Inlet Tube Research Articles

Mass transfer enhancement and flow field simulations for a Venturi bubble generator with multiple inlet tubes

The Venturi device plays a crucial role in the Filtered Containment Venting System (FCVS), effectively preventing the release of radioactive pollutants from nuclear reactor containment during overpressure events. Various configurations of the Venturi device yield different outcomes, with mass transfer efficiency being contingent on bubble size and overall gas holdup. This study examines the impact of varying numbers of inlet tubes on the Venturi device, while keeping the inlet area constant. Specifically looking at the gas–liquid flow field, bubble behavior, and mass transfer capacity in the diverging section. By comparing Single Inlet Tube Venturi Device (SITVD) and Double Inlet Tubes Venturi Device (DITVD), we have reached the following conclusions: (1) In SITVD, the liquid flow field is biased to one side, whereas in DITVD, the liquid flows along the axial direction, resulting in bubbles experiencing stronger shear forces. (2) DITVD exhibits a higher overall gas holdup compared to SITVD, with an average axial gas holdup of 0.0317, which is greater than SITVD's 0.0247. However, SITVD's gas holdup is more evenly distributed. (3) SITVD exhibits a higher turbulence dissipation rate of 545 m2/s3, while the bubble residence time in DITVD is comparatively longer. (4) The average bubble diameter of DITVD is 1.116 mm, which is higher than the 1.011 mm of SITVD. (5) DITVD has better mass transfer than SITVD. The proposed structure of the new Venturi scrubber in this study demonstrates a significant enhancement in mass transfer capacity and filtration efficiency. This discovery provides a solid basis for the design and optimization of FCVS, thereby ensuring the safe operation of nuclear reactors.

A Dynamic Ultrasound Phantom with Tissue-Mimicking Mechanical and Acoustic Properties.

Tissue-mimicking phantoms are valuable tools that aid in improving the equipment and training available to medical professionals. However, current phantoms possess limited utility due to their inability to precisely simulate multiple physical properties simultaneously, which is crucial for achieving a system understanding of dynamic human tissues. In this work, novel materials design and fabrication processes to produce various tissue-mimicking materials (TMMs) for skin, adipose, muscle, and soft tissue at a human scale are developed. Target properties (Young's modulus, density, speed of sound, and acoustic attenuation) are first definedfor each TMM based on literature. Each TMM recipe is developed, associated mechanical and acoustic properties are characterized, and the TMMs are confirmed to have comparable mechanical and acoustic properties with the corresponding human tissues. Furthermore, a novel sacrificial core to fabricate a hollow, ellipsoid-shaped bladder phantom complete with inlet and outlet tubes, which allow liquids to flow through and expand this phantom, is adopted. This dynamic bladder phantom with realistic mechanical and acoustic properties to human tissues in combination with the developed skin, soft tissue, and subcutaneous adipose tissue TMMs, culminates in a human scale torso tank and electro-mechanical system that can be systematically utilized for characterizing various medical imaging devices.

Experimental studies on the Hartmann tube

Abstract The effect of tube depth, the separation distance between the tube and nozzle exit, and the nozzle pressure ratio on the characteristics of the flow coming out of the Hartmann tube was studied experimentally. The configuration used in this work consists of an underexpanded sonic jet emanating from a convergent nozzle directed into a closed-ended cylindrical tube of the same diameter (D) as the nozzle exit. The nozzle was operated at two levels of underexpansion corresponding to nozzle pressure ratio (NPR) 3 and 5. The distance (S) from nozzle exit and tube inlet was varied from 0.4D to 4D. Discrete high-amplitude tones (the jet regurgitant, JRG) were produced, only at certain (periodic) intervals (near the shock-cell terminations) of spacing for NPR 3, while for NPR 5 the JRG tones are produced at all points beyond the first shock-cell. For locations other than these, high-frequency tones (screech mode) were observed. The connection between the jet structure and operating modes revealed that the location of standoff shock ahead of the tube with respect to the jet structure plays a dominant role in the observed ‘modes’ rather than the nozzle tube separation. The results reveal that the frequency response of longer tubes in JRG mode approaches their quarter wave frequencies. The high-frequency oscillations observed in the screech mode showed independency with cavity (pipe) depth, contrary to the available literature, the transition between ‘different modes’ oscillation is a function of cavity depth.

