.... a a $1, LIBRARY Michiga- tat: University . -‘- i. '2' , This is to certify that the 2.- -I - " ‘ f .,a'- ,. ' thesis entitled ‘-- r " ‘ . . z I ‘ Sr 1.“ . presented'by T {if I‘aII Steven R. Auvil has been accegted towards fulfillment of tho Lequirements for Major professor i. " 1'" in Br I I W. Wilkinson ' ‘J', , .I .. a 'J I .. "I EEI fu‘ ABSTRACT A GENERAL ANALYSIS OF GAS CENTRIFUGATION WITH EMPHASIS ON THE COUNTERCURRENT PRODUCTION CENTRIFUGE By Steven R. Auvil Contained in this work is a theoretical analysis of gas centrifugation with emphasis on the countercurrent production centri- fuge. The in-depth analysis of gas centrifugation presented in this work significantly updates the previously available literature. Results are presented which allow an evaluation to be made regarding the applicability of the gas centrifuge to perform a given separation. Operating variables such as pressure, temperature, rotational speed, centrifuge diameter and length, throughput, molecular weight differ— ences and diffusion coefficients, among others, are discussed in detail for the gas centrifuge separation of a binary gas mixture. Included as Appendices are user instructions, descriptions and listings of computer programs that are based on this work which can be used to model (predict) the performance of either a stripping, rec- tifying or Zippe type gas centrifuge in a given situation. Results predicted by the programs agree very well to separations presented in the literature for Uranium isotope (as UF6) separations. 9 Steven R. Auvil Throughout the work many significant points are illustrated using the gas pairs SOZ-Nz, SOZ-H2 and UF6 (235 and 238 isotopes). The SOZ-N2 gas pair reflects the desire to determine the aplicability of gas centrifugation as a viable means for removing $02 from power plant stack gas. Equations were developed describing the equilibrium pressure and composition profiles and the times required for the development of these profiles in a simple centrifuge containing a binary gas mixture. This analysis illustrated that, while an 8-inch diameter simple centrifuge charged at 70°F and 1 atmosphere with an SOZ-N2 gas mixture (5000 ppm 802) and then rotated at 20000 RPM would give a 39.4% increase in 50 concentration going from the axis to the wall, 2 it would take approximately #3 seconds for 50% of this separation to occur. The diffusion coefficient for the gas pair was found to be the only factor controlling the rate of separation. With such slow rates of mass transfer, it was concluded that long residence times would be required to realize the separating potential of the counter- current production centrifuge. To meet the demands of high rotational speeds and long resi- dence times, the mechanical limitations (size versus rotational speeds) of the centrifuge were investigated. Expressions were developed for the bursting and whirling speeds of a simply supported cylinder, illustrating that Aluminum and Magnesium alloys were best suited for centrifuge design. It was shown for these alloys that a peripheral speed of 700 ft/sec was well below the bursting speed of an 8-inch diameter rotating cylinder with a l/h-inch wall thickness, Steven R. Auvil r, at this speed the length to diameter ratio has to be less for safe operation under the first whirling speed. Expressions were developed describing the separations in a rcurrent gas centrifuge. It was determined that in the opera- f a countercurrent rectifying (concentrates the heavy species) ipping (concentrates the light species) centrifuge that an m feed rate (OFR) existed, yielding a maximum in the separation (MSF). For the gas pair SOZ-N2 at 70°F and l atmosphere axis re in an 8-inch diameter by 36—inch long rectifying centrifuge ng at 15000 RPM, the OFR was found to be 0.0l69 scfm giving an ' 2.337 at total reflux. If a rich product stream equal to l0% : feed stream is removed the OFR increases to 0.0l9l scfm but as the MSF (rich to lean stream) to l.696 or a reduction of It was shown that the OFR and the MSF could be quickly and 1tely estimated for any gas pair in any size rectifying centri- ut any operating conditions by using the following equations: OFR x a - (diffusivity times pressure, cm2 - atm/sec) (diameter, inches), 2 MSF 3 Exp [b ° (rotational speed, ft/sec) (molecular weight difference) (length/diameter) / (temperature, °F)], a = .0157 and b = 1.013 x 10'5 at total reflux, and Steven R. Auvil a — .0l78 and b = 6.30A x l0_6 where the rich stream is l0% of the feed stream. As is shown inthe above estimating equations, increasing the er but maintaining a constant peripheral speed increases the rectly; however, the reduction in the length to diameter ratio ntially decreases the MSF. Hence, to sustain a given level of tion when scaling up a centrifuge design, the length and diameter ncrease by the same percentage. Also, shown in this work is the hat removal of a product stream rather sharply reduces the MSF less of the centrifuge length. This is an equilibrium limita~ lhiCh was shown to be characteristic of the gas centrifuge. The results of this work clearly indicate that for the gas 502-N2 in the 8-inch diameter centrifuge operated as described the OFR of 0.0l9l scfm and corresponding MSF of l.696 with a tream removal of 0.00l9 scfm (l0% of the feed) cannot be sig- antly altered by any operating parameters. These very small and low MSFs are incompatible with the tremendous volumes of gas 'ed by power plants and the high degree (90+%) of SO2 removal 'ed. Gas centrifugation for this purpose cannot begin to compare ’he many conventional 802 removal processes proposed for removal I from coal fired power plant stack gases. Included as part of the Appendices is a description and analysis experimental program which paralleled this work. The centrifuge n the experimental program, which was modeled after a design by Zippe (University of Virginia, l960), had 3% inches available Steven R. Auvil axial flow, a 7.5-inch diameter, and was rotated at llOOO RPM. [e Zippe's experiments were with UF6 (23S and 238) separations with ; pressures less than O.l mm Hg, the above experimental program carried out at atmospheric pressure with $02-N2 gas mixtures. The est feed rate (limited by equipment design and sampling methods for analysis) which was used was O.l scfm. No measurable separation observed at this feed rate or at any other higher feed rate. The yses presented in this work, which predict Zippe's observations :e accurately, show the separation factor predicted at the above l rate to be only l.OA5. This very low predicted separation factor ‘esponds to $02 concentration differences much too small to be seen 1 the infrared spectrophotometer used for gas analysis. Further- a, the uncontrollable and unpredicatable increase in internal Julence promoted by the Zippe type centrifuge operated at atmospheric sure was expected to have reduced if not destroyed this small, expected, separation factor. The very small feed rate and predicted separation factor, led with the turbulence problems fundamental to the Zippe centri- design when operated at pressures near l atmosphere, resulted in termination of the experimental program. A GENERAL ANALYSIS OF GAS CENTRIFUGATION WITH EMPHASIS ON THE COUNTERCURRENT PRODUCTION CENTRIFUGE By 0 Steven R.iAuvil A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering l97A ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. Bruce W. Ison for his guidance and numerous suggestions during the course 5 work. Appreciation is also extended to the members of my 1 Guidance Committee for their advice and suggestions. The author is indebted to the National Science Foundation ; GK-3423l) and the Division of Engineering Research at Michigan University for providing funds for the purchasing and machining : experimental equipment and other related experimental costs, as computer time. Also, during the period of time from September, through August, I973, the author was very fortunate to have been d financial aid in the form of an NDEA Type IV Traineeship. Special thanks is given to Mr. Don Childs and Mr. Carl Redman e fabrication of parts for the experimental equipment and to hn Shaffer, Division of Air Pollution Control, Department of l Resources, for the many hours spent helping build and run the mental equipment. The understanding and patience of the author's wife, Pam, is ely appreciated. TABLE OF CONTENTS F TABLES . F FIGURE F APPEND INTROD MUDWD STEADY A. B. C D. E. UNSTEA S ICES UCTION Gas Separations in a Gravitational Field Centrifuge Types Scope of Past Work . Scope of This Work . The Organization of the Information Provided in this Work . STATE SIMPLE GAS CENTRIFUGE THEORY Pressure Gradients, Pure Gases Pressure Gradients, Gas Mixtures Mole Fraction Distributions and Simple Separation Factors . Axis Composition in a Simple Centrifuge . A Simple Centrifuge Containing a Stationary Pipe Centered Along the Axis . . . DY STATE SIMPLE GAS CENTRIFUGE THEORY Concentration and Pressure Diffusion Fluxes The Partial Differential Equation for the Unsteady State Operation of the Simple Centrifuge . The Method of Solution Used to Solve the Partial Differential Equation for the Unsteady State Operation of the Simple Centrifuge The Mole Fraction Profiles Developed in the Unsteady State Operation of the Simple Gas Centrifuge for the Gas Pairs 502- N2, SOZ- H2, and UF6 (235,238 Isotopes) Page ll l5 I7 22 26 33 35 40 THE ESTIMATION OF THE BURSTING AND WHIRLING SPEEDS OF A GAS CENTRIFUGE A. Bursting Speed Analysis B. Whirling Speed Analysis THE COUNTERCURRENT PRODUCTION CENTRIFUGE A. Partial Differential Equation for the Counter- current Centrifuge . 8. Velocity Profiles . . . . . . . C. The Solution of the Partial Differential Equa- tion for the Countercurrent Centrifuge D. Analysis of the Analytical (Approximate) Solution . . . . . . E. Computing the Analytical (Approximate) Solution . . . . . . . ANALYSIS OF THE RESULTS OF THE COUNTERCURRENT CENTRIFUGE A. A Comparison of the Numerical and Analytical (Approximate) Solutions . . . . B. The Effect of the Position of the intersection of the Inner and Outer Streams on the Maximum Separation Factor and the Optimum Feed Rate C. The Effect of the Centrifuge Radius on the Maximum Separation Factor and the Optimum Feed Rate . D. The Effect of Rotational Speed and Temperature on the Maximum Separation Factor and the Optimum Feed Rate E. The Effect of Removing a Rich Product Stream on the Maximum Separation Factor and the Optimum Feed Rate Countercurrent Centrifuge CONCLUSIONS lAPHY \TURE F. Simple Power Requirements and Efficiencies of the Page 53 5b 6] 7O 72 76 8O 86 9| 93 93 llO lll llS 119 12A 131 1A2 IAA LIST OF TABLES Simple Centrifuge Separations of the Gas Pairs: SOz-Hz, SOZ-NZ, UF6 (235, 238 Isotopes) at 70°F. Centrifuge Peripheral Speed = 500 ft/sec . . . . ’redicted Transition RPMs for the Gas Pairs SOZ-NZ and SOZ-Hz in a Simple Centrifuge Containing a Stationary Center Pipe . . )perating Conditions and Diffusivities Used in the Unsteady State Analysis of the Gas Pairs SOz- N2, SOZ-Hz and UF6 (235, 238 isotopes) /alues<3f(y-yf)/(ye-yf) x IOO Versus Time at the Axis and the Wall for the Gas Pair SOZ-NZ in the Unsteady State Simple Centrifuge, Radius = A inches lalues of (y-yf)/(ye-yf) x IOO Versus Time at the Axis and the Wall for the Gas Pair SOZ-HZ in the Unsteady State Simple Centrifuge, Radius = A inches alues of (y-yf)/(Ye'Yf) x lOO Versus Time at the Axis and the Wall for the Gas UF6 (235 and 238 isotopes) in the Unsteady State Simple Centrifuge. Radius = A inches and Charge Pressure = 2.l psia otal Pressure Profiles in the Unsteady State Simple Centrifuge for the Gas Pair SOZ—NZ at 70°F and lOOOO RPM. Radius = A inches . . atios of Yield (Ultimate) Strength to Density for Some Common Metals and Their Alloys pper Bounds for W for an Aluminum Alloy Centrifuge A0 Inches Long atios of the Elastic Modulus to Density for Some Common Metals and Their Alloys he Analysis Capabilities of the FORTRAN Program CENTRI . . . . . . . . . . Page 25 28 A0 Al A3 L111 52 6I 66 69 92 aximum Separation Factors and Their Corresponding Feed Rates Found During the Comparison of the Numerical and Analytical (Approximate) Solutions: SOz-NZ lole Fraction, y, and By/Bz Profiles at Various Radial Positions for Laminar and Plug Type Flow With a Feed Rate of 0. Ol25 scfm: 502- N2 lole Fraction, y, and By/az Profiles at Various Radial Positions for Laminar and Plug Type Flow at Feed Rate of 0.02 scfm: SOZ-Nz . Separation Factors for the Gas Pair $02-N2 at Various Ratios of the Rich Stream to the Feed Stream (R/F) as Computed from the Numerical and Analytical (Approximate) Solutions. Feed Rate = 0.02 scfm, Rw = A Inches, Rm/RW = 0.5625, w = lOOOO RPM, L = 36 Inches and Plug Type Flow . . . . . . . . . . . Iaximum Separation Factors and Optimum Feed Rates Obtained for Various Centrifuge Radii for SOz-Nz with Plug Type Flow,L ~36 Inches, Rm/Rw = 0.5625, Peripheral Speeds of 350 and 700 ft/sec, and Total Reflux . . . . . The Effect of Increasing Rotational Speed and Increasing Temperature on the Optimum Feed Rate for the Gas Pair SOZ-Nz. Rw = A Inches, L = 36 Inches, Rm/Rw = 0.5625 and Total Reflux . . . . . . . . . . aximum Separation Factors and Optimum Feed Rates for Various Ratios of the Rich Stream to the Feed Stream (R/F) for the Gas Pair SOZ-Nz, w = lSOOO RPM, Rw = A Inches, L = 36 Inches, Rm/Rw = 0.5625, and Plug Flow . . . . . . . . . . . . . . aximum Separation Factors for Various Centrifuge Lengths at Total Reflux for the Gas Pair 502- N2. Feed Rate = O. Ol69 scfm (optimum), w = 15000 RPM, Rw = A Inches, L = 36 Inches, Rm /RW = 0.5625 and Plug Flow . . . . . . . . . . . . he Power Required to Simply Rotate the Gas Being Fed to a Countercurrent Centrifuge for Different Flow Intersections: Feed Rate = 0.02 scfm, Radius = A Inches, m = lOOOO RPM, Plug Type Flow and Total Reflux . . . . . . . . . . . Page lOO lOl l02 108 11A 117 IZI l27 Simple Power Requirements, Entropy Losses and Resulting Efficiencies for the Separation of the Gas Pair SOZ-Nz at Various Ratios of the Rich Stream to the Feed Stream (R/F). Feed Rate = 0.02 scfm, Rw = A Inches, Rm/RW = 0.5625, w = lOOOO RPM and Plug Type Flow . Operating Variables and Their Effect on the Maximum Separation Factor and the Optimum Feed Rate for the Separation of a Binary Gas Mixture in a Counter- current Rectifying Centrifuge Maximum Separation Factors and Optimum Feed Rates for Various Ratios of the Rich to the Feed Stream (R/F) for the Gas Pairs SOz-Nz and SOz-Hz. Rw = A Inches, Type Flow . w = 150000 RPM, L = 36 Inches, Rm/Rw = 0.5625, Plug Maximum Separation Factors for the Gas Pairs SOZ-Nz and $02-H2 for Various Ratios of the Rich to the Feed Stream. and Plug Type Flow . Data Data Data ‘Data Data Data Data Data Data Data Data Data Data Data (0 = ISOOO RPM, Rw = A Inches, Used to Construct Used Used Used Used Used Used Used Used Used Used Used Used Used t0 t0 t0 t0 t0 to to t0 t0 to to t0 to Construct Construct Construct Construct Construct Construct Construct Construct Construct Construct Construct Construct Construct Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Rm/Rw = 0.5625, 10 . ll 13 1A l5 l6 . 17 Page l29 133 I35 138 1A8 1A9 ]50 ISI l53 l55 155 156 l57 l59 159 160 161 162 Data Used to Construct Figure l8 Data Used to Construct Figure 19 Data Used to Construct Figure 20 Data Used to Construct Figure 21 Calculation of the Diffusivities of the Gas Pairs SOZ'NZ and SOZ-HZ Finite Difference Approximations Used for the Various Partial Derivatives Data Required by the FORTRAN Program USTEADY Subprograms Associated With PrOgram NCENTRI Data Required by the FORTRAN Program NCENTRI Operating Parameters Required by Program CENTRI Subprograms Associated With Program CENTRI A Comparison of Experimental Separations Observed by Beams (A) and Those Computed by CENTRI for the Enrichment of Uranium . . . . . . viii Page 163 16A 165 166 169 17A 176 I97 l99 237 238 2A2 LIST OF FIGURES Page Wall to Axis Pressure Ratios Minus l Versus Peripheral Speed for Various Molecular Weights Between 2 and 100 . . . . . . . . . . . . . . . . . . l3 Wall to Axis Pressure Ratios Versus Peripheral Speed for Various Molecular Weights Between 100 and A00 . . IA Effective Molecular Weights Versus Actual Molecular Weights at Temperatures Between 0 and 300°F . . . . l6 IY(RW) - y(O)]/y(0) Versus y(0) for Values of a . . . . 21 Mole Fraction Profiles Developed in a Simple Centrifuge With a Peripheral Speed of 500 ft/sec for the Gas Pairs: SOZ-HZ, SOZ-NZ, and UF6 (235, 238 Isotopes) at 70°F . . . . . . . . . . . . . . . . 2A 0 Versus k for a Simple Centrifuge Containing a Stationary Center Pipe . . . . . . . . . . . 31 Mole Fraction Profiles for SO , y, Versus Dimensionless Radius, r/Rw (RW = A Inchesi, at Various Times in the Unsteady State Operation of a Simple Centri— fuge Containing $02— H2 at 70°F. RPM: 20000 . . . . A2 [y(axis) - 0. 002]/[ye (axis) — 0.002] Versus Time in the Unsteady State Simple Centrifuge with a Radius of A Inches . . . . . . . . . . . . . A5 The Cross Section of a Rotating Cylinder Illustrating the Radial and Circumferential Stresses, Sr and 59, Acting on a Small Element - - - - . 55 50(m ma O)/(D/gc) Versus Peripheral Speed, so, for k Values Of L90 0.95, and I.OO . . . . . . . . . . 60 Critical Elastic Modulus Over Density, E/(O/QC) VefSUS Peripheral Speed, so, for Length to Diameter Ratlos, 68 L/DO of 3, A, 5, and 6 lure Countercurrent Rectifying Centrifuge Illustrating Inner and Outer Axial Flows and Feed and Product Streams Separation Factor, a, Versus Feed Rate (scfm) for SOZ-Nz With Laminar Velocity Profiles, w = IOOOO RPM, Rw = A Inches, Rm/Rw = 0.5625, L = l8 and 36 Inches, and Total Reflux . . . . . . . . . . . Separation Factor, 0, Versus Feed Rate (scfm) for SOZ—NZ With Plug Type Flow, w = IOOOO RPM, RW = A Inches, Rm/RW = 0.5625, L = l8 and 36 Inches and Total Reflux . . . . . . . . . . . . . Separation Factor, 0, Versus Feed Rate (scfm) for SOZ-NZ With Laminar Velocity Profiles, w = 20000 RPM, Rw = A Inches, Rm/Rw = 0.5625, L = l8 Inches, and Total Reflux . . . . . . . . . . . . . . Separation Factor, 0, Versus Feed Rate (scfm) for SOZ-NZ With Plug Type Flow, w = 20000 RPM, Rw = A Inches, Rm/Rw = 0.5625, L = I8 Inches and Total Reflux Assumed Laminar and Plug Flow Velocity Profiles for a Feed Rate of 0.0125 scfm in a Countercurrent Recti- fying Centrifuge With Rm = 2.25 Inches, RW = A.O Inches, w = 10000 RPM and Total Reflux . Separation Factor, 0, Versus Feed Rate, scfm, for SOZ-HZ, With Plug Type Flow, w = IOOOO RPM, Rw = A Inches, Rm/Rw = 0.5625, L = 36 Inches and Total Reflux Separation Factors, 0, and Optimum Feed Rates Versus the Ratio of the Radius of the Flow Intersection to the Centrifuge Radius, Rm/RW, for SOZ-NZ With Plug Type Flow, w = 10000 RPM, Rw = A Inches, L = 36 Inches and Total Reflux . . . . . . . . . . Maximum Separation Factors Versus Centrifuge Rotational Speed for the Gas Pair SOZ-NZ at 70 and 300°F. Rw = A Inches, L = 36 Inches, Rm/Rw = 0.5625, PIUQ Type Flow and Total Reflux . . . . . . . . Separation Factors Based on the Rich and Lean Streams Versus Feed Rate for Various Ratios of the R1ch Stream to the Feed Stream for the Gas Pair S02-N2. w = 15000 RPM, Rw = A Inches, L = 36 Inches, Rm/RW ~05625 and Plug Type Flow . . . . . . Page 95 96 97 98 10A 107 ll2 Il8 l25 ire Page Optimum Feed Rate Versus Diffusion Coefficient in a Countercurrent Rectifying Centrifuge at 70°F With Rm/Rw = 0.5625, Plug Type Flow and Total Reflux . . . I37 A Schematic of the Zippe Gas Centrifuge . . . . . . 27A Schematic of the Experimental Centrifuge . . . . . . 276 Experimental Centrifuge and Auxiliaries . . . . . . 281 Flow Chart of the Experimental Apparatus . . . . . . 282 xi LIST OF APPENDICES X QUATIONS AND CALCULATED DATA USED TO CONSTRUCT ALL FIGURES . . IFFUSIVITIES FOR THE GAS PAIRS 502- N2, 502- Hz AND UF6 (235,238 ISOTOPES) . . UMERICAL METHOD AND FORTRAN PROGRAM USED TO ANALYZE THE UNSTEADY STATE SIMPLE CENTRIFUGE UMERICAL METHOD AND FORTRAN PROGRAM USED TO ANALYZE THE COUNTERCURRENT RECTIFYING CENTRIFUGE ORTRAN PROGRAM USED TO EVALUATE THE APPROXIMATE ANALYTICAL SOLUTION OF THE COUNTERCURRENT CENTRIFUGE . . ESCRIPTION OF EXPERIMENTAL EQUIPMENT AND ANALYSIS OF ITS PERFORMANCE Page 1A6 I67 I70 I92 23A 27l CHAPTER I INTRODUCTION A. Gas Separations in a Gravitational Field The change in composition of the atmosphere with altitude is known effect of the Earth's gravitational field. The effect acceleration of gravity, 9, is to set up a force, g/gc, per iss. Under the influence of this force, a pressure gradient 1blished for each species according to _ _ .éL 5h pi gC ’ II the partial pressure of Species i, the concentration of species i, and 3" II the height. 9 the ideal gas law and assuming constant temperature and t g/gc, this equation may be integrated giving the ratio partial pressure of gas Species i at any height to that at level, i.e. 9 Pi(h) ~MWigh W07: exp ("m—3:) ’ E II the molecular weight of species i, 30 II the gas constant, and .4 II the absolute temperature. 1 xample, at an altitude of l0,000 ft, assuming a constant tempera- of A0°F, the ratio of the partial pressure of oxygen to that at d level is calculated to be 0.66]. Since the ratio of the partial pressures is dependent on the ular weight of the species, the ratios of the partial pressures fferent species will decrease at different rates. This is illus- d by taking the ratio of the partial pressures of Species i and j height, h, P,(h) P,(0) -(MWi-MWj)gh leh5 = leoi eXp [ RT gC I e ratio of the partial pressure of oxygen to nitrogen is taken as at ground level, then at 10,000 ft, assuming a constant tempera- of AO°F, the ratio reduces to 0.237. In l9l9, Lindeman and Aston (I) suggested that the gas ations caused by the Earth's gravitational field could be dupli- by using a centrifugal field. That is, for separations in a ifugal field the gravitation energy term, gh/gc, could be ced by the equivalent centrifugal energy term, (erZ/ch), w is the rotational speed, and r is the radius of rotation. ating the two energy terms it can be shown that for a radius inches a rotational speed of 22980 RPM is required to create ect identical to that caused by the Earth's gravitational field altitude of 10,000 ft. B. Centrifuge Types One year after the suggestion by Lindeman and Aston, the first ication appeared marking thebeginning of research in the area of centrifugation. The first experiments involving centrifugation directed towards realizing the equilibrium separation as predicted he above equation. However, by 19AO, with several researchers ng demonstrated that the equilibrium separation could be realized, asis was turned to gas centrifuge designs which could be operated inuously separating, to some degree, a gas mixture. The idea of an evaporative centrifuge was introduced by iken (2) in 1922 as a means of realizing the equilibrium separation. type of centrifuge is a very short device (length to diameter 0 ~ O.l) in which a small amount of liquid containing different tile species is placed. As the centrifuge is brought up to ting Speed, the liquid forms a layer at the periphery. Vapor is ed from the centrifuge very slowly via a hollow shaft at the axis. ng as the vapor is drawn off very slowly, an equilibrium condition tablished throughout the vapor in the centrifuge. The vapor that is extracted is supplied by the evaporation of liquid at the periphery which must diffuse along the centrifuge 5 against the centrifugal field. The result is that the vapor off the axis is richer in the lighter species and the remaining Id is richer in the heavier species than the original liquid. er work along these lines was done by Beams and Skarstrom (3) in _,.,_. The idea of the cream separator as applied to gas mixture ‘ations resulted in the concept of a flow-through centrifuge. In type of centrifuge design a gas mixture enters the cylinder at and via a hollow shaft. As the gas traverses the length of the :e (length to diameter ratio ~ 5) the equilibrium composition ile deveIOps. At the other end, two streams are taken off, one the periphery (rich stream) and one near the axis (lean stream). on such devices has been reported by Beams (A), Groth (5), and emaker (6) among others. The third and final centrifuge type to be mentioned here is countercurrent centrifuge. In this centrifuge design two concentric streams flow in opposite axial directions. As a result the inner am becomes richer in the lighter species near its exit and the outer am becomes richer in the heavier Species near its exit. Since pposing flow continually upsets the establishment of an equilibrium sition profile, separations greater than equilibrium can be ved. Work on the development of the countercurrent centrifuge has reported by Beams (A), Groth (5), Kistemaker (6), and Zippe (7), others. The centrifuge used by Beams was the largest reported 9 a length of II 2/3 ft, a radius of 3.75 inches and rotating 000 RPM. While the performance of the evaporative and the flow-through ifuges have a definite place in the overall understanding of gas ifugation, only the countercurrent centrifuge has the potential used as a production device. That is to say, the "best'I either the evaporative or flow-through centrifuge can ever Ve is an equilibrium separation. On the other hand, the rcurrent centrifuge has the capability of yielding separations r than equilibrium. This conclusion is supported in the litera- where a major portion of the work that has been done in fuge research has been done using the countercurrent centrifuge eparations reported exceeding corresponding equilibrium values. C. Scope of Past Work The suggestion by Lindeman and Aston that centrifugation was a Ile means of separating gas mixtures reflected their desire to I new technique to separate mixtures of isotopes. The simple 1ical nature and higher expected separations of the gas centrifuge Ipared to complex diffusion processes made it a definite avenue 'estigation for the difficult problem of isotope separation. Con- Itly, almost all of the research which has been published deals :he separation of isotopes of such elements as Fluorine, Chlorine, 1e, Krypton, Xenon, Selenium and Uranium. The few gas mixtures ined, that were not isotopes, were used merely as test gases and e basis of the experimental work. Until l95l, all of the investigations of the gas centrifuge xperimental in nature with the published results consisting of ntrifuge design, operating conditions and the separations ed. Usually the results were compared to those predicted by brium calculations. In 1951, however, the first and only avail- heoretical analysis of gas centrifugation was presented by (8). The analysis was contained in a collection of theories vailable for various processes which could be used in the ge-scale separation of uranium isotopes. Equations resulting from en's analysis of the countercurrent centrifuge, although containing eral approximations, were used to predict (with reasonable success) erimental results in future published experimental works. A few estigators modified Cohen's equations slightly, while others lained differences between their experimental results and those dicted by assumptions that were incorporated in Cohen's analysis. At this point in time, 1951, with the demand for fissionable nium beginning to grow very quickly, investigations involving the .centrifuge were being entirely directed toward its applicability an efficient means of enriching Uranium. In this country, much such work was, and still is, Sponsored and funded by the Atomic rgy Commission. The results of this change in emphasis and moti- ion was a steady decline in the number of publications available general review, varying from a few publications of demonstration e work at the University of Virginia in the late l950$ to the early OS, to no publications currently. 