Feed Velocity Research Articles

Performance analyses on a concentrated photovoltaics driven direct contact membrane distillation coupled system

For concentrated photovoltaics (CPV), most of incident solar energy is wasted as heat. If the heat is not removed in time, the junction temperature of CPV will rise and eventually worsen the performance. To remove and harvest the waste heat, a direct contact membrane distillation (DCMD) is coupled to CPV for fresh water generation in this study. Based on the first and second laws of thermodynamics, key performance indicators for each component within the coupled system and the whole coupled system are derived. The basic performance features of the coupled system are revealed. The utmost power density, utmost energy efficiency and utmost exergy efficiency of the coupled system allow to be, 35.37%, 35.37% and 5.37% greater than that of single CPV, respectively. After optimizing electric current of CPV, the effects of solar irradiance, operating temperature of CPV, temperature and flow velocity of feed and permeate sides and thickness of membrane on the coupled system performance are studied via extensive parametric studies. In addition, monthly average performance of the proposed coupled system from January to October in 2021 in Ningbo, China is investigated. The results acquired may be beneficial to design and run such a real CPV-DCMD coupled system.

Experimental validation of an adsorbent-agnostic artificial neural network (ANN) framework for the design and optimization of cyclic adsorption processes

The efficacy of an adsorbent agnostic machine-learning surrogate model for rapid design and optimization of a Skarstrom cycle vacuum swing adsorption (VSA) process is experimentally validated. The surrogate model is trained to predict the process performance using adsorbent features that include hypothetical Langmuir adsorption isotherm parameters, particle density, porosity and bed voidage, and process variables such as pressure, step duration and feed velocity. The training data was generated from a detailed process model for 20,000 unique combinations of the training variables. The model shows high accuracy of Radj2>0.99 for predicting key performance parameters such as product purity, recovery and productivity. The ability of this surrogate to predict the experimental performance for the purification of O2 from air on two adsorbents, namely 13X and LiX zeolites, was studied. Two separate multi-objective optimization studies, to maximize purity and recovery, and to maximize productivity and purity were performed. For these optimization studies, the volumetrically measured isotherms of N2 and O2 were used as inputs to the surrogate model. Note that these isotherms were not a part of the dataset used to train the model. Nine points were chosen from the Parteo curves and the corresponding decision variables were used as set-points in a two-column lab-scale rig. The average difference between the calculated and experimentally measured purity, recovery and productivity was 3%, 5% and 9%, respectively. This study provides the necessary confidence to use surrogate-based process models for adsorbent screening and adsorption process optimization.

Non-catalytic direct conversion of CH4 and CO2 into high-quality syngas

Both partial oxidation and CO2 reforming can theoretically be used to produce syngas from natural gas. In this work, the possibility of combining the two methods is explored by the use of computational fluid dynamics with detailed chemical reaction mechanisms. It is showed that methane partial oxidation reactor being used industrially can be directly used to transfer CH4 and CO2 into syngas. The effect of the ratio, feed velocity and preheating temperature of the reactants are examined. The optimal overall ratio of reactants is 5:4:1 for CH4 to O2 to CO2, and the optimal preheating temperature and inlet velocity is 923 K and 200–300 m/s, respectively. The conversions of CH4 and CO2 are 99.88% and 46.85%, respectively, and the yields of CO and H2 are 90.8% and 65.13%, respectively. Cooling down the product can easily obtain the target product (syngas) from water vapor. These results suggest an economically competitive way to use the two greenhouse gases for the production of syngas.

Investigating different types of fluids in pulsating heat pipes for improving heat transfer in an air-cooled heat exchanger

Air-cooled heat exchangers are utilized in various industries to cool process fluids. To have a better heat transfer, fins are used in the structure of exchangers. In this study, a set of experiments were performed to investigate the improvement in the amount of heat transfer in an air-cooled heat exchanger. For this purpose, a real case study was conducted on air-coolers in a natural gas compressor station; afterward, the finned tubes were replaced by pulsating heat pipes (PHPs), and a different type of fluid was applied to the pulsating heat pipes. The laboratory system was set up to measure the overall heat transfer coefficient under two conditions involving either finned tubes or a PHP with R134, R22, and filled acetone fluids. Finally, the results were verified by the theory method. The results showed that all the three fluid agents (R134, R22, and Acetone) with natural convection of 9.94 m/s, an inlet fluid velocity of 5.54 m/s, and an average feed velocity were able to enhance the heat transfer coefficient by 10, 13, and 11%, respectively. In addition, compared with finned tubes, R134, R22, and Acetone improved the natural convection heat transfer coefficient by 22.2, 18.6, and 21.7%, respectively.

