Mathematical modeling of slamming in a single-piston pump inside the Ocean Grazer

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Bachelor Integration Project

Final report

Mathematical modeling of slamming in a single-piston pump inside the Ocean

Grazer

Author:

Rens Folkertsma

Supervisor:

Dr. Antonis Vakis

Groningen, June 2018

Industrial Engineering and Management Faculty of Science and Engineering

University of Groningen

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Abstract

This research is focused on the Ocean Grazer, which is a novel ocean energy collection and storage device, designed to extract and store multiple forms of ocean energy. The core technology of the Ocean Grazer is a multi-piston power take-off wave energy converter (MP²PTO WEC), which consists of several multi-piston pump units. Single-piston pumps constitute the basic building block for these multi-piston pumps and, eventually, the MP²PTO WEC.

The Single Piston Pump (SPP) is analyzed in order to better understand the slamming phenomenon which occurs. This is a pressure surge or wave caused when water in motion is forced to stop or change direction suddenly. In the SPP this occurs when a valve closes suddenly and oscillations propagate in the fluid column.

A general equation which describes the slamming force is derived. This allows one to relate the slamming phenomenon to the system’s efficiency. A high efficiency, obtained by closing the valve directly, results in a large slamming amplitude. On the other hand, the slamming effect can be minimized when the valve is closed smoothly, however this results is a low efficiency. It is key to find the right balance between the slamming occurring in the piston and the desired system’s efficiency. This balance can be tuned by adjusting one variable in the general slamming force equation and consequently the effect on the slamming amplitude can be observed.

Finally, a mathematical model is designed to simulate the slamming which occurs in the SPP at the Ocean Grazer. The SPP is classified as a spring-damper system; therefore, slamming can be simulated by applying a step force to a harmonic oscillator. The mathematical solution of this equation is implemented in the mathematical model. Furthermore, the mass, damping and stiffness of the system are determined for different situations and also implemented in the model. The results show that the slamming force pattern generated by the mathematical model matches the experimental measurements with sufficient degree of agreement.

Keywords—Ocean Grazer, Slamming, Single Piston Pump, Simulation model

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Introduction

Energy, and more specifically renewable energy, is a hot topic nowadays. Renewable energy sources are being used more and more often, but several disadvantages come along with this. Energy that for example comes from wind is, as mentioned, strongly dependent on the weather. Therefore, reliability is an issue. Furthermore, it is hard to produce renewable energy in the quantities that are available by fossil fuel resources. It can thus be said that energy generation lacks a method that is not dependent on fossil fuels, independent of the weather and which can be applied on large scale. A solution to this problem is the Ocean Grazer: a hybrid, modular and scalable renewable energy capture and storage device.

This research is focused on the Ocean Grazer. The Ocean Grazer is a novel ocean energy collection and storage device, designed to extract and store multiple forms of ocean energy.

The Ocean Grazer differs from other wave energy devices by its adaptability that allows for an efficient energy extraction of heights of various waves and by its ability to provide a predictable and stable energy output, on demand. The device is currently being developed by the University of Groningen.

The Ocean Grazer combines wave energy converter technology with on-site energy storage and wind turbines to harvest renewable energy off shore. Each hybrid device delivers significantly higher and continuous power output compared to separate wave or wind technology solutions. Lossless storage systems smoothen the intermittent power output of wind and wave energy converters, offering energy buffering capabilities to satisfy energy market demand and maximize revenue. The integration of technologies and infrastructure leads to a cost-efficient solution for harvesting multi-source offshore renewable energy [Ocean Grazer, 2018].

The Ocean Grazer is a multi-piston power take-off wave energy converter (MP²PTO WEC). The system, shown in figure 1.1, aims to create a pressure difference in the working fluid which circulates between the two reservoirs. This pressure difference is defined as the hydraulic head (H). When there is an energy need, this pressure difference can be transformed into electricity via a turbine (T). The buoys (B) will follow the motion of an incoming wave and as a result the hydraulic pumps (P) start to move the working fluid column during the upstroke. In this manner, the working fluid can be pumped to the upper reservoir. The working fluid is stored as potential energy which can be transformed into electricity [Vakis and Anagnostopoulos, 2016].

The MP²PTO consists of several single-piston pump units. These single-piston pumps (SPP) will constitute the basic building block for the multi-piston and, eventually, the

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Figure 1.1: The MP²PTO WEC (a) and multi-piston pump (b) [Vakis and Anagnostopoulos, 2016]

A description of the single-piston pump is given in figure 1.2. This figure shows all factors of importance, namely the mass of the buoy (mb) and the mass of the piston (mp), the spring constant K and the damping coefficient C. This indicates that the SPP can be seen as a spring-damper system.

Figure 1.2: Dynamical model of the single-piston pump [Vakis and Anagnostopoulos, 2016]

The SSP allows the water to move through the fluid column in two di- rections. The top piston surface com- prises two flaps that can open and close passively with the flow. Check valves control the flow from the cylinder to each reservoir. During the upstroke, the piston flaps close and the check valves open, allowing working fluid to enter the cylinder from the lower reservoir and discharge into the upper. For the downstroke, the opposite procedure holds, allowing the piston to sink into the stationary fluid column.

This research will analyze the SPP in order to better understand the dy- namic behavior of the total system.

Overall, the use of multiple pump units is designed to extract almost all of the energy available in an incoming wave and can transfer this energy to the stored working fluid with a high efficiency.

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There are, however, some limitations. One of them is the fact that it is observed that slamming may occur in the SPP. The slamming phenomenon is a pressure surge or wave caused when water in motion is forced to stop or change direction suddenly, a momentum change. This is also called a water hammer, which commonly occurs when a valve closes suddenly and a pressure wave (oscillations) propagates in the fluid column. These oscillations can cause major problems and will reduce the durability of the valves.

In the case of the SPP, the direction of the water flow will change at the end of each up- or down stroke. It has been observed that slamming may occur when the piston switches between the up- and down strokes. This is indicated in figure 1.3, which shows the force pattern during an up- and down stroke.

Figure 1.3: Force pattern from simulation model [Zaharia, 2018]

In boxes 1 and 3 oscillations can be seen. These oscillations indicate the slamming phenomenon. The oscillations found in box 1 have a lower frequency than the frequency seen in box 3, while the amplitude of the oscillations are larger. Furthermore it can be noticed that the frequency of the oscillations decreases during time. This indicates an under-damped case, or ζ < 1.

To be able to learn from this phenomena, a mathematical model is required. By means of a mathematical model the effects and causes of slamming can be identified and analyzed.

