Upward-Facing Plates

In detached flow, the mixing of the boundary layer fluid and the ambient fluid is driven by the component of gravity normal to the surface. Since this mixing dominates the heat exchange, the heat transfer from a tilted upward-facing plate can be obtained from the equation for a horizontal plate by replacing g by g cos For the flow to remain detached, however, g cos must remain positive. Since Nu, < Raj 3, it follows directly that, in detached flow,... 

Horizontal Heated Upward-Facing Plates ( = 0°) With Uniform T and T>,... 

FIGURE 4.10 Definition sketch for natural convection on a horizontal upward-facing plate of arbitrary planform. Only the top heated surface of area A is heated. 

Plates at Arbitrary Angle of Tilt. The previous sections provide equations from which to compute the total heat transfer from vertical plates ( = 90°), horizontal upward-facing plates ( = 0°), and horizontal downward-facing surfaces ( = 180°). These equations are the basis for obtaining the heat transfer from tilted plates. 

For a wide isothermal plate at any angle of tilt, first compute the heat transfer from Eq. 4.33 (vertical plate) but with g replaced by g sin then compute the heat transfer from Eq. 4.40 (downward-facing plate) with g replaced by g(0, -cos Qmax, then compute the heat transfer from Eq. 4.39 (upward-facing plate) with g replaced by g(0, cos Qmax, and take the maximum of the three heat transfer rates. It is important to take maximum heat transfer rather than the maximum Nusselt number, because the Nusselt numbers are based on different length scales. 

Horizontal Upward-Facing Plate (Fig. 4.41b). Based on the similarity solution of Cheng and Chang [48], the average heat transfer from an isothermal plate in Darcian flow is given by... 

The Nusselt analysis for a vertical plate may be adapted to an inclined upward-facing plate (inclined at an angle a with respect to the horizontal) by replacing g in Eq. 14.10 with (g sin a). This simple substitution is valid except for plates that are near horizontal (i.e., a <... 

In pipe distributors, the pressure drop requited for good gas distribution is 30% of the bed pressure drop for upward facing holes, but only 10% for downward facing ones. The pressure drop calculation and the recommended hole density are the same as for a perforated plate. To maintain good gas distribution within the header system, it is recommended the relation... 

Physical arrangement of equipment Horizontal plates facing upward Horizontal plates facing downward Vertical plates more than 2 ft high Vertical plates less than 2 ft high Vertical pipes more than 1 ft high Horizontal pipes K 0.38 0.20 0.30 0.28 atmospheric pressure, normal conditions for more exact equations, see Perry.5... 

Equation for C,. A number of experiments at different Prandtl numbers (mostly on tilted plates) have been carried out that permit the function C,(Q to be modeled. Observation has also revealed that there are two patterns of turbulent flow detached and attached. Attached flow, where the flow sticks to the body contour, is best exemplified by the flow on a vertical plate, and the C,(Q, = C, (90°) applying in this situation is denoted Cv,. Detached flow, where turbulent eddies rise away from the heated surface, is best exemplified by the flow on a horizontal upward-facing (heated) plate, and the C,(Q = C,(0°) applying in this situation is denoted Cf. The first step in establishing the C,(Q function has been to model how Cf and CJ depend on the Prandtl number the equations for these quantities (justified later) that have been found to best fit currently available data are... 

FIGURE 4.11 Comparison of Eq. 4.39 to data for upward-facing heated plates of various planform in air. 

K. Kitamura and F. Kimura, Heat Transfer and Fluid Flow of Natural Convection Adjacent to Upward-Facing Horizontal Plates, Int. J. Heat Mass Transfer (38/17) 3149-3159,1995. 

Nimmo and Leppert [111] used a Nusselt-type analysis to study laminar film condensation on a finite, upward-facing horizontal plate, assuming that the condensate flow was driven by the hydrostatic pressure gradient due to changes in condensate film thickness from the center of the plate to the edges. They arrived at the following approximate expression for the mean Nusselt number ... 

Chiou and Chang [113] analyzed laminar film condensation on a horizontal upward-facing disk and found that their model overpredicts the theoretical result from Eq. 14.98 by about 25 percent. They attribute this to the fact that on a disk, the surface area increases radially outward, and this makes the condensate film thinner than on a rectangular plate, as studied by Nimmo and Leppert [111]. 

