Sealing devices with elements formed from elastic materials, which can be arranged between the inner and outer

pipes, are used to seal the interpipe space and eliminate gaps between movable components, including safety devices in flow-

ing-well fittings. The elastic elements of these devices are characterized by different shapes, but are reliable only within lim-

ited ranges of application.

To improve the sealing reliability of the interpipe space, the AOOT VNIIneftemash has developed a new device [1],

a critical component of which is a conical-cylindrical spring positioned between the shoulder of the conical housing and a

self-sealing ring (Fig. 1). Under pressure, the sealing ring in the well is displaced along the conical surface of the housing

and compresses the spring, which prevents “run-out” of the sealing ring through the gap being sealed, reducing this gap to

zero (Fig. 1b). When the pressure drops, the spring forces the sealing outward (Fig. 1a).

Use of a spring with a rectangular coil section is dictated by the need to eliminate the sealed gap between the inner

surface of the outer pipe and the sealing device. As compared with a spring with a circular; section (under similar loads), this

spring develops a considerably greater force, and there is considerably less displacement.

With respect to working conditions, the spring combines the functions of a compression and torsional spring; here,

the spring material is in a complex stress-strain state. Structural features of the conical-cylindrical spring and its operating

conditions with a forcibly varying coil geometry as the spring is depressed along the conical surface do not make it possible

to use familiar methods of spring analysis. A procedure that establishes the order and method of force and strength analysis

of these springs was therefore developed.

The following data can be assumed as known for analysis of the spring:

• properties of spring material: ultimate strength σu; modulus of longitudinal elasticity E; shear modulus G; max-

imum relative elongation δ; coefficients f1 and f2 of sliding friction of spring material over polished steel (conical surface of

inner pipe column), and over non-machined steel (inner surface of outer pipe), respectively;

• properties of sealing-ring material: coefficient of sliding friction of sliding-ring material over polished steel f3and over non-machined steel f4;

• structural parameters of spring: outside diameter do; number of effective coils n; width a of blank; height h of

blank; slope γ of inner surface; height hi of coils; coil spacing t; maximum internal diameters d0i of unloaded spring; coil

diameters dci with spring under compression prior to contact of the coils (minimum inside coil diameter; average coil diam-

eters davi; length H0 of spring in free state; length H1 of predeformed spring; and, length H3 of spring at maximum defor-

mation;

• structural parameters of pipe and sealing ring, defining load on spring:maximum Dop and minimum dop(based on tolerances) inside diameter of outer pipe; inside diameter ds0 of sealing ring; outside diameter dso of sealing ring;

length lsi of sealing ring based on inside diameter; and, length lso of sealing ring based on outside diameter;

• maximum effective pressure p in the well; and

• yield point σyp of material in the outer pipe.

Chemical and Petroleum Engineering, Vol. 38, Nos. 5–6, 2002

METHOD OF ANALYZING CONICAL-CYLINDRICAL

SPRINGS OF SEALING DEVICES IN PIPE WELL SPACE

R. E. Gazarov and Yu. V. Zaslavskii UDC 62-272:622.24.001.24

OIL AND GAS EQUIPMENT

Open Joint-Stock Company VNIIneftemash. Translated from Khimicheskoe i Neftegazovoe Mashinostroenie, No. 5,

pp. 17–19, May, 2002.

0009-2355/02/0506-0269$27.00 ©2002 Plenum Publishing Corporation 269

The spring is subject to force analysis to determine the contact pressure on the inner surface of the outer pipe under

the maximum effective pressure in the well, and to confirm self-braking of the spring as the effective pressure drops. Figure2

shows the computational diagram.

The specific load on the ith coil of the spring [2]

where Ii = ai hi3/12 is the axial moment of inertia of the ith rectangular coil of the spring, and ai and hi are, respectively, the

width and height of the ith spring coil.

The compression force of the ith spring coil [2]

Si = KπdciPi,

where K = α/2π is the contact factor (at a contact angle α = 122°35′, K = 0.34 [2]).

For the last coil of the spring (structural assumed a half coil),the factor 0.5 should be introduced additionally to the

formula for Si when determining the compression force.

The total compression force of the spring SΣ = ∑Si.

The total friction force of the spring, which is governed by the elasticity Ffr = f1SΣ, and its projection onto the axis

of the spring Ffr* = Ffr /cosγ.

The compressing force due to the effective pressure, which can be transmitted onto the axis of the spring through

the sealing ring

= π−

− −

pD d f d l f D lop

2si2

si si op so

4 2 23 4 .

SD d p f d l p f D l p

sop2

si2

si si op so=π −

−π

−π

=( )

4 2 23 4

PEI d d

di

i i i

i

=−8 0

4( )

,c

c

270

1234567

89

p0

p0

H0 H

0

Dop Dop

p0 p0

p

p

a b

Fig. 1. Sealing device for interpipe well space:a) with equal or atmospheric pressure p0in internal column of pipes and in interpipe space; b) under pressure p in internal column

of pipes,which exceeds pressure p0 in interpipe space above self-sealing ring; 1) interpipe

space; 2) shoulder-stop; 3, 7) outer and inner columns of pipes,respectively; 4) conical

surface of internal pipe column; 5) helical compression spring; 6) self-sealing ring;

8) opening (channel into interpipe space); 9) clip-ring stop.

The force of the compression elasticity of the spring, which moves against the conical housing, is then determined.

The stiffness of the ith coil of the spring

where λi is the flexibility of the ith coil, and ∆i is the coefficient of the effect of the geometry of the ith coil of the spring on

the elastic properties.