Furthermore it enables one to simulate the slamming which occurs. In the end, this makes it possible to reduce the effect of slamming in the single-piston pump and as a result increase the performance of the MP²PTO and thus the total system. Therefore, this research aims to develop a simple mathematical model of slamming occurring in a single piston pump.

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Chapter 2

Problem Determination

2.1 Problem definition

The problem can be defined as follows: at the Ocean Grazer there is no technology to simulate slamming (water hammer pulses), which occur in their piston pumps. In previous experiments, it has been observed that slamming may occur when the piston switches between the up- and down strokes. Namely, the acting forces on the system, measured by force sensors, showed fluctuations in their behaviour, which indicates slamming. By a mathematical model the causes of slamming may be identified. Therefore, a mathematical model to simulate the slamming force pattern has to be designed.

2.2 System description

After having analyzed the problem context, it is possible to define the system, subsystems and elements that will be considered in this research. This research focuses on the single piston pump, used at the Ocean Grazer. Despite the fact that the piston pump is just a small part of the Ocean Grazer, it is taken as the system for this research. It is assumed that the behaviour of the SPP does not change when the pump is integrated into the total, multi-piston system. The SPP is seen as a column filled with a fluid, water in the case of the Ocean Grazer. The piston acts as a valve, which can regulate the passage of water.

This valve is passive and fixed inside the piston. The system’s input acts on this valve and determines the output. The input is an oscillating motion. As can be seen in figure 1.2, the piston follows the movement of the buoy (blue line). This displacement causes movement of the water inside the piston. This oscillating motion is of specific interest for this research since it causes slamming. Therefore, the oscillating motion is defined as the input of the system. At the end of each stroke, slamming appears, as is explained in the previous section. This slamming is defined as the output of the system, as a result of the oscillating motion acting on the valve.

Figure 2.1: System description

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2.3 Stakeholder analysis

The problem owner in the project is the supervisor. He is asking for a mathematical model of water hammer pulses. As the leader of the Ocean Grazer Research Team, this project contributes to his research, which declares his interest.

The Ocean Grazer is still under development, so much details are not known yet. The Ocean Grazer project is initiated and run from within the research institute ENTEG of the Faculty of Science and Engineering, University of Groningen. It is not commercially available yet, therefore, no external stakeholders are involved.

It can however be argued that there are some internal stakeholders. For instance, the Ocean Grazer project leader is a stakeholder. This person is in the end responsible for the deliverable, and also for the decisions made in the process. This stakeholder therefore has much power. Besides this, the project leader has also much interest in the process, because the process of designing the system should be efficiently and smoothly. In the end it is of course the goal to make the system commercially available. This is the main objective of the project leader. However, the project leader should take into account the broader picture and, therefore, cannot be concerned with all details. That is where the supervisor of the project appears.

Figure 2.2: Stakeholder mapping For this project, which is just a

small piece of the total Ocean Grazer system, the supervisor has more power and interest compared to the project leader. The supervisor is able to set out the direction of this particular research, the project leader definitely is in- terested in the project but follows the developments from a larger distance.

The stakeholders and the problem owner in this research do not con- flict with each other; in the end all have the same goal and, therefore, agree on the stated problem. After this analysis, it is known how important the identified stakeholders are, and how to cope with each of them. In figure 2.2 the stakeholders are placed in the different categories.

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2.4 Goal and scope

The goal of this research is to develop a mathematical model which generates the slamming force pattern which occurs in the single piston pump inside the Ocean Grazer. Slamming can be simulated by applying a step force to a harmonic oscillator. The single piston pump used inside the Ocean Grazer is defined as a harmonic oscillator. The motion of a harmonic oscillator, or in our case the motion of the single piston pump, can be described by an already existing formula, as is presented in equation 2.1. Several important characteristics of a single piston pump can be determined from this equation, such as the damping coefficient (C) and the stiffness coefficient (K) of the system, but also the mass (m) participating in the slamming.

F(t) = md²x

dt² + Cdx

dx+ Kx (2.1)

The scope of the research is to determine the mass and the damping and stiffness coefficients. The calculated coefficients can be validated by comparing the model to the experimentally obtained values. If all variables in equation 2.1 are determined, one is able to generate the pump’s characteristics and eventually to develop the mathematical model.

2.5 Research questions

Now that the goal and scope are clear, and the stakeholders are identified, it is possible to derive the research question. For this research there will be one main research question and several sub questions. The main research question relates to the goal of the research.

Considering the defined goal the following main research question emerges:

How to simulate the slamming phenomenon in a single-piston pump using a mathematical model?

Sub questions:

1. How to determine the mass difference per situation and how does it influences the frequency?

2. How to determine the factors which account for the stiffness of the system?

3. How to determine the factors which account for the damping of the system?

4. How does changing parameters influence the slamming pattern?

5. How to optimize the model’s result and to match these with experiments?

6. How to interpret the derived results?

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2.6 Risk analysis

Every project contains an element of risk. Timescales; project management; resource availability; technology; the research environment - any of these may be subject to an unplanned occurrence and therefore to risk [Grant et al., 2013].

In the initial stage, several uncertain factors, called elements of risk, were indicated. For instance, there were uncertainties relating the resource availability. Since the Ocean Grazer is still under development, much information is not publicly available. Although, as a researcher on the Ocean Grazer it is possible to get access to relevant reports. This was, however, still a small amount of information and there were no similar researches conducted, so the research availability was considered as an element of risk.

To avoid time related risks, it was considered as important to follow a strict planning.

Therefore, in case of delays or other unexpected events, the planning was adjusted directly.

It was a hard demand that the final deliverable was handed in at the deadline. If the project was not finished before this deadline, there were serious aftereffects. Consequently, this was also considered as an uncertain factor.

A table of risks, as presented in Table 2.1, is a useful tool. In this, each risk is outlined and a score is given for its likelihood and severity. By multiplying the two scores you can gauge how important each potential risk is to your project and therefore plan appropriate mitigating action [Grant et al., 2013].

Table 2.1: Table of risks

Risk Probability (P) Severity (S) Risk Score

1 = low 5 = high ( P x S )

Project not finished on time 2 4 8

Lack of resources 3 3 9

To avoid time-related risks, a flexible planning was designed with some hard deadlines. In this manner the impact of unexpected setbacks was minimized. Furthermore, during the research, the planning was compared to the planning and progress of fellow students, as a check. This ensured that the project was finished on time without stressful moments.

