Linear profile

Using a log-linear profile of the wind speed, and assuming a surface roughness length of about 0.3 m, u is estimated from the 10-meter wind speed, u,o, as ... 

This section considers three special cases. The first is a flat velocity profile that can result from an extreme form of fluid rheology. The second is a linear profile that results from relative motion between adjacent solid surfaces. The third special case is for motionless mixers where the velocity profile is very complex, but its net effects can sometimes be approximated for reaction engineering purposes. 

For flow parallel to an electrode, a maximum in the value of the mass-transfer rate occurs at the leading edge of the electrode. This is not only the case in flow over a flat plate, but also in pipes, annuli, and channels. In all these cases, the parallel velocity component in the mass-transfer boundary layer is practically a linear function of the distance to the electrode. Even though the parallel velocity profile over the hydrodynamic boundary layer (of thickness h) or over the duct diameter (with equivalent diameter de) is parabolic or more complicated, a linear profile within the diffusion layer (of thickness 8d) may be assumed. This is justified by the extreme thinness of the diffusion layer in liquids of high Schmidt number ... 

Figure 19 (a) Peak melting temperature as a function of the branch content in ethylene-octene copolymers (labelled -O, and symbol —B (symbol, ) and -P (symbol, A) are for ethylene-butene and ethylene-propylene copolymers, respectively) and obtained from homogeneous metallocene catalysts show a linear profile, (b) Ziegler-Natta ethylene-octene copolymers do not show a linear relationship between peak melting point and branch content [125]. Reproduced from Kim and Phillips [125]. Reprinted with permission of John Wiley Sons, Inc. 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

As a special case, we consider a linear concentration profile along the x-axis C(x) = a0 + atx. Since the second derivative of C(x) of such a profile is zero, diffusion leaves the concentrations along the x-axis unchanged. In other words, a linear profile is a steady-state solution of Eq. 18-14 (dC/dt = 0). Yet, the fact that C is constant does not mean that the flux is zero as well. In fact, inserting the linear profile into Fick s first law (Eq. 18-6) yields ... 

If we switch from a situation with uniform chemical potential (A/i0 = 0) to a situation in which on one side a different but constant PQ is established, a transient occurs during which the homogeneous stoichiometry profile changes to an approximately linear profile (see chemical polarization, see Appendix 3). As long as the electrode reactions are fast, the emf measured at such a sample is always determined by the invariant boundary values of the oxygen potential (ju0,ju0 + Aju0) but, owing to the internal virtually neutral short-circuit, lower than the Nernst-value. The result is, instead of Eq. (20),56 57 now... 

The boundary conditions for the depolarization are the same as in Appendix B.l, but 7=0. The initial condition here is the final linear profile in Appendix B.l with ibeing the polarization current. The result for the concentration is... 

As shown in Table 6.17, the hardness—compression force profile for Drug A tablets is linear across the range of compression forces studied. This linear profile is attributed to the properties inherent in the brittle and ductile excipients chosen for this formulation. In contrast, the hardness—compression force profile for Drug B shows a plateau in tablet hardness at higher compression forces (> 11 kN), as listed in Table 6.18. 

A general mathematical formulation and a detailed analysis of the dynamic behavior of this mass-transport induced N-NDR oscillations were given by Koper and Sluyters [8, 65]. The concentration of the electroactive species at the electrode decreases owing to the electron-transfer reaction and increases due to diffusion. For the mathematical description of diffusion, Koper and Sluyters [65] invoke a linear diffusion layer approximation, that is, it is assumed that there is a diffusion layer of constant thickness, and the concentration profile across the diffusion layer adjusts instantaneously to a linear profile. Thus, they arrive at the following dimensionless set of equations for the double layer potential, [Pg.117]

Note, for highly non-linear profiles of state variables, switching from continuous to discrete or from discrete to continuous using linear interpolation technique may not be efficient and non-linear interpolation technique may need to be employed. 

