Hydraulic Transients for Liquids

CORPORATE CRITERIA

HYDRAULIC TRANSIENTS FOR LIQUIDSDoc. n. CR-COR-ENG_PRC-105-E

Rev. 01Date 15/11/13

Pag. 1 di 39

CORPORATE CRITERIA

HYDRAULIC TRANSIENTS FOR LIQUIDS CR-COR-ENG_PRC-105-E28/10/13

01

First IssueV. Buonocore

AUSG. Poni

PROTECA. Cipelli

ENG

DateRevisionDescription of RevisionPreparedCheckedApproved

This document is the property of Saipem SpA. All rights reserved.

Revision Summary

28/10/13

01

First IssueV. Buonocore

AUSG. Poni

PROTECA. Cipelli

ENG

DateRevisionDescription of RevisionPreparedCheckedApproved

INDEX41SCOPE AND PURPOSE

42REFERENCE DOCUMENTS

42.1INTERNATIONAL STANDARDS

42.2REFERENCES

53DEFINItions

53.1Terms

53.2Acronyms

53.3Symbols

74ACTIVITIES DESCRIPTION

74.1Background and Field of application

74.1.1Background

74.1.2Main Outcome and objectives

74.1.3Consolidated fields of application

84.1.4Analytic approach

94.2Hydraulic Facets and Simplified Method

94.2.1Period of Resonance and Wavespeed

104.2.2Bulk Modulus

104.2.3Surge pressure due to Valve closure

134.2.4Valve Closure time

134.2.5Acceptance Criteria

144.2.6Surge pressure and cavitation

174.3Guideline in building an advanced simulation

174.3.1General Calculation Bases

174.3.2Simulation Time Step

184.3.3Fluid properties definition

184.3.4Piping mechanical properties

194.3.5Pumps properties

214.3.6Valves

254.4Pressure Output and Mitigations Strategies

254.4.1General

254.4.2Results with no cavitation

264.4.3Results with cavities generation and collapse

314.5Forces Calculations

314.5.1Foreword

314.5.2Unbalanced forces

324.5.3Surge Loads: Simplified method

324.6Surge Loads from software simulations

324.6.1Results with no cavitation

324.6.2Results with cavitation

325APPENDIX

325.1Equations For Transient Simulations

1 SCOPE AND PURPOSEThe purpose of this Corporate Criteria is to define the guidelines for the hydraulic transient analyses for liquid systems (Surge Analysis).

Hydraulic transient analyses can be conducted by mean of preliminary calculations or by using advanced simulation software, according to the complexity, the criticality of the systems and the required detail level.

This Corporate Criteria aims to provide the basic principles and theoretical elements to enable the designer to:

Identify the field of application of surge analyses

Understand the physical problems related to the hydraulic transients

Conduct a simplified surge analysis

Find the guidelines to conduct a detailed surge analysis by using a simulation software

Assess the numerical results reliability, in particular for advanced software simulation.

This Corporate Criteria applies to the engineering and construction projects of interest of the Saipem Group.

2 REFERENCE DOCUMENTS2.1 INTERNATIONAL STANDARDS[1] Process Piping ASME B31.3 2012Process Piping

[2] EN 13480-3: 2012Metallic industrial piping - Part 3: Design and calculation

2.2 REFERENCESCrane Flow of fluids through valves, fittings and pipeDouglas, Gasiorek, Swaffield Fluid Mechanics3 DEFINItions3.1 TermsSurge AnalysisAnalysis dedicated to evaluate effects of Hydraulic Transients for liquid systems

Hydraulic Transient Whenever Hydraulic steady state flow is perturbed due to an operational variation or an upset

CvBy definition is the volume flow in US gallons per minute of water at a temperature of 60 Fahrenheit with a pressure drop across the valve of 1 psi. It describes the relationship between the pressure drop across the valve, and the corresponding flow rate.

KvSame as CV, but in SI, is defined as the volume flow in m3/h of water at a temperature of between 5 and 40 Celsius, with a pressure drop across the valve of 1 bar.

Water hammerWater hammer (or, more generally, fluid hammer) is a pressure surge or wave caused when a fluid (usually a liquid but sometimes also a gas) in motion is forced to stop suddenly (momentum change).

3.2 Acronyms DLFDynamic Load Factor

ESDEmergency Shut Down

EPCEngineering Procurement and Construction

GREGlass Reinforced Epoxy

GRPGlass Reinforced Plastic

LNGLiquefied Natural Gas

LPGLiquefied Petroleum Gas

UNBUnbalanced Force

PSMaximum Allowable Pressure according to Ref. [2]

SGSpecific Gravity

SIInternational System of Units

3.3 Symbols

SymbolDefinitionUnit of measure

APipe internal Aream2

cspeed of sound (or elastic wavespeed) in a fluidm/s

CPump TorqueN*m

dPipe internal diameterMm

EModulus of elasticity of the pipe (Young)GPa

fFrequency of the 3-phase power supplyHz

SymbolDefinitionUnit of measure

FForceN

gGravity accelerationm/s2

HPump HeadM

JPump InertiaKg*m2

kFluid compressibility, or Bulk ModulusGPa

l CALClength of a pipe portionM

LPipe LengthM

NMotor Synchronous Speedrevolution per minute

nPump Speedrevolution per minute

pnumber of pole pairs of the 3-phase motor-

pcCalculation Pressure according to [2]Bar(a)

PPressureBar(g)

PSHUTOFFPump Shut off PressureBar(g)

QVolumetric flowm3/h

RAvgaverage rate of closure-

RMaxmaximum rate of valve area closure-

tPipe wall thicknessmm

TCValve closure times

TC*Effective valve closure times

t CALCTime it takes the wave to travel along a pipe portions

V0Average fluid velocity at steady statem/s

VCavCavity Volumem3

WPump Hydraulic PowerW

W*Pump Mechanical PowerW

Greek letters

SymbolDefinitionUnit of measure

Amplification Factor-

F

Surge Load (Unbalanced Force)N

P

Pressure Dropbar

tTripTime associated to a pump trips

Efficiency-

fluid densityKg/m3

Pipe natural resonance periods

4 ACTIVITIES DESCRIPTION4.1 Background and Field of application4.1.1 Background

Undertaking Surge analysis for the distinctive systems presented in this section, is often a contractual requirement within the EPC contracts. Nevertheless, further than any contractual requirement, there are a number of study cases in which surge analysis is not only, and not any longer, interpreted as a check / verification task, but embodies a real positive contribute in the multidisciplinary design, since the Front End engineering. Surge analysis outcome definitely may influence the process engineering, and may impact on the piping supports and civil structures design as well. For this reason, this activity has reached higher criticality degree through the years, and is now considered almost as a standard practice in the engineering workflow.A deep knowledge in this field has spread also among Customers, and it is now a key task to build a specific know how, founded on shared and clear principles, allowing to create and offer a standard Corporate engineering service for this discipline.4.1.2 Main Outcome and objectivesGenerally, surge analyses are performed to assess the effects of sudden changes in hydraulic steady state, leading to potential overpressure or water hammer consequent to cavity collapse.