Rotating flow characteristics in centrifugal high gradient magnetic separation and its effect on particle capture behavior

Centrifugal high gradient magnetic separation (CHGMS) concentrates weakly magnetic minerals at high selectivity, and the rotating flow in the separating area of the CHGMS separator significantly affects its separation performance. In this work, a simulation model based on finite element method was built up to simulate the rotating flow, and the effects of two most important operating parameters affecting the flow, i.e., matrix rotation speed and feed velocity, on the matrix capture behavior were analyzed through a pilot-scale CHGMS separator in concentrating a fine ilmenite ore. The COMSOL Multiphysics simulation of the rotating flow indicates that the slurry in the separating area forms into a rotating flow as a result of the rotation of matrix, and its flow trace length of unit volume slurry prolongs with increase in the matrix rotation speed and reduces with increase in the feed velocity, thereby affecting the collision probability of matrix to particles. It was found that the slurry velocities on the upstream and downstream of magnetic wire in the matrix are significantly higher than those on the right and left sides. There is a large radial velocity gradient in the separating area, which produces a fluid shearing stress to particles in the slurry, thus improves the loosing state of particles and the capture selectivity of matrix; this velocity gradient increases with increase in the matrix rotation speed and decreases with increase in the feed velocity. The capture experiments of matrix to ilmenite particles indicate that the increased matrix rotation speed has a more significant effect on the capture behavior than that of the feed velocity, and these results are basically consistent with those of simulations. This investigation provides an important reference for the optimum development and application of CHGMS technology.

Pulsating high-gradient magnetic separation of chalcopyrite and talc

The selective flotation of chalcopyrite from talc in practice is currently difficult and consumes a large amount of depressors, resulting in high production and environmental costs. The pulsating high-gradient magnetic separation (PHGMS) method has a strong potential for the effective separation of chalcopyrite and talc, based on their difference in magnetic susceptibility. In this work, the effects of magnetic induction, pulsating frequency and feed velocity of a pliot-scale PHGMS separator on the separation performance for pure chalcopyrite and talc minerals were investigated. The results indicate that the PHGMS achieves a selective separation of chalcopyrite from talc; and under the optimized conditions, a chalcopyrite concentrate of 30.22% Cu grade with 75.79% chalcopyrite recovery at talc content as low as 5.77% was produced from a chalcopyrite-talc mixture assaying 10.69% Cu and 66.67% talc. It was found that the chalcopyrite recovery for the ultra-fine particles fraction below 30 µm is lower than those of coarser fractions, as a result of the increased hydrodynamic drag over the magnetic force acting upon the ultra-fine particle. Entrainment may also occur between ultra-fine talc and chalcopyrite particles, which reduces the Cu grade for such ultra-fine particles in the copper concentrate. The PHGMS method may provide an environment-friendly and low-cost technology for the separation of chalcopyrite and talc.

Experimental and numerical study on thermodynamic characteristics of a vacuum membrane distillation system based on mechanical vapor recompression for sulfuric acid waste

Energy conservation and emission reduction in the field of industrial wastewater treatment have attracted worldwide attention. This study focused on the concentration and recovery of industrial sulfuric acid waste by a vacuum membrane distillation system based on mechanical vapor recompression (VMD-MVR). Mathematical models were built considering the mass and energy conservation principles and relevant thermal equilibrium theory, and an experimental platform, which was suitable for sulfuric acid solution, with excellent corrosion resistance was also constructed. Initially, the performance of VMD-MVR from the energetic and exergetic standpoints was evaluated based on the experimental data. Furthermore, the effects of critical parameters on compression and condensation heat transfer in VMD-MVR were simulated and revealed, and exergy destruction analysis was conducted using various selected parameters. The obtained experimental results indicated that VMD-MVR exhibited a membrane flux of 1.7 kg·m−2·h−1, separation efficiency of 99.9%, and performance coefficient (COP) of 7.88 under the following conditions: feed concentration: 5%; feed temperature: 353.0 K; feed velocity: 1.6 m·s−1; permeate side pressure: 45 kPa; heat transfer temperature difference (ΔThex): 2 K; and compressor frequency: 40 Hz. The exergy input, exergy destruction, exergy output, and exergy efficiency were 4.95 kW, 4.691 kW, 0.259 kW, and 5.24%, respectively. Moreover, according to the numerical simulation results, with a decrease in the feed concentration and ΔThex or with an increase in the feed temperature, feed velocity, and permeate side pressure, the compression ratio and exergy destruction in VMD-MVR decreased, whereas the COP increased.