0. Scope of This Work Even though research on the subject began in the 19205, still sing from the available literature is a general analysis of the centrifuge. An analysis in which a thorough study of the counter- rent gas centrifuge is made, detailing important operating variables, e restrictions, allowable flow rates, and any other fundamental itations that may exist, is needed. With such an analysis, a stion such as, ”May the gas centrifuge be considered as a viable i to separate sulfur dioxide from flue gas?” could be answered :tly. However, as stated above, information to answer such a :ion is simply not available in the unclassified literature. 2, it was with this goal in mind that the analyses presented in work were made. Throughout the body of this work important points are illus- :d by using the gas pairs SO SO 'H and UF6 (235 and 238 2'N2’ 2 2 Ipes). The sulfur dioxide-hydrogen combination was chosen reflect- he current interest in removing $02 from stack gases. The r dioxide—hydrogen combination was used as a variation to give a Iixture with a very low density at standard conditions (at low oncentrations), large diffusion coefficient and large molecular t difference. The gasified Uranium isotopes (as UF6)’ on the hand, represent an opposite extreme having a relatively large ty at standard conditions, low self-diffusion coefficient and molecular weight difference. The use of any additional gas for illustration was found unnecessary, since the analysis con- , and the use of the above three gas pairs concurred, that no I properties exist for particular gas pairs. Paralleling this work from its beginning was an experimental m which was initiated to demonstrate the performance of a (7) type gas centrifuge, when used to separate the gas pair at one atmosphere and 70°F. The centrifuge constructed was hes long and 8 inches in diameter, operating at Speeds up to RPM. The experimental program, however, was terminated after ed efforts failed to yield any measurable separations. Faced this situation, an even more exhaustive effort was undertaken derstand the gas centrifuge in order to determine whether the design was in fact simply not applicable to the situation and/or er there existed fundamental reasons for not observing a measurable ration. It was found that the information and conclusions contained ughout the body of this work, coupled with basic problems associ- with the ”soundness“ of the Zippe gas centrifuge design, indeed ided more than sufficient reasons to expect that no measurable rations would be seen. A complete description and analysis of experimental program may be found in Appendix F. E. The Organization of the Information Provided in This Work The analysis of gas centrifugation put forth in the body of work is contained in six chapters. The analysis begins in er II with the simple equilibrium centrifuge. Although such a ifuge is defined to be simply a sealed rotating cylinder containing mixture, the analysis illustrates the magnitude of pressure ients and equilibrium separations, setting the ground work neces- for the analysis of production centrifuges. In Chapter III the e equilibrium centrifuge is analyzed with respect to the times ired for the equilibrium composition profiles to develop. With id of this section a feel for the expected performance and limi- ns of the gas centrifuge begin to develop. Furthermore, it was Iished that while the degree of separation varies with different airs, all gas pairs behave similarly in the gas centrifuge. In Chapter IV attention is turned to actual limitations of centrifuge itself. That is, how fast may the centrifuge be rotated out bursting and/or entering into a critical mode of vibration h dould destroy it? These analyses gave the necessary perspective 1 considering production centrifuges which require not only high ational speeds, but length to diameter ratios greater than four. With all the ground work laid in the previous sections, the Iysis of the countercurrent production centrifuge is undertaken in >ter V. The necessary equations are developed and a numerical and ‘oximate analytical solution are developed. Conclusions that may Irawn immediately regarding the solutions are put forth and ained. Having developed the necessary equations and drawn several iminary conclusions, the countercurrent rectifying centrifuge pduces a product stream richer in the heavy species) is analyzed Ireat detail in Section VI. Changes in performance with various IgeS in operating conditions are illustrated and explained. Chapter VII concludes this work by illustrating perhaps the most significant aspects limiting the use of a gas centrifuge. :ral example calculations are used to illustrate the magnitude of e limitations. Also included in this section is a list of the lts contained in this work. However, this list is not meant to omplete, since the entire text contains numerous pertinent points examples which lose their significance when taken out of context Simplified to enter into a table or list. l0 Appendices following the text contain such things as calculated sed to construct figures in the text and complete descriptions eral computer programs that may be used to analyze, with great , the expected performance of the gas centrifuge. Also included appendix is a description of the experimental program which eled this work, as referenced above. CHAPTER H STEADY STATE SIMPLE GAS CENTRIFUGE THEORY A. Pressure Gradients, Pure Gases A simple gas centrifuge shall be defined as a closed rotating ler containing an isothermal pure gas or gas mixture. The gas as uniformly at the same speed as the cylinder, i.e., w rad/sec, 15 no radial or axial motions. To prevent radial flow a pressure :nt is established in the radial direction balancing the centrifu- . 2 . Irce per unIt mass (m r/gc) created by the rotatIon of the gas. 5, 2 w r 9c O) .0 3 DI~ Q) '1 0 = mass density (lbm/ft3), P = pressure (lbf/ftz), and c = gravitational constant, 32.17A lbm/lbf ft/sec2 Using the ideal gas law and assuming a pure gas with molecular MW, the pressure gradient equation can be written as 1. OP _ MWer ST‘ RT gC ’ .0 upon integration gives a P(O) is the pressure at the axis, r = 0. 1e cylinder wall (r = RW) the quantity mRW represents the inner peripheral speed of the cylinder, s(ft/sec), i.e., MW 52) P(wall)/P(axis) = exp (ZRTg c Figures I and 2 illustrate values of the ratio, P(walI)/P(axis), Inable in a simple gas centrifuge operating at different peripheral Is and containing pure gases of different molecular weights at Data used to construct the Figures are found in Appendix A. in be seen from the equation for P(wall)/P(axis), the limiting : of the ratio, as 5 and/or MW go to zero, is one. Hence, in order ‘Stinguish small changes from one, Figure 1 contains a plot of l)/P(axis) - l, due to the small molecular weights (2-100), :as Figure 2 contains a plot of simply P(walI)/P(axis) with :ular weights between ICC and A00. To illustrate the use of Figures l and 2, consider Sulfur de (MW = 6A) in a simple gas centrifuge operating at 70°F with a Iheral speed of 500 ft/sec. From Figure I, P(waII)/P(axis) - l is as .36 (actual value = .3547)- Since the ratio MW/T appears in the pressure equation, a cor- on to the true molecular weight can be used to determine pressure ’5» P(Wall)/P(axis), at temperatures other than those at 70°F I in Figures 1 and 2. That is, OI I g 3 5 : : i I l I n 1 200 A00 600 800 IOOO IZOO IAOO Peripheral speed, s(ft/sec) Figure I.--Wall to Axis Pressure Ratios Minus I Versus Peripheral for Various Molecular Weights Between 2 and 100. II II II II II wN— OOU1 OOO lAOO 200 I400 600 800 1000 1200 Peripheral speed , s (ft/sec) Figure 2.--Wall to Axis Pressure Ratios Versus Peripheral Speed rious Molecular Weights Between I00 and A00. IS MWe = thrue [530/(A60 + °F)], re MWe is the effective molecular weight to be used in place of true molecular weight. Figure 3 contains a plot of the corrected ecular weight versus the true molecular weight with temperature ) as a parameter. Data used to construct the Figure are found in endix A. For example, had the above illustration been worked at °F, a MWe = AA.6 is read from Figure 3 and from Figure l, P(walI)/ xis) - l = .2A (actual value = .2358). The fact that increasing temperature can be thought of as decreasing the effective magni- e of the molecular weight of a gas species will serve as a useful cept in the analysis of gas centrifugation. B. Pressure Gradients, Gas Mixtures If a perfect mixture of gas species is placed in a Simple centrifuge a pressure gradient will be established for each specie For example, for ording to the equation developed in Section A. cies i we have integrating as before gives 2 Pi(r) MWIer W = EXP (Tm—9:) total pressure at any point r is then defined by 0°F AO°F 70°F I00°F IAO°F 180°F 220°r 260°F 300°r —‘3'LO-nm0.no-m , I I I 1.0 '00 200 300 Aoo Actual Molecular Weight, MWa Figure 3.--Effective Molecular Weights Versus Actual Molecular ths at Temperatures Between 0 and 300°F. were n = number of species. Due to the different molecular weights of each species, the 1dividual partial pressures increase at different rates proceeding ’om the axis to the wall. Thus, the ratio of species 1 and j at the (is will differ from their ratios at any other position. This fact 5 illustrated by taking the ratios of the partial pressures of necies i and j, i.e., Pi(r) P.(0) (MWi-MW.)w2r2 _ I PJ.(r) ‘ leoi °X° ' 2RT gC ' 1us, if MWi is greater than MW., the gas mixture near the wall will i more concentrated in the heavier (ith) species. With this observation in mind, all further analyses embodied I this work will be restricted to binary gas mixtures. This is not >indicate that the concepts developed are restricted to only binary xtures, but was done simply to simplify the equations and simplify 1e methods by which they may be analyzed. C. Mole Fraction Distributions and Simple Separation Factors In a binary gas mixture (species I and 2) the mole fraction (y) ' the heavy species (taken as species 1) in a simple gas centrifuge In be written in terms of the partial pressures of the species, 1.e., l8 Mwlwzrz PI(°' exp ( 2RT gc y(r) = , 2r2 MW (13er I (r) = y P2(0) (MWZ-MWI)w2r2 ‘+ mew [WT—l y(O) is the mole fraction at the axis, then P2(0) = [l-y(0)l P(O), and P](0) = Y(0) P(0)- Istitution of this information into the mole fraction equation and rranging gives 2 y(O) eXP (Ag-l y(r) = 2 , y(O) [exp (fig-9 - I] + I 2 A _ (MWI-MW2)w = AMW wz RT gC RT 9c 5 expression allows one to compute the mole fraction profile in a ple gas centrifuge given A and y(O). l9 A separation factor for the simple gas centrifuge may be 'ined as the ratio of species I to 2 at the wall to the ratio of :cies l to 2 at the axis, i.e., y(RW) ]_y(0) a — . . l-y R lel in terms of the mole fraction distribution given above, AR2 d = ]_y(0) y(0) em (——2 I , ”of AR 2 AR 2 on> [exp 1,—W>—11 + 1 - y exp 1,—W> ch simplifies to ARW2 a — exp ( 2 ) expression for y(Rw) can now be written in terms of a, 1 e , _ (DA/(0) y(Rw) _ leI ld—II + I ' Since the expression for u is identical to that developed for ratio P(wall)/P(axis) for a pure gas, except that molecular weight ference appears in the exponential instead of an actual molecular ght, Figures 1 and 2 can be used to estimate values of d by just stituting molecular weight difference. For example, conSIder a ture of Hydrogen (MW=2) and Sulfur Dioxide (MW=6A) at 70°F In a ple centrifuge with a peripheral speed of 500 ft/sec. From FIgure I 20 with MW=62, a - l is found to be .3A (actual value = .3Al9). The value of a - l at a different temperature, say 300°F, can be found by first using Figure 3 which gives an effective AMW=A3.2 and then using Figure I giving a - I = .22 (actual value = .2198). An important aspect to be noted in the expression for a is that it is a function of molecular weight difference, not the ratio of the molecular weights or the molecular weight difference divided by the sum of molecular weights as in diffusion processes. This gives the gas centrifuge an important advantage for separating heavy gases, hence the interest in the field of gas centrifugation for the separation of Uranium isotopes (5, 6, 7, and 8) (AMW=3) when gasified as Uranium Hexafloride (MW=352l- Although a is a fundamental quantity depicting the separating potential of a gas pair, a more tangible quantity is the increase of Y(Rw) over the value of y(O) as a function of a, i.e., - l w _ 01 y(o) ' W0) (01-1) +1 Figure A contains a plot of [y(RW) - y(O)]/y(O) versus y(O) for various values of a. Data used to construct Figure A are found in Appendix A. AS can be seen in Figure A, the greatest increase in the value of YiRw) over the value of y(O) occurs at low concentrations [y(O) 5-0.3] regardless of the value of q. The curve in Figure A for a very large value of 0 corresponds to y(Rw) x1 for all values of y(O). 21 A.0~ a. 01 = very large a ‘b. It: 20.0 C. LL: 5.0 d. It: 2.0 I) e. (1: I.5 3.0 " f. 11.: 1.2 c 2.0 " |.O .. d e f -0 I 1 I i I t t .—= 0.0 01 02 O3 O.A 0.5 06 O7 08 0.9 I 0 Figure A.--[y(Rw) - y(O)]/y(0) Versus y(O) for Values of u. 22 D. Axis Composition in a Simple Centrifuge In Section C a relationship was given between y(Rw) and y(O) as a function of a, a quantity which only depends on the molecular weight difference of the gas pair, temperature, and the peripheral speed. This expression, however, lacks usability when applied to a real situation. For example, if a simple gas centrifuge is charged with a gas of known composition, yf, and then rotated until the steady state mole fraction distribution has developed, the mole fraction profile cannot be computed. Knowing the value of y(O), however, would allow the calculation of the mole fraction profile [giving y(Rw)]. It should be pointed out that if y(Rw) was known instead of y(O) the profile could also be computed; however, analytically computing y(O) is an easier task. If B moles of gas had been used to charge the simple gas centrifuge, then the following relationship representing conservation of mass must be satisfied: R R W B = 2nL (JW pIrdr + ZnL L) pzrdr , where 0] 80d 92 are the molar densities of species I and 2 respectively, in the mixture. 0r, using the ideal gas law and the pressure ratio equations developed previously, 2 2 211L P (0) R Q r 2nL P2(O) R er w l w °=—‘R‘T__“(0 ”pi 2'r°r+ RT '0 exp' 2 ) rdr , 23 where Q] = 9 Q2 = 3 and L = length of the simple centrifuge. Also, the conservation of species I must be maintained, thus 2nt P,(0) R Q r2 _ w l ny — RT () exp ( 2 ) rdr . Integrating and solving for yf gives P](O)Iexp(QlR£)-l]/Ql y = 2 . f P](O)[exp(Q]RIi/2)-ll/Q]'tP2(O)[eXp(Q2RW/2)'ll/Q2 Substituting for P](O) and P2(O) in terms of P(O) and y(O) gives the following expression for y(O): Ql Y(O) = Q i l + y1c (El-C ' I) 2 where 2 exp (02Rw /2) - l C = exp (QIRWZ/Z) * 1 Using this expression, Figure 5 was constructed showing the ole fraction profiles developed in a simple centrifuge for the following .uoom um AmmaouOmb wmm .mmmv on: cam NZINOm .NIINom “mc_md mmo o u _mcoca_cod m Lu_3 om:m_cucou o_oE_m m c_ vooo_m>mo mm__m . c to; oom\ue oo 2A Qua C0_uumum m_O 11. m w0 Umwam 3m\c .m:_vmc mmo_c0_mcme_o z m oc:m_u mco ”to b ““0 3 who b mno :.0 mco - . r n n . . q . . . n . . . . N o _.o q u q . q . u u n u 0.0 q IJ N—.O Sr it 9.0 D II‘ o .1 9.0 .W t m. a om.o m e e. .. m. ...U . 1A T a o x . m a 7.. mmw: co_uomce 0.05 n > .Amoa000m_ wmm .mmmv mu: .6 .. NN.o mom co_uumcm 0.05 n > .NZINom .o .. Now co_uumcm o_OE n > .NIINOm .n t W a . 1U MN.O N00 0 N > co_u_moqe00 omcmcu m I. .11 AU . I I 1 25 three gas paIrs: SOZ-Hz, SOZ-N2 and UF6 The composition of the charge (yf) in the SOZ-H2 and $02-N2 pairs is (235, 238 isotopes) at 70°F. 0.002 mole fraction $02 (a typical flue gas composition), whereas yf is 0.002 for the UF6 (235 isotope). Data used to construct this figure are found in Appendix A. The centrifuge operating speed is 500 ft/sec. Table 1 contains a list of some observations taken from Figure 5 for comparison. As expected (due to the large AMW) the SOZ-H2 mixture separates best as indicated by the separation factors. Also, due to the radial pressure gradient, the point at which the mole fraction profile crosses the value of y1: does not occur at the equal area position (5 = 353.5 ft/sec). Instead, the crossing point is 358 ft/sec for SOZ-HZ, 360 ft/sec for SOZ-NZ, and 398 ft/sec for UF6 (235, 238 isotopes). The large molecular weight of the UF6 (larger pressure gradient) pushes the crossing point closest to the periphery. Also worth noting is that-the values of y(O) and y(RW) are not equal distances on either side of yf, and the mole fraction profile becomes flat as TABLE I.--Simple Centrifuge Separations of the Gas Pairs: SO -H 50 -N F (235, 238 isotopes) at 70°F. Centrifuge 2 16$peed = 500 ft/sec. , u 2 Peripfiera yf-y(0) y(Rwl-yf Gas Pair yf I y(O) I y(RW) ——:a:—-—- -—-:§:—-— a SOZ'H2 0.002 .00172 .00231 .IAO .I55 1.3AI9 SOZ‘NZ 0.002 .00183 .00217 .085 .085 1.1862 UF6 0.002 .002018 .001989 -.0090 -.0053 .9859 (235, 238) 26 the axis approached. This situation exists since as the axis is approached the centrifugal force diminishes and the concentration decreases simply to make up for what has moved to the wall. 0n theotherhand, near the periphery where the difference in the centrifu- gal forces acting on the heavy and light species is growing farther apart, the mole fraction profile increases quite rapidly. E. A Simple Centrifuge Containing a Stationary Pipe Centered Along the Axis Before leaving this chapter it is worth mentioning that the experimental gas centrifuge enrichment of the Uranium isotopes has been accomplished by using centrifuges containing no internals and to a lesser extent by centrifuges containing a stationary pipe cen- tered along the axis. Such work had been done by Zippe (7) and Groth (5). The question then at this point is what effect does the stationary center pipe have on the pressure ratios and separation factors found for the simple centrifuge containing no center pipe? To answer this question the following assumptions must be made: I. The centrifuge is sufficiently long so that end effects may be neglected. 2. The radial pressure changes are small enough so that viscosity changes can be neglected. This is a good assumption at pressures around 1 atmosphere and lower. 3. The flow between the cylinders is laminar. 0f the assumptions, number 3 requires the closest analysis. Since, if the tangential flow is not laminar, the mixing associated with turbulent flow would destroy the desired concentration profiles. 27 Intuitively, because of the centrifugal forces involved, one may conclude that laminar flow is favored. That is, the gas near the wall will not tend to move inward because the centrifugal forces are greater near the wall than near the axis. Likewise, gas near the axis will not tend to move outward because of the higher centrifugal forces on the particles it would have to replace. Hence, shear rates higher than those encountered in normal gas flows in a pipe are expected to be possible. H. Schlichting (9) determined the transition (changing from laminar to turbulent flow) Reynold's number, wOszp/u, giving the transition RPM, for tangential flow between a stationary inner cylinder (pipe) and a rotating outer cylinder. From the results of his work, it was determined that for a ratio of pipe to centrifuge diameter of 1/16, the transition Reynold's number is 20x10°, while for a ratio of 1/2 the transition Reynold's number is 16x10°. Table 2 contains all the parameters and their values required to compute the transition RPMs of an 8-inch diameter centrifuge containing either SOZ-N2 or SOZ-H2 gas mixtures. Since in practice one would expect the center pipe to be small in diameter as compared to the centrifuge, a center pipe diameter of 1/2 inch is used in Table 2. AS can be seen from Table 2, the transition RPMs at 70°F and 1 atmosphere are rather small (~2790 and ~2080) for the gas pair 50 -N with SO2 mole fractions of 0.002 and 0.2. The reduction in 2 2 the transition RPM going from an $02 mole fraction of 0.002 to 0.2 reflects the change in density of the gas (viscosity only changes by 6.3% whereas the density changes 25.5%). This is especially true for 28 TABLE 2.--Predicted TranSition RPMs for the Gas Pairs so -N and $02-H2 in a Simple Centrifuge Containing a Stationary Center Pipe. Value Parameter SOZ-N2 $02-H2 ch (center pipe), inches l/A l/A R , inches A A w 2 w R p ( ° w ) trans 20x10“ 20x10° T. OR 530 530 P (average), psia lA.696 lA.696 Mole fraction SO2 0.002 0.2 0.002 0.2 p, lbm/ft3 0.0725 0.091 0.0055 0.0372 U, Cp 0.0175 0.016A 0.0090 0.0096 mo (trans), RPM 2788 2082 18900 2980 the gas pair S02 and 2980 for $02 change of 576%). -H2 where the transition Reynold's numbers are ~18900 mole fractions of 0.002 and 0.2, respectively (density The transition RPMs, however, could be Increased by either increasing the temperature or decreasing the pressure. The latter is the case for Uranium isotope separation where low pressures (approximately l/IO atmosphere or less) are necessary to keep the UF6 vaporized for feed Stock and at the centrifuge wall. At normal pressures (near 1 atm) and the high rotational speeds necessary in gas centrifugation, however, the operation of a centrifuge with a center Pipe does not seem very promising due to the problem of turbulence. 29 Nevertheless, with assumptions 1, 2 and 3 the equation of motion for the tangential flow of the gas in the annular space can be written as d 1 d 2 _ 217'? d‘r‘lrwll-O where w = O at r = R (center pipe), and w = m at r = R . This CP 0 w equation can be integrated, giving w R where C = (k?IYk , and k = Rep/Rw' Substituting this expression for w in the pressure equation derived in Chapter II-A gives (for a pure gas) 2 2 ._L ch = MW 02 (k Rw _ _g + 1 ) r PC dr RT gC rA r2 szwz where P = P (R ) at r = kR . c c cp w Integrating gives the following expression for the ratio of the pressure at the wall to that at the center pipe, i.e., 2 _ AR (1/k2 — k2 + A In k) (1/k2 + k2 - 2) ’ where A = MWwOZ/RT gC as before. 30 Remembering that without a center pipe In P(Rw)/P(O) = allows the following comparative expression to be written: In PC(RW)/PC(RCp) = e In P(RW)/P(O) , or PM PM6 CW =[ W] Pc(ch' P(Ol ’ where (I/k2 — k2 + A In k) (k2 + 1/I<2 — 2) The limit of 0 can easily be seen to be 1 and 0 as k approaches its limits of O and I. Since the above expression is valid for either species In a binary gas mixture, a relation can also be written for the separation factors, i.e., Figure 6 contains a plot of 0 Versus k. The calculated data used to construct the figure can be found in Appendix A- For a given simple centrifuge containing a center pipe (given k): the value Of 9 3i ~ Figure 6.