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Rigid filters made of ceramic or metal are widely used to remove solid particles from hot gases at temperature above 260 °C in the petrochemical and coal industries. Pulse-jet cleaning of fine dust from rigid filter candles plays a critical role in the long-term operation of these filters. In this study, an experimental apparatus was fabricated to investigate the behavior of a 2050 mm filter candle, which included monitoring the variation of pressure dynamic characteristics over time and observing the release of dust layers that allowed an analysis of the cleaning performance of ISO 12103-1 test dusts with different particle size distributions. These results showed the release behavior of these dusts could be divided into five stages: radial expansion, axial crack, flaky release, irregular disruption and secondary deposition. The cleaning performance of smaller sized dust particles was less efficient as compared with larger sized dust particles under the same operating conditions primarily because large, flaky-shaped dust aggregates formed during the first three stages were easily broken into smaller, dispersed fragments during irregular disruption that forced more particles back to the filter surface during secondary deposition. Also, a “low-pressure and long-pulse width” cleaning method improved the cleaning efficiency of the A1 ultrafine test dust from 81.4% to 95.9%.

Fibrous materials are subjected to compression in many industrial processes, such as natural gas filtration at high pressures. A number of investigations have been conducted to study the relationship among compression load, pore size and permeability. However, the effect of structural parameters on the filtration performance of compressed media has not been reported. In our study, the variation of packing density, pore size distribution and contact angle for both oleophilic and oleophobic glass fibrous materials with different degrees of compression were experimentally measured. Furthermore, the filtration performance of compressed filters such as wet pressure drop, penetration, saturation, and liquid distribution are investigated. The results showed that the contact angle of the filter material did not change within the compression range of 55%, but the liquid holding capacity was improved, which was due to the increase of fiber surface area per unit volume with the increase of compression amount. After compression, the pore size distribution shifted to small pores, leading to an increase in jump pressure drop. For multi-layer filters, the channel pressure drop was affected by the variations of the thickness, packing density and pore size of filter material. In addition, the saturation depended on the interaction between the fiber surface area per unit volume and the interstitial velocity, which also determined the liquid distribution in the filter. And for single-layer filters, the diffusion mechanism was weakened after compression, resulting in an increase in the penetration for droplets smaller than 500 nm.

The performance of coalescing filter cartridges (of four different specifications) after a 410-day field operation for the high-pressure natural gas purification has been evaluated in this study. Our evaluation of field-tested filter cartridges showed the increase of the filter pressure drop by a factor of 5.28 and the decrease of the filtration efficiency at various degrees, resulting in 50.3 % loss in the filter quality factor (on average). The above evidenced the performance degradation of coalescing filter cartridges after the field application. Through the basic property characterization of composite media in tested filter cartridges, it was found the observed performance degradation of tested coalescing filter cartridges was attributed to the increase in the pore size of the coalescing layer (0.4–30.4 %), and the decrease of both pore size (2.3–23.7 %) and thickness (3.3–28.7 %) in the drainage layer. Via the performance evaluation of composite media composed of only coalescing and drainage layers used in both fresh and tested cartridges, it is concluded that the coalescing filter cartridges with composite media having both oleophobic coalescing and drainage layers, and high tensile and compression strengths, are excellent candidates for the stable and reliable operation of natural gas filtration at high pressure.

Sodium lignosulfonate (SL) and silane coupling agent (SCA) were used to modify the fibrous filter elements to remove the useless binder and improve the strength, and their effects on the microstructures and properties were investigated. SL reduced the slurry viscosity to remove the binder present in the pores. The bulk density of the filter element was decreased after modification by SL, but the flexural strength was hardly affected. SCA enabled more Si–O bonds to be formed to enhance the adhesion between the binder and fibers. After further modification by SCA, the binder content at the crossing points between the fibers and the bonding strength between the binder and fibers were improved so that the flexural strength was significantly increased. Furthermore, owing to the decrease in the surface tension after adding SL, more binder bridges were formed on the upper surface of the filter element, affecting the gas permeability.

