English translation
Etude sur Simulateur des Régimes Transitoires des Concentrations dans une Installation de Diffusion Gazeuse
Complete English translation of the original French-language document (49 pages).
PRIME MINISTER — ATOMIC ENERGY COMMISSION
ANALOG SIMULATION STUDY OF CONCENTRATION TRANSIENTS IN A GASEOUS DIFFUSION PLANT
by P. DELAROUSSE, C. TROUVE, R. JACQUES
CEA Report No. 2010 Nuclear Studies Centre of Saclay
Abstract (French/English bilingual in source)
CEA 2010 — DELAROUSSE P., TROUVE C., JACQUES R. ANALOG SIMULATION STUDY OF CONCENTRATION TRANSIENTS IN A GASEOUS DIFFUSION PLANT (1961). Summary. — The transient behaviour of a gaseous diffusion cascade is represented in approximate form by a finite-difference differential system. The original analog hardware that made it possible to simulate this system is described. A series of examples illustrates the various problems that have been solved with this apparatus.
CEA 2010 — DELAROUSSE P., TROUVE C., JACQUES R. ANALOG SIMULATION OF CONCENTRATION TRANSIENTS IN A GASEOUS DIFFUSION PLANT (1961). Summary. — A finite difference system is used to describe concentration transients in a gaseous diffusion plant for uranium isotope separation. The equipment used in this study is described and examples are given to illustrate the problems which have been solved with it.
CEA Report No. 2010 Uranium Isotope Separation Studies Department
ANALOG SIMULATION STUDY OF CONCENTRATION TRANSIENTS IN A GASEOUS DIFFUSION PLANT
by P. DELAROUSSE, C. TROUVE and R. JACQUES
— 1962 —
I — INTRODUCTION
Isotope separation plants are characterised by long equilibrium times and high operating costs. It is therefore of interest to be able to determine in advance the evolution of concentrations and to derive from it optimum procedures both for start-up and for plant operation.
It is possible to solve on an arithmetic computer the equations that define the transient concentration regimes. The use of a simulator allows greater flexibility, obtained however at the expense of precision.
The system of finite-difference differential equations that describes concentration evolution exactly is difficult to handle. The calculation procedure used consists of grouping several equations together so as to arrive at a differential system of reasonable order.
The simulator used is capable of handling thirty-five linear differential equations. The computing elements are derived from fire-control computers built by the Compagnie de Télégraphie sans Fil (CSF).
— 2 —
The precision of the apparatus is of the order of 2 % for steady-state regimes and ranges from 2 % to 10 % for equilibrium rise times, depending on the configuration of the installation being simulated.
The simulator is designed as a very flexible tool capable of representing the various types of arrangements encountered in the gaseous diffusion process:
- Separation cascades with one or more stage sizes, with feed at a fixed point.
- Cascades with one or more stage sizes connected to an infinite reservoir.
- Chains of cascades joined by loops covering either a small number of separation stages or a half-cascade.
- Chains of cascades joined by junctions incorporating reservoirs and bleed-off lines.
- Chains of cascades with multiple feed points.
For each of these arrangements, the simulator makes it possible in particular to examine the following problems:
- Effect of a disturbance on concentrations, and compensation of that disturbance.
- Realisation of a transition following a predetermined optimum process.
- Start-up of a cascade and operation in perfusion mode. Calculation of a start-up schedule.
Simulators built on the same principles may be useful for studies of the commissioning of chemical plants, in cases where such studies appear justified by the risk of production time losses.
— 3 —
II — DESCRIPTION OF THE SIMULATOR
A — Equations of the gaseous diffusion cascade
The equations representing the transient concentration regime in a gaseous diffusion cascade are described by COHEN (1) and MONTROLL and NEWELL (2).
