Calculations of heat transfer rates in cylindrical furnaces, making rigorous allowance for axial radiant heat flux, were carried out, and the results were used to evaluate existing approximate methods of calculation. A Fortran program for the IBM 709 was written to calculate the radiantconvective heat fluxes for gas-filled cylindrical furnaces using the Hottel-Cohen method of zoned analysis and making use of the direct interchange areas for cylindrical furnaces calculated by H. Errku.
The major assumptions made in the formulation of the program are that there are no radial temperature gradients or gas recirculation, i.e., plug flow of the combustion gases is assumed in the furnace.
The heat transfer rates were calculated for eighteen sets of conditions. The design parameters investigated were length-to-diameter ratios of two and four and diameters of two and four feet. The operational variables studies were sink temperatures of 1460°K and 2460°K, and enthalpy feed rates of 100×10⁴, 50×10⁴ or 25×10⁴, and 6.25×10⁴ Btu/hr·cu.ft. The furnace walls were treated as heat sinks with an emissivity of 0.8 and the end walls as refractories with an emissivity of 0.5. The fuel was assumed to have a composition corresponding to (CH₂) _n and to be instantaneously burned with a stoichiometric amount of air. No flame luminosity was assumed.
From the results it was found that 90% of the net heat flux to any wall zone originated in the gas zone contiguous with it and the adjacent gas or end zoned.
The results of the computer runs were compared with two approximate methods of calculating furnace performance. One method, which formulates the heat flux neglecting the net radiation along the axis of the furnace but allowing for the axial gas temperature gradient, was found to give heat fluxes from 9.1% lower at high firing rates to as much as 12% higher at lowest firing rates. The error in heat flux distribution along the furnace was larger still, ranging from +24% to -12%.
A completely stirred furnace approximation with the heat-transfer temperature equal to the exit temperature gave total fluxes which ranged from 1.2% to 18.7% lower than the computer calculations at high and low firing rate, respectively.
The overall efficiencies of the eighteen runs were correlated as a function of dimensionless parameters defining the design and operational variables within 3%. Also, corrections to be added to the exit gas temperature to give the “effective” heat transfer temperature of a well-stirred chamber of equivalent performance were calculated and correlated as a function of a dimensionless parameter. The magnitude of the correction ranged from 570°F to 60°F. Either correlation gives a mean of predicting furnace performance for furnace where radial temperature gradients and gas recirculation are unimportant.
The efficiencies of the plug-flow furnaces studied here and of the well-stirred furnaces bracket that of real furnaces. Consequently, the present calculations indicate the range of possible error in estimating the thermal efficiency of furnaces by various simple but inaccurate models.

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The physical data taking part in heat transfer vary with temperature, and some to a large extent. Therefore, the estimation of temperature distribution in packed bed extending over wide temperature cannot be obtained accurately, unless the variation of physical data is known. We took these into consideration, and discussed possible mathematical ways which are related to heat transfer of such a nonlinear type.
For those assumptions that axial diffusion and temperature difference between fluid and particle are neglected, we made the following two models.
i) Q=0, and λ_er/K' is a linear function of the cross sectional average temperature.
ii) Q, λ_er/K', and λ_er/U are arbitrary functions of local temperature.
Then, model i) can be expressed by these fundamental equations (9)-(11), (Eq.(13)-15) are of their dimensionless form). This problem is not difficult to solve analytically if some approximation could be permitted. We obtained these approximate analytical solutions (30)-(32). Improvement in these equations is projected by the term F (s)(Eq.(28)) and the value of F (s) can be drawn on a graph (Fig. 1) with axial distance Aζ²₁s and alienate parameter (B/A)θ_m0. Furthermore, in order to compare the projected F (s) with that of linear model, three curves are shown in Fig. 2. The curve (I) is for the case of (B/A)θ_m0=5, (II) is for (B/A)θ_m0=-0.833 and (III) for (B/A)θ_m0=0 taking A' the arithmetic mean value of k (s).
The equation (34)-(36) are the fundamental equations of model ii). These equations can be developed to (37)-(43). The solution of these equations can not be obtained by any analytical method. The method only remained for us would be numerical calculation. Then, equation (34) is transformed to difference equation (44). The situation changes depending upon whether Δz/Δr²k (t) is less than 1/2 or not.
Case 1)Δz/Δr²k (t)≤1/2. If we denote the incremental value of t as Eq.(45), the equation (44) may be solved as (46). The boundary condition defined by Eq.(36) also gives (47)-(49).
Case 2)Δz/Δr²k (t)<1/2. We recall here to Neumann's method. If we denote Δ²_rt as the equation (53) and prescribe the boundary condition in the same way from case 1), a set of simultaneous linear algebral equation (55) for each step in z is obtained.
Applying these solutions, we computed a temperature distribution for an example, and showed their results in Fig.3.