It should be noted that, regarding the resource availability, several experts are working on the Ocean Grazer. During the research, this was absolutely helpful. For instance, the authors of previous theses, were eager to explain uncertainties and to discuss obtained results. In this manner, the required information could be gained.

In conclusion, the potential elements of risk, indicated in the beginning phase of the research, were both tackled during the research, such that it did not affect the progress of the research.

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Chapter 3

Research Design

3.1 Choice of cycle

The most prominent cycle in this case is the design cycle. The output of this cycle is a construct that can afterwards be applied to fulfill a goal. An approach and methods are applied to come to an outline model, which is validated in the end. The construct that is desired here is a model to simulate slamming. The design cycle gives a clear procedure on how to construct the deliverable. This first design has to be evaluated thoroughly such that it fulfills the wishes of all stakeholders. When the outcome of the cycle is validated, the final deliverable can be constructed.

Figure 3.1: Design vs rigor cycle

In addition, the rigor cycle is also of importance for this design. It must be clarified that the rigor cycle will not be applied directly. The output of this cycle is knowledge, and that does not comply with the goal in this design case. However, output of this cycle may be used in order to fully understand the phenomenon.

For the construction of the model, knowledge is needed. This can be taken from the output of the rigor cycle. In this problem, the application of the rigor cycle can be seen as a cycle within a cycle; the gained knowledge is used as a tool to design the deliverable. In fact, the rigor cycle can be seen as input in the design cycle in order to solve the problem optimally. Furthermore, the rigor cycle has an additional function. It is also applicable to gain validation and knowledge from field experts. By means of expert meetings, provisional designs can be critically assessed and evaluated. This is a strong validation tool, because a step beyond the design phase is considered to improve the design.

To conclude, it can be stated that the rigor cycle is not as important as the design cycle, but above all a powerful tool that contributes to it.

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3.2 Methods

Literature research

Some of the sub questions could be answered by means of literature research. Since the single-piston pump acts as a spring-damper system, much useful literature exists. This is a well-known problem in physics; many experts have worked out this problem. A literature analysis will give more insights in how to determine the relevant coefficients. With doing this, the current status quo becomes clear and a starting point for the project arises. In short, investigating the existing body of knowledge is an important first step in answering the final research question proposed.

Interviews with experts

Another important data acquisition method is conduction interviews with field experts.

There are many other researchers working on the Ocean Grazer. Some conducted comparable researches, which makes it interesting and useful to talk to them.

Mathematical modelling

Currently, there are research papers available that try to determine a systemâĂŹs coefficients with mathematical models. This provides a framework for the design of a mathematical model. A thorough literature research is necessary and should result in the development of a model that takes the main characteristics for slamming into account.

Eventually, the final model should be able to simulate water hammer pulses accurately.

This means that the slamming force pattern generated by the model should match the experimentally obtained force pattern.

3.3 Planning of the research

In order to complete the research within the available time, a planning was constructed, which can be found in Appendix A. The planning was constantly updated during the research to ensure an efficient time use.

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Chapter 4

Data from experiments

4.1 Experimental setup

At the University of Groningen a prototype of the Ocean Grazer is built, where all experiments are executed. The experimental data used in this research is obtained by K.

Paapst, since this is the main part of his thesis. In this chapter the data obtained by these experiments will be discussed. Both this research and the conducted experiments are based on the settings of this prototype. The prototype is visualized in figure 4.2, whereas an additional graphical representation is presented in figure 4.1.

Figure 4.1: A graphical representation of the experimental setup [Zaharia, 2018]

In figure 4.2, the different piping sections are highlighted with red. The first section, from the lower reservoir to the horizontal pipes, cannot be changed. The water will always flow through these pipes; no other route is possible. In the second section, the horizontal pipes, one can change the settings. Namely, in this section it is possible to choose between two pipe combinations, each with different diameter. In the report, the pipe combination with largest diameter is referred to as big pipes, whereas the other pipe combination is referred to as small pipes. In section two it should be determined which piping combination will be used, either small, big or big plus small. In the third section, the vertical pipes straight to the upper reservoir, only the middle pipe is operational, while the other two smaller pipes are closed, because there is no piston nor check valve placed here. So, for this research, only the middle pipe will be taken into account.

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Figure 4.2: Ocean Grazer prototype as built in the Water Hall [van Rooij, 2015]

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Table 4.1: Situations experiment Situation Stroke Pipes

1 up small

2 down small

3 up big

4 down big

5 up big+small

6 down big+small

The experiment was ran with several different settings, to cover all possible cases. As is explained above, it is possible to run the simulations with small pipes, big pipes and a combination of both. In addition, the oscillations differ also for the up and downstroke.

So, in total there are six different situations which are taken into consideration. The different situations are presented in table 4.1. The mathematical model is adjusted for each situation, such that the experiment and model will match as best as possible.

In total the experiment was ran thirteen times, each time with different settings. The experiments taken into consideration for this research are presented in 4.2. An overview of all experiments conducted is added in appendix B. The frequency of the motor was constant during these three sessions. Furthermore, in these sessions the amount of strokes is quite large. In the other sessions, the amount of strokes was much smaller, namely ten strokes per session. A larger amount of strokes makes it easier to get accurate results relating the ratio between the increasing hydraulic head and the corresponding frequencies. Therefore it is decided to focus on experiments 2.2.11, 2.2.12 and 2.2.13. In these experiments both the up and downstroke are measured, so the data for all the situations described in 4.1 is obtained by these three experiments. As the experimental setup is explained, it is possible to discuss the results from the experiment.

Table 4.2: Settings experiment

Experiment number Amount of strokes Piping Initial hydraulic head [m]

2.2.11 36 small 2.460

2.2.12 36 big 2.467

2.2.13 36 big+small 2.469

4.2 Force pattern: small pipes

In this section the results obtained in experiment 2.2.11 are discussed. In this experiment only the small pipes were used. This means, the water is pumped from the lower reservoir through piping section 1, the small-diameter pipes, and piping section 3, where the check valve and the piston pipe are located, to the upper reservoir.

The experiment generates a force pattern as presented in figure 4.3. Here only three out of 36 strokes are displayed, where each up and downstroke is indicated. The figure identifies oscillations in each up and down stroke, which is called slamming.

The oscillations that are observed in figure 4.3, are caused by the change of momentum.

There is a certain shock when the upstroke starts, because first only the mass of the piston was pulling on the rod, whereas at the beginning of the upstroke the whole mass of the water is added.

