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Approximation Error RACKETT EQUATION RADAR RADIAL COMPRESSOR RADIAL ENERGY FLOWS RADIAL FANS RADIAL GAP SIZE RADIATION RADIATION ABSORPTION METHOD RADIATION BETWEEN PARALLEL PLATES RADIATION DIFFUSION APPROXIMATION, FOR COMBINED RADIATION AND CONDUCTION RADIATION DOSIMETRY RADIATION DRYING Radiation from semi-transparent oxide particles in thermal spraying Radiation heat transfer in a supersonic nozzle of a solid-propellant rocket engine Radiation heat transfer in solid-propellant rocket engines RADIATION IN ENCLOSURES Radiation in nanomanufacturing Radiation in production of carbon fibers Radiation of an isothermal plane-parallel layer Radiation of isothermal volumes of scattering medium: An error of the diffusion model Radiation of nonisothermal layer of scattering medium RADIATION SHIELDS RADIATION TO FURNACE TUBES Radiation transfer between the surfaces through a non-participating medium Radiation transfer in emitting, absorbing and scattering media Radiation transfer in combustion chambers Radiation transfer problems in nature and engineering Radiation transfer theory and the computational methods Radiation-turbulence interaction Radiative boundary layer Radiative cooling and solidification of core melt droplets Radiative cooling of particle flow in vacuum RADIATIVE DIFFUSION Radiative effects in semi-transparent liquid containing gas bubbles Radiative equilibrium in plane-parallel layer RADIATIVE EXCHANGE Radiative exchange between an isothermal gas and surrounding walls RADIATIVE HEAT FLUX Radiative heat transfer Radiative heat transfer in moving media RADIATIVE HEAT TRANSFER, IN POROUS MEDIA Radiative properties of gas bubbles in semi-transparent medium Radiative properties of metal particles in infrared and microwave spectral ranges Radiative properties of micro- and nanostructures Radiative properties of particles and fibers (theoretical analysis) Radiative properties of polydisperse systems of independent particles Radiative properties of semi-transparent fibers at arbitrary illumination Radiative properties of semi-transparent particles Radiative properties of single particles and fibers: the hypothesis of independent scattering and the Mie theory Radiative properties of soot particles Radiative properties of water droplets in near infrared RADIATIVE SPECTRAL INTENSITY Radiative transfer equation Radiative transfer equation: a general formulation Radiative transfer for coupled atmosphere and ocean systems Radiative transfer in combustion phenomena affected by radiation Radiative transfer in combustion systems Radiative Transfer in Coupled Atmosphere and Ocean Systems: Impact of Surface Roughness on Remotely Sensed Radiances Radiative transfer in coupled atmosphere and ocean systems: the adding and doubling method Radiative Transfer in Coupled Atmosphere and Ocean Systems: the Discrete Ordinate Method RADIATIVE TRANSFER IN COUPLED ATMOSPHERE AND OCEAN SYSTEMS: THE SUCCESSIVE ORDER OF SCATTERING METHOD Radiative transfer in glass production Radiative transfer in laminar flames Radiative transfer in laser processing Radiative transfer in medical laser treatment RADIATIVE TRANSFER IN MULTIDIMENSIONAL PROBLEMS: A COMBINED COMPUTATIONAL MODEL Radiative transfer in space applications Radiative transfer in the atmosphere Radiative transfer in turbulent flames Radiative transfer in two-phase combustion Radiative-conductive heat transfer in dispersed materials Radiative-conductive heat transfer in foam insulations RADIO FREQUENCY HEATING RADIO FREQUENCY, RF, DRYING RADIO WAVES RADIOACTIVE DECAY RADIUM RADIUS, HYDRAULIC RADON RAE RAFFINATE PHASE RAINBOW VOLUMIC VELOCIMETRY RAINFALL RAMAN SPECTROSCOPY RAMJET ENGINES RANDOM PROCESSES RANKINE CYCLE RANKINE DEGREE RANKINE VORTEX RANKINE, WJM RAOULT'S AND DALTON'S LAW RAOULT'S LAW RAREFACTION RAREFACTION WAVE RAREFIED GAS