Although the studies on the heat transfer in non-Newtonian flow are increasing in recent years, the law of similarity of the heat transfer is not yet established. It is true that the previous investigators tried to find correlation between the experimental data on the heat transfer and various dimensionless numbers, but there is observed no consistency between them.
In this paper, the author describes the law of similarity of the heat transfer in non-Newtonian flow which he has derived, using the fundamental equation. The results may be summarized as follows:
(1) Pseudoplastic and dilatant fluid:
(a) Natural Convection
Nu=f1(Gr, Pr₂)
(b) Forced convection
Nu=f2(Re, Pr₁):
(2) Bingham fluid:
(a) Natural convection
Nu=f3(Gr, cGr, Pr, cP_r)
(b) Forced convection
Nu=f4(Re, cRe, Pr, cP_r)

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The equations which represent the head-separation factors of the dual-temperature-exchange columns are derived and the optimum operating conditions for them as well as the design of the columns to bring about the following four possible cascade schemes are discussed.
Cascade Schemes:
Non-Mixing cascade (Cf. Fig. 3)
The stripped output from the second stage enters the top of the enriching column together with the enriched output from the first stage.
Scheme 1: Gas is circulated.
Scheme 2: Liquid is circulated.
Mixing cascade (Cf. Fig. 4)
The stripped output from the second stage enters the top of the stripping column.
Scheme 3 Gas is circulated.
Scheme 4: Liquid is circulated.
Assumptions:
(1) Only the low-deuterium-content region is made the subject of study.
(2) Liquid-phase catalysts may be employed in these cascades, although no catalyst will be necessary in water-H2 S system.
(3) The cross sectional area of each column is proportional to its gas throughput.
(4) The stage variables (γ, N, n, δ, Tl, Th, etc.) are constant throughout each of the entire cascades.
Summaries of the results obtained are as follows:
(1) The head separation factors β are given by the following equations,
Scheme 1:
Scheme 2:
Scheme 3:
Scheme 4:
(2) The total volumes of the enriching and stripping columns, V, are given by:
Scheme 1:
Scheme 2:
Scheme 3:
Scheme 4:
(3) Some numerical calculations are carried out for each of the cascade schemes and the following results are obtained.
a) There is an optimum condition for the minimum total volume of the cascade, and it depends upon the variables of each stage.
b) The liquid circulating cascade is preferable to the gas circulating cascade.
c) If the cost of feed may be left out of consideration, the cascade of Scheme 4 will be the most preferable, as shown in Table 1.

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When gas absorption is performed in an irrigated packed tower, the following conditions may be assumed:
i) Flowing characteristics of liquid are specified by the mean flow velocity, u, and the apparent mixing coefficient, E.
ii) The rate of mass transfer across the interphase is proportional to the difference between the bulk concentration in the gas and that in the liquid stream, and the capacity coefficient is constant throughout the tower.
iii) The composition of the gas is uniform in the tower.
Eq. (1) is the basic differential equation governing the concentration gradient in the flowing liquid, where, in the boundary conditions, Eq. (2) or Eq. (3) is to be applied. Thus, when the mixing coefficient, E, in the tower, is small, Eq. (5), and when large, Eq. (4) is derived from Eq. (1). Eq. (5) may be replaced by the approximation, Eq. (6). The degree of absorption calculated by means of Eq. (5) is illustrated in Fig. 1, as a function of dimensionless groups (kLa/Ht)θ and M. kLa is the modified iquid-side capacity coefficient, obtained, from Eq. (4) or Eq. (5), by taking into account the mixing characteristics of liquid flowing through the tower.
Fig. 2 shows the experimental apparatus for determining kLa. The total holdup Ht and the mixing coefficient E of the packed tower used in this study have been measured, in our preliminary experiments, by the direct weighing method and the residence time curve method, which have determined the correlations of Ht and E to u, as shown by Eqs. (9) and (10). (Cf. our previous paper³⁾).
In this tower, water flows countercurrently to CO₂ gas which is transferred at 15°C.
Fig. 4 shows the results of these experiments, as well as kLa calculated from the experimental data obtained by Hikita et al1), and Onda et al²⁾, as shown in the table, by making use of Eq. (11) or Eq. (12) and general correlation of Ht and E to u, Eqs., (14), (15) and (16). The correlation of kLa to Reynolds numbcr, (dpuρ/μ), is expressed by Eq. (17).
The results of this study indicate that the absorption of CO₂ in water is rapid and that when a short packed tower is used, the error due to the longitudinal mixing of liquid has a serious effect on the measuring of kLa of this system, and when a long packed tower is used, the same is true of the error due to the high degree of absorption, as suggested by Eq. (5) or Fig. 1. For determining kLa or predicting the degree of absorption, it is more reasonable to use Eq. (5) than Eq. (7).

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Where the relationship between the equilibrium adsorbate content of the adsorbent and that of the gas was not linear, the differential equation for unsteady state adsorption was solved by numerical calculation and the method was compared with the graphical one.

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