Applying the adequate assumptions and the consideration by W.E. Ranz, theoretical formulas for the effective thermal conductivities k_e in packed beds were obtained as follows.
k_g: thermal conductivity for the fluid
k_e⁰: effective thermal conductivity in case of non-flowing
k_s: thermal conductivity for the solid
h_rs: heat transfer coefficient by thermal radiation, solid to solid, eq. (9)
h_rυ: heat transfer coefficient by thermal radiation, void to void, eq. (10)
α: factor for lateral mixing by W.E. Ranz
β: average distance between centers of solid layers in packed bed/average diameter of solid
ε: fraction void
φ: effective thickness of fluid film in the void/average diameter of solid
Then, the experimental data previously reported were analysed and the necessary data for estimation of k_e were shown in Table 1, Fig. 10 and Fig. 11.

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The characteristics of the air lift pump for water under various conditions have been previously reported. During the past three years many experiments have been conducted with this pump relative to the use of many kinds of steel and glass pipes for soap suds, light oil, spindle oil, ammonium sulphate solution, muddy water, sandy water and wood powder water. The effects of physical properties of liquids on the characteristics of this pump have shown that the pump is nonsensitive to these properties on the whole.
The main experimental results are as follows:
1) Effects of surface tension between 35 and 80dyne/cm are negligible. (Fig. 1)
2) Effects of specific density between 1 and 1.2g/cc are also negligible, if lifted quantity of liquid is expressed by volume. (Fig. 1, 6 & 7)
3) Static viscosity makes lifted quantity decrease at larger flow quantity, but rather improves the characteristics at smaller flow. In the usual region between the optimum point (the max. efficiency)and the maximum liquid point, the next farmula is obtained. (Fig. 2-5)
where
4) The characteristics of three phase flow, which contains solid particles, are inferior. (Fig. 6, 7)
5) Mean lifted velocity υ_m=(Q+V₀)/A is a little lower than 1m/s at the optimum point, and is about 2m/s at the maximum liquid point in all cases.
6) Instantaneous slip velocities between liquid and air are obtained by the salt water (electrical) method, and the slip is below 0.5m/s at the optimum point and about 1m/s at the maximum point. (Table 3)

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The studies of centrifugal permeability test were carried out with a commercial basket type centrifuge, with its diameter, 0.3m, and filtering area, A=1.46×10^-1m² as computed from the cylindrical part of the basket. Filtering medium used was cotton filter cloth which was inserted into the basket, forming a bag such as closely fitting the inner shape of it. And to compare the operation characteristics with those of pressure filtration, constant filtration was performed by a bomb filter (A=3.68×10^-3m²). Slurries of diatomaceous earth and silica powder were fed for making the filter cake through which the permeability test was performed.
Results obtained are summarized as follows:
1) Plotting of permeability rate dV/dθ against pressure P which is caluculated on the basis of centrifugal force shows that the equation (13) is significant in solving the problem of centrifugal filtration as illustrated in Fig. 8 and 13. However, if the useless part other than the effective filtering area of filter cloth in the centrifuge is not covered with water proof material, the curves are drawn in different manner. The deviation of the curves from lineality may have been caused by leakage of the feed water through the undesirable part outside of the filtering area considered here.
2) It seems that the penetrating transfer of particles throughout the layer of the cake occurs during the period of filtration or permeation in the centrifuge. Fig. 10 is a photograph of the surface of dry cake which is formed by two step filtration; first by filtering through diatomaceous earth and then successively through active carbon, the white part against the black back ground indicating microparticles of diatomaceous earth brought by transfer from the inner to the surface of the cake.
This is also confirmed by measuring the particle size distribution along the depth of the cake. since such a phenomenon can not be found in pressure filtration, this seems to be a special operation characteristic in centrifugal filtration.
3) Specific resistance obtained by centrifugal permeability is far larger than that by pressure filtration. For instance, specific resistance of diatomaceous earth for the former is α=4.5×10¹⁰m/kg against α=1.5×10¹⁰m/kg for that of the latter. This is also due to the difference of structure of particle arrangement in the cake.
4) Specific resistance of centrifugal permeability increases with the lapse of operating time as shown in Fig. 15. This tendency may be related to transferring of microsize particles through the cake.

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The Chemical reactions were classified according to their reaction types, and taking the molal ratio of reactants as their ratio in the corresponding chemical reaction equation, for each type of the reaction, the equilibrium conversions x were fixed on certain values, and the corresponding values of of mole fraction equilibrium constants K_N were calculated. The results of calculations were plotted on the special coordinate in which the ordinate was log K_N and the abscissa was logx for the range of x<0.5, and log(1-x) for the range of x>0.5; thus obtaining the curve which extended straight between both the ends of the coordinate.
Fig. 1 thus obtained can be universally applied to the liquid phase reactions as well as to the gaseous reactions and, furthermore, values of x of the backward (reverse) reactions can easily be obtained only by changing the sign of log K_N and reading the curve of the forward reaction and then substracting thus obtained value x from 1. The value 1-x is the equilibrium conversion of the backward reaction required.

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