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To tackle the growing demand for grid-scale energy storage, the ALPHEUS project proposes a novel low-head pumped hydro storage system aimed for coastal application in countries where the topography does not allow for traditional high-head storage. This system consists of a reversible pump-turbine technology with two contra-rotating runners coupled to their respective axial-flux motor-generators as well as a dedicated control, optimising for energy balancing and the provision of ancillary services. To better understand the integration and dynamic interaction of the individual components of the plant and to allow for the simulation of a wide variety of operating conditions and scenarios, this research aims at developing a system model coupling the hydraulic, mechanical and electrical components. Numerical results are compared and verified based on CFD simulations. While some inaccuracies have to be expected, the comparison shows an overall good match with only minor deviations in dynamic behaviour and steady state results.
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System model development and numerical simulation of low-head
pumped hydro storage
J.P. Hoffstaedt & A.J. Laguna
Department of Maritime and Transport Technology, Delft University of Technology, Delft, the Netherlands
J.Fahlbeck & H. Nilsson
Department of Mechanics and Maritime Sciences, Division of Fluid Dynamics, Chalmers University of
Technology, Gothenburg, Sweden
ABSTRACT: To tackle the growing demand for grid-scale energy storage, the ALPHEUS project proposes a
novel low-head pumped hydro storage system aimed for coastal application in countries where the topography
does not allow for traditional high-head storage. This system consists of a reversible pump-turbine technology
with two contra-rotating runners coupled to their respective axial-flux motor-generators as well as a dedicated
control, optimising for energy balancing and the provision of ancillary services. To better understand the inte-
gration and dynamic interaction of the individual components of the plant and to allow for the simulation of a
wide variety of operating conditions and scenarios, this research aims at developing a system model coupling
the hydraulic, mechanical and electrical components. Numerical results are compared and verified based on
CFD simulations. While some inaccuracies have to be expected, the comparison shows an overall good match
with only minor deviations in dynamic behaviour and steady state results.
With electricity generation being one of the major
contributors to the emission of greenhouse gases,
countries worldwide are rapidly increasing their share
of renewable electricity sources. Depending on the re-
gion, solar PV and wind turbines are considered the
preferred solution to achieve this aim. Such typically
inverter coupled renewable generators introduce inter-
mittency and a reduction in spinning reserves to the
electrical grids designed for centralised power plants.
To balance the increasing mismatch between demand
and supply as well as to provide crucial ancillary ser-
vices (AS) to stabilise the grid, large-scale energy
storage is required.
While pumped hydro storage (PHS) is responsible
for the overwhelming majority of energy storage to-
day (REN21 2021), it is traditionally applied in sce-
narios with high-head differences between the upper
and lower reservoir. To enable the technology for us-
age in regions without sufficient elevation changes,
the ALPHEUS project proposes a novel system aimed
at low- and ultra low-head (2 - 20 m) scenarios. For
such a system it has been proposed that axial-flow re-
versible pump-turbines (RPT) coupled to axial-flux
permanent magnet synchronous machines utilising
variable speed drives is one of the most promising
solutions (Hoffstaedt et al. 2022). Additionally, util-
ising two counter-rotating opposed to a single runner
bears the potential to increase efficiency, operating
range and reduce the overall size of the pump-turbine
(Qudaih et al. 2020).
Aimed at coastal applications, the proposed sys-
tem can utilise the sea as the upper reservoir with the
lower reservoir being separated by a dam. The pro-
totype developed as part of the project uses two run-
ners with a diameter of 6 m each coupled to their in-
dividual motor-generator placed in a hub adjacent to
the RPT. With a design power rating of 10 MW for
each pump-turbines set, multiple sets can be utlised
to reach the desired capacity. The lower reservoir is
scaled to fulfill the required storage in the timescale
of hours to days.
To maximise the potential of a low-head PHS plant
contributing to grid stability through energy balanc-
ing and AS, reduced switching times between pump
and turbine mode and rapid power ramp rates are de-
sired. The transient behaviour of the proposed low-
head PHS plant is, however, not yet well understood.
To investigate its dynamic behaviour as well as the in-
teraction and influence of the individual components,
experiments and numerical simulations are conducted
within the ALPHEUS project. Experiments and CFD
simulations provide high accuracy results but are
also time and resource intensive and hence limit the
amount of scenarios that can be tested. The goal of
this research is to develop a comprehensive system
model covering the more relevant dynamics of the
proposed low-head PHS system while significantly
reducing computational resources required.