The internal structure of a separation stage is shown in Fig. 1. A set of stages of the same type and size constitutes a constant cascade. A gaseous diffusion plant is made up of several constant cascades (Fig. 2). The interconnections and the ends of the cascades are represented schematically in Fig. 3. The notation used is as follows:
| Symbol | Definition |
|---|---|
| L | Diffused mass flow rate |
| H | Mass capacity of a stage |
| Nₙ, N’ₙ, N”ₙ | Light-isotope concentration for the incoming, enriched, and depleted flows at stage n |
| P, W | Mass flow rates of the product and waste withdrawals |
| Nₚ, Nw | Concentrations of the rich (product) and lean (waste) withdrawals |
| F, Nf | Mass flow rate and concentration of the feed |
| U | Light-isotope transport at stage n |
The enrichment formula for a single stage is written:
$$N’_n = \alpha \frac{N_n}{1 - N_n} \tag{1}$$
which, for low concentrations, becomes:
$$N’_n \approx (1 + \varepsilon) N_n \tag{2}$$
— 4 —
The concentration behaviour is described by:
$$2L \frac{d}{dt} N_n = \alpha N_n (1 - N_n) - \varepsilon (N_{n+1} - N_n) + \frac{\partial U}{\partial n} \bigg|_{N_n} \tag{4}$$
With the notation:
$$K_{0n} = U_n / 2L \tag{5}$$
$$U^* = P / 2L \quad \text{(enriching cascade section)} \tag{6}$$
$$= -W / 2L \quad \text{(depleting cascade section)} \tag{7}$$
The boundary conditions vary according to the types of junctions and terminations used (Fig. 3). At the rich and lean ends one has:
$$(\cdots) = \gamma N \tag{8}$$
B — Simulation equations
In order to reduce the number of equations to be handled, the following approximation is introduced. The cascade is divided into sections of C stages, and the equations of the stages within each section are added term by term:
$$H \frac{d}{dt} \sum_{n=j}^{n=j+C} N_n = 2L \frac{d}{dt} \sum_{n=j}^{n=j+C} \cdots \tag{9}$$
This equation is that of a fictitious reservoir of mean concentration N̄ⱼ:
$$N_{j,0} \tag{10}$$
If ωᵢ₋₁ and ωᵢ denote the normalised light-isotope transports on either side of this reservoir, the equation becomes:
$$2L \frac{d}{dt} N_i = \omega_{i-1} - \omega_i \tag{11}$$
— 5 —
The light-isotope transport between two fictitious reservoirs separated by X stages is obtained from the differential equation of cascades (Ref. 1):
$$\omega = \frac{1}{4} \varepsilon (1 + \gamma)^2 (1 - r^2) \tag{12}$$
where the variable f is defined implicitly by:
$$\frac{N - N_2}{N_1 - N_2} = e^{2,\delta(1+\gamma),r,\cdots} \cdot \frac{N_1 - N_2 - \cdots}{N_1 - N_2} \tag{13}$$
With the notation:
$$\omega^2 = \cdots \tag{14}$$
This system takes into account the nonlinearities of the problem. An approximate expression is given below:
$$\omega \approx \frac{\varepsilon}{N} \cdot N_i \tag{15}$$
The term Nᵢ₊₁ Nᵢ can be neglected in the region of low concentrations.
Thus, an isotope-separation cascade is simulated by a system of separating sections without stage retention, and fictitious reservoirs each grouping C stages.
The approximation on steady-state regimes arises from the passage from equation (14) to equation (15), and the approximation on the transient regime is essentially due to the use of formula (10).
— 6 —
A judicious choice of reservoir locations and of fictitious section boundaries makes it possible to improve precision.
Table 1 gives, as an example, the simulation equations for a 100-stage cascade fed at stage 15 and with a reservoir at the top of the cascade.
Reservoirs 1, 2, 3, …, i, …, 10 correspond to levels 5, 15, 25, …, (10(i−1) + 5), …, 95. The top reservoir has a mass capacity R, and the following notation is used:
$$r = R/L, \quad h = H/L, \quad \gamma = P / 2\varepsilon L, \quad \gamma’ = W / 2\varepsilon L$$
with
$$\varepsilon = 10 \quad \text{[i.e. } C = 10 \text{ stages per fictitious section]}$$
— 7 —
TABLE I
Simulation equations for a constant 100-stage cascade equipped with a top reservoir and fed at stage 15.
