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### PhD Thesis: Free Convective Heat Transfer in a Rotating Annulus

 Posted By: Martin King Date: Thu, 22 May 2003, 3:59 p.m.

Below is the abstract to the thesis and a link to the complete copy. The thesis was defended successfully on 20 March 2003 at the Department of Mechanical Engineering, University of Bath, U.K. It was jointly supervised by Prof. J.M. Owen and Dr M. Wilson (I bear the sole responsibility of errors in the thesis) and examined by Dr D.A.S. Rees (Bath) and Dr C.A. Long (Sussex).

N.B. 1. The figures are each in separate pdf files. The file thesis.pdf has pages shifted slightly upward (thesis.ps is good). I apologise if these cause inconveniences. I was rushing to finish my thesis and didn't have time to extract the figures and insert them directly in latex. I could fix it but I have more important new problems now. 2. Please kindly send me an email if you download a copy. I would just like to know how popular (or unpopular) it is. 3. In the rare event that you may want to cite it, the citation should be: King, M.P., 2003, Convective heat transfer in a rotating annulus, University of Bath, U.K.

Convective heat transfer in a rotating annulus

Martin P. King Submitted for the degree of Ph.D. of the University of Bath 2003

Abstract

The free convection heat transfer in a sealed rotating annulus is investigated. The annulus has a heated outer cylinder and a cooled inner cylinder. Flow is induced by buoyancy in the centrifugal field and affected by the Coriolis force. In addition to fundamental understanding of free convection in rotating enclosures, the problem is specifically relevant to flow in compressors and turbine-discs of gas-turbines.

The heat transfer efficiency, characterised by Nusselt number-Rayleigh number scaling laws $Nu \sim Ra^\gamma$, is the main focus of the thesis. Both numerical simulations and simple phenomenological models with dimensional analyses have been employed to address the problem. The latter method is also used to study heat transfer in a cavity with axial throughflow.

It is found that the influence of rotational Reynolds number, which is associated with the Coriolis force, in reducing heat transfer is weak if the value of $\beta \Delta T$ cannot be varied over a large range. The Nusselt number is strongly dependent only on Rayleigh number. Simulations do not reproduce measured heat transfer results by Bohn {\sl et al.} (1995) for rotating annuli, instead the results are closer to those for Rayleigh-B\'enard convection. Rayleigh-B\'enard convection is the stationary `counterpart' of free convective flow in rotating annulus, and its understanding has enjoyed much progress in recent years (e.g. Grossmann \& Lohse, 2000 \& 2001). Results from Rayleigh-B\'enard convection are applied carefully for comparisons with and interpretations of thecomputational results obtained in the current study. An explanation is also suggested in this context for the results of Bohn {\sl et al.}

From the results obtained in the current study and in the literature, it is remarkable to note that scaling laws for apparently very different convective heat transfer situations (such as orientations and shapes of the enclosures, cavities or flat plates and even whether they are stationary or rotating) are not vastly different. The typical $\gamma$ values, for example, vary between $1/5$ and $1/2$ inclusive. The orders of magnitude for heat transfer are similar; the effect is especially noticeable if a large range of Rayleigh number is considered.

Simple phenomenological models (inspired by Castaing {\sl et al.}, 1989) and dimensional analyses are used to derive many scaling laws. This type of study enables a more unified understanding of the flow processes and parameters that give rise to the scaling laws for different convective flows.

For cavities with axial throughflow, different covection regimes with their corresponding scaling laws were also derived using similar theoretical models and dimensional considerations. The results are summarised in a regime diagram of $Re_z$ vs. $Ra_z$ delineating different types of convection and scaling laws.