FPMC DRAFT GEOMETRIC OPTIMIZATION OF A HYDRAULIC MOTOR ROTARY VALVE

Transcription

1 Proceedings of the ASME/BATH 213 Symposium on Fluid Power & Motion Control FPMC213 October 6-9, 213, Sarasota, Florida, USA DRAFT FPMC GEOMETRIC OPTIMIZATION OF A HYDRAULIC MOTOR ROTARY VALVE Hao Tian Department of Mechanical Engineering University of Minnesota Minneapolis, MN, USA tianx156@umn.edu James D. Van de Ven Assistant Professor Department of Mechanical Engineering University of Minnesota Minneapolis, MN, USA vandeven@umn.edu ABSTRACT The performance of a hydraulic motor is strongly influenced by the timing, leakage, and friction of the valves connecting to the pressure and tank ports. For the application of a linkage-based hydraulic piston motor, a novel clearance sealed cylindrical rotary valve is introduced. A model of the cylinder pressure dynamics is formulated from bulk modulus definition and solved using a variable-step Runge-Kutta stiff solver. Energy loss equations are developed and used to aid the creation of objective functions. A geometric optimization of the motoring efficiency is conducted based on coupled relationships: including throttling across the transitioning and fully open valve ports, viscous friction, and leakage, but excluding piston related losses. A simple genetic algorithm is used to obtain optimized geometry, and a theoretical system efficiency greater than 97% is achieved. A multi-objective subpopulation genetic algorithm is then used to generate Pareto Optimal solutions for different combinations of efficiency and output power. A sensitivity study of the valve timing and rotational frequency shows the starting angle of the pressure port and the width of the rotor orifice dominate the performance while the system frequency has negligible effects. NOMENCLATURE : Start angle of the pressure port on sleeve (degrees) : Finish angle of the pressure port on sleeve (degrees) : Start angle of the tank port on sleeve (degrees) : Finish angle of the tank port on sleeve (degrees) : Arc angle of the orifice on rotor (degrees) : Leading edge angle of the orifice on rotor (degrees) : Trailing edge angle of the orifice on rotor (degrees) : Total number of discretized points in 18 o L: Length of the sleeve (m) W: Width of the sleeve (m) H: Height of the orifice in axial direction (m) h: Clearance between rotor and sleeve (m) p: System pressure (Pa) p : Tank/atmosphere pressure (Pa) p h : Supplied pressure (Pa) P: Total dissipative power (W) P c: Viscous friction power (W) P p: Leakage power (W) P t : Throttling power (W) P: Pressure port notation T: Tank port notation O: Rotor orifice notation : Bulk modulus at atmospheric pressure (N/m 2 ) : Effective bulk modules (N/m 2 ) V: System volume (m 3 ) V * : Dead volume (m 3 ) V min : Minimization vector q: Total flow rate (m 3 /s) : Intake flow rate (m 3 /s) : Exhaust flow rate (m 3 /s) l: Subdomain length in y axis (m) w: Subdomain width in x axis (m) v: Subdomain linear constant velocity in y axis (m/s) l i: ith dissipative path E loss: Total dissipative energy per cycle (J) E me : Total mechanical energy output per cycle (J) E * me: Theoretical mechanical energy output per cycle (J) E fluid: Total fluidic energy input per cycle (J) : Dynamic viscosity (Pa s) : Discharge coefficient : Instantaneous orifice area (m 2 ) : Heat capacity ratio R: Volumetric percentage of dissolved air in fluid : Objective function i f: Rotational frequency : Geometrical vector 1 Copyright 213 by ASME

