DESIGN OPTIMIZATION STUDY ON A CONTAINERSHIP PROPULSION SYSTEM

Transcription

1 DESIGN OPTIMIZATION STUDY ON A CONTAINERSHIP PROPULSION SYSTEM Brian Cuneo Thomas McKenney Morgan Parker ME 555 Final Report April 19, 2010 ABSTRACT This study develops an optimization algorithm to explore the tradeoff between fuel consumption and engine room volume of a direct drive containership. Standard regression formulas, first principles analysis and new regression formulas from published manufacturer data are used to formulate a model. This model is constrained by the data used in the individual regression formulas, physical constraints and manufacturing capabilities. Each of the subsystems of the total algorithm, hull, propeller and engine are validated and tested independently to demonstrate feasible solutions. The combined system uses a sequential approach, hull-propeller-engine, exchanging vectors of interacting variables to produce an integrated Pareto front between fuel consumption and engine room volume. A test case is run through the algorithm and the results are examined. With additional data pertaining to routes, fuel prices and cargo rates, a ship designer could implement this model to find an optimal propulsion system solution for a given ship speed and displacement. This solution would be subject to scrutiny if the optimum lies on the subsystem model constraint boundaries, implying different regression models are required.

2 Table of Contents 1 Design Problem Statement Nomenclature Hull Optimization Subsystem (Thomas McKenney) Mathematical Model Objective Function Constraints Design Variables and Parameters Model Summary Model Analysis Optimization Study Global Optimality Constraint Activity Case Study Parametric Study Volume Parametric Study Ship Speed Parametric Study Discussion of Results Propeller Optimization Subsystem (Brian Cuneo) Mathematical Model Objective Function Constraints Design Variables and Parameters Model Summary Model Analysis Constraint Activity Page 2

3 4.3 Numerical Analysis Optimization Study Case Study Introduction Global Optimality and Constraint Activity Parametric Study Discussion of Results Engine Optimization Subsystem (Morgan Parker) Mathematical Model Objective Function Constraints Feasibility Model Summary Model Analysis Boundedness Constraint Activity Optimization Study Implementation Results Model Validation Parametric Studies Results Discussion System Integration Study Subsystem Tradeoffs Methodology System Optimization Results Comparison to Subsystem Optimization Page 3

4 6.5 Integrated System Parametric Study Conclusions Bibliography Appendix A Hull Code Hull Optimization Code Hull Objective Function Hull Constraint Function Appendix B Propeller Code Propeller Optimization Code Propeller Objective Function Propeller Constraint Function Appendix C Engine Code Engine Optimization Code Engine Objective Function Engine Constraint Function Page 4

5 1 Design Problem Statement Containerships are a vital component of the world s economy. Over 95% of the world s goods are transported by sea. With this fact in mind, it can be concluded that an optimized containership design could provide a major advantage in the industry. Figure 1.1 shows an example of what a containership looks like. For this project, the containership s propulsion system was optimized. A ship s propulsion system can be divided into three main subsystems including the hull, propeller, and engine. Thomas worked on the hull subsystem; Brian worked on the propeller subsystem; and Morgan worked on the engine subsystem. These distinct systems are linked through a few vital parameters. Independently a ship hull is optimized for speed, volume, resistance and stability. Propellers are optimized for a combination of thrust, open water efficiency and vibration. Marine engines are optimized based on power, fuel consumption, size, weight and revolutions. Several parameters, such as cargo weight/volume and speed, were set based on typical containership values. This project will use well- optimize a documented methods from a variety of sources to create algorithms that can independently hull form, propeller and engine. Once these algorithms are linked, they will share key variables to find a global optimum. This optimum will target fuel the consumption and engine room volume tradeoff. Figure 1.1: Emma Maersk Containership ( There are many trade-offs and competing goals in the ship design process. Some of these include maximizing useable volume while minimizing resistance. Another trade-off is picking an engine that meets the power and rpm requirements while maintaining low fuel consumption. It is also important to maximize the propeller efficiency while ensuring proper thrust characteristics. All these trade-offs and more are aspects of the ship design process. This project focused on the specific trade-offs between the hull, engine, and propeller of a ship to determine the optimal combination. The optimization at the individual levels was based on analytical models that have been used for decades in the marine industry. Page 5

