Maximum Rod Research Articles

Developing and modeling of the “Laguna Verde” BWR CRDA benchmark

Reactivity initiated accidents (RIA) and design basis transients are one of the most important aspects related to nuclear power reactor safety. These events are re-evaluated whenever core alterations (modifications) are made as part of the nuclear safety analysis performed to a new design. These modifications usually include, but are not limited to, power upgrades, longer cycles, new fuel assembly and control rod designs, etc. The results obtained are compared with pre-established bounding analysis values to see if the new core design fulfills the requirements of safety constraints imposed on the design. The control rod drop accident (CRDA) is the design basis transient for the reactivity events of BWR technology. The CRDA is a very localized event depending on the control rod insertion position and the fuel assemblies surrounding the control rod falling from the core. A numerical benchmark was developed based on the CRDA RIA design basis accident to further asses the performance of coupled 3D neutron kinetics/thermal-hydraulics codes. The CRDA in a BWR is a mostly neutronic driven event. This benchmark is based on a real operating nuclear power plant — unit 1 of the Laguna Verde (LV1) nuclear power plant (NPP). The definition of the benchmark is presented briefly together with the benchmark specifications. Some of the cross-sections were modified in order to make the maximum control rod worth greater than one dollar. The transient is initiated at steady-state by dropping the control rod with maximum worth at full speed. The “Laguna Verde” (LV1) BWR CRDA transient benchmark is calculated using two coupled codes: TRAC-BF1/NEM and TRAC-BF1/ENTRÉE. Neutron kinetics and thermal hydraulics models were developed for both codes. Comparison of the obtained results is presented along with some discussion of the sensitivity of results to some modeling assumptions.

Formation, Thermal Stability and Mechanical Properties of Ca-Based Bulk Glassy Alloys

New bulk glassy alloys were formed in the Ca–Mg–Cu system by the copper mold casting method. The maximum rod diameter (dmax) for the formation of a glassy phase was 2 mm for Ca67Mg19Cu14 and above 4 mm for Ca57Mg19Cu24. The glass transition temperature (Tg), crystallization temperature (Tx), supercooled liquid region (ΔTx=Tx−Tg) and reduced glass transition temperature (Tg⁄Tm) are 387 K, 407 K, 20 K and 0.60, respectively, for the former alloy and 404 K, 440 K, 36 K and 0.64, respectively, for the latter alloy. There is a tendency for dmax to increase with increasing ΔTx and Tg⁄Tm. Young’s modulus (E) and compressive fracture strength are 38 GPa and 545 MPa, respectively, for the Ca57Mg19Cu24 alloy rod with a diameter of 2 mm. The success of synthesizing bulk glassy alloys in the simple metal (Ca) base system makes it important as a basic alloy system for examining fundamental chemical and physical properties of bulk glassy alloys.

Formation and Thermal Stability of Ca-Mg-Ag-Cu Bulk Glassy Alloys

Calcium-based bulk glassy alloys in Ca–Mg–Ag and Ca–Mg–Ag–Cu systems were produced by a metallic mold casting method. The maximum rod diameter (dmax) for the formation of a glassy phase was 4 mm for Ca60Mg20Ag20 and 7 mm for Ca60Mg20Ag10Cu10. The glass transition temperature (Tg), crystallization temperature (Tx), temperature interval of supercooled liquid region (ΔTx=Tx−Tg) and melting temperature (Tm) were 401 K, 435 K, 35 K and 654 K, respectively, for the former alloy and 398 K, 428 K, 30 K and 624 K, respectively, for the latter alloy. The simultaneous addition of Ag and Cu in the Ca–Mg–(Ag, Cu) system causes a steep decrease in Tm and an increase in the reduced glass transition temperature (Tg⁄Tm), leading to an increase in the glass-forming ability.

