Miniature Pressure Sensor Research Articles

Wide-range, durable, and adaptable miniature pressure sensor based on planar capacitance

Capacitive pressure sensor (CPS) is widely used in the field of industrial equipment, because of the merits of fast dynamic response and high resolution. However, the traditional laminated CPS makes it difficult to achieve a wide detection limit in a small size, and this structure is susceptible to electromagnetic interference. Here we developed a miniature planar capacitive pressure sensor (MPCPS) with high performance, which can realize the response to external touching stimuli through the deformation of the packaging material and the change of the equivalent resistance. A metal shielding layer was added under the insulating substrate to effectively isolate the external interference. The thickness of the sensor is about 200 μm, and the diameter of the core sensing area is less than 1 mm. Two types of electrodes with different shapes were designed, among which the spiral electrode MPCPS (S-MPCPS) has better performance than the linear electrode MPCPS. The S-MPCPS has a sensitivity of 99.2% MPa−1 in the low-pressure range (0–0.1 MPa), fast response (20 ms), wide detection limit (>1 MPa), and high durability (>2000 cycles). In addition, MPCPS is proven to have good resistance to high temperature and oil contamination. Finally, practical applications such as contact pressure measuring on the meshing surface of spur gears and mechanical gripper clamping force monitoring were successfully demonstrated. These results shed light on the potential application of the MPCPS in the pressure detection of industrial equipment.

Photothermal-Responsive Lightweight Hydrogel Actuator Loaded with Polydopamine-Modified Hollow Glass Microspheres.

Aquatic actuators based on the light-to-work conversion are of paramount significance for the development of cutting-edge fields including robots, micromachines, and intelligent systems. Herein, we report the design and synthesis of near-infrared light-driven hydrogel actuators through loading with lightweight polydopamine-modified hollow glass microspheres (PDA-HGMPs) into responsive poly(N-isopropylacrylamide) (PNIPAM) hydrogels. These PDA-HGMPs can not only function as an excellent photothermal agent but also accelerate the swelling/desewlling of hydrogels due to their reconstruction for polymer gel skeleton, which speeds up the response rate of hydrogel actuators. The resulting hydrogel actuator shows controlled movements under light illumination, including complex self-propellant and floating/sinking motions. As the proof-of-concept demonstrations, a self-sensing robot is conceptualized by integrating the PDA-HGMP-containing hydrogel actuator with an ultrathin and miniature pressure sensor. Hopefully, this work can offer some important insights into the research of smart aquatic soft actuators, paving the way to the potential applications in emerging fields including micromachines and intelligent systems.

Performance Evaluation of Micro Automatic Pressure Measurement Sensor for Enhanced Accuracy

The major objective of this research is to design sensitive components, conversion components, and various sensor circuits to achieve the miniaturization design for more accurate measurements. This article conducts performance testing on the designed miniaturized pressure sensor to determine whether it meets the qualified standards. The response time of designed sensor results increases with the increase of pressure under experimental conditions of different pressure application values (5MPa to 50MPa). The detection accuracy of the micro automatic pressure measurement sensor designed in this paper can reach 0.0452\%; the average pressure is 0.00364%, and the insulation resistance is 68.44 megohms, which meets reliability requirements. The sensitivity is 0.0582%; the nonlinearity is 0.0741%; the hysteresis is 0.0266%; the repeatability of 0.0625% meets the qualification standard for this instrument. Still, compared with traditional sensors, the sensor reduces the response time of results by about 60%. However, the author conducted the detection in an ideal environment. The actual working environment of sensors is relatively good. Therefore, the detection results obtained in this article may have some errors compared to the actual situation, and further analysis and testing are needed to optimize the performance of the designed sensor.

Rotation-traction manipulation induced intradiskal pressure changes in cervical spine-an in vitro study.