6 The main focus was to integrate these individual models to obtain a global optimization for ship propulsion. 2 Nomenclature Molded Volume [m 3 ] 1+k 1 Form Factor [-] A BT Transverse Bulb Area [m 2 ] A E /A O Propeller Expanded Area Ratio [-] A P Piston Area [m 2 ] A T Immersed Transverse Transom Area [m 2 ] A X Max. Transverse Underwater Area [m 2 ] B Maximum Beam [m] B cyl Cylinder Bore [m] C B Block Coefficient [-] C F Frictional Resistance Coefficient [-] C M Midship Coefficient [-] C P Prismatic Coefficient [-] C R Residuary Resistance Coefficient [-] C WP Waterplane Coefficient [-] D Depth [m] D P Propeller Diameter [m] DP Delivered Power [kw] ERV Engine Room Volume [m 3 ] EW Engine Weight [MT] FC Fuel Consumption[MT/h] F N Froude Number [-] g Gravitational Constant [m/s 2 ] H B Vertical Center of Bulb Area [m] i Number of Cylinders [-] J Advance Coefficient [-] K Cavitation Constant [-] K Q Thrust Coefficient [-] K T Thrust Coefficient [-] L Length on Waterline [m] LCB Longitudinal Center of Buoyancy [m] LCG Longitudinal Center of Gravity [m] L R L s Length of the Run [m] Length of Stroke [m] n Propeller Revolutions per Second [1/s] P,BMEP Brake Mean Effective Pressure [Pa] P/D Pitch-Diameter Ratio [-] P 0 Pressure at Propeller Hub [-] P E Engine Effective Power [kw] P V Water Vapor Pressure [-] Q Propeller Torque [kn-m] R A Model-Ship Correlation Resistance [N] R APP Appendage Resistance [N] R B Bulbous Bow Resistance [N] R Bare Bare Hull Resistance [N] R F Frictional Resistance [N] R T Required Thrust [-] R Total Total Resistance [N] R TR Immersed Transom Resistance [N] R W Wave Resistance [N] S APP Wetted Area of Appendages [m 2 ] SFC Specific Fuel Consumption [g/kwh] T Propeller Thrust [-] t Thrust Deduction fraction [-] T m Average Draft [m] V Ship Speed [m/s] V A Speed of Advance [m/s] w Taylor wake fraction [-] Z Number of Blades [-] Δ Displacement [MT] η 0 Propeller Efficiency [-] μ Kinematic Viscosity [m 2 /s] ρ Seawater Density [kg/m 3 ] Page 6

7 3 Hull Optimization Subsystem (Thomas McKenney) The main goal in hull optimization is to minimize the resistance or drag of the vessel as it travels through the water, while maintaining a specified displacement. Lower resistance will lead to a smaller power requirement, which translates to the use of a smaller engine. Although there are basic guidelines for reducing resistance, there are certain restrictions and considerations that are required to produce a valid ship design. In general, the longer and more slender a ship s hull is the less resistance there is. Making the beam or width of a ship smaller is a good way of reducing resistance. But there are some consequences if the beam becomes too small or the ship becomes too long. These include stability issues, freeboard requirements, and reduction in useable volume for cargo. 3.1 Mathematical Model The objective of the model is to minimize resistance. There are many resistance models that could be used for this project. Most resistance models are analytical and based on a series of experiments on a certain type of hull. To ensure that the model is accurate for any given ship, certain similarities are required. This evaluation is conducted by determining coefficients such as the length-to-beam ratio, beam-to-draft ratio, or the block coefficient, which describes the underwater hull form. This project will focus on a basic hull form, used mainly for container ships. One of the most common resistance models used for these types of ships is the Holtrop and Mennen model. This method is based on regression analysis of model and full-scale tests of commercial cargo and tanker vessels Objective Function The objective function is based on the Holtrop and Mennen model. All derivations in this section are from the papers entitled An Approximate Power Prediction Method by J. Holtrop and G.G.J. Mennen published in 1982 and A Statistical Re-Analysis of Resistance and propulsion Data by J. Holtrop published in The objective function is the resistance equation provided in this paper. The total resistance of a ship is expressed in Equation 1 below. = Equation 1 The form factor of the hull uses a prediction formula that is shown as Equation 2 below. Page 7

8 1+ = { } Equation 2 The form factor formula includes the parameter L R, which is the length of the run according to Equation 3. = Equation 3 The coefficient c 12 is defined by the following equations depending on the draft to length ratio (T/L). Draft is the vertical distance from the keel or bottom of the ship to the waterline. =. h >0.05 Equation 4 = h 0.02< <0.05 Equation 5 = h <0.02 Equation 6 In Equation 4, Equation 5, and Equation 6 the average molded draft is defined as T. The coefficient c 13 accounts for the shape of the afterbody and is a function of the coefficient C Stern that has a value based on Table 3.1. = Equation 7 Afterbody Form V-shaped sections -10 Normal section shape 0 U-shaped sections with Hogner stern 10 Table 3.1: CStern Value Table C Stern Page 8

9 The wetted area of the hull can be approximately found using Equation 8. = Equation 8 The appendage resistance can be determined using Equation / = Equation 9 Table 3.2 below outlines the approximate values for (1+k 2 ) for given streamlined flow-oriented appendages. These were determined using resistance tests with bare and appended ship models. Approximate 1+k 2 values Rudder behind Skeg Rudder Behind Stern Twin-Screw Balance Rudders 2.8 Shaft Brackets 3.0 Skeg Strut Bossings 3.0 Hull Bossings 2.0 Shafts Stabilizer Fins 2.8 Dome 2.7 Bilge Keels 1.4 Table 3.2: Approximate 1+k2 Value Table The equivalent 1+k 2 value for all appendages is calculated using Equation = 1+ Equation 10 The wave resistance is determined using Equation 11. = exp { + cos } Equation 11 The following equations express the coefficients included in Equation 11. Page 9