Aqueous-phase reforming of glycerol over Pt-Co catalyst: Effect of process variables

This study examined the influence of process variables (glycerol concentration in feed, coupled temperature/pressure and space velocity) in the catalytic performance in the APR of glycerol over 0.3Pt/CoAl catalyst in a continuous fixed-bed reactor in order to maximize the production of H2. The effect of glycerol concentration in the feed was studied from 5 to 20 wt%, the coupled temperature/pressure varied from 225 °C/25 bar to 260 °C/50 bar and the spatial velocity was changed from 0.68 to 17 h-1. Our results reflected that H2 production was favored at higher reaction temperature/pressure (3.62 vs. 2.49 molH2/molGly-converted, at the most severe and mild conditions, respectively), lower WHSV (3.89 vs. 1.27 molH2/molGly-converted, at the lowest and highest space velocity, respectively) and more diluted feedstocks (3.95 vs. 1.44 molH2/molGly-converted, at the most diluted and concentrated freestreams, respectively). A threshold value at 10 wt% glycerol was found for the ratio of dehydrogenation to dehydration liquid products. The post-reaction catalyst was also characterized by several techniques, showing that Co leaching was the major drawback, especially at the mildest operation conditions, while carbonaceous deposits are negligible.

Characterization of the use of vacuum enhancement in commercial pilot-scale air gap membrane distillation modules with different designs

The use of vacuum enhancement in air gap membrane distillation (AGMD) has been shown to improve the permeate production and the thermal efficiency. This work presents the characterization of that improvement for different operating conditions (input temperatures and feed flow rate) in three commercial multi-envelope AGMD modules with different internal designs. Modules had different number and length of internal circulation channels, yielding different feed velocities and residence times, and thus allowing to investigate the role of these. Experiments were performed with simulated seawater in solar membrane distillation pilot plants, and the decrease of permeate flux (PFlux) and specific thermal energy consumption (STEC) were analysed for each operating condition. The greatest impact of vacuum enhancement in the performance improvement was observed in the modules with the longest residence time operated at low hot temperature. The lower the driving force, the more important the role of vacuum enhancement in the improvement of vapour diffusion was, independently of feed velocity. Increases of PFlux up to 88% and reductions of STEC down to 70% compared to AGMD operation were quantified. Moreover, vacuum enhancement led to the reduction of hydraulic pressure drop, decreasing 26% the specific electrical consumption.

Simulation analysis and experimental validation of conveying device in uniform rrushed straw throwing and seed-sowing Machines using CFD-DEM coupled approach

• No-till farming is being increasingly used worldwide for more efficient crops. • In crop rotation, the straw of a crop can be used for the next cycle. • We developed a planter that performs even crushed straw coverage. • The coverage and planter performance are verified in simulations and implementation. • We provide the optimal parameters for even coverage and meeting operation standards. In agricultural fields with full straw retention, shredded straw tends to be thrown unevenly. To achieve uniform crushed straw throwing in dense fields of rice stubble, and high-quality straw mulching in dense fields, the conveying device of a no-till planter to perform uniform throwing of crushed straw stubble on fields and determine the device operation mechanism was analyzed. Using kinematics and dynamics analyses of particles of crushed rice straw during lifting and dispersion, a flexible-body model of rod-shaped and agglomerate-shaped crushed straw and a coupling model that included the mechanical structure of the device were developed. By integrating computational fluid dynamics and the discrete element method, the gas–solid coupling theory in numerical simulations and motion analysis of crushed straw particles was used to determine the flow field and motion characteristics under different parameters. The simulation focused on the conveying performance of different pipelines under four scenarios with varying throwing blade speed, conveying volume, pipeline diameter, and initial feeding velocity. The influence of these parameters on both the conveying device performance and their own variation was determined. Actual operations that are then compared with test results to achieve optimal operation were demonstrated. By adopting the Box-Behnken design, the throwing speed of crushed straw was set to the evaluation index a response surface test was performed on the throwing blade speed, conveying volume of crushed straw, and pipeline diameter. The optimization results showed that the predicted optimal parameters were 2300 rpm throwing blade speed, 1.4 kg/s conveying volume, and 220 mm pipeline diameter. The simulated planter achieved an optimized throwing speed of 14.6 m/s. Then, a field test verification was conducted, and the results were averaged. The planter achieved a throwing speed of 14.8 m/s with the same optimal parameters. The straw lifting and throwing device provides high conveying performance and satisfies requirements for real-time operation. This study provides a theoretical basis and supporting data for innovating seeding technologies and optimizing equipment to achieve uniform crushed straw-throwing in dense fields of rice stubble.