Despite existing modelling approaches for hy-
dropower and PHS, there is a lack of research on
models tailored to low-head scenarios, utilising vari-
able speed operation and integrating two individual
runners. Coupling individual components such as the
conduit, drivetrain and control into a system model
further allows for the investigation of the dynamic in-
teraction of various components. The choice of mod-
elling approaches for these components is determined
by weighing up accuracy and performance require-
ments. Here, the characteristics of the proposed sys-
tem should be considered. An example for this would
be the shift towards lower heads resulting in higher
mass flow rates for the same desired power. This shift
means inertial effects are more important during sud-
den changes to the flow rate potentially leading to a
higher risk of high amplitude pressure waves com-
monly known water hammer effects.
This paper presents the system model developed
before applying it to a model-scale version of the sys-
tem proposed by ALPHEUS to be used for a later ex-
perimental validation of the project. The results are
compared to CFD simulations done by Fahlbeck et al.
(2022b). For this, two cases are evaluated. A start-up
sequence and rapid change of operating points.
2.1 Flow model inside the conduit
Several approaches exist to model the conduit of
PHS plants depending on the required accuracy and
computational performance. Approaches that neglect
compressibility of the fluid and elasticity of the con-
duit are computationally efficient and can cover the
basic dynamics of the system. However, if transients
are to be investigated these effects have to be con-
sidered (Mohanpurkar et al. 2018). Hence, for this
model a one-dimensional compressible flow approach
is chosen. These coupled partial differential equations
shown in Eq. 1 and Eq. 2 (Stecki and Davis 1986) are
based on the fundamental equations of conservation
of mass and momentum and are also known as water
hammer equations.
∂t =UH
∂x a2
∂x (1)
∂t =UU
∂x gH
∂x S(2)
Here His the pressure head, Uthe fluid velocity,
athe effective pressure wave velocity in the fluid, g
the gravitational acceleration, Dthe conduit diameter
and Sthe friction losses. If the pressure wave veloc-
ity in the pipe is much greater than the fluid veloc-
ity and the conduit is cylindrical the convective terms
given by U∂H
∂x and U U
∂x can be neglected (Sharma
and Kumar 2014). The speed of pressure waves within
the conduit depends on the stiffness of the pipe and
the compressibility of the liquid (Ghidaoui 2004). In
a stiff conduit of a hydroelectric power plant with
potentially some gas bubbles forming due to cavita-
tion caused by pressure transients, the pressure wave
velocity can be assumed to be between 1000 and
1480 m/s. In the scenario considered, the fluid ve-
locity will not exceed 6 m/s.
Conical pipe sections such as draft tubes can be
modelled using an approximation based on equiva-
lent pipe elements. Alternatively, changes to the ve-
locity head have to be considered in the momentum
equation and the continuity equation is extended as
shown in Eq. 3 (Adamkowski 2003) with Abeing the
cross-sectional area of the conduit. Additional expan-
sion and contraction losses should be considered.
∂t =a2
∂x a2
gUln A
∂x (3)
While only considering steady friction is sufficient
to estimate the initial magnitude of a pressure tran-
sient in the system, an additional unsteady friction
term will improve the simulated dissipation of such a
pressure wave over time (Chaudhry 2014). The equa-
tions resulting from these changes and adapted to rep-
resent flow rates rather than fluid velocities are shown
in Eq. 4 and Eq. 5.
∂t =a2
∂x (4)
∂t =gA∂H
∂x SsSu(5)
The additional Q,A,Ssand Suare volumetric flow
rate, cross-sectional area of the conduit, steady and
unsteady friction losses. The steady friction losses
can be calculated using the Darcy-Weisbach for-
mulations using the friction factor fas shown in
Eq. 6 (Riasi et al. 2010). For the unsteady friction
losses, a one-coefficient model, k, of an instanta-
neous acceleration-based method is used as shown
in Eq. 7 (Chaudhry 2014). The fluid accelerations
that induce these losses, are calculated from the av-
eraged cross-sectional values. With values between
0.015 and 0.06 m/s2the coefficient khas been shown
to closely match experimental results.
