Tar Dew Point Research Articles

Sewage sludge and digestate gasification in an atmospheric fluidized bed gasifier

The gasification of sewage sludge (SS) and digestate was investigated in a pilot-scale fluidized bed gasifier with an output of 100 kWt. The treatment of these by-products is an ongoing challenge for sustainable development. SS and digestate are most commonly used as fertilizers. However, regulations restrict their use, mainly because of the content of heavy metals, pathogens and bacteria. Gasification of these by-products instead of application to agricultural land seems to be more efficient, as the syngas can subsequently be used for combined heat and power (CHP) generation. A series of measurements were carried out to get a better understanding of the gasification process of these fuels and to study the effects of gasifying agent on the syngas composition, particulate matter (PM) and tar. The produced syngas and tar were analyzed using a gas chromatograph with mass spectrometry (GC–MS). The results showed that no ash slagging was observed and therefore it is feasible to operate digestate and SS gasification at 750°C. The lower heating value (LHV) of the syngas from digestate and SS with air as the gasifying agent is comparable, 4.06 MJ·Nm−3 for digestate and 4.11 MJ·Nm−3 for SS. The addition of steam had a positive effect on the amount of tar and the tar dew point, which was below 150°C. Tar reduction in digestate was 5037.3 mg·Nm−3 to 3566.3 mg·Nm−3 and in SS 7447.7 mg·Nm−3 to 3390.3 mg·Nm−3. Furthermore, the concentrations of the individual tar compounds were determined and subsequently divided into tar classes.

Fe-doped olivine and char for in-bed elimination of gasification tars in an air-blown fluidised bed reactor coupled with oxidative hot gas filtration

Gasification experiments were carried out in a pilot scale fluid bed reactor operated under allothermal mode and low fluidisation regime with iron-doped olivine and char as catalyst for in-situ tar abatement.The catalyst combination resulted in a reduction of 50% in the overall tar yield with respect to the reference values. Furthermore, the integration of an oxidative Hot Gas Filtration unit downstream the gasification reactor led to a further reduction in overall tar yield and relatively clean gas was obtained (approx. 1 g/Nm3, benzene-free). The tar dew point of the resulting producer gas was estimated to 80 °C, only 40 °C above the threshold value recommended for its valorisation in standard internal combustion engines. Moreover, catalyst elutriation and char hold-up took place to a large extent inside the reactor. The analysis of catalyst samples at different Time-On-Stream (TOS) revealed: (i) a considerable loss of iron oxides during the first hour of test because of the interparticle mechanical attrition (mostly surface abrasion) and partial reduction of hematite to magnetite and wustite but, stable composition at higher TOS, (ii) the loss of the iron oxide coverage of Fe/olivine particles and the formation of agglomerates with increasing TOS and, (iii) the amount of carbon deposited in the surface of the Fe/olivine particles increased with TOS, but in any case, these carbon deposits can be completely oxidized above 650 °C.

Catalytic reforming of volatiles from pyrolysis of biomass components over a novel Ni/magnesium slag catalyst

A novel magnesium slag (MS) supported nickel-based catalysts (Ni/MS) was prepared and tested in catalytic reforming of volatiles derived from the pyrolysis of biomass components (cellulose, hemicellulose and lignin) and two biomass feedstocks (pine and straw). The physical and chemical properties of catalysts were characterized by XRD, TPR, BET, TEM, TPO-TGA and Raman spectroscopy. Ni/MS catalyst showed a higher tar removal efficiency than conventional Ni/γ-Al2O3 under the same Ni loading. The tar conversion of cellulose hemicellulose and lignin reached 94.8 %, 92.0 % and 89.4 %, respectively over Ni/MS catalyst. In addition, the lignin content of the biomass feedstock appreciably affected the yield and dew point of tar. The highest increment in gas and H2 yield (increased by 39.0 % and 106.4 %, respectively) was observed in cellulose. The highest decrement in tar dew point (from 166.4 ℃ to 53.0 ℃) was observed in hemicellulose. The excellent tar removal of Ni/MS was attributed to the metallic Ni and Ni-Fe alloy on catalyst surface. Besides, the MgO and Fe2O3 phases in the support can retard the carbon deposition during the reaction. The results of this study are expected to provide insightful guidance for the application of Ni/MS catalysts for pyrolysis/gasification of biomass.

