Ind. Eng. Chem. Prod. Res. Dev. lQ85, 2 4 , 283-289
Table 111. Supported Catalyst Parameters: 1.5-Order Model (Eq 7) figure half-life supplying rate const. parameter T ~ ! ~ b, h-l min k., cm/s cat. data 0.946 26.3 Montedison 7 0.0150 2 0.0153 0.70 35.5 PureChem Series 700 Improved Montedison
3
0.0332
0.269
92.3
come from a 1-L semibatch run in propylene liquid pool with 675 mL (STP)of hydrogen added (Jacobson, 1982). The first two listed catalysts are quite similar when analyzed by the 1.5-order kinetic model. There is no significant difference in their rate constants. The somewhat newer PureChem catalyst has a longer half-life. If the older Montedison catalyst were used in a liquid pool run, the calculated yield would be only 12 500 instead of 16000 in 5 h reported for PureChem. With an 18% shorter half-life, this catalyst would achieve 22% lower yield at 5 h. Obviously, catalyst half-life is a very important consideration. On the other hand, the improved Montedison catalyst has higher values for both the rate constant and the half-life, so it achieves considerablyhigher yields than the other two in Table 111. (It also has the advantage of higher Isotactic Index, but that comparison is outside the scope of this study.) Note that if the Series 700 catalyst half-life parameter b were adjusted to the 8 atm pressure hexane slurry condition, it would be 0.32 h-l, or close to that determined for improved Montedison catalyst.
Conclusions A kinetic model based on 1.5-order catalyst decay for propylene polymerization with a supported T i c 4 catalyst has been compared at the same temperature with two other models from the literature. At reaction times from zero to 2 h, a first-order decay model provides the most accurate correlation. The 1.5-order model has greater accuracy than the first-order model in correlating yield profiles at longer reaction times (over 2 h). A 1.5-model has about the same accuracy as a second-order model at long reaction times and somewhat less accuracy at short times. The 1.5-order model provides a simpler, more direct
,
283
data analysis than either of the other models. Examination of the rate constant and half-life as a function of temperature demonstrates that, when the rate equation is based on 1.5-order catalyst decay, the activation is close t~ the value of 14500 determined energy for k,, Ek, by Natta and Pasquon (1959) for unsupported TiC1,. We conclude from this that the 1.5-order kinetic model may be successful in separating the intrinsic activity of a stereoregular site from the deactivation mechanism. This observation is consistent with the proposition that deactivation proceeds by decreasing the number of sites. The 1.5-order or Amoco model is versatile in correlating both hydrocarbon slurry runs and liquid pool runs, at quite different operating conditions. From the reaction rate, estimates of heat and mass transfer at operating conditions are readily calculated. Acknowledgment P. C. Barbe is acknowledged for providing helpful editorial suggestions. C. H. Lin is acknowledged for providing insight into classical deactivation kinetics. The authors thank Amoco Chemicals for permission to publish. Registry No. Propylene, 115-07-1.
Literature Cited Barbe, P. C. Himont, Inc., Ferrara, Italy, personal communication, April 14, 1983. Barbe, P. C.; Noristi, L. 1984 AIChE Natlonal Meeting, Anaheim, CA; Paper No. Ed, May 21, 1984. Brockmeler. N. F.; Rogan, J. B. AIChE Symp. Ser. No. 160 1978, 72, 28. DiDrusco. G.; Luclani, L. J . Appl. fo/ym. Sci. 1981, 36, 95. Dol, Y.; Murata, M.; Yano, K.; Keil, T. Ind. Eng. Chem. Prod. Res. D e v . lg82, 27, 580. Galll, P. Proceedings IUPAC Macro '82, 28th Macromol. symposium, Amherst, MA, July 12-16, 1982. Galll, P.; Luciani, L.; Cecchln, 0. Angew. Macro. Chem. 1981, 9 4 , 63. Jacobson, F. I . FToceedlngs IUPAC Macro '82, 28th Macromol. Symposium, Amherst, MA, July 12-16, 1982. Lucianl, L.; Barbe, P. C.; Slmonazzi, T. Symposium on New Processes and Technoiody of the Overseas Chemical Industries, Centennial Meeting of the American Chemical Soclety, New York, April 3-9, 1978. Natta, G.; Pasquon, I. "Advances In Catalysis"; Academlc Press: New York, 1959; Vol. 11, pp 21-23. Suzuki, E.; Tamura, M.; Doi, Y.; Keii, T. Macromol. Chem. 1979, 180, 2235. Wu, J. C.; Kuo, C. I.; Chien, J. C. W. Proceedings IUPAC Macro '82, 28th Macromoi. Symposium, Amherst, MA, July 12-16, 1982.
