Effectiveness factor flat plate

Effect of geometric factors As shown in Section 2.4.3.1, Lienhard and Dhir (1973b) expressed the minimum dimension, L, of a horizontal flat-plate heater in terms of the dimensionless ratio L/ d. For ordinary liquids they found that the CHF is constant as long as L/ d > 3 (Eq. 2-128a). Otherwise, the CHF depends on the actual number of vapor jets (Lienhard and Dhir, 1973b),... 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities Extension to Reactions Other than First-Order and Various Catalyst Geometries. The analysis developed in Section 12.3.1.3 may be extended in relatively simple straightforward fashion to other integer-order rate expressions and to other catalyst geometries such as flat plates and cylinders. Some of the key results from such extensions are treated briefly below. 

For a flat-plate porous particle of diffusion-path length L (and infinite extent in other directions), and with only one face permeable to diffusing reactant gas A, obtain an expression for tj, the particle effectiveness factor defined by equation 8.5-5, based on the following... 

Fig. 3.3. Effectiveness factors for flat plate, cylinder and sphere...

Figure 6 shows the effectiveness factor for any of the three different pellet shapes as a function of the generalized Thiele modulus p. It is obvious that for larger Thiele moduli (i.e. p > 3) all curves can be described with acceptable accuracy by a common asymptote t] — 1 / p. The largest deviation between the solutions for the individual shapes occurs around p x 1. However, even for the extremely different geometries of the flat plate and the sphere, the deviation of the efficiency... 

Figure 7. Effectiveness factor tj as a function of the generalized Thiele modulus pn for different reaction orders. Influence of intraphase diffusion on the effective reaction rate (isothermal, irreversible reaction in a flat plate).

In Fig. 7 the effectiveness factor is shown as a function of the generalized Thiele modulus pn for different reaction orders (flat plate). From this figure, it is obvious that, except for the case of a zero order reaction, the curves agree quite well over the entire range of interest. The asymptotic solution t = l/ pn is valid for any reaction order and for values of the modulus pn > 3. 

A plot of the effectiveness factor from cq 53 against the Weisz modulus 1ppn from cq 58 gives the curve depicted in Fig. 8 for a first order reaction (flat plate). On the basis of this diagram, the effectiveness factor can be determined easily once the effective reaction rate and the effective diffusivity arc known. 

Roberts and Satterfield [87, 88] analyzed this type of reaction. On the basis of numerical calculations for a flat plate, these authors presented a solution in the form of effectiveness factor diagrams, from which the effectiveness factor can be determined as a function of the Weisz modulus as well as an additional parameter Kp s which considers the influence of the different adsorption constants and effective diffusivities of the various species [91], The constant K involved in this parameter is defined as follows ... 

Dispersion. Dispersion or London-van der Waals forces are ubiquitous. The most rigorous calculations of such forces are based on an analysis of the macroscopic electrodynamic properties of the interacting media. However, such a full description is exceptionally demanding both computationally and in terms of the physical property data required. For engineering applications there is a need to adopt a procedure for calculation which accurately represents the results of modem theory yet has more modest computational and data needs. An efficient approach is to use an effective Lifshitz-Hamaker constant for flat plates with a Hamaker geometric factor [9]. Effective Lifshitz-Hamaker constants may be calculated from readily available optical and dielectric data [10]. 

An important application of transpiration cooling is that of plane stagnation flow, as illustrated in Fig. 12-8. Solutions for the influence of transpiration on heat transfer in the neighborhood of such a stagnation line have also been worked out in Ref. 3, and the results are shown in Fig. 12-9. As would be expected, gas injection or suction can exert a significant effect on the temperature recovery factor for flow over a flat plate. These effects are indicated in Fig. 12-10, where the recovery factor r is defined in the conventional way as... 

Fig. 12-10 Effects of fluid injection on recovery factor for flow over a flat plate, according to Ref 3.

Instead of the numerical factor 4.0 in Equation 7.10, hydrodynamic theory predicts a factor near 6.0 for the effective boundary layer thickness adjacent to a flat plate (both numbers increase about 3% per 10°C Schlichting and Gersten, 2003). However, wind tunnel measurements under an appropriate turbulence intensity, as well as field measurements, indicate that 4.0 is more suitable for leaves. This divergence from theory relates to the relatively small size of leaves, their irregular shape, leaf curl, leaf flutter, and, most important, the high turbulence intensity under field conditions. Moreover, the dependency of 6bl on /° 5, which applies to large flat surfaces, does... 

Set up the equations necessary to calculate the effectiveness factor for a flat-plate catalyst pellet in which the following isothermal reaction takes place ... 

To evaluate the effectiveness factor for a first-order, isobaric, nonisothermal, flat plate catalyst pellet, the material and energy balances must be solved simultaneously. As shown previously, the mole balance in a slab is given by ... 

Find an expression for the overall effectiveness factor of a first-order isothermal reaction in a flat plate catalyst pellet. 

The derivation of this equation can be found in various advanced texts, for example, those of Warren [G.30] and James [G.7]. It applies to a polycrystalline specimen, made up of randomly oriented grains, in the form of a flat plate of effectively infinite thickness, making equal angles with the incident and diffracted beams and completely filling the incident beam at all angles 6. The second factor in square brackets, containing F, p, and 0, will be recognized as Eq. (4-19), the approximate equation for relative line intensities in a Debye-Scherrer pattern. 

Many numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in coeduits under conditions of fully developed or developing concentration or velocity profiles. Skellaed31 provides a particularly good summary of these results. The laminar boundary layer model has been extended to predict tha effects of high mass transfer flux on the mass transfer coefficient from a flat plate. The results of this work ate shown in Fig. 2.4-2 and. in com rest to the other theories, iedicate a Schmith number dependence of Ihe correction factor. 

Effectiveness Factors for Reversible Reactions The vast majority of the literature dealing with catalyst effectiveness factors presumes the reactions to be irreversible. However, in some cases it is possible to extend the analysis to certain reversible reactions. First-order reversible reactions have been treated for various catalyst geometries. For flat-plate geometry where only one side of the plate is exposed to reactant gases, one may proceed as in previous subsections to show that for mechanistic equations of the form... 

For zeolites, the following relationship can be used to directly estimate the concentration dependence of the effectiveness factor for a first-order reaction (involving single-component diffusion) in a flat plate (Ruthven, 1972) ... 

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