Thermodynamic approach to gas analysis in gasification systems

Gas analysis is the primary method for performance evaluation in gasification systems. Ratio between CO and CO2 as well as the methane content, give valuable information about the quality of the ongoing gasification process. Furthermore, the heating value of the gas is the fundamental parameter for the calculation of cold gas efficiency. The most common way to perform gas analysis consists of using gas chromatography, this method provides high precision but lacks some fundamental points that would make it a perfect match for gasification systems: robustness to tar and soot pollution, capability to operate with wet hot gases above tar dew point and fast responsiveness. This work lays the basis of a different approach for gas analysis. The proposed methodology uses a Junker-Boys calorimeter, the basic calorimetric approach is hybridized with the boundary conditions imposed by gasification, enabling users to define and solve a system of linear equations. This system uses the few data that can be collected from the calorimeter and the gasifier then it calculates the gas composition. An experimental campaign is used to validate the proposed approach. Results showed a good fit between the composition calculated with the calorimetric approach and the one evaluated through standard methods.

Experimental study on electrostatic removal of high-carbon particle in high temperature coal pyrolysis gas

A high-temperature electrostatic precipitator (ESP) presents a good solution for hot gas cleaning, which can remove fly ash from pyrolysis gas at temperatures higher than the tar dew point. In this paper, the characteristics of negative DC corona discharge in air and simulated coal pyrolysis gas were studied. The removal of coal pyrolysis furnace fly ash (ash A) was investigated and compared with that of coal-fired power plant fly ash (ash B) in ESP with a temperature ranging from 300 K to 900 K. The current density of simulated gas was higher than that of air under the same discharge voltage and at different temperatures. The simulated gas also had a higher spark voltage and a lower onset voltage compared with air. The fractional collection efficiency of ash A was lower for particles with diameters of larger than 0.1 µm at high temperature, compared with ash B. A lower collection efficiency in simulated gas was obtained for particles with diameters of less than 0.1 µm compared with air. The collection efficiency of submicron particles in simulated gas was usually higher than it in air, especially for particles with diameters of less than 0.04 µm. In simulated gas, the overall collection efficiency of ash A was obviously lower than that of ash B, especially at high temperature. From 300 K to 700 K, the collection efficiencies of both ash samples were as high as above 93%, but the collection efficiency of ash A in simulated gas decreased to 78.7% at 900 K.

CO2 gasification of biomass: The effect of lime concentration in a fluidised bed

Fluidised bed (FB) technology can be advantageously used for the gasification of solid fuels. Calcined carbonate materials, such as limestone or dolomite, can be used directly in the fluidised bed of the gasification reactor to reform tars in situ and to enhance carbon conversion and cold gas efficiency of the gasification process. However, they exhibit poorer mechanical stability, they tend to be attrited and carried over from the reactor, and their catalytic activity decreases over time. Therefore, a portion of the material has to be replenished continuously or periodically. To decrease the amount of carbonate material that has to be replenished, a lower amount of lime (calcined limestone or dolomite) can be used in the FB, diluted by a mechanically robust material, such as silica sand or olivine. According to the literature, even concentrations in the order of 10–35% wt. of carbonate material in the FB of silica sand have a substantial effect on the decomposition of tars during steam or air gasification. However, the effect of the concentration of lime in the FB has not yet been described for CO2 gasification. In this paper, we focus on the effect of the ratio of calcined dolomitic limestone and silica sand in the FB (0%, 25%, 50% and 100% vol. of dolomitic limestone) for CO2+O2 gasification of biomass and compare it with H2O+O2 gasification at the temperature of 850 °C. The experiments were performed in a semi-autothermal spouting FB reactor, gasifying 1.4 kg h−1 of woody biomass. The effects of the concentration of dolomitic lime in the fluidised bed differed for H2O+O2 and CO2+O2 gasification. When gasifying with H2O+O2, optimal results were found with 50% vol. (35% wt.) lime in the FB, when the yield of tar was similar to the use of pure lime in the FB. When gasifying with CO2+O2, a substantial decrease in tar yield was observed when using 50% vol. (35% wt.) lime in the FB (compared to the use of pure silica sand); nevertheless, the use of undiluted lime in the FB remains the best option to attain minimal tar yield when gasifying with a CO2+O2 gasifying agent. In this case, the tar yield was decreased 8.7-fold, and the tar dew point was decreased by 124 °C (to 71 °C) compared to the non-catalysed case with silica sand being the fluidised bed of the gasifier.