Objective: Evaluate the effect of rotation-traction manipulation on intradiskal pressure in human cervical spine specimen with different force and duration parameters, and compare the intradiskal pressure changes between rotation-traction manipulation and traction. Methods: Seven human cervical spine specimens were included in this study. The intradiskal pressure was measured by miniature pressure sensor implanting in the nucleus pulposus. rotation-traction manipulation and cervical spine traction were simulated using the MTS biomechanical machine. Varied thrust forces (50N, 150N, and 250N) and durations (0.05s, 0.1s, and 0.15s) were applied during rotation-traction manipulation with Intradiscal pressure recorded in the neutral position, rotation-anteflexion position, preloading, and thrusting phases. Futuremore, we documented changes in intradiscal pressure during cervical spine traction with different loading forces (50N, 150N, and 250N). And a comparative analysis was performed to discern the impact on intradiscal pressure between manipulation and traction. Results: Manipulation application induced a significant reduction in intradiscal pressure during preloading and thrusting phases for each cervical intervertebral disc (p < 0.05). When adjusting thrust parameters, a discernible decrease in intradiscal pressure was observed with increasing thrust force, and the variations between different thrust forces were statistically significant (p < 0.05). Conversely, changes in duration did not yield a significant impact on intradiscal pressure (p > 0.05). Additionally, after traction with varying loading forces (50N, 150N, 250N), a noteworthy decrease in intradiscal pressure was observed (p < 0.05). And a comparative analysis revealed that rotation-traction manipulation more markedly reduced intradiscal pressure compared to traction alone (p < 0.05). Conclusion: Both rotation-traction manipulation and cervical spine traction can reduce intradiscal pressure, exhibiting a positive correlation with force. Notably, manipulation elicits more pronounced and immediate decompression effect, contributing a potential biomechanical rationale for its therapeutic efficacy.

A versatile and customizable low-cost printed multipass microrheometer for high-throughput polymers rheological experimentation

We report the design of a multipass microrheometer that can be fully customizable at a low cost and reasonable time, which allows us to perform experiments rapidly and in a broad range of shear rates (i.e., from 0.1 to 100 s−1), using small amounts of material (i.e., just some milligrams). Additionally, the low-cost approach opens for an easy parallelization of the setup that makes it suitable for high-throughput rheological experimentation of polymer melts (HT-Rheo-E). The novel rheometer consists of a microchannel (i.e., a microcapillary or a microslit) in which the fluid flows driven by two controlled millimetric pistons (diameter of 2 mm). Two piezoelectric miniaturized pressure sensors are placed at the microchannel entrance and exit to record the pressure drop across the capillary during the motion. The current work reports the design of the rheometer with two different cross sections of the microchannel, i.e., circular and rectangular, and measurements of the shear viscosity with a Newtonian and a non-Newtonian polymer over a wide range of shear rates using less than 1 g of sample. We demonstrate that the current multipass microrheometer can measure viscoelastic properties of polymers by applying an oscillatory flow. The printed setup is of potential interest for applications in quality control in industrial production, in natural systems (such as starch-based mixtures) and academic research where rapid and repeated measurements using limited milligrams of polymer are required (e.g., biological systems).

All-Silica diaphragm-based optical fiber fabry-perot pressure sensor fabricated by CO2 laser melting capillary end face

An all-silica Diaphragm-based miniature optical fiber pressure sensor based on the Fabry-Perot (FP) interferometric principle is proposed. The FP structure is composed of a single-mode fiber and a fiber capillary end face diaphragm by a CO2 laser welding. The fabrication process and parameter control of all-silicon diaphragm are introduced in detail, and the pressure performance test system of FP is set up. The experimental results show that the fiber optic FP sensor has good sensitivity and linearity to air pressure, the sensitivity can reach 9.48 pm/kPa when the pressure ranges from 0 to 200 kPa, and the linearity is 0.998. All-silica diaphragm-based fiber FP pressure sensor fabricated by CO2 laser melting capillary end face is more compact, easy to produce, and has great potential in the extreme harsh environment.

Silver Nanoflakes-Enhanced Anisotropic Hybrid Composites for Integratable Pressure Sensors.

Flexible pressure sensors based on polymer elastomers filled with conductive fillers show great advantages in their applications in flexible electronic devices. However, integratable high-sensitivity pressure sensors remain understudied. This work improves the conductivity and sensitivity of PDMS-Fe/Ni piezoresistive composites by introducing silver flakes and magnetic-assisted alignment techniques. As secondary fillers, silver flakes with high aspect ratios enhance the conductive percolation network in composites. Meanwhile, a magnetic field aligns ferromagnetic particles to further improve the conductivity and sensitivity of composites. The resistivity of the composite decreases sharply by 1000 times within a tiny compression strain of 1%, indicating excellent sensing performance. On the basis of this, we demonstrate an integratable miniature pressure sensor with a small size (2 × 2 × 1 mm), high sensitivity (0.966 kPa-1), and wide sensing range (200 kPa). Finally, we develop a flexible E-skin system with 5 × 5 integratable sensor units to detect pressure distribution, which shows rapid real-time response, high resolution, and high sensitivity.

