Showing papers on "Nusselt number published in 1975"

Electro-convection in a dielectric liquid layer subjected to unipolar injection

Abstract: The problem of electric charge convection in a dielectric liquid layer of high ionic purity, when subjected to unipolar injection, is in many ways analogous to that of thermal convection in a horizontal fluid layer heated from below, although no formal analogy can be established. The problem treated is intrinsically more nonlinear than the thermal problem. We consider two asymptotic states of convection: one where the whole motion is dominated by viscosity, and one where inertial effects dominate. In each state, two or three spatial regions are distinguished. From the approximate equations that hold in the different regions, information about the variation of the different quantities with distance from the injector is obtained, and further approximations permit us to establish the dependence of the current density ratio I/I0 (called the electric Nusselt number) on the stability parameter T = M2R = eϕ0/Kρν, and on 1/R = ν/Kϕ0, which is an equivalent Prandtl number (e is the permittivity, ρ the fluid density, K the mobility, ν the kinematic viscosity, and ϕ0 the applied voltage). In the viscous state, the analysis gives I/I0 ∞ T½; in the inertial state the law I/I0 ∞ (T/R)1/4 = M½ is obtained. Since M is independent of the applied voltage, the latter law shows the saturation in the electric Nusselt number observed in earlier experiments. The transition in the states is associated with a transition number (MR)T [gap ] 30, which is an electric Reynolds number, related to an ordinary Reynolds number of about 10.The experimental results, obtained in liquids of very different viscosities and dielectric constants, verify these theoretical predictions; further, they yield more precise numerical coefficients. As for the transition criteria, the experiments confirm that the viscous and inertial effects are of the same order when Re [gap ] 10. It was also possible to determine roughly the limits of the viscous and inertial states. The viscous analysis remains valid up to a Reynolds number of about 1; the inertial state can be considered valid down to a Reynolds number of 60. Schlieren observations show that the motion has the structure of very stable hexagonal cells at applied voltages just above the critical voltage, which are transformed into unstable filaments when the voltage is increased further. At even higher voltages, the motion finally breaks down into turbulence. It may be of interest to point out that, when M < 3, the electric Nusselt number approaches 1, which is equivalent to the situation in thermal convection at low Prandtl numbers.

Thermoconvective instabilities in a horizontal porous layer

Abstract: Experimental investigations of natural convection in a porous layer placed between two horizontal and isothermal plane surfaces have revealed a new type of convection as the Rayleigh number Ra* increases: fluctuating convection. A numerical study carried out on a two-dimensional model in order to simulate this phenomenon shows that, apart from the influence of the Rayleigh number, the aspect ratio A (length/height) of a vortical cell is the most important parameter for the occurrence of this type of convection. These quasi-periodic fluctuations induce important variations in the temperature field and in the streamlines. The total heat transport, as defined by the Nusselt number Nu*, varies within limits which may be separated by 80% of the mean value. Using the Galerkin method it is possible to deduce the conditions for the onset of convection from a state of pure conduction and also to define the critical conditions for the development of fluctuating convection from another perturbed state. A physical interpretation of the results is given for each type of convection. The results seem to agree with the experimental and numerical results obtained by different authors.

Free convection in low-temperature gaseous helium

Abstract: Free convection has been studied in gaseous helium at low temperatures in a cylindrical vessel of aspect ratio (diameterlheight) 2·5. Compared with measurements in fluids at room temperature the present arrangement has the advantages of small size, a short time constant and improved accuracy. As the Rayleigh number was varied from 60 to 2 × 109, the Nusselt number rose from 1 to 69, obeying the relation Nu = 0·173Ra0·2800±0·0005 over the upper four decades of Ra. The critical Rayleigh number was 1630, but the conditions of the experiment did not allow reliable measurements at such low values of Ra. The very high sensitivity within a given experiment showed the presence of several ‘discrete transitions’, which were often step like and not merely a change of gradient as reported by other workers. The largest of these, at Ra = 3 · 105, involved a drop in heat flux of some 6% and was somewhat hysteretic. The temperature fluctuations increased markedly as the step was crossed.

Natural convection in a porous medium - Effects of confinement, variable permeability, and thermal boundary conditions

Abstract: Two-dimensional numerical calculations are reported for natural convection of a fluid in a porous, horizontal layer heated from below. Effects of the following parameters are examined: rigid (impermeable) and constant-pressure (permeable) upper boundaries; isothermal and uniform heat flux at the lower boundary; and permeabilities which are constant, or which vary with depth to simulate compaction of a porous medium or property variations of real fluids within the medium. Steady-state results are presented for the heat flux distribution on the upper surface, as well as for flow and temperature fields in the interior.

Modal equations for cellular convection

Abstract: We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possible useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of 'modal equations' is analyzed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.

Numerical simulation of two-dimensional compressible convection

Abstract: A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

Thermodynamic Effects on Developed Cavitation

Abstract: The results of an investigation of thermodynamic effects are presented. Distributions of temperature and pressure in a developed cavity were measured for zero- and quarter-caliber ogives. A semiempirical entrainment theory was developed to correlate the measured temperature depression in the cavity. This theory correlates the maximum temperature depression expressed in dimensionless form as the Jakob number in terms of the dimensionless numbers of Nusselt, Reynolds, Froude, and Peclet, and dimensionless cavity length, L/D. The results show that in general, the temperature depression increases with L/D and temperature and the cavitation number based on measured cavity pressure is a function of L/D for a given model contour, independent of the thermodynamic effect.