10 = /. 90. Equation 12 = /. h / <0.11 Equation 13 = h 0.11< / <0.25 Equation 14 = / h / >0.25 Equation 15 =exp 1.89 Equation 16 =1 0.8 / Equation 17 = / h / <12 Equation 18 = h / >12 Equation 19 = / / / Equation 20 = h <0.80 Equation 21 = h >0.80 Equation 22 = exp.1 Equation 23 Page 10

11 = / / 8.0 /2.36 Equation 24 = 0.9 Equation 25 The half angle of entrance, i E, is the angle of the waterline at the bow in degrees with reference to the center plane. It can be approximated using Equation 26. =1+89exp { / /. 100 /. } Equation 26 =0.56. /{ h } Equation 27 The additional resistance due to the presence of a bulbous bow near the surface is determined using Equation 28. =0.11exp 3. / 1+ Equation 28 =0.56 / 1.5h Equation 29 = / h Equation 30 Similarly, the additional pressure resistance due to the immersed transom can be determined using Equation 31. =0.5 Equation 31 = h <5 Equation 32 Page 11

12 =0 h 5 Equation 33 = / 2 / + Equation 34 The model-ship correlation resistance can be approximated by Equation 35. =1/2 Equation 35 = / Equation 36 = / h / 0.04 Equation 37 =0.04 h / > Constraints There are numerous constraints that were be considered for this optimization problem. These constraints can be grouped into physical constraints and practical constraints. Physical constraints would include a minimum draft to navigate a canal or enter a harbor or a maximum beam or length to be able to transit the Panama Canal. Practical constraints would include requiring a certain beam to ensure stability or dimensions that provide adequate freeboard. There is a third type of constraint for this particular problem. There are restrictions of the resistance model, which are based on the types of hull forms used to develop the model. All constraints used in this problem are provided below. T 15 m (Draft limit for Port of Los Angeles and Panama Canal) L 366 m (Length limit for Panama Canal) B 49 m (Beam limit for Panama Canal) 0.0 / L WL 2.0 (Speed to Length Ratio Criteria for Holtrop Model) 0.01 V/ gl 0.55 (Froude Number Criteria for Holtrop Model) Page 12

13 2.1 B/T 4.0 (Beam to Draft Ratio Criteria for Holtrop Model) 0.55 / L A 0.85 (Prismatic Coefficient Criteria for Holtrop Model) 3.9 L WL /B 14.9 (Length to Beam Ratio Criteria for Holtrop Model) D T 4 (U.S. Coast Guard Required Freeboard) GM T 0.5 (U.S. Coast Guard Wind Heel Stability Requirement) B/D 1.65 (Additional Stability Requirement) C B 0 (Block Coefficient Lower Bound) L B T C B = (Volume Equality Constraint) The variables were also bounded at the lower end with values of zero. None of the dimensions of the ship can be negative. The length, beam, and draft have upper bounds based on access to ports and canals. The upper bound of the block coefficient is one, because it is a ratio and can only be between zero and one. The depth is defined as the vertical distance from the keel to the main deck. The depth has a lower bound from the required freeboard constraint. The upper bound was set for well boundedness as 50 m in the optimizer. The optimizer will never output a value this high mainly because the depth would like to be minimized by the stability requirement. The U.S. Coast Guard Wind Heel Stability Requirement is based on some basic naval architecture principles and regression equations. The details of the GM T calculations are provided below. = / = + =0.7 = = = Page 13

14 = + = Design Variables and Parameters The design variables for this optimization define the basic dimensions and shape of the ship hull. From these variables, approximate calculations can be completed to determine design considerations and determine if a design is feasible. The list of design variables is provided below. T, Mean Draft L, Length on Waterline B, Maximum Beam C B, Block Coefficient D, Depth The design parameters also play an important role in this optimization and are listed below. Also provided are example values or ranges for the parameters. V S, Speed of the Ship [ m/s], Molded Volume [10, ,000 m 3 ] C WP, Waterplane Coefficient [ ] C M, Midship Coefficient [ ] LCB, Longitudinal Center of Buoyancy [±5% from amidships] A TR, Submerged Transom Area [0 30 m 2 ] C STERN, Stern Shape [-25 10] S APP, Appendage Area [0 100 m 2 ] A BT, Transverse Area of Bulb [0 50 m 2 ] H B, Center of Bulb Area [0 10 m] Page 14

15 3.1.4 Model Summary Objective Function: max =,, ,, + +, +,, Subject to: = 15 0 = = 49 0 = L WL 0 = L WL =0.01 gl WL 0 = gl WL =2.1 0 = = = =3.9 0 = =4 + 0 =0.5 0 = = 0 h = =0 Page 15

