A Moehwald HDA (HDA is a German acronym: Hydraulischer Druckanstieg: hydraulic pressure increase) injection quantity and rate measuring unit is used to investigate injection rates obtained with a fast-acting, preproduction diesel solenoid injector. Experimental parametric variations are performed to determine their impact on measured injection rate traces. A pilot–main injection strategy is investigated for various dwell times; these preproduction injectors can operate with very short dwell times with distinct pilot and main injection events. Dwell influences the main injection rate shape. A comparison between a diesel-like fuel and a gasoline-like fuel shows that injection rates are comparable for a single injection but dramatically different for multiple injections with short dwells.
Development and optimization of diesel engines require detailed understanding and control of the fuel injection and mixture preparation processes. Advances in injector hardware such as direct acting piezo actuators or pressure-balanced, fast-acting solenoid valves create new possibilities for advanced fuel injection strategies, particularly those with close coupling between pilot and main injections. In order to understand how these strategies affect mixture preparation and combustion, and to support modeling work, it is of great interest to measure the rate of injection. This work is concerned with the measurement of several preproduction solenoid injectors using a Moehwald HDA injection quantity and rate measuring unit. It is of interest to characterize the injector rate shapes in terms of their important features and repeatability, and also to understand the effects of some simple but possibly important parameters such as digital signal filtering, temperature, axial clamping force, high pressure line length, and pull-up time. Also, the variance between injectors is of interest, as is the injection rate behavior in multiple injection strategies. A detailed analysis of the causes for the observed trends is not within the scope of this work.
where is the mass of fluid in the chamber, is the volume of the chamber, is the time, is the chamber pressure is the fluid density, and is the speed of sound in the fluid.
So, for a given chamber volume, the mass flow rate and cumulative mass can be determined if the time-dependent pressure and pressure-dependent speed of sound are known. Whereas Zeuch assumed a constant value for the compressibility of his fuel , in the HDA, the speed of sound is measured directly. Hence, uncertainties in fluid properties, such as density and bulk modulus, do not adversely affect the accuracy of the results.
A schematic diagram of the HDA is shown in Fig. 1. The cylindrical chamber has a volume of 128 ml, and its walls are temperature controlled via an external heated recirculator that pumps heat transfer fluid through the chamber walls. A piezoresistive pressure sensor is mounted on the cylinder wall halfway between the top and the bottom of the chamber. The pressure signal is passed through an analog anti-aliasing filter with a cutoff frequency of 25 kHz before being digitized with 16-bit resolution at a rate of 100 kHz. Additionally, the digital processing of the pressure signal accounts for the temperature dependence of the pressure sensor sensitivity, as well as nonlinearities in the sensor output. Finally, the pressure signal is digitally filtered using one of several raised-cosine, finite-impulse-response (FIR) filters; several combinations of cutoff frequency and roll-off factor can be selected in the Moehwald software. Available cutoff frequencies range from 3 kHz to 10 kHz.
To measure the speed of sound, an electronic pulse is sent to a piezoceramic sensor, which creates a pressure wave that travels through the fluid in the chamber, is reflected at the top of the chamber, and detected by the same sensor upon its return. A clock speed of 75 MHz yields a timing resolution of 13.33 ns, and quantization noise is reported to be on the order of 0.1 m/s . Neglecting any uncertainty in the chamber height, the total uncertainty in the speed of sound is estimated to be 0.25 m/s, which is typically less than 0.025% of the measured value.
Figure 2 depicts the measurement process for a single injection event. Before the injection starts, an average chamber pressure (P1) is taken over a suitable period and the speed of sound is measured (c1). The desired excitation current is then sent to the injector solenoid, and fuel is injected into the chamber. After the injection has completed, a second average pressure (P2) speed of sound measurement is made (c2). The time-averaged pressures P1 and P2 are used as the limits of integration in Eq. (2) to determine the total injected fuel mass. The sound speed measurements are averaged with an averaging window of either 20 or 200 individual injection events (this can be configured in the Moehwald software). For this work, 20 injection events are used for this averaging, as no significant difference could be observed between the two settings. Since the speed of sound cannot be measured continuously, a linear interpolation of the two time-averaged values is applied. The resulting interpolated sound speeds are used with Eqs. (1) and (2) to calculate the injection rate and the total injected fuel mass.
