CHINESE JOURNAL OF MECHANICAL ENGINEERING
Vol. 23,aNo. 1,a2010
·7·
DOI: 10.3901/CJME.2010.01.007, available online at www.cjmenet.com; www.cjmenet.com.cn
Leakage Prediction Method for Contacting Mechanical Seals with Parallel Faces
SUN Jianjun1, 2, *, WEI Long3, FENG Xiu3, and GU Boqin4
1 College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China
2 Department of Engineering Mechanics, Southeast University, Nanjing 210096, China
3 Fluid Sealing Technique Development Center, Nanjing College of Chemical Technology, Nanjing 210048, China
4 College of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing 210009, China
Received February 18, 2009; revised December 7, 2009; accepted December 14, 2009; published electronically December 17, 2009
Abstract: Since the beginning of the 20th century, many researches on the sealing characteristic of mechanical seals were carried out
broadly and in depth by various methods and some leakage models were built. But due to the lack of the way to characterize the main
factors of influence on the leakage, most of the early researches were based on the assumptions that the seal faces topography and the
frictional conditions were invariant. In the early built models, the effect of the surface topography change of the seal face on the leakage
rate was neglected. Based on the fractal theory, the contact of end faces of the rotary and stationary rings was simplified to be the
contact of a rough surface and an ideal rigid smooth surface, and the contact interface’s cavity size-distribution function as well as the
fractal characteristic of the cavity profile curve was discussed. By analyzing the influence of abrasion on the seal face topography and
the leakage channel, the time-correlation leakage prediction model of mechanical seals based on the fractal theory was established and
the method for predicting the leakage rate of mechanical seals with parallel plane was proposed. The values of the leakage rate predicted
theoretically are similar to the measured values of the leakage rate in the model test and in situ test. The experimental results indicate
that the leakage rate of mechanical seals is a transient value. The surface topography of the end faces of the seal rings and its change
during the frictional wear of mechanical seals can be accurately characterized by the fractal parameters. Under the work conditions of
changeless frictional mechanism, the fractal parameters measured or calculated based on the accelerated testing equation can be used to
predict the leakage rate of mechanical seal in service. The proposed research provides the basis for determining the leakage state and
predicting working life of mechanical seal.
Key words: mechanical seal, leakage rate, fractal theory, frictional wear, accelerated test model
1
Introduction
The most direct failure of mechanical seal is that the
leakage rate is bigger than the maximum allowable one in
the working life. However, the leakage rate and the service
life of mechanical seals are very difficult to be predicted
accurately, which may lead to the unnecessary replacement
of mechanical seals or the unacceptable leakage rate. The
leakage of mechanical seals often causes serious accidents,
such as fire hazard, explosion, intoxication, environment
contamination, and so on. Then it is not only necessary but
also urgent to investigate the leakage prediction methods of
mechanical seals.
Since the advent of mechanical seals in 1885, the
solutions to the leakage problem have continuously been
explored. Based on the assumptions that the fluid in the
clearance between mechanical seal end faces abided by the
* Corresponding author. E-mail: [email protected]
This project is supported by China Postdoctoral Science Foundation
(Grant No. 20070410323), Jiangsu Provincial Planned Projects for
Postdoctoral Research Funds of China (Grant No. 0701001C), and
Jiangsu Provincial Planned Projects for Fostering Talents of Six
Scientific Fields of China (Grant No. 07-D-027)
hydrokinetics law and the flow is in the stable laminar flow
regime, the leakage model of mechanical seals with the
ideal smooth parallel faces was established by HEINZE[1].
Because the film thickness in the clearance between
mechanical seal end faces is not constant and the shape of
the actual clearance is more different from that of the
assumed parallel clearance, there exit some deviation
between the test results and theoretical calculation values[2].
In the 1950s, the surface roughness of mechanical seals was
measured and researched by MAYER[2]. His research
results indicate that the surface topography of the end faces
of mechanical seals has especially significant influence on
the performance of mechanical seals. Then the leakage
models of mechanical seals under the boundary regime and
under the mixed friction regime were built. It was
considered by LEBECK[3] that the factors, such as the face
load, mismachining tolerance, caused the end face of the
soft ring to form the roughness or even the wave amplitude
that is larger than the roughness. And the leakage model of
mechanical seals with the end faces of the wave clearance
was built. In China, the radial leakage models of the fluid
in the clearance between mechanical seal faces were also
built by PENG, et al[4] and SUN, et al[5].
