Preference Modeling and Optimization
Fuel Injector Cavitation Study
ME6105
Homework 4
Eric Danish
James Seaton
Presented to: Dr Chris Paredis
Date: May 11, 2007
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Preference Modeling and Optimization - Assignment | ME 6105, Assignments of Mechanical Engineering
Material Type: Assignment; Professor: Paredis; Class: Modeling&Simulation-Dsgn; Subject: Mechanical Engineering; University: Georgia Institute of Technology-Main Campus; Term: Summer 2007;
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Preference Modeling and Optimization
Fuel Injector Cavitation Study
ME
Homework 4
Eric Danish
[email protected]
James Seaton
[email protected]
Presented to: Dr Chris Paredis
Date: May 11, 2007
Revisit the Decision Situation Identified in HW
This study concentrated specifically on optimization of an electronic unit fuel injector (EUI) used on the 710 model series diesel electric locomotives produced by Electro Motive Diesel (EMD). A diagram of the injector is shown in Figure 1.
Figure 1: Cross Sectional View of Electronic Unit Injector
Currently, there is an issue occurring in the pumping chamber of this injector during operation. Cavitation occurs when the liquid experiences significant low pressure and gaseous voids are formed. When these bubbles collapse, a pressure wave is produced along with high local temperatures, which can cause damage. For reliable performance, it is necessary to maintain positive chamber pressure to avoid these phenomena. While preventing cavitation, it is also important to examine performance and emissions. The injector must have the capacity to deliver enough fuel to the engine under any operating conditions. In harsh operating environments, a locomotive can increase power past the nominal output, so the injector must be able to accommodate this change. The setup of the control valve in the injector allows a variable quantity to injected, which is only limited by volume. Finally, emissions requirements mandated by the EPA must be met by all new engines. Fuel injectors play a dominant role in meeting these new emissions requirements. The current configuration of components in this system affords alterations to be made within feed-line and feed-port diameter, plunger speed profile according to camshaft lift, and pumping chamber geometries to yield parameters of interest in
Part Geometry
Pressure Flow Rate
Cavitation
Economics
Costs
Reliability (^) Performance
Overall Utility
Tooling Change
Fluid Properties
Dimensional Accuracy
Figure 2: Modified Influence Diagram
In order to achieve the design objectives, the system must be modeled in relation to decision alternatives. An energy-based model of the fuel injector during an injection event helped to predict performance and the occurrence of cavitation. This type of model is ideal to effectively describe system tradeoffs and overall performance. It does not model three dimensional effects such as turbulence, vorticity, or shocks, but in general will result in a basic description of the system. For example, the hydraulic model of the pumping chamber includes velocity and pressure distributions of the flows passing through it. A few assumptions are made to simplify this problem. Regardless of these assumptions, an overall understanding of the dominating factors for this system will be gained. The initial assessment of the design problem was a good start. We modified some of our initial objective to those shown above, which were better to quantify. The overall objective of eliminating cavitation was investigated and modeled thoroughly, whereby the parameters listed above were varied and modeled under uncertainty to find a robust design. Modeling injection quantity proved to be somewhat more difficult. A typical injection quantity is on the same order of magnitude of the leakage quantities for most models, and unfortunately, the model became unstable if leakage quantities were set too low. This result could be evaluated in the future with further refinement to the model. Initially, plunger velocity was held constant due to the fact that it requires camshaft and engine speed changes, but it was found to influence the chamber pressure so
much that changes were investigated. During the process, supply pressure variation was also investigated.
Dymola Model Changes
Our model underwent many changes since the last homework. We began by removing the influence of supply flow on the system, which previously dominated the analysis. The conductance values of all of the valves had a very large influence. After these values were increased, supply flow no longer influenced the system pressure once it reached the threshold to keep it up at adequate levels. In addition, the layouts of the pumping chamber and high pressure pumps were changed to adjust the logic and performance in general. Next, the control valve was dropping the pressure through it by about 20 psi, which was way too much. Currently, it is about 0.5 psi. In the injection line, the needle valve was changed to experience a progressive opening. Previously, pressure was venting immediately, making it difficult to keeping building pressure during an injection event. Finally, the pumping chamber model was simplified, excluding unknown parameters and correcting the reference positions for motion.
