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Research Papers

Use of Fin Equation to Calculate Nusselt Numbers for Rotating Disks

[+] Author and Article Information
Hui Tang

Department of Mechanical Engineering,
University of Bath,
Bath, BA2 7AY, UK
e-mail: h.tang2@bath.ac.uk

Tony Shardlow

Department of Mathematical Sciences,
University of Bath,
Bath, BA2 7AY, UK

J. Michael Owen

Department of Mechanical Engineering,
University of Bath,
Bath, BA2 7AY, UK

Manuscript received June 26, 2015; final manuscript received July 29, 2015; published online September 23, 2015. Editor: Kenneth C. Hall.

J. Turbomach 137(12), 121003 (Sep 23, 2015) (10 pages) Paper No: TURBO-15-1123; doi: 10.1115/1.4031355 History: Received June 26, 2015; Revised July 29, 2015

Conduction in thin disks can be modeled using the fin equation, and there are analytical solutions of this equation for a circular disk with a constant heat-transfer coefficient. However, convection (particularly free convection) in rotating-disk systems is a conjugate problem: the heat transfer in the fluid and the solid are coupled, and the relative effects of conduction and convection are related to the Biot number,  Bi, which in turn is related to the Nusselt number. In principle, if the radial distribution of the disk temperature is known then Bi  can be determined numerically. But the determination of heat flux from temperature measurements is an example of an inverse problem where small uncertainties in the temperatures can create large uncertainties in the computed heat flux. In this paper, Bayesian statistics are applied to the inverse solution of the circular fin equation to produce reliable estimates of Bi for rotating disks, and numerical experiments using simulated noisy temperature measurements are used to demonstrate the effectiveness of the Bayesian method. Using published experimental temperature measurements, the method is also applied to the conjugate problem of buoyancy-induced flow in the cavity between corotating compressor disks.

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Figures

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Fig. 1

Simplified diagram of high-pressure compressor rotor

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Fig. 2

Simplified diagram of axial throughflow in an isothermal rotating cavity

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Fig. 3

Experimental multicavity rig used by Atkins and Kanjirakkad [16] (dimensions in mm)

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Fig. 4

Radial variation of Θ, from Atkins and Kanjirakkad [16]

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Fig. 5

Effect of Bi on theoretical variation of θ v. x  for xa=0.3

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Fig. 6

Comparison between Bayesian method and curve-fitting methods for Bi = 100x5: (a) temperature distribution and (b) Bi distribution

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Fig. 7

Comparison between Bayesian method and curve-fitting methods for Bi = 100x2: (a) temperature distribution and (b) Bi distribution

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Fig. 8

Distributions of temperature and Nusselt numbers for Ro ≈ 5: (a) temperature distribution (symbols denote measurements; curves show computations) and (b) Nusselt number distributions (curves show computations; shading shows 95% confidence intervals)

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Fig. 9

Distributions of temperature and Nusselt numbers for Ro ≈ 1: (a) temperature distribution (symbols denote measurements; curves show computations) and (b) Nusselt number distributions (curves show computations; shading shows 95% confidence intervals)

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Fig. 10

Distributions of temperature and Nusselt numbers for Ro ≈ 0.6: (a) temperature distribution (symbols denote measurements; curves show computations) and (b) Nusselt number distributions (curves show computations; shading shows 95% confidence intervals)

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Fig. 11

Distributions of temperature and Nusselt numbers for Ro ≈ 0.3: (a) temperature distribution (symbols denote measurements; curves show computations) and (b) Nusselt number distributions (curves show computations; shading shows 95% confidence intervals)

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