Dimensionless group correlations

I 8.1 can (seldom) be measured, but dimensionless group correlations are available... 

Empirical dimensionless group correlations have been used in the scale-up process. In particular, the correlation for the inside film heat transfer coefficient for agitated, jacketed vessels has been employed for the scale-up to a larger vessel. Reaction calorimeters are often used to give some indication of heat transfer coefficients compared to water in the same unit. Correlation for plant heat transfer is of the general form... 

A number of excellent compilations of dimensionless group correlations are available in the literature (see Levich [12], Selman and Tobias [14], and Poulson [17]). These compilations are not reproduced here, but the correlations for a number of common hydrodynamic geometries are listed in Table 1. Because these equations are empirical in nature, they are valid only for the employed ranges in Re and Sc. 

In summary, while a number of dimensionless group correlations have been proposed to transfer mass-transfer and corrosion-rate data from one hydrodynamic geometry to another, all of these correlations are system-specific and are not generally valid for use outside of the ranges of conditions (Re, Sc, T) for which they have been derived. Clearly, a universal correlation is lacking (and is perhaps unattainable), particularly one that can be used to correlate mass-transfer geometries at elevated temperatures. Again, we emphasize that the development of correlations of this type is vitally important for the quantitative comparison of different series of experimental data. 

It is particularly important in the case of dimensionless group correlation, in order to provide a meaningfuJ value of the constants K, a and b in the expression Sh= where Re is the Reynolds number, 5c is the Schmidt number and Sh is the Sherwood number. 

From the above discussion, it can be seen that the mass transport for a given electrode-electrolyte geometry and electrolyte hydrodynamics may be related to the process parameters by a suitable dimensionless group correlation, e.g., mass transport to the rotating disc in laminar flow conditions may be described by the equation (see footnote, page 23) ... 

Example Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below ... 

It has been found that these dimensionless groups may be correlated well by an equation of the type... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

Heat Transfer In general, the fluid mechanics of the film on the mixer side of the heat transfer surface is a function of what happens at that surface rather than the fluid mechanics going on around the impeller zone. The impeller largely provides flow across and adjacent to the heat-transfer surface and that is the major consideration of the heat-transfer result obtained. Many of the correlations are in terms of traditional dimensionless groups in heat transfer, while the impeller performance is often expressed as the impeller Reynolds number. 

The inverse of the Bodenstein number is eD i/u dp, sometimes referred to as the intensity of dispersion. Himmelblau and Bischoff [5], Levenspiel [3], and Wen and Fan [6] have derived correlations of the Peclet number versus Reynolds number. Wen and Fan [6] have summarized the correlations for straight pipes, fixed and fluidized beds, and bubble towers. The correlations involve the following dimensionless groups ... 

Forced convection heat transfer has been measured under widely differing conditions, and using the dimensionless groups makes correlation of the experimental... 

The rotating disc and rotating cylinder have been successfully applied in the laboratory to study the effect of flow on corrosion rates and are much easier to use than actual pipelines and other real geometries. The results of these tests can now be correlated to geometries likely to be found in pipes, pumps, bends, etc. in plant by use of dimensionless group analysis. There-... 

Roy et al. (R3) define the critical solids holdup as the maximum quantity of solids that can be held in suspension in an agitated liquid. They present measurements of this factor for various values of gas velocity, gas distribution, solid-particle size, liquid surface tension, liquid viscosity, and a solid-liquid wettability parameter, and they propose the following two correlations in terms of dimensionless groups containing these parameters ... 

Convective heat transfer to fluid inside circular tubes depends on three dimensionless groups the Reynolds number. Re = pdtu/ii, the Prandtl number, Pr = Cpiilk where k is the thermal conductivity, and the length-to-diameter ratio, L/D. These groups can be combined into the Graetz number, Gz = RePr4/L. The most commonly used correlations for the inside heat transfer coefficient are... 

Xu et al. [124] numerically computed the adiabatic temperature rise in a micro channel due to viscous heating and expressed their results by a correlation based on dimensionless groups. They introduced a dimensionless temperature rise AT = AT/Tjgf with a reference temperature of 1 K. The correlation they found is given by... 

Van Winkle et al. (1972) have published an empirical correlation for the plate efficiency which can be used to predict plate efficiencies for binary systems. Their correlation uses dimensionless groups that include those system variables and plate parameters that are known to affect plate efficiency. They give two equations, the simplest, and that which they consider the most accurate, is given below. The data used to derive the correlation covered both bubble-cap and sieve plates. 

Dimensional analysis of the variables characteristic of mass transfer under flow conditions suggests that the following dimensionless groups are appropriate for correlating mass transfer data. 

The dimensionless group hD/k is called the Nusselt number, /VNu, and the group Cp i/k is the Prandtl number, NPl. The group DVp/p is the familiar Reynolds number, NEe, encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as... 

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