In an experiment designed to reveal surface features, a sample of rolled aluminum 2024 sheet (dimensions 100 X 40 X 4 mm) was placed in a 250-mL beaker in such a way that it was immersed in aerated 3% NaCl solution to a level about 30 mm from the top of the specimen. The effect of aeration created a splash zone over the portion of the surface that was not immersed. During the course of exposure, a portion of the immersed region in the center of the upward-facing surface became covered with gas bubbles and suffered a higher level of attack than the rest of the immersed surface. After 24 h, the plate was removed from the solution. Figure 4.2 shows the specimen and the areas where the surface profiles were measured in diagrammatic form. 

If a particle presents a face inclined at an angle to the direction of motion of the fluid, it may be subjected to an upward lift due to the aerofoil effect. In Figure 5.13 a flat plate is shown at an angle to a stream of fluid flowing horizontally. The fluid pressure acts normally at the surface and thus produces forces with vertical and horizontal components... 

For shapes which have an appreciable fraction of area over which the normal component of the buoyancy force is directed away from the surface (e.g., a heated horizontal plate facing upward) the turbulent regime where SF oc (Ray occurs at low Ra. For example, the horizontal plates of Fig. 10.3 have only one side exposed, the side for which the normal component of buoyancy is directed away from the plate. Here the power relation applies at Ra > 10. For horizontal cylinders, on the other hand, such a relationship is exhibited for Ra > 5 X 10. For spheres in Fig. 10.2, there is no indication of this transition even at Ra = 10. The critical Rayleigh number, Ra/, above which the j-power relationship applies is correlated by... 

More recently, Simon and Dasgupta (110) used a parallel-plate wet denuder in which the scrubber liquid is made to flow down two closely spaced silica-coated parallel plates facing each other with the air sample flowing upward in the gap. Excellent collection efficiency (90% + ) was established for S02 even at flow rates of 15 L/min. The denuder effluent was preconcentrated and chromatographed as in reference 108. On the basis of signal-to-noise data observed at S02 test concentrations of 19 pptrv, the calculated LOD of the system is on the order of 0.5 pptrv. 

The experiments of Ref. 44 have produced the following correlations for constant heat flux on a horizontal plate. For the heated surface facing upward,... 

For an inclined plate with heated surface facing upward the empirical correlations become more complicated. For angles between -15 and -75° a suitable correlation is... 

Heated plate facing upward or cooled plate facing downward... 

Heated plate facing downward or cooled plate acing upward... 

Repeat Prob. 7-51 for the heated plate facing upward. 

The most common configuration for analytical IEF is the horizontal polyacrylamide slab gel. Gels are cast on glass plates or specially treated plastic sheets with one exposed face. They are placed on cooling platforms and run with the exposed gel face upward. Electrolyte strips, saturated with 1 N phosphoric acid at the anode and 1 N sodium hydroxide at the cathode, are placed directly on the exposed surface of the IEF gel. Contact between the... 

Consider the natural-convection equations available. Heat-transfer coefficients for natural convection may be calculated using the equations presented below. These equations are also valid for horizontal plates or discs. For horizontal plates facing upward which are heated or for plates facing downward which are cooled, the equations are applicable directly. For heated plates facing downward or cooled plates facing upward, the heat-transfer coefficients obtained should be multiplied by 0.5. 

The third example of field emission from Si-based nanowires is from the aligned SiC nanowires. The field emission measurements [68] were carried out in a vacuum chamber at a pressure of 5 x 10 Torr at room temperature. An oriented SiC nanowire array, which was used as the cathode, was stuck to a stainless steel substrate by silver paste with the bottom end of the nanowires facing upward. A copper plate with a diameter of 1 cm, mounted on a precision linear feedthrough, was used as the anode. Field emission current densities of 10 pA cm were observed at applied fields of 0.7-1.5 V pm and current densities of 10 mA cm were realized at applied fields as low as 2.5-3.5 V pm , as shown in Figure 10.35. These results represent one of the lowest fields ever reported for any field-emitting materials at technologically useful current densities. We attributed this emission... 

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