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Figure 4.3: Force pattern upstroke, small pipes

Moreover, the force in the upstroke is much larger than the force in the downstroke. The force acting on the rod is about 1000 Newton during the upstroke. In the downstroke the equilibrium force is about 200 Newton. At this point, there is no mass acting on the piston from the water column anymore: only the mass of the piston is hanging on the rod. So, since this force is determined by the mass times the gravitational constant, the downstroke force should be around 250 Newton. The difference between the 200 and the 250 Newton is probably caused by a valley which occurs in the downstroke pattern. The valve in the piston has a smaller surface area than the rest of the piping in the system, due to area restrictions. This prevents the water from flowing fast enough through the piston valve; the cable is moving downwards faster than the piston itself, which results in a lower force acting on the piston.

Figure 4.4: Experiment: small pipes, upstroke

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It is, however, hard to draw further conclusions from such a force pattern; it is better to isolate the waves properly such that only the first, linear part of each stroke is selected.

Presented in figures 4.4 and 4.5 are the force patterns corresponding to the small pipes of the up and downstroke, respectively. For both cases, just the relevant parts of the waves are visualized. In other words, the waves are isolated such that only the linear part is taken into consideration.

Figure 4.5: Experiment: small pipes, downstroke

To be able to learn from the generated force waves, the data should be shifted from the time domain to the frequency domain. From there it is easier to recognize the dominant frequencies. This domain shift can be achieved by a fast Fourier transform (FFT). FFT is an algorithm that samples a signal over a period of time and divides it into its frequency components. For the FFT, only the relevant parts of the experimentally obtained curve will be taken into consideration. This means that the wave length and period should be selected properly. The FFT transformation is illustrated in figure 4.6. Over the time period measured in the figure, the signal contains one distinct dominant frequency, which is indicated in the figure. This peak in amplitude indicates a natural frequency of 1.06 Hz for the small pipes in the upstroke.

Figure 4.6: Frequency plot of isolated waves upstroke, small pipes

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The obtained natural frequency is valuable for further calculations, but first the value should be validated. To validate the obtained natural frequency, the force pattern, as presented in figure 4.4, should be critically analyzed. When the peaks of each period are indicated, the total time for a given number of periods can be calculated. From here the frequency of the wave can be calculated. This calculation, among the highlighted maximums, is visualized in figure 4.7.

Figure 4.7: Frequency from force pattern

This calculation suggests a frequency of 0.95 Hz. This is slightly smaller than the natural frequency which follows from the FFT. The reason for this difference is caused by the selected data for the FFT. Such a transformation requires a large data set in order to generate accurate results. However, since only the first linear parts of the wave is taken into consideration, the isolated wave length which is used for the FFT is very small. Therefore the FFT does not produce perfect results; it approximates the natural frequency rather than perfectly generate the natural frequency. After discussion it follows that this is a limitation of the FFT method. Normally the transformation is run for longer data sets instead of the small sample which is used in this research. It is allowed to run the FFT for smaller samples, but it affects the accuracy of the results.

In figure 4.6, the point previous to the peak, at x = 0.94, emphasizes the large interval between each different point. This however cannot be decreased, since the step interval, and thus the accuracy, is determined by the sampling frequency and the length of the isolated waves. The sampling frequency is fixed on 200 Hz and the length of the isolated waves cannot be increased, so consequently a higher accuracy cannot be achieved.

The use of the FFT is validated, since the results are comparable to some extend. We take the difference between the FFT result and the analytically obtained frequency for granted, as the difference is explained in the paragraph above.

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4.3 Force pattern: big pipes

In this section the results obtained in experiment 2.2.12 are discussed. In this experiment only the big pipes were used. The experiment generated again a force pattern consisting of 36 strokes, where in figures 4.8 and 4.9 only the relevant parts of the up and downstroke are showed.

Figure 4.8: Experiment: big pipes, upstroke

Noticeable in figure 4.8 is the second peak of the upstroke force pattern. This peak does not correspond to the linear pattern of the wave, although the other peaks do follow this linear pattern. The cause of this non-matching behaviour is probably an error-measurement, because all other peaks just follow the damped, linear pattern. Moreover, the oscillations are quickly damped as the amplitude of the waves is approaching zero after just seven periods. There are no deviant observations regarding the force pattern of the downstroke observed.

Figure 4.9: Experiment: big pipes, downstroke

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4.4 Force pattern: big + small pipes

In this section the results obtained in experiment 2.2.13 are discussed. In this experiment both the big as well as the small pipes were used. The experiment generated again a force pattern consisting of 36 strokes, where in figures 4.10 and 4.11 only the relevant parts of the up and downstroke are showed.

Figure 4.10: Experiment: big + small pipes, upstroke

For the upstroke no deviant observations are found. The waves are rapidly approaching a steady state and the first peaks do follow a linear pattern. However, for the downstroke an interesting trend can be identified. Namely, the steady state of the waves follow a slightly decreasing line. This drop is also indicated in figure 4.11. This is the valley in the force pattern which is explained earlier on.

Figure 4.11: Experiment: big + small pipes, downstroke

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Chapter 5

Research Methodology

As is observed in the experiments, slamming occurs in the single piston pump. To be able to develop a mathematical model to simulate this slamming, all parameters of importance should be known. As is stated before, the SPP can be seen as a spring-damper system.

Therefore the following general equation can be used:

d²x

dt² + 2ζω0

dx

dx+ ω₀²x= 0 (5.1)

Equation 5.1 can however only be applied if the system under consideration shows linear behaviour. For the SPP this is not the case, which can be clearly observed if the experimental data is examined. The force pattern has many fluctuations as well as some strange valleys. There are however options to tackle this non-linearity hurdle, which will be explained in the next sections.

5.1 Determination of coefficients

First, it should be clear how the coefficients will be determined. Relevant parameters will be the damping and stiffness coefficients of the oscillation. It is not possible to directly identify these coefficients from the system’s force pattern (figure 1.3), because there are different sub-parameters which in total determines the coefficient. For instance relating the stiffness: the coefficient consists of the sub-parameters Krod (stiffness of rod), the connections, the stiffness of the water column, but also the movement of the frame, since this is not completely rigid. The same holds for the damping coefficient: it consists of wall friction, bends in the piping, and of course the damping of the water column. It is not in the scope of this research to investigate the particular influence of all these sub- parameters. Instead, the damping and stiffness coefficients can be approximated by two different procedures, which will be presented in the next two subsections.

5.1.1 Coefficient determination: procedure 1

Data regarding the slamming force is available from the experiment, as was presented in previous chapter. These data identify oscillations in each up and down stroke. The data should be shifted from the time domain to the frequency domain because from there it is easier to recognize the dominant frequencies. This domain shift can be achieved by applying Fast Fourier Transform (FFT).

