Reference velocities

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

All variables in the system can be expressed in reduced form. Velocity can be expressed as tt = where Uq is a fixed reference velocity and j/ is the dimensionless reduced velocity. Because time, /q, is the quotient of length, Lq, and velocity, Uq, the equation can be manipulated to yield... 

Cross-sectionai area. The combustor cross section can be determined by dividing the volumetric flow at the combustor inlet by a reference velocity which has been selected as being appropriate for the particular turbine conditions on the basis of proven performance in a similar engine. Another basis for selecting a combustor cross section comes from correlations of thermal loading per unit cross section. Thermal loading is proportional to the primary zone air flow because fuel and air mixtures are near stoichiometric in all combustors. 

Manufacturers of thermal anemometers provide small rigs for their calibration. They typically consist of a nozzle, an air supply unit, and a regulating valve. The probe is placed into the nozzle jet. The reference velocity is calculated from the nozzle upstream pressure and nozzle characteristics. Due to its small size, this type of rig can be used only for hot-wire or other thermal anemometers. ... 

The calibration air flows through a thin tube. The probe is placed at the exit of the tube. When the tube is long enough and the tube flow is laminar, the reference velocity for calibration can be calculated from the theoretical, fully developed laminar velocity profile. 

Pressure drop is calculated by k x velocity pressure, where k is the resistance factor for the above items other than duct friction and is found from Table 29.1 or similar references. Velocity pressure is found by density x velocity, taking standard density at 1.2kg/m°, or from Table 29.2. 

Since the mean velocities in the y- and z-directions are zero, we will set the reference velocities in these directions to zero. 

For example, we may choose to as the average volume velocity, to = (c, V,) v,-. In more general terms, we may define to by X Pi vi > with Y Pi = 1 The s are weighting factors. If we formulate Eqn. (4.72) for two different reference velocities, to and to", and take into account the partial molar volumes (V,) which are not independent of each other (Gibbs-Duhem relation), we obtain after some algebraic rearrangements [H. Schmalzried (1981)] the quite general expression... 

To up-scale the previous model to the macroscale we make use of a formal homogenization procedure based on asymptotic expansions in terms of a perturbation parameter e which quantify the ratio between the meso and macroscales. To describe the physics properly, the coefficients must be scaled. Further, denoting vref and T>ref reference velocity and diffusion coefficint, and... 

By introducing a reference length l and a reference velocity V, we may define the nondi-mensional quantities... 

I/n Gas velocity component normal V Reference velocity, defined by... 

FIGURE 4 An example of reference velocities. Descriptions of diffusion imply reference to a velocity relative to the system s mass or volume. While the mass often has a nonzero velocity, the volume often shows no velocity. Hence, diffusion is best referred to the volume s average velocity. 

The following discussion is restricted to two-dimensional, steady, incompressible, constant-property flow. For simplicity, the body forces are neglected. The effects of body forces are considered in the chapter on natural convection. To nondimensionalize the appropriately reduced form of the governing equations from Tables 6.1 -6.3, we select a characteristic length L, a reference velocity a reference temperature... 

The reference velocity is taken as the freestream velocity u, which is, of course, constant for flow over a flat plate. Hence, the conditions on this initial line are as follows ... 

For the reasons given previously in this chapter, the reference velocity is taken... 

The mass flow of component i, p,v is a vector showing the flow of a component relative to a motionless coordinate system. On the other hand, diffusion flow shows the transport of a component relative to a coordinate system moving at the reference velocity vr. The diffusion flow relative to the center-of-mass velocity v (or mass average velocity) is... 

The Maxwell-Stefan equations do not depend on choice of the reference velocity, and therefore they are a proper starting point for other descriptions of multicomponent diffusion. For ideal gas mixtures, diffusivities /, and D u are... 

The first viscosity is the laminar viscosity i/. This is the only true flow characteristic and it defines the true Re3molds number of the flow Rcreai = ULjv where U and L are a reference velocity and length of the flow respectively. 

Transference numbers are quantities which are treated in the thermodynamics of irreversible processes. In a continuous system, the average velocity Vi of a species i related to a reference velocity w, describes the diffusional motion of the species i. The diffusion current density Ji represents in moles/cm sec the flow of species i in unit time perpendicular to a surface of unit area which by itself is moving with velocity... 

With k = 2, the reference velocity is equal to the velocity of solvent component 2, cj = Vj, and one obtains analogously... 

Since the total electric current density I is independent of the reference velocity... 

If after termination of a Hittorf transference experiment the changes in concentration in the electrode compartments are referred to a fixed weight of solvent, the reference velocity co = v is equal to... 

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