Critical Value Ra Research Articles

Experimental study on the initial thermal plumes produced by small heating plates in water

Herein, the evolution of initial thermal plumes (TPs) from small heating plates in water was investigated using near-infrared (NIR) imaging. The imaging system comprised a telecentric uniform illumination system and an NIR camera. The system employed the temperature dependence of the absorbance of water at a wavelength of 1150 nm for the measurements. Two-dimensional absorbance images of a 10 mm thick water cell were captured. A 1 mm thick circular or square steel plate was placed at the bottom and heated via induction. The images revealed that the thickness of the thermal boundary layer, followed by the rise in the TP, increased linearly with time, and several patterns of the TP appeared depending on the heat-flux-based Rayleigh number, Ra*. For a small Ra*, a TP with a mushroom-shaped cap was formed. At the critical value of Ra*, the boundary-layer flow on the plate surface was destabilized, leading to a bifurcated TP and an umbrella-shaped cap above it. These results were validated by comparing them with those of numerical simulations. Furthermore, the relationships between the ascent speed of the TP, plate size, and heat flux were assessed quantitatively.

Analysis of multiple steady states for natural convection of Newtonian fluids in a square enclosure

Multiple steady states are investigated for natural convection of fluids in a square enclosure with non-isothermally hot bottom wall, isothermally cold side walls, and thermally insulated top wall. A robust computation scheme involving steady-state governing equations has been developed to compute the steady states as a function of Rayleigh number (102 ⩽ Ra ⩽ 106) for two different Prandtl numbers (Pr = 0.026 and 0.1). Penalty Galerkin finite element method with Newton–Raphson solver is employed for the solution of the governing equations, while the solution branches are initiated by varying initial guess to the Newton–Raphson solver. In this context, a dual-perturbation scheme involving perturbations of the boundary conditions and various process parameters has been designed leading to the rich spectrum of the symmetric and asymmetric solution branches for the current symmetric problem. It is found that multiple steady states occur beyond a critical value of Ra, which depends on the magnitude of Pr. In addition to the basic solution branch (corresponding to the solutions obtained via uniform initial guesses), nineteen additional solution branches (six symmetric and thirteen asymmetric) are obtained for Pr = 0.026, while four additional solution branches (two symmetric and two asymmetric) are obtained for Pr = 0.1. The solution branches are associated with a wide spectrum of flow structures (24 distinct types for Pr = 0.026 excluding the reflection symmetric mirror images of the asymmetric solutions), which are reported for the first time. The flow structures lead to various heating scenarios within the enclosure resulting in a significant variation of heat transfer rates (more than 50%). The current results are important for the practical applications. The spectrum of the possible scenarios revealed in this work can be pivotal to design the optimal processes based on the process requirement (targeted heating or enhanced heating rates).

Characteristics of temperature fluctuation in two-dimensional turbulent Rayleigh-Bénard convection

We study the characteristics of temperature fluctuation in two-dimensional turbulent Rayleigh–Bénard convection in a square cavity by direct numerical simulations. The Rayleigh number range is 1 × 108 ≤ Ra ≤ 1 × 1013, and the Prandtl number is selected as Pr = 0.7 and Pr = 4.3. It is found that the temperature fluctuation profiles with respect to Ra exhibit two different distribution patterns. In the thermal boundary layer, the normalized fluctuation θ rms/θ rms,max is independent of Ra and a power law relation is identified, i.e., θ rms/θ rms,max∼ (z / δ)0.99 ± 0.01, where z / δ is a dimensionless distance to the boundary (δ is the thickness of thermal boundary layer). Out of the boundary layer, when Ra ≤ 5 × 109, the profiles of θ rms/θ rms,max descend, then ascend, and finally drop dramatically as z/δ increases. While for Ra ≥ 1 × 1010, the profiles continuously decrease and finally overlap with each other. The two different characteristics of temperature fluctuations are closely related to the formation of stable large-scale circulations and corner rolls. Besides, there is a critical value of Ra indicating the transition, beyond which the fluctuation 〈 θ rms〉 V has a power law dependence on Ra, given by 〈 θ rms〉 V ∼ Ra −0.14 ± 0.01.