Performance increasement in shell-and-tube heat exchangers reinforced with dimpled tubes: A correlation-based approach

Shell-and-tube heat exchangers (STHEs) are pivotal components in various industries, facilitating efficient heat transfer across fluid streams. The simulation of STHEs often demands a substantial number of computational cells, significantly when augmenting heat transfer surfaces by incorporating additional tubes or dimples. This study introduces a novel approach to mitigate the computational cost associated with STHE simulations. Correlations are designed to predict heat transfer within and outside the tubes, both in the presence and absence of dimples. These correlations are developed using the P-NTU thermal analysis method and a turbulence model that is carefully selected to align with the specific characteristics of STHEs. Following the validation of our approach, the correlations are applied to analyze the performance of various baffle arrangements within STHEs. Precisely, segmental, disk and donut baffle configurations, each featuring elliptical and teardrop dimples on the tube surfaces, are investigated. Furthermore, the impact of variations in shell and tube inlet mass flow rates on the overall performance of the heat exchanger is explored. The results reveal compelling findings. Elliptical dimples contribute to a remarkable enhancement in the performance evaluation criteria (PEC), yielding improvements of 36.7 % and 19.7 % for segmental as well as disk and donut baffle arrangements, respectively. Similarly, teardrop dimples exhibit their effectiveness, elevating the PEC by 51.5 % and 32.6 % in these same configurations under the specified working conditions. These findings contribute to the optimization of heat exchanger designs in diverse applications, fostering greater energy efficiency and environmental sustainability.

CFD‐ANN coupling model simulation of gas–solid feeding design for ternary biomass mixtures in bubbling fluidized bed gasifier

AbstractThe performance of continuous feeding fluidized bed reactors is significantly influenced by their design. These reactors can effectively operate with a wide range of biomass mixtures. Therefore, it is imperative to carefully design the gas distributor plate and solid tube inlet to ensure stable fluidization and uniform distribution of fluidizing gas and solid particles within the reactor. This study investigated the impact of gas–solid feeder design in bubbling fluidized bed gasifier for biomass mixtures on system hydrodynamics, employing a computational fluid dynamics–artificial neural network (CFD‐ANN) coupling model to achieve more realistic simulations. A 2k factorial experimental design was adopted to inquire the impact of gas and solid feeding systems. The responses under investigation included the gas–solid mixing index and the solid residence time, both of which hold pivotal roles in specific chemical processes related to biomass utilization in fluidized bed technology. All cases were successfully simulated, and the results uncovered that the position and length of the solid inlet tube wielded a significant influence on reactor performance, particularly concerning solid residence time. Furthermore, the designs of the gas distributor were identified as critical factors capable of enhancing system turbulence and mixing. In summary, the results showed the potential for enhancing reactor performance through the optimization of gas–solid feeding systems and underscored the efficacy of the ANN drag model in simulating continuous biomass gasification systems.

Computational Fluid Dynamics Simulation of Filling a Hydrogen Type 3 Tank at a Constant Mass Flow Rate

Hydrogen storage in high-pressure tanks can be performed with different filling strategies. Many studies have been carried out on supplies with increasing pressure rates. The present work aims to carry out CFD numerical simulations, using Ansys Fluent®, in a type 3 tank of 70 MPa normal working pressure (NWP) using a constant flow rate, to analyze the influence of inlet key parameters such as initial temperature, mass flow rate, and material properties on the evolution of temperature, pressure, and velocity. From the analysis of the results, it was possible to discover linear relationships between the increase in the total equilibrium temperature and the final hydrogen temperature, as well as a linear increasing relationship between inlet and final temperatures when the equilibrium temperature was fixed. Considering fully adiabatic walls resulted in a significant increase in temperature with no predictable pattern. The difference between total and static temperatures found in the inlet tube decreased with the decrease in mass flow rate and subsequently the Mach number. The choice of a polymer for the tank lining led to higher temperatures when compared to an aluminum lining.