Роль интраоперационного внутрилоханочного давления в течении послеоперационного периода перкутанной нефролитотрипсии у больных с мочекаменной болезнью.

Introduction. Percutaneous and endourological surgeries are the most common interventions for nephrolytiasis in urologists’ practice. During the operation, it is important to maintain the right balance between sufficient flow of irrigation fluid for adequate visualization and safe intrarenal pressure. Intraoperative increase in intrarenal pressure is associated with the development of purulent-inflammatory complications in the postoperative period. The aim of the study. To substantiate the expediency of intraoperative monitoring of intrarenal pressure, as well as to assess the risks of postoperative complications after percutaneous nephrolithotripsy. Materials and methods. The study included 250 patients with staghorn kidney stones. In the first group (n=120) percutaneous nephrolithotripsy (PCNL) was performed according to the standard procedure with a puncture canal diameter of 30-32 Сh. The second group included 130 patients who underwent percutaneous nephrolithotripsy using a mini nephroscope (mPCNL) with a working channel diameter of 16 Ch. To achieve the aim of our study, we have developed a method for determining intrapelvic pressure, implemented using a miniature pressure sensor, which is installed directly into the pelvis at the time of the calycopelvical system puncture. During the surgery, the intrapelvic pressure was constantly monitored according to the method we proposed. Results. The lowest values of intrapelvic pressure were recorded when performing PCNL with the installation of an Amplatz sheath exceeding the size of the nephroscope by 2 or more Ch. When performing mPNLT, the installation of a ureteral catheter allows minimizing the increase in intrapelvic pressure. Hyperthermia in the postoperative period significantly depends on an increased intrarenal pressure and the presence of an infectious agent in the renal system. Separately performed urine cultures from the pelvic and bladder with obstruction of even minimal severity may have significant differences. In patients of mini-PNLT group intraoperative intrapelvic pressure was significantly higher (p≤0.05). The frequency of fever and exacerbation of calculous pyelonephritis is significantly higher in group of standart PNLT (p &lt; 0.05), which is explained by greater traumatization of the renal parenchyma when creating a working fistula. Conclusion. The proposed method allows the most accurate intrasurgery monitoring of intrarenal pressure during minimally invasive surgical treatment of kidney stones in order to minimize the frequency of infectious and inflammatory complications.

Nano-optomechanical Resonators for Sensitive Pressure Sensing.

Nanomechanical sensors made from suspended graphene are sensitive to pressure changes. However, these devices typically function by obtaining an electrical signal based on the static displacement of a suspended graphene membrane and so, in practice, have limited sensitivity and operational range. The present work demonstrates an optomechanical Au/graphene membrane-based gas pressure sensor with ultrahigh sensitivity. This sensor comprises a suspended Au/graphene membrane appended to a section of hollow-core fiber to form a sealed Fabry-Pérot cavity. In contrast to conventional nanomechanical pressure sensors, pressure changes are monitored via resonant sensing with an optical readout. A miniature pressure sensor based on this principle was able to detect an ultrasmall pressure difference of 1 × 10-7 mbar in the ultrahigh-vacuum region with a pressure range of 4.1 × 10-5 to 8.3 × 10-6 mbar. Furthermore, this pressure sensor can work over an extended pressure range of 7 × 10-6 mbar to 1000 mbar at room temperature, outperforming commercial pressure sensors. Similar results were obtained using both the fundamental and higher-order resonant frequencies but with the latter providing improved sensitivity. This sensor has a wide range of potential applications, including indoor navigation, altitude monitoring, and motion detection.

