Dimensionless correlations

Convection Heat Transfer. Convective heat transfer occurs when heat is transferred from a soHd surface to a moving fluid owing to the temperature difference between the soHd and fluid. Convective heat transfer depends on several factors, such as temperature difference between soHd and fluid, fluid velocity, fluid thermal conductivity, turbulence level of the moving fluid, surface roughness of the soHd surface, etc. Owing to the complex nature of convective heat transfer, experimental tests are often needed to determine the convective heat-transfer performance of a given system. Such experimental data are often presented in the form of dimensionless correlations. 

The design approach is particularly feasible for those reactions in which chemical and pore diffusion rates are most important. For flow related phenomena semi-empirical, dimensionless correlations must be relied on. Therefore in this book scale-up will be used in the more general sense with the airri of using methods that are fundamentally based wherever feasible. 

Fig. 4.2.14 Dimensionless correlation between the minimum shear rate at which the shear viscosity can be obtained via MRI, ymjn and the velocity resolution, These data were obtained using an aqueous polyacrylamide solution at two different volume flow rates, or mean velocities, w.

Patience et al. (1992) developed a dimensionless correlation for the mean slip factor between the gas and solid by using solid suspension data from various small laboratory beds. The proposed correlation relates the slip to the Froude number based on the bed diameter. It remains to be seen if the correlation will hold at Froude numbers typical of large beds and if other dimensionless factors are important. 

Predictability of Film Resistance. The magnitude of film resistance can be estimated from dimensionless correlations such as Eq. 24. Thus an observed rate approximately equal to the calculated rate suggests that film resistance controls. 

The mass transfer between phases is, of course, the very basis for most of the diffusional operations of chemical engineering. A considerable amount of experimental and empirical work has been done in connection with interphase mass transfer because of its practical importance an excellent and complete survey of this subject may be found in the text book of Sherwood and Pigford (S9, Chap. Ill), where dimensionless correlations for mass transfer coefficients in systems of various shapes are assembled. 

For practical purposes, heat-transfer engineers often use empirical or semi-empirical correlations to predict h values. These formulations are usually based on the dimensionless numbers described before. In this case, the appropriate formulation should be used to prevent significant errors. If dimensionless correlations are applicable under conditions of gas extraction, then heat-transfer coefficients can be determined from these correlations and the influence of parameter variations may be derived also from them. 

Dimensionless correlations based on momentum-or energy balances, and using the Lockhart and Martinelli parameter, Xu that is the ratio of the single-phase pressure drops at the same velocities ... 

Dimensionless correlations based on momentum-or energy-balances and using the Lockhart and Martinelli parameter, Xc, or some similar parameter. The practical relevance of such correlations is limited since their use requires the knowledge of the single-phase pressure drop of both gas and liquid furthermore, the influence of the geometry of the bed is not always well described by these single-phase pressure drops alone. 

Dimensionless correlations based on dimensional analysis and on a qualitative analysis of the two-phase flow. 

Pipeline Contactors The correlation for droplet diameter based on power/mass is similar to that for two-fluid nozzles. The dimensionless correlation is... 

More recent literature regarding generalized correlational efforts for gas holdup is adequately reviewed by Tsuchiya and Nakanishi [Chem. Eng Sci., 47(13/14), 3347 (1992)] and Sotelo etal. [Inf. Chem. Eng., 34(1), 82-90 (1994)]. Sotelo et al. (op. cit.) have developed a dimensionless correlation for gas holdup that includes the effect of gas and liquid viscosity and density, interfacial tension, and diffuser pore diameter. For systems that deviate significantly from the waterlike liquids for which Fig. 14-104 is applicable, their correlation (the fourth numbered equation in the paper) should be used to obtain a more accurate estimate of gas holdup. Mersmann (op. cit.) and Deckwer et al. (op. cit.) should also be consulted. 

One of the main disadvantages of the above-mentioned relationship is that it does not account for the variation of density (p), viscosity (rj), and solute diffusivity (Dvt) with cB- On the contrary, the use of dimensionless Sh, Re, and Sc numbers has been traditionally regarded as the most accurate mode to deal with such variations. The empirical dimensionless correlations reported in Table III, except for the one established by Kraaijeveld et al. (1995),... 

Heat transfer and its counterpart diffusion mass transfer are in principle not correlated with a scale or a dimension. On a molecular level, long-range dimensional effects are not effective and will not affect the molecular carriers of heat. One could say that physical processes are dimensionless. This is essentially the background of the so-called Buckingham theorem, also known as the n-theorem. This theorem states that a product of dimensionless numbers can be used to describe a process. The dimensionless numbers can be derived from the dimensional numbers which describe the process (for example, viscosity, density, diameter, rotational speed). The amount of dimensionless numbers is equal to the number of dimensional numbers minus their basic dimensions (mass, length, time and temperature). This procedure is the background for the development of Nusselt correlations in heat transfer problems. It is important to note that in fluid dynamics especially laminar flow and turbulent flow cannot be described by the same set of dimensionless correlations because in laminar flow the density can be neglected whereas in turbulent flow the viscosity has a minor influence [144], This is the most severe problem for the scale-up of laminar micro results to turbulent macro results. 