Study on surface integrity of DD5 nickel-based single crystal super-alloy in creep-feed grinding

In order to control the forming surface quality of signal crystal turbine blade tenon teeth in the creep feed grinding, the influence of the creep feed grinding parameters on the grinding surface integrity of DD5 nickel-based single crystal superalloy was investigated via orthogonal experiment. The results showed that the surface roughness along vertical grinding direction was ranged at 0.56-0.74 μm at the grinding wheel speed range of 15-30 m/s, feeding velocity range of 120-210 mm/min and grinding depth range of 0.1-0.7 mm, and the surface roughness in the grinding direction is about 1/5 of that in the vertical grinding direction. The surface topography and texture results showed that there were the obvious grooves and ridges on the grinding surface caused by the grain ploughing and scratching, the length and height of grooves and ridges on the grinding surface changed obviously under different processing parameters, and the three-dimensional topography of the grinding surface fluctuated obviously. The length of grooves and ridges along the grinding direction were sensitive to the speed of grinding wheel, waviness of grooves and ridges along the vertical grinding direction were sensitive to the grinding depth and workpiece feed rate. The different degrees of work hardening effect were presented at the grinding surface, the biggest work hardening effect achieved at 11.6%, and the maximum depth of work hardening effect was 110 μm. The distinct plastic deformation appeared at the grinding surface. The γ phase presented slip deformation along the grinding direction with various degrees, and the γ' phase presented skewing, twisting, broken and fracture, the maximum depth of plastic deformation was 2.92 μm. The work hardening effect of DD5 creep feed grinding mainly due to the plastic deformation degree at the grinding surface. The experimental conclusions provided theoretical guidance for DD5 signal crystal turbine blade tenon teeth.

Simulation and Experimental Investigations on the Sugarcane Cutting Mechanism and Effects of Influence Factors on the Cutting Quality of Small Sugarcane Harvesters under Vibration Excitations

In order to study the sugarcane cutting mechanism and the effects of influence factors on the sugarcane cutting quality (SCQ) of small sugarcane harvesters in hilly areas, sugarcane cutting experiments and simulations were done through a self-developed sugarcane harvester experiment platform (SHEP) and the finite element analysis (FEA) method. The comprehensive cutting quality evaluating value (CCQEV) was calculated through the number of sugarcane cracks, the crack thickness, and the crack length to evaluate the SCQ. Effects of the amplitude and the frequency of the axial cutter vibration, the cutter rotation velocity, the sugarcane feeding velocity, and the cutter installing angle on the CCQEV were studied. Effects of interaction between the amplitude and the frequency of the axial cutter vibration and that between the axial cutter vibration amplitude and the cutter rotation velocity on the CCQEV were also studied. The sugarcane cutting mechanism was studied through analysis of cutting force signals, the high-speed photographing result in the sugarcane cutting process, and FEA simulations of the sugarcane cutting process, which verified discoveries obtained through sugarcane cutting experiments. This research laid the foundation for the development of small sugarcane harvesters with a good SCQ in hilly areas.

Observation of the melt dynamics during laser remote fusion cutting

Laser remote fusion cutting (RFC) offers a method for laser fusion cutting of metal sheets without need for a cutting gas. It is closely related to laser keyhole welding, which can transition into RFC for suitable parameters, such that the pressures in the process zone lead to a melt ejection on the bottom of the metal sheet. The mechanisms leading up to the melt expulsion are not fully understood yet and need further investigation. In this work, the cutting front is observed through a borosilicate glass sheet using a high-speed imaging system. In the setup, the glass sheet and a sample metal sheet are clamped together and the process zone is set directly at the interface between metal and glass. This setup allows for the observation of the melt dynamics adjacent to the glass sheet. In the experiments, the keyhole formation and the cutting front are investigated using a 6 kW disk laser and different feed velocities for cutting.