Aluminum silicate fiber-based ceramic filter elements were prepared using different drying methods and calcination temperatures. The effects of these parameters on the performance, including the microstructure, bulk density, shrinkage, flexural strength, and pressure drop, were investigated. The results indicated that there were differences in the microstructures of the filter elements fabricated via three drying methods and that the microstructure significantly affected the flexural strength and initial pressure drop. Compared with blast and vacuum drying, microwave drying inhibited the migration of the binder, improving the structural uniformity and mechanical strength. The filter element effectively removed particles with a size of >1 μm and exhibited a good regeneration performance. Furthermore, owing to the crystallization of fibers after calcination at 1000 °C, the density and shrinkage of the sample were significantly increased and the initial pressure drop was reduced. The mechanical strength was significantly reduced after calcination at 800 and 1000 °C. As the calcination temperature increased, the filter element gradually became brittle. Therefore, the optimal calcination temperature of 600 °C was important for obtaining an excellent filter element with a high flexural strength and low pressure drop.

Ceramic filters have been widely used in industrial engineering, such as the Shell coal gasification process (SCGP) and Integrated gasification combined cycle (IGCC), where the performance of the ceramic filter is achieved by the pulse jet. Residual pressure drop and gas consumption are directly related to reverse-flow pulse (RFP) pressure. However, in the process of operation, the RFP pressure is too large and gas consumption is too high. In this study, the effects of RFP pressures on filter’s cleaning efficiency, residual pressure drop, and gas consumption were investigated on a ceramic filter. Within a certain range, the cleaning efficiency gradually increased with increased RFP pressure. When the RFP pressure reached a certain value, the cleaning efficiency did not increase with increased pressure, showing a quadratic relationship between cleaning efficiency and RFP pressure. The residual pressure drop and RFP pressure were also in a quadratic relationship. Besides, the gas consumption increased linearly as increased RFP pressure according to the theoretical model. Based on the research results, a multi-objective optimization model of a ceramic filter was established with the cleaning efficiency as a constraint condition, gas consumption and residual pressure drop as the optimization objectives. A fuzzy decision-making method was used to solve the optimization model and calculate the residual pressure drop and gas consumption, from which the optimal RFP pressure was obtained.

A low-pressure plasma technology was applied for the first time to modify the surface of filter media used for separation of liquid particles from air streams without affecting their bulk properties. The effect of surface wettability on the filtration performance and internal liquid distribution of glass microfiber filters was investigated experimentally. It was found that typical jump and channel characteristics existed in the evolution of both pressure drop and penetration of the filters, which means that in the jump stage of pressure drop, the penetration of oleophobic filters decreased steeply with loading time and the rate of decrease was greater in low-efficiency grade media. However, the variation trends of penetration were opposite in high- and low-efficiency grade oleophilic filters; that is, increased and decreased, respectively. In the channel stage of pressure drop, the penetration increased gradually with loading time in the oleophilic filters but changed little in the oleophobic ones. The oleophobic filters demonstrated lower jump pressure drop, film thickness, equilibrium saturation and loss of efficient fibers compared with their oleophilic counterparts. Furthermore, the oleophobic filters exhibited a remarkable advantage in overall filtration performance due to their comparable pressure drop and significantly higher filtration efficiency at steady state compared with their oleophilic counterparts. It was demonstrated that the most penetrating particle size remained almost constant for filters with the same structure but different wettabilities, although the filtration efficiency for particles across the whole size range was improved in oleophobic filters. Additionally, the dependence of filter performance on the surface modification method was also explored using an oleophilic medium as the control. The results showed that different approaches were capable of changing the media wettability to various degrees. Compared with the unmodified filter or those modified only by nanoparticles or fluoride copolymer, the filters modified by the combination of nanoparticles and fluoride copolymer had the overall best filtration performance.

There exists large space to save energy of high-sulfur natural gas purification process. The multi-objective optimization problem has been investigated to effectively reduce the total comprehensive energy consumption and further improve the production rate of purified gas. A steady-state simulation model of high-sulfur natural gas purification process has been set up by using ProMax. Seven key operating parameters of the purification process have been determined based on the analysis of comprehensive energy consumption distribution. To solve the problem that the process model does not converge in some conditions, back-propagation (BP) neural network has been applied to substitute the simulation model to predict the relative parameters in the optimization model. The uniform design method and the table U21 (107) have been applied to design the experiment points for training and testing BP model. High prediction accuracy can be achieved by using the BP model. Non-dominated sorting genetic algorithm-II has been developed to optimize the two objectives, and 100 Pareto optimal solutions have been obtained. Three optimal points have been selected and evaluated further. The results demonstrate that the total comprehensive energy consumption is reduced by 13.4% and the production rate of purified gas is improved by 0.2% under the optimized operating conditions.

