Modelling Water Hammer Effects in Rising Pipeline Systems Using the PKP Method and the MOC

Abstract

Water hammer is a critical transient phenomenon in pumping systems, occurring when a sudden change in flow velocity generates pressure waves propagating along the pipeline. This study focuses on the dynamic response of a long rising pipeline subjected to an emergency pump shutdown, with particular emphasis on the sudden release and propagation of hydraulic energy in the form of pressure waves. Such scenarios are typical for mine dewatering and water supply systems with high elevation differences. Two numerical approaches were investigated: the Method of Characteristics (MOC) implemented in TSNet as a reference model, and the Train Analogy Method (PKP) implemented in MATLAB R2024b/Simulink, where the fluid is represented as discrete masses connected by elastic links, enabling the inclusion of pump and motor dynamics. Simulations were performed for two configurations: first–with a check valve installed only at the pump discharge and second–with a check valve at the pump discharge and in the middle of the pipeline. The results demonstrate that both models capture the essential features of water hammer: a sharp initial pressure drop, the formation of transient waves, and pressure oscillations with decreasing amplitude. These oscillations reflect the propagation and gradual dissipation of hydraulic energy stored in the moving fluid, primarily due to frictional and elastic effects within the pipeline. The presence of a check valve accelerates the attenuation of oscillations, effectively reducing the impact of returning waves on the downstream pipeline. The novelty of this study lies in the use of the PKP method to simulate transient flow and energy exchange in long rising pipelines with dynamic pump behavior. The method offers a physically intuitive and modular approach that enables the modelling of local flow phenomena, pressure wave propagation, and system components such as pump–motor inertia and check valves. This makes PKP a valuable tool for investigating complex water hammer scenarios, as it enables the analysis of pressure wave propagation and damping, providing insight into the scale and evolution of energy released during sudden operational incidents, such as an emergency pump shutdown. The close agreement between the PKP and MOC results confirms that the PKP method implemented in Simulink is a reliable tool for predicting transient pressure behavior in hydraulic installations and supports its use for further validation and dynamic system analysis.

1. Introduction

The water hammer phenomenon occurs when the velocity of a fluid in a closed pipe changes suddenly, usually due to rapid valve closure or pump shutdown. This abrupt change generates a pressure wave that travels through the fluid and reflects at system boundaries, such as valves, tanks, or closed ends.

If the transient is strong, the resulting overpressure can cause pipeline failure or equipment damage.

There are two main classic approaches to describe water hammer behavior:

• The rigid water column model assumes that the fluid is incompressible and the pipe walls are perfectly rigid. Under these conditions, the basic governing equation is as follows:

  $${\displaystyle \frac{H_1-H_2}{L}}={\displaystyle \frac{f{Q}^2}{2gD{A}^2}}+{\displaystyle \frac{1}{gA}}{\displaystyle \frac{dQ}{dt}},$$

  (1)

  where H₁ and H₂ are hydraulic heads at two points, L is the length of the pipe, f is the Darcy-Weisbach friction factor, Q is the volumetric flow rate, D is the diameter of the pipe, A is the cross-sectional area and g is the gravitational acceleration. When the flow rate does not change over time, the equation reduces to the classic Darcy-Weisbach formula [1].
• The elastic model includes the compressibility of the fluid and the elasticity of the pipe wall. The pressure wave then travels at a finite speed that depends on both the fluid and the pipe properties. The wave speed a is given by:

  $$a={\left({\displaystyle \frac{1}{\frac{1}{E_f}+\frac{D}{e{E}_p}}}\right)}^{-1/2}$$

  (2)

  where E_f is the bulk modulus of the fluid, E_p is the Young’s modulus of the pipe material, D is the internal diameter and e is the thickness of the pipe wall [1].

Figure 1 illustrates how the geometric layout and structural characteristics of the pipeline affect the propagation speed of the pressure waves during transient flow conditions.

The relation between the change in velocity and the resulting pressure increase is expressed by the Joukowsky equation:

$$\mathsf{\Delta }p=\rho a\mathsf{\Delta }\mathsf{\upsilon }$$

(3)

where Δp is the pressure rise, ρ is the fluid density, a is the wave velocity, and Δv is the change in flow velocity.

An important parameter is the characteristic time of the transient wave, denoted TM, which represents the round-trip time of the wave in the pipeline:

$$TM={\displaystyle \frac{2L}{a}}$$

(4)

This time determines whether a valve closure is classified as rapid or slow. If the closure time is less than TM, the transient is considered rapid and results in a full water hammer effect [1].

In practical applications, these equations are used in time-domain simulations to assess pressure transients and wave propagation. Numerical methods such as the Method of Characteristics (MOC) are widely applied. The behavior can also be modelled in MATLAB/Simulink using differential blocks and real-time valve or pump scenarios. Such simulations are useful for understanding the dynamic pressure response and for designing protective systems.

