Dimensionless numbers Rayleigh

Dimensional analysis is a method for producing dimensionless numbers that completely describe the process. The analysis should be carried out before the measurements are made because dimensionless numbers essentially condense the frame in which the measurements are performed and evaluated. The method can be applied even when the equations governing the process are not known. Dimensional analytical procedure was first proposed by Rayleigh in 1915. ... 

Riri = modified Rayleigh number, dimensionless number used to determine the onset of convection in heated fluids S = surface area t = time... 

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... 

There are two dimensionless numbers that characterize the stability of the configuration shown in Figure 9.10b. We have seen the Rayleigh number before but now formulate it in terms of the temperature difference, AT, and the layer thickness, d ... 

In the Rayleigh method of carrying out a dimensional analysis the dependent variable is assumed to be proportional to the product of the independent variables raised to different powers. By equating dimensions, the number of independent dimensionless groups and one set of their possible forms can be obtained. By way of illustration two examples may be considered. 

Secondary atomization, the breakup of the drops first formed, has been studied by Littaye (11C), who assumes a necessary criterion that the drag forces exceed the inertia forces. Ohnesorge (17C) makes use of the principles of mechanical similarity by introducing dimensionless coefficients to help explain jet breakup. Above certain well defined numbers, the jet completely atomizes at the nozzle. Lower values indicate the formation of a jet which disintegrates, owing to helical vibrations which later change into Rayleigh vibrations. 

Horizontal Reactors. Horizontal reactor flow may involve both transverse and longitudinal rolls, as well as time-periodic flows. Insights into these phenomena may be gained from previous analysis of idealized, analogous systems, as well as from recent experiments and computations. Analytical studies of flow between two plates of infinite size differentially heated from below (180) and horizontal channel flow (181) indicate that the development of transverse and longitudinal rolls depends on the relative and absolute magnitudes of the dimensionless Rayleigh and Reynolds numbers, Ra and Re, as well as the aspect ratio. 

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,... 

Table 12-3. Rayleigh number at the neutral stability point (a = 0) as a function of the dimensionless wave number...

Important Dimensionless Groups. The average Nusselt number for steady-state heat transfer from the body shown in Fig. 4.1a depends on the dimensionless groups that arise in the nondimensionalized equations of motion and their boundary conditions [78]. With Tw and T constant, the only dimensionless groups that appear in the boundary conditions are those associated with the body shape. Provided the simplified equations (Eqs. 4.5 1.7) are valid, the only other dimensionless groups are the Rayleigh and Prandtl numbers. Thus, for a given body shape,... 

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). 

Rayleigh showed that if the cells are to form, then the vertical adverse temperature gradient must be sufficiently large that a particular dimensionless parameter proportional to the magnitude of the gradient exceed a critical value. We now term this parameter the Rayleigh number... 

Rayleigh number (Ra) - A dimensionless quantity used in fluid mechanics, defined by Ra = Pga s.Tplx a, where I is length, g is acceleration of gravity, a is cubic expansion coefficient, T is temperature, p is density, q is viscosity, and a is thermal dif-fusivity. [2]... 

Dimensionless Critical Wave Number for the Rayleigh Problem... 

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT < AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number... 

Formulas for heat convection coefficients he can be found from available empirical correlation and/or theoretical relations and are expressed in terms of dimensional analysis with the dimensionless parameters Nusselt number Nu, Rayleigh number Ra, Grashof number Gr, Prandtl number Pr, and Reynolds number Re, which are defined as follows ... 

Those engineering disciplines concerned with fluid flow, such as aeronautical, civil, and mechanical, have used dimensional analysis to good effect for the past hundred years. Their success is largely attributable to the fact that fluid flow requires only three fundamental dimensions and generates a limited number of dimensionless parameters. Those engineering disciplines can still use the Rayleigh indices method, which is, essentially, a hand calculation, to derive the dimensionless parameters. 

Dimensional analytical procedure was first introduced by Lord Rayleigh in 1915 (71). It is a process whereby the variables pertinent to a physical problem are systematically organized into dimensionless groups/numbers. Assuming the absence of chemical reaction and heat transfer, the following influencing and independent variables are pertinent to the wet-granulation process (72) ... 

Consider a single-component fluid in a rectangular container with no free interface, as shown in Figure 1. We can heat it from the top, bottom or side. Will all three configurations cause fluid motion No. If we heat from above, the fluid is stable, meaning no fluid motion will occur. If we heat from the side, there will be convection, whose magnitude will be determined by several factors. We combine these factors into a dimensionless quantity called the Rayleigh number... 

For gravity driven free convection the velocity v is not known a priori therefore one has to use relations with a dimensionless group containing the acceleration of gravity. This is expressed by the Grashof number Gr, or by the Rayleigh number Ra ... 

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