Free-Convection Heat Transfer on a Vertical Flat Plate

Consider the vertical flat plate shown in Fig. 7-1. When the plate is heated, a free-convection boundary layer is formed, as shown. The velocity profile in this boundary layer is quite unlike the velocity profile in a forced-convection boundary layer. At the wall the velocity is zero because of the no-slip condition it increases to some maximum value and then decreases to zero at the edge of the boundary layer since the free-stream conditions are at rest in the free-convection system. The initial boundary-layer development is laminar but at 

To analyze the heat-transfer problem, we must first obtain the differential equation of motion for the boundary layer. For this purpose we choose the jc coordinate along the plate and the y coordinate perpendicular to the plate as in the analyses of Chap. 5. The only new force which must be considered in the derivation is the weight of the element of fluid. As before, we equate the sum of the external forces in the x direction to the change in momentum flux through the control volume dx dy. There results 

In other words, the change in pressure over a height dx is equal to the weight per unit area of the fluid element. Substituting Eq. (7-2) into Eq. (7-1) gives 

The density difference p - p may be expressed in terms of the volume coefficient of expansion p, defined by 

This is the equation of motion for the free-convection boundary layer. Notice that the solution for the velocity profile demands a knowledge of the temperature distribution. The energy equation for the free-convection system is the same as that for a forced-convection system at low velocity  

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FREE-CONVECTION HEAT TRANSFER ON A VERTICAL FLAT PLATE... 

Free-convection heat transfer on a vertical flat plate 329... 

The foregoing analysis of free-convection heat transfer on a vertical flat plate is the simplest case that may be treated mathematically, and it has served to introduce the new dimensionless variable, the Grashof number, which is important in all free-convection problems. But as in some forced-convection problems, experimental measurements must be relied upon to obtain relations for heat transfer in other circumstances. These circumstances are usually those in which it is difficult to predict temperature and velocity profiles analytically. Turbulent free convection is an important example, just as is turbulent forced convection, of a problem area in which experimental data are necessary however, the problem is more acute with free-convection flow systems than with forced-convection systems because the velocities are usually so small that they are very difficult to measure. Despite the experimental difficulties, velocity measurements have been performed using hydrogen-bubble techniques [26], hot-wire anemometry [28], and quartz-fiber anemometers. Temperature field measurements have been obtained through the use of the Zehnder-Mach interferometer. The laser anemometer [29] is particularly useful for free-convection measurements because it does not disturb the flow field. 

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