·8·
YSUN Jianjun, et al: Leakage Prediction Method for Contacting Mechanical Seals with Parallel FacesY
These models perfected the theory of mechanical seals to
some extent and played a very important role in
engineering practice. But most of the early researches were
based on the assumptions that the seal faces profile of the
seal rings and the frictional conditions were invariant. In
the early built models, the effect of the surface topography
change of the end faces on the leakage rate was neglected,
and only the initial roughness and waviness of the seal
faces were considered. The relationship between the
leakage rate and the working time was not taken into
consideration. Actually, both the seal faces profile of the
seal rings and the gap composed by the two seal faces were
changed continually during the life period of mechanical
seals. Therefore, a precise calculation of the leakage rate
using the present models is very difficult. Because the
roughness Ra expressed by the profile arithmetic average
deviation refers to that the mean arithmetical value of
profile set over absolute value within the sample length has
the scale correlativity, which cannot reflect the unstable
stochastic characteristic of the topography of seal faces of
the rotary and the stationary rings and is unable to describe
the leakage channel change between the contacting end
faces accurately. The fractal theory provides one new
scientific thinking mode and powerful mathematical means
for characterizing the surface topography of mechanical
seal faces and describing its change law[6].
In this paper, the fractal theory was introduced and the
change of the surface topography of seal rings as well as
the leakage channel with the fractal parameters was
investigated. A time-correlated leakage prediction model of
contacting parallel end face mechanical seal based on the
fractal parameters was established, and some experiments
were carried out to confirm the accuracy of the presented
model.
2
Fractal Characterization of Leakage
Channel
The machined surface and the abrasion surface formed
during the working process of mechanical seal are both
rough, but they have the fractal characteristic[7–9] and can
be simulated by Weierstrass-Mandelbrot(W-M) function
method[10]. There are contacts and cavities between the seal
faces of the rotary and stationary rings. In order to describe
the leakage channel between two seal faces by fractal
geometry, the real contact area and the cavity area of
contacting end faces must be found out firstly.
2.1 Size-distribution of contact spots
At present, the fractal geometry theory is a more mature
method for the study of rough surfaces contact. M-B fractal
model was proposed by MAJUMDAR, et al[11], based on
the M-B fractal function, namely, the simplification from
the contact of the rotary and stationary seal faces to the
contact of an equivalent rough surface and an ideal rigid
smooth surface, as shown in Fig. 1.
Fig. 1. M-B fractal model for the contact
of the rotary and stationary seal faces
Based on Ref. [11], and considered all micro contact
spots in the area (0, AL], the contact spots area distribution
function of the seal faces of the rotary and stationary rings
can be expressed by[12]
ì D (2－D ) / 2 D / 2 - ( D + 2) / 2
AL A
, 0 < A < AL ,
ï y
n( A) = í 2
ïî 0,　AL < A < +¥.
(1)
Where n(A) is the size-distribution of contact spots area, D
is the fractal dimension of a surface profile, y is the
modified coefficient, A is the area of the micro contact spot,
and AL is the area of the largest contact spot. The modified
coefficient y is related to Ar/AL, where Ar is the true contact
area.
2.2
True contact area and cavity area between seal
interfaces under axial load
The deformability of micro contact spots of the true
contact area has very tremendous influence on the bearing
capacity of contact surface. Based on Johnson’s findings[13],
the relationship between the load and the contact area of the
elastic contact spot of asperities, the elastic-plastic contact
spot as well as the plastic contact spot of mechanical seal
faces can be given as follows:
Fe ( A) =
Fep ( A) =
4 π
EG ( D -1) A(3- D ) / 2 ,
3
é
æ π G ( D -1) A(1- D ) 2
2A
Ej 0 ê 2 + ln çç
3
3j 0
êë
è
Fp ( A) = EK j 0 A.
(2)
öù
÷÷ ú ,
ø úû
(3)
(4)
Where Fe(A) is the load on the elastic contacting spot, Fep(A)
is the load on the elastic-plastic contacting spot, Fp(A) is
the load on the plastic contacting spot, E is the elastic
modulus; G is a characteristic length scale of the surface,
j0 is the material property constant and K is a ratio of the
hardness H to the yield strength sy of the soft material.