[The table contains the differential equations for the fictitious reservoir concentrations. Due to the mathematical encoding in the source, the equations have the following general structure:]
For the top reservoir (capacity r):
$$r \frac{d N_{\text{top}}}{dt} = \cdots$$
For intermediate fictitious reservoirs i (i = 3 to 10, i ≠ feed reservoir):
$$h \frac{d N_i}{dt} = \omega_{i-1} - \omega_i - h \varepsilon \left(1 + f\right) \left(\omega_{i-1} + \omega_i\right)$$
For the feed reservoir (i = 2, at stage 15):
$$h \frac{d N_2}{dt} = \cdots \quad \text{[includes feed term } f \cdot (1-\omega’) \cdot f(N_f \cdot N_i) \text{]}$$
(Note: several equation terms in the source text are partially corrupted by the OCR/text-extraction process; the structure above reflects the correct physical interpretation.)
— 8 —
C — Machine equations
As in any analog computation, a certain number of variable changes are necessary in order to make the variation ranges of the problem variables match those permitted for the representative quantities generated by the computer. These variable changes will not be described in detail in this report.
D — Equipment
The computation process is electromechanical. The quantities on which calculations are performed are represented by electrical voltages or by mechanical rotation angles (for example in the drive of the variable capacitors of the multiplier circuits).
The computation chains operate at a frequency of 472 kc/s and consist of fixed or variable capacitors, inductors, and feeders. The integrators used require a transformation of high-frequency currents into direct currents. The classical scheme of an integrating computing amplifier then reappears. Once integrated, the direct current generates a proportional high-frequency voltage through a variable impedance coupled to a motor servo-controlled by the integrating amplifier. A detailed study of this high-frequency current computation process has been carried out by UFFLER (3). Several examples of circuits are given in Fig. 4 (a and b).
E — Description of the simulator
Each machine equation is implemented by means of functional blocks arranged in horizontal rows on the five racks that make up the machine.
— 9 —
TABLE 2 — FUNCTIONAL SIMULATION BLOCKS
| Block Name | Symbol | Characteristics |
|---|---|---|
| Multiplication cell | Variable ratio from 0.16 to 0.2 | |
| Variable ratio from 0 to −0.1 | ||
| Variable ratio from −1/8 to 1 | ||
| 6 coupled multiplication cells | ||
| Variable ratio from 0.16 to 0.66 | ||
| 2-input addition cells | Ratios: −1, +1, −1, +1/3 | |
| 4-input addition cells | Ratios: +1, −1, +1, −1, +1/3 | |
| Integrator | Gain 10 | |
| Repeater | Ratio 6 |
— 10 —
Table 2 gives the representations of the various computing blocks. These blocks are interconnected by self-compensating junctions incorporating a test point that allows measurement of the corresponding voltage by means of a setting block. A control panel provides display, run, and hold (freeze) positions. The complete apparatus is shown in Fig. 5 (a and b).
An example of a wiring diagram, for the cascade described in Table 1, is shown in Fig. 6.
III — PRECISION
Various comparisons have been carried out between machine results and numerical calculations, drawn either from Ref. (1) or performed specifically on an arithmetic computer. The numerical values relating to each example are given with the corresponding figures.
A — Comparison by calculation
A 512-stage cascade was simulated with 4 fictitious reservoirs each grouping 128 stage capacities, arranged regularly. The numerical values are:
| Parameter | Value |
|---|---|
| Normalised rich withdrawal | γ = 0.38 |
| Normalised lean withdrawal | γ’ = 2.50 |
| Separation coefficient | ε = 0.002146 |
| Enriching stages | 379 |
| Depleting stages | 133 |
The results for the significant concentrations are given below. The relative error does not exceed:
— 11 —
| Arithmetic calculation | Simulation | |
|---|---|---|
| Nₚ | 2.00 % | 1.98 % |
| N₁₃₃ | 0.71 % = Nf | 0.99 Nf |
| 0.51 % | 0.51 % |
As a second example, the importance of the approximation made on the light-isotope transport in a plant section operating at high concentration was examined. The term −Nᵢ · Nᵢ₊₁ of equation (15) is no longer negligible.
The section comprises 250 stages between two fictitious reservoirs, with the numerical values:
$$\omega = 1, \quad \omega’ = 1, \quad N_1 = 0.30, \quad N_{\omega+1} = 0.40$$
The exact calculation using expression (12) gives ψ = 0.93 × 10⁻³. The approximation (15) gives ψ = 0.96 × 10⁻³; the error therefore reaches 4 % in this case.
