Natural Convection Models

The above theoretical approaches apply estimating MTCs for the forced convection of fluid flowing parallel to a surface. Typically, the fluid forcing process is external to the fluid body. In the case of natural or free convection fluid motion occurs because of gravitational forces (i.e., g = 9.81 m/s ) acting upon fluid density differences within (i.e., internal to) regions of the fluid. Temperature differences across fluid boundary layers are a major factor enhancing chemical mass transport in these locales. Concentration differences may be present as well, and these produce density differences that also drive internal fluid motion (i.e., free convection). 

The phenomenon of free convection results in nature, primarily from the fact that when the fluid is heated, the density (usually) decreases the warmer fluid portions move upward. This process is dramatically evident in rural areas on sunny days with low to no-wind when the soil surface is significantly hotter than the air above. The air at the soil surface becomes heated and rises vertically, producing velocity updrafts that carry the chemical vapor and the fine aerosol particles, laden with adsorbed chemical fractions, upward into the atmospheric boundary layer. When accompanied by lateral surface winds, the combined processes produce a very turbulent boundary layer and numerically large MTCs. This section will outline the major aspects of the theory of natural convection using elementary free convection concepts. Details are presented in Chapter 10 of Transport Phenomena (Bird et al., 2002). 

Thermal-free convection heat transfer. The basic parameter that regulates a fluid response to temperature variations is the thermal coefficient of volume expansion at constant pressure it is defined as 

Middleman (1998) presents the natural convection generated velocity profile in a fluid between two vertical parallel plates separated by distance w, meters, across which a temperature gradient AT/w, in °K/m, exist. The fluid region near the hot plate experiences an upward flow and that adjacent to the cold plate, a downward flow. The maximum velocity in each region is 

A more realistic and useful, but nevertheless simple, natural connection geometric shape is a single vertical plate with one face in contact with the fluid. It approximates the side of a building, a vertical cliff, steep mountain side or a similar feature in nature. An analysis of the heat flux for this situation has resulted in the following equation for the heat-transfer coefficient, h in W/m s  

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The effect of natural convection can be illustrated by considering the following simplified model. The reaction rates and physical parameters are constant except for the density which is given as ... 

Figure 10. Comparison of cup average conversion predicted by horizontally lumped model (natural convection included) and axisymmetric model (natural convection neglected) for a = 0, 5 and 10.

For simplicity of the model, it is assumed that the natural convection, radiation, and ionic wind effect are ignored. The ignorance of the radiation loss from the spark channel during the discharge may be reasonable, because the radiation heat loss is found to be negligibly small in the previous studies [5,6]. The amount of heat transfer from the flame kernel to the spark electrodes, whose temperature is 300 K, is estimated by Fourier s law between the electrode surface and an adjacent cell. 

Experimental gas-solid mass-transfer data have been obtained for naphthalene in CO2 to develop correlations for mass-transfer coefficients [Lim, Holder, and Shah, Am. Chem. Soc. Symp. Ser, 406, 379 (1989)]. The mass-transfer coefficient increases dramatically near the critical point, goes through a maximum, and then decreases gradually. The strong natural convection at SCF conditions leads to higher mass-transfer rates than in liquid solvents. A comprehensive mass-transfer model has been developed for SCF extraction from an aqueous phase to CO2 in countercurrent columns [Seibert and Moosberg, Sep. Sci. Techrwl, 23, 2049 (1988) Brunner, op. cit.]. 

Figure 2.24 Comparison of experimental results of bubble period with predictions of a model involving different mechanisms (a) nucleate boiling only (b) nucleate boiling and natural convection (c) nucleate boiling, natural convection, and microlayer evaporation. (From Judd, 1989. Copyright 1989 by American Society of Mechanical Engineers, New York. Reprinted with permission.)...

A boiling heat transfer model incorporating nucleate boiling, natural convection, and microlayer evaporation was formulated as... 

Judd (1989) interpreted experimental results of Ibrahim and Judd (1985), in which the bubble period first increased and then decreased as subcooling varied over the range 0 < (7 t - Tm) < 15°C (27°F), by means of a comprehensive model incorporating the contributions of nucleate boiling, natural convection, and microlayer evaporation components. The mechanism responsible for the nucleation of bubbles at exactly the frequency required at each level of subcooling is the subject of their continuing research. 

One can model this system in principle using the mass- and ener -balance equations written with the equations of Chapter 8 with flow in the j direction and diSiision in the jc direction to obtain profiles of Cj z, x) and T z, x). However, the student can see immediately that this will be a very complex mathematical problem to solve because there are many species (at least 30 for natural gas flames), but the problem will be made even more complex because of natural convection. Since the temperature in the flame varies from... 

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. 

Maximum Release. The analytical model described above assumes that the liquid phase is completely stagnant. While this may be true in an ideal laboratory experiment where a small sample can be kept isothermal at a specified temperature, in large scale systems where non-isothermal conditions exist, both natural convection and molecular diffusion will contribute to mass transfer. This combined effect, which is often very difficult to assess quantitatively, will result in an increase in fission-product release rate. Therefore, in making reactor safety analyses, it is desirable to be able to estimate the maximum release under all possible conditions. 

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. 

By introduction of a typical value for D0, 10 r> cm2 s 1, it is seen that the value of 8 after, for example, 5 seconds amounts to 0.1 mm. At times larger than 10-20 seconds, natural convection begins to interfere and the assumption of linear diffusion as the only means of mass transport is no longer strictly valid. At times larger than approximately 1 minute, the deviations from pure diffusion are so serious and unpredictable that the current observed experimentally cannot be related to a practical theoretical model. 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

Kumar, S. Mathematical modelling of natural convection in fire—A state of the art review of the field modelling of variable density turbulent flow. Fire and Materials, 1983. 7, 1-24. 

In order to. illustrate how natural convection in a vertical channel can be analyzed, attention will be given to flow through a wide rectangular channel, i.e., to laminar, two-dimensional flow in a plane channel as shown in Fig. 8.15. This type of flow is a good model of a number of flows of practical importance. 

Turbulent natural convective flows can also be analyzed by numerically solving the governing equations together with some form of turbulence model. This is... 

Henkes, R.A.W.M. and Hoogendoom. C.J., Comparison of Turbullence Models for the Natural Convection Boundary Layer Along a Heated Vertical Plate , Int. J. Heat Mass Transfer, Vol. 32, pp. 157-169, 1989. 

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