DYNAMICS RATE-CONTROLLED CONSTRAINED EQUILIBRIUM Ray effects and false scattering Ray optics and wave effects in radiation propagation Rayleigh equation, for bubble growth Rayleigh equation, for droplet formation Rayleigh formula Rayleigh law of scattering Rayleigh number Rayleigh scattering Rayleigh, Lord (1842-1919) Rayleigh-Gans scattering Rayleigh-Taylor instability REACTING GAS FLOW REACTION TURBINES REACTIVE CONTAMINANT TRANSPORT REACTOR PHYSICS Real gaseous spectra REATTACHMENT REATTACHMENT, OF BOUNDARY LAYER Reaumur Degree REBOILERS RECIPROCATING COMPRESSOR RECIRCULATION RECIRCULATION BOILERS RECONSTRUCTED WAVEFRONTS RECOVERY COEFFICIENT RECOVERY TEMPERATURE RECTANGULAR CHANNEL RECTANGULAR CYLINDERS RECTANGULAR DUCTS RECTANGULAR STENOTIC MODELS RECUPERATIVE HEAT EXCHANGERS REDLICH-KWONG EQUATION REDOX REACTIONS REDUCED GRAVITY CONDITIONS REDUCED INSTRUCTION SET COMPUTER, RISC REDUCED PROPERTIES REFINING REFLECTANCE REFLECTION COEFFICIENT (REFLECTANCE) REFLECTION COEFFICIENTS FOR EARTH'S SURFACE REFLECTIVITY REFLOOD REFLUX CONDENSATION REFLUX CONDENSER REFLUX RATIOS REFORMING REFRACTION REFRACTIVE INDEX REFRACTIVE INDICES FOR GASES AND LIQUIDS REFRACTORY MATERIALS, FOR ELECTRIC FURNACES REFRIGERANTS REFRIGERATION REGENERATIVE BURNER REGENERATIVE FEED HEATING REGENERATIVE GAS TURBINE REGENERATIVE HEAT EXCHANGERS REGULAR REGIME OF DRYING REHEATING REICHARDT'S FORMULA, FOR VELOCITY DISTRIBUTION IN TUBES REIMANN'S INTEGRAL REINER-RIVLIN FLUID RELATIVE HUMIDITY RELATIVE MOLAR MASS RELATIVE PERMEABILITY RELATIVE POWER DEMAND, RPD RELATIVE ROUGHNESS RELAXATION TIME RENEWABLE ENERGY RENEWABLE ENERGY SOURCES RESIDUAL ENTHALPY RESIDUAL GIBBS ENERGY RESINS RESISTANCE HEATING RESISTANCE THERMOMETERS RESISTANCE THERMOMETRY RESISTANCE, ELECTRICAL RESISTIVITY, ELECTRICAL RESONANCE FLUORESCENCE RETENTATE RETENTION INDEX RETROGRADE CONDENSATION RETURN TO NUCLEATE BOILING REVERSE OSMOSIS REVERSED HEAT ENGINE CYCLES REVERSIBILITY PRINCIPLE REVERSIBLE PROCESSES REWETTING REWETTING OF HOT SURFACES REYNOLDS ANALOGY Reynolds Number REYNOLDS NUMBER, CRITICAL, IN TUBES REYNOLDS STRESS REYNOLDS STRESS TRANSPORT MODELS REYNOLDS' AVERAGING REYNOLDS' EQUATIONS REYNOLDS, OSBORNE (1842-1912) RHEOLOGY RHEOMETERS RHEOPEPTIC FLUIDS RHODAMINE RICCATTY-BESSEL FUNCTIONS RICHARDSON NUMBER RIDEAL-ELEY MODEL, FOR HETEROGENEOUS CATALYSIS RIEDEL-PLANK-MILLER EQUATION RIEMAN WAVES RIGHT-ANGLE TRIANGULAR ENCLOSURE RIGID-WALLED CHANNEL RISC. REDUCED INSTRUCTION SET COMPUTER RISK ANALYSIS TECHNIQUES RISK ASSESSMENT ROASTING ROCKET PROPELLANTS ROCKETS ROD BAFFLES ROD BUNDLE TESTS ROD BUNDLES, FLOW IN ROD BUNDLES, HEAT TRANSFER IN ROD BUNDLES, PARALLEL FLOW IN ROD CLIMBING ROD-STABILIZED LAMINAR PREMIXED FLAME RODRIGUES FORMULA ROHRSCHNEIDER CONSTANT ROLL MOMENT ROLL WAVES ROOTS TYPE COMPRESSOR ROSIN-RAMMLER ROSIN-RAMMLER SIZE DISTRIBUTION ROSSBY NUMBER ROSSELAND COEFFICIENT ROTAMETERS ROTARY ATOMIZERS ROTARY DRYERS ROTARY KILNS ROTARY REGENERATORS ROTATED TUBE BANKS ROTATING CHANNEL WITH RIBS ROTATING CYLINDERS, CRITICAL SPEED ROTATING CYLINDERS, FLOW BETWEEN ROTATING CYLINDERS, FLOW OVER ROTATING DISC CONTACTOR ROTATING DISC SYSTEMS, APPLICATIONS ROTATING DISC SYSTEMS, BASIC PHENOMENA ROTATING DUCT SYSTEMS, ORTHOGONAL, HEAT TRANSFER IN ROTATING DUCT SYSTEMS, PARALLEL, HEAT TRANSFER IN ROTATING FLOW IN A POROUS LAYER ROTATING FLOW PASSAGE ROTATING PIPE FLOW ROTATING SURFACES ROTATIONAL DISCONTINUITIES Rotational Rayleigh number ROTATIONAL REYNOLDS NUMBERS ROUGH CHANNELS, FRICTION FACTOR IN ROUGH SURFACE FRICTION FACTORS ROUGH SURFACES ROUGH TUBES ROUGH TUBES, FLOW IN ROUGH TUBES, HEAT TRANSFER IN ROUGHNESS FACTORS ROYAL ACADEMY OF ENGINEERING, RAE ROYAL SOCIETY OF CHEMISTRY ROYAL SOCIETY, RS RS RSC RUBBER RUMFORD, COUNT, BENJAMIN THOMPSON RUSHTON TURBINE
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ROTATING DUCT SYSTEMS, PARALLEL, HEAT TRANSFER IN