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For the FFT, only the relevant parts of the experimentally obtained curve will be taken into consideration. This means that the wave length and period should be selected properly.

Afterwards, the selected period can be shifted to the frequency domain (FD). In the FD graph there will be a peak in amplitude. This peak corresponds to the natural frequency (ω0) of the wave. From the natural frequency one is able to obtain the stiffness, following equation 5.2.

ω0 = s

K

m (5.2)

Furthermore, based on the natural frequency, damping can be determined by the half- power bandwidth method. Half-power bandwidth is defined as the ratio of the frequency range between the two half power points to the natural frequency. In this way all relevant coefficients can be determined.

5.1.2 Coefficient determination: procedure 2

Moreover, it is also possible to derive the coefficients analytically. In this way, the general equation of a spring-damper system should be approximated by summing several linear functions, such that it corresponds to the original nonlinear equation. This could be executed in the following manner:

d²x

dt² + 2ζω0

dx

dx + ω0²x ≈^XA_isin (ωi+ θ) (5.3) The right hand side of equation 5.3 is a new designed linear function which accurately approximates the general equation of a spring-damper system.

5.1.3 Conclusion

Conducting the calculations corresponding to the analytic determination of the spring damper system will become extremely complex. High-level knowledge relating system dynamics is required. This is out of the scope of this research. Therefore, after discussion, it was decided to skip this method. Instead, it was decided to focus on the first, linear part of the oscillations, rather than on the whole force period corresponding to one stroke.

In this way it is also possible to simulate the oscillations, by implementing the expressions which belong to a spring-damper system in a mathematical model. The advantage of this procedure is that the required steps are more straightforward to conduct, compered to the analytic determination of the nonlinear spring-damper equation.

Furthermore, as was indicated before, by means of a mathematical model slamming can be simulated. To achieve an accurate simulation, the model should be based on the differential equation which describes the slamming. Since there is only focused on the first part of the oscillations, the part which shows linear behaviour, the non-linearity hurdle is tackled and the general equation of a spring-damper system can be used, as is presented in equations 2.1 and 5.1.

There are three unknowns which should be determined. Namely, the mass, the damping and the stiffness of the system. The mass can be calculated since the pipe details and the fluid characteristics are known. From the FFT, the natural frequency appears and eventually the stiffness of the system can be calculated. Subsequently, the damping could

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The ζ will be obtained by the half-power bandwidth method. In this way one is able to determine the damping coefficient of the system. So, a procedure for each relevant coefficient is proposed. These procedures need to be translated into mathematical expressions in order to successfully implement it in our model. In the next sections these mathematical expressions will be presented and discussed.

5.2 Validation proposed theory

The equations obtained in previous work, and recommended to use in our research, should be evaluated to ensure it is justified to implement these in our research.

In previous research [Zaharia, 2018], it was suggested that in the case of the SPP, there are two springs and dampers that should be taken into consideration. The first is the spring and damper in the rod, second comes from the water column. The equations introduced below are found in tuned liquid dampers literature ([Mondal et al., 2014, Malekghasemi et al., 2015]. In the case of the experimental setup the rod stiffness factor is defined as:

K_rod= πR²_rEst

L_r , (5.5)

where Rr is the radius of the rod, Est the Young’s modulus of the material used (steel), and Lr is the length of the rod. The damping coefficient C is defined as:

C_rod= 2ζrod

q

m_pK_rod, (5.6)

where ζrod is the rod damping ratio, and mp is the mass of the piston. The spring that can be found in the water column is determined by:

Kwater = 2ρAg, (5.7)

where ρ is the density of the working fluid (water), A is the surface area of the pipe and g is the gravitational constant. The damper coefficient for the water column can be determined with the following equation:

C_water = 2mfω_fζ_f, (5.8)

where mf is the mass of the fluid, ωf the natural frequency of the liquid damper, and ζf

the damping ratio.

The natural frequency, in this case, can be calculated using the following equation:

ω_f = r2g

l , (5.9)

where l is the length of the piping. The mass of the fluid can simply be determined by adding all the pipe section with their length and diameter, or:

mf =^XρAl. (5.10)

All the required equations are provided and one should be able to implement these in the mathematical model. Unfortunately, this never has generated any realistic outcome. The generated force pattern did not match the experiment at all. In section 6.1 there is further elaborated on this observation.

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After a validation, it appears that much other parameters were neglected. For instance, the stiffness coefficient K does not consist of just the stiffness of the rod and the water column. It should also comprise the (small) movement of the frame and all the transfer mechanisms, such as carabiners, cable thimbles, roof pulleys etc. The same holds for the determination of the damping coefficient. Again there was assumed that this coefficient was a combination of just the damping of the rod and the water column. This cannot hold since the system’s damping also depends on the wall friction, minor losses due to the bends, or a suddenly change of piping diameter.

To calculate the spring and damper, it should be kept in mind that these are placed in series. This means that the stiffness coefficient K should be calculated in the following manner:

K= 1

1

K1 + _K¹₂ + ... + _K¹_n, (5.11) and the damper coefficient C should be determined as:

C = C1+ C2+ ... + Cn, (5.12)

where in both equations the n stand for the number of springs or capacitors. As is explained above, the number of springs and capacitors is definitely larger than two. Probably, it should consists of three of four components to ensure a realistic prediction. It follows that the use of the above equations does not cover all aspects involved. There should be further research on how to determine the total stiffness and damping coefficients analytically.

Furthermore, it should be noted that both ζrod and ζf are unknown. No formulas are found to express these ratios. By means of the experiment a damping ratio was obtained, however this cannot be specified into two separated damping ratios; it just produced the total damping ratio.

5.3 Approach to solution

Instead of using the equations provided above, another approach will be applied. Rewriting equation 5.2 and equation 5.4, gives to the following expressions, respectively:

K = mω₀² (5.13)

C = 2ζ√

mK (5.14)

All pipe details are known, so the mass, in both expressions, is calculated using equation 5.10. It should be noted that for the mass in the downstroke a reduction factor applies.

This reduction factor is required since not all mass in the downstroke is oscillating due to the limited throughput area of the piston valve. The throughput area is depicted in figure 5.1, where the holes indicate the points where the water can flow through.

From several calculations it follows that this reduction factor equals 0.52. This number is determined such that the frequency of the downstroke corresponds to the initial frequency, where the stiffness coefficient K is the same for both the up and downstroke.