1.1. Mechanism and Importance of Water Hammer in Pressure Systems

The water hammer (WH) occurs when the velocity of the fluid changes suddenly. This occurs during fast valve closure, pump stop, or load rejection. The event generates pressure waves moving along the pipeline. These waves can deform pipe walls, cause fatigue, or damage equipment.

The Joukowsky equation links the rise in pressure with the density of the fluid, the speed of the wave, and the change in velocity. It does not include damping or unsteady effects. Urbanowicz et al. [2] showed that unsteady wall shear stress dominates during WH and contributes to fatigue. Di Nucci et al. [3] introduced a damped wave model incorporating a dimensionless parameter Λ, which quantifies the ratio between inertial wave propagation and dissipative effects such as viscous or turbulent damping. Low Λ values indicate strong attenuation of the pressure wave, typical for rough-walled or highly viscous systems. In the article [4], the authors demonstrated that gradual-then-fast valve closure reduces pressure peaks and vibration.

WH is a safety and diagnostic factor. In water networks, pressure surges indicate air pockets, stuck valves, or pump faults [1]. In hydroelectric systems, WH during load rejection can drop pressure below vapor pressure, causing cavitation [5]. The fluid–structure interaction (FSI) modifies the speed and amplitude of the wave [6]. In mining pipelines, WH can rupture pipes and flood wells [7].

1.2. Dynamic Effects: Transient Pressures, Cavitation, Damage

WH generates strong pressure transients that interact with material defects. In HDPE pipes, surges increase the J-integral at the crack tips, accelerating failure [8]. The growth of the crack depends on the orientation relative to the propagation of the waves.

The geometry of the outlet manifold affects the transient pressure. Abdulameer et al. [9] showed that asymmetric layouts without surge protection create non-uniform flow, high residual pressures, and air pocket formation. These conditions promote cavitation and hydraulic separation.

Steel pipelines with corrosion or elliptical defects fail at transient peaks [10]. Numerical studies confirmed that the geometry of the defect and the shape of the pressure waveform affect the risk of rupture. WH must be assessed with fracture mechanics to ensure structural safety.

1.3. Material and Structural Properties

The material and structure of the pipe determine the amplitude and damping of WH. Kandil et al. [11] compared steel, copper, uPVC, PPR, and GRP pipes. Metals produced higher peak pressures (copper ~ 15.5 bar) and low strain (~110 µε). Plastics showed lower pressures (uPVC ~ 12.5 bar) but high strain (>1600 µε). There is a trade-off between damping and deformation.

Flexible elements can also add additional damping. Rubber bypass tubes reduced WH peaks by 40–62% and reduced wave speed to 63 m/s [12]. Flexible liners inside steel pipes decreased pressure by 55% [13].

Temperature changes influence wave speed and cavitation. Higher water and pipe temperature reduce fluid density and increase vapor risk [14]. Steel wave speed peaks near 50 °C; HDPE speed decreases with temperature. Poor support or anchoring increases dynamic deformation and the risk of failure.

1.4. Influence of Pipeline Geometry and Elevation on Water Hammer Propagation

Pipeline layout and elevation profile shape propagation of WH. Inclinations, bends, and supports change wave reflection and damping. Xin et al. [15] showed that the valve angle and steep slopes amplify WH. Shi et al. [16] found that upward slopes promote vapor accumulation and CIWH onset; downward slopes stabilize flow.

Flexible supports smooth pressure pulses [17]. Small inclinations (2–3°) can create local high pressures [18]. Vertical shafts in mining pipelines produce high static head, requiring staged closure and surge mitigation. Offshore risers with sharp bends experience WH–motion interaction and fatigue [19]. Horizontal pipes can store energy before condensation-induced WH release [20].

1.5. Advanced Numerical and Simulation Methods

Accurate prediction of WH in systems with elevation changes requires advanced modelling. Neuhaus [21] used Simulink to show that elevation amplifies pressure peaks during pump trips. Naik et al. [22] applied second-order finite-difference schemes and IMC-tuned controllers for better damping. Geometry-optimized surge tanks reduce amplitudes by up to 40% [23].

CFD resolves local effects and phase change. Vertical pipes reached 70 MPa during CIWH, compared to 6 MPa in horizontal pipes [24]. Wan et al. [25] simulated cavity collapse with the Schnerr–Sauer model, confirming elevation-dependent amplification. Hybrid models such as Lee–Hughes-Duffey better predict condensation [26].

Structural analysis with iFEM tracks pipe deformation in real-time [27]. Small reductions in wall thickness at high elevation increase the probability of failure by >50% [28]. WH signals can detect subsidence in buried pipelines [29]. Swing flex check valves reduce WH by 33% [30].