In the case of multiple, closely spaced injection events (i.e., a pilot–main–post injection strategy), the pressure is averaged between individual injection events to compute the injected mass of each injection event. The second sound speed measurement is made after the completion of the last injection event, and linear interpolation is performed as described above. After the second sound speed measurement, the pressure relief valve is opened and the pressure chamber returns to the desired starting pressure.
The HDA is calibrated via a gravimetric procedure under specific testing conditions determined by Moehwald. The calibration is then checked with a number of other tests. The total injected mass deviates slightly from the value measured with a precision balance for these measurement points, but Moehwald ensures that each and every HDA has nearly the same deviations from the gravimetrically measured values at each of these points. An evaluation of the factory calibration was not within the scope of this project.
The analysis of multiple cycles yields statistical information about a particular set of operating conditions. This includes mean injected mass and standard deviation of injected mass, among other quantities. For the current work, no difference was observed between the mean values and standard deviations when the statistics were taken over either 50 or 100 cycles. For this reason, each set of statistical data presented in this paper represents 50 injections' worth of data. The mass flow rate curves shown in this work are ensemble-averaged over these same 50 cycles.
In addition to the injection rate, the injection current is measured using a Pearson Model 411 Current Monitor placed around the injector cable. The resulting solenoid current sense signal is digitized at 100 kHz.
Fuel Injection System.
The test bench setup, including the common rail, HDA, and control systems, is shown in Fig. 3. A personal computer (PC) is used to communicate with the injector driver and the HDA control module. Fuel is supplied by a small feed pump to the high pressure, three-piston pump, after which it flows into a production common rail. Rail pressure is controlled via modulation of the metering valve in the high pressure pump.
The injectors employed in this study are preproduction solenoid injectors with minisac nozzles. These are fast-acting injectors with pressure-balanced nozzle control valves; their advantageous rapid dynamic response will be the subject of future studies. For this study, three nominally identical injectors are used; they are referred to as injectors A, B, and C. Every injector nozzle has seven evenly spaced holes, each with an outlet diameter of 139 μm and conicity of 1.5. The included angle is 149 deg. A Genotec injector driver controls the rail pressure and sends the electric command signals to the injector solenoid. Measured rail pressures just before the time of injection are typically within 5 bar of the set point and the cycle-to-cycle standard deviation is typically less than 5 bar. A mixture of 58 vol. % heptamethylnonane and 42 vol. % n-hexadecane (DPRF58) is used as the working fluid for these investigations unless otherwise stated.
Electrical resistance heaters are placed around the line between the injector and the rail and the portion of the rail surrounding the port to which the injector is connected. The power dissipated by the heaters is controlled by a variable transformer. Additionally, the upper portion of the injector body is heated using coils of elastic tubing through which the heat transfer fluid used to condition the HDA is pumped. In this way, the fuel and injector temperature can be varied; these temperatures are monitored using thermocouples placed on the fuel line and on the upper portion of the injector body. For some experiments, the line between the rail and the injector is not heated. The rail temperature (Trail) is measured by a thermocouple at the high pressure fuel line approximately 50 mm upstream of the injector. The high pressure line is a rigid steel tube and has a length of approximately 250 mm. It is used in each experiment except for one, in which a flexible high pressure line and its adapter are used; their combined lengths are approximately 570 mm.
The injector is inserted into an adapter, which is in turn inserted into a plastic sleeve in the top of the HDA's measurement chamber. The tip of the injector protrudes through the adapter and into the measurement chamber, and is sealed with an O-ring that is squeezed radially against the injector tip and axially against the HDA chamber outer wall. Consequently, a large axial force is not required to seal the interface between the injector and the HDA, in contrast to the typical injector sealing arrangement used in an engine.
where is the axial force, is the tightening torque, is the pitch diameter of the screw, is the radial angle (60 deg), is the lead angle, and is the coefficient of friction in the threads (0.14).
The injector and its adapter are held together with this clamping mechanism and are isolated from the measurement chamber with a specially designed, mechanically decoupled mounting mechanism on the top of the measurement chamber.