2DA (6)
∂t + Sign(Q)a
To solve the conduit equations, the partial differ-
ential equations are transformed to ordinary differ-
ential equations using a central schemed finite dif-
ference method. In order to accurately simulate hy-
draulic transients within the system, the states of pres-
sure and flow rates should be known across the whole
space and time. Practically, the spatial dimension is
discretised and the temporal steps solved via a nu-
merical integration scheme. Results are obtained si-
multaneously along the spatial dimension and con-
secutively for the temporal steps. The resulting grid
of nodes is illustrated in figure 1. The temporal incre-
ments are indicated by the subscript iand the spatial
by j.
Figure 1: Node grid used for spatial and temporal discretization.
Using the central finite difference method the ∂Qi,j
and ∂Hi,j
∂x terms can now be replaced by Qi,j+1 Qi,j1
and Hi,j+1 Hi,j1
2∆xwith xreferring to the spatial dis-
tance between nodes. The finite difference method is
chosen due to its simplicity and ease of implemen-
tation when applied to sets of coupled equation with
one of the disadvantages being the requirement of a
regular node grid (Benito et al. 2001).
2.2 Boundary conditions
To solve for the transient-state pressure head and flow
rate at the boundaries of the conduit and to the two
contra-rotating runners, Eq. 4 and Eq. 5 need to be
simultaneously solved with the boundary conditions
imposed. To develop these boundary conditions, the
method of characteristics is used resulting in alge-
braic equations describing the relationship between
pressure head and flow rate. The basic characteristic
equations for upstream and downstream boundaries
as described by Chaudhry (2014) are shown in Eq. 8
and Eq. 9.
Qi,j =Qi,j+1 gA
aHi,j+1 f
2DA tQi,j+1 |Qi,j +1|
aHi,j (8)
Qi,j =Qi,j1+gA
2DA tQi,j1|Qi,j 1|
aHi,j (9)
If the upstream and downstream boundaries repre-
sent the connection to a reservoir, the entrance losses
can be accounted for using Eq. 10 and Eq. 11. Here he
are the entrance head losses to the conduit, keis the
entrance loss coefficient and Hres the total head at the
Hi,j =Hres he(11)
If the reservoir is assumed to be large enough that
travelling pressure waves would be entirely absorbed,
non-reflecting boundaries can be implemented. To do
so, the terms accounting for the pressure difference
between the boundary and the neighbouring node
are removed. In a friction-less system this leaves the
boundary flow rate to be equal to the flow rate at the
neighbouring node.
If characteristic equations are used, the spatial and
temporal steps have to be chosen correctly to ensure a
stable solution and hence accurately represent a pres-
sure wave propagating through the conduit. Neglect-
ing non-linear terms the Courant condition shown in
Eq. 12 can be used to determine the appropriate spa-
tial distance between nodes for a given time step or
vice versa (Ghidaoui et al. 2005). E.g. assuming a
maximum wave velocity of 1480 m/s combined with
a spatial distance between nodes of 0.148 m would re-
sult in 0.1 ms timesteps. To reach a sufficient level of
accuracy a minimum node density is required. In an
iterative process the spatial distance can be reduced
until no further improvements are achieved.
To fully develop the boundaries at the turbine as
well as to couple the conduit model to the other
sub-models, characterising the hydrodynamic perfor-
mance of the rotors is required.
2.3 Coupling to drivetrain and motor-generators
A variable speed power take-off is chosen due to its
improved flexibility and efficiencies across a wider
range of operating conditions (Vasudevan et al. 2021).
The proposed system uses direct-drive transmissions
Figure 2: Modelling Domain for CFD comparison, including hub, struts and runners. (Fahlbeck et al. 2022b)
coupling the two runners to their individual motor-
generators. Neglecting flexibility in the drivetrain
components, the angular velocities can be assumed
constant along the driveshaft between runners and
motor-generators. This allows to model each drive-
train as an individual rigid body with an equivalent
rotational inertia to account for the different rotating
parts leading to Eq. (13) and Eq. (14) (Leithead and
Connor 2000).
dt=τh1 τg1 Df1ω1(13)
dt=τh2 τg2 Df2ω2(14)
Here we have J1,2as the rotational inertias, ω1,2as
the angular velocities, τh1,2as hydraulic torques, τg1,2
as generator torques and Df1, 2 as a viscous damp-
ing torque coefficient to account for torque losses.