Thermal management of transverse magnetic source effects on nanofluid natural convection in a wavy porous enclosure

Thermal transport of nanofluid natural convection in a wavy porous enclosure exposed to an external and uniform magnetic source is investigated numerically. Numerous pertinent factors in terms of Darcy (Da = 10−4–10−2), Hartmann (Ha = 0–40), Rayleigh (Ra = 104–107), Prandtl (Pr = 0.71–7) and undulation (n = 3) numbers, in addition to wave amplitudes (A = 0.025–0.1) and particle volume concentration ( $$\phi$$ = 0, 2 and 4%) have been investigated. The Brinkmann–Forchheimer extended Darcy model is utilized, and the governing equations are solved by employing our own finite difference ADI-based program. Code accuracy was successfully validated with the open literature. The results revealed that for Ra > 105 and Da < 10−3, the magnetic field does not play a substantial role in the convective thermal energy, while at high Ha and low Ra, the intensity of conduction increases. The surface waviness and Darcy number both have significant effects on the heat transfer suppression if insulation is desired. Furthermore, a critical value of Ra = 105 is observed whereby the mean Nusselt number decreases despite particle volume concentration, particularly at high Hartmann values.

Thermal convection in the porous methane-soaked regolith in Titan: Finite amplitude convection

Titan is the only body, beside the Earth, where liquid is present on the surface. This paper is aimed to show the properties of possible convection in a porous regolith on Titan. In our previous work (Czechowski, L., Kossacki, K.J. [2009]. Icarus 202, 599–607) we showed, that the Rayleigh number Ra can exceed its critical value Ra c. Hence, the convective motion of liquid filling pores in the regolith is likely for Titan relevant parameters. In the present work we investigate the properties of finite amplitude convection, i.e. for Ra > Ra cr. We study the basic properties of the steady state solution, the Nusselt number, the density of the heat flow and the average temperatures. Evolution of the convection is also considered. We conclude that any reasonable thermal model of Titan’s regolith should take into account the possibility of the considered convection. We discuss also possibility of identification of this convection (or its consequences in the form of evaporates) by the Cassini and possible future spacecrafts.

Natural convection and heat transfer in attics subject to periodic thermal forcing

The heat transfer through the attics of buildings under realistic thermal forcing has been considered in this study. A periodic temperature boundary condition is applied on the sloping walls of the attic to show the basic flow features in the attic space over diurnal cycles. The numerical results reveal that, during the daytime heating stage, the flow in the attic space is stratified; whereas at the night-time cooling stage, the flow becomes unstable. A symmetrical solution is seen for relatively low Rayleigh numbers. However, as the Ra gradually increases, a transition occurs at a critical value of Ra. Above this critical value, an asymmetrical solution exhibiting a pitchfork bifurcation arises at the night-time. It is also found that the calculated heat transfer rate at the night-time cooling stage is much higher than that during the daytime heating stage.

Effect of pressure work and viscous dissipation in the analysis of the Rayleigh–Bénard problem

The classical Rayleigh–Bénard problem in an infinitely wide horizontal fluid layer with isothermal boundaries heated from below is revisited. The effects of pressure work and viscous dissipation are taken into account in the energy balance. A linear analysis is performed in order to obtain the conditions of marginal stability and the critical values of the wave number a and of the Rayleigh number Ra for the onset of convective rolls. Mechanical boundary conditions are considered such that the boundaries are both rigid, or both stress-free, or the upper stress-free and the lower rigid. It is shown that the critical value of Ra may be significantly affected by the contribution of pressure work, mainly through the functional dependence on the Gebhart number and on a thermodynamic Rayleigh number. While the pressure work term affects the critical conditions determined through the linear analysis, the viscous dissipation term plays no role in this analysis being a higher order effect. A nonlinear analysis is performed showing that the superadiabatic Rayleigh number replaces Ra in the functional dependence of the excess Nusselt number. Finally, a reasoning is proposed to show how the results obtained may be used as a test on the most appropriate formulation of the Oberbeck–Boussinesq model.