1.6. Experimental Studies and Applications

Experiments clarify how geometry and operation affect WH. In steam pipelines, CIWH controls the diameter, pressure, and water velocity [20]. Low-height passive cooling systems show four-phase CIWH: interface formation, Kelvin–Helmholtz waves, bubble collapse, and pressure peak [31].

In hydropower, WH modifies torque and can reduce grid stability [32]. WH reflection coefficients estimate free vibration and damping without a complete modal analysis [33].

Practical mitigation methods include bypasses, air chambers, and surge tanks. A bypass with check valve reduced WH peaks by 33% [34]. Air chambers in pump-rising systems prevent vacuum and overpressure if volume and orifice size are optimized. In long water supply pipelines, combined surge tank and control valve cut overpressure from 174 m to 146 m [35].

In mining, slurry pipelines with higher velocity and concentration experience 79% higher peaks [36]. Laboratory and field studies support simulation-based WH protection.

1.7. Purpose and Scope of the Research

The water hammer is one of the most basic and dangerous dynamic phenomena occurring in fluid transport systems. This phenomenon is caused by a sudden change in flow velocity, most often due to rapid valve closure or emergency pump shutdown. In the technical literature, the most commonly analyzed case is that of a hydraulic shock in a horizontal pipeline when the end valve is closed abruptly. However, in real systems, especially those with large height differences, the course and effects of the shock can be much more complex.

This article analyses the case of an emergency shutdown of a pump in a system that pumps fluid through a long pipeline to a high elevation. In such configurations, the greatest threat is not the maximum pressure that occurs at the end of the installation, but the phenomenon of a water hammer occurring directly behind the pump. This occurs as a result of a reverse flow wave, which, in the absence of appropriate safeguards, can lead to serious consequences such as damage to pipelines, fittings, and connections.

Such systems are typical, for example, in mine drainage systems as well as in some petroleum product transmission systems and water supply networks located in mountainous areas. As an example for analysis, an emergency shutdown of a centrifugal pump supplying a pipeline with a length of L = 2000 m and a geometric head of H_g = 200 m was selected.

The analysis took into account a number of important factors that influenced the development of the water hammer:

• inertia of the pump-motor assembly,
• flow and power characteristics of the pump at variable speeds,
• dynamic properties of the fluid and pipeline,
• pressure losses in the pipeline and fittings,
• dynamic properties of additional system components.

Wave phenomena in pipelines with variable diameters, varying wall thicknesses, and made of different materials are difficult to describe accurately using classical analytical equations. Therefore, numerical simulation methods were used for the analysis.

The calculations were performed in parallel using two independent simulation environments. The first model was developed in the MATLAB/Simulink environment using the PKP method, which involves discretization of mass and mass and momentum balance equations in a variable geometry system. The second approach uses the TSNet Python 3.11.13 package, which is designed to perform transient simulations in water distribution networks and implements the method of characteristics (MOC) for solving unsteady flow problems. This method is based on the direct solution of the mass and momentum conservation equations along the propagation characteristics of pressure waves. Both methods–PKP and MOC–differ in their model assumptions and numerical structure, and their detailed description can be found in the following sections of the article.

2. Methods

2.1. Methodology for Water Hammer Analysis-MOC

Modelling of water hammer phenomena in TSNet is based on the classic Method of Characteristics (MOC), which assumes a constant propagation speed of acoustic waves in fluids. The simulation involves mapping the propagation of hydraulic disturbances, such as sudden valve closure or pump shutdown, along the water supply network pipes, assuming one-dimensional compressible flow in slightly deformable pipes.

The physical model is based on a hyperbolic system of partial differential equations describing the behavior of mass and momentum in the pipe:

$${\displaystyle \frac{\partial H}{\partial t}}+{\displaystyle \frac{a^2}{gA}}{\displaystyle \frac{\partial Q}{\partial x}} =0,$$

(5)

$${\displaystyle \frac{\partial Q}{\partial t}}+gA{\displaystyle \frac{\partial H}{\partial x}}+RQ\left|Q\right|=0,$$

(6)

where

• H(x, t)—total elevation gain of the pipeline [m],
• Q(x, t)—flow rate [m³/s],
• A—cross-sectional area [m²],
• a—wave velocity [m/s],
• g—gravity [m/s²],
• R—hydraulic resistance of the segment.

The characteristic method transforms the above system into ordinary differential equations along the characteristics C⁺ (Equation (7)) and C⁻ (Equation (8)), which correspond to lines in the x–t space along which the disturbance propagates in the direction and opposite to the direction of wave propagation, respectively:

$${\displaystyle \frac{\partial Q}{\partial t}}+gA{\displaystyle \frac{\partial H}{\partial x}}+RQ\left|Q\right|=0.$$

(7)

$${\displaystyle \frac{\partial Q}{\partial t}}-gA{\displaystyle \frac{\partial H}{\partial x}}+RQ\left|Q\right|=0.$$

(8)

This provides a numerically stable and unambiguous description of changes in pressure and velocity over time, which can be implemented in the form of a difference scheme along discretized pipes in accordance with Courant’s condition.