The conduit model is coupled to both of these driv-
etrains via the hydraulic torques determined by flow
rate, pressure drop across the individual runners h1,2,
runner efficiencies η1,2, fluid density ρand their re-
spective angular velocities as shown in Eq. (15).
In return the angular velocities of both runners
combined with the flow rate determine the pressure
drop across them as well as their efficiencies illus-
trated in Eq. (16) and Eq. (17). This relationship is
determined through the aforementioned characterisa-
tion of the runners. Such a characterisation can be
based on a range of CFD simulations or experimen-
tal results of the particular rotor design under steady-
state conditions. By doing a multi-variant regression
on such data, the pressure drop across, as well as the
efficiency of both rotors, can be expressed as a func-
tion of flow rate and both angular velocities.
τh1,2 =ρgQh1,2η1,2
h1,2=f(Q, ω1, ω2)(16)
η1,2=f(Q, ω1, ω2)(17)
The governor controlling the angular velocities of
the runners is coupled through the generator torques.
If an additional control variable is required, a valve
that is able to regulate the flow from the conduits to
the runners can be added to the system.
Following, the results of two scenarios, applied to the
model outlined in the previous section, are presented.
The methodology and setup are explained before the
sensitivity using varying node densities for the finite
difference method are compared. Lastly the results are
compared to CFD simulations.
3.1 Methodology & system properties
The setup and initial conditions for the results used
for comparing the model and CFD simulations are
based on a scaled down version of the system with
a runner diameter of 27.6 cm. On both sides of the
pump-turbines, draft tubes increase the pipe diameter
to 50 cm connecting the upper and lower reservoir.
The static head difference between these reservoirs is
6.45 m. Since the model results are compared with the
results of CFD simulations done by Fahlbeck et al.
(2022b), the same domain is used for the numerical
model. This domain is shown in figure 2.
Aside from the initial conditions of pressure and
flow rate across the conduit, the main input to the
model are the static heads at the upper and lower
reservoir. These are based on the static head differ-
ence expected in the lab. Major and minor head losses
outside of the domain are included. These consist of
entrance and exit losses from the lower and upper
reservoir, a pipe bend, a fixed and a variable valve. For
both scenarios the fixed and variable valve are consid-
ered fully open. Additionally the major losses for an
additional 1 m of pipe upstream of the pump-turbine
and 15.5 m downstream of it are applied. Coefficients
are chosen to match the CFD simulations. The param-
eters used for the simulations are shown in table 1.
Table 1: Model parameters used for the simulations.
Static head upper reservoir Hures 9.7 m
Static head lower reservoir Hlres 3.25 m
Pressure wave velocity a1200 m/s
Surface roughness e0.05 mm
Unsteady loss coefficient k0.04 m/s2
Minor loss coefficients
Entrance 0.45 -
Exit 1 -
Bend 0.2 -
Fixed Valve (fully open) 0.4 -
Variable Valve (fully open) 0.39 -
To compare the model against the CFD results and
ensure that no tuning of parameters is required for
individual cases to match, two scenarios are anal-
ysed. Between these cases no changes are made to the
model setup. The first is a start-up sequence and the
second a change of operating point. The CFD simu-
lations were done with both a high- and low-fidelity
model with one order of magnitude difference in num-
ber of mesh cells. The comparison is based on the
results of the high-fidelity model. The methodology
and boundaries used for the CFD simulations are de-
scribed in further detail by Fahlbeck et al. (2021) and
Fahlbeck et al. (2022a).
3.2 Start-up sequence
To start up the system in pump mode, the chosen case
accelerates the runners from an initial 916 RPM to
their final operating point at 1502 RPM over a period
of 3 s. At this initial angular velocity the net head pro-
duced by the pump-turbine matches the gross head of
the system resulting in zero flow. The angular veloc-
ity of the two runners for the sequence is shown in
figure 3.
Figure 3: Angular velocity during start-up.
To determine the required node density, this se-
quence is first simulated with 20, 40 and 60 nodes.
The resulting conduit flow rate for modelling the sys-
tem with these varying node densities can be found
in figure 4. There is no relevant difference between
the 40 and 60 node model. When reduced to 20
nodes, minor differences in the dynamics are notice-
able. However, in steady state all solutions converge.
