Characteristic lengths

Let us consider the flow in a narrow gap between two large flat plates, as shown in Figure 5.19, where L is a characteristic length in the a and y directions and h is the characteristic gap height so that /z < L. It is reasonable to assume that in this flow field il c iq, Vy. Tlierefore for an incompressible Newtonian fluid with a constant viscosity of q, components of the equation of motion are reduced (Middleman, 1977), as... 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

Similarly, let the lengths be expressed in terms of some characteristic length, the most suitable being the most frequently occurring length L (i.e. that corresponding to the maximum in Fig. 1.14). Thus, let... 

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. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

L Characteristic length m R.. Infinite shear viscosity (Bingham plastics) Pa s... 

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 ... 

In these expressions, B = ZJd, Nps = dVp/EE, Np r = dVn/Eii, where d = some characteristic length such as dp for packed towers or T for spray towers. Ep and Er are the longitudinal dispersion coefficients, which must ultimately be deter-... 

Dust explosibility characteristics Length and shape of relief pipe if existent... 

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 ... 

---