Natural convection motion, equation

It is well known that a hot piece of material will cool faster when air is blown or forced by the object. When the fluid outside the solid surface is in forced or natural convective motion, we express the rate of heat transfer from the solid to the fluid, or vice versa, by the following equation ... 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

In this section we derive the equation of motion that governs the natural convection flow in laminar boundary layer. The conservation of mass and energy equations derived in Chapter 6 for forced convection are also applicable for natural convection, but tlie momentum equation needs to be modified to incorporate buoyancy. 

Here, u represents a characteristic velocity of the flow and usotmd is the speed of sound in the fluid at the same temperature and pressure. It may be noted that usound for air at room temperature and atmospheric pressure is approximately 300 m/s, whereas the same quantity for liquids such as water at 20°C is approximately 1500 m/s. Thus the motion of liquids will, in practice, rarely ever be influenced by compressibility effects. For nonisothermal systems, the density will vary with the temperature, and this can be quite important because it is the source of buoyancy-driven motions, which are known as natural convection flows. Even in this case, however, it is frequently possible to neglect the variations of density in the continuity equation. We will return to this issue of how to treat the density in nonisothermal flows later in the book. 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

Equations (12—168)—(12 170) are known as the Boussinesq equations of motion and will form the basis for the natural convection stability analyses in this chapter. In fact, the Boussinesq approximation has been used in much of the published theoretical work on natural convection flows. Although one should expect quantitative deviations from the Boussinesq predictions for systems in which the temperature differences are large (greater than 10°C-20°C), it is likely that the Boussinesq equations remain qualitatively useful over a considerably larger range of temperature differences. In any case, although the Boussinesq equations represent a very substantial simplification of the exact equations, the essential property of coupling between the thermal and velocity fields is preserved, and, even in the Boussinesq approximation, the solution of natural convection problems is more complicated than the forced convection heat transfer problems that we encountered earlier. 

We saw above that the concentration gradient at an electrode will be linear with respect to the spatial coordinate perpendicular to the electrode surface if the anode/cathode cell were operated at a constant current density and if the fluid velocity were zero. In actuality, there will always be some bulk liquid electrolyte stirring during current flow, either an imposed forced convection velocity or a natural convection fluid motion due to changes in the reacting species concentration and fluid density near the electrode surface. In electrochemical systems with fluid flow, the mass transfer and hydrodynamic fluid flow equations are coupled and the solution of the relevant differential equations is often a formidable task, involving complex mathematical and/or numerical solution techniques. The concept of a stagnant diffusion layer or Nemst layer parallel and adjacent to the electrode surface is often used to simplify the analysis of convective mass transfer in... 

The above equation of motion is used for setting up problems in natural convection when the ambient temperature T may be defined. 

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... 

The first term is the pressure at some reference point (xref, yref, zKf) in the ambient fluid far from the body the second term is the hydrostatic pressure of this ambient fluid and the third term, Pd, is the pressure component associated with dynamics of the flow. Substituting Eq. 4.1 for P into the full equations of motion, we find that Pd simply replaces P in the x and y momentum equations. But in the z momentum equation there arises the additional term pM(z)g, representing the hydrostatic pressure gradient force. The local gravitational body force pg also appears in the z momentum equation, and the imbalance between these two forces, represented by the difference p (z)g- pg, is the driving force of natural convection. By introducing P = -(l/p)(dp/d7),> and y= lp(dpldP)r, this imbalance, called the buoyancy force, can be expressed by... 

Thus the appropriate equations of motion for natural convection [107] are produced by introducing Eq. 4.1 for P into the full equations of motion and then introducing Eq. 4.3 or 4.4 for the term (p - p)g. These equations are then further simplified by taking each of the properties p, p, Cp, and k as constant at their respective values, p0, Po, c,0, and k0, evaluated at Tf. 

With terms III-V in Eq. 4.7 deleted, Eqs. 4.5-4.7, together with the x and y momentum equations, constitute the simplified equations of motion appropriate to natural convection problems. For constant T . and 7U, the boundary conditions on these equations are 0 = 1 and u = v = w = 0 on the body and 0 = 0 far from the body. Steady-state laminar solutions to these equations are those that are obtained after setting the time partials (i.e., terms containing partial derivatives with respect to t ) in the equations equal to zero. Steady-state turbulent... 

In thermally non-homogeneous supercritical fluids, very intense convective motion can occur [Ij. Moreovei thermal transport measurements report a very fast heat transport although the heat diffusivity is extremely small. In 1985, experiments were performed in a sounding rocket in which the bulk temperature followed the wall temperature with a very short time delay [11]. This implies that instead of a critical slowing down of heat transport, an adiabatic critical speeding up was observed, although this was not interpreted as such at that time. In 1990 the thermo-compressive nature of this phenomenon was explained in a pure thermodynamic approach in which the phenomenon has been called adiabatic effect [12]. Based on a semi-hydrodynamic method [13] and numerically solved Navier-Stokes equations for a Van der Waals fluid [14], the speeding effect is called the piston effecf. The piston effect can be observed in the very close vicinity of the critical point and has some remarkable properties [1, 15] ... 

Natural convection heat transfer occurs when a solid surface is in contact with a gas or liquid which is at a different temperature from the surface. Density differences in the ffuid arising from the heating process provide the buoyancy force required to move the ffuid. Free or natural convection is observed as a result of the motion of the fluid. An example of heat transfer by natural convection is a hot radiator used for heating a room. Cold air encountering the radiator is heated and rises in natural convection because of buoyancy forces. The theoretical derivation of equations for natural convection heat-transfer coefficients requires the solution of motion and energy equations. 

Heat transfer by convection occurs in liquids and gases where there is a velocity field caused by extorted fluid motion or by natural fluid motion caused by a difference in density. The former case involves forced convection, and the latter case free convection. Combined convection occurs when both forced and free convection are present. The convection coefficient of surface heat transfer, a, defining the heat exchange in the contact boundary layer between fluid and soUd, is determined. Coefficient or is often expressed by equations containing criteria numbers, such as those of Nusselt (Nu), Prandtl (Pr), Reynolds (Re) and Grashof(Gr) ... 

Convection is the transfer of energy by conduction and radiation in moving, fluid media. The motion of the fluid is an essential part of convective heat transfer. A key step in calculating the rate of heat transfer by convection is the calculation of the heat-transfer coefficient. This section focuses on the estimation of heat-transfer coefficients for natural and forced convection. The conservation equations for mass, momentum, and energy, as presented in Sec. 6, can be used to calculate the rate of convective heat transfer. Our approach in this section is to rely on correlations. 

In order to determine the heat transfer rate by convection, the temperature distribution in the thermal boundary layer needs to be known. This temperature distribution depends on the nature of the fluid motion or the velocity field, and this is determined by solving the energy equation along with the mass and momentum equations for specific flow geometry. 

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