Temperature profile natural convection

The wall boundary condition applies to a solid tube without transpiration. The centerline boundary condition assumes S5anmetry in the radial direction. It is consistent with the assumption of an axis5Tnmetric velocity profile without concentration or temperature gradients in the 0-direction. This boundary condition is by no means inevitable since gradients in the 0-direction can arise from natural convection. However, it is desirable to avoid 0-dependency since appropriate design methods are generally lacking. 

Let us examine methanol. Its flashpoint temperature is 12 to 16 °C (285-289 K) or, say, 15 °C. If this is in an open cup, then the concentration near the surface is Xl = 6.7 %. Performed under normal room temperatures of, say, 25 °C, the temperature profile would be as in Figure 6.2. This must be the case because heat must be added from the air to cause this evaporated fuel vapor at the surface. This decrease in temperature of an evaporating surface below its environment is sometimes referred to as evaporative cooling. If the convective heat transfer coefficient, typical of natural convection, is,... 

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... 

Dimensionless temperature 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 ... 

As a first example of low-density heat transfer let us consider the two parallel infinite plates shown in Fig. 12-14. The plates are maintained at different temperatures and separated by a gaseous medium. Let us neglect natural-convection effects. If the gas density is sufficiently high so that A — 0, a linear temperature profile through the gas will be experienced as shown for the case of A. As the gas density is lowered, the larger mean free paths require a greater distance from the heat-transfer surfaces in order for the gas to accommodate to the surface temperatures. The anticipated temperature profiles are shown in... 

The velocity and temperature profiles for natural convection over a vertical hot plate are also shown in Fig. 9 -6. Note lhat as in forced convection, the thickness df the boundary layer increases in the flow direction. Unlike forced convection, however, the fluid velocity is zero at the outer edge of the velocity boundary layer as well as at the surface of the plate. This is expected since the fluid beyond the boundary layer is motionless. Thus, the fluid velocity increases with distance from the surface, reaches a maximum, and gradually decreases to zero at a distance sufflciently far from (be surface. At the. surface, the fluid temperature is equal to the plate temperature, and gradually decreases to the temperature of the surrounding fluid at a distance sufficiently far from the surface, as shown in the figure. In the case of cold surfaces, the shape of the velocity and temperature profiles remains the same but their direction is reversed. 

A 12-cm-wide and 18-cm-high vertical hot surface in 30 C air is lo be cooled by a heat sink v/ilh equally spaced fins of rectangular profile (Fig. 9-20). The. fins are 0.1 cm thick and 18 cm long in the vertical direction and have a height of 2.4 cm from the base. Determine the optimum fin spacing and the rate of heat i transfer by natural convection from the heat sink if the base temperature is 80°C. ... 

The simulations were performed assuming that the flow is laminar. Additionally, the contact angle is assumed to be known. The initial velocity is assumed to be zero everywhere in the domain. The initial fluid temperature profile is taken to be linear in the natural convection thermal boundary layer and the thermal boundary layer thickness, 5j, is evaluated using the correlation for the turbulent natural convection on a horizontal plate as, Jj. =1. 4(vfiCil ... 

Note that the distributions are cardinally different from those over smooth surfaces like in the previous cases. Again, two parts of the profiles have to be discussed. Over the top SCS s level z = h = 6 m, the wind velocity distributions grow monotonically in the case of a strong wind the temperature diminishes, as a rule. Few cases where the wind velocity diminishes over the SCS are characterized by a weak external wind so that the horizontal forced convection is perhaps comparative with the intense natural convective motion rising up from the heated and wetted air layer within SCS. 

Consider the natural convection for Pr > 1 from a vertical plate at a temperature Tw iu an ambient at temperature T . Evaluate the local heat transfer for the following three cases, (a) linear temperature, parabolic velocity profiles, (b) parabolic temperature, linear velocity profiles, (c) parabolic temperature and velocity profiles. Compare the results with Eq. (5.104). 

When strong temperature gradients exist, natural convection may be primarily induced thermally or both heat and mass transfer may play comparable roles. In these cases the situation is more complex, because the number of parameters increases. In liquid metals and semiconductors the Schmidt number, v/Dl, is several orders of magnitude greater than the Prandtl number, v/a, and this enables one to solve for the concentration profile in a rather general way without great difficulty as will be discussed next. 

Fig. 7.4-2 Possible temperature profiles of the liquid in the tubes of a natural convection evaporator...

An important heat-transfer system occurring in process engineering is that in which heat is being transferred from a hot vertical plate to a gas or liquid adjacent to it by natural convection. The fluid is not moving by forced convection but only by natural or free convection. In Fig. 4.7-1 the vertical flat plate is heated and the free-convection boundary layer is formed. The velocity profile differs from that in a forced-convection system in that the velocity at the wall is zero and also is zero at the other edge of the boundary layer since the free-stream velocity is zero for natural convection. The boundary layer initially is laminar as shown, but at some distance from the leading edge it starts to become turbulent. The wall temperature is T K and the bulk temperature T. ... 

The steady laminar flow of a liquid through a heated cylindrical pipe has a parabolic velocity profile if natural convection effects, and variation of physical properties with temperature are neglected [4], If the fluid entering the heated section is at a uniform temperature (Ti) and the wall is maintained at a crmstant temperature (T ), develop Graetz s solution by neglecting the thermal conductivity in the axial directiOTi. 

Note that we have divided the Darcy velocity by fractional porosity in the last step to have true velocity. The (8p/Bx) term was previously expressed in terms of (BT/Bx) and (Bxi/3x) (see Eq. (2.122)). Equation (2.139) applies to both thermal convection, where the convection is driven by (BT/Bx) as well as natural convection where flow is driven by (BT/Bx) and (BXi/Bx). As was stated before, convection may weaken or enhance composition variation. Figure 2.33 provides a simple explanation of the change in composition due to convection. In this figure, the diagram on the right (Fig. 2.33a) shows the composition variation vs. depth with zero convection at x = 0 assuming that = 0, and that and are not functions of temperature (see Eq. (2.125)). The thin line shows zero vertical compositional grading. Now allow for small values of pv (proportional to z) as shown by thick line B. Assume that pv is identically zero. Because of convection, the composition profile A cannot stay the same, otherwise the material balance for component 1... 

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... 

Figure 6.30 (a) Velocity and (b) temperature profiles for natural convection of a fluid between two vertical plates maintained at different temperatures. Results from FEMLAB (www.comsol.com). 

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