Analytical solutions have been made on important problems in heat transfer that should be taken into consideration for the design purpose of multistage-fluidized bed-type heat-exchangers and reactors.
When the results of the present investigation are considered in combination with the contents of the previous paper5) concerning the conditions of stable fluidization, the design of the above-mentioned heat exchangers and the calculations of heat transfer in the reactors would become possible.
Transient heating of solid particle in the multi-stage fluidized beds has been considered under conditions that the tower employed is adiabatic and that the heat capacity of each bed is constant. Thus basic equation (1) resulting from heat balance around an arbitrary i-th stage has been solved by the method of Laplace transformation. The results of the solution are given in Eqs. (4) to (6) and the experimental results are illustrated in Fig. (8).
Fig. (2) shows a cross sectional view of the apparatus which consists of the triple-bed unit in a 7cm I.D. steel tube, equipped with downspouts (1.27cm I.D.) each inserted in an orifice-type plate with a hole (0.7cm in diameter) at its bottom. The heights of the downspouts each projecting above 1st, 2nd and 3rd plates are 10, 7 and 5cm, respectively. The temperature of air entering above each plate has been measured by means of a high-velocity thermocouple.
In steady-state heat transfer, when the effect of heat losses through the tube wall and the flange surface is taken into consideration, together with the heat of reaction and heat balances around each plate, we have simultaneous 2nd order difference equations. Solving these fundamental equations by means of matrix calculus we obtain Eq. (12). These results are rigorous solutions applicable to the case where the effect of radiant-heat transmission between successive plates is negrected. When heat exchange is carried out without heat of reaction, Q_r becomes 0. Regarding each plate as an ideal one, Eq. (7) is derived.
In Table 3, Runs No. 1 to 9 indicate the data obtained when hot fluid is cooled through fluidized bed, and Runs No. 10 to 13 show those obtained when cold fluid is heated. Theoretical values of yi have agreed satisfactorily with the experimental data.
Graphical solutions for the units in both the cases, where the radiant-heat transmission is neglected and where it is taken into consideration, are illustrated in Figs. 3 and 4, respectively, in order to compare them with the analytical calculus. To these graphical solutions, the principle of Tiller's method³⁾ has been applied. The graphical calculus is convenient in obtaining the required number of plates.

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Although the underflow concentration of a continuous thickener can be calculated by Roberts' equation, ⁵⁾
(22)
the constant, k_R, has to be determined by the batch settling experiment for each slurry in question. The experiments conducted by Comings and his collaborates³⁾ revealed that the underflow concentrations of a continuous thickener are affected not only by the retention time in the compression zone (as Roberts states) but by the height of the zone. Therefore, the Roberts' constant may be expressed in terms of the pertinent properties of the slurry as well as its initial height.
A fundamental equation for the compression stage in the batch settling of well-dispersed slurries whose initial concentration is the critical one at which the compression stage begins, has been derived. The void fraction of any portion of the slurry is assumed to become its equilibrium as soon as the pressure is exerted on the portion.
where has been linealized toandis defined.
The fundamental equation is solved for the initial and boundary conditions (13-16), to get a generalized batch settling equation, for the compression stage, in the form of
(19)
(20)
(21)
Using these equations and the result of batch settling experiment for one appropriate initial height, the batch settling curve for other initial heights can be predicted with considerable accuracy (Table 2).
Roberts' constant, k_R, can be expressed in terms of c_v⁺ and the initial height of the slurry, as folliows,
(25)
where c_v⁺ represents the characteristics of the slurry, and consists of two parts, viz. the equilibrium relationship ε^*=f(p_s⁺) and the modified permeability which can be measured by the compressionpermeability tester and by one batch settling experiment, respectively. (Table 1 & 2). Since c_v⁺ is dependent of H0, the product k_RH₀ is approximately constant.
From the analysis made for the batch settling of slurry, the relationships among the depth of the compression zone, the underflow concentrations, the retention time and the underflow rate of a continuous thickener is expressed by a set of equations,
(29)
Fig. 7, 8 and 9 are the samples of the relationships calculated from the set of Eqs. (29). When comparing the lower right portion of Fig. 8 with Comings' graph (Fig. 10), we can observe the essential similarity between them, though the values are not equal, due to the difference of samples employed.

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Previously, Tanaka¹⁾ proposed a chart of a closed circuit grinding plant. The idea was based upon the assumption that the sharpness of the classifier was perfect. This fictitious condition, however, did not apply to practice.
In this paper an assumption is made that the partial collection efficiency of a classifier is
and on this assumption the "Charts" (Figs. 4, 5, 6 and 7) have been drawn up. By these, one can easily obtain the values of several variables, such as circulating load, cut-off size of the classifier, residue of the products, etc.
Based on these "Charts" (Figs. 4, etc.) and Rittinger's Law, we have got Figs. 8, 9, etc., which show the increase of the product rate vs. circulating load and the performance of the classifier at several values of R_P, l and cl.
Several relations of closed circuit where n is not 1 are shown in Figs. 13, 14, 15 and 16. These showthat the capacity of the mill increases with the increase of circulating load and that reformation of the classifier at n<1 is greater than at n>1.

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The flow patterns of agitated liquid in the cylindrical mixing vessel without baffles were measured by the photographic method. for the range extending from the laminar to the so-called transitional flow region.
As the authors have already made clear the flow patterns of liquid and the discharging performance of impellers, for the turbulent flow range¹⁾, they are now able to discuss these over the wide range of Reynolds Number.
Some of the experimental results are shown in Figs. 4, 5 and 6.
The results revealed by the authors on the distribution of liquid velocity are as follows:
In the range where Reynolds Number is very small, the liquid velocity is considerably high in the neighborhood of the impeller, which, however, rapidly decreases as the distance from the impeller increases. On the other hand, in the range where Reynolds Number is big, the liquid velocity is kept high by the transmission of the momentum, even in a place far from the impeller, since the secondary circulation flow occurs as shown in Fig. 6 (c) and the liquid flow becomes turbulent, keeping the velocity distribution nearly uniform throughout the vessel.
Of the discharge flow from the tip of the impeller blades, which causes the secondary circulation, the non-dimensional quantities, N_q1 and N_P/N_q1, are defined as in the previous report¹⁾. These values have been calculated and the characteristic curves of discharging performance obtained for various types of impellers (Cf. Table 1), as shown in Fig. 7.
Furthermore, the relations among flow pattern, discharging performance and power consumption (N_P-R_e relation) are discussed.

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