When the vapor rate is small, the vapor passes sporadically through some of the holes of the plate even if the plate is horizontal. As the vapor rate increases, the part that the vapor passes through becomes wider. For the effective operation of the perforated plate column, it is necessary that the vapor rises uniformly through holes all over the plate.
Minimum allowable vapor velocity to satisfy the above condition is shown by the following equation,
It was found that the slope of the plate had little to do with the uniform passage of vapor, so long as the slope of the plate was not too large. Nomenclature
H =hold-up on the plate [g/cm²]
R' =overflow rate [kg/hr]
V' =gas or vapor rate [kg/hr]
z =height of weir [cm]
Δz =liquid head over the weir by the Francis' Eq. [cm]
c =coefificient of discharge
d =hole diameter [cm]
f =effective foaming fraction
g =acceleration due to gravity [cm/sec2]
h =foaming height [cm]
i =slope of the plate (tangent of angle to horizontal plane)
Δp =total pressure drop through a tray [g/cm²]
Δpd =pressure drop through a dry plate [g/cm²]
Δp_w =wet pressure drop (=Δp-Δpd) [g/cm²]
Δp_σ =pressure drop due to the surface tension acting on the hole [g/cm²]
u =gas or vapor velocity through the hole [cm/sec]
ρ_g, ρ_l=gas or vapor and liquid densities [g/cm³]
σ =surface tension [dyne/cm]
φ =equivalent length of plate in the direction of slope [cm]

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The batch and continuous settling phenomena of the suspensions were studied, using magnesium hydroxide and dicyandiamide. The thickeners used for the experiments were 1 and 0.5(m) in diameter.
The following conclusions were arrived at:
1. The fundamental similitude relationships between the bactch and continuous settling held, as far as the upward current in the thickener was negligible.
2. The general relation between the radius of the thickener and the rate of overflow at the critical loading was shown as:
in which the second term in the right hand bracket expresses the effect of upward current in the thickener.
In an ideal case, where there occurred no upward current at all, i.e. r0=rf, the above relation might be expressd as follows:
This equation corresponds to the widely known formula derived by Coe and Clevenger, et. al.
3. Assuming the rate of thickening in the compression zone, not the rate of subsidence of suspensions, for both the cases of the batch and continuous settling, to be:
the depth of the thickening zone in the thikener would be calculated by the following equation:
which may be computed either by the numerical or graphical integration.
As we obtained n 2, from our experiments with magnesium hydroxide, the above equation may be integrated as:
Comparing the values calculated from this equation with those from the experiments, we confirmed that this equation might be available in the practical designing of the depth of the thickener. When n=1, the above equation would be expressed as:
This equation corresponds to the equation derived by Roberts using data from Coe's experiments.

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(I) The concept of the reversible countercurrent evaporater was proposed; its model pattern was fixed; the relationship between the heat quantity and the temperature of the heat input or output was made clear.
When the heat source available is limited to the heat reservoir of the normal surrounding temperature T0, the work to be required for the operation of the reversible evaporater, was shown graphically, using the heat input and output distribution curve.
The theoretical equivalent numbers of effects of the reversible evaporater was defined as εr=Qr/wr/σ, where Qr is the total heat input and σ is the thermal efficiency of the steam turbine. The value of εr increases rapidly as the feed solution becomes dilute, which is shown in Fig. 6.
(II) The work of vapor compression 'W' required for the thermocompression evaporation decreases as the evaporater approaches the reversible evaporater. The deviation of 'W' from the work to be required for the reversible evaporater was shown graphically, and the analysis of the graph clarified that effective means to cut down the work of compression are as the following: (1) the reduction of the temperature difference of heat transfer at the stills, (2) the application of two or more vessels and compressors in series, instead of a single vessel and one compressor, (3) the improvement of compressors in the adiabatic compression efficiency and (4) the recovery of work from the heat quantity which is the conversion of the compression work.
(III) The work balance analysis of the thermocompression evaporater for the recovery of NaCl crystal from its dilute aqueous solution was performed, with the following factors changing separately: (1) temperature difference of heat transfer at the stills, (2) application of two or more evaporating vessels and comressors in series, (3) efficiency of irreversible vapor compression cycle to Carnot's reverse cycle, (4) the recovery of work from the excess steam, (5) exit-temperature of feed from the heat exchanger, (6) evaporation pressure, (7) heat losses from the apparatus. This analysis made clear quantitatively the effect of each factor to cut down the work of compression.

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