Abstract

Introduction

Historically water hammer was first studied for systems with instantly closing or opening valves. The first person to describe this effect was the Russian scientist Joukowski, who is responsible for the formula for pressure rise, which bears his name. The theory of water hammer was further expanded by the works of Allievi and Gibson, who are both responsible for the calculating the pressure response due to linear stroking valves2. However most of these equations are written for incompressible flow and do not apply to compressible flow, especially sonic flow. To date, a large amount of work has been performed on the subject, and there are many commercially available water hammer software packages that will solve almost any problem1.

The purpose of this paper is to show the importance of the need to combine the line dynamics model with the solenoid valve model, since the time history of the valve poppet directly controls the resulting pressure rise. In doing so, we will present a simplified approach to combining the upstream line with the valve. The approach presented will show how to reduce the feedline from a Partial Differential Equation (PDE) to an Ordinary Differential Equation (ODE)4,5. This is done by reducing the feedline to a lumped one-dimensional model. It will be shown that by the proper selection of parameters for the lumped model, this simple model can then provide accurate results, which compare well to the commonly used water hammer calculation method, “The Method of Characteristics” (MOC). This paper will also present parametric studies showing the effects of different flow control orifices.

The modeling steps for the valve will also be discussed. The valve model is important since it is the valve that triggers the transient event. Many attempts to model water hammer have assumed a linear closing or opening time for the valve. This is usually not the case since the main poppet in a pilot operated valve is moved by differential pressure. As the valve is closing, the differential pressure across the poppet increases, causing the valve to usually start out moving slow, but then slam shut. It is important to have the time history of the main poppet for this reason.

Water hammer pressure rise is determined by valve closing time. The Joukowski formula, equation (1), can be used for fast closing valves. A fast closing valve closes in t<2*L/c. This means the valve is fully closed by the time the pressure wave returns to the valve. The pressure rise for a slowly closing valve can be found from the rigid column theory, equation (2). A slow closing valve closes in t>5(2*L/c)6. The need for numerical simulation arises when the simplified formulas for fast or slow closing valves no longer apply. For a typical launch vehicle, with a 30 ft helium feedline, the time for a wave to travel from the valve and return in helium is .018 seconds. The solenoid valve is pilot operated by a 3 way valve. The main poppet closing time is typically .025 seconds. This closing time cannot be considered fast or slow, so a numerical solution is required, which will be the main topic of discussion of this paper.

Modeling Approach

The dynamic model was created with the following goals:

• Determine the effects of upstream pressure spikes with different flow orifices
• Determine the effects of the different flow orifices on poppet closing times
• Compare the results from a simple 1 DOF Simulink model of the line dynamics with the MOC

The system to be modelled is shown in Figure 1. The following is a list of the modelling assumptions:

• Flow is adiabatic
• Small pressure variations (density is constant)
• 1D axisymmetric flow
• Wave speed is constant
• Mach Number <<1
• Constant pipe diameter
• Pipe is rigid
• Pipe is straight, no bends or tees
• Friction modeled using Darcy-Weisbach

The upstream line was modelled as a lumped parameter fluid element using Simulink. This lumped model uses one resistance, inertance, and capacitance element to model the line. The schematic of this single fluid element is shown in Figure 2. The math model of this fluid element is shown in Figure 3. The derivations of the constants shown in Figure 3 are listed in equations (3a), (3b), and (3c). This fluid element presented is a suitable model for linking to the solenoid valve, since the fixed tank pressure can be applied at Pa, and the flow demand from the valve can be applied at Qb. The flow demand at Qb is calculated after Pb is used to determine the flow through the valve6. This arrangement also allows fluid elements to be linked in series if more DOF are desired. The sequence of the components is arbitrary and can be changed if the line terminations are changed.

The use of a simplified feedline allows replacing a system having an infinite number of degrees of freedom with a system having only one degree of freedom. The lumped model has the same surge impedance as the MOC model, as is shown in equations (3d) thru (3i). This means that the lumped model should accurately predict sudden valve closing events7. The natural frequency of the lumped model is 17 Hz, compared to 27 Hz for the MOC model. We will artificially increase the lumped model natural frequency to match the MOC frequency of 27 Hz, by using a factor of ?/2. This can be accomplished by reducing the pipe length by 2/?. This will help predict more accurate results for the slower valve closing times. It should be noted that changing the pipe length will not reduce the surge impedance. This can be realized by inspection of the equations for surge impedance, inertance, and capacitance.

An MOC program was created in Mathcad. The program was based on the example in reference [8] and [9]. This program was modified for compressible flow. It was modified to include the change in friction factor f with Reynolds Number. This model was also modified to calculate the sonic and subsonic compressible flow through the valve end of the line. These same capabilities were also included in the lumped Simulink line model, so that a fair comparison could be made between the two models. The MOC Model is based on the wave equations (4a) and (4b).

The solenoid valve is a 2 way pilot operated valve. The pilot valve used is a 3 way, 2 position direct acting solenoid valve. The main poppet is held closed by the poppet return spring and the pressure P2. When the pilot valve is de-energized P2 is the same as P1. When the pilot valve is energized P2 is vented down to ambient pressure, which then allows the inlet pressure to open the main poppet. The Simulink model of the valve is more involved than the line model. The valve model uses the compressible flow equation for the two fixed orifices, the pilot valve and the flow control orifice. The main poppet creates a variable area orifice as its lifts off the main seat. Simulink models the thermodynamics of the volume behind the poppet (P2), and the volume between the seat and the outlet orifice (P3). The effects of seal friction are included in the model. This dynamic model calculates the motion history of the valve poppet by integrating the force balance equation for the poppet. All three pressures P1, P2, and P3 are included in the poppet force balance, as is shown in equation (5). A summary of the parameters used in the model are listed in Table 2.