The final purpose of this work is to examine and interpret the measured rate shapes for a variety of injection schedules. However, here, we report primarily on how some simple parameter variations affect the measured rate shapes and, in some cases, the measured injected mass. These variations include the following:
digital FIR filter cutoff frequency
temperature of the rail, high pressure fuel line, and measurement chamber
axial clamping force on the injector
the length of the high pressure fuel line between the rail and the injector
the pull-up time for the solenoid current
Finally, three of these injectors are compared with one another for the case of a pilot–main injection strategy for several different dwell times. This provides information about injector-to-injector variance and highlights some of the capabilities of the injectors used in this study.
In general, it is desired to recreate conditions as close to those encountered during engine operation as possible. For all of the results shown here, the injection pressure (Prail) is held constant at 800 bar. The axial clamping force on the injector is 14.30 kN unless otherwise stated, as this is the clamping force that is used when the injector is used in our engine. Chamber backpressure (as measured before each injection event) is maintained at 5.6 MPa. The chamber temperature (Tchamber) values reported are represented by the temperature of the heat transfer fluid measured just before it enters the chamber. Injection events (and thus measurements) are triggered at a rate of 2.5 Hz. The pull-up current and holding current for this injector are nominally 18 A and 12 A, respectively. Unless otherwise stated, the pull-up time for each injection event is set to 250 μs. The hold time makes up the balance of the actuation times reported here. In the case of multiple injections, the dwell time δt is defined as the delay between the point at which the solenoid current begins to drop off at the end of injector actuation and the point at which the solenoid current begins to rise for the next injection event; it is an energizing dwell time.
Figure 4 shows the solenoid current and rate of injection (ensemble averaged over 50 individual injection events) for a single injection with an actuation time of 900 μs in. The rate of injection was determined using a digital filter cutoff frequency of 10 kHz. The lighter shaded region around the mean rate of injection curve represents two standard deviations above and below the mean. It demonstrates that the shape of the injection rate curve, including its details, is highly repeatable. All injection rate traces in this work are similarly repeatable. A time of zero represents the start of the actuation current; this convention is maintained for all of the results shown. The total injected mass (averaged over 50 injection events) for this case is 29.24 mg/str and the standard deviation is 0.17 mg/str.
Examination of the rate of injection curve reveals some small negative excursions before the rate of injection increases significantly. This phenomenon has also been observed by other researchers examining rates of injection with solenoid injectors using both Zeuch-based measurement techniques [4,5] and Bosch or long tubes [6–8]. Needle lift data taken from a solenoid injector by Kastengren et al. show a small initial needle lift before any significant lift occurs . We attribute the negative excursion in the injection rate traces the increase in chamber volume (or, alternatively, the backflow of fluid from the measurement chamber into the nozzle) created by this initial needle motion before fuel begins to flow out from the nozzle and into the chamber. Momentum flux measurements taken by Manin et al.  indicate that the hydraulic start of injection coincides with the zero-crossing of the injection rate signal as it begins to increase significantly. This is assumed to be the case for the current study. The time of the hydraulic start of injection is determined by linear interpolation around the zero crossing.
The rate of injection increases rapidly at the beginning of the injection, but the rate of increase decreases suddenly as the rate approaches 15 g/s. Results reported by Postrioti et al. suggest that this behavior may change as injection pressure changes . Later, approximately 0.7 ms after the start of actuation and 0.35 ms after the hydraulic start of injection, the rate of increase in injection rate again decreases. The rate of injection then continues to increase nonmonotonically until it reaches its maximum 350 μs after the end of actuation. The subsequent decrease in injection rate occurs slowly at first and then more rapidly. After the apparent hydraulic end of injection, there exist both positive and negative excursions. It is unknown if these are measurement errors due to pressure fluctuations in the measurement chamber, mechanical vibrations, injector needle bounce, or some combination of these. In the current version of the Moehwald software, no correction is made to account for the increase in fluid temperature resulting from the dissipation of the kinetic energy that is introduced into the chamber; this also contributes to uncertainty, especially at the end of the injection event. Future optical spray measurements will likely provide more information about whether or not more fuel exits the nozzle after the end of the main injection phase.
Digital Filter Cutoff Frequency.
In order to illustrate the effects of changing the filter cutoff frequency (fc), data are taken with each available combination of fc and filter roll-off factor β; smaller values of β result in a more rapid decrease in the filter frequency response about fc. Figure 5 shows the rate shapes measured for a single injection for each of these filter settings.