Numerical simulation of the 3D behavior of thermal buoyant airflows in pentahedral spaces

A numerical study of three-dimensional natural convection in an attic space with heated horizontal base and cooled upper walls is presented. Every previous study pertinent to this subject as of today has assumed that the flow in attics is two-dimensional and restricted to triangular cavities. This problem is examined for fixed aspect ratios holding width to height of 2 and depth to height of 3.33 and Rayleigh numbers ranging from 10 4 to 8 × 10 5. The coupled system of conservation equations, subject to the proper boundary conditions, along with the equation of state assuming the air behaves as a perfect gas are solved with the finite volume method. In the conservation equations, the second-order-accurate QUICK scheme was used for the discretization of the convective terms and the SIMPLE scheme for the pressure-velocity coupling. It is categorically found that the flow in the attic is 3D. From the physics of the problem, two steady-state solutions are possible. The symmetrical solution prevails for relatively low Rayleigh numbers. However, as the Ra is gradually increased, a transition occurs at a critical value Ra C. Above this value of Ra C, an asymmetrical solution exhibiting a pitchfork bifurcation arises and eventually becomes steady. Results are presented detailing the occurrence of the pitchfork bifurcation and the resulting flow patterns are described.

Oscillatory and nonoscillatory behavior of a simple model for cool flames, Sal'nikov's reaction, P → A → B, occurring in a spherical batch reactor with varying intensities of natural convection

When cool flames, or indeed any exothermic chemical reaction, occur in a fluid inside an unstirred vessel, the heat from the reaction often induces temperature gradients and consequently motion, i.e., natural convection. The intensity of the resulting flow is governed by the Rayleigh number (Ra). This work simulates numerically the behavior of Sal'nikov's reaction, P → A → B, under the influence of natural convection in an unstirred spherical reactor. This reaction is the simplest to exhibit the thermokinetic oscillations characterizing cool flames. The behavior of this system can be represented on a three-dimensional regime diagram, whose axes are ratios of the characteristic timescales ( τ) for chemical reaction, diffusion (of both heat and mass), and natural convection. Previous work has identified a region of oscillations on this diagram in the purely diffusive limit, when Ra = 0. This work extends this analysis to the general 3D space, where diffusion and natural convection are both important. A region in which oscillations are observed has been found for fixed values of the first-order rate constants for Sal'nikov's reaction. There is a distinct change in the shape of the region of oscillations around the critical value of Ra ∼ 500, when natural convection becomes important. When diffusion dominates transport (Ra < 500), the boundaries between oscillatory and nonoscillatory solutions are largely independent of the ratio of timescales τ step 2 / τ convection and agree well with the values found previously in the purely diffusive limit. When natural convection is important (Ra > 500), the oscillations occur over a wider range of parameters than is the case for a diffusive system. The presence of natural convection also leads to various, more complex behaviors than are seen in the diffusive or well-mixed limits. A region in the regime diagram was found where the oscillations in temperature and the concentration of A have small amplitudes and a frequency that is quite different from those generated in a well-mixed system. It is possible that these oscillations are caused by natural convection, i.e., are not thermokinetic oscillations produced by the chemical reaction. It was also found that sometimes the oscillations in the temperature and the concentration of A are in phase; more generally they are in anti-phase. The evolution of nonoscillatory behavior with relatively small increases in temperature was found to be always fairly similar, regardless of the intensity of natural convection. The shape of the temperature profile along the vertical axis of the reactor did, however, change with the intensity of natural convection. Finally, the nonoscillatory solutions with a large rise in temperature in the presence of natural convection were found to be very much like those seen in the purely diffusive limit for small times, due to the relatively long induction time (∼3.5 s in a vessel with diameter 0.1 m) for the onset of natural convection.

The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: Moderate heterogeneity

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are now studied analytically for the case of moderate heterogeneity (rather than the weak heterogeneity previously studied), for the case of a square box where the properties vary in a piecewise constant or linear fashion, with conducting impermeable top and bottom boundaries and insulating impermeable sidewalls. In order to allow for the moderate heterogeneity the order of the Galerkin expansion employed has been increased, and the expansion of a determinant of high order has been avoided by the use of a least squares methodology to find the critical value of the Rayleigh number Ra. It is found that the effects of permeability heterogeneity and conductivity heterogeneity each cause a reduction in the critical value of Ra in all cases, and the effects of horizontal and vertical heterogeneity are still approximately additive.

Application of a non-linear local analysis method for the problem of mixed convection instability

We consider the problem of laminar mixed convection flow between parallel, vertical and uniformly heated plates where the governing dimensionless parameters are the Prandtl, Rayleigh and Reynolds numbers. Using the method based on the centre manifold theorem which was derived from the general theory of dynamical systems, we reduce a three-dimensional simplified model of ordinary differential amplitude equations emanating from the original Navier-Stokes system of the problem in the vicinity of a trivial stationary solution. We have found that when the forcing parameter, the Rayleigh number, increases beyond the critical value Ra s , the stationary solution is a pitchfork bifurcation point of the system.