Treatment of Volatile Compounds from Municipal Solid Waste Pyrolysis to Obtain High Quality Syngas: Effect of Various Scrubbing Devices

Volatile compounds produced from pyrolysis of municipal solid waste (MSW) are treated in a scrubbing system, which includes a hot char filter, a condenser cooled with ice water, a cooking oil scrubber, and a Na2CO3 solution scrubber, aimed at obtaining high quality syngas. The performances of each treatment device are evaluated at a laboratory scale. The results indicate that hot char filtration is able to capture 71% of particulates and 32% of the tar, and, thanks to its catalytic activity, syngas yield increased significantly from 0.21 to 0.39 m3N/kgMSW, and the molar percentage of combustible fractions in the syngas increased by 35%. The condenser can capture 90%, 85%, 87%, and 90% of evaporated Na, K, Ca, and Mg, respectively; 50% of particulates and 98% of the tar in the syngas. The cooking oil scrubber collected a major portion of tar, with 144 mg/m3N remaining in the syngas after scrubbing, and the dew point of the tar decreased below 50 °C. The scrubber filled with Na2CO3 solution of 5 wt% running at 70 °C and liquid/gas (L/G) ratio of 6 reduced the concentrations of tar and H2S in syngas to 38.5 mg/m3N and 16 ppm, respectively. Consequently, the tar dew point decreased to a level of about 25–30 °C, then acceptable for downstream applications, such as feeding of a gas engine. The results provide detailed information on the cleaning effects of various syngas cleaning units, their reciprocal influence and contributions to the total removal efficiency, which helps in choosing suitable options for volatile treatment to obtain high quality syngas from MSW pyrolysis.

EGR control on operation of a tar tolerant HCCI engine with simulated syngas from biomass

In combined heat and power plants operated with biomass syngas, the removal of condensible tars is a necessary but expensive step (up to one third of the installation and maintenance costs). This step is required because the syngas has to be cooled down to avoid knocking in the spark ignition engines traditionally used in such plants. To remove the tar condensation problem, we developed an alternative system based on an Homogeneous Charge Compression Ignition (HCCI) engine operated at intake temperatures above the tar dew point. To address the challenge of power derating of such engine setups, the current paper focuses on the application of Exhaust Gas Recirculation (EGR) as a control parameter that would indirectly allow the improvement of the engine performance. Based on a conservative estimate of tar dew points, HCCI combustion was studied at an intake temperature of 250°C using synthetic biomass syngas and synthetic EGR on a mono-cylinder HCCI engine operated at 1000 RPM. The effects of charge dilution, thermal and kinetic damping due to the EGR gases were also analysed to understand their main effects. The use of EGR successfully increased the maximum achievable Indicated Mean Effective Pressure from 2.5bar at EGR=0% up to 3.3bar at EGR=25%, through damping the maximum pressure rise rate and allowing higher equivalence ratios.

The characteristic of biomass gas tar removal by a quench coupled with adsorption process

The effective removal of tar is one of the main obstacles to the large-scale industrialization of biomass gasification process. A quench coupled with multiple adsorption technology (QCADT) for the tar stagewise removal was developed and patented. Based on an experimental device of QCADT, the tar removal behavior as well as the removal mechanism was measured and analyzed respectively. The results show that, the tar has been removed cascadedly in accordance with the idea of QCADT, providing a residual tar concentration of less than 10 mg/Nm3. However, the measured results represent boundary differences to the tar classification in QCADT baesd on IUPAC (International Union of Pure and Applied Chemistry) pore size classification, leading to the residual heavier tar moving to the downstream stage and forming a competitive adsorption. Mesoporous and microporous units can hardly lower the dew point of the heavy tar. Mesopores play dominant role in the medium tar removal, which is the most difficult to be removed. The results of the tar dew point prove the effectiveness of the QCADT based on the correspondence of the tar classification, the tar molecular size and the pore size of the adsorbent.

