Showing papers on "Nusselt number published in 1982"

The effects of significant viscosity variation on convective heat transport in water-saturated porous media

Abstract: Weakly nonlinear theory and finite-difference calculations are used to describe steadystate and oscillatory convective heat transport in water-saturated porous media. Two-dimensional rolls in a rectangular region are considered when the imposed temperature difference between the horizontal boundaries is as large as 200 K, corresponding to a viscosity ratio of about 6·5. The lowest-order weakly nonlinear results indicate that the variation of the Nusselt number with the ratio of the actual Rayleigh number to the corresponding critical value R/Rc, is independent of the temperature difference for the range considered. Results for the Nusselt number obtained from finite-difference solutions contain a weak dependence on temperature difference which increases with the magnitude of R/Rc. When R/Rc = 8 the constantviscosity convection pattern is steady, while those with temperature differences of 100 and 200 K are found to oscillate.

Direct numerical simulation of laminar and turbulent Bénard convection

Abstract: The TURBIT-3 computer code has been used for the direct numerical simulation of Benard convection in an infinite plane channel filled with air. The method is based on the three-dimensional non-steady-state equations for the conservation of mass, momentum and enthalpy. Subgrid-scale models of turbulence are not required, as calculations with different grids show that the spatial resolution of grids with about 322 × 16 nodes provides sufficient accuracy for Rayleigh numbers up to Ra = 3·8 × 105. Hence this simulation model contains no tuning parameters.The simulations start from nearly random initial conditions. This has been found to be essential for calculating flow patterns and statistical data insensitive to grid parameters and agreeing with experimental experience. The numerical results show the theoretically predicted ‘skewed varicose’ instability at Ra = 4000. Warm and cold ‘blobs’ are identified as causing temperature-gradient reversals for all the high Rayleigh numbers under consideration. The calculated wavelengths and the corresponding flow regimes observed in the transition range confirm the stability maps determined theoretically. In the turbulent range the wavelengths agree qualitatively with low-aspect-ratio experiments. Accordingly, the Nusselt numbers lie at the upper end of the scatter band of experimental data, as these also depend on the aspect ratio. Appropriately normalized, the velocity and temperature fluctuation peaks are independent of the Rayleigh number. The vertical profiles agree largely with experimental data and, especially in case of temperature statistics, exhibit comparable or less scatter.

Heat transfer in flow through a porous medium bounded by an infinite vertical plate under the action of a magnetic field

Abstract: An analysis of steady two-dimensional flow, of an electrically conducting fluid through a porous medium, occupying a semi-infinite region of the space bounded by an infinite, vertical and porous limiting surface under the action of a transverse magnetic field is considered. Approximate solutions have been derived for the velocity and temperature fields and the rate of heat transfer (Nusselt number). The variations of the velocity and Nusselt number are shown on graphs.

The effects of a strongly temperature-dependent viscosity on slow flow past a hot sphere

Abstract: This work determines analytically the drag on, and heat flux out of, a hot sphere that translates steadily in a fluid of strongly temperature-dependent viscosity. There is no dissipative heating. The essentials are illustrated by an exact solution for the flow induced by slowly squeezing two parallel planes together. The lower plane is hot and stationary; the upper is cold and advances in a direction normal to itself at uniform speed U. The gap is completely filled by a fluid of strongly temperature-dependent viscosity. We find the temperature and velocity profiles, and determine the Nusselt number N and Peklet number P as functions of the normal force D on the lower plane. The large viscosity variation tries to concentrate the flow into a relatively thin softened layer in which the viscosity is of order its value μ0 at the hot plane. In the limit of infinite viscosity ratio (fixed P), it succeeds (lubrication limit): if P [Lt ] 1, the width of the softened layer is determined by conduction and D ∝ μ0U; but D ∝ μ0U4 when forced convection is important. If P → ∞ (fixed viscosity ratio), the softened layer is so thin that it chokes, and all the deformation occurs outside the thermal layer in the fluid of uniform viscosity μ∞ (Stokes limit); then D ∝ μ∞U. These mechanisms appear as three distinct legs in our plot of log P against log D. There are similar transitions in the plot of log N against log D. The solution gives an estimate of the drag on a sphere. We test this estimate against an analytical solution for the sphere in the lubrication limit. Then we extend the solution to cover power-law fluids, and apply it to a model (by Marsh) of magma transport beneath island-arc volcanoes. The results suggest that the magma covers the first 50 km of its ascent by an isoviscous mechanism, with the lubrication mechanism operating in the remaining 50 km. To open a fresh pathway from the source to the surface takes about 106 years and uses about 1027 erg.

Heat transfer by free convection between two parallel flat plates

Abstract: A few numerical analyses of the free convection between two heated parallel plates have been carried out without using the boundary-layer approximation. In this paper, an approach that can determine the flow rate with consideration of the pressure drop is proposed. As an example of the calculation, the method is applied to Kettleborough's model. In addition, a comparison is made with Kettleborough's and Aihara's results.