The equivalent elastic modulus of the contacting surfaces
holds[14]
CHINESE JOURNAL OF MECHANICAL ENGINEERING
æ 1 - n12 1 - n 22
E = 2 çç
+
E2
è E1
-1
ö
÷÷ ,
ø
where E1 and E2 are the elastic modulus of hard material
and soft material rings; n1 and n 2 are Poisson’s ratios of
hard material and soft material rings, respectively.
The relationship between the total load on the contact
area and the true contact area can be expressed by
Fg =
AL
òA
c
Fe ( A)n( A)dA +
Ape
ò0
Ac
òA
pe
Fep ( A)n( A)dA +
Fp ( A)n( A)dA.
(5)
Integrating Eq. (5), we can obtain the function of Ar:
*
Ar = Aa Ar* ( pg* , D, Ac* , Ape
, j 0 ),
where
Ar = Ar Aa ,
*
pg*
= Fg EAa ,
Ac*
= Ac Aa , and
Aa is the apparent contact area; pg* is the normalized end
face pressure, Ac* is the normalized critical area
demarcating the elastic and plastic regimes and Ape* is the
normalized critical area demarcating the plastic and
elastic-plastic regimes. Where Ac is the critical area
demarcating the elastic and plastic regimes which is given
by
2 /( D -1)
,
(7)
and Ape is the critical area demarcating the plastic and
elastic-plastic regimes which is given by
æ π
Ape = G 2 ç
ç 30j
0
è
ö
÷÷
ø
ì D (2 - D ) / 2 D / 2 -( D + 2) / 2
AsL As
, 0 < As < AsL ,
ï y
n( As ) = í 2
ïî0,
AsL < As < +¥ ,
(10)
æ π x ö ls
ls
zs ( x) = G D -1ls2 - D cos ç
÷, - < x < ,
2
è ls ø 2
(11)
As =
.
Cavity size-distribution function and profile
fractal curve for contact surface
When the seal faces of the rotary and stationary rings of
mechanical seals contact mutually, the higher micro convex
bodies in two rough faces have the actual contact, and
which withstand the axial closing force acting on the
mechanical seals. In fact, speaking of two surface profiles,
(12)
2.4 Influence of abrasion time on leakage channel
The size and shape of leakage channel change along with
the changing of seal face topography. There is the parallel
sliding frictional wear characteristic under the mechanical
seal operating. Thus, the changing of fractal dimension D
and the scale amplitude G of the seal face follows the rule
of parallel sliding frictional abrasion [15], namely,
(8)
(9)
D （2-D）/ 2
y
AsL .
2-D
Where As is a micro cavity area in contact face，AsL is the
biggest of As, zs is the height of cavity and As is the sum of
the micro cavity areas in contact face. According to the
M-B model, the hemline width of the micro cavity contour
curve in contact interface ls has the relation ls=As1/2.
2 /( D -1)
Eq. (6) indicates that Ar can be evaluated as a function of
Ar*, pg*, D, Ac*, Ape* and j0 .
Along with the changes of unit load on seal face, the true
contact area between the contact interfaces Ar will change.
If the apparent contact area is Aa, then the cavity area As can
be given by
As=Aa–Ar.
while the seal surfaces formed the micro convex body
contact spots, also formed the micro cavities between the
rotary and stationary rings, as shown in Fig. 1.
Regarding the identical rough surface, in all length scales,
there are the exclusive D and G. The micro cavity size
distribution function between rough contact interfaces n(As),
the micro cavity profile curve, and the total area of all
micro cavity of contact interface have the similar
expressing form to the micro convex body contact spot:
(6)
*
Ape
= Ape Aa . Ar* is the non-dimensional true contact area,
æ
ö
4
Ac = G 2 ç
ç 3 π Kj ÷÷
0 ø
è
·9·
D = D(t ),
(13)
G = G (t ).
(14)
After contact interface having been loaded and worn, the
profile of individual leakage channel can be expressed in
the following form:
æπ x ö
ls
ls
zs ( x, t ) = G D (t ) -1 (t )ls2 - D (t ) cos ç
÷, - < x < ,
2
2
è ls ø
where t is the operating time of mechanical seals.
Leakage channel model is shown in Fig. 2.
2.3
Fig. 2.