The network model for analyzing the water hammer phenomenon was prepared in EPANET 2.2 and saved in a text file format that is used to store water supply network data (*.inp). Then it was loaded into TSNet as a Transient Model object, which maps the network topology along with its hydraulic elements: junctions, pipes, pumps, and tanks/reservoirs.

Each pipe in the model is treated as a separate pressure wave propagation domain and is discretized into sections that satisfy the Courant condition. The simulation is carried out in two stages:

• Initialization of steady-state conditions

In the first stage, TSNet calculates the distribution of flows and pressures in the network in a steady state using the EPANET environment calculation procedures. The results obtained constitute the initial conditions for the simulation of unsteady flows.

2.

Transient flow analysis

In the model, a transient event is defined, such as pump shutdown or valve position change. The change in boundary conditions in the network initiates a pressure wave that propagates along the pipeline. The calculations are performed in successive time steps, and the simulation parameters, duration, time step, and wave propagation speed, depend on the settings adopted in the Python code and the model configuration in TSNet.

At each time step, TSNet calculates the flow and pressure values along the network. The results can then be converted into pressure time histories at selected nodes, enabling analysis of the system dynamics at any point in the network.

2.2. Methodology for Water Hammer Analysis-The PKP Method

The PKP method was proposed by Skowroński in his book ‘Pump Systems’ [37] and website ‘iPumps.EU’ [38]. In this method, see Figure 2, each wagon represents a fluid element with a specific mass. The elastic connections between the wagons represent the elasticity of the pipe and the fluid, and the flow resistance is described as friction forces. In this method, emphasis is placed on mechanical modelling of the entire pumping system, including dynamic simulation of the operation of the pump, motor, valves and fittings, as well as modelling of the varying properties of pipelines in different sections.

In the case under consideration, when the locomotive (pump) pushes the train, the system remains in equilibrium, and the elastic forces between the carriages balance the resistance forces. When the locomotive suddenly stops (pump shutdown), the carriages initially continue to move due to inertia. Then, the change in the pushing force of the locomotive in the first carriage is transferred to the next one, generating a wave that corresponds to the pressure wave in the pipeline.

The movement of fluid in a pipeline can be described using many small fluid elements (equivalent to carriages) with a length ΔL_i and a mass m_i.

Each such element, see Figure 3, is subject to forces: pressure at the beginning and end, gravity, and the force resulting from friction against the walls.

Based on Newton’s second law, for each element of the fluid, we can write a general formula for acceleration d²x/dt² in the Form (9).

$$\sum F=m·{\displaystyle \frac{d^{2}x}{d{t}^2}} .$$

(9)

The sum of forces acting on a fluid element is described by Formula (10).

$$F=F_1-F_2-F_t-F_g$$

(10)

where

• F₁, F₂—forces at the end of the elements [N],
• F_t—friction [N],
• F_g—gravity [N].

After substituting Equation (10) into (9) and taking into account the basic formula for mass, we obtain the equation of motion of the element, using Pressure (11).

$$L·A·\rho ·{\displaystyle \frac{d^{2}x}{d{t}^2}}=A·\left(p_1-p_2-p_t-p_g\right).$$

(11)

Finally, after dividing both sides by cross-sectional area A and rearranging, we obtain the formula for the acceleration of the element of the Form (12).

$${\displaystyle \frac{d^{2}x}{d{t}^2}}={\displaystyle \frac{1}{\mathsf{\Delta }L·\rho }}·\left(p_1-p_2-p_t-p_g\right).$$

(12)

It should be noted that, for the first fluid element, directly behind the pump, the pressure p₁ corresponds to the pump discharge pressure.

The remaining pressures between the elements p are determined on the basis of the volumetric elasticity of the fluid element KV and the mutual change in the position of the elements ΔV, taking into account changes in the diameter of the pipeline, according to Formula (13).

$$p={\displaystyle \frac{\mathsf{\Delta }V}{KV}} .$$

(13)

The change in the volume of the element results from the compressibility of the fluid and the elasticity of the pipeline. The relationship between the change in volume under the influence of pressure is described by Formula (14).

$$\mathsf{\Delta }V=V_0·\left({\displaystyle \frac{d}{E_p·e}}+{\displaystyle \frac{1}{E_f}}\right)·p ,$$

(14)

where

• ΔV—change in fluid volume in segment [m³],
• E_p—elastic modulus of pipe material [Pa],
• E_f—elastic modulus of fluid [Pa],
• d—pipe diameter [m],
• e—wall thickness [m],
• p—pressure between the fluid elements [Pa].