Differential self-adapting drilling machine with three degrees of freedom

A critical factor for engineering progress both in surface and underground mining of extremely hard minerals is the expensive and labor-hungry process of drilling. A top priority task is to create highproduction and energy-efficient drill rigs. It is only possible to improve greatly the drilling process performance through automation of the drilling machine mechanisms which effect the drill column with certain force and kinematic parameters to implement a certain mode of drilling. Actually, the rotary percussive drill rigs mostly possess three mutually independent dynamic systems—axial drive, rotary drive and percussion mechanism. These systems never coordinate their drive signals with the drill column response when interacting with rock mass, which leads to overloads, unwanted vibrations and other adverse effects. Improvement of productivity and energy efficiency of drill rigs is impossible without automation. This article examines the mechanism of a differential self-adapting rotary percussive drilling machine with programmable drilling mode control. The mechanism and its operation are extensively described. The proposed layout of the differential self-adapting drilling machine with three degrees of freedom, with feedback between the drilling column and the axial drive, rotary drive and percussion mechanism toward the automated control of the feed force and velocity, as well as the impact energy and frequency, enables neutralization of adverse effects in rotary percussion drilling.

Performance analysis of a vacuum membrane distillation system coupled with heat pump for sulfuric acid solution treatment

This paper proposed a vacuum membrane distillation system coupled with heat pump (VMD-HP) for the treatment of sulfuric acid solution, in which the evaporator was employed to cool secondary vapor from the VMD process while the condenser was used to heat feed sulfuric acid solution. Mathematical models were first established in view of mass and energy conservation principles, and then validated through a conventional VMD experimental system with polytetrafluoroethylene (PTFE) membrane material. Furthermore, the effects of operating parameters on system performance were investigated comprehensively. The results showed that the combined system could achieve the evaporation process with lower temperature of 318.15 K–333.15 K. Decreasing feed concentration or increasing feed temperature, feed velocity and permeate side pressure, was effective to reduce the specific heating energy consumption (SHEC) and elevate the performance coefficient (COP). Taking the treatment concentration of 20% for instance, feed temperature, feed flow rate and evaporation rate were fixed at 333.15 K, 2 t•h−1and 1 t•h−1, respectively. The SHEC and COP for the VMD-HP system were 181.47 kWh•t−1 and 3.44, respectively. The specifics evaporation cost was found to be 15.56 $•t−1, with a reduction amplitude of 58.74% than that of conventional VMD system.

CFD-PBM modelling of tailings flocculation in a lab-scale gravity thickener

Flocculation occurring in a feedwell plays a critical role in tailings slurry thickening process, which is complicated and significantly influenced by flow characteristics. This work presents a numerical approach to explore the effect of flow characteristics on flocculation performance. It combines an aggregation kernel and a breakage kernel, used to describe the polymer-bridging flocculation kinetics, with a conventional Computational Fluid Dynamics-Population Balance Model (CFD-PBM) coupling to model the complex flocculation-thickening behavior in a lab-scale gravity thickener. The solid-liquid phase interaction is described by an Euler-Euler approach with a modified Schiller-Naumann drag model. The turbulence of liquid phase is resolved by the RNG k-ε turbulence model, while the solid kinematic eddy viscosity is described by a dispersed phase zero equation model. The capability of this proposed model is validated by a good agreement between experimental and predicted results in terms of single-liquid velocity and floc size distribution. The momentum and turbulence dissipation rates are investigated in and around the feedwell over a wide range of feed velocities, showing that the momentum and turbulence dissipation rates have a positive correlation with feed velocity. The momentum and turbulence dissipation rates decrease with the increase in scaled depth in the feedwell. The formation of large vortexes in the feedwell may cause a locally low turbulence dissipation rate. A reasonable increase of feed velocity favours the flocculation, however, an excessive feed velocity can cause a decrease in mean floc size. Modelling tailings flocculation is of great significance for understanding the flocculation behavior and revealing the effect of flow characteristics on flocculation performance.