Axial rod growth and age estimation of the sea pen, Halipteris willemoesi Kölliker

Halipteria willemoesi is a large octocoral commonly found in the Bering Sea. It is a member of a ubiquitous group of benthic cnidarians called sea pens (Octocorallia: Pennatulacea). Sea pens have a skeletal structure, the axial rod, that in cross section exhibits growth rings. Pairs of adjacent rings, or ring couplets, were assumed to be annuli and were used to estimate the age and growth of H. willemoesi. Twelve axial rods, extracted from H. willemoesi collected in the Bering Sea, were selected to represent small (25-29 cm total length), medium (97-130 cm TL) and large (152-167 cm TL) colonies. Each rod resembled a tapered dowel; the thickest part (0.90-6.75 mm in diameter) was at about 5-10% of total length from the base tip, the distal part was more gradually tapered than was the base. The number of ring couplets increased with rod size indicating their utility in estimating age and growth. Estimated age among rods was based on couplet counts at the thickest part of each rod: the average estimated age (±SE) was 7.1 ±0.7, 19.3 ±0.5, and 44.3 ±2.0 yr for small, medium and large-size rods, respectively. Based on these estimated ages, average growth rate in total length was 3.9 ± 0.2, 6.1 ± 0.3, and 3 6 ± 0.I cm yr -1 for small, medium, and large-size colonies. The average annual increase in maximum rod diameter among all colonies was 0.145 ± 0.003 SE mm yr -1 ; therefore, age prediction from maximum rod diameter was calculated (estimated age (yr) = 7.0 * (maximum rod diameter, mm) -0.2; R 2 = 0.99). At maximize diameter, the average couplet width was relatively constant among the three colony sizes (0.072 ± 0.05 mm). X-ray diffraction and electron microprobe analyses revealed that the inorganic portion of the rod is composed of a high-magnesium calcite. Radiometric validation of these age and growth rate estimates was attempted, but high amounts of exogenous 210 Pb precluded using the disequilibria of 210 Pb: 226 Ra. Instead, 210 Pb activities were measured in a series of cores extracted along the axial rod. These activities ranged from 0.691 ± 0.036 (SE) to 2.76 ± 0.13 dpm g -1 , but there was no pattern of decay along the length of the rod; therefore, the growth rates and corresponding ages could not be validated. Based on estimated age from ring couplet counts, growth in total rod length is slow at first, fastest at medium size, and slows toward maximum size, with an estimated longevity approaching 50 yr.

Development of a Transient Boiling Transition Analysis Method Based on a Film Flow Model

A new single-channel, transient boiling transition (BT) prediction method based on a film flow model has been developed for a core thermal-hydraulic code. This method could predict onset and location of dryout and rewetting under transient conditions mechanically based on the dryout criterion and with consideration of the spacer effect. The developed method was applied to analysis of steady-state and transient BT experiments using BWR fuel bundle mockups for verification. Comparisons between calculated results and experimental data showed that the developed method tended to predict occurrence of rewetting earlier, however, onset time of BT and maximum rod surface temperature were well predicted within 0.6 s and 20°C, respectively. Moreover, it was confirmed that consideration of the spacer effect on liquid film flow rate on the rod surface was required to predict dryout phenomena accurately under transient conditions.

Synthesis and characterization of rigid-chain liquid-crystalline polymers containing laterally attached side rods

A series of rigid-chain polymers containing different concentrations of laterally attached side rods was synthesized. These polymers exhibited liquid crystallinity even up to a maximum side rod concentration of 20 mol %. The crystallinity of the polymers, however, decreased with an increase in the side-rod concentration. These polymers had lower dielectric constants compared with their parent polymers, that is, similar polymers, but without laterally attached side rods. A dielectric constant of 2.6 can be achieved by incorporating 10 mol % of laterally attached side rods, which is 0.5 lower than that of its parent polymer. The reduction of dielectric constant may be attributed to low crystallinity as well as the less dense packing structure of the polymers induced by the incorporation of laterally attached side rods. This series of polymers also had good thermal stability. © 2001 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 39: 1288–1294, 2001