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Introduction

When a straight duct rotates about an arbitrary axis, rotation influences an internal pressure-driven flow via Coriolis and centripetal forces. These forces must be included in the momentum conservation equations in order to predict the resulting flow field as a precurser to the evaluation of heat transfer. Coriolis forces generate relative vorticity in the duct via a vector product coupling of the angular velocity and the flow field. In other words complex secondary flows may be generated. With unheated flow, the centripetal force field is hydrostatic. However, with heated flow, where the density of the fluid is temperature dependent, a centripetal buoyancy force is created analogous to natural convection in the earth’s gravitational field. The result of this is a mechanism which causes the warmer fluid (i.e., less dense) to move towards the axis of rotation. Again this will modify the nonrotating flow field and the heat transfer. A detailed description of how Coriolis and centripetal buoyancy terms may be incorporated into the momentum conservation equations is given by Morris (1981).

Although an infinite variety of rotating duct systems may be envisaged, reflecting the relative location of the duct to the axis of rotation, two important configurations have been seriously researched. These are commonly referred to as "parallel-mode" and "orthogonal-mode" rotation, respectively. With parallel-mode rotation the axis of the duct is parallel to, but displaced from the axis of rotation. Similarly, with orthogonal-mode rotation the axis of the duct and the axis of rotation are mutually perpendicular. In this entry heat transfer with parallel-mode rotation will be discussed. The case of orthogonal-mode rotation is discussed in the previous entry.

Parallel-Mode Rotation

Figure 1 shows a typical straight duct rotating in the parallel-mode. Theoretical and experimental studies of heat transfer with this geometry have been extensively reviewed by Morris (1980) for laminar and turbulent flow and readers are referred to this text for full details. A brief summary of heat transfer data for circular-sectioned ducts follows by way of illustration of the general results.

