Phenomenological models of the transient processes of diesel spray tip penetration

The multiple-injection strategy that has been used widely in diesel engines usually features a short duration for each injection pulse, which makes the start-of-injection (SOI) and end-of-injection (EOI) transients increasingly important for sprays in an injection event. Owing to the needle movement, spray developments during the transient processes are quite different from the spray at the quasi-steady state. In this paper, considering the sac pressurization processes during the SOI transients and effects of ''entrainment wave'' after the EOI, a theoretical zero-dimensional (0-D) model for the entire development processes of spray tip penetration is deduced. Then, the model is validated against the experimental spray data using a constant volume chamber and high-speed shadowgraphy. The model and experimental results demonstrate that the spray tip penetration has a t 3/2 dependence at the initial stage of injection rather than the t dependence suggested by the Hiroyasu model. Later, the spray tip penetration has a t 3/4 dependence owing to the spray breakup, a t 1/2 dependence with the completion of sac pressurization, and a ( t - t i ) 1/4 dependence after two injection durations from the SOI 1 .


Introduction
Spray tip penetration (S tip ) is one of the most important spray characteristics for diesel engines. As a result, accurate and quick calculation of S tip is important for optimization of the spray combustion system. Because of its simplicity and ability to clearly reflect the effects of various design parameters, the 0-D model has been widely used in model-based optimization of diesel spray combustion systems. With the assumption that the diesel spray is approximated as a gas jet, Wakuri et al. [1] developed a model for S tip based on the theory of momentum conservation. They found that S tip is proportion to the square root of time t 0.5 . The t 0.5 dependence was verified by many studies [2]. Hiroyasu et al. [3] demonstrated that S tip is initially proportional to the time t and then proportional to t 0.5 . They derived a correlation based on Levich's jet disintegration theory [4]. Recently, the model constants of Hiroyasu's model were modified by Arai [5] for higher fuel injection pressures. Naber and Siebers [6] modified the Wakuri model [1] by considering a turbulent two-phase jet. According to their theoretical derivation, the dependence of S tip upon the time gradually transitions from t to t 0.5 , which confirms the correctness of Hiroyasu's model [3].
In the derivation processes of the above models, an idealized rectangular fuel injection rate profile was assumed [1][2][3][4][5][6], and the increase of sac pressure during the start-of-injection (SOI) transients and the termination of fuel momentum supply after the end-of-injection (EOI) were not considered. Recently, the above models were found to give an unacceptable prediction of S tip during the SOI transients and after the EOI [7,8]. These limitations seem to be acceptable when using a single long-pulsed fuel injection [1][2][3][4][5][6], where the SOI transients account for a relatively small proportion of the entire injection processes and the spray usually impinges upon the wall before the EOI. For modern diesel engines employing multiple-injection strategies, the SOI transients play an important role in the entire injection processes and spray evolution after the EOI can not be ignored, especially for the low temperature combustion that features in the early fuel injection. Therefore, further studies about S tip during the SOI transients and after the EOI are needed. Kostas et al. [9] studied S tip during the initial 0.5 ms after SOI, and they found that S tip follows an empirical correlation S tip (t) = At 1.5 during the SOI transients. They suggested using their correlation in conjunction with Hiroyasu's correlation to describe the whole S tip behaviour. Following the t 1.5 dependence suggested by Kostas et al. [9], Taşkiran et al. [10] modified the empirical correlation for the initial S tip by considering the effects of injection pressure and ambient density. They confirmed that S tip has a t 1.5 dependence at the injection startup. Recently, the authors of this paper proposed a theoretical model for S tip during the SOI transients [8]. They reported that S tip is proportional to t 1.5 at the acceleration stage, t 0.75 or t 1 at the transition stage, and t 0.5 at the quasi-steady stage. They found that the model prediction results are in great agreement with the experimental data. There are also many studies on spray evolution after the EOI. For example, Musculus et al. [11] developed a 1-D model for S tip during the EOI transients. They reported that S tip gradually transitions to be proportional to t 0.25 after two injection durations. Liu et al. [12] developed a 0-D model for S tip during the EOI transients. They confirmed that S tip is proportional to t 0.25 when the time is much longer than the injection duration. The time dependence t 0.25 has also been observed for a small quantity diesel spray [13] and water-jet [14]. The authors of this paper modified the fuel injection speed in Liu's model to the fuel injection pressure and made the changed model continuous with the Hiroyasu model at the time of two injection durations, and proposed a model for spray tail penetration based on the discrete control volume method [7]. Recently, Zhou et al. [15] studied the spray evolution during the EOI transient and compared various models developed so far. They reported that only the present authors' model [7] exhibits a trend close to the experimental results. As reviewed in the above paragraph, although S tip during the SOI and EOI transients have been studied to some extent, so far no models are summarized to predict the entire development processes of S tip . More unfortunately, there are almost no data that can be used to develop and validate such a model. This is because the existing studies about the SOI and EOI transients are independent of each other. One of the objectives of this paper is to summarize the present authors' previous work on the transient diesel spray [7,8] and further develop a model for the entire development processes of S tip . Another objective is to provide experimental data to verify the newly developed model. Figure 1(a) shows the conceptual models for the entire development process of S tip when t p is smaller than t b . As can be seen, there are four key time points in the entire development process of S tip : the sac pressurization time (t p ), breakup time (t b ), injection duration (t i ) and two injection durations (2t i ). Five stages can be divided according to these four time points: the acceleration stage, transition stage 1, quasi-steady stage, transition stage 2 and decelerating stage. (1) At the acceleration stage (0 < t ≤ t p ), the sac pressurization processes have not been completed and the liquid core is intact from the nozzle exit to the spray tip. During this time, the nozzle exit fuel velocity gradually increases and the increased fuel momentum at the nozzle exit would be easily transferred to the spray tip. As a result, S tip exhibits an acceleration behavior during this stage.