Thermoregulation and the determinants of heat transfer in Colias butterflies.

Abstract: As a means of exploring behavioral and morphological adaptations for thermoregulation in Colias butterflies, convective heat transfer coefficients of real and model butterflies were measured in a wind tunnel as a function of wind speed and body orientation (yaw angle). Results are reported in terms of a dimensionless heat transfer coefficient (Nusselt number, Nu) and a dimensionless wind speed (Reynolds number, Re), for a wind speed range typical of that experienced by basking Colias in the field. The resultant Nusselt-Reynolds (Nu-Re) plots thus indicate the rates of heat transfer by forced convection as a function of wind speed for particular model geometries.For Reynolds numbers throughout the measured range, Nusselt numbers for C. eurytheme butterflies are consistently lower than those for long cylinders, and are independent of yaw angle. There is significant variation among individual butterflies in heat transfer coefficients throughout the Re range. Model butterflies without artificial fur have Nu-Re relations similar to those for cylinders. Heat transfer in these models depends upon yaw angle, with higher heat transfer at intermediate yaw angles (30-60°); these yaw effects increase with increasing Reynolds number. Models with artificial fur, like real Colias, have Nusselt numbers which are consistently lower than those for models without fur at given Reynolds numbers throughout the Re range. Unlike real Colias, however, the models with fur do show yaw angle effects similar to those for models without fur.The independence of heat loss from yaw angle for real Colias is consistent with field observations indicating no behavioral orientation to wind direction. The presence of fur on the models reduces heat loss but does not affect yaw dependence. The large individual variation in heat transfer coefficients among butterflies is probably due to differences in fur characteristics rather than to differences in wing morphology.Finally, a physical model of a butterfly was constructed which accurately simulates the body temperatures of basking Colias in the field for a variety of radiation and wind velocity conditions. The success of the butterfly simulator in mimicking Colias thermal characteristics confirms our preliminary understanding of the physical bases for and heat transfer mechanisms underlying thermoregulatory adaptations in these butterflies.

Heat transport and temporal evolution of fluid flow near the Rayleigh-Bénard instability in cylindrical containers

Abstract: First this paper describes in detail an apparatus for heat-transport measurements in shallow horizontal layers of fluid at low temperatures Then high-precision results of convective heat transport as a function of the Rayleigh number R are presented for cylindrical cells of aspect ratio L = 208,472 and 57 The present paper concentrates on the long-time behaviour of Boussinesq systems Non-Boussinesq effects, transient effects near the convective onset, and time-dependent states are described elsewhere (Walden & Ahlers 1981 Ahlers et al 1981 Ahlers 1980b and references therein) The measurements show that the convective onset near the critical Rayleigh number Rc is sharp within the experimental resolution of about 01 % of the Nusselt number N even in laterally finite containers Values of R and of the initial slopes of N(R), are obtained and compared with predictions for different flow patterns Over a wider range of R and for L = 57 and 472, N was found within experimental resolution to be a unique, continuous function of R For L = 208, hysteretic transitions are revealed by N(R) near R ≈ 3 and R ≈ 10 For L = 472, the effect of impulsive heating was studied and revealed complicated, long-lived, but surprisingly repro- ducible transients

Laminar natural convection along vertical square ducts

Abstract: Laminar natural convection of air along a vertical square duct open at both ends has been investigated for constant temperature and constant heat flux boundary conditions on the duct walls. The velocity of entering air at the bottom of the duct is assumed uniform at atmospheric pressure or the acceleration head of the entering fluid. The three-dimensional differential equations for momentum and energy have been solved numerically by a strongly implicit finite-difference procedure. The axial development of flow and heat transfer rates are presented, A curve fit for the mean heat transfer coefficient in the Rayleigh number range of 10 to 1000 results in the following relations: Constant wall temperature: Num = 0.512Ra0.285 Constant heat flux: Num = 0.53 Ra0.3 Comparison of the computed heat transfer rates with the measurements of W. Elen-bass for constant wall temperature square ducts show very good agreement over the entire Rayleigh number range of investigation.

Turbulent Heat Transfer in the Entrance Region of a Tube

Abstract: Correlation of the data for heat transfer between a fluid in turbulent flow and the entrance region of a tube is made for the entrance shapes normally used in heat exchangers. Equations representing the variation of the “average heat transfer” with tube length, Reynolds number, and Prandtl number are suggested.

Heat Transfer around Tubes in Staggered Tube Banks

Abstract: An experimental study was conducted to investigate the heat transfer and flow around tubes in staggered tube banks in a cross flow of air. The cylinder spacings examined Cy/d x Cx/d are 1.6 x 1.6 and 1.2 x 1.2 where Cx and Cy denote the cylinder spacings along and normal to the upstream uniform flow direction, and d the cylinder diameter, respectively. Main results obtained are as follows. In case of 1.2 x 1.2, the mean Nusselt numbers for the second and subsequent cylinders are almost equal to each other and much higher than that for the first cylinder. In case of 1.6 x 1.6, the third cylinder exhibits the maximum heat transfer rate. This may be due to the high velocity of an oncoming flow to that along with its high turbulence intensity.