The linear profile of molecular weight versus conversion... 

J (length element), the variational problem is obviously analogous to finding the curve of minimum length between two terminals, which is the line in the x-c-plane. Hence the linear profile of the steady state refers to a minimum entropy production. It is also straightforward to show that this is stable. 

In the SFA experiments, normal (P ) and shear forces on the film are measured as the plates are sheared past each other at velocity (v). The effective viscosity (p) is then calculated from experimental measurements of the shear stress (f) and the applied shear rate (y), which is determined using the measured plate separation h) and assuming a linear profile between plates. 

Linear profiles are the simplest profiles to use for powder compressions. Typically, a sawtooth or v-shaped profile is used where the punch is extended at a constant velocity and retracts at a constant velocity. In theory, during a sawtooth profile, the punch reverses its motion instantaneously between the compression and a decompression strokes. At low speeds (e.g.. <1 mm/sec), the hydraulic response system can easily accommodate this discontinuity. However, at high speeds (>100mm/.sec), the control system may show a small lag in the position-time waveform (<10 milliseconds) as it attempts to rapidly reverse the direction of punch. The sawtooth waveform is commonly used for more fundamental compression studies (e.g.. Heckel analysis), where the desired powder volume reduction is proportional to time. It is also u.seful when evaluating instrument performance during factory acceptance testing. 

The assumption of a linear profile between the neighboring nodes offers the simplest approximation of the gradient at the face lying between those nodes. For example, the gradient of 0 at face e can be written ... 

The simplest continuous injection profile is a linear profile, i.e., a linear concentration gradient [23]. So, let us assume as the boundary condition of the problem a trapezoidal injection profile with a linear ramp ending in a plateau of constant concentration (see Figure 7.5). This profile is the classical boundary condition for the strong solvent in linear gradient elution (see Chapter 2, Section 2.1.4). Its equation is... 

Example 7.2 is solved again with a linear profile in x as the initial condition.[l] The dimensionless temperature profile is governed by ... 

The linear profile approximation of the property gradient is second order and widely used for the evaluation of the diffusive fluxes in (12.68). For the diffusive flux at position e, the approximation is then written as ... 

In reversed-phase and ion-exchange chromatography it is a common procedure to run a gradient scouting run if the conditions for a successful separation are unknown. Such a mn is performed from 10 to 100% B solvent (stronger solvent) with a linear profile. (As already explained, a 100% A mobile phase is often not recommended in reversed-phase chromatography because the alkyl chains are collapsed and equilibration with... 

The temporal evolution of formic acid in the double layer [Eq. (15a)] is governed by the rate of the oxidation of formic acid via the direct (v direct) and indirect (Vpoison) paths and its replenishment by diffusion. For the latter it was again assumed that the concentration profile across the diffusion layer relaxes instantaneously to a linear profile. 5 denotes the diffusion-layer thickness (which is assumed to be constant), DpAthe diffusion constant of formic acid, and CpAthe bulk concentration of formic acid. A... 

Figure 4.2a shows an ideally linear profile of concentration X. Then if Xq is the mean value of X for the portion of bar shown, the two triangles have exactly equal areas. But the entropy of the sample is controlled by R In Z as shown in Figure 4.2b. If the profile of X is linear, the profile of In X is curved the areas of the two stippled portions are now not equal and the integrated area under the curve is less than that under the horizontal line. In more direct terms, if we take a portion of bar in which component R is homogeneous and push some of the R up to one end, we are imposing a degree of order on R—we are decreasing its entropy the entropy of R in a graded concentration is slightly less than in a uniform concentration of the same total quantity in the same space.

We note that, in spite of the superficial similarity to a unidirectional flow, the solution for uo is actually quite different from the linear profile of a simple, planar shear flow. This is not at all surprising for an arbitrary ratio of the cylinder radii. However, we should expect that the approximation to a simple shear flow should improve as the gap width becomes... 

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