Model formulation
(2) At the transition stage 1 (t p ≤ t ≤ t b ), the sac pressurization processes has been completed but S tip is still smaller than the breakup length. At this point, S tip exhibits a linear dependence upon time, which is consistent with the Hiroyasu model before spray breakup.
(3) At the quasi-steady stage (t b ≤ t ≤ t i ), since the breakup length is obviously shorter than the spray penetration length, S tip during this stage is mainly controlled by fuel-air mixing. As a result, the late-injected fuel finds it is difficult to catch up to the early-injected fuel and S tip exhibits a deceleration behavior. (4) At the transition stage 2 (t i ≤ t ≤ 2t i ), a disturbance of increased air entrainment termed the ''entrainment wave'' gradually travels downstream from the nozzle outlet to the spray tip. Since the ''entrainment wave'' has not arrived, the spray tip still remains at the quasi -steady stage before 2t i and the same model in the quasi-steady stage can be used. (5) At the decelerating stage (2t i ≤ t), the whole spray enters into the decelerating state and the total momentum at the spray tip decreases with the time elapsing. As a result, the velocity of the spray tip is further reduced during this stage. Figure 1(b) shows the conceptual models of the entire development processes of S tip when t p is larger than t b . S tip behavior at the acceleration stage, quasi-steady stage, transition stage 2 and decelerating stage is consistent with the situation when t p is smaller than t b . At the transition stage 1 (t b ≤ t ≤t p ), the sac pressurization processes are still incomplete but S tip is longer than the breakup length. The fuel moving inside the liquid core penetrates to the breakup length with negligibly small velocity loss, then the fuel moves to the spray tip with a gradually decreased velocity after breakup owing to the momentum exchange between the ambient air and fuel. As a result, although the nozzle exit fuel velocity still increases during this stage, S tip exhibits a slight deceleration behavior. As discussed in the above paragraphs, the Hiroyasu model [3] can be used at the quasi-steady stage and the transition stage 2. Based on the relationship among sac pressure, needle lift and time after start of injection (aSOI), the authors of this paper proposed a theoretical model for S tip at the acceleration stage and the transition stage 1 [8]. Moreover, a theoretical model for S tip at the decelerating stage is developed by the present authors [7]. Making the models in [3,[7][8] (1) is suitable when t p is smaller than t b , while Equation (2) is suitable when t p is larger than t b . It can be seen that both sides of Equations (1) and (2) can satisfy the conservation of dimensions, which verifies the correctness of the models. According to Equations (1) and (2), S tip shows a t 1.5 dependence upon time at the acceleration stage, t 1 or t 0.75 dependence at the transition stage 1, t 0.5 dependence at the quasi-steady stage and transition stage 2, and (t-t i ) 0.25 dependence at the decelerating stage. Since the exponent gradually decreases from 1.5 to 0.25, the spray tip experiences an acceleration process at the injection startup and then moves downstream at a gradually reduced speed.