Leakage channel model
(15)
·10·
3
YSUN Jianjun, et al: Leakage Prediction Method for Contacting Mechanical Seals with Parallel FacesY
Dp * =
Fractal Model of Prediction Leakage
Regarding the contact mechanical seals with hard
material rings and the soft material rings, in order to build
the time-correlated leakage prediction model based on
fractal theory, the following basic suppositions are made.
The leakage of fluid through the clearance between
mechanical seal interfaces may be regarded as the stable
laminar flow of incompressible viscous fluid in the leakage
channel.
The contact interface of mechanical seal may be
regarded as the contact between a rough surface and a
smooth surface. The micro cavities have the different sizes
and are distributed stochastically on the contact interface.
The surface fractal characteristic is unity statistically. It
is not necessary to consider the mutual function between
the neighboring micro contact spots in the contacting
process, the strengthening function of elastic-plastic
contacts, and the changing of the hardness of materials
along with attrition depth.
In the working process, the unit load on seal face and the
frictional abrasion have no influence on the distribution of
micro cavities on the contact interface. The changes of fluid
coherency in seal clearance, the spin of fluid, and the
changes of curvature of seal rings can be neglected.
Based on Navier-Stokes equation, the volume flow
(leakage rate) q through a single leakage channel is
q = ò vr dzdx =
A
1 dp ls / 2
[zs (x, t )]3dx,
12h dr ò-ls 2
(16)
where vr is velocity of fluid-flowing along diameter
direction of seal ring, dp/dr is pressure gradient along
diameter direction of seal ring, and h is dynamic viscosity
of fluid flowing through contact interface.
By substituting Eq. (15) into Eq. (16), the volume
leakage rate in entire seal face on the action of end face
load Fg becomes
Q=
AsL
ò0
qn( As ) dAs =
p2 - p1
´
18ph (r2 - r1 )
D (t ) 2
D (t )y [2 - D (t )]/ 2 G 3[ D (t ) -1] AsL
AsL
ò0
As[5- 4 D (t )]/ 2 dAs =
(p2 - p1 )D(t )G 3[ D (t ) -1] [2 - D (t )]/ 2 [7 -3 D (t )] 2
y
AsL
,
9ph (r2 - r1 ) [7 - 4 D (t )]
( p2 - p1 ) Aa1/ 2
,
hv
G* =
B* =
G
,
Aa1/ 2
r2 - r1
Aa1 2
.
Where v is relativity sliding velocity for seal faces, Q* is
the normalized leakage rate, Dp* is the normalized pressure
difference, G* is the normalized scale amplitude and B* is
the normalized seal width. With the normalized variables,
Eq. (17) can be rewritten as
Q* =
Dp
*
3 D (t ) -5
[ D(t )] 2 [G* (t )]3[ D (t ) -1]
9πB* [7 - 4 D (t )]
æ 2 - D (t ) ö
çy 2 ÷
ç
÷
è
ø
{[2 - D(t )] éë1 - Ar* ( pg* , D, Ac* , Ape* ,j0 )ùû}
Q
,
Aa v
7 -3 D (t )
2
.
´
(18)
Eq. (18) is the time-correlated fractal geometry model of
leakage prediction for mechanical seal, which describes the
relationship among leakage rate Q*, normalized pressure
Dp*, fractal dimension D, normalized scale amplitude G*,
normalized seal width B* and non-dimensional true contact
area Ar*. Because there is frictional abrasion on the contact
interface of mechanical seal, the surface topography
parameter changes unceasingly, this causes its leakage rate
no longer a constant, but a non-stable value which changes
along with the working time.
4
Experimental Procedure and Data
Analysis
Two set of GY-70 mechanical seals were tested severally
under two sorts of different working conditions. The
leakage rate and seal face profile of soft material ring were
measured periodically. In order to validate the correctness
of time-correlation leakage rate prediction model based on
fractal theory, the measured values were compared with the
calculated values. Experimental apparatus is displayed in
Fig. 3.
(17)
where p1 and p2 are inside and outboard medium pressures.
r1 and r2 are inside radii and outer radii seal face,
respectively.
For ease in analysis, the variables in Eq. (17) are
normalized as follows:
Q* =
3 D ( t ) -5
2
Fig. 3.