The 20 node model takes about 28 % of the computa-
tional time to solve compared to the 60 node model.
For all further results the 40 node model is used. This
results in a spatial distance between nodes of around
13.3 cm. Together with a pressure wave velocity of
1200 m/s the simulation time step is chosen at about
The comparison of flow rates during the start-up
sequence between the 40 node model and CFD re-
sults is shown in figure 5. At the final operating points
the results are closely matched with a difference of
only 1.3 %. However, the flow rate of the CFD sim-
ulation is slightly more stable in steady state. At the
initial acceleration of runner two and the consequent
increase in pump head over the RPT, the flow rate
in the proposed model reacts slower compared to the
CFD results. This marginally slower reaction time to
changes in the pump head is also visible after runner
Figure 4: Flow rate for varying node density.
two reaches its final angular velocity and before run-
ner one is accelerated. After the acceleration of runner
one, the flow rate of the model overshoots before con-
Figure 5: Flow rate and pressure head over the computational
domain of model and CFD simulation - start up sequence.
The pressure head over the domain is shown in the
same figure. The head in the CFD results clearly rises
faster, reaches a higher maximum during the acceler-
ation of runner one and remains higher in steady state.
At the final operating point the head over the CFD do-
main reaches around 7.4 m compared to 7.16 m over
the proposed model. This reflects a 3.4 % difference.
Since the final flow rates and the static heads are
similar for both results, the overall losses should be
roughly equivalent. However, the reduced head over
the model domain indicates that the losses outside of
the domain are slightly lower compared to the CFD
and the losses within the domain slightly higher. The
slower response of the model flow rate to changes
in pump head could also be caused by these higher
steady or unsteady losses within the domain. Inaccu-
racies may be introduced through the use of equiva-
lent friction factors and cross-sections along the con-
duit as well as the use of a static pressure wave veloc-
3.3 Change of operating point
To ensure consistent and repeatable modelling results
for varying scenarios and operating conditions, a sec-
ond case is compared. This second case simulates a
rapid change between operating points. This particu-
lar scenario starts at the final operating point of the
start-up sequence, before reducing the angular veloc-
ity of runner two down to 1129 RPM as shown in fig-
ure 6. This corresponds with a sudden reduction in
power by about 22 % over a period of 1 s.
Figure 6: Angular velocity during change of operating point.
Figure 7: Flow rate and pressure head over the computational do-
main of model and CFD simulation - change of operating point.
The resulting flow rate and pressure head over the
domain from the model and CFD simulation can be
found in figure 7. The flow rate in the model starts
at a lower operating point and overshoots after the de-
crease in angular velocity of runner two and the corre-
sponding reduction in pump head. It reacts marginally
faster than in the CFD results. This further supports
the conclusion drawn from the first case that the dif-
ference in dynamic reaction may be caused by slightly
overestimated losses within the domain. Aligned with
the first case the pressure head over the domain is
about 2.6 % higher in the CFD results.
After runner two reaches the final operating point,
the flow rate of the CFD simulation stabilises while
the flow rate in the model keeps increasing resulting in
a difference of about 6.2 % at the final operating point.
Such a divergence over time is most probably caused
by changes in the pump head. Since the head is deter-
mined through the runner characterisation, the inaccu-
racy in the model is likely introduced by a mismatch
between characterisation and actual pump head.
A system model to investigate the dynamics and com-
ponent interaction of a low-head pumped hydro stor-
age plant has been proposed. While using the model
compared to CFD simulations can decrease signifi-
cantly the required computational resources, a com-
promise in the results accuracy is expected. One of
the major differences is that the model is based on
a one dimensional approach. This means that veloc-
ity and pressure values are averaged in the cross-
sections perpendicular to the flow. Consequently, lo-
calised pressure variations are not covered which are
crucial when predicting effects such as cavitation. Ad-
ditionally non-axial fluid velocities are not included
potentially leading to inaccurate loss approximations.
The comparison of the model results using varying
node densities has shown that, for the setup used in
the simulations, no further improvements to accuracy
can be achieved beyond 40 nodes and the consequent
spatial distance of around 13.3 cm. The comparison
of the results to CFD simulations demonstrated that a
simplified one dimensional model can produce satis-
factory results covering the relevant dynamics of the
system. In steady state the resulting flow rates and
pressure heads of the model are within a small de-
viation of the CFD results.