Numerical investigation of no-load startup in a high-head Francis turbine: Insights into flow instabilities and energy dissipation

The presented paper numerically investigates the internal flow behaviors and energy dissipation during the no-load startup process toward a Francis turbine. Passive runner rotation is implemented through the angular momentum balance equation accompanied by dynamic mesh technology and user defined function. Three phases of rotational speed are identified: stationary, rapid increase, and slow increase. Head exhibits a monotonic decrease, rapid rise and fall, and eventual fluctuation. Flow rate shows quasi-linear increase. The pressure fluctuations in the vaneless region are primarily dominated by the frequencies induced by Rotor-Stator Interaction and a broad frequency range below 50 Hz, and below 30 Hz in the draft tube. Runner inlet experiences positive to negative incidence angles, causing intense flow separation and unstable structures. Draft tube exhibits large-scale recirculation and evolving vortex structures. Energy loss analysis based on the entropy production method highlights the runner and draft tube as primary contributors. The energy loss within the runner exhibits an initial increase, subsequent decrease, and then a rise again during the stationary and rapid speed increase phases. While the draft tube shows a rapid increase during the phase of rapid speed increase. Turbulent fluctuations significantly contribute to entropy production loss, with trends matching total entropy production. Maximum energy loss locations correspond to runner inlet and draft tube wall, emphasizing the importance of unstable flow and vortex generation. This study establishes foundational insights into unstable hydrodynamics and energy dissipation modes during hydraulic turbine no-load startup, paving the way for further research.

Numerical investigation of flow boiling behavior in a vertical tube of a shell‐and‐tube phase change heat storage unit

AbstractPhase Change Materials (PCMs) demonstrate significant potential as latent heat storage systems for direct steam generation. This investigation explores the heat transfer dynamics of an upward‐flowing heat transfer fluid (HTF) boiling within a vertical tube, concurrently examining the solidification process of PCM outside the tube. A two‐dimensional model of a shell and tube heat storage device is established for this purpose. The simulation employs the Volume of Fluid (VOF) model coupled with the Lee model to simulate HTF boiling inside the tube. Concurrently, the enthalpy‐porosity model is utilized to analyze the PCM solidification process. The findings reveal that PCM solidification and heat discharge are influenced by the temperature and thermal conductivity of the HTF, leading to uneven solidified PCM thickness along the tube's length over time. Additionally, variations in tube length, inlet velocity, and inlet temperature impact both the vapor quality of HTF and the thickness of solidified PCM. Tube length correlates with boiling time, affecting the accumulation of steam quality throughout the coupling process. Inlet velocity dictates the heat transfer of HTF with PCM within a limited distance, and insufficient heat exchange resulting in the cessation of HTF boiling. Inlet temperature determines the appropriate device size, lower inlet temperatures necessitate longer tubes to extend the heat charging duration until boiling initiation.

Empirical Investigation of Thermal Features of Phase Change Material as Thermal Storage System

This paper presents an empirical investigation of the melting processes of phase change material (PCM) in a tube and shell heating exchanger utilizing solar thermal energy. The experiments were conducted outdoors in Al-Mussaib city, Babylon, Iraq (Lat 32.5° North, long 44.3° East), utilizing a north-south (N-S) collector oriented at a tilt angle of 32.5° from the horizontal. This work utilized pure Iraqi black paraffin with 12 kg as PCM. The main empirical rig involves a vacuum tube solar collector (VTSC) utilizing a tube and shell heating exchanger to melt PCM in the shell regime. Different rates of water flow inside the internal heating exchanger's tube, namely (3.5, 5, and 8.5 L/m), were utilized for each season from August 2022 to January 2023. The empirical findings concluded that the internal tube inlet and surrounding temps significantly influence the melting process compared to the rates of water flow. It was found that the melting process of PCM needs about (3 to 4) hours in the summer season. Still, winter needs a longer time, about 14 to 16 hours. Increasing surrounding temp and solar radiation reduces the melting time of phase-change material. The findings of the present work were compared with the earlier investigations and agree with increasing the present work by 23.5 %. The findings from the present study are of practical importance for industries requiring storage materials to improve thermal storage system performance.

Reducing temperature difference of a direct ammonia tubular SOFC to 1K

In this study, a three-dimensional direct ammonia solid oxide fuel cell (DA-SOFC) was numerically modeled to investigate the effects of ammonia inlet flow velocity, tube structure, and catalyst filling on the internal temperature distribution of the cell. The results show that shortening the length of the ammonia inlet tube and increasing the inlet flow velocity leads to an increase in the temperature difference inside the cell. By perforating the ammonia inlet tube, the temperature difference can be further reduced, and the low-temperature zone inside the cell gradually decreases with the perforated area percentage increasing. In addition, placing ammonia decomposition catalyst inside the perforated inlet tube can further improve the temperature difference between the electrodes inside the cell, reducing it from 30 K to about 1 K. Furthermore, the temperature distribution pattern inside the fuel cell with metal supports was investigated, and it was found that the metal support can better export the heat generated inside the fuel cell. This study provides a new idea to improve the temperature distribution of DA-SOFC and enhance the stability and reliability of cell operation.