Experimental apparatus
CHINESE JOURNAL OF MECHANICAL ENGINEERING
4.1 Testing conditions
The test samples are rotary and stationary rings of GY-70
mechanical seal, which are made of YG-8 hard alloy and
carbon-graphite, respectively, and their key parameters are
listed in Table 1. Stationary and rotary rings are shown in
Fig. 4. Two pairs of seal members were tested.
Table 1.
Parameters of test samples of mechanical seals
Rotary ring
Stationary ring
Material
Parameter
YG-8
Carbon-graphite
Inside diameter of seal face d1/m
0.062
0.068
Outer diameter of seal face d2/m
0.082
0.079
Balance coefficient b
0.83
—
600
14 500
Hardness H/MPa
Elastic ratio E/ GPa
Density r/(kg•m–3)
Poisson’s ratio n
Fractal dimension of seal face
Scale amplitude of seal face
G/nm
0. 24
D
—
—
30
20
1 783.5
·11·
After a period of time of operating, we take out the seal to
scour and dry, measure the seal face profile and weigh the
quality of the soft material ring for the mechanical seal.
Write down leakage rate, and calculate the fractal
parameters with having measured surface profile and wear
value with quality difference of the soft material ring.
Duplicate above steps.
By substituting the fractal parameters of seal face
topography and the wear value into Eq. (18), the theory
prediction leakage rate can be obtained. Fig. 5 shows the
relational curve between the measured leakage rate, the
calculated value and the working time of test sample 1#
under working condition A, and those of test sample 2#
under working condition B.
0. 29
1＃: 1.588 9
2＃: 1.587 1
1＃: 17.957
2＃: 12.646
Fig. 5. Relationship curve between the measured leakage rate
and working time under working conditions A and B
Fig. 4. Mechanical seals for test
The tests were conducted under the working conditions
of A or B, respectively. The sealed medium was water with
the density of 1 000 kg/m3. The water temperature was 302
K and its viscosity was 1.005 mPa • s. In the experimental
process, the pressure differences between inner and outer
radii of seal face Δp were 0.4 MPa and 0.5 MPa, and the
rotational speed remained at 3 kr/min. For each sample,
two increasing face pressures, namely, 0.5 MPa and 0.55
MPa, were exerted, and the surface topography and wear
rate were measured at each face pressure level.
4.2
Experimental procedure and analysis to test
results
Before the experiment, firstly survey the seal faces
profile, and the structural sizes, then put the mechanical
seal soft material ring into the drying oven to carry on
drying, and then use the analytical balance to carry on the
quality weighing, and calculate the seal face fractal
parameters and the material density. After that, put the
surveyed mechanical seal into the testing machine, then
adjust the operational parameters, and conduct the
experiment to measure leakage rate for the mechanical seal.
In the initial period of working, the calculated value and
the measured value of leakage rate of the mechanical seals
were very big. But along with the running time passing, the
seal faces were run in, the end face fractal dimension D
increased gradually. These caused the bearing surface area
to increase, and the cavity area in contact interfaces
changes to a small one, which led to a reduction of the
cavity cross-sectional area that formed the leakage channels,
and a very quick increasing of the resistance while fluid
flowed through the seal faces, thus leakage rate reduced
rapidly. When the surface became very smooth, namely
while D was bigger, along with D increasing, the
cross-sectional area of leakage channel became smaller,
which led to a slow increasing of the resistance to fluid
flow through the seal interface. In a stable attritional state
in a quite long period of time, the leakage rate remained
basically the same. The predicted value of leakage rate and
the measured value were basically consistent. The
measured value was smaller than the predicted result, but
this was in the identical magnitude.
It can be also seen from Fig. 5 that under the same end
face pressures, the leakage rate of test sample 1# is small
because of a small medium pressure acting on the sample
·12·
YSUN Jianjun, et al: Leakage Prediction Method for Contacting Mechanical Seals with Parallel FacesY
1# and the leakage rate of test sample 2# is big due to a big
medium pressure. When leakage rate stays slight, the
frictional heat between the seal faces is difficult to release
rapidly, which causes the mechanical seal faces abrasion to
speed up. From then on the leakage rate will increase
gradually. During the initial testing time, the leakage rate of
test sample 2# is bigger than that of test sample 1#, and the
curve of the leakage rate of test sample 1# is moving more
gently than that of test sample 2#. As far as the end faces
topography is concerned, the surface dimension of test
sample 1# changes more quickly than that of test sample 2#,
as shows in Fig. 6.
