I
G. P. KOTELEWSKY 801 Court St., Honesdale, Pa.
Gasket Reactions and Sealing Pressures in Vessels What maximum variable pressure may be sealed by a gasket in a high pressure vessel? The answer is found by calculation developed in the following analysis
Tto find o the this problem it is necessary gasket reaction in a presSOLVE
sure condition. However, it is hard to calculate the gasket reaction in a bolted connection, because applied pressure changes the relation between the gasket load and bolt load. Three kinds of gasket reaction may occur-elastic reaction when the gasket is not overstressed by the bolt and its mass stays within elastic limits; plastic reaction when it is overstressed; and springy reaction when it is fully enclosed and resists compression like a confined liquid
where A , is the cross-sectional area of a gasket. This area must be distinguished from the surface of the gasket, which actually touches the flange and is less than A,. Accordingly, Y, is the stress in mass of gasket, which must be not beyond the yield point of gasket material. After internal pressure P is applied, the bolt load is given by a new expression :
W, = AgYP
(5).
Plastic reaction is unsatisfactory for sealing where operating conditions are varied. Releasing internal pressure in the vessel increases crushing of the gasket, so that when pressure is applied again, leakage occurs. In this analysis, it is assumed that the surface of the gasket has yielded into the imperlections of flanged faces which now are usually serrated (Z), but that thickness (mass) of the gasket remains undeformedi.e., stressed within elastic limits only. Generally, it is also assumed that elastic or springy gasket reaction and bolt load vary with change of the vessel’s internal pressure. T h e bolt load at atmospheric conditions is equal to the gasket reaction:
+ AhP
internal pressure is applied is equal to the change of bolt length plus change in the deflection of both flanges: Tp
-
T,
=
AT
=
AL
4-2Af
(4)
where T , is gasket seating thickness in the atmospheric condition and T, is gasket thickness in the pressure condition. Using the expression for modulus elasticity and taking into account that flange deflection is proportioned to the force, Af = CAW, one gets:
(2)
where
the pressure gasket reaction, is unknown. It is assumed that the nuts are in the same location at atmospheric and applied pressures (no additional tightening) With this assumption, the change of gasket mass stress, caused by the pressure action only, is
AJ,,
I
AY = Y a - Y1,
(3)
The internal pressure reduces the gasket stress, but this change depends, also, on other elements of bolted joint. T h e factors that affect AY are considered below. A change of gasket thickness after
where C is a deflection factor of the flange. According to Whal and Lobo (7) this factor may be expressed as:
c = -MR2 +Eft3
0.239 In a tG
(6)
Equation 6 includes both bending and shear deflections of a circular plate with a central hole, where the inner edge is fixed (built in) and the outer edge is loaded along the edge. VOL. 51, NO. 8
AUGUST 1959
949
4 0 ::
0.9
0.09
0.8
0.08
0.6
006
05
005
r
tially constant. The general expression for the change of bolt load is obtained by combining Equations 7 and 8.
& w
0 W
W
0.4
0.04
s
Figure 1. Stress 2nd b e n d i n g :oefficients are Functions of cir:ular plate ratio, R /r
-R
r
I n Equation 6, coefficient M is a function of ratio a = R/r; this function is shown in Figure 1, which shows also the stress coefficient, K , as a function of the same ratio, a , Curves M and K in Figure 1 are plotted on the basis of Whal and Lobo test data (7, 8 ) . T h e general expression for stress in a plate is S = KW/P where W is force and t is thickness of the plate. This expression, in connection with the plotted curve for K in Figure 1, might be used for a simplified flange stress calculation. However, comparing K with its counterpart Y (6, pp. 27, 39) shows that K is inconsistent with Y , although both are plotted as functions of the same ratio, Rlr. Coefficient K in Figure 1 is increasing, but Y:on the contrary, is decreasing with the ratio R / r (= A/B)( 6 , p. 39). T h e above inconsistency suggests that it might be advisable to review the method of flange calculation. Equations 1, 2, and 3 show that the change of bolt load caused by pressure is AW = W,