Hydrodynamic instability in fluid layers with uniform volumetric energy sources

Abstract: Linear and energy theory stability criteria are presented for fluid layers of infinite horizontal extent heated internally by a uniform volumetric energy source. The thermal coupling between the layer and its environment is modeled by a general mixed boundary condition in both the conduction state and the disturbance temperature. Rigid-rigid, free-free, free-rigid, and rigid-free boundaries are considered in the computations. For a fixed ratio of upper surface Biot number to that at the lower surface, decreasing the Biot number is strictly destabilizing for both linear and energy theory criteria. A region of possible subcritical instability is found; its size is strongly dependent on Biot number and becomes small for small values of lower surface Biot number and large Biot number ratio. For two rigid surfaces and an upper and lower surface Biot number of 47.5, mean energy transport measurementswithin the convecting layer indicate a critical Reyleigh number close to that predicted by linear theory. Subcritical instability is not observed when finite amplitude disturbances are introduced at a Rayleigh number between the critical values predicted by the linear theory and the energy theory.

Natural convection of a heat-generating fluid within closed vertical cylinders and spheres

Abstract: Steady natural convective flow fields were numerically and experimentally characterized for 0.7 Prandtl number fluids having constant, uniformly distributed, internal heat sources. The bounding isothermal walls containing the fluid were considered to be either a sphere or a cylinder of finite height. An instrumented cylinder containing radioactive tritium gas was used to demonstrate experimental and analytical agreement for local temperatures over a range of Grashof numbers. For the spherical geometry, a generalized correlation was obtained for the surface-averaged Nusselt number as a function of a modified Grashof number.

Shallow Laminar Flows over Rough Granular Surfaces

Abstract: Several investigations have established that at high values of relative roughness mean depths of liquid in laminar shear flow over rough surfaces are greater than the corresponding depths on a smooth surface. In this investigation the nature of the flow field in the vicinity of isolated roughness elements fixed to a smooth base is demonstrated using a flow visualization technique. Velocity profiles measured with an optical velocity meter are shown to be governed by the Nusselt equation, but surface velocity is a function of relative roughness and is less than the corresponding value for a smooth surface. An analytical model is proposed to explain an empirical relationship between friction coefficient, relative roughness, and Reynolds number. The equation developed is used to calculate the product of the coefficient of drag of a sphere resting on a plane surface and the tip Reynolds number and also to demonstrate the influence of velocity distribution on the drag coefficient.

Solidification and Melting of Materials Subject to Convection and Radiation

Abstract: One-dimensional heat transfer in a finite region of solid or liquid with phase change associated with radiative and convective boundary conditions at the fixed boundary surface is solved using both Blot's variational and Goodman's integral methods. The total solidification time and the rates of solidification obtained based on the present approximate analyses agree very well with the numerical solutions of an earlier work.

Convective Energy Transfer in Fur

Abstract: Motion of the surrounding air increases the transfer of energy from an animal. To apply thermal modeling techniques, one has to know of the importance of convection within the fur relative to conduction and radiation (see Birkebak, 1966; Birkebak et al., 1966; Gates and Porter, 1969).

High Rayleigh number convection in enclosed fluid layers with internal heat sources

Abstract: An experimental study of high Rayleigh number natural convection with internal heat sources has been conducted in a horizontal fluid layer with an insulated lower boundary and a constant-temperature upper boundary Average heat transfer coefficients at the upper surface were obtained for Rayleigh numbers up to 217 x 10/sup 12/ A review of currently available heat transfer correlations for internally heated fluid layers is also presented In this review, the correlations are compared using a common definition of Rayleigh number and Nusselt number (GRA)

Turbulent natural convection of liquid deuterium, hydrogen, and nitrogen within enclosed vessels

Abstract: Quasi-steady natural convection of liquid deuterium, hydrogen, and nitrogen within a sphere, hemisphere, horizontal cylinder, and vertical cylinder has been studied experimentally for the case of a nearly uniform wall temperature. A single expression relating the Nusselt and Rayleigh numbers, Nu = 0·104Ra0·352, fits the deuterium and nitrogen data over the range 7 × 108

Heat-transfer in a taylor vortex flow

Abstract: The characteristics of heat transfer affected by Taylor vortex motion were investigated experimentally by the use of electrochemical method and theoretically with the aid of the nonlinear theory suggested by J. T. Stuart. The present non-linear theory, which considers the effect of the fundamentals of disturbances on heat transfer, especially in the wide gap problem, is also applicable to the heat-transfer problem in the range of 1

Viscous Dissipation Effects on Developing Laminar Flow in Adiabatic and Heated Tubes

Abstract: A finite-difference solution is presented for the problem of laminar developing flow in a circular tube, allowing for viscous dissipation and viscosity variation with temperature. Predictions of temperature and pressure under adiabatic conditions compare reasonably with published experimental and analytical results. Polak's postulation of a hot slip zone is considered. Heat transfer, with constant wall heat flux, is also studied and it is concluded, with reference to some recent experimental data, that it is not necessary to postulate viscous dissipation as being a substantial aiding effect. Predicted Nusselt numbers are, in fact, reduced by allowing for the effect.

Modal Analysis of Convection in a Rotating Fluid

Abstract: The Boussinesq equations for convection in a plane layer of fluid rotating about a vertical axis are expanded in the planform functions of linear theory. The case where only one horizontal mode is retained is studied in detail for a wide range of parameters and for free and rigid boundaries. For large Rayleigh numbers numerical techniques are used and the results compared with an asymptotic solution. For large Rayleigh numbers, moderate Prandtl numbers and rigid boundaries, steady solutions are found which display non-monotonic dependence of beat flux on rotation rate even when the horizontal wavenumber is fixed. This behavior is suggestive of Rossby's experimental results and the appearance of Ekrman-like boundary layers in the solution seems to confirm his proposed explanation. At lower Rayleigh numbers, the effect diminishes markedly and an appeal to a variable horizontal wavenumber still seems needed. However, the heat flux varies monotonically with rotation rate when the boundaries are free,...