Effects of the Overall Length and OD on Opposing-pulse-jet Cleaning for Pleated Filter Cartridges

Opposing-pulse-jet technology has been proposed as a solution for regenerating filtration media by minimizing the incomplete cleaning of pleated filter cartridges. In this study, we investigated the effects of a pleated filter cartridge’s overall length and outer diameter (OD) on the performance of opposing-pulse-jet cleaning via numerical modeling. For each pleated filter cartridge, the delay time, Δt (defined as the delay in launching the secondary nozzle), was varied to analyze the intensity and the uniformity of the static pressure distribution in the core of the cartridge. It was found that two opposing jet flows collided much more intensely in the core when the length of the overall cartridge, L, was short, and the OD was small. For a given L, the pressure pulse’s performance was maximized by varying Δt. The pulse intensity and uniformity in the filter core can be represented as a bimodal (double-peaked) function of Δt. In general, the pulse intensity was greater when Δt > 0 s than when Δt < 0 s (when the secondary jet was launched before the primary jet).


INTRODUCTION
Dust collectors have been applied in various industrial settings for the control of particulate emission over the decades in order to meet the gradually more stringent dust emission regulation. A fabric collector is the most efficient and versatile among all the types of collectors. In the fabric collector, dust-laden airflow passes through a fabric filtration element and dust cake quickly grows on the air entrance surface of the fabric element. Because of the advantage of large filtration area in unit space, pleated filter cartridges are recently favored over baghouse filters (Zhang and Wang, 2011;Chen and Chen, 2017a, b;Chen et al., 2017). The pressure drop across the filter element increases with the growth of dust cake. To save the energy consumption of air movers, the periodical cleaning of the fabric filter element is required to regenerate the filtration media. One of online techniques for the regeneration of filter elements, named as reverse pulse-flow/jet cleaning, is to apply pressurized air flow, preferred to be pulsed, in the opposite direction of filtration flow (Tien, 2012;Tarleton, 2015). Pleated filter cartridges however have less elasticity than baghouse filters because of the fabric material, pleated structure and having large surface area to be cleaned. As a result, the issue of incomplete cleaning was more serious when using pleated filter elements as compared to using baghouse filters (Li et al., 2015;Chen and Chen, 2017a). The quality of the cleaning gets worse when filtering dust in humid air (e.g., in the underground mine tunnels) or sticking dust (e.g., the carbon black dust in rubber plants) (Li et al., 2017;Li et al., 2019a, b).
Various methods have been proposed to improve the effectiveness of reverse pulse-jet cleaning for pleated filter cartridges. To effectively transfer more dynamic pressure into static pressure from the cartridge opening to its inside, Shim et al. (2017) investigated a dual-slit injector to achieve better cleaning performance and to considerably increase the cleaning interval. Chen (2016, 2017b) proposed a new cleaning scheme using multiple pulse jets in one cleaning cycle to improve the cleaning performance without the increase of compressed gas tank pressure. The multi-pulsing jet cleaning increased the number of cleaning actions and local medium acceleration at the top sections of pleated filter cartridges. Chen and Chen (2017a) further investigated an annular-slit nozzle for the reverse pulse-jet cleaning and found it could produce a favorable effect on the local cleaning efficiency by significantly increasing the local filter pressure drop at the top sections of pleated filter cartridge. The optimization of pulse-jet cleaning performance via the variation of the Wu et al.,Aerosol and Air Quality Research,xxxx 7 pleat shape, nozzle-opening/tube area and jet distance has also been investigated (Qian et al., 2014;Chen et al., 2017;Qian et al., 2018). Despite the above methods to improve the uniformity of pulse-jet cleaning, the issue of incomplete cleaning has not been satisfactorily resolved. The opposingpulse-jet cleaning (Li et al., 2019b), formally named as colliding-pulse-jet cleaning, was recently proposed and experimentally evidenced to improve the cleaning intensity and uniformity for a pleated filter cartridge. In the above method, a second nozzle is installed at the cartridge base in addition to the primary cleaning nozzle which is typically installed in front of the filter cartridge opening. By equipping with a second cleaning nozzle, an additional pulse flow could be launched during the cleaning cycle. The timing and intensity of the second pulse jet can be varied to enhance the cleaning quality of pleated filter cartridges. The mechanism of the opposing-pulse-jet cleaning was further studied with numerical modeling (Li et al., 2020). It was found that the achievement of effective jet-flow collision and wide area of filtration media covered by moving the flow collision location in the core of pleated filter cartridge in the opposing-pulsejet cleaning is the key to obtain improved cleaning efficiency compared with that attained by the classical pulse-jet cleaning (with only the primary nozzle). However, various factors, for examples, the cartridge design variables and operational parameters, that will influence the performance in the opposing-pulse-jet cleaning were difficult to explore by the experiments.
