Circumferential angle

A coordinate system that is natural for the conical channel can be established as illustrated in right-hand panel of Fig. 5.20. The origin of the new coordinate system begins on the tube wall at the entrance of the conical section. The x coordinate aligns with the surface of the tube wall and the y coordinate measures the distance across the channel and is normal to the tube wall. The 4> coordinate measures the circumferential angle around the conical... 

A certain direction in space is determined by two angular coordinates ft and p, Fig. 5.3. ft is the polar anglemeasured outwards from the surface normal ft = 0) and is the circumferential angle with an arbitrarily assumed position for p = 0. The radiative flux, that falls on a small area d.4n at a distance r from the surface element dA, perpendicular to the radiation direction, Fig. 5.4, is proportional to the solid angle element... 

Fig. 5.3 Spherical coordinates of the point P distance from origin r, polar angle ft, circumferential angle p...

Fig. 5.4 Radiative flux d24> into a solid angle element do) in the direction of the polar angle ft and the circumferential angle p...

In most cases, the spectral intensity Lx only depends on the polar angle j3 and not on the circumferential angle ip. We then obtain the more simple relationship... 

Example 5.1 The spectral intensity L of radiation emitted by a body shall not depend on the circumferential angle ip and can be approximated by the function... 

In general, the directional emissivities s x(X,P,ip,T) and s (f3,ip,T) do not depend on the circumferential angle ip. Integration over all solid angles in the hemisphere, which, according to Table 5.4, leads from s x to A and from s to s, is then simplified. This produces... 

In view of the experimental difficulties a theory for radiation properties is desirable. The classical theory of electromagnetic waves from J.C. Maxwell (1864), links the emissivity e x with the so-called optical constants of the material, the refractive index n and the extinction coefficient k, that can be combined into a complex refractive index n = n — ik. The optical constants depend on the temperature, the wavelength and electrical properties, in particular the electrical resistivity re of the material. In addition, the theory delivers, in the form of Fresnel s equations, an explicit dependence of the emissivity on the polar angle / , whilst no dependence on the circumferential angle ip appears, as isotropy has been assumed. 

Emission or reflection from diffuse gray or black surface Cone angle 0 Circumferential angle < > Wavelength X sin 0= R 12 < > = 2 nR Fi,- = R ... 

Isotropic scattering Cone angle 0 Circumferential angle cos 0=1— 27 = 2nR ... 

Laser Dopple Velocity (LDV) measurements in the venturi premix duct outlet therefore were carried out. These tests, which were run at cold conditions at the design point with the same Reynolds number, show a very symmetric and even velocity profile at all circumferential angles in the duct outlet. The measured RMS values, 30-35% of the mean velocity, also showed that a high turbulence level exists in the venturi. This will enhance the mixing and fuel vaporization process. 

The channel depth reduces along the flight flank. If circumferential angle 0 starts at the beginning of the flight flank the channel depth as a function of angle 0 can be written as ... 

Figure 10.15 illustrates how the channel depth varies with circumferential angle 0. The axial coordinate 1 is related to the circumferential angle 0 by ... 

Figure 17. Strain energy density variation In dependence on the circumferential angle ijj.

FIGURE 8.22 Channel depth for a double-flighted corotatmg self-wiping twin-screw extruder, (a) Channel depth as a function of circumferential angle, (b) Channel depth as a function of distance across channel. 

This relation is not easy to determine from Figure 8.21, but one must consult the work of Booy (1978). We can now determine the channel depth as a function of the circumferential angle, 0. Referring to Figure 8.22 we And H 6) to be... 

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