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The effects of thermoelastic and wear on the leakage of compressible gases in shaft seals

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Analysis are presented for a compressible gas flow across shaft seals. The leakage flow rate is developed analytically for a sealing gap due to eccentricity, misalignment of a shaft, sinusoidal wavy surfaces for both bodies, thermoelastic deformation on the edge of the body, and wear on the mating surfaces. A pressure distribution is determined as a power series based on the simplified nonlinear Reynolds equation. The restriction to exclude the thermoelastic effects was presented. Analytical results indicate that the thermoelastic distortion may dominate the seal performance when the eccentricity is large and width of the seal small at high speeds.

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e 0 :

Shaft eccentricity atz=0

\(\bar h\) :

r 1r 2, Mean sealing gap atz=0

\(\tilde h\) :

Sealing gap variation due to wavy surfaces

\(\tilde h_t \) :

Sealing gap variation due to the eccentricity and tilted shaft

m :

Exponent determined by the fluid property

n :

Number of waves

r :


R g :

Gas constant

t :


T :


w :

Wear coefficient

β :

T/T 1

β A :

T u/T1

γ :

Angle of tilt

ɛ :

\({{\gamma L} \mathord{\left/ {\vphantom {{\gamma L} {\bar h}}} \right. \kern-\nulldelimiterspace} {\bar h}}\), Tilt parameter of the shaft seal

ɛ 0 :

\({{e_0 } \mathord{\left/ {\vphantom {{e_0 } {\bar h}}} \right. \kern-\nulldelimiterspace} {\bar h}}\), Eccentricity ratio atz=0


2π/λ, Wave number



Λ t :

T 1/Tr

Λ u :

T u/Tr

ξ l :

\(\left| {\tilde h_l \left| {\bar h} \right.} \right.\)

ξ u :

\(\left| {\tilde h_u \left| {\bar h} \right.} \right.\)


Angle between the line of centers

\(\tilde \Phi _t \) :

Ω l t +Φ l

\(\tilde \Phi _u \) :

Ω u tΦ u


Attitude angle

Ω l :

ϰ l C l

Ω u :

ϰ u (c u +U)

l :

Lower body

r :

Reference conditions

u :

Upper body


  1. Bryant, M.D. and Kim, C.K., 1987, “Estimating the Leakage of Compressible Gases Through Face Seals Rotating at High Speeds, Part I-Geometric Effect on Sealing Gap”, Submitted to ASME.

  2. Burton, R.A., 1975, “Large Disturbance Solutions for Initially Flat, Frictionally Heated Thermoelastically Deformed Surfaces”, J. of Lub. Tech., pp. 539–545.

  3. Burton, R.A. and Heckmann, S.R., 1975, “Effects of Shear and Wear on Instabilities Caused by Frictional Heating in a Seal-Like Configuration”, Preprint Number 75-LC-1B-2.

  4. Burton, R.A., Kilaparti, S.R., and Nerlikar, V., 1973, “A limiting Stationary Configuration with Partially Contacting Surfaces, Wear, Vol. 24, pp. 199–206.

  5. Burton, R.A. and Nerlikar, V., 1974, “Effect of initial Surface Curvature on Frictionally Excited Thermoelastic Phenomena”, Wear, Vol. 27, pp. 195–207.

  6. Burton, R.A. and Nerlikar, V., 1975, “Clearance, Leakage and Contact Temperature in a Thermoelastically Deformed Seal-Like Configuration”, J. of Lub. Tech., pp. 546–551.

  7. Burwell, J.T., and Strang, C.D., 1952, “On the Empirical Law of Adhesive Wear”, J. of Applied Physics, Vol. 23, No. 1, pp. 18–28.

  8. Constantinescu, V.N., 1969, “Gas Lubrication”, ASME.

  9. Kim, C.K., 1988, “Estimation of the Leakage for Compressible Gases in High-Speed Shaft Seals”, KSME Journal, Vol. 2, No. 1, pp. 3–8.

  10. Kim, C.K. and Burton, R.A., 1986, “Thermoelastic Boundary Condition for Laminar Flows”, North Carolina State University, Raleigh, N.C.

  11. Sassenfeld, H. and Walther, A., 1954, “Gleitlagerberechnungen”, VMI-Forschungsheft 441.

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Correspondence to Chung Kyun Kim.

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Kim, C.K. The effects of thermoelastic and wear on the leakage of compressible gases in shaft seals. KSME Journal 2, 133–139 (1988). https://doi.org/10.1007/BF02953673

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Key Words

  • Shaft Seal
  • Sealing Gap
  • Thermoelastic Effects
  • Leakage Flow Rate
  • Surface Waveness