Natural-convection dimensionless

Commercially available microdisc electrodes of radii 0.6-70/xm may be used for steady-state measurements without problems associated with natural convection. Dimensionless rate constants for spherical and microdisc electrode were interpolated from the working curves of Alden and Compton (1997a). 

The secondary flows from natural convection can become larger than the primary flow, so it seems likely that the secondary flows might become turbulent or nonsteady. Shown in Tables 1 and 2 are the dimensionless groups at the inlet and outlet, based on cup-average quantities, as well as the Reynolds numbers for the primary and secondary flows (Reynolds numbers defined in terms of the respective total mass flowrate, the viscosity and the ratio of tube perimeter to tube area). 

Dimensional analysis shows that, in the treatment of natural convection, the dimensionless Grashof number, which represents the ratio of buoyancy to viscous forces, is often important. The definition of the Grashof number, Gr, is... 

In Fig. 10.4 the sphere diameter, terminal velocity, and temperature difference each appear in only one dimensionless group. The effect of natural convection on is smaller at Pr = 10 because the region over which the buoyancy force acts is much thinner than for Pr = 1. As Pr oo the effect should disappear altogether. For Pr = 0, numerical solutions (W7) show effects about 50% larger than for Pr = 1. 

The relationships developed from field measurements have been made dimensionless with the assumptions that v = 1.33 x 10 m /s and AijO = 2.6 x 10 m /s to facilitate comparisons between relations and avoid dimensional problems. They are given in Table 9.2. The early measurements were to investigate the loss of water from the reservoirs of the Colorado River in the United States, and the later measurements were designed to investigate heat loss from heated water bodies. A revelation occurred in 1969, when Shulyakovskyi brought in buoyancy forces as related to natural convection to explain the heat loss from heated water at low wind velocities. This was picked up by Ryan and Harleman (1973), who realized that natural convection could explain the need for a constant term in front of the relationship for gas film coefficient, as had been found by Brady et al. (1969), Kohler (1954), Rymsha and Dochenko (1958), and Shulyakovskyi (1969). Finally, Adams et al. (1990) rectified... 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

Spalding (51, 55, 60) presents, in dimensionless form, data on natural convection for kerosine, gas oil, petrol, and heavy naphtha burning from a 1.5-inch sphere. He suggests the following empirical relation (see Equation 8) for a range of transfer numbers, B, of 0.25 to 3 ... 

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 ... 

Dimensionless velocity profiles in natural convective boundary layer on a vertical plate for various values of Prandtl number. 

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,... 

There are two types of convection, free and forced (Holman, 2009 Incropera et al., 2007 Kreith and Bohn, 2007). Free (natural) convection occurs when the heat transferred from a leaf causes the air outside the unstirred layer to warm, expand, and thus to decrease in density this more buoyant warmer air then moves upward and thereby moves heat away from the leaf. Forced convection, caused by wind, can also remove the heated air outside the boundary layer. As the wind speed increases, more and more heat is dissipated by forced convection relative to free convection. However, even at a very low wind speed of 0.10 m s-1, forced convection dominates free convection as a means of heat loss from most leaves (0.10 m s-1 = 0.36 km hour-1 = 0.22 mile hour-1). We can therefore generally assume that heat is conducted across the boundary layer adjacent to a leaf and then is removed by forced convection in the surrounding turbulent air. In this section, we examine some general characteristics of wind, paying particular attention to the air boundary layers adjacent to plant parts, and introduce certain dimensionless numbers that can help indicate whether forced or free convection should dominate. We conclude with an estimate of the heat conduction/convection for a leaf. 

Derive the governing equations of natural convection, and obtain the dimensionless Grashof number by nondimensionalizing them,... 

The dimensionless parameter in the brackets represents the natural convection effects, and is called the Grashof number Gr, ... 

Having learned the dimensionless numbers associated with forced convection we now proceed to those for natural convection. 

Here, we recall that dimensionless numbers are composed only of independent quantities, and note that V is not an independent quantity for natural convection. Consequently, Eqs. (5.135), (5.153), and (5.163) may describe natural convection after they are combined and made independent of V. For example, the following combination of... 

Having learned the functional (implicit) relation among the dimensionless numbers of forced convection and of natural convection, we proceed to Chapter 6 for explicit relations among these numbers... 

What are the dimensionless numbers of combined forced-natural convection ... 

Consider the natural convection from a horizontal cylinder rotating with an angular frequency to (Fig. 5P-9). The peripheral surface temperature of the cylinder is Tm and the ambient temperature is To,. The diameter of the cylinder is D. Assuming that the natural convection resulting from rotation and that from gravity can be superimposed, express the Nusselt number in terras of the appropriate dimensionless numbers. 

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... 

Depending on the geometry and orientation of the object, pick the numerical value of Cq from Table 6.4 and compute the fundamental dimensionless number for natural convection,... 

In this section the full equations of motion for the external problem sketched in Fig. 4.1a are simplified by using approximations appropriate to natural convection, and the resulting equations are nondimensionalized to bring to light the important dimensionless groups. Although... 

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