Showing papers on "Nusselt number published in 1969"

A study of Bénard convection with and without rotation

Abstract: An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.

Evaluation of internal heat-transfer coefficients for impingement-cooled turbine airfoils.

Abstract: Although internal impingement cooling of the leading edge of gas-turbine airfoils has been shown to be effective, previously available heat-transfer data are not generally applicable to present-day turbine designs because of the unique geometry requirements. An experimental system which allows the ready acquisition of heat-transfer data necessary for thermal design of turbine airfoils is described. A cold-flow model is developed, and the measurement of local heat-transfer coefficients on a full-size model is accomplished by considering Joulean dissipation in very thin platinum strips bonded to the model. Heattransfer results are given which show the dependence of Nusselt number on Reynolds number, geometry, and chordwise location on the inside leading-edge region of the airfoil. Dimensionless correlations are presented which allow the designer to predict heat transfer for impingement cooling in these geometries for the range of parameters tested.

On Howard's upper bound for heat transport by turbulent convection

Abstract: The variational problem introduced by Howard (1963) for the derivation of an upper bound on heat transport by convection in a layer heated from below is analyzed for the case in which the equation of continuity is added as constraint for the velocity field. Howard's conjecture that the maximizing solution of the Euler equations is characterized by a single horizontal wave-number is shown to be true only for a limited range of the Rayleigh number, Ra. A new class of solutions with a multiple boundary-layer structure is derived. The upper bound for the Nusselt number, Nu, given by these solutions is Nu ≤ (Ra/1035)½ in the limit when the Rayleigh number tends to infinity. The comparison of the maximizing solution with experimental observations by Malkus (1954a) and Deardorff & Willis (1967) emphasizes the similarity pointed out by Howard.

Natural Convection between Concentric Vertical Cylinders

Abstract: The motion of a fluid in the closed annular cavity formed by two concentric vertical cylinders and two horizontal planes has been analyzed by a numerical solution of the equations of motion and energy using a high‐speed digital computer. The motion is generated by a radial density gradient caused by the thermal boundary conditions which are, typically: inner cylinder at a (dimensionless) temperature of unity; outer cylinder at a temperature of zero; horizontal boundaries adiabatic. The fluid is assumed to have constant thermodynamic and transport properties except for the density, which is temperature‐dependent in the buoyancy term of the equation of vertical motion (the Boussinesq approximation); the flow is assumed to be axisymmetric. The equations of time‐dependent motion have been solved, so that both transient and steady‐state solutions are obtained. The parameters of the problem, and the respective ranges of values which have been considered, are: Rayleigh number (based on gap width) up to 2 × 105; Prandtl number 0.5 to 5; radius ratio 1 to 4; aspect ratio (cavity height/gap) 1 to 20. At moderate Rayleigh numbers the motion consists of a single cell (i.e., torus), while at higher Rayleigh numbers the onset of a multicellular motion is observed. The local and average Nusselt numbers, of interest in determining the insulating value of an annular cavity, have been obtained.

Transport of heat and mass between liquids and spherical particles in an agitated tank

Abstract: Data are reported for heat transfer from water to melting ice spheres and for mass transfer in the case of dissolving spheres of pivalic acid suspended in water agitated in a stirred vessel. The transport coefficients are found to depend on agitator power input but not on agitator design, in agreement with the Kolmogoroff theory. These experimental results are used with others in the literature to develop a correlation involving Nusselt and Prandtl or Schmidt numbers together with a dimensionless group involving agitation power. The correlation is essentially independent of solid-liquid density ratio in the range 0.8 to 1.25, and in this range the gravity group also appears to be unimportant.

The effect of initial conditions and lateral boundaries on convection

Abstract: A theoretical analysis of two-dimensional, finite-amplitude, thermal convection is made for a fluid which has an infinite Prandtl number. The vertical velocity disturbance is expanded in a double Fourier series which satisfies the horizontal and lateral boundary conditions. The resulting coupled sets of non-linear differential equations are solved numerically. It is found that for a particular Rayleigh number the number and size of the convection cells that form depend upon the ratio of the distance between the lateral boundaries to the depth of the fluid layer and on the initial conditions. The steady-state solutions are not unique and the solution for which the heat transport is a maximum is not necessarily the solution that results. Where there are no lateral boundaries, the lateral edges of the cells tend to tilt and the Nusselt number increases slightly.

Thermal Entry for Low Reynolds Number Turbulent Flow

Abstract: Thermal entry problem solution for low Reynolds number turbulent gas flow based on Reynolds number dependent velocity profile

Some Gasdynamic Problems in the Flow of Condensing Vapors

Abstract: Some Gasdynamic Problems in the Flow of Condensing Vapors. The general problem of the flow of a wet vapor, with or without an inert diluent is formulated under the assumption that the liquid phase is finely divided and dispersed throughout the gaseous component in droplets whose radii are nearly constant in any local region. The processes of momentum transfer, heat transfer between phases are assumed to take place according to Stokes law and Nusselt number of unity, respectively. The mass transfer process is treated as diffusion governed in the presence of an inert diluent and kinetic governed for two phases of a pure substance. The physical understanding of such problems, in contrast with those of conventional gas dynamics, rests largely in the role played by the relaxation times or equilibration lengths associated with these three processes. Consequently, both simple and coupled relaxation processes are examined rather carefully by specific examples. Subsequently, the problem of near-equilibrium flow in a nozzle with phase change is solved under the small-slip approximation. The structure of the normal shock in a pure substance is investigated and reveals three rather distinct zones: the gasdynamic shock, the vapor relaxation zone, and the thermal and velocity equilibration zone. The three-dimensional steady flow of the two-phase condensing continuum is formulated according to first order perturbation theory, and the structure of waves in such supersonic flow is examined. Finally, the attenuation of sound in fogs is formulated and solved accounting for the important effects of phase change as well as the viscous damping and heat transfer which have been included in previous analyses.