After these changes, actually solving the model became difficult. A stiff system had developed due to the large range of certain parameters with a fast acting system. Most importantly, when injection pressure began to build quickly, the model would fail. Various stiff solvers were investigated and had better results. Unfortunately, these solvers were seemingly unsupported by ModelCenter. Eventually, a proper number of samples with the Dassl solver were found. By having a much larger number of sample points, the sharp increases in parameters became more gradual, keeping the model from failing. Finally, the result was a model that produced characteristic injection pressures and could experience cavitation if the wrong combination of parameters was utilized. Updated screenshots of the Dymola model are included below.
Figure 5: High Pressure Pump Model Layout
Figure 6: Fluid Control Model Layout
Figure 7: Injection Line Model Layout
Figure 8: Feed Line Model Layout
Elicit Preferences
In order to examine the trade-offs that exist with each design alternative, we developed a utility function. The utility function was a result of three objectives and their corresponding measurable attributes mentioned in detail above. Our preferences for these objectives were determined through elicitation.
Our preference for maximum injection pressure was analyzed first. The minimum acceptable pressure was set at 17000 psi. This is a typical injection pressure of previous generation injectors and was used to meet previous emissions regulations. The upper bound was set at 22000 psi, which pushes the ability of unit injection technology. Newer injection technologies have pressures beyond this, but 22000 psi is meaningful goal for this style of injector. These system limits were set as bounds for the individual utility function for injection pressure. Then, to evaluate preferences under uncertainty, the repeated question asked was, “How much would you be willing to pay for a gamble in which you earn $1 if the maximum injection pressure was greater than ____ psi, and $0 if it wasn’t?” The Utility function resulting from our responses is shown below in a plot. The plot is risk-adverse, reflecting a strong preference for higher injection pressures. If our result is closer to the upper end, the injector technology will have a longer useful life due to its emissions performance.
Maximum Injection Pressure Utility
0
1
17000 18000 19000 20000 21000 22000 Injection Pressure, psi
Figure 9: Maximum Injection Pressure Utility
a potential to greatly reduce pressure pulsation waves through the system during plunger movement.
Minimum Chamber Pressure Utility
0
1
5 15 25 35 45 55 65 75 Chamber Pressure, psi
Figure 11: Minimum Chamber Pressure Utility
a = 8.1708929509783125E+ b = 9.0204099139647803E- c = -2.7893686662854776E+ d = 7.5616922304958506E-
Figure 12: ZunZun Function of Chamber Pressure
Finally, the preference for maximum quantity of injected fuel was analyzed. From typical engine performance, a lower bound was set at 7.0E-6 m^3. An attainable upper bound according to space constraints of the injector body was set at 1.5E-5 m^3. These system limits were set as bounds for the individual utility function for injected volume. Then, to evaluate preferences under uncertainty, the repeated question asked was, “How much would you be willing to pay for a gamble in which you earn $1 if the maximum injected quantity was greater than ____ m^3 , and $0 if it wasn’t?” The Utility function resulting from our responses is shown below in a plot. It reflects a strong risk-seeking preference near the lower end. It is necessary to be able to inject a bit more than the nominal full output. This leaves room for uncertainty and overloading conditions. The injector has room to include more injected volume, but it is likely unnecessary near the upper end, which is why the curve is risk-neutral in this section.
Maximum Injected Quantity Utility
0
1
7.00E-06 9.00E-06 1.10E-05 1.30E-05 1.50E- Injected Quantity, m ^
Figure 13: Maximum Injected Quantity Utility
The overall utility is a combination of the determined individual utility functions, and takes the basic form of
subjected to the constraint of
Where m, y and z represent chamber pressure, injection quantity and injection pressure respectively for this project. The coefficients of the utility functions were determined through an iterative process of elicitation. These k values are indicative of willingness for tradeoffs in the overall performance of the model in a variety of situations.
The first step was to set up utilities in order of preference. In this particular study, we want to place the most emphasis on injection quantity, followed by injection pressure, and chamber pressure. Our end result is fuel delivery, so while intermediate dynamics are important, we consider the system output to be somewhat higher. Next, we went through the elicitation process, and asked the following set of questions.
