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

Presentation Overview

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Alternate approaches to simulating Hydraulic Line Dynamics

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Comparison of Models and Simulation Results using the alternate approaches

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Conclusions and Recommendations

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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.

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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.

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Equivalent Differential Equations Using Method of Characteristics

(1 / c) . dp/dt + 

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dv/dt + f(t) = 0 valid on the characteristic given by: dx / dt = c and - (1 / c) . dp/dt + 

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dv/dt + f(t) = 0 valid on the characteristic given by: dx / dt = -c where c = velocity of sound in fluid = (K /  )

0.5

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Characteristic Lines in the x- t Plane

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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.

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Comparison of Models and Results

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EASY5 Model Using Component PW

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EASY5 Model Using Continuous Line Model Approach

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Details of New Submodel for Line Dynamics

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Data Used for Simulations Parameter Value Length of line upstream of valve Length of line downstream of valve Diameter of line upstream of valve Diameter of line downstream of valve Valve area change command 1420.8 in 1420.8 in 1.0 in 1.0 in 0.785 in 2 at t



0.005 sec to 0 in 2 at t



0.006

sec, varying linearly betweeen 0.005 sec to 0.006 sec 0.0002 sec 5 in 3 Valve time constant (TC VM) Lumped volume upstream of valve (VOLVM) Fluid Resistive Pipe: Length Resistive Pipe : Absolute Roughness Skydrol at 76 deg F, no entrained air, with the following calculated properties: Density = 9.430E-05 lb f -sec 2 / in 4 Bulk modulus = 3.634 E+05 psi Kinematic viscosity = 0.02423 in 2 / sec Speed of sound in fluid = 62080 in/sec 1 in 0.08 in (controls initial flow to 161 lbs/min)

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Pressure Transients Using Component PW

Normalized Pressure Downstream of Valve Normalized Pressure Upstream of Valve Arun K. Trikha 13

Pressure Transients Using Component Time Delays

Normalized Pressure Downstream of Valve Normalized Pressure Upstream of Valve Arun K. Trikha 14

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.

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Conclusions and Recommendations

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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.

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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.

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