1. Improved Simulation of Hydraulic System Pressure Transients Using EASY5 Dr. Arun K. Trikha Associate Technical Fellow The Boeing Company (206) 655-0826 Presented at the 2000 EASY5 User Conference May 17, 2000

2. Arun K. Trikha 2 Presentation Overview Alternate approaches to simulating Hydraulic Line Dynamics Comparison of Models and Simulation Results using the alternate approaches Conclusions and Recommendations

3. Arun K. Trikha 3 Alternate Approaches to Simulating Hydraulic System Line Dynamics Approach 1 (Lumped Line Model Approach): Divide a line into many sections, each of which can be assumed to have a uniform pressure within it. Use continuity equation to calculate rate of change of pressure within each section Use momentum equation to calculate the rate of change of flow from one section to the next section. This approach results in solution of ordinary differential equations and is the approach used in EASY5 Hydraulic Library components PW and PX. Approach 2 (Continuous Line Model Approach) Work directly with the continuous line model which represents the continuity and the momentum equations as partial differential equations. Use Method of Characteristics for solving partial differential equations The implementation of this inherently more accurate approach by using standard EASY5 components is discussed in this presentation.

4. Arun K. Trikha 4 One-Dimensional Model of Hydraulic Line Dynamics The Continuity Equation is: (1/K).  p /  t +  v /  x = 0 and the Momentum equation is:  p /  x + .  v /  t + f(t) = 0 where: x = coordinate in axial direction of the line t = time p = pressure v = fluid velocity f(t) = pressure drop per unit length (including frequency-dependent friction effects)  = fluid density K = bulk modulus of fluid With proper selection of f(t), the above equations are equivalent to linearized two-dimensional Navier-Stokes equations.

5. Arun K. Trikha 5 Equivalent Differential Equations Using Method of Characteristics (1 / c). dp/dt + . dv/dt + f(t) = 0 valid on the characteristic given by: dx / dt = c and - (1 / c). dp/dt + . dv/dt + f(t) = 0 valid on the characteristic given by: dx / dt = -c where c = velocity of sound in fluid = (K /  ) 0.5

6. Arun K. Trikha 6 Characteristic Lines in the x- t Plane

7. Arun K. Trikha 7 First Order Finite Difference Approximations to Differential Equations along Characteristic Lines (1 / c).(p N - p R ) + . (v N - v R ) + 0.5 (f N + f R ).  t = 0. x N - x R = c (t N - t R ) - (1 / c).(p N - p S ) + . (v N - v S ) + 0.5 (f N + f S ).  t = 0. x S - x N = c (t N - t S ) Note that if point N is at the current time, points R and S are at time  t in the past. The continuous time delay component CD (in EASY5) can be used to keep track of the variable values in the past.

8. Arun K. Trikha 8 Comparison of Models and Results

9. Arun K. Trikha 9 EASY5 Model Using Component PW

10. Arun K. Trikha 10 EASY5 Model Using Continuous Line Model Approach

11. Arun K. Trikha 11 Details of New Submodel for Line Dynamics

12. Arun K. Trikha 12 Data Used for Simulations

13. Arun K. Trikha 13 Pressure Transients Using Component PW Normalized Pressure Downstream of Valve Normalized Pressure Upstream of Valve

14. Arun K. Trikha 14 Pressure Transients Using Component Time Delays Normalized Pressure Downstream of Valve Normalized Pressure Upstream of Valve

15. Arun K. Trikha 15 Comparison of Results When using component PW, there are significant high frequency pressure ripples superimposed on the primary pressure transients. The frequencies of these extraneous pressure ripples are proportional to the no. of pipe sections and their amplitudes are inversely proportional to the same. With the continuous line model approach using time delays, there are no significant high frequency pressure ripples superimposed on the primary pressure transients. The no. of sections affects only the accuracy of the pressure drop. The calculated pressure wave amplitude and period are significantly closer to the closed form solution when using the time delay approach. For the simulated system, the computation time using the time delays approach was only 10 percent of that required when using component PW.

16. Arun K. Trikha 16 Conclusions and Recommendations Working directly with the continuous line model for hydraulic line dynamics, by using appropriate time delays, provides significantly better results than the lumped line model implemented in component PW. It is recommended that the hydraulic line submodel presented here be packaged as a new EASY5 component for ease of use. Note: This recommendation is being implemented.