16 3.2 Model Analysis Before attempting to implement the optimization problem, it is important to evaluate the objective function and constraints to see if any information about the problem can be extracted. A common method used to evaluate models is monotonicity analysis. This analysis can be used to validate that the problem is well bounded with respect to every variable as well as determine possibly active constraints. The application of monotonicity analysis for optimization problems is only possible under certain conditions. For resistance optimization, the monotonicity of the objective function is unknown. This is because the total resistance is a combination of different types of resistance that incorporate the variables with various monotonicities. It cannot be determined if the objective function is increasing or decreasing with respect to any of the variables. Although the monotonicity of the objective function cannot be completed, the constraints can still be evaluated to prove well boundedness of the problem. Monotonicity analysis was completed for all constraints. Each variable has at least one upper and lower bound. This was determined by showing that there are both increasing and decreasing constraints with respect to every variable. Table 3.3 below shows the monotonicity table for all the constraints. The plus sign signifies that the constraint is increasing with respect to the variable. The minus sign signifies that the constraint is decreasing with respect to the variable. The dots signify that the variable is not included in the given constraint. The stars after the plus or minus signs signify that the given constraint is active with respect to that variable. Most of the variables were present in multiple constraints, which mean that active constraints could not be readily determined. Page 16

17 Table 3.3: Monotonicity Table for Constraints The block coefficient and depth are the two variables that the most information can be determined from the monotonicity analysis. The block coefficient is bounded by inequality constraint 15 (GM T stability constraint). Although inequality constraint 17 bounds the block coefficient at the lower bound, it will never reached this bound because the equality constraint requires a certain volume value, which cannot be achieved when the block coefficient is zero. The depth variable is not present in the objective function. It is, however, a very important dimension of a ship and was used for many calculations. The depth plays a role in stability calculations as well as freeboard requirements. It can be seen in Table 3.3 that the depth was constrained by inequality constraint 14, which is the required freeboard constraint. This constraint was active with respect to the depth because the freeboard should be pushed to its minimum based on the other constraints. At least one of inequality constraints 15 and 16 was also active with respect to depth. Due to many of the variables having multiple increasing and decreasing roles in the constraints, it was worth evaluating the constraints further to determine if any are redundant. This can be very difficult when there are more than one or two variables because of the design space in multiple dimensions. Page 17

18 Some basic conclusions were made for a few constraints that seem to be related. After evaluating inequality constraints 4 through 7, there seems to be possible redundancies. It can be determined that inequality constraint 4 was not needed because inequality constraint 6 reached its lower bound first. The same can be concluded for the upper bounds in inequality constraints 5 and 7. Inequality constraint 7 was not needed because inequality constraint 5 reached its upper bound first. It is very difficult to determine any additional information from the monotonicity analysis. It can be concluded that the optimization problem is well bounded and should output valid optimal results. 3.3 Optimization Study Due to the fact that the resistance objective function was smooth and could be calculated very fast, MATLAB was used as the optimization tool. The fmincon function was used to implement the gradient based method used to determine the optimal solution. Three MATLAB files were generated: one that calculated the objective function, one that had all the constraints, and a third that ran the optimization. These files are included in Appendix A. The results of the optimization problem mainly focus on the trends of how the principle dimensions of the ship change as both the speed and volume vary, which will be discussed further in the next section. One optimal solution for this problem would not be that meaningful. The test values used as parameters were decided based on similar ship data. If an actual design of a ship was being completed, more detailed information would be required to set the parameter values. This is why the main focus for the results analysis was on the parametric study completed. The two most influential parameters were the speed and volume of the ship. Both studies produced general trends that are logical based on engineering judgment. The specific changes in the variables were more interesting as well as their association with which constraints were active for all the parameter values. One of the most interesting and unexpected occurrences is how the active constraints changed as the parameter values were altered. In order to fully understand the design space and what factors impacted the optimal solution, certain tests were completed. The following subsections include example results for certain situations including determining global optimally, constraint activity, as well as a case study that was completed using the same values for all subsystems. Page 18

19 3.3.1 Global Optimality Due to the smooth nature of the objective function and the constraints, it was determined that a global optimum could be obtained using a gradient based or line search method. To verify these assumptions, the model was started at various points in the design space. The results show that regardless of the starting point, the final optimal solution is the same. This can be seen in Table 3.4 below. Various starting points from the lower and upper bounds of all the variables were used. The same resulting optimal solution proves that a global optimum can be found using the gradient based method utilized in MATLAB. If all the resulting optimal solutions were not the same, this would lead to the conclusion that there are multiple local optimums. Table 3.4: Optimal Solution for Various Starting Points Constraint Activity Based on general naval architecture principles, it was hypothesized that the active constraints for this problem would be the constraints associated with stability. This is because the resistance model can reduce the resistance dramatically by making the ship narrower. The stability of the ship is directly related to the beam or width of the ship. From these two statements, it could be concluded that the stability requirements would most likely be the active constraints for this problem. Monotonicity analysis also indicted that the stability constraints would most likely be active, at least for certain variables. The two main stability requirements are inequality constraints 14 and 15. Inequality constraint 14 sets a required value for the freeboard (vertical distance from the waterline to the main deck). This value would most likely be pushed to its limit because of the depth s role in stability calculations. As the freeboard increases, the depth also increases. The overall center of gravity of the ship usually increases Page 19