The corrosion product scale formation mechanism of X80 steel in the supercritical (SC) CO2-H2O binary phase containing flue gases and NaCl impurities at 8 MPa and 60 °C was investigated. The formation process of corrosion product scale was significantly different in the SC CO2-saturated aqueous and water-saturated SC CO2 phases. In the SC CO2-saturated aqueous phase, an amorphous product scale was formed initially, then a crystalline product scale was precipitated. The formation process of the corrosion product scale in the saturated-water SC CO2 phase proceeded in just the opposite way. In the water-saturated SC CO2 phase, the inner scale had better protectiveness to the substrate than the outer scale, but it was just the opposite in the SC CO2-saturated aqueous phase. The severity of pitting in the SC CO2-saturated aqueous phase was much greater than the pitting corrosion in the water-saturated SC CO2 phase.

In this study, the effects of pressure on the optical sensor in an on-line particle measurement system were investigated in detail based on the quantum theory and kinetic theory of gas. The results show that pressure alters the divergent light angle by altering the density and refractive index of the gas, so that the focused beam position is offset from the center of the aerosol tube; this causes the intensity per cross-section area of the optical measurement volume (OMV) to change, creating a significant deviation in the measurement result relative to the actual sample. Theoretical and experimental dynamic models were established based on the geometric optics theory and the geometric mathematical relation to analyze these variations. The results show that OMV size increases as pressure and divergent light angle increase. During an actual particle measurement process, the dynamic models proposed here can be used to effectively adjust and reconstruct the optical sensor, which can substantially improve the precision and accuracy of the measurement results.

Fibrous coalescing filters are used to collect droplets from air streams and are widely applied in the process industry but liquid surface tension affects the filter performance. In this work, the wet pressure drop, saturation, liquid distribution, and filtration efficiency of both oleophilic and oleophobic filters were investigated using four liquids with different surface tensions but similar densities and viscosities to produce aerosols. The results showed that an increase in surface tension accelerated the evolution of the wet pressure drop of filters. The jump pressure drop increased significantly while the excess channel pressure drop was constant with growing surface tension. The average saturation in the channel layers decreased as the surface tension increased and the decrease rate was much greater in oleophobic than oleophilic filters. The liquid distribution was also influenced by the surface tension. In oleophilic filters, higher surface tension led to a smaller number of channels and a larger size per channel. The channel boundaries were difficult to distinguish when the relative wettability between the liquid and filter media was very high or very low. Both oleophilic and oleophobic filters showed higher filtration efficiency at steady state in both sub-micrometer and micrometer droplet-size ranges when separating liquids with a larger surface tension. Additionally, superoleophilic and superoleophobic filters were prepared through a dip-coating method. When compared with untreated (oleophilic) and superoleophilic filters, the superoleophobic filters had the best overall performance for each liquid type due to the greatly reduced liquid hold-up.

Oil aerosol usually causes air pollution, health issues, and corrosion to equipment. The removal of aerosol oil particles from the air is a crucial process in industrial production and daily life. Although fibrous filters have been a widely used material for the separation of oil aerosol from the air, it is still a challenge to separate submicrometer aerosol oil particles with both high filtration efficiency and low resistance. Herein, we report a novel approach to markedly reduce the pressure drop of a fibrous filter and simultaneously increase its aerosol filtration efficiency, only by surface treatment to make the filter have in-plane alternating superoleophilic and superoleophobic patterns. We used a spraying method to prepare superoleophobic and superoleophilic patterns on the filter. The best filtration results were achieved when two layers of the patterned filters that have superoleophobic and superoleophilic strips (both width, 5 mm) were stacked in a way that the opposite wetting surfaces contacted each other between the layers. The filter showed a much-reduced filtration resistance and the pressure drop (4.16 kPa) at the pseudo-steady state being at least 45% lower when compared to the two-layer controls with a homogeneous surface wettability (i.e., untreated surface, superoleophobicity, and superoleophilicity). It also showed higher filtration efficiency (98.37% for small oil mists and 99.99% for large oil mists) and over two times higher quality factor (0.99 kPa-1 for small oil mists and 2.27 kPa-1 for large oil mists). The asymmetric wettability leads to the formation of unobstructed channels for the air stream to penetrate through the filter matrix, leading to a low resistance with improved oil capture efficiency. The pattern strip width showed an effect on filtration performance. This unexpected finding may provide a novel approach to designing high-performance, low energy consumption, and long-life coalescence filters.

