Abstract
The analysis of the heat distribution between a stationary pin and rotating ring was considered. To solution of the governing quasi-stationary heat conductivity equation the finite Fourier transform was used. The convective cooling from outer and internal surface of the ring as the boundary conditions were considered. The ring surface temperature, the average temperature on the ring surface and the heat distribution coefficient for the studied system were determined. The numerical results for the temperatures and heat distribution coefficient which demonstrated the effects of the Biot number and internal radius of the ring on them, were presented.
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Abbreviations
- A :
-
Area of the heating zone (m2)
- Bi=hR/K :
-
Dimensionless Biot number associated with the external surface of the ring
- Bi 0=h 0 R/K :
-
Dimensionless Biot number associated with the internal surface of the ring
- B i′=h′R/K :
-
Dimensionless Biot number associated with the sides of the ring
- h :
-
Heat transfer coefficient on external surface of the ring (Wm−2K−1)
- h 0 :
-
Heat transfer coefficient on the internal surface of the ring, (Wm−2K−1)
- h′:
-
Heat transfer coefficient on the sides of the ring (Wm−2K−1)
- k :
-
Thermal diffusivity (m2s−1)
- K :
-
Thermal conductivity (Wm−1 K−1)
- Pe=ω R 2/k :
-
Dimensionless Peclet number
- q :
-
Intensity of the heat flow into the ring (Wm−2)
- Q=qA :
-
Total rate of friction heat directed into the ring supply from area A (W)
- r :
-
Radial coordinate (m)
- R :
-
External radius (m)
- R 0 :
-
Internal radius (m)
- T :
-
Temperature rise (*C)
- T *=KT/(qR):
-
Dimensionless temperature
- T * *=T */θ 0 :
-
Dimensionless temperature
- T * * max=T * * (1, 2θ 0):
-
Maximum dimensionless temperature
- T a :
-
Average temperatures over areas, A (*C)
- T * a =KT a /(qR):
-
Dimensionless average temperature
- A 1 :
-
Cross-section area (m2)
- Bi 1=h 1 R 1/K 1 :
-
Dimensionless Biot number
- h 1 :
-
Surface heat transfer coefficient (Wm−2 K−1)
- K 1 :
-
Thermal conductivity (Wm−1 K−1)
- l :
-
Total length (m)
- l *=l/R 1 :
-
Dimensionless length
- P :
-
Total load (N)
- q 1 :
-
Intensities of heat flow (Wm−2)
- Q 1=q 1 A 1 :
-
Total rates of friction heat directed into the pin (W)
- R 1 :
-
Radius of the cross-section area (m)
- T 1 :
-
Temperature on the sliding end (*C)
- T * 1=K 1 T 1/(q 1 R 1):
-
Dimensionless temperature
- :
-
Other quantities
- f :
-
Dimensionless friction coefficient
- K *=K/K 1 :
-
Dimensionless heat conductivity
- I k (·):
-
Modified Bessel function of the first kind of the order k (k=1,2)
- K k (·):
-
Modified Bessel function of the second kind of the order k (k=1,2)
- δ:
-
Half thickness
- Δ=(δ/R):
-
Dimensionless half thickness
- γ=Q/Q 1 :
-
Dimensionless heat distribution coefficient
- η=Q/(Q+Q 1):
-
Dimensionless heat distribution coefficient
- \(\varphi = {\sqrt {2Bi_{1}}}l^{*}\) :
-
Dimensionless coefficient
- θ:
-
Circumferential coordinate
- θ 0 :
-
Half of the contact angle
- ρ=r/R :
-
Dimensionless radial coordinate
- ρ 0=R 0/R :
-
Dimensionless internal radius
- σ=B i′/Δ:
-
Dimensionless parameter
- ω:
-
Rotational speed
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Yevtushenko, A., Tolstoj-Sienkiewicz, J. Distribution of Friction Heat Between a Stationary Pin and Rotating Ring. Arch Appl Mech 76, 33–47 (2006). https://doi.org/10.1007/s00419-006-0003-2
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DOI: https://doi.org/10.1007/s00419-006-0003-2