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Research Papers: Applications

Analysis of Surface Temperatures Within Heat Flux Bands During Constant Acceleration Including Deceleration to Halt

[+] Author and Article Information
Thierry A. Blanchet

Department of Mechanical, Aerospace
and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180
e-mail: blanct@rpi.edu

Shane P. Lenihan

Department of Mechanical, Aerospace
and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received May 26, 2016; final manuscript received August 8, 2016; published online January 10, 2017. Assoc. Editor: Sinan Muftu.

J. Tribol 139(3), 031102 (Jan 10, 2017) (10 pages) Paper No: TRIB-16-1175; doi: 10.1115/1.4034532 History: Received May 26, 2016; Revised August 08, 2016

In the modeling of a uniformly distributed band heat flux region experiencing constant acceleration from rest over a half-space surface, it is found that the maximum surface temperature at the instantaneous speed and the corresponding Peclet number are already well approximated by the long-established steady-state constant-speed models very soon after the moment the flux region clears the overlap of its original footprint at the initiation of motion. During startup when the flux still overlaps its original footprint, maximum temperature at any instant given the level of flux is well approximated by a simple one-dimensional conduction problem with a correspondingly stationary heat flux initiating at time zero. The above acceleration behaviors are observed regardless of whether the uniform flux is constant or Coulombic (proportional to instantaneous speed as frictional heating), though during the initial startup the maximum temperature rise in the Coulombic case is only two-thirds that of the constant flux case. The case of constant deceleration was additionally modeled, where at the eventual instant of halt, the maximum temperature in the case of constant flux was found to be directly proportional to the rate of deceleration to the 1/4 power, whereas in the case of Coulombic flux it was found that maximum temperature was instead inversely proportional to the rate of deceleration.

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References

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Figures

Grahic Jump Location
Fig. 1

Uniform heat flux q˙ (unit: W/m2) within a long (in/out of page) band region of breadth λ that is instantaneously centered about the origin on a stationary half-space and traversing at speed v to the left over its surface, with (x,z) as the position of interest for temperature rise

Grahic Jump Location
Fig. 2

Line heat rate input Q˙ (unit: W) distributed uniformly over long line length B (in/out of page) traversing to the left over a stationary half-space. At arbitrary earlier time t′, it is at surface position x′ with speed v′, while at final time t of interest for temperature rise at half-space position (x,z), it is at surface position xf′ with speed v.

Grahic Jump Location
Fig. 3

Temperature rise as a function of surface position within a band of uniform constant flux while accelerating at a high rate a* = 3 × 106 through various Peclet numbers: (a) T*, (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge, and (c) T*/Pe highlighting low Pe start-up behavior with a flat-top value ∼ 2/πa*  = 0.00046

Grahic Jump Location
Fig. 4

Temperature rise as a function of surface position within a band of uniform constant flux while accelerating at a lower rate a*= 0.3 × 106 through various Peclet numbers: (a) T*, (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge, and (c) T*/Pe highlighting low Pe start-up behavior with a flat-top value ∼ 2/πa*  = 0.00146

Grahic Jump Location
Fig. 5

Temperature rise as a function of surface position within a band of uniform Coulombic flux while accelerating at a high rate a* = 3 × 106 through various Peclet numbers: (a) T*, (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge, and (c) T*/Pe highlighting low Pe start-up behavior with a flat-top value ∼ (2/3)2/πa*  = 0.00031

Grahic Jump Location
Fig. 6

Temperature rise as a function of surface position within a band of uniform Coulombic flux while accelerating at a lower rate a* = 0.3 × 106 through various Peclet numbers: (a) T*, (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge, and (c) T*/Pe highlighting low Pe start-up behavior with a flat-top value ∼ (2/3)2/πa*  = 0.00097

Grahic Jump Location
Fig. 7

Temperature rise as a function of surface position within a band of uniform constant flux while decelerating at a high rate a* = 3 × 106 through various Peclet numbers: (a) T* directly highlighting low Pe halt behavior with a value ∼0.78/a*1/4  = 0.019 near the trailing edge, (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge, and (c) T*/Pe no longer highlighting any particular asymptotic behavior in deceleration cases

Grahic Jump Location
Fig. 8

Temperature rise as a function of surface position within a band of uniform constant flux while decelerating at a lower rate a* = 0.3 × 106 through various Peclet numbers: (a) T* directly highlighting low Pe halt behavior with a value ∼0.78/a*1/4  = 0.0335 near the trailing edge and (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge

Grahic Jump Location
Fig. 9

Temperature rise as a function of surface position within a band of uniform Coulombic flux while decelerating at a high rate a* = 3 × 106 through various Peclet numbers: (a) T* no longer directly highlighting low Pe decelerating halt behavior if flux is instead Coulombic, (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge, and (c) T*/Pe no longer highlighting any particular asymptotic behavior in deceleration cases

Grahic Jump Location
Fig. 10

Temperature rise as a function of surface position within a band of uniform Coulombic flux while decelerating at a lower rate a* = 0.3 × 106 through various Peclet numbers: (a) T* no longer directly highlighting low Pe decelerating halt behavior if flux is instead Coulombic, (b) T*Pe highlighting asymptotic high Pe behavior with a value ∼1/π  = 0.564 at the trailing edge, and (c) T*/Pe no longer highlighting any particular asymptotic behavior in deceleration cases

Grahic Jump Location
Fig. 11

Alternate nondimensional representation of temperature rise T*alt as a function of position within a band of uniform flux which is capable of directly highlighting low Pe decelerating halt behavior in the case of Coulombic heat: (a) high rate deceleration a* = 3 × 106 showing a maximum value ∼1.05 a*1/4  = 44 at the trailing edge and (b) lower rate deceleration a* = 0.3 × 106 showing a maximum value ∼1.05 a*1/4 = 24.5 at the trailing edge

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