Table 5.1: Reduction factor calculations

Frequency [Hz] Mass [kg] Reduction factor [-]

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Namely, it was known, from FFT analysis, that the frequency in the upstroke was 2.0 Hz and for the downstroke 4.4 Hz, in the big piping situation. Suppose the stiffness coefficient K remains unchanged, then the following expression holds:

ω₁

ω2 =^rm₂ m1

, (5.15)

where m1 and m2 corresponds to the mass participating in the up - and downstroke, and ω denotes the corresponding frequencies.

Figure 5.1: Piston valve sketch

Filling in the numbers, where the mass of the downstroke, m2, is calculated including the reduction coefficient, the outcomes are almost similar to each other. This validates the use of the reduction factor.

The natural frequency ω²0 is determined by applying the fast Fourier transform algorithm to the data obtained from the experiment. This generates a similar plot as is presented in figure 4.6, with clearly exposed peaks corresponding to the natural frequency.

The damping ratio ζ is obtained by the half-power bandwidth method and can directly be used in the calculations. Using this procedure, both the stiffness and damping coefficients could be calculated. The only part left is an accurate description of the slamming force pattern, which will be presented in the next section.

5.4 Step response

The slamming force pattern will be simulated by applying a step force to a harmonic oscillator. The step response is the response of the system to a unit step input. For example, if the system is defined as:

md²x dt² + cdx

dx+ kx = y (5.16)

and y is considered to be the input, then the unit step response is the solution to:

md²x dt² + cdx

dx+ kx = δ(t). (5.17)

In the case ζ < 1 and a unit step input with x(0) = 0, the solution to equation 5.1, which describes our system, is:

x(t) = 1 − e^−ζω⁰^tsin^p1 − ζ² ω₀t+ ϕ

sin(ϕ) , (5.18)

with phase ϕ given by cos ϕ = ζ [Bloch, 2013, Serway and Jewett, 2018].

Knowing how the system responds to a sudden input is important because large and pos- sibly fast deviations from the long term steady state may have extreme effects on the system’s behaviour [Golnaraghi and Kuo, 2010]. In our model, equation 5.18 is implemented in order to create a realistic simulation of the slamming.

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Let us consider the graphical representation of the prototype as presented in figure 4.1.

The force sensor is located close to the motor, so it only measures the force required to lift the piston. Due to the location of the force sensor, the force only consists of the displacement force-component. This force, called the slamming force, is defined as follows:

F_slamming = Kx (5.19)

Since it is only the force acting on the rod which is measured, the acceleration and ve- locity components are not included in this force determination. Comparing the model’s prediction to the experiment validates this decision.

The solution proposed in equation 5.18 is dimensionless. In order to implement it properly into our model, it should be transformed into the displacement of the piston, xp, such that it can be used in equation 5.19. This is achieved in the following manner:

x_p = F

Kx(t) (5.20)

Now, the displacement of the piston xp is defined, such that the general solution to the step response is specified for our model. Additionally, another force equation should be introduced, namely the hydraulic force. This force is known as the pressure on an object submerged in a fluid times the area:

F_hydraulic= ρgHπr², (5.21)

where H is the initial hydraulic head and r is the radius of the water column. The total force pattern, as presented in figure 4.3, follows a step function, where the step-amplitude is about 1000 Newton for the upstroke and about 200 Newton for the downstroke.

Due to this step-amplitude, there should be a certain offset introduced, which is called β further on. Physically, this offset just defines the equilibrium or steady state value for either the up or downstroke. It is required in order to ensure the model will match the experiment. This offset is determined by an iterative process and differs per situation. The reason this offset is not a constant value has to do with the experiment’s settings. The results from the experiment needs to be calibrated; this calibration changes every time and therefore the offset will change as well. For instance, if we compare the experimentally obtained data of previous research (figure 1.3) with the experiments recently conducted (figure 4.3), one can observe that the offset in both situations differ much. Namely, in the experiments of [Zaharia, 2018], the step-amplitude in the downstroke even became negative. This has just to do with the offset and therefore this needs to be included in the slamming force equation.

Moreover, there is also found a multiplication factor α. This factor controls the amplitude of the waves produced by the model. Without this value, the amplitude of the waves largely exceeds the amplitude of the experimentally obtained pattern. Again, this factor is determined by an iterative process and differs per situation. Physically, this multiplication factor elaborates on the working of the valves used. Namely, suppose a valve closes directly after switching from stroke. Then, the water in motion is forced to stop or change direction suddenly, which causes a large slamming amplitude. These oscillations can cause major problems and will reduce the durability of the valves. If the valve is closed slowly and smoothly, the slamming will be minor. On the contrary, this negatively affects the efficiency of the system. Namely, if the valve is closed really slow, the leakage will be high, reducing the system’s efficiency.

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It is key to find the right balance between the slamming occurring in the piston and the desired system’s efficiency. This balance can be tuned by adjusting α: when α approaches one, the valve closes almost directly, resulting in a large slamming amplitude. On the other hand, if α approaches zero, the valve is closed smoothly, resulting in minor slamming amplitude. This reasoning is summarized in table 5.2.

Table 5.2: Interpretation of multiplication factor α

α Consequence

Close to 0 Valve closes smoothly: minor slamming, low efficiency Close to 1 Valve closes directly: large slamming, high efficiency

Taken into consideration the factors discussed above, the general slamming force pattern is given by:

F_slamming = α(Kxp) + β, (5.22)

or equivalently:

Fslamming= α(ρgHπr²x(t)) + β (5.23)

This equation enables one to generate an accurate simulation of the slamming force. As will be presented in chapter 6, the produced force pattern, following equation 5.23, sufficiently matches the experiment.

5.5 Physical meaning and validation

When the mathematical model is designed following the proposed structure above, the physical meaning of the results should also be explained. For different phenomena the influence on the system should be explained. For instance, how does a large slamming force influence the system’s performance? From here recommendations can be composed.

Besides the physical meaning, the derived coefficients should also be validated. In this stage the experimentally obtained values should be compared to the derived coefficients to conclude if the results make sense. The determination of the coefficients, such that the model matches the experiment, is the first step, where in the second step the results must be evaluated. Throughout the sections there will be elaborated on the physical meaning of the findings. In addition, the findings will be validated directly, in order to prevent putting effort into something which makes no sense.

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Chapter 6

Results and discussion

In this chapter all the obtained results will be presented. The plots are obtained by MATLAB, where several codes are designed to generate the required output. The codes are attached in appendix C. By means of a mathematical model, designed in MATLAB, slamming is simulated and compared to the experiment. Different types of results will be discussed: first the outcome of the model for the small piping situation, second the big piping combination followed by the big + small piping combination. Lastly, the hydraulic head and the influence of the hydraulic head on the frequency will be discussed.