Table 1. The lubricating medium between the seal faces is
N46 lubrication oil.
Fig. 7 is the typical relationship between the fractal
dimension D and the working time t at four different levels
of unit load on seal face. At the early stage of the frictional
wear experiment, the end faces are rough. Hence the wear
amount is larger and D is smaller. With the frictional wear
going on, the end faces become smooth and D increases.
When t = 175–200 min, D reaches the maximum value, and
the end faces are the smoothest. Hereafter, D reduces
slowly, and the wear goes on steadily for a long time.
Finally, D reduces rapidly and the frictional wear
aggravates. When pg is large, the running-in time might be
shortened and D would reach the maximum value most
rapidly.
Fig. 6. Relationship between D and t
for test samples 1# and 2#
5 Engineering Application
Application of leakage prediction theory mainly includes
two aspects. One is to obtain leakage rate by surveying the
seal face topography used on the device. Another is,
according to the material characteristic, to seek for the
changing rule of the seal face topography, and to predict
the lifetime of mechanical seal based on the leakage rate. It
is infeasible to acquire the changing data of the seal face
topography at any point of time in the production field, for
the machine have to been stopped to unload the mechanical
seals now and then. In this paper, the changing of
mechanical seal face topography was researched by the
accelerated test mode[12]. And based on the research, the
leakage rate of the mechanical seals used on the diesel oil
pump of a petrochemical corporation was predicted.
5.1 Changing rule of seal face topography
In order to obtain surface topography fractal parameters
of mechanical seals having worked a certain time, the
relationship between the change of fractal parameters of
contact interfaces and the working time was investigated by
HDM-2 friction abrasion testing machine. The material and
the sizes of seal faces of the test samples are listed in
Fig. 7. Relationship between fractal dimension and
working time under different unit load on seal face
Fig. 8 illustrates the change of the scale amplitude G
with working time t. Because the unit load on seal face pg
has very small effect on the scale amplitude, only G-t curve
corresponding to pg=0.48 MPa is given.
Fig. 8.
Relationship between scale amplitude
and working time
Except for the running-in period, the G-t curve is similar
·13·
CHINESE JOURNAL OF MECHANICAL ENGINEERING
to the bathtub curve. At the early stage of frictional wear,
the wave crests on the surface of the hard material ring not
only are abraded but also scuff the surface of the soft
material ring. Thereupon, the surface of the soft material
ring becomes rough and G increases. Hereafter, the surface
of the soft material ring fits that of the hard material ring,
and they mate well each other gradually. The surface of the
soft material ring becomes more and more smooth and G
decreases. In the subsequent stage, G remains a constant for
a long time. But if G is too small, the contact surfaces
would be too smooth to form a necessary gap between
interfaces to contain enough lubricant. In this situation, the
abrasion would be aggravated reversely.
According to Ref. [12], there is
Kv =
WH d t Aa H
=
,
SFg
vtFg
(19)
where Kv is the coefficient of wearing, and dt is the wearing
capacity at the thickness direction of soft material ring.
The variation of Kv with working time t under different
unit loads on seal face is shown in Fig. 9.
Fig. 9. Relationship between coefficient of wear and
working time under different unit loads on seal face
It can be seen that the coefficient of wear increases with
increasing working time in the running-in stage, while it
decreases continually in the normal wear stage.
In the stable frictional wear stage, the Kv-t relational
Eq. (20) is gotten by regression
2
æ
ö
K v = ç 7.25 ´ 10-8t +
- bk ÷ ´ 10-6 ,
2.25t + 950
è
ø
(20)
when pg=0.48 MPa, bk=5×10–4; when pg=0.63 MPa,
bk=4.666 7×10–4; when pg=0.80 MPa, bk=4.333 3×10–4;
when pg=1.00 MPa, bk=3.666 7×10–4.
5.2 Accelerated test
It can be found from Fig. 7 and Fig. 8 that the seal face
topography has some certain changing rules in the process
of accelerated test. And Fig. 9 is the change rule of
coefficient of wear along with the time. The working time
under the real wok condition, which is corresponding to the
testing time, was obtained based on the similarity theory.