In particular, simulations are typically but not exclusively conducted to determine:

Surge pressure raising in the context of safety shut down scenarios, whose simplicity or complexity may require or not the use of advanced simulation software

Pressure trends consequent to quick depressurization scenario due to general power failure

Pressure trends consequent to operative scenario that may involve a transient (such as operating pump switch over, or start up procedure), whose effects are deemed worth investigating, due to considerable diameters or pump size involved.

Surge forces acting on pipe runs (input for piping flexibility analysis and stress analysis)The purpose of the surge analysis study is therefore aimed to find mitigations and technical solutions whenever the outcome is not acceptable in terms of surge pressure, or surge forces, or vacuum generation in the system.

4.1.3 Consolidated fields of applicationReflecting the above, Surge Analyses are typically conducted for:

1. Off-site product transfer system: a. export systems for refined or cryogenic products to loading terminals (Jetty berths or offshore terminal);b. import systems for refined or cryogenic products from off-site terminals (Jetty berths or offshore terminal);2. Transfer pipelines for liquids (refined products or water)3. Wide distribution networks: cooling water systems (once through or cooling tower circuits)

4.1.4 Analytic approach

Any time liquid flows into pipes, it is necessary to see if surge analysis is to be performed. The main parameters to be considered are:

Piping length

Diameter

Devices that could generate surge (pumps, automatic valves, etc.).

In absence of specific contractual requirements, one analytical approach to determine whether a surge analysis study should be undertaken or not, can be based on the ratio of two parameters: the pipes period of resonance (see definition in 4.2.1), and the time TC (see 4.2.3) associated to the event responsible of the transient (typically the valve closure time).

TC / (Quick Closure (advanced simulation required

1< TC / (simplified method

TC / (Slow Closure

TC / ,

(Not critical

For ratio TC / higher than 5, the system can be considered not critical, and surge analysis can be omitted. The ratio TC / ranging from 1 to 5 represents all those cases for which the line extension can be critical compared to the hypothetical valve closure time or, in other words, all the cases for which the closure time is still quick, compared to the line extension, and may generate appreciable surge pressure, whose magnitude may be worth assessing more in detail. For these cases, a preliminary surge analysis based on the formulas given in 4.2, can be useful to assess whether maximum expected pressure according to Equation 8, is below the admissible, or if an advanced simulations is required to find any mitigation.

4.2 Hydraulic Facets and Simplified Method

Some basic elements relevant to the theory of water hammer are summarized in this section, in order to achieve the basis to evaluate all the physical topics, and the potential criticality of the analyzed systems. The information given in this section should enable the designer to proceed in a preliminary surge analysis calculation for simple systems.

4.2.1 Period of Resonance and Wavespeed

The Period of Resonance of the basic system represented in Fig. 1 is defined as:

Equation 1

= 2L / c [sec]Where

L [m] is the pipe extension from the source of the perturbation up to the reflection point (the check valve at pump discharge) c [m/s] is the speed of sound through the pipe with the considered liquid (or elastic wavespeed). Any perturbation generating in the system, will then propagate along the pipe at speed c.

For an unconfined liquid, Wavespeed can be evaluated as follows:

Equation 2

[m/sec]Where:

k [GPa] is the Fluid compressibility, or Bulk Modulus (see 4.2.2) [kg/m3] is the Fluid densityIn a pipe of a given geometry, Wavespeed becomes:Equation 3

[m/sec]Where

E [GPa] is the Pipe Young Modulus

d [mm] is the Pipe internal diameter

t [mm] is the Pipe wall thickness

The ratio

quantifies the contribute related to pipe radial deformability (squeezing effect). The stiffer the pipe walls are, the quicker the wave travels (c increases when E and / or t increase). Furthermore, c decreases when pipe diameter increases.

4.2.2 Bulk Modulus

The Bulk Modulus is an indicator of the fluid compressibility; it is by definition the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. It can be practically evaluated assessing the change of density deriving from a change in pressure for a specified fluid (Equation 4) under the following transformations: Isentropic -> most conservative (higher) value of k Isothermal -> intermediate values of k

Adiabatic -> less conservative (lower) value of k

Equation 4

[GPa]

Where: P0, P1, 0 and 1 are evaluated upstream (0) and downstream (1) of a pressure drop, fixed by the user, in one of the previous mentioned conditions, with the support of a process simulator.

P0 and P1 [kPa] are the Pressure values (g or a is irrelevant)

0 and 1 are the fluid Density values (the increment is non-dimensional)

Fluid Bulk Modulus reference values are anyhow available from literature for the most common fluids, such as Water (2.2 GPa).

Tab. 1 - Bulk Modulus typical values

Liquid Temperature Bulk Modulus of Elasticity Density

(oC)(GPa)(kg/m3)

Fresh Water 202.20998

Salt Water 152.271,025

Mineral Oils 251.5 to 1.9860 to 890

Kerosene 201.3800

Methanol 201.0790

Methane (LNG)-1620.8 1.4450

4.2.3 Surge pressure due to Valve closure

A classic and well known sample available from literature for the water hammer theory is the problem of the valve closure in the elementary system represented in Fig. 1.

Whenever a fluid column travelling in a pipe is arrested due to the closure of a valve, a surge pressure generates upstream of the valve, consequent to the variation of the kinetic energy. This wave travels backwards along the pipe, then reverses again, oscillating back and forth (packing effect) with a characteristic periodic trend with crests and troughs.

Fig. 1 Wave generation and reflection

The reflection point for the elementary system is the check valve installed at the pump discharge. However, all discontinuity point in complex networks represent sources of wave reflection (tie in points or outlet branches, boundary or intermediate reservoirs, change in diameter, etc.). In this case, wave interference produces complex pressure patterns. The time it takes the wave to go forth and back in the pipe is one of key parameter to determine surge pressure. As mentioned in 4.1.4, valve closure time TC, compared to the Period of Resonance, can be classified as: Instantaneous (theoretical case)

Quick

Slow

4.2.3.1 Instantaneous arrest (theoretical case)

The maximum theoretical overpressure is given by Joukowsky formula:

Equation 5

[Bar]

Where: V0 [m/s] is the fluid velocity at steady state. [kg/m3] is the fluid densityFig. 2 Sudden (instantaneous) closure

This is only a theoretical case, and can be considered as if the whole liquid column would be instantaneously stopped. The typical pressure pattern is represented in Fig. 2. It can be noticed that deviation from steady state given by Equation 5 are to be intended as both positive (pressure increase) and negative (pressure reduction).4.2.3.2 Quick valve closureIn case of quick closure time, TC is less than the Period of Resonance (TC / ). The maximum surge pressure value is the same as the instantaneous arrest, but the qualitative wave pattern is different (Fig. 3).Fig. 3 Quick Closure

4.2.3.3 Slow ClosureIn case of slow closure, TC is more than the Period of Resonance (TC / ). Maximum surge pressure is:

Equation 6

[Bar]The ratio ( / TC) evaluates how much the actual surge pressure is mitigated, compared to the quick closure case. Valve closure time relaxation means lower P.