--6 Versus k for a Simple Centrifuge Containing a Dtationary Center Pipe. 32 can be taken from Figure 6. This value of 6 can then be used to compute the pressure ratios and/or the simple separation factors by raising the corresponding value taken from Figure l or 2 for a simple centrifuge without a center pipe to the 8 power. CHAPTER ill UNSTEADY STATE SIMPLE GAS CENTRIFUGE THEORY A. Concentration and Pressure Diffusion Fluxes In Chapter II the steady state mole fraction profiles for a binary, perfect gas mixture were derived and analyzed from a pressure gradient approach. This approach made use of the fact that the centrifugal force, which is tending to pile all the gas up at the periphery, acts to a greater degree on the heavier species giving rise to the mole fraction profile. From a molecular point of View, however, what the steady state operation implies is a no net flux condition. That is, at any radial position r, the flux of say, species l, by diffusion is equal and Opposite to the flux of species l caused by the centrigual field tending to pile it up at the wall. In terms of transport properties, the sum of the terms describing ordinary concentration diffusion and pressure diffusion is zero. From Bird, Stewart, and Lightfoot, ILEEEEQLEJEEEEIEEIE (IQ), for a perfect gas mixture the one dimensional molar flux of species 1 for concentration is given by =—_21E_D r-B—y, Fly RT 12 Sr and the one dimensional molar flux of SPQCleS ' for pressure diffusion is given by 33 F DEED (L I. lp RT RT 12 MilI p where Fly and Flp are expressed in lb-mole/ftZ-sec, D12 = D21 is the diffusion coefficient, ftZ/sec, and V] is the partial molar volume of species l, ft3/lb-mole. Using the following relations: __..= ————-= —l-for a perfect gas mixture, l where p] = the mass density of species 1, and 2 %;-= pm r in the simple centrifuge, 9c where D = the mass density of the mixture, allows F1p to be written as F _-2_TTP_ MD wzrz (2H1) lp RT RT l2 9 D c l P P MWI Since p = RT'[Y MW] + (i—y) MWZ] and p1 = —E?—* , F1p may be further reduced to _ ZWP 2 _ lp ‘ ‘RT D12 Ar Y(' Y)’ (MWl - MW2)w where A ~ RT 9 In the steady state operation of the simple gas centrifuge, where the two diffusional fluxes are equal and opposite in sign, the foiiowing expression results: Integrating and using the fact that at r = O, y = y(O), gives y(r) = y(0) exp (ArZ/Z) y(O) [exp (ArZ/Zl-l] + l which is identical to the expression derived in Chapter II-C. B. The Partial Differential Equation forthe Unsteady State Operation of the Simple Centrifuge The analysis of the unsteady State operation of the simple centrifuge will provide insight as to how fast the equilibrium mole fraction profile develops, hence yielding information about the ability of the centrifugal forces to perform a separation. This simpler analysis, which is being carried out before the analysis of gas centrifuges with axial flows, will give a very good indication of the magnitude of the flows that may be expected and the centrifuge lengths required to perform a separation. Consider a simple centrifuge that has been filled with a perfect binary gas mixture (total pressure = P0) with composition yf (heavy species) and then quickly brought up to rotational speed. Certainly from a practical point of View, it would be impossible to bring the machine instantly up to speed nor would it be possible to 36 overcome the inertia of the gas (without perhaps the aid of radial baffles, etc.) so that angular velocity gradients did not develop. Nevertheless, the above hypothetical situation is proposed for the purpose of this analysis. The first thing to occur will be a rapid bulk radial flow of gas to the wall. When this flow essentially Stops, diffusional flow will establish the equilibrium mole fraction profile. Starting from this point in time (when the gas is still of uniform composition, yf) the unsteady state problem will be solved. By using the diffusional flux equations developed in Chapter III-A, the equation of continuity for the heavy species in the rotating cylinder is given by lo; 0) 27 (LY) flea {—30 [Arzyu—w — r gl1) = o. t "a? RT 12 ar At this point it is worthwhile to note that the product PDl2 can be taken as a constant. That is, in expressions for the diffusivity (binary mixture), resulting from kinetic theory and corresponding states arguments, at low pressures, the diffusivity was found to vary inversely with pressure to the first power. Such expressions are given by, among others, Hirschfelder, Bird and Spotz (ll). Taking the necessary partial derivatives and simplifying, the equation of continuity becomes 2 _ v- 2J-o. r 3y 8P _ _ _ _ 35+ y E+ DIZP [2 Ay(l y) + [(Ar(l 2y) 1 ._.... coo; 1‘< o) o; The boundary conditions are that there is no flux at the axis or the wall as expressed by the following equations: 37 at r = o, -1-= o , and at r = RN,%%-= Ar Y(i - y). The initial condition is that the mole fraction is uniform at the value Yf- As the equilibrium mole fraction profile develops, the pressure profile also changes due to changing centrifugal forces coupled with the fact that conservation of mass must be maintained. This gives rise to the partial derivative of pressure with respect to time. The pressure distribution can be computed from the pressure gradient equation developed in Chapter II-A with the mass density of the gas mixture used in place of a pure gas density. That is, 8P sz 5?-= RT 9c (AMW y + sz) r , and at r = 0, P = P(O). The pressure at the axis, P(O), may be computed by using the fact that the centrifuge contains a fixed amount of mass. Thus, the expression for P(O) can be written as B/L P(O) = , 2n RW fifo Prdr where B/L are the moles per foot in the centrifuge, i.e., B/L = 2N P /RT. 0 38 C. The Method of Solution Used to Solve the Partial Differential Equation for the Unsteady State Operation of the Simple Centrifuge Due to the coupling of P and y and hence noniinearity of the partial differential equations which had to be solved, a numerical solution was sought. The numerical approach chosen was to solve the partial differential equations implicitly. That is, all partial deriva- tives were approximated with fourth order finite difference formulas at the next increment in time, resulting in a set of simultaneous equations. To obtain the numerical solution a general FORTRAN program, UNSTEADY, aided by the scientific subroutine 0NED|AG to solve the simultaneous equations, was written. The program was constructed so that it could be used to analyze any gas pair in any size centrifuge operating at any rotational speed. A complete listing of the finite difference formulas used to approximate the various partial derivatives and a complete listing of the FORTRAN program and subprogram can be found in Appendix C. The following is a brief outline of the caicuiational steps used in the program: 1. Input operating conditions. 2. Divide the radius (Rw = A inches) into 2] equally spaced grid points. The time increments were chosen so the PDIZAT 2 0.5 ibf. Using more radial and/or time increments gave essentially no improvement in the solutions. 3. Compute the initial pressure profile using the uniform composition y . The axis pressure is such that the known moles per foot are maintained. 39 A. Compute the equilibrium mole fraction profile so that a comparison can be made to the actual mole fraction profile as a function of time. 5. Based on past information, estimate the values of P and y at the next position in time. This must be done because of the nonlinearity of the equations. 6. Compute the partials, BP/Bt, at each grid point. 7. Using finite difference approximations for the partials of y, an equation is written for each grid point at the next position in time. This results in 2] simultaneous equations which are solved for the new values of y. 8. Using the new values of y, the finite difference approxi- mations are written for the pressure equation at each grid point resulting in 2i simultaneous equations. The equa- tions are solved with the axis pressure arbitrarily taken at i.0. 9. Using the new values of P, the integration for the total moles present is performed which gives the axis pressure. The correct pressure profile can be obtained by multiply- ing all the values of P by the axis pressure. IO. Due to the nonlinear nature of the equations, which requires the estimation of P and y at a new position in time (step 5), statements 6 through 9 are executed twice during each incremental move ahead in time. More itera- tions did not significantly affect the results. il. Output results and increment time. If the time interval over which the calculations were desired is exceeded, calculations are complete. Otherwise, return to step 5. The program was executed on a Control Data Corporation 6500 digital COmputer and required approximately i5 seconds of central processor time to march iOO time increments. c I .( . 1+0 D. The Mole Fraction Profiles Developed in the Unsteady State Operation of the Simple Gas Centrifuge for the Three Gas Pairs: SOO-Nm SOZ-HZ, and us, (235, 238 isotopes) With the aid of the above equations the separation of the two gas pairs, SO -N2 and SOZ-HZ, was analyzed in the unsteady state simple 2 centrifuge at temperatures of 70° and 300°F with centrifuge RPMs of IOOOO and 20000. Also, a mixture of Uranium isotopes (as UF6) was analyzed at a temperature of 80°F and a centrifuge RPM of 20000. Table 3 contains a complete tabulation of the operating conditions and diffusivities used. Values of the diffusivities were either calcula- ted or found in the literature. All information regarding the diffusivities may be found in Appendix B. TABLE 3. Operating Conditions and Diffusivities Used in the Unsteady State Analysis of the Gas Pairs SOZ-NZ, SOZ-H2 and UF6 (235, 238 Isotopes). Gas Pairs <1 Charge pressure, psia lA.696 lA.696 Z-I Charge composition, y 0.002(S02) 0.002 (802) 0.002(U235) Temperatures, °F 70, 300 70, 300 80 Centrifuge radius, inches A.0 “-0 “-0 Centrifuge RPM 10000, 20000 10000, 20000 20000 Molecular weights 6A, 28 6A, 2 349, 352 Diffusivities, cmZ/sec @ 70°F and lA.696 psia OLI3A6 0;§§82 0.TT;6 @ 80°F and 2.l psia ___ @ 300°F and lA.696 psia 0.2590 0-9869 AI Figure 7 illustrates the changing mole fraction profiles with time for the gas pair SOZ-H2 at 70°F and a centrifuge rotation of 20000 RPM. The profiles given are for times of O, 3.6, 7.2 and IA.A seconds. Also included is the equilibrium mole fraction profile, i.e., the profile established after a long time period. The curves in Figure 7 illustrate the expected fact that the profile develops much quicker near the wall than near the axis. This fact is further illustrated by Tables A, 5 and 6 which contain values of (y-yf)/(ye-yf) at the axis and the wall for each of the pairs and all the operating TABLE A.--Values of (y-yf)/ye-yf) x IOO Versus Time at the Axis and the Wall for the Gas Pair SOz-N2 in the Unsteady State Simple Centrifuge, Radius = A inches. 70°F 70°F 300°F 300°F Time and IOOOO RPM and 20000 RPM and IOOOO RPM and 20000 RPM (sec) axis wall axis wall axis wall axis wall 9 8.7 39.9 9.7 39.2 16.6 52.5 17.8 52.0 18 I8.A 54.2 20.3 53.7 39.7 68.8 36.6 68.7 27 27.9 63.5 30.5 63.2 51.2 78.5 53.2 78.5 36 37.2 70.3 00.1 70.2 6A.A 8A.9 66.A 85.0 45 A5.9 75.5 99.0 75.5 79.5 89.2 76.1 89.0 5A 53.8 79.7 56.9 79.9 81.8 92.3 83.2 92.5 63 60.8 83.1 63.8 83.3 -- __ __ '- 72 66.9 85.9 69.7 86.2 -- __ ‘— '- 81 72.2 88.2 70.8 88.5 -— -- -- -- 90 76.7 90.l 79.0 90.A -- __ “ “ 99 80.5 91.7 82.6 92-0 " " A2 .oooom n 2am m >Umoumc: ecu :_ mme_h m:o_cm> um .Nom Lo» mo__¢0cm co_uomtu m_ozii.m ocsm_m NIiNom mc_c_mucou om:m_cu:mo o_QE_m m #0 co_umcmao bump : u say 3m\c .m:_pmx mmm_co_mcoE_o msmco> .> 3x\c .m:_nmt mmm_c0_mcoe_o o._ m.o w.o m.o 0.0 m.o :.o m. . . _ . . . . . . u u a . . . omtm_ >Lo> oom m.m oom o n u .. :N.o mN.o N .uaom um .Amosoc_ w_.o N.o A3 TABLE S.--Values of (y-yf)/(ye-yf) x IOO Versus Time at the Axis and the Wall for the Gas Pair SOZ-H2 in the Unsteady State Simple Centrifuge, Radius = A inches. 70°F 70°F 300°F 300°F Time and IOOOO RPM and 20000 RPM and IOOOO RPM and 20000 RPM (sec) axis wall axis wall axis wall axis wall .9 —- —- -— -- 6.3 30.9 6.7 30.0 1.8 6.8 35.8 7.0 30.6 13.3 07.8 10.0 07.0 2.7 -- -— —— —— 20.3 56.5 21.2 55.7 3.6 10.3 09.0 15.5 07.9 27.2 63.1 28.3 62.0 0.5 —- -- —— —— 30.0 68.3 35.2 67.7 5.0 21.8 57.9 23.3 56.9 00.5 72.6 01.8 72.0 6.3 -— —— —— —- 06.7 76.1 08.0 75.7 7.2 29.2 60.5 31.0 63.6 52.5 79.2 53.7 78.8 8.1 -— —— —- -- 57.8 81.8 58.9 81.5 9.0 36.0 69.8 38.3 69.0 62.6 80.0 63.7 83.8 9.9 —— —— -— -— 67.0 86.0 68.0 85.8 10.8 03.3 70.0 05.2 73.0 70.8 87.7 71.8 87.5 11,7 -_ -- __ —- 70.3 89.2 75.2 89.0 12.6 09.8 77.6 51.6 77.1 77.0 90.5 78.2 90.3 13.5 -— —- —- —— 80.1 91.6 80.8 91.5 10.0 55.7 80.6 57.0 80.2 -- -- -- —— 16.2 61.0 83.2 62 7 82.8 —- —— -- -- 18.0 65.8 85.0 67-0 85.1 -— -~ " ‘— 19.8 70 1 87.3 71 5 87 1 -- __ " " 21.6 73.9 88.9 75.2 88.7 __ " " " 23.0 77.2 90.3 78.0 90.2 __ __ " " 25.2 80.2 91.6 81.2 91.5 -- __ " " 00 E 6.--Values of (y-yfl/(Ve‘yf) x IOO Versus Time at the Axis and the Wall for the Gas UF6 (235 and 238 isotopes) in the Unsteady State Simple Centrifuge. Radius = A inches and I Charge Pressure = 2.l psia. Time 80°F and 20000 RPM (sec) Axis Wall 7.2 36.6 30.9 10.1. 52.2 00.2 21.6 60.2 53.7 28.8 73.7 60.7 36.0 8l.l 66.I A3.2 86.8 70.2 50.A 9I.3 73.A 57.6 9A.8 75.8 itions investigated. For a better comparison of the gas pairs, re 8 contains plots of (y-yf)/(ye-yf) at the axis versus time at of the temperatures investigated. The curves for SOZ-N2 and $02-H2 are at a centrifuge rotation of IOOOO RPM, whereas the e for the UF6 is at 20000 RPM. The results presented here are only for a centrifuge having a eter of 8 inches. As shown in Chapter II, varying the radius, maintaining the same peripheral speed, does not affect the magni- of the equilibrium separation. However, in the case of the eady state operation of the simple centrifuge, maintaining the peripheral speed implies the same flux due to pressure diffusion given dimensionless radius, r/Rw. Hence, since increasing the .mmcoc_ : mo m:_emm m :u_3 6m:m_cucmu . ii. ecnmw o_ae_m oumum >emmumc: 650 c_ 0E_H msmcm> HNoo.o . Am_XmV0>_\HNo o i Am_xmv>~ w a . . . o 0.9: 0.0m 0.8 0.2 0.8 0.8 0.3 o om 0 ON 0 .2 0 vii " n n u u n n u . . A! m_mq omw.q. mo 11 omLmLo _m_u_c_ cm £u_3 mmm: Lem o>cso .21m MP oooom 6cm doom um AmoaouOm_ wmm .mmwv mu: .6 : o.o~ w 2% 89: 2.6 008m 06 NTNom .6 r W Eng 0000. new mock um NIiNom .u _ 2.2 082 6:6 a 8m 66 N -Nom .6 .0 Mu .. N N .. 0.3 m 2mm oooo_.ucm doom um zi Om .m m MN I / a m. 9 mw .. 0.8 m. S D I: _ . a m n.o.om 0 ”w 0 m_ma mmm.:_ mo omLMLO _m_u_c_ cm £u_3 Now Lo» mew n 6cm 6 .n .m mm>csu 06 dius allows more mass to be contained in the centrifuge (all other erating conditions being equal), the time for the separation to occur st necessarily increase. Nevertheless, it was felt that sticking 1 just one size served the present purpose of establishing and putting 1to perspective key operating parameters and their effect on the dial fluxes. From the above information, three observations may be made garding operating parameters which control the rate of separation the simple gas centrifuge: I. As illustrated with the gas pairs SO N2 and SOZ-H2 in bles A and 5, increasing the rotational speed from IOOOO to 20000 2- M had very little effect on the time required for the mole fraction ofiles to develop. This is expected since increasing the rotational eed increases the flux due to pressure diffusion (depends on the tational speed squared) which correspondingly increases the magni- e of the separation as established in Chapter II-C. The fact that reasing the rotational speed moves the gas contained in the centri- e nearer the wall (steeper pressure profile), leaves less gas near axis. Hence, as can be seen in Tables A and 5, at 20000 RPM as pared to IOOOO RPM, the composition approaches equilibrium slightly ter at the axis. In lieu of the effect of rotational speed, the effect of using harge composition (yf) of 0.002 mole fraction on the times for separations to occur as presented in Tables A, 5 and 6 may be luated. If a function 0 is defined such that y = (byf at every ial position and every point in time, the partial differential 07 tion for the continuity of Species I in the unsteady state simple rifuge may be rewritten as PD i%1r20(1—¢y,) ~ r €811 = 0 , e at time = O, 0 = l for all r. The only remaining dependence on the value of yf is in the pressure, P, and the term A(l-¢yf), 1 are the same terms which were affected by changing the rota- ai Speed. Changing the values of yf changes the average molecular weight 1e gas in the centrifuge which changes the steepness of the pres- profile (the greater the molecular weight, the steeper the lie). But, as shown above, for a given charge pressure (fixed 1t of gas) the steepness of the pressure profile had very little :t on the time required for the equilibrium separation to occur. At early times (t close to 0) when ¢ is very nearly equal to all radial positions, the magnitude of A(I-¢yf) ~ A (I-yf) is ainly determined by the value of yf. But, as pointed out above, I ging the magnitude of A(l-yf) changes the flux due to pressure sion which correspondingly changes the magnitude of the equilibrium ation. Hence, no effect on the time for the separation to occur pected. After some time has passed, however, the quantity A(l-¢yf) not only be changing with radial position but also with time. The um radial change that may take place in the term l-cpyf is at ibrium. As illustrated in Chapter II-C, the composition at the y(Rw), based on the axis composition, y(O), at equilibrium is by I——'— .2 A8 Y(R ) = “(3m . w yI0)Ia I) + I Tere a is the simple separation factor, exp (ARWZ/Z). Rearranging 1e above equation, the following ratio may be obtained: l-yiRw) 1—y0) = y(01(oc-1)+1 r small values of y(O), regardless of the value of a, the ratio is ‘ar I. On the other hand, for values of y(O) near I, the ratio >proaches l/d. The values of I/a for the gas pairs SOZ-N2 and )2-H2 at 70°F and 20000 RPM (R2 = A inches) are 0.717 and 0.56A, aspectively. At 80°F the value of l/a for UF6 (235 and 238 isotopes) s 0.973. Hence, even for large mole fractions the radial change in '¢yf at equilibrium is not very significant and the term A(l—¢yf) iy be replaced by A(l-yf) for all times. Thus, it would be expected that using the charge composition : 0.002 mole fraction had very little effect on the times for the quilibrium separation to take place and did not bias the times given 1 Tables A, 5 and 6. To substantiate this conclusion, consider the 5 pair SOZ-N2 at 70°F and 20000 RPM (Rw = A inches) in the unsteady ate operation of the simple centrifuge. For charge mole fractions $02 of 0.002, 0.2, 0.5, 0.75 and 0.9, the values of the quantity ‘yf)/ye-yf) x 100 at the axis after 81 seconds are 7A.8, 75.1, .I, 76.8 and 77.1, respectively. 2. As illustrated in Tables A and 5, temperature has a sub— antial effect on the time required for the mole fraction profiles to A9 evelop. The effect of changing the temperature is of an indirect ature, since increasing the temperature from 70°F to 300°F almost oubles the diffusivity (~l.9 times), which in turn almost halves he time for mass transfer to take place. It should be remembered, owever, as illustrated in Chapter II-A, that increasing the tempera- ure reduces the effective value of AMW (decreases flux due to pressure iffusion), hence decreasing the magnitude of the separation. Along hese same lines it should be noted that the ratio of the diffusivities imes pressure for SOZ-N2 at 70°F to the corresponding value for JF6 at 80°F is 8.3 (Appendix B). This would appear to indicate that if it required A9 seconds in the case of $02-N2 for the dimensionless axis composition [(y-yf)/(ye-yf)] to move within 50% of its equilibrium Ialue, it would require A07 seconds for the corresponding situation to )ccur for UF6 due to the much lower radial diffusion flux in the case >f UF6' However, the actual time calculated for UF6 is only I} seconds. The key to this apparent inconsistency is that not only was 'he charge pressure in the case of the UF6 l/7 that of the SOZ-N2 less total mass present) but at 20000 RPM the much greater molecular eight of UF6 further reduces the ratio of the pressures at the axis 0 0.02A. Correcting for such a reduction in the mass present at the xis reduces the expected time from A07 seconds to approximately l0 econds. 3. Perhaps the most important is the fact that the times equired for mass transfer to take place in the gas centrifuge are elatively large when compared to times for mass transfer to take lace in conventional gas separation. For example, consider the gas 50 502-N2 at 70°F, IOOOO RPM and t = 0 in the unsteady state simple rifuge, the flux due to concentration diffusion is zero and the due to pressure diffusion is A.ll x l0-iO lb moles/sec/ft2 with 0.002 and r = 2.9 inches. At future times the net flux, of se, becomes smaller as the flux due to the concentration gradient tes a greater and greater part of the flux due to pressure diffu- In a gas centrifuge with axial flow, the effect of the small es is to requTre long residence times. With these three points the following conclusions can be drawn rding the production gas centrifuge with a feed and product stream: 1. To make the best use of the small fluxes the device should axial countercurrent flow. That is, the device should have an r axial flow in one direction and an annular outer flow in the site direction. Their combined effect would be to continually t the establishment of an equilibrium profile, allowing the fullest ization of the flux by pressure diffusion. Also, separations ter than those in the simple centrifuge could be realized. 2. As observed in 2 above, the magnitude of the fluxes due ressure diffusion are very small. In the production centrifuge means very large residence times will be required for a separa- to take place. The significance of this aspect is that the ughput rates will correspondingly have to be very small. This t is certainly not in line with the use of the gas centrifuge to ess flue gas where the tremendous volumes of gas to be processed ssarily dictate high throughput rates. Although increasing the S t 'I r1. I . SI ze of the centrifuge (increasing the length and/or diameter) rather 1an decreasing the throughput rate is also a means of increasing the :sidence time, certain mechanical limitations on the size of the :ntrifuge as presented in Chapter IV prohibit this from being a :asonable approach. It was shown above that increasing the tempera- ire and hence, increasing the diffusivity, correspondingly increases 1e radial flux rate, and to a certain degree, provides a means of :creasing the required residence time. 3. The magnitude of the centrifuge's rotation and the large composition had very little effect on the time required to .tablish the mole fraction profiles as shown above. Thus, these two rameters can be expected to have very little effect on the residence me required in the production centrifuge, but, of course, have a bstantial effect on the magnitude of the separation. One final point to be mentioned in this chapter is that garding the magnitude of the pressure changes as the mole fraction ofile developed in the unsteady state simple centrifuge. This servation will give insight regarding the importance of including rivatives of pressure with respect to distance in the axial direction en analyzing the production centrifuge. Table 7 contains the total essure profile at t = 0 and at equilibrium for the gas pair SOZ-N2 70°F and iOOOO RPM. Table 7 illustrates that the change in the essure profile is very small for charge compositions of 0.002, and 2 mole fraction of $02. For two heavy gases like UF6 (U235, U238) th a small molecular weight difference the change in the pressure )file is essentially, for practical purposes, nonexistent. 52 LE 7.--Total Pressure Profiles in the Unsteady State Simple Centri- fuge for the Gas Pair SO -N at 70°F and IOOOO RPM. Radius = A inches. 2 2 Total Pressures (psia) r/Rw yf = 0.002 y1: = 0.2 t=0 t=oo t=0 t=°° 0.0 IA.22A IA.22A IA.IO6 IA.I08 O.l IA.233 IA.233 IALII7 lA.II9 0.2 IA.287 IA.287 IA.I52 IA.I5A 0.3 IA.307 IA.307 IA.2I0 IA.2I2 0.A IA.372 IA.372 IA.29I IA.293 0.5 lA.A57 IA.A57 lA.396 lA.397 0.6 IA.S6O IA.560 IA.525 IA.525 0.7 lA.68A lA.68A lA.680 lA.680 0.8 1A.827 lA.827 lA.860 IA.86I 0.9 IA.992 IA.992 I5.067 I5.068 l.0 I50I78 l5.l78 I5.302 l5.303 CHAPTER IV THE ESTIMATION OF THE BURSTING AND WHIRLING SPEEDS OF A GAS CENTRIFUGE As determined in the previous chapter, large rotational speeds long residence times are necessary for significant separations to r in a gas centrifuge. It was also pointed out that for a produc- centrifuge (feed and product streams) the residence time could ontrolled by the size of the centrifuge and/or the feed rate. :ver, as will be shown in this section, the choice of the rota- al speeds and the ratios of length to diameter of a centrifuge not arbitrary, but must lie in a certain range dictated by the rial of which the centrifuge is constructed. As the rotational speed of a given centrifuge is increased, a d will be reached where either the centrifuge will burst or enter itical vibrational mode which will destroy it if it is left to er at this speed. It is possible for both the bursting and ling speeds to be identical. The bursting Speed can be simply thought of as a speed at which centrifugal forces acting on the wall of the centrifuge exceed the d (ultimate) strength of the metal (alloy) of which it is composed. whirling speed, however, is a slightly more complex phenomena. is, as the centrifuge is rotating its centerline will not 53 SA tly coincide with the geometric centerline due to eccentricities, ations and other causes. Hence, due to the inertia of the cylinder, :entrifugal forces will produce a bending moment tending to act the centerline of rotation even farther from the geometric erline. These centrifugal forces which are proportional to the tional speed squared and to the deflection are countered only by elastic forces of the cylinder. Thus, as the rotational speed 1creased a value will be reached where the elastic forces cannot ter the centrifugal forces, and if the centrifuge is left to ate at this speed, it will destroy itself. A. Bursting Speed AnalysisI To determine the intensities of the stresses in a cylinder ting about its axis a few simple assumptions are made: I. The material of which it is composed is isotropic. 