6.1 Model outcome: small pipes

Firstly, the model for the small piping situation will be introduced. This experiment is conducted with an initial hydraulic head of 2.460 meters. For all situations the first up or downstroke is depicted, such that the initial hydraulic head is still valid to use in the calculations. Furthermore, the radius of the water column, as used in equation 5.21, is 0.10 meters. This radius is constant during all experiments. All other parameters are presented in table 6.1.

Table 6.1: Parameters mathematical model: small pipes upstroke downstroke unit

m 161.0 45.7 kg

K 7600 34901 N/m

C 199.1 80.8 kg/s

f 1.1 4.4 Hz

ζ model 0.100 0.027 -

ζ experiment 0.108 0.034 -

α 0.310 -0.059 -

β 810 242 -

Fhydraulic 758.9 758.9 N

The mass difference in the up and downstroke follows from the different pipes used. For the downstroke only the pipes above the piston valve are taken into consideration. This is defined as the upper part of section 3 of the prototype presented in figure 4.2.

K and C are calculated by equations 5.13 and 5.14, respectively. The frequency follows

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The same procedure is applied to determine α and β. The hydraulic force is calculated using equation 5.21. The physical meaning of Fhydraulic will be explained later on. The following plots, using the parameters as presented in table 6.1, are generated by executing the model:

Figure 6.1: Model vs experiment: small pipes, upstroke

Figure 6.2: Model vs experiment: small pipes, downstroke

It can be stated that the model and the experiment show similar behaviour, such that the degree of agreement is acceptable. After several waves, the experimental curve will deflect slightly, but this phenomenon is declared; this deviation is taken for granted.

During the project it is decided to outsource the half-power bandwidth method, as proposed to calculate the damping ratio, to the research which focuses on the experiment, to prevent double work. The results of this method were made available for this research.

Considering the damping ratios as presented in the table, it can be argued that the numbers are reasonable. The difference between the numbers is about 8% for the upstroke,

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versus 26% for the downstroke, which seems a large error. However, this difference is caused by the lack of accuracy of the fast Fourier Transform method. The experimentally obtained damping ratio is calculated using the half-power bandwidth method. Half-power bandwidth is defined as the ratio of the frequency range between the two half power points to the natural frequency, where the natural frequency is obtained by conducting a FFT analysis. In a previous section of this research it is already explained that the use of this FFT method comes along with accuracy limitations. This explains the relatively large difference between ζmodel and ζexperiment.

Not only in this situation, but also in the other situations the damping ratios differ much.

It is tried to find the cause of these large differences. According to the data from the experiment, the damping ratio is calculated per stroke. The calculated damping ratios are fluctuating much over time, as depicted in figure 6.3. For instance, the damping ratio for stroke 36 is more than twice as large as the damping ratio for stroke 33. This is a curious difference; however it is not in the scope of this research to recalculate or verify the damping ratios of the experiment. Although, the experimentally obtained damping ratios are not reliable to a high degree and thus potentially large differences between the model and the experiment, in terms of damping ratios, cannot be avoided.

Figure 6.3: Damping ratio per stroke (experiment 2.2.13)

It should be mentioned that the multiplication factor α is negative for the downstroke.

This is explainable because the downstroke pattern follows exactly the opposite curve of the upstroke. The upstroke pattern is called (positive) overshoot, whereas the downstroke pattern is called negative overshoot, which declares the minus sign in front of the multiplication factor.

Furthermore, the physical meaning of Fhydraulicshould be discussed. According to various experiments conducted previously on the prototype of the Ocean Grazer, the hydraulic force always accounted for the difference between the equilibrium or steady state value of the up and downstroke. This is visualized in figure 6.4. For example, if the equilibrium value of the upstroke equals 1000 Newton and the value of the downstroke equals 200 Newton, the hydraulic force turned out to be about 1000 - 200 = 800 Newton.

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Figure 6.4: Force plot with explanation

If we compare this trend to our model, the theory does not perfectly match the expectations. Namely, the difference between both equilibrium values for the first stroke is approximately 1030 - 190 = 840 Newton, although Fhydraulicis calculated to be about 760 Newton. For this calculation a hydraulic head of 2.460 meters is used, which is the value obtained from the experiment (table 4.2). It is unclear why the experimental data differ from the original trend where Fhydraulic accounts for the difference in equilibrium values.

A possible reason can be a measurement error in the determination of the hydraulic head.

For instance, if the hydraulic head is measured to be 2.730 meters, it perfectly matches the theory, because then Fhydraulic equals 840 Newton. However, this is then a measurement error which occurs in each situation, which does not sound likely. The observation is thoroughly discussed, but no rigid explanation is found. From this perspective the experiment did not match the expectations.

The last noticeable parameter is the frequency of the upstroke. A frequency comparison for each situation will be made in section 6.5; for now only the upstroke frequency for the small piping situation is taken into consideration. A frequency of 1.1 Hz is rather low, especially since the participating mass is relatively small in this situation. However, another quick view on table 6.1 reveals us a very low K value as well. The frequency depends on this stiffness coefficient K, so the low frequency is a consequence of this small coefficient K.

An in depth discussion on the stiffness coefficient K is performed. The cause of the low K should lay in using the small pipes (earlier on defined as piping section 2), since the upper part (section 3) and the first part of the piping (section 1) are used in all situations.

As is previously stated, the stiffness coefficient K consists of multiple variables. In the initial stage, just the stiffness of the rod and the water column were taken into consideration. It was calculated that, using equations 5.5, 5.7 and 5.11, the stiffness coefficient K in this case equaled 615 N/m, which resulted in a frequency of 0.265 Hz. So, the system’s stiffness coefficient was excessively small and, consequently, the frequency was also too small, compared to the experiment.

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This observation was the reason to investigate the stiffness coefficient. After further ana- lyzing the pipes, it is observed that rubber, which is used in between different pipe sections to absorb small piping movements, also affects the stiffness. It causes the rigidness of the pipe because it enables a pipe to slightly expend and deform. In addition, the small piping route has significantly more bends compared to the big piping combination, which also affects the rigidness.