Received for review May 14, 1984 Revised manuscript receiued October 18, 1984 Accepted February 4, 1985
Hydrodynamics of Rubber Seals for Reciprocating Motion Artur Karasrklewlcz Technical University of W a r s w, Warsaw, Poland
In this paper the elements of the inverse theory of hydrodynamic lubrication of seals and the formulas determining out-, in-, and net-leakage of the rubber seals for the reciprocating motion are presented. The reasons of divergencies between theoretical and experimental leakage have been analyzed. Based upon our analysis and upon the results of our research, a modified formula for determining leakage of rubber seals has been proposed.
Introduction One of the more significant theoretical achievements of sealing technology is a so-called theory of hydrodynamic lubrication of flexible seals which was proposed by Blok (1963). The merit of that theory is a possibility to explain the problems of leakage, lubrication, friction, and formu0196-4321/85/1224-0283$01.50/0
lating practical recommendationsas far as both design and exploitation are concerned. However, it could be asserted that only a few experimental research programs have verified this theory. This paper presents a continuation of the research described in the author's work (Karaszkiewicz,1982) where, 0 1985 American Chemical Society
284
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985
where p' is the gradient of pressure at the point of inflection of the pressure distribution curve p(x) which is different at the out-stroke (p'J and the in-stroke (p'J of shaft (see Figure 1). Equation 3 and the linear distribution of flow velocity in the section of height ho give the formula defining the leakage q for a single stroke of length L (in the period of duration of a single stroke)
P a '
v=O
where D is the diameter of the shaft. From eq 4 when p ' = pr1(see Figure l),we have q l ; let us name it an outleakage-the leakage during out-stroke of the shaft, and when p' = p $ we have similarly an in-leakage during the in-stroke. The difference between the out-leakage q1 and the in-leakage q2 results in the net leakage qN for a double stroke
; h,=O
Figure 1. The hydrodynamic mechanism of the seal leakage: (a,b) pressure distribution in slot; (c) contact pressure distribution; (d) contact pressure distribution and pressure distribution at entrance of slot.
adopting the results of many years of investigations of the author and his co-workers on leakage and on distribution of contact pressure, it was attempted to estimate to what extent the leakage formula, which results from hydrodynamics of seals, can be applied.
Hydrodynamics of Flexible Seals The mechanism of lubrication and leakage of seals for reciprocating motion results from the theoretical work of Blok. Muller's work (1965) can be considered to be the first experiments confirming the above-mentionedtheory. Blok adapted to the characteristics of the flexible seals a so-called inverse theory of hydrodynamic lubrication introduced by Grubin in the 1940's. The inverse theory of hydrodynamic lubrication of seals-hydrodynamics of seals-is based on the Reynolds equation, which in the case of a bearing reads
where dpldx is the gradient of pressure in the slot, 7 is the dynamic viscosity, u is the sliding velocity, h is the height of the slot, and ho is the height of the slot where dp/dx = 0. The classical problem of lubrication is to determine the distribution of pressure which will permit one to calculate a load on a bearing while the profile of the slot is given. Concerning seals (see Figure l),an inverse problem is brought in question; i.e., the distribution of pressure in the slot between a seal and an element being sealed is given. Instead, the profile of the slot is searched, and particularly a section of the height hoin which the linear distribution of flow velocity appears and where obviously dpldx = 0. At such assumptions a differential equation (1)becomes an algebraic one, since dpldx is replaced by the known function p ' ( x )
The solution of eq 2 obtained by Blok determines the height ho of the slot (film thickness) h o = ( i ; ) 0.5 (3)
If q2 > q1 we have a case of a so-called negative net leakage, which as it can be assumed, was recorded first by Denny (1958). In order to apply the eq 3,4, and 5, the values of P ' ~and pr2should be known. Values of gradients ptl and p i are practically estimated on the basis of the distribution of the contact pressure u(x) (see Figure IC). In accordance with Blok's thesis, the distribution of pressure except "entry" zone and "exit" zone, equal to the distribution of contact pressure as the lifting (deformation) of a rubber seal resulting from a hydrodynamic reaction, amounts maximally to a few micrometers for the regime of hydraulic power. A practically unknown difference between the courses of the pressure and the contact pressure distribution curves in the above-mentioned "entry" and "exit" zones creates an essential problem. The hydrodynamic reaction resulting from the profile of the slot causes the curve of distribution of pressure to have a specific point of inflection which is situated in the point of intersection in the contact pressure distribution curve (see Figure Id). Differences of distributions as discussed above quite often appear to be the reason for a misinterpretation of Blok's thesis on the correspondence of the distributions. The fact that the problem of determination of the value of gradient p' has not been solved so far intensifies this misinterpretation. Equation 5 has been verified by comparing the experimental leakage qN to the calculated leakage of O-seals in the author's work (1978) and Wyszynsky's work (1980). Maximum values uIl and d2as estimated on the basis of the experimental contact pressure distributions have been introduced as the gradients p '1 and p r Zto eq 5. As it results from these comparisons, the calculated leakage qN has differed from the experimental one for some seal regimes. Considerable differences amounting even up to 200-300% have occurred, particularly at the low value of the parameter vu. Reasons of Divergencies between Theoretical and Experimental Leakage Object of Research. A simplified assumption that the gradient p' of the pressure distribution at the point of inflection of the curve p ( x ) equals the maximum gradient d of contact pressure distribution a(x) can supposedly be treated as the first reason of the above mentioned divergencies. The following theses have resulted from the analysis of the nature of the contact pressure distribution