Simulation Results

Six simulations were run. Three were run with Simulink and three were run with the MOC program. The MOC program was written in Mathcad. All of the runs are shown in Figure 5 through Figure 10. The Simulink models were run using a Runge-Kutta 4th Order Model with a time step of .000001 second. This small time step was required due to the small time constant for the 2.0 in3 volume between the main seat and the flow control orifice. This is the volume for P2. The MOC simulations were run with 10 reaches and a time step of .001 second. For each run the flow was started at zero and the valve was opened to its largest CdA. The flow was allowed to reach steady state, and after all the opening transients had disappeared, the valve closing sequence started at time = 2.5 seconds. All of the simulations were run with the highest inlet pressure of 4,500 psi, which is worst case for pressure transients. The results are shown in Table 3.

The closing time from the Simulink runs was used as the closing time for the corresponding MOC simulation runs. The shape of the closing profile from the Simulink model was taken from the valve flow profile. This is required since the poppet motion is not controlling the flow until it closes to a smaller equivalent orifice than the downstream flow control orifice. Due to the flow being controlled by two orifices in series, this is necessary. The shape of the closing profile from the Simulink model was duplicated by the use of a closing function that was then used in the MOC model10. The closing function is shown in equation (6). This allows for the use of other than linear closing profiles, such as concave or convex closing curves. Each shape curve is controlled by one exponent, m, in the curve equation. These curves are shown in Figure 4. The error listed in the last column is calculated for the Simulink results. The error was calculated by comparing this result to the MOC results, and assuming that the MOC result is correct since it has 10 DOF in its model and should be more accurate.

The max closing pressure from the simulations is compared to the sudden closing pressure rise and the slow closing pressure rise as shown in Table 3. All of the calculated pressures are between the limits for fast closing and slow closing, as expected. It can be seen from the results that the closing time decreased with increasing flow control orifice size. This is attributed to the fact that piston pressure, P2, has to be higher to match P3, since P3 is higher when the flow orifice is smaller. The pressure spikes increased with orifice size due to the increase in flow rate. On average, the pressure rise was 74% of the fast closing Joukowski pressure. The time plots for P1 show that both the MOC and Simulink models have the same natural frequency. The Simulink model shows good correlation with the MOC model, with only slight overestimation of P1 in all cases. The largest deviation between the two models was only 6.7%.

Conclusions

This simulation provides an ideal tool for evaluating pneumatic water hammer effects in propellant tank pressurization lines. The results from the simple Simulink line model matched the MOC model. The largest error was 6.7%. Thus, a continuous system can be reduced to a much simpler model with accurate results. This holds true as long as the frequencies involved are less than the natural frequency of the 1 DOF line model. Due to the versatility of Simulink, many components today are being modeled using Simulink. It is of great benefit to have a Simulink line dynamics model, which can easily be integrated into any other Simulink component model. The analysis presented is useful for determining the detrimental effects of pneumatic water hammer in both the pressurization line and the flow control valve.

The following conclusions are a result of this investigation:

• 1) The 1 DOF line model will closely match the MOC results if they have the same surge impedance and natural frequency. This has been shown by the results of this investigation, as listed in Table 3.
• 2) The pressure rise decreased with decreasing flow control orifice size, as expected due to reduced flow.
• 3) The closing time decreased with increasing orifice size.
• 4) Since the acoustic waves travel slower in gas, the time for a valve to be considered as “slow closing” is longer, as compared to a liquid. However the spikes are usually higher with a liquid due to the much higher density.
• 5) The elasticity of the pipe is not as significant when using a gas due to the smaller pressure spikes1.
• 6) The poppet motion is not a good indication of the equivalent CdA when the solenoid valve has a flow control orifice in its outlet, resulting in two orifices in series.

References

• 1) Nguyen, H. V., “Pressure Surge Analysis In The Feedlines Of An X-Vehicle,” JPC AIAA 97-3303.
• 2) Gibson, A. H., “Water Hammer in Hydraulic Pipelines,” Archibald Constable, London, UK, 1908.
• 3) Ghidaoui, Mohamed S., “A Review of Water Hammer Theory and Practice,” Applied Mechanics Reviews, January 2005, Vol. 58, pp 49-76.
• 4) Blackburn, J. F., “Fluid Power Control,” pp 134-137.
• 5) Keller, G. R., “Hydraulic System Analysis,” pp 65-73.
• 6) Karney, B. W., Ruus, E., “Charts for Water Hammer in Pipelines Resulting from Valve Closure from Full Opening Only,” Canadian Journal of Engineering, no. 12, pp 241-264, 1985.
• 7) Paynter, H. M., “Fluid Transients in Engineering Systems,” Handbook of Fluid Dynamics, pp 20-1 to 20-26.
• 8) Wylie, E. B. and Streeter, V.L., “Applications of Fluid Mechanics,” pp 500-514.
• 9) Kohlenberg, M. W. and Wood, R. K., “Pressure and Flow Transient Response Prediction For Power Plant Piping Networks,” SIMULATION, 63:4, pp 235-238, October 1994.
• 10) Provenza, P. G., Baroni, F. and Aguerre, R. J., “The Closing Function in The Water Hammer Modelling,” Latin America Applied Research, vol. 41, no. 1, pp 43-47, 2011.