20 as the depth increases. A higher center of gravity translates to a less stable ship, which is taken into account in inequality constraint 15. The GM T is a value that determines the upright stability of the ship. A GM T value greater than zero means that the ship is stable. That value is usually increased based on additional heel caused by wind. This is the only constraint that involves every variable. Another major driver in this problem was the volume equality constraint, which is directly related to displacement. If this constraint was not included, the optimizer would simply reduce the dimensions of the ship, which would in turn reduce the resistance. This is not a meaningful result because ships are designed for a purpose. In most cases their purpose involves carrying a specific amount of cargo. This equality constraint only allows the hull form to change size while maintaining the same volume or displacement. After running the optimizer for varies conditions, the hypothesis made earlier in regards to the stability constraints being active was generally correct. There was, however, an occurrence that was not predicted. Other than the two stability constraints, there were other active constraints. The two other constraints encountered were restrictions set by the model. This means that the optimizer wanted to go outside the ranges that the model was valid for. In order to determine exactly how these active constraints were limiting the optimal solution, the model constraints were removed and the optimizer was run without them. The new result led to larger values for all the dimensions and a decrease in the block coefficient. This means that the ship overall became larger, but the underwater shape was not as full. Although this does have a better resistance, the shape of the hull no longer matches the shapes used for the model, which makes that result invalid. Table 3.5 shows example outputs with and without the model constraints. It can be concluded that if a wider range of hull form options is desired, another resistance model would be required. Table 3.5: Optimal Solutions With and Without Model Constraints Based on constraint activity analysis, it can be concluded that there will always be at least one active constraint for this individual subsystem. This means that all optimums are boundary solutions. Interior optimums do occur, however, during the system integration. The details of the results of the system integration will be discussed later in this report. Page 20

21 3.3.3 Case Study A case study was completed using the same parameter values for all subsystems. This was completed so the final integrated results could be compared to the individual optimal solutions obtained from each subsystem. The main parameters that were set include the ship speed, which was set at 18 knots, and the molded volume, which was set at 75,000 cubic meters. Both values are typical for containerships and produce valid results from all subsystems. The results of the case study are provided below in Table 3.6. Table 3.6: Optimal Solution of Case Study The results of this case study show ship dimensions closer to their upper bounds. This is mainly due to the large parameter values used for volume and speed. It can also be seen that the two active constraints are the required freeboard and stability requirement. This shows that for the parameter values selected, the model is obtaining a true optimum within the model limits and the typical constraints are active. 3.4 Parametric Study A parametric study was completed for this project. The two parameters that were evaluated were the ship speed and molded volume. The optimal results were evaluated as these two parameters were varied within a reasonable range. The active constraints were also evaluated as these parameters changed. Because detailed information on a specific ship was not used for this project, one optimal solution could not be obtained. The parametric study does show how the optimal hull would change as key parameters such as speed and volume change Volume Parametric Study The first parameter that was varied for this study was the molded volume. The molded volume is the volume of the hull under the waterline. This value can be multiplied by the density of water to obtain a displacement, or weight, of the ship. The volume was varied between 20,000 and 40,000 m 3. This range corresponds to a typical medium-sized containership. Although volume is being considered as a parameter for this study, it is truly an equality constraint in the optimization. Because it is an equality constraint, it must always be met by the optimal result. Equality constraints like these should be evaluated at various values to fully understand their impacts. Table 3.7 shows the results of the Page 21

22 parametric study for volume. The table shows the volume value and its associated displacement, the optimal solution with resistance value, and the active constraints for each solution. Table 3.7: Parametric Study Results for the Volume Parameter To help evaluate the results of the parametric study, a series of graphs were producedfor the resistance of each solution as well as one for each variable. Figure 3.1 through Figure 3.3 shows the graphs of the parametric study for volume. Figure 3.1: Resistance and Draft Curves for Volume Parametric Study Page 22

23 Figure 3.2: Length and Beam Curves for Volume Parametric Study Figure 3.3: Depth and Block Coefficient Curves for Volume Parametric Study It can be seen from the results of the parametric study that most of the variables increase as the volume increases. The only variable that does not increase is the block coefficient, which remains constant. The general trend of the principle dimensions correspond to the increased volume requirement. To increase the molded volume, the dimensions of the ship must increase to accommodate the added volume. From the resistance curve, it can also be seen that the resistance increases linearly with volume. This also makes sense because the added volume will cause the resistance to increase. The draft, length, and beam curves are relatively linear as the volume varies. The block coefficient remained constant around a value of This variable most likely remained constant due to a lower block coefficient always translating to a lower resistance. The value of 0.54 was the lowest value allowed by the active constraint, which was the prismatic coefficient lower limit. Due to the block coefficient not changing, the remaining variables would then have to be increased to meet the changing volume requirement. The one variable that had very unexpected results was the depth. When the freeboard constraint is active, the depth should follow the same trend as the draft, but at higher values. During this parametric study, the freeboard constraint was only active for the first five values for the volume. The lower Page 23