Ref. [12] has given the frictional abrasion accelerated test
Eq. (21) of the end faces of mechanical seals in which the
soft material ring is made of graphite:
t2 æ n 2 ö
=ç ÷
t1 è n1 ø
2.84
æ Eu 2 ö
ç
÷
è Eu1 ø
-1.496 4
æ Gˆ 2 ö
çç
÷
ˆ ÷
è G1 ø
-3.243
*
d t2 pg1 Ar1
.
d t1 pg2 Ar2*
(21)
Where Eu is Euler number and Ĝ is duty parameter.
Subscripts 1 and 2 refer to two sorts of working conditions
of mechanical seal.
The surface topography parameter of mechanical seals
after a period of operating time under different wok
conditions by using this equation could be obtained. At the
same time, the corresponding operating time to some
certain topography parameters of mechanical seals could be
obtained.
Under the condition of invariable friction mechanics,
according to the frictional abrasion accelerated test
equation of the end faces of mechanical seals, the testing
time could be shortened by the means of increasing the unit
load on end face, changing the testing media and rotational
speed of rotary ring, and so forth.
5.3 Applied conditions
The prerequisite to using the data in Fig. 7 and Fig. 8 is
that the friction regime in the operating process of
mechanical seals and in the process of accelerated test
should be the same one.
The conditions of accelerated test are as follows: testing
medium is N46 hydraulic oil, and its density r=872 kg/m3.
Medium pressure difference between inner and outer radii
of seal face Dp=80 Pa. The temperature of testing medium
qs=335 K and the dynamical viscosity of testing medium
h=17.11 mPa•s, which can be calculated by Refs. [16–17].
Working rotational speed n=3 kr/min. The unit loads on end
face are 0.47, 0.63, 0.71, 0.80 and 1.0 MPa, respectively.
The test of the single factor 5 level trial was carried out.
Using the following formula:
hn (r2 - r1 )
Gˆ =
,
Fg
(22)
Ĝ of the wok condition parameters of the 5 level unit load
on end face obtained are from 0.913´10–6 to 4.291´10–7,
respectively. From Ref. [18], in the process of accelerated
test, the friction regime of mechanical seals is the mixing
friction.
Work conditions are as follows: unit load on seal face
pg=1.1 MPa; working rotational speed n=3 kr/min; medium
pressure differential between inner and outer radii of seal
·14·
YSUN Jianjun, et al: Leakage Prediction Method for Contacting Mechanical Seals with Parallel FacesY
0.24
pressure Dp=0.8 MPa, by substituting the allowable leakage
rate into Eqs. (6) and (18), the end face fractal dimension
D=1.59. Using Eq. (6), Ar*=0.263 0. The wearing capacity
dt=2.98 mm, which could be obtained by Eq. (21). And the
coefficient of wear was got from Eq. (20). As seal face unit
load pg=1.1 MPa, bk=0.000 21. By substituting D=1.59 and
Ar*=0.263 0 into Eqs. (19)–(21), the theoretical lifetime of
mechanical seals t=10 100 h under the condition of
maximum allowable leakage rate.
For type 108 mechanical seal in diesel oil pump, their
predicted lifetime t=10 100 h, and their real service time is
11 500 h, which means that this mechanical seal has
over-served 1 400 h. There are invisible incipient faults
because the leaked diesel oil vapor will diffuse
continuously around the pump, and the density of the vapor
would increase and even reach the explosive limit range.
Therefore, it is of great necessity to predict the leakage rate
properly.
0.29
6 Conclusions
faces Dp=0.8 MPa. Testing medium is diesel oil, whose
density r=850 kg/m3 and viscosity h=3.145 mPa•s. The
key parameters of 108 mechanical seals used in diesel oil
pump are listed in Table 2.
Table 2.
Key parameters of 108 mechanical
seals in diesel oil pump
Parameter
Rotary ring (hard material ring)
Material
Mechanical
parameter
Value
YG-8
Carbon
graphite
30
Stationary ring (soft material ring)
Hardness of soft material ring H/MPa
Density of soft material ring
r2/ (kg • m–3)
Elastic ratio of hard material ring
E1/ GPa
Elastic ratio of soft material ring
E2/ GPa
Poisson’s ratio of the material of hard
material ring n1
1 780
600
20
Poisson’s ratio of the material of soft
material ring n 2
Structural
parameter
Yield strength of the material of soft
material ring sy/MPa
Seal face size d2/mm, d1/mm
Coefficient of balance b
Initial fractal dimension of soft
material ring D
Initial scale amplitude of soft
material ring G/nm
50
82.54, 72.80
1.2
1.5607
2.03
By calculating, Ĝ=0.715´10–7 (in the sector 5´10–8<Ĝ <
1´10–6). It indicates that the friction regime of mechanical
seals is the mixing friction in practical working. Therefore,
the data got from the accelerated test can be used in the
prediction of leakage rate and lifetime evaluation of type
108 mechanical seals in diesel oil pump.