The pressure loss of a pipeline element is calculated using the Darcy-Weisbach formula from Equation (15).

$$p_t=\lambda ·{\displaystyle \frac{\mathsf{\Delta }L}{d}}·{\displaystyle \frac{{\rho v}^2}{2}} ,$$

(15)

where

• p_t—frictional pressure loss [Pa],
• λ—Darcy friction factor,
• ΔL—length of a pipeline segment [m],
• d—pipe internal diameter [m],
• ρ—fluid density [kg/m³]
• v—fluid element velocity [m/s].

The pressure caused by the difference in elevation between the end and the beginning of the element is determined by Formula (16).

$$p_g=\left(R{z}_2-R{z}_1\right)·\rho ·g.$$

(16)

In the calculations, at this stage of system modelling, the phenomenon of fluid evaporation at pressures below the vapor pressure was not taken into account.

The presented method is based on mechanical wave theory. The mechanical approach to the phenomenon of water hammer is particularly useful for modelling the operation of pumping systems, where the dynamics of pumps, drives, and fittings play an important role.

2.3. The Analyzed Pumping System

A pumping system with a gradually rising pipeline was selected for the analysis of water hammer. Such systems are often found, for example, in mine drainage. The analyzed case is shown in Figure 4. The system consists of a motor, a pump, and a pipeline. Two variants of the system were considered. In the first variant, a single check valve is installed at the beginning of the pipeline, Figure 4a. In the second variant, two valves are installed, at the beginning and in the middle of the pipeline, Figure 4b.

In order to model dynamic phenomena, the pipeline was divided into 10 sections numbered from 1 to 10, as shown in Figure 4. The pipeline analyzed with a length of L = 2000 m, a geometric head H_g = 200 m, and a diameter of DN355 mm is made of PE100 plastic. Each two consecutive pipeline elements are constructed of pipes with nominal pressure, starting from the pump: PN32, PN25, PN20, PN16, PN12.5. Local losses caused by joints and fittings were taken into account by adding constant loss coefficients ζ = 1 for each segment.

The system is equipped with a WPS200M/5 centrifugal pump with a Sh355 H4D electric motor with a power of 315 kW. The properties of the pump are described by the flow and power characteristics shown in Figure 5.

Table 1. Parameters of the WPS200M/5 pump and Sh355 H4D electric motor.

Pump WPS200M/5	Parameter	Value	Unit
Discharge	Q	280	m3/h
Head	H	235	m
Power	P	225	kW
Rotational speed	n	1480	rpm
Electric motor Sh355 H4D	Parameter	Value	Unit
Power	P	315	kW
Voltage	U	6000	V
Power factor	cosφ	0.85	-
Rotational speed	n	1486	rpm

summarizes the pump and motor parameters.

The motor torque at variable speed is described using the standard Kloss formula.

The remaining system and fluid parameters are summarized in

Table 2. System and fluid parameters.

Parameter	Symbol	Value	Unit	Notes
Fluid density	ρ	1134	kg/m3	Water used in the simulation
Gravitational acceleration	g	9.81	m/s2	Standard value
Elastic modulus of fluid	Ef	2.03 × 109	Pa	Compressibility of water
Elastic modulus of pipe (PE100)	Ep	3.20 × 109	Pa	Stiffness of pipeline material
Poisson’s ratio of pipe	ν	0.4	–	For volumetric stiffness calculation
Darcy friction factor	λ	0.021	–	Assumed uniform

.

2.4. Model Calculation System–PKP Method

The hydraulic system was modelled in the MATLAB R2024b/Simulink environment using the railway analogy method (PKP), in which the fluid was represented as a series of discrete mass elements connected by elastic bonds. This approach enables the analysis of the propagation of pressure waves in transient states and the representation of the impact of pump, drive, and valve dynamics on the operation of the system. The block diagram of the model shown in Figure 4a,b is presented in Figure 6.

The model consists of the following (six) blocks: Pump, Section_0, Pipe_1, Section_2, Pipe_2, and PSV_valve.

The geometric and material data of the pipeline, presented in Table 2, are converted in a MATLAB batch file into parameters of PKP model elements. For each segment, the following are determined: cross-sectional area A_i, mass of the fluid element M_i, and volume stiffness of the fluid-pipe system KV_i. A fragment of the MATLAB code corresponding to the definition of a single segment is shown in Figure 7.

All other pipeline elements (L12–L15, L21–L25) are defined in the same way. Each mass element in the block diagram (Figure 6) corresponds to a set of parameters A_i, M_i, KV_i in the code. Similarly, the safety valve parameters and pump characteristics used by the Pump and PSV_valve blocks in Simulink are defined in the batch file—Figure 8.