Typical qualitative pressure pattern is represented in Figure 4.Fig. 4 Slow Closure

It is worth noting that in slow closure, P does not depend any more on the fluid Bulk Modulus and on pipe Young Modulus (elastic properties of fluid and pipe); P is only affected by fluid density, closure time, and pipe geometry (including pipe thickness, that influences V0).Finally, it should be noted that above figures are referred to the upstream valve side. A rarefaction wave generates downstream of the closing valve as well; the resulting depressurization could potentially lead to cavity volumes generation, whose effect should be assessed (see 4.2.6).

4.2.4 Valve Closure time

The correct evaluation of the input parameters and the correct use of the equations shown in 4.2.3 are the basis of a correct preliminary surge analysis.

It has to be highlighted that Equation 5 gives very conservative figures, and is generally not used to determine the design pressure of a pipeline. On the other hand, Equation 6 must be used in a careful way. In fact, the surge pressure accuracy depends on the correct evaluation of valve closure time.

In real practice, during valve closure, only a portion of the stroke is effective in stopping the fluid; basically, the valve may close through most of its stroke without significantly reducing the fluid flow, and concentrating almost the whole kinetic energy variation in the last run.

Hence, surge pressure should be assessed considering not the overall closure time, but rather the effective closure time TC*.

It is common practice to consider:

0.2 TC TC* 0.5 TC

Equations 6 becomes:

Equation 7

[Bar]4.2.5 Acceptance CriteriaSurge evaluated by Equation 7 represents how much pressure deviates from steady state. Maximum pressure reached during the transient shall be:Equation 8

[Bar(g)]

Where PSHUTOFF is the system shutoff pressure (maximum pressure at no flow condition).

Acceptance criteria shall be according to the applicable standard, as described herein.4.2.5.1 ASME B31.3 standardMaximum pressure should be compared to the Design Pressure of the piping system, whenever the ASME B31.3 standard is applicable.

Subject to the Clients approval, it is permissible to exceed the pressure rating or the allowable stress for pressure design at the temperature of the increased condition by not more than:(a) 33% for no more than 10 hr at any one time and no more than 100 hr/yr; or

(b) 20% for no more than 50 hr at any one time and no more than 500 hr/yr.The effects of such variations shall be determined by the designer to be safe over the service life of the piping system by methods acceptable to the Client.When the variation is self-limiting (e.g., due to a pressure relieving event), and lasts no more than 50 hr at any one time and not more than 500 hr/year, it is permissible to exceed the pressure rating or the allowable stress for pressure design at the temperature of the increased condition by not more than 20%.4.2.5.2 EN 13480:3 standardWhenever EN 13480:3 Code is applicable, no mention is made about possibility to exceed the Maximum Allowable Pressure PS. Hence acceptance condition shall be:

According to EN 13480:3 Sect 12.3.3, in the context of the piping flexibility analysis, some allowance is made on the stress values due to sustained and occasional or exceptional loads intended as the sum of: primary stresses due to calculation pressure, pc stress due to the resultant moment from weight and other sustained mechanical loads (piping dead weight, fluid weight); stress due to the resultant moment from occasional loads (among which dynamic loadings due to safety valve operations and dynamic shock forces due to water hammer are mentioned) or exceptional loads.However, flexibility analysis and stress calculation are not within the scope of this standard, this task being performed by piping stress function. Surge analysis aim is rather to find:

the PMAX (reasonably calculation pressure pc can be taken equal to PMAX) that should be one input for the primary stress calculation.

The surge forces on pipe runs, that concur to determine the occasional loads4.2.6 Surge pressure and cavitation

4.2.6.1 Physical interpretationThe term surge pressure can be misleading, because it can be implicitly associated to the overpressure only, consequent for instance to valve closure.

As anticipated at the end of 4.2.3, a rarefaction wave generates downstream of the closing valve, propagating downstream of valve with the sound speed, and periodic trend; the resulting depressurization effect could potentially lead to have the fluid vapour pressure reached at the actual temperature condition, causing vapour pocket generation (or vacuum pocket, according to the liquid handled). Fig. 5 Cavity Volumes downstream of closing valves

This cavity volume generation is responsible of the so called liquid column separation. This condition is by nature instable: cavity volumes will tend to collapse in a relative short time, with impulsive pressure spikes consequent to the liquid columns hammering each other.

It has to be noted that the same risk potentially exist at the upstream of the valve (though the threat in this case is much less likely). As it appears from Fig. 2, Fig. 3, and Fig. 4 in 4.2.3, pressure contributes are to be evaluated as both positive and negative deviation from steady state pressure.Therefore, if positive surge pressure can lead to exceed the design, or the allowable limits, negative surge pressure can be dangerous as well. Effects should be taken into account since the early hydraulic design, and during the surge analysis itself, especially in advanced software simulations.

Generally, cavity volumes generates:

Downstream of closing valves (or any device with a restriction of area) whose pressure drop leads pressure to interfere with vapour fluid pressure.

At the network highest points, in a depressurization scenario.

Physical interpretation of cavity volumes generation and collapse can be perceived from Fig. 6. In particular, three phases can be distinguished:

cavity volumes generation and build up (expansion) cavity volumes contraction (the fluid columns do reverse flow)

Cavity Collapse, originating impulsive surge pressure due to hammering of liquid surfaces

Fig. 6 Cavity Volumes Generation and Collapse

Quantitative approach of hammering effects is not simple. The theoretical basis to evaluate surge pressure is analog to Equation 5 of 4.2.3.1, Instantaneous arrest:

Equation 9

[Bar]

Where VImpact is the relative velocity of the fluid columns at impact time. Software for hydraulic transient are usually able to evaluate this component, and generally results are conservative as far as pressure is involved.

4.3 Guideline in building an advanced simulation

4.3.1 General Calculation Bases

The following is a list of the most important mathematical, thermodynamic and mechanical assumptions, on which surge analyses are commonly based, that will be deemed implicitly valid for the dissertations included in the following sections.

