# ROTATING DUCT SYSTEMS, ORTHOGONAL, HEAT TRANSFER IN

### Introduction

In the following entry, Rotating Duct Systems, Parallel, Heat transfer In, it is explained that rotation of a straight duct influences an internal pressure-driven flow via Coriolis and centripetal buoyancy interactions with consequential modifications to the otherwise forced convection heat transfer between the duct wall and the flow.

### Orthogonal-Mode Rotation

In this entry, these effects are reviewed for the special case where the duct is constrained to rotate about an axis which is perpendicular to its axis of symmetry. This rotating configuration is referred to as orthogonal-mode rotation and is illustrated in Figure 1.

Coriolis forces, in this case, act as a source term for the generation of a cross stream secondary flow as indicated in the figure. Thus a mechanism exists to cause relatively cool fluid from the central area of the tube to move towards the trailing edge. If the duct wall is heated this suggests that Coriolis forces tend to induce a relatively better cooled trailing edge region. With significant density variations, associated with heated flow, a centripetal buoyancy effect is produced analogous to the earth's gravitation field with a stationary vertical duct. Thus buoyancy effects will vary according to the direction of flow through the duct. In other words, buoyancy will change direction according to whether the flow is radially outward or inward. In the present article, only radially outward flow is considered for reasons of space.

### Developed Laminar Flow

Mori and Nakayama (1968) analyzed laminar fully developed heat transfer with a uniformly heated tube of circular cross section using a momentum integral technique. These authors did not include buoyancy in their analysis and characterized Coriolis effects using a rotational form of the Reynolds Number, Jd (Ωd2ρ/η) where the velocity term involves the product of the tube diameter and the angular velocity. They did not allow for circumferential variations in heat transfer due to the Coriolis induced secondary flow owing to their choice of thermal boundary conditions, but recommendations for estimating a mean heat transfer coefficient over the circumference were proposed. Depending on the relative value of the rotational Reynolds number with respect to the through flow Reynolds number and the fluid Prandtl Number, two sets of recommendations were made. Figure 2 illustrates the typical enhancement for a uniformly heated tube which resulted from the analysis. Here "enhancement" is defined as the ratio of the developed Nusselt Number obtained with rotation compared to the nonrotating value for a specified Reynolds number. The authors repeated the analysis for a constant wall temperature and found that the relative enhancement in heat transfer was insensitive to the thermal boundary condition.

The equations recommended are likely to give useful results for a mean circumferential heat transfer coefficient but will become less reliable if significant buoyancy effects are present since buoyancy was not included in the analysis. The equations derived do not permit the circumferential variation in heat transfer, due to the Coriolis driven secondary flow, to be evaluated. A number of experimental studies have been reported for laminar heat or mass transfer for this rotating geometry, prior to the early 1980's. Unfortunately the data is not conclusive, probably due to the difficulty of the experimental regime. A review of all data up to 1980 is given by Morris (1981).

### Developing and Developed Tbrbulent Flow

Turbulent flow in orthogonally rotating ducts has been the subject of considerable theoretical and experimental study since about 1980 due to its importance for the internal cooling of gas turbine rotor blades. Morris and Ayhan (1979) first demonstrated experimentally that the combined influence of Coriolis forces and centripetal buoyancy could produce regions of impaired heat transfer, relative to the stationary duct case, on the leading edge of a circular tube. This was important from the design viewpoint in turbine blades since local hot spots might occur on the leading edge. Morris (1981) reviews virtually all published work prior to 1980 and a number of early results are available in this text.

An examination of the momentum and energy conservation equations for this rotating geometry suggests that the Nusselt number, evaluated at a specified location, z, in the duct, will be functionally related to the usual nondimensional groups which control forced convection (i.e. the through flow Reynolds number, Re, the Prandtl number of the fluid, Pr) together with two additional terms which arise from the Coriolis forces and the centripetal buoyancy. The nondimensional group which derives from the Coriolis term is the Rossby Number. Ro, which is the ratio of the coolant velocity along the duct to an angular speed term involving the product of the angular velocity with the tube diameter. Thus

The second additional term is a so-called buoyancy parameter, Bu, defined as

where Tw,z and Tb,z refer to the local wall and fluid bulk temperatures, respectively.