Modelling and analysis of micro-grinding surface generation of hard brittle material machined by micro abrasive tools with helical chip pocket

The abrasive particles on the surface of micro abrasive tool are very small and tightly distributed, which is prone to appear debris blockage and accelerate tool wear, so the service life and performance of existing micro abrasive tools have become one of the key factors restricting the engineering application of micro-grinding technology. Compared to ordinary cylindrical micro abrasive tools, the micro abrasive tool with micro helical chip pocket (MAT-HCP) presents excellent performance in facilitating debris discharge and improving wear resistance, but the existence of helical chip pocket will significantly affect grains distribution, grinding contact area and surface generated mechanism in the micro-grinding process. In this work, the micro-grinding surface topography model of hard brittle materials obtained with MAT-HCP is built by considering grain size and distribution, dynamic impact, helical chip pocket structures, transient chip thickness and hard brittle material removal mechanism. The micro- grinding experiments are carried out to analyze surface morphology characteristics and surface generated mechanism of sapphire material machined by MAT-HCP. The contrast of simulated and experimental results showed that the average predicted error of contour supported ductility domain ratio and surface roughness respectively can be confined to be 10 % and 5%, which indicates that this surface topography predicted model can succeed in capturing the micro-grinding surface characteristics of sapphire material fabricated by MAT-HCP. Moreover, experimental results disclosed that the increase of pitch length will make MAT-HCP harder to realize the plastic region machining for sapphire, and this novel micro abrasive tool is potential to obtain better surface quality at high feed velocity compared with ordinary micro abrasive tools. The research results are of great significance for extending the service life of micro abrasive tools and promoting engineering application of micro-grinding technology.

Vacuum pressure swing adsorption process with carbon molecular sieve for CO2 separation from biogas

A 3-bed 6-step vacuum pressure swing adsorption (VPSA) process for biogas upgrading using carbon molecular sieve (CMS) adsorbent is here studied both experimentally and numerically. The selected CMS adsorbent, with micropore size distribution in the range of 0.3−0.65 nm, obtained a high kinetic selectivity of 163 but a low equilibrium selectivity of 2.4 for CO2/CH4. A biogas mixture containing 55 % CH4 balanced with CO2 was effectively upgraded to a 98 % CH4 purity product with a CH4 recovery higher than 83 % using a desorption pressure at 0.1 bar and a light product purge-to-feed (P/F) ratio of 0.1. Meanwhile, VPSA process experiments indicated that a lower feed velocity with a longer cycle time achieved much higher CH4 purity with a slight decrease of CH4 recovery. A lower desorption pressure more efficiently improved the degree of regeneration of the CMS adsorbent than increasing the P/F ratio. A non-isothermal mathematical process model, deployed a macropore-micropore bi-linear driving force mass transfer model, was validated by comparison to both dynamic column breakthrough and VPSA process experiments.

Binary gas dehydration using molecular sieve 4A within the pressure-vacuum swing adsorption

This research studied the separation of a binary mixture gas. The separation process was using a pressure vacuum swing adsorption (PVSA) and molecular sieve 4A as adsorbents. The binary mixture gas in this research was nitrogen and water. The research consists of an experiment and numerical model. The laboratory scale experiment was focused on behavior of the water removal and an adsorption capability of adsorbents, The numerical model was used to perform parametric study of the adsorption performance that affected by gas feed velocity, pressure and temperature. The water removal characteristic was employed by the Freundlich isotherm. The experiment shows that the molecular sieve 4A within the PVSA system reached the adsorption efficiency of 94.79% within 6 hours. The absorption decreased along the number of consecutive adsorption cycles. The adsorption modeling shows that gas velocity and working temperature have major effect on the adsorption performance. Increasing pressure improves adsorption performance. However, the results show that the threshold pressure is 3 MPa, which is the point that pressure is not affect the performance any further. The optimize condition in this paper was located at: velocity gas feed of 0.0170 m/s, pressure 3 MPa and temperature 323.15 K.

Buckling instability on a rotating plane

The initial circular shape of a rope coiling on a rotating plane loses axial symmetry and deforms continuously into two rotational order shapes, the hypotrochoid and epitrochoid. Shape transitions were experimentally induced by changing either the feeding velocity or the plane frequency. Varying the former caused the bending energy per unit length to increase significantly at the hypotrochoid-epitrochoid transition, a feature reminiscent of first-order phase transitions. Varying the latter produced comparably significant increases at both the hypotrochoid-epitrochoid transition and the epitrochoid-circle transition. In the limit of the perfectly flexible rope, the solutions to the equations of motion give either a hypotrochoid or an epitrochoid, depending on the tension relative to the inertial force. In the phase diagram, hypotrochoids emerge in the regime where the height is large, the feeding velocity is high and the plane frequency is low. Circles appear at all heights when the feeding velocity is low and the plane frequency is high. Epitrochoids emerge as an intermediate shape.