Transient regime
The cascade studied comprises 160 stages. A first simulation was carried out with 4 fictitious reservoirs, a second with 2 fictitious reservoirs, and these results were compared with the direct calculation of the equilibrium rise according to Ref. (1). Fig. 7 (a and b) gives the results of this comparison. In general terms, the simulation is satisfactory when no more than 40 stage capacities are grouped per fictitious reservoir. The correction coefficient for the equilibrium rise time is then close to 0.90.
— 12 —
B — Examination of results produced by the machine
Long constant cascade with feed at mid-point (examination of steady-state regime). Fig. 8 gives the calculated and simulated concentration profiles and a table of steady-state concentration values, for comparison purposes. The constant cascade comprises 1400 stages. The feed is placed at stage 700 and the simulation is carried out with 35 fictitious reservoirs distributed regularly. A fairly large withdrawal was imposed in order to place the test in an unfavourable case. The relative error does not exceed 2 %.
Fig. 9 gives the comparison with the calculation for the approach to steady-state under total reflux of a 100-stage constant cascade. The relative error is close to 2.5 %.
Schematic two-stage-size installation. Figs. 10 (a and b) describe the behaviour of a plant grouping two cascades of different sizes, for disturbances of various origins.
The response of the rich outlet concentration N₂ takes the form:
$$\Delta N_2 = A(\Delta N_F) + B(\Delta \sigma)$$
where the perturbation acts on the feed and by taking stages out of service.
The response of concentration N₂ to a unit step in the feed concentration Nf (Fig. 10a) was examined, as well as the effect of taking 30 stages out of service in
— 13 —
the part of the plant operating at low concentration and 60 stages in the other part (Fig. 10b). The approximation is satisfactory in this example.
Loop between two cascades. In addition to the arrangements corresponding to a constant cascade equipped with an enriching section and a depleting section, or to a constant cascade mounted on an infinite reservoir, it is possible to represent on the simulator chains of constant cascades joined to one another by loops covering either a small number of separation stages or a half-cascade.
Fig. 11 (a and b) describes the case of a set of two constant cascades mounted on an infinite reservoir and joined by a loop covering one half-cascade. The evolution of the concentration profile was determined as a function of the loop flow rate for a constant rich withdrawal. The rigorous calculation was also carried out separately for the top concentration.
In conclusion, the precision, although different for each particular case, can be estimated as follows:
- Better than 2 % for steady-state regimes, for sections grouping up to 200 stages.
- Of the order of 2 % to 10 % for transient regimes, with fictitious reservoirs representing up to 50 stage capacities.
If the simulation requires larger fictitious reservoirs, a correction coefficient must be introduced for time, the uncorrected simulation leading to equilibrium times that are too long.
IV — TYPES OF PROBLEMS THAT CAN BE SOLVED ON THE SIMULATOR
The preceding chapter described examples of simple problems
— 14 —
solved on the simulator:
- Approach to equilibrium of plants previously filled with gas at natural isotopic abundance, under various operating modes: total reflux, withdrawal, or perfusion.
- Transient concentration regime caused by a disturbance of a withdrawal stream, of the feed, or by a change in the plant structure due, for example, to taking a group of stages out of service.
All of these problems can be treated for the following arrangements, to which extreme or intermediate reservoirs may be added:
- Long cascade with one or more stage sizes, with an enriching section and a depleting section.
- Long cascade with one or more stage sizes mounted on an infinite reservoir.
- Chains of cascades joined by a short or long loop.
[page 14: schematic line diagrams of cascade arrangements appear in the source — enriching/depleting cascade symbol, cascade-on-reservoir symbol, and chained cascades with loop symbol]
Thanks to the “hold” (freeze) position of the control selector, it is possible to vary various coefficients during the course of the problem; in particular, the withdrawals can be made to follow step-wise curves.
— 15 —
This makes it possible to tackle complicated problems such as:
- Compensation of a disturbance.
- Realisation of a predetermined transition.
- Optimisation of a start-up.
An example of the technique employed in each of the preceding cases is given below.