2 INTRODUCTION In hydraulic motor applications, the cylinder valves are of great importance. The transition speed, timing, leakage, and friction of the valves impact the efficiency, flow ripple, and reliability of the system. The majority of variable displacement hydraulic piston motors are of the axial piston architecture, in which the piston block rotate relative to a stationary valve plate. For an adjustable linkage pump/motor currently under development [1], the piston chambers remain stationary and thus an alternative valve solution is needed. In this paper, a novel mechanically driven cylindrical rotary motor valve is presented. In this new valve design, the fluid in the motor cylinder flows through a continuously rotating valve rotor to pressure and tank ports, located radially outwards from the rotor. The radial flow design allows precise control of the clearance between the rotor and valve sleeve. Research in piston pump and motor valves generally falls into three categories: system dynamics analysis, geometry investigation, or specific design problems. In formulating the dynamic system pressure, Refs. [2-11] all develop system equations from the classic control volume theory in one way or another. To better grasp the physics inside the clearances, Bergada et al. demonstrated the process of using the 1D Reynolds Equation in deriving the governing equation [2]. Wang introduced the pressure carryover concept and developed a mathematical relationship between the valve geometry and the volumetric efficiency [3]. Wu modeled the cross-sectional area of orifice and studied the flow equation for small openings. In studying the relationship between valve geometry and performance [4], Manring compared different slot shapes and demonstrated that constant area and linearly varying areas have advantages over quadratic ones [5]. Harrison and Edge demonstrated the implementation of heavily damped check valves to reduce the flow ripple of an axial piston pump [6]. Manring and Zhang investigated a trapped-volume design for better volumetric efficiency [7]. Bergada et al. studied the impact of plate tilting, clearance, rotation speed, and timing on system pressure and torque [8]. In the design process, numerical and computer aided engineering approaches have been applied. Through the software program CASPAR, Seeniraj and Ivantysynova designed a valve plate for low noise and better volumetric efficiency[9]. Mandal et al. developed a numerical model considering effect of flow inertia to optimize the valve plate pre-compression and the barrel kidney angle design [1]. These prior works have primarily focused on the influence of valves in pumping applications, and thus there is a need to better understand the impact of valve timing on motor operation. Furthermore, the geometry of the cylindrical rotary valve introduces unique design parameters warranting study. The objective of this research is to find the optimal valve timing and geometric parameters to maximize efficiency and mechanical energy output per cycle. The first part of the paper focuses on developing the system of equations to described the fluid pressure in the cylinder using a lumped parameter bulk modulus formulation and setting up a detailed energy loss model. This model is then used as the objective functions of a multi-objective optimization. The optimization results are then presented and discussed, followed by a valve timing and rotational frequency sensitivity study. VALVE GEOMETRY A simplified schematic of a single cylinder of the hydraulic motor circuit is presented in Fig. 1. During the power stroke, the piston moves from top dead center (TDC) to bottom dead center (BDC) with the pressure valve open. Near BDC the pressure valve closes and the tank valve opens. During the exhaust stroke the tank valve remains open and then closes near TDC. While the operation of the pressure valve and tank valve can be considered separately, they are often integrated, as in the cylindrical rotary valve, and timed to the rotation of the crankshaft. Figure 1. Control volume of the rotary valve. The core components of the cylindrical hydraulic rotary valve are a rotor and a sleeve, as illustrated in Fig. 2 (a). The valve rotor contains two radial orifices, for pressure balancing, while the pressure and tank connections are created through paired ports located on the sleeve. Hydrodynamic lubrication surfaces are created and maintained between rotor and sleeve surfaces through continuous rotation. With reference to Fig. 2 (b), when the rotor rotates in one direction, i.e. clock-wise, the orifice on the rotor will line up with the paired ports on the sleeve periodically and thus creates two alternating pathways between cylinder and either the pressure or tank port. 2 Copyright 213 by ASME