Effects of high peak power microwaves on the retina of the Rhesus monkey

We studied the retinal effects of 1.25 GHz high peak power microwaves in Rhesus monkeys. Preexposure fundus photographs, retinal angiograms, and electroretinograms (ERG) were obtained to screen for normal ocular structure and function and, after exposure, as endpoints of the study. Histopathology of the retina was an additional endpoint. Seventeen monkeys were randomly assigned to receive sham exposure or pulsed microwave exposures. Microwaves were delivered anteriorly to the face at 0, 4.3, 8.4, or 20.2 W/kg spatially and temporally averaged retinal specific absorption rates (R-SAR). The pulse characteristics were 1.04 MW ( approximately 1.30 MW/kg temporal peak R-SAR), 5.59 micros pulse length at 0, 0.59, 1. 18, and 2.79 Hz pulse repetition rates. Exposure was 4 h per day and 3 days per week for 3 weeks, for a total of nine exposures. The preexposure and postexposure fundus pictures and angiograms were all within normal limits. The response of cone photoreceptors to light flash was enhanced in monkeys exposed at 8.4 or 20.2 W/kg R-SAR, but not in monkeys exposed at 4.3 W/kg R-SAR. Scotopic (rod) response, maximum (combined cone and rod) response, and Naka-Rushton R(max) and log K of scotopic b-waves were all within normal range. Retinal histopathology revealed the presence of enhanced glycogen storage in photoreceptors among sham (2/5), 8.4 W/kg (3/3), and 20.2 W/kg (2/5) exposed monkeys, while enhanced glycogen storage was not observed in the 4.3 W/kg (0/4) exposed group. Supranormal cone photoreceptor b-wave was R-SAR dependent and may be an early indicator of mild injury. However no evidence of degenerative changes and ERG depression was seen. We concluded that retinal injury is very unlikely at 4 W/kg. Functional changes that occur at higher R-SAR are probably reversible since we saw no evidence of histopathologic correlation with ERG changes. Bioelectromagnetics 21:439-454, 2000. Published 2000 Wiley-Liss, Inc.

Dynamic surface displacement measurement in 1-3 and 1-1-3 piezocomposites

The sensitivity of 1-3 piezocomposites can be improved by the addition of a third phase or interlayer around the piezoceramic rods which results in a composite with 1-1-3 connectivity. In the current investigation, dynamic surface displacements of both 1-3 and 1-1-3 piezocomposites were measured using a laser Doppler heterodyne interferometer. Dynamic displacement profiles showing the nonuniform deformation of the piezoelectric rod and the nonpiezoelectric polymer were obtained at 10 Hz, 100 Hz, and 1.0 kHz. Measurements of the displacement at the center of the piezoelectric rod were performed over the frequency range from 10 Hz to 27 kHz. Three different interlayer conditions were considered. The experimental measurements demonstrated that both the surface displacement profile and the maximum rod displacement varied with frequency and interlayer properties. The dynamic displacement profiles were also compared with previously measured static displacement profiles.

Anchor rod forces and maximum bending moments in sheet pile walls using the factored strength approach

Recently, the limit states approach using factored strength has been recommended in geotechnical design. Some recent research has indicated that the application of limit states design using recommended load and strength factors leads to conservative designs compared with the conventional methods. In this study the influence of sheet pile wall geometry, type of water pressure distribution, and different methods of analysis on the maximum bending moment and achor rod force are presented. Recommendations are made to make the factored strength design compatible with conventional design. Key words: factored strength, working stress design, ultimate limit state design, anchored sheet pile wall, bending moment, anchor rod force.

THE STRUCTURE OF PHYCOBILISOMES IN MUTANTS OF Synechococcus sp. STRAIN PCC 7942 DEVOID OF SPECIFIC LINKER POLYPEPTIDES

Abstract— The effect of elimination of the 30, 33 and 9 kDa phycobilisome rod‐linker polypeptides on rod length was examined by electron microscopy of phycobilisomes isolated from wild‐type Synechococcus sp. strain PCC 7942 and from genetically engineered mutants with lesions in the genes encoding the rod‐linker polypeptides. The maximum rod length in the absence of the 33 kDa linker polypeptide was two phycocyanin hexamers, whereas rods with up to five hexamers were found in the mutant strain lacking the 30 kDa linker polypeptide. Elimination of the 9 kDa linker polypeptide did not have a significant effect on rod length. Finally, mutants lacking either the 30 or 33 kDa rod‐associated linker polypeptides had an increased number of rods that terminated with a phycocyanin trimer. These observations are discussed with respect to the role of the linker polypeptides in the biosynthesis of the rod substructure.

Experimental Study of the Outside Vapor Deposition Process

Outside vapor deposition has been studied experimentally, and the deposition process parameters have been examined. The system consisted of an industrial-type burner and a cylindrical bait rod. The following aspects of this process were studied: 1) the temperature profile of the bait rod surface, 2) the deposition efficiency measured in terms of the mass of deposited SiO2, and 3) the size distribution of the deposited particles. The maximum bait rod surface temperature was, as expected, at the stagnation point. The temperature on the surface dropped rapidly along the axis of the bait rod away from the stagnation point; this is expected to influence the thermophoretic deposition rate. The particle size distribution shows that the particle diameters are near 0.2 μm and above.