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985 285
curves, particularly of 0-seals and from the effect of the seal profile on the pressure distribution: (a) p < is unrelated to the sealed pressure p s and smaller than dl;(b) p i increases with the pressure pa approaching d2for higher pressure pa; (c) p’l and p’2 increase while the value of parameter qv decreases. Getting beyond the unknown range of applicability of hydrodynamic theory can be considered the second reason of the above divergencies. Taking into consideration that eq 3 is based upon classic hydrodynamics, one can expect that below some value of qvlp’, this formula ceases to correspond with experiment. In such a case the thickness ho of film resulting from eq 3 is equal to 1-2 pm, which is comparable to the height of roughness of surface of the sealed element (R, = 0.16-0.63 pm). Based on the results of our research on leakage of 0-seals (1982) the following formula has been obtained, which satisfactorily proves conformity with experiment (*lo%)
(z)]
A
testedseals
hand pump-instrument f o r leakage measuring
a) ~
,
d5.7
1-
+9N
I
f +% b)
lRHD85
\ R H O 70 Ci
(6)
Figure 2. The scheme of the stand for measuring the leakage (a); (b) housing with additional closing chambers; (c) tested seals.
Equation 6 is a product of eq 5 and of the dimensionless correction factor
attempt to estimate the effect of the parameter qv on the gradient p’. The second object of the research has been to establish the precise conditions under which the negative leakage takes place. The author is of the opinion that this essential property of seals has not been sufficiently examined so far. Research Stand. Conditions of Research. Figure 2 shows the stand used for measuring the leakage. The shaft which simulates a piston rod or a plunger is sealed by means of two identical seals which are mounted in the pressure chamber. The shaft is brought in motion by a hydraulic cylinder which permits variation of the velocity. The oil temperature and the temperature of the surface of the shaft are maintained by circulating the liquid through a thermostat. The pressure of oil in the chamber between the seals is obtained with a special pump which makes it possible to measure the leakage as the loss or rise of the volume of the oil in the chamber. The leakage q N of a single seal for the double stroke of the shaft (diameter 40 mm, stroke length L = 500 mm, R, = 0.22 pm) has been calculated based on the loss or the rise of the oil volume during at least several dozen double strokes, at the condition of stable pressure, velocity (out- and in-stroke), and temperature. The measuring of the out-leakage q1 is worked out similarly to the measuring of qN, but with the simultaneous removing of the oil from the surface of the shaft during its out-stroke, then qz = 0; consequently, q1 = q N in accordance with eq 5. The addition of a closing chamber filled with the oil under atmospheric pressure to the housing of seals (see Figure 2b) creates the conditions when q2 > q l , i.e., the occurrence of the negative net leakage qN. Referring to the hydraulic power, the part of a closing chamber is performed by the space between seals and wipe-seals in the hydraulic cylinder. In the research, the leakage for U- and 0-seals shown in the Figure 2c has been measured. In order to obtain a wide range of qv, hydraulic oil of high viscosity has been used. The effect of the pressure on the dynamic viscosity q (Pa s) is defined by 1= qt exp(0.027~~) (8) where qt is the dynamic viscosity q at the given tempera-
where fl is a constant which depends on the dimensions of seals and rubber hardness, P a mm0.5. The values of gradients and P ’ ~for eq 6 have been so selected to obtain conformity to the experiment1 data for the value of the parameter qv exceeding 20-30 Pa mm. According to the idea of the author’s paper (1982), the factor a takes into account a variation of leakage resulting from the change of seal regime, Le., passing from hydrodynamic lubrication to a boundary or a mixed type lubrication. At the values of qv > 5-15 Pa mm, a = 1;below these values the factor a is considerably decreased, while the film thickness ho I 1-2 pm, which is comparable to the height of surface roughness of a sealed shaft (R, = 0.22 pm). As the factor a is an average of the in-stroke and the out-stroke, it is imperfect. Undoubtfully there exist different factors for these strokes, alfor the out-stroke and azfor the in-stroke. The factor a should be treated as the first step in making the leakage formula more precise, or broadly speaking, in explaining the phenomena of leakage and lubrication of seals. Referring to the thesis on the relation p’(qv),it can be considered that the factor a is not only a function of transition from hydrodynamic lubrication to boundary lubrication, but it also includes a variation of gradients p ’ depending on the parameter qv. It is obviously impossible to separate these two functions at the present stage of knowledge, but we can try to estimate the effect of the parameter q v on the variation of pr1and p t Z ,if we know the functions of out-leakage ql(qv) from the experiments. Gradient p’can be calculated from the transformed eq 4 P‘= 8 TDL 4-2 (7)
$&)
Using eq 7, one should realize the possibility of increasing faults caused by the fact that q is raised to the second power. Consequently, the first object of the research work presented in this paper has been the above-mentioned
286
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985
ture and atmospheric pressure, and ps is the system pressure, MPa.
