Normalized distance from wall

Gas fluctuating velocity y Normal distance from wall... 

Cheremisinoff and Davis (1979) relaxed these two assumptions by using a correlation developed by Cohen and Hanratty (1968) for the interfacial shear stress, using von Karman s and Deissler s eddy viscosity expressions for solving the liquid-phase momentum equations while still using the hydraulic diameter concept for the gas phase. They assumed, however, that the velocity profile is a function only of the radius, r, or the normal distance from the wall, y, and that the shear stress is constant, t = tw. ... 

In this equation, z is the normal distance from the adsorbent surface, p is the particle density, U(z) the fluid - fluid potential and V(z) the wall potential, all at position z. Eq. 6 therefore defines the total integral heat of adsorption at any pressure point on a model isotherm calculated by DFT. 

When a fluid flows past a stationary wall, the fluid adheres to the wall at the interface between the solid and the fluid. Therefore, the local velocity v of the fluid at the interface is zero. At some distance y normal to and displaced from the wall, the velocity of the fluid is finite. Therefore, there is a velocity variation from point to point in the flowing fluid. This causes a velocity field in which the velocity is a function of the normal distance from the wall, i.e., v = /(y). If y = 0 at the wall, u = 0, and v increases with y. The rate of change of velocity with respect to distance is the velocity gradient ... 

The project of vortex zone replacement is shown in Figure 12. The technological parameters of replacement management is determined by the parameters of pneumatic fan, the details are as follows ) The center angle corresponding to the arc AC is 60°, 2) The vertical distance of AB is 3.0 meters, 3) The horizontal distance of BC is 2.0 meters, 4) The height of airtight wall and replacements equals the roof of upper roadway, 5) The normal distance from pneumatic fan to BC is 5.0 meters, 6) Pneumatic fan is fixed on the posterior column of end support, and 5 meters away from the upper beam, 7) The type of pneumatic... 

Minimum distance from nearest cylinder to fire wall should normally be 1.5 m except as qualified. 

Figure 1.25 shows the boundary layer that develops over a flat plate placed in, and aligned parallel to, the fluid having a uniform velocity upstream of the plate. Flow over the wall of a pipe or tube is similar but eventually the boundary layer reaches the centre-line. Although most of the change in the velocity component vx parallel to the wall takes place over a short distance from the wall, it does continue to rise and tends gradually to the value vx in the fluid distant from the wall (the free stream). Consequently, if a boundary layer thickness is to be defined it has to be done in some arbitrary but useful way. The normal definition of the boundary layer thickness is that it is the distance from the solid boundary to the location where vx has risen to 99 per cent of the free stream velocity v . The locus of such points is shown in Figure 1.25. It should be appreciated that this is a time averaged distance the thickness of the boundary layer fluctuates owing to the velocity fluctuations.

Fig. 6. Schematic illustration of the stopped-flow magnetic tweezers experiments to follow single chromatin fiber assembly, (a) Flow diagram of how the experiment was performed, (b) A blow-up of the cuvette, with the bead attached to its side note that the DNA tether is not normal to the wall of the cuvette because of the position of the external magnet, i.e., the z direction is out of the plane of the video frame, (c) A schematic explaining the calculation of the distance traveled by the bead across the videoscreen. The X- and y-coordinates of the bead position on each successive video frame are used to calculate the projected traveled distance, (d) The actual shortening of the fiber can be calculated from the projected shortening (travel of bead across screen) and the cosine of the angle theta.

Different types of distributions correspond to different operating modes. The two most frequently used operating modes are normal mode and steric and hyperlayer mode (Reschiglian et al., 2005). The normal mode of separation is active for particles < 1 pm and the steric and hyperlayer modes are applicable to particles > 1 pm. In the normal mode as macromolecules or particles that constitute the sample are driven by the field toward the accumulation wall, their concentration increases with decreasing distance from the wall. This creates a concentration gradient that causes sample diffusion away from the wall. Retention time in the normal FFF is therefore... 

The amount adsorbed at each pressure can be obtained by integrating the equilibrium density profile p(r) and subtracting the quantity of adsorptive which would be present in the absence of wall forces p0(r). The integration limits depend on the pore geometry, for example, in the case of slit-shape pores, the densities are only functions of the normal distance (z) from the wall thus,... 