The objective of this study is to investigate the effects of cartridge length and diameter, the major variables of a filter element, on the cleaning quality of the opposing-pulse-jet cleaning via the numerical modeling. The delay time (Δt, defined as the delay in time to launch the pulsed jet from the second nozzle relative to the timing for initiating the jet flow by the primary one) was also varied to study the intensity and uniformity of static pressure distribution in the core of pleated filter cartridges studied.

NUMERICAL MODELING
A typical computational domain is shown in Fig. 1. The computational domain used for this modeling is deduced from the test system described in the work (Li et al., 2019b). The experimental dust collector has the filtration chamber dimensions of 1225 mm (width) × 750 mm (depth) × 1550 mm (height) with only one pleated filter cartridge installed vertically. Due to axisymmetric configuration of pleated filter cartridges, one district pleat was used in our modeling. To keep the axisymmetric feature, the box-type filtration chamber was simplified as a cylindrical one keeping the same cross-sectional area in the horizontal plane. The round inlet and outlet for the experimental filtration chamber was also modeled as annular ones with the same opening areas as the round ones.
The primary cleaning nozzle (with the diameter of 24 mm) is installed over the opening of pleated filter cartridge. The secondary cleaning nozzle, which is the same type and size as the primary one, is placed at the base of pleated cartridge. The filter cartridge studied in the previous experiment has the length L of 660 mm and outer diameter OD of 320 mm. To investigate how the cartridge length and diameter affects the cleaning performance of the opposing-pulse-jet process, the length and diameter of pleated cartridges were varied: L = 1000, 1350, 1700, 2000, 2300, 350 mm, respectively (referenced but not limited to Chinese standard JB/T 10341-2014; MIIT, 2014). The pleat height and number were kept the same as the experiment one. The media thickness and permeability were kept as 0.6 mm and 2.0 × 10 11 m -2 , respectively. ANSYS Meshing R.19.0 was applied to generate hexahedral meshes in the computational domain. Among all the studied ones, the case with the cartridge having L = 660 mm and OD = 160 mm had the minimal number of ~0.08 M elements and the one with L = 2300 mm and OD = 320 mm had the maximal number of ~0.43 M elements. ANSYS Fluent R.19.0 was used to calculate the flow and pressure fields in the computational domain. The flow and pressure fields during the media regeneration phase of the filtration system was assumed to be unsteady, compressible and turbulent. The realizable k-ε model was elected for turbulent flow modeling. The detail of realizable k-ε program can be found in the Fluent Theory Guide (ANSYS Inc., 2018a). The flow in the porous media in ANSYS Fluent are modeled by the addition of a momentum source term to Navier-Stokes equations (ANSYS Inc., 2018b).
The boundary conditions applied in this study are also given in Fig. 1. The pressure inlet boundary condition was applied at the inlets of two nozzles. For all cases in our study were assumed to conduct under the so-called "offline" cleaning mode (i.e., without the filtration phase occurred in the practical operation of dust collector). Under the above cleaning, the pressure at the dust collector top inlet and bottom outlet were close to the ambient pressure. The pressure boundary condition was also applied at the top inlet and bottom outlet of the dust collector. The reference pressure of 101.325 kPa was set at the dust collector inlet. The no-slip condition was applied for all solid walls with a standard wall function for the turbulent flow modeling. The time step size was set at 0.0002 s.
The transient pressure for the cleaning nozzles was specified using a user-defined function (UDF). The pressure value for the primary nozzle inlet in the modeling was obtained from the experiment study of opposing-pulse-jet cleaning with the pulse duration of 0.05 s and no delay time, Δt = 0 s (Li et al., 2019b). The fitted function of the nozzle inlet pressure P (kPa) with the time t (s) was: The pressure value for the secondary nozzle was set the same as the primary one. The time to delay the launch of jet from the secondary nozzle (Δt, defined as the delay time) was varied in the range of -0.150 to +0.150 s. Note that a negative value in the delay time refers to the fact that the secondary jet will be initiated earlier than the primary jet. For the convenience, the time to initiate the first cleaning jet (either from the primary or secondary nozzle) was set at t = 0.002 s.