Development of morphological types and distribution patterns of amacrine cells immunoreactive to tyrosine hydroxylase in the cat retina

Using an antibody against tyrosine hydroxylase on newborn to 30-day kitten retinas, we have been able to follow the development of the dopaminergic amacrine cells of the cat retina by light-microscopical investigations of retinal wholemounts. The Type 1 or large Toh+ amacrine cells described by others (Oyster et al., 1985; Törk & Stone, 1979) and named A18 from a Golgi study (Kolb et al., 1981), is at birth (P1) an immature neuron with a small cell body and two or three simple thick radiating dendrites stratifying in stratum 1 with many of the dendrites ending in enlarged growth cones. With increasing postnatal age, the cell body size increases from 12.5 microns diameter to reach 15.5 microns diameter at P30. The dendritic fields also increase in size and complexity. At P1, cells of the central area exhibit dendritic appendages which then develop progressively until at P13 (after eye opening) they are part of rudimentary rings and by P30 the dendritic plexus of Toh+ dendrites and rings in stratum 1, typical of the adult cells, are complete. Toh+ stained processes with growth cones that run deep in stratum 5 of the inner plexiform layer and processes passing to the outer plexiform layer first become apparent at P1 in cells of central inferior retina but not till after P13 are these processes clearly expressed. At P1, the total number of Toh+ Type 1 cells is approximately 4000 and this number remains unchanged to the adult retina. However, the retina increases in size over the P1-P30 stage and thus the mean density of Type 1 Toh+ cells decreases from 30/mm2 at P1 to 18/mm2 at P30. The maximum density of Type 1 Toh+ cells occurs in central retina 2-4 mm superior temporal to the area centralis, corresponding to the maximum rod photoreceptor concentration. A second type of small Toh+ amacrine cell can be visualized at P1. This Type 2 cell is characterized by a much smaller cell body than Type 1 cells (9 microns diameter), and with faintly stained dendrites located in stratum 3 of the inner plexiform layer. During later postnatal days, Type 2 cells gradually become unstainable and only few are still seen in far peripheral retina by P23. Type 2 Toh+ cells from a total population of 40,000 cells at P1 with their highest density occurring in peripheral retina. By P13, they cannot be seen in central retina and are reduced to a total population of cells staining for the antibody of 7400 cells in far peripheral retina.(ABSTRACT TRUNCATED AT 400 WORDS)

LWR fuel rod behavior during severe accidents

Chemical interactions between UO 2 fuel and Zircaloy cladding up to 2350°C are described. UO 2/Zircaloy single effects tests have been performed with short LWR fuel rod segments in inert gas and under oxidizing conditions. The reaction kinetics of molten Zircaloy cladding with solid UO 2 fuel has been investigated with UO 2 crucibles containing molten Zircaloy. The UO 2/Zircaloy reactions obey parabolic rate laws. The oxygen uptake by solid Zircaloy due to chemical interaction with UO 2 occurs nearly as quickly as that from the reaction with steam or oxygen. To study the competing effects of the external and internal cladding oxidation under realistic boundary conditions and the influence of the uncontrolled temperature escalation due to the exothermic steam/Zircaloy reaction on the maximum cladding temperature, single rod and bundle experiments have been performed. Electrically heated fuel rod simulators, including absorber rod material (Ag, In, Cd alloy), guide tubes and grid spacers are used. The maximum measured cladding temperature during the temperature escalation was about 2200°C. The failure temperature of the absorber rods and the extent of bundle damage depends on the guide tube material (Zircaloy or stainless steel) and varies between 1200 and 1350°C. The molten materials and liquid reaction products can relocate and form large coherent lumps on solidification, which may result in complete blockage of the fuel rod bundle cross section. In the future, 7 × 7 bundle experiments of 2 m overall length will be performed in the new CORA facility to study, in addition, the influence of quenching on fuel rod integrity.