3 = () (2) Furthermore, from the bulk modulus equation, the influence of flow into and out of the system on pressure variation can expressed as Eqn. (3). = () () (3) where q is the sum of flow rates into and out of the cylinder. Note that the negative sign is omitted as positive flow is defined as into the cylinder. Then the two equations are combined to form the governing equation of the pressure change in the cylinder considering both the piston displacement and flow rates into and out of the cylinder. (a) Valve rotor shaft and sleeve assembly = () + () () (4) The bulk modulus formulation above, can be compared to the result of mass-flow continuity or control volume formulation, which is constituted by the equation of state and is commonly used by other researchers [13]: = + (5) (b) Cross section view Figure 2. Hydraulic motor valve geometry. CYLINDER PRESSURE DYNAMICS There are multiple ways to derive the governing equation capturing the pressure dynamics in the motor cylinder. One intuitive and straight forward approach starts directly from the definition of the bulk modulus [12]: = where is the bulk modulus, V is the cylinder volume and p is the pressure. By rearranging Eqn. (1) we can relate the pressure variation and fluid compressibility. And if only time dependent behavior is considered and the system pressure is assumed lumped, the partial derivative operator can be dropped: (1) After two steps of simple mathematical manipulation, the two formulations are of the exact same form, which validates the approach. Next, constitutive equations are needed to define the instantaneous flow rates through the valve as well as the effective bulk modulus. From Bernoulli s equation, the flow rate, valve port area A v, and pressure drop across the valve can be related as: = 2 where C q is the discharge coefficient and ρ is the fluid density [12]. There exist a significant number of bulk modulus models, with reviews provided in [14, 15]. In this work, the effective bulk modulus relationship developed by Cho et al. will be used, which includes the effect of entrained air in the hydraulic oil [16]: ( ) + = +( ) (6) (7) 3 Copyright 213 by ASME

4 where p o is the reference pressure, R is the entrained air volume fraction at the reference pressure, and the ratio of specific heats,, is set to 1.4 for an adiabatic process. ENERGY DISSIPATION MODELS During valve operation, energy is dissipated through viscous friction, leakage, and throttling of the working fluid. To design a rotary valve with good performance and facilitate later optimization work, the details of these losses will now be developed. At an infinitesimal scale, the rotor to sleeve interface can be modeled as two parallel plates with a relative velocity and a pressure difference between the ends driving leakage, as shown in Fig. 3. Figure 3. Parallel plate flow. In this figure, the bottom plate, representing the sleeve, is held fixed while the top plate, representing the rotor, moves at a constant speed,, in the positive y direction. The pressure gradient is along the y axis and is assumed negligible in both the x and z directions. The infinitesimal power loss due to Poiseuille leakage flow can be then expressed as: = = 12 where μ is dynamic viscosity. The infinitesimal power dissipated due to viscous fluid shearing can be expressed through Newton s law of viscosity as: == Note that Couette flow also exists between the plates, but this does not results in an energy loss. By integrating Eqns. (8) and (9) across the solution domain, the instantaneous power loss is expressed as: (8) (9) = = 12 where w is the width of a subdomain tangent to the pressure gradient, which is equal to the port length or width, l is the length of a subdomain normal to the pressure gradient, which is the leakage path and h is the clearance height. Another significant loss term is in the form of fluid throttling into and out of the system. This power loss can be described as: = A final energy loss term is through the compressibility of the fluid. However, since we have already included the effective bulk modulus equation in the pressure dynamics, the resultant power loss is accounted for in the throttling energy loss term. In all, the cumulative power loss at each instant of time is a summation of all forms of energy dissipation described above. The energy loss per cycle, can be described by a time integration of the power summation. The total mechanical energy output, can be defined as a time integration of the product of the pressure difference and the volume time derivative. The total fluid energy input, is defined as the integral of the pressure difference between the pressure and tank ports times the actual flow rate with respect to time: () = + + = = = () ( ) ( ) (11) (12) (13) (14) (15) (16) DISSIPATION PATHS DEFINITION A tedious but important part of analyzing the system energy losses is to develop the correct energy dissipative paths within the cycle. With these paths properly defined, the instantaneous power consumption can be obtained by summing all the power loss through every possible path. Then by integrating the dissipative power throughout each angle as the rotor rotates continuously from to 18 degrees, the total energy loss for each cycle can be calculated. For easier determination of these paths, half the sleeve is unwrapped and discretized into 8 artificial subdomains (from 1 st to NX th point) based on the relative location between the orifice on the rotor and the ports on sleeve, shown in Figure 4. = = (1) 4 Copyright 213 by ASME