1. Models are based on Newtonian fluids. A Newtonian fluid is a fluid whose shear stress arising from its flow are proportional to the local strain rate (the rate of change of its deformation over time). The constant of proportionality is the viscosity.2. Simulated systems are Monophase (only liquid). Contribution of vapourized liquid is not within the surge analysis inclusion. At the time this document is written, multi-phase models are deemed not appropriate to develop surge analyses, in particular for complex networks.3. In the advanced simulation models, Momentum and Continuity equations (se 5.1, Equations For Transient Simulations) are solved numerically for pressure and flowrate within the defined domain. In principle, for an infinitesimal fluid control volume:a. balance along longitudinal axis gives the momentum equationb. The rate of increase of mass of a control volume of fluid is equal to the net mass flowrate entering the volume (continuity equation).4. Simulations for liquid systems are usually carried out in iso-thermal and adiabatic conditions. No heat exchange is allowed with the external environment as surge time scale is maximum few seconds while heat transfer time scale is often minutes. Notwithstanding some software are able to offer thermal models, capable to study temperature dynamics of pump units and valves, or to determine temperature trend with distance in a pipeline. Suggestion is to start with isothermal mode and add a more complex thermal mode after model is running well, only if strictly needed.5. One-Dimensional Domain Models are used to perform surge analyses; pipe singular points are described in x, y and z coordinates, giving the network geometry; calculated pressure and flowrate are to be referred to the pipeline longitudinal axis only; no pressure variation is computed along the pipe cross section.6. Surge analysis determines fluid internal pressure trends, not the pipe radial or longitudinal stress, nor the pipe deformation / elongation. Nevertheless, surge analysis outcome can be used as an input for the piping stress analysis.7. The linear elastic model is assumed for the pipe material (steel or plastic pipes); any dissipative contribute associated to the radial and longitudinal deformation of pipe is excluded, as well as any non-linear or non-elastic contribute; pipe walls friction is the only (minor) contribute to the phenomenon reduction with time.

Despite the above assumption can be perceived as considerable restrictions, they are commonly accepted as necessary approximations to enable for the simulation of hydraulic transients in systems getting more and more complex.

The sensibility, experience and awareness grown through the experience, together with the strategies given in this criteria, will enable the designer to recognize, with competent eye, which limitation are associated to those approximations, and hence applicability range of results.

4.3.2 Simulation Time StepOf primary importance to the accuracy of results produced by the software is the calculation of the time step. This is directly related to the discretization of the system: all pipes in the network are divided into several sections of minimum length, for which pressure and flow results are calculated.In most cases, the maximum time step is generated automatically by the software, based on the pipes lengths defined in the system, but it can be also specified by the User. The automatic generation of the time step is commonly influenced by some factors:

1. The minimum pipe length in the network2. The relative length of pipes in the network (max vs. min length). 3. The time associated to valve closure or spin-up / spin-down time of rotating equipment.

The maximum allowed time step for a calculation is often the time taken for a pressure wave to travel the length of the shortest pipe defined in the network, or a fraction of this value. However, the automatic definition of the maximum time step does not prevent the software to automatically decrease this value during the simulation, if required; for instance, to solve particular transients originated or related to fast components, for which it is required to analyze the dynamics in a smaller time scale. This typically occurs when fast components such as check valves, liquid surge relief valves are parts of a network, and definitely control the dynamics of the network). If a simulation takes a long time to converge, this can be due to a time step definition problem, often related to the non-homogeneous distribution of lengths. A way for the User to remove the influence of shorter lengths of pipe from the calculation of a time step is to merge or extend some of the shorter pipes in the network. It is sometimes useful to do this initially when obtaining preliminary results. The final runs can then be made with the correct pipe lengths for maximum accuracy.

4.3.3 Fluid properties definition

Essential parameters normally required to define fluid properties in the context of an isothermal simulation are, at a given temperature:

The Fluid Density.

The Fluid Bulk Modulus.

The Viscosity (at operating Temperature)

The Fluid Vapour Pressure (at operating Temperature).

Usually, the direct specification of the above parameters is the cheapest way to input fluid properties. Other options can be offered by the software, such as defining API fluids, or even fluid composition in mass fraction, mole percent, or ppm. However, once the composition is given, process simulator are rather useful to determine the above listed properties, if they are not available.

If a thermal model is chosen, properties need to be defined over the entire range of operating pressures and temperatures (in addition to the above listed, Heat capacity and Thermal conductivity are necessary), and the definition of the equation of state is required.

Finally, some software are able to have more than one liquid, and their corresponding properties defined; this opportunity is typically suitable for batched product in crude pipelines, but it is not the focus of the discipline covered in this criteria.

4.3.4 Piping mechanical properties

Pipe elastic properties influence the fluid pressure response during transients. Stiffer pipes (thicker walls, or higher Young Modulus related to the pipe material) generate higher surge pressure, for a given fluid. The correct input leads to a better definition of the elastic response of the system.

In case of steel pipes, wall thickness is according to the piping classes, and Young Modulus is in the range 200 225 GPa.

In case of plastic material piping (GRP, GRE), the pipe thickness and Young Modulus are provided by Vendor. However, for Young Modulus, It can be assumed 10% of the one used for steel pipes, varying from 20 to 25 GPa.

4.3.5 Pumps properties

In this paragraph, the most important parameters for electrical centrifugal type pumps (horizontal or vertical) are reported.

4.3.5.1 Pump Power

Pump hydraulic power is:

Equation 10

[W]Where

Q is the volumetric flow rate [m3/h] g is the Gravity acceleration [m/s2] H is the pump Head [m] is the fluid density [kg/m3]

Considering pump mechanical efficiency , pump mechanical power is:

Equation 11

[W]4.3.5.2 Pump Speed Pump speed is commonly available from mechanical data sheets. Whenever this datum is missing, it can be indirectly determined by assessing the electric motor Synchronous speed. Basing on the expected motor power, it is possible to evaluate, with the help from the electrical function, the number of pole pairs commonly selected for that size.

In electric three phase motors, the Synchronous Speed is given by:

Equation 12

N = 60 f / p

Where: f [Hz] = Frequency of the 3-phase power supply

p = number of pole pairs of the 3-phase motorTab. 2 - rpm versus pole pairsPole pair Numbers123456

Synchronous Speed (at 50 Hz) n [rpm]300015001000750600500

Synchronous Speed (at 60 Hz) n [rpm]360018001200900720600

4.3.5.3 Pump TorqueTo determine the Pump Torque, the following formula can be used:Equation 13

[N*m]Where n is the pump rotational speed [rpm]. W * is the pump mechanical power [W]C is the torque at 100% pump speed (at steady state), and can be considered equal to the resistant torque (see Fig. 7). In case of trip, the resistant torque C will apply to the impeller, causing the pump rotational deceleration. A conservative estimation of the time required for the pump to stop is be the following:

Equation 14

[s]Where J [kg*m2] is the total moment of inertia of the pump (see 4.3.5.4).Considering Equation 11, and Equation 13, the following is valid:

Equation 15

[s]During start up, the applied torque can be higher than the one at 100% pump speed (sample in Fig. 7). For this reason, whenever a pump start up scenario is simulated, it is good practice to consider some increment on the driving torque. The peak in the driving torque may be responsible of potential unwished pressure peaks.

