Governing Equations for Natural Convection

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Contrary to forced convection, which is driven by the external means, natural convection is induced by a density difference or density gradient in the flow field. This density difference may occur due to influence of temperature gradients on the density field, or due to differences in concentration or concentration gradients. Another condition that is necessary for natural convection to occur is body force that is proportional to the density. The most common body force is gravitational force, but the body force resulting from centrifugal or Coriolis forces is also possible. The combined body force and the density difference or gradient will produce buoyancy force within the fluid, which is the driving force behind natural convection. Natural convection can find its application in electronics cooling, the environmental science, manufacturing and materials processing, energy systems, and fire safety, etc. While the flow field for most forced convection problems can be solved independently of the solution for the temperature distribution, the flow field and temperature distribution in natural convection problems cannot be solved separately. A special technique must be developed to handle the coupling between the flow and heat transfer.

Most natural convection problems that one encounters are due to density differences caused by temperature differences. The densities of most liquids and gases decrease with increasing temperature, i.e., \partial \rho /\partial T<0 (with a few exceptions, such as water between 0°C and 4°C). So as to understand the mechanism of natural convection, let us consider the natural convection from a heated vertical flat plate placed in a quiescent fluid. The surface temperature of the flat plate, Tw, is higher than the temperature of the quiescent fluid, {T_{\infty}}. The fluid that is in contact with the heated vertical plate is heated and its density decreases (\partial \rho /\partial T<0). The heated lighter fluid flows up and the fluid from the neighbor moves in. The temperature in a thin layer of the fluid near the heated vertical plate is higher than the quiescent fluid temperature {T_{\infty}} and a thermal boundary layer is formed. As the fluid near the heated vertical plate rises, a velocity boundary layer also forms. Depending on the Prandtl number of the fluid, the velocity boundary layer can be thicker (Pr > 1; see Fig. 1) or thinner (Pr < 1) than the thermal boundary layer. The distributions of temperature and velocity in the boundary layers are also sketched in Fig. 1. Depending on the

 Natural convection over a vertical flat plate (Pr >1)
Figure 1: Natural convection over a vertical flat plate (Pr >1)

temperature difference, the properties of the fluid, and the length of the flat plate, a transition from laminar to turbulent flow, similar to Fig. 5.3, can also occur.

The local heat flux at the flat surface can be described using Newton’s law of cooling

{q}''={{h}_{x}}({{T}_{w}}-{{T}_{\infty }}) \qquad \qquad(1)

where hx is the convective heat transfer coefficient, which depends on flow configurations and the properties of the fluid. The heat transfer coefficient usually also depends on the temperature difference, {{T}_{w}}-{{T}_{\infty }}, which means that the heat flux is not a linear function of the temperature difference. At the surface of the heated plate, the velocity of the fluid is zero due to the non-slip condition. Therefore, conduction is the mechanism of heat transfer from the heated plate to the fluid immediately in contact with the heated plate so the heat transfer can be described by Fourier’s law of conduction:

{q}''=-k{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}} \qquad \qquad(2)

Combining eqs. (1) and (2) yields

{{h}_{x}}=-\frac{k}{{{T}_{w}}-{{T}_{\infty }}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}} \qquad \qquad(3)

which can be used to determine the local heat transfer coefficient. It reveals that the temperature profile in the thermal boundary layer is required to determine the heat transfer coefficient.

While the above discussions are for the case wherein the temperature of the vertical flat plate is higher than that of the quiescent fluid, they are also valid for the case wherein the temperature of the vertical flat plate is lower than that of the quiescent fluid. The only difference in the latter case is that the flow near the vertical plate will be directed downward and the boundary layers will start to form from the top of the flat plate.

Natural convection can also occur in an enclosure. In this case the development of the boundary layer is restricted by the wall of the enclosure. Natural convection in an enclosure is referred to as internal convection, which is different from the external natural convection from a heated or cooled vertical plate. The examples of internal natural convection include buoyancy-induced flow in a room caused by a heater or air-conditioner, electronics cooling in the computer chassis, melting or solidification of phase change materials in an enclosure, and propagation of fire in a room. Both internal and external natural convections are the subjects of discussion in this chapter.

Section 6.2 presents the governing equations for natural convection, which are obtained by simplifying the generalized governing equations presented in Chapter 2, followed by scale analysis of natural convection in Section 6.3. Section 6.4 discusses external natural convection from vertical, inclined, and horizontal flat plates, cylinders, and spheres, as well as free flows including plumes, and wakes. Internal natural convection is the subject of Section 6.5. Natural convection in melting and solidification is discussed in Section 6.6. This chapter is closed by discussion of instability analysis of natural convection.


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