Fluid dynamics, dimensionless groups

Transfer properties, the heat and mass transfer coefficient and friction factor, depend not only on transport and thermodynamic properties but also on the hydro-dynamic behavior of a fluid. The geometry of the system will influence the hydro-dynamic behavior. By reducing the parameters by arranging them into dimensionless groups, we can reduce the number of parameters that have to be varied to correlate any of the transfer properties. For example, the ffiction factor equation. 

It should be noted that for some problems, more than one characteristic time for the flow can be identified [3]. Thus, a second dimensionless group, the Weissenberg number, Wi, is sometimes used in polymeric fluid dynamics. It can be defined as... 

As with viscoelastic fluids in Eq. (100) and plastic viscous fluids in Eq. (106), dynamic similarity involves systems with the same UIL. Both viscoelastic and plastic flow phenomena are then associated with dimensionless groups involving UIL. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

Specification of initial conditions completes the statement of the problem. We emphasize that little can be done analytically (we will refer to what can be done later), and in general numerical solutions are required. The problems between the fluid dynamics and the surfacr tant concentration are coupled nonlinearly and we describe numerical methods and results later. In addition to the nonlinear nature of the problem, certain dimensionless groups are large and make the equations stiff, in the sense that either small time-steps are needed or boundary layer structures need to be resolved. For example, the Peclet numbers tend to be large in applications (see earlier) and solute boundary layers develop near the surface. Also, the bulk concentration constant k which appears in the mass balance equation (39) can be of order 10 - making that equation stiff. These issues are taken up in Section . 

A dimensionless group, used in fluid dynamics, heat and mass transfer A flight of small height, welded to pillars or vanes attached to the conveyor hub Metal strips sometimes welded, or spot-welded, axially to the bowl or beach, or onto a liner thereon. Used sometimes in place of grooves... 

---