A — Compensation of a disturbance
Fig. 12(a) describes the search for the rich-withdrawal curve that makes it possible to absorb a disturbance caused by taking 60 stages out of service in the middle section of the schematic two-size installation already used for Fig. 10. For this operation, a tolerance band is fixed around the initial steady-state regime, and an attempt is made, by modifying the rich withdrawal, to keep the outlet concentration within this band.
A step-wise curve for the rich withdrawal is thus obtained, which is all the closer to the theoretical curve the narrower the initial tolerance band. By carrying out several trials of the same type, it is possible to progressively narrow the tolerance band and to obtain a good approximation of the theoretical variation curve of the withdrawal.
B — Realisation of a predetermined transition
Using the same technique as for the absorption of a disturbance, it is possible to determine the evolution of the withdrawal that effects the passage from one steady-state concentration regime to another, following a transient calculated in advance and satisfying certain optimisation criteria.
The transition described in Fig. 12 (b and c) concerns the schematic two-size installation already
— 16 —
[page 16: figure only — Figs. 12(a), 12(b), and 12(c) occupy this page, showing step-wise withdrawal curves and concentration transient traces for the disturbance-compensation and predetermined-transition examples]
— 17 —
[page 17: figure only — continuation of graphical results; figures showing concentration profiles or transient curves related to the cascade simulation examples described in Sections III and IV]
— 18 —
[page 18: figure only — additional graphical results; figures showing concentration profiles or transient curves from simulator runs, likely including the start-up optimisation or loop-between-cascades examples]
used for Fig. 10. It achieves the transition from one concentration level to another by seeking to approximate the step-function form as closely as possible, while keeping withdrawal variations within acceptable limits. This objective corresponds to a criterion of the form:
∫ [(N − N*)² + A(ΔY)²] dt → minimum.
This case was also computed on an arithmetic machine, which yielded the theoretical curves shown in Fig. 12 (a, b, and c). In these figures, the horizontal axis is the normalized time T₁ = 10⁻⁵T, where T = Δh, Δh being the stage time: h = H/L.
C — Optimization of a Start-up
A gaseous diffusion plant comprises a very large number of stages, so that its assembly can only be carried out section by section as the equipment is delivered. The total assembly time is long compared with the concentration equilibration times. Accordingly, before the entire plant has been built, sections of varying size are available and ready for production.
The problem of managing the start-up consists in determining the partial ramp-up times for each section, their optimum utilization, the stocks of natural-assay product required, and the stocks of partially enriched product that will accumulate during this period together with their optimum reuse. This determination is based on a final objective — for example, producing as rapidly as possible a given quantity of highly enriched product.
[Page 17]
Such an optimization involves a series of problems of the types already described above. On the simulator, all of these operations can be carried out rapidly with immediate visualization of the results.
Fig. 13 gives an example of the search for equilibration times in perfusion mode for a cascade section filled initially with natural-assay product. The fastest way to move from one concentration level to another is to start in perfusion mode until the desired level is reached, followed by the gradual establishment of withdrawal so as to maintain that level constant. Furthermore, the simulator makes it possible to determine the perfusion mode that minimizes a given criterion. This criterion can, for example, relate the time to reach equilibrium and the quantity of natural product consumed, so that the computed mode achieves a compromise between these two factors.
V — DEVELOPMENT OF A START-UP SCHEDULE
A constant cascade of 640 stages is considered, assimilable to the low-enrichment plant of a gaseous diffusion complex. It is assumed that this plant is delivered in 4 sub-cascades of 160 stages each, each one comprising a bleed and a top reservoir. The sub-cascades were simulated by means of 4 fictitious reservoirs; the bleed and the reservoir were simulated by a fictitious reservoir equivalent to 160 stages. The delivery time for each sub-cascade is three months.
The objective is to produce, as rapidly as possible, enriched gas at concentration N such that:
N/N₀ = [value specified in the original]
[Page 18]
Moreover, for the depleted rejection stream of concentration N_w, the following conditions are imposed:
- Either withdraw such that N_w/N_F ≥ 0.95 — which allows the rejection to be reused
- Or withdraw such that N_w/N_F ≥ 0.70 — which constitutes a rejection of zero value
Achievement of the objective requires delivery of the low-enrichment plant beginning with the highest sub-cascade.