24 portion of the depth curve does show the same trend as the draft variable, but becomes very non-linear after the freeboard constraint is no longer active. This non-linearity could correspond to the fact that the depth could be multiple values if unconstrained by the freeboard requirements. The resulting depth values could have been determinedd by the values that meet the stability requirements the best. As mentioned previously in this report, the prismatic coefficient constraint was active for most of the results. For this parametric study, it was active for all solutions. This means that the optimal solution is being pushed to the limits of the type of hull form used to develop the Holtrop model Ship Speed Parametric Study The second parameter that was varied for this study was ship speed. The ship speed is often determined by the owner of the vessel and depends on the value of the cargo and the distance that the ship is required to travel. Choosing the design speed of a ship is a very important decision and drives a large portion of the rest of the design. Containerships usually travel faster than other cargo carriers such as oil tankers because of the type of goods they carry. Typical speeds of containerships are between 20 and 25 knots. To fully understand how the design changes depending on speed, a full range of values from one to 25 knots was used for this parametric study. Table 3.8 shows the results of the parametric study for ship speed. The table shows the ship speed in knots, the optimal solution with resistance value, and the active constraints for each solution. Table 3.8: Parametric Study Results for the Ship Speed Parameter Page 24

25 To help evaluate the results of the parametric study, a series of graphs were producedfor the resistance of each solution as well as one for each variable. Figure 3.4 through Figure 3.6 are the graphs of the parametric study for ship speed. Figure 3.4: Resistance and Draft Curves for Ship Speed Parametric Study Figure 3.5: Length and Beam Curves for Ship Speed Parametric Study Figure 3.6: Depth and Block Coefficient Curves for Ship Speed Parametric Study The resistance curve shows the basic relationship between speed and resistance for ships. Also, the variables seem to change dramatically at certain speed values. These changes can be related to the active constraints for the optimal solutions as the speed changes. The first trend in the active constraints is that the freeboard and stability constraints are active from one knot to 20 knots. These Page 25

26 two constraints do not have a major impact in the change in dimension values though. The freeboard and stability constraints do affect the results from 20 to 25 knots because they are no longer active. The prismatic coefficient constraint then becomes active, which means that the optimal ship is pushing the limits of the model. This changeover in active constraints made the draft and depth decrease and the length and beam increase slightly. The block coefficient remains constant for this range, which is similar to the volume parametric study when the prismatic coefficient constraint was active. The major changes in the results occur when the length to beam ratio constraint and the draft upper limit constraints were active. An initial trend can be seen for the first two speed values, but is stopped when the length to beam ratio constraint became active. This constraint being active caused all variables to remain relatively constant. This occurs because with the length to beam ratio being set, the values of length and beam do not change much. With the length and beam not changing, the draft and depth must remain at the same values also to maintain the required volume. Between 9 and 13 knots the draft upper limit constraint became active. This in turn set the draft and depth, which translated to the length and beam not varying that much to maintain the required volume. At around 13 knots, all variables change dramatically. At this point, only the freeboard and stability constraints were active. In general, as the speed increases a more slender hull form would have better resistance. This means that the length would increase and the beam would decrease. Draft would also decrease as speed increased to have better resistance. The block coefficient would decrease to generate a more slender hull. This trend can be seen in the results, but to a dramatic degree. It can be seen that changing active constraints play a major role in the optimal solution. The Holtrop model seems to play a restrictive role in finding the true optimal solutions. It can be seen that between 18 and 25 knots that the solution is constrained by the model limits. It is important to note that when the Holtrop model was developed, the ships were not designed to go at higher speeds greater than 20 knots. Because none of the hulls used for the model were designed to go this fast, it can be concluded that these hulls might not be the optimal designs for these higher speeds. This idea is reinforced by the results of this parametric study. The active constraints at these higher speeds are related to the limits of the model, not the typical freeboard and stability requirements. 3.5 Discussion of Results The results of the parametric studies show how important active constraints are in the resulting optimal solutions. Although the predicted trends could be seen in the resulting data, the optimal solutions were Page 26

27 restricted by other considerations. The results of this optimization study show that a better model incorporating present day considerations such as larger volumes and higher speeds should be used to determine the true optimal designs. It can be seen that optimal solutions are moving towards finer hulls under the waterline. The Holtrop model was developed in the early 1980s and was revised over the next decade. The revisions came from evaluations of inconsistencies with the model and additional tests were completed to change the model. These revisions did not continue into the 1990s and further. For future work in this area, it is recommended that another model be used for resistance calculations such as the Hollenbach method. Basic fundamentals in ship design are still proven important by the results as the freeboard and stability constraints played a major role in the optimal solutions. It was predicted that the model would want to make the ship as slender as possible. If the stability constraint was not included, the ship would be very long, narrow and very unstable. The stability constraint allowed the optimal solution to be as slender as possible while still maintaining proper stability. A stability check is always an essential part of the design of a ship. This optimizer automates this design step and iterates many designs to find the best possible solution as opposed to simply a feasible one. The freeboard requirement is also very important. Freeboard is required to protect the ship from being swamped by having water come over the main deck. Also, freeboard is closely related to reserve buoyancy. Reserve buoyancy is important because if the ship was damaged and took on water, the added buoyancy from a higher waterline would counteract the flooding and stabilize the ship. The optimization could be improved by adding additional constraints such as maneuvering or seakeeping requirements. A hull could also be optimized for maneuvering and seakeeping instead of considering them constraints. A tradeoff between resistance, maneuvering, and seakeeping would be an additional and more complete analysis to complete a hull optimization. Maneuvering and seakeeping are much harder to model than resistance and have no empirical models that simplify the analysis. In most cases these two calculations are completed using highly nonlinear and complicated models. Simplified constraints could be developed, but would not incorporate the full extent of these calculations. Page 27