5.4 Engineering application
The fractal parameters of the seal face of type 108
mechanical seals are displayed in Table 2. According to
Eqs. (5) and (6), when end face unit load pg=1.1 MPa, Ar*=
0.260 75. By Eq. (18), the theoretical leakage rate of type
108 mechanical seal can be calculated.
According to the fugitive escape amount control criterion
of Society of Tribologists and Lubrication Engineers[19],
and American Petroleum Institute[20], the allowable leakage
rate of the mechanical seals working in diesel oil medium is
0.706 cm3/h. It can clearly be seen that type 108
mechanical seals are failure:
3D (t ) -5
Dp* [ D (t )] 2 [G* (t )]3[ D ( t ) -1]
Q=
9π B* [7 - 4 D (t )]
æ 2- D (t ) ö
çy 2 ÷
ç
÷
è
ø
{[2 - D(t )] éë1 - Ar* ( pg* , D, Ac* , Ape* ,j0 )ùû}
7 - 3 D (t )
2
3 D (t ) - 5
2
AaV ´
= 2.95 cm 3 / h.
As unit load on seal face pg=1.1 MPa, and medium
(1) To predict the leakage rate for mechanical seals has
the important influence on the safety operation of
machinery. The research results show that the relationship
between the fractal dimension and the work time is similar
to a reverse bathtub curve. The single leakage channel in
the end faces of mechanical seals can be described as a
cosine function. The leakage rate is related to D, G*, pg*,
Dp* and the material property, and it is a transient state
value. The investigated result of mechanical seal, which
was obtained in analog device or diesel oil pump, show that
the leakage prediction model of mechanical seals based on
fractal theory is correct.
(2) For the given real work time, the work time under the
simulated condition can be calculated by the frictional
abrasion accelerated test equation. From the accelerated test
curve D-t, the corresponding D and G under the end face
unit load pg can be obtained. Then the leakage rate can be
predicted using the leakage rate predicted model of
mechanical seals based on the fractal theory. Also for the
given Q, the corresponding working time of mechanical
seals under the real work condition can be evaluated by the
leakage rate formula, the accelerated test curve, the formula
which described the relationship of Kv-t, and frictional
abrasion accelerated test equation.
(3) The leakage prediction fractal model of mechanical
seals is proposed on the assumptions of the contact between
the rigid ideal smooth plane and the rough surface, and the
small length of flowing channels. In order to consider the
influence of taper and radial waviness simultaneously, it is
necessary to establish the three dimensional rough surfaces
contact model. The quantitative relationship between the
fractal dimension of mechanical seal face and the working
time should be established on basis of the statistical data,
and therefore, the massive experimental researches still
have to be done.
CHINESE JOURNAL OF MECHANICAL ENGINEERING
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Biographical notes
SUN Jianjun, born in 1965, is currently a postdoctor in Southeast
University, China, and a professor in Nanjing Forestry University,
China. He received his PhD degree from Nanjing University of
Technology, China, in 2006. His research interests include fluid
sealing technology, chemical engineering machinery, mechanical
layout and design theory.
Tel: +86-25-83427795; E-mail: [email protected]
WEI Long, born in 1972, is currently a PhD candidate in Nanjing
University of Technology, China, and an associate professor in
Nanjing College of Chemical Technology, China. His research
interests include fluid sealing technology, chemical engineering
machinery, mechanical layout and design theory.
Tel: +86-25-58371314; E-mail: [email protected]
FENG Xiu, born in 1975, is currently a PhD and works in Nanjing
College of Chemical Technology, China. His research interests
include fluid sealing technology, chemical engineering machinery,
mechanical layout and design theory.
E-mail: [email protected]
GU Boqin, born in 1957, is currently a professor and a PhD
candidate supervisor in Nanjing University of Technology, China.
His main research interests include fluid sealing technology,
chemical engineering machinery, and sealing materials.
E-mail: [email protected]