The dust cake on the entrance surface of pleated cartridges was not considered in this modeling to focus our research on exploring the action of opposing jets in the cleaning process. The deformation of filtration media in the cleaning process was also neglected.
Our model in the case of L = 660 mm and OD = 320 mm was verified by the comparison of calculated static pressure with the measured one in the core of pleated filter cartridge (Li et al., 2020).

Effect of Filter Length (1) On the Evolution of Static Pressure
Shown in Fig. 2 is the evolution of static pressure contour and flow streamline when the opposing-pulse-jet cleaning was operated with the delay time Δt = +0.025 s. The filter cartridges under the study have the overall length L = 660, 1350, 2000 mm, and outer diameter OD = 320 mm. At the start of the primary jet, the flow entered into the core of pleated cartridge moving towards its base. At the same time, the same jet flow drove the surrounding air moving into the cartridge core. As the secondary jet nozzle initiated, the secondary jet flow impinged with the primary one in the cartridge core. The flow impinging position moved upward from the cartridge base due to the continuous operation of the secondary jet and then stopped temporarily in the region near ~2/3 of the pleated cartridge length (measured from its base). The cleaning-flow impingement ended when the primary flow stopped.
By examining the cleaning-flow impingement location in all the shown cases with different cartridge lengths, it is found that the cleaning-flow collision time was very close, from t = ~0.04 s to ~0.12 s. The static pressure contour evidenced a relative high pressure (more than ~500 Pa) was developed in a narrow core region in the case with a short cartridge (L = 660 mm; Fig. 2(a)) compared to the high pressure in other cases. At the time t = 0.085 s, when the position of flow collision was relatively steady, the high-pressure region (marked in dashed box in Fig. 2(a) occupied ~1/3 of the cartridge length. While, in the case with the cartridge having 7 L = 1350 mm ( Fig. 2(b)), the high-pressure region occupied ~1/2 of the cartridge length while the static pressure in the region was lower than that in the case with the one having L = 660 mm. For the case of the cartridge having L = 2000 mm ( Fig. 2(c)), the high-pressure region occupied ~3/5 of the cartridge length and the static pressure was even lower than that in the case of L = 1350 mm. It is thus concluded that more filter surface of pleated cartridges could be covered as the overall length of pleated cartridges was increased while keeping the cartridge OD constant although the static pressure in the high-pressure region was reduced (given the assumption of the same cleaning-flow rate).
(2) On the Pressure Evolution at Selected Observation Points Shown in Fig. 3  observation points, P1-P9, selected in the pleated cartridges with the overall length L = 660, 1350, 2000 mm while keeping the outer diameter OD = 320 mm (when the opposing-pulsejet cleaning was operated at various delay times Δt). Also shown in Fig. 1, the selected P1-P9 points are located at the middle of inner filtration surface, and evenly distributed in the distance between P1 and P9 (in the longitudinal direction). The P1 is located at 100 mm from the cartridge opening and the P9 is located at 100 mm above the cartridge base. Under the opposing-pulse-jet cleaning, three events were observed resulting in the increase of static pressure at all the obervation points in a pleated filter cartridge. One event is due to the initiation of the first jet flow (either from the primary or sendary nozzle, as illustrated in Fig. 3(a)). Another event is due to the air accumulation in the cartridge core, resulting from the continuous operation of cleaning flow. The other one is due to the impingment of two jet flow under a suitable delay time operation. Note that, in some cases, the first event might not be visible in the static pressure plot given in Fig. 3. It is because of the merging of the first and second events.
