The Characteristic Lengths

The solutions that arise can be understood in terms of a set of characteristic lengths. (For a steady state system where the concentrations vary with distance, characteristic lengths play the same part as characteristic times in ordinary reaction kinetics.) The lengths are summarised in Table 1 and illustrated in figure 2. 

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Equation 74 is shown graphically ia Figure 19a for a given set of conditions. Curves such as these cannot be directly used for design, however, because the Peclet number contains the tower height as a characteristic dimension. Therefore, new Peclet numbers are defined containing as the characteristic length. These relate to the conventional Pe as... 

For rectangular ducts Kays and Clark (Stanford Univ, Dept. Mech. Eng. Tech. Rep. 14, Aug. 6, 1953) published relationships for headng and cooling of air in rectangular ducts of various aspect rados. For most noncircular ducts Eqs. (5-39) and (5-40) may be used if the equivalent diameter (= 4 X free area/wetted perimeter) is used as the characteristic length. See also Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. 

Values of and m for various configurations are hsted in Table 5-5. The characteristic length is used in both the Nusselt and the Reynolds numbers, and the properties are evaluated at the film temperature = (tio + G)/2. The velocity in the Reynolds number is the undisturbed free-stream velocity. 

Oztnrk et al. (1987) developed a new correlation on the basis of a modification of the Aldta-Yoshida correlation suggested hy Nakanoh and Yoshida (1980). In addition, the hnhhle diameter rather than the colnmn diam-eterwas used as the characteristic length as the colnmn diameter has little influence on k a. The value of was assumed to he approximately constant (dfc = 0.003 m). The correlation was obtained hy nonlinear regression is as follows ... 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

Here the characteristic length, /, the reciprocal real part of the transfer constant according to Eq. (23-12), is analogous to the data in Section 24.4.2 ... 

The characteristic resistance Z is therefore the lowest value of the grounding resistance. The path constant /a = I R G = is the characteristic length of the infinite ground. 

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

The characteristic length L denotes the pipe diameter or the hydraulic diameter djjyj = 4A/F A is the cross-sectional area and P is the wet periphery). If the cross-section is not circular, or in the case of a plane, the length is measured in the flow direction. 

This equation does not incorporate the characteristic length L hence the wail height has no influence on heat transfer. 

Fully developed nonisothermal flow may also be similar at different Reynolds numbers, Prandtl numbers, and Schmidt numbers. The Archimedes number will, on the other hand, always be an important parameter. Figure 12.30 shows a number of model experiments performed in three geometrically identical models with the heights 0.53 m, 1.60 m, and 4.75 m." Sixteen experiments carried out in the rotxms at different Archimedes numbers and Reynolds numbers show that the general flow pattern (jet trajectory of a cold jet from a circular opening in the wall) is a function of the Archimedes number but independent of the Reynolds number. The characteristic length and velocity in Fig. 12.30 are defined as = 4WH/ 2W + IH) and u = where W is... 

One particularly important feature of the plate heat exchanger is that the turbulence induced by the troughs reduces the Reynolds number at which the flow becomes laminar. If the characteristic length dimension in the Reynolds number is taken as twice the average gap between plates, the Re number at which the flow becomes laminar varies from about 100 to 400, according to the type of plate. 

A particular fluid flow problem must have an associated characteristic length L and characteristic velocity V. These values may be more or less arbitrarily specified, with the only constraint being that they represent some typical scales. For example, if the problem involves a flow past a sphere, L could be the diameter of the sphere and V could be the velocity of the fluid at infinity. The characteristic length and characteristic velocity also fix a characteristic time scale T = L/V. 

The solution of this equation is in the form of a Bessel function 32. Again, the characteristic length of the cylinder may be defined as the ratio of its volume to its surface area in this case, L = rcJ2. It may be seen in Figure 10.13 that, when the effectiveness factor rj is plotted against the normalised Thiele modulus, the curve for the cylinder lies between the curves for the slab and the sphere. Furthermore, for these three particles, the effectiveness factor is not critically dependent on shape. 

Equations (3.11) and (3.12) show that the friction factor of a rectangular micro-channel is determined by two dimensionless groups (1) the Reynolds number that is defined by channel depth, and (2) the channel aspect ratio. It is essential that the introduction of a hydraulic diameter as the characteristic length scale does not allow for the reduction of the number of dimensionless groups to one. We obtain... 

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