Capillary phenomena. Part 24.—Properties of fluid bridges between solid rods in a gravitational field

Properties are predicted in dimensionless terms for fluid bridges between vertically aligned rods in a gravitational field. Meridian curves, which on rotation give the bridges, have been obtained for reduced rod radii R= 0.5–10.0. Actual rod radii are given by r=aR, where the capillary constant a is ca. 3.8 mm for water bridges in air and 8 mm for silicon at its freezing point in an inert atmosphere subjected to terrestrial gravity. Properties such as the pressure difference across the bridge interface and the force on the upper rod have been obtained for cases of varying fluid volume at several fixed rod-separation distances and of varying rod separation at fixed volumes. Stability limits as well as equilibrium paths are discussed. The results also provide a basis for analysing ideal float-zone refining, e.g. a silicon zone formed from its own solid: in these cases there are limitations on the fluid volume and on the angle of inclination of the fluid interface needed for solidification of a rod of purified material of uniform radius. Taking a growth angle of 11° for silicon, the maximum rod radius that can be grown from a zone of melt is ca. 40 mm.

Design of High-Performance Fuel Pin Simulators for Thermodynamic Experiments with Nuclear Fuel Elements

Investigations of the thermodynamic behavior of reactor fuel elements require out-of-pile experiments to be carried out on fuel element mockups made up of electrical heater rods. The results of these experiments depend strongly on the similarity of thermodynamic behavior between heater rods applied and nuclear fuel rods to be simulated.Typical requirements for the heater rods that simulate the nuclear fuel rods of interest are, for example, heat flux density and the associated heat flux density distribution in case of nonuniform coolant conditions and heat capacity. Because of the various modes of heat production in nuclear fuel rods, electrically heated rods in experiments are able to only partially meet these requirements. A type I heater with a nickel-chromium conductor, maximum rod power up to 340 W/cm at cladding temperatures up to 1200 K, and a type II heater with a tantalum-tungsten conductor, rod powers up to 1000 W/cm at cladding temperatures of 1200 K, were examined experimentally in a liquid sodium flow and showed lifetimes up to 10 h and more. They can be fabricated with different geometrical dimensions (e.g., diameters, heated and unheated lengths) and varying axial heat production.

The distribution of rods and cones in the retina of the cat (Felis domesticus).

AbstractThe distribution of rods and cones in the cat retina was studied by light microscopy. The rods and cones were counted from the area centralis to the temporal periphery in photomicrographs of transverse sections through the inner segments. A 16% correction for the effect of tissue shrinkage was applied to the densities obtained in fixed‐dehydrated tissue, by comparing these counts with those obtained from similar retinal areas in fresh tissue.The cone distribution was characterized by a steep increase in cone density centrally, which was elongated in the nasotemporal axis, and coincided approximately with the central increase in ganglion cell density (Stone, '65). Cone density in the area centralis peaked at 26,000–27,000/mm2, and fell to a plateau of 4000/mm2 in the periphery and to less than 3000/mm2 near the ora serrata. The density of rods was greater than the density of cones in all regions. The rods reached a maximum density of 460,000/mm2 at an eccentricity of 10–15°, in a region which completely surrounded the central cone elevation. Centrally, the rod distribution was characterized by a sharp fall in density, reaching a low of 275,000/mm2 at the point of peak cone density. Peripherally, there was a plateau of high rod density at above 400,000/mm2, out to about 30° temporal eccentricity. Beyond this plateau, rod density steadily fell, and reached a low of 250,000/mm2 near the ora serrata. The rod/cone ratio reached a low value of 10.5–11.0 in the central region of maximum cone density. It then rose steeply to reach a plateau of about 65 in the periphery, while near the ora serrata there were 100 rods for each cone.The rods were arranged in well defined rows, and each rod was surrounded by six other rods. The cones were interspersed throughout the rod mosaic and were less regularly arranged than the rods. In fresh tissue, rod outer segment diameters ranged from 1.0 μ to 1.6 μ. rod diameters were smallest in the region of maximum rod density and increased, along with the decrease in rod density, both in the periphery and in the area centralis. Two thirds of the central decrease in rod density from the pericentral peak could not be accounted for by the increase in the percentage of space occupied by cones, and was consistent with a central increase in rod diameter.Receptor‐ganglion cell convergence was estimated by comparing receptor densities with the ganglion cell densities of Stone ('65). Cone‐ganglion cell convergence reached a minimum in the area centralis at the cone density peak, probably indicating the retinal region in which photopic acuity is at an optimum. Minimum rod‐ganglion cell convergence, as estimated from the rod/large ganglion cell ratio, reached a minimum at about 7.5° central to the peak of rod density, and at about the same eccentricity as the human scotopic acuity optimum.