-
W, = AhP
- AYA,
(7)
and Equations 5 and 7 that change of gasket stress when presssure is applied is
where B is the total deflection factor,
Gasket cross area, A, Bolt c r o s area, At, Hydrostatic pressure area, Ah Bolt length (between nuts), L Flange thickness, t Modulus of elasticity of bolts and flange,
E
Calculation of Equation 8 for standard flange, 24 inches, 600 lb. Outer radius (radius of bolt circle), R Inner radius, r Ratio R / r M 0.021 (Figure 1) Gasket width, A'
950
'/2
inch
8.25 inches 4 inches
11,500,000 15,000,000 inch 1.5
x
10-8
Substituting the above data in Equation 8 gives 1'
=
1.5 X 10-8 X 416 P 0.125 106 1.5 x 10-8 X 36 Is 624P 0.835 54
+
+
This calculation shows that T / E R in Equation 8 is insignificant compared with BAR and in this case amounts to about only 1.5% (0.835/54). I t appears that for most conventional flange connections, with small gasket thickness (less than inch) T / E , may be neglected, compared with BAR. Equation 8 acquires a simple form : hp AY = A A*
From Equations 8a and 7 one finds that bolt load, when pressure is applied, remains unchanged :
w,- w,= 0
16.5 inches 11.5 inches 1.47
416 sq. inches
30,000,000
Shear modulus of flange. G Modulus of elasticity of gasket (soft iron), E, Gasket thickness, T Coefficient B in Equation 9 calculated
(9)
and Cis from Equation 6
3.14 X 23 X 0.5 = 36 sq. inches 24 X 2.304 = 55 sq. inches
From Equation 10 it is seen that increment AW, as result of internal pressure action, is always a positive value. If total deflection factor B = 0 (practically it is impossible), then A W = AhP. When T / E yis negligible compared with BA,, increment A W is practical1)- equal to zero. Theoretically, it was not clear how bolt load is changed with pressure---e.g., increment AW may be negative (7). This means that bolt load decreases after pressure is applied. Blick ( I ) , however, points out that more theoretical work must be done to determine the bolt load incremenr in the pressure condition. T h e above analysis shows that the bolt load change is positive and in particular cases (metallic gaskets) may be near zero. If the change of gasket reaction is known, it is possible to find maximum sealing pressure in a vessel within elastic limits of gasket material. For this, Equation 8 or 8a must be connected with the sealing characteristics of the gasket material. An empirical gasket factor? m, has been suggested ( 6 ) for the minimum gasket compression, which is necessary for the sealing. The gasket (mass) stress in pressure condition must be as a minimum :
This result is in good agreement with the fact that pressure does not change the bolt load, which remains substan-
INDUSTRIAL AND ENGINEERING CHEMISTRY
b
Y, = 2m;t -
(11)
N
where b /.?-is a correlation factor between contact area and cross-sectional area of gasket. Combining Equation 11 with 3 and 8 gives the following expression for sealing pressure : AY =
Y, - Y, = Y, - 2 m P --b
.v
=
For the case where approximated Equation 8a may be used (metallic gaskets), it becomes:
P =
Y, b 2m -
iV
A + A-P ,
(12a)
Equations 12 and 12a make i t possible to check permissible internal pressure which may be applied in a vessel.
GASKET S E A L I N G PRESSURE Conclusion Sealing Pressure Depends on Gasket Material and Character of Surface Reqd. Bolt Load,” Point, P.S.I.
(4)
Gasket Material
w,
Y , Yield
Gasket Factor
=
w,
Pounds, Eq. 1
Max. Pressure, P.S.I. Eq. 12a Serr. Unserr.
Soft aluminum 4.0 9,000 324,000 662 576 Soft copper or brass 4.75 13,000 470,000 930 795 Iron or soft steel 5.5 19,000 685,000 1320 1100 a Available bolt load for 24 inches, 600 lb. flange = 55 X 20,000 = 1,100,000 pounds.
For example, if the above data are used for the 24-inch, 600 lb. standard flange, the table shows results of calculations on Equation 12a. Ah
- 416 -
- _-
under outer and inner rings, A , hydrostatic pressure area, A h
147 sq. inches
P = 2 X 19’000 5.5 147 4
= 2780 p.s.i.
+36
b =
for unserrated surface
l / ~
In the table, atmospheric seating stress,
Y,, is assumed to be the “Yield Point” of gasket material on the basis of the following properties of the metals ( 4 ) . Yield Point, Modulus, X 108
x
P.S.I.
P.S.I.
Brinell No.