Indeed, suppose for example that enriched gas at concentration N_P such that N_P/N_F = 2.857 can be produced with only two sub-cascades: it would be inadvisable, having raised the concentration profile of these 2 sub-cascades, to let it fall again by attaching an upper sub-cascade. It is more logical to plan — through self-feed policies, for example — for the production of enriched gas at N_P/N_F = 2.857 using 1 or 2 sub-cascades, to which the lower sub-cascades are added one by one without disturbing the established profiles and while maintaining production at a constant concentration.
In addition to this delivery policy, the following operating assumptions were made:
- As soon as each sub-cascade is delivered, it is placed under observation at total reflux for an unspecified but equal duration for all.
- At each assembly stage, the feed point may be shifted. Feed stations exist every 40 stages.
A self-feed policy makes it possible to produce enriched gas at N_P/N_F = 2.857 with the first sub-cascade.
The optimum concentration for the first withdrawal plateau of this sub-cascade was determined, the optimum corresponding to a maximum of production at N_P/N_F = 2.857. The various depleted rejections from this sub-cascade constitute a stock with a mean assay of N_w/N_F = 0.98 per unit time. Once the concentration profile is properly established for the first sub-cascade, the simulator shows that the second sub-cascade — previously held at total reflux — can immediately be connected and withdrawals can begin at the chosen values without appreciable loss of head concentration. The same applies to the third and fourth sub-cascades. Withdrawals were chosen so as to yield a depleted rejection such that N_w/N_F = 0.7, i.e., of no further use.
On schedule Fig. 14a the evolution of the top and tail concentrations of each sub-cascade over the course of this start-up is shown. In Fig. 14b the assembly policy is schematized and the balance of available stocks and natural gas consumed at the end of the start-up is presented.
[Page 20]
In Figs. 14a and 14b, the gas quantities Q and the withdrawals P and W have been normalized, in the usual manner, with respect to 2ΔL. This normalization coefficient is not given in the report. The intention was simply to show in detail one of the possible applications of the simulator.
VII — CONCLUSION
This report has presented a technique for representing the transient behavior of a gaseous diffusion cascade by means of simple equations in limited number. The analog hardware that enabled these equations to be processed has been described, and the validity of the necessary approximations has been determined from numerical examples.
In particular, the comparison with arithmetic computation proved very satisfactory and identifies the simulator as a valuable instrument for working out operating schedules, start-up plans, and control procedures.
The equations used yield approximate steady-state and transient behavior and are valid only for low concentrations. However, equations applicable to high concentrations — which would be straightforward to program with a few nonlinear analog elements — have also been provided. More elaborate hardware, capable of representing the basic equations that have been established, would even make it possible to dispense with all approximations on steady-state behavior. The enriching sections, whose behavior is governed by system (12), would then be simulated by function generators or solving circuits. Such circuits have now reached a high degree of precision owing to their use in fire-control computers.
[Page 21]
The small number of integrators required to represent the fictitious reservoirs would then make the drift problem — one of the difficulties in the analog solution of transient systems with a large number of equations — less significant.
Devices of the same type can be constructed for all processes involving a large number of identical elements assembled in cascade, as found in most isotopic separation systems. They can also be envisaged as auxiliaries in the operation of distillation columns or of various production units.
Manuscript received 26 January 1962.
[Page 22] — LIST OF FIGURES
Figure 1 — Internal arrangement of separation stages
Figure 2 — Possible layout of a gaseous diffusion plant
Figure 3 — Notation
Figures 4a and 4b — Computation circuits using high-frequency currents
Figures 5a and 5b — Photographs of the simulator
Figure 6 — Diagram of the simulation setup for a constant cascade of 100 stages
Figures 7a and 7b — Constant cascade of 160 stages
Figure 8 — Curves and comparison table of equilibration times at total reflux obtained using several simulation models
Figure 9 — Constant cascade of 1400 stages with central feed — Concentration profile
Figure 10 — Constant cascade of 100 stages operated with an infinite reservoir
Figures 10a and 10b — Perturbations in a schematic two-size plant
Figures 11a and 11b — Long junction between two constant cascades
Figures 12a, 12b, and 12c — Concentration regulation in a schematic two-size plant
Figure 13 — Perfusion-mode operation of a constant cascade of 280 stages
Figures 14a and 14b — Implementation of a start-up schedule
[Page 23] — REFERENCES
(1) K. COHEN The theory of isotope separation as applied to the large-scale production of U McGraw-Hill Book Co., Inc. (1951)
(2) E.W. MONTROLL and G.F. NEWELL “Unsteady-state separation performance of cascades” Journal of Applied Physics 23, 184–194 (Feb. 1952)
(3) H.J. UFFLER “Sur un nouveau procédé de calcul par courants de haute fréquence” Annales de Radio-Électricité XI, no. 45 (July 1956)
[Pages 24–36: figures only]
Fig. 1 — Internal arrangement of separation stages. (Schematic showing diffuser, compressor, separator stages in a cascade with labeled flows.)