The use of a two-phase expander in place of the throttling valve can improve the organic double-flash cycle (ODFC) thermal efficiencies. This work compares the thermoeconomics of a regenerative ODFC (RODFC), an expander-ODFC (EODFC) and a double expander-ODFC (DEODFC). The evaporation and flash temperatures of these cycles are optimized to maximize the net power outputs for geofluid temperatures of 120–180 °C. The results show that the two-phase expander in EODFC leads to the lowest optimal first-stage flash temperature with the highest vapor mass flow rate for the high-pressure turbine. The power generated by the two-phase expanders in DEODFC is less than that generated by the expander in EODFC but the power generated by the turbines is 5.9–8.4% higher. DEODFC produces the maximum net power, followed by EODFC and RODFC for each geofluid temperature. The exergy losses during evaporation are reduced by up to 41% by regeneration in RODFC, EODFC and DEODFC. The two-phase expanders increase the exergy losses during power generation but decrease the losses during throttling and condensation; thus, the exergy efficiency of DEODFC increases to 32.8–46.8%. The levelized cost of electricity for DEODFC is 15–33% lower than for ODFC due to the great increase in the net power output.

The effect of pore size on filtration performance of coalescing filters was investigated experimentally, using four oleophilic and three oleophobic glass-fiber filter materials. The filters were fabricated into sandwiches composed of a varying number of layers, each of the same filter materials. The results showed that both trend and extent of variation of the excess channel pressure drop with number of layers were dominated by the pore size. As the pore size decreased, the excess channel pressure drop exhibited the increasing, stable, and decreasing trends with increasing number of layers for oleophobic filters, respectively, while it kept growing with number of layers for oleophilic filters. The jump pressure drop controlled by liquid film was inversely proportional to the mean pore size, with an empirical fit equation established to allow quantitative analysis of relationship between film thickness and mean pore size for oleophilic filters. The pore size was a dominant factor in improving steady-state filtration efficiency, as a large-mean-pore-size filter was unable to efficiently separate sub-micrometer-sized droplets even with a high number of layers. Conversely, it was found that re-entrainment of micrometer-sized droplets was prone to occur in small-mean-pore-size oleophobic and large-mean-pore-size oleophilic filters, resulting in the pronounced reduction of steady-state filtration efficiency. Based on quality factors at steady state, a filter with large mean pore size was much superior in separation of micrometer-sized droplets, while a filter with moderate rather than smallest mean pore size, should be applied to separate sub-micrometer-sized droplets.

Diesel spray and combustion in a constant-volume engine cylinder was simulated by a large eddy simulation (LES) approach coupling with a multicomponent vapourisation (MCV) modelling. The simulation focused on the inclusion of the interaction between fuel spray and gas-phase turbulence flow at the sub-grid scale. The LES was based on the dynamic structure sub-grid model, and an additional source term was added to the filtered momentum equation to account for the effect of drop motion on the gas-phase turbulence. The multicomponent drop vapourisation modelling was based on the continuous thermodynamics approach using a gamma distribution to describe the complex diesel fuel composition and was capable of predicting a more complex drop vapourisation process. The effect of gas-phase turbulence flow on the fuel drop vapourisation process was evaluated through the solution of the gas-phase moments of the distribution in the present LES framework. A non-evaporative spray in a constant-volume engine cylinder was f...

Two-stage filtration units including pre-filtration and coalescence filtration play an important role in the quality control of high-sulfur natural gas, because micron-sized particles can significantly influence the gas sweetening process. Filtration failure usually occurs in operation, because of the mismatch between the two types of filter cartridge in the filtration unit. To overcome this drawback, the performance of the coalescing filter cartridge must be improved and a matching pre-filter cartridge need to be designed. One approach for accomplishing this is to spray an atomizing adhesive onto the fibrous layers of the filter cartridge to enhance its integrated filtration performance. In this study, a pleated protective layer and a hydrophobic drainage layer were designed for the coalescing filter cartridge to prolong the operation cycle and quickly drain the captured liquid. Testing results showed that the new-type of coalescing filter cartridge exhibited 75% decrease in its steady pressure drop over that of the original filter and the filtration efficiency reached 99.99% for liquid droplets with diameter greater than 0.3 µm. The structure of pre-filter cartridge was also redesigned to match the performance of the new coalescing filter cartridge. As a result, the operational loading of the improved two-stage filter cartridges achieved the desired performance. Field testing of the new filter combination showed that the acid gas filtration unit produced a more reasonable pressure drop and a higher rate of captured liquid than with the conventional filter.