28 4 Propeller Optimization Subsystem (Brian Cuneo) To achieve the maximum fuel efficiency for the hull-propeller-engine system, the propeller efficiency was maximized. The final propeller design provided the necessary thrust to meet the design speed of the ship. The propulsion system of a ship can have many forms however for the design of this system choices were limited to Wageningen B-Series Propellers. B-Series Propellers have become very popular for ships with fixed pitch propellers due to the variety of blade number, pitch to diameter ratios, and expanded area ratios that are available. Design variables for B-Series propellers include speed of advance, expanded area ratio, pitch to diameter ratio, and the number of propeller blades. The main input parameters for the optimization problem include the thrust required to maintain design speed, the diameter that fits under the hull and the rpm and torque provided by the engine. The interaction between hull, propeller and engine introduces trade-offs that must be made if all subsystems are to be optimized for maximum fuel efficiency. The diameter of the prop is restricted by the hull. A larger diameter propeller increases propeller efficiency, however, the hull draft is restricted by port depths and stability issues. Also the input shaft rpm of the engine influences the maximum diameter that can be used for the propeller due to cavitation concerns that is a function of propeller blade tip speeds. 4.1 Mathematical Model The optimal propeller design for fuel efficiency is to maximize the propeller efficiency behind the hull of the ship. This optimization is dependent on coefficients of torque and thrust, which are determined by the hull shape and the properties of the engine. For B-Series propellers a model has been developed by Bernitsas and Ray. Propeller optimization must meet the requirements for thrust to meet the speed that the owner has specified using the power that is delivered by the engine. Figure 4.1 displays a graph of the objective function versus the advance coefficient. This graph is for a fixed blade number (4) and fixed expanded area ratio (0.6) with different lines representing pitch to diameter ratios. Many assumptions were made in this model to allow for simplification of calculations while still producing meaningful results. First, the Taylor wake fraction, and thrust deduction coefficient are considered constant for all iterations of the hull. While this is not completley accurate, because the same hull type and clearances are used for all runs, the results are reasonable as the two coefficients would change very little between cases. Another main assumption is that no efficiencies are used Page 28

29 between the propeller and engine. As there is no reduction gear used this becomes more accurate, but the missing efficiencies would have the same effect on all iterations of the optimization code. So this may affect the final value of the objective function, but the optimum design variables will be the same. Figure 4.1: Wageningen B-Series Chart Objective Function The standard mathematical model for the optimization problem can be written as follows in Equation 38, where η 0 is a function of K T, K Q, and J as shown in Equation 39. The values for K T and K Q in terms of the design variables are found using experimental results. The experimental data gives coefficients and exponents to Equation 40 and Equation 41, which can be found in K T, K Q and Efficiency Curves for the Wageningen B-Series Propellers and in the code implementation shown in Appendix B. 0 =,,, Equation 38 = 2 =,,, Equation 39 Equation 40 Page 29

30 4.1.2 Constraints =,,, Equation 41 The model is constrained by several physical and practical constraints. The diameter is constrained to being less that a constant, a, determined by the hull shape and necessary hull clearances shown in Equation 42. The advance coefficient design variable is defined by the speed of advance, the shaft revolutions per second, and the propeller diameter set by the ship speed, hull form, and engine revolution per second as shown in Equation 43. The thrust from the propeller is related to the required thrust to make the ship speed by Equation 46. Equation 42 = Equation 43 = = Equation 44 2 Equation 45 = 1 Equation 46 For the model, to ensure that the propeller rpm and diameter are constant in the dimensionless coefficients, Equation 43, Equation 44, and Equation 45 are combined into two non-linear constraints which are shown below. 0= Equation 47 Page 30

31 2 0 Equation 48 Equation 48 is an inequality constraint because the power needed to overcome the resistance may be less than the max power that is supplied by the engine. Equation 48 is not used as a constraint because the engine was matched to the necessary thrust. The model can also be used to maximize the speed for a given engine. If this is the case, Equation 48 is active. Another problem when dealing with propeller efficiency optimization is cavitation concerns. The following constraint is placed on the blade expanded area ratio to prevent cavitation based on the propeller diameter, the water pressure at the propeller hub, and the thrust provided by the propeller Equation 49 Where P 0 is the static pressure at the propeller hub, P V is the vapor pressure of water, and K is a constant depending on ship type for a single screw vessel K is 0.2. (Van Manen & Van Oossanen, 1988) The following six constraints are practical constraints required by the Wageningen B-Series Propellers. The first two practical constraints are for the blade number which must be an integer value. The next two constraints are required for the expanded area ratio of the B-Series propeller. Outside of the range given for expanded area ratio the experimental data for the thrust coefficient and torque coefficient is no longer reliable. The final requirements by the Wageningen B-Series model are placed on the pitch to diameter ratio. Again, outside of the given range the experimental data equations are no longer reliable. 2 <0 Equation 50 8<0 Equation 51 Page 31