Fig. 7 Driving and resistant torque (sample)

4.3.5.4 Pumps InertiaThe moment of inertia (impeller + motor) is the capability of the pump to keep the system pressurized after the pump has been switched off for any reason (operations, trip, power failure). Similarly, it influences the delay by which steady state pressure is reached after the pump has been switched on.

Equation 16

[kg m2]Higher moments of inertia means slower depressurization time profile, and consequent lower risk of vapour cavity generation. Lower moment of inertia means steeper time depressurization profile, with higher risks of cavity generation.

Following empirical formulas assess pump inertia for impeller and electric motor, in case data are not available from the pump vendor; estimated values are conservative (lower values for inertia than available recorded data).

Equation 17

Equation 18

In some cases, the Inertia datum in mechanical data sheet is expressed as PD2 (or PD2). In this cases, the moment of inertia is PD2 / 4.

In general, moments of inertia for vertical pumps are lower than horizontal centrifugal type ones. Without any vendor data, moment of inertia for vertical pumps can be divided by 2 as a first prudential approximation.4.3.6 Valves

4.3.6.1 Operating ValvesThe change in setting of an actuated valve (mainly ESD Valves or ON-OFF valves) is responsible for a hydraulic transient (change in the hydraulic steady state). This event can be simulated by mean of the two following steps: Definition of the valve Cv characteristic curve (from the in-built library or creating a user defined one) Definition of the time evolution of the valve status (e.g. from wide open to complete closure in a certain time);

the Cv represent the flow through the valve under the unit pressure drop. CV is by definition in imperial units; the equivalent in SI is the Kv:

Equation 19

[m3/h / bar ]

Where:

Q is the volumetric flow rate [m3/h]

P is the valve pressure drop [bar]

SG is the fluid specific gravity.

Equation 20

Kv = Cv / 1.16

The valve coefficient (Kv or Cv) input in building the model must be coherent with the units chosen in the simulation.

Common Valve characteristic types are: Equal percentage

Linear

Quick Opening

In general, ball and butterfly valves are roughly associated to equal percentage curves (though some difference is present), while globe valves are linear type.

Fig. 8 - Valve characteristics (typicals)

4.3.6.2 Check ValvesCheck valves are used to prevent back flow.

For swing check valves or spring check valves, closure time can be fractions of a second. They are very effective to protect pumps, but can be associated to shock actuation.

Counterweighted valves with hydraulic control are used as an alternative: they allow for a certain back flow, but are very useful during pumps start up (automatic sequence can be easily implemented).

4.3.6.3 Liquid Surge Relief ValveThese tools are very effective in cutting the pressure pattern at the limit imposed by the design.

Surge relief valves are commonly installed at the end of long pipeline (kilometers), for which a closure time of some seconds is often quick (see 4.2.3).

Surge relief valves are effective because pressure can be kept at the allowable limit with a small volumetric liquid release concentrated in a short relief time. On the other hand, the relief flow rate can reach considerable percentage of the steady state.

4.3.6.4 Vacuum BreakersVacuum breakers are typically installed in water circuits in order to prevent pressure from falling below atmospheric pressure. They are cheap and can be engineered in size and number with some redundancy.

The typical values for inlet and outlet air flow (inbreathing and out breathing charts) according to the valve size are available from commercial catalogues.

Vacuum breakers are very effective in contrasting cavity volumes generation and consequent collapse, leading to water hammer. It is important to locate them at the highest points (at which pressure reaches the minimum values due to the elevation contribute).

It is not uncommon to find them even at pump discharge; often, during a trip or power failure scenario, pressure drop is drastic, and liquid column separation and consequent collapse is likely to occur.

4.3.6.5 Cavitation ModelsAs anticipated in 4.2.6, whenever pressure falls at the vapour pressure, an instability raises due to the fluid columns separation, the buildup of cavity volumes, and the ultimate collapse.This may occur typically:

Downstream of a valve closing too fast

At pump discharge, as a consequence of all pumps arrest (trip or power failure), due to the fluid column inertia

At network highest points, as a consequence of depressurization.

Typical calculation approach available from the software are the following two: Vapour cavitation in punctual definition. Vapour cavitation as volume fraction. 4.3.6.6 Cavitation - Punctual CalculationFig. 9 Cavity Volumes Generation and Collapse

The punctual calculation is performed for every section in which the network is divided into. Basically a virtual flowrate is introduced giving a volumetric balance to the buildup of the cavity volume. The main limit of this approach is that cavity volume in a generic point can potentially exceed the whole pipe inventory. Furthermore, pressure can never fall below limit imposed by vapour pressure.

This is a mathematical approach, whose results however are conservative, as far as pressure estimation is involved. It can be considered flexible, because it is ensures convergence almost regardless to the network complexity.

4.3.6.7 Cavitation Volume Fraction

The volume fraction calculation is performed also for every section in which the network is divided into. But in this case the buildup of the cavity volume is considered responsible of an open channel flow within the pipe. Cavity Volume fraction range is from zero (no cavity) to 1 (no liquid); and cavity volume cannot exceed the pipe inventory. Pressure, on the other hand, may go below vapour pressure limit.Fig. 10 Cavity Volumes Generation and Collapse

This approach gives a more realistic interpretation to the physics of the problem. Nevertheless, simplifications are required in the network definition, since convergence may be difficult for complex networks.

4.4 Pressure Output and Mitigations Strategies

4.4.1 General

The graphical output is the first rough indicator of simulation correctness. Flow and pressure results will have to Confirm expectations in trend in terms of:

positive pressure surge occurring upstream of closing valves

negative pressure surge occurring downstream of closing valve or in case of pump shut down

trend confirmed by preliminary calculations (magnitude, wave period)

Furthermore, it is good practice to check correctness of flowrate balance at nodes.

This analysis may be helpful also for the trouble-shooting phase, since it can give some clues to identify potential errors befallen during the input phase.

4.4.2 Results with no cavitation

Pressure results are of primary importance to identify criticalities:

smooth and regular pressure patterns indicate absence of cavity generation and collapse; no hammering is present. If the detected maximum pressure is within the allowable limit, pressure surge can be borne by the system during the transient, without any recommendation or concerns. Also, surge loads transferred to the piping are of minor entities, and can be usually handled with standard supports.

