High-pressure and flow metering systems often include a positive displacement (PD) pump and a network of piping and process equipment components. Steady-state pulsation, that is periodic or cyclic pressure and flow dynamics, is a common problem in PD pump systems, which can cause high vibration, fatigue failures, frequent maintenance outages, and flow uniformity or product quality problems.
This article describes a methodology to computer model pulsation using a combination of general-purpose and readily available, widely used software.
By James Blanding, Ph.D. and Trey Walters, PE, Applied Flow Technology
Waterhammer and modal analysis have been extensively documented in literature, but these analyses largely fall to the separate disciplines of civil and mechanical engineering respectively. Detailed methodology for pulsation analysis is only now being documented here because this is the first time the “dots are connected” linking these historically unrelated niche specialties.
The following general steps are proposed to computationally model pulsation:
STEP 1. Model the piping and process equipment system.
STEP 2. Determine the natural frequencies and mode shapes of the system.
STEP 3. Identify ‘worst-case’ pump speeds that excite natural frequencies.
STEP 4. Simulate operating pulsation response to PD pump flow forcing.
STEP 5. Modify the system and reanalyze if pulsation is too high.
Among the goals of the remainder of the paper are to answer questions such as, “Why is pulsation so much higher at a lower pump speed?”, and “How does one determine this and gain a thorough understanding of the pulsation behavior in the system?”
STEP 1: Model the Piping and Process Equipment System
The Method of Characteristics (MOC) is one of many approaches found in waterhammer software. In short, the MOC is a ‘lumped parameter’ type modeling technique which means the actual continuous system is discretized into segments. Rather than represent pipes as a whole, each pipe is broken into a series of segments of length, Li, which is generally different for each pipe and vessel. The time-transient simulation is carried out in increments of time, dt, which is fixed for the simulation.
The relationship between these varying length segments, Li, and the fixed time step, dt, is simple but important.
The wavespeed is the speed at which an excitation travels through a medium. In a bulk fluid the wavespeed is equal to the sonic velocity. A common example is the speed of sound through air, but similarly there is a speed of sound through liquids and solids. This wavespeed will vary with a fluid’s density and ‘stiffness’ represented in fluids as the Bulk Modulus, B. In a bulk fluid, a lower density and higher ‘stiffness’ results in a faster wavespeed.
The Bulk Modulus represents the change in a fluid’s volume for any change in pressure, analogous to Young’s Modulus, E, for solids. The less volume change experienced by a fluid for a given pressure change, the ’stiffer’ the fluid and the larger the Bulk Modulus. A smaller Bulk Modulus represents a larger change in volume, representing a more ‘compressible fluid’.
Flow in a pipeline further complicates wavespeed, which is impacted by pipe geometry (D for diameter and e for pipe thickness), the pipe wall’s flexibility (E for the pipe’s modulus of elasticity), and piping support conditions (c1). Generally, wavespeed in pipes will be slower than in a bulk fluid as some energy is lost distorting the pipe.
A suitable segment length and time step is often suggested by the waterhammer program depending on the system’s pipe specifications and fluid properties. It is unlikely that a time step and length segment will perfectly section every pipe in a system, so adjustments are made to each pipe’s length to make the physical system conform to the mathematical grid in the MOC. Similar adjustments could be instead made to wavespeed, but the error introduced to maintain the integrity of the MOC grid is the same. The time step and section length can decrease to reduce this introduced error below an acceptable criterion.
While wavespeed and MOC is general to waterhammer type analysis, pulsation analysis must determine the range of frequency to consider which impacts time step selection. There are three criteria to consider:
1: The ‘base pulsation frequency’ – fb = nN/60
a. This is the pump speed, N, times the number of heads, n, assuming all cylinders are identical and equally spaced around a crank revolution.
2: The expected maximum frequency of interest – fc ≈ 6fb = nN/10
a. One rule of thumb is fc is about six times the base pulsation frequency. Though this rule of thumb is quite arbitrary, the author has found it generally conservative.
b.The time step dt must be less than ½ of 1/fc. This time step is denoted dta
dta = 1/(2fc)
3: The final criterion is the actual time step selected must reduce modeling error to acceptable levels. Again, generally this is usually determined by the waterhammer software.