Assessment of the influence of fuel system elements on the filling coefficient of a gas engine

The purpose of this work is to determine the dependence of the filling factor on the position and burial of the gas supply tube in the inlet manifold of the gas engine. Modeling of gas-dynamic processes occurring during engine intake was carried out by finite element method in the Ansys Fluent engineering analysis software environment. As geometric models for modeling, the part of the engine intake system in which the gas fuel supply tubes were located, which were located in different parts of the intake manifold and had a different amount of burial into the air flow, was considered. Simulations have shown that gas inlet tubes are the source of significant hydraulic losses in the engine inlet system, and as the amount of pipe buria into the inlet manifold cavity increases, these losses increase. Keywords filling ratio, engine intake process modeling, gas engine, hydraulic losses, gas supply elements, intake manifold

Freeze range of a condensing gas propagating in a liquid helium-cooled tube

Understanding air propagation and condensation following a catastrophic vacuum break in particle accelerator beamlines cooled by liquid helium is crucial for maintaining the operational safety of these facilities. Previous experimental investigations on nitrogen gas propagation in both normal liquid helium (He I) and superfluid helium (He II) cooled copper tubes unveiled a nearly exponential deceleration of the gas propagation. A comprehensive theoretical model incorporating gas dynamics, heat transfer, and condensation mechanisms has been developed, which effectively reproduces various key experimental observations. An intriguing phenomenon uncovered in our model simulation is that the gas propagation appears to nearly stop beyond a certain distance from the location where condensation starts. We refer to this distance as the freeze range. In this paper, we present our systematic study of the freeze range at various inlet mass fluxes and tube diameters. We show that the results can be well described by a simple correlation. The underlying physical mechanism that supports this useful correlation is explained. Knowing the freeze range may allow accelerator engineers to develop protocols for controlling frost-layer contamination in the beamline tubes, which is of great practical importance.

Research on the thermal performance of the condenser in a loop heat pipe anti-icing system

Loop Heat Pipes (LHPs) are efficient heat transfer devices, providing a highly efficient and energy-saving thermal control integrated management method for Unmanned Aerial Vehicles (UAVs) with their flexible structural design. In this paper, a conceptual design of LHP anti-icing system was proposed, using LHP to transfer waste heat to wing’s leading edge. The outer surface temperature of wing skin is one of the main indicators to measure the anti-icing effect. As a result, this paper mainly focused on the heat transfer process of the condenser section by experimental and numerical simulation methods. A stainless steel-nickel LHP was fabricated and tested at different conditions. Volume of Fluid (VOF) method and Lee model were adopted to simulate the condensation process. Results showed that increasing the heating power (from 35 W to 60 W) slowed down the condensation and lengthened the two-phase zone. However, when the angle of attack changes within the range of 0°–10°, the liquid-vapor distribution in the tube remains almost unchanged. The average error of surface temperature between experiment and simulation results is 3.6% and 2.5% for the heating power of 180 W and 60 W respectively. Additionally, because the volume of water droplets collected is the largest at the leading edge of the wing, we recommend arranging the LHP condenser tube inlet near the leading edge of the wing and increasing the density of pipe arrangement to achieve better anti-icing effect.

Surge and lifting analysis of internal transmission gas–liquid two-phase flow lifting pipe

The gas–liquid two-phase flow inside large aspect ratio lifting pipes is often characterized by a complex flow pattern, potentially causing coupling vibration of the lifting pipe. And produce a surge effect, aggravating the vibration of the lifting pipe and thus affecting its fatigue life and the lifting performance of the pipe. In this paper, a combination of numerical simulation and experiment is used to investigate the effects of inlet flow rate (Q) and tube diameter (D) on the vibration and lifting performance of the lifting tube. The obtained conclusions are as follows: with an increase in the inlet gas flow rate, the amplitude of the lifting pipe increases, and the lifting flow increases; with an increase in pipe diameter, the amplitude of the lifting pipe decreases and the overall lifting performance increases. According to the numerical simulation and experimental results, when D = 40 mm lifting pipe at the intake flow between Q = 50-60 L/min, the impact of surge effect is the least, and the liquid lifting quality also reaches the best state at this time, The research presented in this paper provides a reference for the selection of parameters for lifting pipes in engineering applications.