5 this case the relative lengths of leakage or viscous friction paths are changing with the valve rotor angle. An example of this situation is shown in Fig. 6. Figure 4. Subdomain(s) definition. To find out which subdomain the rotor orifice belongs to, we define the current subdomain as where the orifice s leading (clockwise most) edge resides. Then we will arrive at three major zones : (1) Neutral Zones Neutral zones (NZ) are the ones in which the rotor orifice is located inside of either the pressure port or the tank port. In this case the relative position between rotor and sleeve does not have impact on the system power loss. An example of this situation is shown in Fig. 5. Figure 6. Transitional zone. For instance, if the rotor orifice is located in TZ1, the length of the circumferential paths can be calculated as follows: = + +1 = +1 And the width of the axial leakage paths are: = +1 = = +1 (3) Incremental Zones Incremental zones (IZ) are the ones in which the rotor orifice is located between the pressure and tank ports. In this case not only the relative lengths of leakage and viscous friction paths change, but also new paths are generated. An example of this situation is shown in Fig. 7. Figure 5. Netural zone. If the rotor orifice is located in NZ1 or NZ2, with reference to Fig. 5, the length of the circumferential paths are: = + +1 = +1 And the widths of the axial paths are: = = +1 = +1 (2) Transitional Zones Transitional zones (TZ) are the ones in which the edge of pressure port or the tank port resides inside the rotor orifice. In Figure 7. Incremental zone. 5 Copyright 213 by ASME

6 For instance, if the rotor orifice is located in IZ1, the length of the circumferential paths are: = +1 = = + 1 And the widths of the axial paths are: = = +1 = +1 = +1 With a fully defined dissipation paths at each given instance of time, the overall system power loss can be expressed in the compact form: () = + + = =5,6,7 =4,5 (17) Where represents the number a particular path, and differentiate the paths between leakage and viscous friction. is the instantaneous viscous friction power at time t, is the instantaneous leakage flow through clearance power at time t, and is the instantaneous throttling through valve orifice power at time t. SOLVE SYSTEM EQUATIONS The cylinder pressure dynamics are nonlinear and time dependent. Due to the complexity, a numerical approach was employed. The equations were solved with a variable time step Runge-Kutta stiff solver (MATLAB ode15s). The system parameters can be found in Table 1. Table1. System constants. Physical meaning Value Unit p h Supplied pressure Pa p Tank pressure Pa Dynamic viscosity.4 Pa s Fluid density 874 kg/m 3 V Cylinder volume at BDC 12 cm 3 V * Dead volume.5v cm 3 R Volumetric air percentage 1% - f System frequency 2 Hz Time step sec Apart from the constants shown in Table 1, there are time dependent parameters as well. As is noticed from Eqn. (4), the volume trajectory and its time derivative are required for each instance of time. Considering a sinusoidal piston motion, these two time dependent parameters are then defined, as presented in Figure 8. Volume/m3 2 x Rotation angle Figure 8. Cylinder volume and volume time derivative at 2 Hz. From the previous inputs and system settings, the system is now fully defined and the system pressure () can be solved for at each time step. After system pressure is obtained, the transient power loss and total energy loss per cycle are found from Eqns. (13) ~ (17). Since the private variables of ode15s in MATLAB are totally isolated from the main function, they cannot be easily retrieved without hindering the calculation efficiency. Instead, the flow rates are found through post process using Eqn. (6) from the solved system pressure. MULTIOBJECTIVE OPTIMIZATION The optimal cylindrical rotary valve geometry was determined through a multi-objective optimization, which is equivalent to finding the Pareto Optimal solution for the following mathematical problem [17]: () = (), (), ().. In this case, the objective functions () = 1,2, are defined as follows: () =1 () =1 = ={,,,,,,,} x volume volume time derivative (18) Where is the difference between unity and system efficiency and is the difference between unity and actual mechanical work output with respect to, the maximum theoretical mechanical energy output of a square wave pressure profile. x is the geometry vector. GENETIC ALGORITHM In order to maximize the efficiency and the mechanical energy output, a solution that satisfies Eqn. (18) is needed. For the multi-objective optimization problem here, one classic approach is to use weighting coefficients to linearize multiple (19) dv/dt/m3/s 6 Copyright 213 by ASME