Influence of Prandtl number on stability of mixed convective flow in a vertical channel filled with a porous medium

Buoyancy opposed mixed convection is considered in a vertical channel filled with an isotropic, porous medium, in which the motion of an incompressible fluid is induced by external pressure gradients and buoyancy forces. The Brinkman-Wooding-extended Darcy model has been used to study the instability mechanisms of the basic flow and its dependence on the Prandtl number (Pr) of the fluid. The stability analysis indicated that for the same Reynolds number (Re), the fully developed base flow was highly unstable for a fluid with high Pr. For a porous medium with a Darcy number (Da) of 10−6 and Pr⩾0.7, two different types of instability, Rayleigh-Taylor (R-T) and buoyant instability, are observed. The R-T instability mode is observed for relatively small values of Re. Further, the results show that for Da=10−5 and Pr&amp;lt;1, the spectrum of the energy profile is abrupt and sudden, whereas the same is smooth when Da=10−6. In the case of R-T instability, the critical value of Ra at low Re is given by −2.47∕Da. Though the R-T mode of instability is independent of Pr, the range of Re that sustains the R-T mode is a function of Pr. It has been found that enhancement of Pr reduces the Re range mentioned above. In contrast to the case of a purely viscous fluid, where the effect of Pr is not significant, in isotropic porous media Pr plays a significant role in characterizing the flow stability. The instability characteristics of zero temperature flux perturbation (BC-I) and zero heat flux perturbation (BC-II) on the boundaries differ significantly in the case of the R-T stability mode. However, both conditions lead to similar results for buoyant stability, except at small values of Re.

The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically for the case of weak heterogeneity. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. For the case of conducting impermeable top and bottom boundaries and a square box, the effects of permeability heterogeneity and conductivity heterogeneity each cause a reduction in the critical value of Ra, while for the case of a tall box there can be either a reduction or an increase.

Temperature oscillations in turbulent Rayleigh-Bénard convection.

A systematic study of temperature oscillations in turbulent thermal convection was carried out in two aspect-ratio-one convection cells filled with water. Temperature correlation functions and local velocity fluctuations were measured over varying Rayleigh numbers and spatial positions across the entire cell. These measurements fully characterize the spatial structure of the temperature oscillation and reveal the mixing and emission dynamics of the thermal plumes near the conducting surface. A sharp transition from a random chaotic state to a correlated turbulent state of finite coherence time is found when the Rayleigh number becomes larger than a critical value Ra(c) approximately equal 5 x 10(7). Above Ra(c) the measured temperature correlation functions show a well-defined oscillation with a finite coherence time. The oscillation period is found to be twice as large as the cell crossing time. The experiment demonstrates how the thermal plumes in a closed cell organize themselves both in space and time and generate coherent oscillations in a turbulent environment.

Onset of coherent oscillations in turbulent Rayleigh-Bénard convection.

We report temperature cross correlation and velocity profile measurements in the aspect-ratio-one convection cell filled with water. A sharp transition from a random chaotic state to a correlated turbulent state of finite coherence time is found when the Rayleigh number becomes larger than a critical value Ra(c) approximately equal to 5 x 10(7). The experiment reveals a unique mechanism for the onset of coherent oscillations in turbulent Rayleigh-Bénard convection.

Linear stability of mixed convection flows in horizontal rectangular channels of finite transversal extension heated from below

We present a study of the linear hydrodynamic stability of purely conductive Poiseuille flow in three-dimensional horizontal rectangular channels uniformly heated from below. Its originality is to modellize the flow using realistic conditions on all the boundaries of the channel, in particular, no-slip conditions on the vertical lateral boundaries. The aim of this paper is to provide quantitative results for the values of the critical dimensionless parameters. When the Rayleigh number Ra is higher than the critical value Ra ∗, the instabilities are shown to be three-dimensional horseshoe-shaped transversal rolls for Reynolds numbers Re smaller than the critical value Re ∗, and longitudinal rolls for Re> Re ∗. The results are given for a wide range of the Prandtl number (10 −6≤ Pr≤10 5) and of the transversal aspect ratio B of the duct (0.1≤ B≤5.4 and B→∞).