A dual-functional micro-focus X-ray source based on carbon nanotube field emission

A micro-focus X-ray source with miniature pressure sensor was developed based on the multi-walled carbon nanotube (MWNT) field emission. The field emitters for both X-ray cathode and the pressure sensor were synthesized by direct chemical vapor deposition growth of MWNTs on catalytic substrates. MWNTs exhibited excellent field emission properties with the threshold field at 10 mA/cm2 of 2.30 V/μm, and the high current density of 10 A/cm2 could be reached at the field of 4.36 V/μm. The field emission electron gun was constructed with the aid of electrostatic simulations on the electron beam trajectories and the focal spot size (FSS) on the target with the demagnification factor of 5. The practical objects, including the semiconductor chip and the ceramic heating element, were tested by the X-ray source with the spatial resolution of less than 35 μm, and the fracture of the ceramic element inside metal case could be clearly detected. The pressure sensor monitored the in-situ pressures inside the source during the processing and after the sealing-off. The pressures in the sealed sources were generally in 10−4 Pa level and could rise to 10−2 Pa level during the operation.

Proof-of-concept of a miniature pressure sensor based on coupled optical WGMs excited in a dielectric microsphere

Optical whispering-gallery mode (WGM) resonators retain increased interest as a key element in various high-resolution sensing and spectroscopic devices in modern optoelectronic, optofluidic and Lab-on-chip technologies. As a theoretical proof-of-concept, we present a new miniature pressure sensor based on coupled WGMs optically excited in a dielectric microsphere placed near a flexible reflective membrane which acts as an ambient pressure sensing element. WGMs are excited by free-space coupling of incoming optical radiation to a spherical microparticle. The distinctive feature of proposed sensor is double WGM excitation by forward and backward propagating radiation upon reflection from a membrane that causes WGMs interference in particle volume. Unlike known WGM-sensors, the proposed design utilize not the resonant frequency shift but the changes in optical intensity of resulting field established in the microsphere which carries information about the exact position of pressure-loaded membrane. The sensitivity of the proposed sensor strongly depends on the quality factor of the working resonance, as well as geometrical and mechanical parameters of flexible reflecting membrane. Important advantages of the proposed sensor are its miniature design and the absence of a mechanical contact of pressure-sensitive element with WGM resonator.

Coupled optical resonances in a dielectric microsphere: physical concept of miniature optical pressure sensor

Coupled optical resonances in a dielectric microsphere: physical concept of miniature optical pressure sensor

Direct measurement and modeling of intraglottal, subglottal, and vocal fold collision pressures during phonation in an individual with a hemilaryngectomy.

The purpose of this paper is to report on the first in vivo application of a recently developed transoral, dual-sensor pressure probe that directly measures intraglottal, subglottal, and vocal fold collision pressures during phonation. Synchronous measurement of intraglottal and subglottal pressures was accomplished using two miniature pressure sensors mounted on the end of the probe and inserted transorally in a 78-year-old male who had previously undergone surgical removal of his right vocal fold for treatment of laryngeal cancer. The endoscopist used one hand to position the custom probe against the surgically medialized scar band that replaced the right vocal fold and used the other hand to position a transoral endoscope to record laryngeal high-speed videoendoscopy of the vibrating left vocal fold contacting the pressure probe. Visualization of the larynx during sustained phonation allowed the endoscopist to place the dual-sensor pressure probe such that the proximal sensor was positioned intraglottally and the distal sensor subglottally. The proximal pressure sensor was verified to be in the strike zone of vocal fold collision during phonation when the intraglottal pressure signal exhibited three characteristics: an impulsive peak at the start of the closed phase, rounded peak during the open phase, and minimum value around zero immediately preceding the impulsive peak of the subsequent phonatory cycle. Numerical voice production modeling was applied to validate model-based predictions of vocal fold collision pressure using kinematic vocal fold measures. The results successfully demonstrated feasibility of in vivo measurement of vocal fold collision pressure in an individual with a hemilaryngectomy, motivating ongoing data collection that is designed to aid in the development of vocal dose measures that incorporate vocal fold impact collision and stresses.

Coherence of wall pressure fluctuations in zero and adverse pressure gradients

The wall pressure fluctuations induced by turbulent boundary layers on a flat plate model were measured with an L-shaped array of Kulite miniature pressure sensors. By installing a NACA airfoil above the plate with an adjustable angle of attack, different adverse pressure gradients were produced on the plate. The streamwise and spanwise coherence and coherence lengths of the wall pressure fluctuations are calculated for zero and adverse pressure gradient flows. The effect of the pressure gradient on the coherence and coherence lengths in both streamwise and spanwise directions is quantified. Based on the results of the present test cases and selected published datasets in zero pressure gradients, covering the range of Reynolds numbers based on the momentum thickness between 3300 and 42000, the Reynolds number dependence of the streamwise wall pressure coherence is discussed and mathematically expressed. A coherence length model for zero pressure gradient boundary layers, taking into account the Reynolds number effect, is proposed based on scaled spectra of the coherence lengths. In comparison with the other published models, the present model achieves a considerable improvement of the prediction accuracy for boundary layer flows, covering a large range of Reynolds numbers. Furthermore, an evaluation of the Corcos model and the Smol’yakov and Tkachenko model for prediction of the off-axis coherence of the fluctuating wall pressure field is made.