2. The length of the cylinder is greater than the radius [length to diameter (L/DO) greater than I]. 3. The principal stresses are in the radial, circum- ferential, and axial directions. A. The plane sections perpendicular to the axis remain so after straining due to the rotation of the cylinder. For a long cylinder this is very nearly correct every— where except near the ends. Figure 9 illustrates a disk formed by the cutting of a cylinder V0 parallel planes Az apart and perpendicular to the axis of the Ider. Also shown is a small element of the disk subtended by the lThis section is composed (in part) of an overview of a . iled analysis dealing with the stresses developed In a rotating 3w cylinder by Morley (l2)- 55 Figure 9.——The Cross Section of a Rotating Cylinder Illustrating .adial and Circumferential Stresses, Sr and Se, Acting on a Small :nt. 56 ngle A0, with width Ar and thickness Az. The volume of the mass .ontained in the element is given by v = r Ar A0 A2, md the centrifugal force acting on this element is given by F = £0er Ar A0 A2 C 9 c Ifter neglecting second order quantities. The quantity 0 is the density f the material. The centrifugal force acting on the element is countered by :he equal and opposite forces due to the radial and circumferential itresses Sr and 36. The force due to these stresses is given by . A0 . A0 5 = A2 [2 Sesin ég-Ar + 2 Srr Sin 7?-- 2(Sr+ASr) ' (r+Ar) Sln 7?] A0 . ior small angles, sin gg-is approximately ez-and the above expreSSIOn ‘educes to F5 = A2 A0 (S Ar - SrAr - r AS!) 0 Equating F and F and taking the limit as Ar, A0, and A2 go to zero C S iives pwzrz rdSr F— gC+SF+—d—r—_ The circumferential stress, or hoop tension as it is sometimes called, is the quantity of interest, since as S8 approaches the yield POI"t 0f 57 1e material, bursting will occur. To determine Se, however, use must i made of the strains taking place as the stresses, Sr’ Se, and 52, ’e applied. The strain in the 0 direction is given by the ratio of 1e stress in that direction to Young's modulus of elasticity (E) minus 1e contraction that takes place due to the r and z direction stresses. Iat is, 80 = [$6 '— P(Sr + SZ)]/E’ Iere p is Poisson's ratio which varies from .25 for steel to .33 for uminum. Likewise the strains in the r and 2 directions are given by er = [Sr - p(Se + SZ)]/E, and ez = [52 - p(sr + $9)]/E. If as assumed, the displacement of points in the disk are Irely radial then the points at a position r move to a position r + q Id the circumferential strain can also be written as _ 20(r+q) — an = 3 e0 T an r ’ Id the radial strain is given by Substituting these expressions in the strain equations and )lv1ng for Sr and Se gives 58 (MD) dCI 9 _ Ep Sr " i1—2p5(1+p‘)'[ p a?" r + 62] and Se I-2p1+p [p r+fi+621 using assumption A (plane sections remain perpendicular after raining) eZ must be constant with respect to r. Thus, the above Dressions for Sr and Se may be substituted into the force balance Jation giving 2 +__9+&_ (l+p)(l2p) r2=0. dr r gC E l-p 2 dr One additional equation must be used to determine e2. This Jation can be written by symmetry considerations, since, midway tWeen the ends 0f the cylinder there can be no net forces in the ial direction. Thus, the following equation can be written R0 = 0 . ZW kw Sz rdr Integrating the second order ordinary differential equation F q and using the boundary conditions plus the above equation gives 2 2 2 R R _ _ 001 (3-5) 2 2 (1+p)(3-213) o w _ (l+p)(12p) ] T 9C8E [II-p? (Rw +Ro ) + Il-pI 2 Il—pl r F e Circumferential stress is then given by 2 2 2 R R (1+2) 2 pm (3-20) 2 2 &_ p r] , s = (R +R 1+ _ 0 9C [Ii-pl o w r2 (I Pl 59 and has its maximum value at Rw’ given by 2 _pi» (3—2) 2 (1—2) 2 SGimaX) _ REE-'T__—T_ R0 + TTTEFT R ' l-p w The above expression can be rewritten in a more general form as 2 S9(maX) = :g_ [(3-2p) + (l-Zp) k2] p/gC A Il-pl (I'D) ’ where 50 = the outer peripheral speed, wRo, and k = Rw/Ro' Using a value of .33 for Poisson's ratio for common materials of the construction of a centrifuge (Aluminum alloys, and Magnesium alloys) is on the side of safety and simplifies the above equation, giving 2 S 0(max) = :g_ (3 + 5k2) p/gC 8 FIgure l0 contains a plot of 80(max)/(p/gc) versus 50 for k values of 0-9, 0-95, and l.0. Data used to construct Figure 10 can be found in Appendix A. At each outer peripheral speed, so, the value Se(maX)/(p/ga must lie below (within a certain safety factor) the value Y/(p/gC) cor- responding to the metal or alloy of which the cylinder in question is composed. The quantity Y is the yield (ultimate) strength of the metal or alloy. Table 8 contains some Values of the ratio of the yield (ultimate) strength to density for some common metals and their alloys, COmPUted from data taken from Perry's Chemical Engineers' Handbook (I3). As can be seen by comparing Figure I0 With alloys in Table 8, 60 l5.09r °0(max)"p'9c’ A '” l 1 1 I L : : = ‘4' 200 300 A00 500 600 700 800 900 I000 IIOO l200 Peripheral Speed, so, ft/sec Figure 'O-"Se(max)/(P/9c) Versus Peripheral Speed, so, for lalues of 0.90, 0.95, and l.00. 61 TABLE 8.--Ratios of Yield (Ultimate) Strength to Density for Some Common Metals and Their Alloys. Y/(p/gc) x 10'5, Alloy Condition 2 2 ft /sec Aluminum + alloy IIOO 99 % cold-rolled—HIA A.65 alloy 2017 Heat treated TA l0.62 alloy 7075 Heat treated T6 19.38 Red Brass (wrought) Cold drawn 1.7A Cartridge Brass Cold rolled 5.A8 Magnesium alloy A280A Extruded 10.85 Carbon Steel Hot rolled 3.97 Wrought Iron (pipe) Hot rolled 2.89 the alloys of Aluminum and Magnesium are the only materials that can safely be used to construct a centrifuge where a typical peripheral Velocity might be 700 ft/sec (R0 = A inches, w = 20000 RPM). B. Whirling Speed Analysis2 To determine the whirling speeds of a centrifuge the following assumptions are made: I. The material of which the centrifuge is composed is Isotropic. 2. The centrifuge can be thought of as a uniform shaft simply Supported at both ends; i.e., neither end can be displaced nor support a bending moment. 2Contained in this section is information taken (in part) from the analyses of buckling shafts by J. P. DenHartog (lA). 62 3. The only thrust felt by the centrifuge will be due to the weight of the top end cap. This is assumed even though the material at the base of the centrifuge must support the weight of the centrifuge above it plus the end cap. Whereas, the material at the top of the centrifuge need only support the end cap. The validity of this assumption will be evaluated later. Let y be the deflection of the shaft from the geometric center- Iine at a distance 2 from one end. The centrifugal force acting on the shaft at this point is given by 2 FC — mw y/gC , . 2 where m = the mass per unit length = 0(RO -RW )0. Since the shaft will be concave to its unstrained position, the sign convention is to consider the bending moment due to the end cap weight as negative, i.e., Fw = -Wy . where W = the weight of the end cap. The total bending moment is then the sum of that due to the centrifugal force and the end cap weight and must be countered by the stiffness of the shaft. This fact may be expressed as follows: 63 vhere E = modulus of elasticity, and I = the moment of inertia of the cross-sectional area about a diameter, 0(R u-R °)/A. o w The boundary conditions as stated in assumption 2 can be written as d2 y=—y=o atz=0,and 2 dz 2 y = 9—1-= O at z = L dz 'he fourth order ordinary differential equation for the deflections ias the complete solution y = A cos clz + B cos C12 + C cosh C22 + D sinh C22 There 2 2 l/2 2 W mm W C = [___+__ +2?! ' 0(51)2 ch' 2 2 2 _ [ W2 + mm ]'/ _ W C ‘ El 2E1 ’ 2 0(Ei)2 96 ind A, B, C, and D are constants. ' 2 2 _ JSing the boundary conditions that y and d y/dz are zero at z — O, 't may be concluded that A = O and C = 0. Applying the boundary :onditions at z = L gives the two equations Iiililllilllli 6A B sin CIL + D sinh c2L = 0 and 2 . 2 . B cI Sin CIL + Dc2 Sinh ch — 0 , Tich may be solved, giving 2 2 . _ (c] + c2 ) D Sinh c2L — 0 and 2 2 . _ (Cl + c2 ) B Sin CIL - 0 . ince the quantity (CIZ + c22) cannot be zero and sinh is only zero Ten its argument is zero, D = 0. If B was also zero, the solution )Uld be the trivial solution, y = 0. To avoid this situation the alues of ClL must be multiples of 0 so that sin CIL = 0, i.e., IL = W, 20, 30, etc. Thus, there exists characteristic values of such that CIL is exact multiples of W and the deflection, y, is idefined. These values of w are the so-called ”whirling speeds'I of 1e shaft. In the remainder of this analysis only the lowest whirling ieed, corresponding to CIL =.W, will be considered, i.e., 2 w2 mwz l/2 W _ n ' 2 gEi' +251—7 A(EI) c L iuaring and rearranging the terms in the above equation gives the >llowing: defini force iihere critii Using resti Wei the 65 At this point the validity of assumption 3 can be tested by iing 0 as the whirling speed obtained had only the centrifugal : been taken into account, i.e., | ch ITI E '—4-“I :14: ...n W has been replaced by zero. Taking the ratio of the two cal speeds gives this expression, an upper limit can be found for W, such that s no smaller than 0.99. The upper bound placed on W by this iction may be written as v.3 0.0199 iiZEi/L2 To evaluate the actual magnitudes of W, consider centrifuges ches long and constructed from one of the alloys of Aluminum typically have modulii of elasticities around I0.2 x lo 12 and densities around O.l lbm/in3. By assuming radii of 2, 3, inches and different wall thickness, several corresponding values he upper bound of W can be computed. These values are presented Die 9. The results in Table 9 indicate how very large end cap ts must be to affect the critical speeds by '%' Furthermore, when large values of W are compared to the various weights of the TABLE 66 9.--Upper Bounds for W for an Aluminum Alloy Centrifuge A0 Inches Long. E = 10.2 x 106 Ibf/inz, p = 0.1 ibm/in3 Radius Wall Thickness Centrifuge Weight W inches) (inches) (lbf) (lbf) 2 l/A 11.8 6511 1/8 6.1 3580 1/16 3.1 1876 1/32 1.6 961 3 l/A 18.1 23AIO 1/8 9.2 12A70 1/16 A.7 6433 1/32 2.3 3267 A I/A 2A.3 57280 1/8 12.A 30020 ]/]6 6.2 15370 1/32 3.1 7775 ifuges the assumption made to neglect the effect of the weight a centrifuge itself is valid. Dropping the special symbols used to differentiate between No critical speeds and hence dropping the term involving W, the Ming general expression may be written: 2 TTA 9 ER = ——- k 8’ L0 "0'3“ (H TT(R °-R h)/A and 00(R 2—R 2) have been substituted for I and m o w o w Ctively. k is the ratio of inner to outer radii as previously ed. The above expression may be further generalized as follows: _ 6A L A 2 E/(O/gcl - (TTRETRE (5;) so i a D0 = the outside diameter, and so = outer peripheral speed. 'e 11 contains a plot of E/(p/gc) versus 50 for L/DO values of , 5, and 6. To construct Figure 11, a value of 0.95 was chosen <. Data used to construct Figure 11 can be found in Appendix A. To operate below the whirling speed a given centrifuge design >r rotational speed must be adjusted so that the value of ’gC) read from Figure 11 is less than the value of E/(p/gc) for naterial of construction (within certain safety limits). Table 10 ains the ratio of the elastic modulus to the density for some )n metals and their alloys. These values are computed from data 1 from Perry's Chemical Engineers' Handbook (13). As can be seen from Table 10, all the metal and alloys, at for Brass, have essentially the same value of E/(p/gC). . the conclusion derived from the bursting analysis, that only inum and Magnesium alloys (of those investigated) were acceptable, not have to be modified. However, what the whirling analysis provide are restrictions on the length to diameter ratio of the 'ifuge. For example, an Aluminum alloy centrifuge with an :h diameter, l/A-inch wall, and rotating at 20000 RPM, must be iches or less in length to maintain a safety factor of 15% below 'irst critical value of L/DO. N U0W\NU$ W O— X AIU\QV\M 68 )0 300 000 500 600 700 800 900 1000 11001200 I l a I 1 Peripheral speed, s ft/sec 0, Figure ll.--Critical Elastic Modulus Over Density, E/(p/gc) ’eripheral Speed, 50, for Length to Diameter Ratios, L/Do of 69 -E IO.--Ratios of the Elastic Modulus to Density for Some Common Metals and Their Alloys. E/(p/gc) x 10‘6, Alloy Condition 2 2 ft /sec ninum + Iloy 1100 99 % cold-rolled-HIA 273.6 Iloy 2017 Heat treated TA 278.7 Iloy 7075 Heat treated T6 276.1 Brass (wrought) Cold drawn 1AA.2 :ridge Brass Cold rolled 139.3 iesium Iloy AZ80A Extruded 268.1 >0n Steel Hot rolled 283-2 ight Iron (pipe) Hot rolled 279-7 To keep the sizes and rotational speeds realistic, the analyses artaken in the remaining chapters of this work will have incorporated :hem the restrictions on the rotational speed and on the length to ieter ratio established in the above two sections (A and B). To Ilitate this, it will further be assumed that the material of itruction of all centrifuge designs is either an Aluminum or iesium alloy. CHAPTER V THE COUNTERCURRENT PRODUCTION CENTRIFUGE Figure 12 illustrates a countercurrent rectifying (enriches heavy species) centrifuge. The feed gas stream enters at the top he centrifuge and proceeds downward in an annular stream, near periphery. As the outer annular stream passes down the centri- it is enriched in the heavy species by the radial flux of the y speciesfrom the inner stream. After the outer stream leaves the rifuge, a portion of it is directed off as the rich product stream the remainder recycled to the centrifuge. The recycle stream rs the bottom of the centrifuge flowing upward in the inner stream. inner stream leaving the top of the centrifuge is the lean product am. Not illustrated is a countercurrent stripping centrifuge. In type of centrifuge the feed would enter the bottom of the centri- flowing upward in the inner stream. A portion of the inner stream t leaves the top of the centrifuge is directed off as the lean uct stream with the remainder recycled to the centrifuge. The cle stream entering the top of the centrifuge would proceed down— in the annular outer stream. The outer stream leaving the bottom he centrifuge would be the rich product stream. 70 71 \ a. Feed stream b. Rich product stream c. Light product stream \ \ —.\ \ —.' I z I / d. Recycle Stream \—————-—_———- ———————-—_—- ‘— \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ I‘— —\ = 5..“ ’— | —__________—_'.r ‘\S§S\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ gure 12.--Countercurrent Rectifying Centrifuge Illustrating Outer Axial Flows and Feed and Product Streams. 72 It is worth mentioning at this point that a rectifying centri- duces a richer heavy product stream than does a stripping ge. The opposite is true if a lean stream containing the aunt of the heavy speices is desired. This result has an Nith the distillation of a binary mixture in a rectifying or 3 column where in centrifugation the heavy species may be to the more volatile component in distillation. A. Partial Differential Equation for the Countercurrent Centrifuge When the differential equations for the simple centrifuge were it was assumed that the gas in the centrifuge was isothermal ted uniformly at the speed of the centrifuge. These assumptions necessary for the development of the equations describing the urrent centrifuge, in addition to the following assumptions: 1. All bulk flows are in the axial direction and are not turbulent. The axial flows are further assumed to only be a function of radial position. 2. All derivatives of pressure with respect to axial position may be ignored. further assumed in assumption 1 is that the axial flow may tween ”plug or rod“ type flow and laminar flow. The importance Ctual shape of the velocity profile and the radial position of rsection of the inner and outer flows will be evaluated in a ction. Equations representing the concentration and pressure diffusion n the radial direction in a gas centrifuge were developed in III-A. They are rewritten here for convenience, where 73 F _ -.___ __ ly RT D12 r 8r he radial flux due to a concentration gradient, and _ ZWP 2 F1P " RT 12 Ar Y("y) ’ 2 = (11wI MW2)w RT gC he radial flux due to a pressure gradient. n the countercurrent centrifuge an expression is also needed 'lux in the axial direction. This flux is made up of the flux 1e bulk flow of the gas minus the flux due to diffusion in the The axial flux may thus be written as ‘ection. PD _ Pv(r)ry _ 12 81 Flz — 22' RT RT r 82] ’ ') Is the axial velocity profile assumed to be a function of isition only. Iith expressions for the fluxes in the radial and axial direc- ie partial differential equation for the continuity of species 1 :eady state operation of the countercurrent centrifuge may be 7A : case with the unsteady state simple centrifuge, the radial onditions for the countercurrent centrifuge are that there . through the axis or the wall. These boundary conditions as :viously are ie axial boundary conditions depend on the type of centrifuge ,idered: rectifying or stripping. Letting Rm be the radial it which the inner and outer flows intersect, the axial .onditions may be stated as follows: 'ying Centrifuge : the top of the centrifuge where the feed enters becoming 1r outer stream, the boundary condition is that y = yf for : interval Rm S—r E’Rw' At the bottom the average composi- e outer stream leaving the centrifuge is given by .21 0 _ _Y_ RT L, PIv(r)y D12 8 lrdr y = r R 211] RT R Pv(r) rdr equation is simply the moles of species 1 leaving the centri- e outer stream divided by the total moles leaving the in the outer stream. After a portion of the outer stream d S COIIIF hour the IIIIJ 75 cted off as the rich product stream, the remaining gas of ition yr is recycled becoming the inner stream. Thus, the y condition at the bottom is that y = yr for all r in the interval 0 §.r :‘Rm ipping Centrifuge At the bottom of the centrifuge where the feed enters becoming er stream, the boundary condition is that Y = yf for all r in the interval 0 §.r S'Rm top of the centrifuge the average composition of the inner leaving the centrifuge is given by 2n Rm 8y ' RT '0 P[v(r)y - D12 SEerr Vi: R 20 m R—TIO P v(r) rdr portion of the inner stream is directed off as the lean product the remaining gas of composition y] is recycled making up the outer stream. Thus, the boundary condition at the top is that = ' ' < < y y, for all r in the Interval Rm _.r.. Rw' With some reflection on the radial fluxes taking place and agnitudes (Chapter 111), two extremes in the operation of the current centrifuge can be visualized. In one case, the rate of put is made very small in hopes of allowing the radial fluxes to antly move the heavy species from the inner to the outer However, with the very small axial bulk flows, axial diffusion se the large axial gradient hoped for. In the other extreme, ate of throughput is made quite large, to eliminate the tal effect of axial diffusion, the very small radial fluxes be able to move the heavy species from the inner to the ream before the end of the centrifuge is reached. Thus, in f these two extremes very little separation would be seen. suggested by this reasoning is that a rate of throughput such that the separation achieved in a countercurrent centri- be maximized. B. Velocity Profiles This work does not deal with the analysis and computation of t shape of the velocity profiles and hence, the exact position low intersection, Rm, of the inner and outer stream in the urrent centrifuge. Instead, two different velocity profiles med: ”plug or rod'l type flow and laminar flow. This approach idered more in line with the fundamental problem of determining s of separations that may be expected along with the magnitude llowable flows in the countercurrent centrifuge. Also, by quite different velocity profiles and different flow inter- , Rm, the importance of knowing the exact shape of the velocity nd the flow intersection position could be evaluated. 0 compute the actual magnitude of an assumed velocity profile, sure (density) profile is needed. By knowing the pressure 77 and the shape of the velocity profile in a stream, the magnitude elocity profile is determined by equating the integral of times velocity, over the area of flow, to the known flow rate tream. To compute the pressure profile, a molecular weight equal to the feed gas was used. The pressure profile is then given by tion derived in Chapter II-A. That is, Ar2 P(r) = P(O) exp '7' , MW w avg A = W , Man9 = Mwlyl: + MW2(1-yf) , and P(O) is the axis pressure. The axis pressure, P(O), in the countercurrent centrifuge is the desired operating conditions. This is unlike the analysis nsteady state simple centrifuge where a special equation was to compute it. IWith the aid of the above pressure equation the velocity pro- y be computed by evaluating the integral ow = the flow rate of the stream, F) = the velocity profile, and and r2 is the region in which the flow takes place. z- -22 78 ig (Rod) Type Flow By definition, the velocity profiles for both the inner and streams are constant with radial position. This result may be sed as vi = constant for all r in the interval 0 5_r E‘Rm , and v = constant for all r in the interval R.i hi R , o m w I. = v = 0 at r = R and v = 0 at r = R . i o m o w :egral for the inner flow may be written as 20 P(O) v. R __ I m 2 Flowi — RT L) exp (A r /2) rdr, after integrating gives the following expression for vi: Flow. RTA Vi = ____J___._.._, 2nP(0)(ea-I) a = A R 2/2. m a similar approach, an expression may be found for v0, i.e., Flow RT A o Vo=—"‘_‘b“‘a" 20 P(0)(e -e ) b= A R2/2. w 79 minar Flow The shape of the inner stream velocity profile will be assumed that for laminar flow in a circular pipe, given by vi(r) = vi(max) [l - (r/Rm)2] this expression in the integral for the inner flow gives 20P(0) vi(max) R m r 2 2 Flowi — —-_——-7§F—_——__ L) [1 (PE) ] exp (A r /2) rdr , after integrating gives the following expression for vi (max): Flow. RT A l vi(max) = ———————-:;'-———— i 20 P(0)[(e -1)/a—I] a is defined as above. The shape of the outer stream velocity profile will be assumed that for laminar flow through an annulus given by v0(max) 2 vo(r) = ———7;-—- [l - (r/Rw) + d ln(r/Rw)] , 2 d = (l-k )/ln(I/k), C = l - d[l-ln(d/2)]/2 , and k = R /R . m w for the outer flow gives this expression in the integral R 2 2 2 2 d = —%Eégl-vo(max) At: "—(r/Rw) + d ln(r/Rw)] exp (A r / ) r r 80 :her complicated integral may be integrated analytically, after iipulation, giving the following expression for vo(max): Flow CRT A o VOW") = W ’ 00 n n = (eb_ea)/b_ea(]_k2) + d 1n(1/k)(-e°-l) "S- 2 b 1 =1 n n! n nd b are as defined above. Since the inner and outer flows oppose one another, regardless shape of the profile, a sign convention was adopted to further the velocity profiles. The top of the centrifuge was considered 0 position in the axial direction (2 = 0), thus fixing the sign downward flowing outer stream as positive and the upward flowing tream as negative. C. The Solution of the Partial Differential Equation for the Countercurrent Centrifuge The partial differential equation for the conservation of the pecies in a countercurrent centrifuge was given above in V-A as 1 a 2 _ 8y1+Pv(r)8_y_82y=O F '07 'AT y"‘y' r 07 P 012 02 322 Jation coupled with its associated boundary conditions may be Iirectly by using a numerical approach. However, to find an :al (approximate) solution the equation must be simplified by :ertain assumptions. An analytical (approximate) solution 81 :ainly desirable, since finding a numerical solution is itself 2r complex problem. ierical Solution A general FORTRAN program, NCENTRI, was written to solve the aquation numerically for the steady state mole fraction profiles countercurrent rectifying centrifuge. (It was found that having ierical solution to just the rectifying centrifuge was sufficient uate the analytical [approximate] solution.) The program is in the sense that it can be used to analyze the separations ad in a countercurrent rectifying centrifuge of any size, for ; pair, and any operating conditions. Also, by giving the option, either the plug type or laminar velocity profile can I in obtaining a numerical solution. A complete description and listing of the FORTRAN program and its ten associated FORTRAN subprograms may be found in X D. Also included in Appendix D is a sample output from pro- ZENTRI. The numerical approach used by the program to find the >0 is best illustrated by the following steps which, in a general I, outline the flow of calculations: 1. Input the gas properties, operating conditions, centrifuge ons, and the number of increments into which the radius and the is to be divided. The program is written so that regardless of .ition of the flow intersection, Rm, the radial distances from s to R and from R to the wall, Rw’ are divided into the same m m of increments. 82 2. Compute the pressure and velocity profiles. 