32 Equation Equation Design Variables and Parameters Equation Equation 55 The optimization design variables for propeller optimization are mainly dimensionless values used to describe the blade shapes and angles. The dimensionless values depend on parameters that are dependent on the other subsystem which induces coupling in the optimization process. These variables are listed below. Speed of Advance (J) Pitch to Diameter Ratio (P/D) Expanded Area Ratio (A E /A O ) Number of Blades (Z) The main parameters that are used in the propeller optimization system are listed below. Required Thrust (R T ) Ship Speed (V S ) Maximum Propeller Diameter (D) Page 32

33 4.1.4 Model Summary Objective Function: 0 =,,, = 2 Where: =,,, Subject To: =,,, h = =0 = = 2 0 = 0 = 2 <0 = 7<0 = = = = Page 33

34 4.2 Model Analysis Before running the optimization code the model was examined for well boundedness by using monotonicity analysis. Due to the complexity of the objective function, monotinicities could not be determined. This required all of the variables to be well bounded in the constraints. Table 4.1 shows the montonicity table for the optimization problem. The table shows that all of the design variables are bounded by the physical limitations of the model used for analyses, so from monotonicity principle one the problem is well bounded. J P/D A E /A 0 Z f h 1 + g g 2 + g 3 - g 4 - g 5 + g 6 - g 7 + g 8 - g 9 + Table 4.1 Monotonicity Table Constraint Activity Activity of the constraints is difficult to determine due to the lack of information surrounding the objective function. The pitch to diameter ratio is bounded by active constraints g 8 and g 9. The speed of advance is constrained by h 1 and g 3. The expanded area ratio is constrained by the conditionally critical set of g 1, g 6, and g 7. The blade number is constrained by the conditionally critical set of g 1, g 4, and g 5. Ane of changing active constraints can be seen in the case study in section 4.4. In the example provided the following constraints are active depending on the blade number being examined: Page 34

35 Blade Number (Z) Active Constraints 3 h 1,g 6 4 h 1,g 6,g 9 5 h 1,g 1 6 h 1,g 1 7 h 1,g 9,g 7 Table 4.2: Constraint Activity 4.3 Numerical Analysis The optimization algorithm was then run for a test case to find an optimal value for a given ship and engine. The optimization algorithm used was a version of an active set algorithm found in MATLAB s fmincon function. The constraints in the optimization method require tradeoffs between the other two subsystems of the hull-propeller-engine optimization problem. Either the required thrust from the hull or the delivered horsepower from the engine can be the factor that most influences the propeller efficiency. A parametric study was done to see the effects of changing the resistance and delivered power. 4.4 Optimization Study Case Study Introduction A case study was analyzed to see if the model successfully found an optimum for realistic parameters for a ship. The case study was done for a preselected volume of 75,000 m 3 and ship speed for 18 knots. The volume and ship speed were used to find an optimum resistance. The optimal resistance was input to the propeller optimization along with the corresponding draft. The following data was used for the optimization: Draft = [m] D = [m] V S = 18 [knots] R T = [kn] t = [-] w = [-] When this data is entered into the optimization code, the following optimums were obtained for each blade number. Page 35

36 Blade Number (Z) J P/D A E /A 0 η Table 4.3: Test Case Results From Table 4.3 the overall optimum is a 3 blade prop with P/D of 1.05, A E /A 0 of 0.3, and operating at J of This combination of design variables results in an η o of With this propeller, the thrust required to overcome the resistance can be accomplished with a delivered power of 28,650 kw. The engine would be required to operate at a speed of rps if no reduction gear is used, which would increase the required power due to losses in gearing efficiency Global Optimality and Constraint Activity Multiple starting points were examined to check for global versus local optimum. Table 4.4 shows a sample of results of starting from multiple points. All runs converged to the same point indicating a global maximum. J 0 P/D 0 (A E /A 0 ) 0 η 0 max Table 4.4: Results for Multiple Starting Points For the optimal case, the constraint activity is examined. The overall optimum is constrained by the model constraint on A E /A O, blade number, and the thrust constraint h 1. Propellers are most efficient with the smallest number of blades, and small expanded area ratios, so these results are to be expected if the cavitation constraint is not active. As the blade number is increased, the cavitation becomes active for blade numbers 5 through 7. How the constraints affect the optimization problem can be examined by relaxing the different constraints. When the equality constraint on thrust is removed, the program can find the optimal efficiency for a given draft. Table 4.5 shows the results of running the optimization program with the Page 36