In case extreme pressure exceeds the allowable limits, then some action are to be taken for mitigation, such as:(a) Increase the valve closure time according to Equation 7. Also the maximum CV and the relevant profile effect on results, but this input often cannot be changed if the surge analysis is conducted during advanced engineering activities. A sample of pressure pattern can be perceived from Fig. 11, in which some alternatives in valve size (different CV) and closure time are comparedFig. 11 Results no cavitation

(b) Installation of a pressure relief system (relief valve and surge vessel of adequate capacity), located as much as possible next to the surge pressure generation point, to limit pressure at the allowable limit;(c) Reduce the delivery flowrate as necessary to meet that limit (this option may be not practical, or even not acceptable for Clients, for obvious reason);(d) Switch off the pumps, e.g. inducing a pump trip in case of valve closure; this can be based:

On an ESD logic activated by High-High pressure signal (2oo3 void is preferred, whenever a high reliability of the loop is required). Pressure transmitter are to be located as much as possible near to the shut-down valve (upstream side).

On a loop activated by the loss of the valve-open position, possibly integrated with a High differential pressure signal across valve. This option need to be further investigated, in terms of safety requirements and reliability. 4.4.3 Results with cavities generation and collapse

In case cavitation is detected, a first smooth portion pattern is followed, from the cavitation collapse, by an irregular trend, with very steep spikes at high frequency and highest gradients in time, and general absence of attenuation even after appreciable time. This may occur: downstream of closing valves as a consequence of the pressure source fallFig. 12 Cavitation sample

4.4.3.1 downstream of closing valvesExcessive Pressure drop downstream of closing valves generates negative surge pressure. Negative effects increase with the distance L DOWNSTREAM of the valve.

Fig. 13 Negative pressure surge downstream of a closing valve

The most important risk is associated to the cavity volumes generating as vapour pressure is reached, with subsequent collapse with liquid hammering: Fig. 14 indicates alternation of cavity volumes and pressure spikes. Fig. 14 Hammering downstream of a closing valve

Fig. 15 Hammering mitigation

This phenomenon can be mitigated by acting on the valve specification: relaxation of closure time is the most effective way to mitigate pressure drop across the valve (see Fig. 15): probably, the effective closure time is too narrow. This may be the cheapest solution if surge analysis is being conducted in check/verification phase, during advanced phases of engineering. If surge analysis is conducted during design phase (early stage), a chance can be in specifying the proper Kv and/or the relevant profile (and hence, touch up the size). As it can be seen from Equation 21, big values of Kv will let fluid pass through with minor pressure drop. It is equivalent to say that if the valve is too big, the effective fluid arrest may take place during the last percentage of the stroke.

Equation 21

4.4.3.2 Fall of the pressure sourcePressure drop due to the fall of the pressure source (depressurization due to pump trip, or power failure) generates cavity at network highest point, due to the elevation contribute associated to pressure. Here following cases are to be distinguished: Cooling Water networks: in once through water network cavity volumes can reach considerable values. The main risk is associated to the potential pipe crush, especially for plastic material pipes. For this reason, minimum vacuum pressure should be declared by pipes manufacturers. Full vacuum requirement is usually deemed not convenient, because some internal reinforcement is required at the internal. The practical and cheapest way to mitigate the problem is the installation of vacuum breakers. Air is let free to enter the system, keeping pressure above the vapour pressure limit (cfr 4.3.6.4). Vacuum breakers are usually cheap and easy maintenance, and gives best results in terms of vacuum prevention, as illustrated in Fig. 16 and Fig. 17.

Fig. 16 pump trip and cavitation

Fig. 17 Mitigation offered by vacuum breakers

In case of water injection systems, no air can be introduced into the network and mitigation solution should be the same as for oils and refined products.

Oils and refined products: since no air can be usually introduced into the system, non-standard solution should be researched, among which:(a) Have the pumps fed by different power sources (electrical driver plus diesel driver): this is efficient during power failure scenarios, but may not help in process shut down scenarios(b) Have pump trip cascaded: this is efficient for operating pumps trip scenarios, but does not relieve to consider effects of power failure scenarios (c) Foresee an accumulator (pressurized vessel able to compensate positive and negative pressure surge with level fluctuation).

(d) Increase as necessary the pump moment of inertia by adding inertia wheels (this option does not find practice up to date, since it negatively impacts on the power consumption at start up and on motor size)However, cavitation in terms of pressure surge is often not critical, because extreme values are in most cases below the allowable limits. The main issue is related to the pressure gradient, that give birth to impulsive forces that may threaten supports stability. This topic is covered in 4.6.2.

Cryogenic products: these are particular cases, because cryogenic liquids vaporize when vapour pressure is reached, and a gas fraction is generated. Hydraulic simulators are typically mono-phase (suitable for stiff liquids, as water), and cannot simulate this liquid to gas transformation. However, if only the pressure results are under investigation, the approach is conservative and can be generally accepted (the gas fraction generating acts as a dumper against cavity collapse, and pressure spikes are consequently milder). Nevertheless, if the axial surge loads are under investigation, and a reduction of cavitation is needed, the same non-standard solution described in the case of refined products are valid.

It should be noted that maximum pressure in case of cavitation can be within the allowable limits, and there would be no issue for the piping integrity; nevertheless, axial loads transferred to the structures may be critical, and some shock phenomena could lead to damage the pipe support. For this reason, whenever possible, it is recommendable to take action in order to reduce cavity collapse to the minimum acceptable. 4.4.3.3 Considerations on cavity volumes

If the simulation is based on the punctual cavity volume calculation, it is possible to estimate the cavity volume gap. With reference to Fig. 9 in 4.3.6.6, known the cavity volume VCav (it is a simulation outcome), the gap between liquid columns will be:

Equation 22

L Cav = V Cav / A

[m]It should be noted that liquid columns are separated by sharp vertical liquid surfaces. Small cavity volumes occurring in big size diameter may have very short L Cav (even fraction of the pipe diameter); this condition may be not realistic under a physical point of view, however it leads to overestimate pressure surge.This may frequently occur in power failure scenarios. Consequently, recommendation is made always to check cavity volumes and cavity gap magnitude.4.5 Forces Calculations

4.5.1 Foreword

In this section, facets are provided to understand principles and theoretical approaches available from literature, to calculate forces generated by fluids contained in a opportunely defined bounded section (the Control Volume).