Convection forcée en milieu poreux: écarts à la loi de Darcy

In an infinite horizontal porous layer differentially heated, with an imposed through-flow, a conductive regime is observed when the filtration Rayleigh number Ra is below a critical value Ra c . Beyond the threshold, time-periodic convective flows occur. Introducing the Forchheimer's and Brinkman's corrections into the Darcy law enhances Ra c , and the two-dimensional travelling waves are studied.

鉛直平行二平板の加熱と冷却による安定密度成層水溶液の混合機構に関する研究 第２報 多層対流層生成過程と吸収合体機構

In this report the evolution of layering and the mechanism of degradation of layers formed from a stably stratified solution between the heated and cooled vertical parallel plates are precisely discussed. The relationship between Ra, f which expresses the dimensionless parameter of temperature fluctuation in the layer and Ra, w which is related to the intensity of the natural convection flow induced by the heated and cooled walls is obtained. When Ra, w is less than 15000, Ra, f increases with the increase of Ra, w and the flow is laminar. When Ra, w is between 15000 and 25000, Ra, f increases rapidly with the increase of Ra, w and the transition to turbulence occurs. The analysis shows that the temperature fluctuation in the layer is attenuated due to the transfer of heat across the interfaces contacting with upper and lower layers and the horizontal convection flow disappears when the critical value of Ra, w is 16800.

Autoignition and cool-flame behaviour in a convective system

When cool flames, or indeed any exothermic chemical reaction, occur in a fluid inside an unstirred vessel, the heat from the reaction often induces temperature gradients and consequently motion, i.e., natural convection. The intensity of the resulting flow is governed by the Rayleigh number (Ra). This work simulates numerically the behavior of Sal'nikov's reaction, P → A → B, under the influence of natural convection in an unstirred spherical reactor. This reaction is the simplest to exhibit the thermokinetic oscillations characterizing cool flames. The behavior of this system can be represented on a three-dimensional regime diagram, whose axes are ratios of the characteristic timescales (τ) for chemical reaction, diffusion (of both heat and mass), and natural convection. Previous work has identified a region of oscillations on this diagram in the purely diffusive limit, when Ra = 0. This work extends this analysis to the general 3D space, where diffusion and natural convection are both important. A region in which oscillations are observed has been found for fixed values of the first-order rate constants for Sal'nikov's reaction. There is a distinct change in the shape of the region of oscillations around the critical value of Ra ∼ 500, when natural convection becomes important. When diffusion dominates transport (Ra < 500), the boundaries between oscillatory and nonoscillatory solutions are largely independent of the ratio of timescales τstep2/τconvection and agree well with the values found previously in the purely diffusive limit. When natural convection is important (Ra > 500), the oscillations occur over a wider range of parameters than is the case for a diffusive system. The presence of natural convection also leads to various, more complex behaviors than are seen in the diffusive or well-mixed limits. A region in the regime diagram was found where the oscillations in temperature and the concentration of A have small amplitudes and a frequency that is quite different from those generated in a well-mixed system. It is possible that these oscillations are caused by natural convection, i.e., are not thermokinetic oscillations produced by the chemical reaction. It was also found that sometimes the oscillations in the temperature and the concentration of A are in phase; more generally they are in anti-phase. The evolution of nonoscillatory behavior with relatively small increases in temperature was found to be always fairly similar, regardless of the intensity of natural convection. The shape of the temperature profile along the vertical axis of the reactor did, however, change with the intensity of natural convection. Finally, the nonoscillatory solutions with a large rise in temperature in the presence of natural convection were found to be very much like those seen in the purely diffusive limit for small times, due to the relatively long induction time (∼3.5 s in a vessel with diameter 0.1 m) for the onset of natural convection.

Kinetics of continuous addition emulsion polymerization

AbstractAn analysis of the kinetics of continuous addition emulsion polymerization is presented. For systems that follow the Smith‐Ewart mechanism, this analysis predicts that the reaction will approach a steady‐state condition if the rate of monomer addition Ra is constant. In the steady state, the rate of polymerization depends on Ra according to a reciprocal relationship, 1/Rp = (1/K) + (1/Ra), where K depends on the number of particles and the propagation rate constant. Above a critical value of Ra, the particles will become saturated with monomer and Rp will be constant and independent of Ra.