Natural Convection and Mixed Flow

The Boussinesq Approximation Analyses of time-steady free convection usually assume that  

density is constant in the continuity and momentum equations, except in the body force term  

density variations are caused only by temperature and composition gradients  

Under these assumptions Eq. (1-1), the Navier-Stokes equation, becomes 

The variable density is expanded in a Taylor series about the density of the fluid far from the body, 

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This model is rather simple, because it neglects possible mixing effects caused by natural convection and convection forced by H2 flow or slider motion and the dependence of impurity diffusion coefficients on the concentrations of other impurities present in the melt. The exact mechanism by which baking influences the concentration of trace impurities is not well understood. However, the use of a prebaking step is considered necessary to achieve high-purity film growth by LPE. 

Horizontal Cylinders. For a heated horizontal cylinder in perpendicular cross flow, the angle of the approaching stream, ( > in Fig. 4.47, greatly affects the heat flow in the mixed convection regime. For ( > = 0 the forced flow assists the natural convection and the dependence of the average Nus-selt number on Re resembles path A in Fig. 4.44. For = 90° there is a sharper transition from natural to forced convection than when 4> = 0, while for opposed flow (( > = 180°) there is a minimum as shown by path B in Fig. 4.44. For a cooled cylinder the same description applies except that ( > is measured from the vertical axis extending upward from the cylinder. 

Equating the Nusselt numbers for pure natural convection and pure forced convection provides a good estimate of the Ra-Re curve along which mixed convection effects are most important, as already discussed. After a careful study of available data, Morgan [198] proposed the following equation for forced convection heat transfer from a cylinder for cross flow in a low-turbulence airstream ... 

FIGURE 4.48 Regimes of forced, mixed, and natural convection for assisting flow over a horizontal circular cylinder. 

FIGURE 4.51 Regimes of natural, forced, and mixed convection for flow through vertical tubes with uniform wall temperature of heat flux, for 10 2 < (Pr DIL) < 1. From Metais and Eckert [190],... 

For prediction of subassembly coolant flow rate and temperature distributions a wide range of coolant flow and thermal convection regimes must be considered including laminar and turbulent flow natural, forced and mixed (forced + natural) convection and steady state and transient reactor conditions. 

The equilibrium between these two effects causes mixed convection and stratified flow. The stratification interface oscillates and may induce severe damage to the neighbouring structures. Furthermore, the support structures at the bottom of the hot pool have to be protected from hot sodium. The behaviour of such a region has been studied, namely for SPXl [8.27] and EFR [8.19]. Studies were mainly conducted through scale model tests, because computations are not yet able to predict the fluctuation characteristics for such complex situations. As the main physical phenomenon of interest is the interaction between buoyancy forces (natural convection) and inertia forces (forced convection from the main pool... 

In general, density stratification, with hotter fluid, vapour or liquid, above colder fluid, acts so as to oppose vertical natural convection and vertical mixing between layers. In the absence of any heat flow, this convective stability applies to both vapour and liquid. 

Convection. Heat transfer by convection arises from the mixing of elements of fluid. If this mixing occurs as a result of density differences as, for example, when a pool of liquid is heated from below, the process is known as natural convection. If the mixing results from eddy movement in the fluid, for example when a fluid flows through a pipe heated on the outside, it is called forced convection. It is important to note that convection requires mixing of fluid elements, and is not governed by temperature difference alone as is the case in conduction and radiation. 

In this section we consider the combustion of premixed gaseous fuel and air mixtures. Consider first the laboratory Bunsen burner, shown in Figure 10-1 1. Natural gas from the gas supply system enters the bottom of the burner, where it is mixed with air, with flow rates adjusted by the gas valve and holes in the bottom of the burner, where air is sucked in by natural convection. The premixed gases travel up the barrel of the burner (a tubular reactor), and, if flows are suitably adjusted and a match has been used to ignite the mixture, a stable flame forms at the top of the tube. 

When the air inlet is shut off in a Bunsen burner, the flame can still be maintained, but it switches from the blue color of a normal CH4 flame to the yellow of a difllision flame. Now pure CH4 flows up the tube, and the O2 from outside the tube is mixed with CH4 just above the tube, with flow of air driven by natural convection below the hot product gas mixture. 

Consideration of the available data for spheres indicates that forced flow correlations are accurate to about 10% for Gq/Re < 0.2. The analogous limit for natural convection is not so well defined, being about 10 at Pr = 0.7 and increasing with Pr. Additional studies of mixed convection are needed to elucidate the physical phenomena and provide correlations. Simultaneous mass... 