Fig. 2 — Possible layout of a gaseous diffusion plant. (Block diagram showing enriching stages in a constant cascade, inter-cascade junctions, feed point F/N_F, rich withdrawal P/N_P, and depleted rejection W/N_W.)
Fig. 3 — Notation. (Diagram defining flow-rate symbols H, L, P, W and concentration symbols N at each node of the cascade, with inter-stage flows and feed/withdrawal points labeled.)
Fig. 4a — Computation circuits using high-frequency currents — Multiplication by a variable. (Circuit schematic with input, transformer elements C₁, C₂, inductances L, and variable-parameter element; note states: L(C₁+C₂)ω² = 1; X = f(θ), where f(θ) is a linear, sinusoidal, or arbitrary (mathematical or empirical) function.)
Fig. 4b — Computation circuits using high-frequency currents — Weighted addition. (Circuit schematic with two inputs, transformer, detector, amplifier, vibrator, motor, and integrator ∫U dt; output is in DC current; HF reference source shown.)
Fig. 5a — General view of the simulator. (Photograph.)
Fig. 5b — Simulator — Adjustment block — Recorders. (Photograph.)
Fig. 6 — Diagram of the simulation setup for a constant cascade of 100 stages brought to equilibrium at total reflux. (Block diagram.)
Fig. 7a — Constant cascade of 160 stages; comparison of equilibration times at total reflux. (Graph of enrichment ΔN_P vs. time in seconds, comparing curves for fictitious-reservoir models (1 reservoir, 2 reservoirs) against the real cascade. Parameters: 160 stages, h = 5 s, δ = 0.002; sub-group sizes of 40 stages; models labeled with number of fictitious reservoirs.)
Fig. 7b — Constant cascade of 160 stages. Adjustment of equilibration times. (Table and curves. Table gives, in hours, values at times 1, 10, 50, 100, 120, 200:
- Calculated value (Cohen solution): 1.09, 1.30, 1.69, 1.84, 1.86, 1.90
- Simulation with 4 reservoirs: 1.12, 1.30, 1.64, 1.80, 1.84, 1.89
- Simulation with 2 reservoirs: 1.17, 1.29, 1.59, 1.76, 1.79, 1.87
Note: correction factor applied to adjust equilibration times to 50% of steady-state value. Graph shows importance of fictitious reservoir (stages vs. 80 stages parameter, values 0.90 and 0.75 noted) on log-time axis (10³ to 10⁵ seconds), curves after adjustment shown.)
Cohen Simulator Calculations (35 fictitious reservoirs)
ε = 0.002
L = 2 kg, P₀ = 232.3 v i ec
W = 674.3
Comparison Table
| Stage rank | Values calculated by stage using Cohen’s formulas (NP) | Values obtained from the simulator | Relative errors |
|---|---|---|---|
| 1/9 | 933 | — | — |
| — | 1140 | 2.0130 | — |
| — | 1.1211 | 1.1172 | −0.4% |
| — | 860 | — | — |
| — | 1.0100 | 1.0130 | +0.3% |
| — | 700 | — | — |
| — | 1.0004 | 1.0040 | +0.4% |
| — | 580 | — | — |
| — | 0.9961 | 1.0030 | +0.7% |
| — | 300 | — | — |
| — | 0.9534 | 0.9560 | +0.3% |
| — | 20 | — | — |
| — | 0.6750 | 0.6610 | −2.6% |
Fig. 8 — Constant cascade of 1400 stages with central feed. Concentration profile.