Comparison of the use of different nanofluids for heat transfer from the outer surface of spiral tubes: Energy, exergy, and exergoeconomic (3E) analysis

Heat exchangers play a crucial role in industrial processes by facilitating the efficient transfer of thermal energy. The thermophysical properties of nanofluids, which consist of nanoparticles dispersed in conventional heat transfer fluids, are enhanced. Their remarkable thermal conductivity and convective heat transfer characteristics have the potential to significantly improve heat transfer processes. In the present investigation, a numerical study has been carried out to investigate the effect of different nanofluids on heat transfer from outer surface of a spiral coil based on the first and second laws of thermodynamics. The numerical solutions are checked against previous experimental and numerical research in the field. Nusselt number, Euler number, exergy efficiency, irreversibility, and equipment cost are studied with respect to concentration, mass flow ratio (Rm), and shell and tube inlet temperature for a variety of nanofluids, including Al2O3/H2O, CuO/H2O, Fe2O3/H2O, and TiO2/H2O. Compared to water, the exergy efficiency is lower while using nanofluids, as shown by the results, especially in higher nanofluid concentrations, and among the different nanofluids, Al2O3–H2O nanofluid was found to have the highest thermodynamics second law efficiency. The exergy destruction rate experiences linear growth by increasing the nanofluid concentration. A higher pressure drop based on the Euler number is expected for nanofluids. In addition, the consumption of nanofluids on the shell side results in a considerable reduction of the purchased equipment cost (PEC) value compared to water. As opposed to the tube-side input temperature, variations in the shell-side inlet temperature found to have a greater impact on the heat exchanger's exergetic performance.

A case study evaluating the performance of a cost-effective optical particle counter coupled with a humidity compensation approach for ambient air monitoring of particulate matter

The spatial monitoring of submicron particles has become an essential issue due to their negative effects on human health. However, the use of high-cost and high-grade measurement instruments is a challenging investment cost. Thus, a cost-effective optical particle counter (OPC), which is improved measurement quality, has become a good candidate. In this study, two cost-effective OPCs, coupled with a heated inlet tube to reduce the effect of humidity on its measurements, were applied in the field to measure ambient PM10, PM2.5, and PM1 over 1 month. Their 1-h-average and 24-h-average data were compared with those obtained from a reference-grade OPC and a beta attenuation monitor (BAM). In particular, the correlations between the measurement data obtained from them, the differences in the mean values of these data, and the relative errors were evaluated. The PM2.5 data obtained from cost-effective OPCs were comparable to those from BAM at even under high humidity conditions, except for rainy days. The data obtained from the cost-effective OPCs also showed good correlations and low relative errors (i.e., < 7%) compared to the reference-grade OPC, with no significant difference in mean values in terms of the PM2.5 and PM1 data. Although the measurement of PM10 by the cost-effective OPCs showed lower quality than PM2.5 and PM1, the relative errors were still acceptable (i.e., < 18%) compared with those in other studies. Therefore, the cost-effective OPC coupled with a heated inlet tube has the potential to serve as a real-time monitoring instrument for ambient PM2.5 and PM1.

Numerical study of an infrared suppression (IRS) device with cooling hole configuration on the mixing tube