3. Output all the input information along with the pressure d velocity profiles. A. Compute special fourth order finite difference equations to used to approximate the first and second partials of y with respect radius at one grid point before, one grid point after and at the id point corresponding to the flow intersection. These special equa- ons are needed due to the changing increment size at the flow tersection. 5. Due to the nonlinear nature of the partial differential uation, the entire mole fraction grid is initialized with values com- ted from a simplified version of the analytical (approximate) solution. 6. By using fourth order finite difference approximations for e partial derivatives appearing in the partial differential equation d the boundary conditions, an equation is written for each grid int. Thus, for example, if the grid consisted of 15 radial grid ints and 21 axial grid points, a set of 315 simultaneous equations 0 be written for the 315 unknown mole fractions at each grid point. 7. Solve simultaneously the set of equations. 8. If the proper option is given, the newly computed mole oximation to the true values and actions are used as a better appr BPS 6 and 7 are repeated a specified number of times. It was found all cases investigated that repeating steps 6 and 7 once was suffi- ent to account for the nonlinearity. 9- Compute the average mole fraction of the lean stream leaving e centrifuge and output it along with the entire mole fraction profile. 83 2. Approximate Analytical Solution The analytical (approximate) solution of the partial differen- tial equation for the countercurrent centrifuge was obtained by following an approach similar to that used by Furry, Jones, and Onsager (15) for the thermal-diffusion column. This approach was also used by Benedict and Pigford (l6) and Cohen (8) in their analysis of Jranium isotope separation by gas centrifugation. The method used depends on the fact that the term 8y/8r is 3f the order of the term Ar y(l-y) (Chapter III), which is a small quantity. Thus, the change in y in the radial direction as compared to the variation of r, P, or P v(r) is small and may be neglected. This assumption, for instance, is certainly more reasonable under total , reflux operating conditions (no products) than when a rich product stream is taken from a rectifying centrifuge. In the total reflux :ase the composition of the leaving Inner stream must equal to feed :Omposition. However, when a rich product stream is taken, the leaving inner stream by necessity of mass balance must be less concentrated than the feed. Consequently, a significant change in y must take place in the radial direction. Nevertheless, by using the above assumption, the partial dif- ferential equation may be integrated with respect to r, giving 2 8 8y FPv(r) = 0 Ar Y(I'Y) — r 5%.+ 52-0 fi—BT——rdr + Constant , Nhere the second order term azy/Bz2 has been neglected- Using the boundary condition that 8A By/Br = O at r = 0, gives: Constant = 0 . The second integration with respect to r requires the following relationship to get a form that may be integrated: R PD w Pv(r) _ 12 By _ " '0 ' RT RT 82' rdr ‘ yPQ ’ where Q is the flow rate of the product stream given by R w Pv(r) rdr 2" '0 RT ’ and yp Is the mole fraction of the product stream. The above equation Is Simply a statement of the mass balance for the heavy species at any axial position with respect to the product stream. The first term on the left-hand side of the above integral may be integrated by parts, giving R P ( ) dr 20y Rw W _X_LQLJ;__= ___ P d 2n (0 RT RT I V(r) r r %j Rw 3y dr L; Pv(r) rdr After substituting for By/Br, the above equation may be rewritten as R 3y 2 _ _ .__ §¥- 0w Pv(r)yrdr = yQ — C] Y(I Y) C3 32 i where = ZLA TA:W rdr L; Pv(r)rdr , and 85 R 20 1 w dr r 2 C3 = 'fi P—D]_2'0 T "0 WWW] - The integral for the mass balance may now be written as _ 3y _ _ _ C5 5; + [yQ Clyll 7)] — yPQ , where C5 = C2 + C3, and EI_ 12 w 2 RT 2 ‘ With all the radial dependence having been integrated, the parameters C], C2, and C3 are constants. The resulting differential equation may then be solved by separating variables, yp Csdy y 2 _ _ ' Integrating gives (yP - yfiw yP — 2yPyf + yf + (VP - yflq tanh (qu) = where q = Q/CI , 1 2 w = [1 — 2q(l-2yP) + qz] / , and u = Cl/CS/Z . 86 0. Analysis of the Analytical (Approximate) Solution For the time being it will be assumed that the following operating parameters are held constant: the shape of the velocity profiles for the inner and outer streams, the flow intersection posi- tion, R , the ratio of the inner to outer flow rates, Fi/Fo’ the pressure, and the temperature. Under these conditions, regardless of the actual magnitude of the inner and outer streams, q is a constant. That is, if the flow rates Fi and F0 are doubled, Fi—Fo’ which equals Q, is doubled as are the maximum velocities of the inner and outer streams. Thus, since 0 is doubled and Cl is doubled due to the doubling of the maximum velocities, q remains unchanged. The quantity u, however, Is very dependent on the magnitude of the flow rates. From above, u may be written as Where the additional subscripts onCI and C3 indicate they were computed for some flow rate, 8. Since C1 is directly related to the flow rates and C is related to the square of the flow rates, u for any other flow 3 rate can be expressed as it: Is u=__._2._._— 2 1: 2032 + C35) Where f is the ratio of the new flow rate to the old flow rate. By taking the derivative of u with respect to f and setting it equal to Zero, the value of f which maximized us can be found, i.e., 2 $1” 0 = (C2 + f C35) C15 2f 0,5035 df 2 2 2(C2 + f C35) giving C 1 35 Since u has a maximum value, so does tanh (qu) for a fixed L. Thus, there exists in the analytical (approximate) solution, as expected intuitively, a flow rate such that the separation is maximized. l. The Effect of Feed Mole Fraction on the Optimum Flow Rate The effect of the feed mole fraction on the optimum flow rate will be very slight. This is due to the small variation of C1 and C3 with a change in the shape of the pressure profile. Feed containing little of the heavy species will give a pressure profile essentially equal to that of the light species and essentially equal to that of the heavy species for feed mole fractions near 1. In either case the magnitude of the velocity profile, having been adjusted to give the Proper flow rates, nearly compensates for any changes in the pressure Profile, hence little effect on C1 and C3. This agrees with the fact that the charge composition had very little effect on the time required in the unsteady state for the equilibrium separation to take place Simple centrifuge (Chapter III-D). 88 2. The Effect of Rotation Speed on the Optimum Flow Rate The effect of increasing the rotational speed on the optimum flow rate will also be very slight, although its effect on the actual separation at this flow rate will be great. The change in separation 15 due to the dependence of C], and likewise u, on the rotational speed squared. Otherwise, changing the rotational speed, as does changing the feed mole fraction, simply changes the shape of the pressure profile. This change, as described above, will very nearly be compensated for by a change in the magnitude of the velocity profile to maintain the proper flow rates. Hence, the effect on the values of CI and C3 caused by the changing pressure profile will be minimal. 3. The Effect of Operating Pressure on the Optimum Flow Rate Operating at optimum flow rate conditions, a doubling of the axis pressure would be compensated for by a halving of the magnitude Of the velocity profile. Thus, there would be no effect on the values of Cl and C3 and hence, on the optimum flow rate. A. The Effect of Diffusivity Times Pressure on the Optimum Flow Rate As shown above, the ratio of the optimum flow rate to some other flow rate, f, is given by Since C2S is directly related to D12P and C35 is inversely related to D P, f is then directly related to DIZP' 12 89 This result is of fundamental importance since it and it alone places a definite restriction on the throughput in the operation of the gas centrifuge. That is, for any gas pair, the optimum flow rate is directly related to the quantity DIZP' Thus, knowing the optimum flow rate for one gas pair, f , the optimum flow for any other gas pair, 1 f2, at the same operating conditions is very nearly equal to the known optimum times the ratio of the diffusivities, i.e., f _ f (Dizplz 2 ‘ i (012M1 The small difference expected between the optimum flow rate given above and the actual optimum is due to the different molecular weights of the new gas pair which would affect the pressure profile. 5- The Effect of Temperature on the Optimum Flow Rate Assuming for the moment that DIZP is related directly to the absolute temperature raised to the 3/2 power (see Appendix B), C2 which is given by 2 2n DI2PRW 2 = RT 2 ’ 2 . . . is related directly to T'/ . C3, which is given by 90 1/2 is related inversely to T , noting that the effect of temperature in the integral is compensated for by the magnitude of the velocity profile. Thus, the ratio of the optimum flow, fn, at a new tempera- ture Tn is very nearly given by Tn 1/2 C20 1/2 T = (—-) (—-) T C 0 30 where the subscript ”0“ refers to the old temperature. However, recalling that CI is given by 2 (MW -MW )w R l 2 w r P = ___.._.___— __ d I RTgC To rdr [211i0 RT v(r) r r) C") shows that while increasing the temperature increases the optimum flow, it has the adverse effect of decreasing Cl (effectively decreas- Ing the molecular weight difference, see Chapter II-A). Since decreasing C decreases u, the optimum separation gets worse. 1 As the analytical (approximate) solutions for the counter- current centrifuge indicates (as does intuition), the longer the centrifuge, the greater the separation. The effect of the shape of the velocity profile and the radial position of the intersection of the inner and outer stream on the Optimum flow and on the separation will be dealt with in the next chapter. Also covered will be the effect of increasing the centrifuge diameter. 91 E. Computing the Analytical (Approximate) Solution A general FORTRAN program, CENTRI, was written to compute the constants and solve the equation resulting from the approximate ana- lytical solution. A detailed description, sample output, and complete listing can be found in Appendix E. The program was written to be used as a working model for the analysis of any gas in a countercurrent centrifuge. Furthermore, the centrifuge may be operated as a rectifying, stripping or combination rectifying-stripping centrifuge. The numerous capabilities of the general program are listed in Table 11. 92 TABLE ll.--The Analysis Capabilities of the FORTRAN Program CENTRI. Aspect Needed Parameter Description and/or Comments Centrifuge types Centrifuge size Centrifuge internals Gas pair Internal flows Optimum flows An option Radius, length, and rotational speed Radius of center pipe 1111,1111 ,T, P(axis), l (f . . D an VISCOSIty 12 Option for type of flow Flow intersection An option Solutions can be found for countercurrent rectifying, or combination rectifying- stripping centrifuges. If a combination rectifying- stripping centrifuge is used, the lengths for each section are needed. If the contrifuge contains a center pipe a laminar angular velocity distribution will be used between the stationary center pipe and the wall. Any gas pair may be analyzed. Any flow profile ranging between plug and laminar may be specified. The flow intersection may be specified or for laminar flow (with an option given) it will be adjusted so that the shear rate of the inner and outer streams at the intersec- tion are equal. Using the ratio of the inner to outer stream flow rates given the optimum flow rates will be computed. CHAPTER VI ANALYSIS OF THE RESULTS OF THE COUNTERCURRENT CENTRIFUGE A. A Comparison of the Numerical and Analytical Approximate) Solutions The comparison of the numerical and analytical (approximate) solutions was made using the gas pair SOZ-N2 ina rectifying counter- current centrifuge. In addition, the following operating conditions were fixed: Temperature, T, 530°R Pressure (axis), P(O), 1A.7 psia yf = 0.005 Centrifuge radius, Rw’ A inches Flow intersection, Rm, 2.25 inches (Rm/Rw = .5625) All numerical calculations were done using 15 radial grid points and 21 axial grid points. In the cases investigated, increasing the number of grid points in either direction, or both, did not sig— nificantly affect the results. Separation factors used are defined as follows: y (I - yf) where yr is the composition at the rich end of the centrifuge. 93 9A 1. Comparison No. 1: w = 10000 RPM, Length = 18 and 36 Inches and Total Reflux Figure 13 contains plots of the separation factor, 0, versus feed rate in standard cubic feet per minute (scfm) computed using the numerical and analytical (approximate) solutions forlaminar velocity profiles. Figure 1A contains plots of the separation factor, 0, versus feed rate (scfm) computed using the numerical and analytical (approxi- mate) solutions for plug type flow. Data used to construct the figures can be found in Appendix A. 2. Comparison No. 2: w = 20000 RPM, Length = 18 inches and Total Reflux Figure 15 contains plots of the separation factor, 0, versus feed rate (scfm) computed using the numerical and analytical (approxi- mate) solutions for laminar velocity profiles. Figure 16 contains plots of the separation factor, 0, versus feed rate (scfm) computed using the numerical and analytical (approximate) solutions for plug type flow. Data used to construct the figures can be found in Appendix A. Total reflux (no products) was used in the comparisons since by mass balance the average composition of the inner stream must equal the average composition of the outer stream at any axial position. This, it was hoped, would be a better approximation of the assumptions made in the process of obtaining an analytical (approximate) solution. That is, that radial changes in y and By/Bz were small and could be neglected at any axial position. Also, total reflux represents the 95 .x:_emm _muoe ace .88.: 8 new m: u a .mNeme 1 355a .86.: a u 3m .28 88. n a .856: .3622, cm:_Emu ;u_3 NziNom LOm AEmomv oumx comm mamco> .o .LOHUmm co_omcmammii.m_ oc:m_u Emom .oumc boom wo.o 30.0 no.0 No.0 _o.o woo.o moo.o :oo.o. m o _ I: u - moco:_ mm H 4 .. .:o_u3~Om AoumE_xocaomv ~mo_u>_mc< m : N mmgoc_ mm H a .co_us_0m _mo_LmE:z moLoc_ w_ u a .co_u:_0m Aoumemxotqomv _mo_u>~mc< I I m ,_.. p ‘401394 uoileJedag mmcoc_ m_ n 4 .co_u:_om ~mo_coE:z 96 83:8 :38. 8m. .885 em 86 M: i a .mNama i seem .885 a u 3m .28 88. n 3 .3o_d oa>e m:_m Lu_3 NziNom co» AEmomV boom boom mamcm> .o .couomu comumtmaomii.:_ mcammm Emom .oumc wood 00.0 :0.0 m0.0 No.0 _o.0 w00.o I 9 I I 1 000.0 :00.0 I I (\l l r m _ u 0.. o ‘Jonoe; uoileJedag mogocw mm H 4 .co_u:_0m Aoume_x0cqomv _mo_u>_mc< moto:_ mm H 4 .co_u:_0m _mu_coE:z moLoc_ w_ n a .co_u:_0m Aoome_xoLaamv _mo_o>_mc< mococm w. n a .co_u:_0m ”mowc6832 -_,__..- -ise‘ 97 ->u_oo_o> cm:_Emu ;u_3 N21N0m cow AEmomV oumm wood mzmcm> .8 .cOuomd c .x:_wom .mHOH 0cm mococ_ m_ n 4 .mmmm.0 n 3x\Em .mmgoc_ a n 2e .zdm OOOON n s .md__eota o_umcmaomii.m_ 0L:m_m Emom .oumc boom . 0w.0 . :omo m@.0 NOM0 _0.0 000.0 000.0 400.0 a u 1 I- n i u u u u d- " M..— t.:._ 1am.— La0._ S 9 w 4.R._ w m. U le._. 13 e 3 m ..m._ .u D LfiO.N comes—0m Aopme_xocaamv _mo_u>_mc< .0 co_u:_0m .mo_coE:z .m .._.N .xa_edm _6uoe 6em mdede. m_ u a .meede_ mNem.o n sa\eM .meede_ a .so_u ease ms_a eu_3 Nz-Nom tee 250060 deem 6660 mamtd> .6 .tdudmu eo_uatmadmi i.0_ oc30_m Emom .oumc 6000 00.0 00.0 :0.0 m0.0 No.0 . . . an n u I w 4.1 n —On On wOuO 0" mouc O 6 #00. L1 98 co_u:_0m Aoume_ iXOLaamv _mo_u>_mc< .0 CO_HD_Om —mu_t_®E32 .m n— " 3a .zda OOOON m; 0.N _.N l lvl 3 p ‘Jonoej UOIlEJEdSS 99 situation where the greatest separations will occur, hence amplifying any differences between the separations computed by the two solutions at a given feed rate. As can be seen from the figures for either laminar or plug type flow, at either lOOOO or 20000 RPM, the numerical solution always gives higher maximum separations. Also, these maximum separations occur at feed rates lower than those corresponding to the maximum separations computed analytically. These observations are listed for convenience in Table l2. it should be noted that even though the curves in each figure do not coincide, their shapes are essentially identical. Also, the sides of the curves at feed rates higher than the optimum agree very well. And, finally, that the numerical and ana- lytical (approximate) solutions agree best, overall, when a plug type flow profile is used. For illustration purposes, the following two tables were con- structed from the results of two selected numerical solutions. Table l3 contains the radial mole fraction, y, and ay/az profiles at the axial mid-point of the 36-inch long centrifuge, rotating at 10000 RPM used in Comparison No. l with a feed rate of 0.0l25 scfm. Also, Table l4 contains the radial mole fraction, y, and By/Bz profiles at the axial mid-point of the lB-inch long Centrifuge, rotation at 20000 RPM used in Comparison No. 2 with a feed rate of 0.02 scfm. The fact that the analytically computed optimum flow rates are larger and the magnitude of the separations are smaller than those computed numerically can be explained in terms of the variance in the mole fraction profile. The analytical (approximate) solution assumes TABLE 12.--Maximum Separation Factors and Their Corresponding Feed Rates Found During the Comparison of the Numerical and Analytical (Approximate) Solutions: SO -N 2 2' Velocity Rotational Speed Solution Feed Rate, Separation Profile and Length scfm Factor, a laminar 10000 RPM numerical 0.0138, 0.0130 1.196, l.h72 l8 and 36 inches plug 10000 RPM numerical 0.0167, 0.0164 1.228, 1.487 18 and 36 inches laminar 20000 RPM numerical 0.0117 2.20A 18 inches plug 20000 RPM numerical 0.0136 2.193 18 inches laminar 10000 RPM analytical 0.01Al, 0.01Al 1.180, 1.392 18 and 36 inches (approximate) plug 10000 RPM analytical 0.0169, 0.0169 1.207, l.h56 18 and 36 inches (approximate) laminar 20000 RPM analytical 0.0141 1.9h7 18 inches (approximate) plug 20000 RPM analytical 0.0168 2.136 18 inches (approximate) 101 TABLE l3.--Mole Fraction, y, and ay/az Profiles at Various Radial Positions for Laminar and Plug Type Flow With a Feed Rate of 0.0125 scfm: SOZ-NZ. 10000 RPM, Length = 36 inches Radius Laminar Velocity Profile Plug Type Flow (1”Ches) yx102 ay/azx103 yxio2 ay/azx103 0.000 .6018 .8022 .5862 .8091 0.321 .6017 .8022 .5863 .8091 0.643 .6015 .8022 .5867 .8089 0.964 .6011 .8023 .5872 .8087 1.286 .6009 .8026 .5880 .8083 1.607 .6010 .8033 .5890 .8079 1.929 .6014 .8046 .5902 .8073 2.250 (Rm) .6026 .8069 .5917 .8068 2.500 .6042 .8096 .5935 .8074 2,750 .6066 .8132 .5961 .8094 3.000 .6099 .8178 .5999 .8128 3.250 .6139 .8235 .6034 .8174 3.500 .6188 .8301 .6081 .8232 3.750 .6245 .8377 .6136 .8302 4.000 .6307 .8460 .6197 .8389 102 TABLE 19.--Mole Fraction, y, and By/az Profiles at Various Radial Positions for Laminar and Plug Type Flow at Feed Rate of 0.02 scfm: SOZ-Nz. 20000 RPM, Length = 18 inches Laminar Velocity Profile Plug Type Flow Radius (1“Ches) yx102 ay/azx103 yx102 3y/32x103 0.000 .7141 .3391 .6922 .3768 0.321 .7125 .3386 .6921 .3767 0.693 .7091 .3375 .6921 .3769 0.969 .7090 .3359 .6920 .3758 1.286 .6979 .3342 .6917 .3751 1.607 .6920 .3327 .6914 .3740 1.929 .6875 .3320 .6909 .3727 2 250 (Rm) .6864 .3329 .6905 .3713 2.500 .6893 .3353 .6938 .3721 2.750 .6964 .3396 .7016 .3755 3.000 .7083 .3461 .7139 .3815 3.250_ .7254 .3549 .7307 .3899 3.500 .7476 .3659 .7521 .4010 3.750 .7797 .3792 .7789 .9197 4.000 .8061 .3999 .8097 .9313 103 that radial changes in By/BZ are small and can be neglected. That is, at any axial position, the rate of change of the composition with axial position is the same all across the radius. However, as was seen in the operation of the unsteady state simple centrifuge (Chapter 111), near the wall the mole fraction profile increased the fastest with time. This is also the case with the countercurrent rectifying centrifuge as shown in Tables 13 and 19, where the composition changes fastest with axial position near the wall. Thus, with this extra capacity of the outer stream, allowed for in the numerical solution, maximum separations would be expected to be larger. In the inner stream, however, where the assumptions imposed by the analytical (approximate) solution are best approximated (almost constant y and By/az), the radial flux is essentially the same as computed by both the analytical(approximate) and numerical solutions. That is, the radial flux is essentially maximized being equal to that caused by pressure diffusion, since almost no radial gradient exists. Hence, to allow for the extra capacity of the outer stream, but yet being restricted to a given rate of flux from the inner stream, the optimum flows computed by the numerical solution would necessarily be expected to be lower. The fact that agreement between the analytical (approximate) and numerical solutions is best for plug type flow can best be explained in terms of the velocity profiles. Figure 17 illustrates the shape of the laminar and plug type velocity profiles for a feed rate of 0.0125 scfm and centrifuge rotational speed of 10000 RPM. 1n the case of laminar flow, the velocity goes to zero in a continuous 109 .xs_wom .mHoe new 2mm 0000— u 3 .moso:_ 0.: n 3m .mococ_ mN.N n Ex £u_3 om:m_cocou mcm>m_uoox “coccsocoHCJOQ m c_ Ewom mN_o.o mo oumm you; w Lo» mo__moca >u_oo~o> 30_u m:_m 6cm cm:_Em4 6683mm<11.m_ mtzmwm 2M\m o._ m.o m.o m.o w.o m.o :.o m.o N.o _.o 0.0 J. u u n u l n .i u . mood > L. _.OO.O 30_w Loose u .1 ..i .F 0.0 zo_m cmcc_ .1 floo.01 Noo.ou : moo.o: 6__cota zo_c m:_a .6 . o__woLa cmcLEma .m : #00 01 _. so..- Des/1; ‘AliaoleA [eixv H, -M‘. ...,.M- _-,_. 105 fashion as the wall is approached. This slower moving region near the wall will allow an even greater concentration buildup when the solution is computed numerically. But, as mentioned above, this increase in allowable separation must be accompanied by a decrease in theflow rates of the inner and outer streams. For plug type flow, however, the velocity profile is assumed constant over to the wall where it immediately goes to zero. With all the gas moving uniformly there is a lesser tendency for a radial mole fraction buildup near the wall, hence, a closer approximation to the assumptions made to obtain the analytical (approximate) solution. However, even for plug flow, as the rotational speed is increased from 10000 to 20000 RPM, the agreement becomes worse. At these speeds the centrifugal forces are becoming so great that the tendency for the heavy species to be ”piled” up at the wall is seen regardless of the shape of the velocity profile. Using feed rates higher than the optimum (increases the mag- nitudes of the velocity profiles) at a given rotational speed, inhibits the buildup of the heavy species near the wall (tends to diminish radial changes in y and ay/Bz). Hence, there exists good agreement between the numerical solutions in this feed range. On the other hand, the opposite is true when feed rates less than the optimum are used. The slow moving gas, which not only allows the composition to build up near the wall, but farther into the outer stream, gives large deviations between the two solutions. It may be concluded, then, by this analysis, that the optimum separation should always be higher than that predicted by the analytical 106 (approximate) solution, but occur at a lower flow rate than predicted analytically. Also that the agreement between the analytical (approxi- mate) solution and the expected results will be determined by how well the operating conditions approximate the assumptions made to obtain the analytical (approximate) solution. That is, velocity profiles which have slow moving regions, high rotational speeds, and large molecular weight differences are typical parameters which enhance radial mole fraction gradients. Large diffusion coefficients, on the other hand, tend to erase a concentration gradient. By making a comparison of the gas pair and the operating conditions desired with the analysis presented above for SOZ-NZ, an estimation of the accuracy of the results predicted analytically can be made. For example, consider the sytem SOZ-H in a countercurrent rectifying centrifuge 2 with the operating conditions as fixed in Comparison No. 1 (length = 36 inches). Though the molecular weight difference is 1.72 times greater, the diffusion coefficient (see Appendix B) is 3.93 times greater; hence fairly good agreement is expected between the analytical (approximate) and numerical solutions. A comparison of the two solu- tions may be found in Figure 18 which illustrates the expected good agreement. One last comparison to be made between the numerical and ana- lytical (approximate) solutions is the effect of removing a product stream. Returning to the gas pair SOZ-N2 with the operating conditions as given in Comparison No. l, agreement betwaen the solutions was found to be quite good at total reflux with a feed rate of 0.02 scfm (higher 107 .30_u ®Q>H m:_m Lu_ .x:_mom _mu0H ucm .mococ_ mm 1 u 3 .NINOm tee Emom .oumm comm mam .mwwm.o u 3m\Em .mmcuc_ a co> .5 .LOHUmw :o_umcmao n 31 .zax oooo. m--.w_ 6tsa_u Emom «mumc wood N.o _.o wo.o wo.o :o.o mo.o No.0 T1111 " u u u u n n u u m._ co_u:_0m Awummeotoamv _mo_u>_mc< .c co_u:~0m ~mo_coE:z .6 L1 o.~ .- N.— + m.