In the time period of only one nozzle jetting (either the primary jet only or the secondary jet only), continuous gradients of static pressure were found along the filter length in the case of short cartridge. However, more observation points were found to stay approximately the same pressure in the case of long cartridges. For examples, in Fig. 3(a4), during the period of time t = 0.18-0.25 s under the delay time operation of Δt = -0.150 s (i.e., in the period that only the primary nozzle was jetting), the static pressure continuously increased along all the points (P1-P9) in the case of the pleated cartridge having L = 660 mm. Under the same operation condition, the static pressure was increased from P1 to P5 and from P1 to P3, and kept approximately constant for the points P5-P9 and for the points P3-P9 in the cartridges of L = 1350 and 2000 mm, respectively. The above trend was also found during the period of time t = 0.05-0.10 s under the operation of delay time Δt = -0.150 s (i.e., only the secondary nozzle was jetting) and during the period of time t = 0.05-0.10 s under the delay time operation of Δt = +0.075 s (i.e., only the primary nozzle was jetting). During the time of only the secondary nozzle jetting, the static pressure in a portion of observation points turned from the negative to positive value (relative to the reference pressure). For example, during the time t = 0-0.12 s and under the delay time operation Δt = -0.150 s, the static pressures at all the observation points were negative for the pleated cartridge of the overall length L = 660 mm (except the positive pressure at the very brief initial time). Note that the negative value in the static pressure simply means that the pressure in the jet-flow collision phase was less than the reference one and its absolute value is positive. However, under the same operational condition, the static pressure was positive at the points P1-P4, and P1-P6 in the cases of pleated cartridges of L = 1350 and 2000 mm, respectively.
Under the delay time Δt = -0.150 s operation, it can be found that, when the jetting from the secondary nozzle launched, the primary cleaning jet has not yet been started, thus the impinging of the primary and secondary jet flows is not possible. Comparing the positive pressure evolution while only the primary nozzle was jetting among the cases of cartridges with different overall length, it revealed that the positive pressure decreased with the increase of overall cartridge length. For example, the peak pressure was ~750 Pa for the cartridge of L = 660 mm, and it decreased to ~550 and ~450 Pa in cases of cartridges of L = 1350 and 2000 mm, respectively. The similar observation on the positive pressure could also be found in the case with the delay time operation of Δt = +0.075 s.
During the jet-flow impinging period (which can be found in the operation of delay time Δt = -0.050, 0.000 and +0.075 s), it is found that in general the achievable static pressure was higher than that during the period of one nozzle jetting only. The above is more obvious for the case with long cartridge. For examples, during the period of time t = 0.07-0.13 s and under the delay time operation of Δt = -0.050 s, the static pressure during the flow impingement were found to be 181, 296, and 309 Pa, which were higher than those in the time period without the flow impingement for the cartridges of L = 660, 1350 and 2000 mm, respectively. Also, during the time period t = 0.08-0.13 s and under the operation of Δt = +0.075 s, the static pressures were found -146, 298 and 366 Pa, which is again higher than that in time period with no flow colliding for the cartridges of L = 660, 1350 and 2000 mm, respectively.

(3) On the Pulse Intensity and Uniformity
The max. static pressures during the reverse pulse-flow cleaning are usually taken as the index to quantitatively represent the cleaning performance (Yan et al., 2013;Li et al., 2015;Yan et al., 2018;Li et al., 2019a). The average and inverse of coefficient of variation (C.V., defined as the ratio of the standard deviation to the mean) of the peak static pressures could be adopted to represent the cleaning intensity and uniformity, respectively. Shown in Fig. 4 is the pulse intensity of opposing pulse jet under the variation of the delay time for the cartridges of overall length, L, 660-2300 mm.
The pulse intensity curve was found to be bimodal (i.e., its intensity first increased and followed with the decrease, and then increased and decreased again) with the variation of delay time Δt in all the cases of L = 660-2300 mm. Generally, the pulse intensity in the cases of Δt ≠ 0 s were found to be higher than those in the case of Δt = 0 s (shown in Fig. 4). The above observation is primarily because of the location of jet-flow collision moved in the cartridge core. The pulse intensity decreased with the increase of overall cartridge length in the cases of Δt ≠ 0. It is simply due to the enlargement of core volume with the increase of cartridge overall length, resulting in the weakening of the pressure buildup. Moreover, the pulse intensity under the delay time Δt < 0 s operation were lower than that under the Δt > 0 s operation. For examples, for the cartridge with L = 660 mm, the pulse intensity was 601 and 655 Pa when Δt was -0.050 s and +0.050 s, respectively. Also, for the cartridge with L = 1350 mm, the pulse intensity was 519 and 597 Pa when Δt = -0.075 s and +0.075 s, respectively. It is because, under the delay time Δt < 0 s operation, the secondary nozzle started first, a portion of the cleaning flow from the second nozzle was easy to leak off through the cartridge opening when the primary cleaning flow was launched. Under the operation of Δt > 0 s, the primary cleaning flow started first, the primary jet flow would not be easy to escape from the cartridge opening and could collide with the secondary jet flow in the full strength. It is also found that both peak pressure intensities (i.e., one in Δt < 0 and the other in Δt > 0) were decreased with the increase of the overall cartridge length and the effect was more pronounced in the case of Δt < 0. It is because the secondary jet flow could not entrain surrounding air effectively Fig. 4. The pulse intensity of opposing pulse-jet as the function of the delay time for the pleated cartridges having the overall length L varied from 660 to 2,300 mm.