Changing values of fc are observed to significantly affect the rate shape at the beginning and end of the injection event. As fc increases, the detected hydraulic start of injection occurs later, but this effect becomes smaller as fc increases. When fc changes from 7 kHz to 10 kHz, the hydraulic start of injection is delayed by only 10 μs; this corresponds to 0.09 crank angle degrees at an engine speed of 1500 r/min. The filter behavior is similar near the end of the injection, except that lower values of fc lead to later apparent ends of injection.
Lower values of fc result in significantly different measured rates of injection during the initial and final transients. The resulting combination of over and under prediction during the transient seems to be the reason that no systematic change in the measured total injected mass is observed as fc is changed. No significant disadvantages are observed with a cutoff frequency of 10 kHz, so this value is used for all other experiments reported here (unless otherwise stated).
Fuel and Chamber Temperature.
As the temperature of the heat transfer fluid is increased and the electrical power delivered to the resistance heaters is increased, the temperature measured at the injector body is typically between 15 and 20 K cooler than the temperature measured either at the high pressure line upstream of the injector or at the heat transfer fluid before it enters the measurement chamber wall. (We chose to keep these two temperatures equal.) The measured injection rate shapes for a single injection are shown in Fig. 6.
The initial phase of the injection is only affected slightly by temperature. The detected hydraulic start of injection changes by only 3 μs over the range of temperatures tested here. The maximum rate of injection increases noticeably with increasing temperature, and the rate of injection decreases sooner as temperature decreases. However, features of the rate shape are largely preserved even as temperature changes. The cumulative effect of this manifests itself in a variation of the total injected mass, which is shown in Fig. 7. Error bars indicate one standard deviation in either direction of the mean. As expected from the rate shapes, the injected mass tends to increase as temperature increases. The most substantial increase occurs between 40 °C and 60 °C; the nonlinear decrease in fuel viscosity with increasing temperature is one factor that likely contributes to this behavior.
Injector Clamping Force.
The tests to determine the effects of injector clamping force on rate shape are performed with the rail and the chamber at room temperature. These temperatures were not observed to change as testing progressed. Before each data point was taken, the hold-down screw was loosened and then torqued to the desired value. Figure 8 shows the observed differences in rate shape for a single injection case.
Dashed lines are used for the zero clamping force and 4.77 kN clamping force traces. All of the other traces lie very close to one another. The fluctuations observed with zero clamping force that are not seen with any other traces are attributed to movement of the injector. This movement is nearly completely attenuated with a clamping force of 4.77 kN, as only a small deviation (near 0.5 ms after the start of actuation) from the expected rate shape is observed for this clamping force. The measured variation in the average injected mass is very small (<1% for clamping forces >0), and no consistent trend in total injected mass is observed, as shown in Fig. 9.
High Pressure Fuel Line Length.
Figure 10 shows that changing the length of the high pressure line between the rail and the injector leads to very subtle differences in the injection rate shapes during the initial phase of the injection. However, more significant differences appear after the end of actuation. Beyond 0.62 ms after the start of actuation, however, the measured rates of injection diverge. The total injected masses are 12.37 mg/str with the 250 mm line and 13.18 mg/str with the 570 mm line. It is apparent that line length can have a significant effect on the total injected mass for a constant actuation duration.
Assuming a speed of sound of 1500 m/s at a pressure of 800 bar, the time required for a pressure wave to travel from the injector to the rail and back through the 250 mm high pressure fuel line is approximately 330 μs. This corresponds to the time between the local minimum in injection rate preceding the main injection event and the point at which the injection rates begin to diverge at approximately 0.62 ms. It is, therefore, conceivable that the pressure waves within the high pressure fuel line can influence the rate of injection; this is discussed briefly in Ref.  and in detail in Ref. . Time-resolved measurement of the fuel pressure and/or needle lift measurements would be helpful in interpreting these data. The detailed modeling efforts such as those presented in Ref.  could also confirm whether or not pressure wave dynamics are responsible for the increase in maximum injection rate with the shorter fuel line. Likewise, more information is needed to explain the differences in the rate shapes at the end of the injection event; both the aforementioned measurements and modeling efforts could provide insight.
Solenoid Current Pull-Up Time.