These parameters control the generation of pressure, power, and torque waveforms of the pump during simulation and define the response of the safety valve when the permissible pressure in the system is exceeded.

The dynamic pump and motor model is stored in the Pump block. The motor is controlled by the supply voltage. The motor torque is described by its mechanical characteristic M(ω). The pump pressure is determined on the basis of its flow characteristic, taking into account conversions for the current flow and rotational speed. The torque on the pump shaft is determined on the basis of its power characteristics, taking into account conversions as for the flow characteristics. The inertia of the rotating assembly consisting of the motor, pump and fluid mass in the flow space of the pump impeller was estimated on the basis of the motor’s moment of inertia with a factor of 1.46.

The basic block for the construction of a long hydraulic line is the block of fluid elements. A diagram of such a block is shown in Figure 9.

The fluid element block consists of: a spring calculation part and a mass motion equation part. The pressure between elements p₁ is calculated using the Sum1 summation block, which determines the volume change ΔV, and the KV volume stiffness coefficient block.

The resulting force F acting on the fluid element is determined using the Sum3 summing block and the Pump_A block. The Sum3 block sums the pressure before and after the element and the pressure caused by gravity and losses. Based on the force F, the Integrator2 integrator integrates the accelerations and determines the velocity of the element. The velocity of the element is used to calculate the pressure losses, which are sent as feedback to the Sum3 summing block. Based on the velocity of the element, the Integrator determines the displacement of the element. The displacement is used as feedback to determine the force on the spring. Additionally, the block contains a Sum8 summing block, which allows the impact of the maximum valves to be taken into account in the calculations.

The block diagram in Figure 6 shows two blocks grouping Pipe_1 and Pipe_2. These blocks are identical and each consists of 5 fluid element blocks inside. Directly behind the pump is a fluid element block called Seg_0. This block has a low mass and does not significantly affect the simulation of the system’s operation. This block allows the pump to be ‘connected’ to a long hydraulic line. In addition, the operation of a check valve is simulated in this block. Between the Pipe_1 and Pipe_2 blocks, there is a fluid element block called Seg_2. This block also has a low mass, similar to the Seg_0 block. It is used to model the operation of a midway check valve. In addition, there is also a safety valve block called PSV_valve. This block was not used in the modelling of the hydraulic shock.

3. Results

3.1. TSNet Results

The simulations were performed in the TSNet environment using the Method of Characteristics (MOC), assuming a wave velocity of a = 650 m/s (value determined for the analyzed hydraulic system based on Equation (2)) and a total calculation time of t_f = 100 s. The analyzed case concerned a sudden pump shutdown in a system equipped with a check valve located directly behind the pump. The results are presented as time histories of pressure levels at selected network node.

Figure 10 shows the time course of the pressure level at a selected node (Junc2–just behind the pump) of the hydraulic system during the pump shutdown simulation. In the first phase of observation, before the pump is shut down, the pressure remains at a constant level corresponding to the established operating conditions of the system. When the pump is turned off, there is a sudden drop in pressure, which leads to values close to zero. This drop is the result of the appearance of the first wave of negative pressure, which is characteristic of the initial phase of the water hammer phenomenon.

Once the minimum pressure is reached, pressure oscillations are observed, which alternate between local maxima and minima in subsequent cycles. These fluctuations result from the propagation and reflection of pressure waves along the pipe. Over time, the amplitude of the oscillations decreases, which is the result of hydraulic losses in the pipeline and the gradual dissipation of the wave energy.

The observed pressure time course reflects the typical response of a hydraulic system. In this case, the pressure wave generated after the pump is turned off can freely travel along the pipe in both directions, causing periodic pressure fluctuations in the tested node.

The observed waveforms confirm the usefulness of the TSNet model for determining transient conditions in simple pumping systems and can be the basis for validation of other calculation methods.

3.2. Results of PKP Method

The simulations were performed in the MATLAB/Simulink environment using a pump system model based on the PKP method. The model includes a pump block with a dynamic drive, discretized pipeline sections represented by liquid elements, and a check valve block. The analysis simulated the shutdown of the pump at a given time, observing the propagation of pressure waves along the pipeline.

The pressure curve shown in Figure 11 indicates that in the steady state the system operates under stable conditions. Following pump shutdown, a transient pressure wave is generated and propagates along the entire pipeline. The recorded response shows cyclic oscillations caused by wave reflections at the pipeline ends and intermediate nodes. The amplitudes of successive peaks gradually decrease over time due to frictional effects and energy dissipation in the discretized liquid elements. The legend in Figure 11 uses the following notation: P0_Seg0 and P2_Seg2 refer to the points located 10 m before Section 1 and Section 2, respectively, while P1.1–P1.5 and P2.1–P2.5 correspond to pressure measurement sections spaced every 200 m along Section 1 and Section 2. A phase shift between the selected nodes is also visible, reflecting the finite propagation speed of the pressure wave. This waveform corresponds to a system with a check valve located directly behind the pump, which allows the wave to travel freely through the entire pipeline.