Analysis of laminar non-Newtonian flow and heat transfer in curved tubes

Abstract: The governing equations for the laminar fully developed flow and heat transfer in curved tubes are solved numerically for a power-law fluid. Results for the velocity and temperature fields, the friction factor, and the Nusselt number are presented for different values of the Dean number, the Prandtl number, and the power-law index. The friction factor results are compared with available experimental data.

Combined Forced and Free Convection in the Entrance Region of an Isothermally Heated Horizontal Pipe

Abstract: Numerical solutions are given without the aid of a large Prandtl number assumption for combined forced and free laminar convection in the entrance region of a horizontal pipe with uniform wall temperature. The steady-state solutions have been obtained from the asymptotic time solutions of the time-dependent equations of momentum and energy with the Poisson equation for pressure. Results are presented for the developing primary and secondary velocity profiles, developing temperature fields, local wall shear stress, and local and average Nusselt numbers, which reveal how the developing flow and heat transfer in the entrance region are affected by the secondary flow due to buoyancy forces.

Internal heat transfer coefficients of porous metals

Abstract: The internal heat transfer coefficients of porous metals have been experimentally determined in order to develop correlations between approximately defined Nusselt and Reynolds numbers. Scaled-up models of porous materials, and actual porous metal specimens, were subjected to countercurrent heat and mass transfer boundary conditions. Solid and gas phase temperatures were measured for both the scaled-up models and the actual porous metal specimens. On the basis of these measurements, the average internal heat transfer coefficient was evaluated, and a correlation between the Nusselt and Reynolds numbers was derived.

Time-dependent solutions of multimode convection equations

Abstract: Truncated modal equations are used to study the time evolution of thermal convection. In the Boussinesq approximation these nonlinear equations are obtained by expanding the fluctuating velocity and temperature fields in a finite set of planforms of the horizontal coordinates. Numerical studies dealing with two or three modes with triad interactions are discussed. Rich time dependence was found in these cases: periodic and aperiodic solutions can be obtained, along with various steady solutions. Three-mode solutions reproduce the qualitative appearance of spoke-pattern convection as observed in experiments at high Prandtl numbers. Though the values of the periods of the time-dependent solutions do not agree with those of the experiments, their variation with Rayleigh number compares favorably. Except at the highest Rayleigh number considered (10,000,000), the theoretical Nusselt numbers agree well with experiment.

Welding, Glazing, and Heat Treating —A dimensional analysis of heat flow

Abstract: Through a rigorous mathematic demonstration and the application of general enthalpy equations, the key dimensionless variables that characterize heat flow during welding, glazing, and heat treating of a workpiece were systematically presented. Both the 2-dimensional and the 3-dimensional heat flows due to a moving heat source were considered. The 2-dimensional heat flow during the welding of thin plates with a stationary, instantaneous heat source was also analyzed. While the primary dimensionless variables such as the dimensionless temperature, the Fourier number, and the dimensionless distances were sufficient to describe the heat flow during heat treating, additional primary dimensionless variables such as the dimensionless heat input, the Stephan number, the dimensionless thermal conductivity, and the dimensionless specific heat were found necessary to define the heat flow during welding and glazing. The validity of such a dimensional analysis was verified by existing analytical solutions. Due to the additional heat flow variables such as the size of the heat source, the size of the workpiece, the surface heat loss, and the freezing range of alloy systems, secondary dimensionless variables including the dimensionless size of the heat source, the dimensionless width and thickness of the workpiece, the Biot number, and the dimensionless liquidus temperature were presented and discussed. The results of heat flow calculations involving both the surface heat treating of a substrate with a square laser beam and the gas tungsten-arc full-penetration welding of 5052 and 2014 aluminum alloys were presented using the dimensionless variables introduced in the present study.

Low Rayleigh Number Thermal Convection in a Vertical Cylinder Filled With Porous Materials and Heated From Below

Abstract: Low Rayleigh number (R < 500) thermal convection is examined experimentally and analytically for a vertical circular cylinder filled with saturated porous materials. The cylinder is heated from below and cooled from above. The upper boundary is permeable to flow. Results are reported for the critical Rayleigh number at the onset of convection (Rc) and the Nusselt (Nu) vs. Rayleigh (R) number correlation. The convective structure is deduced from temperature measurements. The lateral walls have a stabilizing effect and tend to increase Rc relative to the case of a laterally unbounded porous layer. The preferred convective patterns in the cylinder tend to be nonaxisymmetric. The experimental results are in good agreement with analytic predictions. A linear stability analysis is used to calculate Rc and the structure of the convective modes for a range of aspect ratios (radius/height). An integral technique is applied to obtain heat transfer rates for 1 < R/Rc < 5.5. 20 references.