- For what Injection pressure value are we indifferent between (CP 1 ,IQ 1 ,IP 0 ) and (CP 0 ,IQ 1 ,IP’)
- For what Injection pressure value are we indifferent between (CP 0 ,IQ 1 ,IP 0 ) and (CP 0 ,IQ’,IP 1 )
- For what Injection pressure value are we indifferent between (CP 1 ,IQ 0 ,IP 0 ) and (CP 0 ,IQ 0 ,IP’’)
- For what Injection pressure value are we indifferent between (CP 1 ,IQ 0 ,IP 1 ) and (CP 0 ,IQ”,IP 0 )
- For what Injection pressure value are we indifferent between (CP 0 ,IQ 0 ,IP 1 ) and (CP 0 , IQ”’, IP 0 )
- For what Injection pressure value are we indifferent between (CP1/2,IQ1/2,IP1/2) and (CP0.1,IQ0.1,IP0.1)
- Constraint that k values sum to 1
After answering each of these questions individually, the resulting values were determined graphically from the utility charts. Since we were not concerned with exact values from a subjective solicitation, rounded values were obtained. The following chart outlines our responses to each question.
Figure 15: Overall Utility Responses
Figure 16: Overall Utility Coefficient Matrices
The u-values shown above in matrix form were obtained by equating the constant values with those obtained from the tradeoffs. The k-coefficients were obtained by multiplying the inverse matrix of u-values with the right hand side of the equation. The overall utility equation was obtained from the following array of coefficients.
Figure 17: k-Coefficients
The values appeared reasonable in that the first three reflected original ranking of our utilities in order of Injection Quantity, Injection Pressure, and Chamber Pressure.
Figure 19: DOE Setup for Design Space Exploration
First, the utility of injected quantity was entirely influenced by the piston area. Our model kept the distance traveled by the plunger constant since it was directly connected to the velocity as well, so this was the only influence. This was not a definite decision for setting the piston area though because it was influential in each of the utility functions. This parameter was therefore investigated the most to its presence in each of them.
0 10 20 30 40 50 60 70 80 90 100
piston_area
feedline_diameter
feedport_diameter
omega
nozzle_diameter
needle_pclosed
comp_vel
ret_vel
100%
0%
0%
0%
0%
0%
0%
0%
Figure 20: DOE Results for Injected Quantity Utility
The individual utility results for minimum chamber pressure are shown below. Three parameters had almost equal influence over this function, so it was important to examine their respective influences over the other utilities, to determine a tradeoff approach. Note that feed line diameter is not influential over the other utilities below, so it was focused on to improve the minimum chamber pressure. The piston area was most influential in the injected quantity, so it was not varied much for this utility function. The retracting velocity did affect another utility, but was more influential here.
Figure 22: DOE Results for Maximum Injection Pressure
The DOE results for the overall utility function are shown below. Clearly, piston area dominated the overall utility which was expected due to its presence in each of the individual utility functions, and became the starting point for optimization. Our preference for the maximum amount of injected quantity was dominant over its effects on pressure, so this leads our optimization to start at the large end of piston area. Next, the plunger compression and retraction velocities were the most influential. Each had a strong effect on a respective utility, as discussed in more detail above in the individual utility diagrams. The less influential parameters can be fine tuned easier, due to their effects on a limited number of the utility functions. A histogram of the results is included as well. The piston area dominance is illustrated here. The overall utility jumps drastically due to its variation, leading to a scattered result. Fortunately, it gives a strong indication of where to begin optimization.
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
piston_area
comp_vel
ret_vel
needle_pclosed
nozzle_diameter
omega
feedport_diameter
feedline_diameter
81%
8%
8%
2%
1%
1%
0%
0%
Figure 23: DOE Results for Multi-Attribute Utility
Value
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.
O c c u rre n c e s
125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0
um
Figure 24: Histogram Results for Multi-Attribute Utility from DOE
Since the strongest influences were known, the next step was to determine the actual values to start at. Parametric studies were done on the different utility functions to find these values. First, the piston area was varied along its possible inputs, and as predicted, the best starting point was the maximum value. Next, compression velocity was examined. It had some local behavior over the wide range that needed to be optimized. A good place to start appeared around 50 m/s since it was a local maximum, near where our model was set during the design phase. Finally, compression velocity also experienced a local maximum with respect to the overall utility function. The starting point for optimization was selected near 2.25 m/s.