Rise to Equilibrium
ΔN/N
- Cohen calculations (dashed)
- Final profile
Time in hours
t = 0.004, h = 2 sec
| Level reached (% of equilibrium value) | Time per Cohen (seconds) | Time with simulator | Relative error |
|---|---|---|---|
| 30% | 2,600 | 2,600 | 0 |
| 50% | 6,150 | 6,300 | +2.5% |
| 70% | 12,000 | 12,300 | +2.5% |
| 80% | 16,100 | 16,500 | +2.5% |
| 90% | 24,100 | 24,700 | +2.5% |
Fig. 9 — Constant cascade of 100 stages mounted on an infinite reservoir. Total reflux.
Real Plant
Numerical values: ε = 0.0055, α = 0.3636
nf = 180, n₀ = 120
Unit step of Nᴘ
- Arithmetic calculation
- Simulator (24 fictitious reservoirs)
Fig. 10a — Perturbations in a schematic two-size plant.
Stage Bypass
- Arithmetic calculation vs. simulator
- from nf = 60 to nf = 120
- from n₀ = 30 to n₀ = 560 (or 30 to n = 60)
- T′ = 10⁻⁵ T
Fig. 10b — Perturbations in a schematic two-size plant.
Influence of the loop flux P₂/2εL = 0.5
Concentration Profile
| Nᴘ | A/Fsl | Nᴘ/N |
|---|---|---|
| 0.5 | 1.456 | 1.554 |
| 1 | 1.542 | 1.798 |
| 2 | 1.470 | 1.934 |
| 3 | 1.450 | 1.942 |
| 5 | 1.424 | 1.904 |
| 10 | 1.351 | 1.773 |
| 15 | 1.312 | 1.596 |
[page 43: figure only — concentration profile curves for various Nᴘ values plotted against stage rank, showing evolution of notable concentrations as a function of loop flux]
Fig. 11a — Long junction between two constant cascades. Evolution of notable concentrations as a function of loop flux.
Fig. 11b — Long junction between two constant cascades. Regulation of a stage bypass.
Concentration Regulation in a Schematic Two-Size Plant
[page 45: figure only — regulation curves showing:
- Curve with regulation (within tolerance band)
- Curve without regulation
- Tolerance domain (shaded region)
- Variations in withdrawal (soutirage) plotted on x-axis (0.03, 0.06, 0.09)
- y-axis values: 0.04, 0.16, 0.24, 0.32]
Fig. 12a — Regulation of concentrations in a schematic two-size plant.
Realization of a Predetermined Transition
- Ideal curve
- Curve obtained from the simulator
- T′ = 10 T
- T′ = 10⁻⁵ T
(Theoretical curve shown as dashed line)
Withdrawal realizing a pre-determined transition of amplitude ΔNᴘ/Nᶠ = 1
Fig. 12b — Regulation of concentrations in a schematic two-size plant.
Fig. 12c — Regulation of concentrations in a schematic two-size plant.
Operation in Perfusion Mode of a Constant Cascade of 280 Stages
Evolution of concentrations, h = 5 sec
[page 48: figure only — concentration evolution chart showing:
- Sub-cascade 1 alone
- Sub-cascades 1+2
- Sub-cascades 1+2+3
- SC2 (unobstructed)
- Various stage counts: 100 stages, 160 stages
- Concentration values marked at: 0.153(0), 0.085, 0.715, 1.420, 1.001, 5.135
- Base plant n = 42.9, n = 6680
- Time axis marked in months (~3 months) and days
- Final values approaching n = 149, y = 71 at t ~ 61 days]
Fig. 13 — Perfusion operation of a constant cascade of 280 stages. Evolution of concentrations, h = 5 sec.
Implementation of a Startup Planning Schedule
[page 49: figure only — startup planning diagram (Fig. 14a) showing:
- Assembly policy and inventory balance (politique de montage et bilan des stocks)
- Time axis in units up to ~2800
- Multiple stage sub-cascade activation events over time (~3 months)
- Stage count annotations: n = 149, y = 71, 61 days marked
- Various cascade milestones indicated along the time axis]
Fig. 14a — Implementation of a startup planning schedule.
Fig. 14b — Implementation of a startup planning schedule. Assembly policy and inventory balance.