Simulations using the PKP method in the MATLAB/Simulink environment confirmed the typical course of water hammer in a pump pipeline. Switching off the pump initiates a pressure wave that propagates along the entire pipeline, generating pressure peaks and oscillations that gradually decay due to reflections and hydraulic losses. The obtained waveforms are consistent with the theoretical description of the PKP method and provide a reliable basis for model validation against the reference TSNet model.

3.3. Comparison of the PKP and MOC Methods-Validation of Results

A comparative analysis was performed for two variants of modelling the water hammer phenomenon, presented in

Table 3. Comparison of the MOC and the PKP method.

Feature Description	MOC	PKP Method
Theoretical basis	Wave equations (mass and momentum)	Newton’s second law for liquid elements
Model type	Hydraulic model, continuous	Mechanical model, discrete
Discretization	Segments satisfying the Courant condition, N~100	Fluid elements with mass m and length ΔL, N = 10
Consideration of machinery and safety fittings (pump and valve)	Through boundary conditions	Part of the dynamic model
Accuracy in wave analysis	Compared to Bentley Hammer, High accuracy for classic WH	Depends on the number of elements and integrator
Application in validation	Reference model	Comparative model

. The first variant is the TSNet reference model, in which the method of characteristics (MOC) was used to solve the equations of mass and momentum conservation in one-dimensional compressible flow. The second variant is a model developed in the MATLAB/Simulink environment using the railway analogy method (PKP), in which the fluid is represented as a series of discrete masses connected by elastic bonds, while taking into account the dynamic operation of the pump and its drive. In both models, a sudden pump shutdown at t = 40 s was simulated for a pump system with a check valve located directly behind the pump.

For both simulations, for the variant with the check valve located directly behind the pump, the pressure in the pipeline remained at a level corresponding to nominal pump operation until shutdown. Switching off the drive source initiated a pressure wave that propagated along the pipeline, generating distinct pressure peaks and oscillations. Subsequent oscillations gradually decreased in amplitude due to wave reflections at the pipeline ends and hydraulic losses, as illustrated in Figure 12.

Based on Figure 12, key physical parameters such as maximum and minimum pressure values and dominant oscillation periods were identified for each case. In addition, a quantitative comparison was performed to evaluate the accuracy of the PKP method relative to the MOC-based reference. Standard statistical indicators, including RMSE, NRMSE, MAE, and the coefficient of determination (R²), were calculated using time-matched simulation results. All results, both physical and statistical, are summarized in

Table 4. Comparison of physical parameters and statistical indicators for the MOC and PKP methods.

	Method	MOC	PKP
Physical Parameters
Maximum pressure value [bar]		35.46	37.21
Minimum pressure value [bar]		4.08	−5.37 *
Oscillation periods [s]		13	12
Standard statistical indicators
Mean Squared Error		MSE	11.27
Root Mean Square Error		RMSE	3.36
Normalized Root Mean Square Error		NRMSE	0.11
Coefficient of determination		R2	0.94

* The calculated minimum pressure reaches approximately −5.37 bar (gauge), which indicates the occurrence of transient underpressure conditions. This suggests a potential for cavitation, as the absolute pressure may approach vapor pressure levels. The current PKP model is under active development and does not yet include cavitation effects. In reality, the pressure of a liquid in a closed pipeline cannot fall below 0 bar absolute, corresponding to vacuum.

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The TSNet reference model (black line) reproduces the water hammer effect, with oscillations of gradually decreasing amplitude caused by friction and dissipative effects in the pipeline. The PKP method implemented in the Simulink model (colored curves) captures the transient wave propagation along the entire pipeline at multiple points, due to the discrete formulation of the method and the inclusion of fluid element inertia. The quantitative comparison summarized in Table 4 further confirms the good agreement between the two models. Pressure extrema and oscillation periods differ only slightly, while the high coefficient of determination (R² = 0.94) and low NRMSE (0.11) indicate that the PKP method delivers results consistent with the MOC-based reference. This validation demonstrates that the Simulink-based PKP model can be effectively used to predict transient pressure behavior in pumping systems.

4. Transient Analysis of Water Hammer in Pipelines with the Midway Check Valve

In this section, the transient analysis is carried out exclusively using the PKP method in MATLAB/Simulink. As demonstrated in the previous section, the method was validated against the TSNet MOC model for the case of sudden pump shutdown, showing consistent results. Therefore, the PKP approach is further applied here to investigate wave propagation throughout the entire pipeline. Here, the model is used to study the effect of a midway check valve and the propagation of pressure waves along the entire pipeline. The simulation includes pump–motor inertia, variable pump characteristics, fluid and pipe dynamics, pressure losses, and the response of additional fittings.