While there can be improvements made to bet-
ter cover the losses within the conduit and the run-
ner characterisation, overall the results between the
proposed model and CFD simulations are closely
matched. This enables the more efficient model to be
used to investigate system dynamics for varying com-
ponents and operating conditions. To further verify
the results an experimental validation should be con-
This research is part of a project that has received
funding from the European Union’s Horizon 2020 re-
search and innovation programme under grant agree-
ment No. 883553.
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Full-text available
To counteract a potential reduction in grid stability caused by a rapidly growing share of intermittent renewable energy sources within our electrical grids, large scale deployment of energy storage will become indispensable. Pumped hydro storage is widely regarded as the most cost-effective option for this. However, its application is traditionally limited to certain topographic features. Expanding its operating range to low-head scenarios could unlock the potential of widespread deployment in regions where so far it has not yet been feasible. This review aims at giving a multidisciplinary insight on technologies that are applicable for low-head (2-30 m) pumped hydro storage, in terms of design, grid integration, control, and modelling. A general overview and the historical development of pumped hydro storage are presented and trends for further innovation and a shift towards application in low-head scenarios are identified. Key drivers for future deployment and the technological and economic challenges to do so are discussed. Based on these challenges, technologies in the field of pumped hydro storage are reviewed and specifically analysed regarding their fitness for low-head application. This is done for pump and turbine design and configuration, electric machines and control, as well as modelling. Further aspects regarding grid integration are discussed. Among conventional machines, it is found that, for high-flow low-head application, axial flow pump-turbines with variable speed drives are the most suitable. Machines such as Archimedes screws, counter-rotating and rotary positive displacement reversible pump-turbines have potential to emerge as innovative solutions. Coupled axial flux permanent magnet synchronous motor-generators are the most promising electric machines. To ensure grid stability, grid-forming control alongside bulk energy storage with capabilities of providing synthetic inertia next to other ancillary services are required.
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The problem of transient flow of liquid in tapered or expanding pipes is discussed. As such, a mathematical model based on the one-diimensional theory of the unsteady flow is proposed. This paper explains the differences in methods of computation of the considered problem found.
This paper presents a hierarchical classification of linear, distributed parameter models which have been used in the study of rigid, uniform fluid transmission lines. The development of the models from the fundamental equations of fluid dynamics is discussed, and the assumptions necessary in obtaining each model are shown. The solutions to the models are presented in general Laplace forms, and the terms that appear in the solutions are discussed with reference to the initial and subsequent assumptions. The existence of seven distinct forms of models was identified and verified by comparison with a vast number of apparently different models quoted in over three hundred research papers covering various engineering, medical, and biological applications. Significant contributions by various researchers over the last eighty years to the development of distributed parameter models of fluid transmission lines are reviewed. The application of the various models based upon comparisons made in the literature is dealt with, and the effect of the initial assumptions on each model's accuracy is discussed. The aim of the paper is to assist in the identification of areas of research where future efforts should be directed and hopefully to prevent needless duplication of research effort.
The wide spectrum of practical problems that warrant water-hammer modelling has increased the importance of careful formulation of the fundamental equations of water hammer and critical analysis of their assumptions. To this end, this paper reviews the relation between state equations and wave speeds in single as well as multiphase and multicomponent transient flows, formulates the various forms of one- and two-dimensional water-hammer equations and illuminates the assumptions inherent in these equations. The derivation of the one- and two-dimensional water-hammer equations proceeds from the three-dimensional Navier - Stokes equations for a compressible fluid. The governing equations for turbulent water-hammer flows are obtained by applying ensemble averaging to the simplified form of the Navier - Stokes equations for water-hammer problems. Unlike time averaging, ensemble averaging is applicable to unsteady flows where the time scale of the transient is often much smaller than the time scales of the turbulence. Order of magnitude analysis, physical understanding and recent research findings are used throughout the paper to evaluate the accuracy of the assumptions made in the derivation of the one- and two-dimensional water-hammer equations.
Applied hydraulic transients
  • M H Chaudhry
Chaudhry, M. H. (2014). Applied hydraulic transients.