Schuster equations

Complex processes are involved in transmittance and reflectance of scattered radiation, which are theoretically described by Schuster [4]. In an ideal scattering medium all fluxes of light can be summed up as components of two vectors. Vector I stands for the light flux in the direction of the incident light, and the vector J describes the light intensity in the antiparallel direction. With k, the absorption coefficient, and 5", the scattering coefficient, the two Schuster equations are as follows ... 

If a sample is illuminated by a parallel light flux with the intensity /g and there is no scattering in the sample (s = 0) and no fluorescence, the incoming light is reduced in intensity and leaves the sample as transmitted light Ij. The reduction over a distance d can be calculated from the two Schuster equations as ... 

This simple relationship between incident and transmitted light is well known as the Boguert-Lambert-Beer law. This expression renders positive values for Ij < Iq. In case of scattering material like TLC plates, a part of the scattered light is emitted as reflectance J from the plate surface to the top. For the hrst approximation of a parallel incident light beam with the intensity /g, some radiation may be scattered inside the layer and some radiation may be absorbed either by the sample or by the layer itself. According to the Schuster equations and with the abbreviation R (the diffuse reflectance of an infinitely thick layer). 

This is the discontinuous equivalent of the Schuster equation for isotropic scatter. 

Schuster, S.H. and Isenberg, J., Equations of State for Geologic Materials, US Defense Nuclear Agency Report No. DNA 2925Z, Washington, DC, 46 pp., September 1972. 

The nature and the distribution of different types of Fe species in calcined (C) and steamed (S) samples were investigated by means of UV-vis spectroscopy. UV-vis spectra of Fe species were monitored on UV-vis spectrometer GBS CINTRA 303 equipped with a diffuse reflectance attachment with an integrating sphere coated with BaS04 and BaS04 as a reference. The absorption intensity was expressed using the Schuster-Kubelka-Munk equation. 

The products from the acid-catalyzed hydration of a-tertiary alcohols 30 (Meyer-Schuster and Rupe rearrangements) are formed via the mesomeric propargyl-allenyl cation (equation 9) and have been extensively investigated28. 

When 2-ethynyl-2-hydroxyadamantane (31) was treated with 95% formic acid or dilute sulfuric acid only a Meyer-Schuster rearrangement took place to give 95% of 2-(formylmethylene)adamantane (32) (equation 10). No Rupe rearrangement took place29. 

UV-VIS-NIR diffuse reflectance (DR) spectra were measured using a Perkin-Elmer UV-VIS-NIR spectrometer Lambda 19 equipped with a diffuse reflectance attachment with an integrating sphere coated by BaS04. Spectra of sample in 5 mm thick silica cell were recorded in a differential mode with the parent zeolite treated at the same conditions as a reference. For details see Ref. [5], The absorption intensity was calculated from the Schuster-Kubelka-Munk equation F(R ,) = (l-R< )2/2Roo, where R is the diffuse reflectance from a semi-infinite layer and F(R00) is proportional to the absorption coefficient. 

In accordance with the classical theory of propagation of detonation of Chapman [1], Schuster [2], Crussard [4], constructed by analogy with the theory of shock waves of Riemann [5], Hugoniot [6], Rayleigh [7], Rankine [8], assuming the total absence of any dissipative forces (transfer of heat or momentum to the outside, the effect of viscosity or heat conduction in the direction of propagation), the conservation equations may be written as follows ... 

Equations (3) and (4) are formally identical with the earlier Kubelka s hyperbolic solutions of differential equations for forward and backward fluxes (11), although the Chandrasekhar-Klier and Kubelka s theories start from different sets of assumptions and employ different definitions of constants characterizing the scattering and absorption properties of the medium. In Kubelka s theory, the constants a, b, and Y are related to the Schuster-Kubelka-Munk (SIM) absorption K and scattering S coefficients as... 

FIGURE 4 Model for the derivation of the Schuster-Kubelka-Munk equation. 

Many different flux vectors will satisfy this equation. All of those solutions are linear combinations of the independent fluxes in the network. The coefficients to appear in those linear combinations derive from the nullspace or kernel matrix of N denoted by K (Schuster et al. 1999). Reordering J with the independent fluxes on top of the dependent fluxes J one obtains ... 

A very important form of such disturbances is caused by the presence of the wall of the tube containing the packed bed. Vortmeyer and Schuster (1983) have used a variational approach to evaluate the steady two-dimensional velocity profiles for isothermal incompressible flow in rectangular and circular packed beds. They used the continuity equation, Brinkman s equation (1947), and a semiempirical expression for the radial porosity profile in the packed bed to compute these profiles. They were able to show that significant preferential wall flow occurs when the ratio of the channel diameter to the particle diameter becomes sufficiently small. Although their study was done for an idealized situation it has laid the foundation for more detailed studies. Here CFD has definitely contributed to the improvements of theoretical prediction of reactor performance. 

Schuster calculated the molecular co-volume at the b.p. from 0b= Vmb--bb =RTblBby where bb is van der Waals s constant at the b.p. and Bb is the internal pressure. For normal substances, fe6=0 646c, 6=0 236 =0 147Z>c-Unlike Traube s co-volume, Schuster s depends on the nature of the substance. If the degree of association at the critical point is assumed to be 1, that for an abnormal substance was calculated from the observed and normal 0b values as in the case of Traube s equation. 

Schiff s.law of specific heats, 216 law of surface tension, 195 Schroeder on molecular volumes, 20 Schuster s co-volume equation, 24-5, 27, 146... 

SSW] P. Schuster, K. Sigmund, and R. Wolf (1979), Dynamical systems under constant organization. 111 Cooperative and competitive behavior of hypercycles, Journal of Differential Equations 32 357-86. 

Several authors present approximate porosity functions, which involve only the porosity increase towards the wall without any minima in the profile or including only the first minimum. Vortmeyer and Schuster give the following equation for a circular tube ... 

Schuster solution [24] It consists in summarizing the radiation transfer by forward and reverse light flux with respect to the incident light. After derivation, the following equation is obtained ... 

Kubelka-Munk solution [25] While the Schuster solution is for one particle, Kubelka-Munk generalized the absorption and scattering phenomena to the whole sample (K and S) and solved the radiation transfer equation in a different way than Schuster. More information about the derivation is provided in Griffiths and Dahm [23]. The solution is... 

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