Experimental setups
The Bosch long-tube method was used in the present study to measure the fuel injection rate, more details can be found in [16]. A constant volume combustion chamber (CVCC) was employed to measure the spray characteristics. The transient pressure in the CVCC was obtained by a pressure transducer (Kistler 6054B). One injector adapter was mounted on the head of the chamber, while five optical windows of 100 mm diameter can be mounted on the bottom and sides of the chamber. Moreover, the distance from the nozzle outlet to the field of view was changed by using different injector adapters, as shown in Figure 2. In this study, the injector adapter 1 in Figure 2(a) was used to obtain S tip between 0 to 90 mm (i.e. the spray upstream), while the injector adapter 2 in Figure 2(b) was used to obtain S tip between 90 to 180 mm (i.e. the spray downstream). Figure 2(c) illustrates the diffused back-illumination (DBI) setup for imaging the temporal development of spray. An in-house light emission diode (LED) was adopted as the light source. After passing through a diffuser and the spray region, the light was captured by a CMOS camera coupled with a Nikon lens. One can refer to previous studies for more information about the optical arrangements [17]. Details about the imaging parameter settings and the experimental conditions are shown in Table 1.   Figure 3 shows the fuel injection rate profile with 120 MPa fuel injection pressure and 2 ms injection duration. According to the methods proposed in [8,18], the sac pressurization time can be approximated as the time of the rapid increase of the injection rate at the injection startup. In this study, the sac pressurization time is determined to be 0.34 ms based on the ramp-up processes in Figure 3. Figure 4(a) shows the evolutions of S tip under different ambient densities. The solid, half hollow and hollow symbols are used to distinguish the data from SOI to t i , from t i to 2t i , and after 2t i , respectively. Since the momentum exchange between the ambient air and fuel is enhanced at the higher ambient density, the S tip decreases with the ambient density increasing. The spray tip velocity (V tip ) can be calculated by the derivative of S tip , as given in Figure 4(b). It can be seen that V tip increases rapidly during the initial stage of injection and then gradually decreases to a lower value, which is consistent with the analyses of Figure 1(b). During the acceleration stage, the liquid core is relatively intact, and V tip mainly depends on the difference between fuel injection pressure and ambient pressure.

Experimental results
Since the ambient pressure is negligibly small compared to the fuel injection pressure, the effects of ambient density on V tip is relatively small during the acceleration stage. After the acceleration stage, when S tip is longer than the breakup length, the ambient air entrainment rate plays an important role in affecting V tip . As a result, V tip decreases with ambient density increasing after the acceleration stage. In the next section, the data in Figure 4(a) will be used to verify the newly developed model of this paper.

Model evaluation
For the experimental conditions of this study, the sac pressurization time is larger than the breakup time. As a result, Equation (2) is applicable. Owing to the relationship between K 1 and K 2 , there is only one model constant in Equation (2) that should be calibrated to consider the effects of different nozzle geometries. Calibrating K 1 against the experimental S tip data, the optimal K 1 is 0.51 with the determination coefficient R 2 up to 98. 7%, as shown in Figure 5. It can be seen that the calculated results agree very well with the experimental data.   Figure 6 uses the linear coordinates to observe the overall evolution, while the right side uses logarithmic coordinates to better compare the predictability of these two models at the injection startup. The black, dark gray and light gray dots represent the experimental data from SOI to t i , from t i to 2t i (i.e. the transition stage 2) and after 2t i (i.e. the decelerating stage), respectively. Since the sac pressurization processes at the injection startup are not considered in the derivation processes, the Hiroyasu model overpredicts S tip at the initial stage of injection. Moreover, the Hiroyasu model significantly overpredicts S tip after 2t i , this is because that the quasi-steady fuel injection is assumed and the termination of fuel injection and fuel momentum supply is not considered in its derivation processes. Since the sac pressurization processes during the SOI transients and effects of ''entrainment wave'' after the EOI are considered, the newly developed model can predict the entire development processes of S tip . It should be noted that K bt of 28.65 suggested by Hiroyasu et al [3] is used for the newly developed model. As a result, in the logarithmic coordinates, the time of the first turning point of the newly developed model and the Hiroyasu model is consistent.

Conclusions
The model and experimental results clarify four key time points in the entire development processes of S tip : the sac pressurization time (t p ), breakup time (t b ), injection duration (t i ) and two injection durations (2t i ).
The entire development processes of S tip can be summarized into five stages: the acceleration stage (S tip proportion to t 1.5 ), transition stage 1 (t 1 or t 0.75 ), quasi-steady stage (t 0.5 ), transition stage 2 (t 0.5 ) and decelerating stage ((t-t i ) 0.25 ).
The calculated results of the newly developed model agree very well with the experimental data.

Acknowledgments
The supports by the Major International (Regional) Joint Research Project of National Natural Science Foundation of China (52020105009) are gratefully acknowledged.