Laboratory Evidence of Transient Pressure Surge in a Fluid-Filled Fracture as a Potential Driver of Remote Dynamic Earthquake Triggering

Abstract Seismic waves carrying tiny perturbing stresses can trigger earthquakes in geothermal and volcanic regions. The underlying cause of this dynamic triggering is still not well understood. One leading hypothesis is that a sudden increase in the fluid-pore pressure in the fault zone is involved, but the exact physical mechanism is unclear. Here, we report experimental evidence in which a fluid-filled fracture was shown to be able to amplify the pressure of an incoming seismic wave. We built miniature pressure sensors and directly placed them inside a thin fluid-filled fracture to measure the fluid pressure during wave propagation. By varying the fracture aperture from 0.2 to 9.2 mm and sweeping the frequency from 12 to 70 Hz, we observed in the lab that the fluid pressure in the fracture could be amplified up to 25.2 times compared with the incident-wave amplitude. Because an increase of the fluid pressure in a fault can reduce the effective normal stress to allow the fault to slide, our observed transient pressure surge phenomenon may provide the mechanism for earthquake dynamic triggering.

Real-time in vitro measurement of denture-mucosa pressure distribution in a typical edentulous patient with and without implants: Development of a methodology

PurposeTo measure the pressure distribution on the oral mucosa in vitro by comparing the pressure distributions under a complete denture and that of an implant overdenture. Materials and methodAnatomically accurate models and conventional Class I complete denture (CD) were produced and subjected to cyclic loading using a 100 N vertical centric and unilateral masticatory load with the universal testing machine (Instron 3369). Four miniature pressure sensors were positioned at four different locations in the intaglio surface of the denture and recorded pressure at 100 Hz frequency measured during a 10-cycle load at 1 Hz. Testing was repeated in different clinical combinations; CD vs. single implant overdentures (1-IOD), CD vs. two, three and four implant overdentures (2-IOD, 3-IOD vs. 4-IOD). The pressure profile (kPa) of complete dentures were measured and compared to the implant overdenture combinations. Collected data was statically analysed using SPSS and one-way analysis of variance. ResultsThe highest mean pressure was observed in CD group, with the mean mandible buccal ridge pressure value of 212.82 kPa ± 136.9 due to its surface area. There were no statistically significant differences between the group combinations (p = 0.146) but between various locations in the mean pressure recorded across the five denture/overdenture combinations. ConclusionCD experienced large pressure values on mandibular denture. 1-IOD demonstrated the most pressure in comparison to CD where with an increase in the number of implants used, it transformed the denture from being pure-borne mucosa to an implant overdenture, providing support and distributing the pressure amongst the implants.

Design and Fabrication of a New Wearable Pressure Sensor for Blood Pressure Monitoring.

In recent years, research into the field of materials for flexible sensors and fabrication techniques directed towards wearable devices has helped to raise awareness of the need for new sensors with healthcare applicability. Our goal was to create a wearable flexible pressure sensor that could be integrated into a clinically approved blood pressure monitoring device. The sensor is built from a microfluidic channel encapsulated between two polymer layers, one layer being covered by metal transducers and the other being a flexible membrane containing the microfluidic channel, which also acts as a sealant for the structure. The applied external pressure deforms the channel, causing changes in resistance to the microfluidic layer. Electrical characterization has been performed in 5 different configurations, using alternating current (AC) and (DC) direct current measurements. The AC measurements for the fabricated pressure sensor resulted in impedance values at tens of hundreds of kOhm. Our sensor proved to have a high sensitivity for pressure values between 0 and 150 mm Hg, being subjected to repeatable external forces. The novelty presented in our work consists in the unique technological flow for the fabrication of the flexible wearable pressure sensor. The proposed miniaturized pressure sensor will ensure flexibility, low production cost and ease of use. It is made of very sensitive microfluidic elements and biocompatible materials and can be integrated into a wearable cuffless device for continuous blood pressure monitoring.