A Novel Sweep Method for Monopole Mass Spectrometers

The regular mass range of the monopole mass spectrometer can be extended to very high mass numbers if at the end of the regular mass range, the maximum ac rod voltage is kept constant and the dc rod voltage is reduced. During this extended mass range the mass is inversely proportional go this dc voltage. If the ion energy is kept constant the sensitivity stays essentially constant over the whole mass range and the resolution is inversely proportional to the mass. If the ion energy is reduced in proportion to the dc voltage the resolution stays constant, but the sensitivity is reduced. This method of mass scan permits with excellent sensitivity the detection of vacuum system contaminants of very high molecular weight. The total scanning time necessary to cover this wide mass range is much shorter than the time required for a regular scan over the same mass range.

High Volume Pumping with Sucker Rods

Abstract Recent studies have shown that often large fluid volumes can be lifted by sucker rods with great effectiveness from shallow to medium-depth wells. Wherever feasible, significant redaction in cost of elevating a barrel of fluid may be realized by the operator. The advent of longer stroke units with improved geometry, larger speed reducers and bottom-hole pumps, as well as the new high-strength sucker rods, make high volume rod pumping a practical and economical achievement. Past applications of this type generally have been considered only for bottom-hole centrifugal pumps. This paper reviews some field studies and emphasizes the practicality of high volume production with sucker rods. Introduction A century ago the almost universal mechanism for artificially lifting fluid in an oil well was the standard rig-front. A wooden walking beam drove a string of hickory sucker rods, often called "well poles", as many as ten 15-in.- strokes/minute with the maximum tensile stress of the rods about 12,800 psi. The bottom-hole pump was cast iron or brass with the barrel approximately 1 1/4-in. in diameter, and the well depth ranged from 500 to 1,000 ft. The torque capacity of the band wheel and flat-belt speed reducer ran only a few thousand inch-pounds, and the unit's structural capacity was from 1,000 to 1,500 lb. Today, the structure of a modern pumping unit with its 240-in. maximum stroke reaches higher than the wooden drilling and servicing derricks of those early days. Its structural capacity exceeds 47,000 lb, with a torque rating greater than 2.5 million in. lb. The plunger diameter of some bottom-hole pumps (casing-type) runs as high as 5 3/4 -in. Perhaps the greatest improvement is in the sucker rod, which now has a tensile stress of some 140,000 psi. Today practice sucker rod pumping approaches 13,000 ft, and capacities of 5,000 and 6,000 B/D from shallow to medium depths are handled with ease. Volumes as high as 9,500 B/D are entirely practical with modern sucker rod pumping equipment. Comparing the modern sucker-rod system with its counterpart of 100 years ago produces some startling figures. The structural capacity of the modern unit has increased nearly fifty-fold, the torque capacity perhaps a thousand-fold, the area of the bottom-hole pump over 20 times; the stroke length nearly twenty-fold; and maximum rod tensile stress nearly 12 times. With the increased stroke length, even the maximum rate of polished rod travel per minute has increased five or six times. Since all of these are compounding increases, the sucker-rod pumping unit of today has the capacity for producing massive fluid volumes from relatively great depths.

Optimization of Neutron Flux Distribution

A method of optimizing neutron flux distribution to obtain maximum power generation from a given core is presented. Constraining conditions governing maximum heat flux and control rod absorption are taken into consideration. It is shown that the problem is reduced to that of optimal control. In the case of one group model the phase plane technique is effective for an intuitive analysis of the problem. It is verified that the solution so obtained satisfies Pontry-agin's principle. The solution is one of “bang-pang type”. The result is extended to the case of two group model, assuming that the solution will again be of bang-pang type. A comparison is made between the one group and the two group models.

Optimization of Neutron Flux Distribution

A method of optimizing neutron flux distribution to obtain maximum power generation from a given core is presented. Constraining conditions governing maximum heat flux and control rod absorption are taken into consideration. It is shown that the problem is reduced to that of optimal control. In the case of one group model the phase plane technique is effective for an intuitive analysis of the problem. It is verified that the solution so obtained satisfies Pontry-agin's principle. The solution is one of “bang-pang type”. The result is extended to the case of two group model, assuming that the solution will again be of bang-pang type. A comparison is made between the one group and the two group models.