Actually, this approach is traditionally referred to the hydraulic steady state condition: the original purpose was actually to find out static loads generated by the movement of fluid on a containment surface; for instance, the reaction offered by an elbow as a consequence of the flow deviation, or the effect of splashing on a walls. This approach, can be extended anyhow to the transients, to determine fluctuation of such loads or, in other words, their deviation from steady state in portions of pipeline: the so called Surge Loads. Software are currently able to offer this kind of calculation. However, results interpretation is not always immediate and easy, and endorsement must be done carefully. At this purpose, detailed simulations should always be accompanied by the calculations done with preliminary methods described in this section, to check magnitude, and definitely to have results validated. 4.5.2 Unbalanced forces

During a transient, pressure wave generated in the fluid travels through the pipe. In the context of this dynamic response, Surge Loads are caused by the differential pressures detected in the system, creating unbalanced forces in the piping.

The frequent task is to determine surge loads acting on the piping and transferred to the structure through pipe supports.

In first approximation, the surge load on a fixed point can be seen as the unbalanced pressure component acting on the associated straight pipe run, as represented in Fig. 18. In this case, the Control Volume is the pipe section from A to B. During steady state, this unbalanced component is negligible, but during transients it can reach considerable values.

Broadly speaking, pressure pattern at the two ends of a straight pipe will be almost the same, time-shifted in function of the distance between the two points. In a time plot, the gap between the two pressure curves is the time it takes the wave to cover the distance AB, as represented in Fig. 18. Therefore, in general surge loads are proportional to the span considered for the calculation.

In particular, surge loads are often requested as forces acting on fixed points, given by the product of the unbalanced pressure component and the pipe internal area. Equation 23 is general physical interpretation of the UNBs.

Equation 23

[N]

Where:

PUNB is the unbalanced pressure component [Bar]

A is the pipe internal area. [m2]

The main issue, on the other hand, is the correct and realistic evaluation of the pressure pattern and, definitely, of the PUNB.

Fig. 18 - Surge Loads physical interpretation

If a software simulation has been conducted, time pressure patterns are available for all the network, and determination of surge load is the algebraic and punctual expression of Equation 23, whose interpretation is discussed in 4.6.

4.5.3 Surge Loads: Simplified methodSimplified methods proposed herein allow for surge loads estimation as a consequence of valve closure or trip / failure of all the operating pumps. In both cases, impulsive effects of cavity volumes collapse are not considered. Hence, the designer should be aware that deeper analyses, or at least adequate safety margins are needed, whenever cavity generation and collapse cannot be prevented.

4.5.3.1 Case Of Valve Closure

The simplified method discussed in this paragraph is in compliance with Annex A of EN 13480-3:2012 (Dynamic analysis).In the case of valve closure, the maximum unbalanced load F, in a length of pipe section l Calc, may be calculated as follows: for stiff piping:

Equation 24

[N]

for flexible piping:

Equation 25

[N]

Where:

RMax is the maximum rate of valve area closure (see afterwards); RAvg is the average rate of closure determined by the total closure time (see afterwards). lCalc [m] is the length of the considered pipe section TC* [s] is the valve effective closure time

[kg/m3] is the Fluid density V0 [m/s] is the fluid velocity at steady state A [m2] is the pipe internal areaNo clear indication is given in the EN 13480-3 about what can be considered stiff or flexible.

A preliminary criteria can be based on the following:

steel pipes with diameter higher or equal to 10 can be considered stiff;

plastic material pipes, or steel pipes with diameter lower than 10 can be considered flexible.

However, this approach needs to be shared with piping function and, in case, submitted to Client. Equation 24 and Equation 25 can be interpreted as the maximum surge pressure, calculated by Equation 7, scaled on the calculation length lCalc and extended to the pipe internal area; the unit surge load is the maximum P spread on the total length (P / L).

In calculating the UNBs, factors RMax and RAvg are applied to make allowance for the variation in closure rate throughout the valve stroke and the dynamic nature of the surge load. Interpretation of these factor, and meaning of their ratio, can be seen in Fig. 19. It can be realized that this ratio can be even greater than 1. Reasonable values, under the hypothesis that stroke is linear with time, are the following, though these values are to be assessed case by case:

1.3 for gate valvesRMax / RAvg

1.4for ball or butterfly valvesFig. 19 - Rate of valve area closure

4.5.3.2 Case Of Pumps Trip

In case of trip of all operating pumps, a negative surge Ptrip is generated at the pump discharge and transferred into the system. For this specific case, UNBs may be assessed by calculating the pressure differential pertinent to the considered straight run of pipe, as represented in Fig. 20. Fig. 20 - Surge Loads: simplified method

The differential pressure is a proportion of the surge pressure developed over the piping length under consideration. Considering the pipe branch AB, pressure surge travels from A to B in a time proportional to the span of extension.

The unbalanced pressure can be evaluated, by geometric similitude:

Equation 26

[Bar]

Where:

P Trip is the negative surge pressure generated by the trip [bar]; t Calc is the time it takes the wave to travel from A to B [s]; l Calc is span extension [m]; t Trip is the time associated to the pressure decrease [s]. is an Amplification Factor And UNB is:

Equation 27

[N]

The main topic is how to determine the PTrip and the tCalc . In principle, the former is the pump differential pressure; the latter can be determined by Equation 15 given in 4.3.5.3. Equation 27 can then be seen as:Equation 28

[N]

Similarly to what was discussed in 4.5.3.1 for the ratio RMax / RAvg, an amplification factor is introduced, to consider deviation of the actual pressure profile from the average pressure profile, as illustrated in Fig. 21.

Fig. 21 Amplification factor

Suggested values for the amplification factor are: = 1.3 1.54.6 Surge Loads from software simulations

Surge Loads achieved from advanced software simulations are basically achieved by the same approach observed in Fig. 18, and expressed in Equation 23. Software generally allow for definition of the control volume in which UNB are to be calculated.The main discrimination is related to the effects of cavity volume collapse, because surge pressure is almost instantaneous, as well as the associated surge load. 4.6.1 Results with no cavitation

As far as no cavitation is detected, pressure pattern is regular and smooth, even in presence of a pressure surge. In this case surge load pattern is regular as well, and is typically composed of a first pulse, followed by minor peaks in gradual attenuation.

Fig. 22 Sample of UNB (no cavitation)

This case is normally not critical: generated forces are realistic and reliable, under the approximation of the calculation approach, and can be used as an input for the stress analysis and support design, with no additional mitigations. Simulation outcome is in general quite aligned to preliminary UNBs values estimated with equations shown in 4.5.3.

Fig. 23 UNBs vs. Span (sample)

Surge loads magnitude is proportional to the pipe internal area and to the considered span. The above should find confirmation in preliminary results achieved from Equation 24, Equation 25 and Equation 27, reported in 4.5.3.