Experimental study on catalytic effect of biomass pyrolysis volatile over nickel catalyst supported by waste iron slag

Summary The study aims to analyze catalytic tar destruction, evaluate the activity of the Ni-based catalyst supported by waste iron slag, and obtain clean pyrolysis syngas. The effects of different nickel loadings, catalytic temperatures, and catalyst calcination temperatures on volatile were investigated in order to determine the optimal process condition. The analysis results showed that the iron slag Ni-based catalyst had a relatively low specific surface area. However, it showed an excellent resistance performance to the coke deposition and displayed the high tar removal ability. Moreover, the tar conversion and the yield of syngas were significantly affected by nickel loadings. When the nickel loading reached 3%, the tar dew point was decreased by nearly 100 °C and the tar conversion reached 94.84%. The favorable reaction temperature was about 800 °C based on the consideration of energy consumption and the catalytic performance. Calcination temperature affected tar yield and syngas yield. The application of iron slag in nickel catalyst realized the reutilization of waste materials, indicating significant practical values. Copyright © 2017 John Wiley & Sons, Ltd.

Tar Tolerant HCCI Engine Fuelled with Biomass Syngas: Combustion Control Through EGR

Abstract The filtration of condensable tars is a necessary step before biomass syngas can be used in spark ignition engines. In small scale gasifiers, this step represents one third of the investment cost and create several issues of maintenance. Since tar related problems occur primarily due to tar condensation, an alternative system is being developed where a Homogeneous Charge Compression Ignition (HCCI) engine is operated at intake temperatures above the tar dew point, thereby eliminating the need for tar filtration. Based on a conservative estimate of tar dew points, HCCI combustion has been studied at an intake temperature of 250 °C. This paper presents a comprehensive study of the first HCCI engine operated with syngas. In particular, it addresses a control technique based on Exhaust Gas Recirculation (EGR). Simulated biomass syngas and simulated EGR were used on a 500cc mono-cylinder HCCI engine operated at 1000 RPM. The effects of charge dilution, thermal and kinetic damping as a consequence of EGR and the consequential combustion control have been discussed. In addition to being an effective control, the use of EGR decreases the Maximum Pressure Rise Rate and allows increasing the upper limit of the Indicated Mean Effective Pressure from 2.8 bar at EGR=0% to up to 3.5 bar at EGR=25%, a rise of about 25%.

The effects of biomass syngas composition, moisture, tar loading and operating conditions on the combustion of a tar-tolerant HCCI (Homogeneous Charge Compression Ignition) engine

Syngas, from biomass gasification, needs to be purified of condensible tars before being used as a fuel for spark ignition engines. A novel tar tolerant system is being developed, wherein a (Homogeneous Charge Compression Ignition) HCCI engine is fueled by impure syngas at intake temperatures above the tar dew point, thus avoiding the condensation of tars and its consequent problems. This paper introduces the novel concept and studies the sensitivities of HCCI combustion to key context relevant issues such as the variations in the combustible composition (H2:CO ratio from 30:70 to 55:45%), moisture (upto 12% by volume) and tars (from 3 to 17 g/Nm3) at the intake temperatures of around 250 °C. The effects of intake temperatures (230–270 °C) and pressures (1.0–1.2 bar) are also studied. The experiments are carried out with artificial syngas on a 435 cc mono-cylinder HCCI engine with a compression ratio of 12 and operated at 1500 RPM. Indicated efficiencies about 34% and Indicated Mean Effective Pressures about 2.5 bar were achieved at equivalence ratios between 0.3 and 0.34. In the tested ranges, moisture delayed the combustion while the effect of tars were insignificant.

Moving bed syngas conditioning: Modelling

This paper presents a modelling approach for simulating tars and particulate (dust) removal in a moving bed heat exchange filter (MBHEF) in order to satisfy gas requirements of end-use syngas applications: engines and turbines. The two-dimension, adiabatic, steady-state proposed model accounts for two-phase (gas and solid) and neglects conduction and mass diffusion. Tars condensation is modelled through representative tar class lumps: phenol (class 2), naphthalene (class 4), pyrene (class 5). The model also considers tar concentration influence on the tar dew point. The filtration model is taken from literature. A sensitivity analysis is performed varying the particle size and the superficial gas velocity. Maps of temperature and tars abatement efficiency are presented. The simulation results indicate the feasibility of the use a MBHEF as tars removal equipment benefiting its advantages against others gas-cleaning methods with acceptable pollutant removal efficiencies, ranging 88–94% for ranges studied. Results also point out low gas velocities (0.5-1 m/s) and high particle size (400–700 μm) for reducing operational costs in MBHEFs with compact size.