Experimental values of mass transfer coefficients can be collected as dimensionless correlations. One collection of these correlations is in Table II (Cussler, 1997). Because heat transfer is mathematically so similar to mass transfer, many assert that other correlations can be found by adapting results from the heat transfer literature. While this is sometimes tme, the analogy is frequently overstated because mass transfer coefficients normally apply across... 

These two cases illustrate the use of dimensionless correlations for mass transfer and some of the problems with units in mass transfer. 

Koziol and Mackowiak (55a) found the Kister and Haas correlation to give good agreement with experimental data. They developed a new dimensionless correlation [thus overcoming the need for a dimensional exponent in Eq. (6.27)] for spray regime entrainment at very low liquid rates (0.1-1.5 gpm/in). Their correlation, however, postulates that entrainment rises with tower diameter at the same steep rate at which it rises with hole diameter. This postulate conflicts with the industiy s experience that entrainment does not increase upon tower diameter scale-up. 

Dimensionless Correlation of the Hanging Film Phenomenon Schneider, H., T. Roth, Hochschulkurs Emul-giertechnik, Universitat Karlsruhe 1996, XIII-1 /18 Emulgierverfahren in der Lebensmitte-lindustrie... 

From the dimensionless correlation between the power number and the expanded Froude number (Zlokarnik, 1972), one obtains... 

The mixing time for viscous liquids was examined by Hoogendoorn and Den Hartog (1967). The types of mixers examined in this study are illustrated in Fig. 23. The mixing time was measured by a decoloration and a thermal response technique (see Section IX). In truly viscous flow, the mixing time was inversely proportional to the stirrer speed. The performance of the various mixers were compared using the two dimensionless correlations 02P/(dfp) and pdf/(fid). The turbine and anchor mixers were found to be unsatisfactory for viscous mixing. 

Here, p°° and p°° are standard chemical potentials of reaction groups i and j of the reference composition p° and p° the same of the groups of the inspected composition and Xy and Xji are dimensionless correlation coef ficients. Coefficients Xy usually fall in the range between 0.1 and 0.5. 

Another complication in the use of generalized dimensionless correlations for scale-up of mixing systems lies in the difficulty of establishing an adequate performance parameter. In some cases there may be several different parameters, like conversion and purity, for example, or particle size and catalytic activity the correlations between the different parameters and the agitation system properties may not be the same, and this may make the scale-up more difficult and more arbitrary. 

Except for those described above, all other investigators correlated their data for the liquid holdup in terms of dimensionless correlations. Otake and Okada65 correlated the dynamic liguid—Jroldup for 6.4- through 22-mm spheres by a relation/... 

The dimensionless correlations between the liquid holdup and the gas and liquid Reynolds numbers are also given by Ford6 and Saada.26 Ford6 suggested that for h, < 0.43, a relation... 

Kim et al.56 studied the effects of air and water velocity and particle size ranging from 2.6 through 6.0 mm on the gas and liquid holdups in a two-dimensional column. Based on their data, they reported the following dimensionless correlations for the liquid and gas holdups ... 

Kim et al.56 correlated their data for liquid-phase backmixing in a three-phase rectangular column by a dimensionless correlation ... 

Thin solid films of polymeric materials used in various microelectronic applications are usually commercially produced the spin coating deposition (SCD) process. This paper reports on a comprehensive theoretical study of the fundamental physical mechanisms of polymer thin film formation onto substrates by the SCD process. A mathematical model was used to predict the film thickness and film thickness uniformity as well as the effects of rheological properties, solvent evaporation, substrate surface topography and planarization phenomena. A theoretical expression is shown to provide a universal dimensionless correlation of dry film thickness data in terms of initial viscosity, angular speed, initial volume dispensed, time and two solvent evaporation parameters. 

Figure 5. Permeability autocorrelograms realizations whose dispersivity is in Figure the dimensionless correlation length by varying...

In the limits of large / a trajectory of renormalization group transformation terminates in a fixed point. The dimensionless correlation length 2/ can be used as a measure of the remoteness of a percolation system from a critical point... 

On the assumption that permeate flow does not strongly modify hydrodynamics at walls, k is ordinarily estimated using classical dimensionless correlation for non-porous walls, such as the Leveque equation for laminar flow or the Deissler equation for turbulent flow. More recently, in order to account for specific disturbances that take place at pore entrance when high permeate fluxes prevail, new specific equations have been proposed [3]. 

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