Tangential pressure

Much of the pioneering work on ablative pyrolysis reactors was performed by CNRS at Nancy where extensive basic research has been carried out onto the relationships between pressure, motion and tenperature (23). NREL developed an ablative vortex reactor (see Figure 6 and (14)), in which the biomass is accelerated to supersonic velocities to derive high tangential pressures inside a heated cylinder. Unreacted particles are recycled and the vapours and char fmes leave the reactor axially for collection. Liquid yields of 60 65%wt. on dry feed basis are typically obtained. 

We developed a method when the spinodal can be estimated using the vapour pressure and an interfacial property, namely the maximum of the tangential pressure across the interface ... 

Figures. Spinodal calculated by Eq. 3, using different idealised tangential pressure profiles across the interface, compared to the spinodal given in the literature for helium-3. The solid curve is the vapour pressure curve, the dashed curve is the theoretical spinodal curve, the light grey band represents our results for the triangular pressure profile, the dark grey band represents our results for the rectangular pressure profile.

It is well known that the attraction between two portions of a fluid decreases very rapidly with the distance and may be taken as zero when this distance exceeds a limiting value, the so-called range of molecular action. According to Laplace, surface tension, y, is a force acting tangentially to the interfacial area, which equals the integral of the difference between the external pressure, and the tangential pressure, p. ... 

Now, it is well known that in the inn r part of a fluid the pressure across a plane is the same irrespective of the directions of the plane, that is, Pascal s law holds. But Eq. II.1 shows that surface tension arises from the circumstance that the law does not hold in the transition region of two homogeneous phases. Thus the pressure pr (tangential pressure) across a plane perpendicular to the surface of the liquid is different from that across a plane parallel to the surface of the liquid. The latter may be called the normal pressure and denoted by />N. Since />N is shown to have the same value irrespective of position, it may be denoted simply by p as is done in Eq. II.l... 

In the transition layer the density is uniform in the (x, y) plane which is parallel to the transition layer. Let a small plane of area dS be taken perpendicular to the plane (, y). The pressure across such a plane will be called the tangential pressure. Let the positions of two molecules P and Q be given by the position vectors r-dxlt yv zx) and ra( 2, yt, zt), and let r2—rx be denoted by R. Then the force across LM in the sense of the first definition is given by... 

In any attempt to establish a theory of surface tension, it is very natural to assume, as a first approximation, that the transition from the liquid phase to the gas phase occurs discontinuously at a mathematical plane. As stated in Section V, Fowler introduced this assumption and obtained Eq. V.9 for the surface tension. The nature of this assumption will now be studied more closely by calculating the normal pressure and the tangential pressure />x.14-21 The assumption is stated as... 

The tangential pressure can be calculated either from Eq. VI.6 or from Eq. VI.8. Both equations give the same result in this particular case, yielding... 

The combination of temperature, dwell time, pressure (and cooling) will also depend on whether the sealing processes are platen or rotary, and whether the web is only heated at the point of seal or is preheated. As rotary invariably gives a tangential pressure between two cylinders, the dwell time is extremely short, thereby demanding a higher temperature than that used in a platen process for the same material, even when preheating is employed. 

The quantity pT can be regarded as the tangential pressure , acting in a plane parallel to the interface and tending to decrease an interfacial area. Taking into account that the difference between pT and p is significant only within the discontinuity surface, the Bakker equation can be written as... 

CP in a system with a tangential pressure-driven shear flow, mimicking the electrodialysis situation [10]. 

The maximum tangential pressure gradient occurs when there is no net output from the extruder. The maximum pressure gradient can be determined from ... 

When p approaches unity, ge/Rb approaches the tangential pressure gradient in the flat plate case. The angular shear rate is determined from the first derivative of the angular velocity this leads to ... 

Figure 7.104 Flow rate versus tangential pressure gradient for cylindrical and flat plate system flow rate is per unit axial width...

The flow rate expressions can be expressed in terms of cross-channel (x) and down-channel coordinates by considering that the tangential pressure gradient g, can be expressed as ... 

Equations 10.61 through 10.66 allow the determination of the tangential pressure gradient, which, in turn, allows the determination of the leakage flow through the calender gap. 

With the matched screw configuration, the tangential pressure profiles in the two screws are symmetrical. Thus, there will be little interscrew material transfer. In the staggered screw configuration, however, the tangential pressure profiles are non-symmetrical because the flights are 180° offset. When the flight of the left screw is... 

From their work, Gubbins et al beheved that the normal and tangential pressures play different roles in observed phenomena in confined nanophases. And it is not profitable to define a single scalar effective pressure for the confined nanophase. 

As the whole of a surfactant micelle has the intrinsic nature of an interfacial complex, we must assume, of course, that the pressure within the hydrocarbon core varies from point to point, and is generally anisotropic, except at the very center, implying that the pressure tensor components Py and Pn in the tangential and normal directions, respectively, are, as a rule, different. Thus, there is no natural definition of P. Nonetheless, it turns out to be convenient to put P equal to the average tangential pressure within the micelle core, that is. 

Here, denotes / for pure tangential pressure flow (A-> x), which is discussed in a separate subsection below. Values of XPn,K) have been tabulated by Prasanth and Shenoy [11]. 

Figure 6 shows a typical plot of versus Q for c = 0.8 and = 0.4, The plot shows a horizontal asymptote corresponding to the pure tangential pressure flow limit and A- c ) and a vertical asymptote corresponding to... 

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