Statistical theory of thermal convection at large Prandtl number.

Abstract: Three statistical theories of turbulence are applied to thermal convection between infinite slippery plates in the limit of infinite Prandtl number. The range of Rayleigh number R investigated is 657 ≤ R ≤ 1.25 × 104. The theories, which are compared in a series of numerical calculations, are the direct interaction approximation, the quasilinear approximation, and the quasinormal approximation. The direct interaction approximation gives results for the evolution Nusselt number in good agreement with some simple exact solutions to the problem. The flow predicted by this method is very persistent. Turbulence first appears just at critical R and its intensity gradually increases with increasing R. The quasinormal approximation gives satisfactory results for R ≲ 2000, but some of the initial value problems lead to an unphysical negative temperature autocorrelation spectrum for larger R. In none of the initial value problems for R > 2 × 103 did the quasinormal procedure yield a sensible stationary state. The q...

Effect of Vibration on Heat Transfer From Spheres

Abstract: The effect of vibration on free and forced convection heat transfer from spheres was investigated. Test spheres made of copper were subjected to sinusoidal vibration in the vertical plane, this being perpendicular to the direction of airstream in the case of forced convection studies. In free convection studies the amplitude of vibration was varied from 4 mm to 25.5 mm and the frequency of vibration from 150 cpm to 930 cpm. It was found that the effect of vibration on Nusselt number was negligible for values of vibrational Reynolds number less than 200. For values of vibrational Reynolds number greater than 200, the vibration increased the heat transfer coefficient considerably and values of heat transfer coefficients as high as seven times the free convection values without vibration were obtained. The following correlations were obtained for heat transfer from spheres to air: free convection without vibration, NNu = 2 + 0.401 (NGr)0.25 for 4 × 103 < NGr < 6 × 104 and free convection with vibration: hvho = 0.83 (NRe)v0.5(a/D)0.1(NGr)0.251.28 In the case of forced convection studies with vibration, the amplitude of vibrations varied between 4 mm and 12.4 mm, and the frequency of vibration from 200 cpm to 1600 cpm. The flow velocity was varied from 24.5 ft/sec to 84 ft/sec. The results in the absence of vibration could be represented by: NNu = 0.304 (NRe )0.56 or NNu = 2 + 0.222 (NRe )0.587 in the range 6 × 103 < NRe < 3.3 × 104 . Nusselt numbers were not found to be affected by the imposition of vibrational velocity even as high as 19.6 percent of the flow velocity.

Heat Transfer in the Oscillating Turbulent Boundary Layer

Abstract: Measurements of local heat transfer coefficients in the fully established oscillating turbulent boundary layer over a flat plate are reported. In the range of frequencies from 0.1 to 200 cps and amplitudes from 8 to 92 percent of the freestream mean velocity increases in local Nusselt numbers of 3 to 5 percent were found. It is concluded that substantial increases in local coefficients sometimes reported in oscillating flows of low standing wave ratio may be traced to reduced transition Reynolds numbers.Copyright © 1969 by ASME

The operation of a rotating, wickless heat pipe

Abstract: : A Nusselt type analysis was performed during laminar film condensation on the inside of a rotating truncated cone. This analysis was employed to determine the condensing limit of a wickless heat pipe, rotating about its longitudinal axis. Performance characteristics including the effects of geometry, rotational speed, and the characteristics of fluid are given. A comparison is made between the condensing, boiling, sonic, and entrainment limits for a given heat pipe geometry.

The Use of Pitot-Static Tubes and Hot-Film Anemometers in Dilute Polymer Solutions

Abstract: The operation of Pitot-static tubes in dilute polyacrylamide-water solutions was verified experimentally by using an independent method to measure the velocity. Measurement of the heat transfer characteristics of cylindrical hot-film anemometer probes in these solutions revealed reduced heat transfer in addition to the following anomalous results. In certain velocity ranges, the Nusselt number was independent of velocity. At high velocities, a form of vortex shedding from the probe occurred, and the Nusselt number was proportional to the square-root of velocity. The dependence of the Nusselt number on the angle of incidence was, for the most concentrated solution, opposite to that for pure water. The above results indicate that these probes cannot be used to measure velocity in certain cases, and interpretation of turbulence measurements would seem to be open to question. Calibration of a conical hot-film probe revealed only slightly reduced heat transfer. Measurement of the one-dimensional velocity spectrum in a polymer solution indicated that the cylindrical probe gave a power-law dependence on the wave number at high wave numbers; whereas the data obtained with the conical probe was qualitatively similar to Newtonian spectral data.

Laminar forced convection in elliptic ducts

Abstract: The problem of laminar forced convection heat transfer in short elliptical ducts with (i) uniform wall temperature and (ii) prescribed wall heat flux is examined in detail with the well known Leveque theory of linear velocity profile near the wall. Moreover, consideration is given to the variation of the slope of the linear velocity profile with the position on the duct wall. A correction factor for the temperature dependent viscosity is included. Expressions for the local and average Nusselt numbers and wall temperatures are obtained. For the case of constant heat flux the Nusselt numbers are higher than for constant wall temperature. The results corresponding to the classical Graetz and Purday problems are deduced as special cases.

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

Abstract: A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.