In this section, a comparison is made for a single injection case for two pull-up times: 250 μs (the standard pull-up time used in all other tests) and 350 μs. Injection rate data for this comparison are shown in Fig. 11; the total actuation time is held constant. The rate shapes for both cases are nearly identical except for the final phase of the injection, during which the rate for the shorter pull-up time decreases slightly more rapidly than for the longer pull-up time. The total injected masses are 10.93 mg and 11.30 mg for the pull-up times of 250 μs and 350 μs, respectively.
Different Injectors and Multiple Injections.
For this section, a simple pilot–main injection strategy is investigated and a variation of the dwell time is performed. These tests are repeated with three injectors to demonstrate possible differences between injectors of the same type. The actuation schedules, each of which consists of the pilot injection actuation time, a dwell time δt, and a main injection actuation time, are shown in Table 1, as well as the measured pilot injection masses. The main injection energizing times are taken from engine testing with injector A at a constant load and depend strongly on dwell time.
The rates of injection for all injectors and all actuation schedules are shown in Fig. 12. The injection command signals are shown below each set of injection rate profiles.
For all dwell times, the injection rates of the pilot injections are nearly identical. There are slight differences in the maximum rates between the injectors. For a dwell time of 1200 μs, the rates of injection for all three injectors increase at similar rates at the beginning of the main injection. However, injector A is characterized by lower maximum injection rates during the main injection, whereas injectors B and C are similar to one another. All three injectors behave similarly during the end-of-injection transient. At a dwell time of 400 μs, the rising edge of the main injection is steeper than for a dwell time of 1200 μs. However, the rate of injection momentarily stops increasing around 20 g/s, after which the rate of injection increases toward its maximum. The tops of the rate traces are generally flatter than for 1200 μs, and the maximum rates of injection are lower. This is an example of how changing the dwell time can be used to shape the rate of injection. As before, injector A's maximum rate of injection is lower than for the other two injectors, which have maximum rates of injection that are similar to each other. For a dwell time of 100 μs, the two injection events are very close to each other, but are indeed still distinct from one another. This very close spacing is made possible by the injectors' pressure-balanced control valves. For injectors B and C, the rising edge of the main injection is the steepest of all the data shown, and the ramp-up in the main injection is again interrupted. Injector A's rate of injection also increases very rapidly for a short time, but the rate then begins to increase at a slower rate and a boot-shaped profile is not observed. In fact, this rate of injection decreases slowly during the quasi-steady portion of the injection. The apparent ends of injection are different; the first zero crossings on the falling edge of the main injection lie as far as 150 μs from one another.
In order to investigate possible effects of changing fuel properties, experiments were performed with a blend of 75 vol. % n-heptane and 25 vol. % isooctane (PRF25). Properties of the mixtures are not available for the injection pressures employed in this work, but properties of the normal alkanes n-heptane and n-hexadecane are given in Table 2 for a pressure of 800 bar and a temperature of 360 K [11–14]. These properties represent conditions in the injector rather than in the nozzle or the measurement chamber.
Figure 13 compares rates of injection for a single injection with the reference, DPRF58, and with the PRF25 for a fixed actuation duration. Some slight differences are noticed during the initial ramp-up of the injection rates, but the most significant difference is observed during the ramp-down: the injection rate for PRF25 ramps down slightly later than for DPRF58. These changes are not negligible, but they are comparable in magnitude to those obtained by changing the line length (cf. Fig. 10).
An increased amount of fluctuation is present in the PRF25 injection rate signal and is clearly visible after the apparent end of injection. In the case of the pilot–main injection strategy, these large pressure fluctuations complicate the interpretation of the measured rate traces, particularly for the main injection. A lower filter cutoff frequency of 6 kHz was chosen to mitigate these effects with PRF25 at the cost of some dynamic response. Figure 14 shows a comparison between these two fuels for the injection schedules listed in Table 3.
The peak pilot injection rates are higher for DPRF58. This may be due in part to the stronger filtering of the PRF25 signal. The main injection events exhibit much more substantial differences: for PRF25, peak rates of injection are approximately 50% lower and injection durations are noticeably shorter. The rate of injection decreases for PRF25 after reaching its peak early during the main injection; the opposite is typically true for DPRF58. The mass of fuel injected during the main injection is approximately 40% higher for DPRF58 than for PRF25 for these injection schedules.
