Characteristic scales pressure

Using the system (9.15-9.17) we determine the distribution of velocity, temperature and pressure within the liquid and vapor domains. We render the equations dimensionless by the following characteristic scales l,o for velocity, 7l,o for temperature, Pl,o for density, Pl,q for pressure, Pl,o Lo f " force and L for length... 

Assuming steady state in Eqs. (10.8-10.10) and (10.18-10.20), we obtain the system of equations, which determines steady regimes of the flow in the heated miero-channel. We introduce values of density p = pp.o, velocity , length = L, temperature r = Ti 0, pressure AP = Pl,o - Pg,oo and enthalpy /Jlg as characteristic scales. The dimensionless variables are defined as follows ... 

In Equation 5.2.2, it is assumed that acoustic wavelengths A are large when compared with any of the characteristic scales of the flow (A L) and that the measurement point r is far from the source region (r /l). The previous expression provides the sormd pressure in the far-field for a compact source, but it can be used indifferently for premixed or nonpremixed flames [30]. 

The other method is the velocity head method. The term V2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V2/2g. Thus II is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to pV2 where V is a characteristic velocity of the flow. 

The key step in developing a reduced coordinate system lies in identifying characteristic scaling parameters. In the case of gases, these characteristic parameters are the critical pressure, the critical volume, and the critical temperature. We seek similar scaling parameters for chromatography. 

Following our usual custom, we now nondimensionalize. The physically obvious characteristic scales are the length scales for variations of the velocity and perturbation pressure in the x and v directions, and the characteristic magnitude of the velocity in the x direction,... 

In most cases one is interested in fluid flows at scales that are much larger than the distance between the molecules. The value of the molecular mean free path in air at room temperature and 1 atm of pressure is A = 6.7 x 10-8 m and in water A = 2.5 x 10-10 m. When the Knudsen number - defined as the ratio of the molecular mean-free-path to a characteristic length scale of the flow (e.g. the size of the smallest eddies) - is small, the fluid can be described as a continuous medium in motion. In this continuum approximation the flow can be characterized by the velocity field v(x, t) representing the instantaneous velocity of infinitesimal fluid elements at time t and at position x. Fluid elements represent small volumes of fluid that are much smaller than the smallest characteristic scale of the flow, but sufficiently large to contain a large number of molecules so that a well defined local velocity exists and molecular fluctuations can be neglected. 

Such boundary layers exist in proximity to any interface solid-liquid, liquid-liquid, and liquid-air. In the vicinity of the apparent three-phase contact line (Fig. 3), those boundary layers overlap. The overlapping of boundary layers is the physical phenomenon which results in the existence of surface forces. Let the thickness of the boundary layers be 8. In the vicinity of the three-phase contact line, the thickness of a droplet/ meniscus, h, is small enough, that is, /i 8, and, hence, boundary layers overlap (Fig. 3), which results in the creation of disjoining pressure. A similar situation occurs at a contact of two particles in a liquid (Fig. 4). The abovementioned characteristic scale of boimdary layer thickness, 8 10 cm, determines the characteristic thickness where the disjoining pressure acts. 

In microfluidic systems, very high pressure gradients are generally required to drive and manipulate the fluid flow. Due to the small characteristic scale... 

With this process the main objective is to produce the same interfacial areas per unit volume on both scales, in order to achieve the same mass transfer. The analysis based on turbulence theory has been confirmed by the knowledge gained in practice in the form of the scale-up criterion P/V = const. This applies to dispersing processes in liquid/liquid and gas/liquid systems. Because of the numerous factors that influence the process (e.g., coalescence properties, physical properties of mixtures, anomalous flow characteristics, static pressure, etc.), substantial... 

The following characteristic scales, radius, R, velocity scale. Us, timescale, R/ Us, pressure scale, fiUs/R, and fiUsIR are used to make the continuity and Navier-Stokes equation dimensionless. The dimensionless density, viscosity, thermal conductivity, and heat capacity distributions are defined as... 

Here, the characteristic velocity is U, characteristic length is L, characteristic pressure is fiU/L, and characteristic time is L/U. The characteristic scale of magnetic field intensity is the maximum field density, B. The characteristic scale of electric field is the maximum electric field, E. The dimensionless numbers appearing in the above equations are Reynolds number. Re = and interaction parameter, N = In case of microchannel, / has large value due to high electric field. This indicates that J is little dependent on V. The interaction parameter, N, measures the ratio of electromagnetic force to inertia force. For low Reynolds number and high interaction parameter, the momentum equation can be written as... 

Variable-Area Flow Meters. In variable-head flow meters, the pressure differential varies with flow rate across a constant restriction. In variable-area meters, the differential is maintained constant and the restriction area allowed to change in proportion to the flow rate. A variable-area meter is thus essentially a form of variable orifice. In its most common form, a variable-area meter consists of a tapered tube mounted vertically and containing a float that is free to move in the tube. When flow is introduced into the small diameter bottom end, the float rises to a point of dynamic equiHbrium at which the pressure differential across the float balances the weight of the float less its buoyancy. The shape and weight of the float, the relative diameters of tube and float, and the variation of the tube diameter with elevation all determine the performance characteristics of the meter for a specific set of fluid conditions. A ball float in a conical constant-taper glass tube is the most common design it is widely used in the measurement of low flow rates at essentially constant viscosity. The flow rate is normally deterrnined visually by float position relative to an etched scale on the side of the tube. Such a meter is simple and inexpensive but, with care in manufacture and caHbration, can provide rea dings accurate to within several percent of full-scale flow for either Hquid or gas. 

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