Effect of CaO on tar reforming to hydrogen-enriched gas with in-process CO2 capture in a bubbling fluidized bed biomass steam gasifier

Previous studies showed that calcium oxide (CaO), when added to a biomass steam gasifier, could play the role of both CO2 sorbent and tar reforming catalyst, and thereby produce more hydrogen. However, most of these works focused on the former role with little attention to tar reforming aspect of CaO. Therefore, this work aims primarily at studying the tar reforming effect of in-bed CaO. To this end, an in-depth analysis of the effect of CaO on tar yield and composition is presented. The present work also studies the role of CaO as a CO2 sorbent to enhance hydrogen production from steam gasification of biomass in a bubbling fluidized bed. The influence of different operating parameters, temperature (T) and steam to biomass (S/B) ratio, as well as the effect of using in-bed CaO on gas and tar production is investigated. Results show that the maximum H2 and minimum CO2 concentration of 63.07% and 18.68%, respectively are obtained at T = 650 °C and S/B = 3.41. The maximum H2 yield of 256.81 ml g−1-biomass was obtained at T = 700 °C and S/B = 3.41, at which the minimum tar content of 6.45 g N m−3 was also received. Compared to a bed of sand alone, a 20% higher H2 concentration, an almost double H2 yield and a 67% reduction in tar content were obtained when a bed of CaO was used. Moreover, shifting the tar species from higher to fewer ring structures as a result of in-bed CaO can reduce tar dew point by 11 °C and tar carcinogenic potential by almost 60%.

Syngas cleaning with nano-structured micro-porous ion exchange polymers in biomass gasification using a novel downdraft gasifier

Sulphonated nano-structured micro-porous ion exchange polymers, known as sulphonated PolyHIPE Polymers (s-PHPs) were used in syngas cleaning to investigate their impact on tar composition, concentration and dew point depression during the gasification of fuel cane bagasse as a model biomass. The results showed that the s-PHPs used as a secondary syngas treatment system, was highly effective at adsorbing and reducing the concentration of all class of tars in syngas by 95%–80% which resulted in tar dew point depression from 90 °C to 73 °C. It was shown that tars underwent chemical reactions within s-PHPs, indicating that tar diffusion from syngas was driven by chemical potential. It was also observed that s-PHPs also captured ash forming elements from syngas. The use of s-PHPs in gasification as well as in an integrated thermochemical biorefinery technology is discussed since the tar loaded s-PHPs can be used as natural herbicides in the form of soil additives to enhance the biomass growth and crop yield.

Detailed process modeling of a wood gasification combined heat and power plant

We provide the first process simulation able to depict a complete life cycle inventory of a biomass gasification combined heat and power (CHP) plant. The model predicts the detailed mass and energy balances, minor compounds emissions (such as NOx, SOx, aromatics, etc.). It is based on a phenomenological approach for dryer, reactors (gasification and combustion) and scrubber modeling. Process units are modeled with Aspen Plus® models completed with dedicated Fortran sub-models when more details are required. The gasifier is a Dual Fluidized Bed (DFB). It is decomposed in three sections, describing the three main mechanisms occurring in the reactor: wood pyrolysis, secondary reactions and char combustion. At the outlet of the gasifier, the complete composition of the syngas is predicted, which includes water, permanent gases, inorganics, particles and tars (phenol, benzene, toluene, styrene, indene, naphthalene, acenaphthylene, anthracene, phenanthrene, pyrene). Simulation results were validated against measurements from a pilot plant and other literature data. A wet conventional syngas cleaning system for gasification plant, including cyclone, catalytic tar cracker, syngas cooler, bag filters, water scrubber and wastewater treatment was considered and modeled. The complexity of tar composition allows a reliable determination of tar dew point. At the outlet of the cleaning system, syngas is burned in gas engines. Gaseous emissions (NOx, SOx, etc.) of the gas engines and the DFB combustor are also handled by the model as well as ashes and liquid waste. The predicted electrical and thermal efficiencies are 27 and 39% respectively.