Effect of Ankle Motion and Tensile Stress at the Achilles Tendon on the Contact Pressure Between the Achilles Tendon and the Calcaneus

Impingement between the Achilles tendon and the posterosuperior prominence of the calcaneus is considered to be a cause of insertional Achilles tendinopathy. The corresponding treatment intends to reduce tensile stress from calf muscles and avoid hyper-dorsiflexion of the ankle joint for decreasing the contact pressure; however, no study has reported on whether these treatments can decrease impingement. Thus, this study investigated the hypothesis that the tensile stress of the Achilles tendon and ankle motion affect the contact pressure between the Achilles tendon and the posterosuperior prominence of the calcaneus. Six fresh-frozen cadaveric lower leg specimens were procured. Each specimen was set to a custom foot-loading frame and loaded with a ground reaction force of 40 N and a tensile load of 70 N along the Achilles tendon. The contact pressure between the Achilles tendon and the posterosuperior prominence of the calcaneus was measured using a miniature pressure sensor under different tensile loadings of the Achilles tendon at the neutral ankle position. Similarly, the contact pressures during the ankle motion from a neutral position to maximum dorsiflexion were measured. The tensile load of the Achilles tendon and ankle motion affected the contact pressure between the Achilles tendon and the posterosuperior prominence of the calcaneus. The contact pressure increased with tensile load or ankle dorsiflexion. Conditions with increasing the tensile load of the Achilles tendon or under ankle dorsiflexion increase the contact pressure between the Achilles tendon and the posterosuperior prominence of the calcaneus.

Biomedical Catheters With Integrated Miniature Piezoresistive Pressure Sensors: A Review

Biomedical catheters are thin hollow tubes inserted into the human body to accurately measure various physiological parameters during invasive surgical procedures and diagnostic methods. Sensors realized using micro-/nano-electro-mechanical systems (MEMS/NEMS) technology are integrated with catheters to measure blood pressures, flows, pH- and glucose levels, and temperature. Of these physiological parameters, pressure sensing is of significant importance in identifying and treating various biomedical conditions. Piezoresistive technique is a widely investigated and preferred sensing mechanism to realize miniature sensors for its numerous advantages. In this paper, we critically review biomedical catheters as well as miniature piezoresistive pressure sensors developed for catheters. First, the evolution of catheters and their applications in measuring physiological parameters are discussed in detail. Next, the progress of piezoresistive pressure sensors developed for integration with catheters are described under three broad categories by considering various aspects such as diaphragm shape, material & size, piezoresistor type & material, readout type, fabrication processes, salient features of the device, and packaging techniques. A detailed section is devoted to alternative recent piezoresistive materials, including silicon nanowire (SNW), carbon nanotube (CNT), and Graphene. Finally, the process of catheterization and testing of biomedical catheters with integrated pressure sensors in the clinical environment are elaborated.

Data reduction of friction factor, permeability and inertial coefficient for a compressible gas flow through a milli-regenerator

A regenerator of a Stirling machine alternately absorbs and releases heat from and to the working fluid which allows to recycle rejected heat during theoretical isochoric processes. This work focuses on a milli-regenerator fabricated with a multiple jet molding process. The regenerator is a porous medium filled with a dense pillar matrix. The pillars have a geometrical lens shape. Two metallic layers (chromium and copper) are deposited on the polymer pillars to increase heat transfer inside the regenerator. We performed experiments on different milli-regenerators corresponding to three porosities (ε = 0.80, 0.85 and 0.90) under nitrogen steady and oscillating compressible flows (oscillating Reynolds number in the range 0 &lt; Reω &lt; 60 and Reynolds number based on the hydraulic diameter ReDh,max&lt;6000) for different temperature gradients (ΔT &lt; 100°C). Temperature, velocity and pressure experimental measurements are performed with microthermocouples (type K with 7,6 µm diameter), hotwires and miniature pressure sensors, respectively. We identified a threeterm composite correlation equation for the friction factor based on a Darcy-Forchheimer flow model that best-fit the experimental data. In steady and oscillating flows permeabilities and inertial coefficients are of the same magnitude order. Inertial coefficients decrease when the porosities increase.