~ gfi m._ r... p ‘301395 uoineJedag 3 108 than the optimum). Table 15 illustrates the effect on this agreement when various percentages of the feed stream are taken off as product. As it has been pointed out, the assumption that changes in y in the radial direction could be neglected, which was used to obtain the analytical (approximate) solution, must necessarily be violated when a product stream is taken off. Since the product stream is richer in the heavy species than is the feed, the leaving inner stream must necessarily be leaner than the feed. Although the error does become greater as the value of R/F is increased from 0 to 0.2, it is not large (maximum of -9.7%). This is due to the rather rapid decrease in the separation factor (-l9.5% at R/F = .2) which, of course, decreases the expected composition difference finfl5/5.2% at R/F= 0.2) between the leaving lean stream and the entering feed. Hence, without the expected TABLE 15.--Separation Factors for the Gas Pair SOz-Nz at Various Ratios of the Rich Stream to the Feed Stream (R/F) as Computed from the Numerical and Analytical (Approximate) Solutions. Feed Rate = 0.02 scfm, RW = 9 inches, Rm/RW = 0.5625, w = 10000 RPM, L = 36 Inches, and Plug Type Flow. Separation Factors Average Mole Fraction R/F . Analytical 8 Error of the Leaving inner Numerical (Approximate) Stream y = 0 005 Solution . ’ f ' Solution 0.00 1.470 1.448 —1.5 0.00500 0.05 1.361 1.327 -2.5 0.00991 0.10 1.289 1.239 -3.5 0.00989 0.15 1.227 1.175 -9.2 0.00979 0.20 1.189 1.128 -9.7 0.00979 —._..__.—-—.—.—-——u .. as“ -H' H"" T‘ 109 composition differences the agreement between the numerical and analytical (approximate) solutions is maintained at approximately the level seen at total reflux. Discussion regarding the rapid decrease in separation with product removed will be presented in Chapter Vl-E. Several areas of disagreement and their causes have been pointed out between the numerical and analytical (approximate) solu- tions. These points of disagreement should be taken into account when investigating the potential of a gas centrifuge design and especially when comparing calculated results to experimental data. From this point on in this work, however, with attention being returned to the overall goal of analyzing the gas centrifuge and determining its key parameters, the analytical (approximate) solution will be used. The analytical (approximate) solution exhibits properties similar to the numerical solution and is certainly more convenient to evaluate. Although it should be remembered that while certain phenomena and trends can be predicted by the analytical (approximate) solution, their actual magnitudes may be in slight error. With reference to Table 12, it should be noted that the dif- ference between the maximum separations obtained using a laminar or Plug type velocity profile differ at most by 9%. Also, the optimum feed rates differ at most by 16%. This is true for both the numerical and analytical (approximate) solutions. Furthermore, while this agreement as taken from Table 12 was obtained for $02-N even better 2! agreement is seen for SOZ-HZ. Hence, as long as it can be shown that the velocity profiles lie between laminar and plug type flow in an 110 actual centrifuge, knowing the exact shape of the profiles is not necessary. However, due to the better agreement between the numerical and analytical (approximate) solutions when plug type flow is assumed, it will be used in the remainder of this work. Also, when possible, feed rates greater than the optimum will be used. B. The Effect of the Position of the Intersection of the Inner and Outer Streams on the Maximum Separation Factor and the Optimum Feed Rate The separation in the countercurrent centrifuge depends on the heavy species moving from the inner to the outer stream. This movement is due to the centrifugal force in the form of pressure diffusion. With the flow intersection very near the axis, the inner stream is subjected to very little centrifugal force. Thus, the radial flux from the inner stream is reduced giving poorer separation factors. Also, as the flow intersection moves nearer to the axis the decreasing area available for the inner stream means higher velocities for a given mass flow rate. This has the effect of reducing the optimum feed rate. 0n the other hand, moving the flow intersection nearer the wall allows a greater portion of the inner stream to be subjected to larger centrifugal forces. This increases the maximum separation factor, and with the increase in area for flow, increases the optimum flow rate. The fact that the outer stream is now moving faster does not hinder the separation since it (outer stream) serves merely to carry away the heavy species coming from the inner stream. 111 Figure 19 was constructed illustrating the effect of varying the flow intersection position. The following operating conditions were used: Gas pair = SOZ-NZ, Temperature = 530°R, Pressure (axis), P(O) = 19.7 psia, Yf = 0.005. Centrifuge radius = 9 inches, Centrifuge length = 36 inches, Centrifuge RPM = 10000, Plug type flow and total reflux. Data used to construct Figure 19 may be found in Appendix A. As expected, the maximum separation factor becomes poorer and the optimum feed rate less as the intersection is moved toward the axis. Increasing Rm/RW from 0.125 to 0.875 increases the separation factor by 32% while increasing the optimum feed rate by 137%. . . 1 C. The Effect of the Centrifuge Radius on the Maximum 1 Separation Factor and the Optimum Feed Rate 1 l Analysis of the simple centrifuge indicated that as long as the peripheral speed was maintained constant, the simple separation factor did not change by changing the size of the centrifuge (Chapter 11-8). If this was true for the countercurrent centrifuge it would be a definite ”plus” factor. Since, by increasing the radius, and thereby increasing the area for flow of each stream, the optimum feed rate would increase. Also, by maintaining a constant peripheral speed, one is not operating any nearer the bursting point and by increasing the 112 3 .x:_mox .mu0h ccm mucus. mm H A .mococw q u m .zam 0000— u 3 .30_m oo>H 0:_m £u_3 N21N0m co» .3m\Em .m3_0mm mmam_cpco0 0:“ OH co_uoomcmuc. 30_m 050 $0 m:_nmm use 00 o_umm ocu mamco> moumm coon Eaempoo 0cm .6 .mcouomu co_umcmqom11.m_ oc30_u 3x\Ea o._ me we To 6.0 me a; ma Nd _.o 0.0. mood l a u u u n u n u c _ m £1 a i 05.0.. ..N._ e t m x w 30.0.. 1m; w 6 J r m... m m. m U U 0N0.0 . 1:0.— d D. P. 0 D m mN0.o r .im._ W moumc 000$ E:E_uqo 0c_m0 .0 1T0._ a Emom No.0 00 oumc 000% m 0c_m0 .m ll3 radius (decreasing L/D), the first whirling speed is increased (Chapter IV, A and B). To analyze the effect of different radii on the maximum sepa- ration factor and the optimum feed rate the following operating conditions were used: Gas pair = SOZ-NZ, Temperature = 530°R, Pressure (axis), P(O) = 19.7 psia, yf = 0.005, Centrifuge length = 36 inches, Rm/Rw = 0.5625, Plug type flow and total reflux. Peripheral speeds of 350 and 700 feet/sec were used for the different radii investigated. Table 16 contains the results of varying the radius from 2 to 7 inches for the two peripheral speeds. The undesirable reduction in the maximum separation factor accompanying an increase in the radial size of the centrifuge can be explained with some reflection on the factors which create a maximum Separation factor. The optimum feed rate is such that the gas is moving axially fast enough to prevent excessive axial diffusion, but slow enough so that the heavy species may move from the inner to the outer stream. Thus, by doubling the radius, but maintaining a constant value of R /R , the optimum feed rate must necessarily increase to m w avoid excessive axial diffusion. The increase, however, is not by 9 times even though 9 times the area is available for flow. The reason for a smaller increase in the feed rate is that for the gas 119 TABLE l6.--Maximum Separation Factors and Optimum Feed Rates Obtained for Various Centrifuge Radii for SOZ-NZ With Plug Type Flow, L = 36 Inches, Rm/Rw = 0.5625, Peripheral Speeds of 350 and 700 ft/sec, and Total Reflux. Peripheral Speed = Peripheral Speed = Centrifuge 350 ft/sec 700 ft/sec radius (inches) Separation Optimum Feed Separation Optimum Feed Factor Rate (scfm) Factor Rate (scfm) 2 2.120 0.0089 20.815 0.0089 3 1.651 0.0127 7.581 0.0126 9 1.956 0.0169 9.562 0.0169 5 1.351 0.0211 3.368 0.0211 6 1.285 0.0253 2.753 0.0253 7 1.290 0.0295 2.382 0.0295 molecules to move radially to the same dimensionless radial position (r/RW), they must travel twice the distance. Thus, a longer residence time (smaller flow rate) is required. Using theSe ideas and given the optimum feed rate for a particular centrifuge diameter, the optimum feed rate can be computed for any other diameter for the same operating conditions by the ”simple mindedll expression Area'} (Radius | = _‘—_'— Feed Feed (Area Radius' ’ where the first multiplier accounts for the increased area for flow and the second multiplier accounts for the increased radial distance that must be traveled. Thus, for example doubling the radius of the 115 countercurrent centrifuge only doubles the optimum feed rate. This is illustrated in Table 16 where the values contained therein were computed using the analytical (approximate) solution. Remembering that the flux due to pressure diffusion was given by ._ P 2 _ FIP — Zn RT DIZAr (l y)y , explains the reduction in the maximum separation factor. Since the value of Rm/Rw was maintained constant, the radial flux at the flow intersection, regardless of the centrifuge radius, is essentially a constant (assuming small mole fraction changes) for a given peripheral speed. Hence, while the flux from the inner to the outer stream is essentially constant, the throughput for optimum conditions, as shown above, must increase. Thus, the degree of separation must necessarily become worse when increasing the radius but maintaining a constant peripheral speed. These points are illustrated in Table 16. D. The Effect of Rotational Speed and Temperature on the Maximum Separation Factor and the Optimum Feed Rate In Chapter V-D-2 it was pointed out by analyzing the analytical (approximate) solution that changing the rotation speed would have very little effect on the optimum feed rate. This was expected from the analysis of the unsteady state simple centrifuge where it was found that while increasing the rotational speed improved the separa- tion, the time required for the equilibrium mole fraction profile to develop essentially did not vary. 116 Increasing the temperature, however, did significantly reduce the time required for the equilibrium mole fraction profile to develop in the unsteady state simple centrifuge. The reduction in the time required is due to the strong dependence of the diffusion coefficient on temperature (~1.5 power, Appendix B). In the countercurrent gas centrifuge, increasing the diffusivity via increasing the temperature is expected to increase the optimum feed rate. To clarify the effect of changing the diffusivity, consider for a moment a hypothetical situation in which the diffusivity of a gas pair in a countercurrent gas centrifuge operating at optimum condition could be doubled without altering the temperature. Doubling the diffu- sivity would necessarily double the magnitude of the net radial flux. However, it would also double the effect of axial diffusion. Hence, under this new condition, the optimum flow rates would have to be twice as great in order to compensate for twice the radial flux, yet keeping the ratio of axial flow to axial diffusion the same. The maximum separation factor would thus also be the same. Increasing the diffusion coefficient via increasing the temperature, while giving higher optimum flow rates as discussed above, does not maintain the same maximum separation factor. This is due to the fact that the increase in temperature effectively decreases the molecular weight difference of the species as shown in Chapter ll-A. These points are illustrated with the following set of operating conditions: Gas pa1r = SOZ-NZ, y1: = 0.005, 117 Axis pressure, P(O) = 19.7 psia, Centrifuge radius = 9 inches, Centrifuge length = 36 inches, Rm/Rw = 0.5625, Plug type flow and total reflux. Table 17 contains the optimum feed rates at temperatures of 70 and 300°F for centrifuge speeds ranging between 10000 and 20000 RPM. Also included is the estimated increase in the optimum feed rates by raising the ratio of the temperatures to the 1/2 power and multiplying by the optimum feed rate at 70°F (from Chapter V-D). Figure 20 illus- trates the effect of increasing the rotational speed and the temperature TABLE l7.--The Effect of Increasing Rotational Speed and Increasing Temperature on the Optimum Feed Rate for the Gas Pair SO —N . Rw = 9 Inches, L = 36 Inches, Rm/Rw = 0.5625, Plug Type Flow and Total Reflux. Optimum Feed Rates (scfm) w, RPM Temperature 70°F 300°F 300°F (estimated) 10000 0.01686 0.02019 0.02269 12000 0.01686 0.02019 0.02269 19000 0.01685 0.02018 0.02263 16000 0.01685 0.02018 0.02263 18000 0.01689 0.02017 0.02263 20000 0.01683 0.02015 0.02261 118 6:6 go: 063 9.: .036 1 352m .8605 6m 1 a .6685 a u 3m .0 com 666 E «Swim; :88 wow ozu co» vmoam _mco_umpox 603mmcucou mamcm> mLOuomu co_umcm00m EmE_xmz .0N “Lama: 0m ccmm . . 11 .0 20m .vooam :o_pmuom oooq_ 1-1 0000— p ‘JOlDEJ uoileJedas wnwixew 119 on the maximum separation factor. Data used to construct Figure 20 can be found in Appendix A. I As can be seen from Table 17 the reduction in the optimum flow rate going from 10000 to 20000 RPM is less than 0.2% at either 70 0F 300°F. However, increasing the temperature from 70 to 300°F increases the optimum feed rate by approximately 39%. The fact that the estima- tion of the increase in the optimum feed rate is only 20% is due to a further dependence of the diffusion coefficient on temperature through the collision integral (see Appendix B). Had this factor been taken into account the estimate of the optimum feed rate at 300°F would have been 0.02269 scfm (10000 RPM) which is identical to that computed analytically. Increasing the temperature from 70 to 300°F has the effect of decreasing the flux due to pressure diffusion by approximately 30%, which has a very significant effect on the separation factor. The reduction in the magnitude of the separation factor going from 70 to 300°F is as great as 37% at a centrifuge rotation speed of 20000 RPM. This, of course, explains why even though the diffusivity increases by 92.9% in going from 70 to 300°F that the optimum feed rate needed only to increase by 39%. E. The Effect of Removing a Rich Product Stream on the Maximum Separation Factor and the Optimum Feed Rate Consider a countercurrent rectifying centrifuge operating at total reflux with the optimum feed rate. As has been discussed previ— ously, the optimum feed rate exists since a very slow flow rate allows ______________.._..[ 120 axial diffusion to destroy the separation and a high flow rate does not allow enough residence time for the heavy species to move from the inner to the outer stream. Both cases yield poor separations. By maintaining the optimum feed rate, but now removing a rich product stream, means the inner stream must necessarily be moving slower than the optimum. Hence, on this basis alone the degree of separation must fall. If the ratio of the rich stream to the feed stream is held constant, the feed rate can be increased until the inner stream flow rate is equal to its value at the total reflux condition above. Since it is the inner stream which is controlling the separation a maximum separation should be realized at or very near this feed rate. To illustrate these points the following operating conditions are used: Gas pair = SOz-NZ, Yf = 0.005, Axis pressure, P(O) = 19.7 psia, Temperature = 530°R, Centrifuge radius = 9 inches, Centrifuge length = 36 inches, Rm/RW = 0.5625. Centrifuge speed = 15000 RPM, Plug type flow. Table 18 contains the maximum separation factors based on the feed and lean streams and the optimum feed rate for various ratios of the rich to the feed stream. i 121 TABLE 18.--Maximum Separation Factors and Optimum Feed Rates for Various Ratios of the Rich Stream to the Feed Stream (R/F) for the Gas Pair SOz-NZ, w = 15000 RPM, Rw = 9 Inches, L = 36 inches, Rm/Rw = 0.5625, and Plug Flow. Separation Factors Inner Stream Outer Stream R/F Flow Rate Feed Rate (scfm) (scfm) Based on Based on Feed Stream Lean Stream 0.00 0.0169 0.0169 2.337 2.337 0.05 0.0170 0.0179 1.860 1.997 0.10 0.0172 0.0191 1.586 1.696 0.15 0.0173 0.0209 1.913 1.523 0.20 0.0175 0.0219 1.296 1.900 The separation factor in Table 18 which is based on the composition of the rich (yr) and lean (y!) streams was computed by using the follow- ing equations: yfFEED - yrRICH y (1-y1) Y] = -—*—-IEEfi~—--- and a = ZT:;~Y -—7a-— where FEED, RICH and LEAN are the flow rates of the feed, rich and lean streams, respectively. The fact that the maximum separation factor drops rapidly as a rich product stream is removed can be explained in terms of the radial fluxes. At total reflux, mass balance requires the average compositions of the inner and outer streams to be equal at every axial position. Furthermore, at total reflux the maximum separation is only restricted by the length of the centrifuge. This is illustrated in 122 in Table 19 which contains the maximum separation factors at total reflux for different centrifuge lengths using the operating condi- tions given above. When a rich product stream is removed the average composition of the inner stream must by mass balance be less than the 1 average composition of the outer stream. However, the problem that arises here is just how much less can the average inner stream compo- sition be. Consider, for a moment, the top of the centrifuge where the outer stream is the entering feed. Using the above operating condi- tions, the composion of the outer stream at this point is yf = 0.005. From Chapter II-B (simple centrifuge) it was determined that knowing the axis composition, the equilibrium mole fraction profile could be computed by using the following expression: TABLE 19.--Maximum Separation Factors for Various Centrifuge Lengths at Total Reflux for the Gas Pair SOZ-Nz, Feed Rate = 0.0169 scfm (optimum), w = 15000 RPM, Rw = 9 Inches, L = 36 inches, Rm/Rw = 0.5625, and Plug Flow. Separation Factor Centrifuge Length (Based on Feed) 0.0 1.000 7.2 1.185 14.4 1.404 21.6 1.664 28.8 1.972 36.0 2.337 93.2 2.769 50.4 3.282 57.6 3.889 64.8 9.608 72.0 5.461 123 01 NW where a = exp (Ar2/2). However, since the composition is known at the flow intersection, Rm = 2.25 inches, to be yf = 0.005, the composition at the axis can be found by rearranging the expressions giving Yr Y(O) = 01 _ _yfl(OLI‘-1) - Using the operating conditions given above, y(O) is found to be 0.00971. As commented on in Chapter III, the equilibrium mole fraction profile results from a no net flux condition. That is, the concen- tration and pressure diffusion fluxes are equal in magnitude but opposite in direction. Herein lies the reason for the rapid decrease in the separation as a product stream is removed, since the mole frac- tion profile of the leaving inner stream cannot ever be any less in composition than the equilibrium mole fraction profile. (This is, of course, true at any axial position but ismost easily seen at the top, since the feed composition is known.) Furthermore, the compo- sition profile of the leaving stream can only approach the equilibrium profile, since the closer it gets the smaller the net radial flux and hence, the smaller the separation. The leaving inner stream mole fraction profile is in essence trapped between the feed composition and equilibrium composition profile. This situation is much like the case of a binary distillation (assume constant molar overflow) in a rectifying column containing a 129 fixed number of trays. If the equilibrium curve lies very close to the ”95° lir1e,'I then removal of any overhead product causes a signifi- cant decrease in the separation per tray at the base of the column. Hence, as compared to total reflux, the overhead composition would be markedly reduced. Further discussions regarding the limitations imposed on gas centrifugation by the results presented above will be contained in Chapter VII. However, Figure 21 is included in this section to further illustrate the reduction in the maximum separation factor with product removal and also how the maximum separation factor varies with feed rate at a given ratio of the rich to the feed stream. The operating cenditions used are as given above and the data used to construct Figure 21 may be found in Appendix A. F. Simple Power Requirements and Efficiencies of the Countercurrent Centrifuge Before leaving this chapter attention will be turned for the moment to the development of expressions for the power required to rotate the gas as it is being fed through the centrifuge and the thermodynamic energy (entropy) required for the separation. This information will provide further insight when comparing the gas centri- fuge to conventional gas separating equipment. The power that must be supplied to rotate the gas can be computed by integrating the product of the kinetic energy and the mass flow rate prOfile over the radius of flow, i.e., 125 Emom Emom Emom .zo_u mask 00.6 6:6 6N6m.0 u :0 co» Emocpm 0600 ocu ou Emocum :u_m ms“ .260 006_ n 3 .N2-N06 c_66 660 6 LOu.v mumm Ummm mDmL®> mEmmLum CQMJ *0 cm £u_m ocu ge\Ee .6666:_ 6m 1 0 .66600_ a n 26 mo momumm m:o_cm> 00 006mm mLOHomu compacmamm11._m oc:m_u opmc new» E:E_000\oumc noon 0.0_ 0.6 0.6 0.0 0.6 0.N 0._ 6.0 6.0 0.0 6.0 N.0 _.0 .1.“"““ n " “unuunuu “ o._ L.N._ 1.: S 9 d . E 46_m 1. m. . U film—.13 P. 3 m -10.N U D m_N0.0 u a .0N.0 .. N.N _m_0.0 1 u .0_.0 me_0.0 n a .60.0 5066 0600.0 1 u .0 u u\e .6 0- 0.N 126 R 2 2 2H w w r Power — KT MWa L) [Pv(r)] 2 9c dr , where MWa is the average molecular weight of the gas. For plug type flow, the integral is easily evaluated giving 111111 P(0)012 Zea Power = _—-%T_—_—_ -v.[ mA - 37-(ea-111 9C l A + v [(R 2eb — ea)/A - 3— (8b 0 eail o w A2 where a = A RmZ/Z, b= A 02/2, and w MW w2 -_a_._ RTgC The minus sign appears in front of vi to account for the fact that the inner flow is taken as negative. For laminar flow the integral becomes difficult to evaluate analytically and a numerical integration tech- nique such as the trapazoidal rule may instead be used, i.e., TiMWawZP(O) n+1 A rf 3 ri Power = ———ET§:———-i=l [e —~§——— v(ri_]) ri-l + e 2 v(ri)r 3 Ar 1117) where the subscript 1 refers to incremental radial positions of which there are n + 1. The power computed above is by no means the power required to operate the centrifuge where losses such as driver inefficiency, 127 windage losses, and bearing losses would be far greater in magnitude. However, the above expressions do give an idea as to how much power is actually required to simply rotate the gas. For example, from Chapter VI-B above, in which the effects of varying the flow inter- section was investigated, change in power requirements to rotate the gas can also be observed. The effect on the power requirements are presented in Table 20 for operating the conditions as stated in Chapter VI-B and with a feed rate of 0.02 scfm. As can be seen from Table 20 the simple power requirements vary from 1.75 watts/scfm 629 watts/lb-mole) at Rm/RW = 0.125 to 9.25 watts/scfm (1527 watts/ lb-mole) at Rm/Rw = 0.875. The increase, of course, being due to TABLE 20.--The Power Required to Simply Rotate the Gas Being Fed to a Countercurrent Centrifuge for Different Flow Inter- sections: Feed Rate = 0.02 scfm, Radius = 9 Inches, w = 10000 RPM, Plug Type Flow and Total Reflux. Rm/RW Power (Watts) 0.125 0.035 0.250 0.038 0.375 0.093 0.500 0-051 0.5625 0.055 0.625 0.060 0 750 0 072 0 875 O 085 128 more gas being near the periphery (higher angular velocities) as Rm/Rw approaches 1. For an ideal binary gas mixture, the entropy lost when the gases are separated at a given temperature and pressure is given by AS = Rly 1n y + (17y) 1n (l'Yll 1 where R is the gas constant. In the case of the countercurrent recti- fying centrifuge where there are feed, rich and lean streams the entropy lost in the separation is then given by AsS =R(R|CH[yr1n yr +(I-yrlln11‘vr11 + LEAN [yI 1n yI + (l-yI) In (l-ylll - FEED [yf 1n yf + (l-yf) In (l-yflll where all streams are referenced to the same temperature and pressure. Pointed out in Chapter Vl-A above was the fact that error between the analytical (approximate) and numerical solutions did not significantly change when a rich product stream was removed. This was found to be due to a rather rapid decrease in separation accompany- ing the removal of a rich product stream keeping the leaving inner stream composition at approximately the feed composition. With this good agreement, even though the analytical (approximate) solution assumes very little change in the mole fraction profile radially, a lean stream composition (yI) was C0mPUted by using the exPress'On given in Chapter VI-E, i.e-1 129 = yfFEED - yr RICH y1 LEAN ' This expresson cannot be in great error since yr computed both numerically and analytically are in good agreement (Chapter VI-A). Using the above expression to compute a lean stream composi- tion from the analytical (approximate) solution allows the calculation of the entropy lost in a separation and an efficiency based on the simple power requirements. Entropy losses and the resulting effi- ciencies for the operating conditions specified in Chapter VI-A are contained in Table 21. The gradual decrease in simple power required is due to less and less gas being recycled. On the other hand, the entropy lost must TABLE 21.