The equivalent beam length for the radiation from gas masses to their boundary surfaces as given by Hottel and Eckert results in good approximation when the gas bodies are small in size. However, the error is inevitable whem the gas body is as large as some furnaces including modern boiler furnaces. An exact calculation was practically impossible because the calculation was so tedious. In this paper is presented a simple method of calculation under the usual assumptions that the gas is homogeneous in its temperature and composition and that the radiation follows Beer's law, i.e.,
ε_g=1-e-Al
where
ε_g=gas emissivity,
A=coefficient of absorption,
l=path length,
e=base of hyperbolic logarithms.
Then, the emissivity of the wedge-shaped gas mass, as shown in Fig. 2, with height h and cross section 1/2 r²Δψ-an element of a cylindrical volume-to the infinitesimal a rea df can be obtained by the following equation given by E. Schmidt:
where
By integration, the emissivity at the corners of the surface bc in the rectangular parallelepiped b×c×h is given by the following equation:
The integration can be performed by transforming Φ(Z) into the following equation:
Φ(Z)=K₀-K₁Z+K₂Z²-K₃Z³
The results of the integration are summarized in Fig. 5 for various shapes and sizes of rectangular parallelepipeds. To find the emissivity at any point on the base surface of any rectangular parallelepiped, divide the parallelepiped into four parallelepipeds by two planes passing through the point. Then the emissivity for the four parallelepipeds to the point can readily be obtained from Fig. 5. The sum of the four emissivities is the emissivity in question.

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On the basis of the well-known empirical relations between physical properties, such as vapor pressure, viscosity, surface tension, latent heat of vaporization and density of the saturated liquids, and temperature, the following general correlations were obtained and simple methods were presented for the general prediction of the properties with engineering accuracies.
(1) The fraction of a property based on the difference of the values (or its logarithmic values) at the normal boiling point and the critical point is a general function of the similar fraction of absolute temperature (or its reciprocal), as generally shown by Eq. 1.11.
(2) Selecting the units of the properties so as to vanish at the critical point or the boiling point in the fractional equations, the general correlations of the certain functions (τ, s) of the properties with T/T_c or T/T_b were obtained. If the critical temperature, T_c, or the normal boiling temperature, T_b, is known and a value of the property at some one temperature is available, values at any temperature can easily be predicted using a chart (Fig. 1·7), as shown in numerous examples.
(3) Combining the τ function of any property with that of density, the reduced equation of state for saturated liquids (Eq. 2·5) and the simple generalized correlations of the properties with density in a reduced form with respect to normal boiling point (Fig. 2·3-2·6) were obtained. These correlations may present simple methods for general prediction of the properties in such cases as when the critical temperatures are not available in mixtures or in high-molecular liquids, as shown in numerous examples. The data required are values of the property at a certain temperature and densities at that temperature and at the temperatures concerned.
Some extensive conclusions, of scientific and engineering interest, are suggested.

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The present paper deals with what we have studied about the heat transfer between the gas flowing through the granular layer and the wall surface of the following types of the apparatvs: (1) the annular type, where heat transfer occurs both through the inner and the outer walls, and (2) the outer-catalyst type, which is also annular, but in which heat transfer occurs only through the inner wall, the outer wall being thermally insulated.
The theoretical temperature distributions were calculated mathematically, and Eqs. (13) and (17) were obtained for the annular type and the outer-catalyst type, respectively. Fig 1, and Fig. 2 represent the axial temperature distribution, when no chemical reaction takes place calculated by Eqs. (8) and (19) respectively, while Fig. 3 and Fig. 4 represent the same calculated byEqs. (13) and (17), taking the temperature of the wall and the inlet gas as equals.
In Fig. 5 maximum temperature rises in the following types of apparatus are shown in comparison: (1) the annular type, (2) the outer-catalyst type, and (3) the cylindrical type. Since the figure shows the effects of various factors upon the maximum temperature rises for these three types, when the maximum temperature in one type is experimentally determined, those for the other two types wlll be successfully predicted, facilitating us in designing the apparatus of different types.
Applying the similar method to our study previously reported (Chem. Eng., Japan, 12 58 (1948)), the heat transfer in the annular type layer was experimentally studied.
λ, 1/b(=h'R/λ) and h were calculated, and they were compared with those for the simple cylindrical type. The results were as follows;
(1) λ/k can be represented by Eqs. (25) and (26) like in the cylindrical type.
(2) b can also be represented by Eq. (29) like in the cylindrical type, taking the equivalent dia.as (D_eq.=D₂-D₁). In the previous paper b was represented by Eq. (30), which was not perfectly nondimensional; but now we have found that Eq. (29) can also be applied in the case of the cylindrical type.
(3) For (h_∞D/k), Eqs. (31) and (32) were obtained; they were found to be about 30-50% greater than those for the cylindrical type in the conditions, Re_k=100 and d_k/D_eq.=0.1-0.03.

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The work balance equation for the flow system in a steady state was directly derived, and the premises and approximations in its derivation were clarified.

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