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It was considered that a gaseous species A dissolves into the liquid phase and then reacts reversibly with species B according to the mechanism A+B⇔2C, that is a mol of the reaction product C is reversibly produced which is more frequently the rate controlling step than the case of A+Bg⇔2C. The effect of the second order reversible reaction on the rate of gas absorption for the unsteady state diffusion theory has been computed numerically by the NEAC-2203 digital computer. To solve the non-linear partial differential equation, it was converted to dimensionless form through various parameters and the solutions for the wide ranges of the parameters K and q were computed for the case of the same diffusivities of each species. The computation results were compared with previous investigations, and the following results were obtained.
1. The fundamental differential equations were solved in the conditions of M=4, y=0.1 and y∞=4.
Then, the dimensionless concentration gradients of a, b and c in the liquid phase were given as functions of dimensionless distance y and time θ.
These numerical calculations have shown good convergence and stability.
2. By comparison of the β-values which are calculated by the numerical and analytical method in the case of the gas absorption with pseudo 1st order & 2nd order irreversible reaction, the accuracy and convergency of the calculating method were confirmed to be fine. And our assumptions that the reaction with K=100 and q=100 corresponds to the infinite irreversible were proved to be sufficient.
3. In the gas absorption with 2nd order reversible chemical reaction, the reaction coefficients have been calculated for arbitrary value of K and q. It was confirmed that the reaction has become psudo 1st order for the larger values of q than 20 and irreversible for the larger values of K than 20, respectively. Further, the empirical Eq.(22) which denotes each values within 10% accuracy have been derived and the correction factor ψ(γ')(23) has also been shown in Fig.8 to give the correct values of β'.

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In this paper, continuous salt-metal extraction involved in the LMFR fuel reprocessing process is treated, putting particular attention on the analysis and formulation of the extraction process in the oxidation section of the FPS removal column. For the oxidation section the parallef-flow operation is recommended and is formulated as equations (8)-(19) to give column behavior and design method (sections 3, 5, and 6). This formulation is also extended to include the operation of bench contactors (section 7).
The basic equations show that the column is divided into two zones of oxidation and equilibration under the conditions chosen in the process. Equation (17) is solved numerically by NEAC-2203 to know the concentration pattern of transferring components in the column (section 6). These solutions turn out to show a good agreement with the behavior of transferring components in the bench contactor (figures 5 and 6). Relative role of the components as oxidants to the others is made clear as shown in figure 6. The concentration patterns given in the figure show where a given species becomes oxidant to the others. Influence of initial concentration of Mg in the Bistream is also computed (figure 7).

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Methods of predicting the extent of chemical reaction or mass transfer are presented for onedimensionalaxial flow systems accompanying longitudinal dispersion, chemical reaction and othertransport phenomena. Four typical cases are treated as follows.
Case 1 Two-phase mass-transfer operation with fast irreversible second-order reaction:
Basic differential equation, Eq.(4), for the column operation are reduced to those for simplemass-transfer case by defining the nondimensional terms included in Eq.(5) as Eq.(6). Previoustheoretical analysis is indicated entirely applicable to this new situation by defining those terms inthis manner.
Case 2 Two-phase mass-transfer operation with slow first-order irreversible reaction:
With negligible reactant concentration in the liquid bulk of phase Y this case is indicatedequivalent to that of one-dimensional homogeneous phase reactor accompanying simple first-orderreaction. When the reactant concentration in phase Y has a finite value, analytical solution of thebasic rate equation is complicated in its form and not convenient for design purpose (see appendix).
Case 3 Homogeneous-phase nonisothermal chemical reactor of back-flow model type:
Chemical reactors of the type of back-flow model, which assumes multi-cell type column withinterstage mixing of fluid between adjacent perfectly mixed cells, is treated in this section and thenext following.
For homogeneous case the operation is shown in Fig. 2. Eq.(11)-(16) are obtained for heat andmass balances, and used to calculate the values at the 1st stage from the values of the N-th stage, which must be fixed at first. Sample calculations for the case of first order reaction are shown in Fig. 3 to 6; Fig. 3 and 4 show the effect of interstage mixing, reaction rate, and wall heat transfer; Fig. 5, 6 show the effect of the number of division. These are calculated by a digital computerHIPAC-101. Solutions by this method are exact, when interstage mixing really exists, and are goodapproximate solution for the diffusion model with enough higher number of division, because thedifferential equations of the diffusion model (Eq.(17), (18)) can be written in a finite differenceform same as Eq.(11)-(16). P-B has a relation expressed by Eq.(19) for heat and mass.Case 4 Two-phase nonisothermal cell-model reactor with interstage mixing:
Operation is shown in Fig. 7. When the rate processes taking place in the dispersed phase isfirst order, this phase can be regarded as the second continuous phase under some restriction. Therefore the basic equations are Eq.(22)-(30), where interstage mixing flow of phase Y (dispersed)is neglected, since usually the hold-up of this phase is rather small. As an example, countercurrent nitration of toluene by mixed acids is treated and shown in Fig. 8. In this case, 6 stages are enough to convert 90% of toluene to mononitrotoluene. These methods are enoughuseful for design purposes in such complicated cases.

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