Developed Laminar Flow

With developed laminar flow, a numerical solution for the Nusselt Number for the special case of very high Prandtl Numbers, Woods and Morris (1981), has been shown to be quite satisfactory even when applied to fluids with a Prandtl number as low as unity. Figure 2 shows a comparison of this high Prandtl number solution for heat transfer enhancement with experimental data obtained with glycerol, water and air. In this context enhancement is defined as the ratio of the developed Nusselt number with rotation, NuR,∞ with the stationary counterpart, Nuo,∞. This figure is recommended for assessing the effect of rotation with gases and liquids. Note that the effect of Coriolis force vanishes identically and rotation influences heat transfer via a centripetal buoyancy mechanism alone. The enhancement in heat transfer relative to the stationary case is dependent on the product of the through flow Reynolds Number, Re, a rotational Rayleigh number Rar, (which uses the product of the axial wall temperature gradient and the tube radius as a characteristic temperature difference in the customary definition) and the Prandtl number, Pr.

Parallel mode rotating geometry.

Figure 1. Parallel mode rotating geometry.

Fully developed laminar flow.

Figure 2. Fully developed laminar flow.

Developing Laminar Flow

With developing flow, Coriolis forces are produced due to an interaction between the axial velocity gradient and the rotation. In this region, when the heating rates are moderate, buoyancy probably has a lesser influence and rotation may be quantified nondimensionally in terms of a rotational Reynolds number, Ja, (Ωd2ρ/4η) which may be thought of as a ratio of Coriolis to viscous forces. Morris and Woods (1978) have presented experimental data for two parallel-mode rotation geometries and recommended the following empirical correlations for the mean rotating Nusselt number, NuR,m

These equations were based on experiments with Ja values up to 150. At high rotational speeds, where buoyancy will become more important, it is not expected that these correlations will be particularly good. Figure 3 compares the actual test data with these correlations. Further details are available from Morris (1981).

Developed Turbulent Flow

With turbulent flow rotation still modifies the flow field to produce heat transfer enhancement. Nakayama (1968) used a momentum integral analysis to calculate developed turbulent Nusselt number and the following equations for determining this enhancement in heat transfer were proposed.

Case 1: Gas-like fluids with Pr close to unity.

where

Case 2: Liquid-like fluids with Pr > 1

where

Developing Turbulent Flow

As with laminar flow, the effect of buoyancy has been found to be small in the turbulent entrance regions, see Morris (1981).

Mean heat transfer with developing flow.

Figure 3. Mean heat transfer with developing flow.

Mean heat transfer with developing flow.

Figure 4. Mean heat transfer with developing flow.

A relatively small number of experimental studies have been reported for air and Morris and Woods (1978) proposed the following correlation, very similar to that suggested for laminar flow, with which to estimate the mean rotating Nusselt number:

Figure 4 compares this equation with their actual experimental data. The data appears to be relatively insensitive to the aspect ratio of the tube and also the eccentricly. There is evidence, however, that this equation can overpredict the enhancement in heat transfer when compared to some data, see Morris (1981).

REFERENCES

Nakayama, W., (1968) Forced convective heat transfer in a straight pipe rotating about a parallel axis (turbulent region) Int. J. Heat Mass Trans., 11, 1185. DOI: 10.1016/0017-9310(68)90034-3

Morris, W. D. and Woods, J. L. (1978) Heat transfer in the entrance region of tubes that rotate about a parallel axis, J. Mech. Eng. Sci., 20–6, 1185 .

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.

Woods, J. L., and Morris, W. D. (1980) A study of heat transfer in a rotating cylindrical tube., Trans. A.S.M.E., J. Heat Trans., 102, 4, 612.

References

  1. Nakayama, W., (1968) Forced convective heat transfer in a straight pipe rotating about a parallel axis (turbulent region) Int. J. Heat Mass Trans., 11, 1185. DOI: 10.1016/0017-9310(68)90034-3
  2. Morris, W. D. and Woods, J. L. (1978) Heat transfer in the entrance region of tubes that rotate about a parallel axis, J. Mech. Eng. Sci., 20–6, 1185 DOI: 10.1243/JMES_JOUR_1978_020_057_02.
  3. 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.
  4. Woods, J. L., and Morris, W. D. (1980) A study of heat transfer in a rotating cylindrical tube., Trans. A.S.M.E., J. Heat Trans., 102, 4, 612.

Following from:

Channel Flow
TUBES, SINGLE-PHASE HEAT TRANSFER IN

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ROTATING DISC SYSTEMS, BASIC PHENOMENA
ROTATING DUCT SYSTEMS, ORTHOGONAL, HEAT TRANSFER IN

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