--Simple Power Requirements, Entropy Losses and Resulting Efficiencies for the Separation of the Gas Pair SOz-NZ at Various Ratios of the Rich Stream t0 the Feed Stream (R/F). Feed Rate = 0.02 scfm, Rw = 9 inches, Rm/Rw = 0.5625, m = 10000 RPM and Plug Type Flow. Separation Simple Power EntrOpy Lost Eff' . R/F Factor Required (watts) (watts) 1c1ency 0.00 1.998 0.0550 0.0 0.0 — -9 0.05 1.327 0.0544 1.30x10 5 2.90x10 - —9 0.10 1.239 0.0539 1.51x10 5 2.8lx10 — -9 0.15 1.175 0.0534 1.32x10 5 2.97x10 — —4 0.20 1.128 0.0529 1.02x10 5 1.92x10 130 be equal to zero for total reflux (R/F = O) and also for no recycle (R/F = 1.0), hence there exists a maximum in the entropy lost column. The fact that there exists a maximum in the efficiency column is by itself significant, but loses its relevance when the absolute values of the efficiencies are considered. Furthermore, these already very small efficiencies would be further reduced (perhaps by magnitudes) if the entries in the entropy lost column were divided by the power required by the centrifuge driver instead of the simple power required. CHAPTER VII CONCLUSIONS The purpose of this work was to analyze the countercurrent gas centrifuge and determine its applicability for the separation of common gas mixtures in industrial situations with emphasis on the viability of gas centrifugation as a means for removing $0 from power 2 plant stack gas. Analyses of the steady and unsteady state simple centrifuge (Chapters 11 and 111) provided information regarding the equilibrium separations possible for different gas species, and the times required for such separations to develop. It was found that while large molecular weight differences and high rotational speeds controlled the magnitude of the separation, the diffusivity, as would be expected, directly controlled the time required for the separation to occur. The limitations of gas centrifuges began to be seen at this point, with times required for the axis composition to move to within 50% of its equilibrium value, being 99.5 seconds for $02 in N2 and 12.6 seconds for $02 in H at 70°F in an 8-inch diameter centrifuge rota~ 2 ting at 10000 RPM. (Both the ratio of the times required and the diffusivities for the two gas pairs are 3.93.) The small radial flux in the gas centrifuge indicated long residence times would be required to realize the separating potential of the countercurrent production centrifuge. The mechanical limitations 131 132 (size and rotational speeds) were investigated by developing expres- sions which allowed the calculation of both the bursting speed and the first whirling speed of a rotating cylinder (Chapter IV). it was shown that only alloys of Aluminum or Magnesium (as compared to other common metals) were acceptable for the construction of a centrifuge. Furthermore, while it was shown that a peripheral speed of 700 ft/sec was well below the bursting speed, the length to diameter ratio had to be less than 5 for safe operation under the first whirling speed. The equation of continuity for the heavy species in the counter- current centrifuge was developed and solved both numerically and, by making certain assumptions, analytically (Chapter V). It was shown that many different operating variables could be used to somewhat alter the maximum separation factor and optimum feed rate for the separation of a gas pair in a countercurrent centrifuge. Table 22 contains a list of many operating variables and their qualitative effect on the maximum separation factor and the optimum feed rate. At the optimum feed rate, the actual magnitude of the separation for a given gas pair was found to be controlled by their molecular weight difference, the operating temperature, and the length and the rota- tional speed of the centrifuge. The optimum feed rate, on the other hand, which may be more important when deciding on the applicability of the gas centrifuge, was found to be directly related to the diffu— sion coefficient for the gas pair and the diameter of the centrifuge. That is, all things being held constant in the operation of the countercurrent rectifying centrifuge, switching to a new gas pair having twice the diffusion coefficient will double the optimum feed TABLE 22.--0perating Variables and Their Effect on the Maximum Sepa- ration Factor and the Optimum Feed Rate for the Separation of a Binary Gas Mixture in a Countercurrent Rectifying Centrifuge. Operating Variable Changed Given the maximum separation factor and optimum feed rate at a given set of conditions, the qualitative changes expected by varying indi- vidual operating conditions are: Using the analytical solution instead of the numerical solution Increase centrifuge radius (constant peripheral speed) Increase centrifuge length Going from plug to laminar flow Moving flow intersection closer to the wall Increase rotational speed Increase molecular weight difference Increase temperature Increase axis pressure Optimum Maximum Sepa- Feed Rate ration Factor Decreases Increases Increases Decreases No effect Increases Decreases Decreases Increases Increases No effect Increases No effect Increases Increases Decreases No effect No effect 139 rate. The fact that the new gas pair will have a different molecular weight and thus a different pressure profile will be compensated for by the magnitude of the velocity profile. Likewise, doubling the diameter of the centrifuge but maintaining the same peripheral speed doubles the optimum feed rate. In this case, however, the positive effect of increased feed rate is negated by a rapid reduction in the maximum separation factor (Chapter VI, Table 16). This is due to the fact that while doubling the diameter at a constant peripheral speed (constant radial flux) and constant Rm/Rw doubles the throughput, twice the centrifuge length will be required (twice the moles to be moved from the inner to the outer stream) to obtain the same separa- tion factor. Hence, if acceptable separations are shown to be possible in a 2-inch diameter by IO-inch long centrifuge at a given peripheral speed, increasing the diameter to 12 inches to obtain 6 times the throughput means the length must be at least 72 inches to sustain the desired level of separation. That is, when scaling up a centrifuge, L/D must be constant. It was also shown that neither the optimum feed rate nor the maximum separation factor is affected by the operating pressure. As an illustration, at 70°F and 1 atmosphere the gas pair 50 -N has a diffusion coefficient of 0.1395 cmZ/sec while the gas 2 2 pair $02—112 has a diffusion coefficient of 0.5282 cmZ/sec. Table 23 contains the maximum separation factors along with the optimum feed rates for the above two gas pairs for several ratios of the rich to the feed stream. 135 TABLE 23.--Maximum Separation Factors and Optimum Feed Rates for Various Ratios of the Rich to the Feed Stream (R/F) for the Gas Pairs $02-N2 and SOz-Hz. w = 15000 RPM, Rw = 9 Inches, L = 36 inches, Rm/Rw = 0.5625, Plug Type Flow. SOZ'N2 SOz-H2 Maximum Optimum Maximum Optimum R/F Separation Feed Separation Feed Factor Rate, Factor Rate, scfm (Rich to Feed) scfm (Rich to Feed) Estimated Actual 0.00 2.337 0.0169 9.270 0.0663 0.0662 0.05 1.860 0.0179 2.792 0.0703 0.0709 0.10 1.586 0.0191 2.089 0.0750 0.0750 0.15 1.913 0.0209 1.739 0.0801 0.0802 0.20 1.296 0.0219 1.519 0.0859 0.0860 The following operating conditions were used: y1: = 0.005 Temperature = 530°R Axis pressure = 19.7 psia Centrifuge radius = 9 inches Centrifuge length = 36 inches Rm/Rw = 0.5625 6 = 15000 RPM and Plug type flow. The estimated optimum feed rates for the gas pair $02-H2 were computed by multiplying the respective optimum feed rates for the gas pair SOZ-N2 by the ratio of the two diffusivities. The excellent agreement between the estimated and actual optimum feed rates illustrates the unimportance of the very different molecular weights of the feed gas (2.3 versus 28.2) on the optimum feed rate. 136 Since almost all common gas pairs have diffusion coefficients ranging between 0.05 and 1.5 cm2/sec at 70°F and 1 atmosphere (Perry's Chemical Engineers'Handbook [13]), the possibility of an optimum feed rate being as great as 1/2 scfm is very remote for an 8-inch diameter centrifuge. While increasing the centrifuge diameter was shown to increase (linearly) the optimum feed rate, the separation factor was found to decrease unless the peripheral speed is increased. However, the increase in driver power requirements resulting from an increase in the centrifuge size and rotational speed would certainly make the only significant means on increasing the optimum feed rate less appealing. Figure 22 contains, for convenience, a plot of estimated optimum flows (scfm) versus diffusivities for different centrifuge sizes at an operating temperature of 70°F with Rm/Rw = 0.5625, plug type flow and total reflux. Coupled with the small optimum feed rates to the gas centri- fuge smaller than hoped for maximum separation factors are predicted. And to make the situation worse, when a rich product stream amounting to only 5% of the feed stream is removed, the maximum separation factor decreases dramatically. As discussed in Chapter Vi, this rapid decrease in the maximum separation factor with product removal is fundamental and unavoidable and perhaps the most serious limitation of the gas centrifuge. Increasing the length of the centrifuge at a constant diameter (increases L/D) gives very large separation factors at total reflux as illustrated in Table 19 (Chapter V1) for the gas pairs SOZ-N2 and SOZ-H2 using the operating conditions stated above. At total reflux .Xo_coe .oooe oeo 3o_e oose 00.6 .mN6m.0 n ze\Ee eo_z cooe oo ooac_eoeoo mc_>u_uoom ucoccsocou:300 m c_ u:o_o_wmoOQ co_m:mm_o msmco> oumm tool E:E_00011.NN oc:0_m oom\NEo .moom um “co_o_wmoou co_mjmm_0 w._ 0.. 0.. N._ 0._ 0.0 0.0 0.0 N.0 0.0 p q q q u . u . . - ..wo.o 137 :0_.0 . ..N_.0 mogoc_ w n mogoc_ 0 n mocuc_ 0 n mogoc_ N n 333 .NN.0 3 10:13:13: I'OJJU'O .0N.0 wjos ‘3193 pea; wnw11do 138 the difference in lengths (36 and 72 inches) significantly affects the maximum separation factor with l3h% increase for SO -N2 and 327% 2 increase for SOZ-HZ. However, by removing l0% of the feed streams as a rich product stream, the maximum separation factor increases by only l0.6% for SOZ-N2 and only lh.5% for SOz-H2 by doubling the length. This observation is expected since regardless of the length, the restriction on the composition of the leaving inner stream is always present. That is, in the case of the countercurrent rectifying centrifuge, the mole fraction profile of the leaving inner stream is trapped between the feed composition and the equilibrium mole fraction profile. Increasing the length merely increases the degree of close- ness that the leaving inner stream profile may get to the equilibrium profile. Thus, there is no advantage to having a very long centrifuge, which necessarily must operate above the first whirling speed, if anything other than a very minute quantity of product is to be removed. TABLE 24.--Maximum Separation Factors for the Gas Pairs SOZ-NZ and $02-H2 for Various Ratios of the Rich to the Feed Stream. w = l5000 RPM, Rw = 4 Inches, Rm/Rw = 0.5625, and Plug Type Flow. Maximum Separation Factors R/F SOZ-N2 SOZ-H2 Length, inches 36 72 36 72 0.00 2 337 5.46l h.270 l8.23h 0.05 l.860 2.h30 2.7h2 3.738 O.lO l.586 l.75h 2.360 2.703 O.lS l.4l3 l.h68 1.733 1.80h 0.20 1.296 1.315 1.5M 1.536 139 At this point the effect of all the operating parameters on the optimum feed rate (OFR) and the corresponding maximum separation factor (MSF) may be summarized in equation form based on the above analyses. These equations may be used to obtain a quick estimate of the OFR and the MSF for any gas pair in any size centrifuge given the calculated performance of any other gas pair in a similar type centri- fuge. The estimating equations are (difoSiVIty times pressurenew) (diameternew) OFR ' . Id) (diameterold) 22 new OFRold . (diffusivity times pressureO , and AMW (rotational speed )2 ) new new AMW01d MF 2 S exp ln(MSF01d . 2 (rotational speedold) (L/D)new ' (temperatureold)l (L/D)old (temperaturenew)J For example, using the performance computed for the gas pair SOZ-N2 at total reflux and with a rich to feed stream ratio of 0.] (Table 23), the equations may be simplified to OFR 3 a ' (diffusivity times pressure, cm2 °atm/sec) - (diameter, inches), and )2 NSF 3 exp[b ' (rotational speed, ft/sec ' (AMW) ‘ (L/D) / (temperature, °R)] , 140 5 where a = .0l57 and b = l.013 x IO- at total reflux, and .0178 and b = 6.30% x 10-6 where the ratio of the rich to 0: II the feed stream is 0.]. It should be remembered that throughout the analysis of the counter- current centrifuge it was assumed that the internal velocities were only in the axial direction. It was further assumed that these velocities were only a function of radial position. This represents an ideal situation giving the greatest separations. Since, if bulk movements existed, for example, in the radial direction (short cir- cuiting) the separations would necessarily decrease. Thus, if the gas centrifuge is decided to have potential in a given situation, more research is needed to analyze the velocity profiles detailing, if possible, how to make them closest resemble the ideal velocity profiles. Embodied in this work is a rather in-depth study of the gas centrifuge. Although the applicability of the gas centrifuge is not as general as had been hoped, its limitations as a gas separating device are now realized. For example, the gas centrifuge cannot be considered a viable means for separating $02 from power plant stack gas. The millions of standard cubic feet of gas produced by power plants and the high degree of SO2 removal required are incompatible with the very small optimum feed rates (0.0l69 scfm for an 8-inch diameter centrifuge) and small separations predicted for the gas centrifuge under ideal conditions. On the other hand, in the case of the enrichment of Uranium where the small predicted optimum flows . and the separation factors are relatively large as compared to lhl those obtained by other enriching techniques, the gas centrifuge is quite attractive. In a situation where the gas centrifuge may be applicable, the computational tools, in the form of the general FORTRAN computer programs, NCENTRI and CENTRI, plus the analysis of all operating variables affecting the performance of the gas centrifuge, which are contained in this work, should prove to be a valuable aid to the investigator. BIBLIOGRAPHY “+2 BIBLIOGRAPHY Lindeman and Aston, Phil. Mag., 37: 530 (l9l9). Mulliken, J. Am. Chem. Soc., Ah: I033 (I922). Beams and Skarstrom, Phys. Rev., 56: 266 (I939). Beams, Snoddy, and Kuhlthau, Conference Paper P/723, Peaceful Uses of Atomic Energy (I958). Groth, Beyerle, Nann, and Welge, Conference Paper P/l807, Peaceful Uses of Atomic Energy (I958). Kistemaker, Los, and Veldhuyzen, Conference Paper P/ll2l, Peaceful Uses of Atomic Energy (I958). Zippe, 0R0 3l5 (I960). Cohen, The Theory of Isotope Separation, McGraw Hill Co., Inc., New York (l95l). Schlichting, Boundary Layer Theory, McGraw Hill, New York (I955), P- 357- Bird, Stewart, and Lightfoot, Transport Phenomena, John Wiley 8 Sons, Inc., New York (I966), pp. 567-56 . Hirshfelder, Bird, and Spots, Trans. Am. Soc. Mech. Engrs., 7i, 92l (I999). Morley, Strength of Materials, Longmans, Green and Co., Ltd., New York I926 , pp. 359—372- Perry, Chilton, and Kirkpartick, Perry's Chemical Engineers' Handbook, McGraw Hill Co., Inc., New York (I963), pp. 23—3l, 23-h7. Hartog, Advanced Strength of Materials, McGraw Hill Co., Inc., New York (1952), pp. 291-297. Furry, Jones, and Onsager, Phys. Rev., 55: 1083 (I939). Benedict and Pigford, Nuclear Chemical Engineering, McGraw Hill Co., Inc., New York (I957 . Wilke and Lee, Ind. Eng. Chem., #7, l253 (I955). IA3 NOMENCLATURE _ 2 A — (MwI Mw2)w /RTgC D12 = diffusion coefficient for a binary gas mixture E = modulus of elasticity er, ee, ez = strains in the radial, angular and axial directions in a rotating cylinder Fly’ Flp = flux of species I due to concentration and pressure diffusion, respectively 9(9C)= acceleration due to gravity, 32.I7H ft/secZ (32.17u lbmft/lbf/seczl h = height i,j = subscripts denoting species i and j k = R /R cp w L = length MW = molecular weight P = pressure 2 Qi = Mwiw /RT/gC . 2 o R = gas constant, lO.73l5 lbf/In 'ft3/lb mole/ R R , R , R = radius of centerpipe, inner radius and outer radius, cp W respectively r = radial position 3 = entropy S , 5 , S = stresses in the radial, angular, and axial directions in r O z a rotating cylinder S, s = inner and outer peripheral speeds lhh I IAS absolute temperature time partial molar volume Young's modulus mole fraction of species I (heavy species) and also used as the deflection in the whirling speed analysis axial position separation factor Y(Rwl/[l-y(Rw)] ° [l-y(0)]/y(0) = exp (ARWZ/Z) for the simple centrifuge density viscosity angular speed APPENDIX A EQUATIONS AND CALCULATED DATA USED TO CONSTRUCT ALL FIGURES 1A6 APPENDIX A EQUATIONS AND CALCULATED DATA USED TO CONSTRUCT ALL FIGURES A. Calculated Data Used to Construct Figure I: Wall to Axis Pressure Ratios Minus I vs. Peripheral Speed for Various Molecular Weights Between 2 and IOO and Figure 2: Wall to Axis Pressure Ratios vs. Peripheral Speed for Various Molecular Weights Between IOO and 500 The equation used to calculate the data is MW 52 P(wall)/P(axis) = exp (ififéz) where MW = molecular weight 5 = peripheral speed, ft/sec R = gas constant, 10.731469 1bTc ft3/in2Ib-mole °R T = absolute temperature, 530°R 2 gravitational constant, 32.l74 lbm/Ibf ft/sec gc Units check: 2 lb ' ftz l inzlb-mole°R I ftz I lbfsec m lb-mole I sec2 . 1bf ft3 1 Ihh 1n2 °R l Ibmft Table A-l contains values of wall to axis pressure ratios minus I at various peripheral speeds and for molecular weights between 2 and IOO. Table A-2 contains values of wall to axis pressure ratios at vari- ous peripheral speedsand for molecular weights between IOO and A00. Th7 ‘ Ih8 mom.:_ Nwmw.m ONm—.¢ omwm._ m_mmm. ON:_M. mmw:_. m_omo. oom— Jmmm.w mmmN.m _mom.N Nmom._ wNNmm. wowmm. :m_m_. mquo. oo__ wwmo.m mmmm.m ONN_.N _wm_._ dm_©:. :mwom. Nmmmo. wowmo. coo— momo.m Nm_:.N m:_m._ mwmqm. mwmmm. m_oo_. wmmmo. NN_mo. oom _wmm.m m_:o._ NNNO._ wmmmo. _m:mN. N_mw_. ommoo. meNo. oow mmmm._ mmo_._ _m@:N. wqomq. mmqom. mqmmo. wmnqo. Nkw_o. oon wmmmm. MFNNN. mmoom. ON:_m. mmm:_. moomo. :mqmo. mmm_o. ooo wmwom. :m_m:. mNmNm. :mwom. Nmmmo. wquo. ooqmo. mmmoo. oom _m:mm. _m:mm. owmm_. N_mm_. omNoo. NwOMO. Omm_o. momoo. oo: Nmmw_. mmm:_. mwno_. moomo. :mqmo. NNm_o. wmmoo. mzmoo. oom mwwmo. omwmo. :mmqo. Nwmo. Omm_o. Nmmoo. owmoo. Nm_oo. CON oo_n3z ownzz omu3z OJHBZ ONHBZ o_u3z mnzz NHBZ oom\uw Amy pmmam _ 1 Am_va¢\A__m3vm _mcm£a_cmm ._ mc:m_m uojcumcou Ou vow: mumoui._i< m4m = 0.00172 3' W yr (”238) = “998 2 2 R Q R Q] W = 3.33909, 2 W = 3.311028 2 2 y(O)(U235) = y(0)(U238) - 1 = 0.002018 The profiles were then computed by the following equation: (R ) = y(O) exp(Ar2/2) y w y(Ollexp(Ar2/2)-l] + 1 2 where A = tflL——. RTg c A I SO -H ——— 2 2 2(02 _ A 2. $02 N2 T-Ef 2w 3. UF6 (235, 238 isotopes) —A§ 2w The calculated data used Table A-5. l53 1.176011 x 10' 6.830770 x 10'7 6 secz/ft2 secz/ft2 5.6923116 x 10'8 secZ/ft2 to construct Figure 5 are found in TABLE A-5.--Data Used to Construct Figure 5. SO -H UF6 (235, 238 Dimegjéoflless 2 2 2 2 isotopes) (r/RW) y, SO2 y, SO2 y, U235 0.0 .00l830 .OOI720 .0020l80 O.l .OOI833 .00l725 .0020l78 0.2 .00l803 .00l700 .0020l69 0.3 .OOI858 .00l776 .0020l55 0.0 .OOI88I .001802 .0020I35 0.5 .00l909 .OOI85] .0020109 0.6 .OOI906 .OOI9IZ .0020078 0.7 .001989 .001986 .002000] 0.8 .00200l .002076 .00l9998 . 0.9 .OOZIOI .002182 .0019950 l.O .002170 .002307 .0019896 150 E. Calculated Data Used to Construct Figure 6: 0Versus k for a Simple Centrifuge Containing a Stationery Center Pipe The equation used to compute 0 was (1/k2 — k2 + 0lnk) e: (1/k2 + 02 — 2) Calculations were done varying k between its limits 0 and I. Table A-6 contains the data used to construct Figure 6. F. Calculated Data Used to Construct Figure 7: Mole Fraction Profiles for 502, y, Versus Dimensionless Radius r/R _(Rw=0 inches), at Various Times in the Unsteady State Operation of a Simple Centrifuge Containing $07-H2 at 70°F, RPM = 20000 The equations used to compute the data were the equations describing the conservation of $02, the total conservation of moles and the total pressure equation. These equations are described in detail in Appendix C and will not be repeated here. The data, however, repre- senting the solution of these equations, and which was used to construct Figure 7, can be found in Table A-7. G. Calculated Data Used to Construct Figure IO: §fl(max)/(p/gc) Versus Peripheral Speed, 50’ for k Values of 0.90, 0.95, and l.OO The equation that was used to calculate the data is 2 L...) :0. (3 + 0 , p/gC 8 = the maximum circumferential stress (hoop tension), where 50(max) . 2 lbf/In , 155 TABLE A-6.--Data Used to Construct Figure 6. k e k e 0.00 1.0000 0.55 0.3805 0 05 0.9709 0.60 0.3291 0 10 0.9262 0.65 0.2803 0 15 0.8673 0.70 0.2338 0 20 0.8039 0.75 0.1897 0.25 0.7390 0.80 0.1078 0.30 0.6700 0.85 0.1080 0.35 0.6111 0.90 0.0701 0.00 0.5098 0.95 0.0302 0.05 0.0909 1.00 0.0000 0 50 0.0300 TABLE A-7.--Ca1cu1ated Data Used to Construct Figure 7. Mole Fractions of $02x102 R/Rw I;::) 3.6 7.2 10.0 very large 0 0.1921 0.1801 0.1700 0.1081 0 05 0.1921 0.1801 0.1705 0.1083 0 10 0.1921 0.1801 0.1707 0.1089 0 15 0.1921 0.1801 0.1709 0.1500 0 20 0.1921 0.1802 0.1710 0.1515 0.25 0.1921 0.1803 0.1719 0.1535 0.30 0.1921 0.1800 0.1727 0.1559 0.35 0.1921 0.1805 0.1738 0.1588 0.00 0.1921 0.1808 0.1751 0.1623 0.05 0.1922 0.1853 0.1768 0.1663 0 50 0.1922 0.1859 0.1789 0.1708 0 55 0.1920 0.1869 0.1816 0.1761 0 60 0.1927 0.1880 0.1809 0.1819 0 65 0.1932 0.1900 0.1889 0.1886 0 70 0.1903 0.1932 0.1939 0.1960 0.75 0.1960 0.1970 0.1998 0.2003 0.80 0.1988 0.2020 0.2069 0.2136 0.85 0.2031 0.2085 0.2152 0.2239 0.90 0.2090 0.2168 0.2251 0.2350 0.95 0.2181 0.2270 0.2366 0.2082 1.00 0.2297 0.2395 0.2099 0.2620 156 p = metal or alloy density, lbm/m3 s = outer peripheral speed, wRO, ft/sec, and = inner radius/outer radius, Rw/Ro. Calculations were done varying 50 from 200 to 1200 ft/sec for k values of 0.90, 0.95 and 1.00. Table A-8 contains the data used to construct Figure 10. H. Calculated Data Used to Construct Figure 11: Critical Elastic Modulus over Density, E/(p/gc), Versus Peripheral Speed, so, for Length to Diameter Ratios, £190, of 3, 0, 5 and 6 The equation that was used to calculate the data is 60 L 0 2 E/(O/gc) - ____2_—0' (5”) SO , )N o where E = the elastic modu1us, Ibf/inz, p = metal or alloy density, lbm/in3, TABLE A-8.--Data Used to Construct Figure 10. /(D/gc) x 1075, ftZ/sec2 5 so, (max) ft/sec k = 0.9 k = .95 k = 1.0 200 .353 .376 .000 300 .793 .805 .900 000 1.01 1.50 1.60 500 2 20 2 35 2.50 600 3.17 3.38 3.60 700 0.32 0 60 0.90 800 5.60 6.01 6.00 900 7.10 7.61 8.10 1000 8.81 9 39 10.0 1100 1.07 1 10 12.1 1200 1.27 1.35 10.0 157 s0 = outer peripheral speed, wRO, ft/sec L/D0 = length todiameter ratio, and k = inner radius/outer radius, Rw/Ro. To perform the calculations a value of 0.95 was used for k. This simplified the above expression to L 0 2 E/(p/gC) = 0.30535 (—D—) s - o Calculations were done varying 50 from 200 to 1200 ft/sec for L/DO values of 3, 0, 5, and 6. Table A-9 contains the data used to construct Figure 11. TABLE A-9.--Data Used to Construct Figure 11. E/(D/gc) x 10_6, ftZ/sec2 ° L/D = 3 L/D = 0 L/D = 5 L/D = 6 O O O O 200 1.12 3.50 8.63 17.9 300 2.52 7.96 19.0 00.3 000 0.08 10.1 30.5 71.6 500 6.99 22.1 50.0 112. 600 10.1 31.8 77.7 161. 700 13.7 03.3 106. 219. 800 17.9 56.6 138. 286. 900 22.7 71.6 175. 363. 1000 28.0 88.0 216. 008. 1100 33.8 107. 261. 502. 1200 00.3 127. 311. 605. 158 1. Calculated Data Used to Construct Figure 13: Separation Factor, 0, Versus Feed Rate lscfm) for $02:fl2 with Laminar Velocity Profiles, w = 10000 RPM, Rw = 0 Inches, Rm/RW:: 0.5625, L = 18 and 36 Inches and Total Reflux The separation factor used is defined as where yr is the composition at the rich end of the centrifuge. Table A-lO contains the data used to construct Figure 13. J. Calculated Data Used to Construct Figure 10: Separation Factor, 0, Versus Feed Rate (567%) for SOZZyzwithPlug Type Flow, 01 = 10000 RPM, RW = 0 Inches, Rm/Rw = 0.5625, L = 18 and 36 Inches and Total Reflux The separation factor is defined as y (I-yf) “'TTTT T’ where yr is the composition at the rich end of the centrifuge. Table A-ll contains the data used to construct Figure 10. K. Calculated Data Used to Construct Figure 15: Separation Factor, 0, Versus Feed Rate (chm) for $02j§7 withLaminar Velocity Profiles, w = 20000 RPM, Rw = 0 Inches, Rm/Rw = 0.5625, L = 18 Inches and Total Reflux The separation factor used is defined as y (1- yf) where yr is the composition at the rich end of the centrifuge. Table A-12 contains data used to construct Figure 15. 159 TABLE A-10.