In conduction, heat is conducted by the transfer of energy of motion between adjacent molecules in a liquid, gas, or solid. In a gas, atoms transfer energy to one another through molecular collisions. In metallic solids, the process of energy transfer via free electrons is also important. In convection, heat is transferred by bulk transport and mixing of macroscopic fluid elements. Recall that there can be forced convection, where the fluid is forced to flow via mechanical means, or natural (free) convection, where density differences cause fluid elements to flow. Since convection is found only in fluids, we will deal with it on only a limited basis. Radiation differs from conduction and convection in that no medium is needed for its propagation. As a result, the form of Eq. (4.1) is inappropriate for describing radiative heat transfer. Radiation is... 

In all of these tests, the specimens, 75 mm square and up to 25 mm thick, are exposed to radiant heat with and without a pilot flame(s). Decomposition takes place inside a closed cabinet of 0.51 m3. There is no control of the air flow or oxygen concentration through the fire zone and the effluent is mixed by natural convection, as it accumulates within the closed cabinet. Gases are sampled using probes mounted in the center of the cabinet. 

The buoyancy forces that arise as the result of the temperature differences and which cause the fluid flow in free convection also exist when there is a forced flow. The effects of these buoyancy forces are, however, usually negligible when there is a forced flow. In some cases, however, these buoyancy forces do have a significant influence on the flow and consequently on the heat transfer rate. In such cases, the flow about the body is a combination or mixture of forced and free convection as indicated in Fig. 9.1 and such flows are referred to as combined or mixed forced and free (or natural) convection. 

Purely forced, purely free, and mixed convective regions in assisting flow over a vertical plate. (Based on results obtained by Patel K., Armaly B.F., and Chen T.S., Transition from Turbulent Natural to Turbulent Forced Convection Adjacent to an Isothermal Vertical Plate , ASME HTD, Vol. 324, pp. 51-56, 1996. With permission.)... 

The book provides a comprehensive coverage of the subject giving a full discussion of forced, natural, and mixed convection including some discussion of turbulent natural and mixed convection. A comprehensive discussion of convective heat transfer in porous media flows and of condensation heat transfer is also provided. The book contains a large number of worked examples that illustrate the use of the derived results. All chapters in the book also contain an extensive set of problems. 

Mixed Forced and Natural Convection Natural convection is commonly assisted or opposed by forced flow. These situations are discussed, e.g., by Mills (Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 340) and Raithby and Hollands (Chap. 4 of Rohsenow, Hartnett, and Cho, Handbook of Heat Transfer, 3d ed., McGraw-Hill, 1998, p. 4.73). 

In a cold wall reactor, the convection regime is mixed. On the one hand, the temperature gradient between the substrate and the walls of the reactor tends to establish a system of natural convection (laminar flow). On the other hand, the flow of gas induces a forced convection (turbulent flow). A laminar flow is necessary to ensure a good uniformity of the epitaxial film thickness. 

Combined forced and free convection at a vertical flat plate, where the forced convection velocity is in the same direction as the natural convection flow (the so-called assisting mixed convection case). Here, researchers have combined Sherwood numbers for the pure forced and natural convection cases in the following way [15, 24-26] ... 

Introduction. For the problem depicted in Fig. 4.44, the heat transfer by pure forced convection would increase monotonically with Reynolds number along the curve shown. The heat transfer by pure natural convection from the same surface for various Ra is denoted by the horizontal lines in the figure. If Re is slowly increased from zero in the real problem, the measured values of Nu would at first follow the natural convection curve, since the superimposed forced convection velocities are too feeble to affect the heat transfer. If the forced convection assists the natural convection, the Nu curve in Fig. 4.44 will break upward along path A at larger Re and approach the pure forced convection curve from above. If the flows are opposed, Nu passes through a minimum along path B in Fig. 4.44 and approaches the forced convection curve from below. Mixed convection occurs when the heat transfer is significantly different from that for either pure natural convection or pure forced convection. 

Uniform Heat Flux. For laminar flow in a horizontal tube where uniform heat flux is applied at the outer boundary of the tube, the bulk temperature Tb, increases linearly in the axial direction. To maintain the heat flow to the fluid, the wall temperature must remain higher than the fluid temperature, and under these conditions a fully developed natural convection motion becomes established in which velocity and temperature gradients become independent of the axial location. Because the fully developed Nusselt number for laminar pure forced convection is small (Nuf —> 4.36), the buoyancy-induced mixing motion can greatly enhance the heat transfer. 

As the film thickens further, turbulence will develop in the condensate film, and the heat transfer mechanism then undergoes a significant change, since the heat is transferred across the condensate film by turbulent mixing as well as by molecular conduction. For gravity-dominated flow (i.e., natural convection), the transition from laminar-wavy flow to turbulent flow occurs at film Reynolds numbers of about 1600 [18]. 

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