All variables which affect the total stiffness of the system are enlisted below:

• Stiffness of the rod (Krod)

• Stiffness of the water column (Kwater)

• Small movement of the frame

• Carabiners (transfer mechanism)

• Cable thimbles (transfer mechanism)

• Roof pulleys (transfer mechanism)

• Rubber between pipe connections

• Number of bends

It is suggested that one of the variables listed above dominates the determination of the total stiffness coefficient, and therefore the contribution of other variables to the total stiffness could be assumed negligible. For example, suppose it turns out that the frame deforms significantly through the agency of the force acting on it. This has an enormous effect on the system’s stiffness, which makes this variable the most important one to take into consideration for further calculations. At this moment, however, it is uncertain which variable dominates, but further work could focus on this thought.

In [Thon, 2014], an analysis on an u-tube with an inner diameter change is performed.

This research can be compared to the prototype when the small piping combination is used. Namely, in that situation there is also an inner diameter change. According to the research, the force balance consists of three different terms on the right hand side. The firsts one represents the losses due to wall friction, the second one gives the gravitational acceleration according to the height difference in the two legs of the u-tube. This is the same as for the u-tube with a constant inner diameter, or in our case, the prototype when the big piping combination is used. However, there is an extra term added in the case of an u-tube with an inner diameter change, which accounts for an unknown pressure change across the coupling. This term denotes the friction losses in the cone-shaped coupling between large and small diameter tubes. It represents the losses due to the drag from turbulent eddies. These eddies arise from the sudden contraction or expansion of the flowing area.

So, it can be stated that because of the inner diameter change in the case of the small piping section, an extra force-component is added which affects both the damping and stiffness. This is an arguable explanation for the large difference in stiffness coefficient K.

Comparing the damping ratios of each situation also confirms this: in the small piping combination the damping is significant higher than the other two situations. Thus, it is suggested that the extra force-component, which only occurs in the small piping situation, causes a decrease in the stiffness coefficient and eventually accounts for the lower frequency.

This declares the obtained results.

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All parameters used in order to generate an accurate force pattern are discussed. In figure 6.5, it is shown how these parameters are implemented into the MATLAB code.

Figure 6.5: Parameters implemented in MATLAB code

The small piping section is discussed in detail. The procedure as explained in this section holds also for the two other situations. Therefore, the other two situations will be discussed briefly.

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6.2 Model outcome: big pipes

Secondly, the model for the big piping situation will be introduced. This experiment is conducted with an initial hydraulic head of 2.467 meters. The parameters are presented in table 6.2.

Table 6.2: Parameters mathematical model: big pipes upstroke downstroke unit

m 211.0 45.7 kg

K 34901 34901 N/m

C 402.7 93.4 kg/s

f 2.0 4.4 Hz

ζ model 0.073 0.028 -

α 0.180 -0.061 -

β 898 248 -

F_hydraulic 758.9 758.9 N

As is mentioned previously, the second peak of the upstroke of the experimentally obtained force pattern is remarkable, as can be seen in figure 6.6. This peak does not correspond to the linear pattern of the wave. Since the model matches the experiment for all other peaks, it can be stated that the second peak is an error-measurement, as was suggested before.

Figure 6.6: Model vs experiment: big pipes, upstroke

Moreover, the stiffness coefficient K is exactly the same for the up and downstroke. This is caused by the mass reduction coefficient as explained in section 5.2. This coefficient ensures that the frequency ratio is proportional to the square root of the inverse mass ratio (equation 5.15). This declares the constant stiffness coefficient in this situation.

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Figure 6.7: Model vs experiment: big pipes, downstroke

Finally, there is a relatively large difference between the damping ratios for the upstroke.

From the experiment a damping ratio of 0.036 is obtained, although the damping ratio used in the model is 0.073, which is twice as large. However, both waves follow a similar pattern. As was explained in section 6.1, these large differences between the model and the experiment, in terms of damping ratios, cannot be avoided.

6.3 Model outcome: big + small pipes

Lastly, the model for the big + small piping situation will be introduced. This experiment is conducted with an initial hydraulic head of 2.469 meters. The parameters are presented in table 6.3.

Table 6.3: Parameters mathematical model: big + small pipes upstroke downstroke unit

m 248.3 45.7 kg

K 44000 34901 N/m

C 495.8 68.2 kg/s

f 2.1 4.4 Hz

ζ model 0.075 0.027 -

α 0.210 -0.060 -

β 865 245 -

Fhydraulic 758.9 758.9 N

It should be noted that Fhydraulic is the same for each situation. This is due to the fact that the initial hydraulic head is almost identical in each situation. As all other variables are exactly identical, Fhydraulic does not change.

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Figure 6.8: Model vs experiment: big + small pipes, upstroke

Again, a significant difference between the damping ratios for the upstroke is spotted.

Considering figure 6.8, this could make sense this time, as the model and experiment does not match nicely in the beginning. Although, more likely, the reason for this deviation is the nonlinear pattern at the middle of the experiment. It seems that the transition from the second to the third wave is not following the usual pattern, resulting in an aberration between the model and the experiment.

Figure 6.9: Model vs experiment: big + small pipes, downstroke

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6.4 Hydraulic head

Figure 6.10: δHup and δHlow defined The hydraulic head is defined

as the difference between the water levels in the upper and lower reservoir. In figure 6.10 the situation is sketched. In section 1 of the figure, the initial situation is sketched. In the second section, the situation after several strokes is sketched, where the water is moved from the lower to the upper reservoir.

The horizontal area is defined as Aup for the upper reservoir and as Alow for the lower reservoir. Please note that δHlow is measured in the opposite direction of δHup, as is indicated by the red arrow, such that both numbers are positive. The increase of the hydraulic head per stroke is the sum of the change of the water level in the upper and lower reservoir, which is defined as:

δH = δHup+ δHlow (6.1)

Moreover, the total volume of the water remains constant, and therefore the volume change in both reservoirs should be equal, which is defined as:

δH_upA_up= δHlowA_low (6.2)

Rewriting 6.2 leads to:

δH_up= δHlow

A_low

A_up (6.3)

Combining equations 6.1 and 6.3 gives:

δH = δHlow(1 +A_low

A_up) (6.4)

From equation 6.4 the expression for the change of hydraulic head in the lower reservoir follows:

δH_low= δH

1 +^A_A^low_up (6.5)

And similarly, the expression for the change of hydraulic head in the upper reservoir:

δHup= δHlow

A_low

A_up (6.6)

Thus, for every stroke, the hydraulic head increases by δH. As an initial condition, let the hydraulic head be located at Hinitial. Over time, the hydraulic head is defined as:

H(i)new = Hinitial+ δH(i), (6.7)

where i ranges from zero to the total number of strokes.

Mathematical modeling of slamming in a single-piston pump inside the Ocean Grazer

Bachelor Integration Project

Final report