10
23-44
l6
50-170
Soft aluminum (pure) 5-21 Brass (various) Soft steel (pure iron) 19-30
105
13.5-21
70-90
The Yield Point n ~ S tbe a w ~ m e das minimum Stress for the gasket seating surface to “yield” gasket material into imperfections of the flange face and as maximum allowable stress in the mass of the gasket. Such a combination is possible because gasket effective surface is less than gasket total cross area. Equation 12a may be applied for a connection, with a thin metallic gasket. Where gasket thickness is greater (ringtype gaskets) and modulus of elasticity of gasket is low (rubber or plastic), T / E , becomes more significant, and Equation 12 should be used for calculation. Equation 12 was derived without restriction for any particular type of joints. Therefore, it is also applicable for special high pressure closures like Multilok. Calculation of Sealing Pressure. Inner radius of closure, r Iron gasket thickness Width, N b/N Total cross section area of gasket
11.5 inches (same as for 24-inch flange in table) l / 8 inch 0.25 inch (serrated)
‘/4
‘
sure, which is represented in Equation 12. This method makes it possible to determine maximum sealing pressure in a vessel, which may be maintained by a gasket (within its elastic limits) in flanged connections and in special closures for high pressure vessels. This calculation is especially important when the high internal pressure varies. Nomenclature = cross-sectional area of bolts, sq. inches A , = cross-sectional area of gasket (cutting parallel to plane of gasket), sq. inches Ah = hydrostatic pressure area, sq. inches a = ratio = R/r B, C, K , M , --Y = coefficients = effective seating width of gasket, b inches (6, p. 33)
Ab
for serrated surface
N
36 sq. inches
O n Equation 12a:
z - 36 - 11.6
By analyzing the system of forces among internal vessel pressure, bolt load, and gasket reaction a method was developed for calculating sealing pres-
Comparing this result with calculated data for the conventional flanged connection (see table), it is Seen that special closure makes it possible to double the sealing pressure in 24-inch connection size, at the same gasket. For larger size this ratio increases-e.g., at 41-inch inside diameter, the Multilok closure seals about three times more pressure than the 41-inch flange connection. I t is known from practice that soft metallic gaskets ((flow” if overstress&, showing the plastic property of material. Inasmuch as Y , is used as a maximum allowable stress within elastic limits, any additional tightening of nuts in the pressure condition would overstress the gasket after release of pressure. For instance, if the nuts were tightened to keep 1100-p.s.i. pressure with a soft aluminum gasket, the gasket Stress after pressure release would be as in Equation 12a, for an unserrated surface:
y,
=
(?-&!+ %)
= 17,200 p.s.i.
Such stress is beyond the Yield Point for the assumed soft aluminum gasket. The gasket would lose its springy prope r t ~ ,affecting the seal. It would be better to measure the seating stress of gasket; the method of bolt load measurement previously worked out (3) may also be applied for gasket stress measurements in the atmospheric condition. The above did not cover the temperature effect on the gasket reaction, which requires a special analysis. Internal temperature of the vessel expands the gasket-Le., increases reaction-but at the same time the bolt elongation decreases it, and in some cases these effects balance each other.
Eb
=
Of
Of
psi. E , = modulus of elasticity of flange, psi. E , = modulus of elasticity of gasket, p.s.i. f = flange deflection, inches G = shear modulus of flange, p.s.i. L = length of bolt between nuts, inches factor rn = gasket N = width of gasket, inches = internal pressure, R = Outer radius, inches = inner radius, inches = stress T = thickness of gasket, inches t = thickness of plate (flange), inches W = force = bolt load in atmospheric condition, p s i . W , = bolt load in pressure condition, pounds Y , = gasket stress in atmospheric condition, p.s.i. Y, = gasket stress in pressure condition, p i .
s
w,
literature Cited (1) Blick, R. G., Petrol. R e h e r . 32, 129 (January 1953). Processing 12, 96 (2) (March 1957). R., (3) Cam, L. H. Power (February 1946). (4) Marks, L;,S., “Mechanical Engineers’ Handbook, 5th ed., pp. 397-8, 1951. (5) Meyer, R., Chem. Eng. 63, 212 (November 1956). (6) Taylor Forge & Pipe Works, Catalog 501, “Nozzles, Welding Necks, Large Diameter Flanges.” (7) Whal, A. M.$ Lobo, G., Jr.7 Trans. Am. SOC.Mech. Engrs. 52,29 (1930). (8) Whal, A. M., Way, S., “S:;ess and Deflection of Circular Plates, ASME publ., p. 7, 1953. RECEIVED for review January 31, 1958 ACCEPTED January 26, 1959 VOL. 51, NO.r8
AUGUST 1959
951