--Data Used to Construct Figure 13. Analytical (Approximate) Feed Numerical Solution Feed . Rate Separation Factor Rate SOlUF'on (scfm) _ . (scfm) SeparatIOn Factor L=18 Inches L=36 Inches L=18 Inches L=36 inches 0.005 1.133 1.325 0.0006 1.103 1.216 0.0075 1.167 1.000 0.0072 1.100 1.308 0.01 1.187 1.056 0.009 1.162 1.350 0.01125 1.192 1.072 0.0113 1.175 1.381 0.0125 1.195 1.072 0.0101 1.180 1.392 0.01375 1.196 1.071 0.0176 1.175 1.381 0.015 1.195 1.067 0.0221 1.162 1.350 0.02 1.183 1.025 0.0276 1.100 1.308 0.03 1.106 1.325 0.0305 1.123 1.261 0.00 1 117 1.252 0.0031 1.103 1.216 0.0538 1.080 1.176 TABLE A.11--Data Used to Construct Figure 10. Analytical (Approximate) Feed Numerical Solution Feed S l t' Rate Separation Factor Rate 0 u Ion (scfm) . . (scfm) Separation Factor L=18 inches L=36 inches L=18 inches L=36 inches 0.005 1.138 1.317 0.0055 1.118 1.209 0.0075 1.117 1.398 0.0086 1.165 1.356 0.010 1.203 1.009 0.0108 1.186 1.007 0.01125 1.213 1.065 0.0135 1.201 1.003 0.0125 1.219 1.076 0.0169 1.207 1.056 0.01375 1.220 1.082 0.0211 1.201 1.003 0.015 1.226 1.085 0.0263 1.186 1.007 0.0175 1.227 ~- 0.0329 1.165 1.356 0.020 1.220 1.070 0.0012 1.101 1.302 0.030 1.190 1.397 0.0515 1.118 1.209 0.000 1.162 1.327 0.0603 1.097 1.202 160 TABLE A-12.--Data Used to Construct Figure 15. Feed . . Feed Analytical (Approxi- Rate Ngmerlcal Soéutlon Rate mate) Solution (scfm) eparation actor (scfm) Separation Factor 0.005 1.870 0.0006 1.083 . 0.0075 2.080 0.0058 1.596 0.010 2.187 0.0072 1.717 0.01125 2.203 0.0090 1.831 0.0125 2.201 0.0113 1.916 0.015 2.159 0.0101 1.907 0.020 2.008 0.0177 1.916 0.030 1.723 0.0221 1.831 0.000 1.501 0.0276 1.717 0.0305 1.596 0.0031 1.083 0.0539 1.387 L. Calculated Data Used to Construct Figure 16: Separation Factor, 0, Versus Feed Rat671scfm) for 502:02 with Plug Type Flow, Rw = 0 Inches, Rm/Rw = 0.5625, w = 20000 RPM, L = 18 Inches 3nd Total Reflux The separation factor used is defined as y (I - yf) (1 - y 1 yf where yr is the composition at the rich end of the centrifuge. Table A'13 contains data used to construct Figure 16. 161 TABLE A-13.--Data Used to Construct Figure 16. 8:8: Numerical Solution :::: Ana$ztécaéoffippgzxi- (scfm) Separat1on Factor (scfm) Separation Factor 0.005 1.821 0.0055 1.567 0.0075 2.022 0.0069 1.703 0.010 2.101 0.0086 1.851 0.0125 2.189 0.0108 1.992 0.01375 2.193 0.0135 2.097 0.015 2.187 0.0168 2.136 0.020 2.102 0.0210 2.097 0.030 1.868 0.0263 1.992 0.000 1.683 0.0329 1.851 0.0011 1.703 0.0510 1.567 0.0602 1.051 M. Calculated Data to Construct Figure 17: Assumed Laminar and Plug Flow Velocity Profiles for a Feed Rate of 0.0125 scfm in aCountercurrent Rectifying Centrifuge with Rm = 2.25 Inches, Rw==0.0 Inches, w = 10000 RPM and Total Reflux The magnitude of the profiles was computed so that the follow— ing integral was satisfied: where Flow = the flow rate of the stream in lb-moles/sec, P = the pressure profile, v(r) = the velocity profile, and r] and r2 are the radial boundaries for the flow. With the top 0f the centrifuge chosan as the zero axial position the sign of the upward moving inner stream is taken as negative. Table A—10 contains data used to construct Figure 17. 162 TABLE A-l0.--Data Used to Construct Figure 17. Velocities (ft/sec) Radius (Inches) Laminar ProfilexlO2 Plug Type ProfilexlO2 0.00 -0.000 -0.201 0.321 -0.395 -0.201 0.603 -0.371 -0.201 0.960 -0.329 -0.201 0.286 -0.272 -0.201 1.607 -0.198 -0.201 1.929 -0.107 -0.201 2.25 0.0 0.0 2.50 0.072 0.90 2.75 0.116 0.90 3-00 0.135 0.90 3-25 0.131 0.90 3.50 0.107 0.90 3.75 0.063 0.90 0.00 0.0 0.0 N. Calculated Data Used to Construct Figure 18: Separation Factor, 0, Versus Feed Rate, scfm, for SOZ-HZ with Plug Type Flow, w = 10000 RPM, Rw = 0 Inches, Rm/Rw = 0.5625, L = 36 Inches and Total Reflux The separation factor used is defined as (1- yf) where yr is the composition at the rich end of the centrifuge. Table A-15 contains the data used to construct Figure 18. 163 TABLE A-15.--Data Used to Construct Figure 18. :5: NW” 1.31 50‘“ 1°” :3: “31.114183010311163“— (scfm) Separat1on Factor (scfm) Separation Factor 0.02 1.605 0.022 1.065 0.00 1.863 0.027 1.572 0.05 1.910 0.030 1.688 0.055 1.918 0.002 1.796 0.06 1.919 0.053 1.876 0.065 1.910 0.066 1.906 0.07 1.900 0.083 1.876 0.08 1.870 0.103 1.796 0.10 1.790 0.129 1.688 0.15 1.602 0.162 1.572 0. Calculated Data Used to Construct Figure 19: Separation Factors, a, and Optimum Feed Rates Versus the Ratio of the Radius of the Flow Intersection to the Centrifuge Radius, _m/Rw, for S02- N2 with Plug Type Flow, w = 10000 RPM, Rw = 0 Inches, =36 Inches, and Total Reflux The separation factor used is defined as where yr is the composition at the rich end of the centrifuge. Table A-16 contains the data used to construct Figure 19. 160 TABLE A-16.--Data Used to Construct Figure 19. Separation Factor Separation Factor Optimum Feed m at 8.733152? °f at $2.082“ 1...... 0.125 1.173 .232 0.0093 0.25 1.250 .298 0.0116 0.375 1.333 .361 0.0138 0.500 1.011 .020 0.0159 0.5625 1.008 .056 0.0169 0.625 1.085 .089 0.0179 0.750 1.556 .556 0.0200 0.875 1.622 .626 0.0220 P. Calculated Data Used to Construct Figure 20: Maximum Separation Factors Versus Centrifuge Rotational Speed for the Gas Pair SOszz at 70°F and 300°F. L = 3 The separation factor used is defined as Rw = 0 Inches, Inches, RméBw = 0.5625, Plug Type Flow and Total Reflux (1 - yf) Yr where yr is the composition at the rich end of the centrifuge. Table A-l7 contains the data used to construct Figure 20. 165 TABLE A-17.--Data Used to Construct Figure 20. w, RPM Temperature = 70°F Temperature = 300°F Maximum Separation Factor Maximum Separation Factor 10000 1.056 1.299 12000 1.719 1.058 10000 2.093 1.672 16000 2.629 1-959 18000 3.009 2.305 20000 0.562 2.870 Q. Calculated Data Used to Construct Figure 21: Separation Factors Based on the Rich and Lean Streams Versus Feed Rate for Various Ratios of the Rich Stream to the Feed Stream for the Gas Pair SOZ:N2. w = 15000 RPM, R“ = 0 Inches, L = 36 Inches, Rmfifiw = 0.5625, and Plug Type Flow The separation factor used is defined as y (I -y]) where yr is the rich stream composition and yI is the lean stream composition. Table A-18 contains the data used to construct Figure 21. TABLE A-18.--Data Used to Construct Figure 21. 166 Feed Rate/ Optimum Feed Separation Factors Rate R/F = 0.0 R/F = 0.05 R/F = 0.10 R/F = 0.20 0.1678 1.319 1.300 1.281 1.230 0.2097 1.006 1.370 1.300 1.269 0.2621 1.517 1.061 1.006 1.300 0.3277 1.652 1.560 1.076 1.335 0.0096 1.810 1.667 1.505 1.360 0.5120 1.991 1.772 1.607 1.379 0.6000 2.161 1.863 1.655 1.391 0.8000 2.289 1.925 1.686 1.398 1.0000 2.337 1.907 1.696 1.000 1.2500 2.289 1.925 1.686 1.398 1.5625 2.161 1.863 1.655 1.391 1.9531 1.991 1.772 1.607 1.379 2.0010 1.810 1.667 1.505 1.360 3.0518 1.652 1.560 1.076 1.335 3.8107 1.517 1.061 1.006 1.300 0.7608 1.006 1.370 1.300 1.269 5.9605 1.319 1.300 1.281 1.230 APPENDIX B DIFFUSIVITIES FOR THE GAS PAIRS SOZ-Nz, SOZ-H2 AND UF6 (235, 238 ISOTOPES) 167 APPENDIX B DIFFUSIVITIES FOR THE GAS PAIRS SOZ-NZ, $02-H2 AND UF6 (235, 238 ISOTOPES) The diffusivities for the gas pairs $02-N2 and SOZ-H2 were computed by using the Wilke and Lee (17) modification of the equation by Hirschfelder, Bird and Spotz (11). The equation used was BTI'5 /1/Mw + 1/Mw 0 = 1 2 where D = gas diffusivity, cmZ/sec, 1, B (10.7 — 2.06 /1/MwI + 1/Mw2) x 10' , T = temperature, °K, MW],Mw2= molecular weights, P = absolute pressure, atm, (r)+(r) r12 = O l 2 O 2 = collision diameter, angstroms, ID = collision integral a function of kT/EIZ’ and e1 E2 0 €12/k = (~E9 (—10 = force constant, K. Values of the individual force constants and collision diameters were taken from a table prepared by Wilke and Lee (17). The collision integral was evaluated by using a table prepared by Hirschfelder, Bird and Spotz (11). 168 169 Table B-1 contains all the parameters along with intermediate values used in the calculations. All units are defined above. The value used for the self diffusion of UF6 was taken from Benedict and Pigford, Nuclear Chemical Engineering (16). At 80°F, Do is given as 6.6 x 10_7 g-mole/cm/sec. Using the ideal gas law and a pressure of 1/7 atm gives D = .1135 cm2/sec. TABLE B-l.--Ca1culation of the Diffusivities of the Gas Pairs $02-N2 and SO -H . 2 2 Parameter SOZ-N2 SOZ-H2 temperature 290.0, 022.2 290.0, 022.2 MW], MWZ 60, 28 60, 2 i/l/MwI + l/Mw2 0.2266 0.71807 -0 -0 B 10.1026 x 10 8.9335 X 10 . .681 0.290, 2.968 00)], (r0)2 0 290. 3 . 86 3.629 r12 3 9 EI/k’ EZ/k 252, 91-5 252. 33.3 kT/El2 1.9391, 2.7806 3.2103, 0.6091 1 0.5033, 0.0806 0.0659, 0.0282 D D 0.1306, 0.2590 0.5282, 0.9869 APPENDIX C NUMERICAL METHOD AND FORTRAN PROGRAM USED TO ANALYZE THE UNSTEADY STATE SIMPLE CENTRIFUGE 170 APPENDIX C NUMERICAL METHOD AND FORTRAN PROGRAM USED TO ANALYZE THE UNSTEADY STATE SIMPLE CENTRIFUGE As established in Chapter III-B, the following three equations define the unsteady state simple centrifuge. A. The Equation of Continuity for Species 1: _ _1_ 39.2-9.1. _ p _+ y a_t_+ 0120 [2A y(i—y)+[Ar(12y) r1Br arzj - 0, 3y- _ 3V- _ = where 5?-— O at r - 0 and §?-— A(1 y) y r at r Rw' In Appendix A, Section A, it was determined that the units on A were ft_2. This allows a quick units check to be made on the above equation as follows: 1b By 25 f 1 P 5Y-and y 8t ’ . 2 sec ’ in M Y(I'Y); 7, ft 3y 1 [Ar (1-2y) ' l/rl ——‘; —-— , 8r ftz 2 1 . §——-; ~——- and by using the fact that DIZP has unlts 8r2 ft ftZ/sec - lbf/in2 completes the check. 171 172 B. The Total Pressure Equation: 01> P2 0) 8r RTgC (AMWy + MW 2) r’ where P = P(O) at r = O. The units for this equation were checked in Appendix A, Section 1. C. The Conservation of Moles Equation: R _ 20P(0) w B/L — RT 10 Prdr , where B/L are the moles/ft charged. The units of the left and right sides of this equation can easily be seen to be equal. The initial value of y was given as the composition of the Charge, yf. This allowed the initial pressure profile to be computed analytically as P(r)/P(O) = exp (A rZ/Z) , B where P(O) = /L 2 , and 211 A R _ 1 §?7K [exp (-——§u—) 1 0.12 ) = + NW A RT gC (AW yr 2 With the values of y and P initialized at time zero, future values were computed using an implicit numerical approach. That is, all par- tial derivatives were approximated at the next increment in time which 173 resulted in a set of simultaneous equations which were then solved using the scientific subroutine ONEDIAG. Table C-l contains a collection of the finite difference approximations used for the various partial derivatives. The sub- scripts used are all referenced to the point about which the expansions are made, I. From Table C-l, for example, the approximation used for the boundary condition 8y/3r = O at the axis (subscript 1) is [T2101 — 1,—0y(1) +1.5 y(2) - y2(3) + yf:)1/Dr = o, where the use of the fictitious point y(O) assumes the continuity of the function through the axis (method of images). For the approxima- tion of the partial derivative By/Bt, four time grid points were used, giving the following formula (see Table C-l): J— y: [Ex/(j) ~3y (J-l) +1.5 y (J-Z) - -Y(—3—311/Dt. Due to the nonlinar nature of the partial differential equa- tions the values of y and P had to be estimated at the next position in time. 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HKkaA n x mA 0» 00 % mOHmA H 2 0A Oh 00 A A .amo wmoz va ..... myqamq paapao oh ammo mA uzAhnommam mAIA ..... \ ooooooIOfiOH \A OHoH" w oAmVNPOO w QPQO \ .....oromoA \A oAAAH A aAAVAPOQ V «H40 AoAthoo voAvAhOD .AQZAQZA» «Aozvm oAQZVN ZOAmZMEAQ oAQIA AQOHWA ohmquA .kmoz A» .m .N AFDOUZOU mzAhDoamDm om m¢ c¢ mm on mm mA cA UUU APPENDIX E FORTRAN PROGRAM USED TO EVALUATE THE APPROXIMATE ANALYTICAL SOLUTION OF THE COUNTERCURRENT CENTRIFUGE 23h APPENDIX E FORTRAN PROGRAM USED TO EVALUATE THE APPROXIMATE ANALYTICAL SOLUTION OF THE COUNTERCURRENT CENTRIFUGE As established in Section V-B, the equation of continuity for Species I in the countercurrent centrifuge is 2 PV(F) gl_ay = 0 D12 P z 822 I 3 2 _ _ §y_ —- §?-[Ar y(l y) r 3r] + where v(r) is the axial velocity profile assumed to be only a function of r. In Chapter V-C-Z the solution was found to be tan uw = _ _ , yP ZYny + y1c + (yp yf)q where q = Q/C ’ w = [l - 2q(l - ZYP) + q C II cl/c /2 = cl/(c2 + c3)/2, R ZNA w r _ .... P r rdr, l RT 0 rdr [O V( ) (‘3 I 2N PDIZRw 2 ‘ RT 2 235 236 3 RT PD 0 2 Q = RITE [OW P v(r) rdr, yf = the feed composition (2 = O), and yP = the product composition at the other end (2 = L). To evaluate the integrals and then solve the nonlinear equation to get the product composition a general FORTRAN program, CENTRI, was written. The program is general in the sense that it can be used to analyze virtually any binary gas separation in any type of countercurrent centrifuge with any internal flow arrangement. Table E-l contains a list of the operating parameters required by program CENTRI. Since the program can handle such things as the presence of a center pipe (laminar angular velocity gradient); velocity profiles ranging between, and including, plug type and laminar velocity profiles; and the possibility of both the inner and outer streams being annular and located near the wall, all integrations are carried out numerically using the trapazoidal rule with IOOO radial increments. Techniques used to establish the magnitudes of the velocity profiles are as described in Chapter V-B. With reference to Table E-l, the following calculations are undertaken when computing the separations in either a rectifying, Stripping, or stripping-rectifying centrifuge. Table E-2 contains a description of the subroutines needed by CENTRI. 237 TABLE E-l.--0perating Parameters Required by Program CENTRI. Parameter Units Description WMI lbm/lb-mole Molecular weight of the heavy species WMZ lbm/Ib-mole Molecular weight of the light species PO lbf/inz Absolute axis pressure T °R Absolute operating temperature DP lbm/ft/sec Diffusivity times the mass density at the operating temperature VIS lbm/ft/sec Gas viscosity (This completes the first data card.) FORMAT (6Fl0.0) YF None SCFMD scfm SCFMU scfm OMEGA RPM VPOWER None Feed gas mole fraction Inner stream flow rate -9 Outer stream flow rate Centrifuge rotational speed VPOWER can range between 0 (plug type flow) and I (laminar type flow (This completes the second data card.) FORMAT (6FI0.0) 'RIS inches RI inches RM inches Rw inches H inches If RIS = 0, then no center pipe is assumed. If RIS is not zero then RIS is taken as the center pipe radius The innermost position of the inner stream (does not have to coincide with the axis or center pipe radius) Flow intersection between the inner and outer streams Centrifuge radius Centrifuge length (This completes the third data card.) FORMAT (6I0.0) NOPT None IDIV None ITOPT None LOPT None (This completes the fourth If NOPT = 0, then the flow intersection is left as read in. If NOPT = l and laminar flow is specified the flow intersection is adjusted so that the inner and outer streams have equal shear rates at the intersection. Must be I or greater. It allows the separations at various axial positions to be computed, i.e., dh - H/IDIV. If ITOPT = 0, then a negative height, H, causes calculations for for a stripping centrifuge to be performed, a positive height defines a rectifying centrifuge. If ITOPT — l, a stripping-rectifying centrifuge is assumed with the stripping section data read first. May be used only if ITOPT = 0. If LOPT = 0, then the flows given are used. If LOPT = I, then the ratio of the inner to the outer stream is maintained, but their magnitudes are adjusted to give the maximum separation. and final data card ) FORMAT (AIS) 238 TABLE E-2.--Subprograms Associated With Program CENTRI. Subprogram Duties PRESSU Computes the average pressure of the inner and outer streams. MAXVEL Computes the maximum velocity of the inner and the outer streams . TWO Used to compute the separations when the centrifuge contains both stripping and rectifying sections. CONVERG A scientific subroutine used to find a root of a non- linear equation after the root has been bracketed. SWITCH Used by CONVERG to update the closeness of the bracket. CHECK Used by CONVERG to speed convergence when the guesses bracketing the root are far apart. FASTC A short version of CONVERG used by TWO to converge on the composition at the feed locations. l. Rectifying centrifuge: This is the most easily handled case. After computing the constants in the above analytical solution, for any height, the true value of yP is bracketed by guessing values of yf and I.0. Using these guesses, subroutine CONVERG quicky converges to the true value of yP. Having computed the value of yP, the lean stream composition is computed using the expression _ SCFMU*(YF - YP) + SCFMD*YP Vi ’ SCFMD 2. Stripping centrifuge: In this type of centrifuge the feed _____________________ enters as the inner stream. A portion of the leaving inner stream is removed as the lean product stream with the remainder recycled as the 239 entering outer stream. The leaving outer stream is the rich product stream. By guessing a value of the lean stream composition, y], the rich stream composition can be computed by mass balanCe. = SCFMD*(YF - YL) + SCFMU*YL r SCFMU This value, however, must also agree with that calculated by the ana- lytical (approximate) solution based on the guessed lean product stream composition. Using 0.0 and yf as guesses to bracket the true lean stream composition, subroutine CONVERG is used to converge to the true value of y]. 3. Stripping-rectifying centrifuge: After computing the constants in the analytical (approximate) solution for both the strip- ping and the rectifying sections the following is the flow of calculations used to arrive at a solution: A. Bracket the true inner stream composition just past the feed position by guessing; yf/IO and (yf + 9)/IO. B. This represents the composition of the feed stream to the stripping section. Using the ideas in “2” above, the composition of the lean product stream leaving the stripper and the composition of the outer stream passing the feed posi- tion can be computed. C. The composition of the outer stream passing the feed position represents the feed to the rectifying section. Using the ideas in ”I” above, the rich product stream composition r—i 240 leaving the rectifier and the composition of the inner stream just before the addition of the feed can be computed. D. A mass balance is then written about the feed position. This is done by finding the composition after the mixing of the inner stream leaving the rectifier and the incoming feed occurs. If this composition, which is the stripper feed composition, matches the assumed value of this composition, calculations are complete. Otherwise, subroutine FASTC is used to provide a better approximation of the composition of the stripper feed and calculations are returned to step I'B“ above. While program CENTRI was used to provide many of the results embodied in the preceding work, a further example of its applicability is included here. Beams (A) published experimental results which were obtained when separating UF6 (23S and 238 isotopes) in a countercurrent rectifying centrifuge. The following are the operating conditions which were included with the experimental results: MW = 352 l mwz = 349 Temperature = 626.A°R Axis pressure: a value was not given, but was assumed to be 0.] psia. As shown in Chapter V—D-3, the actual pressure does not affect the separation. Feed comosition, yf = 0.9928 (U238) DIZQ = l.7AA x lo.5 lbm/ft/sec (computed from Beam's results) 24] Centrifuge radius = 3.673 inches Inner flow radius = l.9l2 inches Centrifuge length = I36 inches Due to the length of his centrifuge it was assumed that laminar flow would best approximate the velocity profiles in his centrifuge. Furthermore, the flow intersection was adjusted so that the shear rates of the inner and outer streams were equal at the flow inter- section. Table E-3 contains the feed and product stream flow rates for his various experimental runs along with the separations he observed and computed theoretically. Also included in Table E-3 are the theoretical separations and the ratio of the feed rate to the optimum feed rate computed by CENTRI. As can be seen in Table E-3, the agreement between the experimental results and those computed by CENTRI are rather good. While a complete listing and description of all the parameters computed and printed by CENTRI will not be given here in the text, a sample output follows. The sample output is for the separation of the gas pair SOZ-N2 in a countercurrent rectifying centrifuge which fully illustrates and defines all parameters computed. Output for a strip- ping centrifuge is identical and that for a stripper-rectifier combination consists of two pages detailing the parameters for each section. Following the sample output is a complete listing of CENTRI and its associated subroutines. 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A OZH aox oN Ax OOPmdu NZHPDOmmDm (‘5) PA LDC) W4 UUUUU APPENDIX F DESCRIPTION OF EXPERIMENTAL EQUIPMENT AND ANALYSIS OF ITS PERFORMANCE 27I APPENDIX F DESCRIPTION OF EXPERIMENTAL EQUIPMENT AND ANALYSIS OF ITS PERFORMANCE As part of an overall analysis of gas centrifugation it was deemed necessary that an experimental program to be undertaken paralleling an extensive theoretical analysis. It was hoped, with the aid of the experimental device, that assumptions regarding internal flow profiles could be evaluated as part of an overall comparison between experimental and calculated results. A. Background German isotope separation efforts in the early l9hOs were concentrated on short bowl centrifuges (L/D N 5) operating in a vacuum to reduce friction and hence pOWer requirements. The short bowls were able to operate at higher rotational speeds (larger pressure diffusion) without crossing the first whirling speed (Chapter IV). The German centrifuge developments were demonstrated in both Russia and the United States (University of Virginia) by Zippe (7). Publi- cation of current work and, to a large degree, past work, which detailed the design aspects of the gas centrifuge are classified due to However, unclassified its potential use in the enrichment of Uranium. publications of Zippe's demonstration devices at the University of 272 273 Virginia are available. These publications served as a basis for the construction of an experimental centrifuge. The centrifuges built and operated by Zippe were small in size (up to A inches in diameter and up to 20 inches in length) with very small (milligrams/sec) feed rates. It was realized that for the concept of gas centrifugation to be extended into the realm of such things as flue gas desulfurization that feed rates would have to be increased significantly and consequently also the size of the centri- fuges used by Zippe, Nevertheless, Zippe's work represented the best available information to be used as an experimental guide. B. Centrifuge Design Figure F-l contains a schematic diagram illustrating the prin- ciples of Zippe's design. It should be noted immediately that the design contains a stationary center pipe which is subject to the results presented in Chapter II-E. Feed gas flows into the center of the rotating bowl (A) through one of the concentric center pipes. The feed gas is accelera- ted to spin with the bowl by the momentum of the surrounding gas which obtains its energy by shearing with the centrifuge wall. Drag on the stationary pitot tube (0) located at the top of the bowl tends to move the gas inward toward the axis, while the rotating disk (N) at the bottom of the bowl tends to move the gas near the axis, slowed due to shear with the stationary center pipe, outward to the wall. The effect is to create a countercurrent flow with an outer gas stream 271+ L ‘— Centrifuge rotor Flexible steel needle Rotor shaft Concave plate to support needle Housing Steel balls to support housing and provide for lateral motion Centering magnets Support spring Hollow permanent magnet Rotor hollow permanent magnet Feed gas tube Heavy product gas Light product gas Shield plate Gas scoops Drive motor armature Motor field winding Screw-type grooves for molecular pump Figure F-l.--A Schematic of the Zippe Gas Centrifuge. 275 moving up along the periphery to the t0p pitot tube (0) where it is directed downward in the inner stream. The tOp pitot tube (0) also serves to extract a portion of outer stream (L) which is richer in heavier species. On the other hand, the stationary pitot tube (0) at the bottom shielded by the rotating baffle (N) extracts a portion of the inner stream (M) which is leaner in the heavier Species. The entire bowl spins in a vacuum chamber with the vacuum being produced (after start up) by a molecular pump (R). Support for the bowl is concentrated on a needle bearing running in damped bearing housing (D). Vertical alignment of the rotating cylinder is maintained by a magnetic bearing (I, J) at the tOp. Other bearing components shown are used to further keep the centrifuge centered and reduce vibrational problems. To eliminate the need for any mechanical connections the armature (P, a disk) of the induction drive motor is attached directly to the centrifuge bottom end cap. The field windings (Q) for the induction motor are attached to the vacuum housing directly under the armature. C. Modifications Made in Zippe‘s Design When Zippe's centrifuge design was scaled up to a centrifuge design having a length of 40 inches and a diameter of 8 inches, several modifications of his basic design were necessary. Figure F-Z contains a schematic diagram of the modified form of the Zippe gas centrifuge which was constructed. The device, made of high tensile 007 r~7~mcm mmm cocmcwe_ .o mucoEocsmmoE mcnmmoco cow mcouoeocms 0H .u mcouoEOHOm .m m m x . { I > ..a 1 p m Emocum poncOLQ Lo_m Emmcum uozooco coop co_p_m0oeoo ascex mo mmm comm couoc mm:w_cucou Ill