The evolution of saturation and liquid distribution in the filtration process of oleophobic coalescence filters were investigated. The effect of a drainage layer on the oleophobic filters with different layers was also evaluated experimentally. It was found that the wet pressure drop, saturation and liquid distribution were varied by assembling a drainage layer outside of the coalescing layer. For the filters with a non-wettable drainage layer, the pressure drop developed in the same fashion as the filters without a drainage layer. A secondary rise in the pressure drop occurred in the filtration process of the filters with a wettable drainage layer. For the filters with the same layers, assembling a drainage layer increased the flow resistance to liquid in the channels, resulting in an increase in the single channel size but a constant in the number of channels per layer. Moreover, the feature of channels in the coalescing layer was affected in varying degrees by the drainage layer, according to the location of filter materials in the filters. It was demonstrated that the middle filter materials were less susceptible to the drainage layer. In addition, there was less effect of a drainage layer on the liquid flow in the filters with more layers.

Yield and quality, as the two most important output variables of high-sulfur gas (HSG) sweetening process, are affected by the operating parameters. The HSG sweetening process involves more than ten operating parameters, so the relationship between the parameters and output variables is complex, non-linear and strong coupling. This paper tries to use data mining methods to explore this relationship and apply it to optimize the yield and quality. First, a ten (inputs)-to-three (outputs) model is established by extreme learning machine (ELM). Then, a preference-driven multi-objective optimization algorithm is used to maximize the yield while ensuring that the concentration of carbon dioxide (CO2) and the concentration of hydrogen sulfide (H2S) in the treated gas are close to but not exceeding 3% and 4 ppm respectively. The proposed method is validated in a HSG purification plant in southwest China. A set of 3044 production data is collected and randomly divided into 80 and 20% for training and testing. The results show that the established ELM model is in good agreement with the actual operation data. The maximum deviation of mean square error (MSE), mean absolute error (MAE) and average absolute deviation percent (AAD %) of the predictions in three scenarios are 0.2047, 0.3177 and 7.91% respectively. Moreover, the optimization based on the obtained ELM model is also validated. In particular, the H2S concentration and CO2 concentration in the treated gas are significantly higher than those before optimization, but have not exceeding the limits. Thus, the consumptions of energy and amine solvents decreased, while the yield increased.

Two-phase closed thermosyphons (TPCTs) are simple and efficient heat exchangers. They have been explored for use in the renewable energy resource utilization marker and low grade thermal energy heat recovery systems. A transient simulation model for a TPCT was established by SINDA/FLUINT with low global warming potential Freon R245fa as working fluid. The TPCT was manufactured from a 40 mm inner diameter (Di), 3 m long smooth copper tube with a wall thickness of 2 mm. It consists of the evaporator, adiabatic and condenser sections with 1 m long for each section. The evaporator section was immersed in a water bath and the condenser section was cooled by recycled water. The effects of water bath temperature Tb and inlet temperature of cooling water Tcw,i were investigated by experimental and simulation methods. The results show the heat transfer rate Q and overall heat transfer coefficient U increase with the increase of Tb, and the decrease of Tcw,i. Good agreement between experimental and simulation results confirms the model is accurate and reliable. The influence of filling ratio (FR) and Di on the performance of TPCT was also investigated. The optimum FR for Di of 30 mm, 40 mm and 50 mm are 15%, 15% and 25% respectively. Moreover performances of 60–150 m long TPCTs were investigated based on the developed model.