Results and Discussion The relations between the net leakage q N and the sealed pressure ps for U-seals at the velocity u up to 200 mm/s and temperature t = 20,50, and 80 "C are shown in Figure 3. The net leakage, which has been measured without closing the chamber (see Figure 2a) in the conditions when in-leakage q2 is smaller than or at least equal to the outleakage q1 (q2I ql), has been designated as qNa. As it could have been assumed, the net leakage qNa is higher than the positive net leakage q N The balance qNa = q N taking place at high pressure depends on the velocity, i.e., the higher pressure and the higher velocity of reaching of the balance. A considerable value of the negative net leakage q N at low pressure, which should be acknowledged as a positive property of the seals, results from the substantial inleakage q2 and from the low level of out-leakage. As the course of the curve qN(ps) indicates, the net leakage q N = 0 depends on the velocity and on the pressure, i.e., the lower the velocity, the lower the pressure. The dependence disappears at the temperature 80 "C (see Figure 3c) when the viscosity is low. According to eq 5 , when u1 = u2, leakage q N = 0 occurs at the equal gradients P ' ~= p;. The dependence q N = O(p,,u) proves that the equality P ' ~= pr2also depends on the velocity and confirms the submitted thesis about the dependence between p'and u. The courses of the curves qN(ps) show that, at the pressure p s = 0, q N will represent a definite negative value as opposed to q N a which at ps = 0 equals zero. It is a result of the high value of the inleakage q2. Figures 4a, and b (Figure 4b is an enlargement of a part of Figure 4a) present the relations between net leakage q N and parameter q v for U-seals under the conditions of different pressures ps = 1-15 MPa. The net leakage q N a (when yz < ql) is growing along with the increase of qu in accordance with the known rule resulting from the hydrodynamics of seals supported by the experimental data. Relations shown in Figure 4 indicate that this rule is to some extent limited. Positive net leakage q N for higher pressure (exceeding 5 MPa) is diminishing while qu is increasing, but a t some range of qu only. It is not contradictory to the rules resulting from eq 5 , but as can be supposed, it is caused by the effect of vu on gradient p'. Relations q N ( p e ) for low pressures up to 5 MPa have a linear nature over qu = 10 Pa mm. Figures 5a, b, and c present the relations between net leakage q N , qNa, out-leakage ql, and the sealed pressure for the 0-seal at velocities up to 200 mm/s and at temperatures t = 20,50,80 "C. In conformity with eq 5, in-leakage q2 = q1 - q N is presented in Figures 5a, and b as segments of the ordinate between the curves of leakage q l ( p s )and qN(ps) which depend on the pressure ps. At high pressures above 10 MPa we obtain equalization of the leakage q1 = q N = q N a and in-leakage q2 = 0, resulting from gradient p i -*m or factor u2 0. The course of the curves qN(ps) indicate that at the pressure ps = 0, the net leakage q N will equal zero, caused by the equation P ' ~= p i , and is a consequence of the symmetry of contact pressure distribution and the symmetry of profile of the 0-seal, both of which occur a t the pressure pB= 0 only. The author is not acquainted with any references on the experimentally verified negative net leakage of 0-seals. This fact is probably due to several reasons. On the basis of analysis that has been carried out until now on the contact pressure distribution of 0-seals, it is concluded that p \ > P ' ~when p s > 0, which according to eq 5 eliminates
+
50
1
3
mm
0
-50
-
9h
a
-250
w
?, "3
0
56
!dl
b
I
tlO
C
-+
Figure 3. The relations between the net leakage qN and the sealed pressure p s for U-seal: (a) t = 20 "C, qzo = 0.470 Pa s; (b) t = 50 "C, qso = 0.080 Pa s; (c) t = 80 "C; qm = 0.024 Pa s.
the existence of any negative net leakage qN. However in this analysis, the effect of parameter qu on p'has not been taken into consideration, as well as the difference of profiles of contact pressure and pressure distribution a t the entrance of a slot between the seal and the shaft surface. The next reason is the fact that the range of pressure where
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 2, 1985 287 + 20
