fluid film
U
V
axial flow velocity of the sealed medium around
the rotating ring
axial flow velocity of the sealed medium around
the stationary ring
Greek symbols
α
heat transfer coefficient from the rings to the
sealed medium
heat transfer coefficient from the fluid film to the
αf
end face of the rings
β
separation angle of the two deformed end faces
δ
ε1
ε
λ
λm
μ
ν
τ
ω
heat distribution ratio
correction factor, ε1=2
Poisson’s ratio
coefficient of heat conductivity of the rings
coefficient of heat conductivity of the sealed
medium
dynamic viscosity
kinematic viscosity
coefficient of thermal expansion
rotational speed
REFERENCES
1. Etsion I. A new conception of zero-leakage non-contacting face seal. Journal of Tribology, 1984; 106(3):
338-343.
2. Zeus D. Viscous friction in small gaps calculations for non-contacting liquid or gas lubricated end face seals.
Tribology Transactions, 1990; 33(3): 454-462.
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