Discussion and Conclusions
The Moehwald HDA has demonstrated its ability to measure the rate of injection of individual injection events with very good repeatability. The parameter variations performed here are of practical importance when considering the measurement of rates of injection, mounting the injector, and how the injector is to be controlled. Raised cosine filters, although typically intended for shaping of digital data pulses , have been applied here for the purpose of analog signal conditioning. Varying the cutoff frequency primarily affects the detected hydraulic start and apparent end of injection, and this should be taken into account when selecting a cutoff frequency.
Changing the chamber and fuel temperatures does not affect the details of the injection rate shape to any significant extent, and for intermediate temperatures, the mass is not sensitive to temperature. It may be more practical to perform testing at an intermediate temperature, although a noticeable difference has been observed in the measured total injected mass between 40 and 60 °C. Some axial clamping force is necessary to hold the injector tightly in its adapter, but once this is ensured, the measured injection rate and total injected mass do not change to any significant extent with changing clamping force. High pressure line length does affect total injected mass and rate shape for a given actuation time. Care was taken with the 250 mm line to use the same line that is used when the injector is installed in the engine, so as to ensure transferability of the results when analyzing behavior observed within the engine. The limited data and scope of this preliminary work do not provide a detailed explanation of the reasons for the changes in rate shape observed with changing line length. A change in pull-up time from 250 μs to 350 μs does not dramatically alter the rate shape, but it does change the behavior of the transient at the end of the injection event and slightly affects the total injected mass.
With a pilot and main injection strategy, changing the dwell time can have a dramatic effect on the shape of the main injection event, including the maximum rate of injection. With a fast acting solenoid injector like the ones used in this study, dwell times as short as 100 μs can be employed while maintaining two separate injection events. This likely has significant implications for the mixture preparation and combustion processes, but it remains to be seen if the behavior of the injector in the engine is the same as in the HDA. With these preproduction injectors, substantial injector-to-injector differences have been observed in the rates of injection. These differences become more pronounced, particularly for very short dwell times. This could potentially be an issue when considering the application of very short dwells in production engines, and understanding the reasons for the discrepancies between the injectors will require a better understanding of the dynamic effects inside the injector.
The two fuels used in this work are different in terms of their physical properties (one can infer this from Table 2), but for a single injection, the differences in the rate shapes are modest. This changes dramatically for multiple injection strategies, and is likely related to hydrodynamics within the injector. If these phenomena are similar in nature to fluid hammer (as discussed in Ref. ), then the dynamic pressure is proportional to the product of sound speed and fluid density. For the normal alkanes in Table 2, this product is approximately 33% higher for n-hexadecane. It is therefore hypothesized that for DPRF58, the amplitude of pressure waves within the injector is higher, which may force the needle open to a greater extent than with PRF25, thus leading to increased rates of injection. Because the maximum needle lift is higher, the needle takes longer to close, and thus the injection duration is longer. The combination of these two effects is attributed to the 40–50% increase in main injection quantity for a given actuation time for the DPRF58 fuel.
The authors thank Kan Zha and Jeremie Dernotte for their help in reviewing the manuscript and Julien Manin for some very helpful discussions.
Support for this work was provided by the United States Department of Energy (Office of Vehicle Technologies) and by General Motors Corporation (Agreement No. FI083070326). This work was performed at the Combustion Research Facility of Sandia National Laboratories in Livermore, California. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04- 94AL85000.
speed of sound of the fluid in the HDA chamber
pitch diameter of screw used to apply axial clamping force on the injector
injection rate; rate at which mass is added to the HDA chamber
time derivative of HDA chamber pressure
derivative of HDA chamber pressure with respect to density of fluid in the chamber
- DPRF58 =
mixture of 58 vol. % heptamethylnonane and 42 vol. % n-hexadecane
axial injector clamping force
- FIR =
- HDA =
German acronym for hydraulic pressure increase: Hydraulischer Druckanstieg; name for injection quantity and rate measuring device
- m =
mass of fluid in the HDA chamber
HDA chamber pressure
- PC =
- PRF25 =
mixture of 75 vol. % n-heptane and 25 vol. % isooctane
HDA chamber volume
radial angle of the screw used to apply axial clamping force on the injector
lead angle of the screw used to apply axial clamping force on the injector
coefficient of friction in the threads of the screw used to apply axial clamping force on the injector
density of fluid in the HDA chamber
tightening torque that determines axial clamping force