The coefficient ∆i is determined from the theory of twisting of a straight beam. Table 1 presents values of the coefficients

∆i based on Reshetov’s data [3] (as a function of the ratio hi /ai) (other ∆i values are determined by interpolation). For the last

incomplete coil of the spring, the factor 0.5 should be entered additionally to the formula for ci when the stiffness is determined.

The force of elasticity, which develops in the ith coil,

Felasi = ciδi,

where δi is the deflection of the ith coil at maximum deformation (δi = t – ai for the full coils,and is determined from the

spring’s drawing for the incomplete coil).

cGa

di

i

i

i i

= =∆

1 4

3λ av

,

271

3

2

1

Felas 3

Felas 2

Felas 1

S3

S2S1

γ

Fig. 2. Diagram used for force analysis of spring.

F*fr 2

Felas Σ

Ffr 1

ScΣ

Ss

Sp

γ

Fig. 3. Computational diagram for determination of contact pressure on

inner surface of pipe.

The total force of spring elasticity FelasΣ = ∑Felasi.

The set of forces acting on the spring is analyzed to determine the contact pressure on the side of the spring against

the inner surface of the pipe (Fig. 3). Let us write the following system of statics equations:

Sp = ScΣcosγ – Ffr1sinγ; Ss = Sp2 + FelasΣ + Ffr1cosγ;

Ffr2 = Spƒ2; Ffr1 = ScΣƒ1.

Solving this system of equations simultaneously, we obtain

Sp = ScΣcosγ – ScΣƒ1sinγ = ScΣ(cosγ – ƒ1sinγ);

Ss = Spƒ2 + FelasΣ + ScΣƒ1cosγ,hence,

ScΣ = (Ss – Spƒ2 – FelasΣ)/(ƒ1cosγ); Sp = (Ss – Spƒ2 – FelasΣ)(cosγ – ƒ1sinγ) / (ƒ1cosγ).

After simultaneous solution of the last two equations in terms of Sp and corresponding transformations,we can write

Sp = (Ss – FelasΣ)(cosγ – ƒ1sinγ) / [(ƒ1 + ƒ2)cosγ – ƒ1ƒ2sinγ] = (Ss – FelasΣ)(1 – ƒ1tanγ) / (ƒ1 + ƒ2 – ƒ1ƒ2tanγ).

Since the product ƒ1ƒ2tanγ << 1, we finally have

The contact pressure of the spring on the pipe

where w = 0.5(amini + ai–1) is the width of the ring of contact of the last coil of the spring.

The value obtained for the contact pressure can be compared with the allowable value for the outer pipe; moreover,

the condition q ≤ [σ]cm = 0.8σyp.

We can then confirm the condition of the spring’s self-braking on the conical mandrel when the effective pressure

drops-off.

The tightening forces

Ft = FelasΣ + ∑(Sisinγ) – ∑(Siƒ1cosγ);

Ft = FelasΣ + SΣ(sinγ – ƒ1cosγ).

qS

d w=

πp

op,

SS F f

f fps elas=

− −+

( )( tan ).Σ 1 1

1 2

γ

272

TABLE 1

hi /ai 1.0 1.5 1.75 2.0 2.5 3.0

∆i 5.567 2.670 2.086 1.713 1.256 0.995

The condition,which excludes self-braking:Ft > 0.

Spring strength is analyzed in the following sequence.

The maximum relative elongation of the spring in %:

δmax = (lf – l0)100/ lf = (Dop – do)100/Dop,

where lf and l0 are the final and initial lengths of the spring. The condition δmax ≤ δ should be fulfilled.

Maximum bending and torsional stresses develop in the last incomplete coil of the spring, which should also be con-

firmed.

Maximum bending stresses in the coil [2]:

σmaxi = 2.37E∆di ymaxi /dci2,

where ∆di = d0(i–1) – d0i is the difference in the diameters,and ymaxi = 0.5hi is the distance to the neutral line.

Maximum torsional stresses in a coil

where νi is a coefficient that takes into account the influence exerted by the geometry of the ith coil of the spring on the

strength properties.

The coefficient νi is determined from the theory of the twisting of a straight beam. Table 2 presents νi values based

on Reshetov’s data [3] (as a function of the ratio hi /ai) (other values are determined by interpolation).

The equivalent stresses in a coil

The value obtained for the equivalent stress is compared with the ultimate strength of the spring material. The con-

dition σeq < σu should be fulfilled.

The spring characteristics are established from the following relationships:

• the force of the spring during preliminary deformation [4]

where D is the average diameter of the spring (the average diameter of the median coil D = dm) is adopted for a spring with

an inner conical bore, and λ1 = H0 – H1 is the shortening of the spring under the force P1.

• the force of the spring at maximum deformation

P3 = FelasΣ.

PGa

D ni

11

4

35 57=

λ.

,

σ σ τeq = +max max .2 23

τνmax ,=

F d

h a

i i

i i i

elas av

2 2

273

TABLE 2

hi /ai 1.0 1.5 1.75 2.0 2.5 3.0

νi 0.208 0.231 0.239 0.246 0.258 0.267

The procedure that we have outlined is used in developing safety devices for type PFA well fittings at pressures to

105 MPa. These safety devices were manufactured by the company ÉMK-ATOMMASh (Volgodonsk),and have successful-

ly passed acceptance tests.

REFERENCES

1. Russian Federation Patent No. 2002030, Device for sealing the interpipe space of wells (1993).

2. V. I. Feodos’ev, Selected Problems and Questions Concerning Material Strength [in Russian],Nauka,Moscow

(1973).

3. D. N. Reshetov, Machine Components[in Russian],Mashinostroenie, Moscow (1989).

4. P. I. Orlov, Fundamentals of Design[in Russian],Book 2,Mashinostroenie, Moscow (1988).

274