Leakage Prediction Method for Contacting Mechanical Seals with Parallel Faces

SUN Jianjun , WEI Long, FENG Xiu, and GU Boqin

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

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 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].

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 Raexpressed 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]

Where n(A) is the size-distribution of contact spots area, D is the fractal dimension of a surface profile, ψ is the modified coefficient, A is the area of the micro contact spot,and ALis the area of the largest contact spot. The modified coefficient ψ is related to Ar/AL, where Aris 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:

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,?0is the material property constant and K is a ratio of the hardness H to the yield strength σyof the soft material.

The equivalent elastic modulus of the contacting surfaces holds[14]

where E1and E2are the elastic modulus of hard material and soft material rings;1ν and2ν 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

Integrating Eq. (5), we can obtain the function of Ar:

and Apeis the critical area demarcating the plastic and elastic-plastic regimes which is given by

Eq. (6) indicates that Arcan be evaluated as a function ofand

Along with the changes of unit load on seal face, the true contact area between the contact interfaces Arwill change.If the apparent contact area is Aa, then the cavity area Ascan be given by

2.3 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,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:

Where Asis a micro cavity area in contact face，AsLis the biggest of As, zsis the height of cavity and Asis 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 lshas the relation ls=As1/2.

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,

After contact interface having been loaded and worn, the profile of individual leakage channel can be expressed in the following form:

where t is the operating time of mechanical seals.

Leakage channel model is shown in Fig. 2.

Fig. 2. Leakage channel model

3 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

where vris velocity of fluid-flowing along diameter direction of seal ring, dp/dr is pressure gradient along diameter direction of seal ring, and η 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 Fgbecomes

where p1and p2are inside and outboard medium pressures.r1and r2are inside radii and outer radii seal face,respectively.

For ease in analysis, the variables in Eq. (17) are normalized as follows:

Where v is relativity sliding velocity for seal faces, Q*is the normalized leakage rate, Δp*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

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 Δp*, 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.

Fig. 3. Experimental apparatus

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

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 ?p 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.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.

Fig. 5. Relationship curve between the measured leakage rate and working time under working conditions A and B

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 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.

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 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 pgis large, the running-in time might be shortened and D would reach the maximum value most rapidly.

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 pghas 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 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

where Kvis the coefficient of wearing, and δtis the wearing capacity at the thickness direction of soft material ring.

The variation of Kvwith 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

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:

Where Euis Euler number and ? 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 ρ=872 kg/m3.Medium pressure difference between inner and outer radii of seal face Δp=80 Pa. The temperature of testing medium θs=335 K and the dynamical viscosity of testing medium η=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:

? of the wok condition parameters of the 5 level unit load on end face obtained are from 0.913×10–6to 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 faces Δp=0.8 MPa. Testing medium is diesel oil, whose density ρ=850 kg/m3and viscosity η=3.145 mPa·s. The key parameters of 108 mechanical seals used in diesel oil pump are listed in Table 2.

By calculating, ?=0.715×10–7(in the sector 5×10–8＜? ＜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:

As unit load on seal face pg=1.1 MPa, and medium pressure Δp=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 δt=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.

6 Conclusions

(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*,Δp*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 pgcan 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.

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