Figure 13 shows the characteristic response of a hydraulic system equipped with the midway check valve when the pump is turned off. In the initial phase, corresponding to the steady state of the pump, the pressures at individual points in the pipeline remain constant, and both check valves remain open. The legend uses the same notation as in Figure 11: P0_Seg0 and P2_Seg2 denote points 10 m before the main pipeline sections, while P1.1–P1.5 and P2.1–P2.5 represent pressure measurement points spaced every 200 m along Section 1 and Section 2, respectively.

Shutting down the pump at t = 40 s and the presence of a check valve midway strongly affects the propagation of pressure waves. The check valve in the middle of the pipeline blocks the fluid flow towards the pump, leading to significantly lower local oscillations in the inlet section, between the pump and the valve. In this zone, repeated reflections generate low-amplitude pressure changes. In the outlet section, downstream of the midway valve, oscillations remain at a similar level to the system without the valve.

The midway check limits energy transmission to the other half of the pipeline, resulting in lower pressure peaks and faster damping of subsequent waves compared to the system without the valve (see Figure 12). Overall, the results confirm that the midway check valve reduces pressure increases in the pump-valve section from 38 bar to 8 bar, effectively protecting this part of the system from excessive pressure increases. In both cases, the maximum pressures in the valve-outlet section of the pipeline remain the same at 23 bar. The PKP model accurately reflects the dynamics of complex pump systems and demonstrates its usefulness in analyzing the impact of midway check valves on water hammer. Compared to a single-valve solution, the midway check valve significantly reduces the maximum amplitude of pressure oscillations and limits the occurrence of large peaks in the pipeline.

5. Conclusions

The study investigated the water hammer phenomenon in a pump pipeline using two numerical approaches: the reference Method of Characteristics (MOC) implemented in the TSNet package, and the PKP Method developed in MATLAB/Simulink. Based on the analysis, the following conclusions can be drawn:

• The PKP method was successfully validated against the TSNet MOC model for the case of sudden pump shutdown, demonstrating good agreement in oscillation amplitude and frequency. The quantitative comparison (Table 4) confirms this alignment, with a high coefficient of determination (R² = 0.94) and a low normalized root mean square error (NRMSE = 0.11). This confirms that PKP can be reliably applied to the modelling of water hammer phenomena.
• The PKP approach is flexible and allows for the incorporation of additional system characteristics, such as pump–motor inertia, variable pump curves, pipeline dynamics, and the influence of auxiliary fittings. This makes it a practical tool not only for validation purposes but also for extended investigations of transient behavior in pumping systems.
• A sudden pump shutdown initiates a pressure wave that propagates along the entire pipeline, regardless of the presence or position of a midway check valve.
• Applying a midway check valve maintains the oscillations in the upstream section of the system while accelerating damping and reducing the amplitude of the waves in the downstream section.
• The TSNet model produces waveforms with clearly identifiable pressure peaks and gradually decaying oscillations, in line with the classical description of hydraulic shock.
• The Simulink PKP model reproduces these overall trends, while additionally capturing local fluctuations resulting from the discrete nature of the method and the dynamic response of the pump.
• The PKP method extends the scope of analysis beyond waveforms, enabling the study of local fluid dynamics and the effects of pump operation. In this way, it complements the MOC approach and can support both research applications and the design and safety assessment of hydraulic systems.
• The PKP method in its present form does not account for cavitation phenomena. Since the approach is based on a purely mechanical representation of the fluid, it does not capture phase changes or the collapse of vapor cavities. This limitation may lead to underestimation of extreme pressure drops and local mechanical risks in sensitive parts of the system. Further work is underway to extend the PKP method to include cavitation modelling.
• Additionally, the model does not incorporate viscoelastic effects in pipe materials or the inertia of mechanical components. Valve dynamics are represented as massless, and friction is treated in a simplified manner. These assumptions may lead to conservative pressure estimates, which, from an engineering standpoint, help maintain a higher safety margin in system design.
• The PKP method provides a simple yet flexible framework for modelling transient hydraulic events using a purely mechanical analogy. Its modular structure allows for intuitive integration of mechanical components and boundary elements, and facilitates dynamic system analysis beyond traditional fluid-focused approaches.
• The PKP method is currently being developed further. Future enhancements will focus on modelling valve inertia and counterweights, incorporating moments of inertia of rotating components, and improving friction representation. The inclusion of non-Newtonian fluid behavior is also being considered to better reflect real operating conditions in mining and industrial applications.