Natural convection Rayleigh number

Each of these is a ratio of a convective transfer rate to the corresponding diffusion rate of transfer. Dimensionless analysis indicates that, for fixed geometry and constant properties, the Nusselt number and the Sherwood number depend on the Reynolds number (forced convection), Rayleigh number (natural convection), flow characteristics, Prandtl number (heat transfer), and Schmidt number (mass transfer). 

One can describe these phenomena through the Reynolds number (forced convection) and Rayleigh number (natural convection), but the reader can see immediately that the situations are so complicated that correlations in elementary texts on fluid flow are not easily applicable to predicting flame behavior. Reactive flows are among the most complex problems in modem engineering. 

In the first case (Figure 8a), the side walls are adiabatic, and the reactor height (2 cm) is low enough to make natural convection unimportant. The fluid-particle trajectories are not perturbed, except for the gas expansion at the beginning of the reactor that is caused by the thermal expansion of the cold gas upon approaching the hot susceptor. On the basis of the mean temperature, the effective Rayleigh number, Rat, is 596, which is less than the Rayleigh number of 1844 necessary for the existence of a two-dimensional, stable, steady-state solution with flow in the transverse direction that was computed for equivalent Boussinesq conditions (188). 

When a liquid warms up, its density decreases, which results in buoyancy and an ascendant flow is induced. Thus, a reactive liquid will flow upwards in the center of a container and flow downwards at the walls, where it cools this flow is called natural convection. Thus, at the wall, heat exchange may occur to a certain degree. This situation may correspond to a stirred tank reactor after loss of agitation. The exact mathematical description requires the simultaneous solution of heat and impulse transfer equations. Nevertheless, it is possible to use a simplified approach based on physical similitude. The mode of heat transfer within a fluid can be characterized by a dimensionless criterion, the Rayleigh number (Ra). As the Reynolds number does for forced convection, the Rayleigh number characterizes the flow regime in natural convection ... 

As previously discussed, there are two limiting cases for natural convective flow through a vertical channel. One of these occurs when /W is large and the Rayleigh number is low. Under these circumstances all the fluid will be heated to very near the wall temperature within a relatively short distance up the channel and a type of fully developed flow will exist in which the velocity profile is not changing with Z and in which the dimensionless cross-stream velocity component, V, is essentially zero, i.e., in this limiting solution ... 

Natural convection occurs when a fluid is in contact with a solid surface of different temperature. Temperature differences create the density gradients that drive natural or free convection. In addition to the Nusselt number mentioned above, the key dimensionless parameters for natural convection include the Rayleigh number Ra = p AT gx3/ va and the Prandtl number Pr = v/a. The properties appearing in Ra and Pr include the volumetric coefficient of expansion p (K-1) the difference AT between the surface (Ts) and free stream (Te) temperatures (K or °C) the acceleration of gravity g(m/s2) a characteristic dimension x of the surface (m) the kinematic viscosity v(m2/s) and the thermal diffusivity a(m2/s). The volumetric coefficient of expansion for an ideal gas is p = 1/T, where T is absolute temperature. For a given geometry,... 

Natural convection heal transfer correlations are usually expressed in terms of the Rayleigh number raised to a constant n multiplied by another constant C, both of which are determined experimentally. 

To determine llie Rayleigh number, we need to know the surface temperature of the glass, which is not available. Therefore, it is clear lhat the soiulion will require a trial-and-error approach. Assuming the glass cover temperature to be 40°C, the Rayleigh number, the Nusselt number, the convection heat transfer coefficteniy and the rate of natural convection heat tiansfer from the glass cover to th e ambient air are determined to be... 

The correlations for the Nussell number Nu = hL /k in natural convection are expressed in terms of the Rayleigh number defined as... 

It is important to note here the conceptual difference between the Reynolds number of forced convection and the Rayleigh number of natural convection. Re results from the nondimensionalized momentum (of forced convection) which is uncoupled from thermal energy of incompressible (and constant property) fluids. On the other - hand, Ra characterizes the coupling (through buoyancy) of momentum to energy.16... 

In Chapter 5, following some dimensional arguments, we learned that the independent dimensionless numbers characterizing buoyancy driven flows are the Rayleigh number and the Prandtl number (Ra, Pr), and the heat transfer in (Nusselt number Nu for) natural convection is governed by... 

Rayleigh number Ra Gr Fr = PtfP A TV pa Modified Grashof number (see interpretations for Gr and Pr) Natural convection... 

When both bottom and top surfaces are maintained at constant temperatures and there is internal generation, there is a superposition of the horizontal layer problem discussed in the section on natural convection within enclosures and the internal generation problem previously described. These are characterized by the external Rayleigh number defined in the section on natural convection within enclosures and the internal Rayleigh number defined in Fig. 4.40a. The dependence of the layer stability on these parameters has been discussed by Ning et al. [208]. The heat transfer at the top and bottom surfaces has been estimated for these conditions by Baker et al. [13], Suo-Anttila and Catton [276], and Cheung [51],... 

M. R. Fand, E. W. Morris, and M. Lum, Natural Convection Heat Transfer From Horizontal Cylinders to Air, Water and Silicone Oils for Rayleigh Numbers Between 3 x 102 and 2 x 107, Int J. Heat Mass Transfer (20) 1173-1184,1977. 

G. Hesse and E. M. Sparrow, Low Rayleigh Number Natural Convection Heat Transfer From High-Temperature Wires to Gases, Int. J. Heat Mass Transfer (17) 796-798,1974. 

M. E. Weber, P. Astrauskas, and S. Petsalis, Natural Convection Mass Transfer to Nonspherical Objects at High Rayleigh Number, Canadian J. Chem. Eng. (62) 68-72,1984. 

M. M. Yovanovich and K. Jafarpur, Models of Laminar Natural Convection From Vertical and Horizontal Isothermal Cuboids for All Prandtl Numbers and All Rayleigh Numbers Below 1011, ASME HTD (264) 111-126,1993. 

If either the monomer or the polymer, or both, are liquid natural convection, caused by the heat liberated by the exothermic reaction, can occur. Consider first the case when the monomer is liquid and the polymer is solid (cf. Section 1). We will discuss separately upward and downward propagating fronts. If the front propagates upward, then the chemical reaction heats the monomer from below which reminds of the classical Rayleigh-Benard problem. If the Rayleigh number is sufficiently large, then the planar front loses its stability and stationary natural convection above the front occurs. For descending planar fronts there is no such convective instability. An approximate analytical approach allows one to find stability conditions for the propagating reaction front and to determine the modes which appear when the front loses stability [22]. 

For vertical plates and natural convection, the Nusselt number is some empirical function of Prandtl and Rayleigh numbers, Nul = f (Pr,RaL), such as [1] ... 

The flux between two lateral walls caused by the nonuniformity of the ion concentration profiles in the layers adjacent to the electrodes is of the same nature as the heat convection arising while the bottom wall is heated [2]. In the latter case a disturbance of the steady state occurs if the Rayleigh number reachs a certain (critical) value (Ra = gPd AT/vx, where P is the coefficient of bulk heat expansion, d is the distance between the walls, AT the increment of temperature, v the dynamic viscosity, and X the thermal diffusivity) the liquid transforms into a new state with a periodic cell structure in such a way that the circulation in the interior of each cell has an opposite direction compared to that of the adjacent one. According to previous evaluations [53] the critical Rayleigh number in the case of lateral rigid walls is about 1700. 

Dixit HN, Babu V (2006) Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method. Int J Heat Mass Transfer 49 727-739... 

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