4.6.2 Results with cavitation

4.6.2.1 General Discussion

UNBs output in case of simulation with cavitation (punctual calculation) is similar to the pressure trends described in 4.4.3: in Fig. 24, the first smooth portion pattern (with no cavitation) is related to the stable solution. At the start of cavitation, forces trend becomes irregular, with very steep spikes at high frequency, and substantial absence of attenuation even after appreciable time (field with cavitation). Fig. 24 Sample of UNB with cavitation

Some peculiarities are the following: UNB application time in the stable solution field, though depending on the span, is appreciable, and UNB is distinctively directional Load time application in the instable solution field is in the magnitude of fractions of a second (decimals or centesimal of a seconds, according to the accuracy of the software solution), and UNBs are random distributed UNBs generated in the stable solution fields generally reflects pressure trends associated to physical event originator of the transient (pump trip or pressure raise due to valve closure), and are linear with the calculation span UNBs generated in the instable solution fields are no longer proportional to the span, and tend rather to the same value regardless to the calculation length. UNBs can be several times (up to 3 - 4 times) the ones achieved in the stable solution field.4.6.2.2 MitigationsAscertained that main criticality related to the UNBs magnitude is the piping support verification, due to the impulsive nature of the UNBs, finding a mitigation to the above may be not an easy task. Following are common sense proposals:

Whenever possible, any process mitigation effective in contrasting cavity volume growth, among the several listed in 4.4.3, would be likewise effective in reducing UNBs The increase of pipe supports capacity (or specification of special shock absorber), despite not a cheap solution, is sometimes deemed the most reliable solution: the pipe supports can be designed, or adequately enhanced, to cope with impulsive loads. To make allowance for the dynamic effects associated to the forces generated during the transient, a Dynamic Load Factor may be applied also. DLF is assigned by Piping stress function, basing on the load application time and the system natural frequency.

4.6.2.3 A critical approach

However, a separated discussion should be dedicated to soft fluids. As anticipated, cavity calculation is the mathematical approach used by the software to simulate liquids at, or below, vapour pressure. UNBs are then the consequence of this mathematical response.Hence, if the mathematical outcome comes from a realistic interpretation of the transient, UNBs shall be considered faithful as well. If the outcome is based on a partial or incomplete representation of the fluid behavior, this could lead to a critical evaluation of results instead.Evaluation of Cavity volume magnitude may a good clue. Reminding the topics discussed in 4.3.6.6 for the Punctual Calculation, the outcome of 4.4.3.3 may constitute a good basis for this kind of discussion.Minor V Cav and very limited L Cav suggest, in particular for soft fluids, that collapse calculation with punctual cavitation model may be misleading, if used to determine UNBs. Cavity volumes will rather accumulate in pockets at the top of the pipe, and the hammering of two sharp vertical liquid surfaces is an unlikely event thereby. Fig. 25 Sample of UNB with cavitation

Furthermore, hydraulic mono-phase simulators do not consider the vacuum cavity as an actual gas pocket, and consequently omit the dumping effects of vapour phase during cavity volume contraction.

The approach of traditional simulators can be considered conservative on one side, but further analyses may be engaged to assess cavity volume effect during transients, particularly on surge loads, if results are deemed not realistic.

Though it is not the scope of this standard to give a sharp and definite answer on this topic, a semi quantitative approach could be as represented in Fig. 25, where an adequate amplification coefficient is applied to the UNB coming from stable solution. This would cover most of (but not all) UNB in the instable field. The bases on which this method can be developed under a quantitative point of view cannot be in the scope of this standard, due to objective limits. However, dedicated evaluation should be dedicated in particular to:

determine the critical scenario in each specific project

understand UNBs sensibility to the input, or to the several software calculation settings4.6.2.4 An alternative approach

One alternative approach is to find a software able to reproduce multi-phase status better than traditional hydraulic simulators (that are more suitable for stiff fluids). Despite there is no wide experimental evidence demonstrating accuracy of UNBs prediction up to date, this could be a chance to achieve more realistic pressure patterns, and hence, more reasonable UNBs values.5 APPENDIX5.1 Equations For Transient Simulations

Transient Module solves hammer equations in the pipes.

Momentum and continuity equations are solved numerically for pressures and flowrates.

For a given fluid volume balancing of forces depends on hydraulic conditions, as indicated below:

where:

p is the pressure

u is fluid velocity

D is the internal diameter

A is the internal Area of the pipe

G is the weight of the fluid

F is a friction coefficient (Fanning)

is the fluid density

L is the length of the considered volume (short)

Balance along longitudinal axis gives:

p A pA ((p/(x) AL ((u/(t) AL gAL sin- (2f L /D)uuA = 0

and hence the momentum equation:

(1)

The Fanning friction factor f is a function of the Reynolds number and the relative pipe roughness r:

The rate of increase of mass of a control volume of fluid is equal to the net mass flowrate into the volume. i.e, the continuity equation:

(2)

EMBED Equation.3

L

c

c

P

t

P0

=2L/c

t 0

P = c Vo

P = c Vo

P

t

P0

=2L/c

=2L/c

TC

P = c Vo

P = c Vo

P = c Vo

P

t

P0

=2L/c

Tc

2T

3T

4T

P = c Vo ( / Tc )

Cavity Volume contraction

Liquid Phase

Cavity Volume expansion

Liquid Phase

Cavity Volume

Cavity Volume

Generation / Expansion

Contraction

Overpressure

Collapse (hammering)

Cavity Volume

Generation

Expansion

L Pipe

L Pipe

VCAV VPIPE P PVAP

VCAV VPIPE P = PVAP

L Cav

Generation

Expansion

L Pipe

L Pipe

VVAP 0 P PVAP

VVAP VPIPE P PVAP

L DOWNSTREAM

P

t

PA

PB

PUNB

PB

PA

Fixed Point

PUNB

Loop

Loop

A

B

l Calc

TC

A

t

50%

100%

RAvg

Cross section area

RMax

Butterfly Valves

Stroke \ average closure rate

Valve Open Area

Rate of valve closure (Max)

TC

TC

A

t

50%

100%

RMax

RAvg

Cross section area

Gate Valves

TC

A

t

50%

100%

RAvg

Cross section area

RMax

Ball Valves

Ball Valves

PB

PA

P

t

PA

PB

PUNB

Fixed Point

PUNB

Loop

Loop

t Calc = l Calc / c

TTrip

Ptrip

A

B

l Calc

P

t

tTrip

Ptrip

Average (calculated)

Real profile

Amplification

Stable Solution

Instable Solution

pA

(p+ ((p/(x)L) A

u2A

u2A

(u/(t

G

L

(2f L /D)uuA

(frictional pressure drop)

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Linear

Equal percentage

Valve Setting

% of maximum flow

Foglio1

Valve settingQuick openingLinearEqual percentage

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0.236203

0.354305

0.472408

0.5815011

0.6876016

0.7927022

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1100100100

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Foglio1

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Equal percentage

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% of maximum flow

Foglio2

Foglio3

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