STEP 2: Determine the Acoustic Natural Frequencies and Mode Shapes of the System
The worst-case condition in pulsation occurs when excitation frequencies coincide with system natural frequencies. There are various ways to determine the natural frequencies of a system and to quantify their relative severity or importance. In mechanical vibration, a common technique in Experimental Modal Analysis (EMA) is to impact a structure with a hammer, instrumented to measure force. The hammer inputs a wideband energy. Depending on where the impact is made, the structure then vibrates at all or many of its natural frequencies across some range of interest. An analogous analytical method is proposed here called Computational Modal Analysis (CMA).
The ‘computational hammer’ is a sudden, brief flow strike applied at the pump. This strike allows the user to determine the important natural frequencies of the system. Modal software allows the user to filter the high frequencies above fc and use a Frequency Response Function (FRF) to generate Bode plots that indicate acoustic natural frequencies. The amplitude of the Bode plot at various frequencies indicate the severity of expected pulsation, which can be used to identify pump speeds in the next step. Figure 1 provides an example overlaying this analysis at multiple locations, identifying 33.2 Hz as a concerning frequency.
STEP 3: Identify ‘Worst-Case’ Pump Speeds that Excite Natural Frequencies
For the most severe frequencies, the various pump speeds for which integer multiples align with the natural frequencies are calculated as follows:
where the natural frequencies, fni, identified by the Bode plot, and the run-speed multiples, j, determine the pump speed.
In this example, one can evaluate the speed which excite these frequencies at various harmonic multiples, in this case 221.6 RPM. Figure 2 shows other speeds that could excite this frequency as well, or speeds for a less concerning frequency.
Depending on the characteristics of the PD pump (stroke, cross-sectional area, number of heads), the flow forcing behavior can be determined for the problematic speed. An example flow forcing plot is shown in Figure 3.
STEP 4: Simulate Operating Pulsation Response to PD Pump Flow Forcing
The flow forcing behavior can be specified in the waterhammer model to determine the specific pressure response at the problematic speed. Figure 4 compares the pressure response at identified speed (221 RPM) and a higher speed (266 RPM). This figure demonstrates that pulsation at frequencies closer to the system’s acoustic natural frequencies, as is the case with 221 RPM, result in a much higher pulsation response.
Included on Figure 4 is the same pressure plot at a different location in the system. The lower amplitude indicates pulsation is less severe at this second point, consistent with the anti-node/node of sinusoidal pulsation.
STEP 5: Modify the System and Reanalyze if Pulsation is Too High
The final step is to consider various design modifications if pulsation is too high for the ‘worst-case’ pump speeds. Changes to the system will adjust its natural frequency, ideally avoiding excitation at the necessary operating speeds. A common approach is to add an orifice or dampener at a strategic location. After making the change, Steps 1-4 should be repeated to identify any new concerning frequencies and corresponding pump speeds.
The content presented here is a truncated summary of the technical paper “Pulsation Analysis In Positive Displacement Pump Systems Using Waterhammer, Modal And Animation Software.”
ABOUT THE AUTHORS
James M. Blanding is (Retired) Principal Consultant for the DuPont Company. He received B.S. (1976), M.S. (1977) and Ph.D. (1985) degrees (Mechanical Engineering) from Virginia Polytechnic Institute, where he also served on the faculty teaching Mechanical Vibration and other undergraduate courses. Dr. Blanding’s expertise is in vibration and diagnostic testing using modal analysis, high-speed data acquisition and experimental stress analysis. He specializes also in computer modeling and simulation of process fluid system pulsation and transients, as well as reciprocating compressor and pump check valve dynamics and performance.
Trey Walters, P.E., is the founder and President of Applied Flow Technology Corporation in Colorado Springs, Colorado, USA. AFT develops simulation software for fluid transfer systems. At AFT Mr. Walters has developed software in the areas of incompressible and compressible pipe flow, waterhammer, slurry systems, and pump system optimization. He is responsible for performing and managing thermal/fluid system consulting projects for numerous industrial applications including power, oil and gas, chemicals and mining. He actively teaches customer training seminars around the world. Mr. Walters founded AFT in 1993. Mr. Walters holds a BSME (1985) and MSME (1986), both from the University of California, Santa Barbara.