Conventional filters for oil-mist filtration chiefly use fibrous materials, which have inadequate efficiency for small oil mists (e.g., size < 1 μm). Thicker filter mats are applied to increase filtration efficiency, but they not only consume more filter materials but also increase the energy cost due to increased air flow resistance. In this study, we have for the first time demonstrated a novel concept of improving the filtration efficiency for the separation of small oil mists from air without apparently increasing the air flow resistance. Using a commercial glass fibre filter as a model, we prepared a superoleophobic surface through a wet chemical coating method using a perfluoroalkyl acrylic copolymer as the material. The coating shows very little effect on the porous feature and air permeability of the fibrous filter. The superoleophobic treatment shows a considerable increase in the filtration efficiency. For the filter of thickness 1.12 mm, the filtration efficiency for small oil mists increases from 96.40% to 99.44%, whereas the pressure drop is increased by only 6% but the downstream oil mist content is reduced by 85%. For large oil-mists, the superoleophobic filter has almost 100% filtration efficiency with downstream mist subsided close to zero. A “bounce-collide-drain” mechanism was proposed to explain the improvement of the filtration performance. Such a novel concept may have great significance in the development of high efficiency, low energy consumption gas–liquid coalescing filters.

Different drying conditions and three types of porous fibrous ceramics with different contents and dispersions of the silica-based binder were used to prepare catalytic filter elements (CFEs) for the simultaneous removal of NOx and dust, and their effects on the catalytic performances were also investigated. The results indicated that the content and dispersion of the silica-based binder had a significant effect on the catalytic activity of the CFE. The removal of useless binder in the pores and the inhibition of binder agglomeration could improve the NO conversion. Compared with traditional electric heating blast drying, microwave drying effectively inhibited the migration of titanium sol so that the catalyst in the prepared CFE had good uniformity and dispersion. Meanwhile, microwave drying changed the morphology, crystal grain, and pore structure of the catalyst support (TiO2), ultimately increasing the specific surface area. Furthermore, this study also determined that the optimal average output power for microwave drying was 385 W. With a filtration velocity of 1 m/min, NO and NH3 inlet concentrations of 600 ppm, and an O2 content of 5 vol%, the prepared CFE achieved more than 98% NO conversion in the temperature range of 250 to 350 °C, and its highest conversion of 100% was achieved at 275–325 °C.

Two-phase closed thermosyphons (TPCTs) are simple, efficient, and low cost heat exchangers. They have been explored for use in the renewable energy resource utilization marker and low grade thermal energy heat recovery systems. Freon R134a has been extensively used in refrigeration systems and researched as a working fluid of TPCTs; however; it has high global warming potential and operating pressure. In this paper, an experimental investigation of the performance of TPCTs charged with eight working fluids: R134a, R601, R245fa, R600a, R1234ze, R152a, R245fa/R152a, and R601/R245fa have been carried out. The experimental results showed that R245fa/R152a offered the best performance in maximum heat transfer rate. R134a outperformed the other pure working fluids, while R600a and R1234ze had close performances to that of R134a. R245fa showed marginal improvement at higher operating temperatures. The predictions of six evaporation heat transfer coefficients (HTCs) correlations, including Imura, Shiraishi, Labuntsov, Kutateladze, Cooper, and Rohsenow were compared with the experimental results. In the five constant coefficients and powers correlations, the Shiraishi and Cooper correlations had superior accuracy. The coefficients and powers of the Rohsenow correlations fitted based on the experimental data, while they had the best accuracy. Nusselt and Hashimoto-Kaminaga correlations were chosen to predict the condensation HTCs. Both of them tend to over-predict the condensation HTCs in low heat fluxes while under-predicting in high heat fluxes. The experimental results had greater agreement with Hashimoto and Kaminaga correlations.

The effect of filtration pressure on the particle collection performance of fibrous filter media was systematically investigated. The experiments were carried out in the filtration pressure range of 0.33–3 atm and using test particles in the electrical mobility size range of 20–400 nm. Two fibrous filter media: one made of glass fibers (GF) and the other made of metal fibers (SF), were selected for these tests. Our study shows that, at a face velocity of 0.1 m/s, by increasing the filtration pressure from 0.33 to 3 atm, the most penetrating particle size (MPPS) was slightly decreased, and the particle penetration was increased from 0.8% to 14% for media GF, and from 6.7% to 32.1% for media SF. In comparison to data calculated for existing single-fiber models, it was found that, for diffusive particles, the best fitting of available experimental data (i.e., those collected in this study and reported in others) was found under the assumption that the −1/3 power for the Peclet number is included in the single-fiber efficiency formula. Lastly, a semi-empirical single-fiber efficiency for diffusive particles is proposed to describe the reported experimental data.