7 objective functions as a new single scalar function, in which case a simple genetic algorithm can be applied. By defining different weighting coefficients, the solution space can be searched in different directions: () = () + () (2) where () is the new objective function that linearly combines and, with the weighting coefficients and. The advantage of the weighted objective function approach is only requiring one objective function to describe a solution, but carries the disadvantage of not being able to obtain the full Pareto Optimal set, which is often of great importance to engineering evaluation and decision making. Chang et al. presented a genetic algorithm scheme that integrates the subpopulation evolution with weighted objective functions [18]. The two significant benefits of sub-population genetic algorithm (SPGA) are: 1) a Pareto Optimal front is obtained since each differently weighted sub-population can search in a different direction of the solution space and 2) high genetic versatility is be maintained during parallel calculations, preventing premature convergence, where one strong individual dominates the entire population. In this optimization process, the simple GA is first applied to get optimized geometry and then the SPGA is used to get the Pareto Optimal front. A flow chart demonstrating the process using the simple GA with the two objective functions in this problem is shown in Fig. 9. The detailed procedure of this optimization process is described as follows: 1. An initial population is randomly generated between the upper and lower bounds. 2. Each individual of the total population runs through one of the two objective functions and the new weighted objective value is defined from Eqn. (2). 3. Qualified individuals are selected based on their fitness level. 4. Selected individuals are involved in the crossover and mutate processes, the results of which will become the population of the next generation. 5. The generation counter is checked and the process is terminated if the maximum number of generations is met. As for SPGA, the major difference lies in the first step, where the total populations are divided into N equally sized groups and the new objective function, (), of nth subpopulation is then assigned with unique weights based on the following equation suggest by Chang et al. [18]: {, } ={ +1,1 +1 } (21) Both algorithms start from randomized initial populations. A total population size of 12 individuals is used for SGA and 3 individuals are used for the SPGA, divided into 3 subpopulations. For both methods, the maximum number of generation is set to be 3 and the upper (UB) and lower bounds (LB) of the geometry vector x are defined in Table 2. Pres. port start Table 2. Optimization boundaries. Pres. Tank port port Orifice Sleeve Port end start width length length Clearance height ( ) ( ) ( ) ( ) L (m) H (m) h (m) LB UB RESULTS The optimization results shown in Table 3 are obtained from the SGA coupled with equally weighted scalar function, that is = =.5. The efficiency of this solution, defined by Eqn. (19), is 98.4% and the maximum mechanical energy output is 76.2 J per cycle. Pres. port start Table 3. Optimized results from SGA. Pres. Tank port port Orifice Sleeve Port end start width length length Clearance height ( ) ( ) ( ) ( ) L (m) H (m) h (m) The behavior of the optimized geometry is presented in Fig. 1. The negative sign in Fig.1 (c) represents an outward flow from the system. Figure 9. Simple GA flow chart. 7 Copyright 213 by ASME

8 x 1-4 (a) Valve open area (a) Best individual Area/m2 P(t)/Pa q(t)/m 3 /s Power/W Pres. port Tank port Time/s x 1 6 x 1-4 (b) System Pressure Time/s (c) Flow Rate Pres. port Tank port Time/s (d) System power loss Throttling Viscous Leakage Time/s Figure 1. Optimized geometry results. The solution presented above is only one from a large solution space, determined by equally weighting the system efficiency and mechanical output. To find the Pareto Optimal solution sets, the SPGA is used. Figure 11 (a) shows the tracking of best results within each sub-population for the SPGA. After 3 generations, the algorithm is nearly converged to the optimal solution. The fitness of Pareto Optimal sets of 3 th generation for objective functions 1 against 2 are shown in Fig. 11 (b). Obj Sub-population Generation Obj2 Figure 11. Results from SPGA. DISCUSSION The results of the SGA are promising with an efficiency of 98.4%. The pressure and flow behavior of the optimized solution, shown in Fig. 1, match the expected behavior of sharp rises and falls in the cylinder pressure and valve transitions near points of zero flow. This behavior is accomplished with a shorter duration open time of the pressure port compared to the tank port, allowing for compression and decompression of fluid in the cylinder. As observed in Fig. 1 (c), there is a slight flow reversal through the tank port as it is opening. While this flow reversal is undesirable, it occurs when the cylinder pressure is low and thus generates a relatively small throttling power loss. As this work focuses on optimizing the valve timing, a logical question arises about the sensitivity of the system performance to timing variation. To verify this, a timing sensitivity study was carried out. By holding other parameters constant while only changing one port/orifice parameter at a time, the results presented in Table 4 are generated. Variable ( ) (b) Pareto front Table 4. Valve timing sensitivity. Angle Input Energy change energy loss Efficiency ( ) (J) (J) η (%) Optimal Var Var Var Var Var From Table 4, the total loss of energy per cycle and efficiency are most sensitive to variation of the opening angle of the pressure port and the rotor orifice arc angle but 8 Copyright 213 by ASME