Hot walls of the infrared suppression (IRS) device intensify the infrared signature of warships, which is detrimental for attaining infrared stealth. In this work, to effectively reduce the wall temperature, hole cooling technology is innovatively adopted on a marine IRS device. Circular holes are arranged on the mixing tube of the device. Numerical simulations are conducted to predict the temperature and flow distribution. The current research comprehensively analyzes the effects of hole position, number of hole rows, and number of holes on the wall temperature of the mixing tube and heat shield. The air suction rate and the exit temperature of the whole device are considered as well. The results indicate that the air suction and exit temperature of the device do not significantly change with the hole arrangement parameters. As the hole position moves towards the mixing tube outlet, the average temperature of the mixing tube continuously increases, while the maximum temperature of the mixing tube exhibits a peak-and-valley pattern. When the holes with large diameters are located near the mixing tube inlet, a backflow occurs between the mixing tube and the heat shield, causing the heat shield temperature to rise appreciably. Increasing the number of hole rows can reduce the mixing tube temperature, but excessive hole rows lead to an elevation in the heat shield temperature. The optimal configuration for the number of hole rows is found to be 2 with a hole diameter of 80 mm. With the increment in hole number, the average temperature of the mixing tube rises due to the accelerated disappearance of the streamwise vortices behind the holes; however, the maximum temperature of the mixing tube shows a decreasing-then-increasing trend owing to that more streamwise vortices are created. This study paves the way for the practical application of hole cooling technology on marine IRS devices. The findings can provide instructive suggestions for the arrangement of cooling holes.

The effect of inlet design and developing flow on local mass transfer in electrolyzers

The mass transfer performance of electrolyzers can vary significantly depending on the type of inlet that is used. Furthermore, significant local effects can occur if the flow is not distributed well by the inlet. In this work, these effects are investigated using segmented electrodes. With a tube inlet, the formation of a jet results in regions of low and high mass transfer. With a conic inlet, significant flow channeling to the sides of the electrolyzer is observed. These local effects were reduced by using a divider inlet, the addition of a calming section, or a turbulence promoter. The type of inlet also determines the flow regime: with the extended conic configuration the flow is developed laminar, whereas with the tube inlet the flow is developing turbulent. The cross-sectional area ratio of the inlet to the channel was found to be an important parameter in predicting the local mass transfer behavior.

Optimizing performance of water-cooled photovoltaic-thermal modules: A 3D numerical approach

To evaluate and improve the efficiency of photovoltaic solar modules connected with linear pipes for water supply, a three-dimensional numerical simulation is created and simulated via commercial software (Ansys-Fluent). The optimization utilizes the principles of the 1st and 2nd laws of thermodynamics by employing the Response Surface Method (RSM). Various design parameters, including the coolant inlet velocity, tube diameter, panel dimensions, and solar radiation intensity, are systematically varied to investigate their impacts on energetic and exergitic efficiencies and destroyed exergy. The relationship between the design parameters and the system responses is validated through the development of a predictive model. Both single and multi-objective optimizations are performed using the predictive model to optimize the thermal and electrical productivity under different scenarios. The findings indicate the significance of the thermal exergy effectiveness, as evidenced by its low P-value for all solar system responses, indicating its crucial role in the predictive model. For single-objective optimization, the desirability is equal to 1 in cases where only heat transfer efficiency, whole energy effectiveness, or thermal exergy efficiency is maximized or only destroyed exergy is minimized. The improvements in energy and exergy efficiencies range from 3.55% to 69.13%, with the amount of destroyed exergy reduced by 81.47% compared to the base case. For multi-objective optimization, desirability values exceeding 0.829 and 0.655 are obtained for single and multi-objective scenarios, respectively, indicating that the expected performance is within desirable limits. The findings provide valuable insights for designing high-efficiency photovoltaic/thermal systems and addressing their challenges and limitations.

CFD simulation of a spiral-channel dust separator for removal of black powder from natural gas pipelines

A spiral-channel dust separator was investigated experimentally and numerically for its efficiency to separate black powder from natural gas pipelines. Effects of gas flowrates and concentration of black powder on separation efficiency and pressure drop were studied. The size distribution (0.15–5 µm) of the black powder particles was measured. Efficiency for the finest particles (mean 0.327 µm) reached 86%, compared to < 1% for a high-efficiency Stairmand cyclone of equal throughput capacity. The separation efficiency reached spiral-channel dust separator 100% for particles > 1 µm. Results of CFD simulations are reported for velocity and pressure fields in the channels, particle separation contours, and local enrichment within the channels. Calculations showed that the radial contraction and diversion of the gas flow at the separator's outlet generated highly decontaminated air above the clean air inlet tube. The distance between the separator outlet and the cleaned gas inlet has been identified as a significant design parameter due to its impact on the separation efficiency. Modeling results showed further that reducing the swirling angle of the channels leads to a decrease in pressure drop with minimal effect on separation efficiency. An evaluation of the RNG k-ε code is provided as well.