Dimensionless numbers Graetz

In order to obtain the (mixing-cup and surface) concentration and temperature profiles, the solution of Equations 8.52 and 8.55 requires the calculation of the dimensionless transfer coefficients Sh and Nw, which are in general functions of the Graetz parameters and of the Damkohler number. Correlations for these dimensionless numbers are available in the literature (e.g., Shah and London [39]), for several conditions regarding the degree of profile development, degree of hydrodynamic flow development, and boundary condition at the wall. These features are briefly discussed in the following. 

A dimensionless number closely related to the Graetz number is the Peclet number ... 

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... 

Analysis of the various stages in mold filling is based on two dimensionless criteria the Graetz Number, Gz, which is a measure of the ratio of convective to conductive heat exchange ... 

For known values of the parameters in the kinetic equation for a specific reactive mix, it is easy to calculate the dimensionless factors y and v. Then the flow pattern in the mold filling process is completely determined by the dimensionless Da and Gz Numbers and the boundary conditions. The Damkohler Number characterizes the ratio of the rates of chemical reaction and convective heat transfer and the Graetz Number is a measure of the ratio of the convective heat flux due to a moving liquid to the heat flux due to the conductivity of the liquid. 

Growth limitation factor for substrate Dimensionless 12.13 Graetz number Dimensionless 5.36... 

O THE GRAETZ AND PECLET NUMBERS. Two other dimensionless groups are commonly used in place of the Fourier number in treating heat transfer to fluids. The Graetz number is defined by the equation... 

The Graetz eigenfunctions and eigenvalues are not widely tabulated. However, Brown (1960) has computed a large number of eigenvalues suitable for most problems. The final result can be written in dimensionless form as... 

Colton et al. used a Graetz-type solution to calculate a log-mean feed-side Sherwood mindin (SHf = kfh/S), where A is the chaiind height) for huninar flow in a flat duct with permeable walls as a function of the wail Sherwood number (Shw, = and the dimensionless length of the dialyzer. This analysis... 

The concentration profile as estimated by Equations 8.34 is plotted in Figure 8.4, along with the numerical solution of Equation 8.16, subject to the boundary condition (8.20). The axial variation of the mixing-cup concentration (expressed in terms of the dimensionless Graetz parameter) is shown for several values of the Damkohler number (covering kinetic and mass transfer control). 

Gz is known as the Graetz number, for this problem we can think of it as a dimensionless fiow rate. Equation 4.19 then becomes... 

Figure 4.2. Dimensionless pressure drop n as a function of Graetz number (dimensionless flow rate) for flow through a plane channel with wall cooling. The horizontal line shows three intersections, corresponding to the three values of Gz that satisfy Equation 4.27 at fixed B for a given n.

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