Fig. 5.
The peak static pressure distribution on the filter inner surface in opposing cleaning mode (with their best Δt) and primary-jet-only cleaning mode with the filter length varying from 660 to 2300 mm.
due to the close cartridge base.Under the operation Δt = 0 s, the pulse intensity achieved for filter cartridges with long overall length (although the high-pressure region due to the jet flow colliding was enlarged, shown in Fig. 3) was actually greater than that achievable in short cartridges. The above observation is different from that under the operation of Δt ≠ 0 s. It is because, under the operation of Δt = 0 s, the primary and secondary jet flow started and ended at the same time and the position of the jet collision was not moved.
It is also found that the optimal delay time corresponding to the maximal pulse intensity changed with the overall cartridge length. The optimal Δt were +0.025, +0.038, +0.038, +0.050, +0.063 and +0.063 s in L = 660, 1000, 1350, 1700, 2000, and 2300 mm, respectively. It is primarily because the increased distance between two nozzles (as a result of the increase of overall cartridge length) requires increased time for the pressure buildup in the filter cartridge core (due to the start of the second jet flow). Shown in Fig. 5 is the peak static pressure distribution on the inner filter surface in the "opposing" cleaning mode (under the optimal Δt) and "primary-nozzle jetting only" cleaning mode for the cartridges with the overall length varying from 660 to 2300 mm. It is evidenced that the static pressure at all observation points in the "opposing" cleaning mode were higher than those in the "primary-nozzle jetting only" mode. The obvious increase of static pressure was observed in the middle of the cartridge, and the less increase in the pressure in the region near the cartridge opening and base. Notice that, for the cartridges with the overall length L ≥ 1700 mm, no obvious pressure increase in the "opposing" cleaning mode were found in the region with a distance less than 400 mm from the cartridge opening (compared to that in the "primary-nozzle jetting only" mode). It is because in the region close to the cartridge opening, which is far from the secondary nozzle, the energy is not easy to be accumulated even by the jet-flow impingement.
Shown in Fig. 6 is the pulse uniformity in the opposing-pulse-jet cleaning as the function of delay time Δt for the cartridges with the overall length L from 660 to 2300 mm. Under the operation of Δt < 0 s, the pulse uniformity corresponded well with the pulse intensity, i.e., the greater the pulse intensity the higher the value of C.V. -1 (representing better pulse uniformity). Under the delay time Δt > 0 s, the pulse uniformity corresponded well with the pulse intensity only for the cartridge with L = 660 mm and the pulse uniformity increased with the increase of Δt for the cartridges with L = 1000-2300 mm. The above finding is mainly due to the fact that the jet collision could not cover the region near the cartridge ends of long pleated cartridges. The pulse Fig. 6. The pulse uniformity under the opposing pulse-jet cleaning as the function of the delay time for the pleated cartridges with the overall length varied from 660 to 2,300 mm.
uniformity was the worst for all cartridges when Δt = 0 s. It is because that the location of the jet collision would not move and pressure energy buildup could only cover a small section of the inner filter surface.

Effect of Cartridge Outer Diameter (OD)
(1) On the Evolution of Static Pressure Shown in Fig. 7 is the evolution of static pressure contour and flow streamlines when the opposing-pulse-jet cleaning was operated at the delay time Δt = +0.025 s for the cartridges with OD = 200 and 350 mm while keeping L = 660 mm. For the comparison, the data of the cartridge with L = 660 mm and OD = 320 mm could be found in Fig. 2(a). For the cartridge with small OD, the jet-flow collision was more intensified (because of small core volume), covering more percentage area under high static pressure. As the OD increased, the core volume of the cartridges increased, leading to less static pressure buildup and less area of the filter surface covered by the jet-flow collision.