Fig. 26. Schematic design of field flow fractionation (FFF) analysis. A sample is transported along the flow channels by a carrier stream after injection and focusing into the injector zone. Depending on the type and strength of the perpendicular field, a separation of molecules or particles takes place the field drives the sample components towards the so-called accumulation wall. Diffusive forces counteract this field resulting in discrete layers of analyte components while the parabolic flow profile in the flow channels elutes the various analyte components according to their mean distance from the accumulation wall. This is called normal mode . Particles larger than approximately 1 pm elute in inverse order hydrodynamic lift forces induce steric effects the larger particles cannot get sufficiently close to the accumulation wall and, therefore, elute quicker than smaller ones this is called steric mode . In asymmetrical-flow FFF, the accumulation wall is a mechanically supported frit or filter which lets the solvent pass the carrier stream separates asymmetrically into the eluting flow and the permeate flow which creates the (asymmetrical) flow field...

This is about as far as thermodynamics takes us. In order to find the profiles statistical theory or simulations are required. By way of illustration a density distribution is given in fig. 1.38. The picture shows the r.h.s. of the distribution. The number density and the distance from the centre of the pore are normalized to become dimensionless (p/ ) is the density averaged over the pore. For (pjv) = 0.2 most adsorption is in the monolayer, for solvent structure force develops. )... 

Figure 11. The orientational order parameter for benzene relative to the wall in a slit pore is wn here as a function of distance from the wall. The order parameter is defined by S=<3 cos 0 - 1)12 where 6 is the angle between the benzene symmetry axis and the normal to the surface (S=l when the molecule lies parallel to the surfece). From Ref. [42], J. Chem. Phys. 99 (1993) 5405-5417.

The planar symmetric pore consists of two parallel walls with the distance H between them which infinitely range into the x- and j/-direction of the pore-fixed coordinate system. The 2-axis stands perpendicularly on the i-j/-plane as the normale of both walls. The cylinder pore model places its j/-axis as the rotational axis. The z-axis stands perpendicularly on the pore wall as in slit-like pores and runs through the middle of the pore. Hence the x- differs from the y-axis inside the cylinder pore in opposite to the slit-like pore. This fact turns out to be important even for the adsorption of fluids which consists of non-spherical particles. 

A simple graphical illustration of a follows from (1.25). As shown in Fig. 1.7 the ratio X/a is the distance from the wall at which the tangent to the temperature profile crosses the = t F line. The length of X/a is of the magnitude of the (thermal) boundary layer thickness which will be calculated in sections 3.5 and 3.7.1 and which is normally a bit larger than X/a. A thin boundary layer indicates good heat transfer whilst a thick layer leads to small values of a. 

In many technical applications, for example flows in channels, the velocity profile close to the wall is only dependent on the distance from the wall. Indicating the velocity parallel to the wall with wx and the coordinate normal to the wall by y, then wx(y), whilst the other velocity components disappear, wy = wz = 0. This type of flow is known as stratified flow. In steady-state, laminar flows with vanishing pressure gradients, the momentum equation (3.98) is simplified to... 

It has been shown that there exists a continuous change in the physical behavior of the turbulent momentum boundary layer with the distance from the wall. The turbulent boundary layer is normally divided into several regions and sub-layers. It is noted that the most important region for heat and mass transfer is the inner region of the boundary layer, since it constitutes the major part of the resistance to the transfer rates. This inner region determines approximately 10 — 20% of the total boundary layer thickness, and the velocity distribution in this region follows simple relationships expressed in the inner variables as defined in sect 1.3.4. 

In solid-liquid systems the size and shape of the baffles are important design parameters. The standard baffling is illustrated in Fig 7.1. As the solid concentration increases and the viscosity becomes high, narrower baffles (approximately 1/24T) placed a distance from the wall, should be used. This design is normally employed to avoid permanent settling of particles in the low velocity zones. In some processes such fillets (settled particles) can nevertheless be advantageous for the power consumption. 

The force and torque exerted on a solid particle were obtained in the form of a power series with respect to RJl, where is the particle radius and I is the distance from the center of the particle to the wall. Lorentz derived an asymptotic expression for the motion of a sphere along the normal... 

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