9 insensitive to the variation of the ending angle of the pressure port. For example, as seen in this table, a -4 degrees change of the opening angle of the pressure port,, results a 3.4% lower efficiency and 212% higher energy loss per cycle than the optimal timing. Another issue that needs to be addressed is the solution sensitivity to the operation frequency. The optimization was performed at a cycle frequency of 2 Hz, yet operating at other frequencies is often required. Figure 12 shows the variation of results when the operation frequency is shifted away from optimization frequency (2 Hz) to 1 Hz or 3 Hz. The performance metrics for these alternate frequencies are presented in Table 5. These results are quite promising in that the maximum variations of efficiency and output work per cycle are within.7%, which are surprisingly well for a 5% variation of rotational speed. Flow rate/m3/s Power/W Pressure/Pa x 1 6 (a) System pressure Rotation angle 1Hz 2Hz 3Hz x 1-4 (b) Absolute flow rate 1Hz-Pres. port 1 1Hz-Tank port 2Hz-Pres. port 5 2Hz-Tank port 3Hz-Pres. port 3Hz-Tank port Rotation angle (c) Overall power loss 1Hz 2Hz 3Hz Rotation angle Figure 12. Results comparison between optimal frequency and working frequencies. Table 5. Rotation frequency sensitivity. Frequency Efficiency Input energy Net output energy (Hz) η (%) (J) (J) It is also of interest to compare the results of the SGA and SPGA. Displayed in Table 6 are the results of the SGA and one chromosome of the Pareto Optimal solutions from the SPGA sub-population that has = =.5, which is the same as the weighting in the SGA. Table 6. Results comparison between SGA and SPGA. No. of No. of sub-pop. Energy individuals individuals Efficiency loss n/n (%) (J) SGA 12 NA SPGA From Table 6, the fitness values of the SGA are slightly better because of the limitation of 1 individuals in each subpopulation of the SPGA, compared to 12 individuals in the SGA. The small population size in the SPGA is further highlighted by the fact that not every sub-populations can return a feasible result. As shown in Fig. 11 (b), the Pareto front only has 12 points while the number of sub-population is 3. While more individual can be used in each sub-population, this is a trade-off at the expense of computational time. Apart from the drawbacks, the benefit of using SPGA is the availability of Pareto Optimal solution set, from which the system performance can be predicted from the two (or more) conflicting objectives. As shown in Fig.11 (b), the system efficiency goes up when objective 1 goes down and vise-versa. In some problems where the objectives are well understood and user is only interested in finding a feasible solution of particular weightings of objectives but not in Pareto front, SGA is suitable for fast calculation. CONCLUSION In this paper, dynamic system equations are derived for the system pressure within a motor cylinder. System power losses are studied and detailed energy dissipation paths are determined. A simple genetic algorithm is used to obtain an optimal valve timing and geometry. A SPGA is used to obtain the full Pareto Optimal front. Theoretically, more than 97% efficiency is achieved. A sensitivity study of the valve timing shows dramatic effects of the opening angle of the pressure port and the total length of the rotor orifice on the system energy loss and efficiency. A comparison between optimal frequency and working frequencies indicates that despite a 5% variation of operating frequency, the variations of efficiency and output work are within.7%. ACKNOWLEDGMENTS This work is supported by the National Science Foundation under grant number EFRI REFERENCES [1] Wilhelm, S., and Van de Ven, J., 211, Synthesis of a Variable Displacement Linkage for a Hydraulic Transformer, 9 Copyright 213 by ASME

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