Frank radius

The sympathetic system also innervates vascular smooth muscle and regulates the radius of the blood vessels. All types of blood vessels except capillaries are innervated however, the most densely innervated vessels include arterioles and veins. An increase in sympathetic stimulation of vascular smooth muscle causes vasoconstriction and a decrease in stimulation causes vasodilation. Constriction of arterioles causes an increase in TPR and therefore MAP. Constriction of veins causes an increase in venous return (VR) which increases end-diastolic volume (EDV), SV (Frank-Starling law of the heart), CO, and MAP. 

The Helfrich-Prost model was extended in a pair of papers by Ou-Yang and Liu.181182 These authors draw an explicit analogy between tilted chiral lipid bilayers and cholesteric liquid crystals. The main significance of this analogy is that the two-dimensional membrane elastic constants of Eq. (5) can be interpreted in terms of the three-dimensional Frank constants of a liquid crystal. In particular, the kHp term that favors membrane twist in Eq. (5) corresponds to the term in the Frank free energy that favors a helical pitch in a cholesteric liquid crystal. Consistent with this analogy, the authors point out that the typical radius of lipid tubules and helical ribbons is similar to the typical pitch of cholesteric liquid crystals. In addition, they use the three-dimensional liquid crystal approach to derive the structure of helical ribbons in mathematical detail. Their results are consistent with the three conclusions from the Helfrich-Prost model outlined above. 

Calculate the radius of a spherical pile of cotton gauze saturated with cottonseed oil to cause ignition in an environment with an air temperature ( /],) of 35 °C and 100 °C. Assume perfect heat transfer between the gauze surface and the air. The gauze was found to follow the Frank-Kamenetskii ignition model, i.e. 

When the reactor is scaled up to 60 cm radius, however, the operating point is between the two curves. This means that the reaction can be safely run at 50°C in a well-agitated process vessel of 60 cm radius with the heat transfer coefficient as stated above becauseerating point is below the Semenov curve. In case the agitation is lost, however, the Frank-Kamenetskii curve becomes the better predictor of runaway temperatures, and because the operating point is above this curve, the estimate is that the reaction will run away. The calculation of the Frank-Kamentskii method is available in ASTME-1231 [166]. 

The more recent Thomas model [209] comprises elements of both the Semenov and Frank-Kamenetskii models in that there is a nonuniform temperature distribution in the liquid and a steep temperature gradient at the wall. Case C in Figure 3.20 shows a temperature distribution curve from self-heating for the Thomas model. The appropriate model (Semenov, Frank-Kamenetskii, or Thomas) is determined by the ratio of the heat removal from the vessel and the thermal conductivity in the vessel. This ratio is determined by the Biot number (Nm) which has been described previously as hx/X, in which h is the film heat transfer coefficient to the surroundings (air, cooling mantle, etc.), x is the distance such as the radius of the vessel, and X is the effective thermal conductivity. 

As mentioned before and assuming the vahdity of the continuum elasticity theory at the dislocation core, F. C. Frank derived the expression for the characteristic radius of a hollow core (Frank, 1951) ... 

Calcns of the critical radius for Tetryl, based on the Merzhanov Friedman treatments are compared in Table 2. Agreement is quite good. However, note that, as expected, critical radii based on the Frank-Kamenetskii-Chambre treatment (steady-state conditions) are considerably smaller than those computed via the hot spot approach. For comparison, Eq 14 (based on Ref 7) gives a r = 2.76 x 10"3cm for a Tetryl sphere at 700 K, and acr = 2.37cm at 445°K, in close agreement with Merzhanov... 

In order to achieve passive safety with reactive material, the radius of the reactor tube is designed to be small to avoid any thermal explosion inside the tube. Using the Frank-Kamenetskii approach (see Chapter 13), the radius remains below the critical radius. Thus, even assuming a purely conductive heat transfer mechanism, corresponding to a worst case, no instable temperature profile can develop inside the reaction mass. The reactor can be shut down and restarted safely. 

The critical heat release rate following the Frank-Kamenetskii theory (see Section 13.4), which describes the passive behavior of the reactor without fluid circulation, when heat transfer occurs by thermal conduction only. The critical heat release rate is the highest power that does not lead to a thermal explosion and varies with the inverse of the squared radius ... 

For the treatment of practical cases, it is often necessary to assess other shapes other than a slab, infinite cylinder, or sphere. In such a case, it is possible to calculate the Frank-Kamenetskii criterion for some commonly used shapes. For a cylinder of radius r and height h, the critical value of the Frank-Kamenetskii criterion is given by [7]... 

In the specific case of chemical drums with a height equal to three times the radius, the Frank-Kamenetskii criterion is 8cri, = 2.37 [1]. A cube with a side length 2ro, can be converted to its thermally equivalent sphere. The Semenov number then becomes... 

In Table 13.2, the best approximation of the cube is obtained with a sphere of radius rsph = 1.16-r0. The Frank-Kamenetskii number then is 2.5 for a cube with a side length 2 r0. 

A fourth attempt could be to use a smaller tank, such as a tank with a diameter of only 1 m that would lead to a stable situation in the frame of the Frank-Kamenetskii model the radius of 0.5 m is smaller than the critical radius of 0.603 m. But this solution means building a new tank. 

The possibility that the core of a dislocation could be empty was first recognised by Frank [1], If the strain energy density arising from a dislocation is sufficiently large, it may become energetically favourable to remove the material near the core and place it in an unstrained environment far from any dislocations. This process creates additional surface area around the core of the dislocation. The equilibrium radius of the hollow core of a screw dislocation is given by... 

The most conceptually attractive model for these solutions is to consider that the organization of water resembles that in the clathrate hydrates (p. 225), the structure being based on pentagonal dodecahedra of hydrogen bonded water molecules (Glew and Moelwyn-Hughes, 1953). This model receives some support from the observation that there is an optimum molecular radius of 4-5 x 10 8 cm for solubility of apolar solutes in water (Franks and Reid, 1973). 

The radius a of the onions in the intermediate shear-rate regime of lyotropic smectics depends on shear rate, scaling roughly as a A similar texture size scaling rule is found in nematics (see Section 10.2.7) there it reflects a balance of shear stress r y against Frank elastic stress. In smectics, the two important elastic constants B and Ki have differing... 

Fig. 1 Qualitative cross section through the potential energy surface, along JT active vibration Qa, Definition of the JT parameters - the JT stabilisation energy, Eji, the warping barrier, A, the JT radius, iJjx, the energy of the vertical Frank-Condon transition, fc...

Frank, his theory of electrostriction, 190 Free eneigies of hydration dependence on radius. 54 how to get them, described, 53 Free eneigies of solvation, described, 53 Friedman... 

Figure 6 The volume free energy of homogeneous nucleation of undercooled water at —40°C as a function of the cluster radius at —40°C (see Equation (2)). According to Equation (3),r = 1.85 nm. The critical nucleus then contains ca. 200 molecules. The value for a was obtained from the nucleation data of Michelmore Franks ...

Dislocation multiplication occurs when the dislocations are made to move during deformation. One possible multiplication mechanism is the Frank-Read source. Suppose that a dislocation is pinned at two points, which are a distance / apart, as shown in Figure 12.21a. Under the action of an applied stress the dislocation will bow out. The radius of curvature R is related to the applied shear stress. To. 

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