Because the distance between the primary and secondary nozzles was kept the same in this part of study, the position for the jet-flow collision occurred in the location close to the middle of the cartridges with the increase of the cartridge outer diameter. For example, at the timing t = 0.085 s, the location of the jet-flow collision was at the distance of ~2/5 of the cartridge length from the opening of the cartridge having the OD = 200 mm. The jet collision location was nearly at the middle of the filter cartridge having the OD = 350 mm. It is because, for small OD cartridges, the pressure energy was quickly built up in the core region and the secondary jet flow required high energy to reach the core region. (2) On the Pulse Intensity and Uniformity Shown in Fig. 8 is the average pulse intensity as the function of delay time Δt for the pleated cartridges varying the outer diameter OD from 160 to 350 mm. The bimodal trend on the average pulse intensity by varying the delay time Δt was found for all the studied filter cartridges. For all the studied case, the pulse intensity peaked in the range of Δt = -0.100 to -0.050 s and Δt = +0.025 to +0.100 s. For the cartridges with OD ≤ 260 mm, the pulse intensity could reach 1000 Pa with a suitable Δt operation. For the ones with OD ≤ 200 mm, the pulse intensity could even reach 1500 Pa. However, the pulse intensity was ~400 Pa for the ones with OD = 350 mm. According to the previous cleaning work (Sievert and Löffler, 1989;Mukhopadhyay, 2010), the minimum pulse pressure to achieve a good fabric-cleaning efficiency is 400-500 Pa. Thus, the OD = 350 mm would be considered to exceed the upper limit for the pulse-jet cleaning studied and the OD upper limit shall be between 320 and 350 mm. In addition, the optimal delay time Δt was found to be +0.025 s, which is independent from the OD of the pleated cartridges.
The peak pressure distribution on the filter inner surface cleaned by the "opposing pulse jet" mode (under the optimal Δt = +0.025 s operation) and by the "primary-nozzle jetting only" mode for the pleated filter cartridges with the outer diameter OD = 160-350 mm and overall length L = 660 mm are shown in Fig. 9. The static pressures were found to be increased when cleaned by the "opposing pulse jet" mode compared to that by the "primary-nozzle jetting only" mode for all the shown cartridges, more significantly for ones with small ODs. Fig. 10 shows the pulse uniformity on the inner filter surface of pleated cartridges with the OD = 160-350 mm under the opposing-pulse-jet cleaning. The pulse uniformities were obviously influenced by the delay time Δt for all the studied cartridges. The bimodal trend was found in the pulse uniformity curve by varying the delay time. For the viewpoint of pulse uniformity, the optimal delay time Δt was in the range of +0.025-0.100 s for the cartridges with OD = 160-320 mm.

CONCLUSION
This parametric study numerically investigated the effect of the overall length, L, and the outer diameter, OD, of pleated Fig. 9. The peak pressure distribution on the filter inner surface cleaned by the opposing pulse-jet mode (the optimal delay time Δt = +0.025s) and by the primary-nozzle jetting only mode for the cartridges with the OD = 160~350 mm while kept the overall length L = 660 mm. filter cartridges on an opposing-pulse-jet cleaning process operated with various delay times. We focused on the intensity and the uniformity of the static pressure distribution in the core of the cartridges. Our findings are summarized as follows: 1. The impingement of the primary and secondary jet flows with various Ls and ODs: a) The impingement of the flows from the primary and secondary nozzles intensifies when the filter cartridge possesses either a short L or a small OD. b) The jet-flow collision generates a gradient of relatively high static pressure, which is concentrated in a small region of the cartridge core, covering a minor fraction of the filter's inner surface, for short pleated cartridges. c) As the L increases, the static pressure generated by the jet-flow collision decreases while the cleaning area-the pressurized region-increases. d) When the OD is small, the generated high-staticpressure gradient occupies the majority of the core space. As the OD increases (with a fixed L), both the intensity of the pressure and the size of the pressurized region decrease. 2. The delay time, Δt, significantly affects the opposingpulse-jet cleaning of pleated filter cartridges with various Ls and ODs: a) Optimal cleaning performance can only be achieved when Δt falls within a suitable range of values. b) The pressure pulse's intensity and uniformity can be depicted as a bimodal function of Δt. In general, the pulse intensity is greater when Δt > 0 s than when Δt < 0 s. c) The optimal Δt increases with the L (when the OD is fixed). However, it remains consistent between cartridges of the same L with various ODs.