Characterisation of tars from biomass gasification: Effect of the operating conditions

Tars formation remains the main technical hurdle in the development of biomass gasification at commercial scale, due to its associated operating problems (condensation, catalyst deactivation, polymerisation). This work aims to study the effect of the operating conditions (biomass/air ratio, temperature and gasifying agent) on the production and the technical and environmental hazard (tar dew point and carcinogenic potential) of tars produced in a small-scale drop-tube gasifier, dealcoholised marc of grape being used as biomass fuel. An analytical HPLC method has been used for the detection of polycyclic aromatic hydrocarbons (PAHs), benzene, toluene, ethyl-benzene and xylenes (BTEX), phenol, and pyridine. Results show that an increase in the relative biomass/air ratio, a decrease in temperature, and higher steam content lead to a higher tar production. BTEX have been found as the main constituents of tars (60–70% wt.), whereas among PAHs, the most significant are the lighter ones (naphthalene, acenaphtylene, acenaphtene). Phenol production is favoured at lower temperatures or/and higher steam content (around 50% wt. of the tar mixture at 750 °C and steam gasification). A progressive aromatisation of tars has been observed when increasing the temperature, the effect of steam addition on tar composition being not significant at temperatures higher than ∼1000 °C.

Effect of CaO on tar production and dew point depression during gasification of fuel cane bagasse in a novel downdraft gasifier

Granular calcium oxide was used in syngas conditioning to investigate its impact on tar composition, concentration, dew point and syngas yield during gasification of fuel cane bagasse (FCB). The results showed that the use of 2, 3 and 6wt.% in-bed CaO promoted the conversion of Class1, 4 and 5 tars to Class 3 tars to varying degrees. Overall, this resulted in a decrease in tar yield ranging from 16 to 35%, a decrease in the tar concentration in syngas of 44–80%, an increase in syngas yield of 17–37%, and a decrease in tar dew point of 37–60°C as the CaO concentration was increased from 2 to 6wt.% .

Occurrence, composition and dew point of tars produced during gasification of fuel cane bagasse in a downdraft gasifier

Abstract Gasification of pelletised fuel cane bagasse a waste residue from an energy crop known as fuel cane was investigated to evaluate the potential of fuelling solid oxide fuel cells (SOFCs) with the raw fuel gas produced. Tars produced during gasification of the bagasse in a 50 kWe air-blown downdraft autothermal gasifier were collected, quantified and characterised and the tar dew point evaluated. The concentration of tar collected was 376 ± 27 mg m−3 of dry syngas (at 273 K, 101 kPa), emphasising the efficiency of the tar cracking reactions in the oxidation zone of the gasifier. However, although tar production was low, the typical mixture of tar compounds produced exhibited a high tar dew point of 90 ± 5 °C and was dominated by Class 2 and 5 tars which condense readily even at low concentrations. Additionally Class 1 tars had a mass fraction of 8% of the total tar produced. Therefore the calculated tar dew point underestimates the actual tar dew point and a high potential for fouling of SOFC anodes exists. Consequently primary or secondary gas cleaning treatment measures targeting the production or occurrence of Class 1, 2 and 5 tars will be essential for long term operation of SOFC power generating systems fuelled by raw fuel gas from fuel cane bagasse.

Influence of mean gas residence time in the bubbling fluidised bed on the performance of a 100-kW dual fluidised bed steam gasifier

In this study, the influence of mean gas residence time in the bubbling fluidised bed, τf, on the performance of a pilot scale 100 kW dual fluidised bed gasifier was experimentally investigated. In the experiments, inert greywacke river sand was used as the bed material, and wood pellets of radiata pine sawdust were used as the feedstock. The gasification temperature and the steam to biomass ratio were set at 730–750°C and 0.8 kg/kgdry, respectively. The τf was varied using three different bed material inventories in the gasification reactor of 5.8, 9.1 and 12.1 kg with the corresponding τf being calculated to be 0.16, 0.19 and 0.21 s, respectively. The results of this study show that the overall performance of the dual fluidised bed steam gasifier increased with τf. The key findings are: (1) the total tar concentration in the producer gas was reduced by 24% (except C3 tar) with the τf increasing from 0.16 to 0.21 s, thus the tar dew point decreased by 16°C, from 188°C to 172°C; (2) the carbon conversion rate and the cold gas efficiency increased by 8% and 3%, respectively, as the τf was increased in the above range and (3) the water–gas shift reaction appeared to play a more important role with the increase of τf.