We expect therefore that the local Nusselt number, NuR,z, at some specified position along the duct will have functional form

and the function, Ψ, will be also dependent on the circumferential location at a give z-value.

Subsequent to 1980, a number of theoretical and experimental studies have been undertaken, see for example Wagner et al. (1989), Iacovides and Launder (1990), Taylor et al. (1991) and Morris and Salemi (1991) have attempted to evaluate the appropriate functional form of the above equation. The references cited in these papers give a full coverage of the total literature currently available with radially outward, radially inward flow and also the effect of internal ribbing.

The works of Wagner et al. (1989) and Morris and Salemi (1992) confirm that Coriolis forces enhance heat transfer on the trailing edge of smooth walled circular ducts relative to the case resulting from a similar value of through flow Reynolds number. Additionally, as the buoyancy parameter, Bu, is increased at fixed values of the Reynolds number and Rossby number, the heat transfer also increases. On the leading edge it is evident that rotation initially impairs heat transfer relative to the stationary duct case. However, at higher rotational speeds this reduction seems to be arrested with subsequent improvement of the relative heat transfer.

Morris and Chang (1994), in an as yet unpublished experimental study, have attempted to correlate their own new data for a smooth-walled circular tube and the data of Wagner et al. (1989) using a particular mathematical structure for Ψ based on physical reasoning concerning the combination of Coriolis and buoyancy interactions.

They have proposed that the heat transfer ratio at different axial locations downstream in the duct may be expressed as

where Nuo,z is the zero speed Nusselt number obtained at the same Reynolds number as the rotating case and Φ is some function. The argument of the function, Φ is a combined Coriolis/buoyancy parameter.

Figure 3 shows the heat transfer ratio data produced by Morris and Chang (1994) plotted against the combined Coriolis/buoyancy parameter proposed. In this figure data from Wagner et al. (1989) is also compared at an axial location equivalent to 10.7 effective diameters downstream of entry to the duct. Although some data scatter is evident the proposed correlating function appears to have merit. Thus this figure may be used to estimate the level of heat transfer ratio expected at a particular operating condition.

#### REFERENCES

Iacovides, H. and Launder, B. E. (1990) Parametric and numerical studies of fully-developed flow and heat transfer in rotating rectangular passages., ASME Gas Turbine and Aeroengine Congress and Exposition., Brussels, Belgium.

Mori, Y. and Nakayama, W. (1968) Convective heat transfer in rotating radial circular tubes (1st Report-Laminar region)., Int. J. Heat and Mass Trans, ii, 1027.

Morris, W. D. (1981) Heat Transfer and Fluid Flow in Rotating Cooling Channels, Research Monograph, Research Studies Press, A Division of J. Wiley and Sons, Ltd., ISBN 0471101214, 1–228.

Morris, W. D. and Ayhan, T. (1979) Observations on the influence of rotation on heat transfer in the coolant channels of gas turbine rotor blades, Proc. Inst. Meek Eng., 193, 21, 303.

Morris, W. D. and S. W. Chang (1994) Unpublished data at the time of writing.

Morris, W. D. and Saiemi, R. (1992) An attempt to experimentally uncouple the effect of coriolis and buoyancy forces on heat transfer in smooth tubes which rotate in the orthogonal mode, trans. A.S.M.E., Journal of Turbomachinery, 114, 858–864.

Taylor, C, Xia, J. Y., Medwell, J. O, and Morris, W. D. (1991) Finite element modeling of flow and heat transfer in turbine blade cooling, Proc. Conf. on Turbomachinery., Inst. Mech. Eng., London.

Wagner, J. H. Johnson. B. V., and Hajek. T. J., (1989) Heat transfer in rotating passages with smooth walls and radial outward flow, ASME Gas Turbine and Aeroengine Congress and Exposition., Brussels, Belgium.